src/HOL/Divides.thy
author haftmann
Sun Oct 16 09:31:06 2016 +0200 (2016-10-16)
changeset 64246 15d1ee6e847b
parent 64244 e7102c40783c
child 64250 0cde0b4d4cb5
permissions -rw-r--r--
eliminated irregular aliasses
paulson@3366
     1
(*  Title:      HOL/Divides.thy
paulson@3366
     2
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
paulson@6865
     3
    Copyright   1999  University of Cambridge
huffman@18154
     4
*)
paulson@3366
     5
wenzelm@60758
     6
section \<open>The division operators div and mod\<close>
paulson@3366
     7
nipkow@15131
     8
theory Divides
haftmann@58778
     9
imports Parity
nipkow@15131
    10
begin
paulson@3366
    11
wenzelm@60758
    12
subsection \<open>Abstract division in commutative semirings.\<close>
haftmann@25942
    13
haftmann@64164
    14
class semiring_div = semidom + semiring_modulo +
haftmann@64164
    15
  assumes div_by_0: "a div 0 = 0"
haftmann@63950
    16
    and div_0: "0 div a = 0"
haftmann@27651
    17
    and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"
haftmann@30930
    18
    and div_mult_mult1 [simp]: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b"
haftmann@25942
    19
begin
haftmann@25942
    20
haftmann@60517
    21
subclass algebraic_semidom
haftmann@60353
    22
proof
haftmann@60353
    23
  fix b a
haftmann@60353
    24
  assume "b \<noteq> 0"
haftmann@60353
    25
  then show "a * b div b = a"
haftmann@63950
    26
    using div_mult_self1 [of b 0 a] by (simp add: ac_simps div_0)
haftmann@63950
    27
qed (simp add: div_by_0)
haftmann@58953
    28
haftmann@63950
    29
text \<open>@{const divide} and @{const modulo}\<close>
haftmann@26100
    30
haftmann@26062
    31
lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
haftmann@64242
    32
  by (simp add: div_mult_mod_eq)
haftmann@26062
    33
haftmann@26062
    34
lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"
haftmann@64242
    35
  by (simp add: mult_div_mod_eq)
haftmann@26062
    36
haftmann@27651
    37
lemma mod_by_0 [simp]: "a mod 0 = a"
haftmann@64242
    38
  using div_mult_mod_eq [of a zero] by simp
haftmann@27651
    39
haftmann@27651
    40
lemma mod_0 [simp]: "0 mod a = 0"
haftmann@64242
    41
  using div_mult_mod_eq [of zero a] div_0 by simp
haftmann@27651
    42
haftmann@27651
    43
lemma div_mult_self2 [simp]:
haftmann@27651
    44
  assumes "b \<noteq> 0"
haftmann@27651
    45
  shows "(a + b * c) div b = c + a div b"
haftmann@57512
    46
  using assms div_mult_self1 [of b a c] by (simp add: mult.commute)
haftmann@26100
    47
haftmann@54221
    48
lemma div_mult_self3 [simp]:
haftmann@54221
    49
  assumes "b \<noteq> 0"
haftmann@54221
    50
  shows "(c * b + a) div b = c + a div b"
haftmann@54221
    51
  using assms by (simp add: add.commute)
haftmann@54221
    52
haftmann@54221
    53
lemma div_mult_self4 [simp]:
haftmann@54221
    54
  assumes "b \<noteq> 0"
haftmann@54221
    55
  shows "(b * c + a) div b = c + a div b"
haftmann@54221
    56
  using assms by (simp add: add.commute)
haftmann@54221
    57
haftmann@27651
    58
lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"
haftmann@27651
    59
proof (cases "b = 0")
haftmann@27651
    60
  case True then show ?thesis by simp
haftmann@27651
    61
next
haftmann@27651
    62
  case False
haftmann@27651
    63
  have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"
haftmann@64242
    64
    by (simp add: div_mult_mod_eq)
haftmann@27651
    65
  also from False div_mult_self1 [of b a c] have
haftmann@27651
    66
    "\<dots> = (c + a div b) * b + (a + c * b) mod b"
nipkow@29667
    67
      by (simp add: algebra_simps)
haftmann@27651
    68
  finally have "a = a div b * b + (a + c * b) mod b"
haftmann@57512
    69
    by (simp add: add.commute [of a] add.assoc distrib_right)
haftmann@27651
    70
  then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
haftmann@64242
    71
    by (simp add: div_mult_mod_eq)
haftmann@27651
    72
  then show ?thesis by simp
haftmann@27651
    73
qed
haftmann@27651
    74
lp15@60562
    75
lemma mod_mult_self2 [simp]:
haftmann@54221
    76
  "(a + b * c) mod b = a mod b"
haftmann@57512
    77
  by (simp add: mult.commute [of b])
haftmann@27651
    78
haftmann@54221
    79
lemma mod_mult_self3 [simp]:
haftmann@54221
    80
  "(c * b + a) mod b = a mod b"
haftmann@54221
    81
  by (simp add: add.commute)
haftmann@54221
    82
haftmann@54221
    83
lemma mod_mult_self4 [simp]:
haftmann@54221
    84
  "(b * c + a) mod b = a mod b"
haftmann@54221
    85
  by (simp add: add.commute)
haftmann@54221
    86
haftmann@60867
    87
lemma mod_mult_self1_is_0 [simp]:
haftmann@60867
    88
  "b * a mod b = 0"
haftmann@27651
    89
  using mod_mult_self2 [of 0 b a] by simp
haftmann@27651
    90
haftmann@60867
    91
lemma mod_mult_self2_is_0 [simp]:
haftmann@60867
    92
  "a * b mod b = 0"
haftmann@27651
    93
  using mod_mult_self1 [of 0 a b] by simp
haftmann@26062
    94
haftmann@60867
    95
lemma mod_by_1 [simp]:
haftmann@60867
    96
  "a mod 1 = 0"
haftmann@27651
    97
proof -
haftmann@64242
    98
  from div_mult_mod_eq [of a one] div_by_1 have "a + a mod 1 = a" by simp
haftmann@27651
    99
  then have "a + a mod 1 = a + 0" by simp
haftmann@27651
   100
  then show ?thesis by (rule add_left_imp_eq)
haftmann@27651
   101
qed
haftmann@27651
   102
haftmann@60867
   103
lemma mod_self [simp]:
haftmann@60867
   104
  "a mod a = 0"
haftmann@27651
   105
  using mod_mult_self2_is_0 [of 1] by simp
haftmann@27651
   106
eberlm@63499
   107
lemma div_add_self1:
haftmann@27651
   108
  assumes "b \<noteq> 0"
haftmann@27651
   109
  shows "(b + a) div b = a div b + 1"
haftmann@57512
   110
  using assms div_mult_self1 [of b a 1] by (simp add: add.commute)
haftmann@26062
   111
eberlm@63499
   112
lemma div_add_self2:
haftmann@27651
   113
  assumes "b \<noteq> 0"
haftmann@27651
   114
  shows "(a + b) div b = a div b + 1"
haftmann@57512
   115
  using assms div_add_self1 [of b a] by (simp add: add.commute)
haftmann@27651
   116
haftmann@27676
   117
lemma mod_add_self1 [simp]:
haftmann@27651
   118
  "(b + a) mod b = a mod b"
haftmann@57512
   119
  using mod_mult_self1 [of a 1 b] by (simp add: add.commute)
haftmann@27651
   120
haftmann@27676
   121
lemma mod_add_self2 [simp]:
haftmann@27651
   122
  "(a + b) mod b = a mod b"
haftmann@27651
   123
  using mod_mult_self1 [of a 1 b] by simp
haftmann@27651
   124
haftmann@58834
   125
lemma dvd_imp_mod_0 [simp]:
haftmann@58834
   126
  assumes "a dvd b"
haftmann@58834
   127
  shows "b mod a = 0"
haftmann@58834
   128
proof -
haftmann@58834
   129
  from assms obtain c where "b = a * c" ..
haftmann@58834
   130
  then have "b mod a = a * c mod a" by simp
haftmann@58834
   131
  then show "b mod a = 0" by simp
haftmann@58834
   132
qed
haftmann@58911
   133
haftmann@58911
   134
lemma mod_eq_0_iff_dvd:
haftmann@58911
   135
  "a mod b = 0 \<longleftrightarrow> b dvd a"
haftmann@58911
   136
proof
haftmann@58911
   137
  assume "b dvd a"
haftmann@58911
   138
  then show "a mod b = 0" by simp
haftmann@58911
   139
next
haftmann@58911
   140
  assume "a mod b = 0"
haftmann@64242
   141
  with div_mult_mod_eq [of a b] have "a div b * b = a" by simp
haftmann@58911
   142
  then have "a = b * (a div b)" by (simp add: ac_simps)
haftmann@58911
   143
  then show "b dvd a" ..
haftmann@58911
   144
qed
haftmann@58911
   145
haftmann@60867
   146
lemma dvd_eq_mod_eq_0 [nitpick_unfold, code]:
haftmann@58834
   147
  "a dvd b \<longleftrightarrow> b mod a = 0"
haftmann@58911
   148
  by (simp add: mod_eq_0_iff_dvd)
haftmann@58911
   149
haftmann@58911
   150
lemma mod_div_trivial [simp]:
haftmann@58911
   151
  "a mod b div b = 0"
huffman@29403
   152
proof (cases "b = 0")
huffman@29403
   153
  assume "b = 0"
huffman@29403
   154
  thus ?thesis by simp
huffman@29403
   155
next
huffman@29403
   156
  assume "b \<noteq> 0"
huffman@29403
   157
  hence "a div b + a mod b div b = (a mod b + a div b * b) div b"
huffman@29403
   158
    by (rule div_mult_self1 [symmetric])
huffman@29403
   159
  also have "\<dots> = a div b"
haftmann@64242
   160
    by (simp only: mod_div_mult_eq)
huffman@29403
   161
  also have "\<dots> = a div b + 0"
huffman@29403
   162
    by simp
huffman@29403
   163
  finally show ?thesis
huffman@29403
   164
    by (rule add_left_imp_eq)
huffman@29403
   165
qed
huffman@29403
   166
haftmann@58911
   167
lemma mod_mod_trivial [simp]:
haftmann@58911
   168
  "a mod b mod b = a mod b"
huffman@29403
   169
proof -
huffman@29403
   170
  have "a mod b mod b = (a mod b + a div b * b) mod b"
huffman@29403
   171
    by (simp only: mod_mult_self1)
huffman@29403
   172
  also have "\<dots> = a mod b"
haftmann@64242
   173
    by (simp only: mod_div_mult_eq)
huffman@29403
   174
  finally show ?thesis .
huffman@29403
   175
qed
huffman@29403
   176
haftmann@58834
   177
lemma dvd_mod_imp_dvd:
haftmann@58834
   178
  assumes "k dvd m mod n" and "k dvd n"
haftmann@58834
   179
  shows "k dvd m"
haftmann@58834
   180
proof -
haftmann@58834
   181
  from assms have "k dvd (m div n) * n + m mod n"
haftmann@58834
   182
    by (simp only: dvd_add dvd_mult)
haftmann@64242
   183
  then show ?thesis by (simp add: div_mult_mod_eq)
haftmann@58834
   184
qed
huffman@30078
   185
wenzelm@60758
   186
text \<open>Addition respects modular equivalence.\<close>
huffman@29403
   187
wenzelm@61799
   188
lemma mod_add_left_eq: \<comment> \<open>FIXME reorient\<close>
haftmann@60867
   189
  "(a + b) mod c = (a mod c + b) mod c"
huffman@29403
   190
proof -
huffman@29403
   191
  have "(a + b) mod c = (a div c * c + a mod c + b) mod c"
haftmann@64242
   192
    by (simp only: div_mult_mod_eq)
huffman@29403
   193
  also have "\<dots> = (a mod c + b + a div c * c) mod c"
haftmann@57514
   194
    by (simp only: ac_simps)
huffman@29403
   195
  also have "\<dots> = (a mod c + b) mod c"
huffman@29403
   196
    by (rule mod_mult_self1)
huffman@29403
   197
  finally show ?thesis .
huffman@29403
   198
qed
huffman@29403
   199
wenzelm@61799
   200
lemma mod_add_right_eq: \<comment> \<open>FIXME reorient\<close>
haftmann@60867
   201
  "(a + b) mod c = (a + b mod c) mod c"
huffman@29403
   202
proof -
huffman@29403
   203
  have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c"
haftmann@64242
   204
    by (simp only: div_mult_mod_eq)
huffman@29403
   205
  also have "\<dots> = (a + b mod c + b div c * c) mod c"
haftmann@57514
   206
    by (simp only: ac_simps)
huffman@29403
   207
  also have "\<dots> = (a + b mod c) mod c"
huffman@29403
   208
    by (rule mod_mult_self1)
huffman@29403
   209
  finally show ?thesis .
huffman@29403
   210
qed
huffman@29403
   211
wenzelm@61799
   212
lemma mod_add_eq: \<comment> \<open>FIXME reorient\<close>
haftmann@60867
   213
  "(a + b) mod c = (a mod c + b mod c) mod c"
huffman@29403
   214
by (rule trans [OF mod_add_left_eq mod_add_right_eq])
huffman@29403
   215
huffman@29403
   216
lemma mod_add_cong:
huffman@29403
   217
  assumes "a mod c = a' mod c"
huffman@29403
   218
  assumes "b mod c = b' mod c"
huffman@29403
   219
  shows "(a + b) mod c = (a' + b') mod c"
huffman@29403
   220
proof -
huffman@29403
   221
  have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c"
huffman@29403
   222
    unfolding assms ..
huffman@29403
   223
  thus ?thesis
huffman@29403
   224
    by (simp only: mod_add_eq [symmetric])
huffman@29403
   225
qed
huffman@29403
   226
wenzelm@60758
   227
text \<open>Multiplication respects modular equivalence.\<close>
huffman@29403
   228
wenzelm@61799
   229
lemma mod_mult_left_eq: \<comment> \<open>FIXME reorient\<close>
haftmann@60867
   230
  "(a * b) mod c = ((a mod c) * b) mod c"
huffman@29403
   231
proof -
huffman@29403
   232
  have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c"
haftmann@64242
   233
    by (simp only: div_mult_mod_eq)
huffman@29403
   234
  also have "\<dots> = (a mod c * b + a div c * b * c) mod c"
nipkow@29667
   235
    by (simp only: algebra_simps)
huffman@29403
   236
  also have "\<dots> = (a mod c * b) mod c"
huffman@29403
   237
    by (rule mod_mult_self1)
huffman@29403
   238
  finally show ?thesis .
huffman@29403
   239
qed
huffman@29403
   240
wenzelm@61799
   241
lemma mod_mult_right_eq: \<comment> \<open>FIXME reorient\<close>
haftmann@60867
   242
  "(a * b) mod c = (a * (b mod c)) mod c"
huffman@29403
   243
proof -
huffman@29403
   244
  have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c"
haftmann@64242
   245
    by (simp only: div_mult_mod_eq)
huffman@29403
   246
  also have "\<dots> = (a * (b mod c) + a * (b div c) * c) mod c"
nipkow@29667
   247
    by (simp only: algebra_simps)
huffman@29403
   248
  also have "\<dots> = (a * (b mod c)) mod c"
huffman@29403
   249
    by (rule mod_mult_self1)
huffman@29403
   250
  finally show ?thesis .
huffman@29403
   251
qed
huffman@29403
   252
wenzelm@61799
   253
lemma mod_mult_eq: \<comment> \<open>FIXME reorient\<close>
haftmann@60867
   254
  "(a * b) mod c = ((a mod c) * (b mod c)) mod c"
huffman@29403
   255
by (rule trans [OF mod_mult_left_eq mod_mult_right_eq])
huffman@29403
   256
huffman@29403
   257
lemma mod_mult_cong:
huffman@29403
   258
  assumes "a mod c = a' mod c"
huffman@29403
   259
  assumes "b mod c = b' mod c"
huffman@29403
   260
  shows "(a * b) mod c = (a' * b') mod c"
huffman@29403
   261
proof -
huffman@29403
   262
  have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c"
huffman@29403
   263
    unfolding assms ..
huffman@29403
   264
  thus ?thesis
huffman@29403
   265
    by (simp only: mod_mult_eq [symmetric])
huffman@29403
   266
qed
huffman@29403
   267
wenzelm@60758
   268
text \<open>Exponentiation respects modular equivalence.\<close>
huffman@47164
   269
haftmann@60867
   270
lemma power_mod: "(a mod b) ^ n mod b = a ^ n mod b"
huffman@47164
   271
apply (induct n, simp_all)
huffman@47164
   272
apply (rule mod_mult_right_eq [THEN trans])
huffman@47164
   273
apply (simp (no_asm_simp))
huffman@47164
   274
apply (rule mod_mult_eq [symmetric])
huffman@47164
   275
done
huffman@47164
   276
huffman@29404
   277
lemma mod_mod_cancel:
huffman@29404
   278
  assumes "c dvd b"
huffman@29404
   279
  shows "a mod b mod c = a mod c"
huffman@29404
   280
proof -
wenzelm@60758
   281
  from \<open>c dvd b\<close> obtain k where "b = c * k"
huffman@29404
   282
    by (rule dvdE)
huffman@29404
   283
  have "a mod b mod c = a mod (c * k) mod c"
wenzelm@60758
   284
    by (simp only: \<open>b = c * k\<close>)
huffman@29404
   285
  also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c"
huffman@29404
   286
    by (simp only: mod_mult_self1)
huffman@29404
   287
  also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c"
haftmann@58786
   288
    by (simp only: ac_simps)
huffman@29404
   289
  also have "\<dots> = a mod c"
haftmann@64242
   290
    by (simp only: div_mult_mod_eq)
huffman@29404
   291
  finally show ?thesis .
huffman@29404
   292
qed
huffman@29404
   293
haftmann@30930
   294
lemma div_mult_mult2 [simp]:
haftmann@30930
   295
  "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
haftmann@57512
   296
  by (drule div_mult_mult1) (simp add: mult.commute)
haftmann@30930
   297
haftmann@30930
   298
lemma div_mult_mult1_if [simp]:
haftmann@30930
   299
  "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"
haftmann@30930
   300
  by simp_all
nipkow@30476
   301
haftmann@30930
   302
lemma mod_mult_mult1:
haftmann@30930
   303
  "(c * a) mod (c * b) = c * (a mod b)"
haftmann@30930
   304
proof (cases "c = 0")
haftmann@30930
   305
  case True then show ?thesis by simp
haftmann@30930
   306
next
haftmann@30930
   307
  case False
haftmann@64242
   308
  from div_mult_mod_eq
haftmann@30930
   309
  have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .
haftmann@30930
   310
  with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)
haftmann@30930
   311
    = c * a + c * (a mod b)" by (simp add: algebra_simps)
haftmann@64242
   312
  with div_mult_mod_eq show ?thesis by simp
haftmann@30930
   313
qed
lp15@60562
   314
haftmann@30930
   315
lemma mod_mult_mult2:
haftmann@30930
   316
  "(a * c) mod (b * c) = (a mod b) * c"
haftmann@57512
   317
  using mod_mult_mult1 [of c a b] by (simp add: mult.commute)
haftmann@30930
   318
huffman@47159
   319
lemma mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)"
huffman@47159
   320
  by (fact mod_mult_mult2 [symmetric])
huffman@47159
   321
huffman@47159
   322
lemma mult_mod_right: "c * (a mod b) = (c * a) mod (c * b)"
huffman@47159
   323
  by (fact mod_mult_mult1 [symmetric])
huffman@47159
   324
huffman@31662
   325
lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)"
huffman@31662
   326
  unfolding dvd_def by (auto simp add: mod_mult_mult1)
huffman@31662
   327
huffman@31662
   328
lemma dvd_mod_iff: "k dvd n \<Longrightarrow> k dvd (m mod n) \<longleftrightarrow> k dvd m"
huffman@31662
   329
by (blast intro: dvd_mod_imp_dvd dvd_mod)
huffman@31662
   330
eberlm@63317
   331
lemma div_div_eq_right:
eberlm@63317
   332
  assumes "c dvd b" "b dvd a"
eberlm@63317
   333
  shows   "a div (b div c) = a div b * c"
eberlm@63317
   334
proof -
eberlm@63317
   335
  from assms have "a div b * c = (a * c) div b"
eberlm@63317
   336
    by (subst dvd_div_mult) simp_all
eberlm@63317
   337
  also from assms have "\<dots> = (a * c) div ((b div c) * c)" by simp
eberlm@63317
   338
  also have "a * c div (b div c * c) = a div (b div c)"
eberlm@63317
   339
    by (cases "c = 0") simp_all
eberlm@63317
   340
  finally show ?thesis ..
eberlm@63317
   341
qed
eberlm@63317
   342
eberlm@63317
   343
lemma div_div_div_same:
eberlm@63317
   344
  assumes "d dvd a" "d dvd b" "b dvd a"
eberlm@63317
   345
  shows   "(a div d) div (b div d) = a div b"
eberlm@63317
   346
  using assms by (subst dvd_div_mult2_eq [symmetric]) simp_all
eberlm@63317
   347
huffman@31661
   348
end
huffman@31661
   349
haftmann@59833
   350
class ring_div = comm_ring_1 + semiring_div
huffman@29405
   351
begin
huffman@29405
   352
haftmann@60353
   353
subclass idom_divide ..
haftmann@36634
   354
wenzelm@60758
   355
text \<open>Negation respects modular equivalence.\<close>
huffman@29405
   356
huffman@29405
   357
lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"
huffman@29405
   358
proof -
huffman@29405
   359
  have "(- a) mod b = (- (a div b * b + a mod b)) mod b"
haftmann@64242
   360
    by (simp only: div_mult_mod_eq)
huffman@29405
   361
  also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b"
haftmann@57514
   362
    by (simp add: ac_simps)
huffman@29405
   363
  also have "\<dots> = (- (a mod b)) mod b"
huffman@29405
   364
    by (rule mod_mult_self1)
huffman@29405
   365
  finally show ?thesis .
huffman@29405
   366
qed
huffman@29405
   367
huffman@29405
   368
lemma mod_minus_cong:
huffman@29405
   369
  assumes "a mod b = a' mod b"
huffman@29405
   370
  shows "(- a) mod b = (- a') mod b"
huffman@29405
   371
proof -
huffman@29405
   372
  have "(- (a mod b)) mod b = (- (a' mod b)) mod b"
huffman@29405
   373
    unfolding assms ..
huffman@29405
   374
  thus ?thesis
huffman@29405
   375
    by (simp only: mod_minus_eq [symmetric])
huffman@29405
   376
qed
huffman@29405
   377
wenzelm@60758
   378
text \<open>Subtraction respects modular equivalence.\<close>
huffman@29405
   379
haftmann@54230
   380
lemma mod_diff_left_eq:
haftmann@54230
   381
  "(a - b) mod c = (a mod c - b) mod c"
haftmann@54230
   382
  using mod_add_cong [of a c "a mod c" "- b" "- b"] by simp
haftmann@54230
   383
haftmann@54230
   384
lemma mod_diff_right_eq:
haftmann@54230
   385
  "(a - b) mod c = (a - b mod c) mod c"
haftmann@54230
   386
  using mod_add_cong [of a c a "- b" "- (b mod c)"] mod_minus_cong [of "b mod c" c b] by simp
haftmann@54230
   387
haftmann@54230
   388
lemma mod_diff_eq:
haftmann@54230
   389
  "(a - b) mod c = (a mod c - b mod c) mod c"
haftmann@54230
   390
  using mod_add_cong [of a c "a mod c" "- b" "- (b mod c)"] mod_minus_cong [of "b mod c" c b] by simp
huffman@29405
   391
huffman@29405
   392
lemma mod_diff_cong:
huffman@29405
   393
  assumes "a mod c = a' mod c"
huffman@29405
   394
  assumes "b mod c = b' mod c"
huffman@29405
   395
  shows "(a - b) mod c = (a' - b') mod c"
haftmann@54230
   396
  using assms mod_add_cong [of a c a' "- b" "- b'"] mod_minus_cong [of b c "b'"] by simp
huffman@29405
   397
nipkow@30180
   398
lemma dvd_neg_div: "y dvd x \<Longrightarrow> -x div y = - (x div y)"
nipkow@30180
   399
apply (case_tac "y = 0") apply simp
nipkow@30180
   400
apply (auto simp add: dvd_def)
nipkow@30180
   401
apply (subgoal_tac "-(y * k) = y * - k")
thomas@57492
   402
 apply (simp only:)
haftmann@64240
   403
 apply (erule nonzero_mult_div_cancel_left)
nipkow@30180
   404
apply simp
nipkow@30180
   405
done
nipkow@30180
   406
nipkow@30180
   407
lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div -y = - (x div y)"
nipkow@30180
   408
apply (case_tac "y = 0") apply simp
nipkow@30180
   409
apply (auto simp add: dvd_def)
nipkow@30180
   410
apply (subgoal_tac "y * k = -y * -k")
haftmann@64240
   411
 apply (erule ssubst, rule nonzero_mult_div_cancel_left)
nipkow@30180
   412
 apply simp
nipkow@30180
   413
apply simp
nipkow@30180
   414
done
nipkow@30180
   415
haftmann@60867
   416
lemma div_diff [simp]:
haftmann@60867
   417
  "z dvd x \<Longrightarrow> z dvd y \<Longrightarrow> (x - y) div z = x div z - y div z"
haftmann@60867
   418
  using div_add [of _ _ "- y"] by (simp add: dvd_neg_div)
nipkow@59380
   419
huffman@47159
   420
lemma div_minus_minus [simp]: "(-a) div (-b) = a div b"
huffman@47159
   421
  using div_mult_mult1 [of "- 1" a b]
huffman@47159
   422
  unfolding neg_equal_0_iff_equal by simp
huffman@47159
   423
huffman@47159
   424
lemma mod_minus_minus [simp]: "(-a) mod (-b) = - (a mod b)"
huffman@47159
   425
  using mod_mult_mult1 [of "- 1" a b] by simp
huffman@47159
   426
huffman@47159
   427
lemma div_minus_right: "a div (-b) = (-a) div b"
huffman@47159
   428
  using div_minus_minus [of "-a" b] by simp
huffman@47159
   429
huffman@47159
   430
lemma mod_minus_right: "a mod (-b) = - ((-a) mod b)"
huffman@47159
   431
  using mod_minus_minus [of "-a" b] by simp
huffman@47159
   432
huffman@47160
   433
lemma div_minus1_right [simp]: "a div (-1) = -a"
huffman@47160
   434
  using div_minus_right [of a 1] by simp
huffman@47160
   435
huffman@47160
   436
lemma mod_minus1_right [simp]: "a mod (-1) = 0"
huffman@47160
   437
  using mod_minus_right [of a 1] by simp
huffman@47160
   438
lp15@60562
   439
lemma minus_mod_self2 [simp]:
haftmann@54221
   440
  "(a - b) mod b = a mod b"
haftmann@54221
   441
  by (simp add: mod_diff_right_eq)
haftmann@54221
   442
lp15@60562
   443
lemma minus_mod_self1 [simp]:
haftmann@54221
   444
  "(b - a) mod b = - a mod b"
haftmann@54230
   445
  using mod_add_self2 [of "- a" b] by simp
haftmann@54221
   446
huffman@29405
   447
end
huffman@29405
   448
haftmann@58778
   449
wenzelm@60758
   450
subsubsection \<open>Parity and division\<close>
haftmann@58778
   451
lp15@60562
   452
class semiring_div_parity = semiring_div + comm_semiring_1_cancel + numeral +
haftmann@54226
   453
  assumes parity: "a mod 2 = 0 \<or> a mod 2 = 1"
haftmann@58786
   454
  assumes one_mod_two_eq_one [simp]: "1 mod 2 = 1"
haftmann@58710
   455
  assumes zero_not_eq_two: "0 \<noteq> 2"
haftmann@54226
   456
begin
haftmann@54226
   457
haftmann@54226
   458
lemma parity_cases [case_names even odd]:
haftmann@54226
   459
  assumes "a mod 2 = 0 \<Longrightarrow> P"
haftmann@54226
   460
  assumes "a mod 2 = 1 \<Longrightarrow> P"
haftmann@54226
   461
  shows P
haftmann@54226
   462
  using assms parity by blast
haftmann@54226
   463
haftmann@58786
   464
lemma one_div_two_eq_zero [simp]:
haftmann@58778
   465
  "1 div 2 = 0"
haftmann@58778
   466
proof (cases "2 = 0")
haftmann@58778
   467
  case True then show ?thesis by simp
haftmann@58778
   468
next
haftmann@58778
   469
  case False
haftmann@64242
   470
  from div_mult_mod_eq have "1 div 2 * 2 + 1 mod 2 = 1" .
haftmann@58778
   471
  with one_mod_two_eq_one have "1 div 2 * 2 + 1 = 1" by simp
haftmann@58953
   472
  then have "1 div 2 * 2 = 0" by (simp add: ac_simps add_left_imp_eq del: mult_eq_0_iff)
haftmann@58953
   473
  then have "1 div 2 = 0 \<or> 2 = 0" by simp
haftmann@58778
   474
  with False show ?thesis by auto
haftmann@58778
   475
qed
haftmann@58778
   476
haftmann@58786
   477
lemma not_mod_2_eq_0_eq_1 [simp]:
haftmann@58786
   478
  "a mod 2 \<noteq> 0 \<longleftrightarrow> a mod 2 = 1"
haftmann@58786
   479
  by (cases a rule: parity_cases) simp_all
haftmann@58786
   480
haftmann@58786
   481
lemma not_mod_2_eq_1_eq_0 [simp]:
haftmann@58786
   482
  "a mod 2 \<noteq> 1 \<longleftrightarrow> a mod 2 = 0"
haftmann@58786
   483
  by (cases a rule: parity_cases) simp_all
haftmann@58786
   484
haftmann@58778
   485
subclass semiring_parity
haftmann@58778
   486
proof (unfold_locales, unfold dvd_eq_mod_eq_0 not_mod_2_eq_0_eq_1)
haftmann@58778
   487
  show "1 mod 2 = 1"
haftmann@58778
   488
    by (fact one_mod_two_eq_one)
haftmann@58778
   489
next
haftmann@58778
   490
  fix a b
haftmann@58778
   491
  assume "a mod 2 = 1"
haftmann@58778
   492
  moreover assume "b mod 2 = 1"
haftmann@58778
   493
  ultimately show "(a + b) mod 2 = 0"
haftmann@58778
   494
    using mod_add_eq [of a b 2] by simp
haftmann@58778
   495
next
haftmann@58778
   496
  fix a b
haftmann@58778
   497
  assume "(a * b) mod 2 = 0"
haftmann@58778
   498
  then have "(a mod 2) * (b mod 2) = 0"
haftmann@58778
   499
    by (cases "a mod 2 = 0") (simp_all add: mod_mult_eq [of a b 2])
haftmann@58778
   500
  then show "a mod 2 = 0 \<or> b mod 2 = 0"
haftmann@58778
   501
    by (rule divisors_zero)
haftmann@58778
   502
next
haftmann@58778
   503
  fix a
haftmann@58778
   504
  assume "a mod 2 = 1"
haftmann@64242
   505
  then have "a = a div 2 * 2 + 1" using div_mult_mod_eq [of a 2] by simp
haftmann@58778
   506
  then show "\<exists>b. a = b + 1" ..
haftmann@58778
   507
qed
haftmann@58778
   508
haftmann@58778
   509
lemma even_iff_mod_2_eq_zero:
haftmann@58778
   510
  "even a \<longleftrightarrow> a mod 2 = 0"
haftmann@58778
   511
  by (fact dvd_eq_mod_eq_0)
haftmann@58778
   512
haftmann@64014
   513
lemma odd_iff_mod_2_eq_one:
haftmann@64014
   514
  "odd a \<longleftrightarrow> a mod 2 = 1"
haftmann@64014
   515
  by (auto simp add: even_iff_mod_2_eq_zero)
haftmann@64014
   516
haftmann@58778
   517
lemma even_succ_div_two [simp]:
haftmann@58778
   518
  "even a \<Longrightarrow> (a + 1) div 2 = a div 2"
haftmann@58778
   519
  by (cases "a = 0") (auto elim!: evenE dest: mult_not_zero)
haftmann@58778
   520
haftmann@58778
   521
lemma odd_succ_div_two [simp]:
haftmann@58778
   522
  "odd a \<Longrightarrow> (a + 1) div 2 = a div 2 + 1"
haftmann@58778
   523
  by (auto elim!: oddE simp add: zero_not_eq_two [symmetric] add.assoc)
haftmann@58778
   524
haftmann@58778
   525
lemma even_two_times_div_two:
haftmann@58778
   526
  "even a \<Longrightarrow> 2 * (a div 2) = a"
haftmann@58778
   527
  by (fact dvd_mult_div_cancel)
haftmann@58778
   528
haftmann@58834
   529
lemma odd_two_times_div_two_succ [simp]:
haftmann@58778
   530
  "odd a \<Longrightarrow> 2 * (a div 2) + 1 = a"
haftmann@64242
   531
  using mult_div_mod_eq [of 2 a] by (simp add: even_iff_mod_2_eq_zero)
haftmann@60868
   532
 
haftmann@54226
   533
end
haftmann@54226
   534
haftmann@25942
   535
wenzelm@60758
   536
subsection \<open>Generic numeral division with a pragmatic type class\<close>
wenzelm@60758
   537
wenzelm@60758
   538
text \<open>
haftmann@53067
   539
  The following type class contains everything necessary to formulate
haftmann@53067
   540
  a division algorithm in ring structures with numerals, restricted
haftmann@53067
   541
  to its positive segments.  This is its primary motiviation, and it
haftmann@53067
   542
  could surely be formulated using a more fine-grained, more algebraic
haftmann@53067
   543
  and less technical class hierarchy.
wenzelm@60758
   544
\<close>
haftmann@53067
   545
lp15@60562
   546
class semiring_numeral_div = semiring_div + comm_semiring_1_cancel + linordered_semidom +
haftmann@59816
   547
  assumes div_less: "0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a div b = 0"
haftmann@53067
   548
    and mod_less: " 0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a mod b = a"
haftmann@53067
   549
    and div_positive: "0 < b \<Longrightarrow> b \<le> a \<Longrightarrow> a div b > 0"
haftmann@53067
   550
    and mod_less_eq_dividend: "0 \<le> a \<Longrightarrow> a mod b \<le> a"
haftmann@53067
   551
    and pos_mod_bound: "0 < b \<Longrightarrow> a mod b < b"
haftmann@53067
   552
    and pos_mod_sign: "0 < b \<Longrightarrow> 0 \<le> a mod b"
haftmann@53067
   553
    and mod_mult2_eq: "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
haftmann@53067
   554
    and div_mult2_eq: "0 \<le> c \<Longrightarrow> a div (b * c) = a div b div c"
haftmann@53067
   555
  assumes discrete: "a < b \<longleftrightarrow> a + 1 \<le> b"
haftmann@61275
   556
  fixes divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"
haftmann@61275
   557
    and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"
haftmann@61275
   558
  assumes divmod_def: "divmod m n = (numeral m div numeral n, numeral m mod numeral n)"
haftmann@61275
   559
    and divmod_step_def: "divmod_step l qr = (let (q, r) = qr
haftmann@61275
   560
    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
haftmann@61275
   561
    else (2 * q, r))"
wenzelm@61799
   562
    \<comment> \<open>These are conceptually definitions but force generated code
haftmann@61275
   563
    to be monomorphic wrt. particular instances of this class which
haftmann@61275
   564
    yields a significant speedup.\<close>
haftmann@61275
   565
haftmann@53067
   566
begin
haftmann@53067
   567
haftmann@54226
   568
subclass semiring_div_parity
haftmann@54226
   569
proof
haftmann@54226
   570
  fix a
haftmann@54226
   571
  show "a mod 2 = 0 \<or> a mod 2 = 1"
haftmann@54226
   572
  proof (rule ccontr)
haftmann@54226
   573
    assume "\<not> (a mod 2 = 0 \<or> a mod 2 = 1)"
haftmann@54226
   574
    then have "a mod 2 \<noteq> 0" and "a mod 2 \<noteq> 1" by simp_all
haftmann@54226
   575
    have "0 < 2" by simp
haftmann@54226
   576
    with pos_mod_bound pos_mod_sign have "0 \<le> a mod 2" "a mod 2 < 2" by simp_all
wenzelm@60758
   577
    with \<open>a mod 2 \<noteq> 0\<close> have "0 < a mod 2" by simp
haftmann@54226
   578
    with discrete have "1 \<le> a mod 2" by simp
wenzelm@60758
   579
    with \<open>a mod 2 \<noteq> 1\<close> have "1 < a mod 2" by simp
haftmann@54226
   580
    with discrete have "2 \<le> a mod 2" by simp
wenzelm@60758
   581
    with \<open>a mod 2 < 2\<close> show False by simp
haftmann@54226
   582
  qed
haftmann@58646
   583
next
haftmann@58646
   584
  show "1 mod 2 = 1"
haftmann@58646
   585
    by (rule mod_less) simp_all
haftmann@58710
   586
next
haftmann@58710
   587
  show "0 \<noteq> 2"
haftmann@58710
   588
    by simp
haftmann@53067
   589
qed
haftmann@53067
   590
haftmann@53067
   591
lemma divmod_digit_1:
haftmann@53067
   592
  assumes "0 \<le> a" "0 < b" and "b \<le> a mod (2 * b)"
haftmann@53067
   593
  shows "2 * (a div (2 * b)) + 1 = a div b" (is "?P")
haftmann@53067
   594
    and "a mod (2 * b) - b = a mod b" (is "?Q")
haftmann@53067
   595
proof -
haftmann@53067
   596
  from assms mod_less_eq_dividend [of a "2 * b"] have "b \<le> a"
haftmann@53067
   597
    by (auto intro: trans)
wenzelm@60758
   598
  with \<open>0 < b\<close> have "0 < a div b" by (auto intro: div_positive)
haftmann@53067
   599
  then have [simp]: "1 \<le> a div b" by (simp add: discrete)
wenzelm@60758
   600
  with \<open>0 < b\<close> have mod_less: "a mod b < b" by (simp add: pos_mod_bound)
wenzelm@63040
   601
  define w where "w = a div b mod 2"
wenzelm@63040
   602
  with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
haftmann@53067
   603
  have mod_w: "a mod (2 * b) = a mod b + b * w"
haftmann@53067
   604
    by (simp add: w_def mod_mult2_eq ac_simps)
haftmann@53067
   605
  from assms w_exhaust have "w = 1"
haftmann@53067
   606
    by (auto simp add: mod_w) (insert mod_less, auto)
haftmann@53067
   607
  with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp
haftmann@53067
   608
  have "2 * (a div (2 * b)) = a div b - w"
haftmann@64246
   609
    by (simp add: w_def div_mult2_eq minus_mod_eq_mult_div ac_simps)
wenzelm@60758
   610
  with \<open>w = 1\<close> have div: "2 * (a div (2 * b)) = a div b - 1" by simp
haftmann@53067
   611
  then show ?P and ?Q
haftmann@60867
   612
    by (simp_all add: div mod add_implies_diff [symmetric])
haftmann@53067
   613
qed
haftmann@53067
   614
haftmann@53067
   615
lemma divmod_digit_0:
haftmann@53067
   616
  assumes "0 < b" and "a mod (2 * b) < b"
haftmann@53067
   617
  shows "2 * (a div (2 * b)) = a div b" (is "?P")
haftmann@53067
   618
    and "a mod (2 * b) = a mod b" (is "?Q")
haftmann@53067
   619
proof -
wenzelm@63040
   620
  define w where "w = a div b mod 2"
wenzelm@63040
   621
  with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
haftmann@53067
   622
  have mod_w: "a mod (2 * b) = a mod b + b * w"
haftmann@53067
   623
    by (simp add: w_def mod_mult2_eq ac_simps)
haftmann@53067
   624
  moreover have "b \<le> a mod b + b"
haftmann@53067
   625
  proof -
wenzelm@60758
   626
    from \<open>0 < b\<close> pos_mod_sign have "0 \<le> a mod b" by blast
haftmann@53067
   627
    then have "0 + b \<le> a mod b + b" by (rule add_right_mono)
haftmann@53067
   628
    then show ?thesis by simp
haftmann@53067
   629
  qed
haftmann@53067
   630
  moreover note assms w_exhaust
haftmann@53067
   631
  ultimately have "w = 0" by auto
haftmann@53067
   632
  with mod_w have mod: "a mod (2 * b) = a mod b" by simp
haftmann@53067
   633
  have "2 * (a div (2 * b)) = a div b - w"
haftmann@64246
   634
    by (simp add: w_def div_mult2_eq minus_mod_eq_mult_div ac_simps)
wenzelm@60758
   635
  with \<open>w = 0\<close> have div: "2 * (a div (2 * b)) = a div b" by simp
haftmann@53067
   636
  then show ?P and ?Q
haftmann@53067
   637
    by (simp_all add: div mod)
haftmann@53067
   638
qed
haftmann@53067
   639
haftmann@60867
   640
lemma fst_divmod:
haftmann@53067
   641
  "fst (divmod m n) = numeral m div numeral n"
haftmann@53067
   642
  by (simp add: divmod_def)
haftmann@53067
   643
haftmann@60867
   644
lemma snd_divmod:
haftmann@53067
   645
  "snd (divmod m n) = numeral m mod numeral n"
haftmann@53067
   646
  by (simp add: divmod_def)
haftmann@53067
   647
wenzelm@60758
   648
text \<open>
haftmann@53067
   649
  This is a formulation of one step (referring to one digit position)
haftmann@53067
   650
  in school-method division: compare the dividend at the current
haftmann@53070
   651
  digit position with the remainder from previous division steps
haftmann@53067
   652
  and evaluate accordingly.
wenzelm@60758
   653
\<close>
haftmann@53067
   654
haftmann@61275
   655
lemma divmod_step_eq [simp]:
haftmann@53067
   656
  "divmod_step l (q, r) = (if numeral l \<le> r
haftmann@53067
   657
    then (2 * q + 1, r - numeral l) else (2 * q, r))"
haftmann@53067
   658
  by (simp add: divmod_step_def)
haftmann@53067
   659
wenzelm@60758
   660
text \<open>
haftmann@53067
   661
  This is a formulation of school-method division.
haftmann@53067
   662
  If the divisor is smaller than the dividend, terminate.
haftmann@53067
   663
  If not, shift the dividend to the right until termination
haftmann@53067
   664
  occurs and then reiterate single division steps in the
haftmann@53067
   665
  opposite direction.
wenzelm@60758
   666
\<close>
haftmann@53067
   667
haftmann@60867
   668
lemma divmod_divmod_step:
haftmann@53067
   669
  "divmod m n = (if m < n then (0, numeral m)
haftmann@53067
   670
    else divmod_step n (divmod m (Num.Bit0 n)))"
haftmann@53067
   671
proof (cases "m < n")
haftmann@53067
   672
  case True then have "numeral m < numeral n" by simp
haftmann@53067
   673
  then show ?thesis
haftmann@60867
   674
    by (simp add: prod_eq_iff div_less mod_less fst_divmod snd_divmod)
haftmann@53067
   675
next
haftmann@53067
   676
  case False
haftmann@53067
   677
  have "divmod m n =
haftmann@53067
   678
    divmod_step n (numeral m div (2 * numeral n),
haftmann@53067
   679
      numeral m mod (2 * numeral n))"
haftmann@53067
   680
  proof (cases "numeral n \<le> numeral m mod (2 * numeral n)")
haftmann@53067
   681
    case True
haftmann@60867
   682
    with divmod_step_eq
haftmann@53067
   683
      have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
haftmann@53067
   684
        (2 * (numeral m div (2 * numeral n)) + 1, numeral m mod (2 * numeral n) - numeral n)"
haftmann@60867
   685
        by simp
haftmann@53067
   686
    moreover from True divmod_digit_1 [of "numeral m" "numeral n"]
haftmann@53067
   687
      have "2 * (numeral m div (2 * numeral n)) + 1 = numeral m div numeral n"
haftmann@53067
   688
      and "numeral m mod (2 * numeral n) - numeral n = numeral m mod numeral n"
haftmann@53067
   689
      by simp_all
haftmann@53067
   690
    ultimately show ?thesis by (simp only: divmod_def)
haftmann@53067
   691
  next
haftmann@53067
   692
    case False then have *: "numeral m mod (2 * numeral n) < numeral n"
haftmann@53067
   693
      by (simp add: not_le)
haftmann@60867
   694
    with divmod_step_eq
haftmann@53067
   695
      have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
haftmann@53067
   696
        (2 * (numeral m div (2 * numeral n)), numeral m mod (2 * numeral n))"
haftmann@60867
   697
        by auto
haftmann@53067
   698
    moreover from * divmod_digit_0 [of "numeral n" "numeral m"]
haftmann@53067
   699
      have "2 * (numeral m div (2 * numeral n)) = numeral m div numeral n"
haftmann@53067
   700
      and "numeral m mod (2 * numeral n) = numeral m mod numeral n"
haftmann@53067
   701
      by (simp_all only: zero_less_numeral)
haftmann@53067
   702
    ultimately show ?thesis by (simp only: divmod_def)
haftmann@53067
   703
  qed
haftmann@53067
   704
  then have "divmod m n =
haftmann@53067
   705
    divmod_step n (numeral m div numeral (Num.Bit0 n),
haftmann@53067
   706
      numeral m mod numeral (Num.Bit0 n))"
lp15@60562
   707
    by (simp only: numeral.simps distrib mult_1)
haftmann@53067
   708
  then have "divmod m n = divmod_step n (divmod m (Num.Bit0 n))"
haftmann@53067
   709
    by (simp add: divmod_def)
haftmann@53067
   710
  with False show ?thesis by simp
haftmann@53067
   711
qed
haftmann@53067
   712
wenzelm@61799
   713
text \<open>The division rewrite proper -- first, trivial results involving \<open>1\<close>\<close>
haftmann@60867
   714
haftmann@61275
   715
lemma divmod_trivial [simp]:
haftmann@60867
   716
  "divmod Num.One Num.One = (numeral Num.One, 0)"
haftmann@60867
   717
  "divmod (Num.Bit0 m) Num.One = (numeral (Num.Bit0 m), 0)"
haftmann@60867
   718
  "divmod (Num.Bit1 m) Num.One = (numeral (Num.Bit1 m), 0)"
haftmann@60867
   719
  "divmod num.One (num.Bit0 n) = (0, Numeral1)"
haftmann@60867
   720
  "divmod num.One (num.Bit1 n) = (0, Numeral1)"
haftmann@60867
   721
  using divmod_divmod_step [of "Num.One"] by (simp_all add: divmod_def)
haftmann@60867
   722
haftmann@60867
   723
text \<open>Division by an even number is a right-shift\<close>
haftmann@58953
   724
haftmann@61275
   725
lemma divmod_cancel [simp]:
haftmann@53069
   726
  "divmod (Num.Bit0 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r))" (is ?P)
haftmann@53069
   727
  "divmod (Num.Bit1 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r + 1))" (is ?Q)
haftmann@53069
   728
proof -
haftmann@53069
   729
  have *: "\<And>q. numeral (Num.Bit0 q) = 2 * numeral q"
haftmann@53069
   730
    "\<And>q. numeral (Num.Bit1 q) = 2 * numeral q + 1"
haftmann@53069
   731
    by (simp_all only: numeral_mult numeral.simps distrib) simp_all
haftmann@53069
   732
  have "1 div 2 = 0" "1 mod 2 = 1" by (auto intro: div_less mod_less)
haftmann@53069
   733
  then show ?P and ?Q
haftmann@60867
   734
    by (simp_all add: fst_divmod snd_divmod prod_eq_iff split_def * [of m] * [of n] mod_mult_mult1
haftmann@60867
   735
      div_mult2_eq [of _ _ 2] mod_mult2_eq [of _ _ 2]
haftmann@60867
   736
      add.commute del: numeral_times_numeral)
haftmann@58953
   737
qed
haftmann@58953
   738
haftmann@60867
   739
text \<open>The really hard work\<close>
haftmann@60867
   740
haftmann@61275
   741
lemma divmod_steps [simp]:
haftmann@60867
   742
  "divmod (num.Bit0 m) (num.Bit1 n) =
haftmann@60867
   743
      (if m \<le> n then (0, numeral (num.Bit0 m))
haftmann@60867
   744
       else divmod_step (num.Bit1 n)
haftmann@60867
   745
             (divmod (num.Bit0 m)
haftmann@60867
   746
               (num.Bit0 (num.Bit1 n))))"
haftmann@60867
   747
  "divmod (num.Bit1 m) (num.Bit1 n) =
haftmann@60867
   748
      (if m < n then (0, numeral (num.Bit1 m))
haftmann@60867
   749
       else divmod_step (num.Bit1 n)
haftmann@60867
   750
             (divmod (num.Bit1 m)
haftmann@60867
   751
               (num.Bit0 (num.Bit1 n))))"
haftmann@60867
   752
  by (simp_all add: divmod_divmod_step)
haftmann@60867
   753
haftmann@61275
   754
lemmas divmod_algorithm_code = divmod_step_eq divmod_trivial divmod_cancel divmod_steps  
haftmann@61275
   755
wenzelm@60758
   756
text \<open>Special case: divisibility\<close>
haftmann@58953
   757
haftmann@58953
   758
definition divides_aux :: "'a \<times> 'a \<Rightarrow> bool"
haftmann@58953
   759
where
haftmann@58953
   760
  "divides_aux qr \<longleftrightarrow> snd qr = 0"
haftmann@58953
   761
haftmann@58953
   762
lemma divides_aux_eq [simp]:
haftmann@58953
   763
  "divides_aux (q, r) \<longleftrightarrow> r = 0"
haftmann@58953
   764
  by (simp add: divides_aux_def)
haftmann@58953
   765
haftmann@58953
   766
lemma dvd_numeral_simp [simp]:
haftmann@58953
   767
  "numeral m dvd numeral n \<longleftrightarrow> divides_aux (divmod n m)"
haftmann@58953
   768
  by (simp add: divmod_def mod_eq_0_iff_dvd)
haftmann@53069
   769
haftmann@60867
   770
text \<open>Generic computation of quotient and remainder\<close>  
haftmann@60867
   771
haftmann@60867
   772
lemma numeral_div_numeral [simp]: 
haftmann@60867
   773
  "numeral k div numeral l = fst (divmod k l)"
haftmann@60867
   774
  by (simp add: fst_divmod)
haftmann@60867
   775
haftmann@60867
   776
lemma numeral_mod_numeral [simp]: 
haftmann@60867
   777
  "numeral k mod numeral l = snd (divmod k l)"
haftmann@60867
   778
  by (simp add: snd_divmod)
haftmann@60867
   779
haftmann@60867
   780
lemma one_div_numeral [simp]:
haftmann@60867
   781
  "1 div numeral n = fst (divmod num.One n)"
haftmann@60867
   782
  by (simp add: fst_divmod)
haftmann@60867
   783
haftmann@60867
   784
lemma one_mod_numeral [simp]:
haftmann@60867
   785
  "1 mod numeral n = snd (divmod num.One n)"
haftmann@60867
   786
  by (simp add: snd_divmod)
haftmann@60867
   787
  
haftmann@53067
   788
end
haftmann@53067
   789
lp15@60562
   790
wenzelm@60758
   791
subsection \<open>Division on @{typ nat}\<close>
wenzelm@60758
   792
haftmann@61433
   793
context
haftmann@61433
   794
begin
haftmann@61433
   795
wenzelm@60758
   796
text \<open>
haftmann@63950
   797
  We define @{const divide} and @{const modulo} on @{typ nat} by means
haftmann@26100
   798
  of a characteristic relation with two input arguments
wenzelm@61076
   799
  @{term "m::nat"}, @{term "n::nat"} and two output arguments
wenzelm@61076
   800
  @{term "q::nat"}(uotient) and @{term "r::nat"}(emainder).
wenzelm@60758
   801
\<close>
haftmann@26100
   802
haftmann@33340
   803
definition divmod_nat_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where
haftmann@33340
   804
  "divmod_nat_rel m n qr \<longleftrightarrow>
haftmann@30923
   805
    m = fst qr * n + snd qr \<and>
haftmann@30923
   806
      (if n = 0 then fst qr = 0 else if n > 0 then 0 \<le> snd qr \<and> snd qr < n else n < snd qr \<and> snd qr \<le> 0)"
haftmann@26100
   807
wenzelm@60758
   808
text \<open>@{const divmod_nat_rel} is total:\<close>
haftmann@26100
   809
haftmann@61433
   810
qualified lemma divmod_nat_rel_ex:
haftmann@33340
   811
  obtains q r where "divmod_nat_rel m n (q, r)"
haftmann@26100
   812
proof (cases "n = 0")
haftmann@30923
   813
  case True  with that show thesis
haftmann@33340
   814
    by (auto simp add: divmod_nat_rel_def)
haftmann@26100
   815
next
haftmann@26100
   816
  case False
haftmann@26100
   817
  have "\<exists>q r. m = q * n + r \<and> r < n"
haftmann@26100
   818
  proof (induct m)
wenzelm@60758
   819
    case 0 with \<open>n \<noteq> 0\<close>
wenzelm@61076
   820
    have "(0::nat) = 0 * n + 0 \<and> 0 < n" by simp
haftmann@26100
   821
    then show ?case by blast
haftmann@26100
   822
  next
haftmann@26100
   823
    case (Suc m) then obtain q' r'
haftmann@26100
   824
      where m: "m = q' * n + r'" and n: "r' < n" by auto
haftmann@26100
   825
    then show ?case proof (cases "Suc r' < n")
haftmann@26100
   826
      case True
haftmann@26100
   827
      from m n have "Suc m = q' * n + Suc r'" by simp
haftmann@26100
   828
      with True show ?thesis by blast
haftmann@26100
   829
    next
haftmann@26100
   830
      case False then have "n \<le> Suc r'" by auto
haftmann@26100
   831
      moreover from n have "Suc r' \<le> n" by auto
haftmann@26100
   832
      ultimately have "n = Suc r'" by auto
haftmann@26100
   833
      with m have "Suc m = Suc q' * n + 0" by simp
wenzelm@60758
   834
      with \<open>n \<noteq> 0\<close> show ?thesis by blast
haftmann@26100
   835
    qed
haftmann@26100
   836
  qed
haftmann@26100
   837
  with that show thesis
wenzelm@60758
   838
    using \<open>n \<noteq> 0\<close> by (auto simp add: divmod_nat_rel_def)
haftmann@26100
   839
qed
haftmann@26100
   840
wenzelm@60758
   841
text \<open>@{const divmod_nat_rel} is injective:\<close>
haftmann@26100
   842
haftmann@61433
   843
qualified lemma divmod_nat_rel_unique:
haftmann@33340
   844
  assumes "divmod_nat_rel m n qr"
haftmann@33340
   845
    and "divmod_nat_rel m n qr'"
haftmann@30923
   846
  shows "qr = qr'"
haftmann@26100
   847
proof (cases "n = 0")
haftmann@26100
   848
  case True with assms show ?thesis
haftmann@30923
   849
    by (cases qr, cases qr')
haftmann@33340
   850
      (simp add: divmod_nat_rel_def)
haftmann@26100
   851
next
haftmann@26100
   852
  case False
wenzelm@61076
   853
  have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q::nat)"
haftmann@26100
   854
  apply (rule leI)
haftmann@26100
   855
  apply (subst less_iff_Suc_add)
haftmann@26100
   856
  apply (auto simp add: add_mult_distrib)
haftmann@26100
   857
  done
wenzelm@60758
   858
  from \<open>n \<noteq> 0\<close> assms have *: "fst qr = fst qr'"
haftmann@33340
   859
    by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)
wenzelm@53374
   860
  with assms have "snd qr = snd qr'"
haftmann@33340
   861
    by (simp add: divmod_nat_rel_def)
wenzelm@53374
   862
  with * show ?thesis by (cases qr, cases qr') simp
haftmann@26100
   863
qed
haftmann@26100
   864
wenzelm@60758
   865
text \<open>
haftmann@26100
   866
  We instantiate divisibility on the natural numbers by
haftmann@33340
   867
  means of @{const divmod_nat_rel}:
wenzelm@60758
   868
\<close>
haftmann@25942
   869
haftmann@61433
   870
qualified definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
haftmann@37767
   871
  "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"
haftmann@30923
   872
haftmann@61433
   873
qualified lemma divmod_nat_rel_divmod_nat:
haftmann@33340
   874
  "divmod_nat_rel m n (divmod_nat m n)"
haftmann@30923
   875
proof -
haftmann@33340
   876
  from divmod_nat_rel_ex
haftmann@33340
   877
    obtain qr where rel: "divmod_nat_rel m n qr" .
haftmann@30923
   878
  then show ?thesis
haftmann@33340
   879
  by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)
haftmann@30923
   880
qed
haftmann@30923
   881
haftmann@61433
   882
qualified lemma divmod_nat_unique:
lp15@60562
   883
  assumes "divmod_nat_rel m n qr"
haftmann@33340
   884
  shows "divmod_nat m n = qr"
haftmann@33340
   885
  using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)
haftmann@26100
   886
haftmann@61433
   887
qualified lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)"
haftmann@61433
   888
  by (simp add: Divides.divmod_nat_unique divmod_nat_rel_def)
haftmann@61433
   889
haftmann@61433
   890
qualified lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)"
haftmann@61433
   891
  by (simp add: Divides.divmod_nat_unique divmod_nat_rel_def)
haftmann@61433
   892
haftmann@61433
   893
qualified lemma divmod_nat_base: "m < n \<Longrightarrow> divmod_nat m n = (0, m)"
haftmann@61433
   894
  by (simp add: divmod_nat_unique divmod_nat_rel_def)
haftmann@61433
   895
haftmann@61433
   896
qualified lemma divmod_nat_step:
haftmann@61433
   897
  assumes "0 < n" and "n \<le> m"
haftmann@61433
   898
  shows "divmod_nat m n = apfst Suc (divmod_nat (m - n) n)"
haftmann@61433
   899
proof (rule divmod_nat_unique)
haftmann@61433
   900
  have "divmod_nat_rel (m - n) n (divmod_nat (m - n) n)"
haftmann@61433
   901
    by (fact divmod_nat_rel_divmod_nat)
haftmann@61433
   902
  then show "divmod_nat_rel m n (apfst Suc (divmod_nat (m - n) n))"
haftmann@61433
   903
    unfolding divmod_nat_rel_def using assms by auto
haftmann@61433
   904
qed
haftmann@61433
   905
haftmann@61433
   906
end
haftmann@61433
   907
  
haftmann@60429
   908
instantiation nat :: semiring_div
haftmann@60352
   909
begin
haftmann@60352
   910
haftmann@60352
   911
definition divide_nat where
haftmann@61433
   912
  div_nat_def: "m div n = fst (Divides.divmod_nat m n)"
haftmann@60352
   913
haftmann@63950
   914
definition modulo_nat where
haftmann@63950
   915
  mod_nat_def: "m mod n = snd (Divides.divmod_nat m n)"
huffman@46551
   916
huffman@46551
   917
lemma fst_divmod_nat [simp]:
haftmann@61433
   918
  "fst (Divides.divmod_nat m n) = m div n"
huffman@46551
   919
  by (simp add: div_nat_def)
huffman@46551
   920
huffman@46551
   921
lemma snd_divmod_nat [simp]:
haftmann@61433
   922
  "snd (Divides.divmod_nat m n) = m mod n"
huffman@46551
   923
  by (simp add: mod_nat_def)
huffman@46551
   924
haftmann@33340
   925
lemma divmod_nat_div_mod:
haftmann@61433
   926
  "Divides.divmod_nat m n = (m div n, m mod n)"
huffman@46551
   927
  by (simp add: prod_eq_iff)
haftmann@26100
   928
huffman@47135
   929
lemma div_nat_unique:
lp15@60562
   930
  assumes "divmod_nat_rel m n (q, r)"
haftmann@26100
   931
  shows "m div n = q"
haftmann@61433
   932
  using assms by (auto dest!: Divides.divmod_nat_unique simp add: prod_eq_iff)
huffman@47135
   933
huffman@47135
   934
lemma mod_nat_unique:
lp15@60562
   935
  assumes "divmod_nat_rel m n (q, r)"
haftmann@26100
   936
  shows "m mod n = r"
haftmann@61433
   937
  using assms by (auto dest!: Divides.divmod_nat_unique simp add: prod_eq_iff)
haftmann@25571
   938
haftmann@33340
   939
lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"
haftmann@61433
   940
  using Divides.divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod)
haftmann@25942
   941
haftmann@63950
   942
text \<open>The ''recursion'' equations for @{const divide} and @{const modulo}\<close>
haftmann@26100
   943
haftmann@26100
   944
lemma div_less [simp]:
haftmann@26100
   945
  fixes m n :: nat
haftmann@26100
   946
  assumes "m < n"
haftmann@26100
   947
  shows "m div n = 0"
haftmann@61433
   948
  using assms Divides.divmod_nat_base by (simp add: prod_eq_iff)
haftmann@25942
   949
haftmann@26100
   950
lemma le_div_geq:
haftmann@26100
   951
  fixes m n :: nat
haftmann@26100
   952
  assumes "0 < n" and "n \<le> m"
haftmann@26100
   953
  shows "m div n = Suc ((m - n) div n)"
haftmann@61433
   954
  using assms Divides.divmod_nat_step by (simp add: prod_eq_iff)
paulson@14267
   955
haftmann@26100
   956
lemma mod_less [simp]:
haftmann@26100
   957
  fixes m n :: nat
haftmann@26100
   958
  assumes "m < n"
haftmann@26100
   959
  shows "m mod n = m"
haftmann@61433
   960
  using assms Divides.divmod_nat_base by (simp add: prod_eq_iff)
haftmann@26100
   961
haftmann@26100
   962
lemma le_mod_geq:
haftmann@26100
   963
  fixes m n :: nat
haftmann@26100
   964
  assumes "n \<le> m"
haftmann@26100
   965
  shows "m mod n = (m - n) mod n"
haftmann@61433
   966
  using assms Divides.divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)
paulson@14267
   967
huffman@47136
   968
instance proof
huffman@47136
   969
  fix m n :: nat
huffman@47136
   970
  show "m div n * n + m mod n = m"
huffman@47136
   971
    using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)
huffman@47136
   972
next
huffman@47136
   973
  fix m n q :: nat
huffman@47136
   974
  assume "n \<noteq> 0"
huffman@47136
   975
  then show "(q + m * n) div n = m + q div n"
huffman@47136
   976
    by (induct m) (simp_all add: le_div_geq)
huffman@47136
   977
next
huffman@47136
   978
  fix m n q :: nat
huffman@47136
   979
  assume "m \<noteq> 0"
huffman@47136
   980
  hence "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)"
huffman@47136
   981
    unfolding divmod_nat_rel_def
nipkow@62390
   982
    by (auto split: if_split_asm, simp_all add: algebra_simps)
huffman@47136
   983
  moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .
huffman@47136
   984
  ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .
huffman@47136
   985
  thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique)
huffman@47136
   986
next
huffman@47136
   987
  fix n :: nat show "n div 0 = 0"
haftmann@61433
   988
    by (simp add: div_nat_def Divides.divmod_nat_zero)
huffman@47136
   989
next
huffman@47136
   990
  fix n :: nat show "0 div n = 0"
haftmann@61433
   991
    by (simp add: div_nat_def Divides.divmod_nat_zero_left)
haftmann@25942
   992
qed
haftmann@26100
   993
haftmann@25942
   994
end
paulson@14267
   995
haftmann@60685
   996
instantiation nat :: normalization_semidom
haftmann@60685
   997
begin
haftmann@60685
   998
haftmann@60685
   999
definition normalize_nat
haftmann@60685
  1000
  where [simp]: "normalize = (id :: nat \<Rightarrow> nat)"
haftmann@60685
  1001
haftmann@60685
  1002
definition unit_factor_nat
haftmann@60685
  1003
  where "unit_factor n = (if n = 0 then 0 else 1 :: nat)"
haftmann@60685
  1004
haftmann@60685
  1005
lemma unit_factor_simps [simp]:
haftmann@60685
  1006
  "unit_factor 0 = (0::nat)"
haftmann@60685
  1007
  "unit_factor (Suc n) = 1"
haftmann@60685
  1008
  by (simp_all add: unit_factor_nat_def)
haftmann@60685
  1009
haftmann@60685
  1010
instance
haftmann@60685
  1011
  by standard (simp_all add: unit_factor_nat_def)
haftmann@60685
  1012
  
haftmann@60685
  1013
end
haftmann@60685
  1014
haftmann@61433
  1015
lemma divmod_nat_if [code]:
haftmann@61433
  1016
  "Divides.divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
haftmann@61433
  1017
    let (q, r) = Divides.divmod_nat (m - n) n in (Suc q, r))"
blanchet@55414
  1018
  by (simp add: prod_eq_iff case_prod_beta not_less le_div_geq le_mod_geq)
haftmann@33361
  1019
haftmann@63950
  1020
text \<open>Simproc for cancelling @{const divide} and @{const modulo}\<close>
haftmann@25942
  1021
wenzelm@51299
  1022
ML_file "~~/src/Provers/Arith/cancel_div_mod.ML"
wenzelm@51299
  1023
wenzelm@60758
  1024
ML \<open>
wenzelm@43594
  1025
structure Cancel_Div_Mod_Nat = Cancel_Div_Mod
wenzelm@41550
  1026
(
haftmann@60352
  1027
  val div_name = @{const_name divide};
haftmann@63950
  1028
  val mod_name = @{const_name modulo};
haftmann@30934
  1029
  val mk_binop = HOLogic.mk_binop;
huffman@48561
  1030
  val mk_plus = HOLogic.mk_binop @{const_name Groups.plus};
huffman@48561
  1031
  val dest_plus = HOLogic.dest_bin @{const_name Groups.plus} HOLogic.natT;
huffman@48561
  1032
  fun mk_sum [] = HOLogic.zero
huffman@48561
  1033
    | mk_sum [t] = t
huffman@48561
  1034
    | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
huffman@48561
  1035
  fun dest_sum tm =
huffman@48561
  1036
    if HOLogic.is_zero tm then []
huffman@48561
  1037
    else
huffman@48561
  1038
      (case try HOLogic.dest_Suc tm of
huffman@48561
  1039
        SOME t => HOLogic.Suc_zero :: dest_sum t
huffman@48561
  1040
      | NONE =>
huffman@48561
  1041
          (case try dest_plus tm of
huffman@48561
  1042
            SOME (t, u) => dest_sum t @ dest_sum u
huffman@48561
  1043
          | NONE => [tm]));
haftmann@25942
  1044
haftmann@30934
  1045
  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
paulson@14267
  1046
haftmann@30934
  1047
  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
haftmann@57514
  1048
    (@{thm add_0_left} :: @{thm add_0_right} :: @{thms ac_simps}))
wenzelm@41550
  1049
)
wenzelm@60758
  1050
\<close>
wenzelm@60758
  1051
wenzelm@60758
  1052
simproc_setup cancel_div_mod_nat ("(m::nat) + n") = \<open>K Cancel_Div_Mod_Nat.proc\<close>
wenzelm@60758
  1053
wenzelm@60758
  1054
wenzelm@60758
  1055
subsubsection \<open>Quotient\<close>
haftmann@26100
  1056
haftmann@26100
  1057
lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
nipkow@29667
  1058
by (simp add: le_div_geq linorder_not_less)
haftmann@26100
  1059
haftmann@26100
  1060
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
nipkow@29667
  1061
by (simp add: div_geq)
haftmann@26100
  1062
haftmann@26100
  1063
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
nipkow@29667
  1064
by simp
haftmann@26100
  1065
haftmann@26100
  1066
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
nipkow@29667
  1067
by simp
haftmann@26100
  1068
haftmann@53066
  1069
lemma div_positive:
haftmann@53066
  1070
  fixes m n :: nat
haftmann@53066
  1071
  assumes "n > 0"
haftmann@53066
  1072
  assumes "m \<ge> n"
haftmann@53066
  1073
  shows "m div n > 0"
haftmann@53066
  1074
proof -
wenzelm@60758
  1075
  from \<open>m \<ge> n\<close> obtain q where "m = n + q"
haftmann@53066
  1076
    by (auto simp add: le_iff_add)
eberlm@63499
  1077
  with \<open>n > 0\<close> show ?thesis by (simp add: div_add_self1)
haftmann@53066
  1078
qed
haftmann@53066
  1079
hoelzl@59000
  1080
lemma div_eq_0_iff: "(a div b::nat) = 0 \<longleftrightarrow> a < b \<or> b = 0"
hoelzl@59000
  1081
  by (metis div_less div_positive div_by_0 gr0I less_numeral_extra(3) not_less)
haftmann@25942
  1082
wenzelm@60758
  1083
subsubsection \<open>Remainder\<close>
haftmann@25942
  1084
haftmann@26100
  1085
lemma mod_less_divisor [simp]:
haftmann@26100
  1086
  fixes m n :: nat
haftmann@26100
  1087
  assumes "n > 0"
haftmann@26100
  1088
  shows "m mod n < (n::nat)"
haftmann@33340
  1089
  using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto
paulson@14267
  1090
haftmann@51173
  1091
lemma mod_Suc_le_divisor [simp]:
haftmann@51173
  1092
  "m mod Suc n \<le> n"
haftmann@51173
  1093
  using mod_less_divisor [of "Suc n" m] by arith
haftmann@51173
  1094
haftmann@26100
  1095
lemma mod_less_eq_dividend [simp]:
haftmann@26100
  1096
  fixes m n :: nat
haftmann@26100
  1097
  shows "m mod n \<le> m"
haftmann@26100
  1098
proof (rule add_leD2)
haftmann@64242
  1099
  from div_mult_mod_eq have "m div n * n + m mod n = m" .
haftmann@26100
  1100
  then show "m div n * n + m mod n \<le> m" by auto
haftmann@26100
  1101
qed
haftmann@26100
  1102
wenzelm@61076
  1103
lemma mod_geq: "\<not> m < (n::nat) \<Longrightarrow> m mod n = (m - n) mod n"
nipkow@29667
  1104
by (simp add: le_mod_geq linorder_not_less)
paulson@14267
  1105
wenzelm@61076
  1106
lemma mod_if: "m mod (n::nat) = (if m < n then m else (m - n) mod n)"
nipkow@29667
  1107
by (simp add: le_mod_geq)
haftmann@26100
  1108
haftmann@64244
  1109
lemma mod_by_Suc_0 [simp]: "m mod Suc 0 = 0"
nipkow@29667
  1110
by (induct m) (simp_all add: mod_geq)
paulson@14267
  1111
nipkow@15439
  1112
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
wenzelm@22718
  1113
  apply (drule mod_less_divisor [where m = m])
wenzelm@22718
  1114
  apply simp
wenzelm@22718
  1115
  done
paulson@14267
  1116
wenzelm@60758
  1117
subsubsection \<open>Quotient and Remainder\<close>
paulson@14267
  1118
haftmann@33340
  1119
lemma divmod_nat_rel_mult1_eq:
bulwahn@46552
  1120
  "divmod_nat_rel b c (q, r)
haftmann@33340
  1121
   \<Longrightarrow> divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"
haftmann@33340
  1122
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
paulson@14267
  1123
haftmann@30923
  1124
lemma div_mult1_eq:
haftmann@30923
  1125
  "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"
huffman@47135
  1126
by (blast intro: divmod_nat_rel_mult1_eq [THEN div_nat_unique] divmod_nat_rel)
paulson@14267
  1127
haftmann@33340
  1128
lemma divmod_nat_rel_add1_eq:
bulwahn@46552
  1129
  "divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br)
haftmann@33340
  1130
   \<Longrightarrow> divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"
haftmann@33340
  1131
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
paulson@14267
  1132
paulson@14267
  1133
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
paulson@14267
  1134
lemma div_add1_eq:
nipkow@25134
  1135
  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
huffman@47135
  1136
by (blast intro: divmod_nat_rel_add1_eq [THEN div_nat_unique] divmod_nat_rel)
paulson@14267
  1137
haftmann@33340
  1138
lemma divmod_nat_rel_mult2_eq:
haftmann@60352
  1139
  assumes "divmod_nat_rel a b (q, r)"
haftmann@60352
  1140
  shows "divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"
haftmann@60352
  1141
proof -
lp15@60562
  1142
  { assume "r < b" and "0 < c"
haftmann@60352
  1143
    then have "b * (q mod c) + r < b * c"
haftmann@60352
  1144
      apply (cut_tac m = q and n = c in mod_less_divisor)
haftmann@60352
  1145
      apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
haftmann@60352
  1146
      apply (erule_tac P = "%x. lhs < rhs x" for lhs rhs in ssubst)
haftmann@60352
  1147
      apply (simp add: add_mult_distrib2)
haftmann@60352
  1148
      done
haftmann@60352
  1149
    then have "r + b * (q mod c) < b * c"
haftmann@60352
  1150
      by (simp add: ac_simps)
haftmann@60352
  1151
  } with assms show ?thesis
haftmann@60352
  1152
    by (auto simp add: divmod_nat_rel_def algebra_simps add_mult_distrib2 [symmetric])
haftmann@60352
  1153
qed
lp15@60562
  1154
blanchet@55085
  1155
lemma div_mult2_eq: "a div (b * c) = (a div b) div (c::nat)"
huffman@47135
  1156
by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique])
paulson@14267
  1157
blanchet@55085
  1158
lemma mod_mult2_eq: "a mod (b * c) = b * (a div b mod c) + a mod (b::nat)"
haftmann@57512
  1159
by (auto simp add: mult.commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique])
paulson@14267
  1160
haftmann@61275
  1161
instantiation nat :: semiring_numeral_div
haftmann@61275
  1162
begin
haftmann@61275
  1163
haftmann@61275
  1164
definition divmod_nat :: "num \<Rightarrow> num \<Rightarrow> nat \<times> nat"
haftmann@61275
  1165
where
haftmann@61275
  1166
  divmod'_nat_def: "divmod_nat m n = (numeral m div numeral n, numeral m mod numeral n)"
haftmann@61275
  1167
haftmann@61275
  1168
definition divmod_step_nat :: "num \<Rightarrow> nat \<times> nat \<Rightarrow> nat \<times> nat"
haftmann@61275
  1169
where
haftmann@61275
  1170
  "divmod_step_nat l qr = (let (q, r) = qr
haftmann@61275
  1171
    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
haftmann@61275
  1172
    else (2 * q, r))"
haftmann@61275
  1173
haftmann@61275
  1174
instance
haftmann@61275
  1175
  by standard (auto intro: div_positive simp add: divmod'_nat_def divmod_step_nat_def mod_mult2_eq div_mult2_eq)
haftmann@61275
  1176
haftmann@61275
  1177
end
haftmann@61275
  1178
haftmann@61275
  1179
declare divmod_algorithm_code [where ?'a = nat, code]
haftmann@61275
  1180
  
paulson@14267
  1181
wenzelm@60758
  1182
subsubsection \<open>Further Facts about Quotient and Remainder\<close>
paulson@14267
  1183
haftmann@64244
  1184
lemma div_by_Suc_0 [simp]:
haftmann@58786
  1185
  "m div Suc 0 = m"
haftmann@58786
  1186
  using div_by_1 [of m] by simp
paulson@14267
  1187
paulson@14267
  1188
(* Monotonicity of div in first argument *)
haftmann@30923
  1189
lemma div_le_mono [rule_format (no_asm)]:
wenzelm@22718
  1190
    "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
paulson@14267
  1191
apply (case_tac "k=0", simp)
paulson@15251
  1192
apply (induct "n" rule: nat_less_induct, clarify)
paulson@14267
  1193
apply (case_tac "n<k")
paulson@14267
  1194
(* 1  case n<k *)
paulson@14267
  1195
apply simp
paulson@14267
  1196
(* 2  case n >= k *)
paulson@14267
  1197
apply (case_tac "m<k")
paulson@14267
  1198
(* 2.1  case m<k *)
paulson@14267
  1199
apply simp
paulson@14267
  1200
(* 2.2  case m>=k *)
nipkow@15439
  1201
apply (simp add: div_geq diff_le_mono)
paulson@14267
  1202
done
paulson@14267
  1203
paulson@14267
  1204
(* Antimonotonicity of div in second argument *)
paulson@14267
  1205
lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
paulson@14267
  1206
apply (subgoal_tac "0<n")
wenzelm@22718
  1207
 prefer 2 apply simp
paulson@15251
  1208
apply (induct_tac k rule: nat_less_induct)
paulson@14267
  1209
apply (rename_tac "k")
paulson@14267
  1210
apply (case_tac "k<n", simp)
paulson@14267
  1211
apply (subgoal_tac "~ (k<m) ")
wenzelm@22718
  1212
 prefer 2 apply simp
paulson@14267
  1213
apply (simp add: div_geq)
paulson@15251
  1214
apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
paulson@14267
  1215
 prefer 2
paulson@14267
  1216
 apply (blast intro: div_le_mono diff_le_mono2)
paulson@14267
  1217
apply (rule le_trans, simp)
nipkow@15439
  1218
apply (simp)
paulson@14267
  1219
done
paulson@14267
  1220
paulson@14267
  1221
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
paulson@14267
  1222
apply (case_tac "n=0", simp)
paulson@14267
  1223
apply (subgoal_tac "m div n \<le> m div 1", simp)
paulson@14267
  1224
apply (rule div_le_mono2)
paulson@14267
  1225
apply (simp_all (no_asm_simp))
paulson@14267
  1226
done
paulson@14267
  1227
wenzelm@22718
  1228
(* Similar for "less than" *)
huffman@47138
  1229
lemma div_less_dividend [simp]:
huffman@47138
  1230
  "\<lbrakk>(1::nat) < n; 0 < m\<rbrakk> \<Longrightarrow> m div n < m"
huffman@47138
  1231
apply (induct m rule: nat_less_induct)
paulson@14267
  1232
apply (rename_tac "m")
paulson@14267
  1233
apply (case_tac "m<n", simp)
paulson@14267
  1234
apply (subgoal_tac "0<n")
wenzelm@22718
  1235
 prefer 2 apply simp
paulson@14267
  1236
apply (simp add: div_geq)
paulson@14267
  1237
apply (case_tac "n<m")
paulson@15251
  1238
 apply (subgoal_tac "(m-n) div n < (m-n) ")
paulson@14267
  1239
  apply (rule impI less_trans_Suc)+
paulson@14267
  1240
apply assumption
nipkow@15439
  1241
  apply (simp_all)
paulson@14267
  1242
done
paulson@14267
  1243
wenzelm@60758
  1244
text\<open>A fact for the mutilated chess board\<close>
paulson@14267
  1245
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
paulson@14267
  1246
apply (case_tac "n=0", simp)
paulson@15251
  1247
apply (induct "m" rule: nat_less_induct)
paulson@14267
  1248
apply (case_tac "Suc (na) <n")
paulson@14267
  1249
(* case Suc(na) < n *)
paulson@14267
  1250
apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
paulson@14267
  1251
(* case n \<le> Suc(na) *)
paulson@16796
  1252
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
nipkow@15439
  1253
apply (auto simp add: Suc_diff_le le_mod_geq)
paulson@14267
  1254
done
paulson@14267
  1255
paulson@14267
  1256
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
nipkow@29667
  1257
by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
paulson@17084
  1258
wenzelm@22718
  1259
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
paulson@14267
  1260
paulson@14267
  1261
(*Loses information, namely we also have r<d provided d is nonzero*)
haftmann@57514
  1262
lemma mod_eqD:
haftmann@57514
  1263
  fixes m d r q :: nat
haftmann@57514
  1264
  assumes "m mod d = r"
haftmann@57514
  1265
  shows "\<exists>q. m = r + q * d"
haftmann@57514
  1266
proof -
haftmann@64242
  1267
  from div_mult_mod_eq obtain q where "q * d + m mod d = m" by blast
haftmann@57514
  1268
  with assms have "m = r + q * d" by simp
haftmann@57514
  1269
  then show ?thesis ..
haftmann@57514
  1270
qed
paulson@14267
  1271
nipkow@13152
  1272
lemma split_div:
nipkow@13189
  1273
 "P(n div k :: nat) =
nipkow@13189
  1274
 ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
nipkow@13189
  1275
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
  1276
proof
nipkow@13189
  1277
  assume P: ?P
nipkow@13189
  1278
  show ?Q
nipkow@13189
  1279
  proof (cases)
nipkow@13189
  1280
    assume "k = 0"
haftmann@27651
  1281
    with P show ?Q by simp
nipkow@13189
  1282
  next
nipkow@13189
  1283
    assume not0: "k \<noteq> 0"
nipkow@13189
  1284
    thus ?Q
nipkow@13189
  1285
    proof (simp, intro allI impI)
nipkow@13189
  1286
      fix i j
nipkow@13189
  1287
      assume n: "n = k*i + j" and j: "j < k"
nipkow@13189
  1288
      show "P i"
nipkow@13189
  1289
      proof (cases)
wenzelm@22718
  1290
        assume "i = 0"
wenzelm@22718
  1291
        with n j P show "P i" by simp
nipkow@13189
  1292
      next
wenzelm@22718
  1293
        assume "i \<noteq> 0"
haftmann@57514
  1294
        with not0 n j P show "P i" by(simp add:ac_simps)
nipkow@13189
  1295
      qed
nipkow@13189
  1296
    qed
nipkow@13189
  1297
  qed
nipkow@13189
  1298
next
nipkow@13189
  1299
  assume Q: ?Q
nipkow@13189
  1300
  show ?P
nipkow@13189
  1301
  proof (cases)
nipkow@13189
  1302
    assume "k = 0"
haftmann@27651
  1303
    with Q show ?P by simp
nipkow@13189
  1304
  next
nipkow@13189
  1305
    assume not0: "k \<noteq> 0"
nipkow@13189
  1306
    with Q have R: ?R by simp
nipkow@13189
  1307
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
  1308
    show ?P by simp
nipkow@13189
  1309
  qed
nipkow@13189
  1310
qed
nipkow@13189
  1311
berghofe@13882
  1312
lemma split_div_lemma:
haftmann@26100
  1313
  assumes "0 < n"
wenzelm@61076
  1314
  shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m::nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@26100
  1315
proof
haftmann@26100
  1316
  assume ?rhs
haftmann@64246
  1317
  with minus_mod_eq_mult_div [symmetric] have nq: "n * q = m - (m mod n)" by simp
haftmann@26100
  1318
  then have A: "n * q \<le> m" by simp
haftmann@26100
  1319
  have "n - (m mod n) > 0" using mod_less_divisor assms by auto
haftmann@26100
  1320
  then have "m < m + (n - (m mod n))" by simp
haftmann@26100
  1321
  then have "m < n + (m - (m mod n))" by simp
haftmann@26100
  1322
  with nq have "m < n + n * q" by simp
haftmann@26100
  1323
  then have B: "m < n * Suc q" by simp
haftmann@26100
  1324
  from A B show ?lhs ..
haftmann@26100
  1325
next
haftmann@26100
  1326
  assume P: ?lhs
haftmann@33340
  1327
  then have "divmod_nat_rel m n (q, m - n * q)"
haftmann@57514
  1328
    unfolding divmod_nat_rel_def by (auto simp add: ac_simps)
haftmann@61433
  1329
  then have "m div n = q"
haftmann@61433
  1330
    by (rule div_nat_unique)
haftmann@30923
  1331
  then show ?rhs by simp
haftmann@26100
  1332
qed
berghofe@13882
  1333
berghofe@13882
  1334
theorem split_div':
berghofe@13882
  1335
  "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
paulson@14267
  1336
   (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
haftmann@61433
  1337
  apply (cases "0 < n")
berghofe@13882
  1338
  apply (simp only: add: split_div_lemma)
haftmann@27651
  1339
  apply simp_all
berghofe@13882
  1340
  done
berghofe@13882
  1341
nipkow@13189
  1342
lemma split_mod:
nipkow@13189
  1343
 "P(n mod k :: nat) =
nipkow@13189
  1344
 ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
nipkow@13189
  1345
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
  1346
proof
nipkow@13189
  1347
  assume P: ?P
nipkow@13189
  1348
  show ?Q
nipkow@13189
  1349
  proof (cases)
nipkow@13189
  1350
    assume "k = 0"
haftmann@27651
  1351
    with P show ?Q by simp
nipkow@13189
  1352
  next
nipkow@13189
  1353
    assume not0: "k \<noteq> 0"
nipkow@13189
  1354
    thus ?Q
nipkow@13189
  1355
    proof (simp, intro allI impI)
nipkow@13189
  1356
      fix i j
nipkow@13189
  1357
      assume "n = k*i + j" "j < k"
haftmann@58786
  1358
      thus "P j" using not0 P by (simp add: ac_simps)
nipkow@13189
  1359
    qed
nipkow@13189
  1360
  qed
nipkow@13189
  1361
next
nipkow@13189
  1362
  assume Q: ?Q
nipkow@13189
  1363
  show ?P
nipkow@13189
  1364
  proof (cases)
nipkow@13189
  1365
    assume "k = 0"
haftmann@27651
  1366
    with Q show ?P by simp
nipkow@13189
  1367
  next
nipkow@13189
  1368
    assume not0: "k \<noteq> 0"
nipkow@13189
  1369
    with Q have R: ?R by simp
nipkow@13189
  1370
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
  1371
    show ?P by simp
nipkow@13189
  1372
  qed
nipkow@13189
  1373
qed
nipkow@13189
  1374
noschinl@52398
  1375
lemma div_eq_dividend_iff: "a \<noteq> 0 \<Longrightarrow> (a :: nat) div b = a \<longleftrightarrow> b = 1"
noschinl@52398
  1376
  apply rule
noschinl@52398
  1377
  apply (cases "b = 0")
noschinl@52398
  1378
  apply simp_all
noschinl@52398
  1379
  apply (metis (full_types) One_nat_def Suc_lessI div_less_dividend less_not_refl3)
noschinl@52398
  1380
  done
noschinl@52398
  1381
haftmann@63417
  1382
lemma (in field_char_0) of_nat_div:
haftmann@63417
  1383
  "of_nat (m div n) = ((of_nat m - of_nat (m mod n)) / of_nat n)"
haftmann@63417
  1384
proof -
haftmann@63417
  1385
  have "of_nat (m div n) = ((of_nat (m div n * n + m mod n) - of_nat (m mod n)) / of_nat n :: 'a)"
haftmann@63417
  1386
    unfolding of_nat_add by (cases "n = 0") simp_all
haftmann@63417
  1387
  then show ?thesis
haftmann@63417
  1388
    by simp
haftmann@63417
  1389
qed
haftmann@63417
  1390
haftmann@22800
  1391
wenzelm@60758
  1392
subsubsection \<open>An ``induction'' law for modulus arithmetic.\<close>
paulson@14640
  1393
paulson@14640
  1394
lemma mod_induct_0:
paulson@14640
  1395
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
  1396
  and base: "P i" and i: "i<p"
paulson@14640
  1397
  shows "P 0"
paulson@14640
  1398
proof (rule ccontr)
paulson@14640
  1399
  assume contra: "\<not>(P 0)"
paulson@14640
  1400
  from i have p: "0<p" by simp
paulson@14640
  1401
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
paulson@14640
  1402
  proof
paulson@14640
  1403
    fix k
paulson@14640
  1404
    show "?A k"
paulson@14640
  1405
    proof (induct k)
wenzelm@61799
  1406
      show "?A 0" by simp  \<comment> "by contradiction"
paulson@14640
  1407
    next
paulson@14640
  1408
      fix n
paulson@14640
  1409
      assume ih: "?A n"
paulson@14640
  1410
      show "?A (Suc n)"
paulson@14640
  1411
      proof (clarsimp)
wenzelm@22718
  1412
        assume y: "P (p - Suc n)"
wenzelm@22718
  1413
        have n: "Suc n < p"
wenzelm@22718
  1414
        proof (rule ccontr)
wenzelm@22718
  1415
          assume "\<not>(Suc n < p)"
wenzelm@22718
  1416
          hence "p - Suc n = 0"
wenzelm@22718
  1417
            by simp
wenzelm@22718
  1418
          with y contra show "False"
wenzelm@22718
  1419
            by simp
wenzelm@22718
  1420
        qed
wenzelm@22718
  1421
        hence n2: "Suc (p - Suc n) = p-n" by arith
wenzelm@22718
  1422
        from p have "p - Suc n < p" by arith
wenzelm@22718
  1423
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
wenzelm@22718
  1424
          by blast
wenzelm@22718
  1425
        show "False"
wenzelm@22718
  1426
        proof (cases "n=0")
wenzelm@22718
  1427
          case True
wenzelm@22718
  1428
          with z n2 contra show ?thesis by simp
wenzelm@22718
  1429
        next
wenzelm@22718
  1430
          case False
wenzelm@22718
  1431
          with p have "p-n < p" by arith
wenzelm@22718
  1432
          with z n2 False ih show ?thesis by simp
wenzelm@22718
  1433
        qed
paulson@14640
  1434
      qed
paulson@14640
  1435
    qed
paulson@14640
  1436
  qed
paulson@14640
  1437
  moreover
paulson@14640
  1438
  from i obtain k where "0<k \<and> i+k=p"
paulson@14640
  1439
    by (blast dest: less_imp_add_positive)
paulson@14640
  1440
  hence "0<k \<and> i=p-k" by auto
paulson@14640
  1441
  moreover
paulson@14640
  1442
  note base
paulson@14640
  1443
  ultimately
paulson@14640
  1444
  show "False" by blast
paulson@14640
  1445
qed
paulson@14640
  1446
paulson@14640
  1447
lemma mod_induct:
paulson@14640
  1448
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
  1449
  and base: "P i" and i: "i<p" and j: "j<p"
paulson@14640
  1450
  shows "P j"
paulson@14640
  1451
proof -
paulson@14640
  1452
  have "\<forall>j<p. P j"
paulson@14640
  1453
  proof
paulson@14640
  1454
    fix j
paulson@14640
  1455
    show "j<p \<longrightarrow> P j" (is "?A j")
paulson@14640
  1456
    proof (induct j)
paulson@14640
  1457
      from step base i show "?A 0"
wenzelm@22718
  1458
        by (auto elim: mod_induct_0)
paulson@14640
  1459
    next
paulson@14640
  1460
      fix k
paulson@14640
  1461
      assume ih: "?A k"
paulson@14640
  1462
      show "?A (Suc k)"
paulson@14640
  1463
      proof
wenzelm@22718
  1464
        assume suc: "Suc k < p"
wenzelm@22718
  1465
        hence k: "k<p" by simp
wenzelm@22718
  1466
        with ih have "P k" ..
wenzelm@22718
  1467
        with step k have "P (Suc k mod p)"
wenzelm@22718
  1468
          by blast
wenzelm@22718
  1469
        moreover
wenzelm@22718
  1470
        from suc have "Suc k mod p = Suc k"
wenzelm@22718
  1471
          by simp
wenzelm@22718
  1472
        ultimately
wenzelm@22718
  1473
        show "P (Suc k)" by simp
paulson@14640
  1474
      qed
paulson@14640
  1475
    qed
paulson@14640
  1476
  qed
paulson@14640
  1477
  with j show ?thesis by blast
paulson@14640
  1478
qed
paulson@14640
  1479
haftmann@33296
  1480
lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"
huffman@47138
  1481
  by (simp add: numeral_2_eq_2 le_div_geq)
huffman@47138
  1482
huffman@47138
  1483
lemma mod2_Suc_Suc [simp]: "Suc (Suc m) mod 2 = m mod 2"
huffman@47138
  1484
  by (simp add: numeral_2_eq_2 le_mod_geq)
haftmann@33296
  1485
haftmann@33296
  1486
lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
huffman@47217
  1487
by (simp add: mult_2 [symmetric])
haftmann@33296
  1488
wenzelm@61076
  1489
lemma mod2_gr_0 [simp]: "0 < (m::nat) mod 2 \<longleftrightarrow> m mod 2 = 1"
haftmann@33296
  1490
proof -
boehmes@35815
  1491
  { fix n :: nat have  "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (cases n) simp_all }
haftmann@33296
  1492
  moreover have "m mod 2 < 2" by simp
haftmann@33296
  1493
  ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" .
haftmann@33296
  1494
  then show ?thesis by auto
haftmann@33296
  1495
qed
haftmann@33296
  1496
wenzelm@60758
  1497
text\<open>These lemmas collapse some needless occurrences of Suc:
haftmann@33296
  1498
    at least three Sucs, since two and fewer are rewritten back to Suc again!
wenzelm@60758
  1499
    We already have some rules to simplify operands smaller than 3.\<close>
haftmann@33296
  1500
haftmann@33296
  1501
lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
haftmann@33296
  1502
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1503
haftmann@33296
  1504
lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
haftmann@33296
  1505
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1506
haftmann@33296
  1507
lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
haftmann@33296
  1508
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1509
haftmann@33296
  1510
lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
haftmann@33296
  1511
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1512
huffman@47108
  1513
lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v
huffman@47108
  1514
lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v
haftmann@33296
  1515
lp15@60562
  1516
lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1"
haftmann@33361
  1517
apply (induct "m")
haftmann@33361
  1518
apply (simp_all add: mod_Suc)
haftmann@33361
  1519
done
haftmann@33361
  1520
huffman@47108
  1521
declare Suc_times_mod_eq [of "numeral w", simp] for w
haftmann@33361
  1522
haftmann@60868
  1523
lemma mod_greater_zero_iff_not_dvd:
haftmann@60868
  1524
  fixes m n :: nat
haftmann@60868
  1525
  shows "m mod n > 0 \<longleftrightarrow> \<not> n dvd m"
haftmann@60868
  1526
  by (simp add: dvd_eq_mod_eq_0)
haftmann@60868
  1527
huffman@47138
  1528
lemma Suc_div_le_mono [simp]: "n div k \<le> (Suc n) div k"
huffman@47138
  1529
by (simp add: div_le_mono)
haftmann@33361
  1530
haftmann@33361
  1531
lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"
haftmann@33361
  1532
by (cases n) simp_all
haftmann@33361
  1533
boehmes@35815
  1534
lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"
boehmes@35815
  1535
proof -
boehmes@35815
  1536
  from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all
lp15@60562
  1537
  from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp
boehmes@35815
  1538
qed
haftmann@33361
  1539
haftmann@33361
  1540
lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"
haftmann@33361
  1541
proof -
haftmann@33361
  1542
  have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp
lp15@60562
  1543
  also have "... = Suc m mod n" by (rule mod_mult_self3)
haftmann@33361
  1544
  finally show ?thesis .
haftmann@33361
  1545
qed
haftmann@33361
  1546
haftmann@33361
  1547
lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"
lp15@60562
  1548
apply (subst mod_Suc [of m])
lp15@60562
  1549
apply (subst mod_Suc [of "m mod n"], simp)
haftmann@33361
  1550
done
haftmann@33361
  1551
huffman@47108
  1552
lemma mod_2_not_eq_zero_eq_one_nat:
huffman@47108
  1553
  fixes n :: nat
huffman@47108
  1554
  shows "n mod 2 \<noteq> 0 \<longleftrightarrow> n mod 2 = 1"
haftmann@58786
  1555
  by (fact not_mod_2_eq_0_eq_1)
lp15@60562
  1556
haftmann@58778
  1557
lemma even_Suc_div_two [simp]:
haftmann@58778
  1558
  "even n \<Longrightarrow> Suc n div 2 = n div 2"
haftmann@58778
  1559
  using even_succ_div_two [of n] by simp
lp15@60562
  1560
haftmann@58778
  1561
lemma odd_Suc_div_two [simp]:
haftmann@58778
  1562
  "odd n \<Longrightarrow> Suc n div 2 = Suc (n div 2)"
haftmann@58778
  1563
  using odd_succ_div_two [of n] by simp
haftmann@58778
  1564
haftmann@58834
  1565
lemma odd_two_times_div_two_nat [simp]:
haftmann@60352
  1566
  assumes "odd n"
haftmann@60352
  1567
  shows "2 * (n div 2) = n - (1 :: nat)"
haftmann@60352
  1568
proof -
haftmann@60352
  1569
  from assms have "2 * (n div 2) + 1 = n"
haftmann@60352
  1570
    by (rule odd_two_times_div_two_succ)
haftmann@60352
  1571
  then have "Suc (2 * (n div 2)) - 1 = n - 1"
haftmann@60352
  1572
    by simp
haftmann@60352
  1573
  then show ?thesis
haftmann@60352
  1574
    by simp
haftmann@60352
  1575
qed
haftmann@58778
  1576
haftmann@58778
  1577
lemma parity_induct [case_names zero even odd]:
haftmann@58778
  1578
  assumes zero: "P 0"
haftmann@58778
  1579
  assumes even: "\<And>n. P n \<Longrightarrow> P (2 * n)"
haftmann@58778
  1580
  assumes odd: "\<And>n. P n \<Longrightarrow> P (Suc (2 * n))"
haftmann@58778
  1581
  shows "P n"
haftmann@58778
  1582
proof (induct n rule: less_induct)
haftmann@58778
  1583
  case (less n)
haftmann@58778
  1584
  show "P n"
haftmann@58778
  1585
  proof (cases "n = 0")
haftmann@58778
  1586
    case True with zero show ?thesis by simp
haftmann@58778
  1587
  next
haftmann@58778
  1588
    case False
haftmann@58778
  1589
    with less have hyp: "P (n div 2)" by simp
haftmann@58778
  1590
    show ?thesis
haftmann@58778
  1591
    proof (cases "even n")
haftmann@58778
  1592
      case True
haftmann@58778
  1593
      with hyp even [of "n div 2"] show ?thesis
haftmann@58834
  1594
        by simp
haftmann@58778
  1595
    next
haftmann@58778
  1596
      case False
lp15@60562
  1597
      with hyp odd [of "n div 2"] show ?thesis
haftmann@58834
  1598
        by simp
haftmann@58778
  1599
    qed
haftmann@58778
  1600
  qed
haftmann@58778
  1601
qed
haftmann@58778
  1602
haftmann@60868
  1603
lemma Suc_0_div_numeral [simp]:
haftmann@60868
  1604
  fixes k l :: num
haftmann@60868
  1605
  shows "Suc 0 div numeral k = fst (divmod Num.One k)"
haftmann@60868
  1606
  by (simp_all add: fst_divmod)
haftmann@60868
  1607
haftmann@60868
  1608
lemma Suc_0_mod_numeral [simp]:
haftmann@60868
  1609
  fixes k l :: num
haftmann@60868
  1610
  shows "Suc 0 mod numeral k = snd (divmod Num.One k)"
haftmann@60868
  1611
  by (simp_all add: snd_divmod)
haftmann@60868
  1612
wenzelm@63145
  1613
lemma cut_eq_simps: \<comment> \<open>rewriting equivalence on \<open>n mod 2 ^ q\<close>\<close>
haftmann@62597
  1614
  fixes m n q :: num
haftmann@62597
  1615
  shows
haftmann@62597
  1616
    "numeral n mod numeral Num.One = (0::nat)
haftmann@62597
  1617
      \<longleftrightarrow> True"
haftmann@62597
  1618
    "numeral (Num.Bit0 n) mod numeral (Num.Bit0 q) = (0::nat)
haftmann@62597
  1619
      \<longleftrightarrow> numeral n mod numeral q = (0::nat)"
haftmann@62597
  1620
    "numeral (Num.Bit1 n) mod numeral (Num.Bit0 q) = (0::nat)
haftmann@62597
  1621
      \<longleftrightarrow> False"
haftmann@62597
  1622
    "numeral m mod numeral Num.One = (numeral n mod numeral Num.One :: nat)
haftmann@62597
  1623
      \<longleftrightarrow> True"
haftmann@62597
  1624
    "numeral Num.One mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q) :: nat)
haftmann@62597
  1625
      \<longleftrightarrow> True"
haftmann@62597
  1626
    "numeral Num.One mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q) :: nat)
haftmann@62597
  1627
      \<longleftrightarrow> False"
haftmann@62597
  1628
    "numeral Num.One mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q) :: nat)
haftmann@62597
  1629
      \<longleftrightarrow> (numeral n mod numeral q :: nat) = 0"
haftmann@62597
  1630
    "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q) :: nat)
haftmann@62597
  1631
      \<longleftrightarrow> False"
haftmann@62597
  1632
    "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q) :: nat)
haftmann@62597
  1633
      \<longleftrightarrow> numeral m mod numeral q = (numeral n mod numeral q :: nat)"
haftmann@62597
  1634
    "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q) :: nat)
haftmann@62597
  1635
      \<longleftrightarrow> False"
haftmann@62597
  1636
    "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q) :: nat)
haftmann@62597
  1637
      \<longleftrightarrow> (numeral m mod numeral q :: nat) = 0"
haftmann@62597
  1638
    "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q) :: nat)
haftmann@62597
  1639
      \<longleftrightarrow> False"
haftmann@62597
  1640
    "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q) :: nat)
haftmann@62597
  1641
      \<longleftrightarrow> numeral m mod numeral q = (numeral n mod numeral q :: nat)"
haftmann@62597
  1642
  by (auto simp add: case_prod_beta Suc_double_not_eq_double double_not_eq_Suc_double)
haftmann@62597
  1643
haftmann@33361
  1644
wenzelm@60758
  1645
subsection \<open>Division on @{typ int}\<close>
haftmann@33361
  1646
wenzelm@61799
  1647
definition divmod_int_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" \<comment> \<open>definition of quotient and remainder\<close>
haftmann@60868
  1648
  where "divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and>
huffman@47139
  1649
    (if 0 < b then 0 \<le> r \<and> r < b else if b < 0 then b < r \<and> r \<le> 0 else q = 0))"
haftmann@33361
  1650
haftmann@33361
  1651
lemma unique_quotient_lemma:
haftmann@60868
  1652
  "b * q' + r' \<le> b * q + r \<Longrightarrow> 0 \<le> r' \<Longrightarrow> r' < b \<Longrightarrow> r < b \<Longrightarrow> q' \<le> (q::int)"
haftmann@33361
  1653
apply (subgoal_tac "r' + b * (q'-q) \<le> r")
haftmann@33361
  1654
 prefer 2 apply (simp add: right_diff_distrib)
haftmann@33361
  1655
apply (subgoal_tac "0 < b * (1 + q - q') ")
haftmann@33361
  1656
apply (erule_tac [2] order_le_less_trans)
webertj@49962
  1657
 prefer 2 apply (simp add: right_diff_distrib distrib_left)
haftmann@33361
  1658
apply (subgoal_tac "b * q' < b * (1 + q) ")
webertj@49962
  1659
 prefer 2 apply (simp add: right_diff_distrib distrib_left)
haftmann@33361
  1660
apply (simp add: mult_less_cancel_left)
haftmann@33361
  1661
done
haftmann@33361
  1662
haftmann@33361
  1663
lemma unique_quotient_lemma_neg:
haftmann@60868
  1664
  "b * q' + r' \<le> b*q + r \<Longrightarrow> r \<le> 0 \<Longrightarrow> b < r \<Longrightarrow> b < r' \<Longrightarrow> q \<le> (q'::int)"
haftmann@60868
  1665
  by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma) auto
haftmann@33361
  1666
haftmann@33361
  1667
lemma unique_quotient:
haftmann@60868
  1668
  "divmod_int_rel a b (q, r) \<Longrightarrow> divmod_int_rel a b (q', r') \<Longrightarrow> q = q'"
nipkow@62390
  1669
apply (simp add: divmod_int_rel_def linorder_neq_iff split: if_split_asm)
haftmann@33361
  1670
apply (blast intro: order_antisym
lp15@60562
  1671
             dest: order_eq_refl [THEN unique_quotient_lemma]
haftmann@33361
  1672
             order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
haftmann@33361
  1673
done
haftmann@33361
  1674
haftmann@33361
  1675
lemma unique_remainder:
haftmann@60868
  1676
  "divmod_int_rel a b (q, r) \<Longrightarrow> divmod_int_rel a b (q', r') \<Longrightarrow> r = r'"
haftmann@33361
  1677
apply (subgoal_tac "q = q'")
haftmann@33361
  1678
 apply (simp add: divmod_int_rel_def)
haftmann@33361
  1679
apply (blast intro: unique_quotient)
haftmann@33361
  1680
done
haftmann@33361
  1681
haftmann@63950
  1682
instantiation int :: modulo
haftmann@60868
  1683
begin
haftmann@60868
  1684
haftmann@60868
  1685
definition divide_int
haftmann@60868
  1686
  where "k div l = (if l = 0 \<or> k = 0 then 0
haftmann@60868
  1687
    else if k > 0 \<and> l > 0 \<or> k < 0 \<and> l < 0
haftmann@60868
  1688
      then int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)
haftmann@60868
  1689
      else
haftmann@60868
  1690
        if l dvd k then - int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)
haftmann@60868
  1691
        else - int (Suc (nat \<bar>k\<bar> div nat \<bar>l\<bar>)))"
haftmann@60868
  1692
haftmann@63950
  1693
definition modulo_int
haftmann@60868
  1694
  where "k mod l = (if l = 0 then k else if l dvd k then 0
haftmann@60868
  1695
    else if k > 0 \<and> l > 0 \<or> k < 0 \<and> l < 0
haftmann@60868
  1696
      then sgn l * int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>)
haftmann@60868
  1697
      else sgn l * (\<bar>l\<bar> - int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>)))"
haftmann@60868
  1698
haftmann@60868
  1699
instance ..      
haftmann@60868
  1700
haftmann@60868
  1701
end
haftmann@60868
  1702
haftmann@60868
  1703
lemma divmod_int_rel:
haftmann@60868
  1704
  "divmod_int_rel k l (k div l, k mod l)"
haftmann@63950
  1705
  unfolding divmod_int_rel_def divide_int_def modulo_int_def
haftmann@60868
  1706
  apply (cases k rule: int_cases3)
lp15@61649
  1707
  apply (simp add: mod_greater_zero_iff_not_dvd not_le algebra_simps)
haftmann@60868
  1708
  apply (cases l rule: int_cases3)
lp15@61649
  1709
  apply (simp add: mod_greater_zero_iff_not_dvd not_le algebra_simps)
lp15@61649
  1710
  apply (simp_all del: of_nat_add of_nat_mult add: mod_greater_zero_iff_not_dvd not_le algebra_simps int_dvd_iff of_nat_add [symmetric] of_nat_mult [symmetric])
haftmann@60868
  1711
  apply (cases l rule: int_cases3)
lp15@61649
  1712
  apply (simp_all del: of_nat_add of_nat_mult add: not_le algebra_simps int_dvd_iff of_nat_add [symmetric] of_nat_mult [symmetric])
wenzelm@41550
  1713
  done
haftmann@33361
  1714
haftmann@60868
  1715
instantiation int :: ring_div
haftmann@60868
  1716
begin
haftmann@60868
  1717
haftmann@60868
  1718
subsubsection \<open>Uniqueness and Monotonicity of Quotients and Remainders\<close>
haftmann@33361
  1719
huffman@47141
  1720
lemma divmod_int_unique:
haftmann@60868
  1721
  assumes "divmod_int_rel k l (q, r)"
haftmann@60868
  1722
  shows div_int_unique: "k div l = q" and mod_int_unique: "k mod l = r"
haftmann@60868
  1723
  using assms divmod_int_rel [of k l]
haftmann@60868
  1724
  using unique_quotient [of k l] unique_remainder [of k l]
haftmann@60868
  1725
  by auto
haftmann@60868
  1726
  
haftmann@60429
  1727
instance
huffman@47141
  1728
proof
huffman@47141
  1729
  fix a b :: int
huffman@47141
  1730
  show "a div b * b + a mod b = a"
haftmann@60868
  1731
    using divmod_int_rel [of a b]
haftmann@57512
  1732
    unfolding divmod_int_rel_def by (simp add: mult.commute)
huffman@47141
  1733
next
huffman@47141
  1734
  fix a b c :: int
huffman@47141
  1735
  assume "b \<noteq> 0"
huffman@47141
  1736
  hence "divmod_int_rel (a + c * b) b (c + a div b, a mod b)"
haftmann@60868
  1737
    using divmod_int_rel [of a b]
huffman@47141
  1738
    unfolding divmod_int_rel_def by (auto simp: algebra_simps)
huffman@47141
  1739
  thus "(a + c * b) div b = c + a div b"
huffman@47141
  1740
    by (rule div_int_unique)
huffman@47141
  1741
next
huffman@47141
  1742
  fix a b c :: int
wenzelm@63834
  1743
  assume c: "c \<noteq> 0"
wenzelm@63834
  1744
  have "\<And>q r. divmod_int_rel a b (q, r)
huffman@47141
  1745
    \<Longrightarrow> divmod_int_rel (c * a) (c * b) (q, c * r)"
huffman@47141
  1746
    unfolding divmod_int_rel_def
wenzelm@63834
  1747
    by (rule linorder_cases [of 0 b])
wenzelm@63834
  1748
      (use c in \<open>auto simp: algebra_simps
huffman@47141
  1749
      mult_less_0_iff zero_less_mult_iff mult_strict_right_mono
wenzelm@63834
  1750
      mult_strict_right_mono_neg zero_le_mult_iff mult_le_0_iff\<close>)
huffman@47141
  1751
  hence "divmod_int_rel (c * a) (c * b) (a div b, c * (a mod b))"
haftmann@60868
  1752
    using divmod_int_rel [of a b] .
huffman@47141
  1753
  thus "(c * a) div (c * b) = a div b"
huffman@47141
  1754
    by (rule div_int_unique)
huffman@47141
  1755
next
huffman@47141
  1756
  fix a :: int show "a div 0 = 0"
huffman@47141
  1757
    by (rule div_int_unique, simp add: divmod_int_rel_def)
huffman@47141
  1758
next
huffman@47141
  1759
  fix a :: int show "0 div a = 0"
huffman@47141
  1760
    by (rule div_int_unique, auto simp add: divmod_int_rel_def)
huffman@47141
  1761
qed
huffman@47141
  1762
haftmann@60429
  1763
end
haftmann@60429
  1764
haftmann@60517
  1765
lemma is_unit_int:
haftmann@60517
  1766
  "is_unit (k::int) \<longleftrightarrow> k = 1 \<or> k = - 1"
haftmann@60517
  1767
  by auto
haftmann@60517
  1768
haftmann@60685
  1769
instantiation int :: normalization_semidom
haftmann@60685
  1770
begin
haftmann@60685
  1771
haftmann@60685
  1772
definition normalize_int
haftmann@60685
  1773
  where [simp]: "normalize = (abs :: int \<Rightarrow> int)"
haftmann@60685
  1774
haftmann@60685
  1775
definition unit_factor_int
haftmann@60685
  1776
  where [simp]: "unit_factor = (sgn :: int \<Rightarrow> int)"
haftmann@60685
  1777
haftmann@60685
  1778
instance
haftmann@60685
  1779
proof
haftmann@60685
  1780
  fix k :: int
haftmann@60685
  1781
  assume "k \<noteq> 0"
haftmann@60685
  1782
  then have "\<bar>sgn k\<bar> = 1"
haftmann@60685
  1783
    by (cases "0::int" k rule: linorder_cases) simp_all
haftmann@60685
  1784
  then show "is_unit (unit_factor k)"
haftmann@60685
  1785
    by simp
haftmann@64240
  1786
qed (simp_all add: sgn_mult mult_sgn_abs)
haftmann@60685
  1787
  
haftmann@60685
  1788
end
haftmann@60685
  1789
  
wenzelm@60758
  1790
text\<open>Basic laws about division and remainder\<close>
haftmann@33361
  1791
haftmann@60868
  1792
lemma zdiv_int: "int (a div b) = int a div int b"
haftmann@60868
  1793
  by (simp add: divide_int_def)
haftmann@60868
  1794
haftmann@60868
  1795
lemma zmod_int: "int (a mod b) = int a mod int b"
haftmann@63950
  1796
  by (simp add: modulo_int_def int_dvd_iff)
haftmann@60868
  1797
  
wenzelm@60758
  1798
text \<open>Tool setup\<close>
wenzelm@60758
  1799
wenzelm@60758
  1800
ML \<open>
wenzelm@43594
  1801
structure Cancel_Div_Mod_Int = Cancel_Div_Mod
wenzelm@41550
  1802
(
haftmann@63950
  1803
  val div_name = @{const_name divide};
haftmann@63950
  1804
  val mod_name = @{const_name modulo};
haftmann@33361
  1805
  val mk_binop = HOLogic.mk_binop;
haftmann@33361
  1806
  val mk_sum = Arith_Data.mk_sum HOLogic.intT;
haftmann@33361
  1807
  val dest_sum = Arith_Data.dest_sum;
haftmann@33361
  1808
huffman@47165
  1809
  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
haftmann@33361
  1810
lp15@60562
  1811
  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
haftmann@59556
  1812
    (@{thm diff_conv_add_uminus} :: @{thms add_0_left add_0_right} @ @{thms ac_simps}))
wenzelm@41550
  1813
)
wenzelm@60758
  1814
\<close>
wenzelm@60758
  1815
wenzelm@60758
  1816
simproc_setup cancel_div_mod_int ("(k::int) + l") = \<open>K Cancel_Div_Mod_Int.proc\<close>
wenzelm@43594
  1817
huffman@47141
  1818
lemma pos_mod_conj: "(0::int) < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b"
haftmann@60868
  1819
  using divmod_int_rel [of a b]
huffman@47141
  1820
  by (auto simp add: divmod_int_rel_def prod_eq_iff)
haftmann@33361
  1821
wenzelm@45607
  1822
lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]
wenzelm@45607
  1823
   and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]
haftmann@33361
  1824
huffman@47141
  1825
lemma neg_mod_conj: "b < (0::int) \<Longrightarrow> a mod b \<le> 0 \<and> b < a mod b"
haftmann@60868
  1826
  using divmod_int_rel [of a b]
huffman@47141
  1827
  by (auto simp add: divmod_int_rel_def prod_eq_iff)
haftmann@33361
  1828
wenzelm@45607
  1829
lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]
wenzelm@45607
  1830
   and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]
haftmann@33361
  1831
haftmann@33361
  1832
wenzelm@60758
  1833
subsubsection \<open>General Properties of div and mod\<close>
haftmann@33361
  1834
haftmann@33361
  1835
lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
huffman@47140
  1836
apply (rule div_int_unique)
haftmann@33361
  1837
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1838
done
haftmann@33361
  1839
haftmann@33361
  1840
lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
huffman@47140
  1841
apply (rule div_int_unique)
haftmann@33361
  1842
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1843
done
haftmann@33361
  1844
haftmann@33361
  1845
lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
huffman@47140
  1846
apply (rule div_int_unique)
haftmann@33361
  1847
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1848
done
haftmann@33361
  1849
haftmann@33361
  1850
(*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
haftmann@33361
  1851
haftmann@33361
  1852
lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
huffman@47140
  1853
apply (rule_tac q = 0 in mod_int_unique)
haftmann@33361
  1854
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1855
done
haftmann@33361
  1856
haftmann@33361
  1857
lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
huffman@47140
  1858
apply (rule_tac q = 0 in mod_int_unique)
haftmann@33361
  1859
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1860
done
haftmann@33361
  1861
haftmann@33361
  1862
lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
huffman@47140
  1863
apply (rule_tac q = "-1" in mod_int_unique)
haftmann@33361
  1864
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1865
done
haftmann@33361
  1866
wenzelm@61799
  1867
text\<open>There is no \<open>mod_neg_pos_trivial\<close>.\<close>
wenzelm@60758
  1868
wenzelm@60758
  1869
wenzelm@60758
  1870
subsubsection \<open>Laws for div and mod with Unary Minus\<close>
haftmann@33361
  1871
haftmann@33361
  1872
lemma zminus1_lemma:
huffman@47139
  1873
     "divmod_int_rel a b (q, r) ==> b \<noteq> 0
lp15@60562
  1874
      ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,
haftmann@33361
  1875
                          if r=0 then 0 else b-r)"
haftmann@33361
  1876
by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)
haftmann@33361
  1877
haftmann@33361
  1878
haftmann@33361
  1879
lemma zdiv_zminus1_eq_if:
lp15@60562
  1880
     "b \<noteq> (0::int)
lp15@60562
  1881
      ==> (-a) div b =
haftmann@33361
  1882
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
haftmann@60868
  1883
by (blast intro: divmod_int_rel [THEN zminus1_lemma, THEN div_int_unique])
haftmann@33361
  1884
haftmann@33361
  1885
lemma zmod_zminus1_eq_if:
haftmann@33361
  1886
     "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
haftmann@33361
  1887
apply (case_tac "b = 0", simp)
haftmann@60868
  1888
apply (blast intro: divmod_int_rel [THEN zminus1_lemma, THEN mod_int_unique])
haftmann@33361
  1889
done
haftmann@33361
  1890
haftmann@33361
  1891
lemma zmod_zminus1_not_zero:
haftmann@33361
  1892
  fixes k l :: int
haftmann@33361
  1893
  shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
haftmann@33361
  1894
  unfolding zmod_zminus1_eq_if by auto
haftmann@33361
  1895
haftmann@33361
  1896
lemma zdiv_zminus2_eq_if:
lp15@60562
  1897
     "b \<noteq> (0::int)
lp15@60562
  1898
      ==> a div (-b) =
haftmann@33361
  1899
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
huffman@47159
  1900
by (simp add: zdiv_zminus1_eq_if div_minus_right)
haftmann@33361
  1901
haftmann@33361
  1902
lemma zmod_zminus2_eq_if:
haftmann@33361
  1903
     "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
huffman@47159
  1904
by (simp add: zmod_zminus1_eq_if mod_minus_right)
haftmann@33361
  1905
haftmann@33361
  1906
lemma zmod_zminus2_not_zero:
haftmann@33361
  1907
  fixes k l :: int
haftmann@33361
  1908
  shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
lp15@60562
  1909
  unfolding zmod_zminus2_eq_if by auto
haftmann@33361
  1910
haftmann@33361
  1911
wenzelm@60758
  1912
subsubsection \<open>Monotonicity in the First Argument (Dividend)\<close>
haftmann@33361
  1913
haftmann@33361
  1914
lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
haftmann@64246
  1915
using mult_div_mod_eq [symmetric, of a b]
haftmann@64246
  1916
using mult_div_mod_eq [symmetric, of a' b]
haftmann@64246
  1917
apply -
haftmann@33361
  1918
apply (rule unique_quotient_lemma)
haftmann@33361
  1919
apply (erule subst)
haftmann@33361
  1920
apply (erule subst, simp_all)
haftmann@33361
  1921
done
haftmann@33361
  1922
haftmann@33361
  1923
lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
haftmann@64246
  1924
using mult_div_mod_eq [symmetric, of a b]
haftmann@64246
  1925
using mult_div_mod_eq [symmetric, of a' b]
haftmann@64246
  1926
apply -
haftmann@33361
  1927
apply (rule unique_quotient_lemma_neg)
haftmann@33361
  1928
apply (erule subst)
haftmann@33361
  1929
apply (erule subst, simp_all)
haftmann@33361
  1930
done
haftmann@33361
  1931
haftmann@33361
  1932
wenzelm@60758
  1933
subsubsection \<open>Monotonicity in the Second Argument (Divisor)\<close>
haftmann@33361
  1934
haftmann@33361
  1935
lemma q_pos_lemma:
haftmann@33361
  1936
     "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
haftmann@33361
  1937
apply (subgoal_tac "0 < b'* (q' + 1) ")
haftmann@33361
  1938
 apply (simp add: zero_less_mult_iff)
webertj@49962
  1939
apply (simp add: distrib_left)
haftmann@33361
  1940
done
haftmann@33361
  1941
haftmann@33361
  1942
lemma zdiv_mono2_lemma:
lp15@60562
  1943
     "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';
lp15@60562
  1944
         r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]
haftmann@33361
  1945
      ==> q \<le> (q'::int)"
lp15@60562
  1946
apply (frule q_pos_lemma, assumption+)
haftmann@33361
  1947
apply (subgoal_tac "b*q < b* (q' + 1) ")
haftmann@33361
  1948
 apply (simp add: mult_less_cancel_left)
haftmann@33361
  1949
apply (subgoal_tac "b*q = r' - r + b'*q'")
haftmann@33361
  1950
 prefer 2 apply simp
webertj@49962
  1951
apply (simp (no_asm_simp) add: distrib_left)
haftmann@57512
  1952
apply (subst add.commute, rule add_less_le_mono, arith)
haftmann@33361
  1953
apply (rule mult_right_mono, auto)
haftmann@33361
  1954
done
haftmann@33361
  1955
haftmann@33361
  1956
lemma zdiv_mono2:
haftmann@33361
  1957
     "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
haftmann@33361
  1958
apply (subgoal_tac "b \<noteq> 0")
haftmann@64246
  1959
  prefer 2 apply arith
haftmann@64246
  1960
using mult_div_mod_eq [symmetric, of a b]
haftmann@64246
  1961
using mult_div_mod_eq [symmetric, of a b']
haftmann@64246
  1962
apply -
haftmann@33361
  1963
apply (rule zdiv_mono2_lemma)
haftmann@33361
  1964
apply (erule subst)
haftmann@33361
  1965
apply (erule subst, simp_all)
haftmann@33361
  1966
done
haftmann@33361
  1967
haftmann@33361
  1968
lemma q_neg_lemma:
haftmann@33361
  1969
     "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
haftmann@33361
  1970
apply (subgoal_tac "b'*q' < 0")
haftmann@33361
  1971
 apply (simp add: mult_less_0_iff, arith)
haftmann@33361
  1972
done
haftmann@33361
  1973
haftmann@33361
  1974
lemma zdiv_mono2_neg_lemma:
lp15@60562
  1975
     "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;
lp15@60562
  1976
         r < b;  0 \<le> r';  0 < b';  b' \<le> b |]
haftmann@33361
  1977
      ==> q' \<le> (q::int)"
lp15@60562
  1978
apply (frule q_neg_lemma, assumption+)
haftmann@33361
  1979
apply (subgoal_tac "b*q' < b* (q + 1) ")
haftmann@33361
  1980
 apply (simp add: mult_less_cancel_left)
webertj@49962
  1981
apply (simp add: distrib_left)
haftmann@33361
  1982
apply (subgoal_tac "b*q' \<le> b'*q'")
haftmann@33361
  1983
 prefer 2 apply (simp add: mult_right_mono_neg, arith)
haftmann@33361
  1984
done
haftmann@33361
  1985
haftmann@33361
  1986
lemma zdiv_mono2_neg:
haftmann@33361
  1987
     "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
haftmann@64246
  1988
using mult_div_mod_eq [symmetric, of a b]
haftmann@64246
  1989
using mult_div_mod_eq [symmetric, of a b']
haftmann@64246
  1990
apply -
haftmann@33361
  1991
apply (rule zdiv_mono2_neg_lemma)
haftmann@33361
  1992
apply (erule subst)
haftmann@33361
  1993
apply (erule subst, simp_all)
haftmann@33361
  1994
done
haftmann@33361
  1995
haftmann@33361
  1996
wenzelm@60758
  1997
subsubsection \<open>More Algebraic Laws for div and mod\<close>
wenzelm@60758
  1998
wenzelm@60758
  1999
text\<open>proving (a*b) div c = a * (b div c) + a * (b mod c)\<close>
haftmann@33361
  2000
haftmann@33361
  2001
lemma zmult1_lemma:
lp15@60562
  2002
     "[| divmod_int_rel b c (q, r) |]
haftmann@33361
  2003
      ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"
haftmann@57514
  2004
by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left ac_simps)
haftmann@33361
  2005
haftmann@33361
  2006
lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
haftmann@33361
  2007
apply (case_tac "c = 0", simp)
haftmann@60868
  2008
apply (blast intro: divmod_int_rel [THEN zmult1_lemma, THEN div_int_unique])
haftmann@33361
  2009
done
haftmann@33361
  2010
wenzelm@60758
  2011
text\<open>proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c)\<close>
haftmann@33361
  2012
haftmann@33361
  2013
lemma zadd1_lemma:
lp15@60562
  2014
     "[| divmod_int_rel a c (aq, ar);  divmod_int_rel b c (bq, br) |]
haftmann@33361
  2015
      ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
webertj@49962
  2016
by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left)
haftmann@33361
  2017
haftmann@33361
  2018
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
haftmann@33361
  2019
lemma zdiv_zadd1_eq:
haftmann@33361
  2020
     "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
haftmann@33361
  2021
apply (case_tac "c = 0", simp)
haftmann@60868
  2022
apply (blast intro: zadd1_lemma [OF divmod_int_rel divmod_int_rel] div_int_unique)
haftmann@33361
  2023
done
haftmann@33361
  2024
haftmann@33361
  2025
lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
haftmann@33361
  2026
by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
haftmann@33361
  2027
haftmann@33361
  2028
(* REVISIT: should this be generalized to all semiring_div types? *)
haftmann@33361
  2029
lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
haftmann@33361
  2030
haftmann@33361
  2031
wenzelm@60758
  2032
subsubsection \<open>Proving  @{term "a div (b * c) = (a div b) div c"}\<close>
haftmann@33361
  2033
haftmann@33361
  2034
(*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
haftmann@33361
  2035
  7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
haftmann@33361
  2036
  to cause particular problems.*)
haftmann@33361
  2037
wenzelm@60758
  2038
text\<open>first, four lemmas to bound the remainder for the cases b<0 and b>0\<close>
haftmann@33361
  2039
blanchet@55085
  2040
lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b * c < b * (q mod c) + r"
haftmann@33361
  2041
apply (subgoal_tac "b * (c - q mod c) < r * 1")
haftmann@33361
  2042
 apply (simp add: algebra_simps)
haftmann@33361
  2043
apply (rule order_le_less_trans)
haftmann@33361
  2044
 apply (erule_tac [2] mult_strict_right_mono)
haftmann@33361
  2045
 apply (rule mult_left_mono_neg)
huffman@35216
  2046
  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)
haftmann@33361
  2047
 apply (simp)
haftmann@33361
  2048
apply (simp)
haftmann@33361
  2049
done
haftmann@33361
  2050
haftmann@33361
  2051
lemma zmult2_lemma_aux2:
haftmann@33361
  2052
     "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
haftmann@33361
  2053
apply (subgoal_tac "b * (q mod c) \<le> 0")
haftmann@33361
  2054
 apply arith
haftmann@33361
  2055
apply (simp add: mult_le_0_iff)
haftmann@33361
  2056
done
haftmann@33361
  2057
haftmann@33361
  2058
lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
haftmann@33361
  2059
apply (subgoal_tac "0 \<le> b * (q mod c) ")
haftmann@33361
  2060
apply arith
haftmann@33361
  2061
apply (simp add: zero_le_mult_iff)
haftmann@33361
  2062
done
haftmann@33361
  2063
haftmann@33361
  2064
lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
haftmann@33361
  2065
apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
haftmann@33361
  2066
 apply (simp add: right_diff_distrib)
haftmann@33361
  2067
apply (rule order_less_le_trans)
haftmann@33361
  2068
 apply (erule mult_strict_right_mono)
haftmann@33361
  2069
 apply (rule_tac [2] mult_left_mono)
haftmann@33361
  2070
  apply simp
huffman@35216
  2071
 using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)
haftmann@33361
  2072
apply simp
haftmann@33361
  2073
done
haftmann@33361
  2074
lp15@60562
  2075
lemma zmult2_lemma: "[| divmod_int_rel a b (q, r); 0 < c |]
haftmann@33361
  2076
      ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"
haftmann@57514
  2077
by (auto simp add: mult.assoc divmod_int_rel_def linorder_neq_iff
lp15@60562
  2078
                   zero_less_mult_iff distrib_left [symmetric]
nipkow@62390
  2079
                   zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: if_split_asm)
haftmann@33361
  2080
haftmann@53068
  2081
lemma zdiv_zmult2_eq:
haftmann@53068
  2082
  fixes a b c :: int
haftmann@53068
  2083
  shows "0 \<le> c \<Longrightarrow> a div (b * c) = (a div b) div c"
haftmann@33361
  2084
apply (case_tac "b = 0", simp)
haftmann@60868
  2085
apply (force simp add: le_less divmod_int_rel [THEN zmult2_lemma, THEN div_int_unique])
haftmann@33361
  2086
done
haftmann@33361
  2087
haftmann@33361
  2088
lemma zmod_zmult2_eq:
haftmann@53068
  2089
  fixes a b c :: int
haftmann@53068
  2090
  shows "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
haftmann@33361
  2091
apply (case_tac "b = 0", simp)
haftmann@60868
  2092
apply (force simp add: le_less divmod_int_rel [THEN zmult2_lemma, THEN mod_int_unique])
haftmann@33361
  2093
done
haftmann@33361
  2094
huffman@47108
  2095
lemma div_pos_geq:
huffman@47108
  2096
  fixes k l :: int
huffman@47108
  2097
  assumes "0 < l" and "l \<le> k"
huffman@47108
  2098
  shows "k div l = (k - l) div l + 1"
huffman@47108
  2099
proof -
huffman@47108
  2100
  have "k = (k - l) + l" by simp
huffman@47108
  2101
  then obtain j where k: "k = j + l" ..
eberlm@63499
  2102
  with assms show ?thesis by (simp add: div_add_self2)
huffman@47108
  2103
qed
huffman@47108
  2104
huffman@47108
  2105
lemma mod_pos_geq:
huffman@47108
  2106
  fixes k l :: int
huffman@47108
  2107
  assumes "0 < l" and "l \<le> k"
huffman@47108
  2108
  shows "k mod l = (k - l) mod l"
huffman@47108
  2109
proof -
huffman@47108
  2110
  have "k = (k - l) + l" by simp
huffman@47108
  2111
  then obtain j where k: "k = j + l" ..
huffman@47108
  2112
  with assms show ?thesis by simp
huffman@47108
  2113
qed
huffman@47108
  2114
haftmann@33361
  2115
wenzelm@60758
  2116
subsubsection \<open>Splitting Rules for div and mod\<close>
wenzelm@60758
  2117
wenzelm@60758
  2118
text\<open>The proofs of the two lemmas below are essentially identical\<close>
haftmann@33361
  2119
haftmann@33361
  2120
lemma split_pos_lemma:
lp15@60562
  2121
 "0<k ==>
haftmann@33361
  2122
    P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
haftmann@33361
  2123
apply (rule iffI, clarify)
lp15@60562
  2124
 apply (erule_tac P="P x y" for x y in rev_mp)
lp15@60562
  2125
 apply (subst mod_add_eq)
lp15@60562
  2126
 apply (subst zdiv_zadd1_eq)
lp15@60562
  2127
 apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)
wenzelm@60758
  2128
txt\<open>converse direction\<close>
lp15@60562
  2129
apply (drule_tac x = "n div k" in spec)
haftmann@33361
  2130
apply (drule_tac x = "n mod k" in spec, simp)
haftmann@33361
  2131
done
haftmann@33361
  2132
haftmann@33361
  2133
lemma split_neg_lemma:
haftmann@33361
  2134
 "k<0 ==>
haftmann@33361
  2135
    P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
haftmann@33361
  2136
apply (rule iffI, clarify)
lp15@60562
  2137
 apply (erule_tac P="P x y" for x y in rev_mp)
lp15@60562
  2138
 apply (subst mod_add_eq)
lp15@60562
  2139
 apply (subst zdiv_zadd1_eq)
lp15@60562
  2140
 apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)
wenzelm@60758
  2141
txt\<open>converse direction\<close>
lp15@60562
  2142
apply (drule_tac x = "n div k" in spec)
haftmann@33361
  2143
apply (drule_tac x = "n mod k" in spec, simp)
haftmann@33361
  2144
done
haftmann@33361
  2145
haftmann@33361
  2146
lemma split_zdiv:
haftmann@33361
  2147
 "P(n div k :: int) =
lp15@60562
  2148
  ((k = 0 --> P 0) &
lp15@60562
  2149
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) &
haftmann@33361
  2150
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
haftmann@33361
  2151
apply (case_tac "k=0", simp)
haftmann@33361
  2152
apply (simp only: linorder_neq_iff)
lp15@60562
  2153
apply (erule disjE)
lp15@60562
  2154
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]
haftmann@33361
  2155
                      split_neg_lemma [of concl: "%x y. P x"])
haftmann@33361
  2156
done
haftmann@33361
  2157
haftmann@33361
  2158
lemma split_zmod:
haftmann@33361
  2159
 "P(n mod k :: int) =
lp15@60562
  2160
  ((k = 0 --> P n) &
lp15@60562
  2161
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) &
haftmann@33361
  2162
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
haftmann@33361
  2163
apply (case_tac "k=0", simp)
haftmann@33361
  2164
apply (simp only: linorder_neq_iff)
lp15@60562
  2165
apply (erule disjE)
lp15@60562
  2166
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]
haftmann@33361
  2167
                      split_neg_lemma [of concl: "%x y. P y"])
haftmann@33361
  2168
done
haftmann@33361
  2169
haftmann@63950
  2170
text \<open>Enable (lin)arith to deal with @{const divide} and @{const modulo}
webertj@33730
  2171
  when these are applied to some constant that is of the form
wenzelm@60758
  2172
  @{term "numeral k"}:\<close>
huffman@47108
  2173
declare split_zdiv [of _ _ "numeral k", arith_split] for k
huffman@47108
  2174
declare split_zmod [of _ _ "numeral k", arith_split] for k
haftmann@33361
  2175
haftmann@33361
  2176
wenzelm@61799
  2177
subsubsection \<open>Computing \<open>div\<close> and \<open>mod\<close> with shifting\<close>
huffman@47166
  2178
huffman@47166
  2179
lemma pos_divmod_int_rel_mult_2:
huffman@47166
  2180
  assumes "0 \<le> b"
huffman@47166
  2181
  assumes "divmod_int_rel a b (q, r)"
huffman@47166
  2182
  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 1 + 2*r)"
huffman@47166
  2183
  using assms unfolding divmod_int_rel_def by auto
huffman@47166
  2184
huffman@47166
  2185
lemma neg_divmod_int_rel_mult_2:
huffman@47166
  2186
  assumes "b \<le> 0"
huffman@47166
  2187
  assumes "divmod_int_rel (a + 1) b (q, r)"
huffman@47166
  2188
  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 2*r - 1)"
huffman@47166
  2189
  using assms unfolding divmod_int_rel_def by auto
haftmann@33361
  2190
wenzelm@60758
  2191
text\<open>computing div by shifting\<close>
haftmann@33361
  2192
haftmann@33361
  2193
lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
haftmann@60868
  2194
  using pos_divmod_int_rel_mult_2 [OF _ divmod_int_rel]
huffman@47166
  2195
  by (rule div_int_unique)
haftmann@33361
  2196
lp15@60562
  2197
lemma neg_zdiv_mult_2:
boehmes@35815
  2198
  assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"
haftmann@60868
  2199
  using neg_divmod_int_rel_mult_2 [OF A divmod_int_rel]
huffman@47166
  2200
  by (rule div_int_unique)
haftmann@33361
  2201
huffman@47108
  2202
(* FIXME: add rules for negative numerals *)
huffman@47108
  2203
lemma zdiv_numeral_Bit0 [simp]:
huffman@47108
  2204
  "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
huffman@47108
  2205
    numeral v div (numeral w :: int)"
huffman@47108
  2206
  unfolding numeral.simps unfolding mult_2 [symmetric]
huffman@47108
  2207
  by (rule div_mult_mult1, simp)
huffman@47108
  2208
huffman@47108
  2209
lemma zdiv_numeral_Bit1 [simp]:
lp15@60562
  2210
  "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =
huffman@47108
  2211
    (numeral v div (numeral w :: int))"
huffman@47108
  2212
  unfolding numeral.simps
haftmann@57512
  2213
  unfolding mult_2 [symmetric] add.commute [of _ 1]
huffman@47108
  2214
  by (rule pos_zdiv_mult_2, simp)
haftmann@33361
  2215
haftmann@33361
  2216
lemma pos_zmod_mult_2:
haftmann@33361
  2217
  fixes a b :: int
haftmann@33361
  2218
  assumes "0 \<le> a"
haftmann@33361
  2219
  shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
haftmann@60868
  2220
  using pos_divmod_int_rel_mult_2 [OF assms divmod_int_rel]
huffman@47166
  2221
  by (rule mod_int_unique)
haftmann@33361
  2222
haftmann@33361
  2223
lemma neg_zmod_mult_2:
haftmann@33361
  2224
  fixes a b :: int
haftmann@33361
  2225
  assumes "a \<le> 0"
haftmann@33361
  2226
  shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
haftmann@60868
  2227
  using neg_divmod_int_rel_mult_2 [OF assms divmod_int_rel]
huffman@47166
  2228
  by (rule mod_int_unique)
haftmann@33361
  2229
huffman@47108
  2230
(* FIXME: add rules for negative numerals *)
huffman@47108
  2231
lemma zmod_numeral_Bit0 [simp]:
lp15@60562
  2232
  "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =
huffman@47108
  2233
    (2::int) * (numeral v mod numeral w)"
huffman@47108
  2234
  unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
huffman@47108
  2235
  unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
huffman@47108
  2236
huffman@47108
  2237
lemma zmod_numeral_Bit1 [simp]:
huffman@47108
  2238
  "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
huffman@47108
  2239
    2 * (numeral v mod numeral w) + (1::int)"
huffman@47108
  2240
  unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
haftmann@57512
  2241
  unfolding mult_2 [symmetric] add.commute [of _ 1]
huffman@47108
  2242
  by (rule pos_zmod_mult_2, simp)
haftmann@33361
  2243
nipkow@39489
  2244
lemma zdiv_eq_0_iff:
nipkow@39489
  2245
 "(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")
nipkow@39489
  2246
proof
nipkow@39489
  2247
  assume ?L
nipkow@39489
  2248
  have "?L \<longrightarrow> ?R" by (rule split_zdiv[THEN iffD2]) simp
wenzelm@60758
  2249
  with \<open>?L\<close> show ?R by blast
nipkow@39489
  2250
next
nipkow@39489
  2251
  assume ?R thus ?L
nipkow@39489
  2252
    by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)
nipkow@39489
  2253
qed
nipkow@39489
  2254
haftmann@63947
  2255
lemma zmod_trival_iff:
haftmann@63947
  2256
  fixes i k :: int
haftmann@63947
  2257
  shows "i mod k = i \<longleftrightarrow> k = 0 \<or> 0 \<le> i \<and> i < k \<or> i \<le> 0 \<and> k < i"
haftmann@63947
  2258
proof -
haftmann@63947
  2259
  have "i mod k = i \<longleftrightarrow> i div k = 0"
haftmann@64242
  2260
    by safe (insert div_mult_mod_eq [of i k], auto)
haftmann@63947
  2261
  with zdiv_eq_0_iff
haftmann@63947
  2262
  show ?thesis
haftmann@63947
  2263
    by simp
haftmann@63947
  2264
qed
nipkow@39489
  2265
wenzelm@60758
  2266
subsubsection \<open>Quotients of Signs\<close>
haftmann@33361
  2267
haftmann@60868
  2268
lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
haftmann@60868
  2269
by (simp add: divide_int_def)
haftmann@60868
  2270
haftmann@60868
  2271
lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
haftmann@63950
  2272
by (simp add: modulo_int_def)
haftmann@60868
  2273
haftmann@33361
  2274
lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
haftmann@33361
  2275
apply (subgoal_tac "a div b \<le> -1", force)
haftmann@33361
  2276
apply (rule order_trans)
haftmann@33361
  2277
apply (rule_tac a' = "-1" in zdiv_mono1)
haftmann@33361
  2278
apply (auto simp add: div_eq_minus1)
haftmann@33361
  2279
done
haftmann@33361
  2280
haftmann@33361
  2281
lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
haftmann@33361
  2282
by (drule zdiv_mono1_neg, auto)
haftmann@33361
  2283
haftmann@33361
  2284
lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
haftmann@33361
  2285
by (drule zdiv_mono1, auto)
haftmann@33361
  2286
wenzelm@61799
  2287
text\<open>Now for some equivalences of the form \<open>a div b >=< 0 \<longleftrightarrow> \<dots>\<close>
wenzelm@61799
  2288
conditional upon the sign of \<open>a\<close> or \<open>b\<close>. There are many more.
wenzelm@60758
  2289
They should all be simp rules unless that causes too much search.\<close>
nipkow@33804
  2290
haftmann@33361
  2291
lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
haftmann@33361
  2292
apply auto
haftmann@33361
  2293
apply (drule_tac [2] zdiv_mono1)
haftmann@33361
  2294
apply (auto simp add: linorder_neq_iff)
haftmann@33361
  2295
apply (simp (no_asm_use) add: linorder_not_less [symmetric])
haftmann@33361
  2296
apply (blast intro: div_neg_pos_less0)
haftmann@33361
  2297
done
haftmann@33361
  2298
haftmann@60868
  2299
lemma pos_imp_zdiv_pos_iff:
haftmann@60868
  2300
  "0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"
haftmann@60868
  2301
using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]
haftmann@60868
  2302
by arith
haftmann@60868
  2303
haftmann@33361
  2304
lemma neg_imp_zdiv_nonneg_iff:
nipkow@33804
  2305
  "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
huffman@47159
  2306
apply (subst div_minus_minus [symmetric])
haftmann@33361
  2307
apply (subst pos_imp_zdiv_nonneg_iff, auto)
haftmann@33361
  2308
done
haftmann@33361
  2309
haftmann@33361
  2310
(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
haftmann@33361
  2311
lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
haftmann@33361
  2312
by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
haftmann@33361
  2313
haftmann@33361
  2314
(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
haftmann@33361
  2315
lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
haftmann@33361
  2316
by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
haftmann@33361
  2317
nipkow@33804
  2318
lemma nonneg1_imp_zdiv_pos_iff:
nipkow@33804
  2319
  "(0::int) <= a \<Longrightarrow> (a div b > 0) = (a >= b & b>0)"
nipkow@33804
  2320
apply rule
nipkow@33804
  2321
 apply rule
nipkow@33804
  2322
  using div_pos_pos_trivial[of a b]apply arith
nipkow@33804
  2323
 apply(cases "b=0")apply simp
nipkow@33804
  2324
 using div_nonneg_neg_le0[of a b]apply arith
nipkow@33804
  2325
using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp
nipkow@33804
  2326
done
nipkow@33804
  2327
nipkow@39489
  2328
lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 ==> m mod k \<le> m"
nipkow@39489
  2329
apply (rule split_zmod[THEN iffD2])
nipkow@44890
  2330
apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)
nipkow@39489
  2331
done
nipkow@39489
  2332
haftmann@60868
  2333
haftmann@60868
  2334
subsubsection \<open>Computation of Division and Remainder\<close>
haftmann@60868
  2335
haftmann@61275
  2336
instantiation int :: semiring_numeral_div
haftmann@61275
  2337
begin
haftmann@61275
  2338
haftmann@61275
  2339
definition divmod_int :: "num \<Rightarrow> num \<Rightarrow> int \<times> int"
haftmann@61275
  2340
where
haftmann@61275
  2341
  "divmod_int m n = (numeral m div numeral n, numeral m mod numeral n)"
haftmann@61275
  2342
haftmann@61275
  2343
definition divmod_step_int :: "num \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int"
haftmann@61275
  2344
where
haftmann@61275
  2345
  "divmod_step_int l qr = (let (q, r) = qr
haftmann@61275
  2346
    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
haftmann@61275
  2347
    else (2 * q, r))"
haftmann@61275
  2348
haftmann@61275
  2349
instance
haftmann@61275
  2350
  by standard (auto intro: zmod_le_nonneg_dividend simp add: divmod_int_def divmod_step_int_def
haftmann@61275
  2351
    pos_imp_zdiv_pos_iff div_pos_pos_trivial mod_pos_pos_trivial zmod_zmult2_eq zdiv_zmult2_eq)
haftmann@61275
  2352
haftmann@61275
  2353
end
haftmann@61275
  2354
haftmann@61275
  2355
declare divmod_algorithm_code [where ?'a = int, code]
lp15@60562
  2356
haftmann@60930
  2357
context
haftmann@60930
  2358
begin
haftmann@60930
  2359
  
haftmann@60930
  2360
qualified definition adjust_div :: "int \<times> int \<Rightarrow> int"
haftmann@60868
  2361
where
haftmann@60868
  2362
  "adjust_div qr = (let (q, r) = qr in q + of_bool (r \<noteq> 0))"
haftmann@60868
  2363
haftmann@60930
  2364
qualified lemma adjust_div_eq [simp, code]:
haftmann@60868
  2365
  "adjust_div (q, r) = q + of_bool (r \<noteq> 0)"
haftmann@60868
  2366
  by (simp add: adjust_div_def)
haftmann@60868
  2367
haftmann@60930
  2368
qualified definition adjust_mod :: "int \<Rightarrow> int \<Rightarrow> int"
haftmann@60868
  2369
where
haftmann@60868
  2370
  [simp]: "adjust_mod l r = (if r = 0 then 0 else l - r)"
haftmann@60868
  2371
haftmann@60868
  2372
lemma minus_numeral_div_numeral [simp]:
haftmann@60868
  2373
  "- numeral m div numeral n = - (adjust_div (divmod m n) :: int)"
haftmann@60868
  2374
proof -
haftmann@60868
  2375
  have "int (fst (divmod m n)) = fst (divmod m n)"
haftmann@60868
  2376
    by (simp only: fst_divmod divide_int_def) auto
haftmann@60868
  2377
  then show ?thesis
haftmann@60868
  2378
    by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
haftmann@60868
  2379
qed
haftmann@60868
  2380
haftmann@60868
  2381
lemma minus_numeral_mod_numeral [simp]:
haftmann@60868
  2382
  "- numeral m mod numeral n = adjust_mod (numeral n) (snd (divmod m n) :: int)"
haftmann@60868
  2383
proof -
haftmann@60868
  2384
  have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
haftmann@63950
  2385
    using that by (simp only: snd_divmod modulo_int_def) auto
haftmann@60868
  2386
  then show ?thesis
haftmann@63950
  2387
    by (auto simp add: split_def Let_def adjust_div_def divides_aux_def modulo_int_def)
haftmann@60868
  2388
qed
haftmann@60868
  2389
haftmann@60868
  2390
lemma numeral_div_minus_numeral [simp]:
haftmann@60868
  2391
  "numeral m div - numeral n = - (adjust_div (divmod m n) :: int)"
haftmann@60868
  2392
proof -
haftmann@60868
  2393
  have "int (fst (divmod m n)) = fst (divmod m n)"
haftmann@60868
  2394
    by (simp only: fst_divmod divide_int_def) auto
haftmann@60868
  2395
  then show ?thesis
haftmann@60868
  2396
    by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
haftmann@60868
  2397
qed
haftmann@60868
  2398
  
haftmann@60868
  2399
lemma numeral_mod_minus_numeral [simp]:
haftmann@60868
  2400
  "numeral m mod - numeral n = - adjust_mod (numeral n) (snd (divmod m n) :: int)"
haftmann@60868
  2401
proof -
haftmann@60868
  2402
  have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
haftmann@63950
  2403
    using that by (simp only: snd_divmod modulo_int_def) auto
haftmann@60868
  2404
  then show ?thesis
haftmann@63950
  2405
    by (auto simp add: split_def Let_def adjust_div_def divides_aux_def modulo_int_def)
haftmann@60868
  2406
qed
haftmann@60868
  2407
haftmann@60868
  2408
lemma minus_one_div_numeral [simp]:
haftmann@60868
  2409
  "- 1 div numeral n = - (adjust_div (divmod Num.One n) :: int)"
haftmann@60868
  2410
  using minus_numeral_div_numeral [of Num.One n] by simp  
haftmann@60868
  2411
haftmann@60868
  2412
lemma minus_one_mod_numeral [simp]:
haftmann@60868
  2413
  "- 1 mod numeral n = adjust_mod (numeral n) (snd (divmod Num.One n) :: int)"
haftmann@60868
  2414
  using minus_numeral_mod_numeral [of Num.One n] by simp
haftmann@60868
  2415
haftmann@60868
  2416
lemma one_div_minus_numeral [simp]:
haftmann@60868
  2417
  "1 div - numeral n = - (adjust_div (divmod Num.One n) :: int)"
haftmann@60868
  2418
  using numeral_div_minus_numeral [of Num.One n] by simp
haftmann@60868
  2419
  
haftmann@60868
  2420
lemma one_mod_minus_numeral [simp]:
haftmann@60868
  2421
  "1 mod - numeral n = - adjust_mod (numeral n) (snd (divmod Num.One n) :: int)"
haftmann@60868
  2422
  using numeral_mod_minus_numeral [of Num.One n] by simp
haftmann@60868
  2423
haftmann@60930
  2424
end
haftmann@60930
  2425
haftmann@60868
  2426
haftmann@60868
  2427
subsubsection \<open>Further properties\<close>
haftmann@60868
  2428
haftmann@60868
  2429
text \<open>Simplify expresions in which div and mod combine numerical constants\<close>
haftmann@60868
  2430
haftmann@60868
  2431
lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q"
haftmann@60868
  2432
  by (rule div_int_unique [of a b q r]) (simp add: divmod_int_rel_def)
haftmann@60868
  2433
haftmann@60868
  2434
lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q"
haftmann@60868
  2435
  by (rule div_int_unique [of a b q r],
haftmann@60868
  2436
    simp add: divmod_int_rel_def)
haftmann@60868
  2437
haftmann@60868
  2438
lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r"
haftmann@60868
  2439
  by (rule mod_int_unique [of a b q r],
haftmann@60868
  2440
    simp add: divmod_int_rel_def)
haftmann@60868
  2441
haftmann@60868
  2442
lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r"
haftmann@60868
  2443
  by (rule mod_int_unique [of a b q r],
haftmann@60868
  2444
    simp add: divmod_int_rel_def)
haftmann@33361
  2445
wenzelm@61944
  2446
lemma abs_div: "(y::int) dvd x \<Longrightarrow> \<bar>x div y\<bar> = \<bar>x\<bar> div \<bar>y\<bar>"
haftmann@33361
  2447
by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
haftmann@33361
  2448
wenzelm@60758
  2449
text\<open>Suggested by Matthias Daum\<close>
haftmann@33361
  2450
lemma int_power_div_base:
haftmann@33361
  2451
     "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
haftmann@33361
  2452
apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
haftmann@33361
  2453
 apply (erule ssubst)
haftmann@33361
  2454
 apply (simp only: power_add)
haftmann@33361
  2455
 apply simp_all
haftmann@33361
  2456
done
haftmann@33361
  2457
wenzelm@60758
  2458
text \<open>by Brian Huffman\<close>
haftmann@33361
  2459
lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
haftmann@33361
  2460
by (rule mod_minus_eq [symmetric])
haftmann@33361
  2461
haftmann@33361
  2462
lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
haftmann@33361
  2463
by (rule mod_diff_left_eq [symmetric])
haftmann@33361
  2464
haftmann@33361
  2465
lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
haftmann@33361
  2466
by (rule mod_diff_right_eq [symmetric])
haftmann@33361
  2467
haftmann@33361
  2468
lemmas zmod_simps =
haftmann@33361
  2469
  mod_add_left_eq  [symmetric]
haftmann@33361
  2470
  mod_add_right_eq [symmetric]
huffman@47142
  2471
  mod_mult_right_eq[symmetric]
haftmann@33361
  2472
  mod_mult_left_eq [symmetric]
huffman@47164
  2473
  power_mod
haftmann@33361
  2474
  zminus_zmod zdiff_zmod_left zdiff_zmod_right
haftmann@33361
  2475
wenzelm@61799
  2476
text \<open>Distributive laws for function \<open>nat\<close>.\<close>
haftmann@33361
  2477
haftmann@33361
  2478
lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
haftmann@33361
  2479
apply (rule linorder_cases [of y 0])
haftmann@33361
  2480
apply (simp add: div_nonneg_neg_le0)
haftmann@33361
  2481
apply simp
haftmann@33361
  2482
apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
haftmann@33361
  2483
done
haftmann@33361
  2484
haftmann@33361
  2485
(*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
haftmann@33361
  2486
lemma nat_mod_distrib:
haftmann@33361
  2487
  "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
haftmann@33361
  2488
apply (case_tac "y = 0", simp)
haftmann@33361
  2489
apply (simp add: nat_eq_iff zmod_int)
haftmann@33361
  2490
done
haftmann@33361
  2491
wenzelm@60758
  2492
text  \<open>transfer setup\<close>
haftmann@33361
  2493
haftmann@33361
  2494
lemma transfer_nat_int_functions:
haftmann@33361
  2495
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
haftmann@33361
  2496
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
haftmann@33361
  2497
  by (auto simp add: nat_div_distrib nat_mod_distrib)
haftmann@33361
  2498
haftmann@33361
  2499
lemma transfer_nat_int_function_closures:
haftmann@33361
  2500
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
haftmann@33361
  2501
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
haftmann@33361
  2502
  apply (cases "y = 0")
haftmann@33361
  2503
  apply (auto simp add: pos_imp_zdiv_nonneg_iff)
haftmann@33361
  2504
  apply (cases "y = 0")
haftmann@33361
  2505
  apply auto
haftmann@33361
  2506
done
haftmann@33361
  2507
haftmann@35644
  2508
declare transfer_morphism_nat_int [transfer add return:
haftmann@33361
  2509
  transfer_nat_int_functions
haftmann@33361
  2510
  transfer_nat_int_function_closures
haftmann@33361
  2511
]
haftmann@33361
  2512
haftmann@33361
  2513
lemma transfer_int_nat_functions:
haftmann@33361
  2514
    "(int x) div (int y) = int (x div y)"
haftmann@33361
  2515
    "(int x) mod (int y) = int (x mod y)"
haftmann@33361
  2516
  by (auto simp add: zdiv_int zmod_int)
haftmann@33361
  2517
<