src/HOL/Divides.thy
author wenzelm
Mon Mar 17 18:37:00 2008 +0100 (2008-03-17)
changeset 26300 03def556e26e
parent 26100 fbc60cd02ae2
child 26748 4d51ddd6aa5c
permissions -rw-r--r--
removed duplicate lemmas;
paulson@3366
     1
(*  Title:      HOL/Divides.thy
paulson@3366
     2
    ID:         $Id$
paulson@3366
     3
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
paulson@6865
     4
    Copyright   1999  University of Cambridge
huffman@18154
     5
*)
paulson@3366
     6
haftmann@26100
     7
header {* The division operators div,  mod and the divides relation dvd *}
paulson@3366
     8
nipkow@15131
     9
theory Divides
haftmann@26100
    10
imports Nat Power Product_Type Wellfounded_Recursion
haftmann@22993
    11
uses "~~/src/Provers/Arith/cancel_div_mod.ML"
nipkow@15131
    12
begin
paulson@3366
    13
haftmann@25942
    14
subsection {* Syntactic division operations *}
haftmann@25942
    15
haftmann@24993
    16
class div = times +
haftmann@25062
    17
  fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
haftmann@25062
    18
  fixes mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)
haftmann@25942
    19
begin
haftmann@21408
    20
haftmann@25942
    21
definition
haftmann@25942
    22
  dvd  :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "dvd" 50)
haftmann@25942
    23
where
haftmann@25942
    24
  [code func del]: "m dvd n \<longleftrightarrow> (\<exists>k. n = m * k)"
haftmann@25942
    25
haftmann@25942
    26
end
haftmann@25942
    27
haftmann@25942
    28
subsection {* Abstract divisibility in commutative semirings. *}
haftmann@25942
    29
haftmann@25942
    30
class semiring_div = comm_semiring_1_cancel + div + 
haftmann@25942
    31
  assumes mod_div_equality: "a div b * b + a mod b = a"
haftmann@25942
    32
    and div_by_0: "a div 0 = 0"
haftmann@25942
    33
    and mult_div: "b \<noteq> 0 \<Longrightarrow> a * b div b = a"
haftmann@25942
    34
begin
haftmann@25942
    35
haftmann@26100
    36
text {* @{const div} and @{const mod} *}
haftmann@26100
    37
haftmann@25942
    38
lemma div_by_1: "a div 1 = a"
haftmann@26062
    39
  using mult_div [of 1 a] zero_neq_one by simp
haftmann@25942
    40
haftmann@25942
    41
lemma mod_by_1: "a mod 1 = 0"
haftmann@25942
    42
proof -
haftmann@25942
    43
  from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp
haftmann@25942
    44
  then have "a + a mod 1 = a + 0" by simp
haftmann@25942
    45
  then show ?thesis by (rule add_left_imp_eq)
haftmann@25942
    46
qed
haftmann@25942
    47
haftmann@25942
    48
lemma mod_by_0: "a mod 0 = a"
haftmann@25942
    49
  using mod_div_equality [of a zero] by simp
haftmann@25942
    50
haftmann@25942
    51
lemma mult_mod: "a * b mod b = 0"
haftmann@25942
    52
proof (cases "b = 0")
haftmann@25942
    53
  case True then show ?thesis by (simp add: mod_by_0)
haftmann@25942
    54
next
haftmann@25942
    55
  case False with mult_div have abb: "a * b div b = a" .
haftmann@25942
    56
  from mod_div_equality have "a * b div b * b + a * b mod b = a * b" .
haftmann@25942
    57
  with abb have "a * b + a * b mod b = a * b + 0" by simp
haftmann@25942
    58
  then show ?thesis by (rule add_left_imp_eq)
haftmann@25942
    59
qed
haftmann@25942
    60
haftmann@25942
    61
lemma mod_self: "a mod a = 0"
haftmann@25942
    62
  using mult_mod [of one] by simp
haftmann@25942
    63
haftmann@25942
    64
lemma div_self: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
haftmann@25942
    65
  using mult_div [of _ one] by simp
haftmann@25942
    66
haftmann@25942
    67
lemma div_0: "0 div a = 0"
haftmann@25942
    68
proof (cases "a = 0")
haftmann@25942
    69
  case True then show ?thesis by (simp add: div_by_0)
haftmann@25942
    70
next
haftmann@25942
    71
  case False with mult_div have "0 * a div a = 0" .
haftmann@25942
    72
  then show ?thesis by simp
haftmann@25942
    73
qed
haftmann@25942
    74
haftmann@25942
    75
lemma mod_0: "0 mod a = 0"
haftmann@25942
    76
  using mod_div_equality [of zero a] div_0 by simp 
haftmann@25942
    77
haftmann@26062
    78
lemma mod_div_equality2: "b * (a div b) + a mod b = a"
haftmann@26062
    79
  unfolding mult_commute [of b]
haftmann@26062
    80
  by (rule mod_div_equality)
haftmann@26062
    81
haftmann@26062
    82
lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
haftmann@26062
    83
  by (simp add: mod_div_equality)
haftmann@26062
    84
haftmann@26062
    85
lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"
haftmann@26062
    86
  by (simp add: mod_div_equality2)
haftmann@26062
    87
haftmann@26100
    88
text {* The @{const dvd} relation *}
haftmann@26100
    89
haftmann@26062
    90
lemma dvdI [intro?]: "a = b * c \<Longrightarrow> b dvd a"
haftmann@26062
    91
  unfolding dvd_def ..
haftmann@26062
    92
haftmann@26062
    93
lemma dvdE [elim?]: "b dvd a \<Longrightarrow> (\<And>c. a = b * c \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@26062
    94
  unfolding dvd_def by blast 
haftmann@26062
    95
haftmann@25942
    96
lemma dvd_def_mod [code func]: "a dvd b \<longleftrightarrow> b mod a = 0"
haftmann@25942
    97
proof
haftmann@25942
    98
  assume "b mod a = 0"
haftmann@25942
    99
  with mod_div_equality [of b a] have "b div a * a = b" by simp
haftmann@25942
   100
  then have "b = a * (b div a)" unfolding mult_commute ..
haftmann@25942
   101
  then have "\<exists>c. b = a * c" ..
haftmann@25942
   102
  then show "a dvd b" unfolding dvd_def .
haftmann@25942
   103
next
haftmann@25942
   104
  assume "a dvd b"
haftmann@25942
   105
  then have "\<exists>c. b = a * c" unfolding dvd_def .
haftmann@25942
   106
  then obtain c where "b = a * c" ..
haftmann@25942
   107
  then have "b mod a = a * c mod a" by simp
haftmann@25942
   108
  then have "b mod a = c * a mod a" by (simp add: mult_commute)
haftmann@25942
   109
  then show "b mod a = 0" by (simp add: mult_mod)
haftmann@25942
   110
qed
haftmann@25942
   111
haftmann@25942
   112
lemma dvd_refl: "a dvd a"
haftmann@25942
   113
  unfolding dvd_def_mod mod_self ..
haftmann@25942
   114
haftmann@25942
   115
lemma dvd_trans:
haftmann@25942
   116
  assumes "a dvd b" and "b dvd c"
haftmann@25942
   117
  shows "a dvd c"
haftmann@25942
   118
proof -
haftmann@25942
   119
  from assms obtain v where "b = a * v" unfolding dvd_def by auto
haftmann@25942
   120
  moreover from assms obtain w where "c = b * w" unfolding dvd_def by auto
haftmann@25942
   121
  ultimately have "c = a * (v * w)" by (simp add: mult_assoc)
haftmann@25942
   122
  then show ?thesis unfolding dvd_def ..
haftmann@25942
   123
qed
haftmann@25942
   124
haftmann@26062
   125
lemma zero_dvd_iff [noatp]: "0 dvd a \<longleftrightarrow> a = 0"
haftmann@25942
   126
  unfolding dvd_def by simp
haftmann@25942
   127
haftmann@25942
   128
lemma dvd_0: "a dvd 0"
haftmann@25942
   129
unfolding dvd_def proof
haftmann@25942
   130
  show "0 = a * 0" by simp
haftmann@25942
   131
qed
haftmann@25942
   132
haftmann@26062
   133
lemma one_dvd: "1 dvd a"
haftmann@26062
   134
  unfolding dvd_def by simp
haftmann@26062
   135
haftmann@26062
   136
lemma dvd_mult: "a dvd c \<Longrightarrow> a dvd (b * c)"
haftmann@26062
   137
  unfolding dvd_def by (blast intro: mult_left_commute)
haftmann@26062
   138
haftmann@26062
   139
lemma dvd_mult2: "a dvd b \<Longrightarrow> a dvd (b * c)"
haftmann@26062
   140
  apply (subst mult_commute)
haftmann@26062
   141
  apply (erule dvd_mult)
haftmann@26062
   142
  done
haftmann@26062
   143
haftmann@26062
   144
lemma dvd_triv_right: "a dvd b * a"
haftmann@26062
   145
  by (rule dvd_mult) (rule dvd_refl)
haftmann@26062
   146
haftmann@26062
   147
lemma dvd_triv_left: "a dvd a * b"
haftmann@26062
   148
  by (rule dvd_mult2) (rule dvd_refl)
haftmann@26062
   149
haftmann@26062
   150
lemma mult_dvd_mono: "a dvd c \<Longrightarrow> b dvd d \<Longrightarrow> a * b dvd c * d"
haftmann@26062
   151
  apply (unfold dvd_def, clarify)
haftmann@26062
   152
  apply (rule_tac x = "k * ka" in exI)
haftmann@26062
   153
  apply (simp add: mult_ac)
haftmann@26062
   154
  done
haftmann@26062
   155
haftmann@26062
   156
lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c"
haftmann@26062
   157
  by (simp add: dvd_def mult_assoc, blast)
haftmann@26062
   158
haftmann@26062
   159
lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c"
haftmann@26062
   160
  unfolding mult_ac [of a] by (rule dvd_mult_left)
haftmann@26062
   161
haftmann@25942
   162
end
haftmann@25942
   163
haftmann@25942
   164
haftmann@26100
   165
subsection {* Division on @{typ nat} *}
haftmann@26100
   166
haftmann@26100
   167
text {*
haftmann@26100
   168
  We define @{const div} and @{const mod} on @{typ nat} by means
haftmann@26100
   169
  of a characteristic relation with two input arguments
haftmann@26100
   170
  @{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments
haftmann@26100
   171
  @{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder).
haftmann@26100
   172
*}
haftmann@26100
   173
haftmann@26100
   174
definition divmod_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where
haftmann@26100
   175
  "divmod_rel m n q r \<longleftrightarrow> m = q * n + r \<and> (if n > 0 then 0 \<le> r \<and> r < n else q = 0)"
haftmann@26100
   176
haftmann@26100
   177
text {* @{const divmod_rel} is total: *}
haftmann@26100
   178
haftmann@26100
   179
lemma divmod_rel_ex:
haftmann@26100
   180
  obtains q r where "divmod_rel m n q r"
haftmann@26100
   181
proof (cases "n = 0")
haftmann@26100
   182
  case True with that show thesis
haftmann@26100
   183
    by (auto simp add: divmod_rel_def)
haftmann@26100
   184
next
haftmann@26100
   185
  case False
haftmann@26100
   186
  have "\<exists>q r. m = q * n + r \<and> r < n"
haftmann@26100
   187
  proof (induct m)
haftmann@26100
   188
    case 0 with `n \<noteq> 0`
haftmann@26100
   189
    have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp
haftmann@26100
   190
    then show ?case by blast
haftmann@26100
   191
  next
haftmann@26100
   192
    case (Suc m) then obtain q' r'
haftmann@26100
   193
      where m: "m = q' * n + r'" and n: "r' < n" by auto
haftmann@26100
   194
    then show ?case proof (cases "Suc r' < n")
haftmann@26100
   195
      case True
haftmann@26100
   196
      from m n have "Suc m = q' * n + Suc r'" by simp
haftmann@26100
   197
      with True show ?thesis by blast
haftmann@26100
   198
    next
haftmann@26100
   199
      case False then have "n \<le> Suc r'" by auto
haftmann@26100
   200
      moreover from n have "Suc r' \<le> n" by auto
haftmann@26100
   201
      ultimately have "n = Suc r'" by auto
haftmann@26100
   202
      with m have "Suc m = Suc q' * n + 0" by simp
haftmann@26100
   203
      with `n \<noteq> 0` show ?thesis by blast
haftmann@26100
   204
    qed
haftmann@26100
   205
  qed
haftmann@26100
   206
  with that show thesis
haftmann@26100
   207
    using `n \<noteq> 0` by (auto simp add: divmod_rel_def)
haftmann@26100
   208
qed
haftmann@26100
   209
haftmann@26100
   210
text {* @{const divmod_rel} is injective: *}
haftmann@26100
   211
haftmann@26100
   212
lemma divmod_rel_unique_div:
haftmann@26100
   213
  assumes "divmod_rel m n q r"
haftmann@26100
   214
    and "divmod_rel m n q' r'"
haftmann@26100
   215
  shows "q = q'"
haftmann@26100
   216
proof (cases "n = 0")
haftmann@26100
   217
  case True with assms show ?thesis
haftmann@26100
   218
    by (simp add: divmod_rel_def)
haftmann@26100
   219
next
haftmann@26100
   220
  case False
haftmann@26100
   221
  have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)"
haftmann@26100
   222
  apply (rule leI)
haftmann@26100
   223
  apply (subst less_iff_Suc_add)
haftmann@26100
   224
  apply (auto simp add: add_mult_distrib)
haftmann@26100
   225
  done
haftmann@26100
   226
  from `n \<noteq> 0` assms show ?thesis
haftmann@26100
   227
    by (auto simp add: divmod_rel_def
haftmann@26100
   228
      intro: order_antisym dest: aux sym)
haftmann@26100
   229
qed
haftmann@26100
   230
haftmann@26100
   231
lemma divmod_rel_unique_mod:
haftmann@26100
   232
  assumes "divmod_rel m n q r"
haftmann@26100
   233
    and "divmod_rel m n q' r'"
haftmann@26100
   234
  shows "r = r'"
haftmann@26100
   235
proof -
haftmann@26100
   236
  from assms have "q = q'" by (rule divmod_rel_unique_div)
haftmann@26100
   237
  with assms show ?thesis by (simp add: divmod_rel_def)
haftmann@26100
   238
qed
haftmann@26100
   239
haftmann@26100
   240
text {*
haftmann@26100
   241
  We instantiate divisibility on the natural numbers by
haftmann@26100
   242
  means of @{const divmod_rel}:
haftmann@26100
   243
*}
haftmann@25942
   244
haftmann@25942
   245
instantiation nat :: semiring_div
haftmann@25571
   246
begin
haftmann@25571
   247
haftmann@26100
   248
definition divmod :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
haftmann@26100
   249
  [code func del]: "divmod m n = (THE (q, r). divmod_rel m n q r)"
haftmann@26100
   250
haftmann@26100
   251
definition div_nat where
haftmann@26100
   252
  "m div n = fst (divmod m n)"
haftmann@26100
   253
haftmann@26100
   254
definition mod_nat where
haftmann@26100
   255
  "m mod n = snd (divmod m n)"
haftmann@25571
   256
haftmann@26100
   257
lemma divmod_div_mod:
haftmann@26100
   258
  "divmod m n = (m div n, m mod n)"
haftmann@26100
   259
  unfolding div_nat_def mod_nat_def by simp
haftmann@26100
   260
haftmann@26100
   261
lemma divmod_eq:
haftmann@26100
   262
  assumes "divmod_rel m n q r" 
haftmann@26100
   263
  shows "divmod m n = (q, r)"
haftmann@26100
   264
  using assms by (auto simp add: divmod_def
haftmann@26100
   265
    dest: divmod_rel_unique_div divmod_rel_unique_mod)
haftmann@25942
   266
haftmann@26100
   267
lemma div_eq:
haftmann@26100
   268
  assumes "divmod_rel m n q r" 
haftmann@26100
   269
  shows "m div n = q"
haftmann@26100
   270
  using assms by (auto dest: divmod_eq simp add: div_nat_def)
haftmann@26100
   271
haftmann@26100
   272
lemma mod_eq:
haftmann@26100
   273
  assumes "divmod_rel m n q r" 
haftmann@26100
   274
  shows "m mod n = r"
haftmann@26100
   275
  using assms by (auto dest: divmod_eq simp add: mod_nat_def)
haftmann@25571
   276
haftmann@26100
   277
lemma divmod_rel: "divmod_rel m n (m div n) (m mod n)"
haftmann@26100
   278
proof -
haftmann@26100
   279
  from divmod_rel_ex
haftmann@26100
   280
    obtain q r where rel: "divmod_rel m n q r" .
haftmann@26100
   281
  moreover with div_eq mod_eq have "m div n = q" and "m mod n = r"
haftmann@26100
   282
    by simp_all
haftmann@26100
   283
  ultimately show ?thesis by simp
haftmann@26100
   284
qed
paulson@14267
   285
haftmann@26100
   286
lemma divmod_zero:
haftmann@26100
   287
  "divmod m 0 = (0, m)"
haftmann@26100
   288
proof -
haftmann@26100
   289
  from divmod_rel [of m 0] show ?thesis
haftmann@26100
   290
    unfolding divmod_div_mod divmod_rel_def by simp
haftmann@26100
   291
qed
haftmann@25942
   292
haftmann@26100
   293
lemma divmod_base:
haftmann@26100
   294
  assumes "m < n"
haftmann@26100
   295
  shows "divmod m n = (0, m)"
haftmann@26100
   296
proof -
haftmann@26100
   297
  from divmod_rel [of m n] show ?thesis
haftmann@26100
   298
    unfolding divmod_div_mod divmod_rel_def
haftmann@26100
   299
    using assms by (cases "m div n = 0")
haftmann@26100
   300
      (auto simp add: gr0_conv_Suc [of "m div n"])
haftmann@26100
   301
qed
haftmann@25942
   302
haftmann@26100
   303
lemma divmod_step:
haftmann@26100
   304
  assumes "0 < n" and "n \<le> m"
haftmann@26100
   305
  shows "divmod m n = (Suc ((m - n) div n), (m - n) mod n)"
haftmann@26100
   306
proof -
haftmann@26100
   307
  from divmod_rel have divmod_m_n: "divmod_rel m n (m div n) (m mod n)" .
haftmann@26100
   308
  with assms have m_div_n: "m div n \<ge> 1"
haftmann@26100
   309
    by (cases "m div n") (auto simp add: divmod_rel_def)
haftmann@26100
   310
  from assms divmod_m_n have "divmod_rel (m - n) n (m div n - 1) (m mod n)"
haftmann@26100
   311
    by (cases "m div n") (auto simp add: divmod_rel_def)
haftmann@26100
   312
  with divmod_eq have "divmod (m - n) n = (m div n - 1, m mod n)" by simp
haftmann@26100
   313
  moreover from divmod_div_mod have "divmod (m - n) n = ((m - n) div n, (m - n) mod n)" .
haftmann@26100
   314
  ultimately have "m div n = Suc ((m - n) div n)"
haftmann@26100
   315
    and "m mod n = (m - n) mod n" using m_div_n by simp_all
haftmann@26100
   316
  then show ?thesis using divmod_div_mod by simp
haftmann@26100
   317
qed
haftmann@25942
   318
wenzelm@26300
   319
text {* The ''recursion'' equations for @{const div} and @{const mod} *}
haftmann@26100
   320
haftmann@26100
   321
lemma div_less [simp]:
haftmann@26100
   322
  fixes m n :: nat
haftmann@26100
   323
  assumes "m < n"
haftmann@26100
   324
  shows "m div n = 0"
haftmann@26100
   325
  using assms divmod_base divmod_div_mod by simp
haftmann@25942
   326
haftmann@26100
   327
lemma le_div_geq:
haftmann@26100
   328
  fixes m n :: nat
haftmann@26100
   329
  assumes "0 < n" and "n \<le> m"
haftmann@26100
   330
  shows "m div n = Suc ((m - n) div n)"
haftmann@26100
   331
  using assms divmod_step divmod_div_mod by simp
paulson@14267
   332
haftmann@26100
   333
lemma mod_less [simp]:
haftmann@26100
   334
  fixes m n :: nat
haftmann@26100
   335
  assumes "m < n"
haftmann@26100
   336
  shows "m mod n = m"
haftmann@26100
   337
  using assms divmod_base divmod_div_mod by simp
haftmann@26100
   338
haftmann@26100
   339
lemma le_mod_geq:
haftmann@26100
   340
  fixes m n :: nat
haftmann@26100
   341
  assumes "n \<le> m"
haftmann@26100
   342
  shows "m mod n = (m - n) mod n"
haftmann@26100
   343
  using assms divmod_step divmod_div_mod by (cases "n = 0") simp_all
paulson@14267
   344
haftmann@25942
   345
instance proof
haftmann@26100
   346
  fix m n :: nat show "m div n * n + m mod n = m"
haftmann@26100
   347
    using divmod_rel [of m n] by (simp add: divmod_rel_def)
haftmann@25942
   348
next
haftmann@26100
   349
  fix n :: nat show "n div 0 = 0"
haftmann@26100
   350
    using divmod_zero divmod_div_mod [of n 0] by simp
haftmann@25942
   351
next
haftmann@26100
   352
  fix m n :: nat assume "n \<noteq> 0" then show "m * n div n = m"
haftmann@25942
   353
    by (induct m) (simp_all add: le_div_geq)
haftmann@25942
   354
qed
haftmann@26100
   355
haftmann@25942
   356
end
paulson@14267
   357
haftmann@26100
   358
text {* Simproc for cancelling @{const div} and @{const mod} *}
haftmann@25942
   359
haftmann@25942
   360
lemmas mod_div_equality = semiring_div_class.times_div_mod_plus_zero_one.mod_div_equality [of "m\<Colon>nat" n, standard]
haftmann@26062
   361
lemmas mod_div_equality2 = mod_div_equality2 [of "n\<Colon>nat" m, standard]
haftmann@26062
   362
lemmas div_mod_equality = div_mod_equality [of "m\<Colon>nat" n k, standard]
haftmann@26062
   363
lemmas div_mod_equality2 = div_mod_equality2 [of "m\<Colon>nat" n k, standard]
haftmann@25942
   364
haftmann@25942
   365
ML {*
haftmann@25942
   366
structure CancelDivModData =
haftmann@25942
   367
struct
haftmann@25942
   368
haftmann@26100
   369
val div_name = @{const_name div};
haftmann@26100
   370
val mod_name = @{const_name mod};
haftmann@25942
   371
val mk_binop = HOLogic.mk_binop;
haftmann@26100
   372
val mk_sum = ArithData.mk_sum;
haftmann@26100
   373
val dest_sum = ArithData.dest_sum;
haftmann@25942
   374
haftmann@25942
   375
(*logic*)
paulson@14267
   376
haftmann@25942
   377
val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}]
haftmann@25942
   378
haftmann@25942
   379
val trans = trans
haftmann@25942
   380
haftmann@25942
   381
val prove_eq_sums =
haftmann@25942
   382
  let val simps = @{thm add_0} :: @{thm add_0_right} :: @{thms add_ac}
haftmann@26100
   383
  in ArithData.prove_conv all_tac (ArithData.simp_all_tac simps) end;
haftmann@25942
   384
haftmann@25942
   385
end;
haftmann@25942
   386
haftmann@25942
   387
structure CancelDivMod = CancelDivModFun(CancelDivModData);
haftmann@25942
   388
haftmann@26100
   389
val cancel_div_mod_proc = Simplifier.simproc @{theory}
haftmann@26100
   390
  "cancel_div_mod" ["(m::nat) + n"] (K CancelDivMod.proc);
haftmann@25942
   391
haftmann@25942
   392
Addsimprocs[cancel_div_mod_proc];
haftmann@25942
   393
*}
haftmann@25942
   394
haftmann@26100
   395
text {* code generator setup *}
haftmann@26100
   396
haftmann@26100
   397
lemma divmod_if [code]: "divmod m n = (if n = 0 \<or> m < n then (0, m) else
haftmann@26100
   398
  let (q, r) = divmod (m - n) n in (Suc q, r))"
haftmann@26100
   399
  by (simp add: divmod_zero divmod_base divmod_step)
haftmann@26100
   400
    (simp add: divmod_div_mod)
haftmann@26100
   401
haftmann@26100
   402
code_modulename SML
haftmann@26100
   403
  Divides Nat
haftmann@26100
   404
haftmann@26100
   405
code_modulename OCaml
haftmann@26100
   406
  Divides Nat
haftmann@26100
   407
haftmann@26100
   408
code_modulename Haskell
haftmann@26100
   409
  Divides Nat
haftmann@26100
   410
haftmann@26100
   411
haftmann@26100
   412
subsubsection {* Quotient *}
haftmann@26100
   413
haftmann@26100
   414
lemmas DIVISION_BY_ZERO_DIV [simp] = div_by_0 [of "a\<Colon>nat", standard]
haftmann@26100
   415
lemmas div_0 [simp] = semiring_div_class.div_0 [of "n\<Colon>nat", standard]
haftmann@26100
   416
haftmann@26100
   417
lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
haftmann@26100
   418
  by (simp add: le_div_geq linorder_not_less)
haftmann@26100
   419
haftmann@26100
   420
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
haftmann@26100
   421
  by (simp add: div_geq)
haftmann@26100
   422
haftmann@26100
   423
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
haftmann@26100
   424
  by (rule mult_div) simp
haftmann@26100
   425
haftmann@26100
   426
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
haftmann@26100
   427
  by (simp add: mult_commute)
haftmann@26100
   428
haftmann@25942
   429
haftmann@25942
   430
subsubsection {* Remainder *}
haftmann@25942
   431
haftmann@25942
   432
lemmas DIVISION_BY_ZERO_MOD [simp] = mod_by_0 [of "a\<Colon>nat", standard]
haftmann@26100
   433
lemmas mod_0 [simp] = semiring_div_class.mod_0 [of "n\<Colon>nat", standard]
haftmann@25942
   434
haftmann@26100
   435
lemma mod_less_divisor [simp]:
haftmann@26100
   436
  fixes m n :: nat
haftmann@26100
   437
  assumes "n > 0"
haftmann@26100
   438
  shows "m mod n < (n::nat)"
haftmann@26100
   439
  using assms divmod_rel unfolding divmod_rel_def by auto
paulson@14267
   440
haftmann@26100
   441
lemma mod_less_eq_dividend [simp]:
haftmann@26100
   442
  fixes m n :: nat
haftmann@26100
   443
  shows "m mod n \<le> m"
haftmann@26100
   444
proof (rule add_leD2)
haftmann@26100
   445
  from mod_div_equality have "m div n * n + m mod n = m" .
haftmann@26100
   446
  then show "m div n * n + m mod n \<le> m" by auto
haftmann@26100
   447
qed
haftmann@26100
   448
haftmann@26100
   449
lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m - n) mod n"
haftmann@25942
   450
  by (simp add: le_mod_geq linorder_not_less)
paulson@14267
   451
haftmann@26100
   452
lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m - n) mod n)"
haftmann@26100
   453
  by (simp add: le_mod_geq)
haftmann@26100
   454
paulson@14267
   455
lemma mod_1 [simp]: "m mod Suc 0 = 0"
wenzelm@22718
   456
  by (induct m) (simp_all add: mod_geq)
paulson@14267
   457
haftmann@25942
   458
lemmas mod_self [simp] = semiring_div_class.mod_self [of "n\<Colon>nat", standard]
paulson@14267
   459
paulson@14267
   460
lemma mod_add_self2 [simp]: "(m+n) mod n = m mod (n::nat)"
wenzelm@22718
   461
  apply (subgoal_tac "(n + m) mod n = (n+m-n) mod n")
wenzelm@22718
   462
   apply (simp add: add_commute)
haftmann@25942
   463
  apply (subst le_mod_geq [symmetric], simp_all)
wenzelm@22718
   464
  done
paulson@14267
   465
paulson@14267
   466
lemma mod_add_self1 [simp]: "(n+m) mod n = m mod (n::nat)"
wenzelm@22718
   467
  by (simp add: add_commute mod_add_self2)
paulson@14267
   468
paulson@14267
   469
lemma mod_mult_self1 [simp]: "(m + k*n) mod n = m mod (n::nat)"
wenzelm@22718
   470
  by (induct k) (simp_all add: add_left_commute [of _ n])
paulson@14267
   471
paulson@14267
   472
lemma mod_mult_self2 [simp]: "(m + n*k) mod n = m mod (n::nat)"
wenzelm@22718
   473
  by (simp add: mult_commute mod_mult_self1)
paulson@14267
   474
haftmann@26100
   475
lemma mod_mult_distrib: "(m mod n) * (k\<Colon>nat) = (m * k) mod (n * k)"
wenzelm@22718
   476
  apply (cases "n = 0", simp)
wenzelm@22718
   477
  apply (cases "k = 0", simp)
wenzelm@22718
   478
  apply (induct m rule: nat_less_induct)
wenzelm@22718
   479
  apply (subst mod_if, simp)
wenzelm@22718
   480
  apply (simp add: mod_geq diff_mult_distrib)
wenzelm@22718
   481
  done
paulson@14267
   482
paulson@14267
   483
lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
wenzelm@22718
   484
  by (simp add: mult_commute [of k] mod_mult_distrib)
paulson@14267
   485
paulson@14267
   486
lemma mod_mult_self_is_0 [simp]: "(m*n) mod n = (0::nat)"
wenzelm@22718
   487
  apply (cases "n = 0", simp)
wenzelm@22718
   488
  apply (induct m, simp)
wenzelm@22718
   489
  apply (rename_tac k)
wenzelm@22718
   490
  apply (cut_tac m = "k * n" and n = n in mod_add_self2)
wenzelm@22718
   491
  apply (simp add: add_commute)
wenzelm@22718
   492
  done
paulson@14267
   493
paulson@14267
   494
lemma mod_mult_self1_is_0 [simp]: "(n*m) mod n = (0::nat)"
wenzelm@22718
   495
  by (simp add: mult_commute mod_mult_self_is_0)
paulson@14267
   496
paulson@14267
   497
(* a simple rearrangement of mod_div_equality: *)
paulson@14267
   498
lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
wenzelm@22718
   499
  by (cut_tac m = m and n = n in mod_div_equality2, arith)
paulson@14267
   500
nipkow@15439
   501
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
wenzelm@22718
   502
  apply (drule mod_less_divisor [where m = m])
wenzelm@22718
   503
  apply simp
wenzelm@22718
   504
  done
paulson@14267
   505
haftmann@26100
   506
subsubsection {* Quotient and Remainder *}
paulson@14267
   507
haftmann@26100
   508
lemma mod_div_decomp:
haftmann@26100
   509
  fixes n k :: nat
haftmann@26100
   510
  obtains m q where "m = n div k" and "q = n mod k"
haftmann@26100
   511
    and "n = m * k + q"
haftmann@26100
   512
proof -
haftmann@26100
   513
  from mod_div_equality have "n = n div k * k + n mod k" by auto
haftmann@26100
   514
  moreover have "n div k = n div k" ..
haftmann@26100
   515
  moreover have "n mod k = n mod k" ..
haftmann@26100
   516
  note that ultimately show thesis by blast
haftmann@26100
   517
qed
paulson@14267
   518
haftmann@26100
   519
lemma divmod_rel_mult1_eq:
haftmann@26100
   520
  "[| divmod_rel b c q r; c > 0 |]
haftmann@26100
   521
   ==> divmod_rel (a*b) c (a*q + a*r div c) (a*r mod c)"
haftmann@26100
   522
by (auto simp add: split_ifs mult_ac divmod_rel_def add_mult_distrib2)
paulson@14267
   523
paulson@14267
   524
lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)"
nipkow@25134
   525
apply (cases "c = 0", simp)
haftmann@26100
   526
apply (blast intro: divmod_rel [THEN divmod_rel_mult1_eq, THEN div_eq])
nipkow@25134
   527
done
paulson@14267
   528
paulson@14267
   529
lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)"
nipkow@25134
   530
apply (cases "c = 0", simp)
haftmann@26100
   531
apply (blast intro: divmod_rel [THEN divmod_rel_mult1_eq, THEN mod_eq])
nipkow@25134
   532
done
paulson@14267
   533
paulson@14267
   534
lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c"
wenzelm@22718
   535
  apply (rule trans)
wenzelm@22718
   536
   apply (rule_tac s = "b*a mod c" in trans)
wenzelm@22718
   537
    apply (rule_tac [2] mod_mult1_eq)
wenzelm@22718
   538
   apply (simp_all add: mult_commute)
wenzelm@22718
   539
  done
paulson@14267
   540
nipkow@25162
   541
lemma mod_mult_distrib_mod:
nipkow@25162
   542
  "(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c"
nipkow@25162
   543
apply (rule mod_mult1_eq' [THEN trans])
nipkow@25162
   544
apply (rule mod_mult1_eq)
nipkow@25162
   545
done
paulson@14267
   546
haftmann@26100
   547
lemma divmod_rel_add1_eq:
haftmann@26100
   548
  "[| divmod_rel a c aq ar; divmod_rel b c bq br;  c > 0 |]
haftmann@26100
   549
   ==> divmod_rel (a + b) c (aq + bq + (ar+br) div c) ((ar + br) mod c)"
haftmann@26100
   550
by (auto simp add: split_ifs mult_ac divmod_rel_def add_mult_distrib2)
paulson@14267
   551
paulson@14267
   552
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
paulson@14267
   553
lemma div_add1_eq:
nipkow@25134
   554
  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
nipkow@25134
   555
apply (cases "c = 0", simp)
haftmann@26100
   556
apply (blast intro: divmod_rel_add1_eq [THEN div_eq] divmod_rel)
nipkow@25134
   557
done
paulson@14267
   558
paulson@14267
   559
lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c"
nipkow@25134
   560
apply (cases "c = 0", simp)
haftmann@26100
   561
apply (blast intro: divmod_rel_add1_eq [THEN mod_eq] divmod_rel)
nipkow@25134
   562
done
paulson@14267
   563
paulson@14267
   564
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
wenzelm@22718
   565
  apply (cut_tac m = q and n = c in mod_less_divisor)
wenzelm@22718
   566
  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
wenzelm@22718
   567
  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
wenzelm@22718
   568
  apply (simp add: add_mult_distrib2)
wenzelm@22718
   569
  done
paulson@10559
   570
haftmann@26100
   571
lemma divmod_rel_mult2_eq: "[| divmod_rel a b q r;  0 < b;  0 < c |]
haftmann@26100
   572
      ==> divmod_rel a (b*c) (q div c) (b*(q mod c) + r)"
haftmann@26100
   573
  by (auto simp add: mult_ac divmod_rel_def add_mult_distrib2 [symmetric] mod_lemma)
paulson@14267
   574
paulson@14267
   575
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
wenzelm@22718
   576
  apply (cases "b = 0", simp)
wenzelm@22718
   577
  apply (cases "c = 0", simp)
haftmann@26100
   578
  apply (force simp add: divmod_rel [THEN divmod_rel_mult2_eq, THEN div_eq])
wenzelm@22718
   579
  done
paulson@14267
   580
paulson@14267
   581
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
wenzelm@22718
   582
  apply (cases "b = 0", simp)
wenzelm@22718
   583
  apply (cases "c = 0", simp)
haftmann@26100
   584
  apply (auto simp add: mult_commute divmod_rel [THEN divmod_rel_mult2_eq, THEN mod_eq])
wenzelm@22718
   585
  done
paulson@14267
   586
paulson@14267
   587
haftmann@25942
   588
subsubsection{*Cancellation of Common Factors in Division*}
paulson@14267
   589
paulson@14267
   590
lemma div_mult_mult_lemma:
wenzelm@22718
   591
    "[| (0::nat) < b;  0 < c |] ==> (c*a) div (c*b) = a div b"
wenzelm@22718
   592
  by (auto simp add: div_mult2_eq)
paulson@14267
   593
paulson@14267
   594
lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b"
wenzelm@22718
   595
  apply (cases "b = 0")
wenzelm@22718
   596
  apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma)
wenzelm@22718
   597
  done
paulson@14267
   598
paulson@14267
   599
lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b"
wenzelm@22718
   600
  apply (drule div_mult_mult1)
wenzelm@22718
   601
  apply (auto simp add: mult_commute)
wenzelm@22718
   602
  done
paulson@14267
   603
paulson@14267
   604
haftmann@25942
   605
subsubsection{*Further Facts about Quotient and Remainder*}
paulson@14267
   606
paulson@14267
   607
lemma div_1 [simp]: "m div Suc 0 = m"
wenzelm@22718
   608
  by (induct m) (simp_all add: div_geq)
paulson@14267
   609
haftmann@25942
   610
lemmas div_self [simp] = semiring_div_class.div_self [of "n\<Colon>nat", standard]
paulson@14267
   611
paulson@14267
   612
lemma div_add_self2: "0<n ==> (m+n) div n = Suc (m div n)"
wenzelm@22718
   613
  apply (subgoal_tac "(n + m) div n = Suc ((n+m-n) div n) ")
wenzelm@22718
   614
   apply (simp add: add_commute)
wenzelm@22718
   615
  apply (subst div_geq [symmetric], simp_all)
wenzelm@22718
   616
  done
paulson@14267
   617
paulson@14267
   618
lemma div_add_self1: "0<n ==> (n+m) div n = Suc (m div n)"
wenzelm@22718
   619
  by (simp add: add_commute div_add_self2)
paulson@14267
   620
paulson@14267
   621
lemma div_mult_self1 [simp]: "!!n::nat. 0<n ==> (m + k*n) div n = k + m div n"
wenzelm@22718
   622
  apply (subst div_add1_eq)
wenzelm@22718
   623
  apply (subst div_mult1_eq, simp)
wenzelm@22718
   624
  done
paulson@14267
   625
paulson@14267
   626
lemma div_mult_self2 [simp]: "0<n ==> (m + n*k) div n = k + m div (n::nat)"
wenzelm@22718
   627
  by (simp add: mult_commute div_mult_self1)
paulson@14267
   628
paulson@14267
   629
paulson@14267
   630
(* Monotonicity of div in first argument *)
paulson@14267
   631
lemma div_le_mono [rule_format (no_asm)]:
wenzelm@22718
   632
    "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
paulson@14267
   633
apply (case_tac "k=0", simp)
paulson@15251
   634
apply (induct "n" rule: nat_less_induct, clarify)
paulson@14267
   635
apply (case_tac "n<k")
paulson@14267
   636
(* 1  case n<k *)
paulson@14267
   637
apply simp
paulson@14267
   638
(* 2  case n >= k *)
paulson@14267
   639
apply (case_tac "m<k")
paulson@14267
   640
(* 2.1  case m<k *)
paulson@14267
   641
apply simp
paulson@14267
   642
(* 2.2  case m>=k *)
nipkow@15439
   643
apply (simp add: div_geq diff_le_mono)
paulson@14267
   644
done
paulson@14267
   645
paulson@14267
   646
(* Antimonotonicity of div in second argument *)
paulson@14267
   647
lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
paulson@14267
   648
apply (subgoal_tac "0<n")
wenzelm@22718
   649
 prefer 2 apply simp
paulson@15251
   650
apply (induct_tac k rule: nat_less_induct)
paulson@14267
   651
apply (rename_tac "k")
paulson@14267
   652
apply (case_tac "k<n", simp)
paulson@14267
   653
apply (subgoal_tac "~ (k<m) ")
wenzelm@22718
   654
 prefer 2 apply simp
paulson@14267
   655
apply (simp add: div_geq)
paulson@15251
   656
apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
paulson@14267
   657
 prefer 2
paulson@14267
   658
 apply (blast intro: div_le_mono diff_le_mono2)
paulson@14267
   659
apply (rule le_trans, simp)
nipkow@15439
   660
apply (simp)
paulson@14267
   661
done
paulson@14267
   662
paulson@14267
   663
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
paulson@14267
   664
apply (case_tac "n=0", simp)
paulson@14267
   665
apply (subgoal_tac "m div n \<le> m div 1", simp)
paulson@14267
   666
apply (rule div_le_mono2)
paulson@14267
   667
apply (simp_all (no_asm_simp))
paulson@14267
   668
done
paulson@14267
   669
wenzelm@22718
   670
(* Similar for "less than" *)
paulson@17085
   671
lemma div_less_dividend [rule_format]:
paulson@14267
   672
     "!!n::nat. 1<n ==> 0 < m --> m div n < m"
paulson@15251
   673
apply (induct_tac m rule: nat_less_induct)
paulson@14267
   674
apply (rename_tac "m")
paulson@14267
   675
apply (case_tac "m<n", simp)
paulson@14267
   676
apply (subgoal_tac "0<n")
wenzelm@22718
   677
 prefer 2 apply simp
paulson@14267
   678
apply (simp add: div_geq)
paulson@14267
   679
apply (case_tac "n<m")
paulson@15251
   680
 apply (subgoal_tac "(m-n) div n < (m-n) ")
paulson@14267
   681
  apply (rule impI less_trans_Suc)+
paulson@14267
   682
apply assumption
nipkow@15439
   683
  apply (simp_all)
paulson@14267
   684
done
paulson@14267
   685
paulson@17085
   686
declare div_less_dividend [simp]
paulson@17085
   687
paulson@14267
   688
text{*A fact for the mutilated chess board*}
paulson@14267
   689
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
paulson@14267
   690
apply (case_tac "n=0", simp)
paulson@15251
   691
apply (induct "m" rule: nat_less_induct)
paulson@14267
   692
apply (case_tac "Suc (na) <n")
paulson@14267
   693
(* case Suc(na) < n *)
paulson@14267
   694
apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
paulson@14267
   695
(* case n \<le> Suc(na) *)
paulson@16796
   696
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
nipkow@15439
   697
apply (auto simp add: Suc_diff_le le_mod_geq)
paulson@14267
   698
done
paulson@14267
   699
paulson@14437
   700
lemma nat_mod_div_trivial [simp]: "m mod n div n = (0 :: nat)"
wenzelm@22718
   701
  by (cases "n = 0") auto
paulson@14437
   702
paulson@14437
   703
lemma nat_mod_mod_trivial [simp]: "m mod n mod n = (m mod n :: nat)"
wenzelm@22718
   704
  by (cases "n = 0") auto
paulson@14437
   705
paulson@14267
   706
haftmann@25942
   707
subsubsection{*The Divides Relation*}
paulson@14267
   708
paulson@14267
   709
lemma dvdI [intro?]: "n = m * k ==> m dvd n"
wenzelm@22718
   710
  unfolding dvd_def by blast
paulson@14267
   711
paulson@14267
   712
lemma dvdE [elim?]: "!!P. [|m dvd n;  !!k. n = m*k ==> P|] ==> P"
wenzelm@22718
   713
  unfolding dvd_def by blast
nipkow@13152
   714
paulson@14267
   715
lemma dvd_0_right [iff]: "m dvd (0::nat)"
wenzelm@22718
   716
  unfolding dvd_def by (blast intro: mult_0_right [symmetric])
paulson@14267
   717
paulson@14267
   718
lemma dvd_0_left: "0 dvd m ==> m = (0::nat)"
wenzelm@22718
   719
  by (force simp add: dvd_def)
paulson@14267
   720
paulson@14267
   721
lemma dvd_0_left_iff [iff]: "(0 dvd (m::nat)) = (m = 0)"
wenzelm@22718
   722
  by (blast intro: dvd_0_left)
paulson@14267
   723
paulson@24286
   724
declare dvd_0_left_iff [noatp]
paulson@24286
   725
paulson@14267
   726
lemma dvd_1_left [iff]: "Suc 0 dvd k"
wenzelm@22718
   727
  unfolding dvd_def by simp
paulson@14267
   728
paulson@14267
   729
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
wenzelm@22718
   730
  by (simp add: dvd_def)
paulson@14267
   731
haftmann@25942
   732
lemmas dvd_refl [simp] = semiring_div_class.dvd_refl [of "m\<Colon>nat", standard]
haftmann@25942
   733
lemmas dvd_trans [trans] = semiring_div_class.dvd_trans [of "m\<Colon>nat" n p, standard]
paulson@14267
   734
paulson@14267
   735
lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
wenzelm@22718
   736
  unfolding dvd_def
wenzelm@22718
   737
  by (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff)
paulson@14267
   738
haftmann@23684
   739
text {* @{term "op dvd"} is a partial order *}
haftmann@23684
   740
haftmann@25942
   741
interpretation dvd: order ["op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> n \<noteq> m"]
haftmann@23684
   742
  by unfold_locales (auto intro: dvd_trans dvd_anti_sym)
haftmann@23684
   743
paulson@14267
   744
lemma dvd_add: "[| k dvd m; k dvd n |] ==> k dvd (m+n :: nat)"
wenzelm@22718
   745
  unfolding dvd_def
wenzelm@22718
   746
  by (blast intro: add_mult_distrib2 [symmetric])
paulson@14267
   747
paulson@14267
   748
lemma dvd_diff: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
wenzelm@22718
   749
  unfolding dvd_def
wenzelm@22718
   750
  by (blast intro: diff_mult_distrib2 [symmetric])
paulson@14267
   751
paulson@14267
   752
lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
wenzelm@22718
   753
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
wenzelm@22718
   754
  apply (blast intro: dvd_add)
wenzelm@22718
   755
  done
paulson@14267
   756
paulson@14267
   757
lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
wenzelm@22718
   758
  by (drule_tac m = m in dvd_diff, auto)
paulson@14267
   759
paulson@14267
   760
lemma dvd_mult: "k dvd n ==> k dvd (m*n :: nat)"
wenzelm@22718
   761
  unfolding dvd_def by (blast intro: mult_left_commute)
paulson@14267
   762
paulson@14267
   763
lemma dvd_mult2: "k dvd m ==> k dvd (m*n :: nat)"
wenzelm@22718
   764
  apply (subst mult_commute)
wenzelm@22718
   765
  apply (erule dvd_mult)
wenzelm@22718
   766
  done
paulson@14267
   767
paulson@17084
   768
lemma dvd_triv_right [iff]: "k dvd (m*k :: nat)"
wenzelm@22718
   769
  by (rule dvd_refl [THEN dvd_mult])
paulson@17084
   770
paulson@17084
   771
lemma dvd_triv_left [iff]: "k dvd (k*m :: nat)"
wenzelm@22718
   772
  by (rule dvd_refl [THEN dvd_mult2])
paulson@14267
   773
paulson@14267
   774
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
wenzelm@22718
   775
  apply (rule iffI)
wenzelm@22718
   776
   apply (erule_tac [2] dvd_add)
wenzelm@22718
   777
   apply (rule_tac [2] dvd_refl)
wenzelm@22718
   778
  apply (subgoal_tac "n = (n+k) -k")
wenzelm@22718
   779
   prefer 2 apply simp
wenzelm@22718
   780
  apply (erule ssubst)
wenzelm@22718
   781
  apply (erule dvd_diff)
wenzelm@22718
   782
  apply (rule dvd_refl)
wenzelm@22718
   783
  done
paulson@14267
   784
paulson@14267
   785
lemma dvd_mod: "!!n::nat. [| f dvd m; f dvd n |] ==> f dvd m mod n"
wenzelm@22718
   786
  unfolding dvd_def
wenzelm@22718
   787
  apply (case_tac "n = 0", auto)
wenzelm@22718
   788
  apply (blast intro: mod_mult_distrib2 [symmetric])
wenzelm@22718
   789
  done
paulson@14267
   790
paulson@14267
   791
lemma dvd_mod_imp_dvd: "[| (k::nat) dvd m mod n;  k dvd n |] ==> k dvd m"
wenzelm@22718
   792
  apply (subgoal_tac "k dvd (m div n) *n + m mod n")
wenzelm@22718
   793
   apply (simp add: mod_div_equality)
wenzelm@22718
   794
  apply (simp only: dvd_add dvd_mult)
wenzelm@22718
   795
  done
paulson@14267
   796
paulson@14267
   797
lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)"
wenzelm@22718
   798
  by (blast intro: dvd_mod_imp_dvd dvd_mod)
paulson@14267
   799
paulson@14267
   800
lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
wenzelm@22718
   801
  unfolding dvd_def
wenzelm@22718
   802
  apply (erule exE)
wenzelm@22718
   803
  apply (simp add: mult_ac)
wenzelm@22718
   804
  done
paulson@14267
   805
paulson@14267
   806
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
wenzelm@22718
   807
  apply auto
wenzelm@22718
   808
   apply (subgoal_tac "m*n dvd m*1")
wenzelm@22718
   809
   apply (drule dvd_mult_cancel, auto)
wenzelm@22718
   810
  done
paulson@14267
   811
paulson@14267
   812
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
wenzelm@22718
   813
  apply (subst mult_commute)
wenzelm@22718
   814
  apply (erule dvd_mult_cancel1)
wenzelm@22718
   815
  done
paulson@14267
   816
paulson@14267
   817
lemma mult_dvd_mono: "[| i dvd m; j dvd n|] ==> i*j dvd (m*n :: nat)"
wenzelm@22718
   818
  apply (unfold dvd_def, clarify)
wenzelm@22718
   819
  apply (rule_tac x = "k*ka" in exI)
wenzelm@22718
   820
  apply (simp add: mult_ac)
wenzelm@22718
   821
  done
paulson@14267
   822
paulson@14267
   823
lemma dvd_mult_left: "(i*j :: nat) dvd k ==> i dvd k"
wenzelm@22718
   824
  by (simp add: dvd_def mult_assoc, blast)
paulson@14267
   825
paulson@14267
   826
lemma dvd_mult_right: "(i*j :: nat) dvd k ==> j dvd k"
wenzelm@22718
   827
  apply (unfold dvd_def, clarify)
wenzelm@22718
   828
  apply (rule_tac x = "i*k" in exI)
wenzelm@22718
   829
  apply (simp add: mult_ac)
wenzelm@22718
   830
  done
paulson@14267
   831
paulson@14267
   832
lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
wenzelm@22718
   833
  apply (unfold dvd_def, clarify)
wenzelm@22718
   834
  apply (simp_all (no_asm_use) add: zero_less_mult_iff)
wenzelm@22718
   835
  apply (erule conjE)
wenzelm@22718
   836
  apply (rule le_trans)
wenzelm@22718
   837
   apply (rule_tac [2] le_refl [THEN mult_le_mono])
wenzelm@22718
   838
   apply (erule_tac [2] Suc_leI, simp)
wenzelm@22718
   839
  done
paulson@14267
   840
haftmann@25942
   841
lemmas dvd_eq_mod_eq_0 = dvd_def_mod [of "k\<Colon>nat" n, standard]
paulson@14267
   842
paulson@14267
   843
lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"
wenzelm@22718
   844
  apply (subgoal_tac "m mod n = 0")
wenzelm@22718
   845
   apply (simp add: mult_div_cancel)
wenzelm@22718
   846
  apply (simp only: dvd_eq_mod_eq_0)
wenzelm@22718
   847
  done
paulson@14267
   848
haftmann@21408
   849
lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n"
wenzelm@22718
   850
  apply (unfold dvd_def)
wenzelm@22718
   851
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
wenzelm@22718
   852
  apply (simp add: power_add)
wenzelm@22718
   853
  done
haftmann@21408
   854
haftmann@26100
   855
lemma mod_add_left_eq: "((a::nat) + b) mod c = (a mod c + b) mod c"
haftmann@26100
   856
  apply (rule trans [symmetric])
haftmann@26100
   857
   apply (rule mod_add1_eq, simp)
haftmann@26100
   858
  apply (rule mod_add1_eq [symmetric])
haftmann@26100
   859
  done
haftmann@26100
   860
haftmann@26100
   861
lemma mod_add_right_eq: "(a+b) mod (c::nat) = (a + (b mod c)) mod c"
haftmann@26100
   862
  apply (rule trans [symmetric])
haftmann@26100
   863
   apply (rule mod_add1_eq, simp)
haftmann@26100
   864
  apply (rule mod_add1_eq [symmetric])
haftmann@26100
   865
  done
haftmann@26100
   866
nipkow@25162
   867
lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat) | n=0)"
wenzelm@22718
   868
  by (induct n) auto
haftmann@21408
   869
haftmann@21408
   870
lemma power_le_dvd [rule_format]: "k^j dvd n --> i\<le>j --> k^i dvd (n::nat)"
wenzelm@22718
   871
  apply (induct j)
wenzelm@22718
   872
   apply (simp_all add: le_Suc_eq)
wenzelm@22718
   873
  apply (blast dest!: dvd_mult_right)
wenzelm@22718
   874
  done
haftmann@21408
   875
haftmann@21408
   876
lemma power_dvd_imp_le: "[|i^m dvd i^n;  (1::nat) < i|] ==> m \<le> n"
wenzelm@22718
   877
  apply (rule power_le_imp_le_exp, assumption)
wenzelm@22718
   878
  apply (erule dvd_imp_le, simp)
wenzelm@22718
   879
  done
haftmann@21408
   880
paulson@14267
   881
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
wenzelm@22718
   882
  by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
paulson@17084
   883
wenzelm@22718
   884
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
paulson@14267
   885
paulson@14267
   886
(*Loses information, namely we also have r<d provided d is nonzero*)
paulson@14267
   887
lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
wenzelm@22718
   888
  apply (cut_tac m = m in mod_div_equality)
wenzelm@22718
   889
  apply (simp only: add_ac)
wenzelm@22718
   890
  apply (blast intro: sym)
wenzelm@22718
   891
  done
paulson@14267
   892
nipkow@13152
   893
lemma split_div:
nipkow@13189
   894
 "P(n div k :: nat) =
nipkow@13189
   895
 ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
nipkow@13189
   896
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
   897
proof
nipkow@13189
   898
  assume P: ?P
nipkow@13189
   899
  show ?Q
nipkow@13189
   900
  proof (cases)
nipkow@13189
   901
    assume "k = 0"
nipkow@13189
   902
    with P show ?Q by(simp add:DIVISION_BY_ZERO_DIV)
nipkow@13189
   903
  next
nipkow@13189
   904
    assume not0: "k \<noteq> 0"
nipkow@13189
   905
    thus ?Q
nipkow@13189
   906
    proof (simp, intro allI impI)
nipkow@13189
   907
      fix i j
nipkow@13189
   908
      assume n: "n = k*i + j" and j: "j < k"
nipkow@13189
   909
      show "P i"
nipkow@13189
   910
      proof (cases)
wenzelm@22718
   911
        assume "i = 0"
wenzelm@22718
   912
        with n j P show "P i" by simp
nipkow@13189
   913
      next
wenzelm@22718
   914
        assume "i \<noteq> 0"
wenzelm@22718
   915
        with not0 n j P show "P i" by(simp add:add_ac)
nipkow@13189
   916
      qed
nipkow@13189
   917
    qed
nipkow@13189
   918
  qed
nipkow@13189
   919
next
nipkow@13189
   920
  assume Q: ?Q
nipkow@13189
   921
  show ?P
nipkow@13189
   922
  proof (cases)
nipkow@13189
   923
    assume "k = 0"
nipkow@13189
   924
    with Q show ?P by(simp add:DIVISION_BY_ZERO_DIV)
nipkow@13189
   925
  next
nipkow@13189
   926
    assume not0: "k \<noteq> 0"
nipkow@13189
   927
    with Q have R: ?R by simp
nipkow@13189
   928
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
   929
    show ?P by simp
nipkow@13189
   930
  qed
nipkow@13189
   931
qed
nipkow@13189
   932
berghofe@13882
   933
lemma split_div_lemma:
haftmann@26100
   934
  assumes "0 < n"
haftmann@26100
   935
  shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m\<Colon>nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@26100
   936
proof
haftmann@26100
   937
  assume ?rhs
haftmann@26100
   938
  with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp
haftmann@26100
   939
  then have A: "n * q \<le> m" by simp
haftmann@26100
   940
  have "n - (m mod n) > 0" using mod_less_divisor assms by auto
haftmann@26100
   941
  then have "m < m + (n - (m mod n))" by simp
haftmann@26100
   942
  then have "m < n + (m - (m mod n))" by simp
haftmann@26100
   943
  with nq have "m < n + n * q" by simp
haftmann@26100
   944
  then have B: "m < n * Suc q" by simp
haftmann@26100
   945
  from A B show ?lhs ..
haftmann@26100
   946
next
haftmann@26100
   947
  assume P: ?lhs
haftmann@26100
   948
  then have "divmod_rel m n q (m - n * q)"
haftmann@26100
   949
    unfolding divmod_rel_def by (auto simp add: mult_ac)
haftmann@26100
   950
  then show ?rhs using divmod_rel by (rule divmod_rel_unique_div)
haftmann@26100
   951
qed
berghofe@13882
   952
berghofe@13882
   953
theorem split_div':
berghofe@13882
   954
  "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
paulson@14267
   955
   (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
berghofe@13882
   956
  apply (case_tac "0 < n")
berghofe@13882
   957
  apply (simp only: add: split_div_lemma)
berghofe@13882
   958
  apply (simp_all add: DIVISION_BY_ZERO_DIV)
berghofe@13882
   959
  done
berghofe@13882
   960
nipkow@13189
   961
lemma split_mod:
nipkow@13189
   962
 "P(n mod k :: nat) =
nipkow@13189
   963
 ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
nipkow@13189
   964
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
   965
proof
nipkow@13189
   966
  assume P: ?P
nipkow@13189
   967
  show ?Q
nipkow@13189
   968
  proof (cases)
nipkow@13189
   969
    assume "k = 0"
nipkow@13189
   970
    with P show ?Q by(simp add:DIVISION_BY_ZERO_MOD)
nipkow@13189
   971
  next
nipkow@13189
   972
    assume not0: "k \<noteq> 0"
nipkow@13189
   973
    thus ?Q
nipkow@13189
   974
    proof (simp, intro allI impI)
nipkow@13189
   975
      fix i j
nipkow@13189
   976
      assume "n = k*i + j" "j < k"
nipkow@13189
   977
      thus "P j" using not0 P by(simp add:add_ac mult_ac)
nipkow@13189
   978
    qed
nipkow@13189
   979
  qed
nipkow@13189
   980
next
nipkow@13189
   981
  assume Q: ?Q
nipkow@13189
   982
  show ?P
nipkow@13189
   983
  proof (cases)
nipkow@13189
   984
    assume "k = 0"
nipkow@13189
   985
    with Q show ?P by(simp add:DIVISION_BY_ZERO_MOD)
nipkow@13189
   986
  next
nipkow@13189
   987
    assume not0: "k \<noteq> 0"
nipkow@13189
   988
    with Q have R: ?R by simp
nipkow@13189
   989
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
   990
    show ?P by simp
nipkow@13189
   991
  qed
nipkow@13189
   992
qed
nipkow@13189
   993
berghofe@13882
   994
theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
berghofe@13882
   995
  apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
berghofe@13882
   996
    subst [OF mod_div_equality [of _ n]])
berghofe@13882
   997
  apply arith
berghofe@13882
   998
  done
berghofe@13882
   999
haftmann@22800
  1000
lemma div_mod_equality':
haftmann@22800
  1001
  fixes m n :: nat
haftmann@22800
  1002
  shows "m div n * n = m - m mod n"
haftmann@22800
  1003
proof -
haftmann@22800
  1004
  have "m mod n \<le> m mod n" ..
haftmann@22800
  1005
  from div_mod_equality have 
haftmann@22800
  1006
    "m div n * n + m mod n - m mod n = m - m mod n" by simp
haftmann@22800
  1007
  with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have
haftmann@22800
  1008
    "m div n * n + (m mod n - m mod n) = m - m mod n"
haftmann@22800
  1009
    by simp
haftmann@22800
  1010
  then show ?thesis by simp
haftmann@22800
  1011
qed
haftmann@22800
  1012
haftmann@22800
  1013
haftmann@25942
  1014
subsubsection {*An ``induction'' law for modulus arithmetic.*}
paulson@14640
  1015
paulson@14640
  1016
lemma mod_induct_0:
paulson@14640
  1017
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
  1018
  and base: "P i" and i: "i<p"
paulson@14640
  1019
  shows "P 0"
paulson@14640
  1020
proof (rule ccontr)
paulson@14640
  1021
  assume contra: "\<not>(P 0)"
paulson@14640
  1022
  from i have p: "0<p" by simp
paulson@14640
  1023
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
paulson@14640
  1024
  proof
paulson@14640
  1025
    fix k
paulson@14640
  1026
    show "?A k"
paulson@14640
  1027
    proof (induct k)
paulson@14640
  1028
      show "?A 0" by simp  -- "by contradiction"
paulson@14640
  1029
    next
paulson@14640
  1030
      fix n
paulson@14640
  1031
      assume ih: "?A n"
paulson@14640
  1032
      show "?A (Suc n)"
paulson@14640
  1033
      proof (clarsimp)
wenzelm@22718
  1034
        assume y: "P (p - Suc n)"
wenzelm@22718
  1035
        have n: "Suc n < p"
wenzelm@22718
  1036
        proof (rule ccontr)
wenzelm@22718
  1037
          assume "\<not>(Suc n < p)"
wenzelm@22718
  1038
          hence "p - Suc n = 0"
wenzelm@22718
  1039
            by simp
wenzelm@22718
  1040
          with y contra show "False"
wenzelm@22718
  1041
            by simp
wenzelm@22718
  1042
        qed
wenzelm@22718
  1043
        hence n2: "Suc (p - Suc n) = p-n" by arith
wenzelm@22718
  1044
        from p have "p - Suc n < p" by arith
wenzelm@22718
  1045
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
wenzelm@22718
  1046
          by blast
wenzelm@22718
  1047
        show "False"
wenzelm@22718
  1048
        proof (cases "n=0")
wenzelm@22718
  1049
          case True
wenzelm@22718
  1050
          with z n2 contra show ?thesis by simp
wenzelm@22718
  1051
        next
wenzelm@22718
  1052
          case False
wenzelm@22718
  1053
          with p have "p-n < p" by arith
wenzelm@22718
  1054
          with z n2 False ih show ?thesis by simp
wenzelm@22718
  1055
        qed
paulson@14640
  1056
      qed
paulson@14640
  1057
    qed
paulson@14640
  1058
  qed
paulson@14640
  1059
  moreover
paulson@14640
  1060
  from i obtain k where "0<k \<and> i+k=p"
paulson@14640
  1061
    by (blast dest: less_imp_add_positive)
paulson@14640
  1062
  hence "0<k \<and> i=p-k" by auto
paulson@14640
  1063
  moreover
paulson@14640
  1064
  note base
paulson@14640
  1065
  ultimately
paulson@14640
  1066
  show "False" by blast
paulson@14640
  1067
qed
paulson@14640
  1068
paulson@14640
  1069
lemma mod_induct:
paulson@14640
  1070
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
  1071
  and base: "P i" and i: "i<p" and j: "j<p"
paulson@14640
  1072
  shows "P j"
paulson@14640
  1073
proof -
paulson@14640
  1074
  have "\<forall>j<p. P j"
paulson@14640
  1075
  proof
paulson@14640
  1076
    fix j
paulson@14640
  1077
    show "j<p \<longrightarrow> P j" (is "?A j")
paulson@14640
  1078
    proof (induct j)
paulson@14640
  1079
      from step base i show "?A 0"
wenzelm@22718
  1080
        by (auto elim: mod_induct_0)
paulson@14640
  1081
    next
paulson@14640
  1082
      fix k
paulson@14640
  1083
      assume ih: "?A k"
paulson@14640
  1084
      show "?A (Suc k)"
paulson@14640
  1085
      proof
wenzelm@22718
  1086
        assume suc: "Suc k < p"
wenzelm@22718
  1087
        hence k: "k<p" by simp
wenzelm@22718
  1088
        with ih have "P k" ..
wenzelm@22718
  1089
        with step k have "P (Suc k mod p)"
wenzelm@22718
  1090
          by blast
wenzelm@22718
  1091
        moreover
wenzelm@22718
  1092
        from suc have "Suc k mod p = Suc k"
wenzelm@22718
  1093
          by simp
wenzelm@22718
  1094
        ultimately
wenzelm@22718
  1095
        show "P (Suc k)" by simp
paulson@14640
  1096
      qed
paulson@14640
  1097
    qed
paulson@14640
  1098
  qed
paulson@14640
  1099
  with j show ?thesis by blast
paulson@14640
  1100
qed
paulson@14640
  1101
paulson@3366
  1102
end