src/HOL/Divides.thy
 author haftmann Mon Jan 09 18:53:06 2017 +0100 (2017-01-09) changeset 64848 c50db2128048 parent 64785 ae0bbc8e45ad child 65556 fcd599570afa permissions -rw-r--r--
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
```     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>More on quotient and remainder\<close>
```
```     7
```
```     8 theory Divides
```
```     9 imports Parity
```
```    10 begin
```
```    11
```
```    12 subsection \<open>Quotient and remainder in integral domains with additional properties\<close>
```
```    13
```
```    14 class semiring_div = semidom_modulo +
```
```    15   assumes div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"
```
```    16     and div_mult_mult1 [simp]: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b"
```
```    17 begin
```
```    18
```
```    19 lemma div_mult_self2 [simp]:
```
```    20   assumes "b \<noteq> 0"
```
```    21   shows "(a + b * c) div b = c + a div b"
```
```    22   using assms div_mult_self1 [of b a c] by (simp add: mult.commute)
```
```    23
```
```    24 lemma div_mult_self3 [simp]:
```
```    25   assumes "b \<noteq> 0"
```
```    26   shows "(c * b + a) div b = c + a div b"
```
```    27   using assms by (simp add: add.commute)
```
```    28
```
```    29 lemma div_mult_self4 [simp]:
```
```    30   assumes "b \<noteq> 0"
```
```    31   shows "(b * c + a) div b = c + a div b"
```
```    32   using assms by (simp add: add.commute)
```
```    33
```
```    34 lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"
```
```    35 proof (cases "b = 0")
```
```    36   case True then show ?thesis by simp
```
```    37 next
```
```    38   case False
```
```    39   have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"
```
```    40     by (simp add: div_mult_mod_eq)
```
```    41   also from False div_mult_self1 [of b a c] have
```
```    42     "\<dots> = (c + a div b) * b + (a + c * b) mod b"
```
```    43       by (simp add: algebra_simps)
```
```    44   finally have "a = a div b * b + (a + c * b) mod b"
```
```    45     by (simp add: add.commute [of a] add.assoc distrib_right)
```
```    46   then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
```
```    47     by (simp add: div_mult_mod_eq)
```
```    48   then show ?thesis by simp
```
```    49 qed
```
```    50
```
```    51 lemma mod_mult_self2 [simp]:
```
```    52   "(a + b * c) mod b = a mod b"
```
```    53   by (simp add: mult.commute [of b])
```
```    54
```
```    55 lemma mod_mult_self3 [simp]:
```
```    56   "(c * b + a) mod b = a mod b"
```
```    57   by (simp add: add.commute)
```
```    58
```
```    59 lemma mod_mult_self4 [simp]:
```
```    60   "(b * c + a) mod b = a mod b"
```
```    61   by (simp add: add.commute)
```
```    62
```
```    63 lemma mod_mult_self1_is_0 [simp]:
```
```    64   "b * a mod b = 0"
```
```    65   using mod_mult_self2 [of 0 b a] by simp
```
```    66
```
```    67 lemma mod_mult_self2_is_0 [simp]:
```
```    68   "a * b mod b = 0"
```
```    69   using mod_mult_self1 [of 0 a b] by simp
```
```    70
```
```    71 lemma div_add_self1:
```
```    72   assumes "b \<noteq> 0"
```
```    73   shows "(b + a) div b = a div b + 1"
```
```    74   using assms div_mult_self1 [of b a 1] by (simp add: add.commute)
```
```    75
```
```    76 lemma div_add_self2:
```
```    77   assumes "b \<noteq> 0"
```
```    78   shows "(a + b) div b = a div b + 1"
```
```    79   using assms div_add_self1 [of b a] by (simp add: add.commute)
```
```    80
```
```    81 lemma mod_add_self1 [simp]:
```
```    82   "(b + a) mod b = a mod b"
```
```    83   using mod_mult_self1 [of a 1 b] by (simp add: add.commute)
```
```    84
```
```    85 lemma mod_add_self2 [simp]:
```
```    86   "(a + b) mod b = a mod b"
```
```    87   using mod_mult_self1 [of a 1 b] by simp
```
```    88
```
```    89 lemma mod_div_trivial [simp]:
```
```    90   "a mod b div b = 0"
```
```    91 proof (cases "b = 0")
```
```    92   assume "b = 0"
```
```    93   thus ?thesis by simp
```
```    94 next
```
```    95   assume "b \<noteq> 0"
```
```    96   hence "a div b + a mod b div b = (a mod b + a div b * b) div b"
```
```    97     by (rule div_mult_self1 [symmetric])
```
```    98   also have "\<dots> = a div b"
```
```    99     by (simp only: mod_div_mult_eq)
```
```   100   also have "\<dots> = a div b + 0"
```
```   101     by simp
```
```   102   finally show ?thesis
```
```   103     by (rule add_left_imp_eq)
```
```   104 qed
```
```   105
```
```   106 lemma mod_mod_trivial [simp]:
```
```   107   "a mod b mod b = a mod b"
```
```   108 proof -
```
```   109   have "a mod b mod b = (a mod b + a div b * b) mod b"
```
```   110     by (simp only: mod_mult_self1)
```
```   111   also have "\<dots> = a mod b"
```
```   112     by (simp only: mod_div_mult_eq)
```
```   113   finally show ?thesis .
```
```   114 qed
```
```   115
```
```   116 lemma mod_mod_cancel:
```
```   117   assumes "c dvd b"
```
```   118   shows "a mod b mod c = a mod c"
```
```   119 proof -
```
```   120   from \<open>c dvd b\<close> obtain k where "b = c * k"
```
```   121     by (rule dvdE)
```
```   122   have "a mod b mod c = a mod (c * k) mod c"
```
```   123     by (simp only: \<open>b = c * k\<close>)
```
```   124   also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c"
```
```   125     by (simp only: mod_mult_self1)
```
```   126   also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c"
```
```   127     by (simp only: ac_simps)
```
```   128   also have "\<dots> = a mod c"
```
```   129     by (simp only: div_mult_mod_eq)
```
```   130   finally show ?thesis .
```
```   131 qed
```
```   132
```
```   133 lemma div_mult_mult2 [simp]:
```
```   134   "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
```
```   135   by (drule div_mult_mult1) (simp add: mult.commute)
```
```   136
```
```   137 lemma div_mult_mult1_if [simp]:
```
```   138   "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"
```
```   139   by simp_all
```
```   140
```
```   141 lemma mod_mult_mult1:
```
```   142   "(c * a) mod (c * b) = c * (a mod b)"
```
```   143 proof (cases "c = 0")
```
```   144   case True then show ?thesis by simp
```
```   145 next
```
```   146   case False
```
```   147   from div_mult_mod_eq
```
```   148   have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .
```
```   149   with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)
```
```   150     = c * a + c * (a mod b)" by (simp add: algebra_simps)
```
```   151   with div_mult_mod_eq show ?thesis by simp
```
```   152 qed
```
```   153
```
```   154 lemma mod_mult_mult2:
```
```   155   "(a * c) mod (b * c) = (a mod b) * c"
```
```   156   using mod_mult_mult1 [of c a b] by (simp add: mult.commute)
```
```   157
```
```   158 lemma mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)"
```
```   159   by (fact mod_mult_mult2 [symmetric])
```
```   160
```
```   161 lemma mult_mod_right: "c * (a mod b) = (c * a) mod (c * b)"
```
```   162   by (fact mod_mult_mult1 [symmetric])
```
```   163
```
```   164 lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)"
```
```   165   unfolding dvd_def by (auto simp add: mod_mult_mult1)
```
```   166
```
```   167 named_theorems mod_simps
```
```   168
```
```   169 text \<open>Addition respects modular equivalence.\<close>
```
```   170
```
```   171 lemma mod_add_left_eq [mod_simps]:
```
```   172   "(a mod c + b) mod c = (a + b) mod c"
```
```   173 proof -
```
```   174   have "(a + b) mod c = (a div c * c + a mod c + b) mod c"
```
```   175     by (simp only: div_mult_mod_eq)
```
```   176   also have "\<dots> = (a mod c + b + a div c * c) mod c"
```
```   177     by (simp only: ac_simps)
```
```   178   also have "\<dots> = (a mod c + b) mod c"
```
```   179     by (rule mod_mult_self1)
```
```   180   finally show ?thesis
```
```   181     by (rule sym)
```
```   182 qed
```
```   183
```
```   184 lemma mod_add_right_eq [mod_simps]:
```
```   185   "(a + b mod c) mod c = (a + b) mod c"
```
```   186   using mod_add_left_eq [of b c a] by (simp add: ac_simps)
```
```   187
```
```   188 lemma mod_add_eq:
```
```   189   "(a mod c + b mod c) mod c = (a + b) mod c"
```
```   190   by (simp add: mod_add_left_eq mod_add_right_eq)
```
```   191
```
```   192 lemma mod_sum_eq [mod_simps]:
```
```   193   "(\<Sum>i\<in>A. f i mod a) mod a = sum f A mod a"
```
```   194 proof (induct A rule: infinite_finite_induct)
```
```   195   case (insert i A)
```
```   196   then have "(\<Sum>i\<in>insert i A. f i mod a) mod a
```
```   197     = (f i mod a + (\<Sum>i\<in>A. f i mod a)) mod a"
```
```   198     by simp
```
```   199   also have "\<dots> = (f i + (\<Sum>i\<in>A. f i mod a) mod a) mod a"
```
```   200     by (simp add: mod_simps)
```
```   201   also have "\<dots> = (f i + (\<Sum>i\<in>A. f i) mod a) mod a"
```
```   202     by (simp add: insert.hyps)
```
```   203   finally show ?case
```
```   204     by (simp add: insert.hyps mod_simps)
```
```   205 qed simp_all
```
```   206
```
```   207 lemma mod_add_cong:
```
```   208   assumes "a mod c = a' mod c"
```
```   209   assumes "b mod c = b' mod c"
```
```   210   shows "(a + b) mod c = (a' + b') mod c"
```
```   211 proof -
```
```   212   have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c"
```
```   213     unfolding assms ..
```
```   214   then show ?thesis
```
```   215     by (simp add: mod_add_eq)
```
```   216 qed
```
```   217
```
```   218 text \<open>Multiplication respects modular equivalence.\<close>
```
```   219
```
```   220 lemma mod_mult_left_eq [mod_simps]:
```
```   221   "((a mod c) * b) mod c = (a * b) mod c"
```
```   222 proof -
```
```   223   have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c"
```
```   224     by (simp only: div_mult_mod_eq)
```
```   225   also have "\<dots> = (a mod c * b + a div c * b * c) mod c"
```
```   226     by (simp only: algebra_simps)
```
```   227   also have "\<dots> = (a mod c * b) mod c"
```
```   228     by (rule mod_mult_self1)
```
```   229   finally show ?thesis
```
```   230     by (rule sym)
```
```   231 qed
```
```   232
```
```   233 lemma mod_mult_right_eq [mod_simps]:
```
```   234   "(a * (b mod c)) mod c = (a * b) mod c"
```
```   235   using mod_mult_left_eq [of b c a] by (simp add: ac_simps)
```
```   236
```
```   237 lemma mod_mult_eq:
```
```   238   "((a mod c) * (b mod c)) mod c = (a * b) mod c"
```
```   239   by (simp add: mod_mult_left_eq mod_mult_right_eq)
```
```   240
```
```   241 lemma mod_prod_eq [mod_simps]:
```
```   242   "(\<Prod>i\<in>A. f i mod a) mod a = prod f A mod a"
```
```   243 proof (induct A rule: infinite_finite_induct)
```
```   244   case (insert i A)
```
```   245   then have "(\<Prod>i\<in>insert i A. f i mod a) mod a
```
```   246     = (f i mod a * (\<Prod>i\<in>A. f i mod a)) mod a"
```
```   247     by simp
```
```   248   also have "\<dots> = (f i * ((\<Prod>i\<in>A. f i mod a) mod a)) mod a"
```
```   249     by (simp add: mod_simps)
```
```   250   also have "\<dots> = (f i * ((\<Prod>i\<in>A. f i) mod a)) mod a"
```
```   251     by (simp add: insert.hyps)
```
```   252   finally show ?case
```
```   253     by (simp add: insert.hyps mod_simps)
```
```   254 qed simp_all
```
```   255
```
```   256 lemma mod_mult_cong:
```
```   257   assumes "a mod c = a' mod c"
```
```   258   assumes "b mod c = b' mod c"
```
```   259   shows "(a * b) mod c = (a' * b') mod c"
```
```   260 proof -
```
```   261   have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c"
```
```   262     unfolding assms ..
```
```   263   then show ?thesis
```
```   264     by (simp add: mod_mult_eq)
```
```   265 qed
```
```   266
```
```   267 text \<open>Exponentiation respects modular equivalence.\<close>
```
```   268
```
```   269 lemma power_mod [mod_simps]:
```
```   270   "((a mod b) ^ n) mod b = (a ^ n) mod b"
```
```   271 proof (induct n)
```
```   272   case 0
```
```   273   then show ?case by simp
```
```   274 next
```
```   275   case (Suc n)
```
```   276   have "(a mod b) ^ Suc n mod b = (a mod b) * ((a mod b) ^ n mod b) mod b"
```
```   277     by (simp add: mod_mult_right_eq)
```
```   278   with Suc show ?case
```
```   279     by (simp add: mod_mult_left_eq mod_mult_right_eq)
```
```   280 qed
```
```   281
```
```   282 end
```
```   283
```
```   284 class ring_div = comm_ring_1 + semiring_div
```
```   285 begin
```
```   286
```
```   287 subclass idom_divide ..
```
```   288
```
```   289 lemma div_minus_minus [simp]: "(- a) div (- b) = a div b"
```
```   290   using div_mult_mult1 [of "- 1" a b] by simp
```
```   291
```
```   292 lemma mod_minus_minus [simp]: "(- a) mod (- b) = - (a mod b)"
```
```   293   using mod_mult_mult1 [of "- 1" a b] by simp
```
```   294
```
```   295 lemma div_minus_right: "a div (- b) = (- a) div b"
```
```   296   using div_minus_minus [of "- a" b] by simp
```
```   297
```
```   298 lemma mod_minus_right: "a mod (- b) = - ((- a) mod b)"
```
```   299   using mod_minus_minus [of "- a" b] by simp
```
```   300
```
```   301 lemma div_minus1_right [simp]: "a div (- 1) = - a"
```
```   302   using div_minus_right [of a 1] by simp
```
```   303
```
```   304 lemma mod_minus1_right [simp]: "a mod (- 1) = 0"
```
```   305   using mod_minus_right [of a 1] by simp
```
```   306
```
```   307 text \<open>Negation respects modular equivalence.\<close>
```
```   308
```
```   309 lemma mod_minus_eq [mod_simps]:
```
```   310   "(- (a mod b)) mod b = (- a) mod b"
```
```   311 proof -
```
```   312   have "(- a) mod b = (- (a div b * b + a mod b)) mod b"
```
```   313     by (simp only: div_mult_mod_eq)
```
```   314   also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b"
```
```   315     by (simp add: ac_simps)
```
```   316   also have "\<dots> = (- (a mod b)) mod b"
```
```   317     by (rule mod_mult_self1)
```
```   318   finally show ?thesis
```
```   319     by (rule sym)
```
```   320 qed
```
```   321
```
```   322 lemma mod_minus_cong:
```
```   323   assumes "a mod b = a' mod b"
```
```   324   shows "(- a) mod b = (- a') mod b"
```
```   325 proof -
```
```   326   have "(- (a mod b)) mod b = (- (a' mod b)) mod b"
```
```   327     unfolding assms ..
```
```   328   then show ?thesis
```
```   329     by (simp add: mod_minus_eq)
```
```   330 qed
```
```   331
```
```   332 text \<open>Subtraction respects modular equivalence.\<close>
```
```   333
```
```   334 lemma mod_diff_left_eq [mod_simps]:
```
```   335   "(a mod c - b) mod c = (a - b) mod c"
```
```   336   using mod_add_cong [of a c "a mod c" "- b" "- b"]
```
```   337   by simp
```
```   338
```
```   339 lemma mod_diff_right_eq [mod_simps]:
```
```   340   "(a - b mod c) mod c = (a - b) mod c"
```
```   341   using mod_add_cong [of a c a "- b" "- (b mod c)"] mod_minus_cong [of "b mod c" c b]
```
```   342   by simp
```
```   343
```
```   344 lemma mod_diff_eq:
```
```   345   "(a mod c - b mod c) mod c = (a - b) mod c"
```
```   346   using mod_add_cong [of a c "a mod c" "- b" "- (b mod c)"] mod_minus_cong [of "b mod c" c b]
```
```   347   by simp
```
```   348
```
```   349 lemma mod_diff_cong:
```
```   350   assumes "a mod c = a' mod c"
```
```   351   assumes "b mod c = b' mod c"
```
```   352   shows "(a - b) mod c = (a' - b') mod c"
```
```   353   using assms mod_add_cong [of a c a' "- b" "- b'"] mod_minus_cong [of b c "b'"]
```
```   354   by simp
```
```   355
```
```   356 lemma minus_mod_self2 [simp]:
```
```   357   "(a - b) mod b = a mod b"
```
```   358   using mod_diff_right_eq [of a b b]
```
```   359   by (simp add: mod_diff_right_eq)
```
```   360
```
```   361 lemma minus_mod_self1 [simp]:
```
```   362   "(b - a) mod b = - a mod b"
```
```   363   using mod_add_self2 [of "- a" b] by simp
```
```   364
```
```   365 end
```
```   366
```
```   367
```
```   368 subsection \<open>Euclidean (semi)rings with cancel rules\<close>
```
```   369
```
```   370 class euclidean_semiring_cancel = euclidean_semiring + semiring_div
```
```   371
```
```   372 class euclidean_ring_cancel = euclidean_ring + ring_div
```
```   373
```
```   374 context unique_euclidean_semiring
```
```   375 begin
```
```   376
```
```   377 subclass euclidean_semiring_cancel
```
```   378 proof
```
```   379   show "(a + c * b) div b = c + a div b" if "b \<noteq> 0" for a b c
```
```   380   proof (cases a b rule: divmod_cases)
```
```   381     case by0
```
```   382     with \<open>b \<noteq> 0\<close> show ?thesis
```
```   383       by simp
```
```   384   next
```
```   385     case (divides q)
```
```   386     then show ?thesis
```
```   387       by (simp add: ac_simps)
```
```   388   next
```
```   389     case (remainder q r)
```
```   390     then show ?thesis
```
```   391       by (auto intro: div_eqI simp add: algebra_simps)
```
```   392   qed
```
```   393 next
```
```   394   show"(c * a) div (c * b) = a div b" if "c \<noteq> 0" for a b c
```
```   395   proof (cases a b rule: divmod_cases)
```
```   396     case by0
```
```   397     then show ?thesis
```
```   398       by simp
```
```   399   next
```
```   400     case (divides q)
```
```   401     with \<open>c \<noteq> 0\<close> show ?thesis
```
```   402       by (simp add: mult.left_commute [of c])
```
```   403   next
```
```   404     case (remainder q r)
```
```   405     from \<open>b \<noteq> 0\<close> \<open>c \<noteq> 0\<close> have "b * c \<noteq> 0"
```
```   406       by simp
```
```   407     from remainder \<open>c \<noteq> 0\<close>
```
```   408     have "uniqueness_constraint (r * c) (b * c)"
```
```   409       and "euclidean_size (r * c) < euclidean_size (b * c)"
```
```   410       by (simp_all add: uniqueness_constraint_mono_mult uniqueness_constraint_mod size_mono_mult)
```
```   411     with remainder show ?thesis
```
```   412       by (auto intro!: div_eqI [of _ "c * (a mod b)"] simp add: algebra_simps)
```
```   413         (use \<open>b * c \<noteq> 0\<close> in simp)
```
```   414   qed
```
```   415 qed
```
```   416
```
```   417 end
```
```   418
```
```   419 context unique_euclidean_ring
```
```   420 begin
```
```   421
```
```   422 subclass euclidean_ring_cancel ..
```
```   423
```
```   424 end
```
```   425
```
```   426
```
```   427 subsection \<open>Parity\<close>
```
```   428
```
```   429 class semiring_div_parity = semiring_div + comm_semiring_1_cancel + numeral +
```
```   430   assumes parity: "a mod 2 = 0 \<or> a mod 2 = 1"
```
```   431   assumes one_mod_two_eq_one [simp]: "1 mod 2 = 1"
```
```   432   assumes zero_not_eq_two: "0 \<noteq> 2"
```
```   433 begin
```
```   434
```
```   435 lemma parity_cases [case_names even odd]:
```
```   436   assumes "a mod 2 = 0 \<Longrightarrow> P"
```
```   437   assumes "a mod 2 = 1 \<Longrightarrow> P"
```
```   438   shows P
```
```   439   using assms parity by blast
```
```   440
```
```   441 lemma one_div_two_eq_zero [simp]:
```
```   442   "1 div 2 = 0"
```
```   443 proof (cases "2 = 0")
```
```   444   case True then show ?thesis by simp
```
```   445 next
```
```   446   case False
```
```   447   from div_mult_mod_eq have "1 div 2 * 2 + 1 mod 2 = 1" .
```
```   448   with one_mod_two_eq_one have "1 div 2 * 2 + 1 = 1" by simp
```
```   449   then have "1 div 2 * 2 = 0" by (simp add: ac_simps add_left_imp_eq del: mult_eq_0_iff)
```
```   450   then have "1 div 2 = 0 \<or> 2 = 0" by simp
```
```   451   with False show ?thesis by auto
```
```   452 qed
```
```   453
```
```   454 lemma not_mod_2_eq_0_eq_1 [simp]:
```
```   455   "a mod 2 \<noteq> 0 \<longleftrightarrow> a mod 2 = 1"
```
```   456   by (cases a rule: parity_cases) simp_all
```
```   457
```
```   458 lemma not_mod_2_eq_1_eq_0 [simp]:
```
```   459   "a mod 2 \<noteq> 1 \<longleftrightarrow> a mod 2 = 0"
```
```   460   by (cases a rule: parity_cases) simp_all
```
```   461
```
```   462 subclass semiring_parity
```
```   463 proof (unfold_locales, unfold dvd_eq_mod_eq_0 not_mod_2_eq_0_eq_1)
```
```   464   show "1 mod 2 = 1"
```
```   465     by (fact one_mod_two_eq_one)
```
```   466 next
```
```   467   fix a b
```
```   468   assume "a mod 2 = 1"
```
```   469   moreover assume "b mod 2 = 1"
```
```   470   ultimately show "(a + b) mod 2 = 0"
```
```   471     using mod_add_eq [of a 2 b] by simp
```
```   472 next
```
```   473   fix a b
```
```   474   assume "(a * b) mod 2 = 0"
```
```   475   then have "(a mod 2) * (b mod 2) mod 2 = 0"
```
```   476     by (simp add: mod_mult_eq)
```
```   477   then have "(a mod 2) * (b mod 2) = 0"
```
```   478     by (cases "a mod 2 = 0") simp_all
```
```   479   then show "a mod 2 = 0 \<or> b mod 2 = 0"
```
```   480     by (rule divisors_zero)
```
```   481 next
```
```   482   fix a
```
```   483   assume "a mod 2 = 1"
```
```   484   then have "a = a div 2 * 2 + 1"
```
```   485     using div_mult_mod_eq [of a 2] by simp
```
```   486   then show "\<exists>b. a = b + 1" ..
```
```   487 qed
```
```   488
```
```   489 lemma even_iff_mod_2_eq_zero:
```
```   490   "even a \<longleftrightarrow> a mod 2 = 0"
```
```   491   by (fact dvd_eq_mod_eq_0)
```
```   492
```
```   493 lemma odd_iff_mod_2_eq_one:
```
```   494   "odd a \<longleftrightarrow> a mod 2 = 1"
```
```   495   by (auto simp add: even_iff_mod_2_eq_zero)
```
```   496
```
```   497 lemma even_succ_div_two [simp]:
```
```   498   "even a \<Longrightarrow> (a + 1) div 2 = a div 2"
```
```   499   by (cases "a = 0") (auto elim!: evenE dest: mult_not_zero)
```
```   500
```
```   501 lemma odd_succ_div_two [simp]:
```
```   502   "odd a \<Longrightarrow> (a + 1) div 2 = a div 2 + 1"
```
```   503   by (auto elim!: oddE simp add: zero_not_eq_two [symmetric] add.assoc)
```
```   504
```
```   505 lemma even_two_times_div_two:
```
```   506   "even a \<Longrightarrow> 2 * (a div 2) = a"
```
```   507   by (fact dvd_mult_div_cancel)
```
```   508
```
```   509 lemma odd_two_times_div_two_succ [simp]:
```
```   510   "odd a \<Longrightarrow> 2 * (a div 2) + 1 = a"
```
```   511   using mult_div_mod_eq [of 2 a] by (simp add: even_iff_mod_2_eq_zero)
```
```   512
```
```   513 end
```
```   514
```
```   515
```
```   516 subsection \<open>Numeral division with a pragmatic type class\<close>
```
```   517
```
```   518 text \<open>
```
```   519   The following type class contains everything necessary to formulate
```
```   520   a division algorithm in ring structures with numerals, restricted
```
```   521   to its positive segments.  This is its primary motiviation, and it
```
```   522   could surely be formulated using a more fine-grained, more algebraic
```
```   523   and less technical class hierarchy.
```
```   524 \<close>
```
```   525
```
```   526 class semiring_numeral_div = semiring_div + comm_semiring_1_cancel + linordered_semidom +
```
```   527   assumes div_less: "0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a div b = 0"
```
```   528     and mod_less: " 0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a mod b = a"
```
```   529     and div_positive: "0 < b \<Longrightarrow> b \<le> a \<Longrightarrow> a div b > 0"
```
```   530     and mod_less_eq_dividend: "0 \<le> a \<Longrightarrow> a mod b \<le> a"
```
```   531     and pos_mod_bound: "0 < b \<Longrightarrow> a mod b < b"
```
```   532     and pos_mod_sign: "0 < b \<Longrightarrow> 0 \<le> a mod b"
```
```   533     and mod_mult2_eq: "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
```
```   534     and div_mult2_eq: "0 \<le> c \<Longrightarrow> a div (b * c) = a div b div c"
```
```   535   assumes discrete: "a < b \<longleftrightarrow> a + 1 \<le> b"
```
```   536   fixes divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"
```
```   537     and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"
```
```   538   assumes divmod_def: "divmod m n = (numeral m div numeral n, numeral m mod numeral n)"
```
```   539     and divmod_step_def: "divmod_step l qr = (let (q, r) = qr
```
```   540     in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
```
```   541     else (2 * q, r))"
```
```   542     \<comment> \<open>These are conceptually definitions but force generated code
```
```   543     to be monomorphic wrt. particular instances of this class which
```
```   544     yields a significant speedup.\<close>
```
```   545
```
```   546 begin
```
```   547
```
```   548 subclass semiring_div_parity
```
```   549 proof
```
```   550   fix a
```
```   551   show "a mod 2 = 0 \<or> a mod 2 = 1"
```
```   552   proof (rule ccontr)
```
```   553     assume "\<not> (a mod 2 = 0 \<or> a mod 2 = 1)"
```
```   554     then have "a mod 2 \<noteq> 0" and "a mod 2 \<noteq> 1" by simp_all
```
```   555     have "0 < 2" by simp
```
```   556     with pos_mod_bound pos_mod_sign have "0 \<le> a mod 2" "a mod 2 < 2" by simp_all
```
```   557     with \<open>a mod 2 \<noteq> 0\<close> have "0 < a mod 2" by simp
```
```   558     with discrete have "1 \<le> a mod 2" by simp
```
```   559     with \<open>a mod 2 \<noteq> 1\<close> have "1 < a mod 2" by simp
```
```   560     with discrete have "2 \<le> a mod 2" by simp
```
```   561     with \<open>a mod 2 < 2\<close> show False by simp
```
```   562   qed
```
```   563 next
```
```   564   show "1 mod 2 = 1"
```
```   565     by (rule mod_less) simp_all
```
```   566 next
```
```   567   show "0 \<noteq> 2"
```
```   568     by simp
```
```   569 qed
```
```   570
```
```   571 lemma divmod_digit_1:
```
```   572   assumes "0 \<le> a" "0 < b" and "b \<le> a mod (2 * b)"
```
```   573   shows "2 * (a div (2 * b)) + 1 = a div b" (is "?P")
```
```   574     and "a mod (2 * b) - b = a mod b" (is "?Q")
```
```   575 proof -
```
```   576   from assms mod_less_eq_dividend [of a "2 * b"] have "b \<le> a"
```
```   577     by (auto intro: trans)
```
```   578   with \<open>0 < b\<close> have "0 < a div b" by (auto intro: div_positive)
```
```   579   then have [simp]: "1 \<le> a div b" by (simp add: discrete)
```
```   580   with \<open>0 < b\<close> have mod_less: "a mod b < b" by (simp add: pos_mod_bound)
```
```   581   define w where "w = a div b mod 2"
```
```   582   with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
```
```   583   have mod_w: "a mod (2 * b) = a mod b + b * w"
```
```   584     by (simp add: w_def mod_mult2_eq ac_simps)
```
```   585   from assms w_exhaust have "w = 1"
```
```   586     by (auto simp add: mod_w) (insert mod_less, auto)
```
```   587   with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp
```
```   588   have "2 * (a div (2 * b)) = a div b - w"
```
```   589     by (simp add: w_def div_mult2_eq minus_mod_eq_mult_div ac_simps)
```
```   590   with \<open>w = 1\<close> have div: "2 * (a div (2 * b)) = a div b - 1" by simp
```
```   591   then show ?P and ?Q
```
```   592     by (simp_all add: div mod add_implies_diff [symmetric])
```
```   593 qed
```
```   594
```
```   595 lemma divmod_digit_0:
```
```   596   assumes "0 < b" and "a mod (2 * b) < b"
```
```   597   shows "2 * (a div (2 * b)) = a div b" (is "?P")
```
```   598     and "a mod (2 * b) = a mod b" (is "?Q")
```
```   599 proof -
```
```   600   define w where "w = a div b mod 2"
```
```   601   with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
```
```   602   have mod_w: "a mod (2 * b) = a mod b + b * w"
```
```   603     by (simp add: w_def mod_mult2_eq ac_simps)
```
```   604   moreover have "b \<le> a mod b + b"
```
```   605   proof -
```
```   606     from \<open>0 < b\<close> pos_mod_sign have "0 \<le> a mod b" by blast
```
```   607     then have "0 + b \<le> a mod b + b" by (rule add_right_mono)
```
```   608     then show ?thesis by simp
```
```   609   qed
```
```   610   moreover note assms w_exhaust
```
```   611   ultimately have "w = 0" by auto
```
```   612   with mod_w have mod: "a mod (2 * b) = a mod b" by simp
```
```   613   have "2 * (a div (2 * b)) = a div b - w"
```
```   614     by (simp add: w_def div_mult2_eq minus_mod_eq_mult_div ac_simps)
```
```   615   with \<open>w = 0\<close> have div: "2 * (a div (2 * b)) = a div b" by simp
```
```   616   then show ?P and ?Q
```
```   617     by (simp_all add: div mod)
```
```   618 qed
```
```   619
```
```   620 lemma fst_divmod:
```
```   621   "fst (divmod m n) = numeral m div numeral n"
```
```   622   by (simp add: divmod_def)
```
```   623
```
```   624 lemma snd_divmod:
```
```   625   "snd (divmod m n) = numeral m mod numeral n"
```
```   626   by (simp add: divmod_def)
```
```   627
```
```   628 text \<open>
```
```   629   This is a formulation of one step (referring to one digit position)
```
```   630   in school-method division: compare the dividend at the current
```
```   631   digit position with the remainder from previous division steps
```
```   632   and evaluate accordingly.
```
```   633 \<close>
```
```   634
```
```   635 lemma divmod_step_eq [simp]:
```
```   636   "divmod_step l (q, r) = (if numeral l \<le> r
```
```   637     then (2 * q + 1, r - numeral l) else (2 * q, r))"
```
```   638   by (simp add: divmod_step_def)
```
```   639
```
```   640 text \<open>
```
```   641   This is a formulation of school-method division.
```
```   642   If the divisor is smaller than the dividend, terminate.
```
```   643   If not, shift the dividend to the right until termination
```
```   644   occurs and then reiterate single division steps in the
```
```   645   opposite direction.
```
```   646 \<close>
```
```   647
```
```   648 lemma divmod_divmod_step:
```
```   649   "divmod m n = (if m < n then (0, numeral m)
```
```   650     else divmod_step n (divmod m (Num.Bit0 n)))"
```
```   651 proof (cases "m < n")
```
```   652   case True then have "numeral m < numeral n" by simp
```
```   653   then show ?thesis
```
```   654     by (simp add: prod_eq_iff div_less mod_less fst_divmod snd_divmod)
```
```   655 next
```
```   656   case False
```
```   657   have "divmod m n =
```
```   658     divmod_step n (numeral m div (2 * numeral n),
```
```   659       numeral m mod (2 * numeral n))"
```
```   660   proof (cases "numeral n \<le> numeral m mod (2 * numeral n)")
```
```   661     case True
```
```   662     with divmod_step_eq
```
```   663       have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
```
```   664         (2 * (numeral m div (2 * numeral n)) + 1, numeral m mod (2 * numeral n) - numeral n)"
```
```   665         by simp
```
```   666     moreover from True divmod_digit_1 [of "numeral m" "numeral n"]
```
```   667       have "2 * (numeral m div (2 * numeral n)) + 1 = numeral m div numeral n"
```
```   668       and "numeral m mod (2 * numeral n) - numeral n = numeral m mod numeral n"
```
```   669       by simp_all
```
```   670     ultimately show ?thesis by (simp only: divmod_def)
```
```   671   next
```
```   672     case False then have *: "numeral m mod (2 * numeral n) < numeral n"
```
```   673       by (simp add: not_le)
```
```   674     with divmod_step_eq
```
```   675       have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
```
```   676         (2 * (numeral m div (2 * numeral n)), numeral m mod (2 * numeral n))"
```
```   677         by auto
```
```   678     moreover from * divmod_digit_0 [of "numeral n" "numeral m"]
```
```   679       have "2 * (numeral m div (2 * numeral n)) = numeral m div numeral n"
```
```   680       and "numeral m mod (2 * numeral n) = numeral m mod numeral n"
```
```   681       by (simp_all only: zero_less_numeral)
```
```   682     ultimately show ?thesis by (simp only: divmod_def)
```
```   683   qed
```
```   684   then have "divmod m n =
```
```   685     divmod_step n (numeral m div numeral (Num.Bit0 n),
```
```   686       numeral m mod numeral (Num.Bit0 n))"
```
```   687     by (simp only: numeral.simps distrib mult_1)
```
```   688   then have "divmod m n = divmod_step n (divmod m (Num.Bit0 n))"
```
```   689     by (simp add: divmod_def)
```
```   690   with False show ?thesis by simp
```
```   691 qed
```
```   692
```
```   693 text \<open>The division rewrite proper -- first, trivial results involving \<open>1\<close>\<close>
```
```   694
```
```   695 lemma divmod_trivial [simp]:
```
```   696   "divmod Num.One Num.One = (numeral Num.One, 0)"
```
```   697   "divmod (Num.Bit0 m) Num.One = (numeral (Num.Bit0 m), 0)"
```
```   698   "divmod (Num.Bit1 m) Num.One = (numeral (Num.Bit1 m), 0)"
```
```   699   "divmod num.One (num.Bit0 n) = (0, Numeral1)"
```
```   700   "divmod num.One (num.Bit1 n) = (0, Numeral1)"
```
```   701   using divmod_divmod_step [of "Num.One"] by (simp_all add: divmod_def)
```
```   702
```
```   703 text \<open>Division by an even number is a right-shift\<close>
```
```   704
```
```   705 lemma divmod_cancel [simp]:
```
```   706   "divmod (Num.Bit0 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r))" (is ?P)
```
```   707   "divmod (Num.Bit1 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r + 1))" (is ?Q)
```
```   708 proof -
```
```   709   have *: "\<And>q. numeral (Num.Bit0 q) = 2 * numeral q"
```
```   710     "\<And>q. numeral (Num.Bit1 q) = 2 * numeral q + 1"
```
```   711     by (simp_all only: numeral_mult numeral.simps distrib) simp_all
```
```   712   have "1 div 2 = 0" "1 mod 2 = 1" by (auto intro: div_less mod_less)
```
```   713   then show ?P and ?Q
```
```   714     by (simp_all add: fst_divmod snd_divmod prod_eq_iff split_def * [of m] * [of n] mod_mult_mult1
```
```   715       div_mult2_eq [of _ _ 2] mod_mult2_eq [of _ _ 2]
```
```   716       add.commute del: numeral_times_numeral)
```
```   717 qed
```
```   718
```
```   719 text \<open>The really hard work\<close>
```
```   720
```
```   721 lemma divmod_steps [simp]:
```
```   722   "divmod (num.Bit0 m) (num.Bit1 n) =
```
```   723       (if m \<le> n then (0, numeral (num.Bit0 m))
```
```   724        else divmod_step (num.Bit1 n)
```
```   725              (divmod (num.Bit0 m)
```
```   726                (num.Bit0 (num.Bit1 n))))"
```
```   727   "divmod (num.Bit1 m) (num.Bit1 n) =
```
```   728       (if m < n then (0, numeral (num.Bit1 m))
```
```   729        else divmod_step (num.Bit1 n)
```
```   730              (divmod (num.Bit1 m)
```
```   731                (num.Bit0 (num.Bit1 n))))"
```
```   732   by (simp_all add: divmod_divmod_step)
```
```   733
```
```   734 lemmas divmod_algorithm_code = divmod_step_eq divmod_trivial divmod_cancel divmod_steps
```
```   735
```
```   736 text \<open>Special case: divisibility\<close>
```
```   737
```
```   738 definition divides_aux :: "'a \<times> 'a \<Rightarrow> bool"
```
```   739 where
```
```   740   "divides_aux qr \<longleftrightarrow> snd qr = 0"
```
```   741
```
```   742 lemma divides_aux_eq [simp]:
```
```   743   "divides_aux (q, r) \<longleftrightarrow> r = 0"
```
```   744   by (simp add: divides_aux_def)
```
```   745
```
```   746 lemma dvd_numeral_simp [simp]:
```
```   747   "numeral m dvd numeral n \<longleftrightarrow> divides_aux (divmod n m)"
```
```   748   by (simp add: divmod_def mod_eq_0_iff_dvd)
```
```   749
```
```   750 text \<open>Generic computation of quotient and remainder\<close>
```
```   751
```
```   752 lemma numeral_div_numeral [simp]:
```
```   753   "numeral k div numeral l = fst (divmod k l)"
```
```   754   by (simp add: fst_divmod)
```
```   755
```
```   756 lemma numeral_mod_numeral [simp]:
```
```   757   "numeral k mod numeral l = snd (divmod k l)"
```
```   758   by (simp add: snd_divmod)
```
```   759
```
```   760 lemma one_div_numeral [simp]:
```
```   761   "1 div numeral n = fst (divmod num.One n)"
```
```   762   by (simp add: fst_divmod)
```
```   763
```
```   764 lemma one_mod_numeral [simp]:
```
```   765   "1 mod numeral n = snd (divmod num.One n)"
```
```   766   by (simp add: snd_divmod)
```
```   767
```
```   768 text \<open>Computing congruences modulo \<open>2 ^ q\<close>\<close>
```
```   769
```
```   770 lemma cong_exp_iff_simps:
```
```   771   "numeral n mod numeral Num.One = 0
```
```   772     \<longleftrightarrow> True"
```
```   773   "numeral (Num.Bit0 n) mod numeral (Num.Bit0 q) = 0
```
```   774     \<longleftrightarrow> numeral n mod numeral q = 0"
```
```   775   "numeral (Num.Bit1 n) mod numeral (Num.Bit0 q) = 0
```
```   776     \<longleftrightarrow> False"
```
```   777   "numeral m mod numeral Num.One = (numeral n mod numeral Num.One)
```
```   778     \<longleftrightarrow> True"
```
```   779   "numeral Num.One mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
```
```   780     \<longleftrightarrow> True"
```
```   781   "numeral Num.One mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
```
```   782     \<longleftrightarrow> False"
```
```   783   "numeral Num.One mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
```
```   784     \<longleftrightarrow> (numeral n mod numeral q) = 0"
```
```   785   "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
```
```   786     \<longleftrightarrow> False"
```
```   787   "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
```
```   788     \<longleftrightarrow> numeral m mod numeral q = (numeral n mod numeral q)"
```
```   789   "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
```
```   790     \<longleftrightarrow> False"
```
```   791   "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
```
```   792     \<longleftrightarrow> (numeral m mod numeral q) = 0"
```
```   793   "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
```
```   794     \<longleftrightarrow> False"
```
```   795   "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
```
```   796     \<longleftrightarrow> numeral m mod numeral q = (numeral n mod numeral q)"
```
```   797   by (auto simp add: case_prod_beta dest: arg_cong [of _ _ even])
```
```   798
```
```   799 end
```
```   800
```
```   801
```
```   802 subsection \<open>Division on @{typ nat}\<close>
```
```   803
```
```   804 context
```
```   805 begin
```
```   806
```
```   807 text \<open>
```
```   808   We define @{const divide} and @{const modulo} on @{typ nat} by means
```
```   809   of a characteristic relation with two input arguments
```
```   810   @{term "m::nat"}, @{term "n::nat"} and two output arguments
```
```   811   @{term "q::nat"}(uotient) and @{term "r::nat"}(emainder).
```
```   812 \<close>
```
```   813
```
```   814 inductive eucl_rel_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool"
```
```   815   where eucl_rel_nat_by0: "eucl_rel_nat m 0 (0, m)"
```
```   816   | eucl_rel_natI: "r < n \<Longrightarrow> m = q * n + r \<Longrightarrow> eucl_rel_nat m n (q, r)"
```
```   817
```
```   818 text \<open>@{const eucl_rel_nat} is total:\<close>
```
```   819
```
```   820 qualified lemma eucl_rel_nat_ex:
```
```   821   obtains q r where "eucl_rel_nat m n (q, r)"
```
```   822 proof (cases "n = 0")
```
```   823   case True
```
```   824   with that eucl_rel_nat_by0 show thesis
```
```   825     by blast
```
```   826 next
```
```   827   case False
```
```   828   have "\<exists>q r. m = q * n + r \<and> r < n"
```
```   829   proof (induct m)
```
```   830     case 0 with \<open>n \<noteq> 0\<close>
```
```   831     have "(0::nat) = 0 * n + 0 \<and> 0 < n" by simp
```
```   832     then show ?case by blast
```
```   833   next
```
```   834     case (Suc m) then obtain q' r'
```
```   835       where m: "m = q' * n + r'" and n: "r' < n" by auto
```
```   836     then show ?case proof (cases "Suc r' < n")
```
```   837       case True
```
```   838       from m n have "Suc m = q' * n + Suc r'" by simp
```
```   839       with True show ?thesis by blast
```
```   840     next
```
```   841       case False then have "n \<le> Suc r'"
```
```   842         by (simp add: not_less)
```
```   843       moreover from n have "Suc r' \<le> n"
```
```   844         by (simp add: Suc_le_eq)
```
```   845       ultimately have "n = Suc r'" by auto
```
```   846       with m have "Suc m = Suc q' * n + 0" by simp
```
```   847       with \<open>n \<noteq> 0\<close> show ?thesis by blast
```
```   848     qed
```
```   849   qed
```
```   850   with that \<open>n \<noteq> 0\<close> eucl_rel_natI show thesis
```
```   851     by blast
```
```   852 qed
```
```   853
```
```   854 text \<open>@{const eucl_rel_nat} is injective:\<close>
```
```   855
```
```   856 qualified lemma eucl_rel_nat_unique_div:
```
```   857   assumes "eucl_rel_nat m n (q, r)"
```
```   858     and "eucl_rel_nat m n (q', r')"
```
```   859   shows "q = q'"
```
```   860 proof (cases "n = 0")
```
```   861   case True with assms show ?thesis
```
```   862     by (auto elim: eucl_rel_nat.cases)
```
```   863 next
```
```   864   case False
```
```   865   have *: "q' \<le> q" if "q' * n + r' = q * n + r" "r < n" for q r q' r' :: nat
```
```   866   proof (rule ccontr)
```
```   867     assume "\<not> q' \<le> q"
```
```   868     then have "q < q'"
```
```   869       by (simp add: not_le)
```
```   870     with that show False
```
```   871       by (auto simp add: less_iff_Suc_add algebra_simps)
```
```   872   qed
```
```   873   from \<open>n \<noteq> 0\<close> assms show ?thesis
```
```   874     by (auto intro: order_antisym elim: eucl_rel_nat.cases dest: * sym split: if_splits)
```
```   875 qed
```
```   876
```
```   877 qualified lemma eucl_rel_nat_unique_mod:
```
```   878   assumes "eucl_rel_nat m n (q, r)"
```
```   879     and "eucl_rel_nat m n (q', r')"
```
```   880   shows "r = r'"
```
```   881 proof -
```
```   882   from assms have "q' = q"
```
```   883     by (auto intro: eucl_rel_nat_unique_div)
```
```   884   with assms show ?thesis
```
```   885     by (auto elim!: eucl_rel_nat.cases)
```
```   886 qed
```
```   887
```
```   888 text \<open>
```
```   889   We instantiate divisibility on the natural numbers by
```
```   890   means of @{const eucl_rel_nat}:
```
```   891 \<close>
```
```   892
```
```   893 qualified definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
```
```   894   "divmod_nat m n = (THE qr. eucl_rel_nat m n qr)"
```
```   895
```
```   896 qualified lemma eucl_rel_nat_divmod_nat:
```
```   897   "eucl_rel_nat m n (divmod_nat m n)"
```
```   898 proof -
```
```   899   from eucl_rel_nat_ex
```
```   900     obtain q r where rel: "eucl_rel_nat m n (q, r)" .
```
```   901   then show ?thesis
```
```   902     by (auto simp add: divmod_nat_def intro: theI
```
```   903       elim: eucl_rel_nat_unique_div eucl_rel_nat_unique_mod)
```
```   904 qed
```
```   905
```
```   906 qualified lemma divmod_nat_unique:
```
```   907   "divmod_nat m n = (q, r)" if "eucl_rel_nat m n (q, r)"
```
```   908   using that
```
```   909   by (auto simp add: divmod_nat_def intro: eucl_rel_nat_divmod_nat elim: eucl_rel_nat_unique_div eucl_rel_nat_unique_mod)
```
```   910
```
```   911 qualified lemma divmod_nat_zero:
```
```   912   "divmod_nat m 0 = (0, m)"
```
```   913   by (rule divmod_nat_unique) (fact eucl_rel_nat_by0)
```
```   914
```
```   915 qualified lemma divmod_nat_zero_left:
```
```   916   "divmod_nat 0 n = (0, 0)"
```
```   917   by (rule divmod_nat_unique)
```
```   918     (cases n, auto intro: eucl_rel_nat_by0 eucl_rel_natI)
```
```   919
```
```   920 qualified lemma divmod_nat_base:
```
```   921   "m < n \<Longrightarrow> divmod_nat m n = (0, m)"
```
```   922   by (rule divmod_nat_unique)
```
```   923     (cases n, auto intro: eucl_rel_nat_by0 eucl_rel_natI)
```
```   924
```
```   925 qualified lemma divmod_nat_step:
```
```   926   assumes "0 < n" and "n \<le> m"
```
```   927   shows "divmod_nat m n =
```
```   928     (Suc (fst (divmod_nat (m - n) n)), snd (divmod_nat (m - n) n))"
```
```   929 proof (rule divmod_nat_unique)
```
```   930   have "eucl_rel_nat (m - n) n (divmod_nat (m - n) n)"
```
```   931     by (fact eucl_rel_nat_divmod_nat)
```
```   932   then show "eucl_rel_nat m n (Suc
```
```   933     (fst (divmod_nat (m - n) n)), snd (divmod_nat (m - n) n))"
```
```   934     using assms
```
```   935       by (auto split: if_splits intro: eucl_rel_natI elim!: eucl_rel_nat.cases simp add: algebra_simps)
```
```   936 qed
```
```   937
```
```   938 end
```
```   939
```
```   940 instantiation nat :: "{semidom_modulo, normalization_semidom}"
```
```   941 begin
```
```   942
```
```   943 definition normalize_nat :: "nat \<Rightarrow> nat"
```
```   944   where [simp]: "normalize = (id :: nat \<Rightarrow> nat)"
```
```   945
```
```   946 definition unit_factor_nat :: "nat \<Rightarrow> nat"
```
```   947   where "unit_factor n = (if n = 0 then 0 else 1 :: nat)"
```
```   948
```
```   949 lemma unit_factor_simps [simp]:
```
```   950   "unit_factor 0 = (0::nat)"
```
```   951   "unit_factor (Suc n) = 1"
```
```   952   by (simp_all add: unit_factor_nat_def)
```
```   953
```
```   954 definition divide_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
```
```   955   where div_nat_def: "m div n = fst (Divides.divmod_nat m n)"
```
```   956
```
```   957 definition modulo_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
```
```   958   where mod_nat_def: "m mod n = snd (Divides.divmod_nat m n)"
```
```   959
```
```   960 lemma fst_divmod_nat [simp]:
```
```   961   "fst (Divides.divmod_nat m n) = m div n"
```
```   962   by (simp add: div_nat_def)
```
```   963
```
```   964 lemma snd_divmod_nat [simp]:
```
```   965   "snd (Divides.divmod_nat m n) = m mod n"
```
```   966   by (simp add: mod_nat_def)
```
```   967
```
```   968 lemma divmod_nat_div_mod:
```
```   969   "Divides.divmod_nat m n = (m div n, m mod n)"
```
```   970   by (simp add: prod_eq_iff)
```
```   971
```
```   972 lemma div_nat_unique:
```
```   973   assumes "eucl_rel_nat m n (q, r)"
```
```   974   shows "m div n = q"
```
```   975   using assms
```
```   976   by (auto dest!: Divides.divmod_nat_unique simp add: prod_eq_iff)
```
```   977
```
```   978 lemma mod_nat_unique:
```
```   979   assumes "eucl_rel_nat m n (q, r)"
```
```   980   shows "m mod n = r"
```
```   981   using assms
```
```   982   by (auto dest!: Divides.divmod_nat_unique simp add: prod_eq_iff)
```
```   983
```
```   984 lemma eucl_rel_nat: "eucl_rel_nat m n (m div n, m mod n)"
```
```   985   using Divides.eucl_rel_nat_divmod_nat
```
```   986   by (simp add: divmod_nat_div_mod)
```
```   987
```
```   988 text \<open>The ''recursion'' equations for @{const divide} and @{const modulo}\<close>
```
```   989
```
```   990 lemma div_less [simp]:
```
```   991   fixes m n :: nat
```
```   992   assumes "m < n"
```
```   993   shows "m div n = 0"
```
```   994   using assms Divides.divmod_nat_base by (simp add: prod_eq_iff)
```
```   995
```
```   996 lemma le_div_geq:
```
```   997   fixes m n :: nat
```
```   998   assumes "0 < n" and "n \<le> m"
```
```   999   shows "m div n = Suc ((m - n) div n)"
```
```  1000   using assms Divides.divmod_nat_step by (simp add: prod_eq_iff)
```
```  1001
```
```  1002 lemma mod_less [simp]:
```
```  1003   fixes m n :: nat
```
```  1004   assumes "m < n"
```
```  1005   shows "m mod n = m"
```
```  1006   using assms Divides.divmod_nat_base by (simp add: prod_eq_iff)
```
```  1007
```
```  1008 lemma le_mod_geq:
```
```  1009   fixes m n :: nat
```
```  1010   assumes "n \<le> m"
```
```  1011   shows "m mod n = (m - n) mod n"
```
```  1012   using assms Divides.divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)
```
```  1013
```
```  1014 lemma mod_less_divisor [simp]:
```
```  1015   fixes m n :: nat
```
```  1016   assumes "n > 0"
```
```  1017   shows "m mod n < n"
```
```  1018   using assms eucl_rel_nat [of m n]
```
```  1019     by (auto elim: eucl_rel_nat.cases)
```
```  1020
```
```  1021 lemma mod_le_divisor [simp]:
```
```  1022   fixes m n :: nat
```
```  1023   assumes "n > 0"
```
```  1024   shows "m mod n \<le> n"
```
```  1025   using assms eucl_rel_nat [of m n]
```
```  1026     by (auto elim: eucl_rel_nat.cases)
```
```  1027
```
```  1028 instance proof
```
```  1029   fix m n :: nat
```
```  1030   show "m div n * n + m mod n = m"
```
```  1031     using eucl_rel_nat [of m n]
```
```  1032     by (auto elim: eucl_rel_nat.cases)
```
```  1033 next
```
```  1034   fix n :: nat show "n div 0 = 0"
```
```  1035     by (simp add: div_nat_def Divides.divmod_nat_zero)
```
```  1036 next
```
```  1037   fix m n :: nat
```
```  1038   assume "n \<noteq> 0"
```
```  1039   then show "m * n div n = m"
```
```  1040     by (auto intro!: eucl_rel_natI div_nat_unique [of _ _ _ 0])
```
```  1041 qed (simp_all add: unit_factor_nat_def)
```
```  1042
```
```  1043 end
```
```  1044
```
```  1045 instance nat :: semiring_div
```
```  1046 proof
```
```  1047   fix m n q :: nat
```
```  1048   assume "n \<noteq> 0"
```
```  1049   then show "(q + m * n) div n = m + q div n"
```
```  1050     by (induct m) (simp_all add: le_div_geq)
```
```  1051 next
```
```  1052   fix m n q :: nat
```
```  1053   assume "m \<noteq> 0"
```
```  1054   show "(m * n) div (m * q) = n div q"
```
```  1055   proof (cases "q = 0")
```
```  1056     case True
```
```  1057     then show ?thesis
```
```  1058       by simp
```
```  1059   next
```
```  1060     case False
```
```  1061     show ?thesis
```
```  1062     proof (rule div_nat_unique [of _ _ _ "m * (n mod q)"])
```
```  1063       show "eucl_rel_nat (m * n) (m * q) (n div q, m * (n mod q))"
```
```  1064         by (rule eucl_rel_natI)
```
```  1065           (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>)
```
```  1066     qed
```
```  1067   qed
```
```  1068 qed
```
```  1069
```
```  1070 lemma div_by_Suc_0 [simp]:
```
```  1071   "m div Suc 0 = m"
```
```  1072   using div_by_1 [of m] by simp
```
```  1073
```
```  1074 lemma mod_by_Suc_0 [simp]:
```
```  1075   "m mod Suc 0 = 0"
```
```  1076   using mod_by_1 [of m] by simp
```
```  1077
```
```  1078 lemma mod_greater_zero_iff_not_dvd:
```
```  1079   fixes m n :: nat
```
```  1080   shows "m mod n > 0 \<longleftrightarrow> \<not> n dvd m"
```
```  1081   by (simp add: dvd_eq_mod_eq_0)
```
```  1082
```
```  1083 instantiation nat :: unique_euclidean_semiring
```
```  1084 begin
```
```  1085
```
```  1086 definition [simp]:
```
```  1087   "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"
```
```  1088
```
```  1089 definition [simp]:
```
```  1090   "uniqueness_constraint_nat = (top :: nat \<Rightarrow> nat \<Rightarrow> bool)"
```
```  1091
```
```  1092 instance
```
```  1093   by standard (use mult_le_mono2 [of 1] in \<open>simp_all add: unit_factor_nat_def mod_greater_zero_iff_not_dvd\<close>)
```
```  1094
```
```  1095 end
```
```  1096
```
```  1097 text \<open>Simproc for cancelling @{const divide} and @{const modulo}\<close>
```
```  1098
```
```  1099 lemma (in semiring_modulo) cancel_div_mod_rules:
```
```  1100   "((a div b) * b + a mod b) + c = a + c"
```
```  1101   "(b * (a div b) + a mod b) + c = a + c"
```
```  1102   by (simp_all add: div_mult_mod_eq mult_div_mod_eq)
```
```  1103
```
```  1104 ML_file "~~/src/Provers/Arith/cancel_div_mod.ML"
```
```  1105
```
```  1106 ML \<open>
```
```  1107 structure Cancel_Div_Mod_Nat = Cancel_Div_Mod
```
```  1108 (
```
```  1109   val div_name = @{const_name divide};
```
```  1110   val mod_name = @{const_name modulo};
```
```  1111   val mk_binop = HOLogic.mk_binop;
```
```  1112   val mk_plus = HOLogic.mk_binop @{const_name Groups.plus};
```
```  1113   val dest_plus = HOLogic.dest_bin @{const_name Groups.plus} HOLogic.natT;
```
```  1114   fun mk_sum [] = HOLogic.zero
```
```  1115     | mk_sum [t] = t
```
```  1116     | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
```
```  1117   fun dest_sum tm =
```
```  1118     if HOLogic.is_zero tm then []
```
```  1119     else
```
```  1120       (case try HOLogic.dest_Suc tm of
```
```  1121         SOME t => HOLogic.Suc_zero :: dest_sum t
```
```  1122       | NONE =>
```
```  1123           (case try dest_plus tm of
```
```  1124             SOME (t, u) => dest_sum t @ dest_sum u
```
```  1125           | NONE => [tm]));
```
```  1126
```
```  1127   val div_mod_eqs = map mk_meta_eq @{thms cancel_div_mod_rules};
```
```  1128
```
```  1129   val prove_eq_sums = Arith_Data.prove_conv2 all_tac
```
```  1130     (Arith_Data.simp_all_tac @{thms add_0_left add_0_right ac_simps})
```
```  1131 )
```
```  1132 \<close>
```
```  1133
```
```  1134 simproc_setup cancel_div_mod_nat ("(m::nat) + n") =
```
```  1135   \<open>K Cancel_Div_Mod_Nat.proc\<close>
```
```  1136
```
```  1137 lemma divmod_nat_if [code]:
```
```  1138   "Divides.divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
```
```  1139     let (q, r) = Divides.divmod_nat (m - n) n in (Suc q, r))"
```
```  1140   by (simp add: prod_eq_iff case_prod_beta not_less le_div_geq le_mod_geq)
```
```  1141
```
```  1142 lemma mod_Suc_eq [mod_simps]:
```
```  1143   "Suc (m mod n) mod n = Suc m mod n"
```
```  1144 proof -
```
```  1145   have "(m mod n + 1) mod n = (m + 1) mod n"
```
```  1146     by (simp only: mod_simps)
```
```  1147   then show ?thesis
```
```  1148     by simp
```
```  1149 qed
```
```  1150
```
```  1151 lemma mod_Suc_Suc_eq [mod_simps]:
```
```  1152   "Suc (Suc (m mod n)) mod n = Suc (Suc m) mod n"
```
```  1153 proof -
```
```  1154   have "(m mod n + 2) mod n = (m + 2) mod n"
```
```  1155     by (simp only: mod_simps)
```
```  1156   then show ?thesis
```
```  1157     by simp
```
```  1158 qed
```
```  1159
```
```  1160
```
```  1161 subsubsection \<open>Quotient\<close>
```
```  1162
```
```  1163 lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
```
```  1164 by (simp add: le_div_geq linorder_not_less)
```
```  1165
```
```  1166 lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
```
```  1167 by (simp add: div_geq)
```
```  1168
```
```  1169 lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
```
```  1170 by simp
```
```  1171
```
```  1172 lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
```
```  1173 by simp
```
```  1174
```
```  1175 lemma div_positive:
```
```  1176   fixes m n :: nat
```
```  1177   assumes "n > 0"
```
```  1178   assumes "m \<ge> n"
```
```  1179   shows "m div n > 0"
```
```  1180 proof -
```
```  1181   from \<open>m \<ge> n\<close> obtain q where "m = n + q"
```
```  1182     by (auto simp add: le_iff_add)
```
```  1183   with \<open>n > 0\<close> show ?thesis by (simp add: div_add_self1)
```
```  1184 qed
```
```  1185
```
```  1186 lemma div_eq_0_iff: "(a div b::nat) = 0 \<longleftrightarrow> a < b \<or> b = 0"
```
```  1187   by auto (metis div_positive less_numeral_extra(3) not_less)
```
```  1188
```
```  1189
```
```  1190 subsubsection \<open>Remainder\<close>
```
```  1191
```
```  1192 lemma mod_Suc_le_divisor [simp]:
```
```  1193   "m mod Suc n \<le> n"
```
```  1194   using mod_less_divisor [of "Suc n" m] by arith
```
```  1195
```
```  1196 lemma mod_less_eq_dividend [simp]:
```
```  1197   fixes m n :: nat
```
```  1198   shows "m mod n \<le> m"
```
```  1199 proof (rule add_leD2)
```
```  1200   from div_mult_mod_eq have "m div n * n + m mod n = m" .
```
```  1201   then show "m div n * n + m mod n \<le> m" by auto
```
```  1202 qed
```
```  1203
```
```  1204 lemma mod_geq: "\<not> m < (n::nat) \<Longrightarrow> m mod n = (m - n) mod n"
```
```  1205 by (simp add: le_mod_geq linorder_not_less)
```
```  1206
```
```  1207 lemma mod_if: "m mod (n::nat) = (if m < n then m else (m - n) mod n)"
```
```  1208 by (simp add: le_mod_geq)
```
```  1209
```
```  1210
```
```  1211 subsubsection \<open>Quotient and Remainder\<close>
```
```  1212
```
```  1213 lemma div_mult1_eq:
```
```  1214   "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"
```
```  1215   by (cases "c = 0")
```
```  1216      (auto simp add: algebra_simps distrib_left [symmetric]
```
```  1217      intro!: div_nat_unique [of _ _ _ "(a * (b mod c)) mod c"] eucl_rel_natI)
```
```  1218
```
```  1219 lemma eucl_rel_nat_add1_eq:
```
```  1220   "eucl_rel_nat a c (aq, ar) \<Longrightarrow> eucl_rel_nat b c (bq, br)
```
```  1221    \<Longrightarrow> eucl_rel_nat (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"
```
```  1222   by (auto simp add: split_ifs algebra_simps elim!: eucl_rel_nat.cases intro: eucl_rel_nat_by0 eucl_rel_natI)
```
```  1223
```
```  1224 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
```
```  1225 lemma div_add1_eq:
```
```  1226   "(a + b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
```
```  1227 by (blast intro: eucl_rel_nat_add1_eq [THEN div_nat_unique] eucl_rel_nat)
```
```  1228
```
```  1229 lemma eucl_rel_nat_mult2_eq:
```
```  1230   assumes "eucl_rel_nat a b (q, r)"
```
```  1231   shows "eucl_rel_nat a (b * c) (q div c, b *(q mod c) + r)"
```
```  1232 proof (cases "c = 0")
```
```  1233   case True
```
```  1234   with assms show ?thesis
```
```  1235     by (auto intro: eucl_rel_nat_by0 elim!: eucl_rel_nat.cases simp add: ac_simps)
```
```  1236 next
```
```  1237   case False
```
```  1238   { assume "r < b"
```
```  1239     with False have "b * (q mod c) + r < b * c"
```
```  1240       apply (cut_tac m = q and n = c in mod_less_divisor)
```
```  1241       apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
```
```  1242       apply (erule_tac P = "%x. lhs < rhs x" for lhs rhs in ssubst)
```
```  1243       apply (simp add: add_mult_distrib2)
```
```  1244       done
```
```  1245     then have "r + b * (q mod c) < b * c"
```
```  1246       by (simp add: ac_simps)
```
```  1247   } with assms False show ?thesis
```
```  1248     by (auto simp add: algebra_simps add_mult_distrib2 [symmetric] elim!: eucl_rel_nat.cases intro: eucl_rel_nat.intros)
```
```  1249 qed
```
```  1250
```
```  1251 lemma div_mult2_eq: "a div (b * c) = (a div b) div (c::nat)"
```
```  1252 by (force simp add: eucl_rel_nat [THEN eucl_rel_nat_mult2_eq, THEN div_nat_unique])
```
```  1253
```
```  1254 lemma mod_mult2_eq: "a mod (b * c) = b * (a div b mod c) + a mod (b::nat)"
```
```  1255 by (auto simp add: mult.commute eucl_rel_nat [THEN eucl_rel_nat_mult2_eq, THEN mod_nat_unique])
```
```  1256
```
```  1257 instantiation nat :: semiring_numeral_div
```
```  1258 begin
```
```  1259
```
```  1260 definition divmod_nat :: "num \<Rightarrow> num \<Rightarrow> nat \<times> nat"
```
```  1261 where
```
```  1262   divmod'_nat_def: "divmod_nat m n = (numeral m div numeral n, numeral m mod numeral n)"
```
```  1263
```
```  1264 definition divmod_step_nat :: "num \<Rightarrow> nat \<times> nat \<Rightarrow> nat \<times> nat"
```
```  1265 where
```
```  1266   "divmod_step_nat l qr = (let (q, r) = qr
```
```  1267     in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
```
```  1268     else (2 * q, r))"
```
```  1269
```
```  1270 instance
```
```  1271   by standard (auto intro: div_positive simp add: divmod'_nat_def divmod_step_nat_def mod_mult2_eq div_mult2_eq)
```
```  1272
```
```  1273 end
```
```  1274
```
```  1275 declare divmod_algorithm_code [where ?'a = nat, code]
```
```  1276
```
```  1277
```
```  1278 subsubsection \<open>Further Facts about Quotient and Remainder\<close>
```
```  1279
```
```  1280 lemma div_le_mono:
```
```  1281   fixes m n k :: nat
```
```  1282   assumes "m \<le> n"
```
```  1283   shows "m div k \<le> n div k"
```
```  1284 proof -
```
```  1285   from assms obtain q where "n = m + q"
```
```  1286     by (auto simp add: le_iff_add)
```
```  1287   then show ?thesis
```
```  1288     by (simp add: div_add1_eq [of m q k])
```
```  1289 qed
```
```  1290
```
```  1291 (* Antimonotonicity of div in second argument *)
```
```  1292 lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
```
```  1293 apply (subgoal_tac "0<n")
```
```  1294  prefer 2 apply simp
```
```  1295 apply (induct_tac k rule: nat_less_induct)
```
```  1296 apply (rename_tac "k")
```
```  1297 apply (case_tac "k<n", simp)
```
```  1298 apply (subgoal_tac "~ (k<m) ")
```
```  1299  prefer 2 apply simp
```
```  1300 apply (simp add: div_geq)
```
```  1301 apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
```
```  1302  prefer 2
```
```  1303  apply (blast intro: div_le_mono diff_le_mono2)
```
```  1304 apply (rule le_trans, simp)
```
```  1305 apply (simp)
```
```  1306 done
```
```  1307
```
```  1308 lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
```
```  1309 apply (case_tac "n=0", simp)
```
```  1310 apply (subgoal_tac "m div n \<le> m div 1", simp)
```
```  1311 apply (rule div_le_mono2)
```
```  1312 apply (simp_all (no_asm_simp))
```
```  1313 done
```
```  1314
```
```  1315 (* Similar for "less than" *)
```
```  1316 lemma div_less_dividend [simp]:
```
```  1317   "\<lbrakk>(1::nat) < n; 0 < m\<rbrakk> \<Longrightarrow> m div n < m"
```
```  1318 apply (induct m rule: nat_less_induct)
```
```  1319 apply (rename_tac "m")
```
```  1320 apply (case_tac "m<n", simp)
```
```  1321 apply (subgoal_tac "0<n")
```
```  1322  prefer 2 apply simp
```
```  1323 apply (simp add: div_geq)
```
```  1324 apply (case_tac "n<m")
```
```  1325  apply (subgoal_tac "(m-n) div n < (m-n) ")
```
```  1326   apply (rule impI less_trans_Suc)+
```
```  1327 apply assumption
```
```  1328   apply (simp_all)
```
```  1329 done
```
```  1330
```
```  1331 text\<open>A fact for the mutilated chess board\<close>
```
```  1332 lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
```
```  1333 apply (case_tac "n=0", simp)
```
```  1334 apply (induct "m" rule: nat_less_induct)
```
```  1335 apply (case_tac "Suc (na) <n")
```
```  1336 (* case Suc(na) < n *)
```
```  1337 apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
```
```  1338 (* case n \<le> Suc(na) *)
```
```  1339 apply (simp add: linorder_not_less le_Suc_eq mod_geq)
```
```  1340 apply (auto simp add: Suc_diff_le le_mod_geq)
```
```  1341 done
```
```  1342
```
```  1343 lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
```
```  1344 by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
```
```  1345
```
```  1346 lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
```
```  1347
```
```  1348 (*Loses information, namely we also have r<d provided d is nonzero*)
```
```  1349 lemma mod_eqD:
```
```  1350   fixes m d r q :: nat
```
```  1351   assumes "m mod d = r"
```
```  1352   shows "\<exists>q. m = r + q * d"
```
```  1353 proof -
```
```  1354   from div_mult_mod_eq obtain q where "q * d + m mod d = m" by blast
```
```  1355   with assms have "m = r + q * d" by simp
```
```  1356   then show ?thesis ..
```
```  1357 qed
```
```  1358
```
```  1359 lemma split_div:
```
```  1360  "P(n div k :: nat) =
```
```  1361  ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
```
```  1362  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
```
```  1363 proof
```
```  1364   assume P: ?P
```
```  1365   show ?Q
```
```  1366   proof (cases)
```
```  1367     assume "k = 0"
```
```  1368     with P show ?Q by simp
```
```  1369   next
```
```  1370     assume not0: "k \<noteq> 0"
```
```  1371     thus ?Q
```
```  1372     proof (simp, intro allI impI)
```
```  1373       fix i j
```
```  1374       assume n: "n = k*i + j" and j: "j < k"
```
```  1375       show "P i"
```
```  1376       proof (cases)
```
```  1377         assume "i = 0"
```
```  1378         with n j P show "P i" by simp
```
```  1379       next
```
```  1380         assume "i \<noteq> 0"
```
```  1381         with not0 n j P show "P i" by(simp add:ac_simps)
```
```  1382       qed
```
```  1383     qed
```
```  1384   qed
```
```  1385 next
```
```  1386   assume Q: ?Q
```
```  1387   show ?P
```
```  1388   proof (cases)
```
```  1389     assume "k = 0"
```
```  1390     with Q show ?P by simp
```
```  1391   next
```
```  1392     assume not0: "k \<noteq> 0"
```
```  1393     with Q have R: ?R by simp
```
```  1394     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
```
```  1395     show ?P by simp
```
```  1396   qed
```
```  1397 qed
```
```  1398
```
```  1399 lemma split_div_lemma:
```
```  1400   assumes "0 < n"
```
```  1401   shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m::nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1402 proof
```
```  1403   assume ?rhs
```
```  1404   with minus_mod_eq_mult_div [symmetric] have nq: "n * q = m - (m mod n)" by simp
```
```  1405   then have A: "n * q \<le> m" by simp
```
```  1406   have "n - (m mod n) > 0" using mod_less_divisor assms by auto
```
```  1407   then have "m < m + (n - (m mod n))" by simp
```
```  1408   then have "m < n + (m - (m mod n))" by simp
```
```  1409   with nq have "m < n + n * q" by simp
```
```  1410   then have B: "m < n * Suc q" by simp
```
```  1411   from A B show ?lhs ..
```
```  1412 next
```
```  1413   assume P: ?lhs
```
```  1414   then have "eucl_rel_nat m n (q, m - n * q)"
```
```  1415     by (auto intro: eucl_rel_natI simp add: ac_simps)
```
```  1416   then have "m div n = q"
```
```  1417     by (rule div_nat_unique)
```
```  1418   then show ?rhs by simp
```
```  1419 qed
```
```  1420
```
```  1421 theorem split_div':
```
```  1422   "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
```
```  1423    (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
```
```  1424   apply (cases "0 < n")
```
```  1425   apply (simp only: add: split_div_lemma)
```
```  1426   apply simp_all
```
```  1427   done
```
```  1428
```
```  1429 lemma split_mod:
```
```  1430  "P(n mod k :: nat) =
```
```  1431  ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
```
```  1432  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
```
```  1433 proof
```
```  1434   assume P: ?P
```
```  1435   show ?Q
```
```  1436   proof (cases)
```
```  1437     assume "k = 0"
```
```  1438     with P show ?Q by simp
```
```  1439   next
```
```  1440     assume not0: "k \<noteq> 0"
```
```  1441     thus ?Q
```
```  1442     proof (simp, intro allI impI)
```
```  1443       fix i j
```
```  1444       assume "n = k*i + j" "j < k"
```
```  1445       thus "P j" using not0 P by (simp add: ac_simps)
```
```  1446     qed
```
```  1447   qed
```
```  1448 next
```
```  1449   assume Q: ?Q
```
```  1450   show ?P
```
```  1451   proof (cases)
```
```  1452     assume "k = 0"
```
```  1453     with Q show ?P by simp
```
```  1454   next
```
```  1455     assume not0: "k \<noteq> 0"
```
```  1456     with Q have R: ?R by simp
```
```  1457     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
```
```  1458     show ?P by simp
```
```  1459   qed
```
```  1460 qed
```
```  1461
```
```  1462 lemma div_eq_dividend_iff: "a \<noteq> 0 \<Longrightarrow> (a :: nat) div b = a \<longleftrightarrow> b = 1"
```
```  1463   apply rule
```
```  1464   apply (cases "b = 0")
```
```  1465   apply simp_all
```
```  1466   apply (metis (full_types) One_nat_def Suc_lessI div_less_dividend less_not_refl3)
```
```  1467   done
```
```  1468
```
```  1469 lemma (in field_char_0) of_nat_div:
```
```  1470   "of_nat (m div n) = ((of_nat m - of_nat (m mod n)) / of_nat n)"
```
```  1471 proof -
```
```  1472   have "of_nat (m div n) = ((of_nat (m div n * n + m mod n) - of_nat (m mod n)) / of_nat n :: 'a)"
```
```  1473     unfolding of_nat_add by (cases "n = 0") simp_all
```
```  1474   then show ?thesis
```
```  1475     by simp
```
```  1476 qed
```
```  1477
```
```  1478
```
```  1479 subsubsection \<open>An ``induction'' law for modulus arithmetic.\<close>
```
```  1480
```
```  1481 lemma mod_induct_0:
```
```  1482   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
```
```  1483   and base: "P i" and i: "i<p"
```
```  1484   shows "P 0"
```
```  1485 proof (rule ccontr)
```
```  1486   assume contra: "\<not>(P 0)"
```
```  1487   from i have p: "0<p" by simp
```
```  1488   have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
```
```  1489   proof
```
```  1490     fix k
```
```  1491     show "?A k"
```
```  1492     proof (induct k)
```
```  1493       show "?A 0" by simp  \<comment> "by contradiction"
```
```  1494     next
```
```  1495       fix n
```
```  1496       assume ih: "?A n"
```
```  1497       show "?A (Suc n)"
```
```  1498       proof (clarsimp)
```
```  1499         assume y: "P (p - Suc n)"
```
```  1500         have n: "Suc n < p"
```
```  1501         proof (rule ccontr)
```
```  1502           assume "\<not>(Suc n < p)"
```
```  1503           hence "p - Suc n = 0"
```
```  1504             by simp
```
```  1505           with y contra show "False"
```
```  1506             by simp
```
```  1507         qed
```
```  1508         hence n2: "Suc (p - Suc n) = p-n" by arith
```
```  1509         from p have "p - Suc n < p" by arith
```
```  1510         with y step have z: "P ((Suc (p - Suc n)) mod p)"
```
```  1511           by blast
```
```  1512         show "False"
```
```  1513         proof (cases "n=0")
```
```  1514           case True
```
```  1515           with z n2 contra show ?thesis by simp
```
```  1516         next
```
```  1517           case False
```
```  1518           with p have "p-n < p" by arith
```
```  1519           with z n2 False ih show ?thesis by simp
```
```  1520         qed
```
```  1521       qed
```
```  1522     qed
```
```  1523   qed
```
```  1524   moreover
```
```  1525   from i obtain k where "0<k \<and> i+k=p"
```
```  1526     by (blast dest: less_imp_add_positive)
```
```  1527   hence "0<k \<and> i=p-k" by auto
```
```  1528   moreover
```
```  1529   note base
```
```  1530   ultimately
```
```  1531   show "False" by blast
```
```  1532 qed
```
```  1533
```
```  1534 lemma mod_induct:
```
```  1535   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
```
```  1536   and base: "P i" and i: "i<p" and j: "j<p"
```
```  1537   shows "P j"
```
```  1538 proof -
```
```  1539   have "\<forall>j<p. P j"
```
```  1540   proof
```
```  1541     fix j
```
```  1542     show "j<p \<longrightarrow> P j" (is "?A j")
```
```  1543     proof (induct j)
```
```  1544       from step base i show "?A 0"
```
```  1545         by (auto elim: mod_induct_0)
```
```  1546     next
```
```  1547       fix k
```
```  1548       assume ih: "?A k"
```
```  1549       show "?A (Suc k)"
```
```  1550       proof
```
```  1551         assume suc: "Suc k < p"
```
```  1552         hence k: "k<p" by simp
```
```  1553         with ih have "P k" ..
```
```  1554         with step k have "P (Suc k mod p)"
```
```  1555           by blast
```
```  1556         moreover
```
```  1557         from suc have "Suc k mod p = Suc k"
```
```  1558           by simp
```
```  1559         ultimately
```
```  1560         show "P (Suc k)" by simp
```
```  1561       qed
```
```  1562     qed
```
```  1563   qed
```
```  1564   with j show ?thesis by blast
```
```  1565 qed
```
```  1566
```
```  1567 lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"
```
```  1568   by (simp add: numeral_2_eq_2 le_div_geq)
```
```  1569
```
```  1570 lemma mod2_Suc_Suc [simp]: "Suc (Suc m) mod 2 = m mod 2"
```
```  1571   by (simp add: numeral_2_eq_2 le_mod_geq)
```
```  1572
```
```  1573 lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
```
```  1574 by (simp add: mult_2 [symmetric])
```
```  1575
```
```  1576 lemma mod2_gr_0 [simp]: "0 < (m::nat) mod 2 \<longleftrightarrow> m mod 2 = 1"
```
```  1577 proof -
```
```  1578   { fix n :: nat have  "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (cases n) simp_all }
```
```  1579   moreover have "m mod 2 < 2" by simp
```
```  1580   ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" .
```
```  1581   then show ?thesis by auto
```
```  1582 qed
```
```  1583
```
```  1584 text\<open>These lemmas collapse some needless occurrences of Suc:
```
```  1585     at least three Sucs, since two and fewer are rewritten back to Suc again!
```
```  1586     We already have some rules to simplify operands smaller than 3.\<close>
```
```  1587
```
```  1588 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
```
```  1589 by (simp add: Suc3_eq_add_3)
```
```  1590
```
```  1591 lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
```
```  1592 by (simp add: Suc3_eq_add_3)
```
```  1593
```
```  1594 lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
```
```  1595 by (simp add: Suc3_eq_add_3)
```
```  1596
```
```  1597 lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
```
```  1598 by (simp add: Suc3_eq_add_3)
```
```  1599
```
```  1600 lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v
```
```  1601 lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v
```
```  1602
```
```  1603 lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1"
```
```  1604 apply (induct "m")
```
```  1605 apply (simp_all add: mod_Suc)
```
```  1606 done
```
```  1607
```
```  1608 declare Suc_times_mod_eq [of "numeral w", simp] for w
```
```  1609
```
```  1610 lemma Suc_div_le_mono [simp]: "n div k \<le> (Suc n) div k"
```
```  1611 by (simp add: div_le_mono)
```
```  1612
```
```  1613 lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"
```
```  1614 by (cases n) simp_all
```
```  1615
```
```  1616 lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"
```
```  1617 proof -
```
```  1618   from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all
```
```  1619   from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp
```
```  1620 qed
```
```  1621
```
```  1622 lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"
```
```  1623 proof -
```
```  1624   have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp
```
```  1625   also have "... = Suc m mod n" by (rule mod_mult_self3)
```
```  1626   finally show ?thesis .
```
```  1627 qed
```
```  1628
```
```  1629 lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"
```
```  1630 apply (subst mod_Suc [of m])
```
```  1631 apply (subst mod_Suc [of "m mod n"], simp)
```
```  1632 done
```
```  1633
```
```  1634 lemma mod_2_not_eq_zero_eq_one_nat:
```
```  1635   fixes n :: nat
```
```  1636   shows "n mod 2 \<noteq> 0 \<longleftrightarrow> n mod 2 = 1"
```
```  1637   by (fact not_mod_2_eq_0_eq_1)
```
```  1638
```
```  1639 lemma even_Suc_div_two [simp]:
```
```  1640   "even n \<Longrightarrow> Suc n div 2 = n div 2"
```
```  1641   using even_succ_div_two [of n] by simp
```
```  1642
```
```  1643 lemma odd_Suc_div_two [simp]:
```
```  1644   "odd n \<Longrightarrow> Suc n div 2 = Suc (n div 2)"
```
```  1645   using odd_succ_div_two [of n] by simp
```
```  1646
```
```  1647 lemma odd_two_times_div_two_nat [simp]:
```
```  1648   assumes "odd n"
```
```  1649   shows "2 * (n div 2) = n - (1 :: nat)"
```
```  1650 proof -
```
```  1651   from assms have "2 * (n div 2) + 1 = n"
```
```  1652     by (rule odd_two_times_div_two_succ)
```
```  1653   then have "Suc (2 * (n div 2)) - 1 = n - 1"
```
```  1654     by simp
```
```  1655   then show ?thesis
```
```  1656     by simp
```
```  1657 qed
```
```  1658
```
```  1659 lemma parity_induct [case_names zero even odd]:
```
```  1660   assumes zero: "P 0"
```
```  1661   assumes even: "\<And>n. P n \<Longrightarrow> P (2 * n)"
```
```  1662   assumes odd: "\<And>n. P n \<Longrightarrow> P (Suc (2 * n))"
```
```  1663   shows "P n"
```
```  1664 proof (induct n rule: less_induct)
```
```  1665   case (less n)
```
```  1666   show "P n"
```
```  1667   proof (cases "n = 0")
```
```  1668     case True with zero show ?thesis by simp
```
```  1669   next
```
```  1670     case False
```
```  1671     with less have hyp: "P (n div 2)" by simp
```
```  1672     show ?thesis
```
```  1673     proof (cases "even n")
```
```  1674       case True
```
```  1675       with hyp even [of "n div 2"] show ?thesis
```
```  1676         by simp
```
```  1677     next
```
```  1678       case False
```
```  1679       with hyp odd [of "n div 2"] show ?thesis
```
```  1680         by simp
```
```  1681     qed
```
```  1682   qed
```
```  1683 qed
```
```  1684
```
```  1685 lemma Suc_0_div_numeral [simp]:
```
```  1686   fixes k l :: num
```
```  1687   shows "Suc 0 div numeral k = fst (divmod Num.One k)"
```
```  1688   by (simp_all add: fst_divmod)
```
```  1689
```
```  1690 lemma Suc_0_mod_numeral [simp]:
```
```  1691   fixes k l :: num
```
```  1692   shows "Suc 0 mod numeral k = snd (divmod Num.One k)"
```
```  1693   by (simp_all add: snd_divmod)
```
```  1694
```
```  1695
```
```  1696 subsection \<open>Division on @{typ int}\<close>
```
```  1697
```
```  1698 context
```
```  1699 begin
```
```  1700
```
```  1701 inductive eucl_rel_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool"
```
```  1702   where eucl_rel_int_by0: "eucl_rel_int k 0 (0, k)"
```
```  1703   | eucl_rel_int_dividesI: "l \<noteq> 0 \<Longrightarrow> k = q * l \<Longrightarrow> eucl_rel_int k l (q, 0)"
```
```  1704   | eucl_rel_int_remainderI: "sgn r = sgn l \<Longrightarrow> \<bar>r\<bar> < \<bar>l\<bar>
```
```  1705       \<Longrightarrow> k = q * l + r \<Longrightarrow> eucl_rel_int k l (q, r)"
```
```  1706
```
```  1707 lemma eucl_rel_int_iff:
```
```  1708   "eucl_rel_int k l (q, r) \<longleftrightarrow>
```
```  1709     k = l * q + r \<and>
```
```  1710      (if 0 < l then 0 \<le> r \<and> r < l else if l < 0 then l < r \<and> r \<le> 0 else q = 0)"
```
```  1711   by (cases "r = 0")
```
```  1712     (auto elim!: eucl_rel_int.cases intro: eucl_rel_int_by0 eucl_rel_int_dividesI eucl_rel_int_remainderI
```
```  1713     simp add: ac_simps sgn_1_pos sgn_1_neg)
```
```  1714
```
```  1715 lemma unique_quotient_lemma:
```
```  1716   "b * q' + r' \<le> b * q + r \<Longrightarrow> 0 \<le> r' \<Longrightarrow> r' < b \<Longrightarrow> r < b \<Longrightarrow> q' \<le> (q::int)"
```
```  1717 apply (subgoal_tac "r' + b * (q'-q) \<le> r")
```
```  1718  prefer 2 apply (simp add: right_diff_distrib)
```
```  1719 apply (subgoal_tac "0 < b * (1 + q - q') ")
```
```  1720 apply (erule_tac [2] order_le_less_trans)
```
```  1721  prefer 2 apply (simp add: right_diff_distrib distrib_left)
```
```  1722 apply (subgoal_tac "b * q' < b * (1 + q) ")
```
```  1723  prefer 2 apply (simp add: right_diff_distrib distrib_left)
```
```  1724 apply (simp add: mult_less_cancel_left)
```
```  1725 done
```
```  1726
```
```  1727 lemma unique_quotient_lemma_neg:
```
```  1728   "b * q' + r' \<le> b*q + r \<Longrightarrow> r \<le> 0 \<Longrightarrow> b < r \<Longrightarrow> b < r' \<Longrightarrow> q \<le> (q'::int)"
```
```  1729   by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma) auto
```
```  1730
```
```  1731 lemma unique_quotient:
```
```  1732   "eucl_rel_int a b (q, r) \<Longrightarrow> eucl_rel_int a b (q', r') \<Longrightarrow> q = q'"
```
```  1733   apply (simp add: eucl_rel_int_iff linorder_neq_iff split: if_split_asm)
```
```  1734   apply (blast intro: order_antisym
```
```  1735     dest: order_eq_refl [THEN unique_quotient_lemma]
```
```  1736     order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
```
```  1737   done
```
```  1738
```
```  1739 lemma unique_remainder:
```
```  1740   "eucl_rel_int a b (q, r) \<Longrightarrow> eucl_rel_int a b (q', r') \<Longrightarrow> r = r'"
```
```  1741 apply (subgoal_tac "q = q'")
```
```  1742  apply (simp add: eucl_rel_int_iff)
```
```  1743 apply (blast intro: unique_quotient)
```
```  1744 done
```
```  1745
```
```  1746 end
```
```  1747
```
```  1748 instantiation int :: "{idom_modulo, normalization_semidom}"
```
```  1749 begin
```
```  1750
```
```  1751 definition normalize_int :: "int \<Rightarrow> int"
```
```  1752   where [simp]: "normalize = (abs :: int \<Rightarrow> int)"
```
```  1753
```
```  1754 definition unit_factor_int :: "int \<Rightarrow> int"
```
```  1755   where [simp]: "unit_factor = (sgn :: int \<Rightarrow> int)"
```
```  1756
```
```  1757 definition divide_int :: "int \<Rightarrow> int \<Rightarrow> int"
```
```  1758   where "k div l = (if l = 0 \<or> k = 0 then 0
```
```  1759     else if k > 0 \<and> l > 0 \<or> k < 0 \<and> l < 0
```
```  1760       then int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)
```
```  1761       else
```
```  1762         if l dvd k then - int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)
```
```  1763         else - int (Suc (nat \<bar>k\<bar> div nat \<bar>l\<bar>)))"
```
```  1764
```
```  1765 definition modulo_int :: "int \<Rightarrow> int \<Rightarrow> int"
```
```  1766   where "k mod l = (if l = 0 then k else if l dvd k then 0
```
```  1767     else if k > 0 \<and> l > 0 \<or> k < 0 \<and> l < 0
```
```  1768       then sgn l * int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>)
```
```  1769       else sgn l * (\<bar>l\<bar> - int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>)))"
```
```  1770
```
```  1771 lemma eucl_rel_int:
```
```  1772   "eucl_rel_int k l (k div l, k mod l)"
```
```  1773 proof (cases k rule: int_cases3)
```
```  1774   case zero
```
```  1775   then show ?thesis
```
```  1776     by (simp add: eucl_rel_int_iff divide_int_def modulo_int_def)
```
```  1777 next
```
```  1778   case (pos n)
```
```  1779   then show ?thesis
```
```  1780     using div_mult_mod_eq [of n]
```
```  1781     by (cases l rule: int_cases3)
```
```  1782       (auto simp del: of_nat_mult of_nat_add
```
```  1783         simp add: mod_greater_zero_iff_not_dvd of_nat_mult [symmetric] of_nat_add [symmetric] algebra_simps
```
```  1784         eucl_rel_int_iff divide_int_def modulo_int_def int_dvd_iff)
```
```  1785 next
```
```  1786   case (neg n)
```
```  1787   then show ?thesis
```
```  1788     using div_mult_mod_eq [of n]
```
```  1789     by (cases l rule: int_cases3)
```
```  1790       (auto simp del: of_nat_mult of_nat_add
```
```  1791         simp add: mod_greater_zero_iff_not_dvd of_nat_mult [symmetric] of_nat_add [symmetric] algebra_simps
```
```  1792         eucl_rel_int_iff divide_int_def modulo_int_def int_dvd_iff)
```
```  1793 qed
```
```  1794
```
```  1795 lemma divmod_int_unique:
```
```  1796   assumes "eucl_rel_int k l (q, r)"
```
```  1797   shows div_int_unique: "k div l = q" and mod_int_unique: "k mod l = r"
```
```  1798   using assms eucl_rel_int [of k l]
```
```  1799   using unique_quotient [of k l] unique_remainder [of k l]
```
```  1800   by auto
```
```  1801
```
```  1802 instance proof
```
```  1803   fix k l :: int
```
```  1804   show "k div l * l + k mod l = k"
```
```  1805     using eucl_rel_int [of k l]
```
```  1806     unfolding eucl_rel_int_iff by (simp add: ac_simps)
```
```  1807 next
```
```  1808   fix k :: int show "k div 0 = 0"
```
```  1809     by (rule div_int_unique, simp add: eucl_rel_int_iff)
```
```  1810 next
```
```  1811   fix k l :: int
```
```  1812   assume "l \<noteq> 0"
```
```  1813   then show "k * l div l = k"
```
```  1814     by (auto simp add: eucl_rel_int_iff ac_simps intro: div_int_unique [of _ _ _ 0])
```
```  1815 qed (auto simp add: sgn_mult mult_sgn_abs abs_eq_iff')
```
```  1816
```
```  1817 end
```
```  1818
```
```  1819 lemma is_unit_int:
```
```  1820   "is_unit (k::int) \<longleftrightarrow> k = 1 \<or> k = - 1"
```
```  1821   by auto
```
```  1822
```
```  1823 lemma zdiv_int:
```
```  1824   "int (a div b) = int a div int b"
```
```  1825   by (simp add: divide_int_def)
```
```  1826
```
```  1827 lemma zmod_int:
```
```  1828   "int (a mod b) = int a mod int b"
```
```  1829   by (simp add: modulo_int_def int_dvd_iff)
```
```  1830
```
```  1831 lemma div_abs_eq_div_nat:
```
```  1832   "\<bar>k\<bar> div \<bar>l\<bar> = int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)"
```
```  1833   by (simp add: divide_int_def)
```
```  1834
```
```  1835 lemma mod_abs_eq_div_nat:
```
```  1836   "\<bar>k\<bar> mod \<bar>l\<bar> = int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>)"
```
```  1837   by (simp add: modulo_int_def dvd_int_unfold_dvd_nat)
```
```  1838
```
```  1839 lemma div_sgn_abs_cancel:
```
```  1840   fixes k l v :: int
```
```  1841   assumes "v \<noteq> 0"
```
```  1842   shows "(sgn v * \<bar>k\<bar>) div (sgn v * \<bar>l\<bar>) = \<bar>k\<bar> div \<bar>l\<bar>"
```
```  1843 proof -
```
```  1844   from assms have "sgn v = - 1 \<or> sgn v = 1"
```
```  1845     by (cases "v \<ge> 0") auto
```
```  1846   then show ?thesis
```
```  1847   using assms unfolding divide_int_def [of "sgn v * \<bar>k\<bar>" "sgn v * \<bar>l\<bar>"]
```
```  1848     by (auto simp add: not_less div_abs_eq_div_nat)
```
```  1849 qed
```
```  1850
```
```  1851 lemma div_eq_sgn_abs:
```
```  1852   fixes k l v :: int
```
```  1853   assumes "sgn k = sgn l"
```
```  1854   shows "k div l = \<bar>k\<bar> div \<bar>l\<bar>"
```
```  1855 proof (cases "l = 0")
```
```  1856   case True
```
```  1857   then show ?thesis
```
```  1858     by simp
```
```  1859 next
```
```  1860   case False
```
```  1861   with assms have "(sgn k * \<bar>k\<bar>) div (sgn l * \<bar>l\<bar>) = \<bar>k\<bar> div \<bar>l\<bar>"
```
```  1862     by (simp add: div_sgn_abs_cancel)
```
```  1863   then show ?thesis
```
```  1864     by (simp add: sgn_mult_abs)
```
```  1865 qed
```
```  1866
```
```  1867 lemma div_dvd_sgn_abs:
```
```  1868   fixes k l :: int
```
```  1869   assumes "l dvd k"
```
```  1870   shows "k div l = (sgn k * sgn l) * (\<bar>k\<bar> div \<bar>l\<bar>)"
```
```  1871 proof (cases "k = 0")
```
```  1872   case True
```
```  1873   then show ?thesis
```
```  1874     by simp
```
```  1875 next
```
```  1876   case False
```
```  1877   show ?thesis
```
```  1878   proof (cases "sgn l = sgn k")
```
```  1879     case True
```
```  1880     then show ?thesis
```
```  1881       by (simp add: div_eq_sgn_abs)
```
```  1882   next
```
```  1883     case False
```
```  1884     with \<open>k \<noteq> 0\<close> assms show ?thesis
```
```  1885       unfolding divide_int_def [of k l]
```
```  1886         by (auto simp add: zdiv_int)
```
```  1887   qed
```
```  1888 qed
```
```  1889
```
```  1890 lemma div_noneq_sgn_abs:
```
```  1891   fixes k l :: int
```
```  1892   assumes "l \<noteq> 0"
```
```  1893   assumes "sgn k \<noteq> sgn l"
```
```  1894   shows "k div l = - (\<bar>k\<bar> div \<bar>l\<bar>) - of_bool (\<not> l dvd k)"
```
```  1895   using assms
```
```  1896   by (simp only: divide_int_def [of k l], auto simp add: not_less zdiv_int)
```
```  1897
```
```  1898 lemma sgn_mod:
```
```  1899   fixes k l :: int
```
```  1900   assumes "l \<noteq> 0" "\<not> l dvd k"
```
```  1901   shows "sgn (k mod l) = sgn l"
```
```  1902 proof -
```
```  1903   from \<open>\<not> l dvd k\<close>
```
```  1904   have "k mod l \<noteq> 0"
```
```  1905     by (simp add: dvd_eq_mod_eq_0)
```
```  1906   show ?thesis
```
```  1907     using \<open>l \<noteq> 0\<close> \<open>\<not> l dvd k\<close>
```
```  1908     unfolding modulo_int_def [of k l]
```
```  1909     by (auto simp add: sgn_1_pos sgn_1_neg mod_greater_zero_iff_not_dvd nat_dvd_iff not_less
```
```  1910       zless_nat_eq_int_zless [symmetric] elim: nonpos_int_cases)
```
```  1911 qed
```
```  1912
```
```  1913 lemma abs_mod_less:
```
```  1914   fixes k l :: int
```
```  1915   assumes "l \<noteq> 0"
```
```  1916   shows "\<bar>k mod l\<bar> < \<bar>l\<bar>"
```
```  1917   using assms unfolding modulo_int_def [of k l]
```
```  1918   by (auto simp add: not_less int_dvd_iff mod_greater_zero_iff_not_dvd elim: pos_int_cases neg_int_cases nonneg_int_cases nonpos_int_cases)
```
```  1919
```
```  1920 instance int :: ring_div
```
```  1921 proof
```
```  1922   fix k l s :: int
```
```  1923   assume "l \<noteq> 0"
```
```  1924   then have "eucl_rel_int (k + s * l) l (s + k div l, k mod l)"
```
```  1925     using eucl_rel_int [of k l]
```
```  1926     unfolding eucl_rel_int_iff by (auto simp: algebra_simps)
```
```  1927   then show "(k + s * l) div l = s + k div l"
```
```  1928     by (rule div_int_unique)
```
```  1929 next
```
```  1930   fix k l s :: int
```
```  1931   assume "s \<noteq> 0"
```
```  1932   have "\<And>q r. eucl_rel_int k l (q, r)
```
```  1933     \<Longrightarrow> eucl_rel_int (s * k) (s * l) (q, s * r)"
```
```  1934     unfolding eucl_rel_int_iff
```
```  1935     by (rule linorder_cases [of 0 l])
```
```  1936       (use \<open>s \<noteq> 0\<close> in \<open>auto simp: algebra_simps
```
```  1937       mult_less_0_iff zero_less_mult_iff mult_strict_right_mono
```
```  1938       mult_strict_right_mono_neg zero_le_mult_iff mult_le_0_iff\<close>)
```
```  1939   then have "eucl_rel_int (s * k) (s * l) (k div l, s * (k mod l))"
```
```  1940     using eucl_rel_int [of k l] .
```
```  1941   then show "(s * k) div (s * l) = k div l"
```
```  1942     by (rule div_int_unique)
```
```  1943 qed
```
```  1944
```
```  1945 ML \<open>
```
```  1946 structure Cancel_Div_Mod_Int = Cancel_Div_Mod
```
```  1947 (
```
```  1948   val div_name = @{const_name divide};
```
```  1949   val mod_name = @{const_name modulo};
```
```  1950   val mk_binop = HOLogic.mk_binop;
```
```  1951   val mk_sum = Arith_Data.mk_sum HOLogic.intT;
```
```  1952   val dest_sum = Arith_Data.dest_sum;
```
```  1953
```
```  1954   val div_mod_eqs = map mk_meta_eq @{thms cancel_div_mod_rules};
```
```  1955
```
```  1956   val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
```
```  1957     @{thms diff_conv_add_uminus add_0_left add_0_right ac_simps})
```
```  1958 )
```
```  1959 \<close>
```
```  1960
```
```  1961 simproc_setup cancel_div_mod_int ("(k::int) + l") =
```
```  1962   \<open>K Cancel_Div_Mod_Int.proc\<close>
```
```  1963
```
```  1964
```
```  1965 text\<open>Basic laws about division and remainder\<close>
```
```  1966
```
```  1967 lemma pos_mod_conj: "(0::int) < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b"
```
```  1968   using eucl_rel_int [of a b]
```
```  1969   by (auto simp add: eucl_rel_int_iff prod_eq_iff)
```
```  1970
```
```  1971 lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]
```
```  1972    and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]
```
```  1973
```
```  1974 lemma neg_mod_conj: "b < (0::int) \<Longrightarrow> a mod b \<le> 0 \<and> b < a mod b"
```
```  1975   using eucl_rel_int [of a b]
```
```  1976   by (auto simp add: eucl_rel_int_iff prod_eq_iff)
```
```  1977
```
```  1978 lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]
```
```  1979    and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]
```
```  1980
```
```  1981
```
```  1982 subsubsection \<open>General Properties of div and mod\<close>
```
```  1983
```
```  1984 lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
```
```  1985 apply (rule div_int_unique)
```
```  1986 apply (auto simp add: eucl_rel_int_iff)
```
```  1987 done
```
```  1988
```
```  1989 lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
```
```  1990 apply (rule div_int_unique)
```
```  1991 apply (auto simp add: eucl_rel_int_iff)
```
```  1992 done
```
```  1993
```
```  1994 lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
```
```  1995 apply (rule div_int_unique)
```
```  1996 apply (auto simp add: eucl_rel_int_iff)
```
```  1997 done
```
```  1998
```
```  1999 (*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
```
```  2000
```
```  2001 lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
```
```  2002 apply (rule_tac q = 0 in mod_int_unique)
```
```  2003 apply (auto simp add: eucl_rel_int_iff)
```
```  2004 done
```
```  2005
```
```  2006 lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
```
```  2007 apply (rule_tac q = 0 in mod_int_unique)
```
```  2008 apply (auto simp add: eucl_rel_int_iff)
```
```  2009 done
```
```  2010
```
```  2011 lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
```
```  2012 apply (rule_tac q = "-1" in mod_int_unique)
```
```  2013 apply (auto simp add: eucl_rel_int_iff)
```
```  2014 done
```
```  2015
```
```  2016 text\<open>There is no \<open>mod_neg_pos_trivial\<close>.\<close>
```
```  2017
```
```  2018
```
```  2019 subsubsection \<open>Laws for div and mod with Unary Minus\<close>
```
```  2020
```
```  2021 lemma zminus1_lemma:
```
```  2022      "eucl_rel_int a b (q, r) ==> b \<noteq> 0
```
```  2023       ==> eucl_rel_int (-a) b (if r=0 then -q else -q - 1,
```
```  2024                           if r=0 then 0 else b-r)"
```
```  2025 by (force simp add: split_ifs eucl_rel_int_iff linorder_neq_iff right_diff_distrib)
```
```  2026
```
```  2027
```
```  2028 lemma zdiv_zminus1_eq_if:
```
```  2029      "b \<noteq> (0::int)
```
```  2030       ==> (-a) div b =
```
```  2031           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
```
```  2032 by (blast intro: eucl_rel_int [THEN zminus1_lemma, THEN div_int_unique])
```
```  2033
```
```  2034 lemma zmod_zminus1_eq_if:
```
```  2035      "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
```
```  2036 apply (case_tac "b = 0", simp)
```
```  2037 apply (blast intro: eucl_rel_int [THEN zminus1_lemma, THEN mod_int_unique])
```
```  2038 done
```
```  2039
```
```  2040 lemma zmod_zminus1_not_zero:
```
```  2041   fixes k l :: int
```
```  2042   shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
```
```  2043   by (simp add: mod_eq_0_iff_dvd)
```
```  2044
```
```  2045 lemma zmod_zminus2_not_zero:
```
```  2046   fixes k l :: int
```
```  2047   shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
```
```  2048   by (simp add: mod_eq_0_iff_dvd)
```
```  2049
```
```  2050 lemma zdiv_zminus2_eq_if:
```
```  2051      "b \<noteq> (0::int)
```
```  2052       ==> a div (-b) =
```
```  2053           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
```
```  2054 by (simp add: zdiv_zminus1_eq_if div_minus_right)
```
```  2055
```
```  2056 lemma zmod_zminus2_eq_if:
```
```  2057      "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
```
```  2058 by (simp add: zmod_zminus1_eq_if mod_minus_right)
```
```  2059
```
```  2060
```
```  2061 subsubsection \<open>Monotonicity in the First Argument (Dividend)\<close>
```
```  2062
```
```  2063 lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
```
```  2064 using mult_div_mod_eq [symmetric, of a b]
```
```  2065 using mult_div_mod_eq [symmetric, of a' b]
```
```  2066 apply -
```
```  2067 apply (rule unique_quotient_lemma)
```
```  2068 apply (erule subst)
```
```  2069 apply (erule subst, simp_all)
```
```  2070 done
```
```  2071
```
```  2072 lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
```
```  2073 using mult_div_mod_eq [symmetric, of a b]
```
```  2074 using mult_div_mod_eq [symmetric, of a' b]
```
```  2075 apply -
```
```  2076 apply (rule unique_quotient_lemma_neg)
```
```  2077 apply (erule subst)
```
```  2078 apply (erule subst, simp_all)
```
```  2079 done
```
```  2080
```
```  2081
```
```  2082 subsubsection \<open>Monotonicity in the Second Argument (Divisor)\<close>
```
```  2083
```
```  2084 lemma q_pos_lemma:
```
```  2085      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
```
```  2086 apply (subgoal_tac "0 < b'* (q' + 1) ")
```
```  2087  apply (simp add: zero_less_mult_iff)
```
```  2088 apply (simp add: distrib_left)
```
```  2089 done
```
```  2090
```
```  2091 lemma zdiv_mono2_lemma:
```
```  2092      "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';
```
```  2093          r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]
```
```  2094       ==> q \<le> (q'::int)"
```
```  2095 apply (frule q_pos_lemma, assumption+)
```
```  2096 apply (subgoal_tac "b*q < b* (q' + 1) ")
```
```  2097  apply (simp add: mult_less_cancel_left)
```
```  2098 apply (subgoal_tac "b*q = r' - r + b'*q'")
```
```  2099  prefer 2 apply simp
```
```  2100 apply (simp (no_asm_simp) add: distrib_left)
```
```  2101 apply (subst add.commute, rule add_less_le_mono, arith)
```
```  2102 apply (rule mult_right_mono, auto)
```
```  2103 done
```
```  2104
```
```  2105 lemma zdiv_mono2:
```
```  2106      "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
```
```  2107 apply (subgoal_tac "b \<noteq> 0")
```
```  2108   prefer 2 apply arith
```
```  2109 using mult_div_mod_eq [symmetric, of a b]
```
```  2110 using mult_div_mod_eq [symmetric, of a b']
```
```  2111 apply -
```
```  2112 apply (rule zdiv_mono2_lemma)
```
```  2113 apply (erule subst)
```
```  2114 apply (erule subst, simp_all)
```
```  2115 done
```
```  2116
```
```  2117 lemma q_neg_lemma:
```
```  2118      "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
```
```  2119 apply (subgoal_tac "b'*q' < 0")
```
```  2120  apply (simp add: mult_less_0_iff, arith)
```
```  2121 done
```
```  2122
```
```  2123 lemma zdiv_mono2_neg_lemma:
```
```  2124      "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;
```
```  2125          r < b;  0 \<le> r';  0 < b';  b' \<le> b |]
```
```  2126       ==> q' \<le> (q::int)"
```
```  2127 apply (frule q_neg_lemma, assumption+)
```
```  2128 apply (subgoal_tac "b*q' < b* (q + 1) ")
```
```  2129  apply (simp add: mult_less_cancel_left)
```
```  2130 apply (simp add: distrib_left)
```
```  2131 apply (subgoal_tac "b*q' \<le> b'*q'")
```
```  2132  prefer 2 apply (simp add: mult_right_mono_neg, arith)
```
```  2133 done
```
```  2134
```
```  2135 lemma zdiv_mono2_neg:
```
```  2136      "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
```
```  2137 using mult_div_mod_eq [symmetric, of a b]
```
```  2138 using mult_div_mod_eq [symmetric, of a b']
```
```  2139 apply -
```
```  2140 apply (rule zdiv_mono2_neg_lemma)
```
```  2141 apply (erule subst)
```
```  2142 apply (erule subst, simp_all)
```
```  2143 done
```
```  2144
```
```  2145
```
```  2146 subsubsection \<open>More Algebraic Laws for div and mod\<close>
```
```  2147
```
```  2148 text\<open>proving (a*b) div c = a * (b div c) + a * (b mod c)\<close>
```
```  2149
```
```  2150 lemma zmult1_lemma:
```
```  2151      "[| eucl_rel_int b c (q, r) |]
```
```  2152       ==> eucl_rel_int (a * b) c (a*q + a*r div c, a*r mod c)"
```
```  2153 by (auto simp add: split_ifs eucl_rel_int_iff linorder_neq_iff distrib_left ac_simps)
```
```  2154
```
```  2155 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
```
```  2156 apply (case_tac "c = 0", simp)
```
```  2157 apply (blast intro: eucl_rel_int [THEN zmult1_lemma, THEN div_int_unique])
```
```  2158 done
```
```  2159
```
```  2160 text\<open>proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c)\<close>
```
```  2161
```
```  2162 lemma zadd1_lemma:
```
```  2163      "[| eucl_rel_int a c (aq, ar);  eucl_rel_int b c (bq, br) |]
```
```  2164       ==> eucl_rel_int (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
```
```  2165 by (force simp add: split_ifs eucl_rel_int_iff linorder_neq_iff distrib_left)
```
```  2166
```
```  2167 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
```
```  2168 lemma zdiv_zadd1_eq:
```
```  2169      "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
```
```  2170 apply (case_tac "c = 0", simp)
```
```  2171 apply (blast intro: zadd1_lemma [OF eucl_rel_int eucl_rel_int] div_int_unique)
```
```  2172 done
```
```  2173
```
```  2174 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
```
```  2175 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
```
```  2176
```
```  2177 (* REVISIT: should this be generalized to all semiring_div types? *)
```
```  2178 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
```
```  2179
```
```  2180
```
```  2181 subsubsection \<open>Proving  @{term "a div (b * c) = (a div b) div c"}\<close>
```
```  2182
```
```  2183 (*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
```
```  2184   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
```
```  2185   to cause particular problems.*)
```
```  2186
```
```  2187 text\<open>first, four lemmas to bound the remainder for the cases b<0 and b>0\<close>
```
```  2188
```
```  2189 lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b * c < b * (q mod c) + r"
```
```  2190 apply (subgoal_tac "b * (c - q mod c) < r * 1")
```
```  2191  apply (simp add: algebra_simps)
```
```  2192 apply (rule order_le_less_trans)
```
```  2193  apply (erule_tac [2] mult_strict_right_mono)
```
```  2194  apply (rule mult_left_mono_neg)
```
```  2195   using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)
```
```  2196  apply (simp)
```
```  2197 apply (simp)
```
```  2198 done
```
```  2199
```
```  2200 lemma zmult2_lemma_aux2:
```
```  2201      "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
```
```  2202 apply (subgoal_tac "b * (q mod c) \<le> 0")
```
```  2203  apply arith
```
```  2204 apply (simp add: mult_le_0_iff)
```
```  2205 done
```
```  2206
```
```  2207 lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
```
```  2208 apply (subgoal_tac "0 \<le> b * (q mod c) ")
```
```  2209 apply arith
```
```  2210 apply (simp add: zero_le_mult_iff)
```
```  2211 done
```
```  2212
```
```  2213 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
```
```  2214 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
```
```  2215  apply (simp add: right_diff_distrib)
```
```  2216 apply (rule order_less_le_trans)
```
```  2217  apply (erule mult_strict_right_mono)
```
```  2218  apply (rule_tac [2] mult_left_mono)
```
```  2219   apply simp
```
```  2220  using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)
```
```  2221 apply simp
```
```  2222 done
```
```  2223
```
```  2224 lemma zmult2_lemma: "[| eucl_rel_int a b (q, r); 0 < c |]
```
```  2225       ==> eucl_rel_int a (b * c) (q div c, b*(q mod c) + r)"
```
```  2226 by (auto simp add: mult.assoc eucl_rel_int_iff linorder_neq_iff
```
```  2227                    zero_less_mult_iff distrib_left [symmetric]
```
```  2228                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: if_split_asm)
```
```  2229
```
```  2230 lemma zdiv_zmult2_eq:
```
```  2231   fixes a b c :: int
```
```  2232   shows "0 \<le> c \<Longrightarrow> a div (b * c) = (a div b) div c"
```
```  2233 apply (case_tac "b = 0", simp)
```
```  2234 apply (force simp add: le_less eucl_rel_int [THEN zmult2_lemma, THEN div_int_unique])
```
```  2235 done
```
```  2236
```
```  2237 lemma zmod_zmult2_eq:
```
```  2238   fixes a b c :: int
```
```  2239   shows "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
```
```  2240 apply (case_tac "b = 0", simp)
```
```  2241 apply (force simp add: le_less eucl_rel_int [THEN zmult2_lemma, THEN mod_int_unique])
```
```  2242 done
```
```  2243
```
```  2244 lemma div_pos_geq:
```
```  2245   fixes k l :: int
```
```  2246   assumes "0 < l" and "l \<le> k"
```
```  2247   shows "k div l = (k - l) div l + 1"
```
```  2248 proof -
```
```  2249   have "k = (k - l) + l" by simp
```
```  2250   then obtain j where k: "k = j + l" ..
```
```  2251   with assms show ?thesis by (simp add: div_add_self2)
```
```  2252 qed
```
```  2253
```
```  2254 lemma mod_pos_geq:
```
```  2255   fixes k l :: int
```
```  2256   assumes "0 < l" and "l \<le> k"
```
```  2257   shows "k mod l = (k - l) mod l"
```
```  2258 proof -
```
```  2259   have "k = (k - l) + l" by simp
```
```  2260   then obtain j where k: "k = j + l" ..
```
```  2261   with assms show ?thesis by simp
```
```  2262 qed
```
```  2263
```
```  2264
```
```  2265 subsubsection \<open>Splitting Rules for div and mod\<close>
```
```  2266
```
```  2267 text\<open>The proofs of the two lemmas below are essentially identical\<close>
```
```  2268
```
```  2269 lemma split_pos_lemma:
```
```  2270  "0<k ==>
```
```  2271     P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
```
```  2272 apply (rule iffI, clarify)
```
```  2273  apply (erule_tac P="P x y" for x y in rev_mp)
```
```  2274  apply (subst mod_add_eq [symmetric])
```
```  2275  apply (subst zdiv_zadd1_eq)
```
```  2276  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)
```
```  2277 txt\<open>converse direction\<close>
```
```  2278 apply (drule_tac x = "n div k" in spec)
```
```  2279 apply (drule_tac x = "n mod k" in spec, simp)
```
```  2280 done
```
```  2281
```
```  2282 lemma split_neg_lemma:
```
```  2283  "k<0 ==>
```
```  2284     P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
```
```  2285 apply (rule iffI, clarify)
```
```  2286  apply (erule_tac P="P x y" for x y in rev_mp)
```
```  2287  apply (subst mod_add_eq [symmetric])
```
```  2288  apply (subst zdiv_zadd1_eq)
```
```  2289  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)
```
```  2290 txt\<open>converse direction\<close>
```
```  2291 apply (drule_tac x = "n div k" in spec)
```
```  2292 apply (drule_tac x = "n mod k" in spec, simp)
```
```  2293 done
```
```  2294
```
```  2295 lemma split_zdiv:
```
```  2296  "P(n div k :: int) =
```
```  2297   ((k = 0 --> P 0) &
```
```  2298    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) &
```
```  2299    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
```
```  2300 apply (case_tac "k=0", simp)
```
```  2301 apply (simp only: linorder_neq_iff)
```
```  2302 apply (erule disjE)
```
```  2303  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]
```
```  2304                       split_neg_lemma [of concl: "%x y. P x"])
```
```  2305 done
```
```  2306
```
```  2307 lemma split_zmod:
```
```  2308  "P(n mod k :: int) =
```
```  2309   ((k = 0 --> P n) &
```
```  2310    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) &
```
```  2311    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
```
```  2312 apply (case_tac "k=0", simp)
```
```  2313 apply (simp only: linorder_neq_iff)
```
```  2314 apply (erule disjE)
```
```  2315  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]
```
```  2316                       split_neg_lemma [of concl: "%x y. P y"])
```
```  2317 done
```
```  2318
```
```  2319 text \<open>Enable (lin)arith to deal with @{const divide} and @{const modulo}
```
```  2320   when these are applied to some constant that is of the form
```
```  2321   @{term "numeral k"}:\<close>
```
```  2322 declare split_zdiv [of _ _ "numeral k", arith_split] for k
```
```  2323 declare split_zmod [of _ _ "numeral k", arith_split] for k
```
```  2324
```
```  2325
```
```  2326 subsubsection \<open>Computing \<open>div\<close> and \<open>mod\<close> with shifting\<close>
```
```  2327
```
```  2328 lemma pos_eucl_rel_int_mult_2:
```
```  2329   assumes "0 \<le> b"
```
```  2330   assumes "eucl_rel_int a b (q, r)"
```
```  2331   shows "eucl_rel_int (1 + 2*a) (2*b) (q, 1 + 2*r)"
```
```  2332   using assms unfolding eucl_rel_int_iff by auto
```
```  2333
```
```  2334 lemma neg_eucl_rel_int_mult_2:
```
```  2335   assumes "b \<le> 0"
```
```  2336   assumes "eucl_rel_int (a + 1) b (q, r)"
```
```  2337   shows "eucl_rel_int (1 + 2*a) (2*b) (q, 2*r - 1)"
```
```  2338   using assms unfolding eucl_rel_int_iff by auto
```
```  2339
```
```  2340 text\<open>computing div by shifting\<close>
```
```  2341
```
```  2342 lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
```
```  2343   using pos_eucl_rel_int_mult_2 [OF _ eucl_rel_int]
```
```  2344   by (rule div_int_unique)
```
```  2345
```
```  2346 lemma neg_zdiv_mult_2:
```
```  2347   assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"
```
```  2348   using neg_eucl_rel_int_mult_2 [OF A eucl_rel_int]
```
```  2349   by (rule div_int_unique)
```
```  2350
```
```  2351 (* FIXME: add rules for negative numerals *)
```
```  2352 lemma zdiv_numeral_Bit0 [simp]:
```
```  2353   "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
```
```  2354     numeral v div (numeral w :: int)"
```
```  2355   unfolding numeral.simps unfolding mult_2 [symmetric]
```
```  2356   by (rule div_mult_mult1, simp)
```
```  2357
```
```  2358 lemma zdiv_numeral_Bit1 [simp]:
```
```  2359   "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =
```
```  2360     (numeral v div (numeral w :: int))"
```
```  2361   unfolding numeral.simps
```
```  2362   unfolding mult_2 [symmetric] add.commute [of _ 1]
```
```  2363   by (rule pos_zdiv_mult_2, simp)
```
```  2364
```
```  2365 lemma pos_zmod_mult_2:
```
```  2366   fixes a b :: int
```
```  2367   assumes "0 \<le> a"
```
```  2368   shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
```
```  2369   using pos_eucl_rel_int_mult_2 [OF assms eucl_rel_int]
```
```  2370   by (rule mod_int_unique)
```
```  2371
```
```  2372 lemma neg_zmod_mult_2:
```
```  2373   fixes a b :: int
```
```  2374   assumes "a \<le> 0"
```
```  2375   shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
```
```  2376   using neg_eucl_rel_int_mult_2 [OF assms eucl_rel_int]
```
```  2377   by (rule mod_int_unique)
```
```  2378
```
```  2379 (* FIXME: add rules for negative numerals *)
```
```  2380 lemma zmod_numeral_Bit0 [simp]:
```
```  2381   "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =
```
```  2382     (2::int) * (numeral v mod numeral w)"
```
```  2383   unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
```
```  2384   unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
```
```  2385
```
```  2386 lemma zmod_numeral_Bit1 [simp]:
```
```  2387   "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
```
```  2388     2 * (numeral v mod numeral w) + (1::int)"
```
```  2389   unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
```
```  2390   unfolding mult_2 [symmetric] add.commute [of _ 1]
```
```  2391   by (rule pos_zmod_mult_2, simp)
```
```  2392
```
```  2393 lemma zdiv_eq_0_iff:
```
```  2394  "(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")
```
```  2395 proof
```
```  2396   assume ?L
```
```  2397   have "?L \<longrightarrow> ?R" by (rule split_zdiv[THEN iffD2]) simp
```
```  2398   with \<open>?L\<close> show ?R by blast
```
```  2399 next
```
```  2400   assume ?R thus ?L
```
```  2401     by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)
```
```  2402 qed
```
```  2403
```
```  2404 lemma zmod_trival_iff:
```
```  2405   fixes i k :: int
```
```  2406   shows "i mod k = i \<longleftrightarrow> k = 0 \<or> 0 \<le> i \<and> i < k \<or> i \<le> 0 \<and> k < i"
```
```  2407 proof -
```
```  2408   have "i mod k = i \<longleftrightarrow> i div k = 0"
```
```  2409     by safe (insert div_mult_mod_eq [of i k], auto)
```
```  2410   with zdiv_eq_0_iff
```
```  2411   show ?thesis
```
```  2412     by simp
```
```  2413 qed
```
```  2414
```
```  2415 instantiation int :: unique_euclidean_ring
```
```  2416 begin
```
```  2417
```
```  2418 definition [simp]:
```
```  2419   "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"
```
```  2420
```
```  2421 definition [simp]:
```
```  2422   "uniqueness_constraint_int (k :: int) l \<longleftrightarrow> unit_factor k = unit_factor l"
```
```  2423
```
```  2424 instance
```
```  2425   by standard
```
```  2426     (use mult_le_mono2 [of 1] in \<open>auto simp add: abs_mult nat_mult_distrib sgn_mod zdiv_eq_0_iff sgn_1_pos sgn_mult split: abs_split\<close>)
```
```  2427
```
```  2428 end
```
```  2429
```
```  2430
```
```  2431 subsubsection \<open>Quotients of Signs\<close>
```
```  2432
```
```  2433 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
```
```  2434 by (simp add: divide_int_def)
```
```  2435
```
```  2436 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
```
```  2437 by (simp add: modulo_int_def)
```
```  2438
```
```  2439 lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
```
```  2440 apply (subgoal_tac "a div b \<le> -1", force)
```
```  2441 apply (rule order_trans)
```
```  2442 apply (rule_tac a' = "-1" in zdiv_mono1)
```
```  2443 apply (auto simp add: div_eq_minus1)
```
```  2444 done
```
```  2445
```
```  2446 lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
```
```  2447 by (drule zdiv_mono1_neg, auto)
```
```  2448
```
```  2449 lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
```
```  2450 by (drule zdiv_mono1, auto)
```
```  2451
```
```  2452 text\<open>Now for some equivalences of the form \<open>a div b >=< 0 \<longleftrightarrow> \<dots>\<close>
```
```  2453 conditional upon the sign of \<open>a\<close> or \<open>b\<close>. There are many more.
```
```  2454 They should all be simp rules unless that causes too much search.\<close>
```
```  2455
```
```  2456 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
```
```  2457 apply auto
```
```  2458 apply (drule_tac [2] zdiv_mono1)
```
```  2459 apply (auto simp add: linorder_neq_iff)
```
```  2460 apply (simp (no_asm_use) add: linorder_not_less [symmetric])
```
```  2461 apply (blast intro: div_neg_pos_less0)
```
```  2462 done
```
```  2463
```
```  2464 lemma pos_imp_zdiv_pos_iff:
```
```  2465   "0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"
```
```  2466 using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]
```
```  2467 by arith
```
```  2468
```
```  2469 lemma neg_imp_zdiv_nonneg_iff:
```
```  2470   "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
```
```  2471 apply (subst div_minus_minus [symmetric])
```
```  2472 apply (subst pos_imp_zdiv_nonneg_iff, auto)
```
```  2473 done
```
```  2474
```
```  2475 (*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
```
```  2476 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
```
```  2477 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
```
```  2478
```
```  2479 (*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
```
```  2480 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
```
```  2481 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
```
```  2482
```
```  2483 lemma nonneg1_imp_zdiv_pos_iff:
```
```  2484   "(0::int) <= a \<Longrightarrow> (a div b > 0) = (a >= b & b>0)"
```
```  2485 apply rule
```
```  2486  apply rule
```
```  2487   using div_pos_pos_trivial[of a b]apply arith
```
```  2488  apply(cases "b=0")apply simp
```
```  2489  using div_nonneg_neg_le0[of a b]apply arith
```
```  2490 using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp
```
```  2491 done
```
```  2492
```
```  2493 lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 ==> m mod k \<le> m"
```
```  2494 apply (rule split_zmod[THEN iffD2])
```
```  2495 apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)
```
```  2496 done
```
```  2497
```
```  2498
```
```  2499 subsubsection \<open>Computation of Division and Remainder\<close>
```
```  2500
```
```  2501 instantiation int :: semiring_numeral_div
```
```  2502 begin
```
```  2503
```
```  2504 definition divmod_int :: "num \<Rightarrow> num \<Rightarrow> int \<times> int"
```
```  2505 where
```
```  2506   "divmod_int m n = (numeral m div numeral n, numeral m mod numeral n)"
```
```  2507
```
```  2508 definition divmod_step_int :: "num \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int"
```
```  2509 where
```
```  2510   "divmod_step_int l qr = (let (q, r) = qr
```
```  2511     in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
```
```  2512     else (2 * q, r))"
```
```  2513
```
```  2514 instance
```
```  2515   by standard (auto intro: zmod_le_nonneg_dividend simp add: divmod_int_def divmod_step_int_def
```
```  2516     pos_imp_zdiv_pos_iff div_pos_pos_trivial mod_pos_pos_trivial zmod_zmult2_eq zdiv_zmult2_eq)
```
```  2517
```
```  2518 end
```
```  2519
```
```  2520 declare divmod_algorithm_code [where ?'a = int, code]
```
```  2521
```
```  2522 context
```
```  2523 begin
```
```  2524
```
```  2525 qualified definition adjust_div :: "int \<times> int \<Rightarrow> int"
```
```  2526 where
```
```  2527   "adjust_div qr = (let (q, r) = qr in q + of_bool (r \<noteq> 0))"
```
```  2528
```
```  2529 qualified lemma adjust_div_eq [simp, code]:
```
```  2530   "adjust_div (q, r) = q + of_bool (r \<noteq> 0)"
```
```  2531   by (simp add: adjust_div_def)
```
```  2532
```
```  2533 qualified definition adjust_mod :: "int \<Rightarrow> int \<Rightarrow> int"
```
```  2534 where
```
```  2535   [simp]: "adjust_mod l r = (if r = 0 then 0 else l - r)"
```
```  2536
```
```  2537 lemma minus_numeral_div_numeral [simp]:
```
```  2538   "- numeral m div numeral n = - (adjust_div (divmod m n) :: int)"
```
```  2539 proof -
```
```  2540   have "int (fst (divmod m n)) = fst (divmod m n)"
```
```  2541     by (simp only: fst_divmod divide_int_def) auto
```
```  2542   then show ?thesis
```
```  2543     by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
```
```  2544 qed
```
```  2545
```
```  2546 lemma minus_numeral_mod_numeral [simp]:
```
```  2547   "- numeral m mod numeral n = adjust_mod (numeral n) (snd (divmod m n) :: int)"
```
```  2548 proof -
```
```  2549   have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
```
```  2550     using that by (simp only: snd_divmod modulo_int_def) auto
```
```  2551   then show ?thesis
```
```  2552     by (auto simp add: split_def Let_def adjust_div_def divides_aux_def modulo_int_def)
```
```  2553 qed
```
```  2554
```
```  2555 lemma numeral_div_minus_numeral [simp]:
```
```  2556   "numeral m div - numeral n = - (adjust_div (divmod m n) :: int)"
```
```  2557 proof -
```
```  2558   have "int (fst (divmod m n)) = fst (divmod m n)"
```
```  2559     by (simp only: fst_divmod divide_int_def) auto
```
```  2560   then show ?thesis
```
```  2561     by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
```
```  2562 qed
```
```  2563
```
```  2564 lemma numeral_mod_minus_numeral [simp]:
```
```  2565   "numeral m mod - numeral n = - adjust_mod (numeral n) (snd (divmod m n) :: int)"
```
```  2566 proof -
```
```  2567   have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
```
```  2568     using that by (simp only: snd_divmod modulo_int_def) auto
```
```  2569   then show ?thesis
```
```  2570     by (auto simp add: split_def Let_def adjust_div_def divides_aux_def modulo_int_def)
```
```  2571 qed
```
```  2572
```
```  2573 lemma minus_one_div_numeral [simp]:
```
```  2574   "- 1 div numeral n = - (adjust_div (divmod Num.One n) :: int)"
```
```  2575   using minus_numeral_div_numeral [of Num.One n] by simp
```
```  2576
```
```  2577 lemma minus_one_mod_numeral [simp]:
```
```  2578   "- 1 mod numeral n = adjust_mod (numeral n) (snd (divmod Num.One n) :: int)"
```
```  2579   using minus_numeral_mod_numeral [of Num.One n] by simp
```
```  2580
```
```  2581 lemma one_div_minus_numeral [simp]:
```
```  2582   "1 div - numeral n = - (adjust_div (divmod Num.One n) :: int)"
```
```  2583   using numeral_div_minus_numeral [of Num.One n] by simp
```
```  2584
```
```  2585 lemma one_mod_minus_numeral [simp]:
```
```  2586   "1 mod - numeral n = - adjust_mod (numeral n) (snd (divmod Num.One n) :: int)"
```
```  2587   using numeral_mod_minus_numeral [of Num.One n] by simp
```
```  2588
```
```  2589 end
```
```  2590
```
```  2591
```
```  2592 subsubsection \<open>Further properties\<close>
```
```  2593
```
```  2594 text \<open>Simplify expresions in which div and mod combine numerical constants\<close>
```
```  2595
```
```  2596 lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q"
```
```  2597   by (rule div_int_unique [of a b q r]) (simp add: eucl_rel_int_iff)
```
```  2598
```
```  2599 lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q"
```
```  2600   by (rule div_int_unique [of a b q r],
```
```  2601     simp add: eucl_rel_int_iff)
```
```  2602
```
```  2603 lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r"
```
```  2604   by (rule mod_int_unique [of a b q r],
```
```  2605     simp add: eucl_rel_int_iff)
```
```  2606
```
```  2607 lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r"
```
```  2608   by (rule mod_int_unique [of a b q r],
```
```  2609     simp add: eucl_rel_int_iff)
```
```  2610
```
```  2611 lemma abs_div: "(y::int) dvd x \<Longrightarrow> \<bar>x div y\<bar> = \<bar>x\<bar> div \<bar>y\<bar>"
```
```  2612 by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
```
```  2613
```
```  2614 text\<open>Suggested by Matthias Daum\<close>
```
```  2615 lemma int_power_div_base:
```
```  2616      "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
```
```  2617 apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
```
```  2618  apply (erule ssubst)
```
```  2619  apply (simp only: power_add)
```
```  2620  apply simp_all
```
```  2621 done
```
```  2622
```
```  2623 text \<open>Distributive laws for function \<open>nat\<close>.\<close>
```
```  2624
```
```  2625 lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
```
```  2626 apply (rule linorder_cases [of y 0])
```
```  2627 apply (simp add: div_nonneg_neg_le0)
```
```  2628 apply simp
```
```  2629 apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
```
```  2630 done
```
```  2631
```
```  2632 (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
```
```  2633 lemma nat_mod_distrib:
```
```  2634   "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
```
```  2635 apply (case_tac "y = 0", simp)
```
```  2636 apply (simp add: nat_eq_iff zmod_int)
```
```  2637 done
```
```  2638
```
```  2639 text  \<open>transfer setup\<close>
```
```  2640
```
```  2641 lemma transfer_nat_int_functions:
```
```  2642     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
```
```  2643     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
```
```  2644   by (auto simp add: nat_div_distrib nat_mod_distrib)
```
```  2645
```
```  2646 lemma transfer_nat_int_function_closures:
```
```  2647     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
```
```  2648     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
```
```  2649   apply (cases "y = 0")
```
```  2650   apply (auto simp add: pos_imp_zdiv_nonneg_iff)
```
```  2651   apply (cases "y = 0")
```
```  2652   apply auto
```
```  2653 done
```
```  2654
```
```  2655 declare transfer_morphism_nat_int [transfer add return:
```
```  2656   transfer_nat_int_functions
```
```  2657   transfer_nat_int_function_closures
```
```  2658 ]
```
```  2659
```
```  2660 lemma transfer_int_nat_functions:
```
```  2661     "(int x) div (int y) = int (x div y)"
```
```  2662     "(int x) mod (int y) = int (x mod y)"
```
```  2663   by (auto simp add: zdiv_int zmod_int)
```
```  2664
```
```  2665 lemma transfer_int_nat_function_closures:
```
```  2666     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"
```
```  2667     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"
```
```  2668   by (simp_all only: is_nat_def transfer_nat_int_function_closures)
```
```  2669
```
```  2670 declare transfer_morphism_int_nat [transfer add return:
```
```  2671   transfer_int_nat_functions
```
```  2672   transfer_int_nat_function_closures
```
```  2673 ]
```
```  2674
```
```  2675 text\<open>Suggested by Matthias Daum\<close>
```
```  2676 lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
```
```  2677 apply (subgoal_tac "nat x div nat k < nat x")
```
```  2678  apply (simp add: nat_div_distrib [symmetric])
```
```  2679 apply (rule Divides.div_less_dividend, simp_all)
```
```  2680 done
```
```  2681
```
```  2682 lemma (in ring_div) mod_eq_dvd_iff:
```
```  2683   "a mod c = b mod c \<longleftrightarrow> c dvd a - b" (is "?P \<longleftrightarrow> ?Q")
```
```  2684 proof
```
```  2685   assume ?P
```
```  2686   then have "(a mod c - b mod c) mod c = 0"
```
```  2687     by simp
```
```  2688   then show ?Q
```
```  2689     by (simp add: dvd_eq_mod_eq_0 mod_simps)
```
```  2690 next
```
```  2691   assume ?Q
```
```  2692   then obtain d where d: "a - b = c * d" ..
```
```  2693   then have "a = c * d + b"
```
```  2694     by (simp add: algebra_simps)
```
```  2695   then show ?P by simp
```
```  2696 qed
```
```  2697
```
```  2698 lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y mod n" and xy:"y \<le> x"
```
```  2699   shows "\<exists>q. x = y + n * q"
```
```  2700 proof-
```
```  2701   from xy have th: "int x - int y = int (x - y)" by simp
```
```  2702   from xyn have "int x mod int n = int y mod int n"
```
```  2703     by (simp add: zmod_int [symmetric])
```
```  2704   hence "int n dvd int x - int y" by (simp only: mod_eq_dvd_iff [symmetric])
```
```  2705   hence "n dvd x - y" by (simp add: th zdvd_int)
```
```  2706   then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
```
```  2707 qed
```
```  2708
```
```  2709 lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"
```
```  2710   (is "?lhs = ?rhs")
```
```  2711 proof
```
```  2712   assume H: "x mod n = y mod n"
```
```  2713   {assume xy: "x \<le> y"
```
```  2714     from H have th: "y mod n = x mod n" by simp
```
```  2715     from nat_mod_eq_lemma[OF th xy] have ?rhs
```
```  2716       apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
```
```  2717   moreover
```
```  2718   {assume xy: "y \<le> x"
```
```  2719     from nat_mod_eq_lemma[OF H xy] have ?rhs
```
```  2720       apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
```
```  2721   ultimately  show ?rhs using linear[of x y] by blast
```
```  2722 next
```
```  2723   assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
```
```  2724   hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
```
```  2725   thus  ?lhs by simp
```
```  2726 qed
```
```  2727
```
```  2728 subsubsection \<open>Dedicated simproc for calculation\<close>
```
```  2729
```
```  2730 text \<open>
```
```  2731   There is space for improvement here: the calculation itself
```
```  2732   could be carried outside the logic, and a generic simproc
```
```  2733   (simplifier setup) for generic calculation would be helpful.
```
```  2734 \<close>
```
```  2735
```
```  2736 simproc_setup numeral_divmod
```
```  2737   ("0 div 0 :: 'a :: semiring_numeral_div" | "0 mod 0 :: 'a :: semiring_numeral_div" |
```
```  2738    "0 div 1 :: 'a :: semiring_numeral_div" | "0 mod 1 :: 'a :: semiring_numeral_div" |
```
```  2739    "0 div - 1 :: int" | "0 mod - 1 :: int" |
```
```  2740    "0 div numeral b :: 'a :: semiring_numeral_div" | "0 mod numeral b :: 'a :: semiring_numeral_div" |
```
```  2741    "0 div - numeral b :: int" | "0 mod - numeral b :: int" |
```
```  2742    "1 div 0 :: 'a :: semiring_numeral_div" | "1 mod 0 :: 'a :: semiring_numeral_div" |
```
```  2743    "1 div 1 :: 'a :: semiring_numeral_div" | "1 mod 1 :: 'a :: semiring_numeral_div" |
```
```  2744    "1 div - 1 :: int" | "1 mod - 1 :: int" |
```
```  2745    "1 div numeral b :: 'a :: semiring_numeral_div" | "1 mod numeral b :: 'a :: semiring_numeral_div" |
```
```  2746    "1 div - numeral b :: int" |"1 mod - numeral b :: int" |
```
```  2747    "- 1 div 0 :: int" | "- 1 mod 0 :: int" | "- 1 div 1 :: int" | "- 1 mod 1 :: int" |
```
```  2748    "- 1 div - 1 :: int" | "- 1 mod - 1 :: int" | "- 1 div numeral b :: int" | "- 1 mod numeral b :: int" |
```
```  2749    "- 1 div - numeral b :: int" | "- 1 mod - numeral b :: int" |
```
```  2750    "numeral a div 0 :: 'a :: semiring_numeral_div" | "numeral a mod 0 :: 'a :: semiring_numeral_div" |
```
```  2751    "numeral a div 1 :: 'a :: semiring_numeral_div" | "numeral a mod 1 :: 'a :: semiring_numeral_div" |
```
```  2752    "numeral a div - 1 :: int" | "numeral a mod - 1 :: int" |
```
```  2753    "numeral a div numeral b :: 'a :: semiring_numeral_div" | "numeral a mod numeral b :: 'a :: semiring_numeral_div" |
```
```  2754    "numeral a div - numeral b :: int" | "numeral a mod - numeral b :: int" |
```
```  2755    "- numeral a div 0 :: int" | "- numeral a mod 0 :: int" |
```
```  2756    "- numeral a div 1 :: int" | "- numeral a mod 1 :: int" |
```
```  2757    "- numeral a div - 1 :: int" | "- numeral a mod - 1 :: int" |
```
```  2758    "- numeral a div numeral b :: int" | "- numeral a mod numeral b :: int" |
```
```  2759    "- numeral a div - numeral b :: int" | "- numeral a mod - numeral b :: int") =
```
```  2760 \<open> let
```
```  2761     val if_cong = the (Code.get_case_cong @{theory} @{const_name If});
```
```  2762     fun successful_rewrite ctxt ct =
```
```  2763       let
```
```  2764         val thm = Simplifier.rewrite ctxt ct
```
```  2765       in if Thm.is_reflexive thm then NONE else SOME thm end;
```
```  2766   in fn phi =>
```
```  2767     let
```
```  2768       val simps = Morphism.fact phi (@{thms div_0 mod_0 div_by_0 mod_by_0 div_by_1 mod_by_1
```
```  2769         one_div_numeral one_mod_numeral minus_one_div_numeral minus_one_mod_numeral
```
```  2770         one_div_minus_numeral one_mod_minus_numeral
```
```  2771         numeral_div_numeral numeral_mod_numeral minus_numeral_div_numeral minus_numeral_mod_numeral
```
```  2772         numeral_div_minus_numeral numeral_mod_minus_numeral
```
```  2773         div_minus_minus mod_minus_minus Divides.adjust_div_eq of_bool_eq one_neq_zero
```
```  2774         numeral_neq_zero neg_equal_0_iff_equal arith_simps arith_special divmod_trivial
```
```  2775         divmod_cancel divmod_steps divmod_step_eq fst_conv snd_conv numeral_One
```
```  2776         case_prod_beta rel_simps Divides.adjust_mod_def div_minus1_right mod_minus1_right
```
```  2777         minus_minus numeral_times_numeral mult_zero_right mult_1_right}
```
```  2778         @ [@{lemma "0 = 0 \<longleftrightarrow> True" by simp}]);
```
```  2779       fun prepare_simpset ctxt = HOL_ss |> Simplifier.simpset_map ctxt
```
```  2780         (Simplifier.add_cong if_cong #> fold Simplifier.add_simp simps)
```
```  2781     in fn ctxt => successful_rewrite (Simplifier.put_simpset (prepare_simpset ctxt) ctxt) end
```
```  2782   end;
```
```  2783 \<close>
```
```  2784
```
```  2785
```
```  2786 subsubsection \<open>Code generation\<close>
```
```  2787
```
```  2788 lemma [code]:
```
```  2789   fixes k :: int
```
```  2790   shows
```
```  2791     "k div 0 = 0"
```
```  2792     "k mod 0 = k"
```
```  2793     "0 div k = 0"
```
```  2794     "0 mod k = 0"
```
```  2795     "k div Int.Pos Num.One = k"
```
```  2796     "k mod Int.Pos Num.One = 0"
```
```  2797     "k div Int.Neg Num.One = - k"
```
```  2798     "k mod Int.Neg Num.One = 0"
```
```  2799     "Int.Pos m div Int.Pos n = (fst (divmod m n) :: int)"
```
```  2800     "Int.Pos m mod Int.Pos n = (snd (divmod m n) :: int)"
```
```  2801     "Int.Neg m div Int.Pos n = - (Divides.adjust_div (divmod m n) :: int)"
```
```  2802     "Int.Neg m mod Int.Pos n = Divides.adjust_mod (Int.Pos n) (snd (divmod m n) :: int)"
```
```  2803     "Int.Pos m div Int.Neg n = - (Divides.adjust_div (divmod m n) :: int)"
```
```  2804     "Int.Pos m mod Int.Neg n = - Divides.adjust_mod (Int.Pos n) (snd (divmod m n) :: int)"
```
```  2805     "Int.Neg m div Int.Neg n = (fst (divmod m n) :: int)"
```
```  2806     "Int.Neg m mod Int.Neg n = - (snd (divmod m n) :: int)"
```
```  2807   by simp_all
```
```  2808
```
```  2809 code_identifier
```
```  2810   code_module Divides \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
```
```  2811
```
```  2812 lemma dvd_eq_mod_eq_0_numeral:
```
```  2813   "numeral x dvd (numeral y :: 'a) \<longleftrightarrow> numeral y mod numeral x = (0 :: 'a::semiring_div)"
```
```  2814   by (fact dvd_eq_mod_eq_0)
```
```  2815
```
```  2816 declare minus_div_mult_eq_mod [symmetric, nitpick_unfold]
```
```  2817
```
```  2818 end
```