src/HOL/GCD.thy
author haftmann
Wed Feb 17 21:51:56 2016 +0100 (2016-02-17)
changeset 62344 759d684c0e60
parent 62343 24106dc44def
child 62345 e66d7841d5a2
permissions -rw-r--r--
pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann@32479
     1
(*  Authors:    Christophe Tabacznyj, Lawrence C. Paulson, Amine Chaieb,
nipkow@31798
     2
                Thomas M. Rasmussen, Jeremy Avigad, Tobias Nipkow
huffman@31706
     3
huffman@31706
     4
haftmann@32479
     5
This file deals with the functions gcd and lcm.  Definitions and
haftmann@32479
     6
lemmas are proved uniformly for the natural numbers and integers.
huffman@31706
     7
huffman@31706
     8
This file combines and revises a number of prior developments.
huffman@31706
     9
huffman@31706
    10
The original theories "GCD" and "Primes" were by Christophe Tabacznyj
wenzelm@58623
    11
and Lawrence C. Paulson, based on @{cite davenport92}. They introduced
huffman@31706
    12
gcd, lcm, and prime for the natural numbers.
huffman@31706
    13
huffman@31706
    14
The original theory "IntPrimes" was by Thomas M. Rasmussen, and
huffman@31706
    15
extended gcd, lcm, primes to the integers. Amine Chaieb provided
huffman@31706
    16
another extension of the notions to the integers, and added a number
huffman@31706
    17
of results to "Primes" and "GCD". IntPrimes also defined and developed
huffman@31706
    18
the congruence relations on the integers. The notion was extended to
berghofe@34915
    19
the natural numbers by Chaieb.
huffman@31706
    20
avigad@32036
    21
Jeremy Avigad combined all of these, made everything uniform for the
avigad@32036
    22
natural numbers and the integers, and added a number of new theorems.
avigad@32036
    23
nipkow@31798
    24
Tobias Nipkow cleaned up a lot.
wenzelm@21256
    25
*)
wenzelm@21256
    26
huffman@31706
    27
wenzelm@60758
    28
section \<open>Greatest common divisor and least common multiple\<close>
wenzelm@21256
    29
wenzelm@21256
    30
theory GCD
lp15@59667
    31
imports Main
huffman@31706
    32
begin
huffman@31706
    33
wenzelm@60758
    34
subsection \<open>GCD and LCM definitions\<close>
huffman@31706
    35
nipkow@31992
    36
class gcd = zero + one + dvd +
wenzelm@41550
    37
  fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
wenzelm@41550
    38
    and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
wenzelm@21256
    39
begin
wenzelm@21256
    40
haftmann@60686
    41
abbreviation coprime :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
haftmann@60686
    42
  where "coprime x y \<equiv> gcd x y = 1"
huffman@31706
    43
huffman@31706
    44
end
huffman@31706
    45
haftmann@60686
    46
class Gcd = gcd +
haftmann@60686
    47
  fixes Gcd :: "'a set \<Rightarrow> 'a"
haftmann@60686
    48
    and Lcm :: "'a set \<Rightarrow> 'a"
haftmann@60686
    49
haftmann@60686
    50
class semiring_gcd = normalization_semidom + gcd +
haftmann@59008
    51
  assumes gcd_dvd1 [iff]: "gcd a b dvd a"
wenzelm@59977
    52
    and gcd_dvd2 [iff]: "gcd a b dvd b"
wenzelm@59977
    53
    and gcd_greatest: "c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> c dvd gcd a b"
haftmann@60686
    54
    and normalize_gcd [simp]: "normalize (gcd a b) = gcd a b"
haftmann@60686
    55
    and lcm_gcd: "lcm a b = normalize (a * b) div gcd a b"
haftmann@60686
    56
begin    
haftmann@60686
    57
haftmann@60689
    58
lemma gcd_greatest_iff [simp]:
haftmann@60686
    59
  "a dvd gcd b c \<longleftrightarrow> a dvd b \<and> a dvd c"
haftmann@60686
    60
  by (blast intro!: gcd_greatest intro: dvd_trans)
haftmann@60686
    61
haftmann@60689
    62
lemma gcd_dvdI1:
haftmann@60689
    63
  "a dvd c \<Longrightarrow> gcd a b dvd c"
haftmann@60689
    64
  by (rule dvd_trans) (rule gcd_dvd1)
haftmann@60689
    65
haftmann@60689
    66
lemma gcd_dvdI2:
haftmann@60689
    67
  "b dvd c \<Longrightarrow> gcd a b dvd c"
haftmann@60689
    68
  by (rule dvd_trans) (rule gcd_dvd2)
haftmann@60689
    69
haftmann@60686
    70
lemma gcd_0_left [simp]:
haftmann@60686
    71
  "gcd 0 a = normalize a"
haftmann@60688
    72
  by (rule associated_eqI) simp_all
haftmann@60686
    73
haftmann@60686
    74
lemma gcd_0_right [simp]:
haftmann@60686
    75
  "gcd a 0 = normalize a"
haftmann@60688
    76
  by (rule associated_eqI) simp_all
haftmann@60686
    77
  
haftmann@60686
    78
lemma gcd_eq_0_iff [simp]:
haftmann@60686
    79
  "gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0" (is "?P \<longleftrightarrow> ?Q")
haftmann@60686
    80
proof
haftmann@60686
    81
  assume ?P then have "0 dvd gcd a b" by simp
haftmann@60686
    82
  then have "0 dvd a" and "0 dvd b" by (blast intro: dvd_trans)+
haftmann@60686
    83
  then show ?Q by simp
haftmann@60686
    84
next
haftmann@60686
    85
  assume ?Q then show ?P by simp
haftmann@60686
    86
qed
haftmann@60686
    87
haftmann@60686
    88
lemma unit_factor_gcd:
haftmann@60686
    89
  "unit_factor (gcd a b) = (if a = 0 \<and> b = 0 then 0 else 1)"
haftmann@60686
    90
proof (cases "gcd a b = 0")
haftmann@60686
    91
  case True then show ?thesis by simp
haftmann@60686
    92
next
haftmann@60686
    93
  case False
haftmann@60686
    94
  have "unit_factor (gcd a b) * normalize (gcd a b) = gcd a b"
haftmann@60686
    95
    by (rule unit_factor_mult_normalize)
haftmann@60686
    96
  then have "unit_factor (gcd a b) * gcd a b = gcd a b"
haftmann@60686
    97
    by simp
haftmann@60686
    98
  then have "unit_factor (gcd a b) * gcd a b div gcd a b = gcd a b div gcd a b"
haftmann@60686
    99
    by simp
haftmann@60686
   100
  with False show ?thesis by simp
haftmann@60686
   101
qed
haftmann@60686
   102
haftmann@60690
   103
lemma is_unit_gcd [simp]:
haftmann@60690
   104
  "is_unit (gcd a b) \<longleftrightarrow> coprime a b"
haftmann@60690
   105
  by (cases "a = 0 \<and> b = 0") (auto simp add: unit_factor_gcd dest: is_unit_unit_factor)
haftmann@60690
   106
wenzelm@61605
   107
sublocale gcd: abel_semigroup gcd
haftmann@60686
   108
proof
haftmann@60686
   109
  fix a b c
haftmann@60686
   110
  show "gcd a b = gcd b a"
haftmann@60688
   111
    by (rule associated_eqI) simp_all
haftmann@60686
   112
  from gcd_dvd1 have "gcd (gcd a b) c dvd a"
haftmann@60686
   113
    by (rule dvd_trans) simp
haftmann@60686
   114
  moreover from gcd_dvd1 have "gcd (gcd a b) c dvd b"
haftmann@60686
   115
    by (rule dvd_trans) simp
haftmann@60686
   116
  ultimately have P1: "gcd (gcd a b) c dvd gcd a (gcd b c)"
haftmann@60686
   117
    by (auto intro!: gcd_greatest)
haftmann@60686
   118
  from gcd_dvd2 have "gcd a (gcd b c) dvd b"
haftmann@60686
   119
    by (rule dvd_trans) simp
haftmann@60686
   120
  moreover from gcd_dvd2 have "gcd a (gcd b c) dvd c"
haftmann@60686
   121
    by (rule dvd_trans) simp
haftmann@60686
   122
  ultimately have P2: "gcd a (gcd b c) dvd gcd (gcd a b) c"
haftmann@60686
   123
    by (auto intro!: gcd_greatest)
haftmann@60688
   124
  from P1 P2 show "gcd (gcd a b) c = gcd a (gcd b c)"
haftmann@60688
   125
    by (rule associated_eqI) simp_all
haftmann@60686
   126
qed
haftmann@60686
   127
haftmann@60686
   128
lemma gcd_self [simp]:
haftmann@60686
   129
  "gcd a a = normalize a"
haftmann@60686
   130
proof -
haftmann@60686
   131
  have "a dvd gcd a a"
haftmann@60686
   132
    by (rule gcd_greatest) simp_all
haftmann@60686
   133
  then show ?thesis
haftmann@60688
   134
    by (auto intro: associated_eqI)
haftmann@60686
   135
qed
haftmann@61913
   136
haftmann@61913
   137
lemma gcd_left_idem [simp]:
haftmann@61913
   138
  "gcd a (gcd a b) = gcd a b"
haftmann@61913
   139
  by (auto intro: associated_eqI)
haftmann@61913
   140
haftmann@61913
   141
lemma gcd_right_idem [simp]:
haftmann@61913
   142
  "gcd (gcd a b) b = gcd a b"
haftmann@61913
   143
  unfolding gcd.commute [of a] gcd.commute [of "gcd b a"] by simp
haftmann@61913
   144
haftmann@60686
   145
lemma coprime_1_left [simp]:
haftmann@60686
   146
  "coprime 1 a"
haftmann@60688
   147
  by (rule associated_eqI) simp_all
haftmann@60686
   148
haftmann@60686
   149
lemma coprime_1_right [simp]:
haftmann@60686
   150
  "coprime a 1"
haftmann@60686
   151
  using coprime_1_left [of a] by (simp add: ac_simps)
haftmann@60686
   152
haftmann@60686
   153
lemma gcd_mult_left:
haftmann@60686
   154
  "gcd (c * a) (c * b) = normalize c * gcd a b"
haftmann@60686
   155
proof (cases "c = 0")
haftmann@60686
   156
  case True then show ?thesis by simp
haftmann@60686
   157
next
haftmann@60686
   158
  case False
haftmann@60686
   159
  then have "c * gcd a b dvd gcd (c * a) (c * b)"
haftmann@60686
   160
    by (auto intro: gcd_greatest)
haftmann@60686
   161
  moreover from calculation False have "gcd (c * a) (c * b) dvd c * gcd a b"
haftmann@60686
   162
    by (metis div_dvd_iff_mult dvd_mult_left gcd_dvd1 gcd_dvd2 gcd_greatest mult_commute)
haftmann@60686
   163
  ultimately have "normalize (gcd (c * a) (c * b)) = normalize (c * gcd a b)"
haftmann@60688
   164
    by (auto intro: associated_eqI)
haftmann@60686
   165
  then show ?thesis by (simp add: normalize_mult)
haftmann@60686
   166
qed
haftmann@60686
   167
haftmann@60686
   168
lemma gcd_mult_right:
haftmann@60686
   169
  "gcd (a * c) (b * c) = gcd b a * normalize c"
haftmann@60686
   170
  using gcd_mult_left [of c a b] by (simp add: ac_simps)
haftmann@60686
   171
haftmann@60686
   172
lemma mult_gcd_left:
haftmann@60686
   173
  "c * gcd a b = unit_factor c * gcd (c * a) (c * b)"
haftmann@60686
   174
  by (simp add: gcd_mult_left mult.assoc [symmetric])
haftmann@60686
   175
haftmann@60686
   176
lemma mult_gcd_right:
haftmann@60686
   177
  "gcd a b * c = gcd (a * c) (b * c) * unit_factor c"
haftmann@60686
   178
  using mult_gcd_left [of c a b] by (simp add: ac_simps)
haftmann@60686
   179
haftmann@60689
   180
lemma dvd_lcm1 [iff]:
haftmann@60686
   181
  "a dvd lcm a b"
haftmann@60686
   182
proof -
haftmann@60686
   183
  have "normalize (a * b) div gcd a b = normalize a * (normalize b div gcd a b)"
haftmann@60686
   184
    by (simp add: lcm_gcd normalize_mult div_mult_swap)
haftmann@60686
   185
  then show ?thesis
haftmann@60686
   186
    by (simp add: lcm_gcd)
haftmann@60686
   187
qed
haftmann@60686
   188
  
haftmann@60689
   189
lemma dvd_lcm2 [iff]:
haftmann@60686
   190
  "b dvd lcm a b"
haftmann@60686
   191
proof -
haftmann@60686
   192
  have "normalize (a * b) div gcd a b = normalize b * (normalize a div gcd a b)"
haftmann@60686
   193
    by (simp add: lcm_gcd normalize_mult div_mult_swap ac_simps)
haftmann@60686
   194
  then show ?thesis
haftmann@60686
   195
    by (simp add: lcm_gcd)
haftmann@60686
   196
qed
haftmann@60686
   197
haftmann@60689
   198
lemma dvd_lcmI1:
haftmann@60689
   199
  "a dvd b \<Longrightarrow> a dvd lcm b c"
haftmann@60689
   200
  by (rule dvd_trans) (assumption, blast) 
haftmann@60689
   201
haftmann@60689
   202
lemma dvd_lcmI2:
haftmann@60689
   203
  "a dvd c \<Longrightarrow> a dvd lcm b c"
haftmann@60689
   204
  by (rule dvd_trans) (assumption, blast)
haftmann@60689
   205
haftmann@60686
   206
lemma lcm_least:
haftmann@60686
   207
  assumes "a dvd c" and "b dvd c"
haftmann@60686
   208
  shows "lcm a b dvd c"
haftmann@60686
   209
proof (cases "c = 0")
haftmann@60686
   210
  case True then show ?thesis by simp
haftmann@60686
   211
next
haftmann@60686
   212
  case False then have U: "is_unit (unit_factor c)" by simp
haftmann@60686
   213
  show ?thesis
haftmann@60686
   214
  proof (cases "gcd a b = 0")
haftmann@60686
   215
    case True with assms show ?thesis by simp
haftmann@60686
   216
  next
haftmann@60686
   217
    case False then have "a \<noteq> 0 \<or> b \<noteq> 0" by simp
haftmann@60686
   218
    with \<open>c \<noteq> 0\<close> assms have "a * b dvd a * c" "a * b dvd c * b"
haftmann@60686
   219
      by (simp_all add: mult_dvd_mono)
haftmann@60686
   220
    then have "normalize (a * b) dvd gcd (a * c) (b * c)"
haftmann@60686
   221
      by (auto intro: gcd_greatest simp add: ac_simps)
haftmann@60686
   222
    then have "normalize (a * b) dvd gcd (a * c) (b * c) * unit_factor c"
haftmann@60686
   223
      using U by (simp add: dvd_mult_unit_iff)
haftmann@60686
   224
    then have "normalize (a * b) dvd gcd a b * c"
haftmann@60686
   225
      by (simp add: mult_gcd_right [of a b c])
haftmann@60686
   226
    then have "normalize (a * b) div gcd a b dvd c"
haftmann@60686
   227
      using False by (simp add: div_dvd_iff_mult ac_simps)
haftmann@60686
   228
    then show ?thesis by (simp add: lcm_gcd)
haftmann@60686
   229
  qed
haftmann@60686
   230
qed
haftmann@60686
   231
haftmann@60689
   232
lemma lcm_least_iff [simp]:
haftmann@60686
   233
  "lcm a b dvd c \<longleftrightarrow> a dvd c \<and> b dvd c"
haftmann@60686
   234
  by (blast intro!: lcm_least intro: dvd_trans)
haftmann@60686
   235
haftmann@60686
   236
lemma normalize_lcm [simp]:
haftmann@60686
   237
  "normalize (lcm a b) = lcm a b"
haftmann@60686
   238
  by (simp add: lcm_gcd dvd_normalize_div)
haftmann@60686
   239
haftmann@60686
   240
lemma lcm_0_left [simp]:
haftmann@60686
   241
  "lcm 0 a = 0"
haftmann@60686
   242
  by (simp add: lcm_gcd)
haftmann@60686
   243
  
haftmann@60686
   244
lemma lcm_0_right [simp]:
haftmann@60686
   245
  "lcm a 0 = 0"
haftmann@60686
   246
  by (simp add: lcm_gcd)
haftmann@60686
   247
haftmann@60686
   248
lemma lcm_eq_0_iff:
haftmann@60686
   249
  "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0" (is "?P \<longleftrightarrow> ?Q")
haftmann@60686
   250
proof
haftmann@60686
   251
  assume ?P then have "0 dvd lcm a b" by simp
haftmann@60686
   252
  then have "0 dvd normalize (a * b) div gcd a b"
haftmann@60686
   253
    by (simp add: lcm_gcd)
haftmann@60686
   254
  then have "0 * gcd a b dvd normalize (a * b)"
haftmann@60686
   255
    using dvd_div_iff_mult [of "gcd a b" _ 0] by (cases "gcd a b = 0") simp_all
haftmann@60686
   256
  then have "normalize (a * b) = 0"
haftmann@60686
   257
    by simp
haftmann@60686
   258
  then show ?Q by simp
haftmann@60686
   259
next
haftmann@60686
   260
  assume ?Q then show ?P by auto
haftmann@60686
   261
qed
haftmann@60686
   262
haftmann@61913
   263
lemma lcm_eq_1_iff [simp]:
haftmann@61913
   264
  "lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b"
haftmann@61913
   265
  by (auto intro: associated_eqI)
haftmann@61913
   266
haftmann@60686
   267
lemma unit_factor_lcm :
haftmann@60686
   268
  "unit_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
haftmann@60686
   269
  by (simp add: unit_factor_gcd dvd_unit_factor_div lcm_gcd)
haftmann@60686
   270
wenzelm@61605
   271
sublocale lcm: abel_semigroup lcm
haftmann@60686
   272
proof
haftmann@60686
   273
  fix a b c
haftmann@60686
   274
  show "lcm a b = lcm b a"
haftmann@60686
   275
    by (simp add: lcm_gcd ac_simps normalize_mult dvd_normalize_div)
haftmann@60688
   276
  have "lcm (lcm a b) c dvd lcm a (lcm b c)"
haftmann@60688
   277
    and "lcm a (lcm b c) dvd lcm (lcm a b) c"
haftmann@60688
   278
    by (auto intro: lcm_least
haftmann@60686
   279
      dvd_trans [of b "lcm b c" "lcm a (lcm b c)"]
haftmann@60686
   280
      dvd_trans [of c "lcm b c" "lcm a (lcm b c)"]
haftmann@60686
   281
      dvd_trans [of a "lcm a b" "lcm (lcm a b) c"]
haftmann@60686
   282
      dvd_trans [of b "lcm a b" "lcm (lcm a b) c"])
haftmann@60686
   283
  then show "lcm (lcm a b) c = lcm a (lcm b c)"
haftmann@60688
   284
    by (rule associated_eqI) simp_all
haftmann@60686
   285
qed
haftmann@60686
   286
haftmann@60686
   287
lemma lcm_self [simp]:
haftmann@60686
   288
  "lcm a a = normalize a"
haftmann@60686
   289
proof -
haftmann@60686
   290
  have "lcm a a dvd a"
haftmann@60686
   291
    by (rule lcm_least) simp_all
haftmann@60686
   292
  then show ?thesis
haftmann@60688
   293
    by (auto intro: associated_eqI)
haftmann@60686
   294
qed
haftmann@60686
   295
haftmann@61913
   296
lemma lcm_left_idem [simp]:
haftmann@61913
   297
  "lcm a (lcm a b) = lcm a b"
haftmann@61913
   298
  by (auto intro: associated_eqI)
haftmann@61913
   299
haftmann@61913
   300
lemma lcm_right_idem [simp]:
haftmann@61913
   301
  "lcm (lcm a b) b = lcm a b"
haftmann@61913
   302
  unfolding lcm.commute [of a] lcm.commute [of "lcm b a"] by simp
haftmann@61913
   303
haftmann@60686
   304
lemma gcd_mult_lcm [simp]:
haftmann@60686
   305
  "gcd a b * lcm a b = normalize a * normalize b"
haftmann@60686
   306
  by (simp add: lcm_gcd normalize_mult)
haftmann@60686
   307
haftmann@60686
   308
lemma lcm_mult_gcd [simp]:
haftmann@60686
   309
  "lcm a b * gcd a b = normalize a * normalize b"
haftmann@60686
   310
  using gcd_mult_lcm [of a b] by (simp add: ac_simps) 
haftmann@60686
   311
haftmann@60686
   312
lemma gcd_lcm:
haftmann@60686
   313
  assumes "a \<noteq> 0" and "b \<noteq> 0"
haftmann@60686
   314
  shows "gcd a b = normalize (a * b) div lcm a b"
haftmann@60686
   315
proof -
haftmann@60686
   316
  from assms have "lcm a b \<noteq> 0"
haftmann@60686
   317
    by (simp add: lcm_eq_0_iff)
haftmann@60686
   318
  have "gcd a b * lcm a b = normalize a * normalize b" by simp
haftmann@60686
   319
  then have "gcd a b * lcm a b div lcm a b = normalize (a * b) div lcm a b"
haftmann@60686
   320
    by (simp_all add: normalize_mult)
haftmann@60686
   321
  with \<open>lcm a b \<noteq> 0\<close> show ?thesis
haftmann@60686
   322
    using nonzero_mult_divide_cancel_right [of "lcm a b" "gcd a b"] by simp
haftmann@60686
   323
qed
haftmann@60686
   324
haftmann@60686
   325
lemma lcm_1_left [simp]:
haftmann@60686
   326
  "lcm 1 a = normalize a"
haftmann@60686
   327
  by (simp add: lcm_gcd)
haftmann@60686
   328
haftmann@60686
   329
lemma lcm_1_right [simp]:
haftmann@60686
   330
  "lcm a 1 = normalize a"
haftmann@60686
   331
  by (simp add: lcm_gcd)
haftmann@60686
   332
  
haftmann@60686
   333
lemma lcm_mult_left:
haftmann@60686
   334
  "lcm (c * a) (c * b) = normalize c * lcm a b"
haftmann@60686
   335
  by (cases "c = 0")
haftmann@60686
   336
    (simp_all add: gcd_mult_right lcm_gcd div_mult_swap normalize_mult ac_simps,
haftmann@60686
   337
      simp add: dvd_div_mult2_eq mult.left_commute [of "normalize c", symmetric])
haftmann@60686
   338
haftmann@60686
   339
lemma lcm_mult_right:
haftmann@60686
   340
  "lcm (a * c) (b * c) = lcm b a * normalize c"
haftmann@60686
   341
  using lcm_mult_left [of c a b] by (simp add: ac_simps)
haftmann@60686
   342
haftmann@60686
   343
lemma mult_lcm_left:
haftmann@60686
   344
  "c * lcm a b = unit_factor c * lcm (c * a) (c * b)"
haftmann@60686
   345
  by (simp add: lcm_mult_left mult.assoc [symmetric])
haftmann@60686
   346
haftmann@60686
   347
lemma mult_lcm_right:
haftmann@60686
   348
  "lcm a b * c = lcm (a * c) (b * c) * unit_factor c"
haftmann@60686
   349
  using mult_lcm_left [of c a b] by (simp add: ac_simps)
haftmann@60686
   350
  
haftmann@60686
   351
end
haftmann@60686
   352
haftmann@60686
   353
class semiring_Gcd = semiring_gcd + Gcd +
haftmann@60686
   354
  assumes Gcd_dvd: "a \<in> A \<Longrightarrow> Gcd A dvd a"
haftmann@60686
   355
    and Gcd_greatest: "(\<And>b. b \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> a dvd Gcd A"
haftmann@60686
   356
    and normalize_Gcd [simp]: "normalize (Gcd A) = Gcd A"
haftmann@60686
   357
begin
haftmann@60686
   358
haftmann@60686
   359
lemma Gcd_empty [simp]:
haftmann@60686
   360
  "Gcd {} = 0"
haftmann@60686
   361
  by (rule dvd_0_left, rule Gcd_greatest) simp
haftmann@60686
   362
haftmann@60686
   363
lemma Gcd_0_iff [simp]:
haftmann@60687
   364
  "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}" (is "?P \<longleftrightarrow> ?Q")
haftmann@60686
   365
proof
haftmann@60686
   366
  assume ?P
haftmann@60686
   367
  show ?Q
haftmann@60686
   368
  proof
haftmann@60686
   369
    fix a
haftmann@60686
   370
    assume "a \<in> A"
haftmann@60686
   371
    then have "Gcd A dvd a" by (rule Gcd_dvd)
haftmann@60687
   372
    with \<open>?P\<close> have "a = 0" by simp
haftmann@60687
   373
    then show "a \<in> {0}" by simp
haftmann@60686
   374
  qed
haftmann@60686
   375
next
haftmann@60686
   376
  assume ?Q
haftmann@60686
   377
  have "0 dvd Gcd A"
haftmann@60686
   378
  proof (rule Gcd_greatest)
haftmann@60686
   379
    fix a
haftmann@60686
   380
    assume "a \<in> A"
haftmann@60687
   381
    with \<open>?Q\<close> have "a = 0" by auto
haftmann@60686
   382
    then show "0 dvd a" by simp
haftmann@60686
   383
  qed
haftmann@60686
   384
  then show ?P by simp
haftmann@60686
   385
qed
haftmann@60686
   386
haftmann@60686
   387
lemma unit_factor_Gcd:
haftmann@60686
   388
  "unit_factor (Gcd A) = (if \<forall>a\<in>A. a = 0 then 0 else 1)"
haftmann@60686
   389
proof (cases "Gcd A = 0")
haftmann@60687
   390
  case True then show ?thesis by auto
haftmann@60686
   391
next
haftmann@60686
   392
  case False
haftmann@60686
   393
  from unit_factor_mult_normalize
haftmann@60686
   394
  have "unit_factor (Gcd A) * normalize (Gcd A) = Gcd A" .
haftmann@60686
   395
  then have "unit_factor (Gcd A) * Gcd A = Gcd A" by simp
haftmann@60686
   396
  then have "unit_factor (Gcd A) * Gcd A div Gcd A = Gcd A div Gcd A" by simp
haftmann@60686
   397
  with False have "unit_factor (Gcd A) = 1" by simp
haftmann@60687
   398
  with False show ?thesis by auto
haftmann@60686
   399
qed
haftmann@60686
   400
haftmann@60686
   401
lemma Gcd_UNIV [simp]:
haftmann@60686
   402
  "Gcd UNIV = 1"
haftmann@60686
   403
  by (rule associated_eqI) (auto intro: Gcd_dvd simp add: unit_factor_Gcd)
haftmann@60686
   404
haftmann@60686
   405
lemma Gcd_eq_1_I:
haftmann@60686
   406
  assumes "is_unit a" and "a \<in> A"
haftmann@60686
   407
  shows "Gcd A = 1"
haftmann@60686
   408
proof -
haftmann@60686
   409
  from assms have "is_unit (Gcd A)"
haftmann@60686
   410
    by (blast intro: Gcd_dvd dvd_unit_imp_unit)
haftmann@60686
   411
  then have "normalize (Gcd A) = 1"
haftmann@60686
   412
    by (rule is_unit_normalize)
haftmann@60686
   413
  then show ?thesis
haftmann@60686
   414
    by simp
haftmann@60686
   415
qed
haftmann@60686
   416
haftmann@60686
   417
lemma Gcd_insert [simp]:
haftmann@60686
   418
  "Gcd (insert a A) = gcd a (Gcd A)"
haftmann@60686
   419
proof -
haftmann@60686
   420
  have "Gcd (insert a A) dvd gcd a (Gcd A)"
haftmann@60686
   421
    by (auto intro: Gcd_dvd Gcd_greatest)
haftmann@60686
   422
  moreover have "gcd a (Gcd A) dvd Gcd (insert a A)"
haftmann@60686
   423
  proof (rule Gcd_greatest)
haftmann@60686
   424
    fix b
haftmann@60686
   425
    assume "b \<in> insert a A"
haftmann@60686
   426
    then show "gcd a (Gcd A) dvd b"
haftmann@60686
   427
    proof
haftmann@60686
   428
      assume "b = a" then show ?thesis by simp
haftmann@60686
   429
    next
haftmann@60686
   430
      assume "b \<in> A"
haftmann@60686
   431
      then have "Gcd A dvd b" by (rule Gcd_dvd)
haftmann@60686
   432
      moreover have "gcd a (Gcd A) dvd Gcd A" by simp
haftmann@60686
   433
      ultimately show ?thesis by (blast intro: dvd_trans)
haftmann@60686
   434
    qed
haftmann@60686
   435
  qed
haftmann@60688
   436
  ultimately show ?thesis
haftmann@60688
   437
    by (auto intro: associated_eqI)
haftmann@60686
   438
qed
haftmann@60686
   439
wenzelm@61799
   440
lemma dvd_Gcd: \<comment> \<open>FIXME remove\<close>
haftmann@60686
   441
  "\<forall>b\<in>A. a dvd b \<Longrightarrow> a dvd Gcd A"
haftmann@60686
   442
  by (blast intro: Gcd_greatest)
haftmann@60686
   443
haftmann@60686
   444
lemma Gcd_set [code_unfold]:
haftmann@60686
   445
  "Gcd (set as) = foldr gcd as 0"
haftmann@60686
   446
  by (induct as) simp_all
haftmann@60686
   447
haftmann@60686
   448
lemma Gcd_image_normalize [simp]:
haftmann@60686
   449
  "Gcd (normalize ` A) = Gcd A"
haftmann@60686
   450
proof -
haftmann@60686
   451
  have "Gcd (normalize ` A) dvd a" if "a \<in> A" for a
haftmann@60686
   452
  proof -
haftmann@60686
   453
    from that obtain B where "A = insert a B" by blast
haftmann@60686
   454
    moreover have " gcd (normalize a) (Gcd (normalize ` B)) dvd normalize a"
haftmann@60686
   455
      by (rule gcd_dvd1)
haftmann@60686
   456
    ultimately show "Gcd (normalize ` A) dvd a"
haftmann@60686
   457
      by simp
haftmann@60686
   458
  qed
haftmann@60688
   459
  then have "Gcd (normalize ` A) dvd Gcd A" and "Gcd A dvd Gcd (normalize ` A)"
haftmann@60688
   460
    by (auto intro!: Gcd_greatest intro: Gcd_dvd)
haftmann@60686
   461
  then show ?thesis
haftmann@60688
   462
    by (auto intro: associated_eqI)
haftmann@60686
   463
qed
haftmann@60686
   464
haftmann@61929
   465
end  
haftmann@61929
   466
haftmann@61929
   467
class semiring_Lcm = semiring_Gcd +
haftmann@61929
   468
  assumes Lcm_Gcd: "Lcm A = Gcd {b. \<forall>a\<in>A. a dvd b}"
haftmann@61929
   469
begin
haftmann@61929
   470
haftmann@61929
   471
lemma dvd_Lcm:
haftmann@61929
   472
  assumes "a \<in> A"
haftmann@61929
   473
  shows "a dvd Lcm A"
haftmann@61929
   474
  using assms by (auto intro: Gcd_greatest simp add: Lcm_Gcd)
haftmann@61929
   475
haftmann@60686
   476
lemma Lcm_least:
haftmann@60686
   477
  assumes "\<And>b. b \<in> A \<Longrightarrow> b dvd a"
haftmann@60686
   478
  shows "Lcm A dvd a"
haftmann@60686
   479
  using assms by (auto intro: Gcd_dvd simp add: Lcm_Gcd)
haftmann@60686
   480
haftmann@60686
   481
lemma normalize_Lcm [simp]:
haftmann@60686
   482
  "normalize (Lcm A) = Lcm A"
haftmann@60686
   483
  by (simp add: Lcm_Gcd)
haftmann@60686
   484
haftmann@60686
   485
lemma unit_factor_Lcm:
haftmann@60686
   486
  "unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
haftmann@60686
   487
proof (cases "Lcm A = 0")
haftmann@60686
   488
  case True then show ?thesis by simp
haftmann@60686
   489
next
haftmann@60686
   490
  case False
haftmann@60686
   491
  with unit_factor_normalize have "unit_factor (normalize (Lcm A)) = 1"
haftmann@60686
   492
    by blast
haftmann@60686
   493
  with False show ?thesis
haftmann@60686
   494
    by simp
haftmann@60686
   495
qed
haftmann@61929
   496
haftmann@61929
   497
lemma Gcd_Lcm:
haftmann@61929
   498
  "Gcd A = Lcm {b. \<forall>a\<in>A. b dvd a}"
haftmann@61929
   499
  by (rule associated_eqI) (auto intro: Gcd_dvd dvd_Lcm Gcd_greatest Lcm_least)
haftmann@61929
   500
 
haftmann@60686
   501
lemma Lcm_empty [simp]:
haftmann@60686
   502
  "Lcm {} = 1"
haftmann@60686
   503
  by (simp add: Lcm_Gcd)
haftmann@60686
   504
haftmann@61929
   505
lemma Lcm_insert [simp]:
haftmann@61929
   506
  "Lcm (insert a A) = lcm a (Lcm A)"
haftmann@61929
   507
proof (rule sym)
haftmann@61929
   508
  have "lcm a (Lcm A) dvd Lcm (insert a A)"
haftmann@61929
   509
    by (auto intro: dvd_Lcm Lcm_least)
haftmann@61929
   510
  moreover have "Lcm (insert a A) dvd lcm a (Lcm A)"
haftmann@61929
   511
  proof (rule Lcm_least)
haftmann@61929
   512
    fix b
haftmann@61929
   513
    assume "b \<in> insert a A"
haftmann@61929
   514
    then show "b dvd lcm a (Lcm A)"
haftmann@61929
   515
    proof
haftmann@61929
   516
      assume "b = a" then show ?thesis by simp
haftmann@61929
   517
    next
haftmann@61929
   518
      assume "b \<in> A"
haftmann@61929
   519
      then have "b dvd Lcm A" by (rule dvd_Lcm)
haftmann@61929
   520
      moreover have "Lcm A dvd lcm a (Lcm A)" by simp
haftmann@61929
   521
      ultimately show ?thesis by (blast intro: dvd_trans)
haftmann@61929
   522
    qed
haftmann@61929
   523
  qed
haftmann@61929
   524
  ultimately show "lcm a (Lcm A) = Lcm (insert a A)"
haftmann@61929
   525
    by (rule associated_eqI) (simp_all add: lcm_eq_0_iff)
haftmann@61929
   526
qed
haftmann@61929
   527
haftmann@60686
   528
lemma Lcm_1_iff [simp]:
haftmann@60686
   529
  "Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)" (is "?P \<longleftrightarrow> ?Q")
haftmann@60686
   530
proof
haftmann@60686
   531
  assume ?P
haftmann@60686
   532
  show ?Q
haftmann@60686
   533
  proof
haftmann@60686
   534
    fix a
haftmann@60686
   535
    assume "a \<in> A"
haftmann@60686
   536
    then have "a dvd Lcm A"
haftmann@60686
   537
      by (rule dvd_Lcm)
haftmann@60686
   538
    with \<open>?P\<close> show "is_unit a"
haftmann@60686
   539
      by simp
haftmann@60686
   540
  qed
haftmann@60686
   541
next
haftmann@60686
   542
  assume ?Q
haftmann@60686
   543
  then have "is_unit (Lcm A)"
haftmann@60686
   544
    by (blast intro: Lcm_least)
haftmann@60686
   545
  then have "normalize (Lcm A) = 1"
haftmann@60686
   546
    by (rule is_unit_normalize)
haftmann@60686
   547
  then show ?P
haftmann@60686
   548
    by simp
haftmann@60686
   549
qed
haftmann@60686
   550
haftmann@60686
   551
lemma Lcm_eq_0_I:
haftmann@60686
   552
  assumes "0 \<in> A"
haftmann@60686
   553
  shows "Lcm A = 0"
haftmann@60686
   554
proof -
haftmann@60686
   555
  from assms have "0 dvd Lcm A"
haftmann@60686
   556
    by (rule dvd_Lcm)
haftmann@60686
   557
  then show ?thesis
haftmann@60686
   558
    by simp
haftmann@60686
   559
qed
haftmann@60686
   560
haftmann@61929
   561
lemma Lcm_UNIV [simp]:
haftmann@61929
   562
  "Lcm UNIV = 0"
haftmann@61929
   563
  by (rule Lcm_eq_0_I) simp
haftmann@60686
   564
haftmann@61929
   565
lemma Lcm_0_iff:
haftmann@61929
   566
  assumes "finite A"
haftmann@61929
   567
  shows "Lcm A = 0 \<longleftrightarrow> 0 \<in> A"
haftmann@61929
   568
proof (cases "A = {}")
haftmann@61929
   569
  case True then show ?thesis by simp
haftmann@61929
   570
next
haftmann@61929
   571
  case False with assms show ?thesis
haftmann@61929
   572
    by (induct A rule: finite_ne_induct)
haftmann@61929
   573
      (auto simp add: lcm_eq_0_iff)
haftmann@60686
   574
qed
haftmann@61929
   575
haftmann@60686
   576
lemma Lcm_set [code_unfold]:
haftmann@60686
   577
  "Lcm (set as) = foldr lcm as 1"
haftmann@60686
   578
  by (induct as) simp_all
haftmann@60686
   579
  
haftmann@60686
   580
end
haftmann@59008
   581
haftmann@59008
   582
class ring_gcd = comm_ring_1 + semiring_gcd
haftmann@59008
   583
huffman@31706
   584
instantiation nat :: gcd
huffman@31706
   585
begin
wenzelm@21256
   586
huffman@31706
   587
fun
huffman@31706
   588
  gcd_nat  :: "nat \<Rightarrow> nat \<Rightarrow> nat"
huffman@31706
   589
where
huffman@31706
   590
  "gcd_nat x y =
huffman@31706
   591
   (if y = 0 then x else gcd y (x mod y))"
huffman@31706
   592
huffman@31706
   593
definition
huffman@31706
   594
  lcm_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
huffman@31706
   595
where
huffman@31706
   596
  "lcm_nat x y = x * y div (gcd x y)"
huffman@31706
   597
huffman@31706
   598
instance proof qed
huffman@31706
   599
huffman@31706
   600
end
huffman@31706
   601
huffman@31706
   602
instantiation int :: gcd
huffman@31706
   603
begin
wenzelm@21256
   604
huffman@31706
   605
definition
huffman@31706
   606
  gcd_int  :: "int \<Rightarrow> int \<Rightarrow> int"
huffman@31706
   607
where
wenzelm@61944
   608
  "gcd_int x y = int (gcd (nat \<bar>x\<bar>) (nat \<bar>y\<bar>))"
haftmann@23687
   609
huffman@31706
   610
definition
huffman@31706
   611
  lcm_int :: "int \<Rightarrow> int \<Rightarrow> int"
huffman@31706
   612
where
wenzelm@61944
   613
  "lcm_int x y = int (lcm (nat \<bar>x\<bar>) (nat \<bar>y\<bar>))"
haftmann@23687
   614
wenzelm@61944
   615
instance ..
huffman@31706
   616
huffman@31706
   617
end
haftmann@23687
   618
haftmann@23687
   619
wenzelm@60758
   620
subsection \<open>Transfer setup\<close>
huffman@31706
   621
huffman@31706
   622
lemma transfer_nat_int_gcd:
huffman@31706
   623
  "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> gcd (nat x) (nat y) = nat (gcd x y)"
huffman@31706
   624
  "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> lcm (nat x) (nat y) = nat (lcm x y)"
haftmann@32479
   625
  unfolding gcd_int_def lcm_int_def
huffman@31706
   626
  by auto
haftmann@23687
   627
huffman@31706
   628
lemma transfer_nat_int_gcd_closures:
huffman@31706
   629
  "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> gcd x y >= 0"
huffman@31706
   630
  "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> lcm x y >= 0"
huffman@31706
   631
  by (auto simp add: gcd_int_def lcm_int_def)
huffman@31706
   632
haftmann@35644
   633
declare transfer_morphism_nat_int[transfer add return:
huffman@31706
   634
    transfer_nat_int_gcd transfer_nat_int_gcd_closures]
huffman@31706
   635
huffman@31706
   636
lemma transfer_int_nat_gcd:
huffman@31706
   637
  "gcd (int x) (int y) = int (gcd x y)"
huffman@31706
   638
  "lcm (int x) (int y) = int (lcm x y)"
haftmann@32479
   639
  by (unfold gcd_int_def lcm_int_def, auto)
huffman@31706
   640
huffman@31706
   641
lemma transfer_int_nat_gcd_closures:
huffman@31706
   642
  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> gcd x y >= 0"
huffman@31706
   643
  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> lcm x y >= 0"
huffman@31706
   644
  by (auto simp add: gcd_int_def lcm_int_def)
huffman@31706
   645
haftmann@35644
   646
declare transfer_morphism_int_nat[transfer add return:
huffman@31706
   647
    transfer_int_nat_gcd transfer_int_nat_gcd_closures]
huffman@31706
   648
huffman@31706
   649
wenzelm@60758
   650
subsection \<open>GCD properties\<close>
huffman@31706
   651
huffman@31706
   652
(* was gcd_induct *)
nipkow@31952
   653
lemma gcd_nat_induct:
haftmann@23687
   654
  fixes m n :: nat
haftmann@23687
   655
  assumes "\<And>m. P m 0"
haftmann@23687
   656
    and "\<And>m n. 0 < n \<Longrightarrow> P n (m mod n) \<Longrightarrow> P m n"
haftmann@23687
   657
  shows "P m n"
huffman@31706
   658
  apply (rule gcd_nat.induct)
huffman@31706
   659
  apply (case_tac "y = 0")
huffman@31706
   660
  using assms apply simp_all
huffman@31706
   661
done
huffman@31706
   662
huffman@31706
   663
(* specific to int *)
huffman@31706
   664
nipkow@31952
   665
lemma gcd_neg1_int [simp]: "gcd (-x::int) y = gcd x y"
huffman@31706
   666
  by (simp add: gcd_int_def)
huffman@31706
   667
nipkow@31952
   668
lemma gcd_neg2_int [simp]: "gcd (x::int) (-y) = gcd x y"
huffman@31706
   669
  by (simp add: gcd_int_def)
huffman@31706
   670
wenzelm@61944
   671
lemma abs_gcd_int[simp]: "\<bar>gcd (x::int) y\<bar> = gcd x y"
nipkow@31813
   672
by(simp add: gcd_int_def)
nipkow@31813
   673
wenzelm@61944
   674
lemma gcd_abs_int: "gcd (x::int) y = gcd \<bar>x\<bar> \<bar>y\<bar>"
nipkow@31813
   675
by (simp add: gcd_int_def)
nipkow@31813
   676
wenzelm@61944
   677
lemma gcd_abs1_int[simp]: "gcd \<bar>x\<bar> (y::int) = gcd x y"
nipkow@31952
   678
by (metis abs_idempotent gcd_abs_int)
nipkow@31813
   679
wenzelm@61944
   680
lemma gcd_abs2_int[simp]: "gcd x \<bar>y::int\<bar> = gcd x y"
nipkow@31952
   681
by (metis abs_idempotent gcd_abs_int)
huffman@31706
   682
nipkow@31952
   683
lemma gcd_cases_int:
huffman@31706
   684
  fixes x :: int and y
huffman@31706
   685
  assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd x y)"
huffman@31706
   686
      and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd x (-y))"
huffman@31706
   687
      and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd (-x) y)"
huffman@31706
   688
      and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd (-x) (-y))"
huffman@31706
   689
  shows "P (gcd x y)"
wenzelm@61944
   690
  by (insert assms, auto, arith)
wenzelm@21256
   691
nipkow@31952
   692
lemma gcd_ge_0_int [simp]: "gcd (x::int) y >= 0"
huffman@31706
   693
  by (simp add: gcd_int_def)
huffman@31706
   694
nipkow@31952
   695
lemma lcm_neg1_int: "lcm (-x::int) y = lcm x y"
huffman@31706
   696
  by (simp add: lcm_int_def)
huffman@31706
   697
nipkow@31952
   698
lemma lcm_neg2_int: "lcm (x::int) (-y) = lcm x y"
huffman@31706
   699
  by (simp add: lcm_int_def)
huffman@31706
   700
wenzelm@61944
   701
lemma lcm_abs_int: "lcm (x::int) y = lcm \<bar>x\<bar> \<bar>y\<bar>"
huffman@31706
   702
  by (simp add: lcm_int_def)
wenzelm@21256
   703
wenzelm@61944
   704
lemma abs_lcm_int [simp]: "\<bar>lcm i j::int\<bar> = lcm i j"
wenzelm@61944
   705
  by (simp add:lcm_int_def)
nipkow@31814
   706
wenzelm@61944
   707
lemma lcm_abs1_int[simp]: "lcm \<bar>x\<bar> (y::int) = lcm x y"
wenzelm@61944
   708
  by (metis abs_idempotent lcm_int_def)
nipkow@31814
   709
wenzelm@61944
   710
lemma lcm_abs2_int[simp]: "lcm x \<bar>y::int\<bar> = lcm x y"
wenzelm@61944
   711
  by (metis abs_idempotent lcm_int_def)
nipkow@31814
   712
nipkow@31952
   713
lemma lcm_cases_int:
huffman@31706
   714
  fixes x :: int and y
huffman@31706
   715
  assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm x y)"
huffman@31706
   716
      and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm x (-y))"
huffman@31706
   717
      and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm (-x) y)"
huffman@31706
   718
      and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm (-x) (-y))"
huffman@31706
   719
  shows "P (lcm x y)"
wenzelm@41550
   720
  using assms by (auto simp add: lcm_neg1_int lcm_neg2_int) arith
huffman@31706
   721
nipkow@31952
   722
lemma lcm_ge_0_int [simp]: "lcm (x::int) y >= 0"
huffman@31706
   723
  by (simp add: lcm_int_def)
huffman@31706
   724
huffman@31706
   725
(* was gcd_0, etc. *)
haftmann@54867
   726
lemma gcd_0_nat: "gcd (x::nat) 0 = x"
haftmann@23687
   727
  by simp
haftmann@23687
   728
huffman@31706
   729
(* was igcd_0, etc. *)
wenzelm@61944
   730
lemma gcd_0_int [simp]: "gcd (x::int) 0 = \<bar>x\<bar>"
huffman@31706
   731
  by (unfold gcd_int_def, auto)
huffman@31706
   732
haftmann@54867
   733
lemma gcd_0_left_nat: "gcd 0 (x::nat) = x"
haftmann@23687
   734
  by simp
haftmann@23687
   735
wenzelm@61944
   736
lemma gcd_0_left_int [simp]: "gcd 0 (x::int) = \<bar>x\<bar>"
huffman@31706
   737
  by (unfold gcd_int_def, auto)
huffman@31706
   738
nipkow@31952
   739
lemma gcd_red_nat: "gcd (x::nat) y = gcd y (x mod y)"
huffman@31706
   740
  by (case_tac "y = 0", auto)
huffman@31706
   741
huffman@31706
   742
(* weaker, but useful for the simplifier *)
huffman@31706
   743
nipkow@31952
   744
lemma gcd_non_0_nat: "y ~= (0::nat) \<Longrightarrow> gcd (x::nat) y = gcd y (x mod y)"
huffman@31706
   745
  by simp
huffman@31706
   746
nipkow@31952
   747
lemma gcd_1_nat [simp]: "gcd (m::nat) 1 = 1"
wenzelm@21263
   748
  by simp
wenzelm@21256
   749
nipkow@31952
   750
lemma gcd_Suc_0 [simp]: "gcd (m::nat) (Suc 0) = Suc 0"
haftmann@60690
   751
  by simp
huffman@31706
   752
nipkow@31952
   753
lemma gcd_1_int [simp]: "gcd (m::int) 1 = 1"
huffman@31706
   754
  by (simp add: gcd_int_def)
huffman@30082
   755
nipkow@31952
   756
lemma gcd_idem_nat: "gcd (x::nat) x = x"
nipkow@31798
   757
by simp
huffman@31706
   758
wenzelm@61944
   759
lemma gcd_idem_int: "gcd (x::int) x = \<bar>x\<bar>"
nipkow@31813
   760
by (auto simp add: gcd_int_def)
huffman@31706
   761
huffman@31706
   762
declare gcd_nat.simps [simp del]
wenzelm@21256
   763
wenzelm@60758
   764
text \<open>
wenzelm@61799
   765
  \medskip @{term "gcd m n"} divides \<open>m\<close> and \<open>n\<close>.  The
wenzelm@21256
   766
  conjunctions don't seem provable separately.
wenzelm@60758
   767
\<close>
wenzelm@21256
   768
haftmann@59008
   769
instance nat :: semiring_gcd
haftmann@59008
   770
proof
haftmann@59008
   771
  fix m n :: nat
haftmann@59008
   772
  show "gcd m n dvd m" and "gcd m n dvd n"
haftmann@59008
   773
  proof (induct m n rule: gcd_nat_induct)
haftmann@59008
   774
    fix m n :: nat
haftmann@59008
   775
    assume "gcd n (m mod n) dvd m mod n" and "gcd n (m mod n) dvd n"
haftmann@59008
   776
    then have "gcd n (m mod n) dvd m"
haftmann@59008
   777
      by (rule dvd_mod_imp_dvd)
haftmann@59008
   778
    moreover assume "0 < n"
haftmann@59008
   779
    ultimately show "gcd m n dvd m"
haftmann@59008
   780
      by (simp add: gcd_non_0_nat)
haftmann@59008
   781
  qed (simp_all add: gcd_0_nat gcd_non_0_nat)
haftmann@59008
   782
next
haftmann@59008
   783
  fix m n k :: nat
haftmann@59008
   784
  assume "k dvd m" and "k dvd n"
haftmann@59008
   785
  then show "k dvd gcd m n"
haftmann@59008
   786
    by (induct m n rule: gcd_nat_induct) (simp_all add: gcd_non_0_nat dvd_mod gcd_0_nat)
haftmann@60686
   787
qed (simp_all add: lcm_nat_def)
lp15@59667
   788
haftmann@59008
   789
instance int :: ring_gcd
haftmann@60686
   790
  by standard
haftmann@60686
   791
    (simp_all add: dvd_int_unfold_dvd_nat gcd_int_def lcm_int_def zdiv_int nat_abs_mult_distrib [symmetric] lcm_gcd gcd_greatest)
lp15@59667
   792
nipkow@31730
   793
lemma dvd_gcd_D1_nat: "k dvd gcd m n \<Longrightarrow> (k::nat) dvd m"
haftmann@59008
   794
  by (metis gcd_dvd1 dvd_trans)
nipkow@31730
   795
nipkow@31730
   796
lemma dvd_gcd_D2_nat: "k dvd gcd m n \<Longrightarrow> (k::nat) dvd n"
haftmann@59008
   797
  by (metis gcd_dvd2 dvd_trans)
nipkow@31730
   798
nipkow@31730
   799
lemma dvd_gcd_D1_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd m"
haftmann@59008
   800
  by (metis gcd_dvd1 dvd_trans)
nipkow@31730
   801
nipkow@31730
   802
lemma dvd_gcd_D2_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd n"
haftmann@59008
   803
  by (metis gcd_dvd2 dvd_trans)
nipkow@31730
   804
nipkow@31952
   805
lemma gcd_le1_nat [simp]: "a \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> a"
huffman@31706
   806
  by (rule dvd_imp_le, auto)
huffman@31706
   807
nipkow@31952
   808
lemma gcd_le2_nat [simp]: "b \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> b"
huffman@31706
   809
  by (rule dvd_imp_le, auto)
huffman@31706
   810
nipkow@31952
   811
lemma gcd_le1_int [simp]: "a > 0 \<Longrightarrow> gcd (a::int) b \<le> a"
huffman@31706
   812
  by (rule zdvd_imp_le, auto)
wenzelm@21256
   813
nipkow@31952
   814
lemma gcd_le2_int [simp]: "b > 0 \<Longrightarrow> gcd (a::int) b \<le> b"
huffman@31706
   815
  by (rule zdvd_imp_le, auto)
huffman@31706
   816
nipkow@31952
   817
lemma gcd_pos_nat [simp]: "(gcd (m::nat) n > 0) = (m ~= 0 | n ~= 0)"
haftmann@62344
   818
  by (insert gcd_eq_0_iff [of m n], arith)
wenzelm@21256
   819
nipkow@31952
   820
lemma gcd_pos_int [simp]: "(gcd (m::int) n > 0) = (m ~= 0 | n ~= 0)"
haftmann@62344
   821
  by (insert gcd_eq_0_iff [of m n], insert gcd_ge_0_int [of m n], arith)
huffman@31706
   822
nipkow@31952
   823
lemma gcd_unique_nat: "(d::nat) dvd a \<and> d dvd b \<and>
huffman@31706
   824
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
huffman@31706
   825
  apply auto
nipkow@33657
   826
  apply (rule dvd_antisym)
haftmann@59008
   827
  apply (erule (1) gcd_greatest)
huffman@31706
   828
  apply auto
huffman@31706
   829
done
wenzelm@21256
   830
nipkow@31952
   831
lemma gcd_unique_int: "d >= 0 & (d::int) dvd a \<and> d dvd b \<and>
huffman@31706
   832
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
nipkow@33657
   833
apply (case_tac "d = 0")
nipkow@33657
   834
 apply simp
nipkow@33657
   835
apply (rule iffI)
nipkow@33657
   836
 apply (rule zdvd_antisym_nonneg)
haftmann@59008
   837
 apply (auto intro: gcd_greatest)
huffman@31706
   838
done
huffman@30082
   839
haftmann@61913
   840
interpretation gcd_nat:
haftmann@62344
   841
  semilattice_neutr_order gcd "0::nat" Rings.dvd "\<lambda>m n. m dvd n \<and> m \<noteq> n"
haftmann@62344
   842
  by standard (auto simp add: gcd_unique_nat [symmetric] intro: dvd_antisym dvd_trans)
nipkow@31798
   843
wenzelm@61944
   844
lemma gcd_proj1_if_dvd_int [simp]: "x dvd y \<Longrightarrow> gcd (x::int) y = \<bar>x\<bar>"
haftmann@54867
   845
  by (metis abs_dvd_iff gcd_0_left_int gcd_abs_int gcd_unique_int)
nipkow@31798
   846
wenzelm@61944
   847
lemma gcd_proj2_if_dvd_int [simp]: "y dvd x \<Longrightarrow> gcd (x::int) y = \<bar>y\<bar>"
haftmann@62344
   848
  by (metis gcd_proj1_if_dvd_int gcd.commute)
nipkow@31798
   849
wenzelm@60758
   850
text \<open>
wenzelm@21256
   851
  \medskip Multiplication laws
wenzelm@60758
   852
\<close>
wenzelm@21256
   853
nipkow@31952
   854
lemma gcd_mult_distrib_nat: "(k::nat) * gcd m n = gcd (k * m) (k * n)"
wenzelm@61799
   855
    \<comment> \<open>@{cite \<open>page 27\<close> davenport92}\<close>
nipkow@31952
   856
  apply (induct m n rule: gcd_nat_induct)
huffman@31706
   857
  apply simp
wenzelm@21256
   858
  apply (case_tac "k = 0")
huffman@45270
   859
  apply (simp_all add: gcd_non_0_nat)
huffman@31706
   860
done
wenzelm@21256
   861
wenzelm@61944
   862
lemma gcd_mult_distrib_int: "\<bar>k::int\<bar> * gcd m n = gcd (k * m) (k * n)"
nipkow@31952
   863
  apply (subst (1 2) gcd_abs_int)
nipkow@31813
   864
  apply (subst (1 2) abs_mult)
nipkow@31952
   865
  apply (rule gcd_mult_distrib_nat [transferred])
huffman@31706
   866
  apply auto
huffman@31706
   867
done
wenzelm@21256
   868
haftmann@60690
   869
context semiring_gcd
haftmann@60690
   870
begin
wenzelm@21256
   871
haftmann@60690
   872
lemma coprime_dvd_mult:
haftmann@60690
   873
  assumes "coprime a b" and "a dvd c * b"
haftmann@60690
   874
  shows "a dvd c"
haftmann@60690
   875
proof (cases "c = 0")
haftmann@60690
   876
  case True then show ?thesis by simp
haftmann@60690
   877
next
haftmann@60690
   878
  case False
haftmann@60690
   879
  then have unit: "is_unit (unit_factor c)" by simp
haftmann@60690
   880
  from \<open>coprime a b\<close> mult_gcd_left [of c a b]
haftmann@60690
   881
  have "gcd (c * a) (c * b) * unit_factor c = c"
haftmann@60690
   882
    by (simp add: ac_simps)
haftmann@60690
   883
  moreover from \<open>a dvd c * b\<close> have "a dvd gcd (c * a) (c * b) * unit_factor c"
haftmann@60690
   884
    by (simp add: dvd_mult_unit_iff unit)
haftmann@60690
   885
  ultimately show ?thesis by simp
haftmann@60690
   886
qed
haftmann@60690
   887
haftmann@62344
   888
lemma coprime_dvd_mult_iff:
haftmann@62344
   889
  assumes "coprime a c"
haftmann@62344
   890
  shows "a dvd b * c \<longleftrightarrow> a dvd b"
haftmann@62344
   891
  using assms by (auto intro: coprime_dvd_mult)
haftmann@60690
   892
haftmann@60690
   893
lemma gcd_mult_cancel:
haftmann@60690
   894
  "coprime c b \<Longrightarrow> gcd (c * a) b = gcd a b"
haftmann@60690
   895
  apply (rule associated_eqI)
haftmann@59008
   896
  apply (rule gcd_greatest)
haftmann@60690
   897
  apply (rule_tac b = c in coprime_dvd_mult)
haftmann@60690
   898
  apply (simp add: gcd.assoc)
haftmann@60690
   899
  apply (simp_all add: ac_simps)
haftmann@60690
   900
  done
wenzelm@21256
   901
haftmann@62344
   902
lemma coprime_crossproduct:
haftmann@62344
   903
  fixes a b c d
haftmann@35368
   904
  assumes "coprime a d" and "coprime b c"
haftmann@62344
   905
  shows "normalize a * normalize c = normalize b * normalize d
haftmann@62344
   906
    \<longleftrightarrow> normalize a = normalize b \<and> normalize c = normalize d" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@35368
   907
proof
haftmann@35368
   908
  assume ?rhs then show ?lhs by simp
haftmann@35368
   909
next
haftmann@35368
   910
  assume ?lhs
haftmann@62344
   911
  from \<open>?lhs\<close> have "normalize a dvd normalize b * normalize d"
haftmann@62344
   912
    by (auto intro: dvdI dest: sym)
haftmann@62344
   913
  with \<open>coprime a d\<close> have "a dvd b"
haftmann@62344
   914
    by (simp add: coprime_dvd_mult_iff normalize_mult [symmetric])
haftmann@62344
   915
  from \<open>?lhs\<close> have "normalize b dvd normalize a * normalize c"
haftmann@62344
   916
    by (auto intro: dvdI dest: sym)
haftmann@62344
   917
  with \<open>coprime b c\<close> have "b dvd a"
haftmann@62344
   918
    by (simp add: coprime_dvd_mult_iff normalize_mult [symmetric])
haftmann@62344
   919
  from \<open>?lhs\<close> have "normalize c dvd normalize d * normalize b"
haftmann@62344
   920
    by (auto intro: dvdI dest: sym simp add: mult.commute)
haftmann@62344
   921
  with \<open>coprime b c\<close> have "c dvd d"
haftmann@62344
   922
    by (simp add: coprime_dvd_mult_iff gcd.commute normalize_mult [symmetric])
haftmann@62344
   923
  from \<open>?lhs\<close> have "normalize d dvd normalize c * normalize a"
haftmann@62344
   924
    by (auto intro: dvdI dest: sym simp add: mult.commute)
haftmann@62344
   925
  with \<open>coprime a d\<close> have "d dvd c"
haftmann@62344
   926
    by (simp add: coprime_dvd_mult_iff gcd.commute normalize_mult [symmetric])
haftmann@62344
   927
  from \<open>a dvd b\<close> \<open>b dvd a\<close> have "normalize a = normalize b"
haftmann@62344
   928
    by (rule associatedI)
haftmann@62344
   929
  moreover from \<open>c dvd d\<close> \<open>d dvd c\<close> have "normalize c = normalize d"
haftmann@62344
   930
    by (rule associatedI)
haftmann@35368
   931
  ultimately show ?rhs ..
haftmann@35368
   932
qed
haftmann@35368
   933
haftmann@62344
   934
end
haftmann@62344
   935
haftmann@62344
   936
lemma coprime_crossproduct_nat:
haftmann@62344
   937
  fixes a b c d :: nat
haftmann@62344
   938
  assumes "coprime a d" and "coprime b c"
haftmann@62344
   939
  shows "a * c = b * d \<longleftrightarrow> a = b \<and> c = d"
haftmann@62344
   940
  using assms coprime_crossproduct [of a d b c] by simp
haftmann@62344
   941
haftmann@35368
   942
lemma coprime_crossproduct_int:
haftmann@35368
   943
  fixes a b c d :: int
haftmann@35368
   944
  assumes "coprime a d" and "coprime b c"
haftmann@35368
   945
  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
   946
  using assms coprime_crossproduct [of a d b c] by simp
haftmann@35368
   947
wenzelm@60758
   948
text \<open>\medskip Addition laws\<close>
wenzelm@21256
   949
nipkow@31952
   950
lemma gcd_add1_nat [simp]: "gcd ((m::nat) + n) n = gcd m n"
huffman@31706
   951
  apply (case_tac "n = 0")
nipkow@31952
   952
  apply (simp_all add: gcd_non_0_nat)
haftmann@62344
   953
  done
huffman@31706
   954
nipkow@31952
   955
lemma gcd_add2_nat [simp]: "gcd (m::nat) (m + n) = gcd m n"
haftmann@62344
   956
  apply (subst (1 2) gcd.commute)
haftmann@57512
   957
  apply (subst add.commute)
huffman@31706
   958
  apply simp
haftmann@62344
   959
  done
huffman@31706
   960
huffman@31706
   961
(* to do: add the other variations? *)
huffman@31706
   962
nipkow@31952
   963
lemma gcd_diff1_nat: "(m::nat) >= n \<Longrightarrow> gcd (m - n) n = gcd m n"
haftmann@62344
   964
  by (subst gcd_add1_nat [symmetric]) auto
huffman@31706
   965
nipkow@31952
   966
lemma gcd_diff2_nat: "(n::nat) >= m \<Longrightarrow> gcd (n - m) n = gcd m n"
haftmann@62344
   967
  apply (subst gcd.commute)
nipkow@31952
   968
  apply (subst gcd_diff1_nat [symmetric])
huffman@31706
   969
  apply auto
haftmann@62344
   970
  apply (subst gcd.commute)
nipkow@31952
   971
  apply (subst gcd_diff1_nat)
huffman@31706
   972
  apply assumption
haftmann@62344
   973
  apply (rule gcd.commute)
haftmann@62344
   974
  done
huffman@31706
   975
nipkow@31952
   976
lemma gcd_non_0_int: "(y::int) > 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
huffman@31706
   977
  apply (frule_tac b = y and a = x in pos_mod_sign)
huffman@31706
   978
  apply (simp del: pos_mod_sign add: gcd_int_def abs_if nat_mod_distrib)
nipkow@31952
   979
  apply (auto simp add: gcd_non_0_nat nat_mod_distrib [symmetric]
huffman@31706
   980
    zmod_zminus1_eq_if)
huffman@31706
   981
  apply (frule_tac a = x in pos_mod_bound)
haftmann@62344
   982
  apply (subst (1 2) gcd.commute)
nipkow@31952
   983
  apply (simp del: pos_mod_bound add: nat_diff_distrib gcd_diff2_nat
huffman@31706
   984
    nat_le_eq_zle)
haftmann@62344
   985
  done
wenzelm@21256
   986
nipkow@31952
   987
lemma gcd_red_int: "gcd (x::int) y = gcd y (x mod y)"
huffman@31706
   988
  apply (case_tac "y = 0")
huffman@31706
   989
  apply force
huffman@31706
   990
  apply (case_tac "y > 0")
nipkow@31952
   991
  apply (subst gcd_non_0_int, auto)
nipkow@31952
   992
  apply (insert gcd_non_0_int [of "-y" "-x"])
huffman@35216
   993
  apply auto
huffman@31706
   994
done
huffman@31706
   995
nipkow@31952
   996
lemma gcd_add1_int [simp]: "gcd ((m::int) + n) n = gcd m n"
haftmann@57512
   997
by (metis gcd_red_int mod_add_self1 add.commute)
huffman@31706
   998
nipkow@31952
   999
lemma gcd_add2_int [simp]: "gcd m ((m::int) + n) = gcd m n"
haftmann@62344
  1000
by (metis gcd_add1_int gcd.commute add.commute)
wenzelm@21256
  1001
nipkow@31952
  1002
lemma gcd_add_mult_nat: "gcd (m::nat) (k * m + n) = gcd m n"
haftmann@62344
  1003
by (metis mod_mult_self3 gcd.commute gcd_red_nat)
wenzelm@21256
  1004
nipkow@31952
  1005
lemma gcd_add_mult_int: "gcd (m::int) (k * m + n) = gcd m n"
haftmann@62344
  1006
by (metis gcd.commute gcd_red_int mod_mult_self1 add.commute)
nipkow@31798
  1007
wenzelm@21256
  1008
huffman@31706
  1009
(* to do: differences, and all variations of addition rules
huffman@31706
  1010
    as simplification rules for nat and int *)
huffman@31706
  1011
haftmann@60689
  1012
lemma gcd_dvd_prod_nat: "gcd (m::nat) n dvd k * n"
haftmann@23687
  1013
  using mult_dvd_mono [of 1] by auto
chaieb@22027
  1014
huffman@31706
  1015
(* to do: add the three variations of these, and for ints? *)
huffman@31706
  1016
nipkow@31992
  1017
lemma finite_divisors_nat[simp]:
nipkow@31992
  1018
  assumes "(m::nat) ~= 0" shows "finite{d. d dvd m}"
nipkow@31734
  1019
proof-
wenzelm@60512
  1020
  have "finite{d. d <= m}"
wenzelm@60512
  1021
    by (blast intro: bounded_nat_set_is_finite)
nipkow@31734
  1022
  from finite_subset[OF _ this] show ?thesis using assms
wenzelm@60512
  1023
    by (metis Collect_mono dvd_imp_le neq0_conv)
nipkow@31734
  1024
qed
nipkow@31734
  1025
nipkow@31995
  1026
lemma finite_divisors_int[simp]:
nipkow@31734
  1027
  assumes "(i::int) ~= 0" shows "finite{d. d dvd i}"
nipkow@31734
  1028
proof-
wenzelm@61944
  1029
  have "{d. \<bar>d\<bar> <= \<bar>i\<bar>} = {- \<bar>i\<bar> .. \<bar>i\<bar>}" by(auto simp:abs_if)
wenzelm@61944
  1030
  hence "finite {d. \<bar>d\<bar> <= \<bar>i\<bar>}" by simp
nipkow@31734
  1031
  from finite_subset[OF _ this] show ?thesis using assms
wenzelm@60512
  1032
    by (simp add: dvd_imp_le_int subset_iff)
nipkow@31734
  1033
qed
nipkow@31734
  1034
nipkow@31995
  1035
lemma Max_divisors_self_nat[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::nat. d dvd n} = n"
nipkow@31995
  1036
apply(rule antisym)
nipkow@44890
  1037
 apply (fastforce intro: Max_le_iff[THEN iffD2] simp: dvd_imp_le)
nipkow@31995
  1038
apply simp
nipkow@31995
  1039
done
nipkow@31995
  1040
wenzelm@61944
  1041
lemma Max_divisors_self_int[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::int. d dvd n} = \<bar>n\<bar>"
nipkow@31995
  1042
apply(rule antisym)
haftmann@44278
  1043
 apply(rule Max_le_iff [THEN iffD2])
haftmann@44278
  1044
  apply (auto intro: abs_le_D1 dvd_imp_le_int)
nipkow@31995
  1045
done
nipkow@31995
  1046
nipkow@31734
  1047
lemma gcd_is_Max_divisors_nat:
nipkow@31734
  1048
  "m ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> gcd (m::nat) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
  1049
apply(rule Max_eqI[THEN sym])
nipkow@31995
  1050
  apply (metis finite_Collect_conjI finite_divisors_nat)
nipkow@31734
  1051
 apply simp
haftmann@62344
  1052
 apply(metis Suc_diff_1 Suc_neq_Zero dvd_imp_le gcd_greatest_iff gcd_pos_nat)
nipkow@31734
  1053
apply simp
nipkow@31734
  1054
done
nipkow@31734
  1055
nipkow@31734
  1056
lemma gcd_is_Max_divisors_int:
nipkow@31734
  1057
  "m ~= 0 ==> n ~= 0 ==> gcd (m::int) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
  1058
apply(rule Max_eqI[THEN sym])
nipkow@31995
  1059
  apply (metis finite_Collect_conjI finite_divisors_int)
nipkow@31734
  1060
 apply simp
haftmann@62344
  1061
 apply (metis gcd_greatest_iff gcd_pos_int zdvd_imp_le)
nipkow@31734
  1062
apply simp
nipkow@31734
  1063
done
nipkow@31734
  1064
haftmann@34030
  1065
lemma gcd_code_int [code]:
haftmann@34030
  1066
  "gcd k l = \<bar>if l = (0::int) then k else gcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
haftmann@34030
  1067
  by (simp add: gcd_int_def nat_mod_distrib gcd_non_0_nat)
haftmann@34030
  1068
chaieb@22027
  1069
wenzelm@60758
  1070
subsection \<open>Coprimality\<close>
huffman@31706
  1071
haftmann@60690
  1072
context semiring_gcd
haftmann@60690
  1073
begin
haftmann@60690
  1074
haftmann@60690
  1075
lemma div_gcd_coprime:
haftmann@60690
  1076
  assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
  1077
  shows "coprime (a div gcd a b) (b div gcd a b)"
wenzelm@22367
  1078
proof -
haftmann@27556
  1079
  let ?g = "gcd a b"
chaieb@22027
  1080
  let ?a' = "a div ?g"
chaieb@22027
  1081
  let ?b' = "b div ?g"
haftmann@27556
  1082
  let ?g' = "gcd ?a' ?b'"
chaieb@22027
  1083
  have dvdg: "?g dvd a" "?g dvd b" by simp_all
chaieb@22027
  1084
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
wenzelm@22367
  1085
  from dvdg dvdg' obtain ka kb ka' kb' where
wenzelm@22367
  1086
      kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
chaieb@22027
  1087
    unfolding dvd_def by blast
haftmann@58834
  1088
  from this [symmetric] have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
haftmann@58834
  1089
    by (simp_all add: mult.assoc mult.left_commute [of "gcd a b"])
wenzelm@22367
  1090
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
wenzelm@22367
  1091
    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
wenzelm@22367
  1092
      dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
huffman@35216
  1093
  have "?g \<noteq> 0" using nz by simp
haftmann@60690
  1094
  moreover from gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
haftmann@60690
  1095
  thm dvd_mult_cancel_left
haftmann@60690
  1096
  ultimately show ?thesis using dvd_times_left_cancel_iff [of "gcd a b" _ 1] by simp
chaieb@22027
  1097
qed
chaieb@22027
  1098
haftmann@60690
  1099
end
haftmann@60690
  1100
nipkow@31952
  1101
lemma coprime_nat: "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
nipkow@31952
  1102
  using gcd_unique_nat[of 1 a b, simplified] by auto
huffman@31706
  1103
nipkow@31952
  1104
lemma coprime_Suc_0_nat:
huffman@31706
  1105
    "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = Suc 0)"
haftmann@60690
  1106
  using coprime_nat by simp
huffman@31706
  1107
nipkow@31952
  1108
lemma coprime_int: "coprime (a::int) b \<longleftrightarrow>
huffman@31706
  1109
    (\<forall>d. d >= 0 \<and> d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
nipkow@31952
  1110
  using gcd_unique_int [of 1 a b]
huffman@31706
  1111
  apply clarsimp
huffman@31706
  1112
  apply (erule subst)
huffman@31706
  1113
  apply (rule iffI)
huffman@31706
  1114
  apply force
lp15@61649
  1115
  using abs_dvd_iff abs_ge_zero apply blast
wenzelm@59807
  1116
  done
huffman@31706
  1117
nipkow@31952
  1118
lemma gcd_coprime_nat:
huffman@31706
  1119
  assumes z: "gcd (a::nat) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
  1120
    b: "b = b' * gcd a b"
huffman@31706
  1121
  shows    "coprime a' b'"
huffman@31706
  1122
huffman@31706
  1123
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
  1124
  apply (erule ssubst)
huffman@31706
  1125
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
  1126
  apply (erule ssubst)
haftmann@62344
  1127
  apply (rule div_gcd_coprime)
wenzelm@41550
  1128
  using z apply force
huffman@31706
  1129
  apply (subst (1) b)
huffman@31706
  1130
  using z apply force
huffman@31706
  1131
  apply (subst (1) a)
huffman@31706
  1132
  using z apply force
wenzelm@41550
  1133
  done
huffman@31706
  1134
nipkow@31952
  1135
lemma gcd_coprime_int:
huffman@31706
  1136
  assumes z: "gcd (a::int) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
  1137
    b: "b = b' * gcd a b"
huffman@31706
  1138
  shows    "coprime a' b'"
huffman@31706
  1139
huffman@31706
  1140
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
  1141
  apply (erule ssubst)
huffman@31706
  1142
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
  1143
  apply (erule ssubst)
haftmann@62344
  1144
  apply (rule div_gcd_coprime)
wenzelm@41550
  1145
  using z apply force
huffman@31706
  1146
  apply (subst (1) b)
huffman@31706
  1147
  using z apply force
huffman@31706
  1148
  apply (subst (1) a)
huffman@31706
  1149
  using z apply force
wenzelm@41550
  1150
  done
huffman@31706
  1151
haftmann@60690
  1152
context semiring_gcd
haftmann@60690
  1153
begin
huffman@31706
  1154
haftmann@60690
  1155
lemma coprime_mult:
haftmann@60690
  1156
  assumes da: "coprime d a" and db: "coprime d b"
haftmann@60690
  1157
  shows "coprime d (a * b)"
haftmann@60690
  1158
  apply (subst gcd.commute)
haftmann@60690
  1159
  using da apply (subst gcd_mult_cancel)
haftmann@60690
  1160
  apply (subst gcd.commute, assumption)
haftmann@60690
  1161
  apply (subst gcd.commute, rule db)
haftmann@60690
  1162
  done
huffman@31706
  1163
haftmann@60690
  1164
end
haftmann@60690
  1165
nipkow@31952
  1166
lemma coprime_lmult_nat:
huffman@31706
  1167
  assumes dab: "coprime (d::nat) (a * b)" shows "coprime d a"
huffman@31706
  1168
proof -
huffman@31706
  1169
  have "gcd d a dvd gcd d (a * b)"
haftmann@59008
  1170
    by (rule gcd_greatest, auto)
huffman@31706
  1171
  with dab show ?thesis
huffman@31706
  1172
    by auto
huffman@31706
  1173
qed
huffman@31706
  1174
nipkow@31952
  1175
lemma coprime_lmult_int:
nipkow@31798
  1176
  assumes "coprime (d::int) (a * b)" shows "coprime d a"
huffman@31706
  1177
proof -
huffman@31706
  1178
  have "gcd d a dvd gcd d (a * b)"
haftmann@59008
  1179
    by (rule gcd_greatest, auto)
nipkow@31798
  1180
  with assms show ?thesis
huffman@31706
  1181
    by auto
huffman@31706
  1182
qed
huffman@31706
  1183
nipkow@31952
  1184
lemma coprime_rmult_nat:
nipkow@31798
  1185
  assumes "coprime (d::nat) (a * b)" shows "coprime d b"
huffman@31706
  1186
proof -
huffman@31706
  1187
  have "gcd d b dvd gcd d (a * b)"
haftmann@59008
  1188
    by (rule gcd_greatest, auto intro: dvd_mult)
nipkow@31798
  1189
  with assms show ?thesis
huffman@31706
  1190
    by auto
huffman@31706
  1191
qed
huffman@31706
  1192
nipkow@31952
  1193
lemma coprime_rmult_int:
huffman@31706
  1194
  assumes dab: "coprime (d::int) (a * b)" shows "coprime d b"
huffman@31706
  1195
proof -
huffman@31706
  1196
  have "gcd d b dvd gcd d (a * b)"
haftmann@59008
  1197
    by (rule gcd_greatest, auto intro: dvd_mult)
huffman@31706
  1198
  with dab show ?thesis
huffman@31706
  1199
    by auto
huffman@31706
  1200
qed
huffman@31706
  1201
nipkow@31952
  1202
lemma coprime_mul_eq_nat: "coprime (d::nat) (a * b) \<longleftrightarrow>
huffman@31706
  1203
    coprime d a \<and>  coprime d b"
nipkow@31952
  1204
  using coprime_rmult_nat[of d a b] coprime_lmult_nat[of d a b]
haftmann@62344
  1205
    coprime_mult [of d a b]
huffman@31706
  1206
  by blast
huffman@31706
  1207
nipkow@31952
  1208
lemma coprime_mul_eq_int: "coprime (d::int) (a * b) \<longleftrightarrow>
huffman@31706
  1209
    coprime d a \<and>  coprime d b"
nipkow@31952
  1210
  using coprime_rmult_int[of d a b] coprime_lmult_int[of d a b]
haftmann@62344
  1211
    coprime_mult [of d a b]
huffman@31706
  1212
  by blast
huffman@31706
  1213
noschinl@52397
  1214
lemma coprime_power_int:
noschinl@52397
  1215
  assumes "0 < n" shows "coprime (a :: int) (b ^ n) \<longleftrightarrow> coprime a b"
noschinl@52397
  1216
  using assms
noschinl@52397
  1217
proof (induct n)
noschinl@52397
  1218
  case (Suc n) then show ?case
noschinl@52397
  1219
    by (cases n) (simp_all add: coprime_mul_eq_int)
noschinl@52397
  1220
qed simp
noschinl@52397
  1221
nipkow@31952
  1222
lemma gcd_coprime_exists_nat:
huffman@31706
  1223
    assumes nz: "gcd (a::nat) b \<noteq> 0"
huffman@31706
  1224
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
  1225
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
  1226
  apply (rule_tac x = "b div gcd a b" in exI)
haftmann@62344
  1227
  using nz apply (auto simp add: div_gcd_coprime dvd_div_mult)
huffman@31706
  1228
done
huffman@31706
  1229
nipkow@31952
  1230
lemma gcd_coprime_exists_int:
huffman@31706
  1231
    assumes nz: "gcd (a::int) b \<noteq> 0"
huffman@31706
  1232
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
  1233
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
  1234
  apply (rule_tac x = "b div gcd a b" in exI)
haftmann@62344
  1235
  using nz apply (auto simp add: div_gcd_coprime)
huffman@31706
  1236
done
huffman@31706
  1237
nipkow@31952
  1238
lemma coprime_exp_nat: "coprime (d::nat) a \<Longrightarrow> coprime d (a^n)"
haftmann@62344
  1239
  by (induct n) (simp_all add: coprime_mult)
huffman@31706
  1240
nipkow@31952
  1241
lemma coprime_exp_int: "coprime (d::int) a \<Longrightarrow> coprime d (a^n)"
haftmann@62344
  1242
  by (induct n) (simp_all add: coprime_mult)
huffman@31706
  1243
haftmann@60690
  1244
context semiring_gcd
haftmann@60690
  1245
begin
haftmann@60690
  1246
haftmann@60690
  1247
lemma coprime_exp_left:
haftmann@60690
  1248
  assumes "coprime a b"
haftmann@60690
  1249
  shows "coprime (a ^ n) b"
haftmann@60690
  1250
  using assms by (induct n) (simp_all add: gcd_mult_cancel)
haftmann@60690
  1251
haftmann@60690
  1252
lemma coprime_exp2:
haftmann@60690
  1253
  assumes "coprime a b"
haftmann@60690
  1254
  shows "coprime (a ^ n) (b ^ m)"
haftmann@60690
  1255
proof (rule coprime_exp_left)
haftmann@60690
  1256
  from assms show "coprime a (b ^ m)"
haftmann@60690
  1257
    by (induct m) (simp_all add: gcd_mult_cancel gcd.commute [of a])
haftmann@60690
  1258
qed
haftmann@60690
  1259
haftmann@60690
  1260
end
haftmann@60690
  1261
haftmann@60690
  1262
lemma gcd_exp_nat:
haftmann@60690
  1263
  "gcd ((a :: nat) ^ n) (b ^ n) = gcd a b ^ n"
haftmann@60690
  1264
proof (cases "a = 0 \<and> b = 0")
haftmann@60690
  1265
  case True then show ?thesis by (cases "n > 0") (simp_all add: zero_power)
haftmann@60690
  1266
next
haftmann@60690
  1267
  case False
haftmann@60690
  1268
  then have "coprime (a div gcd a b) (b div gcd a b)"
haftmann@60690
  1269
    by (auto simp: div_gcd_coprime)
haftmann@60690
  1270
  then have "coprime ((a div gcd a b) ^ n) ((b div gcd a b) ^ n)"
haftmann@60690
  1271
    by (simp add: coprime_exp2)
haftmann@60690
  1272
  then have "gcd ((a div gcd a b)^n * (gcd a b)^n)
huffman@31706
  1273
      ((b div gcd a b)^n * (gcd a b)^n) = (gcd a b)^n"
lp15@60162
  1274
    by (metis gcd_mult_distrib_nat mult.commute mult.right_neutral)
huffman@31706
  1275
  also have "(a div gcd a b)^n * (gcd a b)^n = a^n"
lp15@60162
  1276
    by (metis dvd_div_mult_self gcd_unique_nat power_mult_distrib)
huffman@31706
  1277
  also have "(b div gcd a b)^n * (gcd a b)^n = b^n"
lp15@60162
  1278
    by (metis dvd_div_mult_self gcd_unique_nat power_mult_distrib)
huffman@31706
  1279
  finally show ?thesis .
huffman@31706
  1280
qed
huffman@31706
  1281
nipkow@31952
  1282
lemma gcd_exp_int: "gcd ((a::int)^n) (b^n) = (gcd a b)^n"
nipkow@31952
  1283
  apply (subst (1 2) gcd_abs_int)
huffman@31706
  1284
  apply (subst (1 2) power_abs)
nipkow@31952
  1285
  apply (rule gcd_exp_nat [where n = n, transferred])
huffman@31706
  1286
  apply auto
huffman@31706
  1287
done
huffman@31706
  1288
nipkow@31952
  1289
lemma division_decomp_nat: assumes dc: "(a::nat) dvd b * c"
huffman@31706
  1290
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
  1291
proof-
huffman@31706
  1292
  let ?g = "gcd a b"
huffman@31706
  1293
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
  1294
  moreover
huffman@31706
  1295
  {assume z: "?g \<noteq> 0"
nipkow@31952
  1296
    from gcd_coprime_exists_nat[OF z]
huffman@31706
  1297
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
  1298
      by blast
huffman@31706
  1299
    have thb: "?g dvd b" by auto
huffman@31706
  1300
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
  1301
    with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
huffman@31706
  1302
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
haftmann@57512
  1303
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult.assoc)
huffman@31706
  1304
    with z have th_1: "a' dvd b' * c" by auto
haftmann@62344
  1305
    from coprime_dvd_mult [OF ab'(3)] th_1
haftmann@57512
  1306
    have thc: "a' dvd c" by (subst (asm) mult.commute, blast)
huffman@31706
  1307
    from ab' have "a = ?g*a'" by algebra
huffman@31706
  1308
    with thb thc have ?thesis by blast }
huffman@31706
  1309
  ultimately show ?thesis by blast
huffman@31706
  1310
qed
huffman@31706
  1311
nipkow@31952
  1312
lemma division_decomp_int: assumes dc: "(a::int) dvd b * c"
huffman@31706
  1313
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
  1314
proof-
huffman@31706
  1315
  let ?g = "gcd a b"
huffman@31706
  1316
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
  1317
  moreover
huffman@31706
  1318
  {assume z: "?g \<noteq> 0"
nipkow@31952
  1319
    from gcd_coprime_exists_int[OF z]
huffman@31706
  1320
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
  1321
      by blast
huffman@31706
  1322
    have thb: "?g dvd b" by auto
huffman@31706
  1323
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
  1324
    with dc have th0: "a' dvd b*c"
huffman@31706
  1325
      using dvd_trans[of a' a "b*c"] by simp
huffman@31706
  1326
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
haftmann@60690
  1327
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: ac_simps)
huffman@31706
  1328
    with z have th_1: "a' dvd b' * c" by auto
haftmann@62344
  1329
    from coprime_dvd_mult [OF ab'(3)] th_1
haftmann@57512
  1330
    have thc: "a' dvd c" by (subst (asm) mult.commute, blast)
huffman@31706
  1331
    from ab' have "a = ?g*a'" by algebra
huffman@31706
  1332
    with thb thc have ?thesis by blast }
huffman@31706
  1333
  ultimately show ?thesis by blast
chaieb@27669
  1334
qed
chaieb@27669
  1335
nipkow@31952
  1336
lemma pow_divides_pow_nat:
huffman@31706
  1337
  assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
  1338
  shows "a dvd b"
huffman@31706
  1339
proof-
huffman@31706
  1340
  let ?g = "gcd a b"
huffman@31706
  1341
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
  1342
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
  1343
  moreover
huffman@31706
  1344
  {assume z: "?g \<noteq> 0"
huffman@35216
  1345
    hence zn: "?g ^ n \<noteq> 0" using n by simp
nipkow@31952
  1346
    from gcd_coprime_exists_nat[OF z]
huffman@31706
  1347
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
  1348
      by blast
huffman@31706
  1349
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
  1350
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
  1351
    hence "?g^n*a'^n dvd ?g^n *b'^n"
haftmann@57512
  1352
      by (simp only: power_mult_distrib mult.commute)
haftmann@58787
  1353
    then have th0: "a'^n dvd b'^n"
haftmann@58787
  1354
      using zn by auto
huffman@31706
  1355
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
  1356
    with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
haftmann@57512
  1357
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult.commute)
haftmann@62344
  1358
    from coprime_dvd_mult [OF coprime_exp_nat [OF ab'(3), of m]] th1
haftmann@57512
  1359
    have "a' dvd b'" by (subst (asm) mult.commute, blast)
huffman@31706
  1360
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
  1361
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
  1362
  ultimately show ?thesis by blast
huffman@31706
  1363
qed
huffman@31706
  1364
nipkow@31952
  1365
lemma pow_divides_pow_int:
huffman@31706
  1366
  assumes ab: "(a::int) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
  1367
  shows "a dvd b"
chaieb@27669
  1368
proof-
huffman@31706
  1369
  let ?g = "gcd a b"
huffman@31706
  1370
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
  1371
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
  1372
  moreover
huffman@31706
  1373
  {assume z: "?g \<noteq> 0"
huffman@35216
  1374
    hence zn: "?g ^ n \<noteq> 0" using n by simp
nipkow@31952
  1375
    from gcd_coprime_exists_int[OF z]
huffman@31706
  1376
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
  1377
      by blast
huffman@31706
  1378
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
  1379
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
  1380
    hence "?g^n*a'^n dvd ?g^n *b'^n"
haftmann@57512
  1381
      by (simp only: power_mult_distrib mult.commute)
huffman@31706
  1382
    with zn z n have th0:"a'^n dvd b'^n" by auto
huffman@31706
  1383
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
  1384
    with th0 have "a' dvd b'^n"
huffman@31706
  1385
      using dvd_trans[of a' "a'^n" "b'^n"] by simp
haftmann@60596
  1386
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult.commute)
haftmann@62344
  1387
    from coprime_dvd_mult [OF coprime_exp_int [OF ab'(3), of m]] th1
haftmann@57512
  1388
    have "a' dvd b'" by (subst (asm) mult.commute, blast)
huffman@31706
  1389
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
  1390
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
  1391
  ultimately show ?thesis by blast
huffman@31706
  1392
qed
huffman@31706
  1393
nipkow@31952
  1394
lemma pow_divides_eq_nat [simp]: "n ~= 0 \<Longrightarrow> ((a::nat)^n dvd b^n) = (a dvd b)"
nipkow@31952
  1395
  by (auto intro: pow_divides_pow_nat dvd_power_same)
huffman@31706
  1396
nipkow@31952
  1397
lemma pow_divides_eq_int [simp]: "n ~= 0 \<Longrightarrow> ((a::int)^n dvd b^n) = (a dvd b)"
nipkow@31952
  1398
  by (auto intro: pow_divides_pow_int dvd_power_same)
huffman@31706
  1399
nipkow@31952
  1400
lemma divides_mult_nat:
huffman@31706
  1401
  assumes mr: "(m::nat) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
  1402
  shows "m * n dvd r"
huffman@31706
  1403
proof-
huffman@31706
  1404
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
  1405
    unfolding dvd_def by blast
haftmann@57512
  1406
  from mr n' have "m dvd n'*n" by (simp add: mult.commute)
haftmann@62344
  1407
  hence "m dvd n'" using coprime_dvd_mult_iff [OF mn] by simp
huffman@31706
  1408
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
  1409
  from n' k show ?thesis unfolding dvd_def by auto
huffman@31706
  1410
qed
huffman@31706
  1411
nipkow@31952
  1412
lemma divides_mult_int:
huffman@31706
  1413
  assumes mr: "(m::int) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
  1414
  shows "m * n dvd r"
huffman@31706
  1415
proof-
huffman@31706
  1416
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
  1417
    unfolding dvd_def by blast
haftmann@57512
  1418
  from mr n' have "m dvd n'*n" by (simp add: mult.commute)
haftmann@62344
  1419
  hence "m dvd n'" using coprime_dvd_mult_iff [OF mn] by simp
huffman@31706
  1420
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
  1421
  from n' k show ?thesis unfolding dvd_def by auto
chaieb@27669
  1422
qed
chaieb@27669
  1423
nipkow@31952
  1424
lemma coprime_plus_one_nat [simp]: "coprime ((n::nat) + 1) n"
haftmann@60690
  1425
  by (simp add: gcd.commute del: One_nat_def)
huffman@31706
  1426
nipkow@31952
  1427
lemma coprime_Suc_nat [simp]: "coprime (Suc n) n"
haftmann@60690
  1428
  using coprime_plus_one_nat by simp
huffman@31706
  1429
nipkow@31952
  1430
lemma coprime_plus_one_int [simp]: "coprime ((n::int) + 1) n"
haftmann@60686
  1431
  by (simp add: gcd.commute)
huffman@31706
  1432
nipkow@31952
  1433
lemma coprime_minus_one_nat: "(n::nat) \<noteq> 0 \<Longrightarrow> coprime (n - 1) n"
nipkow@31952
  1434
  using coprime_plus_one_nat [of "n - 1"]
haftmann@62344
  1435
    gcd.commute [of "n - 1" n] by auto
huffman@31706
  1436
nipkow@31952
  1437
lemma coprime_minus_one_int: "coprime ((n::int) - 1) n"
nipkow@31952
  1438
  using coprime_plus_one_int [of "n - 1"]
haftmann@62344
  1439
    gcd.commute [of "n - 1" n] by auto
huffman@31706
  1440
haftmann@62344
  1441
lemma setprod_coprime_nat:
haftmann@62344
  1442
  fixes x :: nat
haftmann@62344
  1443
  shows "(\<And>i. i \<in> A \<Longrightarrow> coprime (f i) x) \<Longrightarrow> coprime (\<Prod>i\<in>A. f i) x"
haftmann@62344
  1444
  by (induct A rule: infinite_finite_induct)
haftmann@62344
  1445
    (auto simp add: gcd_mult_cancel One_nat_def [symmetric] simp del: One_nat_def)
huffman@31706
  1446
haftmann@62344
  1447
lemma setprod_coprime_int:
haftmann@62344
  1448
  fixes x :: int
haftmann@62344
  1449
  shows "(\<And>i. i \<in> A \<Longrightarrow> coprime (f i) x) \<Longrightarrow> coprime (\<Prod>i\<in>A. f i) x"
haftmann@62344
  1450
  by (induct A rule: infinite_finite_induct)
haftmann@62344
  1451
    (auto simp add: gcd_mult_cancel)
huffman@31706
  1452
lp15@60162
  1453
lemma coprime_common_divisor_nat: 
haftmann@60686
  1454
  "coprime (a::nat) b \<Longrightarrow> x dvd a \<Longrightarrow> x dvd b \<Longrightarrow> x = 1"
haftmann@62344
  1455
  by (metis gcd_greatest_iff nat_dvd_1_iff_1)
huffman@31706
  1456
lp15@60162
  1457
lemma coprime_common_divisor_int:
wenzelm@61944
  1458
  "coprime (a::int) b \<Longrightarrow> x dvd a \<Longrightarrow> x dvd b \<Longrightarrow> \<bar>x\<bar> = 1"
haftmann@60686
  1459
  using gcd_greatest_iff [of x a b] by auto
huffman@31706
  1460
lp15@60162
  1461
lemma coprime_divisors_nat:
lp15@60162
  1462
    "(d::int) dvd a \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow> coprime d e"
lp15@60162
  1463
  by (meson coprime_int dvd_trans gcd_dvd1 gcd_dvd2 gcd_ge_0_int)
huffman@31706
  1464
nipkow@31952
  1465
lemma invertible_coprime_nat: "(x::nat) * y mod m = 1 \<Longrightarrow> coprime x m"
haftmann@62344
  1466
by (metis coprime_lmult_nat gcd_1_nat gcd.commute gcd_red_nat)
huffman@31706
  1467
nipkow@31952
  1468
lemma invertible_coprime_int: "(x::int) * y mod m = 1 \<Longrightarrow> coprime x m"
haftmann@62344
  1469
by (metis coprime_lmult_int gcd_1_int gcd.commute gcd_red_int)
huffman@31706
  1470
huffman@31706
  1471
wenzelm@60758
  1472
subsection \<open>Bezout's theorem\<close>
huffman@31706
  1473
huffman@31706
  1474
(* Function bezw returns a pair of witnesses to Bezout's theorem --
huffman@31706
  1475
   see the theorems that follow the definition. *)
huffman@31706
  1476
fun
huffman@31706
  1477
  bezw  :: "nat \<Rightarrow> nat \<Rightarrow> int * int"
huffman@31706
  1478
where
huffman@31706
  1479
  "bezw x y =
huffman@31706
  1480
  (if y = 0 then (1, 0) else
huffman@31706
  1481
      (snd (bezw y (x mod y)),
huffman@31706
  1482
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y)))"
huffman@31706
  1483
huffman@31706
  1484
lemma bezw_0 [simp]: "bezw x 0 = (1, 0)" by simp
huffman@31706
  1485
huffman@31706
  1486
lemma bezw_non_0: "y > 0 \<Longrightarrow> bezw x y = (snd (bezw y (x mod y)),
huffman@31706
  1487
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y))"
huffman@31706
  1488
  by simp
huffman@31706
  1489
huffman@31706
  1490
declare bezw.simps [simp del]
huffman@31706
  1491
huffman@31706
  1492
lemma bezw_aux [rule_format]:
huffman@31706
  1493
    "fst (bezw x y) * int x + snd (bezw x y) * int y = int (gcd x y)"
nipkow@31952
  1494
proof (induct x y rule: gcd_nat_induct)
huffman@31706
  1495
  fix m :: nat
huffman@31706
  1496
  show "fst (bezw m 0) * int m + snd (bezw m 0) * int 0 = int (gcd m 0)"
huffman@31706
  1497
    by auto
huffman@31706
  1498
  next fix m :: nat and n
huffman@31706
  1499
    assume ngt0: "n > 0" and
huffman@31706
  1500
      ih: "fst (bezw n (m mod n)) * int n +
huffman@31706
  1501
        snd (bezw n (m mod n)) * int (m mod n) =
huffman@31706
  1502
        int (gcd n (m mod n))"
huffman@31706
  1503
    thus "fst (bezw m n) * int m + snd (bezw m n) * int n = int (gcd m n)"
nipkow@31952
  1504
      apply (simp add: bezw_non_0 gcd_non_0_nat)
huffman@31706
  1505
      apply (erule subst)
haftmann@36350
  1506
      apply (simp add: field_simps)
huffman@31706
  1507
      apply (subst mod_div_equality [of m n, symmetric])
huffman@31706
  1508
      (* applying simp here undoes the last substitution!
huffman@31706
  1509
         what is procedure cancel_div_mod? *)
hoelzl@58776
  1510
      apply (simp only: NO_MATCH_def field_simps of_nat_add of_nat_mult)
huffman@31706
  1511
      done
huffman@31706
  1512
qed
huffman@31706
  1513
nipkow@31952
  1514
lemma bezout_int:
huffman@31706
  1515
  fixes x y
huffman@31706
  1516
  shows "EX u v. u * (x::int) + v * y = gcd x y"
huffman@31706
  1517
proof -
huffman@31706
  1518
  have bezout_aux: "!!x y. x \<ge> (0::int) \<Longrightarrow> y \<ge> 0 \<Longrightarrow>
huffman@31706
  1519
      EX u v. u * x + v * y = gcd x y"
huffman@31706
  1520
    apply (rule_tac x = "fst (bezw (nat x) (nat y))" in exI)
huffman@31706
  1521
    apply (rule_tac x = "snd (bezw (nat x) (nat y))" in exI)
huffman@31706
  1522
    apply (unfold gcd_int_def)
huffman@31706
  1523
    apply simp
huffman@31706
  1524
    apply (subst bezw_aux [symmetric])
huffman@31706
  1525
    apply auto
huffman@31706
  1526
    done
huffman@31706
  1527
  have "(x \<ge> 0 \<and> y \<ge> 0) | (x \<ge> 0 \<and> y \<le> 0) | (x \<le> 0 \<and> y \<ge> 0) |
huffman@31706
  1528
      (x \<le> 0 \<and> y \<le> 0)"
huffman@31706
  1529
    by auto
huffman@31706
  1530
  moreover have "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> ?thesis"
huffman@31706
  1531
    by (erule (1) bezout_aux)
huffman@31706
  1532
  moreover have "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1533
    apply (insert bezout_aux [of x "-y"])
huffman@31706
  1534
    apply auto
huffman@31706
  1535
    apply (rule_tac x = u in exI)
huffman@31706
  1536
    apply (rule_tac x = "-v" in exI)
nipkow@31952
  1537
    apply (subst gcd_neg2_int [symmetric])
huffman@31706
  1538
    apply auto
huffman@31706
  1539
    done
huffman@31706
  1540
  moreover have "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1541
    apply (insert bezout_aux [of "-x" y])
huffman@31706
  1542
    apply auto
huffman@31706
  1543
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1544
    apply (rule_tac x = v in exI)
nipkow@31952
  1545
    apply (subst gcd_neg1_int [symmetric])
huffman@31706
  1546
    apply auto
huffman@31706
  1547
    done
huffman@31706
  1548
  moreover have "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1549
    apply (insert bezout_aux [of "-x" "-y"])
huffman@31706
  1550
    apply auto
huffman@31706
  1551
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1552
    apply (rule_tac x = "-v" in exI)
nipkow@31952
  1553
    apply (subst gcd_neg1_int [symmetric])
nipkow@31952
  1554
    apply (subst gcd_neg2_int [symmetric])
huffman@31706
  1555
    apply auto
huffman@31706
  1556
    done
huffman@31706
  1557
  ultimately show ?thesis by blast
huffman@31706
  1558
qed
huffman@31706
  1559
wenzelm@60758
  1560
text \<open>versions of Bezout for nat, by Amine Chaieb\<close>
huffman@31706
  1561
huffman@31706
  1562
lemma ind_euclid:
huffman@31706
  1563
  assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
huffman@31706
  1564
  and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
chaieb@27669
  1565
  shows "P a b"
berghofe@34915
  1566
proof(induct "a + b" arbitrary: a b rule: less_induct)
berghofe@34915
  1567
  case less
chaieb@27669
  1568
  have "a = b \<or> a < b \<or> b < a" by arith
chaieb@27669
  1569
  moreover {assume eq: "a= b"
huffman@31706
  1570
    from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq
huffman@31706
  1571
    by simp}
chaieb@27669
  1572
  moreover
chaieb@27669
  1573
  {assume lt: "a < b"
berghofe@34915
  1574
    hence "a + b - a < a + b \<or> a = 0" by arith
chaieb@27669
  1575
    moreover
chaieb@27669
  1576
    {assume "a =0" with z c have "P a b" by blast }
chaieb@27669
  1577
    moreover
berghofe@34915
  1578
    {assume "a + b - a < a + b"
berghofe@34915
  1579
      also have th0: "a + b - a = a + (b - a)" using lt by arith
berghofe@34915
  1580
      finally have "a + (b - a) < a + b" .
berghofe@34915
  1581
      then have "P a (a + (b - a))" by (rule add[rule_format, OF less])
berghofe@34915
  1582
      then have "P a b" by (simp add: th0[symmetric])}
chaieb@27669
  1583
    ultimately have "P a b" by blast}
chaieb@27669
  1584
  moreover
chaieb@27669
  1585
  {assume lt: "a > b"
berghofe@34915
  1586
    hence "b + a - b < a + b \<or> b = 0" by arith
chaieb@27669
  1587
    moreover
chaieb@27669
  1588
    {assume "b =0" with z c have "P a b" by blast }
chaieb@27669
  1589
    moreover
berghofe@34915
  1590
    {assume "b + a - b < a + b"
berghofe@34915
  1591
      also have th0: "b + a - b = b + (a - b)" using lt by arith
berghofe@34915
  1592
      finally have "b + (a - b) < a + b" .
berghofe@34915
  1593
      then have "P b (b + (a - b))" by (rule add[rule_format, OF less])
berghofe@34915
  1594
      then have "P b a" by (simp add: th0[symmetric])
chaieb@27669
  1595
      hence "P a b" using c by blast }
chaieb@27669
  1596
    ultimately have "P a b" by blast}
chaieb@27669
  1597
ultimately  show "P a b" by blast
chaieb@27669
  1598
qed
chaieb@27669
  1599
nipkow@31952
  1600
lemma bezout_lemma_nat:
huffman@31706
  1601
  assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1602
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1603
  shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and>
huffman@31706
  1604
    (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
huffman@31706
  1605
  using ex
huffman@31706
  1606
  apply clarsimp
huffman@35216
  1607
  apply (rule_tac x="d" in exI, simp)
huffman@31706
  1608
  apply (case_tac "a * x = b * y + d" , simp_all)
huffman@31706
  1609
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1610
  apply (rule_tac x="y" in exI)
huffman@31706
  1611
  apply algebra
huffman@31706
  1612
  apply (rule_tac x="x" in exI)
huffman@31706
  1613
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1614
  apply algebra
chaieb@27669
  1615
done
chaieb@27669
  1616
nipkow@31952
  1617
lemma bezout_add_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1618
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1619
  apply(induct a b rule: ind_euclid)
huffman@31706
  1620
  apply blast
huffman@31706
  1621
  apply clarify
huffman@35216
  1622
  apply (rule_tac x="a" in exI, simp)
huffman@31706
  1623
  apply clarsimp
huffman@31706
  1624
  apply (rule_tac x="d" in exI)
huffman@35216
  1625
  apply (case_tac "a * x = b * y + d", simp_all)
huffman@31706
  1626
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1627
  apply (rule_tac x="y" in exI)
huffman@31706
  1628
  apply algebra
huffman@31706
  1629
  apply (rule_tac x="x" in exI)
huffman@31706
  1630
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1631
  apply algebra
chaieb@27669
  1632
done
chaieb@27669
  1633
nipkow@31952
  1634
lemma bezout1_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1635
    (a * x - b * y = d \<or> b * x - a * y = d)"
nipkow@31952
  1636
  using bezout_add_nat[of a b]
huffman@31706
  1637
  apply clarsimp
huffman@31706
  1638
  apply (rule_tac x="d" in exI, simp)
huffman@31706
  1639
  apply (rule_tac x="x" in exI)
huffman@31706
  1640
  apply (rule_tac x="y" in exI)
huffman@31706
  1641
  apply auto
chaieb@27669
  1642
done
chaieb@27669
  1643
nipkow@31952
  1644
lemma bezout_add_strong_nat: assumes nz: "a \<noteq> (0::nat)"
chaieb@27669
  1645
  shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
chaieb@27669
  1646
proof-
huffman@31706
  1647
 from nz have ap: "a > 0" by simp
nipkow@31952
  1648
 from bezout_add_nat[of a b]
huffman@31706
  1649
 have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or>
huffman@31706
  1650
   (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
chaieb@27669
  1651
 moreover
huffman@31706
  1652
    {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
huffman@31706
  1653
     from H have ?thesis by blast }
chaieb@27669
  1654
 moreover
chaieb@27669
  1655
 {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
chaieb@27669
  1656
   {assume b0: "b = 0" with H  have ?thesis by simp}
huffman@31706
  1657
   moreover
chaieb@27669
  1658
   {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
huffman@31706
  1659
     from b dvd_imp_le [OF H(2)] have "d < b \<or> d = b"
huffman@31706
  1660
       by auto
chaieb@27669
  1661
     moreover
chaieb@27669
  1662
     {assume db: "d=b"
wenzelm@41550
  1663
       with nz H have ?thesis apply simp
wenzelm@32960
  1664
         apply (rule exI[where x = b], simp)
wenzelm@32960
  1665
         apply (rule exI[where x = b])
wenzelm@32960
  1666
        by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
chaieb@27669
  1667
    moreover
huffman@31706
  1668
    {assume db: "d < b"
wenzelm@41550
  1669
        {assume "x=0" hence ?thesis using nz H by simp }
wenzelm@32960
  1670
        moreover
wenzelm@32960
  1671
        {assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
wenzelm@32960
  1672
          from db have "d \<le> b - 1" by simp
wenzelm@32960
  1673
          hence "d*b \<le> b*(b - 1)" by simp
wenzelm@32960
  1674
          with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
wenzelm@32960
  1675
          have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
wenzelm@32960
  1676
          from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)"
huffman@31706
  1677
            by simp
wenzelm@32960
  1678
          hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
haftmann@57512
  1679
            by (simp only: mult.assoc distrib_left)
wenzelm@32960
  1680
          hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)"
huffman@31706
  1681
            by algebra
wenzelm@32960
  1682
          hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
wenzelm@32960
  1683
          hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
wenzelm@32960
  1684
            by (simp only: diff_add_assoc[OF dble, of d, symmetric])
wenzelm@32960
  1685
          hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
haftmann@59008
  1686
            by (simp only: diff_mult_distrib2 ac_simps)
wenzelm@32960
  1687
          hence ?thesis using H(1,2)
wenzelm@32960
  1688
            apply -
wenzelm@32960
  1689
            apply (rule exI[where x=d], simp)
wenzelm@32960
  1690
            apply (rule exI[where x="(b - 1) * y"])
wenzelm@32960
  1691
            by (rule exI[where x="x*(b - 1) - d"], simp)}
wenzelm@32960
  1692
        ultimately have ?thesis by blast}
chaieb@27669
  1693
    ultimately have ?thesis by blast}
chaieb@27669
  1694
  ultimately have ?thesis by blast}
chaieb@27669
  1695
 ultimately show ?thesis by blast
chaieb@27669
  1696
qed
chaieb@27669
  1697
nipkow@31952
  1698
lemma bezout_nat: assumes a: "(a::nat) \<noteq> 0"
chaieb@27669
  1699
  shows "\<exists>x y. a * x = b * y + gcd a b"
chaieb@27669
  1700
proof-
chaieb@27669
  1701
  let ?g = "gcd a b"
nipkow@31952
  1702
  from bezout_add_strong_nat[OF a, of b]
chaieb@27669
  1703
  obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
chaieb@27669
  1704
  from d(1,2) have "d dvd ?g" by simp
chaieb@27669
  1705
  then obtain k where k: "?g = d*k" unfolding dvd_def by blast
huffman@31706
  1706
  from d(3) have "a * x * k = (b * y + d) *k " by auto
chaieb@27669
  1707
  hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
chaieb@27669
  1708
  thus ?thesis by blast
chaieb@27669
  1709
qed
chaieb@27669
  1710
huffman@31706
  1711
wenzelm@60758
  1712
subsection \<open>LCM properties\<close>
huffman@31706
  1713
wenzelm@61944
  1714
lemma lcm_altdef_int [code]: "lcm (a::int) b = \<bar>a\<bar> * \<bar>b\<bar> div gcd a b"
haftmann@62344
  1715
  by (simp add: lcm_int_def lcm_nat_def zdiv_int gcd_int_def)
huffman@31706
  1716
nipkow@31952
  1717
lemma prod_gcd_lcm_nat: "(m::nat) * n = gcd m n * lcm m n"
huffman@31706
  1718
  unfolding lcm_nat_def
nipkow@31952
  1719
  by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod_nat])
huffman@31706
  1720
wenzelm@61944
  1721
lemma prod_gcd_lcm_int: "\<bar>m::int\<bar> * \<bar>n\<bar> = gcd m n * lcm m n"
huffman@31706
  1722
  unfolding lcm_int_def gcd_int_def
huffman@31706
  1723
  apply (subst int_mult [symmetric])
nipkow@31952
  1724
  apply (subst prod_gcd_lcm_nat [symmetric])
huffman@31706
  1725
  apply (subst nat_abs_mult_distrib [symmetric])
huffman@31706
  1726
  apply (simp, simp add: abs_mult)
huffman@31706
  1727
done
huffman@31706
  1728
nipkow@31952
  1729
lemma lcm_0_nat [simp]: "lcm (m::nat) 0 = 0"
huffman@31706
  1730
  unfolding lcm_nat_def by simp
huffman@31706
  1731
nipkow@31952
  1732
lemma lcm_0_int [simp]: "lcm (m::int) 0 = 0"
huffman@31706
  1733
  unfolding lcm_int_def by simp
huffman@31706
  1734
nipkow@31952
  1735
lemma lcm_0_left_nat [simp]: "lcm (0::nat) n = 0"
huffman@31706
  1736
  unfolding lcm_nat_def by simp
chaieb@27669
  1737
nipkow@31952
  1738
lemma lcm_0_left_int [simp]: "lcm (0::int) n = 0"
huffman@31706
  1739
  unfolding lcm_int_def by simp
huffman@31706
  1740
nipkow@31952
  1741
lemma lcm_pos_nat:
nipkow@31798
  1742
  "(m::nat) > 0 \<Longrightarrow> n>0 \<Longrightarrow> lcm m n > 0"
nipkow@31952
  1743
by (metis gr0I mult_is_0 prod_gcd_lcm_nat)
chaieb@27669
  1744
nipkow@31952
  1745
lemma lcm_pos_int:
nipkow@31798
  1746
  "(m::int) ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> lcm m n > 0"
nipkow@31952
  1747
  apply (subst lcm_abs_int)
nipkow@31952
  1748
  apply (rule lcm_pos_nat [transferred])
nipkow@31798
  1749
  apply auto
haftmann@62344
  1750
  done
haftmann@23687
  1751
nipkow@31952
  1752
lemma dvd_pos_nat:
haftmann@23687
  1753
  fixes n m :: nat
haftmann@23687
  1754
  assumes "n > 0" and "m dvd n"
haftmann@23687
  1755
  shows "m > 0"
haftmann@62344
  1756
  using assms by (cases m) auto
nipkow@31729
  1757
nipkow@31952
  1758
lemma lcm_unique_nat: "(a::nat) dvd d \<and> b dvd d \<and>
huffman@31706
  1759
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
haftmann@62344
  1760
  by (auto intro: dvd_antisym lcm_least)
chaieb@27568
  1761
nipkow@31952
  1762
lemma lcm_unique_int: "d >= 0 \<and> (a::int) dvd d \<and> b dvd d \<and>
huffman@31706
  1763
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
haftmann@62344
  1764
  using lcm_least zdvd_antisym_nonneg by auto
haftmann@34973
  1765
nipkow@31798
  1766
lemma lcm_proj2_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm x y = y"
huffman@31706
  1767
  apply (rule sym)
nipkow@31952
  1768
  apply (subst lcm_unique_nat [symmetric])
huffman@31706
  1769
  apply auto
huffman@31706
  1770
done
huffman@31706
  1771
wenzelm@61944
  1772
lemma lcm_proj2_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm x y = \<bar>y\<bar>"
huffman@31706
  1773
  apply (rule sym)
nipkow@31952
  1774
  apply (subst lcm_unique_int [symmetric])
huffman@31706
  1775
  apply auto
huffman@31706
  1776
done
huffman@31706
  1777
nipkow@31798
  1778
lemma lcm_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm y x = y"
haftmann@62344
  1779
by (subst lcm.commute, erule lcm_proj2_if_dvd_nat)
huffman@31706
  1780
wenzelm@61944
  1781
lemma lcm_proj1_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm y x = \<bar>y\<bar>"
haftmann@62344
  1782
by (subst lcm.commute, erule lcm_proj2_if_dvd_int)
huffman@31706
  1783
nipkow@31992
  1784
lemma lcm_proj1_iff_nat[simp]: "lcm m n = (m::nat) \<longleftrightarrow> n dvd m"
nipkow@31992
  1785
by (metis lcm_proj1_if_dvd_nat lcm_unique_nat)
nipkow@31992
  1786
nipkow@31992
  1787
lemma lcm_proj2_iff_nat[simp]: "lcm m n = (n::nat) \<longleftrightarrow> m dvd n"
nipkow@31992
  1788
by (metis lcm_proj2_if_dvd_nat lcm_unique_nat)
nipkow@31992
  1789
wenzelm@61944
  1790
lemma lcm_proj1_iff_int[simp]: "lcm m n = \<bar>m::int\<bar> \<longleftrightarrow> n dvd m"
nipkow@31992
  1791
by (metis dvd_abs_iff lcm_proj1_if_dvd_int lcm_unique_int)
nipkow@31992
  1792
wenzelm@61944
  1793
lemma lcm_proj2_iff_int[simp]: "lcm m n = \<bar>n::int\<bar> \<longleftrightarrow> m dvd n"
nipkow@31992
  1794
by (metis dvd_abs_iff lcm_proj2_if_dvd_int lcm_unique_int)
chaieb@27568
  1795
haftmann@61913
  1796
lemma (in semiring_gcd) comp_fun_idem_gcd:
haftmann@61913
  1797
  "comp_fun_idem gcd"
haftmann@61913
  1798
  by standard (simp_all add: fun_eq_iff ac_simps)
haftmann@61913
  1799
haftmann@61913
  1800
lemma (in semiring_gcd) comp_fun_idem_lcm:
haftmann@61913
  1801
  "comp_fun_idem lcm"
haftmann@61913
  1802
  by standard (simp_all add: fun_eq_iff ac_simps)
haftmann@61913
  1803
haftmann@61913
  1804
lemma lcm_1_iff_nat [simp]: "lcm (m::nat) n = 1 \<longleftrightarrow> m=1 \<and> n=1"
haftmann@61913
  1805
  by (simp only: lcm_eq_1_iff) simp
haftmann@61913
  1806
  
haftmann@61913
  1807
lemma lcm_1_iff_int [simp]: "lcm (m::int) n = 1 \<longleftrightarrow> (m=1 \<or> m = -1) \<and> (n=1 \<or> n = -1)"
haftmann@61913
  1808
  by auto
nipkow@31995
  1809
haftmann@34030
  1810
wenzelm@60758
  1811
subsection \<open>The complete divisibility lattice\<close>
nipkow@32112
  1812
wenzelm@60758
  1813
text\<open>Lifting gcd and lcm to sets (Gcd/Lcm).
huffman@45264
  1814
Gcd is defined via Lcm to facilitate the proof that we have a complete lattice.
wenzelm@60758
  1815
\<close>
huffman@45264
  1816
huffman@45264
  1817
instantiation nat :: Gcd
nipkow@32112
  1818
begin
nipkow@32112
  1819
haftmann@62344
  1820
interpretation semilattice_neutr_set lcm "1::nat"
haftmann@62344
  1821
  by standard simp_all
haftmann@54867
  1822
haftmann@61929
  1823
definition
haftmann@61929
  1824
  "Lcm (M::nat set) = (if finite M then F M else 0)"
haftmann@51489
  1825
haftmann@51489
  1826
lemma Lcm_nat_empty:
haftmann@51489
  1827
  "Lcm {} = (1::nat)"
haftmann@60690
  1828
  by (simp add: Lcm_nat_def del: One_nat_def)
haftmann@51489
  1829
haftmann@51489
  1830
lemma Lcm_nat_insert:
haftmann@51489
  1831
  "Lcm (insert n M) = lcm (n::nat) (Lcm M)"
haftmann@61929
  1832
  by (cases "finite M") (auto simp add: Lcm_nat_def simp del: One_nat_def)
haftmann@61929
  1833
haftmann@61929
  1834
lemma Lcm_nat_infinite:
haftmann@61929
  1835
  "infinite M \<Longrightarrow> Lcm M = (0::nat)"
haftmann@61929
  1836
  by (simp add: Lcm_nat_def)
haftmann@61929
  1837
haftmann@61929
  1838
lemma dvd_Lcm_nat [simp]:
haftmann@61929
  1839
  fixes M :: "nat set"
haftmann@61929
  1840
  assumes "m \<in> M"
haftmann@61929
  1841
  shows "m dvd Lcm M"
haftmann@61929
  1842
proof -
haftmann@61929
  1843
  from assms have "insert m M = M" by auto
haftmann@61929
  1844
  moreover have "m dvd Lcm (insert m M)"
haftmann@61929
  1845
    by (simp add: Lcm_nat_insert)
haftmann@61929
  1846
  ultimately show ?thesis by simp
haftmann@61929
  1847
qed
haftmann@61929
  1848
haftmann@61929
  1849
lemma Lcm_dvd_nat [simp]:
haftmann@61929
  1850
  fixes M :: "nat set"
haftmann@61929
  1851
  assumes "\<forall>m\<in>M. m dvd n"
haftmann@61929
  1852
  shows "Lcm M dvd n"
haftmann@61929
  1853
proof (cases "n = 0")
haftmann@61929
  1854
  case True then show ?thesis by simp
haftmann@61929
  1855
next
haftmann@61929
  1856
  case False
haftmann@61929
  1857
  then have "finite {d. d dvd n}" by (rule finite_divisors_nat)
haftmann@61929
  1858
  moreover have "M \<subseteq> {d. d dvd n}" using assms by fast
haftmann@61929
  1859
  ultimately have "finite M" by (rule rev_finite_subset)
haftmann@61929
  1860
  then show ?thesis using assms by (induct M) (simp_all add: Lcm_nat_empty Lcm_nat_insert)
haftmann@61929
  1861
qed
nipkow@32112
  1862
huffman@45264
  1863
definition
huffman@45264
  1864
  "Gcd (M::nat set) = Lcm {d. \<forall>m\<in>M. d dvd m}"
nipkow@32112
  1865
huffman@45264
  1866
instance ..
haftmann@51489
  1867
nipkow@32112
  1868
end
nipkow@32112
  1869
haftmann@61929
  1870
instance nat :: semiring_Gcd
haftmann@61929
  1871
proof
haftmann@61929
  1872
  show "Gcd N dvd n" if "n \<in> N" for N and n :: nat
haftmann@61929
  1873
  using that by (induct N rule: infinite_finite_induct)
haftmann@61929
  1874
    (auto simp add: Gcd_nat_def)
haftmann@61929
  1875
  show "n dvd Gcd N" if "\<And>m. m \<in> N \<Longrightarrow> n dvd m" for N and n :: nat
haftmann@61929
  1876
  using that by (induct N rule: infinite_finite_induct)
haftmann@61929
  1877
    (auto simp add: Gcd_nat_def)
haftmann@61929
  1878
qed simp
nipkow@32112
  1879
haftmann@61929
  1880
instance nat :: semiring_Lcm
haftmann@61929
  1881
proof
haftmann@61929
  1882
  show "Lcm N = Gcd {m. \<forall>n\<in>N. n dvd m}" for N :: "nat set"
haftmann@61929
  1883
    by (rule associated_eqI) (auto intro!: Gcd_dvd Gcd_greatest)
haftmann@61929
  1884
qed
nipkow@32112
  1885
haftmann@61913
  1886
lemma Lcm_eq_0 [simp]:
haftmann@61913
  1887
  "finite (M::nat set) \<Longrightarrow> 0 \<in> M \<Longrightarrow> Lcm M = 0"
haftmann@61913
  1888
  by (rule Lcm_eq_0_I)
haftmann@61913
  1889
haftmann@61913
  1890
lemma Lcm0_iff [simp]:
haftmann@61929
  1891
  fixes M :: "nat set"
haftmann@61929
  1892
  assumes "finite M" and "M \<noteq> {}"
haftmann@61929
  1893
  shows "Lcm M = 0 \<longleftrightarrow> 0 \<in> M"
haftmann@61929
  1894
  using assms by (simp add: Lcm_0_iff)
haftmann@61913
  1895
wenzelm@60758
  1896
text\<open>Alternative characterizations of Gcd:\<close>
nipkow@32112
  1897
nipkow@32112
  1898
lemma Gcd_eq_Max: "finite(M::nat set) \<Longrightarrow> M \<noteq> {} \<Longrightarrow> 0 \<notin> M \<Longrightarrow> Gcd M = Max(\<Inter>m\<in>M. {d. d dvd m})"
nipkow@32112
  1899
apply(rule antisym)
nipkow@32112
  1900
 apply(rule Max_ge)
nipkow@32112
  1901
  apply (metis all_not_in_conv finite_divisors_nat finite_INT)
haftmann@60596
  1902
 apply (simp add: Gcd_dvd)
nipkow@32112
  1903
apply (rule Max_le_iff[THEN iffD2])
nipkow@32112
  1904
  apply (metis all_not_in_conv finite_divisors_nat finite_INT)
nipkow@44890
  1905
 apply fastforce
nipkow@32112
  1906
apply clarsimp
haftmann@60596
  1907
apply (metis Gcd_dvd Max_in dvd_0_left dvd_Gcd dvd_imp_le linorder_antisym_conv3 not_less0)
nipkow@32112
  1908
done
nipkow@32112
  1909
nipkow@32112
  1910
lemma Gcd_remove0_nat: "finite M \<Longrightarrow> Gcd M = Gcd (M - {0::nat})"
nipkow@32112
  1911
apply(induct pred:finite)
nipkow@32112
  1912
 apply simp
nipkow@32112
  1913
apply(case_tac "x=0")
nipkow@32112
  1914
 apply simp
nipkow@32112
  1915
apply(subgoal_tac "insert x F - {0} = insert x (F - {0})")
nipkow@32112
  1916
 apply simp
nipkow@32112
  1917
apply blast
nipkow@32112
  1918
done
nipkow@32112
  1919
nipkow@32112
  1920
lemma Lcm_in_lcm_closed_set_nat:
nipkow@32112
  1921
  "finite M \<Longrightarrow> M \<noteq> {} \<Longrightarrow> ALL m n :: nat. m:M \<longrightarrow> n:M \<longrightarrow> lcm m n : M \<Longrightarrow> Lcm M : M"
nipkow@32112
  1922
apply(induct rule:finite_linorder_min_induct)
nipkow@32112
  1923
 apply simp
nipkow@32112
  1924
apply simp
nipkow@32112
  1925
apply(subgoal_tac "ALL m n :: nat. m:A \<longrightarrow> n:A \<longrightarrow> lcm m n : A")
nipkow@32112
  1926
 apply simp
nipkow@32112
  1927
 apply(case_tac "A={}")
nipkow@32112
  1928
  apply simp
nipkow@32112
  1929
 apply simp
nipkow@32112
  1930
apply (metis lcm_pos_nat lcm_unique_nat linorder_neq_iff nat_dvd_not_less not_less0)
nipkow@32112
  1931
done
nipkow@32112
  1932
nipkow@32112
  1933
lemma Lcm_eq_Max_nat:
nipkow@32112
  1934
  "finite M \<Longrightarrow> M \<noteq> {} \<Longrightarrow> 0 \<notin> M \<Longrightarrow> ALL m n :: nat. m:M \<longrightarrow> n:M \<longrightarrow> lcm m n : M \<Longrightarrow> Lcm M = Max M"
nipkow@32112
  1935
apply(rule antisym)
nipkow@32112
  1936
 apply(rule Max_ge, assumption)
nipkow@32112
  1937
 apply(erule (2) Lcm_in_lcm_closed_set_nat)
nipkow@32112
  1938
apply clarsimp
nipkow@32112
  1939
apply (metis Lcm0_iff dvd_Lcm_nat dvd_imp_le neq0_conv)
nipkow@32112
  1940
done
nipkow@32112
  1941
nipkow@34222
  1942
lemma mult_inj_if_coprime_nat:
nipkow@34222
  1943
  "inj_on f A \<Longrightarrow> inj_on g B \<Longrightarrow> ALL a:A. ALL b:B. coprime (f a) (g b)
nipkow@34222
  1944
   \<Longrightarrow> inj_on (%(a,b). f a * g b::nat) (A \<times> B)"
haftmann@61913
  1945
  by (auto simp add: inj_on_def coprime_crossproduct_nat simp del: One_nat_def)
nipkow@34222
  1946
wenzelm@60758
  1947
text\<open>Nitpick:\<close>
nipkow@34222
  1948
blanchet@41792
  1949
lemma gcd_eq_nitpick_gcd [nitpick_unfold]: "gcd x y = Nitpick.nat_gcd x y"
blanchet@41792
  1950
by (induct x y rule: nat_gcd.induct)
blanchet@41792
  1951
   (simp add: gcd_nat.simps Nitpick.nat_gcd.simps)
blanchet@33197
  1952
blanchet@41792
  1953
lemma lcm_eq_nitpick_lcm [nitpick_unfold]: "lcm x y = Nitpick.nat_lcm x y"
blanchet@33197
  1954
by (simp only: lcm_nat_def Nitpick.nat_lcm_def gcd_eq_nitpick_gcd)
blanchet@33197
  1955
haftmann@54867
  1956
wenzelm@60758
  1957
subsubsection \<open>Setwise gcd and lcm for integers\<close>
huffman@45264
  1958
huffman@45264
  1959
instantiation int :: Gcd
huffman@45264
  1960
begin
huffman@45264
  1961
huffman@45264
  1962
definition
huffman@45264
  1963
  "Lcm M = int (Lcm (nat ` abs ` M))"
huffman@45264
  1964
huffman@45264
  1965
definition
huffman@45264
  1966
  "Gcd M = int (Gcd (nat ` abs ` M))"
huffman@45264
  1967
huffman@45264
  1968
instance ..
haftmann@60686
  1969
wenzelm@21256
  1970
end
huffman@45264
  1971
haftmann@60686
  1972
instance int :: semiring_Gcd
haftmann@60686
  1973
  by standard (auto intro!: Gcd_dvd Gcd_greatest simp add: Gcd_int_def Lcm_int_def int_dvd_iff dvd_int_iff
haftmann@60686
  1974
    dvd_int_unfold_dvd_nat [symmetric])
haftmann@60686
  1975
haftmann@60686
  1976
instance int :: semiring_Lcm
haftmann@60686
  1977
proof
haftmann@60686
  1978
  fix K :: "int set"
haftmann@60686
  1979
  have "{n. \<forall>k\<in>K. nat \<bar>k\<bar> dvd n} = ((\<lambda>k. nat \<bar>k\<bar>) ` {l. \<forall>k\<in>K. k dvd l})"
haftmann@60686
  1980
  proof (rule set_eqI)
haftmann@60686
  1981
    fix n
haftmann@60686
  1982
    have "(\<forall>k\<in>K. nat \<bar>k\<bar> dvd n) \<longleftrightarrow> (\<exists>l. (\<forall>k\<in>K. k dvd l) \<and> n = nat \<bar>l\<bar>)" (is "?P \<longleftrightarrow> ?Q")
haftmann@60686
  1983
    proof
haftmann@60686
  1984
      assume ?P
haftmann@60686
  1985
      then have "(\<forall>k\<in>K. k dvd int n) \<and> n = nat \<bar>int n\<bar>"
haftmann@60686
  1986
        by (auto simp add: dvd_int_unfold_dvd_nat)
haftmann@60686
  1987
      then show ?Q by blast
haftmann@60686
  1988
    next
haftmann@60686
  1989
      assume ?Q then show ?P
haftmann@60686
  1990
        by (auto simp add: dvd_int_unfold_dvd_nat)
haftmann@60686
  1991
    qed
haftmann@60686
  1992
    then show "n \<in> {n. \<forall>k\<in>K. nat \<bar>k\<bar> dvd n} \<longleftrightarrow> n \<in> (\<lambda>k. nat \<bar>k\<bar>) ` {l. \<forall>k\<in>K. k dvd l}"
haftmann@60686
  1993
      by auto
haftmann@60686
  1994
  qed
haftmann@60686
  1995
  then show "Lcm K = Gcd {l. \<forall>k\<in>K. k dvd l}"
haftmann@62344
  1996
    by (simp add: Gcd_int_def Lcm_int_def Lcm_Gcd image_image)
haftmann@60686
  1997
qed
haftmann@60686
  1998
huffman@45264
  1999
lemma Lcm_dvd_int [simp]:
huffman@45264
  2000
  fixes M :: "int set"
huffman@45264
  2001
  assumes "\<forall>m\<in>M. m dvd n" shows "Lcm M dvd n"
haftmann@62344
  2002
  using assms by (auto intro: Lcm_least)
lp15@59667
  2003
haftmann@60596
  2004
Andreas@61856
  2005
subsection \<open>gcd and lcm instances for @{typ integer}\<close>
Andreas@61856
  2006
Andreas@61856
  2007
instantiation integer :: gcd begin
Andreas@61856
  2008
context includes integer.lifting begin
Andreas@61856
  2009
lift_definition gcd_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer" is gcd .
Andreas@61856
  2010
lift_definition lcm_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer" is lcm .
huffman@45264
  2011
end
Andreas@61856
  2012
instance ..
Andreas@61856
  2013
end
Andreas@61856
  2014
lifting_update integer.lifting
Andreas@61856
  2015
lifting_forget integer.lifting
Andreas@61856
  2016
Andreas@61856
  2017
context includes integer.lifting begin
Andreas@61856
  2018
Andreas@61856
  2019
lemma gcd_code_integer [code]:
Andreas@61856
  2020
  "gcd k l = \<bar>if l = (0::integer) then k else gcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
Andreas@61856
  2021
by transfer(fact gcd_code_int)
Andreas@61856
  2022
wenzelm@61944
  2023
lemma lcm_code_integer [code]: "lcm (a::integer) b = \<bar>a\<bar> * \<bar>b\<bar> div gcd a b"
Andreas@61856
  2024
by transfer(fact lcm_altdef_int)
Andreas@61856
  2025
Andreas@61856
  2026
end
Andreas@61856
  2027
Andreas@61856
  2028
code_printing constant "gcd :: integer \<Rightarrow> _"
Andreas@61856
  2029
  \<rightharpoonup> (OCaml) "Big'_int.gcd'_big'_int"
Andreas@61856
  2030
  and (Haskell) "Prelude.gcd"
Andreas@61856
  2031
  and (Scala) "_.gcd'((_)')"
wenzelm@61975
  2032
  \<comment> \<open>There is no gcd operation in the SML standard library, so no code setup for SML\<close>
Andreas@61856
  2033
haftmann@62344
  2034
text \<open>Some code equations\<close>
haftmann@62344
  2035
haftmann@62344
  2036
lemma Lcm_set_nat [code, code_unfold]:
haftmann@62344
  2037
  "Lcm (set ns) = fold lcm ns (1::nat)"
haftmann@62344
  2038
  using Lcm_set [of ns] by (simp_all add: fun_eq_iff ac_simps foldr_fold [symmetric])
haftmann@62344
  2039
haftmann@62344
  2040
lemma Gcd_set_nat [code]:
haftmann@62344
  2041
  "Gcd (set ns) = fold gcd ns (0::nat)"
haftmann@62344
  2042
  using Gcd_set [of ns] by (simp_all add: fun_eq_iff ac_simps foldr_fold [symmetric])
haftmann@62344
  2043
haftmann@62344
  2044
lemma Lcm_set_int [code, code_unfold]:
haftmann@62344
  2045
  "Lcm (set xs) = fold lcm xs (1::int)"
haftmann@62344
  2046
  using Lcm_set [of xs] by (simp_all add: fun_eq_iff ac_simps foldr_fold [symmetric])
haftmann@62344
  2047
haftmann@62344
  2048
lemma Gcd_set_int [code]:
haftmann@62344
  2049
  "Gcd (set xs) = fold gcd xs (0::int)"
haftmann@62344
  2050
  using Gcd_set [of xs] by (simp_all add: fun_eq_iff ac_simps foldr_fold [symmetric])
haftmann@62344
  2051
haftmann@62344
  2052
text \<open>Fact aliasses\<close>
haftmann@62344
  2053
haftmann@62344
  2054
lemma dvd_int_iff: "x dvd y \<longleftrightarrow> nat \<bar>x\<bar> dvd nat \<bar>y\<bar>"
haftmann@62344
  2055
  by (fact dvd_int_unfold_dvd_nat)
haftmann@62344
  2056
haftmann@62344
  2057
lemmas gcd_assoc_nat = gcd.assoc [where ?'a = nat]
haftmann@62344
  2058
lemmas gcd_assoc_int = gcd.assoc [where ?'a = int]
haftmann@62344
  2059
lemmas gcd_commute_nat = gcd.commute [where ?'a = nat]
haftmann@62344
  2060
lemmas gcd_commute_int = gcd.commute [where ?'a = int]
haftmann@62344
  2061
lemmas gcd_left_commute_nat = gcd.left_commute [where ?'a = nat]
haftmann@62344
  2062
lemmas gcd_left_commute_int = gcd.left_commute [where ?'a = int]
haftmann@62344
  2063
lemmas gcd_ac_nat = gcd_assoc_nat gcd_commute_nat gcd_left_commute_nat
haftmann@62344
  2064
lemmas gcd_ac_int = gcd_assoc_int gcd_commute_int gcd_left_commute_int
haftmann@62344
  2065
lemmas gcd_dvd1_nat = gcd_dvd1 [where ?'a = nat]
haftmann@62344
  2066
lemmas gcd_dvd1_int = gcd_dvd1 [where ?'a = int]
haftmann@62344
  2067
lemmas gcd_dvd2_nat = gcd_dvd2 [where ?'a = nat]
haftmann@62344
  2068
lemmas gcd_dvd2_int = gcd_dvd2 [where ?'a = int]
haftmann@62344
  2069
lemmas gcd_greatest_nat = gcd_greatest [where ?'a = nat]
haftmann@62344
  2070
lemmas gcd_greatest_int = gcd_greatest [where ?'a = int]
haftmann@62344
  2071
lemmas gcd_mult_cancel_nat = gcd_mult_cancel [where ?'a = nat] 
haftmann@62344
  2072
lemmas gcd_mult_cancel_int = gcd_mult_cancel [where ?'a = int] 
haftmann@62344
  2073
lemmas gcd_greatest_iff_nat = gcd_greatest_iff [where ?'a = nat]
haftmann@62344
  2074
lemmas gcd_greatest_iff_int = gcd_greatest_iff [where ?'a = int]
haftmann@62344
  2075
lemmas gcd_zero_nat = gcd_eq_0_iff [where ?'a = nat]
haftmann@62344
  2076
lemmas gcd_zero_int = gcd_eq_0_iff [where ?'a = int]
haftmann@62344
  2077
haftmann@62344
  2078
lemmas lcm_assoc_nat = lcm.assoc [where ?'a = nat]
haftmann@62344
  2079
lemmas lcm_assoc_int = lcm.assoc [where ?'a = int]
haftmann@62344
  2080
lemmas lcm_commute_nat = lcm.commute [where ?'a = nat]
haftmann@62344
  2081
lemmas lcm_commute_int = lcm.commute [where ?'a = int]
haftmann@62344
  2082
lemmas lcm_left_commute_nat = lcm.left_commute [where ?'a = nat]
haftmann@62344
  2083
lemmas lcm_left_commute_int = lcm.left_commute [where ?'a = int]
haftmann@62344
  2084
lemmas lcm_ac_nat = lcm_assoc_nat lcm_commute_nat lcm_left_commute_nat
haftmann@62344
  2085
lemmas lcm_ac_int = lcm_assoc_int lcm_commute_int lcm_left_commute_int
haftmann@62344
  2086
lemmas lcm_dvd1_nat = dvd_lcm1 [where ?'a = nat]
haftmann@62344
  2087
lemmas lcm_dvd1_int = dvd_lcm1 [where ?'a = int]
haftmann@62344
  2088
lemmas lcm_dvd2_nat = dvd_lcm2 [where ?'a = nat]
haftmann@62344
  2089
lemmas lcm_dvd2_int = dvd_lcm2 [where ?'a = int]
haftmann@62344
  2090
lemmas lcm_least_nat = lcm_least [where ?'a = nat]
haftmann@62344
  2091
lemmas lcm_least_int = lcm_least [where ?'a = int]
haftmann@62344
  2092
haftmann@62344
  2093
lemma lcm_0_iff_nat [simp]: "lcm (m::nat) n = 0 \<longleftrightarrow> m = 0 \<or> n= 0"
haftmann@62344
  2094
  by (fact lcm_eq_0_iff)
haftmann@62344
  2095
haftmann@62344
  2096
lemma lcm_0_iff_int [simp]: "lcm (m::int) n = 0 \<longleftrightarrow> m = 0 \<or> n = 0"
haftmann@62344
  2097
  by (fact lcm_eq_0_iff)
haftmann@62344
  2098
haftmann@62344
  2099
lemma dvd_lcm_I1_nat [simp]: "(k::nat) dvd m \<Longrightarrow> k dvd lcm m n"
haftmann@62344
  2100
  by (fact dvd_lcmI1)
haftmann@62344
  2101
haftmann@62344
  2102
lemma dvd_lcm_I2_nat [simp]: "(k::nat) dvd n \<Longrightarrow> k dvd lcm m n"
haftmann@62344
  2103
  by (fact dvd_lcmI2)
haftmann@62344
  2104
haftmann@62344
  2105
lemma dvd_lcm_I1_int [simp]: "(i::int) dvd m \<Longrightarrow> i dvd lcm m n"
haftmann@62344
  2106
  by (fact dvd_lcmI1)
haftmann@62344
  2107
haftmann@62344
  2108
lemma dvd_lcm_I2_int[simp]: "(i::int) dvd n \<Longrightarrow> i dvd lcm m n"
haftmann@62344
  2109
  by (fact dvd_lcmI2)
haftmann@62344
  2110
haftmann@62344
  2111
lemmas coprime_mult_nat = coprime_mult [where ?'a = nat]
haftmann@62344
  2112
lemmas coprime_mult_int = coprime_mult [where ?'a = int]
haftmann@62344
  2113
lemmas div_gcd_coprime_nat = div_gcd_coprime [where ?'a = nat]
haftmann@62344
  2114
lemmas div_gcd_coprime_int = div_gcd_coprime [where ?'a = int]
haftmann@62344
  2115
lemmas coprime_dvd_mult_nat = coprime_dvd_mult [where ?'a = nat]
haftmann@62344
  2116
lemmas coprime_dvd_mult_int = coprime_dvd_mult [where ?'a = int]
haftmann@62344
  2117
haftmann@62344
  2118
lemma coprime_dvd_mult_iff_nat: "coprime (k::nat) n \<Longrightarrow>
haftmann@62344
  2119
    (k dvd m * n) = (k dvd m)"
haftmann@62344
  2120
  by (fact coprime_dvd_mult_iff)
haftmann@62344
  2121
haftmann@62344
  2122
lemma coprime_dvd_mult_iff_int: "coprime (k::int) n \<Longrightarrow>
haftmann@62344
  2123
    (k dvd m * n) = (k dvd m)"
haftmann@62344
  2124
  by (fact coprime_dvd_mult_iff)
haftmann@62344
  2125
haftmann@62344
  2126
lemma coprime_exp2_nat [intro]: "coprime (a::nat) b \<Longrightarrow> coprime (a^n) (b^m)"
haftmann@62344
  2127
  by (fact coprime_exp2)
haftmann@62344
  2128
haftmann@62344
  2129
lemma coprime_exp2_int [intro]: "coprime (a::int) b \<Longrightarrow> coprime (a^n) (b^m)"
haftmann@62344
  2130
  by (fact coprime_exp2)
haftmann@62344
  2131
haftmann@62344
  2132
lemmas Gcd_dvd_nat [simp] = Gcd_dvd [where ?'a = nat]
haftmann@62344
  2133
lemmas Gcd_dvd_int [simp] = Gcd_dvd [where ?'a = int]
haftmann@62344
  2134
lemmas dvd_Gcd_nat [simp] = dvd_Gcd [where ?'a = nat]
haftmann@62344
  2135
lemmas dvd_Gcd_int [simp] = dvd_Gcd [where ?'a = int]
haftmann@62344
  2136
lemmas Gcd_empty_nat = Gcd_empty [where ?'a = nat]
haftmann@62344
  2137
lemmas Gcd_empty_int = Gcd_empty [where ?'a = int]
haftmann@62344
  2138
lemmas Gcd_insert_nat = Gcd_insert [where ?'a = nat]
haftmann@62344
  2139
lemmas Gcd_insert_int = Gcd_insert [where ?'a = int]
haftmann@62344
  2140
haftmann@62344
  2141
lemma dvd_Lcm_int [simp]:
haftmann@62344
  2142
  fixes M :: "int set" assumes "m \<in> M" shows "m dvd Lcm M"
haftmann@62344
  2143
  using assms by (fact dvd_Lcm)
haftmann@62344
  2144
haftmann@62344
  2145
lemma Lcm_empty_nat:
haftmann@62344
  2146
  "Lcm {} = (1::nat)"
haftmann@62344
  2147
  by (fact Lcm_empty)
haftmann@62344
  2148
haftmann@62344
  2149
lemma Lcm_empty_int:
haftmann@62344
  2150
  "Lcm {} = (1::int)"
haftmann@62344
  2151
  by (fact Lcm_empty)
haftmann@62344
  2152
haftmann@62344
  2153
lemma Lcm_insert_nat:
haftmann@62344
  2154
  "Lcm (insert (n::nat) N) = lcm n (Lcm N)"
haftmann@62344
  2155
  by (fact Lcm_insert)
haftmann@62344
  2156
haftmann@62344
  2157
lemma Lcm_insert_int:
haftmann@62344
  2158
  "Lcm (insert (n::int) N) = lcm n (Lcm N)"
haftmann@62344
  2159
  by (fact Lcm_insert)
haftmann@62344
  2160
haftmann@62344
  2161
lemma gcd_neg_numeral_1_int [simp]:
haftmann@62344
  2162
  "gcd (- numeral n :: int) x = gcd (numeral n) x"
haftmann@62344
  2163
  by (fact gcd_neg1_int)
haftmann@62344
  2164
haftmann@62344
  2165
lemma gcd_neg_numeral_2_int [simp]:
haftmann@62344
  2166
  "gcd x (- numeral n :: int) = gcd x (numeral n)"
haftmann@62344
  2167
  by (fact gcd_neg2_int)
haftmann@62344
  2168
haftmann@62344
  2169
lemma gcd_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> gcd x y = x"
haftmann@62344
  2170
  by (fact gcd_nat.absorb1)
haftmann@62344
  2171
haftmann@62344
  2172
lemma gcd_proj2_if_dvd_nat [simp]: "(y::nat) dvd x \<Longrightarrow> gcd x y = y"
haftmann@62344
  2173
  by (fact gcd_nat.absorb2)
haftmann@62344
  2174
haftmann@62344
  2175
lemma comp_fun_idem_gcd_nat: "comp_fun_idem (gcd :: nat\<Rightarrow>nat\<Rightarrow>nat)"
haftmann@62344
  2176
  by (fact comp_fun_idem_gcd)
haftmann@62344
  2177
haftmann@62344
  2178
lemma comp_fun_idem_gcd_int: "comp_fun_idem (gcd :: int\<Rightarrow>int\<Rightarrow>int)"
haftmann@62344
  2179
  by (fact comp_fun_idem_gcd)
haftmann@62344
  2180
haftmann@62344
  2181
lemma comp_fun_idem_lcm_nat: "comp_fun_idem (lcm :: nat\<Rightarrow>nat\<Rightarrow>nat)"
haftmann@62344
  2182
  by (fact comp_fun_idem_lcm)
haftmann@62344
  2183
haftmann@62344
  2184
lemma comp_fun_idem_lcm_int: "comp_fun_idem (lcm :: int\<Rightarrow>int\<Rightarrow>int)"
haftmann@62344
  2185
  by (fact comp_fun_idem_lcm)
haftmann@62344
  2186
haftmann@62344
  2187
interpretation dvd:
haftmann@62344
  2188
  order "op dvd" "\<lambda>n m :: nat. n dvd m \<and> m \<noteq> n"
haftmann@62344
  2189
  by standard (auto intro: dvd_refl dvd_trans dvd_antisym)
haftmann@62344
  2190
haftmann@62344
  2191
interpretation gcd_semilattice_nat:
haftmann@62344
  2192
  semilattice_inf gcd Rings.dvd "\<lambda>m n::nat. m dvd n \<and> m \<noteq> n"
haftmann@62344
  2193
  by standard (auto dest: dvd_antisym dvd_trans)
haftmann@62344
  2194
haftmann@62344
  2195
interpretation lcm_semilattice_nat:
haftmann@62344
  2196
  semilattice_sup lcm Rings.dvd "\<lambda>m n::nat. m dvd n \<and> m \<noteq> n"
haftmann@62344
  2197
  by standard simp_all
haftmann@62344
  2198
haftmann@62344
  2199
interpretation gcd_lcm_lattice_nat:
haftmann@62344
  2200
  lattice gcd Rings.dvd "\<lambda>m n::nat. m dvd n \<and> m \<noteq> n" lcm
haftmann@62344
  2201
  ..
haftmann@62344
  2202
haftmann@62344
  2203
interpretation gcd_lcm_complete_lattice_nat:
haftmann@62344
  2204
  complete_lattice Gcd Lcm gcd Rings.dvd "\<lambda>m n. m dvd n \<and> m \<noteq> n" lcm 1 "0::nat"
haftmann@62344
  2205
  by standard (auto simp add: Gcd_nat_def Lcm_nat_empty Lcm_nat_infinite)
haftmann@62344
  2206
Andreas@61856
  2207
end