src/HOL/Divides.thy
 author haftmann Thu Dec 22 10:42:08 2016 +0100 (2016-12-22) changeset 64635 255741c5f862 parent 64630 96015aecfeba child 64715 33d5fa0ce6e5 permissions -rw-r--r--
more uniform div/mod relations
```     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>Quotient and remainder\<close>
```
```     7
```
```     8 theory Divides
```
```     9 imports Parity
```
```    10 begin
```
```    11
```
```    12 subsection \<open>Quotient and remainder in integral domains\<close>
```
```    13
```
```    14 class semidom_modulo = algebraic_semidom + semiring_modulo
```
```    15 begin
```
```    16
```
```    17 lemma mod_0 [simp]: "0 mod a = 0"
```
```    18   using div_mult_mod_eq [of 0 a] by simp
```
```    19
```
```    20 lemma mod_by_0 [simp]: "a mod 0 = a"
```
```    21   using div_mult_mod_eq [of a 0] by simp
```
```    22
```
```    23 lemma mod_by_1 [simp]:
```
```    24   "a mod 1 = 0"
```
```    25 proof -
```
```    26   from div_mult_mod_eq [of a one] div_by_1 have "a + a mod 1 = a" by simp
```
```    27   then have "a + a mod 1 = a + 0" by simp
```
```    28   then show ?thesis by (rule add_left_imp_eq)
```
```    29 qed
```
```    30
```
```    31 lemma mod_self [simp]:
```
```    32   "a mod a = 0"
```
```    33   using div_mult_mod_eq [of a a] by simp
```
```    34
```
```    35 lemma dvd_imp_mod_0 [simp]:
```
```    36   assumes "a dvd b"
```
```    37   shows "b mod a = 0"
```
```    38   using assms minus_div_mult_eq_mod [of b a] by simp
```
```    39
```
```    40 lemma mod_0_imp_dvd:
```
```    41   assumes "a mod b = 0"
```
```    42   shows   "b dvd a"
```
```    43 proof -
```
```    44   have "b dvd ((a div b) * b)" by simp
```
```    45   also have "(a div b) * b = a"
```
```    46     using div_mult_mod_eq [of a b] by (simp add: assms)
```
```    47   finally show ?thesis .
```
```    48 qed
```
```    49
```
```    50 lemma mod_eq_0_iff_dvd:
```
```    51   "a mod b = 0 \<longleftrightarrow> b dvd a"
```
```    52   by (auto intro: mod_0_imp_dvd)
```
```    53
```
```    54 lemma dvd_eq_mod_eq_0 [nitpick_unfold, code]:
```
```    55   "a dvd b \<longleftrightarrow> b mod a = 0"
```
```    56   by (simp add: mod_eq_0_iff_dvd)
```
```    57
```
```    58 lemma dvd_mod_iff:
```
```    59   assumes "c dvd b"
```
```    60   shows "c dvd a mod b \<longleftrightarrow> c dvd a"
```
```    61 proof -
```
```    62   from assms have "(c dvd a mod b) \<longleftrightarrow> (c dvd ((a div b) * b + a mod b))"
```
```    63     by (simp add: dvd_add_right_iff)
```
```    64   also have "(a div b) * b + a mod b = a"
```
```    65     using div_mult_mod_eq [of a b] by simp
```
```    66   finally show ?thesis .
```
```    67 qed
```
```    68
```
```    69 lemma dvd_mod_imp_dvd:
```
```    70   assumes "c dvd a mod b" and "c dvd b"
```
```    71   shows "c dvd a"
```
```    72   using assms dvd_mod_iff [of c b a] by simp
```
```    73
```
```    74 end
```
```    75
```
```    76 class idom_modulo = idom + semidom_modulo
```
```    77 begin
```
```    78
```
```    79 subclass idom_divide ..
```
```    80
```
```    81 lemma div_diff [simp]:
```
```    82   "c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> (a - b) div c = a div c - b div c"
```
```    83   using div_add [of _  _ "- b"] by (simp add: dvd_neg_div)
```
```    84
```
```    85 end
```
```    86
```
```    87
```
```    88 subsection \<open>Quotient and remainder in integral domains with additional properties\<close>
```
```    89
```
```    90 class semiring_div = semidom_modulo +
```
```    91   assumes div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"
```
```    92     and div_mult_mult1 [simp]: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b"
```
```    93 begin
```
```    94
```
```    95 lemma div_mult_self2 [simp]:
```
```    96   assumes "b \<noteq> 0"
```
```    97   shows "(a + b * c) div b = c + a div b"
```
```    98   using assms div_mult_self1 [of b a c] by (simp add: mult.commute)
```
```    99
```
```   100 lemma div_mult_self3 [simp]:
```
```   101   assumes "b \<noteq> 0"
```
```   102   shows "(c * b + a) div b = c + a div b"
```
```   103   using assms by (simp add: add.commute)
```
```   104
```
```   105 lemma div_mult_self4 [simp]:
```
```   106   assumes "b \<noteq> 0"
```
```   107   shows "(b * c + a) div b = c + a div b"
```
```   108   using assms by (simp add: add.commute)
```
```   109
```
```   110 lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"
```
```   111 proof (cases "b = 0")
```
```   112   case True then show ?thesis by simp
```
```   113 next
```
```   114   case False
```
```   115   have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"
```
```   116     by (simp add: div_mult_mod_eq)
```
```   117   also from False div_mult_self1 [of b a c] have
```
```   118     "\<dots> = (c + a div b) * b + (a + c * b) mod b"
```
```   119       by (simp add: algebra_simps)
```
```   120   finally have "a = a div b * b + (a + c * b) mod b"
```
```   121     by (simp add: add.commute [of a] add.assoc distrib_right)
```
```   122   then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
```
```   123     by (simp add: div_mult_mod_eq)
```
```   124   then show ?thesis by simp
```
```   125 qed
```
```   126
```
```   127 lemma mod_mult_self2 [simp]:
```
```   128   "(a + b * c) mod b = a mod b"
```
```   129   by (simp add: mult.commute [of b])
```
```   130
```
```   131 lemma mod_mult_self3 [simp]:
```
```   132   "(c * b + a) mod b = a mod b"
```
```   133   by (simp add: add.commute)
```
```   134
```
```   135 lemma mod_mult_self4 [simp]:
```
```   136   "(b * c + a) mod b = a mod b"
```
```   137   by (simp add: add.commute)
```
```   138
```
```   139 lemma mod_mult_self1_is_0 [simp]:
```
```   140   "b * a mod b = 0"
```
```   141   using mod_mult_self2 [of 0 b a] by simp
```
```   142
```
```   143 lemma mod_mult_self2_is_0 [simp]:
```
```   144   "a * b mod b = 0"
```
```   145   using mod_mult_self1 [of 0 a b] by simp
```
```   146
```
```   147 lemma div_add_self1:
```
```   148   assumes "b \<noteq> 0"
```
```   149   shows "(b + a) div b = a div b + 1"
```
```   150   using assms div_mult_self1 [of b a 1] by (simp add: add.commute)
```
```   151
```
```   152 lemma div_add_self2:
```
```   153   assumes "b \<noteq> 0"
```
```   154   shows "(a + b) div b = a div b + 1"
```
```   155   using assms div_add_self1 [of b a] by (simp add: add.commute)
```
```   156
```
```   157 lemma mod_add_self1 [simp]:
```
```   158   "(b + a) mod b = a mod b"
```
```   159   using mod_mult_self1 [of a 1 b] by (simp add: add.commute)
```
```   160
```
```   161 lemma mod_add_self2 [simp]:
```
```   162   "(a + b) mod b = a mod b"
```
```   163   using mod_mult_self1 [of a 1 b] by simp
```
```   164
```
```   165 lemma mod_div_trivial [simp]:
```
```   166   "a mod b div b = 0"
```
```   167 proof (cases "b = 0")
```
```   168   assume "b = 0"
```
```   169   thus ?thesis by simp
```
```   170 next
```
```   171   assume "b \<noteq> 0"
```
```   172   hence "a div b + a mod b div b = (a mod b + a div b * b) div b"
```
```   173     by (rule div_mult_self1 [symmetric])
```
```   174   also have "\<dots> = a div b"
```
```   175     by (simp only: mod_div_mult_eq)
```
```   176   also have "\<dots> = a div b + 0"
```
```   177     by simp
```
```   178   finally show ?thesis
```
```   179     by (rule add_left_imp_eq)
```
```   180 qed
```
```   181
```
```   182 lemma mod_mod_trivial [simp]:
```
```   183   "a mod b mod b = a mod b"
```
```   184 proof -
```
```   185   have "a mod b mod b = (a mod b + a div b * b) mod b"
```
```   186     by (simp only: mod_mult_self1)
```
```   187   also have "\<dots> = a mod b"
```
```   188     by (simp only: mod_div_mult_eq)
```
```   189   finally show ?thesis .
```
```   190 qed
```
```   191
```
```   192 lemma mod_mod_cancel:
```
```   193   assumes "c dvd b"
```
```   194   shows "a mod b mod c = a mod c"
```
```   195 proof -
```
```   196   from \<open>c dvd b\<close> obtain k where "b = c * k"
```
```   197     by (rule dvdE)
```
```   198   have "a mod b mod c = a mod (c * k) mod c"
```
```   199     by (simp only: \<open>b = c * k\<close>)
```
```   200   also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c"
```
```   201     by (simp only: mod_mult_self1)
```
```   202   also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c"
```
```   203     by (simp only: ac_simps)
```
```   204   also have "\<dots> = a mod c"
```
```   205     by (simp only: div_mult_mod_eq)
```
```   206   finally show ?thesis .
```
```   207 qed
```
```   208
```
```   209 lemma div_mult_mult2 [simp]:
```
```   210   "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
```
```   211   by (drule div_mult_mult1) (simp add: mult.commute)
```
```   212
```
```   213 lemma div_mult_mult1_if [simp]:
```
```   214   "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"
```
```   215   by simp_all
```
```   216
```
```   217 lemma mod_mult_mult1:
```
```   218   "(c * a) mod (c * b) = c * (a mod b)"
```
```   219 proof (cases "c = 0")
```
```   220   case True then show ?thesis by simp
```
```   221 next
```
```   222   case False
```
```   223   from div_mult_mod_eq
```
```   224   have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .
```
```   225   with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)
```
```   226     = c * a + c * (a mod b)" by (simp add: algebra_simps)
```
```   227   with div_mult_mod_eq show ?thesis by simp
```
```   228 qed
```
```   229
```
```   230 lemma mod_mult_mult2:
```
```   231   "(a * c) mod (b * c) = (a mod b) * c"
```
```   232   using mod_mult_mult1 [of c a b] by (simp add: mult.commute)
```
```   233
```
```   234 lemma mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)"
```
```   235   by (fact mod_mult_mult2 [symmetric])
```
```   236
```
```   237 lemma mult_mod_right: "c * (a mod b) = (c * a) mod (c * b)"
```
```   238   by (fact mod_mult_mult1 [symmetric])
```
```   239
```
```   240 lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)"
```
```   241   unfolding dvd_def by (auto simp add: mod_mult_mult1)
```
```   242
```
```   243 named_theorems mod_simps
```
```   244
```
```   245 text \<open>Addition respects modular equivalence.\<close>
```
```   246
```
```   247 lemma mod_add_left_eq [mod_simps]:
```
```   248   "(a mod c + b) mod c = (a + b) mod c"
```
```   249 proof -
```
```   250   have "(a + b) mod c = (a div c * c + a mod c + b) mod c"
```
```   251     by (simp only: div_mult_mod_eq)
```
```   252   also have "\<dots> = (a mod c + b + a div c * c) mod c"
```
```   253     by (simp only: ac_simps)
```
```   254   also have "\<dots> = (a mod c + b) mod c"
```
```   255     by (rule mod_mult_self1)
```
```   256   finally show ?thesis
```
```   257     by (rule sym)
```
```   258 qed
```
```   259
```
```   260 lemma mod_add_right_eq [mod_simps]:
```
```   261   "(a + b mod c) mod c = (a + b) mod c"
```
```   262   using mod_add_left_eq [of b c a] by (simp add: ac_simps)
```
```   263
```
```   264 lemma mod_add_eq:
```
```   265   "(a mod c + b mod c) mod c = (a + b) mod c"
```
```   266   by (simp add: mod_add_left_eq mod_add_right_eq)
```
```   267
```
```   268 lemma mod_sum_eq [mod_simps]:
```
```   269   "(\<Sum>i\<in>A. f i mod a) mod a = sum f A mod a"
```
```   270 proof (induct A rule: infinite_finite_induct)
```
```   271   case (insert i A)
```
```   272   then have "(\<Sum>i\<in>insert i A. f i mod a) mod a
```
```   273     = (f i mod a + (\<Sum>i\<in>A. f i mod a)) mod a"
```
```   274     by simp
```
```   275   also have "\<dots> = (f i + (\<Sum>i\<in>A. f i mod a) mod a) mod a"
```
```   276     by (simp add: mod_simps)
```
```   277   also have "\<dots> = (f i + (\<Sum>i\<in>A. f i) mod a) mod a"
```
```   278     by (simp add: insert.hyps)
```
```   279   finally show ?case
```
```   280     by (simp add: insert.hyps mod_simps)
```
```   281 qed simp_all
```
```   282
```
```   283 lemma mod_add_cong:
```
```   284   assumes "a mod c = a' mod c"
```
```   285   assumes "b mod c = b' mod c"
```
```   286   shows "(a + b) mod c = (a' + b') mod c"
```
```   287 proof -
```
```   288   have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c"
```
```   289     unfolding assms ..
```
```   290   then show ?thesis
```
```   291     by (simp add: mod_add_eq)
```
```   292 qed
```
```   293
```
```   294 text \<open>Multiplication respects modular equivalence.\<close>
```
```   295
```
```   296 lemma mod_mult_left_eq [mod_simps]:
```
```   297   "((a mod c) * b) mod c = (a * b) mod c"
```
```   298 proof -
```
```   299   have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c"
```
```   300     by (simp only: div_mult_mod_eq)
```
```   301   also have "\<dots> = (a mod c * b + a div c * b * c) mod c"
```
```   302     by (simp only: algebra_simps)
```
```   303   also have "\<dots> = (a mod c * b) mod c"
```
```   304     by (rule mod_mult_self1)
```
```   305   finally show ?thesis
```
```   306     by (rule sym)
```
```   307 qed
```
```   308
```
```   309 lemma mod_mult_right_eq [mod_simps]:
```
```   310   "(a * (b mod c)) mod c = (a * b) mod c"
```
```   311   using mod_mult_left_eq [of b c a] by (simp add: ac_simps)
```
```   312
```
```   313 lemma mod_mult_eq:
```
```   314   "((a mod c) * (b mod c)) mod c = (a * b) mod c"
```
```   315   by (simp add: mod_mult_left_eq mod_mult_right_eq)
```
```   316
```
```   317 lemma mod_prod_eq [mod_simps]:
```
```   318   "(\<Prod>i\<in>A. f i mod a) mod a = prod f A mod a"
```
```   319 proof (induct A rule: infinite_finite_induct)
```
```   320   case (insert i A)
```
```   321   then have "(\<Prod>i\<in>insert i A. f i mod a) mod a
```
```   322     = (f i mod a * (\<Prod>i\<in>A. f i mod a)) mod a"
```
```   323     by simp
```
```   324   also have "\<dots> = (f i * ((\<Prod>i\<in>A. f i mod a) mod a)) mod a"
```
```   325     by (simp add: mod_simps)
```
```   326   also have "\<dots> = (f i * ((\<Prod>i\<in>A. f i) mod a)) mod a"
```
```   327     by (simp add: insert.hyps)
```
```   328   finally show ?case
```
```   329     by (simp add: insert.hyps mod_simps)
```
```   330 qed simp_all
```
```   331
```
```   332 lemma mod_mult_cong:
```
```   333   assumes "a mod c = a' mod c"
```
```   334   assumes "b mod c = b' mod c"
```
```   335   shows "(a * b) mod c = (a' * b') mod c"
```
```   336 proof -
```
```   337   have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c"
```
```   338     unfolding assms ..
```
```   339   then show ?thesis
```
```   340     by (simp add: mod_mult_eq)
```
```   341 qed
```
```   342
```
```   343 text \<open>Exponentiation respects modular equivalence.\<close>
```
```   344
```
```   345 lemma power_mod [mod_simps]:
```
```   346   "((a mod b) ^ n) mod b = (a ^ n) mod b"
```
```   347 proof (induct n)
```
```   348   case 0
```
```   349   then show ?case by simp
```
```   350 next
```
```   351   case (Suc n)
```
```   352   have "(a mod b) ^ Suc n mod b = (a mod b) * ((a mod b) ^ n mod b) mod b"
```
```   353     by (simp add: mod_mult_right_eq)
```
```   354   with Suc show ?case
```
```   355     by (simp add: mod_mult_left_eq mod_mult_right_eq)
```
```   356 qed
```
```   357
```
```   358 end
```
```   359
```
```   360 class ring_div = comm_ring_1 + semiring_div
```
```   361 begin
```
```   362
```
```   363 subclass idom_divide ..
```
```   364
```
```   365 lemma div_minus_minus [simp]: "(- a) div (- b) = a div b"
```
```   366   using div_mult_mult1 [of "- 1" a b] by simp
```
```   367
```
```   368 lemma mod_minus_minus [simp]: "(- a) mod (- b) = - (a mod b)"
```
```   369   using mod_mult_mult1 [of "- 1" a b] by simp
```
```   370
```
```   371 lemma div_minus_right: "a div (- b) = (- a) div b"
```
```   372   using div_minus_minus [of "- a" b] by simp
```
```   373
```
```   374 lemma mod_minus_right: "a mod (- b) = - ((- a) mod b)"
```
```   375   using mod_minus_minus [of "- a" b] by simp
```
```   376
```
```   377 lemma div_minus1_right [simp]: "a div (- 1) = - a"
```
```   378   using div_minus_right [of a 1] by simp
```
```   379
```
```   380 lemma mod_minus1_right [simp]: "a mod (- 1) = 0"
```
```   381   using mod_minus_right [of a 1] by simp
```
```   382
```
```   383 text \<open>Negation respects modular equivalence.\<close>
```
```   384
```
```   385 lemma mod_minus_eq [mod_simps]:
```
```   386   "(- (a mod b)) mod b = (- a) mod b"
```
```   387 proof -
```
```   388   have "(- a) mod b = (- (a div b * b + a mod b)) mod b"
```
```   389     by (simp only: div_mult_mod_eq)
```
```   390   also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b"
```
```   391     by (simp add: ac_simps)
```
```   392   also have "\<dots> = (- (a mod b)) mod b"
```
```   393     by (rule mod_mult_self1)
```
```   394   finally show ?thesis
```
```   395     by (rule sym)
```
```   396 qed
```
```   397
```
```   398 lemma mod_minus_cong:
```
```   399   assumes "a mod b = a' mod b"
```
```   400   shows "(- a) mod b = (- a') mod b"
```
```   401 proof -
```
```   402   have "(- (a mod b)) mod b = (- (a' mod b)) mod b"
```
```   403     unfolding assms ..
```
```   404   then show ?thesis
```
```   405     by (simp add: mod_minus_eq)
```
```   406 qed
```
```   407
```
```   408 text \<open>Subtraction respects modular equivalence.\<close>
```
```   409
```
```   410 lemma mod_diff_left_eq [mod_simps]:
```
```   411   "(a mod c - b) mod c = (a - b) mod c"
```
```   412   using mod_add_cong [of a c "a mod c" "- b" "- b"]
```
```   413   by simp
```
```   414
```
```   415 lemma mod_diff_right_eq [mod_simps]:
```
```   416   "(a - b mod c) mod c = (a - b) mod c"
```
```   417   using mod_add_cong [of a c a "- b" "- (b mod c)"] mod_minus_cong [of "b mod c" c b]
```
```   418   by simp
```
```   419
```
```   420 lemma mod_diff_eq:
```
```   421   "(a mod c - b mod c) mod c = (a - b) mod c"
```
```   422   using mod_add_cong [of a c "a mod c" "- b" "- (b mod c)"] mod_minus_cong [of "b mod c" c b]
```
```   423   by simp
```
```   424
```
```   425 lemma mod_diff_cong:
```
```   426   assumes "a mod c = a' mod c"
```
```   427   assumes "b mod c = b' mod c"
```
```   428   shows "(a - b) mod c = (a' - b') mod c"
```
```   429   using assms mod_add_cong [of a c a' "- b" "- b'"] mod_minus_cong [of b c "b'"]
```
```   430   by simp
```
```   431
```
```   432 lemma minus_mod_self2 [simp]:
```
```   433   "(a - b) mod b = a mod b"
```
```   434   using mod_diff_right_eq [of a b b]
```
```   435   by (simp add: mod_diff_right_eq)
```
```   436
```
```   437 lemma minus_mod_self1 [simp]:
```
```   438   "(b - a) mod b = - a mod b"
```
```   439   using mod_add_self2 [of "- a" b] by simp
```
```   440
```
```   441 end
```
```   442
```
```   443
```
```   444 subsection \<open>Parity\<close>
```
```   445
```
```   446 class semiring_div_parity = semiring_div + comm_semiring_1_cancel + numeral +
```
```   447   assumes parity: "a mod 2 = 0 \<or> a mod 2 = 1"
```
```   448   assumes one_mod_two_eq_one [simp]: "1 mod 2 = 1"
```
```   449   assumes zero_not_eq_two: "0 \<noteq> 2"
```
```   450 begin
```
```   451
```
```   452 lemma parity_cases [case_names even odd]:
```
```   453   assumes "a mod 2 = 0 \<Longrightarrow> P"
```
```   454   assumes "a mod 2 = 1 \<Longrightarrow> P"
```
```   455   shows P
```
```   456   using assms parity by blast
```
```   457
```
```   458 lemma one_div_two_eq_zero [simp]:
```
```   459   "1 div 2 = 0"
```
```   460 proof (cases "2 = 0")
```
```   461   case True then show ?thesis by simp
```
```   462 next
```
```   463   case False
```
```   464   from div_mult_mod_eq have "1 div 2 * 2 + 1 mod 2 = 1" .
```
```   465   with one_mod_two_eq_one have "1 div 2 * 2 + 1 = 1" by simp
```
```   466   then have "1 div 2 * 2 = 0" by (simp add: ac_simps add_left_imp_eq del: mult_eq_0_iff)
```
```   467   then have "1 div 2 = 0 \<or> 2 = 0" by simp
```
```   468   with False show ?thesis by auto
```
```   469 qed
```
```   470
```
```   471 lemma not_mod_2_eq_0_eq_1 [simp]:
```
```   472   "a mod 2 \<noteq> 0 \<longleftrightarrow> a mod 2 = 1"
```
```   473   by (cases a rule: parity_cases) simp_all
```
```   474
```
```   475 lemma not_mod_2_eq_1_eq_0 [simp]:
```
```   476   "a mod 2 \<noteq> 1 \<longleftrightarrow> a mod 2 = 0"
```
```   477   by (cases a rule: parity_cases) simp_all
```
```   478
```
```   479 subclass semiring_parity
```
```   480 proof (unfold_locales, unfold dvd_eq_mod_eq_0 not_mod_2_eq_0_eq_1)
```
```   481   show "1 mod 2 = 1"
```
```   482     by (fact one_mod_two_eq_one)
```
```   483 next
```
```   484   fix a b
```
```   485   assume "a mod 2 = 1"
```
```   486   moreover assume "b mod 2 = 1"
```
```   487   ultimately show "(a + b) mod 2 = 0"
```
```   488     using mod_add_eq [of a 2 b] by simp
```
```   489 next
```
```   490   fix a b
```
```   491   assume "(a * b) mod 2 = 0"
```
```   492   then have "(a mod 2) * (b mod 2) mod 2 = 0"
```
```   493     by (simp add: mod_mult_eq)
```
```   494   then have "(a mod 2) * (b mod 2) = 0"
```
```   495     by (cases "a mod 2 = 0") simp_all
```
```   496   then show "a mod 2 = 0 \<or> b mod 2 = 0"
```
```   497     by (rule divisors_zero)
```
```   498 next
```
```   499   fix a
```
```   500   assume "a mod 2 = 1"
```
```   501   then have "a = a div 2 * 2 + 1"
```
```   502     using div_mult_mod_eq [of a 2] by simp
```
```   503   then show "\<exists>b. a = b + 1" ..
```
```   504 qed
```
```   505
```
```   506 lemma even_iff_mod_2_eq_zero:
```
```   507   "even a \<longleftrightarrow> a mod 2 = 0"
```
```   508   by (fact dvd_eq_mod_eq_0)
```
```   509
```
```   510 lemma odd_iff_mod_2_eq_one:
```
```   511   "odd a \<longleftrightarrow> a mod 2 = 1"
```
```   512   by (auto simp add: even_iff_mod_2_eq_zero)
```
```   513
```
```   514 lemma even_succ_div_two [simp]:
```
```   515   "even a \<Longrightarrow> (a + 1) div 2 = a div 2"
```
```   516   by (cases "a = 0") (auto elim!: evenE dest: mult_not_zero)
```
```   517
```
```   518 lemma odd_succ_div_two [simp]:
```
```   519   "odd a \<Longrightarrow> (a + 1) div 2 = a div 2 + 1"
```
```   520   by (auto elim!: oddE simp add: zero_not_eq_two [symmetric] add.assoc)
```
```   521
```
```   522 lemma even_two_times_div_two:
```
```   523   "even a \<Longrightarrow> 2 * (a div 2) = a"
```
```   524   by (fact dvd_mult_div_cancel)
```
```   525
```
```   526 lemma odd_two_times_div_two_succ [simp]:
```
```   527   "odd a \<Longrightarrow> 2 * (a div 2) + 1 = a"
```
```   528   using mult_div_mod_eq [of 2 a] by (simp add: even_iff_mod_2_eq_zero)
```
```   529
```
```   530 end
```
```   531
```
```   532
```
```   533 subsection \<open>Numeral division with a pragmatic type class\<close>
```
```   534
```
```   535 text \<open>
```
```   536   The following type class contains everything necessary to formulate
```
```   537   a division algorithm in ring structures with numerals, restricted
```
```   538   to its positive segments.  This is its primary motiviation, and it
```
```   539   could surely be formulated using a more fine-grained, more algebraic
```
```   540   and less technical class hierarchy.
```
```   541 \<close>
```
```   542
```
```   543 class semiring_numeral_div = semiring_div + comm_semiring_1_cancel + linordered_semidom +
```
```   544   assumes div_less: "0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a div b = 0"
```
```   545     and mod_less: " 0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a mod b = a"
```
```   546     and div_positive: "0 < b \<Longrightarrow> b \<le> a \<Longrightarrow> a div b > 0"
```
```   547     and mod_less_eq_dividend: "0 \<le> a \<Longrightarrow> a mod b \<le> a"
```
```   548     and pos_mod_bound: "0 < b \<Longrightarrow> a mod b < b"
```
```   549     and pos_mod_sign: "0 < b \<Longrightarrow> 0 \<le> a mod b"
```
```   550     and mod_mult2_eq: "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
```
```   551     and div_mult2_eq: "0 \<le> c \<Longrightarrow> a div (b * c) = a div b div c"
```
```   552   assumes discrete: "a < b \<longleftrightarrow> a + 1 \<le> b"
```
```   553   fixes divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"
```
```   554     and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"
```
```   555   assumes divmod_def: "divmod m n = (numeral m div numeral n, numeral m mod numeral n)"
```
```   556     and divmod_step_def: "divmod_step l qr = (let (q, r) = qr
```
```   557     in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
```
```   558     else (2 * q, r))"
```
```   559     \<comment> \<open>These are conceptually definitions but force generated code
```
```   560     to be monomorphic wrt. particular instances of this class which
```
```   561     yields a significant speedup.\<close>
```
```   562
```
```   563 begin
```
```   564
```
```   565 subclass semiring_div_parity
```
```   566 proof
```
```   567   fix a
```
```   568   show "a mod 2 = 0 \<or> a mod 2 = 1"
```
```   569   proof (rule ccontr)
```
```   570     assume "\<not> (a mod 2 = 0 \<or> a mod 2 = 1)"
```
```   571     then have "a mod 2 \<noteq> 0" and "a mod 2 \<noteq> 1" by simp_all
```
```   572     have "0 < 2" by simp
```
```   573     with pos_mod_bound pos_mod_sign have "0 \<le> a mod 2" "a mod 2 < 2" by simp_all
```
```   574     with \<open>a mod 2 \<noteq> 0\<close> have "0 < a mod 2" by simp
```
```   575     with discrete have "1 \<le> a mod 2" by simp
```
```   576     with \<open>a mod 2 \<noteq> 1\<close> have "1 < a mod 2" by simp
```
```   577     with discrete have "2 \<le> a mod 2" by simp
```
```   578     with \<open>a mod 2 < 2\<close> show False by simp
```
```   579   qed
```
```   580 next
```
```   581   show "1 mod 2 = 1"
```
```   582     by (rule mod_less) simp_all
```
```   583 next
```
```   584   show "0 \<noteq> 2"
```
```   585     by simp
```
```   586 qed
```
```   587
```
```   588 lemma divmod_digit_1:
```
```   589   assumes "0 \<le> a" "0 < b" and "b \<le> a mod (2 * b)"
```
```   590   shows "2 * (a div (2 * b)) + 1 = a div b" (is "?P")
```
```   591     and "a mod (2 * b) - b = a mod b" (is "?Q")
```
```   592 proof -
```
```   593   from assms mod_less_eq_dividend [of a "2 * b"] have "b \<le> a"
```
```   594     by (auto intro: trans)
```
```   595   with \<open>0 < b\<close> have "0 < a div b" by (auto intro: div_positive)
```
```   596   then have [simp]: "1 \<le> a div b" by (simp add: discrete)
```
```   597   with \<open>0 < b\<close> have mod_less: "a mod b < b" by (simp add: pos_mod_bound)
```
```   598   define w where "w = a div b mod 2"
```
```   599   with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
```
```   600   have mod_w: "a mod (2 * b) = a mod b + b * w"
```
```   601     by (simp add: w_def mod_mult2_eq ac_simps)
```
```   602   from assms w_exhaust have "w = 1"
```
```   603     by (auto simp add: mod_w) (insert mod_less, auto)
```
```   604   with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp
```
```   605   have "2 * (a div (2 * b)) = a div b - w"
```
```   606     by (simp add: w_def div_mult2_eq minus_mod_eq_mult_div ac_simps)
```
```   607   with \<open>w = 1\<close> have div: "2 * (a div (2 * b)) = a div b - 1" by simp
```
```   608   then show ?P and ?Q
```
```   609     by (simp_all add: div mod add_implies_diff [symmetric])
```
```   610 qed
```
```   611
```
```   612 lemma divmod_digit_0:
```
```   613   assumes "0 < b" and "a mod (2 * b) < b"
```
```   614   shows "2 * (a div (2 * b)) = a div b" (is "?P")
```
```   615     and "a mod (2 * b) = a mod b" (is "?Q")
```
```   616 proof -
```
```   617   define w where "w = a div b mod 2"
```
```   618   with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
```
```   619   have mod_w: "a mod (2 * b) = a mod b + b * w"
```
```   620     by (simp add: w_def mod_mult2_eq ac_simps)
```
```   621   moreover have "b \<le> a mod b + b"
```
```   622   proof -
```
```   623     from \<open>0 < b\<close> pos_mod_sign have "0 \<le> a mod b" by blast
```
```   624     then have "0 + b \<le> a mod b + b" by (rule add_right_mono)
```
```   625     then show ?thesis by simp
```
```   626   qed
```
```   627   moreover note assms w_exhaust
```
```   628   ultimately have "w = 0" by auto
```
```   629   with mod_w have mod: "a mod (2 * b) = a mod b" by simp
```
```   630   have "2 * (a div (2 * b)) = a div b - w"
```
```   631     by (simp add: w_def div_mult2_eq minus_mod_eq_mult_div ac_simps)
```
```   632   with \<open>w = 0\<close> have div: "2 * (a div (2 * b)) = a div b" by simp
```
```   633   then show ?P and ?Q
```
```   634     by (simp_all add: div mod)
```
```   635 qed
```
```   636
```
```   637 lemma fst_divmod:
```
```   638   "fst (divmod m n) = numeral m div numeral n"
```
```   639   by (simp add: divmod_def)
```
```   640
```
```   641 lemma snd_divmod:
```
```   642   "snd (divmod m n) = numeral m mod numeral n"
```
```   643   by (simp add: divmod_def)
```
```   644
```
```   645 text \<open>
```
```   646   This is a formulation of one step (referring to one digit position)
```
```   647   in school-method division: compare the dividend at the current
```
```   648   digit position with the remainder from previous division steps
```
```   649   and evaluate accordingly.
```
```   650 \<close>
```
```   651
```
```   652 lemma divmod_step_eq [simp]:
```
```   653   "divmod_step l (q, r) = (if numeral l \<le> r
```
```   654     then (2 * q + 1, r - numeral l) else (2 * q, r))"
```
```   655   by (simp add: divmod_step_def)
```
```   656
```
```   657 text \<open>
```
```   658   This is a formulation of school-method division.
```
```   659   If the divisor is smaller than the dividend, terminate.
```
```   660   If not, shift the dividend to the right until termination
```
```   661   occurs and then reiterate single division steps in the
```
```   662   opposite direction.
```
```   663 \<close>
```
```   664
```
```   665 lemma divmod_divmod_step:
```
```   666   "divmod m n = (if m < n then (0, numeral m)
```
```   667     else divmod_step n (divmod m (Num.Bit0 n)))"
```
```   668 proof (cases "m < n")
```
```   669   case True then have "numeral m < numeral n" by simp
```
```   670   then show ?thesis
```
```   671     by (simp add: prod_eq_iff div_less mod_less fst_divmod snd_divmod)
```
```   672 next
```
```   673   case False
```
```   674   have "divmod m n =
```
```   675     divmod_step n (numeral m div (2 * numeral n),
```
```   676       numeral m mod (2 * numeral n))"
```
```   677   proof (cases "numeral n \<le> numeral m mod (2 * numeral n)")
```
```   678     case True
```
```   679     with divmod_step_eq
```
```   680       have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
```
```   681         (2 * (numeral m div (2 * numeral n)) + 1, numeral m mod (2 * numeral n) - numeral n)"
```
```   682         by simp
```
```   683     moreover from True divmod_digit_1 [of "numeral m" "numeral n"]
```
```   684       have "2 * (numeral m div (2 * numeral n)) + 1 = numeral m div numeral n"
```
```   685       and "numeral m mod (2 * numeral n) - numeral n = numeral m mod numeral n"
```
```   686       by simp_all
```
```   687     ultimately show ?thesis by (simp only: divmod_def)
```
```   688   next
```
```   689     case False then have *: "numeral m mod (2 * numeral n) < numeral n"
```
```   690       by (simp add: not_le)
```
```   691     with divmod_step_eq
```
```   692       have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
```
```   693         (2 * (numeral m div (2 * numeral n)), numeral m mod (2 * numeral n))"
```
```   694         by auto
```
```   695     moreover from * divmod_digit_0 [of "numeral n" "numeral m"]
```
```   696       have "2 * (numeral m div (2 * numeral n)) = numeral m div numeral n"
```
```   697       and "numeral m mod (2 * numeral n) = numeral m mod numeral n"
```
```   698       by (simp_all only: zero_less_numeral)
```
```   699     ultimately show ?thesis by (simp only: divmod_def)
```
```   700   qed
```
```   701   then have "divmod m n =
```
```   702     divmod_step n (numeral m div numeral (Num.Bit0 n),
```
```   703       numeral m mod numeral (Num.Bit0 n))"
```
```   704     by (simp only: numeral.simps distrib mult_1)
```
```   705   then have "divmod m n = divmod_step n (divmod m (Num.Bit0 n))"
```
```   706     by (simp add: divmod_def)
```
```   707   with False show ?thesis by simp
```
```   708 qed
```
```   709
```
```   710 text \<open>The division rewrite proper -- first, trivial results involving \<open>1\<close>\<close>
```
```   711
```
```   712 lemma divmod_trivial [simp]:
```
```   713   "divmod Num.One Num.One = (numeral Num.One, 0)"
```
```   714   "divmod (Num.Bit0 m) Num.One = (numeral (Num.Bit0 m), 0)"
```
```   715   "divmod (Num.Bit1 m) Num.One = (numeral (Num.Bit1 m), 0)"
```
```   716   "divmod num.One (num.Bit0 n) = (0, Numeral1)"
```
```   717   "divmod num.One (num.Bit1 n) = (0, Numeral1)"
```
```   718   using divmod_divmod_step [of "Num.One"] by (simp_all add: divmod_def)
```
```   719
```
```   720 text \<open>Division by an even number is a right-shift\<close>
```
```   721
```
```   722 lemma divmod_cancel [simp]:
```
```   723   "divmod (Num.Bit0 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r))" (is ?P)
```
```   724   "divmod (Num.Bit1 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r + 1))" (is ?Q)
```
```   725 proof -
```
```   726   have *: "\<And>q. numeral (Num.Bit0 q) = 2 * numeral q"
```
```   727     "\<And>q. numeral (Num.Bit1 q) = 2 * numeral q + 1"
```
```   728     by (simp_all only: numeral_mult numeral.simps distrib) simp_all
```
```   729   have "1 div 2 = 0" "1 mod 2 = 1" by (auto intro: div_less mod_less)
```
```   730   then show ?P and ?Q
```
```   731     by (simp_all add: fst_divmod snd_divmod prod_eq_iff split_def * [of m] * [of n] mod_mult_mult1
```
```   732       div_mult2_eq [of _ _ 2] mod_mult2_eq [of _ _ 2]
```
```   733       add.commute del: numeral_times_numeral)
```
```   734 qed
```
```   735
```
```   736 text \<open>The really hard work\<close>
```
```   737
```
```   738 lemma divmod_steps [simp]:
```
```   739   "divmod (num.Bit0 m) (num.Bit1 n) =
```
```   740       (if m \<le> n then (0, numeral (num.Bit0 m))
```
```   741        else divmod_step (num.Bit1 n)
```
```   742              (divmod (num.Bit0 m)
```
```   743                (num.Bit0 (num.Bit1 n))))"
```
```   744   "divmod (num.Bit1 m) (num.Bit1 n) =
```
```   745       (if m < n then (0, numeral (num.Bit1 m))
```
```   746        else divmod_step (num.Bit1 n)
```
```   747              (divmod (num.Bit1 m)
```
```   748                (num.Bit0 (num.Bit1 n))))"
```
```   749   by (simp_all add: divmod_divmod_step)
```
```   750
```
```   751 lemmas divmod_algorithm_code = divmod_step_eq divmod_trivial divmod_cancel divmod_steps
```
```   752
```
```   753 text \<open>Special case: divisibility\<close>
```
```   754
```
```   755 definition divides_aux :: "'a \<times> 'a \<Rightarrow> bool"
```
```   756 where
```
```   757   "divides_aux qr \<longleftrightarrow> snd qr = 0"
```
```   758
```
```   759 lemma divides_aux_eq [simp]:
```
```   760   "divides_aux (q, r) \<longleftrightarrow> r = 0"
```
```   761   by (simp add: divides_aux_def)
```
```   762
```
```   763 lemma dvd_numeral_simp [simp]:
```
```   764   "numeral m dvd numeral n \<longleftrightarrow> divides_aux (divmod n m)"
```
```   765   by (simp add: divmod_def mod_eq_0_iff_dvd)
```
```   766
```
```   767 text \<open>Generic computation of quotient and remainder\<close>
```
```   768
```
```   769 lemma numeral_div_numeral [simp]:
```
```   770   "numeral k div numeral l = fst (divmod k l)"
```
```   771   by (simp add: fst_divmod)
```
```   772
```
```   773 lemma numeral_mod_numeral [simp]:
```
```   774   "numeral k mod numeral l = snd (divmod k l)"
```
```   775   by (simp add: snd_divmod)
```
```   776
```
```   777 lemma one_div_numeral [simp]:
```
```   778   "1 div numeral n = fst (divmod num.One n)"
```
```   779   by (simp add: fst_divmod)
```
```   780
```
```   781 lemma one_mod_numeral [simp]:
```
```   782   "1 mod numeral n = snd (divmod num.One n)"
```
```   783   by (simp add: snd_divmod)
```
```   784
```
```   785 text \<open>Computing congruences modulo \<open>2 ^ q\<close>\<close>
```
```   786
```
```   787 lemma cong_exp_iff_simps:
```
```   788   "numeral n mod numeral Num.One = 0
```
```   789     \<longleftrightarrow> True"
```
```   790   "numeral (Num.Bit0 n) mod numeral (Num.Bit0 q) = 0
```
```   791     \<longleftrightarrow> numeral n mod numeral q = 0"
```
```   792   "numeral (Num.Bit1 n) mod numeral (Num.Bit0 q) = 0
```
```   793     \<longleftrightarrow> False"
```
```   794   "numeral m mod numeral Num.One = (numeral n mod numeral Num.One)
```
```   795     \<longleftrightarrow> True"
```
```   796   "numeral Num.One mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
```
```   797     \<longleftrightarrow> True"
```
```   798   "numeral Num.One mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
```
```   799     \<longleftrightarrow> False"
```
```   800   "numeral Num.One mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
```
```   801     \<longleftrightarrow> (numeral n mod numeral q) = 0"
```
```   802   "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
```
```   803     \<longleftrightarrow> False"
```
```   804   "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
```
```   805     \<longleftrightarrow> numeral m mod numeral q = (numeral n mod numeral q)"
```
```   806   "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
```
```   807     \<longleftrightarrow> False"
```
```   808   "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
```
```   809     \<longleftrightarrow> (numeral m mod numeral q) = 0"
```
```   810   "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
```
```   811     \<longleftrightarrow> False"
```
```   812   "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
```
```   813     \<longleftrightarrow> numeral m mod numeral q = (numeral n mod numeral q)"
```
```   814   by (auto simp add: case_prod_beta dest: arg_cong [of _ _ even])
```
```   815
```
```   816 end
```
```   817
```
```   818
```
```   819 subsection \<open>Division on @{typ nat}\<close>
```
```   820
```
```   821 context
```
```   822 begin
```
```   823
```
```   824 text \<open>
```
```   825   We define @{const divide} and @{const modulo} on @{typ nat} by means
```
```   826   of a characteristic relation with two input arguments
```
```   827   @{term "m::nat"}, @{term "n::nat"} and two output arguments
```
```   828   @{term "q::nat"}(uotient) and @{term "r::nat"}(emainder).
```
```   829 \<close>
```
```   830
```
```   831 inductive eucl_rel_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool"
```
```   832   where eucl_rel_nat_by0: "eucl_rel_nat m 0 (0, m)"
```
```   833   | eucl_rel_natI: "r < n \<Longrightarrow> m = q * n + r \<Longrightarrow> eucl_rel_nat m n (q, r)"
```
```   834
```
```   835 text \<open>@{const eucl_rel_nat} is total:\<close>
```
```   836
```
```   837 qualified lemma eucl_rel_nat_ex:
```
```   838   obtains q r where "eucl_rel_nat m n (q, r)"
```
```   839 proof (cases "n = 0")
```
```   840   case True
```
```   841   with that eucl_rel_nat_by0 show thesis
```
```   842     by blast
```
```   843 next
```
```   844   case False
```
```   845   have "\<exists>q r. m = q * n + r \<and> r < n"
```
```   846   proof (induct m)
```
```   847     case 0 with \<open>n \<noteq> 0\<close>
```
```   848     have "(0::nat) = 0 * n + 0 \<and> 0 < n" by simp
```
```   849     then show ?case by blast
```
```   850   next
```
```   851     case (Suc m) then obtain q' r'
```
```   852       where m: "m = q' * n + r'" and n: "r' < n" by auto
```
```   853     then show ?case proof (cases "Suc r' < n")
```
```   854       case True
```
```   855       from m n have "Suc m = q' * n + Suc r'" by simp
```
```   856       with True show ?thesis by blast
```
```   857     next
```
```   858       case False then have "n \<le> Suc r'"
```
```   859         by (simp add: not_less)
```
```   860       moreover from n have "Suc r' \<le> n"
```
```   861         by (simp add: Suc_le_eq)
```
```   862       ultimately have "n = Suc r'" by auto
```
```   863       with m have "Suc m = Suc q' * n + 0" by simp
```
```   864       with \<open>n \<noteq> 0\<close> show ?thesis by blast
```
```   865     qed
```
```   866   qed
```
```   867   with that \<open>n \<noteq> 0\<close> eucl_rel_natI show thesis
```
```   868     by blast
```
```   869 qed
```
```   870
```
```   871 text \<open>@{const eucl_rel_nat} is injective:\<close>
```
```   872
```
```   873 qualified lemma eucl_rel_nat_unique_div:
```
```   874   assumes "eucl_rel_nat m n (q, r)"
```
```   875     and "eucl_rel_nat m n (q', r')"
```
```   876   shows "q = q'"
```
```   877 proof (cases "n = 0")
```
```   878   case True with assms show ?thesis
```
```   879     by (auto elim: eucl_rel_nat.cases)
```
```   880 next
```
```   881   case False
```
```   882   have *: "q' \<le> q" if "q' * n + r' = q * n + r" "r < n" for q r q' r' :: nat
```
```   883   proof (rule ccontr)
```
```   884     assume "\<not> q' \<le> q"
```
```   885     then have "q < q'"
```
```   886       by (simp add: not_le)
```
```   887     with that show False
```
```   888       by (auto simp add: less_iff_Suc_add algebra_simps)
```
```   889   qed
```
```   890   from \<open>n \<noteq> 0\<close> assms show ?thesis
```
```   891     by (auto intro: order_antisym elim: eucl_rel_nat.cases dest: * sym split: if_splits)
```
```   892 qed
```
```   893
```
```   894 qualified lemma eucl_rel_nat_unique_mod:
```
```   895   assumes "eucl_rel_nat m n (q, r)"
```
```   896     and "eucl_rel_nat m n (q', r')"
```
```   897   shows "r = r'"
```
```   898 proof -
```
```   899   from assms have "q' = q"
```
```   900     by (auto intro: eucl_rel_nat_unique_div)
```
```   901   with assms show ?thesis
```
```   902     by (auto elim!: eucl_rel_nat.cases)
```
```   903 qed
```
```   904
```
```   905 text \<open>
```
```   906   We instantiate divisibility on the natural numbers by
```
```   907   means of @{const eucl_rel_nat}:
```
```   908 \<close>
```
```   909
```
```   910 qualified definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
```
```   911   "divmod_nat m n = (THE qr. eucl_rel_nat m n qr)"
```
```   912
```
```   913 qualified lemma eucl_rel_nat_divmod_nat:
```
```   914   "eucl_rel_nat m n (divmod_nat m n)"
```
```   915 proof -
```
```   916   from eucl_rel_nat_ex
```
```   917     obtain q r where rel: "eucl_rel_nat m n (q, r)" .
```
```   918   then show ?thesis
```
```   919     by (auto simp add: divmod_nat_def intro: theI
```
```   920       elim: eucl_rel_nat_unique_div eucl_rel_nat_unique_mod)
```
```   921 qed
```
```   922
```
```   923 qualified lemma divmod_nat_unique:
```
```   924   "divmod_nat m n = (q, r)" if "eucl_rel_nat m n (q, r)"
```
```   925   using that
```
```   926   by (auto simp add: divmod_nat_def intro: eucl_rel_nat_divmod_nat elim: eucl_rel_nat_unique_div eucl_rel_nat_unique_mod)
```
```   927
```
```   928 qualified lemma divmod_nat_zero:
```
```   929   "divmod_nat m 0 = (0, m)"
```
```   930   by (rule divmod_nat_unique) (fact eucl_rel_nat_by0)
```
```   931
```
```   932 qualified lemma divmod_nat_zero_left:
```
```   933   "divmod_nat 0 n = (0, 0)"
```
```   934   by (rule divmod_nat_unique)
```
```   935     (cases n, auto intro: eucl_rel_nat_by0 eucl_rel_natI)
```
```   936
```
```   937 qualified lemma divmod_nat_base:
```
```   938   "m < n \<Longrightarrow> divmod_nat m n = (0, m)"
```
```   939   by (rule divmod_nat_unique)
```
```   940     (cases n, auto intro: eucl_rel_nat_by0 eucl_rel_natI)
```
```   941
```
```   942 qualified lemma divmod_nat_step:
```
```   943   assumes "0 < n" and "n \<le> m"
```
```   944   shows "divmod_nat m n =
```
```   945     (Suc (fst (divmod_nat (m - n) n)), snd (divmod_nat (m - n) n))"
```
```   946 proof (rule divmod_nat_unique)
```
```   947   have "eucl_rel_nat (m - n) n (divmod_nat (m - n) n)"
```
```   948     by (fact eucl_rel_nat_divmod_nat)
```
```   949   then show "eucl_rel_nat m n (Suc
```
```   950     (fst (divmod_nat (m - n) n)), snd (divmod_nat (m - n) n))"
```
```   951     using assms
```
```   952       by (auto split: if_splits intro: eucl_rel_natI elim!: eucl_rel_nat.cases simp add: algebra_simps)
```
```   953 qed
```
```   954
```
```   955 end
```
```   956
```
```   957 instantiation nat :: "{semidom_modulo, normalization_semidom}"
```
```   958 begin
```
```   959
```
```   960 definition normalize_nat :: "nat \<Rightarrow> nat"
```
```   961   where [simp]: "normalize = (id :: nat \<Rightarrow> nat)"
```
```   962
```
```   963 definition unit_factor_nat :: "nat \<Rightarrow> nat"
```
```   964   where "unit_factor n = (if n = 0 then 0 else 1 :: nat)"
```
```   965
```
```   966 lemma unit_factor_simps [simp]:
```
```   967   "unit_factor 0 = (0::nat)"
```
```   968   "unit_factor (Suc n) = 1"
```
```   969   by (simp_all add: unit_factor_nat_def)
```
```   970
```
```   971 definition divide_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
```
```   972   where div_nat_def: "m div n = fst (Divides.divmod_nat m n)"
```
```   973
```
```   974 definition modulo_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
```
```   975   where mod_nat_def: "m mod n = snd (Divides.divmod_nat m n)"
```
```   976
```
```   977 lemma fst_divmod_nat [simp]:
```
```   978   "fst (Divides.divmod_nat m n) = m div n"
```
```   979   by (simp add: div_nat_def)
```
```   980
```
```   981 lemma snd_divmod_nat [simp]:
```
```   982   "snd (Divides.divmod_nat m n) = m mod n"
```
```   983   by (simp add: mod_nat_def)
```
```   984
```
```   985 lemma divmod_nat_div_mod:
```
```   986   "Divides.divmod_nat m n = (m div n, m mod n)"
```
```   987   by (simp add: prod_eq_iff)
```
```   988
```
```   989 lemma div_nat_unique:
```
```   990   assumes "eucl_rel_nat m n (q, r)"
```
```   991   shows "m div n = q"
```
```   992   using assms
```
```   993   by (auto dest!: Divides.divmod_nat_unique simp add: prod_eq_iff)
```
```   994
```
```   995 lemma mod_nat_unique:
```
```   996   assumes "eucl_rel_nat m n (q, r)"
```
```   997   shows "m mod n = r"
```
```   998   using assms
```
```   999   by (auto dest!: Divides.divmod_nat_unique simp add: prod_eq_iff)
```
```  1000
```
```  1001 lemma eucl_rel_nat: "eucl_rel_nat m n (m div n, m mod n)"
```
```  1002   using Divides.eucl_rel_nat_divmod_nat
```
```  1003   by (simp add: divmod_nat_div_mod)
```
```  1004
```
```  1005 text \<open>The ''recursion'' equations for @{const divide} and @{const modulo}\<close>
```
```  1006
```
```  1007 lemma div_less [simp]:
```
```  1008   fixes m n :: nat
```
```  1009   assumes "m < n"
```
```  1010   shows "m div n = 0"
```
```  1011   using assms Divides.divmod_nat_base by (simp add: prod_eq_iff)
```
```  1012
```
```  1013 lemma le_div_geq:
```
```  1014   fixes m n :: nat
```
```  1015   assumes "0 < n" and "n \<le> m"
```
```  1016   shows "m div n = Suc ((m - n) div n)"
```
```  1017   using assms Divides.divmod_nat_step by (simp add: prod_eq_iff)
```
```  1018
```
```  1019 lemma mod_less [simp]:
```
```  1020   fixes m n :: nat
```
```  1021   assumes "m < n"
```
```  1022   shows "m mod n = m"
```
```  1023   using assms Divides.divmod_nat_base by (simp add: prod_eq_iff)
```
```  1024
```
```  1025 lemma le_mod_geq:
```
```  1026   fixes m n :: nat
```
```  1027   assumes "n \<le> m"
```
```  1028   shows "m mod n = (m - n) mod n"
```
```  1029   using assms Divides.divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)
```
```  1030
```
```  1031 lemma mod_less_divisor [simp]:
```
```  1032   fixes m n :: nat
```
```  1033   assumes "n > 0"
```
```  1034   shows "m mod n < n"
```
```  1035   using assms eucl_rel_nat [of m n]
```
```  1036     by (auto elim: eucl_rel_nat.cases)
```
```  1037
```
```  1038 lemma mod_le_divisor [simp]:
```
```  1039   fixes m n :: nat
```
```  1040   assumes "n > 0"
```
```  1041   shows "m mod n \<le> n"
```
```  1042   using assms eucl_rel_nat [of m n]
```
```  1043     by (auto elim: eucl_rel_nat.cases)
```
```  1044
```
```  1045 instance proof
```
```  1046   fix m n :: nat
```
```  1047   show "m div n * n + m mod n = m"
```
```  1048     using eucl_rel_nat [of m n]
```
```  1049     by (auto elim: eucl_rel_nat.cases)
```
```  1050 next
```
```  1051   fix n :: nat show "n div 0 = 0"
```
```  1052     by (simp add: div_nat_def Divides.divmod_nat_zero)
```
```  1053 next
```
```  1054   fix m n :: nat
```
```  1055   assume "n \<noteq> 0"
```
```  1056   then show "m * n div n = m"
```
```  1057     by (auto intro!: eucl_rel_natI div_nat_unique [of _ _ _ 0])
```
```  1058 qed (simp_all add: unit_factor_nat_def)
```
```  1059
```
```  1060 end
```
```  1061
```
```  1062 instance nat :: semiring_div
```
```  1063 proof
```
```  1064   fix m n q :: nat
```
```  1065   assume "n \<noteq> 0"
```
```  1066   then show "(q + m * n) div n = m + q div n"
```
```  1067     by (induct m) (simp_all add: le_div_geq)
```
```  1068 next
```
```  1069   fix m n q :: nat
```
```  1070   assume "m \<noteq> 0"
```
```  1071   show "(m * n) div (m * q) = n div q"
```
```  1072   proof (cases "q = 0")
```
```  1073     case True
```
```  1074     then show ?thesis
```
```  1075       by simp
```
```  1076   next
```
```  1077     case False
```
```  1078     show ?thesis
```
```  1079     proof (rule div_nat_unique [of _ _ _ "m * (n mod q)"])
```
```  1080       show "eucl_rel_nat (m * n) (m * q) (n div q, m * (n mod q))"
```
```  1081         by (rule eucl_rel_natI)
```
```  1082           (use \<open>m \<noteq> 0\<close> \<open>q \<noteq> 0\<close> div_mult_mod_eq [of n q] in \<open>auto simp add: algebra_simps distrib_left [symmetric]\<close>)
```
```  1083     qed
```
```  1084   qed
```
```  1085 qed
```
```  1086
```
```  1087 lemma div_by_Suc_0 [simp]:
```
```  1088   "m div Suc 0 = m"
```
```  1089   using div_by_1 [of m] by simp
```
```  1090
```
```  1091 lemma mod_by_Suc_0 [simp]:
```
```  1092   "m mod Suc 0 = 0"
```
```  1093   using mod_by_1 [of m] by simp
```
```  1094
```
```  1095 lemma mod_greater_zero_iff_not_dvd:
```
```  1096   fixes m n :: nat
```
```  1097   shows "m mod n > 0 \<longleftrightarrow> \<not> n dvd m"
```
```  1098   by (simp add: dvd_eq_mod_eq_0)
```
```  1099
```
```  1100 text \<open>Simproc for cancelling @{const divide} and @{const modulo}\<close>
```
```  1101
```
```  1102 lemma (in semiring_modulo) cancel_div_mod_rules:
```
```  1103   "((a div b) * b + a mod b) + c = a + c"
```
```  1104   "(b * (a div b) + a mod b) + c = a + c"
```
```  1105   by (simp_all add: div_mult_mod_eq mult_div_mod_eq)
```
```  1106
```
```  1107 ML_file "~~/src/Provers/Arith/cancel_div_mod.ML"
```
```  1108
```
```  1109 ML \<open>
```
```  1110 structure Cancel_Div_Mod_Nat = Cancel_Div_Mod
```
```  1111 (
```
```  1112   val div_name = @{const_name divide};
```
```  1113   val mod_name = @{const_name modulo};
```
```  1114   val mk_binop = HOLogic.mk_binop;
```
```  1115   val mk_plus = HOLogic.mk_binop @{const_name Groups.plus};
```
```  1116   val dest_plus = HOLogic.dest_bin @{const_name Groups.plus} HOLogic.natT;
```
```  1117   fun mk_sum [] = HOLogic.zero
```
```  1118     | mk_sum [t] = t
```
```  1119     | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
```
```  1120   fun dest_sum tm =
```
```  1121     if HOLogic.is_zero tm then []
```
```  1122     else
```
```  1123       (case try HOLogic.dest_Suc tm of
```
```  1124         SOME t => HOLogic.Suc_zero :: dest_sum t
```
```  1125       | NONE =>
```
```  1126           (case try dest_plus tm of
```
```  1127             SOME (t, u) => dest_sum t @ dest_sum u
```
```  1128           | NONE => [tm]));
```
```  1129
```
```  1130   val div_mod_eqs = map mk_meta_eq @{thms cancel_div_mod_rules};
```
```  1131
```
```  1132   val prove_eq_sums = Arith_Data.prove_conv2 all_tac
```
```  1133     (Arith_Data.simp_all_tac @{thms add_0_left add_0_right ac_simps})
```
```  1134 )
```
```  1135 \<close>
```
```  1136
```
```  1137 simproc_setup cancel_div_mod_nat ("(m::nat) + n") =
```
```  1138   \<open>K Cancel_Div_Mod_Nat.proc\<close>
```
```  1139
```
```  1140 lemma divmod_nat_if [code]:
```
```  1141   "Divides.divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
```
```  1142     let (q, r) = Divides.divmod_nat (m - n) n in (Suc q, r))"
```
```  1143   by (simp add: prod_eq_iff case_prod_beta not_less le_div_geq le_mod_geq)
```
```  1144
```
```  1145 lemma mod_Suc_eq [mod_simps]:
```
```  1146   "Suc (m mod n) mod n = Suc m mod n"
```
```  1147 proof -
```
```  1148   have "(m mod n + 1) mod n = (m + 1) mod n"
```
```  1149     by (simp only: mod_simps)
```
```  1150   then show ?thesis
```
```  1151     by simp
```
```  1152 qed
```
```  1153
```
```  1154 lemma mod_Suc_Suc_eq [mod_simps]:
```
```  1155   "Suc (Suc (m mod n)) mod n = Suc (Suc m) mod n"
```
```  1156 proof -
```
```  1157   have "(m mod n + 2) mod n = (m + 2) mod n"
```
```  1158     by (simp only: mod_simps)
```
```  1159   then show ?thesis
```
```  1160     by simp
```
```  1161 qed
```
```  1162
```
```  1163
```
```  1164 subsubsection \<open>Quotient\<close>
```
```  1165
```
```  1166 lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
```
```  1167 by (simp add: le_div_geq linorder_not_less)
```
```  1168
```
```  1169 lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
```
```  1170 by (simp add: div_geq)
```
```  1171
```
```  1172 lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
```
```  1173 by simp
```
```  1174
```
```  1175 lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
```
```  1176 by simp
```
```  1177
```
```  1178 lemma div_positive:
```
```  1179   fixes m n :: nat
```
```  1180   assumes "n > 0"
```
```  1181   assumes "m \<ge> n"
```
```  1182   shows "m div n > 0"
```
```  1183 proof -
```
```  1184   from \<open>m \<ge> n\<close> obtain q where "m = n + q"
```
```  1185     by (auto simp add: le_iff_add)
```
```  1186   with \<open>n > 0\<close> show ?thesis by (simp add: div_add_self1)
```
```  1187 qed
```
```  1188
```
```  1189 lemma div_eq_0_iff: "(a div b::nat) = 0 \<longleftrightarrow> a < b \<or> b = 0"
```
```  1190   by auto (metis div_positive less_numeral_extra(3) not_less)
```
```  1191
```
```  1192
```
```  1193 subsubsection \<open>Remainder\<close>
```
```  1194
```
```  1195 lemma mod_Suc_le_divisor [simp]:
```
```  1196   "m mod Suc n \<le> n"
```
```  1197   using mod_less_divisor [of "Suc n" m] by arith
```
```  1198
```
```  1199 lemma mod_less_eq_dividend [simp]:
```
```  1200   fixes m n :: nat
```
```  1201   shows "m mod n \<le> m"
```
```  1202 proof (rule add_leD2)
```
```  1203   from div_mult_mod_eq have "m div n * n + m mod n = m" .
```
```  1204   then show "m div n * n + m mod n \<le> m" by auto
```
```  1205 qed
```
```  1206
```
```  1207 lemma mod_geq: "\<not> m < (n::nat) \<Longrightarrow> m mod n = (m - n) mod n"
```
```  1208 by (simp add: le_mod_geq linorder_not_less)
```
```  1209
```
```  1210 lemma mod_if: "m mod (n::nat) = (if m < n then m else (m - n) mod n)"
```
```  1211 by (simp add: le_mod_geq)
```
```  1212
```
```  1213
```
```  1214 subsubsection \<open>Quotient and Remainder\<close>
```
```  1215
```
```  1216 lemma div_mult1_eq:
```
```  1217   "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"
```
```  1218   by (cases "c = 0")
```
```  1219      (auto simp add: algebra_simps distrib_left [symmetric]
```
```  1220      intro!: div_nat_unique [of _ _ _ "(a * (b mod c)) mod c"] eucl_rel_natI)
```
```  1221
```
```  1222 lemma eucl_rel_nat_add1_eq:
```
```  1223   "eucl_rel_nat a c (aq, ar) \<Longrightarrow> eucl_rel_nat b c (bq, br)
```
```  1224    \<Longrightarrow> eucl_rel_nat (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"
```
```  1225   by (auto simp add: split_ifs algebra_simps elim!: eucl_rel_nat.cases intro: eucl_rel_nat_by0 eucl_rel_natI)
```
```  1226
```
```  1227 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
```
```  1228 lemma div_add1_eq:
```
```  1229   "(a + b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
```
```  1230 by (blast intro: eucl_rel_nat_add1_eq [THEN div_nat_unique] eucl_rel_nat)
```
```  1231
```
```  1232 lemma eucl_rel_nat_mult2_eq:
```
```  1233   assumes "eucl_rel_nat a b (q, r)"
```
```  1234   shows "eucl_rel_nat a (b * c) (q div c, b *(q mod c) + r)"
```
```  1235 proof (cases "c = 0")
```
```  1236   case True
```
```  1237   with assms show ?thesis
```
```  1238     by (auto intro: eucl_rel_nat_by0 elim!: eucl_rel_nat.cases simp add: ac_simps)
```
```  1239 next
```
```  1240   case False
```
```  1241   { assume "r < b"
```
```  1242     with False have "b * (q mod c) + r < b * c"
```
```  1243       apply (cut_tac m = q and n = c in mod_less_divisor)
```
```  1244       apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
```
```  1245       apply (erule_tac P = "%x. lhs < rhs x" for lhs rhs in ssubst)
```
```  1246       apply (simp add: add_mult_distrib2)
```
```  1247       done
```
```  1248     then have "r + b * (q mod c) < b * c"
```
```  1249       by (simp add: ac_simps)
```
```  1250   } with assms False show ?thesis
```
```  1251     by (auto simp add: algebra_simps add_mult_distrib2 [symmetric] elim!: eucl_rel_nat.cases intro: eucl_rel_nat.intros)
```
```  1252 qed
```
```  1253
```
```  1254 lemma div_mult2_eq: "a div (b * c) = (a div b) div (c::nat)"
```
```  1255 by (force simp add: eucl_rel_nat [THEN eucl_rel_nat_mult2_eq, THEN div_nat_unique])
```
```  1256
```
```  1257 lemma mod_mult2_eq: "a mod (b * c) = b * (a div b mod c) + a mod (b::nat)"
```
```  1258 by (auto simp add: mult.commute eucl_rel_nat [THEN eucl_rel_nat_mult2_eq, THEN mod_nat_unique])
```
```  1259
```
```  1260 instantiation nat :: semiring_numeral_div
```
```  1261 begin
```
```  1262
```
```  1263 definition divmod_nat :: "num \<Rightarrow> num \<Rightarrow> nat \<times> nat"
```
```  1264 where
```
```  1265   divmod'_nat_def: "divmod_nat m n = (numeral m div numeral n, numeral m mod numeral n)"
```
```  1266
```
```  1267 definition divmod_step_nat :: "num \<Rightarrow> nat \<times> nat \<Rightarrow> nat \<times> nat"
```
```  1268 where
```
```  1269   "divmod_step_nat l qr = (let (q, r) = qr
```
```  1270     in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
```
```  1271     else (2 * q, r))"
```
```  1272
```
```  1273 instance
```
```  1274   by standard (auto intro: div_positive simp add: divmod'_nat_def divmod_step_nat_def mod_mult2_eq div_mult2_eq)
```
```  1275
```
```  1276 end
```
```  1277
```
```  1278 declare divmod_algorithm_code [where ?'a = nat, code]
```
```  1279
```
```  1280
```
```  1281 subsubsection \<open>Further Facts about Quotient and Remainder\<close>
```
```  1282
```
```  1283 lemma div_le_mono:
```
```  1284   fixes m n k :: nat
```
```  1285   assumes "m \<le> n"
```
```  1286   shows "m div k \<le> n div k"
```
```  1287 proof -
```
```  1288   from assms obtain q where "n = m + q"
```
```  1289     by (auto simp add: le_iff_add)
```
```  1290   then show ?thesis
```
```  1291     by (simp add: div_add1_eq [of m q k])
```
```  1292 qed
```
```  1293
```
```  1294 (* Antimonotonicity of div in second argument *)
```
```  1295 lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
```
```  1296 apply (subgoal_tac "0<n")
```
```  1297  prefer 2 apply simp
```
```  1298 apply (induct_tac k rule: nat_less_induct)
```
```  1299 apply (rename_tac "k")
```
```  1300 apply (case_tac "k<n", simp)
```
```  1301 apply (subgoal_tac "~ (k<m) ")
```
```  1302  prefer 2 apply simp
```
```  1303 apply (simp add: div_geq)
```
```  1304 apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
```
```  1305  prefer 2
```
```  1306  apply (blast intro: div_le_mono diff_le_mono2)
```
```  1307 apply (rule le_trans, simp)
```
```  1308 apply (simp)
```
```  1309 done
```
```  1310
```
```  1311 lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
```
```  1312 apply (case_tac "n=0", simp)
```
```  1313 apply (subgoal_tac "m div n \<le> m div 1", simp)
```
```  1314 apply (rule div_le_mono2)
```
```  1315 apply (simp_all (no_asm_simp))
```
```  1316 done
```
```  1317
```
```  1318 (* Similar for "less than" *)
```
```  1319 lemma div_less_dividend [simp]:
```
```  1320   "\<lbrakk>(1::nat) < n; 0 < m\<rbrakk> \<Longrightarrow> m div n < m"
```
```  1321 apply (induct m rule: nat_less_induct)
```
```  1322 apply (rename_tac "m")
```
```  1323 apply (case_tac "m<n", simp)
```
```  1324 apply (subgoal_tac "0<n")
```
```  1325  prefer 2 apply simp
```
```  1326 apply (simp add: div_geq)
```
```  1327 apply (case_tac "n<m")
```
```  1328  apply (subgoal_tac "(m-n) div n < (m-n) ")
```
```  1329   apply (rule impI less_trans_Suc)+
```
```  1330 apply assumption
```
```  1331   apply (simp_all)
```
```  1332 done
```
```  1333
```
```  1334 text\<open>A fact for the mutilated chess board\<close>
```
```  1335 lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
```
```  1336 apply (case_tac "n=0", simp)
```
```  1337 apply (induct "m" rule: nat_less_induct)
```
```  1338 apply (case_tac "Suc (na) <n")
```
```  1339 (* case Suc(na) < n *)
```
```  1340 apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
```
```  1341 (* case n \<le> Suc(na) *)
```
```  1342 apply (simp add: linorder_not_less le_Suc_eq mod_geq)
```
```  1343 apply (auto simp add: Suc_diff_le le_mod_geq)
```
```  1344 done
```
```  1345
```
```  1346 lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
```
```  1347 by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
```
```  1348
```
```  1349 lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
```
```  1350
```
```  1351 (*Loses information, namely we also have r<d provided d is nonzero*)
```
```  1352 lemma mod_eqD:
```
```  1353   fixes m d r q :: nat
```
```  1354   assumes "m mod d = r"
```
```  1355   shows "\<exists>q. m = r + q * d"
```
```  1356 proof -
```
```  1357   from div_mult_mod_eq obtain q where "q * d + m mod d = m" by blast
```
```  1358   with assms have "m = r + q * d" by simp
```
```  1359   then show ?thesis ..
```
```  1360 qed
```
```  1361
```
```  1362 lemma split_div:
```
```  1363  "P(n div k :: nat) =
```
```  1364  ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
```
```  1365  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
```
```  1366 proof
```
```  1367   assume P: ?P
```
```  1368   show ?Q
```
```  1369   proof (cases)
```
```  1370     assume "k = 0"
```
```  1371     with P show ?Q by simp
```
```  1372   next
```
```  1373     assume not0: "k \<noteq> 0"
```
```  1374     thus ?Q
```
```  1375     proof (simp, intro allI impI)
```
```  1376       fix i j
```
```  1377       assume n: "n = k*i + j" and j: "j < k"
```
```  1378       show "P i"
```
```  1379       proof (cases)
```
```  1380         assume "i = 0"
```
```  1381         with n j P show "P i" by simp
```
```  1382       next
```
```  1383         assume "i \<noteq> 0"
```
```  1384         with not0 n j P show "P i" by(simp add:ac_simps)
```
```  1385       qed
```
```  1386     qed
```
```  1387   qed
```
```  1388 next
```
```  1389   assume Q: ?Q
```
```  1390   show ?P
```
```  1391   proof (cases)
```
```  1392     assume "k = 0"
```
```  1393     with Q show ?P by simp
```
```  1394   next
```
```  1395     assume not0: "k \<noteq> 0"
```
```  1396     with Q have R: ?R by simp
```
```  1397     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
```
```  1398     show ?P by simp
```
```  1399   qed
```
```  1400 qed
```
```  1401
```
```  1402 lemma split_div_lemma:
```
```  1403   assumes "0 < n"
```
```  1404   shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m::nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1405 proof
```
```  1406   assume ?rhs
```
```  1407   with minus_mod_eq_mult_div [symmetric] have nq: "n * q = m - (m mod n)" by simp
```
```  1408   then have A: "n * q \<le> m" by simp
```
```  1409   have "n - (m mod n) > 0" using mod_less_divisor assms by auto
```
```  1410   then have "m < m + (n - (m mod n))" by simp
```
```  1411   then have "m < n + (m - (m mod n))" by simp
```
```  1412   with nq have "m < n + n * q" by simp
```
```  1413   then have B: "m < n * Suc q" by simp
```
```  1414   from A B show ?lhs ..
```
```  1415 next
```
```  1416   assume P: ?lhs
```
```  1417   then have "eucl_rel_nat m n (q, m - n * q)"
```
```  1418     by (auto intro: eucl_rel_natI simp add: ac_simps)
```
```  1419   then have "m div n = q"
```
```  1420     by (rule div_nat_unique)
```
```  1421   then show ?rhs by simp
```
```  1422 qed
```
```  1423
```
```  1424 theorem split_div':
```
```  1425   "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
```
```  1426    (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
```
```  1427   apply (cases "0 < n")
```
```  1428   apply (simp only: add: split_div_lemma)
```
```  1429   apply simp_all
```
```  1430   done
```
```  1431
```
```  1432 lemma split_mod:
```
```  1433  "P(n mod k :: nat) =
```
```  1434  ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
```
```  1435  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
```
```  1436 proof
```
```  1437   assume P: ?P
```
```  1438   show ?Q
```
```  1439   proof (cases)
```
```  1440     assume "k = 0"
```
```  1441     with P show ?Q by simp
```
```  1442   next
```
```  1443     assume not0: "k \<noteq> 0"
```
```  1444     thus ?Q
```
```  1445     proof (simp, intro allI impI)
```
```  1446       fix i j
```
```  1447       assume "n = k*i + j" "j < k"
```
```  1448       thus "P j" using not0 P by (simp add: ac_simps)
```
```  1449     qed
```
```  1450   qed
```
```  1451 next
```
```  1452   assume Q: ?Q
```
```  1453   show ?P
```
```  1454   proof (cases)
```
```  1455     assume "k = 0"
```
```  1456     with Q show ?P by simp
```
```  1457   next
```
```  1458     assume not0: "k \<noteq> 0"
```
```  1459     with Q have R: ?R by simp
```
```  1460     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
```
```  1461     show ?P by simp
```
```  1462   qed
```
```  1463 qed
```
```  1464
```
```  1465 lemma div_eq_dividend_iff: "a \<noteq> 0 \<Longrightarrow> (a :: nat) div b = a \<longleftrightarrow> b = 1"
```
```  1466   apply rule
```
```  1467   apply (cases "b = 0")
```
```  1468   apply simp_all
```
```  1469   apply (metis (full_types) One_nat_def Suc_lessI div_less_dividend less_not_refl3)
```
```  1470   done
```
```  1471
```
```  1472 lemma (in field_char_0) of_nat_div:
```
```  1473   "of_nat (m div n) = ((of_nat m - of_nat (m mod n)) / of_nat n)"
```
```  1474 proof -
```
```  1475   have "of_nat (m div n) = ((of_nat (m div n * n + m mod n) - of_nat (m mod n)) / of_nat n :: 'a)"
```
```  1476     unfolding of_nat_add by (cases "n = 0") simp_all
```
```  1477   then show ?thesis
```
```  1478     by simp
```
```  1479 qed
```
```  1480
```
```  1481
```
```  1482 subsubsection \<open>An ``induction'' law for modulus arithmetic.\<close>
```
```  1483
```
```  1484 lemma mod_induct_0:
```
```  1485   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
```
```  1486   and base: "P i" and i: "i<p"
```
```  1487   shows "P 0"
```
```  1488 proof (rule ccontr)
```
```  1489   assume contra: "\<not>(P 0)"
```
```  1490   from i have p: "0<p" by simp
```
```  1491   have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
```
```  1492   proof
```
```  1493     fix k
```
```  1494     show "?A k"
```
```  1495     proof (induct k)
```
```  1496       show "?A 0" by simp  \<comment> "by contradiction"
```
```  1497     next
```
```  1498       fix n
```
```  1499       assume ih: "?A n"
```
```  1500       show "?A (Suc n)"
```
```  1501       proof (clarsimp)
```
```  1502         assume y: "P (p - Suc n)"
```
```  1503         have n: "Suc n < p"
```
```  1504         proof (rule ccontr)
```
```  1505           assume "\<not>(Suc n < p)"
```
```  1506           hence "p - Suc n = 0"
```
```  1507             by simp
```
```  1508           with y contra show "False"
```
```  1509             by simp
```
```  1510         qed
```
```  1511         hence n2: "Suc (p - Suc n) = p-n" by arith
```
```  1512         from p have "p - Suc n < p" by arith
```
```  1513         with y step have z: "P ((Suc (p - Suc n)) mod p)"
```
```  1514           by blast
```
```  1515         show "False"
```
```  1516         proof (cases "n=0")
```
```  1517           case True
```
```  1518           with z n2 contra show ?thesis by simp
```
```  1519         next
```
```  1520           case False
```
```  1521           with p have "p-n < p" by arith
```
```  1522           with z n2 False ih show ?thesis by simp
```
```  1523         qed
```
```  1524       qed
```
```  1525     qed
```
```  1526   qed
```
```  1527   moreover
```
```  1528   from i obtain k where "0<k \<and> i+k=p"
```
```  1529     by (blast dest: less_imp_add_positive)
```
```  1530   hence "0<k \<and> i=p-k" by auto
```
```  1531   moreover
```
```  1532   note base
```
```  1533   ultimately
```
```  1534   show "False" by blast
```
```  1535 qed
```
```  1536
```
```  1537 lemma mod_induct:
```
```  1538   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
```
```  1539   and base: "P i" and i: "i<p" and j: "j<p"
```
```  1540   shows "P j"
```
```  1541 proof -
```
```  1542   have "\<forall>j<p. P j"
```
```  1543   proof
```
```  1544     fix j
```
```  1545     show "j<p \<longrightarrow> P j" (is "?A j")
```
```  1546     proof (induct j)
```
```  1547       from step base i show "?A 0"
```
```  1548         by (auto elim: mod_induct_0)
```
```  1549     next
```
```  1550       fix k
```
```  1551       assume ih: "?A k"
```
```  1552       show "?A (Suc k)"
```
```  1553       proof
```
```  1554         assume suc: "Suc k < p"
```
```  1555         hence k: "k<p" by simp
```
```  1556         with ih have "P k" ..
```
```  1557         with step k have "P (Suc k mod p)"
```
```  1558           by blast
```
```  1559         moreover
```
```  1560         from suc have "Suc k mod p = Suc k"
```
```  1561           by simp
```
```  1562         ultimately
```
```  1563         show "P (Suc k)" by simp
```
```  1564       qed
```
```  1565     qed
```
```  1566   qed
```
```  1567   with j show ?thesis by blast
```
```  1568 qed
```
```  1569
```
```  1570 lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"
```
```  1571   by (simp add: numeral_2_eq_2 le_div_geq)
```
```  1572
```
```  1573 lemma mod2_Suc_Suc [simp]: "Suc (Suc m) mod 2 = m mod 2"
```
```  1574   by (simp add: numeral_2_eq_2 le_mod_geq)
```
```  1575
```
```  1576 lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
```
```  1577 by (simp add: mult_2 [symmetric])
```
```  1578
```
```  1579 lemma mod2_gr_0 [simp]: "0 < (m::nat) mod 2 \<longleftrightarrow> m mod 2 = 1"
```
```  1580 proof -
```
```  1581   { fix n :: nat have  "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (cases n) simp_all }
```
```  1582   moreover have "m mod 2 < 2" by simp
```
```  1583   ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" .
```
```  1584   then show ?thesis by auto
```
```  1585 qed
```
```  1586
```
```  1587 text\<open>These lemmas collapse some needless occurrences of Suc:
```
```  1588     at least three Sucs, since two and fewer are rewritten back to Suc again!
```
```  1589     We already have some rules to simplify operands smaller than 3.\<close>
```
```  1590
```
```  1591 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
```
```  1592 by (simp add: Suc3_eq_add_3)
```
```  1593
```
```  1594 lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
```
```  1595 by (simp add: Suc3_eq_add_3)
```
```  1596
```
```  1597 lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
```
```  1598 by (simp add: Suc3_eq_add_3)
```
```  1599
```
```  1600 lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
```
```  1601 by (simp add: Suc3_eq_add_3)
```
```  1602
```
```  1603 lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v
```
```  1604 lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v
```
```  1605
```
```  1606 lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1"
```
```  1607 apply (induct "m")
```
```  1608 apply (simp_all add: mod_Suc)
```
```  1609 done
```
```  1610
```
```  1611 declare Suc_times_mod_eq [of "numeral w", simp] for w
```
```  1612
```
```  1613 lemma Suc_div_le_mono [simp]: "n div k \<le> (Suc n) div k"
```
```  1614 by (simp add: div_le_mono)
```
```  1615
```
```  1616 lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"
```
```  1617 by (cases n) simp_all
```
```  1618
```
```  1619 lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"
```
```  1620 proof -
```
```  1621   from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all
```
```  1622   from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp
```
```  1623 qed
```
```  1624
```
```  1625 lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"
```
```  1626 proof -
```
```  1627   have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp
```
```  1628   also have "... = Suc m mod n" by (rule mod_mult_self3)
```
```  1629   finally show ?thesis .
```
```  1630 qed
```
```  1631
```
```  1632 lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"
```
```  1633 apply (subst mod_Suc [of m])
```
```  1634 apply (subst mod_Suc [of "m mod n"], simp)
```
```  1635 done
```
```  1636
```
```  1637 lemma mod_2_not_eq_zero_eq_one_nat:
```
```  1638   fixes n :: nat
```
```  1639   shows "n mod 2 \<noteq> 0 \<longleftrightarrow> n mod 2 = 1"
```
```  1640   by (fact not_mod_2_eq_0_eq_1)
```
```  1641
```
```  1642 lemma even_Suc_div_two [simp]:
```
```  1643   "even n \<Longrightarrow> Suc n div 2 = n div 2"
```
```  1644   using even_succ_div_two [of n] by simp
```
```  1645
```
```  1646 lemma odd_Suc_div_two [simp]:
```
```  1647   "odd n \<Longrightarrow> Suc n div 2 = Suc (n div 2)"
```
```  1648   using odd_succ_div_two [of n] by simp
```
```  1649
```
```  1650 lemma odd_two_times_div_two_nat [simp]:
```
```  1651   assumes "odd n"
```
```  1652   shows "2 * (n div 2) = n - (1 :: nat)"
```
```  1653 proof -
```
```  1654   from assms have "2 * (n div 2) + 1 = n"
```
```  1655     by (rule odd_two_times_div_two_succ)
```
```  1656   then have "Suc (2 * (n div 2)) - 1 = n - 1"
```
```  1657     by simp
```
```  1658   then show ?thesis
```
```  1659     by simp
```
```  1660 qed
```
```  1661
```
```  1662 lemma parity_induct [case_names zero even odd]:
```
```  1663   assumes zero: "P 0"
```
```  1664   assumes even: "\<And>n. P n \<Longrightarrow> P (2 * n)"
```
```  1665   assumes odd: "\<And>n. P n \<Longrightarrow> P (Suc (2 * n))"
```
```  1666   shows "P n"
```
```  1667 proof (induct n rule: less_induct)
```
```  1668   case (less n)
```
```  1669   show "P n"
```
```  1670   proof (cases "n = 0")
```
```  1671     case True with zero show ?thesis by simp
```
```  1672   next
```
```  1673     case False
```
```  1674     with less have hyp: "P (n div 2)" by simp
```
```  1675     show ?thesis
```
```  1676     proof (cases "even n")
```
```  1677       case True
```
```  1678       with hyp even [of "n div 2"] show ?thesis
```
```  1679         by simp
```
```  1680     next
```
```  1681       case False
```
```  1682       with hyp odd [of "n div 2"] show ?thesis
```
```  1683         by simp
```
```  1684     qed
```
```  1685   qed
```
```  1686 qed
```
```  1687
```
```  1688 lemma Suc_0_div_numeral [simp]:
```
```  1689   fixes k l :: num
```
```  1690   shows "Suc 0 div numeral k = fst (divmod Num.One k)"
```
```  1691   by (simp_all add: fst_divmod)
```
```  1692
```
```  1693 lemma Suc_0_mod_numeral [simp]:
```
```  1694   fixes k l :: num
```
```  1695   shows "Suc 0 mod numeral k = snd (divmod Num.One k)"
```
```  1696   by (simp_all add: snd_divmod)
```
```  1697
```
```  1698
```
```  1699 subsection \<open>Division on @{typ int}\<close>
```
```  1700
```
```  1701 context
```
```  1702 begin
```
```  1703
```
```  1704 inductive eucl_rel_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool"
```
```  1705   where eucl_rel_int_by0: "eucl_rel_int k 0 (0, k)"
```
```  1706   | eucl_rel_int_dividesI: "l \<noteq> 0 \<Longrightarrow> k = q * l \<Longrightarrow> eucl_rel_int k l (q, 0)"
```
```  1707   | eucl_rel_int_remainderI: "sgn r = sgn l \<Longrightarrow> \<bar>r\<bar> < \<bar>l\<bar>
```
```  1708       \<Longrightarrow> k = q * l + r \<Longrightarrow> eucl_rel_int k l (q, r)"
```
```  1709
```
```  1710 lemma eucl_rel_int_iff:
```
```  1711   "eucl_rel_int k l (q, r) \<longleftrightarrow>
```
```  1712     k = l * q + r \<and>
```
```  1713      (if 0 < l then 0 \<le> r \<and> r < l else if l < 0 then l < r \<and> r \<le> 0 else q = 0)"
```
```  1714   by (cases "r = 0")
```
```  1715     (auto elim!: eucl_rel_int.cases intro: eucl_rel_int_by0 eucl_rel_int_dividesI eucl_rel_int_remainderI
```
```  1716     simp add: ac_simps sgn_1_pos sgn_1_neg)
```
```  1717
```
```  1718 lemma unique_quotient_lemma:
```
```  1719   "b * q' + r' \<le> b * q + r \<Longrightarrow> 0 \<le> r' \<Longrightarrow> r' < b \<Longrightarrow> r < b \<Longrightarrow> q' \<le> (q::int)"
```
```  1720 apply (subgoal_tac "r' + b * (q'-q) \<le> r")
```
```  1721  prefer 2 apply (simp add: right_diff_distrib)
```
```  1722 apply (subgoal_tac "0 < b * (1 + q - q') ")
```
```  1723 apply (erule_tac [2] order_le_less_trans)
```
```  1724  prefer 2 apply (simp add: right_diff_distrib distrib_left)
```
```  1725 apply (subgoal_tac "b * q' < b * (1 + q) ")
```
```  1726  prefer 2 apply (simp add: right_diff_distrib distrib_left)
```
```  1727 apply (simp add: mult_less_cancel_left)
```
```  1728 done
```
```  1729
```
```  1730 lemma unique_quotient_lemma_neg:
```
```  1731   "b * q' + r' \<le> b*q + r \<Longrightarrow> r \<le> 0 \<Longrightarrow> b < r \<Longrightarrow> b < r' \<Longrightarrow> q \<le> (q'::int)"
```
```  1732   by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma) auto
```
```  1733
```
```  1734 lemma unique_quotient:
```
```  1735   "eucl_rel_int a b (q, r) \<Longrightarrow> eucl_rel_int a b (q', r') \<Longrightarrow> q = q'"
```
```  1736   apply (simp add: eucl_rel_int_iff linorder_neq_iff split: if_split_asm)
```
```  1737   apply (blast intro: order_antisym
```
```  1738     dest: order_eq_refl [THEN unique_quotient_lemma]
```
```  1739     order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
```
```  1740   done
```
```  1741
```
```  1742 lemma unique_remainder:
```
```  1743   "eucl_rel_int a b (q, r) \<Longrightarrow> eucl_rel_int a b (q', r') \<Longrightarrow> r = r'"
```
```  1744 apply (subgoal_tac "q = q'")
```
```  1745  apply (simp add: eucl_rel_int_iff)
```
```  1746 apply (blast intro: unique_quotient)
```
```  1747 done
```
```  1748
```
```  1749 end
```
```  1750
```
```  1751 instantiation int :: "{idom_modulo, normalization_semidom}"
```
```  1752 begin
```
```  1753
```
```  1754 definition normalize_int :: "int \<Rightarrow> int"
```
```  1755   where [simp]: "normalize = (abs :: int \<Rightarrow> int)"
```
```  1756
```
```  1757 definition unit_factor_int :: "int \<Rightarrow> int"
```
```  1758   where [simp]: "unit_factor = (sgn :: int \<Rightarrow> int)"
```
```  1759
```
```  1760 definition divide_int :: "int \<Rightarrow> int \<Rightarrow> int"
```
```  1761   where "k div l = (if l = 0 \<or> k = 0 then 0
```
```  1762     else if k > 0 \<and> l > 0 \<or> k < 0 \<and> l < 0
```
```  1763       then int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)
```
```  1764       else
```
```  1765         if l dvd k then - int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)
```
```  1766         else - int (Suc (nat \<bar>k\<bar> div nat \<bar>l\<bar>)))"
```
```  1767
```
```  1768 definition modulo_int :: "int \<Rightarrow> int \<Rightarrow> int"
```
```  1769   where "k mod l = (if l = 0 then k else if l dvd k then 0
```
```  1770     else if k > 0 \<and> l > 0 \<or> k < 0 \<and> l < 0
```
```  1771       then sgn l * int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>)
```
```  1772       else sgn l * (\<bar>l\<bar> - int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>)))"
```
```  1773
```
```  1774 lemma eucl_rel_int:
```
```  1775   "eucl_rel_int k l (k div l, k mod l)"
```
```  1776 proof (cases k rule: int_cases3)
```
```  1777   case zero
```
```  1778   then show ?thesis
```
```  1779     by (simp add: eucl_rel_int_iff divide_int_def modulo_int_def)
```
```  1780 next
```
```  1781   case (pos n)
```
```  1782   then show ?thesis
```
```  1783     using div_mult_mod_eq [of n]
```
```  1784     by (cases l rule: int_cases3)
```
```  1785       (auto simp del: of_nat_mult of_nat_add
```
```  1786         simp add: mod_greater_zero_iff_not_dvd of_nat_mult [symmetric] of_nat_add [symmetric] algebra_simps
```
```  1787         eucl_rel_int_iff divide_int_def modulo_int_def int_dvd_iff)
```
```  1788 next
```
```  1789   case (neg n)
```
```  1790   then show ?thesis
```
```  1791     using div_mult_mod_eq [of n]
```
```  1792     by (cases l rule: int_cases3)
```
```  1793       (auto simp del: of_nat_mult of_nat_add
```
```  1794         simp add: mod_greater_zero_iff_not_dvd of_nat_mult [symmetric] of_nat_add [symmetric] algebra_simps
```
```  1795         eucl_rel_int_iff divide_int_def modulo_int_def int_dvd_iff)
```
```  1796 qed
```
```  1797
```
```  1798 lemma divmod_int_unique:
```
```  1799   assumes "eucl_rel_int k l (q, r)"
```
```  1800   shows div_int_unique: "k div l = q" and mod_int_unique: "k mod l = r"
```
```  1801   using assms eucl_rel_int [of k l]
```
```  1802   using unique_quotient [of k l] unique_remainder [of k l]
```
```  1803   by auto
```
```  1804
```
```  1805 instance proof
```
```  1806   fix k l :: int
```
```  1807   show "k div l * l + k mod l = k"
```
```  1808     using eucl_rel_int [of k l]
```
```  1809     unfolding eucl_rel_int_iff by (simp add: ac_simps)
```
```  1810 next
```
```  1811   fix k :: int show "k div 0 = 0"
```
```  1812     by (rule div_int_unique, simp add: eucl_rel_int_iff)
```
```  1813 next
```
```  1814   fix k l :: int
```
```  1815   assume "l \<noteq> 0"
```
```  1816   then show "k * l div l = k"
```
```  1817     by (auto simp add: eucl_rel_int_iff ac_simps intro: div_int_unique [of _ _ _ 0])
```
```  1818 qed (simp_all add: sgn_mult mult_sgn_abs abs_sgn_eq)
```
```  1819
```
```  1820 end
```
```  1821
```
```  1822 lemma is_unit_int:
```
```  1823   "is_unit (k::int) \<longleftrightarrow> k = 1 \<or> k = - 1"
```
```  1824   by auto
```
```  1825
```
```  1826 instance int :: ring_div
```
```  1827 proof
```
```  1828   fix k l s :: int
```
```  1829   assume "l \<noteq> 0"
```
```  1830   then have "eucl_rel_int (k + s * l) l (s + k div l, k mod l)"
```
```  1831     using eucl_rel_int [of k l]
```
```  1832     unfolding eucl_rel_int_iff by (auto simp: algebra_simps)
```
```  1833   then show "(k + s * l) div l = s + k div l"
```
```  1834     by (rule div_int_unique)
```
```  1835 next
```
```  1836   fix k l s :: int
```
```  1837   assume "s \<noteq> 0"
```
```  1838   have "\<And>q r. eucl_rel_int k l (q, r)
```
```  1839     \<Longrightarrow> eucl_rel_int (s * k) (s * l) (q, s * r)"
```
```  1840     unfolding eucl_rel_int_iff
```
```  1841     by (rule linorder_cases [of 0 l])
```
```  1842       (use \<open>s \<noteq> 0\<close> in \<open>auto simp: algebra_simps
```
```  1843       mult_less_0_iff zero_less_mult_iff mult_strict_right_mono
```
```  1844       mult_strict_right_mono_neg zero_le_mult_iff mult_le_0_iff\<close>)
```
```  1845   then have "eucl_rel_int (s * k) (s * l) (k div l, s * (k mod l))"
```
```  1846     using eucl_rel_int [of k l] .
```
```  1847   then show "(s * k) div (s * l) = k div l"
```
```  1848     by (rule div_int_unique)
```
```  1849 qed
```
```  1850
```
```  1851 ML \<open>
```
```  1852 structure Cancel_Div_Mod_Int = Cancel_Div_Mod
```
```  1853 (
```
```  1854   val div_name = @{const_name divide};
```
```  1855   val mod_name = @{const_name modulo};
```
```  1856   val mk_binop = HOLogic.mk_binop;
```
```  1857   val mk_sum = Arith_Data.mk_sum HOLogic.intT;
```
```  1858   val dest_sum = Arith_Data.dest_sum;
```
```  1859
```
```  1860   val div_mod_eqs = map mk_meta_eq @{thms cancel_div_mod_rules};
```
```  1861
```
```  1862   val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
```
```  1863     @{thms diff_conv_add_uminus add_0_left add_0_right ac_simps})
```
```  1864 )
```
```  1865 \<close>
```
```  1866
```
```  1867 simproc_setup cancel_div_mod_int ("(k::int) + l") =
```
```  1868   \<open>K Cancel_Div_Mod_Int.proc\<close>
```
```  1869
```
```  1870
```
```  1871 text\<open>Basic laws about division and remainder\<close>
```
```  1872
```
```  1873 lemma zdiv_int: "int (a div b) = int a div int b"
```
```  1874   by (simp add: divide_int_def)
```
```  1875
```
```  1876 lemma zmod_int: "int (a mod b) = int a mod int b"
```
```  1877   by (simp add: modulo_int_def int_dvd_iff)
```
```  1878
```
```  1879 lemma pos_mod_conj: "(0::int) < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b"
```
```  1880   using eucl_rel_int [of a b]
```
```  1881   by (auto simp add: eucl_rel_int_iff prod_eq_iff)
```
```  1882
```
```  1883 lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]
```
```  1884    and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]
```
```  1885
```
```  1886 lemma neg_mod_conj: "b < (0::int) \<Longrightarrow> a mod b \<le> 0 \<and> b < a mod b"
```
```  1887   using eucl_rel_int [of a b]
```
```  1888   by (auto simp add: eucl_rel_int_iff prod_eq_iff)
```
```  1889
```
```  1890 lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]
```
```  1891    and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]
```
```  1892
```
```  1893
```
```  1894 subsubsection \<open>General Properties of div and mod\<close>
```
```  1895
```
```  1896 lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
```
```  1897 apply (rule div_int_unique)
```
```  1898 apply (auto simp add: eucl_rel_int_iff)
```
```  1899 done
```
```  1900
```
```  1901 lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
```
```  1902 apply (rule div_int_unique)
```
```  1903 apply (auto simp add: eucl_rel_int_iff)
```
```  1904 done
```
```  1905
```
```  1906 lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
```
```  1907 apply (rule div_int_unique)
```
```  1908 apply (auto simp add: eucl_rel_int_iff)
```
```  1909 done
```
```  1910
```
```  1911 (*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
```
```  1912
```
```  1913 lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
```
```  1914 apply (rule_tac q = 0 in mod_int_unique)
```
```  1915 apply (auto simp add: eucl_rel_int_iff)
```
```  1916 done
```
```  1917
```
```  1918 lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
```
```  1919 apply (rule_tac q = 0 in mod_int_unique)
```
```  1920 apply (auto simp add: eucl_rel_int_iff)
```
```  1921 done
```
```  1922
```
```  1923 lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
```
```  1924 apply (rule_tac q = "-1" in mod_int_unique)
```
```  1925 apply (auto simp add: eucl_rel_int_iff)
```
```  1926 done
```
```  1927
```
```  1928 text\<open>There is no \<open>mod_neg_pos_trivial\<close>.\<close>
```
```  1929
```
```  1930
```
```  1931 subsubsection \<open>Laws for div and mod with Unary Minus\<close>
```
```  1932
```
```  1933 lemma zminus1_lemma:
```
```  1934      "eucl_rel_int a b (q, r) ==> b \<noteq> 0
```
```  1935       ==> eucl_rel_int (-a) b (if r=0 then -q else -q - 1,
```
```  1936                           if r=0 then 0 else b-r)"
```
```  1937 by (force simp add: split_ifs eucl_rel_int_iff linorder_neq_iff right_diff_distrib)
```
```  1938
```
```  1939
```
```  1940 lemma zdiv_zminus1_eq_if:
```
```  1941      "b \<noteq> (0::int)
```
```  1942       ==> (-a) div b =
```
```  1943           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
```
```  1944 by (blast intro: eucl_rel_int [THEN zminus1_lemma, THEN div_int_unique])
```
```  1945
```
```  1946 lemma zmod_zminus1_eq_if:
```
```  1947      "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
```
```  1948 apply (case_tac "b = 0", simp)
```
```  1949 apply (blast intro: eucl_rel_int [THEN zminus1_lemma, THEN mod_int_unique])
```
```  1950 done
```
```  1951
```
```  1952 lemma zmod_zminus1_not_zero:
```
```  1953   fixes k l :: int
```
```  1954   shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
```
```  1955   by (simp add: mod_eq_0_iff_dvd)
```
```  1956
```
```  1957 lemma zmod_zminus2_not_zero:
```
```  1958   fixes k l :: int
```
```  1959   shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
```
```  1960   by (simp add: mod_eq_0_iff_dvd)
```
```  1961
```
```  1962 lemma zdiv_zminus2_eq_if:
```
```  1963      "b \<noteq> (0::int)
```
```  1964       ==> a div (-b) =
```
```  1965           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
```
```  1966 by (simp add: zdiv_zminus1_eq_if div_minus_right)
```
```  1967
```
```  1968 lemma zmod_zminus2_eq_if:
```
```  1969      "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
```
```  1970 by (simp add: zmod_zminus1_eq_if mod_minus_right)
```
```  1971
```
```  1972
```
```  1973 subsubsection \<open>Monotonicity in the First Argument (Dividend)\<close>
```
```  1974
```
```  1975 lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
```
```  1976 using mult_div_mod_eq [symmetric, of a b]
```
```  1977 using mult_div_mod_eq [symmetric, of a' b]
```
```  1978 apply -
```
```  1979 apply (rule unique_quotient_lemma)
```
```  1980 apply (erule subst)
```
```  1981 apply (erule subst, simp_all)
```
```  1982 done
```
```  1983
```
```  1984 lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
```
```  1985 using mult_div_mod_eq [symmetric, of a b]
```
```  1986 using mult_div_mod_eq [symmetric, of a' b]
```
```  1987 apply -
```
```  1988 apply (rule unique_quotient_lemma_neg)
```
```  1989 apply (erule subst)
```
```  1990 apply (erule subst, simp_all)
```
```  1991 done
```
```  1992
```
```  1993
```
```  1994 subsubsection \<open>Monotonicity in the Second Argument (Divisor)\<close>
```
```  1995
```
```  1996 lemma q_pos_lemma:
```
```  1997      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
```
```  1998 apply (subgoal_tac "0 < b'* (q' + 1) ")
```
```  1999  apply (simp add: zero_less_mult_iff)
```
```  2000 apply (simp add: distrib_left)
```
```  2001 done
```
```  2002
```
```  2003 lemma zdiv_mono2_lemma:
```
```  2004      "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';
```
```  2005          r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]
```
```  2006       ==> q \<le> (q'::int)"
```
```  2007 apply (frule q_pos_lemma, assumption+)
```
```  2008 apply (subgoal_tac "b*q < b* (q' + 1) ")
```
```  2009  apply (simp add: mult_less_cancel_left)
```
```  2010 apply (subgoal_tac "b*q = r' - r + b'*q'")
```
```  2011  prefer 2 apply simp
```
```  2012 apply (simp (no_asm_simp) add: distrib_left)
```
```  2013 apply (subst add.commute, rule add_less_le_mono, arith)
```
```  2014 apply (rule mult_right_mono, auto)
```
```  2015 done
```
```  2016
```
```  2017 lemma zdiv_mono2:
```
```  2018      "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
```
```  2019 apply (subgoal_tac "b \<noteq> 0")
```
```  2020   prefer 2 apply arith
```
```  2021 using mult_div_mod_eq [symmetric, of a b]
```
```  2022 using mult_div_mod_eq [symmetric, of a b']
```
```  2023 apply -
```
```  2024 apply (rule zdiv_mono2_lemma)
```
```  2025 apply (erule subst)
```
```  2026 apply (erule subst, simp_all)
```
```  2027 done
```
```  2028
```
```  2029 lemma q_neg_lemma:
```
```  2030      "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
```
```  2031 apply (subgoal_tac "b'*q' < 0")
```
```  2032  apply (simp add: mult_less_0_iff, arith)
```
```  2033 done
```
```  2034
```
```  2035 lemma zdiv_mono2_neg_lemma:
```
```  2036      "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;
```
```  2037          r < b;  0 \<le> r';  0 < b';  b' \<le> b |]
```
```  2038       ==> q' \<le> (q::int)"
```
```  2039 apply (frule q_neg_lemma, assumption+)
```
```  2040 apply (subgoal_tac "b*q' < b* (q + 1) ")
```
```  2041  apply (simp add: mult_less_cancel_left)
```
```  2042 apply (simp add: distrib_left)
```
```  2043 apply (subgoal_tac "b*q' \<le> b'*q'")
```
```  2044  prefer 2 apply (simp add: mult_right_mono_neg, arith)
```
```  2045 done
```
```  2046
```
```  2047 lemma zdiv_mono2_neg:
```
```  2048      "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
```
```  2049 using mult_div_mod_eq [symmetric, of a b]
```
```  2050 using mult_div_mod_eq [symmetric, of a b']
```
```  2051 apply -
```
```  2052 apply (rule zdiv_mono2_neg_lemma)
```
```  2053 apply (erule subst)
```
```  2054 apply (erule subst, simp_all)
```
```  2055 done
```
```  2056
```
```  2057
```
```  2058 subsubsection \<open>More Algebraic Laws for div and mod\<close>
```
```  2059
```
```  2060 text\<open>proving (a*b) div c = a * (b div c) + a * (b mod c)\<close>
```
```  2061
```
```  2062 lemma zmult1_lemma:
```
```  2063      "[| eucl_rel_int b c (q, r) |]
```
```  2064       ==> eucl_rel_int (a * b) c (a*q + a*r div c, a*r mod c)"
```
```  2065 by (auto simp add: split_ifs eucl_rel_int_iff linorder_neq_iff distrib_left ac_simps)
```
```  2066
```
```  2067 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
```
```  2068 apply (case_tac "c = 0", simp)
```
```  2069 apply (blast intro: eucl_rel_int [THEN zmult1_lemma, THEN div_int_unique])
```
```  2070 done
```
```  2071
```
```  2072 text\<open>proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c)\<close>
```
```  2073
```
```  2074 lemma zadd1_lemma:
```
```  2075      "[| eucl_rel_int a c (aq, ar);  eucl_rel_int b c (bq, br) |]
```
```  2076       ==> eucl_rel_int (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
```
```  2077 by (force simp add: split_ifs eucl_rel_int_iff linorder_neq_iff distrib_left)
```
```  2078
```
```  2079 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
```
```  2080 lemma zdiv_zadd1_eq:
```
```  2081      "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
```
```  2082 apply (case_tac "c = 0", simp)
```
```  2083 apply (blast intro: zadd1_lemma [OF eucl_rel_int eucl_rel_int] div_int_unique)
```
```  2084 done
```
```  2085
```
```  2086 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
```
```  2087 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
```
```  2088
```
```  2089 (* REVISIT: should this be generalized to all semiring_div types? *)
```
```  2090 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
```
```  2091
```
```  2092
```
```  2093 subsubsection \<open>Proving  @{term "a div (b * c) = (a div b) div c"}\<close>
```
```  2094
```
```  2095 (*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
```
```  2096   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
```
```  2097   to cause particular problems.*)
```
```  2098
```
```  2099 text\<open>first, four lemmas to bound the remainder for the cases b<0 and b>0\<close>
```
```  2100
```
```  2101 lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b * c < b * (q mod c) + r"
```
```  2102 apply (subgoal_tac "b * (c - q mod c) < r * 1")
```
```  2103  apply (simp add: algebra_simps)
```
```  2104 apply (rule order_le_less_trans)
```
```  2105  apply (erule_tac [2] mult_strict_right_mono)
```
```  2106  apply (rule mult_left_mono_neg)
```
```  2107   using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)
```
```  2108  apply (simp)
```
```  2109 apply (simp)
```
```  2110 done
```
```  2111
```
```  2112 lemma zmult2_lemma_aux2:
```
```  2113      "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
```
```  2114 apply (subgoal_tac "b * (q mod c) \<le> 0")
```
```  2115  apply arith
```
```  2116 apply (simp add: mult_le_0_iff)
```
```  2117 done
```
```  2118
```
```  2119 lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
```
```  2120 apply (subgoal_tac "0 \<le> b * (q mod c) ")
```
```  2121 apply arith
```
```  2122 apply (simp add: zero_le_mult_iff)
```
```  2123 done
```
```  2124
```
```  2125 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
```
```  2126 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
```
```  2127  apply (simp add: right_diff_distrib)
```
```  2128 apply (rule order_less_le_trans)
```
```  2129  apply (erule mult_strict_right_mono)
```
```  2130  apply (rule_tac [2] mult_left_mono)
```
```  2131   apply simp
```
```  2132  using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)
```
```  2133 apply simp
```
```  2134 done
```
```  2135
```
```  2136 lemma zmult2_lemma: "[| eucl_rel_int a b (q, r); 0 < c |]
```
```  2137       ==> eucl_rel_int a (b * c) (q div c, b*(q mod c) + r)"
```
```  2138 by (auto simp add: mult.assoc eucl_rel_int_iff linorder_neq_iff
```
```  2139                    zero_less_mult_iff distrib_left [symmetric]
```
```  2140                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: if_split_asm)
```
```  2141
```
```  2142 lemma zdiv_zmult2_eq:
```
```  2143   fixes a b c :: int
```
```  2144   shows "0 \<le> c \<Longrightarrow> a div (b * c) = (a div b) div c"
```
```  2145 apply (case_tac "b = 0", simp)
```
```  2146 apply (force simp add: le_less eucl_rel_int [THEN zmult2_lemma, THEN div_int_unique])
```
```  2147 done
```
```  2148
```
```  2149 lemma zmod_zmult2_eq:
```
```  2150   fixes a b c :: int
```
```  2151   shows "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
```
```  2152 apply (case_tac "b = 0", simp)
```
```  2153 apply (force simp add: le_less eucl_rel_int [THEN zmult2_lemma, THEN mod_int_unique])
```
```  2154 done
```
```  2155
```
```  2156 lemma div_pos_geq:
```
```  2157   fixes k l :: int
```
```  2158   assumes "0 < l" and "l \<le> k"
```
```  2159   shows "k div l = (k - l) div l + 1"
```
```  2160 proof -
```
```  2161   have "k = (k - l) + l" by simp
```
```  2162   then obtain j where k: "k = j + l" ..
```
```  2163   with assms show ?thesis by (simp add: div_add_self2)
```
```  2164 qed
```
```  2165
```
```  2166 lemma mod_pos_geq:
```
```  2167   fixes k l :: int
```
```  2168   assumes "0 < l" and "l \<le> k"
```
```  2169   shows "k mod l = (k - l) mod l"
```
```  2170 proof -
```
```  2171   have "k = (k - l) + l" by simp
```
```  2172   then obtain j where k: "k = j + l" ..
```
```  2173   with assms show ?thesis by simp
```
```  2174 qed
```
```  2175
```
```  2176
```
```  2177 subsubsection \<open>Splitting Rules for div and mod\<close>
```
```  2178
```
```  2179 text\<open>The proofs of the two lemmas below are essentially identical\<close>
```
```  2180
```
```  2181 lemma split_pos_lemma:
```
```  2182  "0<k ==>
```
```  2183     P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
```
```  2184 apply (rule iffI, clarify)
```
```  2185  apply (erule_tac P="P x y" for x y in rev_mp)
```
```  2186  apply (subst mod_add_eq [symmetric])
```
```  2187  apply (subst zdiv_zadd1_eq)
```
```  2188  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)
```
```  2189 txt\<open>converse direction\<close>
```
```  2190 apply (drule_tac x = "n div k" in spec)
```
```  2191 apply (drule_tac x = "n mod k" in spec, simp)
```
```  2192 done
```
```  2193
```
```  2194 lemma split_neg_lemma:
```
```  2195  "k<0 ==>
```
```  2196     P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
```
```  2197 apply (rule iffI, clarify)
```
```  2198  apply (erule_tac P="P x y" for x y in rev_mp)
```
```  2199  apply (subst mod_add_eq [symmetric])
```
```  2200  apply (subst zdiv_zadd1_eq)
```
```  2201  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)
```
```  2202 txt\<open>converse direction\<close>
```
```  2203 apply (drule_tac x = "n div k" in spec)
```
```  2204 apply (drule_tac x = "n mod k" in spec, simp)
```
```  2205 done
```
```  2206
```
```  2207 lemma split_zdiv:
```
```  2208  "P(n div k :: int) =
```
```  2209   ((k = 0 --> P 0) &
```
```  2210    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) &
```
```  2211    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
```
```  2212 apply (case_tac "k=0", simp)
```
```  2213 apply (simp only: linorder_neq_iff)
```
```  2214 apply (erule disjE)
```
```  2215  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]
```
```  2216                       split_neg_lemma [of concl: "%x y. P x"])
```
```  2217 done
```
```  2218
```
```  2219 lemma split_zmod:
```
```  2220  "P(n mod k :: int) =
```
```  2221   ((k = 0 --> P n) &
```
```  2222    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) &
```
```  2223    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
```
```  2224 apply (case_tac "k=0", simp)
```
```  2225 apply (simp only: linorder_neq_iff)
```
```  2226 apply (erule disjE)
```
```  2227  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]
```
```  2228                       split_neg_lemma [of concl: "%x y. P y"])
```
```  2229 done
```
```  2230
```
```  2231 text \<open>Enable (lin)arith to deal with @{const divide} and @{const modulo}
```
```  2232   when these are applied to some constant that is of the form
```
```  2233   @{term "numeral k"}:\<close>
```
```  2234 declare split_zdiv [of _ _ "numeral k", arith_split] for k
```
```  2235 declare split_zmod [of _ _ "numeral k", arith_split] for k
```
```  2236
```
```  2237
```
```  2238 subsubsection \<open>Computing \<open>div\<close> and \<open>mod\<close> with shifting\<close>
```
```  2239
```
```  2240 lemma pos_eucl_rel_int_mult_2:
```
```  2241   assumes "0 \<le> b"
```
```  2242   assumes "eucl_rel_int a b (q, r)"
```
```  2243   shows "eucl_rel_int (1 + 2*a) (2*b) (q, 1 + 2*r)"
```
```  2244   using assms unfolding eucl_rel_int_iff by auto
```
```  2245
```
```  2246 lemma neg_eucl_rel_int_mult_2:
```
```  2247   assumes "b \<le> 0"
```
```  2248   assumes "eucl_rel_int (a + 1) b (q, r)"
```
```  2249   shows "eucl_rel_int (1 + 2*a) (2*b) (q, 2*r - 1)"
```
```  2250   using assms unfolding eucl_rel_int_iff by auto
```
```  2251
```
```  2252 text\<open>computing div by shifting\<close>
```
```  2253
```
```  2254 lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
```
```  2255   using pos_eucl_rel_int_mult_2 [OF _ eucl_rel_int]
```
```  2256   by (rule div_int_unique)
```
```  2257
```
```  2258 lemma neg_zdiv_mult_2:
```
```  2259   assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"
```
```  2260   using neg_eucl_rel_int_mult_2 [OF A eucl_rel_int]
```
```  2261   by (rule div_int_unique)
```
```  2262
```
```  2263 (* FIXME: add rules for negative numerals *)
```
```  2264 lemma zdiv_numeral_Bit0 [simp]:
```
```  2265   "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
```
```  2266     numeral v div (numeral w :: int)"
```
```  2267   unfolding numeral.simps unfolding mult_2 [symmetric]
```
```  2268   by (rule div_mult_mult1, simp)
```
```  2269
```
```  2270 lemma zdiv_numeral_Bit1 [simp]:
```
```  2271   "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =
```
```  2272     (numeral v div (numeral w :: int))"
```
```  2273   unfolding numeral.simps
```
```  2274   unfolding mult_2 [symmetric] add.commute [of _ 1]
```
```  2275   by (rule pos_zdiv_mult_2, simp)
```
```  2276
```
```  2277 lemma pos_zmod_mult_2:
```
```  2278   fixes a b :: int
```
```  2279   assumes "0 \<le> a"
```
```  2280   shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
```
```  2281   using pos_eucl_rel_int_mult_2 [OF assms eucl_rel_int]
```
```  2282   by (rule mod_int_unique)
```
```  2283
```
```  2284 lemma neg_zmod_mult_2:
```
```  2285   fixes a b :: int
```
```  2286   assumes "a \<le> 0"
```
```  2287   shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
```
```  2288   using neg_eucl_rel_int_mult_2 [OF assms eucl_rel_int]
```
```  2289   by (rule mod_int_unique)
```
```  2290
```
```  2291 (* FIXME: add rules for negative numerals *)
```
```  2292 lemma zmod_numeral_Bit0 [simp]:
```
```  2293   "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =
```
```  2294     (2::int) * (numeral v mod numeral w)"
```
```  2295   unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
```
```  2296   unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
```
```  2297
```
```  2298 lemma zmod_numeral_Bit1 [simp]:
```
```  2299   "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
```
```  2300     2 * (numeral v mod numeral w) + (1::int)"
```
```  2301   unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
```
```  2302   unfolding mult_2 [symmetric] add.commute [of _ 1]
```
```  2303   by (rule pos_zmod_mult_2, simp)
```
```  2304
```
```  2305 lemma zdiv_eq_0_iff:
```
```  2306  "(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")
```
```  2307 proof
```
```  2308   assume ?L
```
```  2309   have "?L \<longrightarrow> ?R" by (rule split_zdiv[THEN iffD2]) simp
```
```  2310   with \<open>?L\<close> show ?R by blast
```
```  2311 next
```
```  2312   assume ?R thus ?L
```
```  2313     by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)
```
```  2314 qed
```
```  2315
```
```  2316 lemma zmod_trival_iff:
```
```  2317   fixes i k :: int
```
```  2318   shows "i mod k = i \<longleftrightarrow> k = 0 \<or> 0 \<le> i \<and> i < k \<or> i \<le> 0 \<and> k < i"
```
```  2319 proof -
```
```  2320   have "i mod k = i \<longleftrightarrow> i div k = 0"
```
```  2321     by safe (insert div_mult_mod_eq [of i k], auto)
```
```  2322   with zdiv_eq_0_iff
```
```  2323   show ?thesis
```
```  2324     by simp
```
```  2325 qed
```
```  2326
```
```  2327 subsubsection \<open>Quotients of Signs\<close>
```
```  2328
```
```  2329 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
```
```  2330 by (simp add: divide_int_def)
```
```  2331
```
```  2332 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
```
```  2333 by (simp add: modulo_int_def)
```
```  2334
```
```  2335 lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
```
```  2336 apply (subgoal_tac "a div b \<le> -1", force)
```
```  2337 apply (rule order_trans)
```
```  2338 apply (rule_tac a' = "-1" in zdiv_mono1)
```
```  2339 apply (auto simp add: div_eq_minus1)
```
```  2340 done
```
```  2341
```
```  2342 lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
```
```  2343 by (drule zdiv_mono1_neg, auto)
```
```  2344
```
```  2345 lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
```
```  2346 by (drule zdiv_mono1, auto)
```
```  2347
```
```  2348 text\<open>Now for some equivalences of the form \<open>a div b >=< 0 \<longleftrightarrow> \<dots>\<close>
```
```  2349 conditional upon the sign of \<open>a\<close> or \<open>b\<close>. There are many more.
```
```  2350 They should all be simp rules unless that causes too much search.\<close>
```
```  2351
```
```  2352 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
```
```  2353 apply auto
```
```  2354 apply (drule_tac [2] zdiv_mono1)
```
```  2355 apply (auto simp add: linorder_neq_iff)
```
```  2356 apply (simp (no_asm_use) add: linorder_not_less [symmetric])
```
```  2357 apply (blast intro: div_neg_pos_less0)
```
```  2358 done
```
```  2359
```
```  2360 lemma pos_imp_zdiv_pos_iff:
```
```  2361   "0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"
```
```  2362 using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]
```
```  2363 by arith
```
```  2364
```
```  2365 lemma neg_imp_zdiv_nonneg_iff:
```
```  2366   "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
```
```  2367 apply (subst div_minus_minus [symmetric])
```
```  2368 apply (subst pos_imp_zdiv_nonneg_iff, auto)
```
```  2369 done
```
```  2370
```
```  2371 (*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
```
```  2372 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
```
```  2373 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
```
```  2374
```
```  2375 (*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
```
```  2376 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
```
```  2377 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
```
```  2378
```
```  2379 lemma nonneg1_imp_zdiv_pos_iff:
```
```  2380   "(0::int) <= a \<Longrightarrow> (a div b > 0) = (a >= b & b>0)"
```
```  2381 apply rule
```
```  2382  apply rule
```
```  2383   using div_pos_pos_trivial[of a b]apply arith
```
```  2384  apply(cases "b=0")apply simp
```
```  2385  using div_nonneg_neg_le0[of a b]apply arith
```
```  2386 using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp
```
```  2387 done
```
```  2388
```
```  2389 lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 ==> m mod k \<le> m"
```
```  2390 apply (rule split_zmod[THEN iffD2])
```
```  2391 apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)
```
```  2392 done
```
```  2393
```
```  2394
```
```  2395 subsubsection \<open>Computation of Division and Remainder\<close>
```
```  2396
```
```  2397 instantiation int :: semiring_numeral_div
```
```  2398 begin
```
```  2399
```
```  2400 definition divmod_int :: "num \<Rightarrow> num \<Rightarrow> int \<times> int"
```
```  2401 where
```
```  2402   "divmod_int m n = (numeral m div numeral n, numeral m mod numeral n)"
```
```  2403
```
```  2404 definition divmod_step_int :: "num \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int"
```
```  2405 where
```
```  2406   "divmod_step_int l qr = (let (q, r) = qr
```
```  2407     in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
```
```  2408     else (2 * q, r))"
```
```  2409
```
```  2410 instance
```
```  2411   by standard (auto intro: zmod_le_nonneg_dividend simp add: divmod_int_def divmod_step_int_def
```
```  2412     pos_imp_zdiv_pos_iff div_pos_pos_trivial mod_pos_pos_trivial zmod_zmult2_eq zdiv_zmult2_eq)
```
```  2413
```
```  2414 end
```
```  2415
```
```  2416 declare divmod_algorithm_code [where ?'a = int, code]
```
```  2417
```
```  2418 context
```
```  2419 begin
```
```  2420
```
```  2421 qualified definition adjust_div :: "int \<times> int \<Rightarrow> int"
```
```  2422 where
```
```  2423   "adjust_div qr = (let (q, r) = qr in q + of_bool (r \<noteq> 0))"
```
```  2424
```
```  2425 qualified lemma adjust_div_eq [simp, code]:
```
```  2426   "adjust_div (q, r) = q + of_bool (r \<noteq> 0)"
```
```  2427   by (simp add: adjust_div_def)
```
```  2428
```
```  2429 qualified definition adjust_mod :: "int \<Rightarrow> int \<Rightarrow> int"
```
```  2430 where
```
```  2431   [simp]: "adjust_mod l r = (if r = 0 then 0 else l - r)"
```
```  2432
```
```  2433 lemma minus_numeral_div_numeral [simp]:
```
```  2434   "- numeral m div numeral n = - (adjust_div (divmod m n) :: int)"
```
```  2435 proof -
```
```  2436   have "int (fst (divmod m n)) = fst (divmod m n)"
```
```  2437     by (simp only: fst_divmod divide_int_def) auto
```
```  2438   then show ?thesis
```
```  2439     by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
```
```  2440 qed
```
```  2441
```
```  2442 lemma minus_numeral_mod_numeral [simp]:
```
```  2443   "- numeral m mod numeral n = adjust_mod (numeral n) (snd (divmod m n) :: int)"
```
```  2444 proof -
```
```  2445   have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
```
```  2446     using that by (simp only: snd_divmod modulo_int_def) auto
```
```  2447   then show ?thesis
```
```  2448     by (auto simp add: split_def Let_def adjust_div_def divides_aux_def modulo_int_def)
```
```  2449 qed
```
```  2450
```
```  2451 lemma numeral_div_minus_numeral [simp]:
```
```  2452   "numeral m div - numeral n = - (adjust_div (divmod m n) :: int)"
```
```  2453 proof -
```
```  2454   have "int (fst (divmod m n)) = fst (divmod m n)"
```
```  2455     by (simp only: fst_divmod divide_int_def) auto
```
```  2456   then show ?thesis
```
```  2457     by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
```
```  2458 qed
```
```  2459
```
```  2460 lemma numeral_mod_minus_numeral [simp]:
```
```  2461   "numeral m mod - numeral n = - adjust_mod (numeral n) (snd (divmod m n) :: int)"
```
```  2462 proof -
```
```  2463   have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
```
```  2464     using that by (simp only: snd_divmod modulo_int_def) auto
```
```  2465   then show ?thesis
```
```  2466     by (auto simp add: split_def Let_def adjust_div_def divides_aux_def modulo_int_def)
```
```  2467 qed
```
```  2468
```
```  2469 lemma minus_one_div_numeral [simp]:
```
```  2470   "- 1 div numeral n = - (adjust_div (divmod Num.One n) :: int)"
```
```  2471   using minus_numeral_div_numeral [of Num.One n] by simp
```
```  2472
```
```  2473 lemma minus_one_mod_numeral [simp]:
```
```  2474   "- 1 mod numeral n = adjust_mod (numeral n) (snd (divmod Num.One n) :: int)"
```
```  2475   using minus_numeral_mod_numeral [of Num.One n] by simp
```
```  2476
```
```  2477 lemma one_div_minus_numeral [simp]:
```
```  2478   "1 div - numeral n = - (adjust_div (divmod Num.One n) :: int)"
```
```  2479   using numeral_div_minus_numeral [of Num.One n] by simp
```
```  2480
```
```  2481 lemma one_mod_minus_numeral [simp]:
```
```  2482   "1 mod - numeral n = - adjust_mod (numeral n) (snd (divmod Num.One n) :: int)"
```
```  2483   using numeral_mod_minus_numeral [of Num.One n] by simp
```
```  2484
```
```  2485 end
```
```  2486
```
```  2487
```
```  2488 subsubsection \<open>Further properties\<close>
```
```  2489
```
```  2490 text \<open>Simplify expresions in which div and mod combine numerical constants\<close>
```
```  2491
```
```  2492 lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q"
```
```  2493   by (rule div_int_unique [of a b q r]) (simp add: eucl_rel_int_iff)
```
```  2494
```
```  2495 lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q"
```
```  2496   by (rule div_int_unique [of a b q r],
```
```  2497     simp add: eucl_rel_int_iff)
```
```  2498
```
```  2499 lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r"
```
```  2500   by (rule mod_int_unique [of a b q r],
```
```  2501     simp add: eucl_rel_int_iff)
```
```  2502
```
```  2503 lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r"
```
```  2504   by (rule mod_int_unique [of a b q r],
```
```  2505     simp add: eucl_rel_int_iff)
```
```  2506
```
```  2507 lemma abs_div: "(y::int) dvd x \<Longrightarrow> \<bar>x div y\<bar> = \<bar>x\<bar> div \<bar>y\<bar>"
```
```  2508 by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
```
```  2509
```
```  2510 text\<open>Suggested by Matthias Daum\<close>
```
```  2511 lemma int_power_div_base:
```
```  2512      "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
```
```  2513 apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
```
```  2514  apply (erule ssubst)
```
```  2515  apply (simp only: power_add)
```
```  2516  apply simp_all
```
```  2517 done
```
```  2518
```
```  2519 text \<open>Distributive laws for function \<open>nat\<close>.\<close>
```
```  2520
```
```  2521 lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
```
```  2522 apply (rule linorder_cases [of y 0])
```
```  2523 apply (simp add: div_nonneg_neg_le0)
```
```  2524 apply simp
```
```  2525 apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
```
```  2526 done
```
```  2527
```
```  2528 (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
```
```  2529 lemma nat_mod_distrib:
```
```  2530   "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
```
```  2531 apply (case_tac "y = 0", simp)
```
```  2532 apply (simp add: nat_eq_iff zmod_int)
```
```  2533 done
```
```  2534
```
```  2535 text  \<open>transfer setup\<close>
```
```  2536
```
```  2537 lemma transfer_nat_int_functions:
```
```  2538     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
```
```  2539     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
```
```  2540   by (auto simp add: nat_div_distrib nat_mod_distrib)
```
```  2541
```
```  2542 lemma transfer_nat_int_function_closures:
```
```  2543     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
```
```  2544     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
```
```  2545   apply (cases "y = 0")
```
```  2546   apply (auto simp add: pos_imp_zdiv_nonneg_iff)
```
```  2547   apply (cases "y = 0")
```
```  2548   apply auto
```
```  2549 done
```
```  2550
```
```  2551 declare transfer_morphism_nat_int [transfer add return:
```
```  2552   transfer_nat_int_functions
```
```  2553   transfer_nat_int_function_closures
```
```  2554 ]
```
```  2555
```
```  2556 lemma transfer_int_nat_functions:
```
```  2557     "(int x) div (int y) = int (x div y)"
```
```  2558     "(int x) mod (int y) = int (x mod y)"
```
```  2559   by (auto simp add: zdiv_int zmod_int)
```
```  2560
```
```  2561 lemma transfer_int_nat_function_closures:
```
```  2562     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"
```
```  2563     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"
```
```  2564   by (simp_all only: is_nat_def transfer_nat_int_function_closures)
```
```  2565
```
```  2566 declare transfer_morphism_int_nat [transfer add return:
```
```  2567   transfer_int_nat_functions
```
```  2568   transfer_int_nat_function_closures
```
```  2569 ]
```
```  2570
```
```  2571 text\<open>Suggested by Matthias Daum\<close>
```
```  2572 lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
```
```  2573 apply (subgoal_tac "nat x div nat k < nat x")
```
```  2574  apply (simp add: nat_div_distrib [symmetric])
```
```  2575 apply (rule Divides.div_less_dividend, simp_all)
```
```  2576 done
```
```  2577
```
```  2578 lemma (in ring_div) mod_eq_dvd_iff:
```
```  2579   "a mod c = b mod c \<longleftrightarrow> c dvd a - b" (is "?P \<longleftrightarrow> ?Q")
```
```  2580 proof
```
```  2581   assume ?P
```
```  2582   then have "(a mod c - b mod c) mod c = 0"
```
```  2583     by simp
```
```  2584   then show ?Q
```
```  2585     by (simp add: dvd_eq_mod_eq_0 mod_simps)
```
```  2586 next
```
```  2587   assume ?Q
```
```  2588   then obtain d where d: "a - b = c * d" ..
```
```  2589   then have "a = c * d + b"
```
```  2590     by (simp add: algebra_simps)
```
```  2591   then show ?P by simp
```
```  2592 qed
```
```  2593
```
```  2594 lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y mod n" and xy:"y \<le> x"
```
```  2595   shows "\<exists>q. x = y + n * q"
```
```  2596 proof-
```
```  2597   from xy have th: "int x - int y = int (x - y)" by simp
```
```  2598   from xyn have "int x mod int n = int y mod int n"
```
```  2599     by (simp add: zmod_int [symmetric])
```
```  2600   hence "int n dvd int x - int y" by (simp only: mod_eq_dvd_iff [symmetric])
```
```  2601   hence "n dvd x - y" by (simp add: th zdvd_int)
```
```  2602   then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
```
```  2603 qed
```
```  2604
```
```  2605 lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"
```
```  2606   (is "?lhs = ?rhs")
```
```  2607 proof
```
```  2608   assume H: "x mod n = y mod n"
```
```  2609   {assume xy: "x \<le> y"
```
```  2610     from H have th: "y mod n = x mod n" by simp
```
```  2611     from nat_mod_eq_lemma[OF th xy] have ?rhs
```
```  2612       apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
```
```  2613   moreover
```
```  2614   {assume xy: "y \<le> x"
```
```  2615     from nat_mod_eq_lemma[OF H xy] have ?rhs
```
```  2616       apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
```
```  2617   ultimately  show ?rhs using linear[of x y] by blast
```
```  2618 next
```
```  2619   assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
```
```  2620   hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
```
```  2621   thus  ?lhs by simp
```
```  2622 qed
```
```  2623
```
```  2624 subsubsection \<open>Dedicated simproc for calculation\<close>
```
```  2625
```
```  2626 text \<open>
```
```  2627   There is space for improvement here: the calculation itself
```
```  2628   could be carried outside the logic, and a generic simproc
```
```  2629   (simplifier setup) for generic calculation would be helpful.
```
```  2630 \<close>
```
```  2631
```
```  2632 simproc_setup numeral_divmod
```
```  2633   ("0 div 0 :: 'a :: semiring_numeral_div" | "0 mod 0 :: 'a :: semiring_numeral_div" |
```
```  2634    "0 div 1 :: 'a :: semiring_numeral_div" | "0 mod 1 :: 'a :: semiring_numeral_div" |
```
```  2635    "0 div - 1 :: int" | "0 mod - 1 :: int" |
```
```  2636    "0 div numeral b :: 'a :: semiring_numeral_div" | "0 mod numeral b :: 'a :: semiring_numeral_div" |
```
```  2637    "0 div - numeral b :: int" | "0 mod - numeral b :: int" |
```
```  2638    "1 div 0 :: 'a :: semiring_numeral_div" | "1 mod 0 :: 'a :: semiring_numeral_div" |
```
```  2639    "1 div 1 :: 'a :: semiring_numeral_div" | "1 mod 1 :: 'a :: semiring_numeral_div" |
```
```  2640    "1 div - 1 :: int" | "1 mod - 1 :: int" |
```
```  2641    "1 div numeral b :: 'a :: semiring_numeral_div" | "1 mod numeral b :: 'a :: semiring_numeral_div" |
```
```  2642    "1 div - numeral b :: int" |"1 mod - numeral b :: int" |
```
```  2643    "- 1 div 0 :: int" | "- 1 mod 0 :: int" | "- 1 div 1 :: int" | "- 1 mod 1 :: int" |
```
```  2644    "- 1 div - 1 :: int" | "- 1 mod - 1 :: int" | "- 1 div numeral b :: int" | "- 1 mod numeral b :: int" |
```
```  2645    "- 1 div - numeral b :: int" | "- 1 mod - numeral b :: int" |
```
```  2646    "numeral a div 0 :: 'a :: semiring_numeral_div" | "numeral a mod 0 :: 'a :: semiring_numeral_div" |
```
```  2647    "numeral a div 1 :: 'a :: semiring_numeral_div" | "numeral a mod 1 :: 'a :: semiring_numeral_div" |
```
```  2648    "numeral a div - 1 :: int" | "numeral a mod - 1 :: int" |
```
```  2649    "numeral a div numeral b :: 'a :: semiring_numeral_div" | "numeral a mod numeral b :: 'a :: semiring_numeral_div" |
```
```  2650    "numeral a div - numeral b :: int" | "numeral a mod - numeral b :: int" |
```
```  2651    "- numeral a div 0 :: int" | "- numeral a mod 0 :: int" |
```
```  2652    "- numeral a div 1 :: int" | "- numeral a mod 1 :: int" |
```
```  2653    "- numeral a div - 1 :: int" | "- numeral a mod - 1 :: int" |
```
```  2654    "- numeral a div numeral b :: int" | "- numeral a mod numeral b :: int" |
```
```  2655    "- numeral a div - numeral b :: int" | "- numeral a mod - numeral b :: int") =
```
```  2656 \<open> let
```
```  2657     val if_cong = the (Code.get_case_cong @{theory} @{const_name If});
```
```  2658     fun successful_rewrite ctxt ct =
```
```  2659       let
```
```  2660         val thm = Simplifier.rewrite ctxt ct
```
```  2661       in if Thm.is_reflexive thm then NONE else SOME thm end;
```
```  2662   in fn phi =>
```
```  2663     let
```
```  2664       val simps = Morphism.fact phi (@{thms div_0 mod_0 div_by_0 mod_by_0 div_by_1 mod_by_1
```
```  2665         one_div_numeral one_mod_numeral minus_one_div_numeral minus_one_mod_numeral
```
```  2666         one_div_minus_numeral one_mod_minus_numeral
```
```  2667         numeral_div_numeral numeral_mod_numeral minus_numeral_div_numeral minus_numeral_mod_numeral
```
```  2668         numeral_div_minus_numeral numeral_mod_minus_numeral
```
```  2669         div_minus_minus mod_minus_minus Divides.adjust_div_eq of_bool_eq one_neq_zero
```
```  2670         numeral_neq_zero neg_equal_0_iff_equal arith_simps arith_special divmod_trivial
```
```  2671         divmod_cancel divmod_steps divmod_step_eq fst_conv snd_conv numeral_One
```
```  2672         case_prod_beta rel_simps Divides.adjust_mod_def div_minus1_right mod_minus1_right
```
```  2673         minus_minus numeral_times_numeral mult_zero_right mult_1_right}
```
```  2674         @ [@{lemma "0 = 0 \<longleftrightarrow> True" by simp}]);
```
```  2675       fun prepare_simpset ctxt = HOL_ss |> Simplifier.simpset_map ctxt
```
```  2676         (Simplifier.add_cong if_cong #> fold Simplifier.add_simp simps)
```
```  2677     in fn ctxt => successful_rewrite (Simplifier.put_simpset (prepare_simpset ctxt) ctxt) end
```
```  2678   end;
```
```  2679 \<close>
```
```  2680
```
```  2681
```
```  2682 subsubsection \<open>Code generation\<close>
```
```  2683
```
```  2684 lemma [code]:
```
```  2685   fixes k :: int
```
```  2686   shows
```
```  2687     "k div 0 = 0"
```
```  2688     "k mod 0 = k"
```
```  2689     "0 div k = 0"
```
```  2690     "0 mod k = 0"
```
```  2691     "k div Int.Pos Num.One = k"
```
```  2692     "k mod Int.Pos Num.One = 0"
```
```  2693     "k div Int.Neg Num.One = - k"
```
```  2694     "k mod Int.Neg Num.One = 0"
```
```  2695     "Int.Pos m div Int.Pos n = (fst (divmod m n) :: int)"
```
```  2696     "Int.Pos m mod Int.Pos n = (snd (divmod m n) :: int)"
```
```  2697     "Int.Neg m div Int.Pos n = - (Divides.adjust_div (divmod m n) :: int)"
```
```  2698     "Int.Neg m mod Int.Pos n = Divides.adjust_mod (Int.Pos n) (snd (divmod m n) :: int)"
```
```  2699     "Int.Pos m div Int.Neg n = - (Divides.adjust_div (divmod m n) :: int)"
```
```  2700     "Int.Pos m mod Int.Neg n = - Divides.adjust_mod (Int.Pos n) (snd (divmod m n) :: int)"
```
```  2701     "Int.Neg m div Int.Neg n = (fst (divmod m n) :: int)"
```
```  2702     "Int.Neg m mod Int.Neg n = - (snd (divmod m n) :: int)"
```
```  2703   by simp_all
```
```  2704
```
```  2705 code_identifier
```
```  2706   code_module Divides \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
```
```  2707
```
```  2708 lemma dvd_eq_mod_eq_0_numeral:
```
```  2709   "numeral x dvd (numeral y :: 'a) \<longleftrightarrow> numeral y mod numeral x = (0 :: 'a::semiring_div)"
```
```  2710   by (fact dvd_eq_mod_eq_0)
```
```  2711
```
```  2712 declare minus_div_mult_eq_mod [symmetric, nitpick_unfold]
```
```  2713
```
```  2714 end
```