src/HOL/GCD.thy
author wenzelm
Sun Sep 18 20:33:48 2016 +0200 (2016-09-18)
changeset 63915 bab633745c7f
parent 63882 018998c00003
child 63924 f91766530e13
permissions -rw-r--r--
tuned proofs;
wenzelm@63489
     1
(*  Title:      HOL/GCD.thy
wenzelm@63489
     2
    Author:     Christophe Tabacznyj
wenzelm@63489
     3
    Author:     Lawrence C. Paulson
wenzelm@63489
     4
    Author:     Amine Chaieb
wenzelm@63489
     5
    Author:     Thomas M. Rasmussen
wenzelm@63489
     6
    Author:     Jeremy Avigad
wenzelm@63489
     7
    Author:     Tobias Nipkow
huffman@31706
     8
haftmann@32479
     9
This file deals with the functions gcd and lcm.  Definitions and
haftmann@32479
    10
lemmas are proved uniformly for the natural numbers and integers.
huffman@31706
    11
huffman@31706
    12
This file combines and revises a number of prior developments.
huffman@31706
    13
huffman@31706
    14
The original theories "GCD" and "Primes" were by Christophe Tabacznyj
wenzelm@58623
    15
and Lawrence C. Paulson, based on @{cite davenport92}. They introduced
huffman@31706
    16
gcd, lcm, and prime for the natural numbers.
huffman@31706
    17
huffman@31706
    18
The original theory "IntPrimes" was by Thomas M. Rasmussen, and
huffman@31706
    19
extended gcd, lcm, primes to the integers. Amine Chaieb provided
huffman@31706
    20
another extension of the notions to the integers, and added a number
huffman@31706
    21
of results to "Primes" and "GCD". IntPrimes also defined and developed
huffman@31706
    22
the congruence relations on the integers. The notion was extended to
berghofe@34915
    23
the natural numbers by Chaieb.
huffman@31706
    24
avigad@32036
    25
Jeremy Avigad combined all of these, made everything uniform for the
avigad@32036
    26
natural numbers and the integers, and added a number of new theorems.
avigad@32036
    27
nipkow@31798
    28
Tobias Nipkow cleaned up a lot.
wenzelm@21256
    29
*)
wenzelm@21256
    30
wenzelm@60758
    31
section \<open>Greatest common divisor and least common multiple\<close>
wenzelm@21256
    32
wenzelm@21256
    33
theory GCD
wenzelm@63489
    34
  imports Main
huffman@31706
    35
begin
huffman@31706
    36
wenzelm@63489
    37
haftmann@62345
    38
subsection \<open>Abstract GCD and LCM\<close>
huffman@31706
    39
nipkow@31992
    40
class gcd = zero + one + dvd +
wenzelm@41550
    41
  fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
wenzelm@41550
    42
    and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
wenzelm@21256
    43
begin
wenzelm@21256
    44
haftmann@60686
    45
abbreviation coprime :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
haftmann@60686
    46
  where "coprime x y \<equiv> gcd x y = 1"
huffman@31706
    47
huffman@31706
    48
end
huffman@31706
    49
haftmann@60686
    50
class Gcd = gcd +
haftmann@63025
    51
  fixes Gcd :: "'a set \<Rightarrow> 'a"
haftmann@63025
    52
    and Lcm :: "'a set \<Rightarrow> 'a"
haftmann@62350
    53
begin
haftmann@62350
    54
haftmann@63025
    55
abbreviation GREATEST_COMMON_DIVISOR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
wenzelm@63489
    56
  where "GREATEST_COMMON_DIVISOR A f \<equiv> Gcd (f ` A)"
haftmann@62350
    57
haftmann@63025
    58
abbreviation LEAST_COMMON_MULTIPLE :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
wenzelm@63489
    59
  where "LEAST_COMMON_MULTIPLE A f \<equiv> Lcm (f ` A)"
haftmann@62350
    60
haftmann@62350
    61
end
haftmann@62350
    62
haftmann@62350
    63
syntax
haftmann@63025
    64
  "_GCD1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3GCD _./ _)" [0, 10] 10)
haftmann@63025
    65
  "_GCD"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3GCD _\<in>_./ _)" [0, 0, 10] 10)
haftmann@63025
    66
  "_LCM1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3LCM _./ _)" [0, 10] 10)
haftmann@63025
    67
  "_LCM"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3LCM _\<in>_./ _)" [0, 0, 10] 10)
haftmann@62350
    68
translations
haftmann@63025
    69
  "GCD x y. B"   \<rightleftharpoons> "GCD x. GCD y. B"
haftmann@63025
    70
  "GCD x. B"     \<rightleftharpoons> "CONST GREATEST_COMMON_DIVISOR CONST UNIV (\<lambda>x. B)"
haftmann@63025
    71
  "GCD x. B"     \<rightleftharpoons> "GCD x \<in> CONST UNIV. B"
haftmann@63025
    72
  "GCD x\<in>A. B"   \<rightleftharpoons> "CONST GREATEST_COMMON_DIVISOR A (\<lambda>x. B)"
haftmann@63025
    73
  "LCM x y. B"   \<rightleftharpoons> "LCM x. LCM y. B"
haftmann@63025
    74
  "LCM x. B"     \<rightleftharpoons> "CONST LEAST_COMMON_MULTIPLE CONST UNIV (\<lambda>x. B)"
haftmann@63025
    75
  "LCM x. B"     \<rightleftharpoons> "LCM x \<in> CONST UNIV. B"
haftmann@63025
    76
  "LCM x\<in>A. B"   \<rightleftharpoons> "CONST LEAST_COMMON_MULTIPLE A (\<lambda>x. B)"
haftmann@62350
    77
haftmann@62350
    78
print_translation \<open>
haftmann@63025
    79
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax GREATEST_COMMON_DIVISOR} @{syntax_const "_GCD"},
haftmann@63025
    80
    Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax LEAST_COMMON_MULTIPLE} @{syntax_const "_LCM"}]
haftmann@62350
    81
\<close> \<comment> \<open>to avoid eta-contraction of body\<close>
haftmann@60686
    82
haftmann@60686
    83
class semiring_gcd = normalization_semidom + gcd +
haftmann@59008
    84
  assumes gcd_dvd1 [iff]: "gcd a b dvd a"
wenzelm@59977
    85
    and gcd_dvd2 [iff]: "gcd a b dvd b"
wenzelm@59977
    86
    and gcd_greatest: "c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> c dvd gcd a b"
haftmann@60686
    87
    and normalize_gcd [simp]: "normalize (gcd a b) = gcd a b"
haftmann@60686
    88
    and lcm_gcd: "lcm a b = normalize (a * b) div gcd a b"
wenzelm@63489
    89
begin
wenzelm@63489
    90
wenzelm@63489
    91
lemma gcd_greatest_iff [simp]: "a dvd gcd b c \<longleftrightarrow> a dvd b \<and> a dvd c"
haftmann@60686
    92
  by (blast intro!: gcd_greatest intro: dvd_trans)
haftmann@60686
    93
wenzelm@63489
    94
lemma gcd_dvdI1: "a dvd c \<Longrightarrow> gcd a b dvd c"
haftmann@60689
    95
  by (rule dvd_trans) (rule gcd_dvd1)
haftmann@60689
    96
wenzelm@63489
    97
lemma gcd_dvdI2: "b dvd c \<Longrightarrow> gcd a b dvd c"
haftmann@60689
    98
  by (rule dvd_trans) (rule gcd_dvd2)
haftmann@60689
    99
wenzelm@63489
   100
lemma dvd_gcdD1: "a dvd gcd b c \<Longrightarrow> a dvd b"
haftmann@62345
   101
  using gcd_dvd1 [of b c] by (blast intro: dvd_trans)
haftmann@62345
   102
wenzelm@63489
   103
lemma dvd_gcdD2: "a dvd gcd b c \<Longrightarrow> a dvd c"
haftmann@62345
   104
  using gcd_dvd2 [of b c] by (blast intro: dvd_trans)
haftmann@62345
   105
wenzelm@63489
   106
lemma gcd_0_left [simp]: "gcd 0 a = normalize a"
haftmann@60688
   107
  by (rule associated_eqI) simp_all
haftmann@60686
   108
wenzelm@63489
   109
lemma gcd_0_right [simp]: "gcd a 0 = normalize a"
haftmann@60688
   110
  by (rule associated_eqI) simp_all
wenzelm@63489
   111
wenzelm@63489
   112
lemma gcd_eq_0_iff [simp]: "gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
wenzelm@63489
   113
  (is "?P \<longleftrightarrow> ?Q")
haftmann@60686
   114
proof
wenzelm@63489
   115
  assume ?P
wenzelm@63489
   116
  then have "0 dvd gcd a b"
wenzelm@63489
   117
    by simp
wenzelm@63489
   118
  then have "0 dvd a" and "0 dvd b"
wenzelm@63489
   119
    by (blast intro: dvd_trans)+
wenzelm@63489
   120
  then show ?Q
wenzelm@63489
   121
    by simp
haftmann@60686
   122
next
wenzelm@63489
   123
  assume ?Q
wenzelm@63489
   124
  then show ?P
wenzelm@63489
   125
    by simp
haftmann@60686
   126
qed
haftmann@60686
   127
wenzelm@63489
   128
lemma unit_factor_gcd: "unit_factor (gcd a b) = (if a = 0 \<and> b = 0 then 0 else 1)"
haftmann@60686
   129
proof (cases "gcd a b = 0")
wenzelm@63489
   130
  case True
wenzelm@63489
   131
  then show ?thesis by simp
haftmann@60686
   132
next
haftmann@60686
   133
  case False
haftmann@60686
   134
  have "unit_factor (gcd a b) * normalize (gcd a b) = gcd a b"
haftmann@60686
   135
    by (rule unit_factor_mult_normalize)
haftmann@60686
   136
  then have "unit_factor (gcd a b) * gcd a b = gcd a b"
haftmann@60686
   137
    by simp
haftmann@60686
   138
  then have "unit_factor (gcd a b) * gcd a b div gcd a b = gcd a b div gcd a b"
haftmann@60686
   139
    by simp
wenzelm@63489
   140
  with False show ?thesis
wenzelm@63489
   141
    by simp
haftmann@60686
   142
qed
haftmann@60686
   143
wenzelm@63489
   144
lemma is_unit_gcd [simp]: "is_unit (gcd a b) \<longleftrightarrow> coprime a b"
haftmann@60690
   145
  by (cases "a = 0 \<and> b = 0") (auto simp add: unit_factor_gcd dest: is_unit_unit_factor)
haftmann@60690
   146
wenzelm@61605
   147
sublocale gcd: abel_semigroup gcd
haftmann@60686
   148
proof
haftmann@60686
   149
  fix a b c
haftmann@60686
   150
  show "gcd a b = gcd b a"
haftmann@60688
   151
    by (rule associated_eqI) simp_all
haftmann@60686
   152
  from gcd_dvd1 have "gcd (gcd a b) c dvd a"
haftmann@60686
   153
    by (rule dvd_trans) simp
haftmann@60686
   154
  moreover from gcd_dvd1 have "gcd (gcd a b) c dvd b"
haftmann@60686
   155
    by (rule dvd_trans) simp
haftmann@60686
   156
  ultimately have P1: "gcd (gcd a b) c dvd gcd a (gcd b c)"
haftmann@60686
   157
    by (auto intro!: gcd_greatest)
haftmann@60686
   158
  from gcd_dvd2 have "gcd a (gcd b c) dvd b"
haftmann@60686
   159
    by (rule dvd_trans) simp
haftmann@60686
   160
  moreover from gcd_dvd2 have "gcd a (gcd b c) dvd c"
haftmann@60686
   161
    by (rule dvd_trans) simp
haftmann@60686
   162
  ultimately have P2: "gcd a (gcd b c) dvd gcd (gcd a b) c"
haftmann@60686
   163
    by (auto intro!: gcd_greatest)
haftmann@60688
   164
  from P1 P2 show "gcd (gcd a b) c = gcd a (gcd b c)"
haftmann@60688
   165
    by (rule associated_eqI) simp_all
haftmann@60686
   166
qed
haftmann@60686
   167
wenzelm@63489
   168
lemma gcd_self [simp]: "gcd a a = normalize a"
haftmann@60686
   169
proof -
haftmann@60686
   170
  have "a dvd gcd a a"
haftmann@60686
   171
    by (rule gcd_greatest) simp_all
haftmann@60686
   172
  then show ?thesis
haftmann@60688
   173
    by (auto intro: associated_eqI)
haftmann@60686
   174
qed
haftmann@61913
   175
wenzelm@63489
   176
lemma gcd_left_idem [simp]: "gcd a (gcd a b) = gcd a b"
haftmann@61913
   177
  by (auto intro: associated_eqI)
haftmann@61913
   178
wenzelm@63489
   179
lemma gcd_right_idem [simp]: "gcd (gcd a b) b = gcd a b"
haftmann@61913
   180
  unfolding gcd.commute [of a] gcd.commute [of "gcd b a"] by simp
haftmann@61913
   181
wenzelm@63489
   182
lemma coprime_1_left [simp]: "coprime 1 a"
haftmann@60688
   183
  by (rule associated_eqI) simp_all
haftmann@60686
   184
wenzelm@63489
   185
lemma coprime_1_right [simp]: "coprime a 1"
haftmann@60686
   186
  using coprime_1_left [of a] by (simp add: ac_simps)
haftmann@60686
   187
wenzelm@63489
   188
lemma gcd_mult_left: "gcd (c * a) (c * b) = normalize c * gcd a b"
haftmann@60686
   189
proof (cases "c = 0")
wenzelm@63489
   190
  case True
wenzelm@63489
   191
  then show ?thesis by simp
haftmann@60686
   192
next
haftmann@60686
   193
  case False
wenzelm@63489
   194
  then have *: "c * gcd a b dvd gcd (c * a) (c * b)"
haftmann@60686
   195
    by (auto intro: gcd_greatest)
wenzelm@63489
   196
  moreover from False * have "gcd (c * a) (c * b) dvd c * gcd a b"
haftmann@60686
   197
    by (metis div_dvd_iff_mult dvd_mult_left gcd_dvd1 gcd_dvd2 gcd_greatest mult_commute)
haftmann@60686
   198
  ultimately have "normalize (gcd (c * a) (c * b)) = normalize (c * gcd a b)"
haftmann@60688
   199
    by (auto intro: associated_eqI)
wenzelm@63489
   200
  then show ?thesis
wenzelm@63489
   201
    by (simp add: normalize_mult)
haftmann@60686
   202
qed
haftmann@60686
   203
wenzelm@63489
   204
lemma gcd_mult_right: "gcd (a * c) (b * c) = gcd b a * normalize c"
haftmann@60686
   205
  using gcd_mult_left [of c a b] by (simp add: ac_simps)
haftmann@60686
   206
wenzelm@63489
   207
lemma mult_gcd_left: "c * gcd a b = unit_factor c * gcd (c * a) (c * b)"
haftmann@60686
   208
  by (simp add: gcd_mult_left mult.assoc [symmetric])
haftmann@60686
   209
wenzelm@63489
   210
lemma mult_gcd_right: "gcd a b * c = gcd (a * c) (b * c) * unit_factor c"
haftmann@60686
   211
  using mult_gcd_left [of c a b] by (simp add: ac_simps)
haftmann@60686
   212
wenzelm@63489
   213
lemma dvd_lcm1 [iff]: "a dvd lcm a b"
haftmann@60686
   214
proof -
haftmann@60686
   215
  have "normalize (a * b) div gcd a b = normalize a * (normalize b div gcd a b)"
haftmann@60686
   216
    by (simp add: lcm_gcd normalize_mult div_mult_swap)
haftmann@60686
   217
  then show ?thesis
haftmann@60686
   218
    by (simp add: lcm_gcd)
haftmann@60686
   219
qed
wenzelm@63489
   220
wenzelm@63489
   221
lemma dvd_lcm2 [iff]: "b dvd lcm a b"
haftmann@60686
   222
proof -
haftmann@60686
   223
  have "normalize (a * b) div gcd a b = normalize b * (normalize a div gcd a b)"
haftmann@60686
   224
    by (simp add: lcm_gcd normalize_mult div_mult_swap ac_simps)
haftmann@60686
   225
  then show ?thesis
haftmann@60686
   226
    by (simp add: lcm_gcd)
haftmann@60686
   227
qed
haftmann@60686
   228
wenzelm@63489
   229
lemma dvd_lcmI1: "a dvd b \<Longrightarrow> a dvd lcm b c"
wenzelm@63489
   230
  by (rule dvd_trans) (assumption, blast)
wenzelm@63489
   231
wenzelm@63489
   232
lemma dvd_lcmI2: "a dvd c \<Longrightarrow> a dvd lcm b c"
haftmann@60689
   233
  by (rule dvd_trans) (assumption, blast)
haftmann@60689
   234
wenzelm@63489
   235
lemma lcm_dvdD1: "lcm a b dvd c \<Longrightarrow> a dvd c"
haftmann@62345
   236
  using dvd_lcm1 [of a b] by (blast intro: dvd_trans)
haftmann@62345
   237
wenzelm@63489
   238
lemma lcm_dvdD2: "lcm a b dvd c \<Longrightarrow> b dvd c"
haftmann@62345
   239
  using dvd_lcm2 [of a b] by (blast intro: dvd_trans)
haftmann@62345
   240
haftmann@60686
   241
lemma lcm_least:
haftmann@60686
   242
  assumes "a dvd c" and "b dvd c"
haftmann@60686
   243
  shows "lcm a b dvd c"
haftmann@60686
   244
proof (cases "c = 0")
wenzelm@63489
   245
  case True
wenzelm@63489
   246
  then show ?thesis by simp
haftmann@60686
   247
next
wenzelm@63489
   248
  case False
wenzelm@63489
   249
  then have *: "is_unit (unit_factor c)"
wenzelm@63489
   250
    by simp
haftmann@60686
   251
  show ?thesis
haftmann@60686
   252
  proof (cases "gcd a b = 0")
wenzelm@63489
   253
    case True
wenzelm@63489
   254
    with assms show ?thesis by simp
haftmann@60686
   255
  next
wenzelm@63489
   256
    case False
wenzelm@63489
   257
    then have "a \<noteq> 0 \<or> b \<noteq> 0"
wenzelm@63489
   258
      by simp
haftmann@60686
   259
    with \<open>c \<noteq> 0\<close> assms have "a * b dvd a * c" "a * b dvd c * b"
haftmann@60686
   260
      by (simp_all add: mult_dvd_mono)
haftmann@60686
   261
    then have "normalize (a * b) dvd gcd (a * c) (b * c)"
haftmann@60686
   262
      by (auto intro: gcd_greatest simp add: ac_simps)
haftmann@60686
   263
    then have "normalize (a * b) dvd gcd (a * c) (b * c) * unit_factor c"
wenzelm@63489
   264
      using * by (simp add: dvd_mult_unit_iff)
haftmann@60686
   265
    then have "normalize (a * b) dvd gcd a b * c"
haftmann@60686
   266
      by (simp add: mult_gcd_right [of a b c])
haftmann@60686
   267
    then have "normalize (a * b) div gcd a b dvd c"
haftmann@60686
   268
      using False by (simp add: div_dvd_iff_mult ac_simps)
wenzelm@63489
   269
    then show ?thesis
wenzelm@63489
   270
      by (simp add: lcm_gcd)
haftmann@60686
   271
  qed
haftmann@60686
   272
qed
haftmann@60686
   273
wenzelm@63489
   274
lemma lcm_least_iff [simp]: "lcm a b dvd c \<longleftrightarrow> a dvd c \<and> b dvd c"
haftmann@60686
   275
  by (blast intro!: lcm_least intro: dvd_trans)
haftmann@60686
   276
wenzelm@63489
   277
lemma normalize_lcm [simp]: "normalize (lcm a b) = lcm a b"
haftmann@60686
   278
  by (simp add: lcm_gcd dvd_normalize_div)
haftmann@60686
   279
wenzelm@63489
   280
lemma lcm_0_left [simp]: "lcm 0 a = 0"
wenzelm@63489
   281
  by (simp add: lcm_gcd)
wenzelm@63489
   282
wenzelm@63489
   283
lemma lcm_0_right [simp]: "lcm a 0 = 0"
haftmann@60686
   284
  by (simp add: lcm_gcd)
wenzelm@63489
   285
wenzelm@63489
   286
lemma lcm_eq_0_iff: "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
wenzelm@63489
   287
  (is "?P \<longleftrightarrow> ?Q")
haftmann@60686
   288
proof
wenzelm@63489
   289
  assume ?P
wenzelm@63489
   290
  then have "0 dvd lcm a b"
wenzelm@63489
   291
    by simp
haftmann@60686
   292
  then have "0 dvd normalize (a * b) div gcd a b"
haftmann@60686
   293
    by (simp add: lcm_gcd)
haftmann@60686
   294
  then have "0 * gcd a b dvd normalize (a * b)"
haftmann@60686
   295
    using dvd_div_iff_mult [of "gcd a b" _ 0] by (cases "gcd a b = 0") simp_all
haftmann@60686
   296
  then have "normalize (a * b) = 0"
haftmann@60686
   297
    by simp
wenzelm@63489
   298
  then show ?Q
wenzelm@63489
   299
    by simp
haftmann@60686
   300
next
wenzelm@63489
   301
  assume ?Q
wenzelm@63489
   302
  then show ?P
wenzelm@63489
   303
    by auto
haftmann@60686
   304
qed
haftmann@60686
   305
wenzelm@63489
   306
lemma lcm_eq_1_iff [simp]: "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"
haftmann@61913
   307
  by (auto intro: associated_eqI)
haftmann@61913
   308
wenzelm@63489
   309
lemma unit_factor_lcm: "unit_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
haftmann@60686
   310
  by (simp add: unit_factor_gcd dvd_unit_factor_div lcm_gcd)
haftmann@60686
   311
wenzelm@61605
   312
sublocale lcm: abel_semigroup lcm
haftmann@60686
   313
proof
haftmann@60686
   314
  fix a b c
haftmann@60686
   315
  show "lcm a b = lcm b a"
haftmann@60686
   316
    by (simp add: lcm_gcd ac_simps normalize_mult dvd_normalize_div)
haftmann@60688
   317
  have "lcm (lcm a b) c dvd lcm a (lcm b c)"
haftmann@60688
   318
    and "lcm a (lcm b c) dvd lcm (lcm a b) c"
haftmann@60688
   319
    by (auto intro: lcm_least
haftmann@60686
   320
      dvd_trans [of b "lcm b c" "lcm a (lcm b c)"]
haftmann@60686
   321
      dvd_trans [of c "lcm b c" "lcm a (lcm b c)"]
haftmann@60686
   322
      dvd_trans [of a "lcm a b" "lcm (lcm a b) c"]
haftmann@60686
   323
      dvd_trans [of b "lcm a b" "lcm (lcm a b) c"])
haftmann@60686
   324
  then show "lcm (lcm a b) c = lcm a (lcm b c)"
haftmann@60688
   325
    by (rule associated_eqI) simp_all
haftmann@60686
   326
qed
haftmann@60686
   327
wenzelm@63489
   328
lemma lcm_self [simp]: "lcm a a = normalize a"
haftmann@60686
   329
proof -
haftmann@60686
   330
  have "lcm a a dvd a"
haftmann@60686
   331
    by (rule lcm_least) simp_all
haftmann@60686
   332
  then show ?thesis
haftmann@60688
   333
    by (auto intro: associated_eqI)
haftmann@60686
   334
qed
haftmann@60686
   335
wenzelm@63489
   336
lemma lcm_left_idem [simp]: "lcm a (lcm a b) = lcm a b"
haftmann@61913
   337
  by (auto intro: associated_eqI)
haftmann@61913
   338
wenzelm@63489
   339
lemma lcm_right_idem [simp]: "lcm (lcm a b) b = lcm a b"
haftmann@61913
   340
  unfolding lcm.commute [of a] lcm.commute [of "lcm b a"] by simp
haftmann@61913
   341
wenzelm@63489
   342
lemma gcd_mult_lcm [simp]: "gcd a b * lcm a b = normalize a * normalize b"
haftmann@60686
   343
  by (simp add: lcm_gcd normalize_mult)
haftmann@60686
   344
wenzelm@63489
   345
lemma lcm_mult_gcd [simp]: "lcm a b * gcd a b = normalize a * normalize b"
wenzelm@63489
   346
  using gcd_mult_lcm [of a b] by (simp add: ac_simps)
haftmann@60686
   347
haftmann@60686
   348
lemma gcd_lcm:
haftmann@60686
   349
  assumes "a \<noteq> 0" and "b \<noteq> 0"
haftmann@60686
   350
  shows "gcd a b = normalize (a * b) div lcm a b"
haftmann@60686
   351
proof -
haftmann@60686
   352
  from assms have "lcm a b \<noteq> 0"
haftmann@60686
   353
    by (simp add: lcm_eq_0_iff)
wenzelm@63489
   354
  have "gcd a b * lcm a b = normalize a * normalize b"
wenzelm@63489
   355
    by simp
haftmann@60686
   356
  then have "gcd a b * lcm a b div lcm a b = normalize (a * b) div lcm a b"
haftmann@60686
   357
    by (simp_all add: normalize_mult)
haftmann@60686
   358
  with \<open>lcm a b \<noteq> 0\<close> show ?thesis
haftmann@60686
   359
    using nonzero_mult_divide_cancel_right [of "lcm a b" "gcd a b"] by simp
haftmann@60686
   360
qed
haftmann@60686
   361
wenzelm@63489
   362
lemma lcm_1_left [simp]: "lcm 1 a = normalize a"
haftmann@60686
   363
  by (simp add: lcm_gcd)
haftmann@60686
   364
wenzelm@63489
   365
lemma lcm_1_right [simp]: "lcm a 1 = normalize a"
haftmann@60686
   366
  by (simp add: lcm_gcd)
wenzelm@63489
   367
wenzelm@63489
   368
lemma lcm_mult_left: "lcm (c * a) (c * b) = normalize c * lcm a b"
haftmann@60686
   369
  by (cases "c = 0")
haftmann@60686
   370
    (simp_all add: gcd_mult_right lcm_gcd div_mult_swap normalize_mult ac_simps,
haftmann@60686
   371
      simp add: dvd_div_mult2_eq mult.left_commute [of "normalize c", symmetric])
haftmann@60686
   372
wenzelm@63489
   373
lemma lcm_mult_right: "lcm (a * c) (b * c) = lcm b a * normalize c"
haftmann@60686
   374
  using lcm_mult_left [of c a b] by (simp add: ac_simps)
haftmann@60686
   375
wenzelm@63489
   376
lemma mult_lcm_left: "c * lcm a b = unit_factor c * lcm (c * a) (c * b)"
haftmann@60686
   377
  by (simp add: lcm_mult_left mult.assoc [symmetric])
haftmann@60686
   378
wenzelm@63489
   379
lemma mult_lcm_right: "lcm a b * c = lcm (a * c) (b * c) * unit_factor c"
haftmann@60686
   380
  using mult_lcm_left [of c a b] by (simp add: ac_simps)
eberlm@62429
   381
eberlm@62429
   382
lemma gcdI:
wenzelm@63489
   383
  assumes "c dvd a" and "c dvd b"
wenzelm@63489
   384
    and greatest: "\<And>d. d dvd a \<Longrightarrow> d dvd b \<Longrightarrow> d dvd c"
eberlm@62429
   385
    and "normalize c = c"
eberlm@62429
   386
  shows "c = gcd a b"
eberlm@62429
   387
  by (rule associated_eqI) (auto simp: assms intro: gcd_greatest)
eberlm@62429
   388
wenzelm@63489
   389
lemma gcd_unique:
wenzelm@63489
   390
  "d dvd a \<and> d dvd b \<and> normalize d = d \<and> (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
eberlm@62429
   391
  by rule (auto intro: gcdI simp: gcd_greatest)
eberlm@62429
   392
eberlm@62429
   393
lemma gcd_dvd_prod: "gcd a b dvd k * b"
eberlm@62429
   394
  using mult_dvd_mono [of 1] by auto
eberlm@62429
   395
eberlm@62429
   396
lemma gcd_proj2_if_dvd: "b dvd a \<Longrightarrow> gcd a b = normalize b"
eberlm@62429
   397
  by (rule gcdI [symmetric]) simp_all
eberlm@62429
   398
eberlm@62429
   399
lemma gcd_proj1_if_dvd: "a dvd b \<Longrightarrow> gcd a b = normalize a"
eberlm@62429
   400
  by (rule gcdI [symmetric]) simp_all
eberlm@62429
   401
eberlm@62429
   402
lemma gcd_proj1_iff: "gcd m n = normalize m \<longleftrightarrow> m dvd n"
eberlm@62429
   403
proof
wenzelm@63489
   404
  assume *: "gcd m n = normalize m"
eberlm@62429
   405
  show "m dvd n"
eberlm@62429
   406
  proof (cases "m = 0")
wenzelm@63489
   407
    case True
wenzelm@63489
   408
    with * show ?thesis by simp
wenzelm@63489
   409
  next
wenzelm@63489
   410
    case [simp]: False
wenzelm@63489
   411
    from * have **: "m = gcd m n * unit_factor m"
eberlm@62429
   412
      by (simp add: unit_eq_div2)
wenzelm@63489
   413
    show ?thesis
wenzelm@63489
   414
      by (subst **) (simp add: mult_unit_dvd_iff)
wenzelm@63489
   415
  qed
eberlm@62429
   416
next
eberlm@62429
   417
  assume "m dvd n"
wenzelm@63489
   418
  then show "gcd m n = normalize m"
wenzelm@63489
   419
    by (rule gcd_proj1_if_dvd)
eberlm@62429
   420
qed
wenzelm@63489
   421
eberlm@62429
   422
lemma gcd_proj2_iff: "gcd m n = normalize n \<longleftrightarrow> n dvd m"
eberlm@62429
   423
  using gcd_proj1_iff [of n m] by (simp add: ac_simps)
eberlm@62429
   424
eberlm@62429
   425
lemma gcd_mult_distrib': "normalize c * gcd a b = gcd (c * a) (c * b)"
eberlm@62429
   426
  by (rule gcdI) (auto simp: normalize_mult gcd_greatest mult_dvd_mono gcd_mult_left[symmetric])
eberlm@62429
   427
wenzelm@63489
   428
lemma gcd_mult_distrib: "k * gcd a b = gcd (k * a) (k * b) * unit_factor k"
eberlm@62429
   429
proof-
eberlm@62429
   430
  have "normalize k * gcd a b = gcd (k * a) (k * b)"
eberlm@62429
   431
    by (simp add: gcd_mult_distrib')
eberlm@62429
   432
  then have "normalize k * gcd a b * unit_factor k = gcd (k * a) (k * b) * unit_factor k"
eberlm@62429
   433
    by simp
eberlm@62429
   434
  then have "normalize k * unit_factor k * gcd a b  = gcd (k * a) (k * b) * unit_factor k"
eberlm@62429
   435
    by (simp only: ac_simps)
eberlm@62429
   436
  then show ?thesis
eberlm@62429
   437
    by simp
eberlm@62429
   438
qed
eberlm@62429
   439
wenzelm@63489
   440
lemma lcm_mult_unit1: "is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c"
eberlm@62429
   441
  by (rule associated_eqI) (simp_all add: mult_unit_dvd_iff dvd_lcmI1)
eberlm@62429
   442
wenzelm@63489
   443
lemma lcm_mult_unit2: "is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c"
eberlm@62429
   444
  using lcm_mult_unit1 [of a c b] by (simp add: ac_simps)
eberlm@62429
   445
eberlm@62429
   446
lemma lcm_div_unit1:
eberlm@62429
   447
  "is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c"
wenzelm@63489
   448
  by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1)
wenzelm@63489
   449
wenzelm@63489
   450
lemma lcm_div_unit2: "is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c"
eberlm@62429
   451
  by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2)
eberlm@62429
   452
wenzelm@63489
   453
lemma normalize_lcm_left [simp]: "lcm (normalize a) b = lcm a b"
eberlm@62429
   454
proof (cases "a = 0")
wenzelm@63489
   455
  case True
wenzelm@63489
   456
  then show ?thesis
eberlm@62429
   457
    by simp
eberlm@62429
   458
next
wenzelm@63489
   459
  case False
wenzelm@63489
   460
  then have "is_unit (unit_factor a)"
eberlm@62429
   461
    by simp
eberlm@62429
   462
  moreover have "normalize a = a div unit_factor a"
eberlm@62429
   463
    by simp
eberlm@62429
   464
  ultimately show ?thesis
eberlm@62429
   465
    by (simp only: lcm_div_unit1)
eberlm@62429
   466
qed
eberlm@62429
   467
wenzelm@63489
   468
lemma normalize_lcm_right [simp]: "lcm a (normalize b) = lcm a b"
eberlm@62429
   469
  using normalize_lcm_left [of b a] by (simp add: ac_simps)
eberlm@62429
   470
eberlm@62429
   471
lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (b * a) c = gcd b c"
eberlm@62429
   472
  apply (rule gcdI)
wenzelm@63489
   473
     apply simp_all
wenzelm@63489
   474
  apply (rule dvd_trans)
wenzelm@63489
   475
   apply (rule gcd_dvd1)
wenzelm@63489
   476
  apply (simp add: unit_simps)
eberlm@62429
   477
  done
eberlm@62429
   478
eberlm@62429
   479
lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c"
wenzelm@63489
   480
  apply (subst gcd.commute)
wenzelm@63489
   481
  apply (subst gcd_mult_unit1)
wenzelm@63489
   482
   apply assumption
wenzelm@63489
   483
  apply (rule gcd.commute)
wenzelm@63489
   484
  done
eberlm@62429
   485
eberlm@62429
   486
lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (b div a) c = gcd b c"
eberlm@62429
   487
  by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1)
eberlm@62429
   488
eberlm@62429
   489
lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c"
eberlm@62429
   490
  by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2)
eberlm@62429
   491
wenzelm@63489
   492
lemma normalize_gcd_left [simp]: "gcd (normalize a) b = gcd a b"
eberlm@62429
   493
proof (cases "a = 0")
wenzelm@63489
   494
  case True
wenzelm@63489
   495
  then show ?thesis
eberlm@62429
   496
    by simp
eberlm@62429
   497
next
wenzelm@63489
   498
  case False
wenzelm@63489
   499
  then have "is_unit (unit_factor a)"
eberlm@62429
   500
    by simp
eberlm@62429
   501
  moreover have "normalize a = a div unit_factor a"
eberlm@62429
   502
    by simp
eberlm@62429
   503
  ultimately show ?thesis
eberlm@62429
   504
    by (simp only: gcd_div_unit1)
eberlm@62429
   505
qed
eberlm@62429
   506
wenzelm@63489
   507
lemma normalize_gcd_right [simp]: "gcd a (normalize b) = gcd a b"
eberlm@62429
   508
  using normalize_gcd_left [of b a] by (simp add: ac_simps)
eberlm@62429
   509
eberlm@62429
   510
lemma comp_fun_idem_gcd: "comp_fun_idem gcd"
eberlm@62429
   511
  by standard (simp_all add: fun_eq_iff ac_simps)
eberlm@62429
   512
eberlm@62429
   513
lemma comp_fun_idem_lcm: "comp_fun_idem lcm"
eberlm@62429
   514
  by standard (simp_all add: fun_eq_iff ac_simps)
eberlm@62429
   515
wenzelm@63489
   516
lemma gcd_dvd_antisym: "gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d"
eberlm@62429
   517
proof (rule gcdI)
wenzelm@63489
   518
  assume *: "gcd a b dvd gcd c d"
wenzelm@63489
   519
    and **: "gcd c d dvd gcd a b"
wenzelm@63489
   520
  have "gcd c d dvd c"
wenzelm@63489
   521
    by simp
wenzelm@63489
   522
  with * show "gcd a b dvd c"
wenzelm@63489
   523
    by (rule dvd_trans)
wenzelm@63489
   524
  have "gcd c d dvd d"
wenzelm@63489
   525
    by simp
wenzelm@63489
   526
  with * show "gcd a b dvd d"
wenzelm@63489
   527
    by (rule dvd_trans)
eberlm@62429
   528
  show "normalize (gcd a b) = gcd a b"
eberlm@62429
   529
    by simp
eberlm@62429
   530
  fix l assume "l dvd c" and "l dvd d"
wenzelm@63489
   531
  then have "l dvd gcd c d"
wenzelm@63489
   532
    by (rule gcd_greatest)
wenzelm@63489
   533
  from this and ** show "l dvd gcd a b"
wenzelm@63489
   534
    by (rule dvd_trans)
eberlm@62429
   535
qed
eberlm@62429
   536
eberlm@62429
   537
lemma coprime_dvd_mult:
eberlm@62429
   538
  assumes "coprime a b" and "a dvd c * b"
eberlm@62429
   539
  shows "a dvd c"
eberlm@62429
   540
proof (cases "c = 0")
wenzelm@63489
   541
  case True
wenzelm@63489
   542
  then show ?thesis by simp
eberlm@62429
   543
next
eberlm@62429
   544
  case False
wenzelm@63489
   545
  then have unit: "is_unit (unit_factor c)"
wenzelm@63489
   546
    by simp
eberlm@62429
   547
  from \<open>coprime a b\<close> mult_gcd_left [of c a b]
eberlm@62429
   548
  have "gcd (c * a) (c * b) * unit_factor c = c"
eberlm@62429
   549
    by (simp add: ac_simps)
eberlm@62429
   550
  moreover from \<open>a dvd c * b\<close> have "a dvd gcd (c * a) (c * b) * unit_factor c"
eberlm@62429
   551
    by (simp add: dvd_mult_unit_iff unit)
wenzelm@63489
   552
  ultimately show ?thesis
wenzelm@63489
   553
    by simp
eberlm@62429
   554
qed
eberlm@62429
   555
wenzelm@63489
   556
lemma coprime_dvd_mult_iff: "coprime a c \<Longrightarrow> a dvd b * c \<longleftrightarrow> a dvd b"
wenzelm@63489
   557
  by (auto intro: coprime_dvd_mult)
wenzelm@63489
   558
wenzelm@63489
   559
lemma gcd_mult_cancel: "coprime c b \<Longrightarrow> gcd (c * a) b = gcd a b"
eberlm@62429
   560
  apply (rule associated_eqI)
wenzelm@63489
   561
     apply (rule gcd_greatest)
wenzelm@63489
   562
      apply (rule_tac b = c in coprime_dvd_mult)
wenzelm@63489
   563
       apply (simp add: gcd.assoc)
wenzelm@63489
   564
       apply (simp_all add: ac_simps)
eberlm@62429
   565
  done
eberlm@62429
   566
eberlm@62429
   567
lemma coprime_crossproduct:
wenzelm@63489
   568
  fixes a b c d :: 'a
eberlm@62429
   569
  assumes "coprime a d" and "coprime b c"
wenzelm@63489
   570
  shows "normalize a * normalize c = normalize b * normalize d \<longleftrightarrow>
wenzelm@63489
   571
    normalize a = normalize b \<and> normalize c = normalize d"
wenzelm@63489
   572
    (is "?lhs \<longleftrightarrow> ?rhs")
eberlm@62429
   573
proof
wenzelm@63489
   574
  assume ?rhs
wenzelm@63489
   575
  then show ?lhs by simp
eberlm@62429
   576
next
eberlm@62429
   577
  assume ?lhs
eberlm@62429
   578
  from \<open>?lhs\<close> have "normalize a dvd normalize b * normalize d"
eberlm@62429
   579
    by (auto intro: dvdI dest: sym)
eberlm@62429
   580
  with \<open>coprime a d\<close> have "a dvd b"
eberlm@62429
   581
    by (simp add: coprime_dvd_mult_iff normalize_mult [symmetric])
eberlm@62429
   582
  from \<open>?lhs\<close> have "normalize b dvd normalize a * normalize c"
eberlm@62429
   583
    by (auto intro: dvdI dest: sym)
eberlm@62429
   584
  with \<open>coprime b c\<close> have "b dvd a"
eberlm@62429
   585
    by (simp add: coprime_dvd_mult_iff normalize_mult [symmetric])
eberlm@62429
   586
  from \<open>?lhs\<close> have "normalize c dvd normalize d * normalize b"
eberlm@62429
   587
    by (auto intro: dvdI dest: sym simp add: mult.commute)
eberlm@62429
   588
  with \<open>coprime b c\<close> have "c dvd d"
eberlm@62429
   589
    by (simp add: coprime_dvd_mult_iff gcd.commute normalize_mult [symmetric])
eberlm@62429
   590
  from \<open>?lhs\<close> have "normalize d dvd normalize c * normalize a"
eberlm@62429
   591
    by (auto intro: dvdI dest: sym simp add: mult.commute)
eberlm@62429
   592
  with \<open>coprime a d\<close> have "d dvd c"
eberlm@62429
   593
    by (simp add: coprime_dvd_mult_iff gcd.commute normalize_mult [symmetric])
eberlm@62429
   594
  from \<open>a dvd b\<close> \<open>b dvd a\<close> have "normalize a = normalize b"
eberlm@62429
   595
    by (rule associatedI)
eberlm@62429
   596
  moreover from \<open>c dvd d\<close> \<open>d dvd c\<close> have "normalize c = normalize d"
eberlm@62429
   597
    by (rule associatedI)
eberlm@62429
   598
  ultimately show ?rhs ..
eberlm@62429
   599
qed
eberlm@62429
   600
eberlm@62429
   601
lemma gcd_add1 [simp]: "gcd (m + n) n = gcd m n"
eberlm@62429
   602
  by (rule gcdI [symmetric]) (simp_all add: dvd_add_left_iff)
eberlm@62429
   603
eberlm@62429
   604
lemma gcd_add2 [simp]: "gcd m (m + n) = gcd m n"
eberlm@62429
   605
  using gcd_add1 [of n m] by (simp add: ac_simps)
eberlm@62429
   606
eberlm@62429
   607
lemma gcd_add_mult: "gcd m (k * m + n) = gcd m n"
eberlm@62429
   608
  by (rule gcdI [symmetric]) (simp_all add: dvd_add_right_iff)
eberlm@62429
   609
wenzelm@63489
   610
lemma coprimeI: "(\<And>l. l dvd a \<Longrightarrow> l dvd b \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd a b = 1"
wenzelm@63489
   611
  by (rule sym, rule gcdI) simp_all
eberlm@62429
   612
eberlm@62429
   613
lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)"
eberlm@62429
   614
  by (auto intro: coprimeI gcd_greatest dvd_gcdD1 dvd_gcdD2)
eberlm@62429
   615
eberlm@62429
   616
lemma div_gcd_coprime:
eberlm@62429
   617
  assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
eberlm@62429
   618
  shows "coprime (a div gcd a b) (b div gcd a b)"
eberlm@62429
   619
proof -
eberlm@62429
   620
  let ?g = "gcd a b"
eberlm@62429
   621
  let ?a' = "a div ?g"
eberlm@62429
   622
  let ?b' = "b div ?g"
eberlm@62429
   623
  let ?g' = "gcd ?a' ?b'"
wenzelm@63489
   624
  have dvdg: "?g dvd a" "?g dvd b"
wenzelm@63489
   625
    by simp_all
wenzelm@63489
   626
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'"
wenzelm@63489
   627
    by simp_all
eberlm@62429
   628
  from dvdg dvdg' obtain ka kb ka' kb' where
wenzelm@63489
   629
    kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
eberlm@62429
   630
    unfolding dvd_def by blast
eberlm@62429
   631
  from this [symmetric] have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
eberlm@62429
   632
    by (simp_all add: mult.assoc mult.left_commute [of "gcd a b"])
eberlm@62429
   633
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
wenzelm@63489
   634
    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)] dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
wenzelm@63489
   635
  have "?g \<noteq> 0"
wenzelm@63489
   636
    using nz by simp
eberlm@62429
   637
  moreover from gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
wenzelm@63489
   638
  ultimately show ?thesis
wenzelm@63489
   639
    using dvd_times_left_cancel_iff [of "gcd a b" _ 1] by simp
eberlm@62429
   640
qed
eberlm@62429
   641
eberlm@62429
   642
eberlm@62429
   643
lemma divides_mult:
eberlm@62429
   644
  assumes "a dvd c" and nr: "b dvd c" and "coprime a b"
eberlm@62429
   645
  shows "a * b dvd c"
wenzelm@63489
   646
proof -
eberlm@62429
   647
  from \<open>b dvd c\<close> obtain b' where"c = b * b'" ..
eberlm@62429
   648
  with \<open>a dvd c\<close> have "a dvd b' * b"
eberlm@62429
   649
    by (simp add: ac_simps)
eberlm@62429
   650
  with \<open>coprime a b\<close> have "a dvd b'"
eberlm@62429
   651
    by (simp add: coprime_dvd_mult_iff)
eberlm@62429
   652
  then obtain a' where "b' = a * a'" ..
eberlm@62429
   653
  with \<open>c = b * b'\<close> have "c = (a * b) * a'"
eberlm@62429
   654
    by (simp add: ac_simps)
eberlm@62429
   655
  then show ?thesis ..
eberlm@62429
   656
qed
eberlm@62429
   657
eberlm@62429
   658
lemma coprime_lmult:
wenzelm@63489
   659
  assumes dab: "gcd d (a * b) = 1"
eberlm@62429
   660
  shows "gcd d a = 1"
eberlm@62429
   661
proof (rule coprimeI)
wenzelm@63489
   662
  fix l
wenzelm@63489
   663
  assume "l dvd d" and "l dvd a"
wenzelm@63489
   664
  then have "l dvd a * b"
wenzelm@63489
   665
    by simp
wenzelm@63489
   666
  with \<open>l dvd d\<close> and dab show "l dvd 1"
wenzelm@63489
   667
    by (auto intro: gcd_greatest)
eberlm@62429
   668
qed
eberlm@62429
   669
eberlm@62429
   670
lemma coprime_rmult:
eberlm@62429
   671
  assumes dab: "gcd d (a * b) = 1"
eberlm@62429
   672
  shows "gcd d b = 1"
eberlm@62429
   673
proof (rule coprimeI)
wenzelm@63489
   674
  fix l
wenzelm@63489
   675
  assume "l dvd d" and "l dvd b"
wenzelm@63489
   676
  then have "l dvd a * b"
wenzelm@63489
   677
    by simp
wenzelm@63489
   678
  with \<open>l dvd d\<close> and dab show "l dvd 1"
wenzelm@63489
   679
    by (auto intro: gcd_greatest)
eberlm@62429
   680
qed
eberlm@62429
   681
eberlm@62429
   682
lemma coprime_mult:
wenzelm@63489
   683
  assumes "coprime d a"
wenzelm@63489
   684
    and "coprime d b"
eberlm@62429
   685
  shows "coprime d (a * b)"
eberlm@62429
   686
  apply (subst gcd.commute)
wenzelm@63489
   687
  using assms(1) apply (subst gcd_mult_cancel)
wenzelm@63489
   688
   apply (subst gcd.commute)
wenzelm@63489
   689
   apply assumption
wenzelm@63489
   690
  apply (subst gcd.commute)
wenzelm@63489
   691
  apply (rule assms(2))
eberlm@62429
   692
  done
eberlm@62429
   693
eberlm@62429
   694
lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1"
wenzelm@63489
   695
  using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b]
wenzelm@63489
   696
  by blast
eberlm@62429
   697
eberlm@62429
   698
lemma gcd_coprime:
wenzelm@63489
   699
  assumes c: "gcd a b \<noteq> 0"
wenzelm@63489
   700
    and a: "a = a' * gcd a b"
wenzelm@63489
   701
    and b: "b = b' * gcd a b"
eberlm@62429
   702
  shows "gcd a' b' = 1"
eberlm@62429
   703
proof -
wenzelm@63489
   704
  from c have "a \<noteq> 0 \<or> b \<noteq> 0"
wenzelm@63489
   705
    by simp
eberlm@62429
   706
  with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" .
wenzelm@63489
   707
  also from assms have "a div gcd a b = a'"
wenzelm@63489
   708
    using dvd_div_eq_mult local.gcd_dvd1 by blast
wenzelm@63489
   709
  also from assms have "b div gcd a b = b'"
wenzelm@63489
   710
    using dvd_div_eq_mult local.gcd_dvd1 by blast
eberlm@62429
   711
  finally show ?thesis .
eberlm@62429
   712
qed
eberlm@62429
   713
eberlm@62429
   714
lemma coprime_power:
eberlm@62429
   715
  assumes "0 < n"
eberlm@62429
   716
  shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1"
wenzelm@63489
   717
  using assms
wenzelm@63489
   718
proof (induct n)
wenzelm@63489
   719
  case 0
wenzelm@63489
   720
  then show ?case by simp
wenzelm@63489
   721
next
wenzelm@63489
   722
  case (Suc n)
wenzelm@63489
   723
  then show ?case
eberlm@62429
   724
    by (cases n) (simp_all add: coprime_mul_eq)
wenzelm@63489
   725
qed
eberlm@62429
   726
eberlm@62429
   727
lemma gcd_coprime_exists:
wenzelm@63489
   728
  assumes "gcd a b \<noteq> 0"
eberlm@62429
   729
  shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1"
eberlm@62429
   730
  apply (rule_tac x = "a div gcd a b" in exI)
eberlm@62429
   731
  apply (rule_tac x = "b div gcd a b" in exI)
wenzelm@63489
   732
  using assms
wenzelm@63489
   733
  apply (auto intro: div_gcd_coprime)
eberlm@62429
   734
  done
eberlm@62429
   735
wenzelm@63489
   736
lemma coprime_exp: "gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1"
wenzelm@63489
   737
  by (induct n) (simp_all add: coprime_mult)
wenzelm@63489
   738
wenzelm@63489
   739
lemma coprime_exp_left: "coprime a b \<Longrightarrow> coprime (a ^ n) b"
wenzelm@63489
   740
  by (induct n) (simp_all add: gcd_mult_cancel)
eberlm@62429
   741
eberlm@62429
   742
lemma coprime_exp2:
eberlm@62429
   743
  assumes "coprime a b"
eberlm@62429
   744
  shows "coprime (a ^ n) (b ^ m)"
eberlm@62429
   745
proof (rule coprime_exp_left)
eberlm@62429
   746
  from assms show "coprime a (b ^ m)"
eberlm@62429
   747
    by (induct m) (simp_all add: gcd_mult_cancel gcd.commute [of a])
eberlm@62429
   748
qed
eberlm@62429
   749
wenzelm@63489
   750
lemma gcd_exp: "gcd (a ^ n) (b ^ n) = gcd a b ^ n"
eberlm@62429
   751
proof (cases "a = 0 \<and> b = 0")
eberlm@62429
   752
  case True
wenzelm@63489
   753
  then show ?thesis
wenzelm@63489
   754
    by (cases n) simp_all
eberlm@62429
   755
next
eberlm@62429
   756
  case False
eberlm@62429
   757
  then have "1 = gcd ((a div gcd a b) ^ n) ((b div gcd a b) ^ n)"
eberlm@62429
   758
    using coprime_exp2[OF div_gcd_coprime[of a b], of n n, symmetric] by simp
wenzelm@63489
   759
  then have "gcd a b ^ n = gcd a b ^ n * \<dots>"
wenzelm@63489
   760
    by simp
eberlm@62429
   761
  also note gcd_mult_distrib
eberlm@62429
   762
  also have "unit_factor (gcd a b ^ n) = 1"
eberlm@62429
   763
    using False by (auto simp add: unit_factor_power unit_factor_gcd)
eberlm@62429
   764
  also have "(gcd a b)^n * (a div gcd a b)^n = a^n"
wenzelm@63489
   765
    apply (subst ac_simps)
wenzelm@63489
   766
    apply (subst div_power)
wenzelm@63489
   767
     apply simp
wenzelm@63489
   768
    apply (rule dvd_div_mult_self)
wenzelm@63489
   769
    apply (rule dvd_power_same)
wenzelm@63489
   770
    apply simp
wenzelm@63489
   771
    done
eberlm@62429
   772
  also have "(gcd a b)^n * (b div gcd a b)^n = b^n"
wenzelm@63489
   773
    apply (subst ac_simps)
wenzelm@63489
   774
    apply (subst div_power)
wenzelm@63489
   775
     apply simp
wenzelm@63489
   776
    apply (rule dvd_div_mult_self)
wenzelm@63489
   777
    apply (rule dvd_power_same)
wenzelm@63489
   778
    apply simp
wenzelm@63489
   779
    done
eberlm@62429
   780
  finally show ?thesis by simp
eberlm@62429
   781
qed
eberlm@62429
   782
wenzelm@63489
   783
lemma coprime_common_divisor: "gcd a b = 1 \<Longrightarrow> a dvd a \<Longrightarrow> a dvd b \<Longrightarrow> is_unit a"
eberlm@62429
   784
  apply (subgoal_tac "a dvd gcd a b")
wenzelm@63489
   785
   apply simp
eberlm@62429
   786
  apply (erule (1) gcd_greatest)
eberlm@62429
   787
  done
eberlm@62429
   788
wenzelm@63489
   789
lemma division_decomp:
wenzelm@63489
   790
  assumes "a dvd b * c"
eberlm@62429
   791
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
eberlm@62429
   792
proof (cases "gcd a b = 0")
wenzelm@63489
   793
  case True
wenzelm@63489
   794
  then have "a = 0 \<and> b = 0"
wenzelm@63489
   795
    by simp
wenzelm@63489
   796
  then have "a = 0 * c \<and> 0 dvd b \<and> c dvd c"
wenzelm@63489
   797
    by simp
eberlm@62429
   798
  then show ?thesis by blast
eberlm@62429
   799
next
wenzelm@63489
   800
  case False
eberlm@62429
   801
  let ?d = "gcd a b"
wenzelm@63489
   802
  from gcd_coprime_exists [OF False]
eberlm@62429
   803
    obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
eberlm@62429
   804
    by blast
wenzelm@63489
   805
  from ab'(1) have "a' dvd a"
wenzelm@63489
   806
    unfolding dvd_def by blast
wenzelm@63489
   807
  with assms have "a' dvd b * c"
wenzelm@63489
   808
    using dvd_trans[of a' a "b*c"] by simp
wenzelm@63489
   809
  from assms ab'(1,2) have "a' * ?d dvd (b' * ?d) * c"
wenzelm@63489
   810
    by simp
wenzelm@63489
   811
  then have "?d * a' dvd ?d * (b' * c)"
wenzelm@63489
   812
    by (simp add: mult_ac)
wenzelm@63489
   813
  with \<open>?d \<noteq> 0\<close> have "a' dvd b' * c"
wenzelm@63489
   814
    by simp
wenzelm@63489
   815
  with coprime_dvd_mult[OF ab'(3)] have "a' dvd c"
wenzelm@63489
   816
    by (subst (asm) ac_simps) blast
wenzelm@63489
   817
  with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c"
wenzelm@63489
   818
    by (simp add: mult_ac)
eberlm@62429
   819
  then show ?thesis by blast
eberlm@62429
   820
qed
eberlm@62429
   821
eberlm@62429
   822
lemma pow_divs_pow:
eberlm@62429
   823
  assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0"
eberlm@62429
   824
  shows "a dvd b"
eberlm@62429
   825
proof (cases "gcd a b = 0")
wenzelm@63489
   826
  case True
eberlm@62429
   827
  then show ?thesis by simp
eberlm@62429
   828
next
wenzelm@63489
   829
  case False
eberlm@62429
   830
  let ?d = "gcd a b"
wenzelm@63489
   831
  from n obtain m where m: "n = Suc m"
wenzelm@63489
   832
    by (cases n) simp_all
wenzelm@63489
   833
  from False have zn: "?d ^ n \<noteq> 0"
wenzelm@63489
   834
    by (rule power_not_zero)
wenzelm@63489
   835
  from gcd_coprime_exists [OF False]
wenzelm@63489
   836
  obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1"
eberlm@62429
   837
    by blast
eberlm@62429
   838
  from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n"
eberlm@62429
   839
    by (simp add: ab'(1,2)[symmetric])
wenzelm@63489
   840
  then have "?d^n * a'^n dvd ?d^n * b'^n"
eberlm@62429
   841
    by (simp only: power_mult_distrib ac_simps)
wenzelm@63489
   842
  with zn have "a'^n dvd b'^n"
wenzelm@63489
   843
    by simp
wenzelm@63489
   844
  then have "a' dvd b'^n"
wenzelm@63489
   845
    using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m)
wenzelm@63489
   846
  then have "a' dvd b'^m * b'"
wenzelm@63489
   847
    by (simp add: m ac_simps)
eberlm@62429
   848
  with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]]
wenzelm@63489
   849
  have "a' dvd b'" by (subst (asm) ac_simps) blast
wenzelm@63489
   850
  then have "a' * ?d dvd b' * ?d"
wenzelm@63489
   851
    by (rule mult_dvd_mono) simp
wenzelm@63489
   852
  with ab'(1,2) show ?thesis
wenzelm@63489
   853
    by simp
eberlm@62429
   854
qed
eberlm@62429
   855
wenzelm@63489
   856
lemma pow_divs_eq [simp]: "n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b"
eberlm@62429
   857
  by (auto intro: pow_divs_pow dvd_power_same)
eberlm@62429
   858
eberlm@62429
   859
lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1"
wenzelm@63489
   860
  by (subst add_commute) simp
wenzelm@63489
   861
wenzelm@63489
   862
lemma setprod_coprime [rule_format]: "(\<forall>i\<in>A. gcd (f i) a = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) a = 1"
wenzelm@63915
   863
  by (induct A rule: infinite_finite_induct) (auto simp add: gcd_mult_cancel)
wenzelm@63489
   864
nipkow@63882
   865
lemma prod_list_coprime: "(\<And>x. x \<in> set xs \<Longrightarrow> coprime x y) \<Longrightarrow> coprime (prod_list xs) y"
wenzelm@63489
   866
  by (induct xs) (simp_all add: gcd_mult_cancel)
wenzelm@63489
   867
wenzelm@63489
   868
lemma coprime_divisors:
eberlm@62429
   869
  assumes "d dvd a" "e dvd b" "gcd a b = 1"
wenzelm@63489
   870
  shows "gcd d e = 1"
eberlm@62429
   871
proof -
eberlm@62429
   872
  from assms obtain k l where "a = d * k" "b = e * l"
eberlm@62429
   873
    unfolding dvd_def by blast
wenzelm@63489
   874
  with assms have "gcd (d * k) (e * l) = 1"
wenzelm@63489
   875
    by simp
wenzelm@63489
   876
  then have "gcd (d * k) e = 1"
wenzelm@63489
   877
    by (rule coprime_lmult)
wenzelm@63489
   878
  also have "gcd (d * k) e = gcd e (d * k)"
wenzelm@63489
   879
    by (simp add: ac_simps)
wenzelm@63489
   880
  finally have "gcd e d = 1"
wenzelm@63489
   881
    by (rule coprime_lmult)
wenzelm@63489
   882
  then show ?thesis
wenzelm@63489
   883
    by (simp add: ac_simps)
eberlm@62429
   884
qed
eberlm@62429
   885
wenzelm@63489
   886
lemma lcm_gcd_prod: "lcm a b * gcd a b = normalize (a * b)"
eberlm@62429
   887
  by (simp add: lcm_gcd)
eberlm@62429
   888
eberlm@62429
   889
declare unit_factor_lcm [simp]
eberlm@62429
   890
eberlm@62429
   891
lemma lcmI:
eberlm@62429
   892
  assumes "a dvd c" and "b dvd c" and "\<And>d. a dvd d \<Longrightarrow> b dvd d \<Longrightarrow> c dvd d"
eberlm@62429
   893
    and "normalize c = c"
eberlm@62429
   894
  shows "c = lcm a b"
eberlm@62429
   895
  by (rule associated_eqI) (auto simp: assms intro: lcm_least)
eberlm@62429
   896
wenzelm@63489
   897
lemma gcd_dvd_lcm [simp]: "gcd a b dvd lcm a b"
eberlm@62429
   898
  using gcd_dvd2 by (rule dvd_lcmI2)
eberlm@62429
   899
eberlm@62429
   900
lemmas lcm_0 = lcm_0_right
eberlm@62429
   901
eberlm@62429
   902
lemma lcm_unique:
wenzelm@63489
   903
  "a dvd d \<and> b dvd d \<and> normalize d = d \<and> (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
eberlm@62429
   904
  by rule (auto intro: lcmI simp: lcm_least lcm_eq_0_iff)
eberlm@62429
   905
wenzelm@63489
   906
lemma lcm_coprime: "gcd a b = 1 \<Longrightarrow> lcm a b = normalize (a * b)"
eberlm@62429
   907
  by (subst lcm_gcd) simp
eberlm@62429
   908
wenzelm@63489
   909
lemma lcm_proj1_if_dvd: "b dvd a \<Longrightarrow> lcm a b = normalize a"
wenzelm@63489
   910
  apply (cases "a = 0")
wenzelm@63489
   911
   apply simp
wenzelm@63489
   912
  apply (rule sym)
wenzelm@63489
   913
  apply (rule lcmI)
wenzelm@63489
   914
     apply simp_all
wenzelm@63489
   915
  done
wenzelm@63489
   916
wenzelm@63489
   917
lemma lcm_proj2_if_dvd: "a dvd b \<Longrightarrow> lcm a b = normalize b"
eberlm@62429
   918
  using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)
eberlm@62429
   919
wenzelm@63489
   920
lemma lcm_proj1_iff: "lcm m n = normalize m \<longleftrightarrow> n dvd m"
eberlm@62429
   921
proof
wenzelm@63489
   922
  assume *: "lcm m n = normalize m"
eberlm@62429
   923
  show "n dvd m"
eberlm@62429
   924
  proof (cases "m = 0")
wenzelm@63489
   925
    case True
wenzelm@63489
   926
    then show ?thesis by simp
wenzelm@63489
   927
  next
wenzelm@63489
   928
    case [simp]: False
wenzelm@63489
   929
    from * have **: "m = lcm m n * unit_factor m"
eberlm@62429
   930
      by (simp add: unit_eq_div2)
wenzelm@63489
   931
    show ?thesis by (subst **) simp
wenzelm@63489
   932
  qed
eberlm@62429
   933
next
eberlm@62429
   934
  assume "n dvd m"
wenzelm@63489
   935
  then show "lcm m n = normalize m"
wenzelm@63489
   936
    by (rule lcm_proj1_if_dvd)
eberlm@62429
   937
qed
eberlm@62429
   938
wenzelm@63489
   939
lemma lcm_proj2_iff: "lcm m n = normalize n \<longleftrightarrow> m dvd n"
eberlm@62429
   940
  using lcm_proj1_iff [of n m] by (simp add: ac_simps)
eberlm@62429
   941
haftmann@60686
   942
end
haftmann@60686
   943
haftmann@62345
   944
class ring_gcd = comm_ring_1 + semiring_gcd
eberlm@62429
   945
begin
eberlm@62429
   946
eberlm@62429
   947
lemma coprime_minus_one: "coprime (n - 1) n"
eberlm@62429
   948
  using coprime_plus_one[of "n - 1"] by (simp add: gcd.commute)
eberlm@62429
   949
wenzelm@63489
   950
lemma gcd_neg1 [simp]: "gcd (-a) b = gcd a b"
wenzelm@63489
   951
  by (rule sym, rule gcdI) (simp_all add: gcd_greatest)
wenzelm@63489
   952
wenzelm@63489
   953
lemma gcd_neg2 [simp]: "gcd a (-b) = gcd a b"
wenzelm@63489
   954
  by (rule sym, rule gcdI) (simp_all add: gcd_greatest)
wenzelm@63489
   955
wenzelm@63489
   956
lemma gcd_neg_numeral_1 [simp]: "gcd (- numeral n) a = gcd (numeral n) a"
eberlm@62429
   957
  by (fact gcd_neg1)
eberlm@62429
   958
wenzelm@63489
   959
lemma gcd_neg_numeral_2 [simp]: "gcd a (- numeral n) = gcd a (numeral n)"
eberlm@62429
   960
  by (fact gcd_neg2)
eberlm@62429
   961
eberlm@62429
   962
lemma gcd_diff1: "gcd (m - n) n = gcd m n"
wenzelm@63489
   963
  by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric], subst gcd_add1, simp)
eberlm@62429
   964
eberlm@62429
   965
lemma gcd_diff2: "gcd (n - m) n = gcd m n"
wenzelm@63489
   966
  by (subst gcd_neg1[symmetric]) (simp only: minus_diff_eq gcd_diff1)
eberlm@62429
   967
eberlm@62429
   968
lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b"
wenzelm@63489
   969
  by (rule sym, rule lcmI) (simp_all add: lcm_least lcm_eq_0_iff)
eberlm@62429
   970
eberlm@62429
   971
lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b"
wenzelm@63489
   972
  by (rule sym, rule lcmI) (simp_all add: lcm_least lcm_eq_0_iff)
eberlm@62429
   973
eberlm@62429
   974
lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a"
eberlm@62429
   975
  by (fact lcm_neg1)
eberlm@62429
   976
eberlm@62429
   977
lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)"
eberlm@62429
   978
  by (fact lcm_neg2)
eberlm@62429
   979
eberlm@62429
   980
end
haftmann@62345
   981
haftmann@60686
   982
class semiring_Gcd = semiring_gcd + Gcd +
haftmann@60686
   983
  assumes Gcd_dvd: "a \<in> A \<Longrightarrow> Gcd A dvd a"
haftmann@60686
   984
    and Gcd_greatest: "(\<And>b. b \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> a dvd Gcd A"
haftmann@60686
   985
    and normalize_Gcd [simp]: "normalize (Gcd A) = Gcd A"
haftmann@62345
   986
  assumes dvd_Lcm: "a \<in> A \<Longrightarrow> a dvd Lcm A"
haftmann@62345
   987
    and Lcm_least: "(\<And>b. b \<in> A \<Longrightarrow> b dvd a) \<Longrightarrow> Lcm A dvd a"
haftmann@62345
   988
    and normalize_Lcm [simp]: "normalize (Lcm A) = Lcm A"
haftmann@60686
   989
begin
haftmann@60686
   990
wenzelm@63489
   991
lemma Lcm_Gcd: "Lcm A = Gcd {b. \<forall>a\<in>A. a dvd b}"
haftmann@62345
   992
  by (rule associated_eqI) (auto intro: Gcd_dvd dvd_Lcm Gcd_greatest Lcm_least)
haftmann@62345
   993
wenzelm@63489
   994
lemma Gcd_Lcm: "Gcd A = Lcm {b. \<forall>a\<in>A. b dvd a}"
haftmann@62345
   995
  by (rule associated_eqI) (auto intro: Gcd_dvd dvd_Lcm Gcd_greatest Lcm_least)
haftmann@62345
   996
wenzelm@63489
   997
lemma Gcd_empty [simp]: "Gcd {} = 0"
haftmann@60686
   998
  by (rule dvd_0_left, rule Gcd_greatest) simp
haftmann@60686
   999
wenzelm@63489
  1000
lemma Lcm_empty [simp]: "Lcm {} = 1"
haftmann@62345
  1001
  by (auto intro: associated_eqI Lcm_least)
haftmann@62345
  1002
wenzelm@63489
  1003
lemma Gcd_insert [simp]: "Gcd (insert a A) = gcd a (Gcd A)"
haftmann@62345
  1004
proof -
haftmann@62345
  1005
  have "Gcd (insert a A) dvd gcd a (Gcd A)"
haftmann@62345
  1006
    by (auto intro: Gcd_dvd Gcd_greatest)
haftmann@62345
  1007
  moreover have "gcd a (Gcd A) dvd Gcd (insert a A)"
haftmann@62345
  1008
  proof (rule Gcd_greatest)
haftmann@62345
  1009
    fix b
haftmann@62345
  1010
    assume "b \<in> insert a A"
haftmann@62345
  1011
    then show "gcd a (Gcd A) dvd b"
haftmann@62345
  1012
    proof
wenzelm@63489
  1013
      assume "b = a"
wenzelm@63489
  1014
      then show ?thesis
wenzelm@63489
  1015
        by simp
haftmann@62345
  1016
    next
haftmann@62345
  1017
      assume "b \<in> A"
wenzelm@63489
  1018
      then have "Gcd A dvd b"
wenzelm@63489
  1019
        by (rule Gcd_dvd)
wenzelm@63489
  1020
      moreover have "gcd a (Gcd A) dvd Gcd A"
wenzelm@63489
  1021
        by simp
wenzelm@63489
  1022
      ultimately show ?thesis
wenzelm@63489
  1023
        by (blast intro: dvd_trans)
haftmann@62345
  1024
    qed
haftmann@62345
  1025
  qed
haftmann@62345
  1026
  ultimately show ?thesis
haftmann@62345
  1027
    by (auto intro: associated_eqI)
haftmann@62345
  1028
qed
haftmann@62345
  1029
wenzelm@63489
  1030
lemma Lcm_insert [simp]: "Lcm (insert a A) = lcm a (Lcm A)"
haftmann@62345
  1031
proof (rule sym)
haftmann@62345
  1032
  have "lcm a (Lcm A) dvd Lcm (insert a A)"
haftmann@62345
  1033
    by (auto intro: dvd_Lcm Lcm_least)
haftmann@62345
  1034
  moreover have "Lcm (insert a A) dvd lcm a (Lcm A)"
haftmann@62345
  1035
  proof (rule Lcm_least)
haftmann@62345
  1036
    fix b
haftmann@62345
  1037
    assume "b \<in> insert a A"
haftmann@62345
  1038
    then show "b dvd lcm a (Lcm A)"
haftmann@62345
  1039
    proof
wenzelm@63489
  1040
      assume "b = a"
wenzelm@63489
  1041
      then show ?thesis by simp
haftmann@62345
  1042
    next
haftmann@62345
  1043
      assume "b \<in> A"
wenzelm@63489
  1044
      then have "b dvd Lcm A"
wenzelm@63489
  1045
        by (rule dvd_Lcm)
wenzelm@63489
  1046
      moreover have "Lcm A dvd lcm a (Lcm A)"
wenzelm@63489
  1047
        by simp
wenzelm@63489
  1048
      ultimately show ?thesis
wenzelm@63489
  1049
        by (blast intro: dvd_trans)
haftmann@62345
  1050
    qed
haftmann@62345
  1051
  qed
haftmann@62345
  1052
  ultimately show "lcm a (Lcm A) = Lcm (insert a A)"
haftmann@62345
  1053
    by (rule associated_eqI) (simp_all add: lcm_eq_0_iff)
haftmann@62345
  1054
qed
haftmann@62345
  1055
eberlm@62429
  1056
lemma LcmI:
wenzelm@63489
  1057
  assumes "\<And>a. a \<in> A \<Longrightarrow> a dvd b"
wenzelm@63489
  1058
    and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> a dvd c) \<Longrightarrow> b dvd c"
wenzelm@63489
  1059
    and "normalize b = b"
wenzelm@63489
  1060
  shows "b = Lcm A"
eberlm@62429
  1061
  by (rule associated_eqI) (auto simp: assms dvd_Lcm intro: Lcm_least)
eberlm@62429
  1062
wenzelm@63489
  1063
lemma Lcm_subset: "A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B"
eberlm@62429
  1064
  by (blast intro: Lcm_least dvd_Lcm)
eberlm@62429
  1065
wenzelm@63489
  1066
lemma Lcm_Un: "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"
eberlm@62429
  1067
  apply (rule lcmI)
wenzelm@63489
  1068
     apply (blast intro: Lcm_subset)
wenzelm@63489
  1069
    apply (blast intro: Lcm_subset)
wenzelm@63489
  1070
   apply (intro Lcm_least ballI, elim UnE)
wenzelm@63489
  1071
    apply (rule dvd_trans, erule dvd_Lcm, assumption)
wenzelm@63489
  1072
   apply (rule dvd_trans, erule dvd_Lcm, assumption)
eberlm@62429
  1073
  apply simp
eberlm@62429
  1074
  done
wenzelm@63489
  1075
wenzelm@63489
  1076
lemma Gcd_0_iff [simp]: "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}"
wenzelm@63489
  1077
  (is "?P \<longleftrightarrow> ?Q")
haftmann@60686
  1078
proof
haftmann@60686
  1079
  assume ?P
haftmann@60686
  1080
  show ?Q
haftmann@60686
  1081
  proof
haftmann@60686
  1082
    fix a
haftmann@60686
  1083
    assume "a \<in> A"
wenzelm@63489
  1084
    then have "Gcd A dvd a"
wenzelm@63489
  1085
      by (rule Gcd_dvd)
wenzelm@63489
  1086
    with \<open>?P\<close> have "a = 0"
wenzelm@63489
  1087
      by simp
wenzelm@63489
  1088
    then show "a \<in> {0}"
wenzelm@63489
  1089
      by simp
haftmann@60686
  1090
  qed
haftmann@60686
  1091
next
haftmann@60686
  1092
  assume ?Q
haftmann@60686
  1093
  have "0 dvd Gcd A"
haftmann@60686
  1094
  proof (rule Gcd_greatest)
haftmann@60686
  1095
    fix a
haftmann@60686
  1096
    assume "a \<in> A"
wenzelm@63489
  1097
    with \<open>?Q\<close> have "a = 0"
wenzelm@63489
  1098
      by auto
wenzelm@63489
  1099
    then show "0 dvd a"
wenzelm@63489
  1100
      by simp
haftmann@60686
  1101
  qed
wenzelm@63489
  1102
  then show ?P
wenzelm@63489
  1103
    by simp
haftmann@60686
  1104
qed
haftmann@60686
  1105
wenzelm@63489
  1106
lemma Lcm_1_iff [simp]: "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)"
wenzelm@63489
  1107
  (is "?P \<longleftrightarrow> ?Q")
haftmann@60686
  1108
proof
haftmann@60686
  1109
  assume ?P
haftmann@60686
  1110
  show ?Q
haftmann@60686
  1111
  proof
haftmann@60686
  1112
    fix a
haftmann@60686
  1113
    assume "a \<in> A"
haftmann@60686
  1114
    then have "a dvd Lcm A"
haftmann@60686
  1115
      by (rule dvd_Lcm)
haftmann@60686
  1116
    with \<open>?P\<close> show "is_unit a"
haftmann@60686
  1117
      by simp
haftmann@60686
  1118
  qed
haftmann@60686
  1119
next
haftmann@60686
  1120
  assume ?Q
haftmann@60686
  1121
  then have "is_unit (Lcm A)"
haftmann@60686
  1122
    by (blast intro: Lcm_least)
haftmann@60686
  1123
  then have "normalize (Lcm A) = 1"
haftmann@60686
  1124
    by (rule is_unit_normalize)
haftmann@60686
  1125
  then show ?P
haftmann@60686
  1126
    by simp
haftmann@60686
  1127
qed
haftmann@60686
  1128
wenzelm@63489
  1129
lemma unit_factor_Lcm: "unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
haftmann@62345
  1130
proof (cases "Lcm A = 0")
wenzelm@63489
  1131
  case True
wenzelm@63489
  1132
  then show ?thesis
wenzelm@63489
  1133
    by simp
haftmann@62345
  1134
next
haftmann@62345
  1135
  case False
haftmann@62345
  1136
  with unit_factor_normalize have "unit_factor (normalize (Lcm A)) = 1"
haftmann@62345
  1137
    by blast
haftmann@62345
  1138
  with False show ?thesis
haftmann@62345
  1139
    by simp
haftmann@62345
  1140
qed
haftmann@62345
  1141
eberlm@62429
  1142
lemma unit_factor_Gcd: "unit_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
wenzelm@63489
  1143
  by (simp add: Gcd_Lcm unit_factor_Lcm)
eberlm@62429
  1144
eberlm@62429
  1145
lemma GcdI:
wenzelm@63489
  1146
  assumes "\<And>a. a \<in> A \<Longrightarrow> b dvd a"
wenzelm@63489
  1147
    and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> c dvd a) \<Longrightarrow> c dvd b"
eberlm@62429
  1148
    and "normalize b = b"
eberlm@62429
  1149
  shows "b = Gcd A"
eberlm@62429
  1150
  by (rule associated_eqI) (auto simp: assms Gcd_dvd intro: Gcd_greatest)
eberlm@62429
  1151
haftmann@62345
  1152
lemma Gcd_eq_1_I:
haftmann@62345
  1153
  assumes "is_unit a" and "a \<in> A"
haftmann@62345
  1154
  shows "Gcd A = 1"
haftmann@62345
  1155
proof -
haftmann@62345
  1156
  from assms have "is_unit (Gcd A)"
haftmann@62345
  1157
    by (blast intro: Gcd_dvd dvd_unit_imp_unit)
haftmann@62345
  1158
  then have "normalize (Gcd A) = 1"
haftmann@62345
  1159
    by (rule is_unit_normalize)
haftmann@62345
  1160
  then show ?thesis
haftmann@62345
  1161
    by simp
haftmann@62345
  1162
qed
haftmann@62345
  1163
haftmann@60686
  1164
lemma Lcm_eq_0_I:
haftmann@60686
  1165
  assumes "0 \<in> A"
haftmann@60686
  1166
  shows "Lcm A = 0"
haftmann@60686
  1167
proof -
haftmann@60686
  1168
  from assms have "0 dvd Lcm A"
haftmann@60686
  1169
    by (rule dvd_Lcm)
haftmann@60686
  1170
  then show ?thesis
haftmann@60686
  1171
    by simp
haftmann@60686
  1172
qed
haftmann@60686
  1173
wenzelm@63489
  1174
lemma Gcd_UNIV [simp]: "Gcd UNIV = 1"
haftmann@62345
  1175
  using dvd_refl by (rule Gcd_eq_1_I) simp
haftmann@62345
  1176
wenzelm@63489
  1177
lemma Lcm_UNIV [simp]: "Lcm UNIV = 0"
haftmann@61929
  1178
  by (rule Lcm_eq_0_I) simp
haftmann@60686
  1179
haftmann@61929
  1180
lemma Lcm_0_iff:
haftmann@61929
  1181
  assumes "finite A"
haftmann@61929
  1182
  shows "Lcm A = 0 \<longleftrightarrow> 0 \<in> A"
haftmann@61929
  1183
proof (cases "A = {}")
wenzelm@63489
  1184
  case True
wenzelm@63489
  1185
  then show ?thesis by simp
haftmann@61929
  1186
next
wenzelm@63489
  1187
  case False
wenzelm@63489
  1188
  with assms show ?thesis
wenzelm@63489
  1189
    by (induct A rule: finite_ne_induct) (auto simp add: lcm_eq_0_iff)
haftmann@60686
  1190
qed
haftmann@61929
  1191
eberlm@62429
  1192
lemma Gcd_finite:
eberlm@62429
  1193
  assumes "finite A"
eberlm@62429
  1194
  shows "Gcd A = Finite_Set.fold gcd 0 A"
eberlm@62429
  1195
  by (induct rule: finite.induct[OF \<open>finite A\<close>])
wenzelm@63489
  1196
    (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd])
eberlm@62429
  1197
eberlm@62429
  1198
lemma Gcd_set [code_unfold]: "Gcd (set as) = foldl gcd 0 as"
wenzelm@63489
  1199
  by (simp add: Gcd_finite comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd]
wenzelm@63489
  1200
      foldl_conv_fold gcd.commute)
eberlm@62429
  1201
eberlm@62429
  1202
lemma Lcm_finite:
eberlm@62429
  1203
  assumes "finite A"
eberlm@62429
  1204
  shows "Lcm A = Finite_Set.fold lcm 1 A"
eberlm@62429
  1205
  by (induct rule: finite.induct[OF \<open>finite A\<close>])
wenzelm@63489
  1206
    (simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm])
wenzelm@63489
  1207
wenzelm@63489
  1208
lemma Lcm_set [code_unfold]: "Lcm (set as) = foldl lcm 1 as"
wenzelm@63489
  1209
  by (simp add: Lcm_finite comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm]
wenzelm@63489
  1210
      foldl_conv_fold lcm.commute)
wenzelm@63489
  1211
wenzelm@63489
  1212
lemma Gcd_image_normalize [simp]: "Gcd (normalize ` A) = Gcd A"
haftmann@62345
  1213
proof -
haftmann@62345
  1214
  have "Gcd (normalize ` A) dvd a" if "a \<in> A" for a
haftmann@62345
  1215
  proof -
wenzelm@63489
  1216
    from that obtain B where "A = insert a B"
wenzelm@63489
  1217
      by blast
haftmann@62350
  1218
    moreover have "gcd (normalize a) (Gcd (normalize ` B)) dvd normalize a"
haftmann@62345
  1219
      by (rule gcd_dvd1)
haftmann@62345
  1220
    ultimately show "Gcd (normalize ` A) dvd a"
haftmann@62345
  1221
      by simp
haftmann@62345
  1222
  qed
haftmann@62345
  1223
  then have "Gcd (normalize ` A) dvd Gcd A" and "Gcd A dvd Gcd (normalize ` A)"
haftmann@62345
  1224
    by (auto intro!: Gcd_greatest intro: Gcd_dvd)
haftmann@62345
  1225
  then show ?thesis
haftmann@62345
  1226
    by (auto intro: associated_eqI)
haftmann@62345
  1227
qed
haftmann@62345
  1228
haftmann@62346
  1229
lemma Gcd_eqI:
haftmann@62346
  1230
  assumes "normalize a = a"
haftmann@62346
  1231
  assumes "\<And>b. b \<in> A \<Longrightarrow> a dvd b"
haftmann@62346
  1232
    and "\<And>c. (\<And>b. b \<in> A \<Longrightarrow> c dvd b) \<Longrightarrow> c dvd a"
haftmann@62346
  1233
  shows "Gcd A = a"
haftmann@62346
  1234
  using assms by (blast intro: associated_eqI Gcd_greatest Gcd_dvd normalize_Gcd)
haftmann@62346
  1235
wenzelm@63489
  1236
lemma dvd_GcdD: "x dvd Gcd A \<Longrightarrow> y \<in> A \<Longrightarrow> x dvd y"
wenzelm@63489
  1237
  using Gcd_dvd dvd_trans by blast
wenzelm@63489
  1238
wenzelm@63489
  1239
lemma dvd_Gcd_iff: "x dvd Gcd A \<longleftrightarrow> (\<forall>y\<in>A. x dvd y)"
eberlm@63359
  1240
  by (blast dest: dvd_GcdD intro: Gcd_greatest)
eberlm@63359
  1241
eberlm@63359
  1242
lemma Gcd_mult: "Gcd (op * c ` A) = normalize c * Gcd A"
eberlm@63359
  1243
proof (cases "c = 0")
wenzelm@63489
  1244
  case True
wenzelm@63489
  1245
  then show ?thesis by auto
wenzelm@63489
  1246
next
eberlm@63359
  1247
  case [simp]: False
eberlm@63359
  1248
  have "Gcd (op * c ` A) div c dvd Gcd A"
eberlm@63359
  1249
    by (intro Gcd_greatest, subst div_dvd_iff_mult)
eberlm@63359
  1250
       (auto intro!: Gcd_greatest Gcd_dvd simp: mult.commute[of _ c])
wenzelm@63489
  1251
  then have "Gcd (op * c ` A) dvd c * Gcd A"
eberlm@63359
  1252
    by (subst (asm) div_dvd_iff_mult) (auto intro: Gcd_greatest simp: mult_ac)
eberlm@63359
  1253
  also have "c * Gcd A = (normalize c * Gcd A) * unit_factor c"
eberlm@63359
  1254
    by (subst unit_factor_mult_normalize [symmetric]) (simp only: mult_ac)
eberlm@63359
  1255
  also have "Gcd (op * c ` A) dvd \<dots> \<longleftrightarrow> Gcd (op * c ` A) dvd normalize c * Gcd A"
eberlm@63359
  1256
    by (simp add: dvd_mult_unit_iff)
eberlm@63359
  1257
  finally have "Gcd (op * c ` A) dvd normalize c * Gcd A" .
eberlm@63359
  1258
  moreover have "normalize c * Gcd A dvd Gcd (op * c ` A)"
eberlm@63359
  1259
    by (intro Gcd_greatest) (auto intro: mult_dvd_mono Gcd_dvd)
eberlm@63359
  1260
  ultimately have "normalize (Gcd (op * c ` A)) = normalize (normalize c * Gcd A)"
eberlm@63359
  1261
    by (rule associatedI)
wenzelm@63489
  1262
  then show ?thesis
wenzelm@63489
  1263
    by (simp add: normalize_mult)
wenzelm@63489
  1264
qed
eberlm@63359
  1265
haftmann@62346
  1266
lemma Lcm_eqI:
haftmann@62346
  1267
  assumes "normalize a = a"
wenzelm@63489
  1268
    and "\<And>b. b \<in> A \<Longrightarrow> b dvd a"
haftmann@62346
  1269
    and "\<And>c. (\<And>b. b \<in> A \<Longrightarrow> b dvd c) \<Longrightarrow> a dvd c"
haftmann@62346
  1270
  shows "Lcm A = a"
haftmann@62346
  1271
  using assms by (blast intro: associated_eqI Lcm_least dvd_Lcm normalize_Lcm)
haftmann@62346
  1272
wenzelm@63489
  1273
lemma Lcm_dvdD: "Lcm A dvd x \<Longrightarrow> y \<in> A \<Longrightarrow> y dvd x"
wenzelm@63489
  1274
  using dvd_Lcm dvd_trans by blast
wenzelm@63489
  1275
wenzelm@63489
  1276
lemma Lcm_dvd_iff: "Lcm A dvd x \<longleftrightarrow> (\<forall>y\<in>A. y dvd x)"
eberlm@63359
  1277
  by (blast dest: Lcm_dvdD intro: Lcm_least)
eberlm@63359
  1278
wenzelm@63489
  1279
lemma Lcm_mult:
eberlm@63359
  1280
  assumes "A \<noteq> {}"
wenzelm@63489
  1281
  shows "Lcm (op * c ` A) = normalize c * Lcm A"
eberlm@63359
  1282
proof (cases "c = 0")
eberlm@63359
  1283
  case True
wenzelm@63489
  1284
  with assms have "op * c ` A = {0}"
wenzelm@63489
  1285
    by auto
wenzelm@63489
  1286
  with True show ?thesis by auto
eberlm@63359
  1287
next
eberlm@63359
  1288
  case [simp]: False
wenzelm@63489
  1289
  from assms obtain x where x: "x \<in> A"
wenzelm@63489
  1290
    by blast
wenzelm@63489
  1291
  have "c dvd c * x"
wenzelm@63489
  1292
    by simp
wenzelm@63489
  1293
  also from x have "c * x dvd Lcm (op * c ` A)"
wenzelm@63489
  1294
    by (intro dvd_Lcm) auto
eberlm@63359
  1295
  finally have dvd: "c dvd Lcm (op * c ` A)" .
eberlm@63359
  1296
eberlm@63359
  1297
  have "Lcm A dvd Lcm (op * c ` A) div c"
eberlm@63359
  1298
    by (intro Lcm_least dvd_mult_imp_div)
wenzelm@63489
  1299
      (auto intro!: Lcm_least dvd_Lcm simp: mult.commute[of _ c])
wenzelm@63489
  1300
  then have "c * Lcm A dvd Lcm (op * c ` A)"
eberlm@63359
  1301
    by (subst (asm) dvd_div_iff_mult) (auto intro!: Lcm_least simp: mult_ac dvd)
eberlm@63359
  1302
  also have "c * Lcm A = (normalize c * Lcm A) * unit_factor c"
eberlm@63359
  1303
    by (subst unit_factor_mult_normalize [symmetric]) (simp only: mult_ac)
eberlm@63359
  1304
  also have "\<dots> dvd Lcm (op * c ` A) \<longleftrightarrow> normalize c * Lcm A dvd Lcm (op * c ` A)"
eberlm@63359
  1305
    by (simp add: mult_unit_dvd_iff)
eberlm@63359
  1306
  finally have "normalize c * Lcm A dvd Lcm (op * c ` A)" .
eberlm@63359
  1307
  moreover have "Lcm (op * c ` A) dvd normalize c * Lcm A"
eberlm@63359
  1308
    by (intro Lcm_least) (auto intro: mult_dvd_mono dvd_Lcm)
eberlm@63359
  1309
  ultimately have "normalize (normalize c * Lcm A) = normalize (Lcm (op * c ` A))"
eberlm@63359
  1310
    by (rule associatedI)
wenzelm@63489
  1311
  then show ?thesis
wenzelm@63489
  1312
    by (simp add: normalize_mult)
eberlm@62429
  1313
qed
eberlm@62429
  1314
wenzelm@63489
  1315
lemma Lcm_no_units: "Lcm A = Lcm (A - {a. is_unit a})"
wenzelm@63489
  1316
proof -
wenzelm@63489
  1317
  have "(A - {a. is_unit a}) \<union> {a\<in>A. is_unit a} = A"
wenzelm@63489
  1318
    by blast
wenzelm@63489
  1319
  then have "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\<in>A. is_unit a})"
wenzelm@63489
  1320
    by (simp add: Lcm_Un [symmetric])
wenzelm@63489
  1321
  also have "Lcm {a\<in>A. is_unit a} = 1"
wenzelm@63489
  1322
    by simp
wenzelm@63489
  1323
  finally show ?thesis
wenzelm@63489
  1324
    by simp
wenzelm@63489
  1325
qed
wenzelm@63489
  1326
wenzelm@63489
  1327
lemma Lcm_0_iff': "Lcm A = 0 \<longleftrightarrow> (\<nexists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))"
eberlm@62429
  1328
  by (metis Lcm_least dvd_0_left dvd_Lcm)
eberlm@62429
  1329
wenzelm@63489
  1330
lemma Lcm_no_multiple: "(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not> a dvd m)) \<Longrightarrow> Lcm A = 0"
eberlm@62429
  1331
  by (auto simp: Lcm_0_iff')
eberlm@62429
  1332
wenzelm@63489
  1333
lemma Lcm_singleton [simp]: "Lcm {a} = normalize a"
eberlm@62429
  1334
  by simp
eberlm@62429
  1335
wenzelm@63489
  1336
lemma Lcm_2 [simp]: "Lcm {a, b} = lcm a b"
eberlm@62429
  1337
  by simp
eberlm@62429
  1338
eberlm@62429
  1339
lemma Lcm_coprime:
wenzelm@63489
  1340
  assumes "finite A"
wenzelm@63489
  1341
    and "A \<noteq> {}"
wenzelm@63489
  1342
    and "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1"
eberlm@62429
  1343
  shows "Lcm A = normalize (\<Prod>A)"
wenzelm@63489
  1344
  using assms
wenzelm@63489
  1345
proof (induct rule: finite_ne_induct)
wenzelm@63489
  1346
  case singleton
wenzelm@63489
  1347
  then show ?case by simp
wenzelm@63489
  1348
next
eberlm@62429
  1349
  case (insert a A)
wenzelm@63489
  1350
  have "Lcm (insert a A) = lcm a (Lcm A)"
wenzelm@63489
  1351
    by simp
wenzelm@63489
  1352
  also from insert have "Lcm A = normalize (\<Prod>A)"
wenzelm@63489
  1353
    by blast
wenzelm@63489
  1354
  also have "lcm a \<dots> = lcm a (\<Prod>A)"
wenzelm@63489
  1355
    by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)
wenzelm@63489
  1356
  also from insert have "gcd a (\<Prod>A) = 1"
wenzelm@63489
  1357
    by (subst gcd.commute, intro setprod_coprime) auto
eberlm@62429
  1358
  with insert have "lcm a (\<Prod>A) = normalize (\<Prod>(insert a A))"
eberlm@62429
  1359
    by (simp add: lcm_coprime)
eberlm@62429
  1360
  finally show ?case .
wenzelm@63489
  1361
qed
wenzelm@63489
  1362
eberlm@62429
  1363
lemma Lcm_coprime':
wenzelm@63489
  1364
  "card A \<noteq> 0 \<Longrightarrow>
wenzelm@63489
  1365
    (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1) \<Longrightarrow>
wenzelm@63489
  1366
    Lcm A = normalize (\<Prod>A)"
eberlm@62429
  1367
  by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)
eberlm@62429
  1368
wenzelm@63489
  1369
lemma Gcd_1: "1 \<in> A \<Longrightarrow> Gcd A = 1"
eberlm@62429
  1370
  by (auto intro!: Gcd_eq_1_I)
eberlm@62429
  1371
eberlm@62429
  1372
lemma Gcd_singleton [simp]: "Gcd {a} = normalize a"
eberlm@62429
  1373
  by simp
eberlm@62429
  1374
wenzelm@63489
  1375
lemma Gcd_2 [simp]: "Gcd {a, b} = gcd a b"
eberlm@62429
  1376
  by simp
eberlm@62429
  1377
eberlm@62429
  1378
wenzelm@63489
  1379
definition pairwise_coprime
wenzelm@63489
  1380
  where "pairwise_coprime A = (\<forall>x y. x \<in> A \<and> y \<in> A \<and> x \<noteq> y \<longrightarrow> coprime x y)"
eberlm@62429
  1381
eberlm@62429
  1382
lemma pairwise_coprimeI [intro?]:
eberlm@62429
  1383
  "(\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> coprime x y) \<Longrightarrow> pairwise_coprime A"
eberlm@62429
  1384
  by (simp add: pairwise_coprime_def)
eberlm@62429
  1385
eberlm@62429
  1386
lemma pairwise_coprimeD:
eberlm@62429
  1387
  "pairwise_coprime A \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> coprime x y"
eberlm@62429
  1388
  by (simp add: pairwise_coprime_def)
eberlm@62429
  1389
eberlm@62429
  1390
lemma pairwise_coprime_subset: "pairwise_coprime A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> pairwise_coprime B"
eberlm@62429
  1391
  by (force simp: pairwise_coprime_def)
eberlm@62429
  1392
haftmann@62350
  1393
end
haftmann@62345
  1394
wenzelm@63489
  1395
haftmann@62345
  1396
subsection \<open>GCD and LCM on @{typ nat} and @{typ int}\<close>
haftmann@59008
  1397
huffman@31706
  1398
instantiation nat :: gcd
huffman@31706
  1399
begin
wenzelm@21256
  1400
haftmann@62345
  1401
fun gcd_nat  :: "nat \<Rightarrow> nat \<Rightarrow> nat"
wenzelm@63489
  1402
  where "gcd_nat x y = (if y = 0 then x else gcd y (x mod y))"
huffman@31706
  1403
haftmann@62345
  1404
definition lcm_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
wenzelm@63489
  1405
  where "lcm_nat x y = x * y div (gcd x y)"
wenzelm@63489
  1406
wenzelm@63489
  1407
instance ..
huffman@31706
  1408
huffman@31706
  1409
end
huffman@31706
  1410
huffman@31706
  1411
instantiation int :: gcd
huffman@31706
  1412
begin
wenzelm@21256
  1413
haftmann@62345
  1414
definition gcd_int  :: "int \<Rightarrow> int \<Rightarrow> int"
haftmann@62345
  1415
  where "gcd_int x y = int (gcd (nat \<bar>x\<bar>) (nat \<bar>y\<bar>))"
haftmann@23687
  1416
haftmann@62345
  1417
definition lcm_int :: "int \<Rightarrow> int \<Rightarrow> int"
haftmann@62345
  1418
  where "lcm_int x y = int (lcm (nat \<bar>x\<bar>) (nat \<bar>y\<bar>))"
haftmann@23687
  1419
wenzelm@61944
  1420
instance ..
huffman@31706
  1421
huffman@31706
  1422
end
haftmann@23687
  1423
haftmann@62345
  1424
text \<open>Transfer setup\<close>
huffman@31706
  1425
huffman@31706
  1426
lemma transfer_nat_int_gcd:
wenzelm@63489
  1427
  "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> gcd (nat x) (nat y) = nat (gcd x y)"
wenzelm@63489
  1428
  "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> lcm (nat x) (nat y) = nat (lcm x y)"
wenzelm@63489
  1429
  for x y :: int
wenzelm@63489
  1430
  unfolding gcd_int_def lcm_int_def by auto
haftmann@23687
  1431
huffman@31706
  1432
lemma transfer_nat_int_gcd_closures:
wenzelm@63489
  1433
  "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> gcd x y \<ge> 0"
wenzelm@63489
  1434
  "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> lcm x y \<ge> 0"
wenzelm@63489
  1435
  for x y :: int
huffman@31706
  1436
  by (auto simp add: gcd_int_def lcm_int_def)
huffman@31706
  1437
wenzelm@63489
  1438
declare transfer_morphism_nat_int
wenzelm@63489
  1439
  [transfer add return: transfer_nat_int_gcd transfer_nat_int_gcd_closures]
huffman@31706
  1440
huffman@31706
  1441
lemma transfer_int_nat_gcd:
huffman@31706
  1442
  "gcd (int x) (int y) = int (gcd x y)"
huffman@31706
  1443
  "lcm (int x) (int y) = int (lcm x y)"
wenzelm@63489
  1444
  by (auto simp: gcd_int_def lcm_int_def)
huffman@31706
  1445
huffman@31706
  1446
lemma transfer_int_nat_gcd_closures:
huffman@31706
  1447
  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> gcd x y >= 0"
huffman@31706
  1448
  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> lcm x y >= 0"
wenzelm@63489
  1449
  by (auto simp: gcd_int_def lcm_int_def)
wenzelm@63489
  1450
wenzelm@63489
  1451
declare transfer_morphism_int_nat
wenzelm@63489
  1452
  [transfer add return: transfer_int_nat_gcd transfer_int_nat_gcd_closures]
huffman@31706
  1453
nipkow@31952
  1454
lemma gcd_nat_induct:
haftmann@23687
  1455
  fixes m n :: nat
haftmann@23687
  1456
  assumes "\<And>m. P m 0"
haftmann@23687
  1457
    and "\<And>m n. 0 < n \<Longrightarrow> P n (m mod n) \<Longrightarrow> P m n"
haftmann@23687
  1458
  shows "P m n"
huffman@31706
  1459
  apply (rule gcd_nat.induct)
huffman@31706
  1460
  apply (case_tac "y = 0")
wenzelm@63489
  1461
  using assms
wenzelm@63489
  1462
   apply simp_all
wenzelm@63489
  1463
  done
wenzelm@63489
  1464
wenzelm@63489
  1465
wenzelm@63489
  1466
text \<open>Specific to \<open>int\<close>.\<close>
wenzelm@63489
  1467
wenzelm@63489
  1468
lemma gcd_eq_int_iff: "gcd k l = int n \<longleftrightarrow> gcd (nat \<bar>k\<bar>) (nat \<bar>l\<bar>) = n"
haftmann@62346
  1469
  by (simp add: gcd_int_def)
haftmann@62346
  1470
wenzelm@63489
  1471
lemma lcm_eq_int_iff: "lcm k l = int n \<longleftrightarrow> lcm (nat \<bar>k\<bar>) (nat \<bar>l\<bar>) = n"
haftmann@62346
  1472
  by (simp add: lcm_int_def)
haftmann@62346
  1473
wenzelm@63489
  1474
lemma gcd_neg1_int [simp]: "gcd (- x) y = gcd x y"
wenzelm@63489
  1475
  for x y :: int
huffman@31706
  1476
  by (simp add: gcd_int_def)
huffman@31706
  1477
wenzelm@63489
  1478
lemma gcd_neg2_int [simp]: "gcd x (- y) = gcd x y"
wenzelm@63489
  1479
  for x y :: int
wenzelm@63489
  1480
  by (simp add: gcd_int_def)
wenzelm@63489
  1481
wenzelm@63489
  1482
lemma abs_gcd_int [simp]: "\<bar>gcd x y\<bar> = gcd x y"
wenzelm@63489
  1483
  for x y :: int
huffman@31706
  1484
  by (simp add: gcd_int_def)
huffman@31706
  1485
wenzelm@63489
  1486
lemma gcd_abs_int: "gcd x y = gcd \<bar>x\<bar> \<bar>y\<bar>"
wenzelm@63489
  1487
  for x y :: int
wenzelm@63489
  1488
  by (simp add: gcd_int_def)
wenzelm@63489
  1489
wenzelm@63489
  1490
lemma gcd_abs1_int [simp]: "gcd \<bar>x\<bar> y = gcd x y"
wenzelm@63489
  1491
  for x y :: int
wenzelm@63489
  1492
  by (metis abs_idempotent gcd_abs_int)
wenzelm@63489
  1493
wenzelm@63489
  1494
lemma gcd_abs2_int [simp]: "gcd x \<bar>y\<bar> = gcd x y"
wenzelm@63489
  1495
  for x y :: int
wenzelm@63489
  1496
  by (metis abs_idempotent gcd_abs_int)
huffman@31706
  1497
nipkow@31952
  1498
lemma gcd_cases_int:
wenzelm@63489
  1499
  fixes x y :: int
wenzelm@63489
  1500
  assumes "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> P (gcd x y)"
wenzelm@63489
  1501
    and "x \<ge> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> P (gcd x (- y))"
wenzelm@63489
  1502
    and "x \<le> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> P (gcd (- x) y)"
wenzelm@63489
  1503
    and "x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> P (gcd (- x) (- y))"
huffman@31706
  1504
  shows "P (gcd x y)"
wenzelm@63489
  1505
  using assms by auto arith
wenzelm@21256
  1506
nipkow@31952
  1507
lemma gcd_ge_0_int [simp]: "gcd (x::int) y >= 0"
wenzelm@63489
  1508
  for x y :: int
huffman@31706
  1509
  by (simp add: gcd_int_def)
huffman@31706
  1510
wenzelm@63489
  1511
lemma lcm_neg1_int: "lcm (- x) y = lcm x y"
wenzelm@63489
  1512
  for x y :: int
huffman@31706
  1513
  by (simp add: lcm_int_def)
huffman@31706
  1514
wenzelm@63489
  1515
lemma lcm_neg2_int: "lcm x (- y) = lcm x y"
wenzelm@63489
  1516
  for x y :: int
huffman@31706
  1517
  by (simp add: lcm_int_def)
huffman@31706
  1518
wenzelm@63489
  1519
lemma lcm_abs_int: "lcm x y = lcm \<bar>x\<bar> \<bar>y\<bar>"
wenzelm@63489
  1520
  for x y :: int
huffman@31706
  1521
  by (simp add: lcm_int_def)
wenzelm@21256
  1522
wenzelm@63489
  1523
lemma abs_lcm_int [simp]: "\<bar>lcm i j\<bar> = lcm i j"
wenzelm@63489
  1524
  for i j :: int
wenzelm@61944
  1525
  by (simp add:lcm_int_def)
nipkow@31814
  1526
wenzelm@63489
  1527
lemma lcm_abs1_int [simp]: "lcm \<bar>x\<bar> y = lcm x y"
wenzelm@63489
  1528
  for x y :: int
wenzelm@61944
  1529
  by (metis abs_idempotent lcm_int_def)
nipkow@31814
  1530
wenzelm@63489
  1531
lemma lcm_abs2_int [simp]: "lcm x \<bar>y\<bar> = lcm x y"
wenzelm@63489
  1532
  for x y :: int
wenzelm@61944
  1533
  by (metis abs_idempotent lcm_int_def)
nipkow@31814
  1534
nipkow@31952
  1535
lemma lcm_cases_int:
wenzelm@63489
  1536
  fixes x y :: int
wenzelm@63489
  1537
  assumes "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> P (lcm x y)"
wenzelm@63489
  1538
    and "x \<ge> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> P (lcm x (- y))"
wenzelm@63489
  1539
    and "x \<le> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> P (lcm (- x) y)"
wenzelm@63489
  1540
    and "x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> P (lcm (- x) (- y))"
huffman@31706
  1541
  shows "P (lcm x y)"
wenzelm@41550
  1542
  using assms by (auto simp add: lcm_neg1_int lcm_neg2_int) arith
huffman@31706
  1543
wenzelm@63489
  1544
lemma lcm_ge_0_int [simp]: "lcm x y \<ge> 0"
wenzelm@63489
  1545
  for x y :: int
huffman@31706
  1546
  by (simp add: lcm_int_def)
huffman@31706
  1547
wenzelm@63489
  1548
lemma gcd_0_nat: "gcd x 0 = x"
wenzelm@63489
  1549
  for x :: nat
haftmann@23687
  1550
  by simp
haftmann@23687
  1551
wenzelm@63489
  1552
lemma gcd_0_int [simp]: "gcd x 0 = \<bar>x\<bar>"
wenzelm@63489
  1553
  for x :: int
wenzelm@63489
  1554
  by (auto simp: gcd_int_def)
wenzelm@63489
  1555
wenzelm@63489
  1556
lemma gcd_0_left_nat: "gcd 0 x = x"
wenzelm@63489
  1557
  for x :: nat
haftmann@23687
  1558
  by simp
haftmann@23687
  1559
wenzelm@63489
  1560
lemma gcd_0_left_int [simp]: "gcd 0 x = \<bar>x\<bar>"
wenzelm@63489
  1561
  for x :: int
wenzelm@63489
  1562
  by (auto simp:gcd_int_def)
wenzelm@63489
  1563
wenzelm@63489
  1564
lemma gcd_red_nat: "gcd x y = gcd y (x mod y)"
wenzelm@63489
  1565
  for x y :: nat
wenzelm@63489
  1566
  by (cases "y = 0") auto
wenzelm@63489
  1567
wenzelm@63489
  1568
wenzelm@63489
  1569
text \<open>Weaker, but useful for the simplifier.\<close>
wenzelm@63489
  1570
wenzelm@63489
  1571
lemma gcd_non_0_nat: "y \<noteq> 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
wenzelm@63489
  1572
  for x y :: nat
wenzelm@21263
  1573
  by simp
wenzelm@21256
  1574
wenzelm@63489
  1575
lemma gcd_1_nat [simp]: "gcd m 1 = 1"
wenzelm@63489
  1576
  for m :: nat
haftmann@60690
  1577
  by simp
huffman@31706
  1578
wenzelm@63489
  1579
lemma gcd_Suc_0 [simp]: "gcd m (Suc 0) = Suc 0"
wenzelm@63489
  1580
  for m :: nat
wenzelm@63489
  1581
  by simp
wenzelm@63489
  1582
wenzelm@63489
  1583
lemma gcd_1_int [simp]: "gcd m 1 = 1"
wenzelm@63489
  1584
  for m :: int
huffman@31706
  1585
  by (simp add: gcd_int_def)
huffman@30082
  1586
wenzelm@63489
  1587
lemma gcd_idem_nat: "gcd x x = x"
wenzelm@63489
  1588
  for x :: nat
wenzelm@63489
  1589
  by simp
wenzelm@63489
  1590
wenzelm@63489
  1591
lemma gcd_idem_int: "gcd x x = \<bar>x\<bar>"
wenzelm@63489
  1592
  for x :: int
wenzelm@63489
  1593
  by (auto simp add: gcd_int_def)
huffman@31706
  1594
huffman@31706
  1595
declare gcd_nat.simps [simp del]
wenzelm@21256
  1596
wenzelm@60758
  1597
text \<open>
wenzelm@63489
  1598
  \<^medskip> @{term "gcd m n"} divides \<open>m\<close> and \<open>n\<close>.
wenzelm@63489
  1599
  The conjunctions don't seem provable separately.
wenzelm@60758
  1600
\<close>
wenzelm@21256
  1601
haftmann@59008
  1602
instance nat :: semiring_gcd
haftmann@59008
  1603
proof
haftmann@59008
  1604
  fix m n :: nat
haftmann@59008
  1605
  show "gcd m n dvd m" and "gcd m n dvd n"
haftmann@59008
  1606
  proof (induct m n rule: gcd_nat_induct)
haftmann@59008
  1607
    fix m n :: nat
wenzelm@63489
  1608
    assume "gcd n (m mod n) dvd m mod n"
wenzelm@63489
  1609
      and "gcd n (m mod n) dvd n"
haftmann@59008
  1610
    then have "gcd n (m mod n) dvd m"
haftmann@59008
  1611
      by (rule dvd_mod_imp_dvd)
haftmann@59008
  1612
    moreover assume "0 < n"
haftmann@59008
  1613
    ultimately show "gcd m n dvd m"
haftmann@59008
  1614
      by (simp add: gcd_non_0_nat)
haftmann@59008
  1615
  qed (simp_all add: gcd_0_nat gcd_non_0_nat)
haftmann@59008
  1616
next
haftmann@59008
  1617
  fix m n k :: nat
haftmann@59008
  1618
  assume "k dvd m" and "k dvd n"
haftmann@59008
  1619
  then show "k dvd gcd m n"
haftmann@59008
  1620
    by (induct m n rule: gcd_nat_induct) (simp_all add: gcd_non_0_nat dvd_mod gcd_0_nat)
haftmann@60686
  1621
qed (simp_all add: lcm_nat_def)
lp15@59667
  1622
haftmann@59008
  1623
instance int :: ring_gcd
haftmann@60686
  1624
  by standard
wenzelm@63489
  1625
    (simp_all add: dvd_int_unfold_dvd_nat gcd_int_def lcm_int_def
wenzelm@63489
  1626
      zdiv_int nat_abs_mult_distrib [symmetric] lcm_gcd gcd_greatest)
wenzelm@63489
  1627
wenzelm@63489
  1628
lemma gcd_le1_nat [simp]: "a \<noteq> 0 \<Longrightarrow> gcd a b \<le> a"
wenzelm@63489
  1629
  for a b :: nat
wenzelm@63489
  1630
  by (rule dvd_imp_le) auto
wenzelm@63489
  1631
wenzelm@63489
  1632
lemma gcd_le2_nat [simp]: "b \<noteq> 0 \<Longrightarrow> gcd a b \<le> b"
wenzelm@63489
  1633
  for a b :: nat
wenzelm@63489
  1634
  by (rule dvd_imp_le) auto
wenzelm@63489
  1635
wenzelm@63489
  1636
lemma gcd_le1_int [simp]: "a > 0 \<Longrightarrow> gcd a b \<le> a"
wenzelm@63489
  1637
  for a b :: int
wenzelm@63489
  1638
  by (rule zdvd_imp_le) auto
wenzelm@63489
  1639
wenzelm@63489
  1640
lemma gcd_le2_int [simp]: "b > 0 \<Longrightarrow> gcd a b \<le> b"
wenzelm@63489
  1641
  for a b :: int
wenzelm@63489
  1642
  by (rule zdvd_imp_le) auto
wenzelm@63489
  1643
wenzelm@63489
  1644
lemma gcd_pos_nat [simp]: "gcd m n > 0 \<longleftrightarrow> m \<noteq> 0 \<or> n \<noteq> 0"
wenzelm@63489
  1645
  for m n :: nat
wenzelm@63489
  1646
  using gcd_eq_0_iff [of m n] by arith
wenzelm@63489
  1647
wenzelm@63489
  1648
lemma gcd_pos_int [simp]: "gcd m n > 0 \<longleftrightarrow> m \<noteq> 0 \<or> n \<noteq> 0"
wenzelm@63489
  1649
  for m n :: int
wenzelm@63489
  1650
  using gcd_eq_0_iff [of m n] gcd_ge_0_int [of m n] by arith
wenzelm@63489
  1651
wenzelm@63489
  1652
lemma gcd_unique_nat: "d dvd a \<and> d dvd b \<and> (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
wenzelm@63489
  1653
  for d a :: nat
huffman@31706
  1654
  apply auto
nipkow@33657
  1655
  apply (rule dvd_antisym)
wenzelm@63489
  1656
   apply (erule (1) gcd_greatest)
huffman@31706
  1657
  apply auto
wenzelm@63489
  1658
  done
wenzelm@63489
  1659
wenzelm@63489
  1660
lemma gcd_unique_int:
wenzelm@63489
  1661
  "d \<ge> 0 \<and> d dvd a \<and> d dvd b \<and> (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
wenzelm@63489
  1662
  for d a :: int
wenzelm@63489
  1663
  apply (cases "d = 0")
wenzelm@63489
  1664
   apply simp
wenzelm@63489
  1665
  apply (rule iffI)
wenzelm@63489
  1666
   apply (rule zdvd_antisym_nonneg)
wenzelm@63489
  1667
      apply (auto intro: gcd_greatest)
wenzelm@63489
  1668
  done
huffman@30082
  1669
haftmann@61913
  1670
interpretation gcd_nat:
haftmann@62344
  1671
  semilattice_neutr_order gcd "0::nat" Rings.dvd "\<lambda>m n. m dvd n \<and> m \<noteq> n"
haftmann@62344
  1672
  by standard (auto simp add: gcd_unique_nat [symmetric] intro: dvd_antisym dvd_trans)
nipkow@31798
  1673
wenzelm@63489
  1674
lemma gcd_proj1_if_dvd_int [simp]: "x dvd y \<Longrightarrow> gcd x y = \<bar>x\<bar>"
wenzelm@63489
  1675
  for x y :: int
haftmann@54867
  1676
  by (metis abs_dvd_iff gcd_0_left_int gcd_abs_int gcd_unique_int)
nipkow@31798
  1677
wenzelm@63489
  1678
lemma gcd_proj2_if_dvd_int [simp]: "y dvd x \<Longrightarrow> gcd x y = \<bar>y\<bar>"
wenzelm@63489
  1679
  for x y :: int
haftmann@62344
  1680
  by (metis gcd_proj1_if_dvd_int gcd.commute)
nipkow@31798
  1681
wenzelm@63489
  1682
wenzelm@63489
  1683
text \<open>\<^medskip> Multiplication laws.\<close>
wenzelm@63489
  1684
wenzelm@63489
  1685
lemma gcd_mult_distrib_nat: "k * gcd m n = gcd (k * m) (k * n)"
wenzelm@63489
  1686
  for k m n :: nat
wenzelm@63489
  1687
  \<comment> \<open>@{cite \<open>page 27\<close> davenport92}\<close>
nipkow@31952
  1688
  apply (induct m n rule: gcd_nat_induct)
wenzelm@63489
  1689
   apply simp
wenzelm@63489
  1690
  apply (cases "k = 0")
wenzelm@63489
  1691
   apply (simp_all add: gcd_non_0_nat)
wenzelm@63489
  1692
  done
wenzelm@63489
  1693
wenzelm@63489
  1694
lemma gcd_mult_distrib_int: "\<bar>k\<bar> * gcd m n = gcd (k * m) (k * n)"
wenzelm@63489
  1695
  for k m n :: int
nipkow@31952
  1696
  apply (subst (1 2) gcd_abs_int)
nipkow@31813
  1697
  apply (subst (1 2) abs_mult)
nipkow@31952
  1698
  apply (rule gcd_mult_distrib_nat [transferred])
wenzelm@63489
  1699
    apply auto
wenzelm@63489
  1700
  done
wenzelm@21256
  1701
haftmann@62344
  1702
lemma coprime_crossproduct_nat:
haftmann@62344
  1703
  fixes a b c d :: nat
haftmann@62344
  1704
  assumes "coprime a d" and "coprime b c"
haftmann@62344
  1705
  shows "a * c = b * d \<longleftrightarrow> a = b \<and> c = d"
haftmann@62344
  1706
  using assms coprime_crossproduct [of a d b c] by simp
haftmann@62344
  1707
haftmann@35368
  1708
lemma coprime_crossproduct_int:
haftmann@35368
  1709
  fixes a b c d :: int
haftmann@35368
  1710
  assumes "coprime a d" and "coprime b c"
haftmann@35368
  1711
  shows "\<bar>a\<bar> * \<bar>c\<bar> = \<bar>b\<bar> * \<bar>d\<bar> \<longleftrightarrow> \<bar>a\<bar> = \<bar>b\<bar> \<and> \<bar>c\<bar> = \<bar>d\<bar>"
haftmann@62344
  1712
  using assms coprime_crossproduct [of a d b c] by simp
haftmann@35368
  1713
wenzelm@63489
  1714
wenzelm@63489
  1715
text \<open>\medskip Addition laws.\<close>
wenzelm@63489
  1716
wenzelm@63489
  1717
(* TODO: add the other variations? *)
wenzelm@63489
  1718
wenzelm@63489
  1719
lemma gcd_diff1_nat: "m \<ge> n \<Longrightarrow> gcd (m - n) n = gcd m n"
wenzelm@63489
  1720
  for m n :: nat
eberlm@62429
  1721
  by (subst gcd_add1 [symmetric]) auto
huffman@31706
  1722
wenzelm@63489
  1723
lemma gcd_diff2_nat: "n \<ge> m \<Longrightarrow> gcd (n - m) n = gcd m n"
wenzelm@63489
  1724
  for m n :: nat
haftmann@62344
  1725
  apply (subst gcd.commute)
nipkow@31952
  1726
  apply (subst gcd_diff1_nat [symmetric])
wenzelm@63489
  1727
   apply auto
haftmann@62344
  1728
  apply (subst gcd.commute)
nipkow@31952
  1729
  apply (subst gcd_diff1_nat)
wenzelm@63489
  1730
   apply assumption
haftmann@62344
  1731
  apply (rule gcd.commute)
haftmann@62344
  1732
  done
huffman@31706
  1733
wenzelm@63489
  1734
lemma gcd_non_0_int: "y > 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
wenzelm@63489
  1735
  for x y :: int
huffman@31706
  1736
  apply (frule_tac b = y and a = x in pos_mod_sign)
huffman@31706
  1737
  apply (simp del: pos_mod_sign add: gcd_int_def abs_if nat_mod_distrib)
wenzelm@63489
  1738
  apply (auto simp add: gcd_non_0_nat nat_mod_distrib [symmetric] zmod_zminus1_eq_if)
huffman@31706
  1739
  apply (frule_tac a = x in pos_mod_bound)
haftmann@62344
  1740
  apply (subst (1 2) gcd.commute)
wenzelm@63489
  1741
  apply (simp del: pos_mod_bound add: nat_diff_distrib gcd_diff2_nat nat_le_eq_zle)
haftmann@62344
  1742
  done
wenzelm@21256
  1743
wenzelm@63489
  1744
lemma gcd_red_int: "gcd x y = gcd y (x mod y)"
wenzelm@63489
  1745
  for x y :: int
wenzelm@63489
  1746
  apply (cases "y = 0")
wenzelm@63489
  1747
   apply force
wenzelm@63489
  1748
  apply (cases "y > 0")
wenzelm@63489
  1749
   apply (subst gcd_non_0_int, auto)
wenzelm@63489
  1750
  apply (insert gcd_non_0_int [of "- y" "- x"])
huffman@35216
  1751
  apply auto
wenzelm@63489
  1752
  done
wenzelm@63489
  1753
wenzelm@63489
  1754
(* TODO: differences, and all variations of addition rules
huffman@31706
  1755
    as simplification rules for nat and int *)
huffman@31706
  1756
wenzelm@63489
  1757
(* TODO: add the three variations of these, and for ints? *)
wenzelm@63489
  1758
wenzelm@63489
  1759
lemma finite_divisors_nat [simp]: (* FIXME move *)
haftmann@62353
  1760
  fixes m :: nat
wenzelm@63489
  1761
  assumes "m > 0"
haftmann@62353
  1762
  shows "finite {d. d dvd m}"
nipkow@31734
  1763
proof-
haftmann@62353
  1764
  from assms have "{d. d dvd m} \<subseteq> {d. d \<le> m}"
haftmann@62353
  1765
    by (auto dest: dvd_imp_le)
haftmann@62353
  1766
  then show ?thesis
haftmann@62353
  1767
    using finite_Collect_le_nat by (rule finite_subset)
nipkow@31734
  1768
qed
nipkow@31734
  1769
haftmann@62353
  1770
lemma finite_divisors_int [simp]:
haftmann@62353
  1771
  fixes i :: int
haftmann@62353
  1772
  assumes "i \<noteq> 0"
haftmann@62353
  1773
  shows "finite {d. d dvd i}"
haftmann@62353
  1774
proof -
haftmann@62353
  1775
  have "{d. \<bar>d\<bar> \<le> \<bar>i\<bar>} = {- \<bar>i\<bar>..\<bar>i\<bar>}"
haftmann@62353
  1776
    by (auto simp: abs_if)
wenzelm@63489
  1777
  then have "finite {d. \<bar>d\<bar> \<le> \<bar>i\<bar>}"
haftmann@62353
  1778
    by simp
wenzelm@63489
  1779
  from finite_subset [OF _ this] show ?thesis
wenzelm@63489
  1780
    using assms by (simp add: dvd_imp_le_int subset_iff)
nipkow@31734
  1781
qed
nipkow@31734
  1782
wenzelm@63489
  1783
lemma Max_divisors_self_nat [simp]: "n \<noteq> 0 \<Longrightarrow> Max {d::nat. d dvd n} = n"
wenzelm@63489
  1784
  apply (rule antisym)
wenzelm@63489
  1785
   apply (fastforce intro: Max_le_iff[THEN iffD2] simp: dvd_imp_le)
wenzelm@63489
  1786
  apply simp
wenzelm@63489
  1787
  done
wenzelm@63489
  1788
wenzelm@63489
  1789
lemma Max_divisors_self_int [simp]: "n \<noteq> 0 \<Longrightarrow> Max {d::int. d dvd n} = \<bar>n\<bar>"
wenzelm@63489
  1790
  apply (rule antisym)
wenzelm@63489
  1791
   apply (rule Max_le_iff [THEN iffD2])
wenzelm@63489
  1792
     apply (auto intro: abs_le_D1 dvd_imp_le_int)
wenzelm@63489
  1793
  done
wenzelm@63489
  1794
wenzelm@63489
  1795
lemma gcd_is_Max_divisors_nat: "m > 0 \<Longrightarrow> n > 0 \<Longrightarrow> gcd m n = Max {d. d dvd m \<and> d dvd n}"
wenzelm@63489
  1796
  for m n :: nat
wenzelm@63489
  1797
  apply (rule Max_eqI[THEN sym])
wenzelm@63489
  1798
    apply (metis finite_Collect_conjI finite_divisors_nat)
wenzelm@63489
  1799
   apply simp
wenzelm@63489
  1800
   apply (metis Suc_diff_1 Suc_neq_Zero dvd_imp_le gcd_greatest_iff gcd_pos_nat)
wenzelm@63489
  1801
  apply simp
wenzelm@63489
  1802
  done
wenzelm@63489
  1803
wenzelm@63489
  1804
lemma gcd_is_Max_divisors_int: "m \<noteq> 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> gcd m n = Max {d. d dvd m \<and> d dvd n}"
wenzelm@63489
  1805
  for m n :: int
wenzelm@63489
  1806
  apply (rule Max_eqI[THEN sym])
wenzelm@63489
  1807
    apply (metis finite_Collect_conjI finite_divisors_int)
wenzelm@63489
  1808
   apply simp
wenzelm@63489
  1809
   apply (metis gcd_greatest_iff gcd_pos_int zdvd_imp_le)
wenzelm@63489
  1810
  apply simp
wenzelm@63489
  1811
  done
wenzelm@63489
  1812
wenzelm@63489
  1813
lemma gcd_code_int [code]: "gcd k l = \<bar>if l = 0 then k else gcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
wenzelm@63489
  1814
  for k l :: int
haftmann@34030
  1815
  by (simp add: gcd_int_def nat_mod_distrib gcd_non_0_nat)
haftmann@34030
  1816
chaieb@22027
  1817
wenzelm@60758
  1818
subsection \<open>Coprimality\<close>
huffman@31706
  1819
wenzelm@63489
  1820
lemma coprime_nat: "coprime a b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
wenzelm@63489
  1821
  for a b :: nat
haftmann@62345
  1822
  using coprime [of a b] by simp
huffman@31706
  1823
wenzelm@63489
  1824
lemma coprime_Suc_0_nat: "coprime a b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = Suc 0)"
wenzelm@63489
  1825
  for a b :: nat
haftmann@60690
  1826
  using coprime_nat by simp
huffman@31706
  1827
wenzelm@63489
  1828
lemma coprime_int: "coprime a b \<longleftrightarrow> (\<forall>d. d \<ge> 0 \<and> d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
wenzelm@63489
  1829
  for a b :: int
nipkow@31952
  1830
  using gcd_unique_int [of 1 a b]
huffman@31706
  1831
  apply clarsimp
huffman@31706
  1832
  apply (erule subst)
huffman@31706
  1833
  apply (rule iffI)
wenzelm@63489
  1834
   apply force
lp15@61649
  1835
  using abs_dvd_iff abs_ge_zero apply blast
wenzelm@59807
  1836
  done
huffman@31706
  1837
wenzelm@63489
  1838
lemma pow_divides_eq_nat [simp]: "n > 0 \<Longrightarrow> a^n dvd b^n \<longleftrightarrow> a dvd b"
wenzelm@63489
  1839
  for a b n :: nat
eberlm@62429
  1840
  using pow_divs_eq[of n] by simp
huffman@31706
  1841
nipkow@31952
  1842
lemma coprime_Suc_nat [simp]: "coprime (Suc n) n"
eberlm@62429
  1843
  using coprime_plus_one[of n] by simp
huffman@31706
  1844
wenzelm@63489
  1845
lemma coprime_minus_one_nat: "n \<noteq> 0 \<Longrightarrow> coprime (n - 1) n"
wenzelm@63489
  1846
  for n :: nat
eberlm@62429
  1847
  using coprime_Suc_nat [of "n - 1"] gcd.commute [of "n - 1" n] by auto
huffman@31706
  1848
wenzelm@63489
  1849
lemma coprime_common_divisor_nat: "coprime a b \<Longrightarrow> x dvd a \<Longrightarrow> x dvd b \<Longrightarrow> x = 1"
wenzelm@63489
  1850
  for a b :: nat
haftmann@62344
  1851
  by (metis gcd_greatest_iff nat_dvd_1_iff_1)
huffman@31706
  1852
wenzelm@63489
  1853
lemma coprime_common_divisor_int: "coprime a b \<Longrightarrow> x dvd a \<Longrightarrow> x dvd b \<Longrightarrow> \<bar>x\<bar> = 1"
wenzelm@63489
  1854
  for a b :: int
haftmann@60686
  1855
  using gcd_greatest_iff [of x a b] by auto
huffman@31706
  1856
wenzelm@63489
  1857
lemma invertible_coprime_nat: "x * y mod m = 1 \<Longrightarrow> coprime x m"
wenzelm@63489
  1858
  for m x y :: nat
wenzelm@63489
  1859
  by (metis coprime_lmult gcd_1_nat gcd.commute gcd_red_nat)
wenzelm@63489
  1860
wenzelm@63489
  1861
lemma invertible_coprime_int: "x * y mod m = 1 \<Longrightarrow> coprime x m"
wenzelm@63489
  1862
  for m x y :: int
wenzelm@63489
  1863
  by (metis coprime_lmult gcd_1_int gcd.commute gcd_red_int)
huffman@31706
  1864
huffman@31706
  1865
wenzelm@60758
  1866
subsection \<open>Bezout's theorem\<close>
huffman@31706
  1867
wenzelm@63489
  1868
text \<open>
wenzelm@63489
  1869
  Function \<open>bezw\<close> returns a pair of witnesses to Bezout's theorem --
wenzelm@63489
  1870
  see the theorems that follow the definition.
wenzelm@63489
  1871
\<close>
wenzelm@63489
  1872
wenzelm@63489
  1873
fun bezw :: "nat \<Rightarrow> nat \<Rightarrow> int * int"
wenzelm@63489
  1874
  where "bezw x y =
wenzelm@63489
  1875
    (if y = 0 then (1, 0)
wenzelm@63489
  1876
     else
huffman@31706
  1877
      (snd (bezw y (x mod y)),
huffman@31706
  1878
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y)))"
huffman@31706
  1879
wenzelm@63489
  1880
lemma bezw_0 [simp]: "bezw x 0 = (1, 0)"
wenzelm@63489
  1881
  by simp
wenzelm@63489
  1882
wenzelm@63489
  1883
lemma bezw_non_0:
wenzelm@63489
  1884
  "y > 0 \<Longrightarrow> bezw x y =
wenzelm@63489
  1885
    (snd (bezw y (x mod y)), fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y))"
huffman@31706
  1886
  by simp
huffman@31706
  1887
huffman@31706
  1888
declare bezw.simps [simp del]
huffman@31706
  1889
wenzelm@63489
  1890
lemma bezw_aux: "fst (bezw x y) * int x + snd (bezw x y) * int y = int (gcd x y)"
nipkow@31952
  1891
proof (induct x y rule: gcd_nat_induct)
huffman@31706
  1892
  fix m :: nat
huffman@31706
  1893
  show "fst (bezw m 0) * int m + snd (bezw m 0) * int 0 = int (gcd m 0)"
huffman@31706
  1894
    by auto
wenzelm@63489
  1895
next
wenzelm@63489
  1896
  fix m n :: nat
wenzelm@63489
  1897
  assume ngt0: "n > 0"
wenzelm@63489
  1898
    and ih: "fst (bezw n (m mod n)) * int n + snd (bezw n (m mod n)) * int (m mod n) =
wenzelm@63489
  1899
      int (gcd n (m mod n))"
wenzelm@63489
  1900
  then show "fst (bezw m n) * int m + snd (bezw m n) * int n = int (gcd m n)"
wenzelm@63489
  1901
    apply (simp add: bezw_non_0 gcd_non_0_nat)
wenzelm@63489
  1902
    apply (erule subst)
wenzelm@63489
  1903
    apply (simp add: field_simps)
wenzelm@63489
  1904
    apply (subst mod_div_equality [of m n, symmetric])
wenzelm@63489
  1905
      (* applying simp here undoes the last substitution! what is procedure cancel_div_mod? *)
wenzelm@63489
  1906
    apply (simp only: NO_MATCH_def field_simps of_nat_add of_nat_mult)
wenzelm@63489
  1907
    done
huffman@31706
  1908
qed
huffman@31706
  1909
wenzelm@63489
  1910
lemma bezout_int: "\<exists>u v. u * x + v * y = gcd x y"
wenzelm@63489
  1911
  for x y :: int
huffman@31706
  1912
proof -
wenzelm@63489
  1913
  have aux: "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> \<exists>u v. u * x + v * y = gcd x y" for x y :: int
huffman@31706
  1914
    apply (rule_tac x = "fst (bezw (nat x) (nat y))" in exI)
huffman@31706
  1915
    apply (rule_tac x = "snd (bezw (nat x) (nat y))" in exI)
huffman@31706
  1916
    apply (unfold gcd_int_def)
huffman@31706
  1917
    apply simp
huffman@31706
  1918
    apply (subst bezw_aux [symmetric])
huffman@31706
  1919
    apply auto
huffman@31706
  1920
    done
wenzelm@63489
  1921
  consider "x \<ge> 0" "y \<ge> 0" | "x \<ge> 0" "y \<le> 0" | "x \<le> 0" "y \<ge> 0" | "x \<le> 0" "y \<le> 0"
wenzelm@63489
  1922
    by atomize_elim auto
wenzelm@63489
  1923
  then show ?thesis
wenzelm@63489
  1924
  proof cases
wenzelm@63489
  1925
    case 1
wenzelm@63489
  1926
    then show ?thesis by (rule aux)
wenzelm@63489
  1927
  next
wenzelm@63489
  1928
    case 2
wenzelm@63489
  1929
    then show ?thesis
wenzelm@63489
  1930
      apply -
wenzelm@63489
  1931
      apply (insert aux [of x "-y"])
wenzelm@63489
  1932
      apply auto
wenzelm@63489
  1933
      apply (rule_tac x = u in exI)
wenzelm@63489
  1934
      apply (rule_tac x = "-v" in exI)
wenzelm@63489
  1935
      apply (subst gcd_neg2_int [symmetric])
wenzelm@63489
  1936
      apply auto
wenzelm@63489
  1937
      done
wenzelm@63489
  1938
  next
wenzelm@63489
  1939
    case 3
wenzelm@63489
  1940
    then show ?thesis
wenzelm@63489
  1941
      apply -
wenzelm@63489
  1942
      apply (insert aux [of "-x" y])
wenzelm@63489
  1943
      apply auto
wenzelm@63489
  1944
      apply (rule_tac x = "-u" in exI)
wenzelm@63489
  1945
      apply (rule_tac x = v in exI)
wenzelm@63489
  1946
      apply (subst gcd_neg1_int [symmetric])
wenzelm@63489
  1947
      apply auto
wenzelm@63489
  1948
      done
wenzelm@63489
  1949
  next
wenzelm@63489
  1950
    case 4
wenzelm@63489
  1951
    then show ?thesis
wenzelm@63489
  1952
      apply -
wenzelm@63489
  1953
      apply (insert aux [of "-x" "-y"])
wenzelm@63489
  1954
      apply auto
wenzelm@63489
  1955
      apply (rule_tac x = "-u" in exI)
wenzelm@63489
  1956
      apply (rule_tac x = "-v" in exI)
wenzelm@63489
  1957
      apply (subst gcd_neg1_int [symmetric])
wenzelm@63489
  1958
      apply (subst gcd_neg2_int [symmetric])
wenzelm@63489
  1959
      apply auto
wenzelm@63489
  1960
      done
wenzelm@63489
  1961
  qed
huffman@31706
  1962
qed
huffman@31706
  1963
wenzelm@63489
  1964
wenzelm@63489
  1965
text \<open>Versions of Bezout for \<open>nat\<close>, by Amine Chaieb.\<close>
huffman@31706
  1966
huffman@31706
  1967
lemma ind_euclid:
wenzelm@63489
  1968
  fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"
wenzelm@63489
  1969
  assumes c: " \<forall>a b. P a b \<longleftrightarrow> P b a"
wenzelm@63489
  1970
    and z: "\<forall>a. P a 0"
wenzelm@63489
  1971
    and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
chaieb@27669
  1972
  shows "P a b"
wenzelm@63489
  1973
proof (induct "a + b" arbitrary: a b rule: less_induct)
berghofe@34915
  1974
  case less
wenzelm@63489
  1975
  consider (eq) "a = b" | (lt) "a < b" "a + b - a < a + b" | "b = 0" | "b + a - b < a + b"
wenzelm@63489
  1976
    by arith
wenzelm@63489
  1977
  show ?case
wenzelm@63489
  1978
  proof (cases a b rule: linorder_cases)
wenzelm@63489
  1979
    case equal
wenzelm@63489
  1980
    with add [rule_format, OF z [rule_format, of a]] show ?thesis by simp
wenzelm@63489
  1981
  next
wenzelm@63489
  1982
    case lt: less
wenzelm@63489
  1983
    then consider "a = 0" | "a + b - a < a + b" by arith
wenzelm@63489
  1984
    then show ?thesis
wenzelm@63489
  1985
    proof cases
wenzelm@63489
  1986
      case 1
wenzelm@63489
  1987
      with z c show ?thesis by blast
wenzelm@63489
  1988
    next
wenzelm@63489
  1989
      case 2
wenzelm@63489
  1990
      also have *: "a + b - a = a + (b - a)" using lt by arith
berghofe@34915
  1991
      finally have "a + (b - a) < a + b" .
wenzelm@63489
  1992
      then have "P a (a + (b - a))" by (rule add [rule_format, OF less])
wenzelm@63489
  1993
      then show ?thesis by (simp add: *[symmetric])
wenzelm@63489
  1994
    qed
wenzelm@63489
  1995
  next
wenzelm@63489
  1996
    case gt: greater
wenzelm@63489
  1997
    then consider "b = 0" | "b + a - b < a + b" by arith
wenzelm@63489
  1998
    then show ?thesis
wenzelm@63489
  1999
    proof cases
wenzelm@63489
  2000
      case 1
wenzelm@63489
  2001
      with z c show ?thesis by blast
wenzelm@63489
  2002
    next
wenzelm@63489
  2003
      case 2
wenzelm@63489
  2004
      also have *: "b + a - b = b + (a - b)" using gt by arith
berghofe@34915
  2005
      finally have "b + (a - b) < a + b" .
wenzelm@63489
  2006
      then have "P b (b + (a - b))" by (rule add [rule_format, OF less])
wenzelm@63489
  2007
      then have "P b a" by (simp add: *[symmetric])
wenzelm@63489
  2008
      with c show ?thesis by blast
wenzelm@63489
  2009
    qed
wenzelm@63489
  2010
  qed
chaieb@27669
  2011
qed
chaieb@27669
  2012
nipkow@31952
  2013
lemma bezout_lemma_nat:
huffman@31706
  2014
  assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  2015
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  2016
  shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and>
huffman@31706
  2017
    (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
huffman@31706
  2018
  using ex
huffman@31706
  2019
  apply clarsimp
wenzelm@63489
  2020
  apply (rule_tac x="d" in exI)
wenzelm@63489
  2021
  apply simp
wenzelm@63489
  2022
  apply (case_tac "a * x = b * y + d")
wenzelm@63489
  2023
   apply simp_all
wenzelm@63489
  2024
   apply (rule_tac x="x + y" in exI)
wenzelm@63489
  2025
   apply (rule_tac x="y" in exI)
wenzelm@63489
  2026
   apply algebra
huffman@31706
  2027
  apply (rule_tac x="x" in exI)
huffman@31706
  2028
  apply (rule_tac x="x + y" in exI)
huffman@31706
  2029
  apply algebra
wenzelm@63489
  2030
  done
chaieb@27669
  2031
nipkow@31952
  2032
lemma bezout_add_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  2033
    (a * x = b * y + d \<or> b * x = a * y + d)"
wenzelm@63489
  2034
  apply (induct a b rule: ind_euclid)
wenzelm@63489
  2035
    apply blast
wenzelm@63489
  2036
   apply clarify
wenzelm@63489
  2037
   apply (rule_tac x="a" in exI)
wenzelm@63489
  2038
   apply simp
huffman@31706
  2039
  apply clarsimp
huffman@31706
  2040
  apply (rule_tac x="d" in exI)
wenzelm@63489
  2041
  apply (case_tac "a * x = b * y + d")
wenzelm@63489
  2042
   apply simp_all
wenzelm@63489
  2043
   apply (rule_tac x="x+y" in exI)
wenzelm@63489
  2044
   apply (rule_tac x="y" in exI)
wenzelm@63489
  2045
   apply algebra
huffman@31706
  2046
  apply (rule_tac x="x" in exI)
huffman@31706
  2047
  apply (rule_tac x="x+y" in exI)
huffman@31706
  2048
  apply algebra
wenzelm@63489
  2049
  done
chaieb@27669
  2050
nipkow@31952
  2051
lemma bezout1_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  2052
    (a * x - b * y = d \<or> b * x - a * y = d)"
nipkow@31952
  2053
  using bezout_add_nat[of a b]
huffman@31706
  2054
  apply clarsimp
wenzelm@63489
  2055
  apply (rule_tac x="d" in exI)
wenzelm@63489
  2056
  apply simp
huffman@31706
  2057
  apply (rule_tac x="x" in exI)
huffman@31706
  2058
  apply (rule_tac x="y" in exI)
huffman@31706
  2059
  apply auto
wenzelm@63489
  2060
  done
wenzelm@63489
  2061
wenzelm@63489
  2062
lemma bezout_add_strong_nat:
wenzelm@63489
  2063
  fixes a b :: nat
wenzelm@63489
  2064
  assumes a: "a \<noteq> 0"
chaieb@27669
  2065
  shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
wenzelm@63489
  2066
proof -
wenzelm@63489
  2067
  consider d x y where "d dvd a" "d dvd b" "a * x = b * y + d"
wenzelm@63489
  2068
    | d x y where "d dvd a" "d dvd b" "b * x = a * y + d"
wenzelm@63489
  2069
    using bezout_add_nat [of a b] by blast
wenzelm@63489
  2070
  then show ?thesis
wenzelm@63489
  2071
  proof cases
wenzelm@63489
  2072
    case 1
wenzelm@63489
  2073
    then show ?thesis by blast
wenzelm@63489
  2074
  next
wenzelm@63489
  2075
    case H: 2
wenzelm@63489
  2076
    show ?thesis
wenzelm@63489
  2077
    proof (cases "b = 0")
wenzelm@63489
  2078
      case True
wenzelm@63489
  2079
      with H show ?thesis by simp
wenzelm@63489
  2080
    next
wenzelm@63489
  2081
      case False
wenzelm@63489
  2082
      then have bp: "b > 0" by simp
wenzelm@63489
  2083
      with dvd_imp_le [OF H(2)] consider "d = b" | "d < b"
wenzelm@63489
  2084
        by atomize_elim auto
wenzelm@63489
  2085
      then show ?thesis
wenzelm@63489
  2086
      proof cases
wenzelm@63489
  2087
        case 1
wenzelm@63489
  2088
        with a H show ?thesis
wenzelm@63489
  2089
          apply simp
wenzelm@63489
  2090
          apply (rule exI[where x = b])
wenzelm@63489
  2091
          apply simp
wenzelm@63489
  2092
          apply (rule exI[where x = b])
wenzelm@63489
  2093
          apply (rule exI[where x = "a - 1"])
wenzelm@63489
  2094
          apply (simp add: diff_mult_distrib2)
wenzelm@63489
  2095
          done
wenzelm@63489
  2096
      next
wenzelm@63489
  2097
        case 2
wenzelm@63489
  2098
        show ?thesis
wenzelm@63489
  2099
        proof (cases "x = 0")
wenzelm@63489
  2100
          case True
wenzelm@63489
  2101
          with a H show ?thesis by simp
wenzelm@63489
  2102
        next
wenzelm@63489
  2103
          case x0: False
wenzelm@63489
  2104
          then have xp: "x > 0" by simp
wenzelm@63489
  2105
          from \<open>d < b\<close> have "d \<le> b - 1" by simp
wenzelm@63489
  2106
          then have "d * b \<le> b * (b - 1)" by simp
wenzelm@63489
  2107
          with xp mult_mono[of "1" "x" "d * b" "b * (b - 1)"]
wenzelm@63489
  2108
          have dble: "d * b \<le> x * b * (b - 1)" using bp by simp
wenzelm@63489
  2109
          from H(3) have "d + (b - 1) * (b * x) = d + (b - 1) * (a * y + d)"
huffman@31706
  2110
            by simp
wenzelm@63489
  2111
          then have "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
haftmann@57512
  2112
            by (simp only: mult.assoc distrib_left)
wenzelm@63489
  2113
          then have "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x * b * (b - 1)"
huffman@31706
  2114
            by algebra
wenzelm@63489
  2115
          then have "a * ((b - 1) * y) = d + x * b * (b - 1) - d * b"
wenzelm@63489
  2116
            using bp by simp
wenzelm@63489
  2117
          then have "a * ((b - 1) * y) = d + (x * b * (b - 1) - d * b)"
wenzelm@32960
  2118
            by (simp only: diff_add_assoc[OF dble, of d, symmetric])
wenzelm@63489
  2119
          then have "a * ((b - 1) * y) = b * (x * (b - 1) - d) + d"
haftmann@59008
  2120
            by (simp only: diff_mult_distrib2 ac_simps)
wenzelm@63489
  2121
          with H(1,2) show ?thesis
wenzelm@32960
  2122
            apply -
wenzelm@63489
  2123
            apply (rule exI [where x = d])
wenzelm@63489
  2124
            apply simp
wenzelm@63489
  2125
            apply (rule exI [where x = "(b - 1) * y"])
wenzelm@63489
  2126
            apply (rule exI [where x = "x * (b - 1) - d"])
wenzelm@63489
  2127
            apply simp
wenzelm@63489
  2128
            done
wenzelm@63489
  2129
        qed
wenzelm@63489
  2130
      qed
wenzelm@63489
  2131
    qed
wenzelm@63489
  2132
  qed
chaieb@27669
  2133
qed
chaieb@27669
  2134
wenzelm@63489
  2135
lemma bezout_nat:
wenzelm@63489
  2136
  fixes a :: nat
wenzelm@63489
  2137
  assumes a: "a \<noteq> 0"
chaieb@27669
  2138
  shows "\<exists>x y. a * x = b * y + gcd a b"
wenzelm@63489
  2139
proof -
wenzelm@63489
  2140
  obtain d x y where d: "d dvd a" "d dvd b" and eq: "a * x = b * y + d"
wenzelm@63489
  2141
    using bezout_add_strong_nat [OF a, of b] by blast
wenzelm@63489
  2142
  from d have "d dvd gcd a b"
wenzelm@63489
  2143
    by simp
wenzelm@63489
  2144
  then obtain k where k: "gcd a b = d * k"
wenzelm@63489
  2145
    unfolding dvd_def by blast
wenzelm@63489
  2146
  from eq have "a * x * k = (b * y + d) * k"
wenzelm@63489
  2147
    by auto
wenzelm@63489
  2148
  then have "a * (x * k) = b * (y * k) + gcd a b"
wenzelm@63489
  2149
    by (algebra add: k)
wenzelm@63489
  2150
  then show ?thesis
wenzelm@63489
  2151
    by blast
chaieb@27669
  2152
qed
chaieb@27669
  2153
huffman@31706
  2154
wenzelm@63489
  2155
subsection \<open>LCM properties on @{typ nat} and @{typ int}\<close>
wenzelm@63489
  2156
wenzelm@63489
  2157
lemma lcm_altdef_int [code]: "lcm a b = \<bar>a\<bar> * \<bar>b\<bar> div gcd a b"
wenzelm@63489
  2158
  for a b :: int
haftmann@62344
  2159
  by (simp add: lcm_int_def lcm_nat_def zdiv_int gcd_int_def)
huffman@31706
  2160
wenzelm@63489
  2161
lemma prod_gcd_lcm_nat: "m * n = gcd m n * lcm m n"
wenzelm@63489
  2162
  for m n :: nat
huffman@31706
  2163
  unfolding lcm_nat_def
eberlm@62429
  2164
  by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod])
huffman@31706
  2165
wenzelm@63489
  2166
lemma prod_gcd_lcm_int: "\<bar>m\<bar> * \<bar>n\<bar> = gcd m n * lcm m n"
wenzelm@63489
  2167
  for m n :: int
huffman@31706
  2168
  unfolding lcm_int_def gcd_int_def
haftmann@62348
  2169
  apply (subst of_nat_mult [symmetric])
nipkow@31952
  2170
  apply (subst prod_gcd_lcm_nat [symmetric])
huffman@31706
  2171
  apply (subst nat_abs_mult_distrib [symmetric])
wenzelm@63489
  2172
  apply (simp add: abs_mult)
wenzelm@63489
  2173
  done
wenzelm@63489
  2174
wenzelm@63489
  2175
lemma lcm_pos_nat: "m > 0 \<Longrightarrow> n > 0 \<Longrightarrow> lcm m n > 0"
wenzelm@63489
  2176
  for m n :: nat
wenzelm@63489
  2177
  by (metis gr0I mult_is_0 prod_gcd_lcm_nat)
wenzelm@63489
  2178
wenzelm@63489
  2179
lemma lcm_pos_int: "m \<noteq> 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> lcm m n > 0"
wenzelm@63489
  2180
  for m n :: int
nipkow@31952
  2181
  apply (subst lcm_abs_int)
nipkow@31952
  2182
  apply (rule lcm_pos_nat [transferred])
wenzelm@63489
  2183
     apply auto
haftmann@62344
  2184
  done
haftmann@23687
  2185
wenzelm@63489
  2186
lemma dvd_pos_nat: "n > 0 \<Longrightarrow> m dvd n \<Longrightarrow> m > 0"  (* FIXME move *)
wenzelm@63489
  2187
  for m n :: nat
wenzelm@63489
  2188
  by (cases m) auto
wenzelm@63489
  2189
wenzelm@63489
  2190
lemma lcm_unique_nat:
wenzelm@63489
  2191
  "a dvd d \<and> b dvd d \<and> (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
wenzelm@63489
  2192
  for a b d :: nat
haftmann@62344
  2193
  by (auto intro: dvd_antisym lcm_least)
chaieb@27568
  2194
wenzelm@63489
  2195
lemma lcm_unique_int:
wenzelm@63489
  2196
  "d \<ge> 0 \<and> a dvd d \<and> b dvd d \<and> (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
wenzelm@63489
  2197
  for a b d :: int
haftmann@62344
  2198
  using lcm_least zdvd_antisym_nonneg by auto
haftmann@34973
  2199
wenzelm@63489
  2200
lemma lcm_proj2_if_dvd_nat [simp]: "x dvd y \<Longrightarrow> lcm x y = y"
wenzelm@63489
  2201
  for x y :: nat
huffman@31706
  2202
  apply (rule sym)
nipkow@31952
  2203
  apply (subst lcm_unique_nat [symmetric])
huffman@31706
  2204
  apply auto
wenzelm@63489
  2205
  done
wenzelm@63489
  2206
wenzelm@63489
  2207
lemma lcm_proj2_if_dvd_int [simp]: "x dvd y \<Longrightarrow> lcm x y = \<bar>y\<bar>"
wenzelm@63489
  2208
  for x y :: int
huffman@31706
  2209
  apply (rule sym)
nipkow@31952
  2210
  apply (subst lcm_unique_int [symmetric])
huffman@31706
  2211
  apply auto
wenzelm@63489
  2212
  done
wenzelm@63489
  2213
wenzelm@63489
  2214
lemma lcm_proj1_if_dvd_nat [simp]: "x dvd y \<Longrightarrow> lcm y x = y"
wenzelm@63489
  2215
  for x y :: nat
wenzelm@63489
  2216
  by (subst lcm.commute) (erule lcm_proj2_if_dvd_nat)
wenzelm@63489
  2217
wenzelm@63489
  2218
lemma lcm_proj1_if_dvd_int [simp]: "x dvd y \<Longrightarrow> lcm y x = \<bar>y\<bar>"
wenzelm@63489
  2219
  for x y :: int
wenzelm@63489
  2220
  by (subst lcm.commute) (erule lcm_proj2_if_dvd_int)
wenzelm@63489
  2221
wenzelm@63489
  2222
lemma lcm_proj1_iff_nat [simp]: "lcm m n = m \<longleftrightarrow> n dvd m"
wenzelm@63489
  2223
  for m n :: nat
wenzelm@63489
  2224
  by (metis lcm_proj1_if_dvd_nat lcm_unique_nat)
wenzelm@63489
  2225
wenzelm@63489
  2226
lemma lcm_proj2_iff_nat [simp]: "lcm m n = n \<longleftrightarrow> m dvd n"
wenzelm@63489
  2227
  for m n :: nat
wenzelm@63489
  2228
  by (metis lcm_proj2_if_dvd_nat lcm_unique_nat)
wenzelm@63489
  2229
wenzelm@63489
  2230
lemma lcm_proj1_iff_int [simp]: "lcm m n = \<bar>m\<bar> \<longleftrightarrow> n dvd m"
wenzelm@63489
  2231
  for m n :: int
wenzelm@63489
  2232
  by (metis dvd_abs_iff lcm_proj1_if_dvd_int lcm_unique_int)
wenzelm@63489
  2233
wenzelm@63489
  2234
lemma lcm_proj2_iff_int [simp]: "lcm m n = \<bar>n\<bar> \<longleftrightarrow> m dvd n"
wenzelm@63489
  2235
  for m n :: int
wenzelm@63489
  2236
  by (metis dvd_abs_iff lcm_proj2_if_dvd_int lcm_unique_int)
wenzelm@63489
  2237
wenzelm@63489
  2238
lemma lcm_1_iff_nat [simp]: "lcm m n = Suc 0 \<longleftrightarrow> m = Suc 0 \<and> n = Suc 0"
wenzelm@63489
  2239
  for m n :: nat
haftmann@62353
  2240
  using lcm_eq_1_iff [of m n] by simp
wenzelm@63489
  2241
wenzelm@63489
  2242
lemma lcm_1_iff_int [simp]: "lcm m n = 1 \<longleftrightarrow> (m = 1 \<or> m = -1) \<and> (n = 1 \<or> n = -1)"
wenzelm@63489
  2243
  for m n :: int
haftmann@61913
  2244
  by auto
nipkow@31995
  2245
haftmann@34030
  2246
haftmann@62345
  2247
subsection \<open>The complete divisibility lattice on @{typ nat} and @{typ int}\<close>
nipkow@32112
  2248
wenzelm@63489
  2249
text \<open>
wenzelm@63489
  2250
  Lifting \<open>gcd\<close> and \<open>lcm\<close> to sets (\<open>Gcd\<close> / \<open>Lcm\<close>).
wenzelm@63489
  2251
  \<open>Gcd\<close> is defined via \<open>Lcm\<close> to facilitate the proof that we have a complete lattice.
wenzelm@60758
  2252
\<close>
huffman@45264
  2253
haftmann@62345
  2254
instantiation nat :: semiring_Gcd
nipkow@32112
  2255
begin
nipkow@32112
  2256
haftmann@62344
  2257
interpretation semilattice_neutr_set lcm "1::nat"
haftmann@62344
  2258
  by standard simp_all
haftmann@54867
  2259
wenzelm@63489
  2260
definition "Lcm M = (if finite M then F M else 0)" for M :: "nat set"
wenzelm@63489
  2261
wenzelm@63489
  2262
lemma Lcm_nat_empty: "Lcm {} = (1::nat)"
haftmann@60690
  2263
  by (simp add: Lcm_nat_def del: One_nat_def)
haftmann@51489
  2264
wenzelm@63489
  2265
lemma Lcm_nat_insert: "Lcm (insert n M) = lcm n (Lcm M)" for n :: nat
haftmann@61929
  2266
  by (cases "finite M") (auto simp add: Lcm_nat_def simp del: One_nat_def)
haftmann@61929
  2267
wenzelm@63489
  2268
lemma Lcm_nat_infinite: "infinite M \<Longrightarrow> Lcm M = 0" for M :: "nat set"
haftmann@61929
  2269
  by (simp add: Lcm_nat_def)
haftmann@61929
  2270
haftmann@61929
  2271
lemma dvd_Lcm_nat [simp]:
haftmann@61929
  2272
  fixes M :: "nat set"
haftmann@61929
  2273
  assumes "m \<in> M"
haftmann@61929
  2274
  shows "m dvd Lcm M"
haftmann@61929
  2275
proof -
wenzelm@63489
  2276
  from assms have "insert m M = M"
wenzelm@63489
  2277
    by auto
haftmann@61929
  2278
  moreover have "m dvd Lcm (insert m M)"
haftmann@61929
  2279
    by (simp add: Lcm_nat_insert)
wenzelm@63489
  2280
  ultimately show ?thesis
wenzelm@63489
  2281
    by simp
haftmann@61929
  2282
qed
haftmann@61929
  2283
haftmann@61929
  2284
lemma Lcm_dvd_nat [simp]:
haftmann@61929
  2285
  fixes M :: "nat set"
haftmann@61929
  2286
  assumes "\<forall>m\<in>M. m dvd n"
haftmann@61929
  2287
  shows "Lcm M dvd n"
haftmann@62353
  2288
proof (cases "n > 0")
wenzelm@63489
  2289
  case False
wenzelm@63489
  2290
  then show ?thesis by simp
haftmann@61929
  2291
next
haftmann@62353
  2292
  case True
wenzelm@63489
  2293
  then have "finite {d. d dvd n}"
wenzelm@63489
  2294
    by (rule finite_divisors_nat)
wenzelm@63489
  2295
  moreover have "M \<subseteq> {d. d dvd n}"
wenzelm@63489
  2296
    using assms by fast
wenzelm@63489
  2297
  ultimately have "finite M"
wenzelm@63489
  2298
    by (rule rev_finite_subset)
wenzelm@63489
  2299
  then show ?thesis
wenzelm@63489
  2300
    using assms by (induct M) (simp_all add: Lcm_nat_empty Lcm_nat_insert)
haftmann@61929
  2301
qed
nipkow@32112
  2302
wenzelm@63489
  2303
definition "Gcd M = Lcm {d. \<forall>m\<in>M. d dvd m}" for M :: "nat set"
wenzelm@63489
  2304
wenzelm@63489
  2305
instance
wenzelm@63489
  2306
proof
wenzelm@63489
  2307
  fix N :: "nat set"
wenzelm@63489
  2308
  fix n :: nat
wenzelm@63489
  2309
  show "Gcd N dvd n" if "n \<in> N"
wenzelm@63489
  2310
    using that by (induct N rule: infinite_finite_induct) (auto simp add: Gcd_nat_def)
wenzelm@63489
  2311
  show "n dvd Gcd N" if "\<And>m. m \<in> N \<Longrightarrow> n dvd m"
wenzelm@63489
  2312
    using that by (induct N rule: infinite_finite_induct) (auto simp add: Gcd_nat_def)
wenzelm@63489
  2313
  show "n dvd Lcm N" if "n \<in> N"
wenzelm@63489
  2314
    using that by (induct N rule: infinite_finite_induct) auto
wenzelm@63489
  2315
  show "Lcm N dvd n" if "\<And>m. m \<in> N \<Longrightarrow> m dvd n"
wenzelm@63489
  2316
    using that by (induct N rule: infinite_finite_induct) auto
wenzelm@63489
  2317
  show "normalize (Gcd N) = Gcd N" and "normalize (Lcm N) = Lcm N"
wenzelm@63489
  2318
    by simp_all
wenzelm@63489
  2319
qed
nipkow@32112
  2320
haftmann@62345
  2321
end
haftmann@61913
  2322
wenzelm@63489
  2323
lemma Gcd_nat_eq_one: "1 \<in> N \<Longrightarrow> Gcd N = 1"
wenzelm@63489
  2324
  for N :: "nat set"
haftmann@62346
  2325
  by (rule Gcd_eq_1_I) auto
haftmann@62346
  2326
wenzelm@63489
  2327
wenzelm@63489
  2328
text \<open>Alternative characterizations of Gcd:\<close>
nipkow@32112
  2329
haftmann@62353
  2330
lemma Gcd_eq_Max:
haftmann@62353
  2331
  fixes M :: "nat set"
haftmann@62353
  2332
  assumes "finite (M::nat set)" and "M \<noteq> {}" and "0 \<notin> M"
haftmann@62353
  2333
  shows "Gcd M = Max (\<Inter>m\<in>M. {d. d dvd m})"
haftmann@62353
  2334
proof (rule antisym)
haftmann@62353
  2335
  from assms obtain m where "m \<in> M" and "m > 0"
haftmann@62353
  2336
    by auto
haftmann@62353
  2337
  from \<open>m > 0\<close> have "finite {d. d dvd m}"
haftmann@62353
  2338
    by (blast intro: finite_divisors_nat)
haftmann@62353
  2339
  with \<open>m \<in> M\<close> have fin: "finite (\<Inter>m\<in>M. {d. d dvd m})"
haftmann@62353
  2340
    by blast
haftmann@62353
  2341
  from fin show "Gcd M \<le> Max (\<Inter>m\<in>M. {d. d dvd m})"
haftmann@62353
  2342
    by (auto intro: Max_ge Gcd_dvd)
haftmann@62353
  2343
  from fin show "Max (\<Inter>m\<in>M. {d. d dvd m}) \<le> Gcd M"
haftmann@62353
  2344
    apply (rule Max.boundedI)
wenzelm@63489
  2345
     apply auto
haftmann@62353
  2346
    apply (meson Gcd_dvd Gcd_greatest \<open>0 < m\<close> \<open>m \<in> M\<close> dvd_imp_le dvd_pos_nat)
haftmann@62353
  2347
    done
haftmann@62353
  2348
qed
nipkow@32112
  2349
wenzelm@63489
  2350
lemma Gcd_remove0_nat: "finite M \<Longrightarrow> Gcd M = Gcd (M - {0})"
wenzelm@63489
  2351
  for M :: "nat set"
wenzelm@63489
  2352
  apply (induct pred: finite)
wenzelm@63489
  2353
   apply simp
wenzelm@63489
  2354
  apply (case_tac "x = 0")
wenzelm@63489
  2355
   apply simp
wenzelm@63489
  2356
  apply (subgoal_tac "insert x F - {0} = insert x (F - {0})")
wenzelm@63489
  2357
   apply simp
wenzelm@63489
  2358
  apply blast
wenzelm@63489
  2359
  done
nipkow@32112
  2360
nipkow@32112
  2361
lemma Lcm_in_lcm_closed_set_nat:
wenzelm@63489
  2362
  "finite M \<Longrightarrow> M \<noteq> {} \<Longrightarrow> \<forall>m n. m \<in> M \<longrightarrow> n \<in> M \<longrightarrow> lcm m n \<in> M \<Longrightarrow> Lcm M \<in> M"
wenzelm@63489
  2363
  for M :: "nat set"
wenzelm@63489
  2364
  apply (induct rule: finite_linorder_min_induct)
wenzelm@63489
  2365
   apply simp
nipkow@32112
  2366
  apply simp
wenzelm@63489
  2367
  apply (subgoal_tac "\<forall>m n. m \<in> A \<longrightarrow> n \<in> A \<longrightarrow> lcm m n \<in> A")
wenzelm@63489
  2368
   apply simp
wenzelm@63489
  2369
   apply(case_tac "A = {}")
wenzelm@63489
  2370
    apply simp
wenzelm@63489
  2371
   apply simp
wenzelm@63489
  2372
  apply (metis lcm_pos_nat lcm_unique_nat linorder_neq_iff nat_dvd_not_less not_less0)
wenzelm@63489
  2373
  done
nipkow@32112
  2374
nipkow@32112
  2375
lemma Lcm_eq_Max_nat:
wenzelm@63489
  2376
  "finite M \<Longrightarrow> M \<noteq> {} \<Longrightarrow> 0 \<notin> M \<Longrightarrow> \<forall>m n. m \<in> M \<longrightarrow> n \<in> M \<longrightarrow> lcm m n \<in> M \<Longrightarrow> Lcm M = Max M"
wenzelm@63489
  2377
  for M :: "nat set"
wenzelm@63489
  2378
  apply (rule antisym)
wenzelm@63489
  2379
   apply (rule Max_ge)
wenzelm@63489
  2380
    apply assumption
wenzelm@63489
  2381
   apply (erule (2) Lcm_in_lcm_closed_set_nat)
wenzelm@63489
  2382
  apply (auto simp add: not_le Lcm_0_iff dvd_imp_le leD le_neq_trans)
wenzelm@63489
  2383
  done
nipkow@32112
  2384
nipkow@34222
  2385
lemma mult_inj_if_coprime_nat:
wenzelm@63489
  2386
  "inj_on f A \<Longrightarrow> inj_on g B \<Longrightarrow> \<forall>a\<in>A. \<forall>b\<in>B. coprime (f a) (g b) \<Longrightarrow>
wenzelm@63489
  2387
    inj_on (\<lambda>(a, b). f a * g b) (A \<times> B)"
wenzelm@63489
  2388
  for f :: "'a \<Rightarrow> nat" and g :: "'b \<Rightarrow> nat"
haftmann@61913
  2389
  by (auto simp add: inj_on_def coprime_crossproduct_nat simp del: One_nat_def)
nipkow@34222
  2390
wenzelm@63489
  2391
wenzelm@63489
  2392
text \<open>Nitpick:\<close>
nipkow@34222
  2393
blanchet@41792
  2394
lemma gcd_eq_nitpick_gcd [nitpick_unfold]: "gcd x y = Nitpick.nat_gcd x y"
wenzelm@63489
  2395
  by (induct x y rule: nat_gcd.induct)
wenzelm@63489
  2396
    (simp add: gcd_nat.simps Nitpick.nat_gcd.simps)
blanchet@33197
  2397
blanchet@41792
  2398
lemma lcm_eq_nitpick_lcm [nitpick_unfold]: "lcm x y = Nitpick.nat_lcm x y"
wenzelm@63489
  2399
  by (simp only: lcm_nat_def Nitpick.nat_lcm_def gcd_eq_nitpick_gcd)
wenzelm@63489
  2400
wenzelm@63489
  2401
wenzelm@63489
  2402
subsubsection \<open>Setwise GCD and LCM for integers\<close>
huffman@45264
  2403
haftmann@62345
  2404
instantiation int :: semiring_Gcd
huffman@45264
  2405
begin
huffman@45264
  2406
wenzelm@63489
  2407
definition "Lcm M = int (LCM m\<in>M. (nat \<circ> abs) m)"
wenzelm@63489
  2408
wenzelm@63489
  2409
definition "Gcd M = int (GCD m\<in>M. (nat \<circ> abs) m)"
wenzelm@63489
  2410
wenzelm@63489
  2411
instance
wenzelm@63489
  2412
  by standard
wenzelm@63489
  2413
    (auto intro!: Gcd_dvd Gcd_greatest simp add: Gcd_int_def
wenzelm@63489
  2414
      Lcm_int_def int_dvd_iff dvd_int_iff dvd_int_unfold_dvd_nat [symmetric])
haftmann@62345
  2415
haftmann@62345
  2416
end
haftmann@62345
  2417
wenzelm@63489
  2418
lemma abs_Gcd [simp]: "\<bar>Gcd K\<bar> = Gcd K"
wenzelm@63489
  2419
  for K :: "int set"
haftmann@62346
  2420
  using normalize_Gcd [of K] by simp
haftmann@62346
  2421
wenzelm@63489
  2422
lemma abs_Lcm [simp]: "\<bar>Lcm K\<bar> = Lcm K"
wenzelm@63489
  2423
  for K :: "int set"
haftmann@62346
  2424
  using normalize_Lcm [of K] by simp
haftmann@62346
  2425
wenzelm@63489
  2426
lemma Gcm_eq_int_iff: "Gcd K = int n \<longleftrightarrow> Gcd ((nat \<circ> abs) ` K) = n"
haftmann@62346
  2427
  by (simp add: Gcd_int_def comp_def image_image)
haftmann@62346
  2428
wenzelm@63489
  2429
lemma Lcm_eq_int_iff: "Lcm K = int n \<longleftrightarrow> Lcm ((nat \<circ> abs) ` K) = n"
haftmann@62346
  2430
  by (simp add: Lcm_int_def comp_def image_image)
haftmann@62346
  2431
haftmann@62345
  2432
haftmann@62345
  2433
subsection \<open>GCD and LCM on @{typ integer}\<close>
haftmann@62345
  2434
haftmann@62345
  2435
instantiation integer :: gcd
haftmann@62345
  2436
begin
haftmann@62345
  2437
haftmann@62345
  2438
context
haftmann@62345
  2439
  includes integer.lifting
haftmann@62345
  2440
begin
haftmann@62345
  2441
wenzelm@63489
  2442
lift_definition gcd_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer" is gcd .
wenzelm@63489
  2443
wenzelm@63489
  2444
lift_definition lcm_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer" is lcm .
haftmann@62345
  2445
haftmann@62345
  2446
end
wenzelm@63489
  2447
huffman@45264
  2448
instance ..
haftmann@60686
  2449
wenzelm@21256
  2450
end
huffman@45264
  2451
Andreas@61856
  2452
lifting_update integer.lifting
Andreas@61856
  2453
lifting_forget integer.lifting
Andreas@61856
  2454
haftmann@62345
  2455
context
haftmann@62345
  2456
  includes integer.lifting
haftmann@62345
  2457
begin
Andreas@61856
  2458
wenzelm@63489
  2459
lemma gcd_code_integer [code]: "gcd k l = \<bar>if l = (0::integer) then k else gcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
haftmann@62345
  2460
  by transfer (fact gcd_code_int)
Andreas@61856
  2461
wenzelm@63489
  2462
lemma lcm_code_integer [code]: "lcm a b = \<bar>a\<bar> * \<bar>b\<bar> div gcd a b"
wenzelm@63489
  2463
  for a b :: integer
haftmann@62345
  2464
  by transfer (fact lcm_altdef_int)
Andreas@61856
  2465
Andreas@61856
  2466
end
Andreas@61856
  2467
wenzelm@63489
  2468
code_printing
wenzelm@63489
  2469
  constant "gcd :: integer \<Rightarrow> _" \<rightharpoonup>
wenzelm@63489
  2470
    (OCaml) "Big'_int.gcd'_big'_int"
Andreas@61856
  2471
  and (Haskell) "Prelude.gcd"
Andreas@61856
  2472
  and (Scala) "_.gcd'((_)')"
wenzelm@61975
  2473
  \<comment> \<open>There is no gcd operation in the SML standard library, so no code setup for SML\<close>
Andreas@61856
  2474
haftmann@62344
  2475
text \<open>Some code equations\<close>
haftmann@62344
  2476
eberlm@62429
  2477
lemmas Lcm_set_nat [code, code_unfold] = Lcm_set[where ?'a = nat]
eberlm@62429
  2478
lemmas Gcd_set_nat [code] = Gcd_set[where ?'a = nat]
eberlm@62429
  2479
lemmas Lcm_set_int [code, code_unfold] = Lcm_set[where ?'a = int]
eberlm@62429
  2480
lemmas Gcd_set_int [code] = Gcd_set[where ?'a = int]
haftmann@62344
  2481
haftmann@62344
  2482
wenzelm@63489
  2483
text \<open>Fact aliases.\<close>
wenzelm@63489
  2484
wenzelm@63489
  2485
lemma lcm_0_iff_nat [simp]: "lcm m n = 0 \<longleftrightarrow> m = 0 \<or> n = 0"
wenzelm@63489
  2486
  for m n :: nat
haftmann@62344
  2487
  by (fact lcm_eq_0_iff)
haftmann@62344
  2488
wenzelm@63489
  2489
lemma lcm_0_iff_int [simp]: "lcm m n = 0 \<longleftrightarrow> m = 0 \<or> n = 0"
wenzelm@63489
  2490
  for m n :: int
haftmann@62344
  2491
  by (fact lcm_eq_0_iff)
haftmann@62344
  2492
wenzelm@63489
  2493
lemma dvd_lcm_I1_nat [simp]: "k dvd m \<Longrightarrow> k dvd lcm m n"
wenzelm@63489
  2494
  for k m n :: nat
haftmann@62344
  2495
  by (fact dvd_lcmI1)
haftmann@62344
  2496
wenzelm@63489
  2497
lemma dvd_lcm_I2_nat [simp]: "k dvd n \<Longrightarrow> k dvd lcm m n"
wenzelm@63489
  2498
  for k m n :: nat
haftmann@62344
  2499
  by (fact dvd_lcmI2)
haftmann@62344
  2500
wenzelm@63489
  2501
lemma dvd_lcm_I1_int [simp]: "i dvd m \<Longrightarrow> i dvd lcm m n"
wenzelm@63489
  2502
  for i m n :: int
haftmann@62344
  2503
  by (fact dvd_lcmI1)
haftmann@62344
  2504
wenzelm@63489
  2505
lemma dvd_lcm_I2_int [simp]: "i dvd n \<Longrightarrow> i dvd lcm m n"
wenzelm@63489
  2506
  for i m n :: int
haftmann@62344
  2507
  by (fact dvd_lcmI2)
haftmann@62344
  2508
wenzelm@63489
  2509
lemma coprime_exp2_nat [intro]: "coprime a b \<Longrightarrow> coprime (a^n) (b^m)"
wenzelm@63489
  2510
  for a b :: nat
haftmann@62344
  2511
  by (fact coprime_exp2)
haftmann@62344
  2512
wenzelm@63489
  2513
lemma coprime_exp2_int [intro]: "coprime a b \<Longrightarrow> coprime (a^n) (b^m)"
wenzelm@63489
  2514
  for a b :: int
haftmann@62344
  2515
  by (fact coprime_exp2)
haftmann@62344
  2516
haftmann@62344
  2517
lemmas Gcd_dvd_nat [simp] = Gcd_dvd [where ?'a = nat]
haftmann@62344
  2518
lemmas Gcd_dvd_int [simp] = Gcd_dvd [where ?'a = int]
haftmann@62353
  2519
lemmas Gcd_greatest_nat [simp] = Gcd_greatest [where ?'a = nat]
haftmann@62353
  2520
lemmas Gcd_greatest_int [simp] = Gcd_greatest [where ?'a = int]
haftmann@62344
  2521
wenzelm@63489
  2522
lemma dvd_Lcm_int [simp]: "m \<in> M \<Longrightarrow> m dvd Lcm M"
wenzelm@63489
  2523
  for M :: "int set"
wenzelm@63489
  2524
  by (fact dvd_Lcm)
wenzelm@63489
  2525
wenzelm@63489
  2526
lemma gcd_neg_numeral_1_int [simp]: "gcd (- numeral n :: int) x = gcd (numeral n) x"
haftmann@62344
  2527
  by (fact gcd_neg1_int)
haftmann@62344
  2528
wenzelm@63489
  2529
lemma gcd_neg_numeral_2_int [simp]: "gcd x (- numeral n :: int) = gcd x (numeral n)"
haftmann@62344
  2530
  by (fact gcd_neg2_int)
haftmann@62344
  2531
wenzelm@63489
  2532
lemma gcd_proj1_if_dvd_nat [simp]: "x dvd y \<Longrightarrow> gcd x y = x"
wenzelm@63489
  2533
  for x y :: nat
haftmann@62344
  2534
  by (fact gcd_nat.absorb1)
haftmann@62344
  2535
wenzelm@63489
  2536
lemma gcd_proj2_if_dvd_nat [simp]: "y dvd x \<Longrightarrow> gcd x y = y"
wenzelm@63489
  2537
  for x y :: nat
haftmann@62344
  2538
  by (fact gcd_nat.absorb2)
haftmann@62344
  2539
haftmann@62353
  2540
lemmas Lcm_eq_0_I_nat [simp] = Lcm_eq_0_I [where ?'a = nat]
haftmann@62353
  2541
lemmas Lcm_0_iff_nat [simp] = Lcm_0_iff [where ?'a = nat]
haftmann@62353
  2542
lemmas Lcm_least_int [simp] = Lcm_least [where ?'a = int]
haftmann@62345
  2543
Andreas@61856
  2544
end