src/HOL/Divides.thy
 author haftmann Sat Jul 09 13:26:16 2016 +0200 (2016-07-09) changeset 63417 c184ec919c70 parent 63317 ca187a9f66da child 63499 9c9a59949887 permissions -rw-r--r--
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
* * *
more rules for setsum, setprod on intervals
```     1 (*  Title:      HOL/Divides.thy
```
```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     3     Copyright   1999  University of Cambridge
```
```     4 *)
```
```     5
```
```     6 section \<open>The division operators div and mod\<close>
```
```     7
```
```     8 theory Divides
```
```     9 imports Parity
```
```    10 begin
```
```    11
```
```    12 subsection \<open>Abstract division in commutative semirings.\<close>
```
```    13
```
```    14 class div = dvd + divide +
```
```    15   fixes mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "mod" 70)
```
```    16
```
```    17 class semiring_div = semidom + div +
```
```    18   assumes mod_div_equality: "a div b * b + a mod b = a"
```
```    19     and div_by_0 [simp]: "a div 0 = 0"
```
```    20     and div_0 [simp]: "0 div a = 0"
```
```    21     and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"
```
```    22     and div_mult_mult1 [simp]: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b"
```
```    23 begin
```
```    24
```
```    25 subclass algebraic_semidom
```
```    26 proof
```
```    27   fix b a
```
```    28   assume "b \<noteq> 0"
```
```    29   then show "a * b div b = a"
```
```    30     using div_mult_self1 [of b 0 a] by (simp add: ac_simps)
```
```    31 qed simp
```
```    32
```
```    33 lemma div_by_1:
```
```    34   "a div 1 = a"
```
```    35   by (fact divide_1)
```
```    36
```
```    37 lemma div_mult_self1_is_id:
```
```    38   "b \<noteq> 0 \<Longrightarrow> b * a div b = a"
```
```    39   by (fact nonzero_mult_divide_cancel_left)
```
```    40
```
```    41 lemma div_mult_self2_is_id:
```
```    42   "b \<noteq> 0 \<Longrightarrow> a * b div b = a"
```
```    43   by (fact nonzero_mult_divide_cancel_right)
```
```    44
```
```    45 text \<open>@{const divide} and @{const mod}\<close>
```
```    46
```
```    47 lemma mod_div_equality2: "b * (a div b) + a mod b = a"
```
```    48   unfolding mult.commute [of b]
```
```    49   by (rule mod_div_equality)
```
```    50
```
```    51 lemma mod_div_equality': "a mod b + a div b * b = a"
```
```    52   using mod_div_equality [of a b]
```
```    53   by (simp only: ac_simps)
```
```    54
```
```    55 lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
```
```    56   by (simp add: mod_div_equality)
```
```    57
```
```    58 lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"
```
```    59   by (simp add: mod_div_equality2)
```
```    60
```
```    61 lemma mod_by_0 [simp]: "a mod 0 = a"
```
```    62   using mod_div_equality [of a zero] by simp
```
```    63
```
```    64 lemma mod_0 [simp]: "0 mod a = 0"
```
```    65   using mod_div_equality [of zero a] div_0 by simp
```
```    66
```
```    67 lemma div_mult_self2 [simp]:
```
```    68   assumes "b \<noteq> 0"
```
```    69   shows "(a + b * c) div b = c + a div b"
```
```    70   using assms div_mult_self1 [of b a c] by (simp add: mult.commute)
```
```    71
```
```    72 lemma div_mult_self3 [simp]:
```
```    73   assumes "b \<noteq> 0"
```
```    74   shows "(c * b + a) div b = c + a div b"
```
```    75   using assms by (simp add: add.commute)
```
```    76
```
```    77 lemma div_mult_self4 [simp]:
```
```    78   assumes "b \<noteq> 0"
```
```    79   shows "(b * c + a) div b = c + a div b"
```
```    80   using assms by (simp add: add.commute)
```
```    81
```
```    82 lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"
```
```    83 proof (cases "b = 0")
```
```    84   case True then show ?thesis by simp
```
```    85 next
```
```    86   case False
```
```    87   have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"
```
```    88     by (simp add: mod_div_equality)
```
```    89   also from False div_mult_self1 [of b a c] have
```
```    90     "\<dots> = (c + a div b) * b + (a + c * b) mod b"
```
```    91       by (simp add: algebra_simps)
```
```    92   finally have "a = a div b * b + (a + c * b) mod b"
```
```    93     by (simp add: add.commute [of a] add.assoc distrib_right)
```
```    94   then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
```
```    95     by (simp add: mod_div_equality)
```
```    96   then show ?thesis by simp
```
```    97 qed
```
```    98
```
```    99 lemma mod_mult_self2 [simp]:
```
```   100   "(a + b * c) mod b = a mod b"
```
```   101   by (simp add: mult.commute [of b])
```
```   102
```
```   103 lemma mod_mult_self3 [simp]:
```
```   104   "(c * b + a) mod b = a mod b"
```
```   105   by (simp add: add.commute)
```
```   106
```
```   107 lemma mod_mult_self4 [simp]:
```
```   108   "(b * c + a) mod b = a mod b"
```
```   109   by (simp add: add.commute)
```
```   110
```
```   111 lemma mod_mult_self1_is_0 [simp]:
```
```   112   "b * a mod b = 0"
```
```   113   using mod_mult_self2 [of 0 b a] by simp
```
```   114
```
```   115 lemma mod_mult_self2_is_0 [simp]:
```
```   116   "a * b mod b = 0"
```
```   117   using mod_mult_self1 [of 0 a b] by simp
```
```   118
```
```   119 lemma mod_by_1 [simp]:
```
```   120   "a mod 1 = 0"
```
```   121 proof -
```
```   122   from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp
```
```   123   then have "a + a mod 1 = a + 0" by simp
```
```   124   then show ?thesis by (rule add_left_imp_eq)
```
```   125 qed
```
```   126
```
```   127 lemma mod_self [simp]:
```
```   128   "a mod a = 0"
```
```   129   using mod_mult_self2_is_0 [of 1] by simp
```
```   130
```
```   131 lemma div_add_self1 [simp]:
```
```   132   assumes "b \<noteq> 0"
```
```   133   shows "(b + a) div b = a div b + 1"
```
```   134   using assms div_mult_self1 [of b a 1] by (simp add: add.commute)
```
```   135
```
```   136 lemma div_add_self2 [simp]:
```
```   137   assumes "b \<noteq> 0"
```
```   138   shows "(a + b) div b = a div b + 1"
```
```   139   using assms div_add_self1 [of b a] by (simp add: add.commute)
```
```   140
```
```   141 lemma mod_add_self1 [simp]:
```
```   142   "(b + a) mod b = a mod b"
```
```   143   using mod_mult_self1 [of a 1 b] by (simp add: add.commute)
```
```   144
```
```   145 lemma mod_add_self2 [simp]:
```
```   146   "(a + b) mod b = a mod b"
```
```   147   using mod_mult_self1 [of a 1 b] by simp
```
```   148
```
```   149 lemma mod_div_decomp:
```
```   150   fixes a b
```
```   151   obtains q r where "q = a div b" and "r = a mod b"
```
```   152     and "a = q * b + r"
```
```   153 proof -
```
```   154   from mod_div_equality have "a = a div b * b + a mod b" by simp
```
```   155   moreover have "a div b = a div b" ..
```
```   156   moreover have "a mod b = a mod b" ..
```
```   157   note that ultimately show thesis by blast
```
```   158 qed
```
```   159
```
```   160 lemma dvd_imp_mod_0 [simp]:
```
```   161   assumes "a dvd b"
```
```   162   shows "b mod a = 0"
```
```   163 proof -
```
```   164   from assms obtain c where "b = a * c" ..
```
```   165   then have "b mod a = a * c mod a" by simp
```
```   166   then show "b mod a = 0" by simp
```
```   167 qed
```
```   168
```
```   169 lemma mod_eq_0_iff_dvd:
```
```   170   "a mod b = 0 \<longleftrightarrow> b dvd a"
```
```   171 proof
```
```   172   assume "b dvd a"
```
```   173   then show "a mod b = 0" by simp
```
```   174 next
```
```   175   assume "a mod b = 0"
```
```   176   with mod_div_equality [of a b] have "a div b * b = a" by simp
```
```   177   then have "a = b * (a div b)" by (simp add: ac_simps)
```
```   178   then show "b dvd a" ..
```
```   179 qed
```
```   180
```
```   181 lemma dvd_eq_mod_eq_0 [nitpick_unfold, code]:
```
```   182   "a dvd b \<longleftrightarrow> b mod a = 0"
```
```   183   by (simp add: mod_eq_0_iff_dvd)
```
```   184
```
```   185 lemma mod_div_trivial [simp]:
```
```   186   "a mod b div b = 0"
```
```   187 proof (cases "b = 0")
```
```   188   assume "b = 0"
```
```   189   thus ?thesis by simp
```
```   190 next
```
```   191   assume "b \<noteq> 0"
```
```   192   hence "a div b + a mod b div b = (a mod b + a div b * b) div b"
```
```   193     by (rule div_mult_self1 [symmetric])
```
```   194   also have "\<dots> = a div b"
```
```   195     by (simp only: mod_div_equality')
```
```   196   also have "\<dots> = a div b + 0"
```
```   197     by simp
```
```   198   finally show ?thesis
```
```   199     by (rule add_left_imp_eq)
```
```   200 qed
```
```   201
```
```   202 lemma mod_mod_trivial [simp]:
```
```   203   "a mod b mod b = a mod b"
```
```   204 proof -
```
```   205   have "a mod b mod b = (a mod b + a div b * b) mod b"
```
```   206     by (simp only: mod_mult_self1)
```
```   207   also have "\<dots> = a mod b"
```
```   208     by (simp only: mod_div_equality')
```
```   209   finally show ?thesis .
```
```   210 qed
```
```   211
```
```   212 lemma dvd_mod_imp_dvd:
```
```   213   assumes "k dvd m mod n" and "k dvd n"
```
```   214   shows "k dvd m"
```
```   215 proof -
```
```   216   from assms have "k dvd (m div n) * n + m mod n"
```
```   217     by (simp only: dvd_add dvd_mult)
```
```   218   then show ?thesis by (simp add: mod_div_equality)
```
```   219 qed
```
```   220
```
```   221 text \<open>Addition respects modular equivalence.\<close>
```
```   222
```
```   223 lemma mod_add_left_eq: \<comment> \<open>FIXME reorient\<close>
```
```   224   "(a + b) mod c = (a mod c + b) mod c"
```
```   225 proof -
```
```   226   have "(a + b) mod c = (a div c * c + a mod c + b) mod c"
```
```   227     by (simp only: mod_div_equality)
```
```   228   also have "\<dots> = (a mod c + b + a div c * c) mod c"
```
```   229     by (simp only: ac_simps)
```
```   230   also have "\<dots> = (a mod c + b) mod c"
```
```   231     by (rule mod_mult_self1)
```
```   232   finally show ?thesis .
```
```   233 qed
```
```   234
```
```   235 lemma mod_add_right_eq: \<comment> \<open>FIXME reorient\<close>
```
```   236   "(a + b) mod c = (a + b mod c) mod c"
```
```   237 proof -
```
```   238   have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c"
```
```   239     by (simp only: mod_div_equality)
```
```   240   also have "\<dots> = (a + b mod c + b div c * c) mod c"
```
```   241     by (simp only: ac_simps)
```
```   242   also have "\<dots> = (a + b mod c) mod c"
```
```   243     by (rule mod_mult_self1)
```
```   244   finally show ?thesis .
```
```   245 qed
```
```   246
```
```   247 lemma mod_add_eq: \<comment> \<open>FIXME reorient\<close>
```
```   248   "(a + b) mod c = (a mod c + b mod c) mod c"
```
```   249 by (rule trans [OF mod_add_left_eq mod_add_right_eq])
```
```   250
```
```   251 lemma mod_add_cong:
```
```   252   assumes "a mod c = a' mod c"
```
```   253   assumes "b mod c = b' mod c"
```
```   254   shows "(a + b) mod c = (a' + b') mod c"
```
```   255 proof -
```
```   256   have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c"
```
```   257     unfolding assms ..
```
```   258   thus ?thesis
```
```   259     by (simp only: mod_add_eq [symmetric])
```
```   260 qed
```
```   261
```
```   262 text \<open>Multiplication respects modular equivalence.\<close>
```
```   263
```
```   264 lemma mod_mult_left_eq: \<comment> \<open>FIXME reorient\<close>
```
```   265   "(a * b) mod c = ((a mod c) * b) mod c"
```
```   266 proof -
```
```   267   have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c"
```
```   268     by (simp only: mod_div_equality)
```
```   269   also have "\<dots> = (a mod c * b + a div c * b * c) mod c"
```
```   270     by (simp only: algebra_simps)
```
```   271   also have "\<dots> = (a mod c * b) mod c"
```
```   272     by (rule mod_mult_self1)
```
```   273   finally show ?thesis .
```
```   274 qed
```
```   275
```
```   276 lemma mod_mult_right_eq: \<comment> \<open>FIXME reorient\<close>
```
```   277   "(a * b) mod c = (a * (b mod c)) mod c"
```
```   278 proof -
```
```   279   have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c"
```
```   280     by (simp only: mod_div_equality)
```
```   281   also have "\<dots> = (a * (b mod c) + a * (b div c) * c) mod c"
```
```   282     by (simp only: algebra_simps)
```
```   283   also have "\<dots> = (a * (b mod c)) mod c"
```
```   284     by (rule mod_mult_self1)
```
```   285   finally show ?thesis .
```
```   286 qed
```
```   287
```
```   288 lemma mod_mult_eq: \<comment> \<open>FIXME reorient\<close>
```
```   289   "(a * b) mod c = ((a mod c) * (b mod c)) mod c"
```
```   290 by (rule trans [OF mod_mult_left_eq mod_mult_right_eq])
```
```   291
```
```   292 lemma mod_mult_cong:
```
```   293   assumes "a mod c = a' mod c"
```
```   294   assumes "b mod c = b' mod c"
```
```   295   shows "(a * b) mod c = (a' * b') mod c"
```
```   296 proof -
```
```   297   have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c"
```
```   298     unfolding assms ..
```
```   299   thus ?thesis
```
```   300     by (simp only: mod_mult_eq [symmetric])
```
```   301 qed
```
```   302
```
```   303 text \<open>Exponentiation respects modular equivalence.\<close>
```
```   304
```
```   305 lemma power_mod: "(a mod b) ^ n mod b = a ^ n mod b"
```
```   306 apply (induct n, simp_all)
```
```   307 apply (rule mod_mult_right_eq [THEN trans])
```
```   308 apply (simp (no_asm_simp))
```
```   309 apply (rule mod_mult_eq [symmetric])
```
```   310 done
```
```   311
```
```   312 lemma mod_mod_cancel:
```
```   313   assumes "c dvd b"
```
```   314   shows "a mod b mod c = a mod c"
```
```   315 proof -
```
```   316   from \<open>c dvd b\<close> obtain k where "b = c * k"
```
```   317     by (rule dvdE)
```
```   318   have "a mod b mod c = a mod (c * k) mod c"
```
```   319     by (simp only: \<open>b = c * k\<close>)
```
```   320   also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c"
```
```   321     by (simp only: mod_mult_self1)
```
```   322   also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c"
```
```   323     by (simp only: ac_simps)
```
```   324   also have "\<dots> = a mod c"
```
```   325     by (simp only: mod_div_equality)
```
```   326   finally show ?thesis .
```
```   327 qed
```
```   328
```
```   329 lemma div_mult_mult2 [simp]:
```
```   330   "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
```
```   331   by (drule div_mult_mult1) (simp add: mult.commute)
```
```   332
```
```   333 lemma div_mult_mult1_if [simp]:
```
```   334   "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"
```
```   335   by simp_all
```
```   336
```
```   337 lemma mod_mult_mult1:
```
```   338   "(c * a) mod (c * b) = c * (a mod b)"
```
```   339 proof (cases "c = 0")
```
```   340   case True then show ?thesis by simp
```
```   341 next
```
```   342   case False
```
```   343   from mod_div_equality
```
```   344   have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .
```
```   345   with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)
```
```   346     = c * a + c * (a mod b)" by (simp add: algebra_simps)
```
```   347   with mod_div_equality show ?thesis by simp
```
```   348 qed
```
```   349
```
```   350 lemma mod_mult_mult2:
```
```   351   "(a * c) mod (b * c) = (a mod b) * c"
```
```   352   using mod_mult_mult1 [of c a b] by (simp add: mult.commute)
```
```   353
```
```   354 lemma mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)"
```
```   355   by (fact mod_mult_mult2 [symmetric])
```
```   356
```
```   357 lemma mult_mod_right: "c * (a mod b) = (c * a) mod (c * b)"
```
```   358   by (fact mod_mult_mult1 [symmetric])
```
```   359
```
```   360 lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)"
```
```   361   unfolding dvd_def by (auto simp add: mod_mult_mult1)
```
```   362
```
```   363 lemma dvd_mod_iff: "k dvd n \<Longrightarrow> k dvd (m mod n) \<longleftrightarrow> k dvd m"
```
```   364 by (blast intro: dvd_mod_imp_dvd dvd_mod)
```
```   365
```
```   366 lemma div_div_eq_right:
```
```   367   assumes "c dvd b" "b dvd a"
```
```   368   shows   "a div (b div c) = a div b * c"
```
```   369 proof -
```
```   370   from assms have "a div b * c = (a * c) div b"
```
```   371     by (subst dvd_div_mult) simp_all
```
```   372   also from assms have "\<dots> = (a * c) div ((b div c) * c)" by simp
```
```   373   also have "a * c div (b div c * c) = a div (b div c)"
```
```   374     by (cases "c = 0") simp_all
```
```   375   finally show ?thesis ..
```
```   376 qed
```
```   377
```
```   378 lemma div_div_div_same:
```
```   379   assumes "d dvd a" "d dvd b" "b dvd a"
```
```   380   shows   "(a div d) div (b div d) = a div b"
```
```   381   using assms by (subst dvd_div_mult2_eq [symmetric]) simp_all
```
```   382
```
```   383 end
```
```   384
```
```   385 class ring_div = comm_ring_1 + semiring_div
```
```   386 begin
```
```   387
```
```   388 subclass idom_divide ..
```
```   389
```
```   390 text \<open>Negation respects modular equivalence.\<close>
```
```   391
```
```   392 lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"
```
```   393 proof -
```
```   394   have "(- a) mod b = (- (a div b * b + a mod b)) mod b"
```
```   395     by (simp only: mod_div_equality)
```
```   396   also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b"
```
```   397     by (simp add: ac_simps)
```
```   398   also have "\<dots> = (- (a mod b)) mod b"
```
```   399     by (rule mod_mult_self1)
```
```   400   finally show ?thesis .
```
```   401 qed
```
```   402
```
```   403 lemma mod_minus_cong:
```
```   404   assumes "a mod b = a' mod b"
```
```   405   shows "(- a) mod b = (- a') mod b"
```
```   406 proof -
```
```   407   have "(- (a mod b)) mod b = (- (a' mod b)) mod b"
```
```   408     unfolding assms ..
```
```   409   thus ?thesis
```
```   410     by (simp only: mod_minus_eq [symmetric])
```
```   411 qed
```
```   412
```
```   413 text \<open>Subtraction respects modular equivalence.\<close>
```
```   414
```
```   415 lemma mod_diff_left_eq:
```
```   416   "(a - b) mod c = (a mod c - b) mod c"
```
```   417   using mod_add_cong [of a c "a mod c" "- b" "- b"] by simp
```
```   418
```
```   419 lemma mod_diff_right_eq:
```
```   420   "(a - b) mod c = (a - b mod c) mod c"
```
```   421   using mod_add_cong [of a c a "- b" "- (b mod c)"] mod_minus_cong [of "b mod c" c b] by simp
```
```   422
```
```   423 lemma mod_diff_eq:
```
```   424   "(a - b) mod c = (a mod c - b mod c) mod c"
```
```   425   using mod_add_cong [of a c "a mod c" "- b" "- (b mod c)"] mod_minus_cong [of "b mod c" c b] by simp
```
```   426
```
```   427 lemma mod_diff_cong:
```
```   428   assumes "a mod c = a' mod c"
```
```   429   assumes "b mod c = b' mod c"
```
```   430   shows "(a - b) mod c = (a' - b') mod c"
```
```   431   using assms mod_add_cong [of a c a' "- b" "- b'"] mod_minus_cong [of b c "b'"] by simp
```
```   432
```
```   433 lemma dvd_neg_div: "y dvd x \<Longrightarrow> -x div y = - (x div y)"
```
```   434 apply (case_tac "y = 0") apply simp
```
```   435 apply (auto simp add: dvd_def)
```
```   436 apply (subgoal_tac "-(y * k) = y * - k")
```
```   437  apply (simp only:)
```
```   438  apply (erule div_mult_self1_is_id)
```
```   439 apply simp
```
```   440 done
```
```   441
```
```   442 lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div -y = - (x div y)"
```
```   443 apply (case_tac "y = 0") apply simp
```
```   444 apply (auto simp add: dvd_def)
```
```   445 apply (subgoal_tac "y * k = -y * -k")
```
```   446  apply (erule ssubst, rule div_mult_self1_is_id)
```
```   447  apply simp
```
```   448 apply simp
```
```   449 done
```
```   450
```
```   451 lemma div_diff [simp]:
```
```   452   "z dvd x \<Longrightarrow> z dvd y \<Longrightarrow> (x - y) div z = x div z - y div z"
```
```   453   using div_add [of _ _ "- y"] by (simp add: dvd_neg_div)
```
```   454
```
```   455 lemma div_minus_minus [simp]: "(-a) div (-b) = a div b"
```
```   456   using div_mult_mult1 [of "- 1" a b]
```
```   457   unfolding neg_equal_0_iff_equal by simp
```
```   458
```
```   459 lemma mod_minus_minus [simp]: "(-a) mod (-b) = - (a mod b)"
```
```   460   using mod_mult_mult1 [of "- 1" a b] by simp
```
```   461
```
```   462 lemma div_minus_right: "a div (-b) = (-a) div b"
```
```   463   using div_minus_minus [of "-a" b] by simp
```
```   464
```
```   465 lemma mod_minus_right: "a mod (-b) = - ((-a) mod b)"
```
```   466   using mod_minus_minus [of "-a" b] by simp
```
```   467
```
```   468 lemma div_minus1_right [simp]: "a div (-1) = -a"
```
```   469   using div_minus_right [of a 1] by simp
```
```   470
```
```   471 lemma mod_minus1_right [simp]: "a mod (-1) = 0"
```
```   472   using mod_minus_right [of a 1] by simp
```
```   473
```
```   474 lemma minus_mod_self2 [simp]:
```
```   475   "(a - b) mod b = a mod b"
```
```   476   by (simp add: mod_diff_right_eq)
```
```   477
```
```   478 lemma minus_mod_self1 [simp]:
```
```   479   "(b - a) mod b = - a mod b"
```
```   480   using mod_add_self2 [of "- a" b] by simp
```
```   481
```
```   482 end
```
```   483
```
```   484
```
```   485 subsubsection \<open>Parity and division\<close>
```
```   486
```
```   487 class semiring_div_parity = semiring_div + comm_semiring_1_cancel + numeral +
```
```   488   assumes parity: "a mod 2 = 0 \<or> a mod 2 = 1"
```
```   489   assumes one_mod_two_eq_one [simp]: "1 mod 2 = 1"
```
```   490   assumes zero_not_eq_two: "0 \<noteq> 2"
```
```   491 begin
```
```   492
```
```   493 lemma parity_cases [case_names even odd]:
```
```   494   assumes "a mod 2 = 0 \<Longrightarrow> P"
```
```   495   assumes "a mod 2 = 1 \<Longrightarrow> P"
```
```   496   shows P
```
```   497   using assms parity by blast
```
```   498
```
```   499 lemma one_div_two_eq_zero [simp]:
```
```   500   "1 div 2 = 0"
```
```   501 proof (cases "2 = 0")
```
```   502   case True then show ?thesis by simp
```
```   503 next
```
```   504   case False
```
```   505   from mod_div_equality have "1 div 2 * 2 + 1 mod 2 = 1" .
```
```   506   with one_mod_two_eq_one have "1 div 2 * 2 + 1 = 1" by simp
```
```   507   then have "1 div 2 * 2 = 0" by (simp add: ac_simps add_left_imp_eq del: mult_eq_0_iff)
```
```   508   then have "1 div 2 = 0 \<or> 2 = 0" by simp
```
```   509   with False show ?thesis by auto
```
```   510 qed
```
```   511
```
```   512 lemma not_mod_2_eq_0_eq_1 [simp]:
```
```   513   "a mod 2 \<noteq> 0 \<longleftrightarrow> a mod 2 = 1"
```
```   514   by (cases a rule: parity_cases) simp_all
```
```   515
```
```   516 lemma not_mod_2_eq_1_eq_0 [simp]:
```
```   517   "a mod 2 \<noteq> 1 \<longleftrightarrow> a mod 2 = 0"
```
```   518   by (cases a rule: parity_cases) simp_all
```
```   519
```
```   520 subclass semiring_parity
```
```   521 proof (unfold_locales, unfold dvd_eq_mod_eq_0 not_mod_2_eq_0_eq_1)
```
```   522   show "1 mod 2 = 1"
```
```   523     by (fact one_mod_two_eq_one)
```
```   524 next
```
```   525   fix a b
```
```   526   assume "a mod 2 = 1"
```
```   527   moreover assume "b mod 2 = 1"
```
```   528   ultimately show "(a + b) mod 2 = 0"
```
```   529     using mod_add_eq [of a b 2] by simp
```
```   530 next
```
```   531   fix a b
```
```   532   assume "(a * b) mod 2 = 0"
```
```   533   then have "(a mod 2) * (b mod 2) = 0"
```
```   534     by (cases "a mod 2 = 0") (simp_all add: mod_mult_eq [of a b 2])
```
```   535   then show "a mod 2 = 0 \<or> b mod 2 = 0"
```
```   536     by (rule divisors_zero)
```
```   537 next
```
```   538   fix a
```
```   539   assume "a mod 2 = 1"
```
```   540   then have "a = a div 2 * 2 + 1" using mod_div_equality [of a 2] by simp
```
```   541   then show "\<exists>b. a = b + 1" ..
```
```   542 qed
```
```   543
```
```   544 lemma even_iff_mod_2_eq_zero:
```
```   545   "even a \<longleftrightarrow> a mod 2 = 0"
```
```   546   by (fact dvd_eq_mod_eq_0)
```
```   547
```
```   548 lemma even_succ_div_two [simp]:
```
```   549   "even a \<Longrightarrow> (a + 1) div 2 = a div 2"
```
```   550   by (cases "a = 0") (auto elim!: evenE dest: mult_not_zero)
```
```   551
```
```   552 lemma odd_succ_div_two [simp]:
```
```   553   "odd a \<Longrightarrow> (a + 1) div 2 = a div 2 + 1"
```
```   554   by (auto elim!: oddE simp add: zero_not_eq_two [symmetric] add.assoc)
```
```   555
```
```   556 lemma even_two_times_div_two:
```
```   557   "even a \<Longrightarrow> 2 * (a div 2) = a"
```
```   558   by (fact dvd_mult_div_cancel)
```
```   559
```
```   560 lemma odd_two_times_div_two_succ [simp]:
```
```   561   "odd a \<Longrightarrow> 2 * (a div 2) + 1 = a"
```
```   562   using mod_div_equality2 [of 2 a] by (simp add: even_iff_mod_2_eq_zero)
```
```   563
```
```   564 end
```
```   565
```
```   566
```
```   567 subsection \<open>Generic numeral division with a pragmatic type class\<close>
```
```   568
```
```   569 text \<open>
```
```   570   The following type class contains everything necessary to formulate
```
```   571   a division algorithm in ring structures with numerals, restricted
```
```   572   to its positive segments.  This is its primary motiviation, and it
```
```   573   could surely be formulated using a more fine-grained, more algebraic
```
```   574   and less technical class hierarchy.
```
```   575 \<close>
```
```   576
```
```   577 class semiring_numeral_div = semiring_div + comm_semiring_1_cancel + linordered_semidom +
```
```   578   assumes div_less: "0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a div b = 0"
```
```   579     and mod_less: " 0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a mod b = a"
```
```   580     and div_positive: "0 < b \<Longrightarrow> b \<le> a \<Longrightarrow> a div b > 0"
```
```   581     and mod_less_eq_dividend: "0 \<le> a \<Longrightarrow> a mod b \<le> a"
```
```   582     and pos_mod_bound: "0 < b \<Longrightarrow> a mod b < b"
```
```   583     and pos_mod_sign: "0 < b \<Longrightarrow> 0 \<le> a mod b"
```
```   584     and mod_mult2_eq: "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
```
```   585     and div_mult2_eq: "0 \<le> c \<Longrightarrow> a div (b * c) = a div b div c"
```
```   586   assumes discrete: "a < b \<longleftrightarrow> a + 1 \<le> b"
```
```   587   fixes divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"
```
```   588     and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"
```
```   589   assumes divmod_def: "divmod m n = (numeral m div numeral n, numeral m mod numeral n)"
```
```   590     and divmod_step_def: "divmod_step l qr = (let (q, r) = qr
```
```   591     in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
```
```   592     else (2 * q, r))"
```
```   593     \<comment> \<open>These are conceptually definitions but force generated code
```
```   594     to be monomorphic wrt. particular instances of this class which
```
```   595     yields a significant speedup.\<close>
```
```   596
```
```   597 begin
```
```   598
```
```   599 lemma mult_div_cancel:
```
```   600   "b * (a div b) = a - a mod b"
```
```   601 proof -
```
```   602   have "b * (a div b) + a mod b = a"
```
```   603     using mod_div_equality [of a b] by (simp add: ac_simps)
```
```   604   then have "b * (a div b) + a mod b - a mod b = a - a mod b"
```
```   605     by simp
```
```   606   then show ?thesis
```
```   607     by simp
```
```   608 qed
```
```   609
```
```   610 subclass semiring_div_parity
```
```   611 proof
```
```   612   fix a
```
```   613   show "a mod 2 = 0 \<or> a mod 2 = 1"
```
```   614   proof (rule ccontr)
```
```   615     assume "\<not> (a mod 2 = 0 \<or> a mod 2 = 1)"
```
```   616     then have "a mod 2 \<noteq> 0" and "a mod 2 \<noteq> 1" by simp_all
```
```   617     have "0 < 2" by simp
```
```   618     with pos_mod_bound pos_mod_sign have "0 \<le> a mod 2" "a mod 2 < 2" by simp_all
```
```   619     with \<open>a mod 2 \<noteq> 0\<close> have "0 < a mod 2" by simp
```
```   620     with discrete have "1 \<le> a mod 2" by simp
```
```   621     with \<open>a mod 2 \<noteq> 1\<close> have "1 < a mod 2" by simp
```
```   622     with discrete have "2 \<le> a mod 2" by simp
```
```   623     with \<open>a mod 2 < 2\<close> show False by simp
```
```   624   qed
```
```   625 next
```
```   626   show "1 mod 2 = 1"
```
```   627     by (rule mod_less) simp_all
```
```   628 next
```
```   629   show "0 \<noteq> 2"
```
```   630     by simp
```
```   631 qed
```
```   632
```
```   633 lemma divmod_digit_1:
```
```   634   assumes "0 \<le> a" "0 < b" and "b \<le> a mod (2 * b)"
```
```   635   shows "2 * (a div (2 * b)) + 1 = a div b" (is "?P")
```
```   636     and "a mod (2 * b) - b = a mod b" (is "?Q")
```
```   637 proof -
```
```   638   from assms mod_less_eq_dividend [of a "2 * b"] have "b \<le> a"
```
```   639     by (auto intro: trans)
```
```   640   with \<open>0 < b\<close> have "0 < a div b" by (auto intro: div_positive)
```
```   641   then have [simp]: "1 \<le> a div b" by (simp add: discrete)
```
```   642   with \<open>0 < b\<close> have mod_less: "a mod b < b" by (simp add: pos_mod_bound)
```
```   643   define w where "w = a div b mod 2"
```
```   644   with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
```
```   645   have mod_w: "a mod (2 * b) = a mod b + b * w"
```
```   646     by (simp add: w_def mod_mult2_eq ac_simps)
```
```   647   from assms w_exhaust have "w = 1"
```
```   648     by (auto simp add: mod_w) (insert mod_less, auto)
```
```   649   with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp
```
```   650   have "2 * (a div (2 * b)) = a div b - w"
```
```   651     by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)
```
```   652   with \<open>w = 1\<close> have div: "2 * (a div (2 * b)) = a div b - 1" by simp
```
```   653   then show ?P and ?Q
```
```   654     by (simp_all add: div mod add_implies_diff [symmetric])
```
```   655 qed
```
```   656
```
```   657 lemma divmod_digit_0:
```
```   658   assumes "0 < b" and "a mod (2 * b) < b"
```
```   659   shows "2 * (a div (2 * b)) = a div b" (is "?P")
```
```   660     and "a mod (2 * b) = a mod b" (is "?Q")
```
```   661 proof -
```
```   662   define w where "w = a div b mod 2"
```
```   663   with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
```
```   664   have mod_w: "a mod (2 * b) = a mod b + b * w"
```
```   665     by (simp add: w_def mod_mult2_eq ac_simps)
```
```   666   moreover have "b \<le> a mod b + b"
```
```   667   proof -
```
```   668     from \<open>0 < b\<close> pos_mod_sign have "0 \<le> a mod b" by blast
```
```   669     then have "0 + b \<le> a mod b + b" by (rule add_right_mono)
```
```   670     then show ?thesis by simp
```
```   671   qed
```
```   672   moreover note assms w_exhaust
```
```   673   ultimately have "w = 0" by auto
```
```   674   with mod_w have mod: "a mod (2 * b) = a mod b" by simp
```
```   675   have "2 * (a div (2 * b)) = a div b - w"
```
```   676     by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)
```
```   677   with \<open>w = 0\<close> have div: "2 * (a div (2 * b)) = a div b" by simp
```
```   678   then show ?P and ?Q
```
```   679     by (simp_all add: div mod)
```
```   680 qed
```
```   681
```
```   682 lemma fst_divmod:
```
```   683   "fst (divmod m n) = numeral m div numeral n"
```
```   684   by (simp add: divmod_def)
```
```   685
```
```   686 lemma snd_divmod:
```
```   687   "snd (divmod m n) = numeral m mod numeral n"
```
```   688   by (simp add: divmod_def)
```
```   689
```
```   690 text \<open>
```
```   691   This is a formulation of one step (referring to one digit position)
```
```   692   in school-method division: compare the dividend at the current
```
```   693   digit position with the remainder from previous division steps
```
```   694   and evaluate accordingly.
```
```   695 \<close>
```
```   696
```
```   697 lemma divmod_step_eq [simp]:
```
```   698   "divmod_step l (q, r) = (if numeral l \<le> r
```
```   699     then (2 * q + 1, r - numeral l) else (2 * q, r))"
```
```   700   by (simp add: divmod_step_def)
```
```   701
```
```   702 text \<open>
```
```   703   This is a formulation of school-method division.
```
```   704   If the divisor is smaller than the dividend, terminate.
```
```   705   If not, shift the dividend to the right until termination
```
```   706   occurs and then reiterate single division steps in the
```
```   707   opposite direction.
```
```   708 \<close>
```
```   709
```
```   710 lemma divmod_divmod_step:
```
```   711   "divmod m n = (if m < n then (0, numeral m)
```
```   712     else divmod_step n (divmod m (Num.Bit0 n)))"
```
```   713 proof (cases "m < n")
```
```   714   case True then have "numeral m < numeral n" by simp
```
```   715   then show ?thesis
```
```   716     by (simp add: prod_eq_iff div_less mod_less fst_divmod snd_divmod)
```
```   717 next
```
```   718   case False
```
```   719   have "divmod m n =
```
```   720     divmod_step n (numeral m div (2 * numeral n),
```
```   721       numeral m mod (2 * numeral n))"
```
```   722   proof (cases "numeral n \<le> numeral m mod (2 * numeral n)")
```
```   723     case True
```
```   724     with divmod_step_eq
```
```   725       have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
```
```   726         (2 * (numeral m div (2 * numeral n)) + 1, numeral m mod (2 * numeral n) - numeral n)"
```
```   727         by simp
```
```   728     moreover from True divmod_digit_1 [of "numeral m" "numeral n"]
```
```   729       have "2 * (numeral m div (2 * numeral n)) + 1 = numeral m div numeral n"
```
```   730       and "numeral m mod (2 * numeral n) - numeral n = numeral m mod numeral n"
```
```   731       by simp_all
```
```   732     ultimately show ?thesis by (simp only: divmod_def)
```
```   733   next
```
```   734     case False then have *: "numeral m mod (2 * numeral n) < numeral n"
```
```   735       by (simp add: not_le)
```
```   736     with divmod_step_eq
```
```   737       have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
```
```   738         (2 * (numeral m div (2 * numeral n)), numeral m mod (2 * numeral n))"
```
```   739         by auto
```
```   740     moreover from * divmod_digit_0 [of "numeral n" "numeral m"]
```
```   741       have "2 * (numeral m div (2 * numeral n)) = numeral m div numeral n"
```
```   742       and "numeral m mod (2 * numeral n) = numeral m mod numeral n"
```
```   743       by (simp_all only: zero_less_numeral)
```
```   744     ultimately show ?thesis by (simp only: divmod_def)
```
```   745   qed
```
```   746   then have "divmod m n =
```
```   747     divmod_step n (numeral m div numeral (Num.Bit0 n),
```
```   748       numeral m mod numeral (Num.Bit0 n))"
```
```   749     by (simp only: numeral.simps distrib mult_1)
```
```   750   then have "divmod m n = divmod_step n (divmod m (Num.Bit0 n))"
```
```   751     by (simp add: divmod_def)
```
```   752   with False show ?thesis by simp
```
```   753 qed
```
```   754
```
```   755 text \<open>The division rewrite proper -- first, trivial results involving \<open>1\<close>\<close>
```
```   756
```
```   757 lemma divmod_trivial [simp]:
```
```   758   "divmod Num.One Num.One = (numeral Num.One, 0)"
```
```   759   "divmod (Num.Bit0 m) Num.One = (numeral (Num.Bit0 m), 0)"
```
```   760   "divmod (Num.Bit1 m) Num.One = (numeral (Num.Bit1 m), 0)"
```
```   761   "divmod num.One (num.Bit0 n) = (0, Numeral1)"
```
```   762   "divmod num.One (num.Bit1 n) = (0, Numeral1)"
```
```   763   using divmod_divmod_step [of "Num.One"] by (simp_all add: divmod_def)
```
```   764
```
```   765 text \<open>Division by an even number is a right-shift\<close>
```
```   766
```
```   767 lemma divmod_cancel [simp]:
```
```   768   "divmod (Num.Bit0 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r))" (is ?P)
```
```   769   "divmod (Num.Bit1 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r + 1))" (is ?Q)
```
```   770 proof -
```
```   771   have *: "\<And>q. numeral (Num.Bit0 q) = 2 * numeral q"
```
```   772     "\<And>q. numeral (Num.Bit1 q) = 2 * numeral q + 1"
```
```   773     by (simp_all only: numeral_mult numeral.simps distrib) simp_all
```
```   774   have "1 div 2 = 0" "1 mod 2 = 1" by (auto intro: div_less mod_less)
```
```   775   then show ?P and ?Q
```
```   776     by (simp_all add: fst_divmod snd_divmod prod_eq_iff split_def * [of m] * [of n] mod_mult_mult1
```
```   777       div_mult2_eq [of _ _ 2] mod_mult2_eq [of _ _ 2]
```
```   778       add.commute del: numeral_times_numeral)
```
```   779 qed
```
```   780
```
```   781 text \<open>The really hard work\<close>
```
```   782
```
```   783 lemma divmod_steps [simp]:
```
```   784   "divmod (num.Bit0 m) (num.Bit1 n) =
```
```   785       (if m \<le> n then (0, numeral (num.Bit0 m))
```
```   786        else divmod_step (num.Bit1 n)
```
```   787              (divmod (num.Bit0 m)
```
```   788                (num.Bit0 (num.Bit1 n))))"
```
```   789   "divmod (num.Bit1 m) (num.Bit1 n) =
```
```   790       (if m < n then (0, numeral (num.Bit1 m))
```
```   791        else divmod_step (num.Bit1 n)
```
```   792              (divmod (num.Bit1 m)
```
```   793                (num.Bit0 (num.Bit1 n))))"
```
```   794   by (simp_all add: divmod_divmod_step)
```
```   795
```
```   796 lemmas divmod_algorithm_code = divmod_step_eq divmod_trivial divmod_cancel divmod_steps
```
```   797
```
```   798 text \<open>Special case: divisibility\<close>
```
```   799
```
```   800 definition divides_aux :: "'a \<times> 'a \<Rightarrow> bool"
```
```   801 where
```
```   802   "divides_aux qr \<longleftrightarrow> snd qr = 0"
```
```   803
```
```   804 lemma divides_aux_eq [simp]:
```
```   805   "divides_aux (q, r) \<longleftrightarrow> r = 0"
```
```   806   by (simp add: divides_aux_def)
```
```   807
```
```   808 lemma dvd_numeral_simp [simp]:
```
```   809   "numeral m dvd numeral n \<longleftrightarrow> divides_aux (divmod n m)"
```
```   810   by (simp add: divmod_def mod_eq_0_iff_dvd)
```
```   811
```
```   812 text \<open>Generic computation of quotient and remainder\<close>
```
```   813
```
```   814 lemma numeral_div_numeral [simp]:
```
```   815   "numeral k div numeral l = fst (divmod k l)"
```
```   816   by (simp add: fst_divmod)
```
```   817
```
```   818 lemma numeral_mod_numeral [simp]:
```
```   819   "numeral k mod numeral l = snd (divmod k l)"
```
```   820   by (simp add: snd_divmod)
```
```   821
```
```   822 lemma one_div_numeral [simp]:
```
```   823   "1 div numeral n = fst (divmod num.One n)"
```
```   824   by (simp add: fst_divmod)
```
```   825
```
```   826 lemma one_mod_numeral [simp]:
```
```   827   "1 mod numeral n = snd (divmod num.One n)"
```
```   828   by (simp add: snd_divmod)
```
```   829
```
```   830 end
```
```   831
```
```   832
```
```   833 subsection \<open>Division on @{typ nat}\<close>
```
```   834
```
```   835 context
```
```   836 begin
```
```   837
```
```   838 text \<open>
```
```   839   We define @{const divide} and @{const mod} on @{typ nat} by means
```
```   840   of a characteristic relation with two input arguments
```
```   841   @{term "m::nat"}, @{term "n::nat"} and two output arguments
```
```   842   @{term "q::nat"}(uotient) and @{term "r::nat"}(emainder).
```
```   843 \<close>
```
```   844
```
```   845 definition divmod_nat_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where
```
```   846   "divmod_nat_rel m n qr \<longleftrightarrow>
```
```   847     m = fst qr * n + snd qr \<and>
```
```   848       (if n = 0 then fst qr = 0 else if n > 0 then 0 \<le> snd qr \<and> snd qr < n else n < snd qr \<and> snd qr \<le> 0)"
```
```   849
```
```   850 text \<open>@{const divmod_nat_rel} is total:\<close>
```
```   851
```
```   852 qualified lemma divmod_nat_rel_ex:
```
```   853   obtains q r where "divmod_nat_rel m n (q, r)"
```
```   854 proof (cases "n = 0")
```
```   855   case True  with that show thesis
```
```   856     by (auto simp add: divmod_nat_rel_def)
```
```   857 next
```
```   858   case False
```
```   859   have "\<exists>q r. m = q * n + r \<and> r < n"
```
```   860   proof (induct m)
```
```   861     case 0 with \<open>n \<noteq> 0\<close>
```
```   862     have "(0::nat) = 0 * n + 0 \<and> 0 < n" by simp
```
```   863     then show ?case by blast
```
```   864   next
```
```   865     case (Suc m) then obtain q' r'
```
```   866       where m: "m = q' * n + r'" and n: "r' < n" by auto
```
```   867     then show ?case proof (cases "Suc r' < n")
```
```   868       case True
```
```   869       from m n have "Suc m = q' * n + Suc r'" by simp
```
```   870       with True show ?thesis by blast
```
```   871     next
```
```   872       case False then have "n \<le> Suc r'" by auto
```
```   873       moreover from n have "Suc r' \<le> n" by auto
```
```   874       ultimately have "n = Suc r'" by auto
```
```   875       with m have "Suc m = Suc q' * n + 0" by simp
```
```   876       with \<open>n \<noteq> 0\<close> show ?thesis by blast
```
```   877     qed
```
```   878   qed
```
```   879   with that show thesis
```
```   880     using \<open>n \<noteq> 0\<close> by (auto simp add: divmod_nat_rel_def)
```
```   881 qed
```
```   882
```
```   883 text \<open>@{const divmod_nat_rel} is injective:\<close>
```
```   884
```
```   885 qualified lemma divmod_nat_rel_unique:
```
```   886   assumes "divmod_nat_rel m n qr"
```
```   887     and "divmod_nat_rel m n qr'"
```
```   888   shows "qr = qr'"
```
```   889 proof (cases "n = 0")
```
```   890   case True with assms show ?thesis
```
```   891     by (cases qr, cases qr')
```
```   892       (simp add: divmod_nat_rel_def)
```
```   893 next
```
```   894   case False
```
```   895   have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q::nat)"
```
```   896   apply (rule leI)
```
```   897   apply (subst less_iff_Suc_add)
```
```   898   apply (auto simp add: add_mult_distrib)
```
```   899   done
```
```   900   from \<open>n \<noteq> 0\<close> assms have *: "fst qr = fst qr'"
```
```   901     by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)
```
```   902   with assms have "snd qr = snd qr'"
```
```   903     by (simp add: divmod_nat_rel_def)
```
```   904   with * show ?thesis by (cases qr, cases qr') simp
```
```   905 qed
```
```   906
```
```   907 text \<open>
```
```   908   We instantiate divisibility on the natural numbers by
```
```   909   means of @{const divmod_nat_rel}:
```
```   910 \<close>
```
```   911
```
```   912 qualified definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
```
```   913   "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"
```
```   914
```
```   915 qualified lemma divmod_nat_rel_divmod_nat:
```
```   916   "divmod_nat_rel m n (divmod_nat m n)"
```
```   917 proof -
```
```   918   from divmod_nat_rel_ex
```
```   919     obtain qr where rel: "divmod_nat_rel m n qr" .
```
```   920   then show ?thesis
```
```   921   by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)
```
```   922 qed
```
```   923
```
```   924 qualified lemma divmod_nat_unique:
```
```   925   assumes "divmod_nat_rel m n qr"
```
```   926   shows "divmod_nat m n = qr"
```
```   927   using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)
```
```   928
```
```   929 qualified lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)"
```
```   930   by (simp add: Divides.divmod_nat_unique divmod_nat_rel_def)
```
```   931
```
```   932 qualified lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)"
```
```   933   by (simp add: Divides.divmod_nat_unique divmod_nat_rel_def)
```
```   934
```
```   935 qualified lemma divmod_nat_base: "m < n \<Longrightarrow> divmod_nat m n = (0, m)"
```
```   936   by (simp add: divmod_nat_unique divmod_nat_rel_def)
```
```   937
```
```   938 qualified lemma divmod_nat_step:
```
```   939   assumes "0 < n" and "n \<le> m"
```
```   940   shows "divmod_nat m n = apfst Suc (divmod_nat (m - n) n)"
```
```   941 proof (rule divmod_nat_unique)
```
```   942   have "divmod_nat_rel (m - n) n (divmod_nat (m - n) n)"
```
```   943     by (fact divmod_nat_rel_divmod_nat)
```
```   944   then show "divmod_nat_rel m n (apfst Suc (divmod_nat (m - n) n))"
```
```   945     unfolding divmod_nat_rel_def using assms by auto
```
```   946 qed
```
```   947
```
```   948 end
```
```   949
```
```   950 instantiation nat :: semiring_div
```
```   951 begin
```
```   952
```
```   953 definition divide_nat where
```
```   954   div_nat_def: "m div n = fst (Divides.divmod_nat m n)"
```
```   955
```
```   956 definition mod_nat where
```
```   957   "m mod n = snd (Divides.divmod_nat m n)"
```
```   958
```
```   959 lemma fst_divmod_nat [simp]:
```
```   960   "fst (Divides.divmod_nat m n) = m div n"
```
```   961   by (simp add: div_nat_def)
```
```   962
```
```   963 lemma snd_divmod_nat [simp]:
```
```   964   "snd (Divides.divmod_nat m n) = m mod n"
```
```   965   by (simp add: mod_nat_def)
```
```   966
```
```   967 lemma divmod_nat_div_mod:
```
```   968   "Divides.divmod_nat m n = (m div n, m mod n)"
```
```   969   by (simp add: prod_eq_iff)
```
```   970
```
```   971 lemma div_nat_unique:
```
```   972   assumes "divmod_nat_rel m n (q, r)"
```
```   973   shows "m div n = q"
```
```   974   using assms by (auto dest!: Divides.divmod_nat_unique simp add: prod_eq_iff)
```
```   975
```
```   976 lemma mod_nat_unique:
```
```   977   assumes "divmod_nat_rel m n (q, r)"
```
```   978   shows "m mod n = r"
```
```   979   using assms by (auto dest!: Divides.divmod_nat_unique simp add: prod_eq_iff)
```
```   980
```
```   981 lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"
```
```   982   using Divides.divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod)
```
```   983
```
```   984 text \<open>The ''recursion'' equations for @{const divide} and @{const mod}\<close>
```
```   985
```
```   986 lemma div_less [simp]:
```
```   987   fixes m n :: nat
```
```   988   assumes "m < n"
```
```   989   shows "m div n = 0"
```
```   990   using assms Divides.divmod_nat_base by (simp add: prod_eq_iff)
```
```   991
```
```   992 lemma le_div_geq:
```
```   993   fixes m n :: nat
```
```   994   assumes "0 < n" and "n \<le> m"
```
```   995   shows "m div n = Suc ((m - n) div n)"
```
```   996   using assms Divides.divmod_nat_step by (simp add: prod_eq_iff)
```
```   997
```
```   998 lemma mod_less [simp]:
```
```   999   fixes m n :: nat
```
```  1000   assumes "m < n"
```
```  1001   shows "m mod n = m"
```
```  1002   using assms Divides.divmod_nat_base by (simp add: prod_eq_iff)
```
```  1003
```
```  1004 lemma le_mod_geq:
```
```  1005   fixes m n :: nat
```
```  1006   assumes "n \<le> m"
```
```  1007   shows "m mod n = (m - n) mod n"
```
```  1008   using assms Divides.divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)
```
```  1009
```
```  1010 instance proof
```
```  1011   fix m n :: nat
```
```  1012   show "m div n * n + m mod n = m"
```
```  1013     using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)
```
```  1014 next
```
```  1015   fix m n q :: nat
```
```  1016   assume "n \<noteq> 0"
```
```  1017   then show "(q + m * n) div n = m + q div n"
```
```  1018     by (induct m) (simp_all add: le_div_geq)
```
```  1019 next
```
```  1020   fix m n q :: nat
```
```  1021   assume "m \<noteq> 0"
```
```  1022   hence "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)"
```
```  1023     unfolding divmod_nat_rel_def
```
```  1024     by (auto split: if_split_asm, simp_all add: algebra_simps)
```
```  1025   moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .
```
```  1026   ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .
```
```  1027   thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique)
```
```  1028 next
```
```  1029   fix n :: nat show "n div 0 = 0"
```
```  1030     by (simp add: div_nat_def Divides.divmod_nat_zero)
```
```  1031 next
```
```  1032   fix n :: nat show "0 div n = 0"
```
```  1033     by (simp add: div_nat_def Divides.divmod_nat_zero_left)
```
```  1034 qed
```
```  1035
```
```  1036 end
```
```  1037
```
```  1038 instantiation nat :: normalization_semidom
```
```  1039 begin
```
```  1040
```
```  1041 definition normalize_nat
```
```  1042   where [simp]: "normalize = (id :: nat \<Rightarrow> nat)"
```
```  1043
```
```  1044 definition unit_factor_nat
```
```  1045   where "unit_factor n = (if n = 0 then 0 else 1 :: nat)"
```
```  1046
```
```  1047 lemma unit_factor_simps [simp]:
```
```  1048   "unit_factor 0 = (0::nat)"
```
```  1049   "unit_factor (Suc n) = 1"
```
```  1050   by (simp_all add: unit_factor_nat_def)
```
```  1051
```
```  1052 instance
```
```  1053   by standard (simp_all add: unit_factor_nat_def)
```
```  1054
```
```  1055 end
```
```  1056
```
```  1057 lemma divmod_nat_if [code]:
```
```  1058   "Divides.divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
```
```  1059     let (q, r) = Divides.divmod_nat (m - n) n in (Suc q, r))"
```
```  1060   by (simp add: prod_eq_iff case_prod_beta not_less le_div_geq le_mod_geq)
```
```  1061
```
```  1062 text \<open>Simproc for cancelling @{const divide} and @{const mod}\<close>
```
```  1063
```
```  1064 ML_file "~~/src/Provers/Arith/cancel_div_mod.ML"
```
```  1065
```
```  1066 ML \<open>
```
```  1067 structure Cancel_Div_Mod_Nat = Cancel_Div_Mod
```
```  1068 (
```
```  1069   val div_name = @{const_name divide};
```
```  1070   val mod_name = @{const_name mod};
```
```  1071   val mk_binop = HOLogic.mk_binop;
```
```  1072   val mk_plus = HOLogic.mk_binop @{const_name Groups.plus};
```
```  1073   val dest_plus = HOLogic.dest_bin @{const_name Groups.plus} HOLogic.natT;
```
```  1074   fun mk_sum [] = HOLogic.zero
```
```  1075     | mk_sum [t] = t
```
```  1076     | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
```
```  1077   fun dest_sum tm =
```
```  1078     if HOLogic.is_zero tm then []
```
```  1079     else
```
```  1080       (case try HOLogic.dest_Suc tm of
```
```  1081         SOME t => HOLogic.Suc_zero :: dest_sum t
```
```  1082       | NONE =>
```
```  1083           (case try dest_plus tm of
```
```  1084             SOME (t, u) => dest_sum t @ dest_sum u
```
```  1085           | NONE => [tm]));
```
```  1086
```
```  1087   val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
```
```  1088
```
```  1089   val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
```
```  1090     (@{thm add_0_left} :: @{thm add_0_right} :: @{thms ac_simps}))
```
```  1091 )
```
```  1092 \<close>
```
```  1093
```
```  1094 simproc_setup cancel_div_mod_nat ("(m::nat) + n") = \<open>K Cancel_Div_Mod_Nat.proc\<close>
```
```  1095
```
```  1096
```
```  1097 subsubsection \<open>Quotient\<close>
```
```  1098
```
```  1099 lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
```
```  1100 by (simp add: le_div_geq linorder_not_less)
```
```  1101
```
```  1102 lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
```
```  1103 by (simp add: div_geq)
```
```  1104
```
```  1105 lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
```
```  1106 by simp
```
```  1107
```
```  1108 lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
```
```  1109 by simp
```
```  1110
```
```  1111 lemma div_positive:
```
```  1112   fixes m n :: nat
```
```  1113   assumes "n > 0"
```
```  1114   assumes "m \<ge> n"
```
```  1115   shows "m div n > 0"
```
```  1116 proof -
```
```  1117   from \<open>m \<ge> n\<close> obtain q where "m = n + q"
```
```  1118     by (auto simp add: le_iff_add)
```
```  1119   with \<open>n > 0\<close> show ?thesis by simp
```
```  1120 qed
```
```  1121
```
```  1122 lemma div_eq_0_iff: "(a div b::nat) = 0 \<longleftrightarrow> a < b \<or> b = 0"
```
```  1123   by (metis div_less div_positive div_by_0 gr0I less_numeral_extra(3) not_less)
```
```  1124
```
```  1125 subsubsection \<open>Remainder\<close>
```
```  1126
```
```  1127 lemma mod_less_divisor [simp]:
```
```  1128   fixes m n :: nat
```
```  1129   assumes "n > 0"
```
```  1130   shows "m mod n < (n::nat)"
```
```  1131   using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto
```
```  1132
```
```  1133 lemma mod_Suc_le_divisor [simp]:
```
```  1134   "m mod Suc n \<le> n"
```
```  1135   using mod_less_divisor [of "Suc n" m] by arith
```
```  1136
```
```  1137 lemma mod_less_eq_dividend [simp]:
```
```  1138   fixes m n :: nat
```
```  1139   shows "m mod n \<le> m"
```
```  1140 proof (rule add_leD2)
```
```  1141   from mod_div_equality have "m div n * n + m mod n = m" .
```
```  1142   then show "m div n * n + m mod n \<le> m" by auto
```
```  1143 qed
```
```  1144
```
```  1145 lemma mod_geq: "\<not> m < (n::nat) \<Longrightarrow> m mod n = (m - n) mod n"
```
```  1146 by (simp add: le_mod_geq linorder_not_less)
```
```  1147
```
```  1148 lemma mod_if: "m mod (n::nat) = (if m < n then m else (m - n) mod n)"
```
```  1149 by (simp add: le_mod_geq)
```
```  1150
```
```  1151 lemma mod_1 [simp]: "m mod Suc 0 = 0"
```
```  1152 by (induct m) (simp_all add: mod_geq)
```
```  1153
```
```  1154 (* a simple rearrangement of mod_div_equality: *)
```
```  1155 lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
```
```  1156   using mod_div_equality2 [of n m] by arith
```
```  1157
```
```  1158 lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
```
```  1159   apply (drule mod_less_divisor [where m = m])
```
```  1160   apply simp
```
```  1161   done
```
```  1162
```
```  1163 subsubsection \<open>Quotient and Remainder\<close>
```
```  1164
```
```  1165 lemma divmod_nat_rel_mult1_eq:
```
```  1166   "divmod_nat_rel b c (q, r)
```
```  1167    \<Longrightarrow> divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"
```
```  1168 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
```
```  1169
```
```  1170 lemma div_mult1_eq:
```
```  1171   "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"
```
```  1172 by (blast intro: divmod_nat_rel_mult1_eq [THEN div_nat_unique] divmod_nat_rel)
```
```  1173
```
```  1174 lemma divmod_nat_rel_add1_eq:
```
```  1175   "divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br)
```
```  1176    \<Longrightarrow> divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"
```
```  1177 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
```
```  1178
```
```  1179 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
```
```  1180 lemma div_add1_eq:
```
```  1181   "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
```
```  1182 by (blast intro: divmod_nat_rel_add1_eq [THEN div_nat_unique] divmod_nat_rel)
```
```  1183
```
```  1184 lemma divmod_nat_rel_mult2_eq:
```
```  1185   assumes "divmod_nat_rel a b (q, r)"
```
```  1186   shows "divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"
```
```  1187 proof -
```
```  1188   { assume "r < b" and "0 < c"
```
```  1189     then have "b * (q mod c) + r < b * c"
```
```  1190       apply (cut_tac m = q and n = c in mod_less_divisor)
```
```  1191       apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
```
```  1192       apply (erule_tac P = "%x. lhs < rhs x" for lhs rhs in ssubst)
```
```  1193       apply (simp add: add_mult_distrib2)
```
```  1194       done
```
```  1195     then have "r + b * (q mod c) < b * c"
```
```  1196       by (simp add: ac_simps)
```
```  1197   } with assms show ?thesis
```
```  1198     by (auto simp add: divmod_nat_rel_def algebra_simps add_mult_distrib2 [symmetric])
```
```  1199 qed
```
```  1200
```
```  1201 lemma div_mult2_eq: "a div (b * c) = (a div b) div (c::nat)"
```
```  1202 by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique])
```
```  1203
```
```  1204 lemma mod_mult2_eq: "a mod (b * c) = b * (a div b mod c) + a mod (b::nat)"
```
```  1205 by (auto simp add: mult.commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique])
```
```  1206
```
```  1207 instantiation nat :: semiring_numeral_div
```
```  1208 begin
```
```  1209
```
```  1210 definition divmod_nat :: "num \<Rightarrow> num \<Rightarrow> nat \<times> nat"
```
```  1211 where
```
```  1212   divmod'_nat_def: "divmod_nat m n = (numeral m div numeral n, numeral m mod numeral n)"
```
```  1213
```
```  1214 definition divmod_step_nat :: "num \<Rightarrow> nat \<times> nat \<Rightarrow> nat \<times> nat"
```
```  1215 where
```
```  1216   "divmod_step_nat l qr = (let (q, r) = qr
```
```  1217     in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
```
```  1218     else (2 * q, r))"
```
```  1219
```
```  1220 instance
```
```  1221   by standard (auto intro: div_positive simp add: divmod'_nat_def divmod_step_nat_def mod_mult2_eq div_mult2_eq)
```
```  1222
```
```  1223 end
```
```  1224
```
```  1225 declare divmod_algorithm_code [where ?'a = nat, code]
```
```  1226
```
```  1227
```
```  1228 subsubsection \<open>Further Facts about Quotient and Remainder\<close>
```
```  1229
```
```  1230 lemma div_1 [simp]:
```
```  1231   "m div Suc 0 = m"
```
```  1232   using div_by_1 [of m] by simp
```
```  1233
```
```  1234 (* Monotonicity of div in first argument *)
```
```  1235 lemma div_le_mono [rule_format (no_asm)]:
```
```  1236     "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
```
```  1237 apply (case_tac "k=0", simp)
```
```  1238 apply (induct "n" rule: nat_less_induct, clarify)
```
```  1239 apply (case_tac "n<k")
```
```  1240 (* 1  case n<k *)
```
```  1241 apply simp
```
```  1242 (* 2  case n >= k *)
```
```  1243 apply (case_tac "m<k")
```
```  1244 (* 2.1  case m<k *)
```
```  1245 apply simp
```
```  1246 (* 2.2  case m>=k *)
```
```  1247 apply (simp add: div_geq diff_le_mono)
```
```  1248 done
```
```  1249
```
```  1250 (* Antimonotonicity of div in second argument *)
```
```  1251 lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
```
```  1252 apply (subgoal_tac "0<n")
```
```  1253  prefer 2 apply simp
```
```  1254 apply (induct_tac k rule: nat_less_induct)
```
```  1255 apply (rename_tac "k")
```
```  1256 apply (case_tac "k<n", simp)
```
```  1257 apply (subgoal_tac "~ (k<m) ")
```
```  1258  prefer 2 apply simp
```
```  1259 apply (simp add: div_geq)
```
```  1260 apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
```
```  1261  prefer 2
```
```  1262  apply (blast intro: div_le_mono diff_le_mono2)
```
```  1263 apply (rule le_trans, simp)
```
```  1264 apply (simp)
```
```  1265 done
```
```  1266
```
```  1267 lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
```
```  1268 apply (case_tac "n=0", simp)
```
```  1269 apply (subgoal_tac "m div n \<le> m div 1", simp)
```
```  1270 apply (rule div_le_mono2)
```
```  1271 apply (simp_all (no_asm_simp))
```
```  1272 done
```
```  1273
```
```  1274 (* Similar for "less than" *)
```
```  1275 lemma div_less_dividend [simp]:
```
```  1276   "\<lbrakk>(1::nat) < n; 0 < m\<rbrakk> \<Longrightarrow> m div n < m"
```
```  1277 apply (induct m rule: nat_less_induct)
```
```  1278 apply (rename_tac "m")
```
```  1279 apply (case_tac "m<n", simp)
```
```  1280 apply (subgoal_tac "0<n")
```
```  1281  prefer 2 apply simp
```
```  1282 apply (simp add: div_geq)
```
```  1283 apply (case_tac "n<m")
```
```  1284  apply (subgoal_tac "(m-n) div n < (m-n) ")
```
```  1285   apply (rule impI less_trans_Suc)+
```
```  1286 apply assumption
```
```  1287   apply (simp_all)
```
```  1288 done
```
```  1289
```
```  1290 text\<open>A fact for the mutilated chess board\<close>
```
```  1291 lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
```
```  1292 apply (case_tac "n=0", simp)
```
```  1293 apply (induct "m" rule: nat_less_induct)
```
```  1294 apply (case_tac "Suc (na) <n")
```
```  1295 (* case Suc(na) < n *)
```
```  1296 apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
```
```  1297 (* case n \<le> Suc(na) *)
```
```  1298 apply (simp add: linorder_not_less le_Suc_eq mod_geq)
```
```  1299 apply (auto simp add: Suc_diff_le le_mod_geq)
```
```  1300 done
```
```  1301
```
```  1302 lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
```
```  1303 by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
```
```  1304
```
```  1305 lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
```
```  1306
```
```  1307 (*Loses information, namely we also have r<d provided d is nonzero*)
```
```  1308 lemma mod_eqD:
```
```  1309   fixes m d r q :: nat
```
```  1310   assumes "m mod d = r"
```
```  1311   shows "\<exists>q. m = r + q * d"
```
```  1312 proof -
```
```  1313   from mod_div_equality obtain q where "q * d + m mod d = m" by blast
```
```  1314   with assms have "m = r + q * d" by simp
```
```  1315   then show ?thesis ..
```
```  1316 qed
```
```  1317
```
```  1318 lemma split_div:
```
```  1319  "P(n div k :: nat) =
```
```  1320  ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
```
```  1321  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
```
```  1322 proof
```
```  1323   assume P: ?P
```
```  1324   show ?Q
```
```  1325   proof (cases)
```
```  1326     assume "k = 0"
```
```  1327     with P show ?Q by simp
```
```  1328   next
```
```  1329     assume not0: "k \<noteq> 0"
```
```  1330     thus ?Q
```
```  1331     proof (simp, intro allI impI)
```
```  1332       fix i j
```
```  1333       assume n: "n = k*i + j" and j: "j < k"
```
```  1334       show "P i"
```
```  1335       proof (cases)
```
```  1336         assume "i = 0"
```
```  1337         with n j P show "P i" by simp
```
```  1338       next
```
```  1339         assume "i \<noteq> 0"
```
```  1340         with not0 n j P show "P i" by(simp add:ac_simps)
```
```  1341       qed
```
```  1342     qed
```
```  1343   qed
```
```  1344 next
```
```  1345   assume Q: ?Q
```
```  1346   show ?P
```
```  1347   proof (cases)
```
```  1348     assume "k = 0"
```
```  1349     with Q show ?P by simp
```
```  1350   next
```
```  1351     assume not0: "k \<noteq> 0"
```
```  1352     with Q have R: ?R by simp
```
```  1353     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
```
```  1354     show ?P by simp
```
```  1355   qed
```
```  1356 qed
```
```  1357
```
```  1358 lemma split_div_lemma:
```
```  1359   assumes "0 < n"
```
```  1360   shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m::nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1361 proof
```
```  1362   assume ?rhs
```
```  1363   with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp
```
```  1364   then have A: "n * q \<le> m" by simp
```
```  1365   have "n - (m mod n) > 0" using mod_less_divisor assms by auto
```
```  1366   then have "m < m + (n - (m mod n))" by simp
```
```  1367   then have "m < n + (m - (m mod n))" by simp
```
```  1368   with nq have "m < n + n * q" by simp
```
```  1369   then have B: "m < n * Suc q" by simp
```
```  1370   from A B show ?lhs ..
```
```  1371 next
```
```  1372   assume P: ?lhs
```
```  1373   then have "divmod_nat_rel m n (q, m - n * q)"
```
```  1374     unfolding divmod_nat_rel_def by (auto simp add: ac_simps)
```
```  1375   then have "m div n = q"
```
```  1376     by (rule div_nat_unique)
```
```  1377   then show ?rhs by simp
```
```  1378 qed
```
```  1379
```
```  1380 theorem split_div':
```
```  1381   "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
```
```  1382    (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
```
```  1383   apply (cases "0 < n")
```
```  1384   apply (simp only: add: split_div_lemma)
```
```  1385   apply simp_all
```
```  1386   done
```
```  1387
```
```  1388 lemma split_mod:
```
```  1389  "P(n mod k :: nat) =
```
```  1390  ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
```
```  1391  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
```
```  1392 proof
```
```  1393   assume P: ?P
```
```  1394   show ?Q
```
```  1395   proof (cases)
```
```  1396     assume "k = 0"
```
```  1397     with P show ?Q by simp
```
```  1398   next
```
```  1399     assume not0: "k \<noteq> 0"
```
```  1400     thus ?Q
```
```  1401     proof (simp, intro allI impI)
```
```  1402       fix i j
```
```  1403       assume "n = k*i + j" "j < k"
```
```  1404       thus "P j" using not0 P by (simp add: ac_simps)
```
```  1405     qed
```
```  1406   qed
```
```  1407 next
```
```  1408   assume Q: ?Q
```
```  1409   show ?P
```
```  1410   proof (cases)
```
```  1411     assume "k = 0"
```
```  1412     with Q show ?P by simp
```
```  1413   next
```
```  1414     assume not0: "k \<noteq> 0"
```
```  1415     with Q have R: ?R by simp
```
```  1416     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
```
```  1417     show ?P by simp
```
```  1418   qed
```
```  1419 qed
```
```  1420
```
```  1421 theorem mod_div_equality' [nitpick_unfold]: "(m::nat) mod n = m - (m div n) * n"
```
```  1422   using mod_div_equality [of m n] by arith
```
```  1423
```
```  1424 lemma div_mod_equality': "(m::nat) div n * n = m - m mod n"
```
```  1425   using mod_div_equality [of m n] by arith
```
```  1426 (* FIXME: very similar to mult_div_cancel *)
```
```  1427
```
```  1428 lemma div_eq_dividend_iff: "a \<noteq> 0 \<Longrightarrow> (a :: nat) div b = a \<longleftrightarrow> b = 1"
```
```  1429   apply rule
```
```  1430   apply (cases "b = 0")
```
```  1431   apply simp_all
```
```  1432   apply (metis (full_types) One_nat_def Suc_lessI div_less_dividend less_not_refl3)
```
```  1433   done
```
```  1434
```
```  1435 lemma (in field_char_0) of_nat_div:
```
```  1436   "of_nat (m div n) = ((of_nat m - of_nat (m mod n)) / of_nat n)"
```
```  1437 proof -
```
```  1438   have "of_nat (m div n) = ((of_nat (m div n * n + m mod n) - of_nat (m mod n)) / of_nat n :: 'a)"
```
```  1439     unfolding of_nat_add by (cases "n = 0") simp_all
```
```  1440   then show ?thesis
```
```  1441     by simp
```
```  1442 qed
```
```  1443
```
```  1444
```
```  1445 subsubsection \<open>An ``induction'' law for modulus arithmetic.\<close>
```
```  1446
```
```  1447 lemma mod_induct_0:
```
```  1448   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
```
```  1449   and base: "P i" and i: "i<p"
```
```  1450   shows "P 0"
```
```  1451 proof (rule ccontr)
```
```  1452   assume contra: "\<not>(P 0)"
```
```  1453   from i have p: "0<p" by simp
```
```  1454   have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
```
```  1455   proof
```
```  1456     fix k
```
```  1457     show "?A k"
```
```  1458     proof (induct k)
```
```  1459       show "?A 0" by simp  \<comment> "by contradiction"
```
```  1460     next
```
```  1461       fix n
```
```  1462       assume ih: "?A n"
```
```  1463       show "?A (Suc n)"
```
```  1464       proof (clarsimp)
```
```  1465         assume y: "P (p - Suc n)"
```
```  1466         have n: "Suc n < p"
```
```  1467         proof (rule ccontr)
```
```  1468           assume "\<not>(Suc n < p)"
```
```  1469           hence "p - Suc n = 0"
```
```  1470             by simp
```
```  1471           with y contra show "False"
```
```  1472             by simp
```
```  1473         qed
```
```  1474         hence n2: "Suc (p - Suc n) = p-n" by arith
```
```  1475         from p have "p - Suc n < p" by arith
```
```  1476         with y step have z: "P ((Suc (p - Suc n)) mod p)"
```
```  1477           by blast
```
```  1478         show "False"
```
```  1479         proof (cases "n=0")
```
```  1480           case True
```
```  1481           with z n2 contra show ?thesis by simp
```
```  1482         next
```
```  1483           case False
```
```  1484           with p have "p-n < p" by arith
```
```  1485           with z n2 False ih show ?thesis by simp
```
```  1486         qed
```
```  1487       qed
```
```  1488     qed
```
```  1489   qed
```
```  1490   moreover
```
```  1491   from i obtain k where "0<k \<and> i+k=p"
```
```  1492     by (blast dest: less_imp_add_positive)
```
```  1493   hence "0<k \<and> i=p-k" by auto
```
```  1494   moreover
```
```  1495   note base
```
```  1496   ultimately
```
```  1497   show "False" by blast
```
```  1498 qed
```
```  1499
```
```  1500 lemma mod_induct:
```
```  1501   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
```
```  1502   and base: "P i" and i: "i<p" and j: "j<p"
```
```  1503   shows "P j"
```
```  1504 proof -
```
```  1505   have "\<forall>j<p. P j"
```
```  1506   proof
```
```  1507     fix j
```
```  1508     show "j<p \<longrightarrow> P j" (is "?A j")
```
```  1509     proof (induct j)
```
```  1510       from step base i show "?A 0"
```
```  1511         by (auto elim: mod_induct_0)
```
```  1512     next
```
```  1513       fix k
```
```  1514       assume ih: "?A k"
```
```  1515       show "?A (Suc k)"
```
```  1516       proof
```
```  1517         assume suc: "Suc k < p"
```
```  1518         hence k: "k<p" by simp
```
```  1519         with ih have "P k" ..
```
```  1520         with step k have "P (Suc k mod p)"
```
```  1521           by blast
```
```  1522         moreover
```
```  1523         from suc have "Suc k mod p = Suc k"
```
```  1524           by simp
```
```  1525         ultimately
```
```  1526         show "P (Suc k)" by simp
```
```  1527       qed
```
```  1528     qed
```
```  1529   qed
```
```  1530   with j show ?thesis by blast
```
```  1531 qed
```
```  1532
```
```  1533 lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"
```
```  1534   by (simp add: numeral_2_eq_2 le_div_geq)
```
```  1535
```
```  1536 lemma mod2_Suc_Suc [simp]: "Suc (Suc m) mod 2 = m mod 2"
```
```  1537   by (simp add: numeral_2_eq_2 le_mod_geq)
```
```  1538
```
```  1539 lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
```
```  1540 by (simp add: mult_2 [symmetric])
```
```  1541
```
```  1542 lemma mod2_gr_0 [simp]: "0 < (m::nat) mod 2 \<longleftrightarrow> m mod 2 = 1"
```
```  1543 proof -
```
```  1544   { fix n :: nat have  "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (cases n) simp_all }
```
```  1545   moreover have "m mod 2 < 2" by simp
```
```  1546   ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" .
```
```  1547   then show ?thesis by auto
```
```  1548 qed
```
```  1549
```
```  1550 text\<open>These lemmas collapse some needless occurrences of Suc:
```
```  1551     at least three Sucs, since two and fewer are rewritten back to Suc again!
```
```  1552     We already have some rules to simplify operands smaller than 3.\<close>
```
```  1553
```
```  1554 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
```
```  1555 by (simp add: Suc3_eq_add_3)
```
```  1556
```
```  1557 lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
```
```  1558 by (simp add: Suc3_eq_add_3)
```
```  1559
```
```  1560 lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
```
```  1561 by (simp add: Suc3_eq_add_3)
```
```  1562
```
```  1563 lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
```
```  1564 by (simp add: Suc3_eq_add_3)
```
```  1565
```
```  1566 lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v
```
```  1567 lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v
```
```  1568
```
```  1569 lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1"
```
```  1570 apply (induct "m")
```
```  1571 apply (simp_all add: mod_Suc)
```
```  1572 done
```
```  1573
```
```  1574 declare Suc_times_mod_eq [of "numeral w", simp] for w
```
```  1575
```
```  1576 lemma mod_greater_zero_iff_not_dvd:
```
```  1577   fixes m n :: nat
```
```  1578   shows "m mod n > 0 \<longleftrightarrow> \<not> n dvd m"
```
```  1579   by (simp add: dvd_eq_mod_eq_0)
```
```  1580
```
```  1581 lemma Suc_div_le_mono [simp]: "n div k \<le> (Suc n) div k"
```
```  1582 by (simp add: div_le_mono)
```
```  1583
```
```  1584 lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"
```
```  1585 by (cases n) simp_all
```
```  1586
```
```  1587 lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"
```
```  1588 proof -
```
```  1589   from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all
```
```  1590   from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp
```
```  1591 qed
```
```  1592
```
```  1593 lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"
```
```  1594 proof -
```
```  1595   have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp
```
```  1596   also have "... = Suc m mod n" by (rule mod_mult_self3)
```
```  1597   finally show ?thesis .
```
```  1598 qed
```
```  1599
```
```  1600 lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"
```
```  1601 apply (subst mod_Suc [of m])
```
```  1602 apply (subst mod_Suc [of "m mod n"], simp)
```
```  1603 done
```
```  1604
```
```  1605 lemma mod_2_not_eq_zero_eq_one_nat:
```
```  1606   fixes n :: nat
```
```  1607   shows "n mod 2 \<noteq> 0 \<longleftrightarrow> n mod 2 = 1"
```
```  1608   by (fact not_mod_2_eq_0_eq_1)
```
```  1609
```
```  1610 lemma even_Suc_div_two [simp]:
```
```  1611   "even n \<Longrightarrow> Suc n div 2 = n div 2"
```
```  1612   using even_succ_div_two [of n] by simp
```
```  1613
```
```  1614 lemma odd_Suc_div_two [simp]:
```
```  1615   "odd n \<Longrightarrow> Suc n div 2 = Suc (n div 2)"
```
```  1616   using odd_succ_div_two [of n] by simp
```
```  1617
```
```  1618 lemma odd_two_times_div_two_nat [simp]:
```
```  1619   assumes "odd n"
```
```  1620   shows "2 * (n div 2) = n - (1 :: nat)"
```
```  1621 proof -
```
```  1622   from assms have "2 * (n div 2) + 1 = n"
```
```  1623     by (rule odd_two_times_div_two_succ)
```
```  1624   then have "Suc (2 * (n div 2)) - 1 = n - 1"
```
```  1625     by simp
```
```  1626   then show ?thesis
```
```  1627     by simp
```
```  1628 qed
```
```  1629
```
```  1630 lemma parity_induct [case_names zero even odd]:
```
```  1631   assumes zero: "P 0"
```
```  1632   assumes even: "\<And>n. P n \<Longrightarrow> P (2 * n)"
```
```  1633   assumes odd: "\<And>n. P n \<Longrightarrow> P (Suc (2 * n))"
```
```  1634   shows "P n"
```
```  1635 proof (induct n rule: less_induct)
```
```  1636   case (less n)
```
```  1637   show "P n"
```
```  1638   proof (cases "n = 0")
```
```  1639     case True with zero show ?thesis by simp
```
```  1640   next
```
```  1641     case False
```
```  1642     with less have hyp: "P (n div 2)" by simp
```
```  1643     show ?thesis
```
```  1644     proof (cases "even n")
```
```  1645       case True
```
```  1646       with hyp even [of "n div 2"] show ?thesis
```
```  1647         by simp
```
```  1648     next
```
```  1649       case False
```
```  1650       with hyp odd [of "n div 2"] show ?thesis
```
```  1651         by simp
```
```  1652     qed
```
```  1653   qed
```
```  1654 qed
```
```  1655
```
```  1656 lemma Suc_0_div_numeral [simp]:
```
```  1657   fixes k l :: num
```
```  1658   shows "Suc 0 div numeral k = fst (divmod Num.One k)"
```
```  1659   by (simp_all add: fst_divmod)
```
```  1660
```
```  1661 lemma Suc_0_mod_numeral [simp]:
```
```  1662   fixes k l :: num
```
```  1663   shows "Suc 0 mod numeral k = snd (divmod Num.One k)"
```
```  1664   by (simp_all add: snd_divmod)
```
```  1665
```
```  1666 lemma cut_eq_simps: \<comment> \<open>rewriting equivalence on \<open>n mod 2 ^ q\<close>\<close>
```
```  1667   fixes m n q :: num
```
```  1668   shows
```
```  1669     "numeral n mod numeral Num.One = (0::nat)
```
```  1670       \<longleftrightarrow> True"
```
```  1671     "numeral (Num.Bit0 n) mod numeral (Num.Bit0 q) = (0::nat)
```
```  1672       \<longleftrightarrow> numeral n mod numeral q = (0::nat)"
```
```  1673     "numeral (Num.Bit1 n) mod numeral (Num.Bit0 q) = (0::nat)
```
```  1674       \<longleftrightarrow> False"
```
```  1675     "numeral m mod numeral Num.One = (numeral n mod numeral Num.One :: nat)
```
```  1676       \<longleftrightarrow> True"
```
```  1677     "numeral Num.One mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q) :: nat)
```
```  1678       \<longleftrightarrow> True"
```
```  1679     "numeral Num.One mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q) :: nat)
```
```  1680       \<longleftrightarrow> False"
```
```  1681     "numeral Num.One mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q) :: nat)
```
```  1682       \<longleftrightarrow> (numeral n mod numeral q :: nat) = 0"
```
```  1683     "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q) :: nat)
```
```  1684       \<longleftrightarrow> False"
```
```  1685     "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q) :: nat)
```
```  1686       \<longleftrightarrow> numeral m mod numeral q = (numeral n mod numeral q :: nat)"
```
```  1687     "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q) :: nat)
```
```  1688       \<longleftrightarrow> False"
```
```  1689     "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q) :: nat)
```
```  1690       \<longleftrightarrow> (numeral m mod numeral q :: nat) = 0"
```
```  1691     "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q) :: nat)
```
```  1692       \<longleftrightarrow> False"
```
```  1693     "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q) :: nat)
```
```  1694       \<longleftrightarrow> numeral m mod numeral q = (numeral n mod numeral q :: nat)"
```
```  1695   by (auto simp add: case_prod_beta Suc_double_not_eq_double double_not_eq_Suc_double)
```
```  1696
```
```  1697
```
```  1698 subsection \<open>Division on @{typ int}\<close>
```
```  1699
```
```  1700 definition divmod_int_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" \<comment> \<open>definition of quotient and remainder\<close>
```
```  1701   where "divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and>
```
```  1702     (if 0 < b then 0 \<le> r \<and> r < b else if b < 0 then b < r \<and> r \<le> 0 else q = 0))"
```
```  1703
```
```  1704 lemma unique_quotient_lemma:
```
```  1705   "b * q' + r' \<le> b * q + r \<Longrightarrow> 0 \<le> r' \<Longrightarrow> r' < b \<Longrightarrow> r < b \<Longrightarrow> q' \<le> (q::int)"
```
```  1706 apply (subgoal_tac "r' + b * (q'-q) \<le> r")
```
```  1707  prefer 2 apply (simp add: right_diff_distrib)
```
```  1708 apply (subgoal_tac "0 < b * (1 + q - q') ")
```
```  1709 apply (erule_tac [2] order_le_less_trans)
```
```  1710  prefer 2 apply (simp add: right_diff_distrib distrib_left)
```
```  1711 apply (subgoal_tac "b * q' < b * (1 + q) ")
```
```  1712  prefer 2 apply (simp add: right_diff_distrib distrib_left)
```
```  1713 apply (simp add: mult_less_cancel_left)
```
```  1714 done
```
```  1715
```
```  1716 lemma unique_quotient_lemma_neg:
```
```  1717   "b * q' + r' \<le> b*q + r \<Longrightarrow> r \<le> 0 \<Longrightarrow> b < r \<Longrightarrow> b < r' \<Longrightarrow> q \<le> (q'::int)"
```
```  1718   by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma) auto
```
```  1719
```
```  1720 lemma unique_quotient:
```
```  1721   "divmod_int_rel a b (q, r) \<Longrightarrow> divmod_int_rel a b (q', r') \<Longrightarrow> q = q'"
```
```  1722 apply (simp add: divmod_int_rel_def linorder_neq_iff split: if_split_asm)
```
```  1723 apply (blast intro: order_antisym
```
```  1724              dest: order_eq_refl [THEN unique_quotient_lemma]
```
```  1725              order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
```
```  1726 done
```
```  1727
```
```  1728 lemma unique_remainder:
```
```  1729   "divmod_int_rel a b (q, r) \<Longrightarrow> divmod_int_rel a b (q', r') \<Longrightarrow> r = r'"
```
```  1730 apply (subgoal_tac "q = q'")
```
```  1731  apply (simp add: divmod_int_rel_def)
```
```  1732 apply (blast intro: unique_quotient)
```
```  1733 done
```
```  1734
```
```  1735 instantiation int :: Divides.div
```
```  1736 begin
```
```  1737
```
```  1738 definition divide_int
```
```  1739   where "k div l = (if l = 0 \<or> k = 0 then 0
```
```  1740     else if k > 0 \<and> l > 0 \<or> k < 0 \<and> l < 0
```
```  1741       then int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)
```
```  1742       else
```
```  1743         if l dvd k then - int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)
```
```  1744         else - int (Suc (nat \<bar>k\<bar> div nat \<bar>l\<bar>)))"
```
```  1745
```
```  1746 definition mod_int
```
```  1747   where "k mod l = (if l = 0 then k else if l dvd k then 0
```
```  1748     else if k > 0 \<and> l > 0 \<or> k < 0 \<and> l < 0
```
```  1749       then sgn l * int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>)
```
```  1750       else sgn l * (\<bar>l\<bar> - int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>)))"
```
```  1751
```
```  1752 instance ..
```
```  1753
```
```  1754 end
```
```  1755
```
```  1756 lemma divmod_int_rel:
```
```  1757   "divmod_int_rel k l (k div l, k mod l)"
```
```  1758   unfolding divmod_int_rel_def divide_int_def mod_int_def
```
```  1759   apply (cases k rule: int_cases3)
```
```  1760   apply (simp add: mod_greater_zero_iff_not_dvd not_le algebra_simps)
```
```  1761   apply (cases l rule: int_cases3)
```
```  1762   apply (simp add: mod_greater_zero_iff_not_dvd not_le algebra_simps)
```
```  1763   apply (simp_all del: of_nat_add of_nat_mult add: mod_greater_zero_iff_not_dvd not_le algebra_simps int_dvd_iff of_nat_add [symmetric] of_nat_mult [symmetric])
```
```  1764   apply (cases l rule: int_cases3)
```
```  1765   apply (simp_all del: of_nat_add of_nat_mult add: not_le algebra_simps int_dvd_iff of_nat_add [symmetric] of_nat_mult [symmetric])
```
```  1766   done
```
```  1767
```
```  1768 instantiation int :: ring_div
```
```  1769 begin
```
```  1770
```
```  1771 subsubsection \<open>Uniqueness and Monotonicity of Quotients and Remainders\<close>
```
```  1772
```
```  1773 lemma divmod_int_unique:
```
```  1774   assumes "divmod_int_rel k l (q, r)"
```
```  1775   shows div_int_unique: "k div l = q" and mod_int_unique: "k mod l = r"
```
```  1776   using assms divmod_int_rel [of k l]
```
```  1777   using unique_quotient [of k l] unique_remainder [of k l]
```
```  1778   by auto
```
```  1779
```
```  1780 instance
```
```  1781 proof
```
```  1782   fix a b :: int
```
```  1783   show "a div b * b + a mod b = a"
```
```  1784     using divmod_int_rel [of a b]
```
```  1785     unfolding divmod_int_rel_def by (simp add: mult.commute)
```
```  1786 next
```
```  1787   fix a b c :: int
```
```  1788   assume "b \<noteq> 0"
```
```  1789   hence "divmod_int_rel (a + c * b) b (c + a div b, a mod b)"
```
```  1790     using divmod_int_rel [of a b]
```
```  1791     unfolding divmod_int_rel_def by (auto simp: algebra_simps)
```
```  1792   thus "(a + c * b) div b = c + a div b"
```
```  1793     by (rule div_int_unique)
```
```  1794 next
```
```  1795   fix a b c :: int
```
```  1796   assume "c \<noteq> 0"
```
```  1797   hence "\<And>q r. divmod_int_rel a b (q, r)
```
```  1798     \<Longrightarrow> divmod_int_rel (c * a) (c * b) (q, c * r)"
```
```  1799     unfolding divmod_int_rel_def
```
```  1800     by - (rule linorder_cases [of 0 b], auto simp: algebra_simps
```
```  1801       mult_less_0_iff zero_less_mult_iff mult_strict_right_mono
```
```  1802       mult_strict_right_mono_neg zero_le_mult_iff mult_le_0_iff)
```
```  1803   hence "divmod_int_rel (c * a) (c * b) (a div b, c * (a mod b))"
```
```  1804     using divmod_int_rel [of a b] .
```
```  1805   thus "(c * a) div (c * b) = a div b"
```
```  1806     by (rule div_int_unique)
```
```  1807 next
```
```  1808   fix a :: int show "a div 0 = 0"
```
```  1809     by (rule div_int_unique, simp add: divmod_int_rel_def)
```
```  1810 next
```
```  1811   fix a :: int show "0 div a = 0"
```
```  1812     by (rule div_int_unique, auto simp add: divmod_int_rel_def)
```
```  1813 qed
```
```  1814
```
```  1815 end
```
```  1816
```
```  1817 lemma is_unit_int:
```
```  1818   "is_unit (k::int) \<longleftrightarrow> k = 1 \<or> k = - 1"
```
```  1819   by auto
```
```  1820
```
```  1821 instantiation int :: normalization_semidom
```
```  1822 begin
```
```  1823
```
```  1824 definition normalize_int
```
```  1825   where [simp]: "normalize = (abs :: int \<Rightarrow> int)"
```
```  1826
```
```  1827 definition unit_factor_int
```
```  1828   where [simp]: "unit_factor = (sgn :: int \<Rightarrow> int)"
```
```  1829
```
```  1830 instance
```
```  1831 proof
```
```  1832   fix k :: int
```
```  1833   assume "k \<noteq> 0"
```
```  1834   then have "\<bar>sgn k\<bar> = 1"
```
```  1835     by (cases "0::int" k rule: linorder_cases) simp_all
```
```  1836   then show "is_unit (unit_factor k)"
```
```  1837     by simp
```
```  1838 qed (simp_all add: sgn_times mult_sgn_abs)
```
```  1839
```
```  1840 end
```
```  1841
```
```  1842 text\<open>Basic laws about division and remainder\<close>
```
```  1843
```
```  1844 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
```
```  1845   by (fact mod_div_equality2 [symmetric])
```
```  1846
```
```  1847 lemma zdiv_int: "int (a div b) = int a div int b"
```
```  1848   by (simp add: divide_int_def)
```
```  1849
```
```  1850 lemma zmod_int: "int (a mod b) = int a mod int b"
```
```  1851   by (simp add: mod_int_def int_dvd_iff)
```
```  1852
```
```  1853 text \<open>Tool setup\<close>
```
```  1854
```
```  1855 ML \<open>
```
```  1856 structure Cancel_Div_Mod_Int = Cancel_Div_Mod
```
```  1857 (
```
```  1858   val div_name = @{const_name Rings.divide};
```
```  1859   val mod_name = @{const_name mod};
```
```  1860   val mk_binop = HOLogic.mk_binop;
```
```  1861   val mk_sum = Arith_Data.mk_sum HOLogic.intT;
```
```  1862   val dest_sum = Arith_Data.dest_sum;
```
```  1863
```
```  1864   val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
```
```  1865
```
```  1866   val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
```
```  1867     (@{thm diff_conv_add_uminus} :: @{thms add_0_left add_0_right} @ @{thms ac_simps}))
```
```  1868 )
```
```  1869 \<close>
```
```  1870
```
```  1871 simproc_setup cancel_div_mod_int ("(k::int) + l") = \<open>K Cancel_Div_Mod_Int.proc\<close>
```
```  1872
```
```  1873 lemma pos_mod_conj: "(0::int) < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b"
```
```  1874   using divmod_int_rel [of a b]
```
```  1875   by (auto simp add: divmod_int_rel_def prod_eq_iff)
```
```  1876
```
```  1877 lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]
```
```  1878    and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]
```
```  1879
```
```  1880 lemma neg_mod_conj: "b < (0::int) \<Longrightarrow> a mod b \<le> 0 \<and> b < a mod b"
```
```  1881   using divmod_int_rel [of a b]
```
```  1882   by (auto simp add: divmod_int_rel_def prod_eq_iff)
```
```  1883
```
```  1884 lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]
```
```  1885    and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]
```
```  1886
```
```  1887
```
```  1888 subsubsection \<open>General Properties of div and mod\<close>
```
```  1889
```
```  1890 lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
```
```  1891 apply (rule div_int_unique)
```
```  1892 apply (auto simp add: divmod_int_rel_def)
```
```  1893 done
```
```  1894
```
```  1895 lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
```
```  1896 apply (rule div_int_unique)
```
```  1897 apply (auto simp add: divmod_int_rel_def)
```
```  1898 done
```
```  1899
```
```  1900 lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
```
```  1901 apply (rule div_int_unique)
```
```  1902 apply (auto simp add: divmod_int_rel_def)
```
```  1903 done
```
```  1904
```
```  1905 (*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
```
```  1906
```
```  1907 lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
```
```  1908 apply (rule_tac q = 0 in mod_int_unique)
```
```  1909 apply (auto simp add: divmod_int_rel_def)
```
```  1910 done
```
```  1911
```
```  1912 lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
```
```  1913 apply (rule_tac q = 0 in mod_int_unique)
```
```  1914 apply (auto simp add: divmod_int_rel_def)
```
```  1915 done
```
```  1916
```
```  1917 lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
```
```  1918 apply (rule_tac q = "-1" in mod_int_unique)
```
```  1919 apply (auto simp add: divmod_int_rel_def)
```
```  1920 done
```
```  1921
```
```  1922 text\<open>There is no \<open>mod_neg_pos_trivial\<close>.\<close>
```
```  1923
```
```  1924
```
```  1925 subsubsection \<open>Laws for div and mod with Unary Minus\<close>
```
```  1926
```
```  1927 lemma zminus1_lemma:
```
```  1928      "divmod_int_rel a b (q, r) ==> b \<noteq> 0
```
```  1929       ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,
```
```  1930                           if r=0 then 0 else b-r)"
```
```  1931 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)
```
```  1932
```
```  1933
```
```  1934 lemma zdiv_zminus1_eq_if:
```
```  1935      "b \<noteq> (0::int)
```
```  1936       ==> (-a) div b =
```
```  1937           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
```
```  1938 by (blast intro: divmod_int_rel [THEN zminus1_lemma, THEN div_int_unique])
```
```  1939
```
```  1940 lemma zmod_zminus1_eq_if:
```
```  1941      "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
```
```  1942 apply (case_tac "b = 0", simp)
```
```  1943 apply (blast intro: divmod_int_rel [THEN zminus1_lemma, THEN mod_int_unique])
```
```  1944 done
```
```  1945
```
```  1946 lemma zmod_zminus1_not_zero:
```
```  1947   fixes k l :: int
```
```  1948   shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
```
```  1949   unfolding zmod_zminus1_eq_if by auto
```
```  1950
```
```  1951 lemma zdiv_zminus2_eq_if:
```
```  1952      "b \<noteq> (0::int)
```
```  1953       ==> a div (-b) =
```
```  1954           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
```
```  1955 by (simp add: zdiv_zminus1_eq_if div_minus_right)
```
```  1956
```
```  1957 lemma zmod_zminus2_eq_if:
```
```  1958      "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
```
```  1959 by (simp add: zmod_zminus1_eq_if mod_minus_right)
```
```  1960
```
```  1961 lemma zmod_zminus2_not_zero:
```
```  1962   fixes k l :: int
```
```  1963   shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
```
```  1964   unfolding zmod_zminus2_eq_if by auto
```
```  1965
```
```  1966
```
```  1967 subsubsection \<open>Monotonicity in the First Argument (Dividend)\<close>
```
```  1968
```
```  1969 lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
```
```  1970 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
```
```  1971 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
```
```  1972 apply (rule unique_quotient_lemma)
```
```  1973 apply (erule subst)
```
```  1974 apply (erule subst, simp_all)
```
```  1975 done
```
```  1976
```
```  1977 lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
```
```  1978 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
```
```  1979 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
```
```  1980 apply (rule unique_quotient_lemma_neg)
```
```  1981 apply (erule subst)
```
```  1982 apply (erule subst, simp_all)
```
```  1983 done
```
```  1984
```
```  1985
```
```  1986 subsubsection \<open>Monotonicity in the Second Argument (Divisor)\<close>
```
```  1987
```
```  1988 lemma q_pos_lemma:
```
```  1989      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
```
```  1990 apply (subgoal_tac "0 < b'* (q' + 1) ")
```
```  1991  apply (simp add: zero_less_mult_iff)
```
```  1992 apply (simp add: distrib_left)
```
```  1993 done
```
```  1994
```
```  1995 lemma zdiv_mono2_lemma:
```
```  1996      "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';
```
```  1997          r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]
```
```  1998       ==> q \<le> (q'::int)"
```
```  1999 apply (frule q_pos_lemma, assumption+)
```
```  2000 apply (subgoal_tac "b*q < b* (q' + 1) ")
```
```  2001  apply (simp add: mult_less_cancel_left)
```
```  2002 apply (subgoal_tac "b*q = r' - r + b'*q'")
```
```  2003  prefer 2 apply simp
```
```  2004 apply (simp (no_asm_simp) add: distrib_left)
```
```  2005 apply (subst add.commute, rule add_less_le_mono, arith)
```
```  2006 apply (rule mult_right_mono, auto)
```
```  2007 done
```
```  2008
```
```  2009 lemma zdiv_mono2:
```
```  2010      "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
```
```  2011 apply (subgoal_tac "b \<noteq> 0")
```
```  2012  prefer 2 apply arith
```
```  2013 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
```
```  2014 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
```
```  2015 apply (rule zdiv_mono2_lemma)
```
```  2016 apply (erule subst)
```
```  2017 apply (erule subst, simp_all)
```
```  2018 done
```
```  2019
```
```  2020 lemma q_neg_lemma:
```
```  2021      "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
```
```  2022 apply (subgoal_tac "b'*q' < 0")
```
```  2023  apply (simp add: mult_less_0_iff, arith)
```
```  2024 done
```
```  2025
```
```  2026 lemma zdiv_mono2_neg_lemma:
```
```  2027      "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;
```
```  2028          r < b;  0 \<le> r';  0 < b';  b' \<le> b |]
```
```  2029       ==> q' \<le> (q::int)"
```
```  2030 apply (frule q_neg_lemma, assumption+)
```
```  2031 apply (subgoal_tac "b*q' < b* (q + 1) ")
```
```  2032  apply (simp add: mult_less_cancel_left)
```
```  2033 apply (simp add: distrib_left)
```
```  2034 apply (subgoal_tac "b*q' \<le> b'*q'")
```
```  2035  prefer 2 apply (simp add: mult_right_mono_neg, arith)
```
```  2036 done
```
```  2037
```
```  2038 lemma zdiv_mono2_neg:
```
```  2039      "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
```
```  2040 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
```
```  2041 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
```
```  2042 apply (rule zdiv_mono2_neg_lemma)
```
```  2043 apply (erule subst)
```
```  2044 apply (erule subst, simp_all)
```
```  2045 done
```
```  2046
```
```  2047
```
```  2048 subsubsection \<open>More Algebraic Laws for div and mod\<close>
```
```  2049
```
```  2050 text\<open>proving (a*b) div c = a * (b div c) + a * (b mod c)\<close>
```
```  2051
```
```  2052 lemma zmult1_lemma:
```
```  2053      "[| divmod_int_rel b c (q, r) |]
```
```  2054       ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"
```
```  2055 by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left ac_simps)
```
```  2056
```
```  2057 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
```
```  2058 apply (case_tac "c = 0", simp)
```
```  2059 apply (blast intro: divmod_int_rel [THEN zmult1_lemma, THEN div_int_unique])
```
```  2060 done
```
```  2061
```
```  2062 text\<open>proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c)\<close>
```
```  2063
```
```  2064 lemma zadd1_lemma:
```
```  2065      "[| divmod_int_rel a c (aq, ar);  divmod_int_rel b c (bq, br) |]
```
```  2066       ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
```
```  2067 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left)
```
```  2068
```
```  2069 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
```
```  2070 lemma zdiv_zadd1_eq:
```
```  2071      "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
```
```  2072 apply (case_tac "c = 0", simp)
```
```  2073 apply (blast intro: zadd1_lemma [OF divmod_int_rel divmod_int_rel] div_int_unique)
```
```  2074 done
```
```  2075
```
```  2076 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
```
```  2077 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
```
```  2078
```
```  2079 (* REVISIT: should this be generalized to all semiring_div types? *)
```
```  2080 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
```
```  2081
```
```  2082 lemma zmod_zdiv_equality' [nitpick_unfold]:
```
```  2083   "(m::int) mod n = m - (m div n) * n"
```
```  2084   using mod_div_equality [of m n] by arith
```
```  2085
```
```  2086
```
```  2087 subsubsection \<open>Proving  @{term "a div (b * c) = (a div b) div c"}\<close>
```
```  2088
```
```  2089 (*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
```
```  2090   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
```
```  2091   to cause particular problems.*)
```
```  2092
```
```  2093 text\<open>first, four lemmas to bound the remainder for the cases b<0 and b>0\<close>
```
```  2094
```
```  2095 lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b * c < b * (q mod c) + r"
```
```  2096 apply (subgoal_tac "b * (c - q mod c) < r * 1")
```
```  2097  apply (simp add: algebra_simps)
```
```  2098 apply (rule order_le_less_trans)
```
```  2099  apply (erule_tac [2] mult_strict_right_mono)
```
```  2100  apply (rule mult_left_mono_neg)
```
```  2101   using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)
```
```  2102  apply (simp)
```
```  2103 apply (simp)
```
```  2104 done
```
```  2105
```
```  2106 lemma zmult2_lemma_aux2:
```
```  2107      "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
```
```  2108 apply (subgoal_tac "b * (q mod c) \<le> 0")
```
```  2109  apply arith
```
```  2110 apply (simp add: mult_le_0_iff)
```
```  2111 done
```
```  2112
```
```  2113 lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
```
```  2114 apply (subgoal_tac "0 \<le> b * (q mod c) ")
```
```  2115 apply arith
```
```  2116 apply (simp add: zero_le_mult_iff)
```
```  2117 done
```
```  2118
```
```  2119 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
```
```  2120 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
```
```  2121  apply (simp add: right_diff_distrib)
```
```  2122 apply (rule order_less_le_trans)
```
```  2123  apply (erule mult_strict_right_mono)
```
```  2124  apply (rule_tac [2] mult_left_mono)
```
```  2125   apply simp
```
```  2126  using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)
```
```  2127 apply simp
```
```  2128 done
```
```  2129
```
```  2130 lemma zmult2_lemma: "[| divmod_int_rel a b (q, r); 0 < c |]
```
```  2131       ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"
```
```  2132 by (auto simp add: mult.assoc divmod_int_rel_def linorder_neq_iff
```
```  2133                    zero_less_mult_iff distrib_left [symmetric]
```
```  2134                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: if_split_asm)
```
```  2135
```
```  2136 lemma zdiv_zmult2_eq:
```
```  2137   fixes a b c :: int
```
```  2138   shows "0 \<le> c \<Longrightarrow> a div (b * c) = (a div b) div c"
```
```  2139 apply (case_tac "b = 0", simp)
```
```  2140 apply (force simp add: le_less divmod_int_rel [THEN zmult2_lemma, THEN div_int_unique])
```
```  2141 done
```
```  2142
```
```  2143 lemma zmod_zmult2_eq:
```
```  2144   fixes a b c :: int
```
```  2145   shows "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
```
```  2146 apply (case_tac "b = 0", simp)
```
```  2147 apply (force simp add: le_less divmod_int_rel [THEN zmult2_lemma, THEN mod_int_unique])
```
```  2148 done
```
```  2149
```
```  2150 lemma div_pos_geq:
```
```  2151   fixes k l :: int
```
```  2152   assumes "0 < l" and "l \<le> k"
```
```  2153   shows "k div l = (k - l) div l + 1"
```
```  2154 proof -
```
```  2155   have "k = (k - l) + l" by simp
```
```  2156   then obtain j where k: "k = j + l" ..
```
```  2157   with assms show ?thesis by simp
```
```  2158 qed
```
```  2159
```
```  2160 lemma mod_pos_geq:
```
```  2161   fixes k l :: int
```
```  2162   assumes "0 < l" and "l \<le> k"
```
```  2163   shows "k mod l = (k - l) mod l"
```
```  2164 proof -
```
```  2165   have "k = (k - l) + l" by simp
```
```  2166   then obtain j where k: "k = j + l" ..
```
```  2167   with assms show ?thesis by simp
```
```  2168 qed
```
```  2169
```
```  2170
```
```  2171 subsubsection \<open>Splitting Rules for div and mod\<close>
```
```  2172
```
```  2173 text\<open>The proofs of the two lemmas below are essentially identical\<close>
```
```  2174
```
```  2175 lemma split_pos_lemma:
```
```  2176  "0<k ==>
```
```  2177     P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
```
```  2178 apply (rule iffI, clarify)
```
```  2179  apply (erule_tac P="P x y" for x y in rev_mp)
```
```  2180  apply (subst mod_add_eq)
```
```  2181  apply (subst zdiv_zadd1_eq)
```
```  2182  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)
```
```  2183 txt\<open>converse direction\<close>
```
```  2184 apply (drule_tac x = "n div k" in spec)
```
```  2185 apply (drule_tac x = "n mod k" in spec, simp)
```
```  2186 done
```
```  2187
```
```  2188 lemma split_neg_lemma:
```
```  2189  "k<0 ==>
```
```  2190     P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
```
```  2191 apply (rule iffI, clarify)
```
```  2192  apply (erule_tac P="P x y" for x y in rev_mp)
```
```  2193  apply (subst mod_add_eq)
```
```  2194  apply (subst zdiv_zadd1_eq)
```
```  2195  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)
```
```  2196 txt\<open>converse direction\<close>
```
```  2197 apply (drule_tac x = "n div k" in spec)
```
```  2198 apply (drule_tac x = "n mod k" in spec, simp)
```
```  2199 done
```
```  2200
```
```  2201 lemma split_zdiv:
```
```  2202  "P(n div k :: int) =
```
```  2203   ((k = 0 --> P 0) &
```
```  2204    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) &
```
```  2205    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
```
```  2206 apply (case_tac "k=0", simp)
```
```  2207 apply (simp only: linorder_neq_iff)
```
```  2208 apply (erule disjE)
```
```  2209  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]
```
```  2210                       split_neg_lemma [of concl: "%x y. P x"])
```
```  2211 done
```
```  2212
```
```  2213 lemma split_zmod:
```
```  2214  "P(n mod k :: int) =
```
```  2215   ((k = 0 --> P n) &
```
```  2216    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) &
```
```  2217    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
```
```  2218 apply (case_tac "k=0", simp)
```
```  2219 apply (simp only: linorder_neq_iff)
```
```  2220 apply (erule disjE)
```
```  2221  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]
```
```  2222                       split_neg_lemma [of concl: "%x y. P y"])
```
```  2223 done
```
```  2224
```
```  2225 text \<open>Enable (lin)arith to deal with @{const divide} and @{const mod}
```
```  2226   when these are applied to some constant that is of the form
```
```  2227   @{term "numeral k"}:\<close>
```
```  2228 declare split_zdiv [of _ _ "numeral k", arith_split] for k
```
```  2229 declare split_zmod [of _ _ "numeral k", arith_split] for k
```
```  2230
```
```  2231
```
```  2232 subsubsection \<open>Computing \<open>div\<close> and \<open>mod\<close> with shifting\<close>
```
```  2233
```
```  2234 lemma pos_divmod_int_rel_mult_2:
```
```  2235   assumes "0 \<le> b"
```
```  2236   assumes "divmod_int_rel a b (q, r)"
```
```  2237   shows "divmod_int_rel (1 + 2*a) (2*b) (q, 1 + 2*r)"
```
```  2238   using assms unfolding divmod_int_rel_def by auto
```
```  2239
```
```  2240 declaration \<open>K (Lin_Arith.add_simps @{thms uminus_numeral_One})\<close>
```
```  2241
```
```  2242 lemma neg_divmod_int_rel_mult_2:
```
```  2243   assumes "b \<le> 0"
```
```  2244   assumes "divmod_int_rel (a + 1) b (q, r)"
```
```  2245   shows "divmod_int_rel (1 + 2*a) (2*b) (q, 2*r - 1)"
```
```  2246   using assms unfolding divmod_int_rel_def by auto
```
```  2247
```
```  2248 text\<open>computing div by shifting\<close>
```
```  2249
```
```  2250 lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
```
```  2251   using pos_divmod_int_rel_mult_2 [OF _ divmod_int_rel]
```
```  2252   by (rule div_int_unique)
```
```  2253
```
```  2254 lemma neg_zdiv_mult_2:
```
```  2255   assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"
```
```  2256   using neg_divmod_int_rel_mult_2 [OF A divmod_int_rel]
```
```  2257   by (rule div_int_unique)
```
```  2258
```
```  2259 (* FIXME: add rules for negative numerals *)
```
```  2260 lemma zdiv_numeral_Bit0 [simp]:
```
```  2261   "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
```
```  2262     numeral v div (numeral w :: int)"
```
```  2263   unfolding numeral.simps unfolding mult_2 [symmetric]
```
```  2264   by (rule div_mult_mult1, simp)
```
```  2265
```
```  2266 lemma zdiv_numeral_Bit1 [simp]:
```
```  2267   "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =
```
```  2268     (numeral v div (numeral w :: int))"
```
```  2269   unfolding numeral.simps
```
```  2270   unfolding mult_2 [symmetric] add.commute [of _ 1]
```
```  2271   by (rule pos_zdiv_mult_2, simp)
```
```  2272
```
```  2273 lemma pos_zmod_mult_2:
```
```  2274   fixes a b :: int
```
```  2275   assumes "0 \<le> a"
```
```  2276   shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
```
```  2277   using pos_divmod_int_rel_mult_2 [OF assms divmod_int_rel]
```
```  2278   by (rule mod_int_unique)
```
```  2279
```
```  2280 lemma neg_zmod_mult_2:
```
```  2281   fixes a b :: int
```
```  2282   assumes "a \<le> 0"
```
```  2283   shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
```
```  2284   using neg_divmod_int_rel_mult_2 [OF assms divmod_int_rel]
```
```  2285   by (rule mod_int_unique)
```
```  2286
```
```  2287 (* FIXME: add rules for negative numerals *)
```
```  2288 lemma zmod_numeral_Bit0 [simp]:
```
```  2289   "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =
```
```  2290     (2::int) * (numeral v mod numeral w)"
```
```  2291   unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
```
```  2292   unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
```
```  2293
```
```  2294 lemma zmod_numeral_Bit1 [simp]:
```
```  2295   "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
```
```  2296     2 * (numeral v mod numeral w) + (1::int)"
```
```  2297   unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
```
```  2298   unfolding mult_2 [symmetric] add.commute [of _ 1]
```
```  2299   by (rule pos_zmod_mult_2, simp)
```
```  2300
```
```  2301 lemma zdiv_eq_0_iff:
```
```  2302  "(i::int) div k = 0 \<longleftrightarrow> k=0 \<or> 0\<le>i \<and> i<k \<or> i\<le>0 \<and> k<i" (is "?L = ?R")
```
```  2303 proof
```
```  2304   assume ?L
```
```  2305   have "?L \<longrightarrow> ?R" by (rule split_zdiv[THEN iffD2]) simp
```
```  2306   with \<open>?L\<close> show ?R by blast
```
```  2307 next
```
```  2308   assume ?R thus ?L
```
```  2309     by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)
```
```  2310 qed
```
```  2311
```
```  2312
```
```  2313 subsubsection \<open>Quotients of Signs\<close>
```
```  2314
```
```  2315 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
```
```  2316 by (simp add: divide_int_def)
```
```  2317
```
```  2318 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
```
```  2319 by (simp add: mod_int_def)
```
```  2320
```
```  2321 lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
```
```  2322 apply (subgoal_tac "a div b \<le> -1", force)
```
```  2323 apply (rule order_trans)
```
```  2324 apply (rule_tac a' = "-1" in zdiv_mono1)
```
```  2325 apply (auto simp add: div_eq_minus1)
```
```  2326 done
```
```  2327
```
```  2328 lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
```
```  2329 by (drule zdiv_mono1_neg, auto)
```
```  2330
```
```  2331 lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
```
```  2332 by (drule zdiv_mono1, auto)
```
```  2333
```
```  2334 text\<open>Now for some equivalences of the form \<open>a div b >=< 0 \<longleftrightarrow> \<dots>\<close>
```
```  2335 conditional upon the sign of \<open>a\<close> or \<open>b\<close>. There are many more.
```
```  2336 They should all be simp rules unless that causes too much search.\<close>
```
```  2337
```
```  2338 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
```
```  2339 apply auto
```
```  2340 apply (drule_tac [2] zdiv_mono1)
```
```  2341 apply (auto simp add: linorder_neq_iff)
```
```  2342 apply (simp (no_asm_use) add: linorder_not_less [symmetric])
```
```  2343 apply (blast intro: div_neg_pos_less0)
```
```  2344 done
```
```  2345
```
```  2346 lemma pos_imp_zdiv_pos_iff:
```
```  2347   "0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"
```
```  2348 using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]
```
```  2349 by arith
```
```  2350
```
```  2351 lemma neg_imp_zdiv_nonneg_iff:
```
```  2352   "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
```
```  2353 apply (subst div_minus_minus [symmetric])
```
```  2354 apply (subst pos_imp_zdiv_nonneg_iff, auto)
```
```  2355 done
```
```  2356
```
```  2357 (*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
```
```  2358 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
```
```  2359 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
```
```  2360
```
```  2361 (*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
```
```  2362 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
```
```  2363 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
```
```  2364
```
```  2365 lemma nonneg1_imp_zdiv_pos_iff:
```
```  2366   "(0::int) <= a \<Longrightarrow> (a div b > 0) = (a >= b & b>0)"
```
```  2367 apply rule
```
```  2368  apply rule
```
```  2369   using div_pos_pos_trivial[of a b]apply arith
```
```  2370  apply(cases "b=0")apply simp
```
```  2371  using div_nonneg_neg_le0[of a b]apply arith
```
```  2372 using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp
```
```  2373 done
```
```  2374
```
```  2375 lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 ==> m mod k \<le> m"
```
```  2376 apply (rule split_zmod[THEN iffD2])
```
```  2377 apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)
```
```  2378 done
```
```  2379
```
```  2380 lemma zmult_div_cancel:
```
```  2381   "(n::int) * (m div n) = m - (m mod n)"
```
```  2382   using zmod_zdiv_equality [where a="m" and b="n"]
```
```  2383   by (simp add: algebra_simps) (* FIXME: generalize *)
```
```  2384
```
```  2385
```
```  2386 subsubsection \<open>Computation of Division and Remainder\<close>
```
```  2387
```
```  2388 instantiation int :: semiring_numeral_div
```
```  2389 begin
```
```  2390
```
```  2391 definition divmod_int :: "num \<Rightarrow> num \<Rightarrow> int \<times> int"
```
```  2392 where
```
```  2393   "divmod_int m n = (numeral m div numeral n, numeral m mod numeral n)"
```
```  2394
```
```  2395 definition divmod_step_int :: "num \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int"
```
```  2396 where
```
```  2397   "divmod_step_int l qr = (let (q, r) = qr
```
```  2398     in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
```
```  2399     else (2 * q, r))"
```
```  2400
```
```  2401 instance
```
```  2402   by standard (auto intro: zmod_le_nonneg_dividend simp add: divmod_int_def divmod_step_int_def
```
```  2403     pos_imp_zdiv_pos_iff div_pos_pos_trivial mod_pos_pos_trivial zmod_zmult2_eq zdiv_zmult2_eq)
```
```  2404
```
```  2405 end
```
```  2406
```
```  2407 declare divmod_algorithm_code [where ?'a = int, code]
```
```  2408
```
```  2409 context
```
```  2410 begin
```
```  2411
```
```  2412 qualified definition adjust_div :: "int \<times> int \<Rightarrow> int"
```
```  2413 where
```
```  2414   "adjust_div qr = (let (q, r) = qr in q + of_bool (r \<noteq> 0))"
```
```  2415
```
```  2416 qualified lemma adjust_div_eq [simp, code]:
```
```  2417   "adjust_div (q, r) = q + of_bool (r \<noteq> 0)"
```
```  2418   by (simp add: adjust_div_def)
```
```  2419
```
```  2420 qualified definition adjust_mod :: "int \<Rightarrow> int \<Rightarrow> int"
```
```  2421 where
```
```  2422   [simp]: "adjust_mod l r = (if r = 0 then 0 else l - r)"
```
```  2423
```
```  2424 lemma minus_numeral_div_numeral [simp]:
```
```  2425   "- numeral m div numeral n = - (adjust_div (divmod m n) :: int)"
```
```  2426 proof -
```
```  2427   have "int (fst (divmod m n)) = fst (divmod m n)"
```
```  2428     by (simp only: fst_divmod divide_int_def) auto
```
```  2429   then show ?thesis
```
```  2430     by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
```
```  2431 qed
```
```  2432
```
```  2433 lemma minus_numeral_mod_numeral [simp]:
```
```  2434   "- numeral m mod numeral n = adjust_mod (numeral n) (snd (divmod m n) :: int)"
```
```  2435 proof -
```
```  2436   have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
```
```  2437     using that by (simp only: snd_divmod mod_int_def) auto
```
```  2438   then show ?thesis
```
```  2439     by (auto simp add: split_def Let_def adjust_div_def divides_aux_def mod_int_def)
```
```  2440 qed
```
```  2441
```
```  2442 lemma numeral_div_minus_numeral [simp]:
```
```  2443   "numeral m div - numeral n = - (adjust_div (divmod m n) :: int)"
```
```  2444 proof -
```
```  2445   have "int (fst (divmod m n)) = fst (divmod m n)"
```
```  2446     by (simp only: fst_divmod divide_int_def) auto
```
```  2447   then show ?thesis
```
```  2448     by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
```
```  2449 qed
```
```  2450
```
```  2451 lemma numeral_mod_minus_numeral [simp]:
```
```  2452   "numeral m mod - numeral n = - adjust_mod (numeral n) (snd (divmod m n) :: int)"
```
```  2453 proof -
```
```  2454   have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
```
```  2455     using that by (simp only: snd_divmod mod_int_def) auto
```
```  2456   then show ?thesis
```
```  2457     by (auto simp add: split_def Let_def adjust_div_def divides_aux_def mod_int_def)
```
```  2458 qed
```
```  2459
```
```  2460 lemma minus_one_div_numeral [simp]:
```
```  2461   "- 1 div numeral n = - (adjust_div (divmod Num.One n) :: int)"
```
```  2462   using minus_numeral_div_numeral [of Num.One n] by simp
```
```  2463
```
```  2464 lemma minus_one_mod_numeral [simp]:
```
```  2465   "- 1 mod numeral n = adjust_mod (numeral n) (snd (divmod Num.One n) :: int)"
```
```  2466   using minus_numeral_mod_numeral [of Num.One n] by simp
```
```  2467
```
```  2468 lemma one_div_minus_numeral [simp]:
```
```  2469   "1 div - numeral n = - (adjust_div (divmod Num.One n) :: int)"
```
```  2470   using numeral_div_minus_numeral [of Num.One n] by simp
```
```  2471
```
```  2472 lemma one_mod_minus_numeral [simp]:
```
```  2473   "1 mod - numeral n = - adjust_mod (numeral n) (snd (divmod Num.One n) :: int)"
```
```  2474   using numeral_mod_minus_numeral [of Num.One n] by simp
```
```  2475
```
```  2476 end
```
```  2477
```
```  2478
```
```  2479 subsubsection \<open>Further properties\<close>
```
```  2480
```
```  2481 text \<open>Simplify expresions in which div and mod combine numerical constants\<close>
```
```  2482
```
```  2483 lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q"
```
```  2484   by (rule div_int_unique [of a b q r]) (simp add: divmod_int_rel_def)
```
```  2485
```
```  2486 lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q"
```
```  2487   by (rule div_int_unique [of a b q r],
```
```  2488     simp add: divmod_int_rel_def)
```
```  2489
```
```  2490 lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r"
```
```  2491   by (rule mod_int_unique [of a b q r],
```
```  2492     simp add: divmod_int_rel_def)
```
```  2493
```
```  2494 lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r"
```
```  2495   by (rule mod_int_unique [of a b q r],
```
```  2496     simp add: divmod_int_rel_def)
```
```  2497
```
```  2498 lemma abs_div: "(y::int) dvd x \<Longrightarrow> \<bar>x div y\<bar> = \<bar>x\<bar> div \<bar>y\<bar>"
```
```  2499 by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
```
```  2500
```
```  2501 text\<open>Suggested by Matthias Daum\<close>
```
```  2502 lemma int_power_div_base:
```
```  2503      "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
```
```  2504 apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
```
```  2505  apply (erule ssubst)
```
```  2506  apply (simp only: power_add)
```
```  2507  apply simp_all
```
```  2508 done
```
```  2509
```
```  2510 text \<open>by Brian Huffman\<close>
```
```  2511 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
```
```  2512 by (rule mod_minus_eq [symmetric])
```
```  2513
```
```  2514 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
```
```  2515 by (rule mod_diff_left_eq [symmetric])
```
```  2516
```
```  2517 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
```
```  2518 by (rule mod_diff_right_eq [symmetric])
```
```  2519
```
```  2520 lemmas zmod_simps =
```
```  2521   mod_add_left_eq  [symmetric]
```
```  2522   mod_add_right_eq [symmetric]
```
```  2523   mod_mult_right_eq[symmetric]
```
```  2524   mod_mult_left_eq [symmetric]
```
```  2525   power_mod
```
```  2526   zminus_zmod zdiff_zmod_left zdiff_zmod_right
```
```  2527
```
```  2528 text \<open>Distributive laws for function \<open>nat\<close>.\<close>
```
```  2529
```
```  2530 lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
```
```  2531 apply (rule linorder_cases [of y 0])
```
```  2532 apply (simp add: div_nonneg_neg_le0)
```
```  2533 apply simp
```
```  2534 apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
```
```  2535 done
```
```  2536
```
```  2537 (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
```
```  2538 lemma nat_mod_distrib:
```
```  2539   "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
```
```  2540 apply (case_tac "y = 0", simp)
```
```  2541 apply (simp add: nat_eq_iff zmod_int)
```
```  2542 done
```
```  2543
```
```  2544 text  \<open>transfer setup\<close>
```
```  2545
```
```  2546 lemma transfer_nat_int_functions:
```
```  2547     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
```
```  2548     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
```
```  2549   by (auto simp add: nat_div_distrib nat_mod_distrib)
```
```  2550
```
```  2551 lemma transfer_nat_int_function_closures:
```
```  2552     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
```
```  2553     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
```
```  2554   apply (cases "y = 0")
```
```  2555   apply (auto simp add: pos_imp_zdiv_nonneg_iff)
```
```  2556   apply (cases "y = 0")
```
```  2557   apply auto
```
```  2558 done
```
```  2559
```
```  2560 declare transfer_morphism_nat_int [transfer add return:
```
```  2561   transfer_nat_int_functions
```
```  2562   transfer_nat_int_function_closures
```
```  2563 ]
```
```  2564
```
```  2565 lemma transfer_int_nat_functions:
```
```  2566     "(int x) div (int y) = int (x div y)"
```
```  2567     "(int x) mod (int y) = int (x mod y)"
```
```  2568   by (auto simp add: zdiv_int zmod_int)
```
```  2569
```
```  2570 lemma transfer_int_nat_function_closures:
```
```  2571     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"
```
```  2572     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"
```
```  2573   by (simp_all only: is_nat_def transfer_nat_int_function_closures)
```
```  2574
```
```  2575 declare transfer_morphism_int_nat [transfer add return:
```
```  2576   transfer_int_nat_functions
```
```  2577   transfer_int_nat_function_closures
```
```  2578 ]
```
```  2579
```
```  2580 text\<open>Suggested by Matthias Daum\<close>
```
```  2581 lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
```
```  2582 apply (subgoal_tac "nat x div nat k < nat x")
```
```  2583  apply (simp add: nat_div_distrib [symmetric])
```
```  2584 apply (rule Divides.div_less_dividend, simp_all)
```
```  2585 done
```
```  2586
```
```  2587 lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"
```
```  2588 proof
```
```  2589   assume H: "x mod n = y mod n"
```
```  2590   hence "x mod n - y mod n = 0" by simp
```
```  2591   hence "(x mod n - y mod n) mod n = 0" by simp
```
```  2592   hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])
```
```  2593   thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)
```
```  2594 next
```
```  2595   assume H: "n dvd x - y"
```
```  2596   then obtain k where k: "x-y = n*k" unfolding dvd_def by blast
```
```  2597   hence "x = n*k + y" by simp
```
```  2598   hence "x mod n = (n*k + y) mod n" by simp
```
```  2599   thus "x mod n = y mod n" by (simp add: mod_add_left_eq)
```
```  2600 qed
```
```  2601
```
```  2602 lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"
```
```  2603   shows "\<exists>q. x = y + n * q"
```
```  2604 proof-
```
```  2605   from xy have th: "int x - int y = int (x - y)" by simp
```
```  2606   from xyn have "int x mod int n = int y mod int n"
```
```  2607     by (simp add: zmod_int [symmetric])
```
```  2608   hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric])
```
```  2609   hence "n dvd x - y" by (simp add: th zdvd_int)
```
```  2610   then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
```
```  2611 qed
```
```  2612
```
```  2613 lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"
```
```  2614   (is "?lhs = ?rhs")
```
```  2615 proof
```
```  2616   assume H: "x mod n = y mod n"
```
```  2617   {assume xy: "x \<le> y"
```
```  2618     from H have th: "y mod n = x mod n" by simp
```
```  2619     from nat_mod_eq_lemma[OF th xy] have ?rhs
```
```  2620       apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
```
```  2621   moreover
```
```  2622   {assume xy: "y \<le> x"
```
```  2623     from nat_mod_eq_lemma[OF H xy] have ?rhs
```
```  2624       apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
```
```  2625   ultimately  show ?rhs using linear[of x y] by blast
```
```  2626 next
```
```  2627   assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
```
```  2628   hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
```
```  2629   thus  ?lhs by simp
```
```  2630 qed
```
```  2631
```
```  2632 subsubsection \<open>Dedicated simproc for calculation\<close>
```
```  2633
```
```  2634 text \<open>
```
```  2635   There is space for improvement here: the calculation itself
```
```  2636   could be carried outside the logic, and a generic simproc
```
```  2637   (simplifier setup) for generic calculation would be helpful.
```
```  2638 \<close>
```
```  2639
```
```  2640 simproc_setup numeral_divmod
```
```  2641   ("0 div 0 :: 'a :: semiring_numeral_div" | "0 mod 0 :: 'a :: semiring_numeral_div" |
```
```  2642    "0 div 1 :: 'a :: semiring_numeral_div" | "0 mod 1 :: 'a :: semiring_numeral_div" |
```
```  2643    "0 div - 1 :: int" | "0 mod - 1 :: int" |
```
```  2644    "0 div numeral b :: 'a :: semiring_numeral_div" | "0 mod numeral b :: 'a :: semiring_numeral_div" |
```
```  2645    "0 div - numeral b :: int" | "0 mod - numeral b :: int" |
```
```  2646    "1 div 0 :: 'a :: semiring_numeral_div" | "1 mod 0 :: 'a :: semiring_numeral_div" |
```
```  2647    "1 div 1 :: 'a :: semiring_numeral_div" | "1 mod 1 :: 'a :: semiring_numeral_div" |
```
```  2648    "1 div - 1 :: int" | "1 mod - 1 :: int" |
```
```  2649    "1 div numeral b :: 'a :: semiring_numeral_div" | "1 mod numeral b :: 'a :: semiring_numeral_div" |
```
```  2650    "1 div - numeral b :: int" |"1 mod - numeral b :: int" |
```
```  2651    "- 1 div 0 :: int" | "- 1 mod 0 :: int" | "- 1 div 1 :: int" | "- 1 mod 1 :: int" |
```
```  2652    "- 1 div - 1 :: int" | "- 1 mod - 1 :: int" | "- 1 div numeral b :: int" | "- 1 mod numeral b :: int" |
```
```  2653    "- 1 div - numeral b :: int" | "- 1 mod - numeral b :: int" |
```
```  2654    "numeral a div 0 :: 'a :: semiring_numeral_div" | "numeral a mod 0 :: 'a :: semiring_numeral_div" |
```
```  2655    "numeral a div 1 :: 'a :: semiring_numeral_div" | "numeral a mod 1 :: 'a :: semiring_numeral_div" |
```
```  2656    "numeral a div - 1 :: int" | "numeral a mod - 1 :: int" |
```
```  2657    "numeral a div numeral b :: 'a :: semiring_numeral_div" | "numeral a mod numeral b :: 'a :: semiring_numeral_div" |
```
```  2658    "numeral a div - numeral b :: int" | "numeral a mod - numeral b :: int" |
```
```  2659    "- numeral a div 0 :: int" | "- numeral a mod 0 :: int" |
```
```  2660    "- numeral a div 1 :: int" | "- numeral a mod 1 :: int" |
```
```  2661    "- numeral a div - 1 :: int" | "- numeral a mod - 1 :: int" |
```
```  2662    "- numeral a div numeral b :: int" | "- numeral a mod numeral b :: int" |
```
```  2663    "- numeral a div - numeral b :: int" | "- numeral a mod - numeral b :: int") =
```
```  2664 \<open> let
```
```  2665     val if_cong = the (Code.get_case_cong @{theory} @{const_name If});
```
```  2666     fun successful_rewrite ctxt ct =
```
```  2667       let
```
```  2668         val thm = Simplifier.rewrite ctxt ct
```
```  2669       in if Thm.is_reflexive thm then NONE else SOME thm end;
```
```  2670   in fn phi =>
```
```  2671     let
```
```  2672       val simps = Morphism.fact phi (@{thms div_0 mod_0 div_by_0 mod_by_0 div_by_1 mod_by_1
```
```  2673         one_div_numeral one_mod_numeral minus_one_div_numeral minus_one_mod_numeral
```
```  2674         one_div_minus_numeral one_mod_minus_numeral
```
```  2675         numeral_div_numeral numeral_mod_numeral minus_numeral_div_numeral minus_numeral_mod_numeral
```
```  2676         numeral_div_minus_numeral numeral_mod_minus_numeral
```
```  2677         div_minus_minus mod_minus_minus Divides.adjust_div_eq of_bool_eq one_neq_zero
```
```  2678         numeral_neq_zero neg_equal_0_iff_equal arith_simps arith_special divmod_trivial
```
```  2679         divmod_cancel divmod_steps divmod_step_eq fst_conv snd_conv numeral_One
```
```  2680         case_prod_beta rel_simps Divides.adjust_mod_def div_minus1_right mod_minus1_right
```
```  2681         minus_minus numeral_times_numeral mult_zero_right mult_1_right}
```
```  2682         @ [@{lemma "0 = 0 \<longleftrightarrow> True" by simp}]);
```
```  2683       fun prepare_simpset ctxt = HOL_ss |> Simplifier.simpset_map ctxt
```
```  2684         (Simplifier.add_cong if_cong #> fold Simplifier.add_simp simps)
```
```  2685     in fn ctxt => successful_rewrite (Simplifier.put_simpset (prepare_simpset ctxt) ctxt) end
```
```  2686   end;
```
```  2687 \<close>
```
```  2688
```
```  2689
```
```  2690 subsubsection \<open>Code generation\<close>
```
```  2691
```
```  2692 lemma [code]:
```
```  2693   fixes k :: int
```
```  2694   shows
```
```  2695     "k div 0 = 0"
```
```  2696     "k mod 0 = k"
```
```  2697     "0 div k = 0"
```
```  2698     "0 mod k = 0"
```
```  2699     "k div Int.Pos Num.One = k"
```
```  2700     "k mod Int.Pos Num.One = 0"
```
```  2701     "k div Int.Neg Num.One = - k"
```
```  2702     "k mod Int.Neg Num.One = 0"
```
```  2703     "Int.Pos m div Int.Pos n = (fst (divmod m n) :: int)"
```
```  2704     "Int.Pos m mod Int.Pos n = (snd (divmod m n) :: int)"
```
```  2705     "Int.Neg m div Int.Pos n = - (Divides.adjust_div (divmod m n) :: int)"
```
```  2706     "Int.Neg m mod Int.Pos n = Divides.adjust_mod (Int.Pos n) (snd (divmod m n) :: int)"
```
```  2707     "Int.Pos m div Int.Neg n = - (Divides.adjust_div (divmod m n) :: int)"
```
```  2708     "Int.Pos m mod Int.Neg n = - Divides.adjust_mod (Int.Pos n) (snd (divmod m n) :: int)"
```
```  2709     "Int.Neg m div Int.Neg n = (fst (divmod m n) :: int)"
```
```  2710     "Int.Neg m mod Int.Neg n = - (snd (divmod m n) :: int)"
```
```  2711   by simp_all
```
```  2712
```
```  2713 code_identifier
```
```  2714   code_module Divides \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
```
```  2715
```
```  2716 lemma dvd_eq_mod_eq_0_numeral:
```
```  2717   "numeral x dvd (numeral y :: 'a) \<longleftrightarrow> numeral y mod numeral x = (0 :: 'a::semiring_div)"
```
```  2718   by (fact dvd_eq_mod_eq_0)
```
```  2719
```
```  2720 end
```