src/HOL/Divides.thy
 author haftmann Thu Oct 29 22:13:09 2009 +0100 (2009-10-29) changeset 33340 a165b97f3658 parent 33318 ddd97d9dfbfb child 33361 1f18de40b43f permissions -rw-r--r--
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
```     1 (*  Title:      HOL/Divides.thy
```
```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     3     Copyright   1999  University of Cambridge
```
```     4 *)
```
```     5
```
```     6 header {* The division operators div and mod *}
```
```     7
```
```     8 theory Divides
```
```     9 imports Nat_Numeral Nat_Transfer
```
```    10 uses "~~/src/Provers/Arith/cancel_div_mod.ML"
```
```    11 begin
```
```    12
```
```    13 subsection {* Syntactic division operations *}
```
```    14
```
```    15 class div = dvd +
```
```    16   fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
```
```    17     and mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)
```
```    18
```
```    19
```
```    20 subsection {* Abstract division in commutative semirings. *}
```
```    21
```
```    22 class semiring_div = comm_semiring_1_cancel + no_zero_divisors + div +
```
```    23   assumes mod_div_equality: "a div b * b + a mod b = a"
```
```    24     and div_by_0 [simp]: "a div 0 = 0"
```
```    25     and div_0 [simp]: "0 div a = 0"
```
```    26     and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"
```
```    27     and div_mult_mult1 [simp]: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b"
```
```    28 begin
```
```    29
```
```    30 text {* @{const div} and @{const mod} *}
```
```    31
```
```    32 lemma mod_div_equality2: "b * (a div b) + a mod b = a"
```
```    33   unfolding mult_commute [of b]
```
```    34   by (rule mod_div_equality)
```
```    35
```
```    36 lemma mod_div_equality': "a mod b + a div b * b = a"
```
```    37   using mod_div_equality [of a b]
```
```    38   by (simp only: add_ac)
```
```    39
```
```    40 lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
```
```    41   by (simp add: mod_div_equality)
```
```    42
```
```    43 lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"
```
```    44   by (simp add: mod_div_equality2)
```
```    45
```
```    46 lemma mod_by_0 [simp]: "a mod 0 = a"
```
```    47   using mod_div_equality [of a zero] by simp
```
```    48
```
```    49 lemma mod_0 [simp]: "0 mod a = 0"
```
```    50   using mod_div_equality [of zero a] div_0 by simp
```
```    51
```
```    52 lemma div_mult_self2 [simp]:
```
```    53   assumes "b \<noteq> 0"
```
```    54   shows "(a + b * c) div b = c + a div b"
```
```    55   using assms div_mult_self1 [of b a c] by (simp add: mult_commute)
```
```    56
```
```    57 lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"
```
```    58 proof (cases "b = 0")
```
```    59   case True then show ?thesis by simp
```
```    60 next
```
```    61   case False
```
```    62   have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"
```
```    63     by (simp add: mod_div_equality)
```
```    64   also from False div_mult_self1 [of b a c] have
```
```    65     "\<dots> = (c + a div b) * b + (a + c * b) mod b"
```
```    66       by (simp add: algebra_simps)
```
```    67   finally have "a = a div b * b + (a + c * b) mod b"
```
```    68     by (simp add: add_commute [of a] add_assoc left_distrib)
```
```    69   then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
```
```    70     by (simp add: mod_div_equality)
```
```    71   then show ?thesis by simp
```
```    72 qed
```
```    73
```
```    74 lemma mod_mult_self2 [simp]: "(a + b * c) mod b = a mod b"
```
```    75   by (simp add: mult_commute [of b])
```
```    76
```
```    77 lemma div_mult_self1_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> b * a div b = a"
```
```    78   using div_mult_self2 [of b 0 a] by simp
```
```    79
```
```    80 lemma div_mult_self2_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> a * b div b = a"
```
```    81   using div_mult_self1 [of b 0 a] by simp
```
```    82
```
```    83 lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0"
```
```    84   using mod_mult_self2 [of 0 b a] by simp
```
```    85
```
```    86 lemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0"
```
```    87   using mod_mult_self1 [of 0 a b] by simp
```
```    88
```
```    89 lemma div_by_1 [simp]: "a div 1 = a"
```
```    90   using div_mult_self2_is_id [of 1 a] zero_neq_one by simp
```
```    91
```
```    92 lemma mod_by_1 [simp]: "a mod 1 = 0"
```
```    93 proof -
```
```    94   from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp
```
```    95   then have "a + a mod 1 = a + 0" by simp
```
```    96   then show ?thesis by (rule add_left_imp_eq)
```
```    97 qed
```
```    98
```
```    99 lemma mod_self [simp]: "a mod a = 0"
```
```   100   using mod_mult_self2_is_0 [of 1] by simp
```
```   101
```
```   102 lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
```
```   103   using div_mult_self2_is_id [of _ 1] by simp
```
```   104
```
```   105 lemma div_add_self1 [simp]:
```
```   106   assumes "b \<noteq> 0"
```
```   107   shows "(b + a) div b = a div b + 1"
```
```   108   using assms div_mult_self1 [of b a 1] by (simp add: add_commute)
```
```   109
```
```   110 lemma div_add_self2 [simp]:
```
```   111   assumes "b \<noteq> 0"
```
```   112   shows "(a + b) div b = a div b + 1"
```
```   113   using assms div_add_self1 [of b a] by (simp add: add_commute)
```
```   114
```
```   115 lemma mod_add_self1 [simp]:
```
```   116   "(b + a) mod b = a mod b"
```
```   117   using mod_mult_self1 [of a 1 b] by (simp add: add_commute)
```
```   118
```
```   119 lemma mod_add_self2 [simp]:
```
```   120   "(a + b) mod b = a mod b"
```
```   121   using mod_mult_self1 [of a 1 b] by simp
```
```   122
```
```   123 lemma mod_div_decomp:
```
```   124   fixes a b
```
```   125   obtains q r where "q = a div b" and "r = a mod b"
```
```   126     and "a = q * b + r"
```
```   127 proof -
```
```   128   from mod_div_equality have "a = a div b * b + a mod b" by simp
```
```   129   moreover have "a div b = a div b" ..
```
```   130   moreover have "a mod b = a mod b" ..
```
```   131   note that ultimately show thesis by blast
```
```   132 qed
```
```   133
```
```   134 lemma dvd_eq_mod_eq_0 [code_unfold]: "a dvd b \<longleftrightarrow> b mod a = 0"
```
```   135 proof
```
```   136   assume "b mod a = 0"
```
```   137   with mod_div_equality [of b a] have "b div a * a = b" by simp
```
```   138   then have "b = a * (b div a)" unfolding mult_commute ..
```
```   139   then have "\<exists>c. b = a * c" ..
```
```   140   then show "a dvd b" unfolding dvd_def .
```
```   141 next
```
```   142   assume "a dvd b"
```
```   143   then have "\<exists>c. b = a * c" unfolding dvd_def .
```
```   144   then obtain c where "b = a * c" ..
```
```   145   then have "b mod a = a * c mod a" by simp
```
```   146   then have "b mod a = c * a mod a" by (simp add: mult_commute)
```
```   147   then show "b mod a = 0" by simp
```
```   148 qed
```
```   149
```
```   150 lemma mod_div_trivial [simp]: "a mod b div b = 0"
```
```   151 proof (cases "b = 0")
```
```   152   assume "b = 0"
```
```   153   thus ?thesis by simp
```
```   154 next
```
```   155   assume "b \<noteq> 0"
```
```   156   hence "a div b + a mod b div b = (a mod b + a div b * b) div b"
```
```   157     by (rule div_mult_self1 [symmetric])
```
```   158   also have "\<dots> = a div b"
```
```   159     by (simp only: mod_div_equality')
```
```   160   also have "\<dots> = a div b + 0"
```
```   161     by simp
```
```   162   finally show ?thesis
```
```   163     by (rule add_left_imp_eq)
```
```   164 qed
```
```   165
```
```   166 lemma mod_mod_trivial [simp]: "a mod b mod b = a mod b"
```
```   167 proof -
```
```   168   have "a mod b mod b = (a mod b + a div b * b) mod b"
```
```   169     by (simp only: mod_mult_self1)
```
```   170   also have "\<dots> = a mod b"
```
```   171     by (simp only: mod_div_equality')
```
```   172   finally show ?thesis .
```
```   173 qed
```
```   174
```
```   175 lemma dvd_imp_mod_0: "a dvd b \<Longrightarrow> b mod a = 0"
```
```   176 by (rule dvd_eq_mod_eq_0[THEN iffD1])
```
```   177
```
```   178 lemma dvd_div_mult_self: "a dvd b \<Longrightarrow> (b div a) * a = b"
```
```   179 by (subst (2) mod_div_equality [of b a, symmetric]) (simp add:dvd_imp_mod_0)
```
```   180
```
```   181 lemma dvd_mult_div_cancel: "a dvd b \<Longrightarrow> a * (b div a) = b"
```
```   182 by (drule dvd_div_mult_self) (simp add: mult_commute)
```
```   183
```
```   184 lemma dvd_div_mult: "a dvd b \<Longrightarrow> (b div a) * c = b * c div a"
```
```   185 apply (cases "a = 0")
```
```   186  apply simp
```
```   187 apply (auto simp: dvd_def mult_assoc)
```
```   188 done
```
```   189
```
```   190 lemma div_dvd_div[simp]:
```
```   191   "a dvd b \<Longrightarrow> a dvd c \<Longrightarrow> (b div a dvd c div a) = (b dvd c)"
```
```   192 apply (cases "a = 0")
```
```   193  apply simp
```
```   194 apply (unfold dvd_def)
```
```   195 apply auto
```
```   196  apply(blast intro:mult_assoc[symmetric])
```
```   197 apply(fastsimp simp add: mult_assoc)
```
```   198 done
```
```   199
```
```   200 lemma dvd_mod_imp_dvd: "[| k dvd m mod n;  k dvd n |] ==> k dvd m"
```
```   201   apply (subgoal_tac "k dvd (m div n) *n + m mod n")
```
```   202    apply (simp add: mod_div_equality)
```
```   203   apply (simp only: dvd_add dvd_mult)
```
```   204   done
```
```   205
```
```   206 text {* Addition respects modular equivalence. *}
```
```   207
```
```   208 lemma mod_add_left_eq: "(a + b) mod c = (a mod c + b) mod c"
```
```   209 proof -
```
```   210   have "(a + b) mod c = (a div c * c + a mod c + b) mod c"
```
```   211     by (simp only: mod_div_equality)
```
```   212   also have "\<dots> = (a mod c + b + a div c * c) mod c"
```
```   213     by (simp only: add_ac)
```
```   214   also have "\<dots> = (a mod c + b) mod c"
```
```   215     by (rule mod_mult_self1)
```
```   216   finally show ?thesis .
```
```   217 qed
```
```   218
```
```   219 lemma mod_add_right_eq: "(a + b) mod c = (a + b mod c) mod c"
```
```   220 proof -
```
```   221   have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c"
```
```   222     by (simp only: mod_div_equality)
```
```   223   also have "\<dots> = (a + b mod c + b div c * c) mod c"
```
```   224     by (simp only: add_ac)
```
```   225   also have "\<dots> = (a + b mod c) mod c"
```
```   226     by (rule mod_mult_self1)
```
```   227   finally show ?thesis .
```
```   228 qed
```
```   229
```
```   230 lemma mod_add_eq: "(a + b) mod c = (a mod c + b mod c) mod c"
```
```   231 by (rule trans [OF mod_add_left_eq mod_add_right_eq])
```
```   232
```
```   233 lemma mod_add_cong:
```
```   234   assumes "a mod c = a' mod c"
```
```   235   assumes "b mod c = b' mod c"
```
```   236   shows "(a + b) mod c = (a' + b') mod c"
```
```   237 proof -
```
```   238   have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c"
```
```   239     unfolding assms ..
```
```   240   thus ?thesis
```
```   241     by (simp only: mod_add_eq [symmetric])
```
```   242 qed
```
```   243
```
```   244 lemma div_add [simp]: "z dvd x \<Longrightarrow> z dvd y
```
```   245   \<Longrightarrow> (x + y) div z = x div z + y div z"
```
```   246 by (cases "z = 0", simp, unfold dvd_def, auto simp add: algebra_simps)
```
```   247
```
```   248 text {* Multiplication respects modular equivalence. *}
```
```   249
```
```   250 lemma mod_mult_left_eq: "(a * b) mod c = ((a mod c) * b) mod c"
```
```   251 proof -
```
```   252   have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c"
```
```   253     by (simp only: mod_div_equality)
```
```   254   also have "\<dots> = (a mod c * b + a div c * b * c) mod c"
```
```   255     by (simp only: algebra_simps)
```
```   256   also have "\<dots> = (a mod c * b) mod c"
```
```   257     by (rule mod_mult_self1)
```
```   258   finally show ?thesis .
```
```   259 qed
```
```   260
```
```   261 lemma mod_mult_right_eq: "(a * b) mod c = (a * (b mod c)) mod c"
```
```   262 proof -
```
```   263   have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c"
```
```   264     by (simp only: mod_div_equality)
```
```   265   also have "\<dots> = (a * (b mod c) + a * (b div c) * c) mod c"
```
```   266     by (simp only: algebra_simps)
```
```   267   also have "\<dots> = (a * (b mod c)) mod c"
```
```   268     by (rule mod_mult_self1)
```
```   269   finally show ?thesis .
```
```   270 qed
```
```   271
```
```   272 lemma mod_mult_eq: "(a * b) mod c = ((a mod c) * (b mod c)) mod c"
```
```   273 by (rule trans [OF mod_mult_left_eq mod_mult_right_eq])
```
```   274
```
```   275 lemma mod_mult_cong:
```
```   276   assumes "a mod c = a' mod c"
```
```   277   assumes "b mod c = b' mod c"
```
```   278   shows "(a * b) mod c = (a' * b') mod c"
```
```   279 proof -
```
```   280   have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c"
```
```   281     unfolding assms ..
```
```   282   thus ?thesis
```
```   283     by (simp only: mod_mult_eq [symmetric])
```
```   284 qed
```
```   285
```
```   286 lemma mod_mod_cancel:
```
```   287   assumes "c dvd b"
```
```   288   shows "a mod b mod c = a mod c"
```
```   289 proof -
```
```   290   from `c dvd b` obtain k where "b = c * k"
```
```   291     by (rule dvdE)
```
```   292   have "a mod b mod c = a mod (c * k) mod c"
```
```   293     by (simp only: `b = c * k`)
```
```   294   also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c"
```
```   295     by (simp only: mod_mult_self1)
```
```   296   also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c"
```
```   297     by (simp only: add_ac mult_ac)
```
```   298   also have "\<dots> = a mod c"
```
```   299     by (simp only: mod_div_equality)
```
```   300   finally show ?thesis .
```
```   301 qed
```
```   302
```
```   303 lemma div_mult_div_if_dvd:
```
```   304   "y dvd x \<Longrightarrow> z dvd w \<Longrightarrow> (x div y) * (w div z) = (x * w) div (y * z)"
```
```   305   apply (cases "y = 0", simp)
```
```   306   apply (cases "z = 0", simp)
```
```   307   apply (auto elim!: dvdE simp add: algebra_simps)
```
```   308   apply (subst mult_assoc [symmetric])
```
```   309   apply (simp add: no_zero_divisors)
```
```   310   done
```
```   311
```
```   312 lemma div_mult_mult2 [simp]:
```
```   313   "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
```
```   314   by (drule div_mult_mult1) (simp add: mult_commute)
```
```   315
```
```   316 lemma div_mult_mult1_if [simp]:
```
```   317   "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"
```
```   318   by simp_all
```
```   319
```
```   320 lemma mod_mult_mult1:
```
```   321   "(c * a) mod (c * b) = c * (a mod b)"
```
```   322 proof (cases "c = 0")
```
```   323   case True then show ?thesis by simp
```
```   324 next
```
```   325   case False
```
```   326   from mod_div_equality
```
```   327   have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .
```
```   328   with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)
```
```   329     = c * a + c * (a mod b)" by (simp add: algebra_simps)
```
```   330   with mod_div_equality show ?thesis by simp
```
```   331 qed
```
```   332
```
```   333 lemma mod_mult_mult2:
```
```   334   "(a * c) mod (b * c) = (a mod b) * c"
```
```   335   using mod_mult_mult1 [of c a b] by (simp add: mult_commute)
```
```   336
```
```   337 lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)"
```
```   338   unfolding dvd_def by (auto simp add: mod_mult_mult1)
```
```   339
```
```   340 lemma dvd_mod_iff: "k dvd n \<Longrightarrow> k dvd (m mod n) \<longleftrightarrow> k dvd m"
```
```   341 by (blast intro: dvd_mod_imp_dvd dvd_mod)
```
```   342
```
```   343 lemma div_power:
```
```   344   "y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n"
```
```   345 apply (induct n)
```
```   346  apply simp
```
```   347 apply(simp add: div_mult_div_if_dvd dvd_power_same)
```
```   348 done
```
```   349
```
```   350 end
```
```   351
```
```   352 class ring_div = semiring_div + idom
```
```   353 begin
```
```   354
```
```   355 text {* Negation respects modular equivalence. *}
```
```   356
```
```   357 lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"
```
```   358 proof -
```
```   359   have "(- a) mod b = (- (a div b * b + a mod b)) mod b"
```
```   360     by (simp only: mod_div_equality)
```
```   361   also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b"
```
```   362     by (simp only: minus_add_distrib minus_mult_left add_ac)
```
```   363   also have "\<dots> = (- (a mod b)) mod b"
```
```   364     by (rule mod_mult_self1)
```
```   365   finally show ?thesis .
```
```   366 qed
```
```   367
```
```   368 lemma mod_minus_cong:
```
```   369   assumes "a mod b = a' mod b"
```
```   370   shows "(- a) mod b = (- a') mod b"
```
```   371 proof -
```
```   372   have "(- (a mod b)) mod b = (- (a' mod b)) mod b"
```
```   373     unfolding assms ..
```
```   374   thus ?thesis
```
```   375     by (simp only: mod_minus_eq [symmetric])
```
```   376 qed
```
```   377
```
```   378 text {* Subtraction respects modular equivalence. *}
```
```   379
```
```   380 lemma mod_diff_left_eq: "(a - b) mod c = (a mod c - b) mod c"
```
```   381   unfolding diff_minus
```
```   382   by (intro mod_add_cong mod_minus_cong) simp_all
```
```   383
```
```   384 lemma mod_diff_right_eq: "(a - b) mod c = (a - b mod c) mod c"
```
```   385   unfolding diff_minus
```
```   386   by (intro mod_add_cong mod_minus_cong) simp_all
```
```   387
```
```   388 lemma mod_diff_eq: "(a - b) mod c = (a mod c - b mod c) mod c"
```
```   389   unfolding diff_minus
```
```   390   by (intro mod_add_cong mod_minus_cong) simp_all
```
```   391
```
```   392 lemma mod_diff_cong:
```
```   393   assumes "a mod c = a' mod c"
```
```   394   assumes "b mod c = b' mod c"
```
```   395   shows "(a - b) mod c = (a' - b') mod c"
```
```   396   unfolding diff_minus using assms
```
```   397   by (intro mod_add_cong mod_minus_cong)
```
```   398
```
```   399 lemma dvd_neg_div: "y dvd x \<Longrightarrow> -x div y = - (x div y)"
```
```   400 apply (case_tac "y = 0") apply simp
```
```   401 apply (auto simp add: dvd_def)
```
```   402 apply (subgoal_tac "-(y * k) = y * - k")
```
```   403  apply (erule ssubst)
```
```   404  apply (erule div_mult_self1_is_id)
```
```   405 apply simp
```
```   406 done
```
```   407
```
```   408 lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div -y = - (x div y)"
```
```   409 apply (case_tac "y = 0") apply simp
```
```   410 apply (auto simp add: dvd_def)
```
```   411 apply (subgoal_tac "y * k = -y * -k")
```
```   412  apply (erule ssubst)
```
```   413  apply (rule div_mult_self1_is_id)
```
```   414  apply simp
```
```   415 apply simp
```
```   416 done
```
```   417
```
```   418 end
```
```   419
```
```   420
```
```   421 subsection {* Division on @{typ nat} *}
```
```   422
```
```   423 text {*
```
```   424   We define @{const div} and @{const mod} on @{typ nat} by means
```
```   425   of a characteristic relation with two input arguments
```
```   426   @{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments
```
```   427   @{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder).
```
```   428 *}
```
```   429
```
```   430 definition divmod_nat_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where
```
```   431   "divmod_nat_rel m n qr \<longleftrightarrow>
```
```   432     m = fst qr * n + snd qr \<and>
```
```   433       (if n = 0 then fst qr = 0 else if n > 0 then 0 \<le> snd qr \<and> snd qr < n else n < snd qr \<and> snd qr \<le> 0)"
```
```   434
```
```   435 text {* @{const divmod_nat_rel} is total: *}
```
```   436
```
```   437 lemma divmod_nat_rel_ex:
```
```   438   obtains q r where "divmod_nat_rel m n (q, r)"
```
```   439 proof (cases "n = 0")
```
```   440   case True  with that show thesis
```
```   441     by (auto simp add: divmod_nat_rel_def)
```
```   442 next
```
```   443   case False
```
```   444   have "\<exists>q r. m = q * n + r \<and> r < n"
```
```   445   proof (induct m)
```
```   446     case 0 with `n \<noteq> 0`
```
```   447     have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp
```
```   448     then show ?case by blast
```
```   449   next
```
```   450     case (Suc m) then obtain q' r'
```
```   451       where m: "m = q' * n + r'" and n: "r' < n" by auto
```
```   452     then show ?case proof (cases "Suc r' < n")
```
```   453       case True
```
```   454       from m n have "Suc m = q' * n + Suc r'" by simp
```
```   455       with True show ?thesis by blast
```
```   456     next
```
```   457       case False then have "n \<le> Suc r'" by auto
```
```   458       moreover from n have "Suc r' \<le> n" by auto
```
```   459       ultimately have "n = Suc r'" by auto
```
```   460       with m have "Suc m = Suc q' * n + 0" by simp
```
```   461       with `n \<noteq> 0` show ?thesis by blast
```
```   462     qed
```
```   463   qed
```
```   464   with that show thesis
```
```   465     using `n \<noteq> 0` by (auto simp add: divmod_nat_rel_def)
```
```   466 qed
```
```   467
```
```   468 text {* @{const divmod_nat_rel} is injective: *}
```
```   469
```
```   470 lemma divmod_nat_rel_unique:
```
```   471   assumes "divmod_nat_rel m n qr"
```
```   472     and "divmod_nat_rel m n qr'"
```
```   473   shows "qr = qr'"
```
```   474 proof (cases "n = 0")
```
```   475   case True with assms show ?thesis
```
```   476     by (cases qr, cases qr')
```
```   477       (simp add: divmod_nat_rel_def)
```
```   478 next
```
```   479   case False
```
```   480   have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)"
```
```   481   apply (rule leI)
```
```   482   apply (subst less_iff_Suc_add)
```
```   483   apply (auto simp add: add_mult_distrib)
```
```   484   done
```
```   485   from `n \<noteq> 0` assms have "fst qr = fst qr'"
```
```   486     by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)
```
```   487   moreover from this assms have "snd qr = snd qr'"
```
```   488     by (simp add: divmod_nat_rel_def)
```
```   489   ultimately show ?thesis by (cases qr, cases qr') simp
```
```   490 qed
```
```   491
```
```   492 text {*
```
```   493   We instantiate divisibility on the natural numbers by
```
```   494   means of @{const divmod_nat_rel}:
```
```   495 *}
```
```   496
```
```   497 instantiation nat :: semiring_div
```
```   498 begin
```
```   499
```
```   500 definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
```
```   501   [code del]: "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"
```
```   502
```
```   503 lemma divmod_nat_rel_divmod_nat:
```
```   504   "divmod_nat_rel m n (divmod_nat m n)"
```
```   505 proof -
```
```   506   from divmod_nat_rel_ex
```
```   507     obtain qr where rel: "divmod_nat_rel m n qr" .
```
```   508   then show ?thesis
```
```   509   by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)
```
```   510 qed
```
```   511
```
```   512 lemma divmod_nat_eq:
```
```   513   assumes "divmod_nat_rel m n qr"
```
```   514   shows "divmod_nat m n = qr"
```
```   515   using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)
```
```   516
```
```   517 definition div_nat where
```
```   518   "m div n = fst (divmod_nat m n)"
```
```   519
```
```   520 definition mod_nat where
```
```   521   "m mod n = snd (divmod_nat m n)"
```
```   522
```
```   523 lemma divmod_nat_div_mod:
```
```   524   "divmod_nat m n = (m div n, m mod n)"
```
```   525   unfolding div_nat_def mod_nat_def by simp
```
```   526
```
```   527 lemma div_eq:
```
```   528   assumes "divmod_nat_rel m n (q, r)"
```
```   529   shows "m div n = q"
```
```   530   using assms by (auto dest: divmod_nat_eq simp add: divmod_nat_div_mod)
```
```   531
```
```   532 lemma mod_eq:
```
```   533   assumes "divmod_nat_rel m n (q, r)"
```
```   534   shows "m mod n = r"
```
```   535   using assms by (auto dest: divmod_nat_eq simp add: divmod_nat_div_mod)
```
```   536
```
```   537 lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"
```
```   538   by (simp add: div_nat_def mod_nat_def divmod_nat_rel_divmod_nat)
```
```   539
```
```   540 lemma divmod_nat_zero:
```
```   541   "divmod_nat m 0 = (0, m)"
```
```   542 proof -
```
```   543   from divmod_nat_rel [of m 0] show ?thesis
```
```   544     unfolding divmod_nat_div_mod divmod_nat_rel_def by simp
```
```   545 qed
```
```   546
```
```   547 lemma divmod_nat_base:
```
```   548   assumes "m < n"
```
```   549   shows "divmod_nat m n = (0, m)"
```
```   550 proof -
```
```   551   from divmod_nat_rel [of m n] show ?thesis
```
```   552     unfolding divmod_nat_div_mod divmod_nat_rel_def
```
```   553     using assms by (cases "m div n = 0")
```
```   554       (auto simp add: gr0_conv_Suc [of "m div n"])
```
```   555 qed
```
```   556
```
```   557 lemma divmod_nat_step:
```
```   558   assumes "0 < n" and "n \<le> m"
```
```   559   shows "divmod_nat m n = (Suc ((m - n) div n), (m - n) mod n)"
```
```   560 proof -
```
```   561   from divmod_nat_rel have divmod_nat_m_n: "divmod_nat_rel m n (m div n, m mod n)" .
```
```   562   with assms have m_div_n: "m div n \<ge> 1"
```
```   563     by (cases "m div n") (auto simp add: divmod_nat_rel_def)
```
```   564   from assms divmod_nat_m_n have "divmod_nat_rel (m - n) n (m div n - Suc 0, m mod n)"
```
```   565     by (cases "m div n") (auto simp add: divmod_nat_rel_def)
```
```   566   with divmod_nat_eq have "divmod_nat (m - n) n = (m div n - Suc 0, m mod n)" by simp
```
```   567   moreover from divmod_nat_div_mod have "divmod_nat (m - n) n = ((m - n) div n, (m - n) mod n)" .
```
```   568   ultimately have "m div n = Suc ((m - n) div n)"
```
```   569     and "m mod n = (m - n) mod n" using m_div_n by simp_all
```
```   570   then show ?thesis using divmod_nat_div_mod by simp
```
```   571 qed
```
```   572
```
```   573 text {* The ''recursion'' equations for @{const div} and @{const mod} *}
```
```   574
```
```   575 lemma div_less [simp]:
```
```   576   fixes m n :: nat
```
```   577   assumes "m < n"
```
```   578   shows "m div n = 0"
```
```   579   using assms divmod_nat_base divmod_nat_div_mod by simp
```
```   580
```
```   581 lemma le_div_geq:
```
```   582   fixes m n :: nat
```
```   583   assumes "0 < n" and "n \<le> m"
```
```   584   shows "m div n = Suc ((m - n) div n)"
```
```   585   using assms divmod_nat_step divmod_nat_div_mod by simp
```
```   586
```
```   587 lemma mod_less [simp]:
```
```   588   fixes m n :: nat
```
```   589   assumes "m < n"
```
```   590   shows "m mod n = m"
```
```   591   using assms divmod_nat_base divmod_nat_div_mod by simp
```
```   592
```
```   593 lemma le_mod_geq:
```
```   594   fixes m n :: nat
```
```   595   assumes "n \<le> m"
```
```   596   shows "m mod n = (m - n) mod n"
```
```   597   using assms divmod_nat_step divmod_nat_div_mod by (cases "n = 0") simp_all
```
```   598
```
```   599 instance proof -
```
```   600   have [simp]: "\<And>n::nat. n div 0 = 0"
```
```   601     by (simp add: div_nat_def divmod_nat_zero)
```
```   602   have [simp]: "\<And>n::nat. 0 div n = 0"
```
```   603   proof -
```
```   604     fix n :: nat
```
```   605     show "0 div n = 0"
```
```   606       by (cases "n = 0") simp_all
```
```   607   qed
```
```   608   show "OFCLASS(nat, semiring_div_class)" proof
```
```   609     fix m n :: nat
```
```   610     show "m div n * n + m mod n = m"
```
```   611       using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)
```
```   612   next
```
```   613     fix m n q :: nat
```
```   614     assume "n \<noteq> 0"
```
```   615     then show "(q + m * n) div n = m + q div n"
```
```   616       by (induct m) (simp_all add: le_div_geq)
```
```   617   next
```
```   618     fix m n q :: nat
```
```   619     assume "m \<noteq> 0"
```
```   620     then show "(m * n) div (m * q) = n div q"
```
```   621     proof (cases "n \<noteq> 0 \<and> q \<noteq> 0")
```
```   622       case False then show ?thesis by auto
```
```   623     next
```
```   624       case True with `m \<noteq> 0`
```
```   625         have "m > 0" and "n > 0" and "q > 0" by auto
```
```   626       then have "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)"
```
```   627         by (auto simp add: divmod_nat_rel_def) (simp_all add: algebra_simps)
```
```   628       moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .
```
```   629       ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .
```
```   630       then show ?thesis by (simp add: div_eq)
```
```   631     qed
```
```   632   qed simp_all
```
```   633 qed
```
```   634
```
```   635 end
```
```   636
```
```   637 text {* Simproc for cancelling @{const div} and @{const mod} *}
```
```   638
```
```   639 ML {*
```
```   640 local
```
```   641
```
```   642 structure CancelDivMod = CancelDivModFun(struct
```
```   643
```
```   644   val div_name = @{const_name div};
```
```   645   val mod_name = @{const_name mod};
```
```   646   val mk_binop = HOLogic.mk_binop;
```
```   647   val mk_sum = Nat_Arith.mk_sum;
```
```   648   val dest_sum = Nat_Arith.dest_sum;
```
```   649
```
```   650   val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
```
```   651
```
```   652   val trans = trans;
```
```   653
```
```   654   val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
```
```   655     (@{thm monoid_add_class.add_0_left} :: @{thm monoid_add_class.add_0_right} :: @{thms add_ac}))
```
```   656
```
```   657 end)
```
```   658
```
```   659 in
```
```   660
```
```   661 val cancel_div_mod_nat_proc = Simplifier.simproc @{theory}
```
```   662   "cancel_div_mod" ["(m::nat) + n"] (K CancelDivMod.proc);
```
```   663
```
```   664 val _ = Addsimprocs [cancel_div_mod_nat_proc];
```
```   665
```
```   666 end
```
```   667 *}
```
```   668
```
```   669 text {* code generator setup *}
```
```   670
```
```   671 lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
```
```   672   let (q, r) = divmod_nat (m - n) n in (Suc q, r))"
```
```   673 by (simp add: divmod_nat_zero divmod_nat_base divmod_nat_step)
```
```   674     (simp add: divmod_nat_div_mod)
```
```   675
```
```   676 code_modulename SML
```
```   677   Divides Nat
```
```   678
```
```   679 code_modulename OCaml
```
```   680   Divides Nat
```
```   681
```
```   682 code_modulename Haskell
```
```   683   Divides Nat
```
```   684
```
```   685
```
```   686 subsubsection {* Quotient *}
```
```   687
```
```   688 lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
```
```   689 by (simp add: le_div_geq linorder_not_less)
```
```   690
```
```   691 lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
```
```   692 by (simp add: div_geq)
```
```   693
```
```   694 lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
```
```   695 by simp
```
```   696
```
```   697 lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
```
```   698 by simp
```
```   699
```
```   700
```
```   701 subsubsection {* Remainder *}
```
```   702
```
```   703 lemma mod_less_divisor [simp]:
```
```   704   fixes m n :: nat
```
```   705   assumes "n > 0"
```
```   706   shows "m mod n < (n::nat)"
```
```   707   using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto
```
```   708
```
```   709 lemma mod_less_eq_dividend [simp]:
```
```   710   fixes m n :: nat
```
```   711   shows "m mod n \<le> m"
```
```   712 proof (rule add_leD2)
```
```   713   from mod_div_equality have "m div n * n + m mod n = m" .
```
```   714   then show "m div n * n + m mod n \<le> m" by auto
```
```   715 qed
```
```   716
```
```   717 lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m - n) mod n"
```
```   718 by (simp add: le_mod_geq linorder_not_less)
```
```   719
```
```   720 lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m - n) mod n)"
```
```   721 by (simp add: le_mod_geq)
```
```   722
```
```   723 lemma mod_1 [simp]: "m mod Suc 0 = 0"
```
```   724 by (induct m) (simp_all add: mod_geq)
```
```   725
```
```   726 lemma mod_mult_distrib: "(m mod n) * (k\<Colon>nat) = (m * k) mod (n * k)"
```
```   727   apply (cases "n = 0", simp)
```
```   728   apply (cases "k = 0", simp)
```
```   729   apply (induct m rule: nat_less_induct)
```
```   730   apply (subst mod_if, simp)
```
```   731   apply (simp add: mod_geq diff_mult_distrib)
```
```   732   done
```
```   733
```
```   734 lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
```
```   735 by (simp add: mult_commute [of k] mod_mult_distrib)
```
```   736
```
```   737 (* a simple rearrangement of mod_div_equality: *)
```
```   738 lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
```
```   739 by (cut_tac a = m and b = n in mod_div_equality2, arith)
```
```   740
```
```   741 lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
```
```   742   apply (drule mod_less_divisor [where m = m])
```
```   743   apply simp
```
```   744   done
```
```   745
```
```   746 subsubsection {* Quotient and Remainder *}
```
```   747
```
```   748 lemma divmod_nat_rel_mult1_eq:
```
```   749   "divmod_nat_rel b c (q, r) \<Longrightarrow> c > 0
```
```   750    \<Longrightarrow> divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"
```
```   751 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
```
```   752
```
```   753 lemma div_mult1_eq:
```
```   754   "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"
```
```   755 apply (cases "c = 0", simp)
```
```   756 apply (blast intro: divmod_nat_rel [THEN divmod_nat_rel_mult1_eq, THEN div_eq])
```
```   757 done
```
```   758
```
```   759 lemma divmod_nat_rel_add1_eq:
```
```   760   "divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br) \<Longrightarrow>  c > 0
```
```   761    \<Longrightarrow> divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"
```
```   762 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
```
```   763
```
```   764 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
```
```   765 lemma div_add1_eq:
```
```   766   "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
```
```   767 apply (cases "c = 0", simp)
```
```   768 apply (blast intro: divmod_nat_rel_add1_eq [THEN div_eq] divmod_nat_rel)
```
```   769 done
```
```   770
```
```   771 lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
```
```   772   apply (cut_tac m = q and n = c in mod_less_divisor)
```
```   773   apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
```
```   774   apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
```
```   775   apply (simp add: add_mult_distrib2)
```
```   776   done
```
```   777
```
```   778 lemma divmod_nat_rel_mult2_eq:
```
```   779   "divmod_nat_rel a b (q, r) \<Longrightarrow> 0 < b \<Longrightarrow> 0 < c
```
```   780    \<Longrightarrow> divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"
```
```   781 by (auto simp add: mult_ac divmod_nat_rel_def add_mult_distrib2 [symmetric] mod_lemma)
```
```   782
```
```   783 lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
```
```   784   apply (cases "b = 0", simp)
```
```   785   apply (cases "c = 0", simp)
```
```   786   apply (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_eq])
```
```   787   done
```
```   788
```
```   789 lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
```
```   790   apply (cases "b = 0", simp)
```
```   791   apply (cases "c = 0", simp)
```
```   792   apply (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_eq])
```
```   793   done
```
```   794
```
```   795
```
```   796 subsubsection{*Further Facts about Quotient and Remainder*}
```
```   797
```
```   798 lemma div_1 [simp]: "m div Suc 0 = m"
```
```   799 by (induct m) (simp_all add: div_geq)
```
```   800
```
```   801
```
```   802 (* Monotonicity of div in first argument *)
```
```   803 lemma div_le_mono [rule_format (no_asm)]:
```
```   804     "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
```
```   805 apply (case_tac "k=0", simp)
```
```   806 apply (induct "n" rule: nat_less_induct, clarify)
```
```   807 apply (case_tac "n<k")
```
```   808 (* 1  case n<k *)
```
```   809 apply simp
```
```   810 (* 2  case n >= k *)
```
```   811 apply (case_tac "m<k")
```
```   812 (* 2.1  case m<k *)
```
```   813 apply simp
```
```   814 (* 2.2  case m>=k *)
```
```   815 apply (simp add: div_geq diff_le_mono)
```
```   816 done
```
```   817
```
```   818 (* Antimonotonicity of div in second argument *)
```
```   819 lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
```
```   820 apply (subgoal_tac "0<n")
```
```   821  prefer 2 apply simp
```
```   822 apply (induct_tac k rule: nat_less_induct)
```
```   823 apply (rename_tac "k")
```
```   824 apply (case_tac "k<n", simp)
```
```   825 apply (subgoal_tac "~ (k<m) ")
```
```   826  prefer 2 apply simp
```
```   827 apply (simp add: div_geq)
```
```   828 apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
```
```   829  prefer 2
```
```   830  apply (blast intro: div_le_mono diff_le_mono2)
```
```   831 apply (rule le_trans, simp)
```
```   832 apply (simp)
```
```   833 done
```
```   834
```
```   835 lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
```
```   836 apply (case_tac "n=0", simp)
```
```   837 apply (subgoal_tac "m div n \<le> m div 1", simp)
```
```   838 apply (rule div_le_mono2)
```
```   839 apply (simp_all (no_asm_simp))
```
```   840 done
```
```   841
```
```   842 (* Similar for "less than" *)
```
```   843 lemma div_less_dividend [rule_format]:
```
```   844      "!!n::nat. 1<n ==> 0 < m --> m div n < m"
```
```   845 apply (induct_tac m rule: nat_less_induct)
```
```   846 apply (rename_tac "m")
```
```   847 apply (case_tac "m<n", simp)
```
```   848 apply (subgoal_tac "0<n")
```
```   849  prefer 2 apply simp
```
```   850 apply (simp add: div_geq)
```
```   851 apply (case_tac "n<m")
```
```   852  apply (subgoal_tac "(m-n) div n < (m-n) ")
```
```   853   apply (rule impI less_trans_Suc)+
```
```   854 apply assumption
```
```   855   apply (simp_all)
```
```   856 done
```
```   857
```
```   858 declare div_less_dividend [simp]
```
```   859
```
```   860 text{*A fact for the mutilated chess board*}
```
```   861 lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
```
```   862 apply (case_tac "n=0", simp)
```
```   863 apply (induct "m" rule: nat_less_induct)
```
```   864 apply (case_tac "Suc (na) <n")
```
```   865 (* case Suc(na) < n *)
```
```   866 apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
```
```   867 (* case n \<le> Suc(na) *)
```
```   868 apply (simp add: linorder_not_less le_Suc_eq mod_geq)
```
```   869 apply (auto simp add: Suc_diff_le le_mod_geq)
```
```   870 done
```
```   871
```
```   872 lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
```
```   873 by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
```
```   874
```
```   875 lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
```
```   876
```
```   877 (*Loses information, namely we also have r<d provided d is nonzero*)
```
```   878 lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
```
```   879   apply (cut_tac a = m in mod_div_equality)
```
```   880   apply (simp only: add_ac)
```
```   881   apply (blast intro: sym)
```
```   882   done
```
```   883
```
```   884 lemma split_div:
```
```   885  "P(n div k :: nat) =
```
```   886  ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
```
```   887  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
```
```   888 proof
```
```   889   assume P: ?P
```
```   890   show ?Q
```
```   891   proof (cases)
```
```   892     assume "k = 0"
```
```   893     with P show ?Q by simp
```
```   894   next
```
```   895     assume not0: "k \<noteq> 0"
```
```   896     thus ?Q
```
```   897     proof (simp, intro allI impI)
```
```   898       fix i j
```
```   899       assume n: "n = k*i + j" and j: "j < k"
```
```   900       show "P i"
```
```   901       proof (cases)
```
```   902         assume "i = 0"
```
```   903         with n j P show "P i" by simp
```
```   904       next
```
```   905         assume "i \<noteq> 0"
```
```   906         with not0 n j P show "P i" by(simp add:add_ac)
```
```   907       qed
```
```   908     qed
```
```   909   qed
```
```   910 next
```
```   911   assume Q: ?Q
```
```   912   show ?P
```
```   913   proof (cases)
```
```   914     assume "k = 0"
```
```   915     with Q show ?P by simp
```
```   916   next
```
```   917     assume not0: "k \<noteq> 0"
```
```   918     with Q have R: ?R by simp
```
```   919     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
```
```   920     show ?P by simp
```
```   921   qed
```
```   922 qed
```
```   923
```
```   924 lemma split_div_lemma:
```
```   925   assumes "0 < n"
```
```   926   shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m\<Colon>nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
```
```   927 proof
```
```   928   assume ?rhs
```
```   929   with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp
```
```   930   then have A: "n * q \<le> m" by simp
```
```   931   have "n - (m mod n) > 0" using mod_less_divisor assms by auto
```
```   932   then have "m < m + (n - (m mod n))" by simp
```
```   933   then have "m < n + (m - (m mod n))" by simp
```
```   934   with nq have "m < n + n * q" by simp
```
```   935   then have B: "m < n * Suc q" by simp
```
```   936   from A B show ?lhs ..
```
```   937 next
```
```   938   assume P: ?lhs
```
```   939   then have "divmod_nat_rel m n (q, m - n * q)"
```
```   940     unfolding divmod_nat_rel_def by (auto simp add: mult_ac)
```
```   941   with divmod_nat_rel_unique divmod_nat_rel [of m n]
```
```   942   have "(q, m - n * q) = (m div n, m mod n)" by auto
```
```   943   then show ?rhs by simp
```
```   944 qed
```
```   945
```
```   946 theorem split_div':
```
```   947   "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
```
```   948    (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
```
```   949   apply (case_tac "0 < n")
```
```   950   apply (simp only: add: split_div_lemma)
```
```   951   apply simp_all
```
```   952   done
```
```   953
```
```   954 lemma split_mod:
```
```   955  "P(n mod k :: nat) =
```
```   956  ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
```
```   957  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
```
```   958 proof
```
```   959   assume P: ?P
```
```   960   show ?Q
```
```   961   proof (cases)
```
```   962     assume "k = 0"
```
```   963     with P show ?Q by simp
```
```   964   next
```
```   965     assume not0: "k \<noteq> 0"
```
```   966     thus ?Q
```
```   967     proof (simp, intro allI impI)
```
```   968       fix i j
```
```   969       assume "n = k*i + j" "j < k"
```
```   970       thus "P j" using not0 P by(simp add:add_ac mult_ac)
```
```   971     qed
```
```   972   qed
```
```   973 next
```
```   974   assume Q: ?Q
```
```   975   show ?P
```
```   976   proof (cases)
```
```   977     assume "k = 0"
```
```   978     with Q show ?P by simp
```
```   979   next
```
```   980     assume not0: "k \<noteq> 0"
```
```   981     with Q have R: ?R by simp
```
```   982     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
```
```   983     show ?P by simp
```
```   984   qed
```
```   985 qed
```
```   986
```
```   987 theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
```
```   988   apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
```
```   989     subst [OF mod_div_equality [of _ n]])
```
```   990   apply arith
```
```   991   done
```
```   992
```
```   993 lemma div_mod_equality':
```
```   994   fixes m n :: nat
```
```   995   shows "m div n * n = m - m mod n"
```
```   996 proof -
```
```   997   have "m mod n \<le> m mod n" ..
```
```   998   from div_mod_equality have
```
```   999     "m div n * n + m mod n - m mod n = m - m mod n" by simp
```
```  1000   with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have
```
```  1001     "m div n * n + (m mod n - m mod n) = m - m mod n"
```
```  1002     by simp
```
```  1003   then show ?thesis by simp
```
```  1004 qed
```
```  1005
```
```  1006
```
```  1007 subsubsection {*An ``induction'' law for modulus arithmetic.*}
```
```  1008
```
```  1009 lemma mod_induct_0:
```
```  1010   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
```
```  1011   and base: "P i" and i: "i<p"
```
```  1012   shows "P 0"
```
```  1013 proof (rule ccontr)
```
```  1014   assume contra: "\<not>(P 0)"
```
```  1015   from i have p: "0<p" by simp
```
```  1016   have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
```
```  1017   proof
```
```  1018     fix k
```
```  1019     show "?A k"
```
```  1020     proof (induct k)
```
```  1021       show "?A 0" by simp  -- "by contradiction"
```
```  1022     next
```
```  1023       fix n
```
```  1024       assume ih: "?A n"
```
```  1025       show "?A (Suc n)"
```
```  1026       proof (clarsimp)
```
```  1027         assume y: "P (p - Suc n)"
```
```  1028         have n: "Suc n < p"
```
```  1029         proof (rule ccontr)
```
```  1030           assume "\<not>(Suc n < p)"
```
```  1031           hence "p - Suc n = 0"
```
```  1032             by simp
```
```  1033           with y contra show "False"
```
```  1034             by simp
```
```  1035         qed
```
```  1036         hence n2: "Suc (p - Suc n) = p-n" by arith
```
```  1037         from p have "p - Suc n < p" by arith
```
```  1038         with y step have z: "P ((Suc (p - Suc n)) mod p)"
```
```  1039           by blast
```
```  1040         show "False"
```
```  1041         proof (cases "n=0")
```
```  1042           case True
```
```  1043           with z n2 contra show ?thesis by simp
```
```  1044         next
```
```  1045           case False
```
```  1046           with p have "p-n < p" by arith
```
```  1047           with z n2 False ih show ?thesis by simp
```
```  1048         qed
```
```  1049       qed
```
```  1050     qed
```
```  1051   qed
```
```  1052   moreover
```
```  1053   from i obtain k where "0<k \<and> i+k=p"
```
```  1054     by (blast dest: less_imp_add_positive)
```
```  1055   hence "0<k \<and> i=p-k" by auto
```
```  1056   moreover
```
```  1057   note base
```
```  1058   ultimately
```
```  1059   show "False" by blast
```
```  1060 qed
```
```  1061
```
```  1062 lemma mod_induct:
```
```  1063   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
```
```  1064   and base: "P i" and i: "i<p" and j: "j<p"
```
```  1065   shows "P j"
```
```  1066 proof -
```
```  1067   have "\<forall>j<p. P j"
```
```  1068   proof
```
```  1069     fix j
```
```  1070     show "j<p \<longrightarrow> P j" (is "?A j")
```
```  1071     proof (induct j)
```
```  1072       from step base i show "?A 0"
```
```  1073         by (auto elim: mod_induct_0)
```
```  1074     next
```
```  1075       fix k
```
```  1076       assume ih: "?A k"
```
```  1077       show "?A (Suc k)"
```
```  1078       proof
```
```  1079         assume suc: "Suc k < p"
```
```  1080         hence k: "k<p" by simp
```
```  1081         with ih have "P k" ..
```
```  1082         with step k have "P (Suc k mod p)"
```
```  1083           by blast
```
```  1084         moreover
```
```  1085         from suc have "Suc k mod p = Suc k"
```
```  1086           by simp
```
```  1087         ultimately
```
```  1088         show "P (Suc k)" by simp
```
```  1089       qed
```
```  1090     qed
```
```  1091   qed
```
```  1092   with j show ?thesis by blast
```
```  1093 qed
```
```  1094
```
```  1095 lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"
```
```  1096 by (auto simp add: numeral_2_eq_2 le_div_geq)
```
```  1097
```
```  1098 lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
```
```  1099 by (simp add: nat_mult_2 [symmetric])
```
```  1100
```
```  1101 lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2"
```
```  1102 apply (subgoal_tac "m mod 2 < 2")
```
```  1103 apply (erule less_2_cases [THEN disjE])
```
```  1104 apply (simp_all (no_asm_simp) add: Let_def mod_Suc nat_1)
```
```  1105 done
```
```  1106
```
```  1107 lemma mod2_gr_0 [simp]: "0 < (m\<Colon>nat) mod 2 \<longleftrightarrow> m mod 2 = 1"
```
```  1108 proof -
```
```  1109   { fix n :: nat have  "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (induct n) simp_all }
```
```  1110   moreover have "m mod 2 < 2" by simp
```
```  1111   ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" .
```
```  1112   then show ?thesis by auto
```
```  1113 qed
```
```  1114
```
```  1115 text{*These lemmas collapse some needless occurrences of Suc:
```
```  1116     at least three Sucs, since two and fewer are rewritten back to Suc again!
```
```  1117     We already have some rules to simplify operands smaller than 3.*}
```
```  1118
```
```  1119 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
```
```  1120 by (simp add: Suc3_eq_add_3)
```
```  1121
```
```  1122 lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
```
```  1123 by (simp add: Suc3_eq_add_3)
```
```  1124
```
```  1125 lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
```
```  1126 by (simp add: Suc3_eq_add_3)
```
```  1127
```
```  1128 lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
```
```  1129 by (simp add: Suc3_eq_add_3)
```
```  1130
```
```  1131 lemmas Suc_div_eq_add3_div_number_of =
```
```  1132     Suc_div_eq_add3_div [of _ "number_of v", standard]
```
```  1133 declare Suc_div_eq_add3_div_number_of [simp]
```
```  1134
```
```  1135 lemmas Suc_mod_eq_add3_mod_number_of =
```
```  1136     Suc_mod_eq_add3_mod [of _ "number_of v", standard]
```
```  1137 declare Suc_mod_eq_add3_mod_number_of [simp]
```
```  1138
```
```  1139 end
```