src/HOL/GCD.thy
author nipkow
Fri Jun 26 19:44:39 2009 +0200 (2009-06-26)
changeset 31814 7c122634da81
parent 31813 4df828bbc411
child 31952 40501bb2d57c
permissions -rw-r--r--
lcm abs lemmas
huffman@31706
     1
(*  Title:      GCD.thy
huffman@31706
     2
    Authors:    Christophe Tabacznyj, Lawrence C. Paulson, Amine Chaieb,
nipkow@31798
     3
                Thomas M. Rasmussen, Jeremy Avigad, Tobias Nipkow
huffman@31706
     4
huffman@31706
     5
huffman@31706
     6
This file deals with the functions gcd and lcm, and properties of
huffman@31706
     7
primes. Definitions and lemmas are proved uniformly for the natural
huffman@31706
     8
numbers and integers.
huffman@31706
     9
huffman@31706
    10
This file combines and revises a number of prior developments.
huffman@31706
    11
huffman@31706
    12
The original theories "GCD" and "Primes" were by Christophe Tabacznyj
huffman@31706
    13
and Lawrence C. Paulson, based on \cite{davenport92}. They introduced
huffman@31706
    14
gcd, lcm, and prime for the natural numbers.
huffman@31706
    15
huffman@31706
    16
The original theory "IntPrimes" was by Thomas M. Rasmussen, and
huffman@31706
    17
extended gcd, lcm, primes to the integers. Amine Chaieb provided
huffman@31706
    18
another extension of the notions to the integers, and added a number
huffman@31706
    19
of results to "Primes" and "GCD". IntPrimes also defined and developed
huffman@31706
    20
the congruence relations on the integers. The notion was extended to
huffman@31706
    21
the natural numbers by Chiaeb.
huffman@31706
    22
nipkow@31798
    23
Tobias Nipkow cleaned up a lot.
wenzelm@21256
    24
*)
wenzelm@21256
    25
huffman@31706
    26
huffman@31706
    27
header {* GCD *}
wenzelm@21256
    28
wenzelm@21256
    29
theory GCD
huffman@31706
    30
imports NatTransfer
huffman@31706
    31
begin
huffman@31706
    32
huffman@31706
    33
declare One_nat_def [simp del]
huffman@31706
    34
huffman@31706
    35
subsection {* gcd *}
huffman@31706
    36
huffman@31706
    37
class gcd = one +
huffman@31706
    38
huffman@31706
    39
fixes
huffman@31706
    40
  gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" and
huffman@31706
    41
  lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
huffman@31706
    42
wenzelm@21256
    43
begin
wenzelm@21256
    44
huffman@31706
    45
abbreviation
huffman@31706
    46
  coprime :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
huffman@31706
    47
where
huffman@31706
    48
  "coprime x y == (gcd x y = 1)"
huffman@31706
    49
huffman@31706
    50
end
huffman@31706
    51
huffman@31706
    52
class prime = one +
huffman@31706
    53
huffman@31706
    54
fixes
huffman@31706
    55
  prime :: "'a \<Rightarrow> bool"
huffman@31706
    56
huffman@31706
    57
huffman@31706
    58
(* definitions for the natural numbers *)
huffman@31706
    59
huffman@31706
    60
instantiation nat :: gcd
huffman@31706
    61
huffman@31706
    62
begin
wenzelm@21256
    63
huffman@31706
    64
fun
huffman@31706
    65
  gcd_nat  :: "nat \<Rightarrow> nat \<Rightarrow> nat"
huffman@31706
    66
where
huffman@31706
    67
  "gcd_nat x y =
huffman@31706
    68
   (if y = 0 then x else gcd y (x mod y))"
huffman@31706
    69
huffman@31706
    70
definition
huffman@31706
    71
  lcm_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
huffman@31706
    72
where
huffman@31706
    73
  "lcm_nat x y = x * y div (gcd x y)"
huffman@31706
    74
huffman@31706
    75
instance proof qed
huffman@31706
    76
huffman@31706
    77
end
huffman@31706
    78
huffman@31706
    79
huffman@31706
    80
instantiation nat :: prime
huffman@31706
    81
huffman@31706
    82
begin
wenzelm@21256
    83
wenzelm@21263
    84
definition
huffman@31706
    85
  prime_nat :: "nat \<Rightarrow> bool"
huffman@31706
    86
where
huffman@31709
    87
  [code del]: "prime_nat p = (1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p))"
huffman@31706
    88
huffman@31706
    89
instance proof qed
haftmann@23687
    90
huffman@31706
    91
end
huffman@31706
    92
haftmann@23687
    93
huffman@31706
    94
(* definitions for the integers *)
huffman@31706
    95
huffman@31706
    96
instantiation int :: gcd
haftmann@23687
    97
huffman@31706
    98
begin
wenzelm@21256
    99
huffman@31706
   100
definition
huffman@31706
   101
  gcd_int  :: "int \<Rightarrow> int \<Rightarrow> int"
huffman@31706
   102
where
huffman@31706
   103
  "gcd_int x y = int (gcd (nat (abs x)) (nat (abs y)))"
haftmann@23687
   104
huffman@31706
   105
definition
huffman@31706
   106
  lcm_int :: "int \<Rightarrow> int \<Rightarrow> int"
huffman@31706
   107
where
huffman@31706
   108
  "lcm_int x y = int (lcm (nat (abs x)) (nat (abs y)))"
haftmann@23687
   109
huffman@31706
   110
instance proof qed
huffman@31706
   111
huffman@31706
   112
end
haftmann@23687
   113
haftmann@23687
   114
huffman@31706
   115
instantiation int :: prime
huffman@31706
   116
huffman@31706
   117
begin
huffman@31706
   118
huffman@31706
   119
definition
huffman@31706
   120
  prime_int :: "int \<Rightarrow> bool"
huffman@31706
   121
where
huffman@31709
   122
  [code del]: "prime_int p = prime (nat p)"
huffman@31706
   123
huffman@31706
   124
instance proof qed
huffman@31706
   125
huffman@31706
   126
end
huffman@31706
   127
huffman@31706
   128
huffman@31706
   129
subsection {* Set up Transfer *}
huffman@31706
   130
huffman@31706
   131
huffman@31706
   132
lemma transfer_nat_int_gcd:
huffman@31706
   133
  "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> gcd (nat x) (nat y) = nat (gcd x y)"
huffman@31706
   134
  "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> lcm (nat x) (nat y) = nat (lcm x y)"
huffman@31706
   135
  "(x::int) >= 0 \<Longrightarrow> prime (nat x) = prime x"
huffman@31706
   136
  unfolding gcd_int_def lcm_int_def prime_int_def
huffman@31706
   137
  by auto
haftmann@23687
   138
huffman@31706
   139
lemma transfer_nat_int_gcd_closures:
huffman@31706
   140
  "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> gcd x y >= 0"
huffman@31706
   141
  "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> lcm x y >= 0"
huffman@31706
   142
  by (auto simp add: gcd_int_def lcm_int_def)
huffman@31706
   143
huffman@31706
   144
declare TransferMorphism_nat_int[transfer add return:
huffman@31706
   145
    transfer_nat_int_gcd transfer_nat_int_gcd_closures]
huffman@31706
   146
huffman@31706
   147
lemma transfer_int_nat_gcd:
huffman@31706
   148
  "gcd (int x) (int y) = int (gcd x y)"
huffman@31706
   149
  "lcm (int x) (int y) = int (lcm x y)"
huffman@31706
   150
  "prime (int x) = prime x"
huffman@31706
   151
  by (unfold gcd_int_def lcm_int_def prime_int_def, auto)
huffman@31706
   152
huffman@31706
   153
lemma transfer_int_nat_gcd_closures:
huffman@31706
   154
  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> gcd x y >= 0"
huffman@31706
   155
  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> lcm x y >= 0"
huffman@31706
   156
  by (auto simp add: gcd_int_def lcm_int_def)
huffman@31706
   157
huffman@31706
   158
declare TransferMorphism_int_nat[transfer add return:
huffman@31706
   159
    transfer_int_nat_gcd transfer_int_nat_gcd_closures]
huffman@31706
   160
huffman@31706
   161
huffman@31706
   162
huffman@31706
   163
subsection {* GCD *}
huffman@31706
   164
huffman@31706
   165
(* was gcd_induct *)
huffman@31706
   166
lemma nat_gcd_induct:
haftmann@23687
   167
  fixes m n :: nat
haftmann@23687
   168
  assumes "\<And>m. P m 0"
haftmann@23687
   169
    and "\<And>m n. 0 < n \<Longrightarrow> P n (m mod n) \<Longrightarrow> P m n"
haftmann@23687
   170
  shows "P m n"
huffman@31706
   171
  apply (rule gcd_nat.induct)
huffman@31706
   172
  apply (case_tac "y = 0")
huffman@31706
   173
  using assms apply simp_all
huffman@31706
   174
done
huffman@31706
   175
huffman@31706
   176
(* specific to int *)
huffman@31706
   177
huffman@31706
   178
lemma int_gcd_neg1 [simp]: "gcd (-x::int) y = gcd x y"
huffman@31706
   179
  by (simp add: gcd_int_def)
huffman@31706
   180
huffman@31706
   181
lemma int_gcd_neg2 [simp]: "gcd (x::int) (-y) = gcd x y"
huffman@31706
   182
  by (simp add: gcd_int_def)
huffman@31706
   183
nipkow@31813
   184
lemma abs_gcd_int[simp]: "abs(gcd (x::int) y) = gcd x y"
nipkow@31813
   185
by(simp add: gcd_int_def)
nipkow@31813
   186
huffman@31706
   187
lemma int_gcd_abs: "gcd (x::int) y = gcd (abs x) (abs y)"
nipkow@31813
   188
by (simp add: gcd_int_def)
nipkow@31813
   189
nipkow@31813
   190
lemma gcd_abs1_int[simp]: "gcd (abs x) (y::int) = gcd x y"
nipkow@31813
   191
by (metis abs_idempotent int_gcd_abs)
nipkow@31813
   192
nipkow@31813
   193
lemma gcd_abs2_int[simp]: "gcd x (abs y::int) = gcd x y"
nipkow@31813
   194
by (metis abs_idempotent int_gcd_abs)
huffman@31706
   195
huffman@31706
   196
lemma int_gcd_cases:
huffman@31706
   197
  fixes x :: int and y
huffman@31706
   198
  assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd x y)"
huffman@31706
   199
      and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd x (-y))"
huffman@31706
   200
      and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd (-x) y)"
huffman@31706
   201
      and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd (-x) (-y))"
huffman@31706
   202
  shows "P (gcd x y)"
huffman@31706
   203
by (insert prems, auto simp add: int_gcd_neg1 int_gcd_neg2, arith)
wenzelm@21256
   204
huffman@31706
   205
lemma int_gcd_ge_0 [simp]: "gcd (x::int) y >= 0"
huffman@31706
   206
  by (simp add: gcd_int_def)
huffman@31706
   207
huffman@31706
   208
lemma int_lcm_neg1: "lcm (-x::int) y = lcm x y"
huffman@31706
   209
  by (simp add: lcm_int_def)
huffman@31706
   210
huffman@31706
   211
lemma int_lcm_neg2: "lcm (x::int) (-y) = lcm x y"
huffman@31706
   212
  by (simp add: lcm_int_def)
huffman@31706
   213
huffman@31706
   214
lemma int_lcm_abs: "lcm (x::int) y = lcm (abs x) (abs y)"
huffman@31706
   215
  by (simp add: lcm_int_def)
wenzelm@21256
   216
nipkow@31814
   217
lemma abs_lcm_int [simp]: "abs (lcm i j::int) = lcm i j"
nipkow@31814
   218
by(simp add:lcm_int_def)
nipkow@31814
   219
nipkow@31814
   220
lemma lcm_abs1_int[simp]: "lcm (abs x) (y::int) = lcm x y"
nipkow@31814
   221
by (metis abs_idempotent lcm_int_def)
nipkow@31814
   222
nipkow@31814
   223
lemma lcm_abs2_int[simp]: "lcm x (abs y::int) = lcm x y"
nipkow@31814
   224
by (metis abs_idempotent lcm_int_def)
nipkow@31814
   225
huffman@31706
   226
lemma int_lcm_cases:
huffman@31706
   227
  fixes x :: int and y
huffman@31706
   228
  assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm x y)"
huffman@31706
   229
      and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm x (-y))"
huffman@31706
   230
      and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm (-x) y)"
huffman@31706
   231
      and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm (-x) (-y))"
huffman@31706
   232
  shows "P (lcm x y)"
huffman@31706
   233
by (insert prems, auto simp add: int_lcm_neg1 int_lcm_neg2, arith)
huffman@31706
   234
huffman@31706
   235
lemma int_lcm_ge_0 [simp]: "lcm (x::int) y >= 0"
huffman@31706
   236
  by (simp add: lcm_int_def)
huffman@31706
   237
huffman@31706
   238
(* was gcd_0, etc. *)
huffman@31706
   239
lemma nat_gcd_0 [simp]: "gcd (x::nat) 0 = x"
haftmann@23687
   240
  by simp
haftmann@23687
   241
huffman@31706
   242
(* was igcd_0, etc. *)
huffman@31706
   243
lemma int_gcd_0 [simp]: "gcd (x::int) 0 = abs x"
huffman@31706
   244
  by (unfold gcd_int_def, auto)
huffman@31706
   245
huffman@31706
   246
lemma nat_gcd_0_left [simp]: "gcd 0 (x::nat) = x"
haftmann@23687
   247
  by simp
haftmann@23687
   248
huffman@31706
   249
lemma int_gcd_0_left [simp]: "gcd 0 (x::int) = abs x"
huffman@31706
   250
  by (unfold gcd_int_def, auto)
huffman@31706
   251
huffman@31706
   252
lemma nat_gcd_red: "gcd (x::nat) y = gcd y (x mod y)"
huffman@31706
   253
  by (case_tac "y = 0", auto)
huffman@31706
   254
huffman@31706
   255
(* weaker, but useful for the simplifier *)
huffman@31706
   256
huffman@31706
   257
lemma nat_gcd_non_0: "y ~= (0::nat) \<Longrightarrow> gcd (x::nat) y = gcd y (x mod y)"
huffman@31706
   258
  by simp
huffman@31706
   259
huffman@31706
   260
lemma nat_gcd_1 [simp]: "gcd (m::nat) 1 = 1"
wenzelm@21263
   261
  by simp
wenzelm@21256
   262
huffman@31706
   263
lemma nat_gcd_Suc_0 [simp]: "gcd (m::nat) (Suc 0) = Suc 0"
huffman@31706
   264
  by (simp add: One_nat_def)
huffman@31706
   265
huffman@31706
   266
lemma int_gcd_1 [simp]: "gcd (m::int) 1 = 1"
huffman@31706
   267
  by (simp add: gcd_int_def)
huffman@30082
   268
nipkow@31798
   269
lemma nat_gcd_idem: "gcd (x::nat) x = x"
nipkow@31798
   270
by simp
huffman@31706
   271
nipkow@31798
   272
lemma int_gcd_idem: "gcd (x::int) x = abs x"
nipkow@31813
   273
by (auto simp add: gcd_int_def)
huffman@31706
   274
huffman@31706
   275
declare gcd_nat.simps [simp del]
wenzelm@21256
   276
wenzelm@21256
   277
text {*
haftmann@27556
   278
  \medskip @{term "gcd m n"} divides @{text m} and @{text n}.  The
wenzelm@21256
   279
  conjunctions don't seem provable separately.
wenzelm@21256
   280
*}
wenzelm@21256
   281
huffman@31706
   282
lemma nat_gcd_dvd1 [iff]: "(gcd (m::nat)) n dvd m"
huffman@31706
   283
  and nat_gcd_dvd2 [iff]: "(gcd m n) dvd n"
huffman@31706
   284
  apply (induct m n rule: nat_gcd_induct)
huffman@31706
   285
  apply (simp_all add: nat_gcd_non_0)
wenzelm@21256
   286
  apply (blast dest: dvd_mod_imp_dvd)
huffman@31706
   287
done
huffman@31706
   288
huffman@31706
   289
lemma int_gcd_dvd1 [iff]: "gcd (x::int) y dvd x"
nipkow@31813
   290
by (metis gcd_int_def int_dvd_iff nat_gcd_dvd1)
wenzelm@21256
   291
huffman@31706
   292
lemma int_gcd_dvd2 [iff]: "gcd (x::int) y dvd y"
nipkow@31813
   293
by (metis gcd_int_def int_dvd_iff nat_gcd_dvd2)
huffman@31706
   294
nipkow@31730
   295
lemma dvd_gcd_D1_nat: "k dvd gcd m n \<Longrightarrow> (k::nat) dvd m"
nipkow@31730
   296
by(metis nat_gcd_dvd1 dvd_trans)
nipkow@31730
   297
nipkow@31730
   298
lemma dvd_gcd_D2_nat: "k dvd gcd m n \<Longrightarrow> (k::nat) dvd n"
nipkow@31730
   299
by(metis nat_gcd_dvd2 dvd_trans)
nipkow@31730
   300
nipkow@31730
   301
lemma dvd_gcd_D1_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd m"
nipkow@31730
   302
by(metis int_gcd_dvd1 dvd_trans)
nipkow@31730
   303
nipkow@31730
   304
lemma dvd_gcd_D2_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd n"
nipkow@31730
   305
by(metis int_gcd_dvd2 dvd_trans)
nipkow@31730
   306
huffman@31706
   307
lemma nat_gcd_le1 [simp]: "a \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> a"
huffman@31706
   308
  by (rule dvd_imp_le, auto)
huffman@31706
   309
huffman@31706
   310
lemma nat_gcd_le2 [simp]: "b \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> b"
huffman@31706
   311
  by (rule dvd_imp_le, auto)
huffman@31706
   312
huffman@31706
   313
lemma int_gcd_le1 [simp]: "a > 0 \<Longrightarrow> gcd (a::int) b \<le> a"
huffman@31706
   314
  by (rule zdvd_imp_le, auto)
wenzelm@21256
   315
huffman@31706
   316
lemma int_gcd_le2 [simp]: "b > 0 \<Longrightarrow> gcd (a::int) b \<le> b"
huffman@31706
   317
  by (rule zdvd_imp_le, auto)
huffman@31706
   318
huffman@31706
   319
lemma nat_gcd_greatest: "(k::nat) dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd gcd m n"
nipkow@31798
   320
by (induct m n rule: nat_gcd_induct) (simp_all add: nat_gcd_non_0 dvd_mod)
huffman@31706
   321
huffman@31706
   322
lemma int_gcd_greatest:
nipkow@31813
   323
  "(k::int) dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd gcd m n"
huffman@31706
   324
  apply (subst int_gcd_abs)
huffman@31706
   325
  apply (subst abs_dvd_iff [symmetric])
huffman@31706
   326
  apply (rule nat_gcd_greatest [transferred])
nipkow@31813
   327
  apply auto
huffman@31706
   328
done
wenzelm@21256
   329
huffman@31706
   330
lemma nat_gcd_greatest_iff [iff]: "(k dvd gcd (m::nat) n) =
huffman@31706
   331
    (k dvd m & k dvd n)"
huffman@31706
   332
  by (blast intro!: nat_gcd_greatest intro: dvd_trans)
huffman@31706
   333
huffman@31706
   334
lemma int_gcd_greatest_iff: "((k::int) dvd gcd m n) = (k dvd m & k dvd n)"
huffman@31706
   335
  by (blast intro!: int_gcd_greatest intro: dvd_trans)
wenzelm@21256
   336
huffman@31706
   337
lemma nat_gcd_zero [simp]: "(gcd (m::nat) n = 0) = (m = 0 & n = 0)"
huffman@31706
   338
  by (simp only: dvd_0_left_iff [symmetric] nat_gcd_greatest_iff)
wenzelm@21256
   339
huffman@31706
   340
lemma int_gcd_zero [simp]: "(gcd (m::int) n = 0) = (m = 0 & n = 0)"
huffman@31706
   341
  by (auto simp add: gcd_int_def)
wenzelm@21256
   342
huffman@31706
   343
lemma nat_gcd_pos [simp]: "(gcd (m::nat) n > 0) = (m ~= 0 | n ~= 0)"
huffman@31706
   344
  by (insert nat_gcd_zero [of m n], arith)
wenzelm@21256
   345
huffman@31706
   346
lemma int_gcd_pos [simp]: "(gcd (m::int) n > 0) = (m ~= 0 | n ~= 0)"
huffman@31706
   347
  by (insert int_gcd_zero [of m n], insert int_gcd_ge_0 [of m n], arith)
huffman@31706
   348
huffman@31706
   349
lemma nat_gcd_commute: "gcd (m::nat) n = gcd n m"
huffman@31706
   350
  by (rule dvd_anti_sym, auto)
haftmann@23687
   351
huffman@31706
   352
lemma int_gcd_commute: "gcd (m::int) n = gcd n m"
huffman@31706
   353
  by (auto simp add: gcd_int_def nat_gcd_commute)
huffman@31706
   354
huffman@31706
   355
lemma nat_gcd_assoc: "gcd (gcd (k::nat) m) n = gcd k (gcd m n)"
huffman@31706
   356
  apply (rule dvd_anti_sym)
huffman@31706
   357
  apply (blast intro: dvd_trans)+
huffman@31706
   358
done
wenzelm@21256
   359
huffman@31706
   360
lemma int_gcd_assoc: "gcd (gcd (k::int) m) n = gcd k (gcd m n)"
huffman@31706
   361
  by (auto simp add: gcd_int_def nat_gcd_assoc)
huffman@31706
   362
nipkow@31766
   363
lemmas nat_gcd_left_commute =
nipkow@31766
   364
  mk_left_commute[of gcd, OF nat_gcd_assoc nat_gcd_commute]
huffman@31706
   365
nipkow@31766
   366
lemmas int_gcd_left_commute =
nipkow@31766
   367
  mk_left_commute[of gcd, OF int_gcd_assoc int_gcd_commute]
huffman@31706
   368
huffman@31706
   369
lemmas nat_gcd_ac = nat_gcd_assoc nat_gcd_commute nat_gcd_left_commute
huffman@31706
   370
  -- {* gcd is an AC-operator *}
wenzelm@21256
   371
huffman@31706
   372
lemmas int_gcd_ac = int_gcd_assoc int_gcd_commute int_gcd_left_commute
huffman@31706
   373
huffman@31706
   374
lemma nat_gcd_unique: "(d::nat) dvd a \<and> d dvd b \<and>
huffman@31706
   375
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
huffman@31706
   376
  apply auto
huffman@31706
   377
  apply (rule dvd_anti_sym)
huffman@31706
   378
  apply (erule (1) nat_gcd_greatest)
huffman@31706
   379
  apply auto
huffman@31706
   380
done
wenzelm@21256
   381
huffman@31706
   382
lemma int_gcd_unique: "d >= 0 & (d::int) dvd a \<and> d dvd b \<and>
huffman@31706
   383
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
huffman@31706
   384
  apply (case_tac "d = 0")
huffman@31706
   385
  apply force
huffman@31706
   386
  apply (rule iffI)
huffman@31706
   387
  apply (rule zdvd_anti_sym)
huffman@31706
   388
  apply arith
huffman@31706
   389
  apply (subst int_gcd_pos)
huffman@31706
   390
  apply clarsimp
huffman@31706
   391
  apply (drule_tac x = "d + 1" in spec)
huffman@31706
   392
  apply (frule zdvd_imp_le)
huffman@31706
   393
  apply (auto intro: int_gcd_greatest)
huffman@31706
   394
done
huffman@30082
   395
nipkow@31798
   396
lemma gcd_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> gcd x y = x"
nipkow@31798
   397
by (metis dvd.eq_iff nat_gcd_unique)
nipkow@31798
   398
nipkow@31798
   399
lemma gcd_proj2_if_dvd_nat [simp]: "(y::nat) dvd x \<Longrightarrow> gcd x y = y"
nipkow@31798
   400
by (metis dvd.eq_iff nat_gcd_unique)
nipkow@31798
   401
nipkow@31798
   402
lemma gcd_proj1_if_dvd_int[simp]: "x dvd y \<Longrightarrow> gcd (x::int) y = abs x"
nipkow@31798
   403
by (metis abs_dvd_iff abs_eq_0 int_gcd_0_left int_gcd_abs int_gcd_unique)
nipkow@31798
   404
nipkow@31798
   405
lemma gcd_proj2_if_dvd_int[simp]: "y dvd x \<Longrightarrow> gcd (x::int) y = abs y"
nipkow@31798
   406
by (metis gcd_proj1_if_dvd_int int_gcd_commute)
nipkow@31798
   407
nipkow@31798
   408
wenzelm@21256
   409
text {*
wenzelm@21256
   410
  \medskip Multiplication laws
wenzelm@21256
   411
*}
wenzelm@21256
   412
huffman@31706
   413
lemma nat_gcd_mult_distrib: "(k::nat) * gcd m n = gcd (k * m) (k * n)"
wenzelm@21256
   414
    -- {* \cite[page 27]{davenport92} *}
huffman@31706
   415
  apply (induct m n rule: nat_gcd_induct)
huffman@31706
   416
  apply simp
wenzelm@21256
   417
  apply (case_tac "k = 0")
huffman@31706
   418
  apply (simp_all add: mod_geq nat_gcd_non_0 mod_mult_distrib2)
huffman@31706
   419
done
wenzelm@21256
   420
huffman@31706
   421
lemma int_gcd_mult_distrib: "abs (k::int) * gcd m n = gcd (k * m) (k * n)"
huffman@31706
   422
  apply (subst (1 2) int_gcd_abs)
nipkow@31813
   423
  apply (subst (1 2) abs_mult)
huffman@31706
   424
  apply (rule nat_gcd_mult_distrib [transferred])
huffman@31706
   425
  apply auto
huffman@31706
   426
done
wenzelm@21256
   427
huffman@31706
   428
lemma nat_coprime_dvd_mult: "coprime (k::nat) n \<Longrightarrow> k dvd m * n \<Longrightarrow> k dvd m"
huffman@31706
   429
  apply (insert nat_gcd_mult_distrib [of m k n])
wenzelm@21256
   430
  apply simp
wenzelm@21256
   431
  apply (erule_tac t = m in ssubst)
wenzelm@21256
   432
  apply simp
wenzelm@21256
   433
  done
wenzelm@21256
   434
huffman@31706
   435
lemma int_coprime_dvd_mult:
nipkow@31813
   436
  "coprime (k::int) n \<Longrightarrow> k dvd m * n \<Longrightarrow> k dvd m"
nipkow@31813
   437
apply (subst abs_dvd_iff [symmetric])
nipkow@31813
   438
apply (subst dvd_abs_iff [symmetric])
nipkow@31813
   439
apply (subst (asm) int_gcd_abs)
nipkow@31813
   440
apply (rule nat_coprime_dvd_mult [transferred])
nipkow@31813
   441
    prefer 4 apply assumption
nipkow@31813
   442
   apply auto
nipkow@31813
   443
apply (subst abs_mult [symmetric], auto)
huffman@31706
   444
done
huffman@31706
   445
huffman@31706
   446
lemma nat_coprime_dvd_mult_iff: "coprime (k::nat) n \<Longrightarrow>
huffman@31706
   447
    (k dvd m * n) = (k dvd m)"
huffman@31706
   448
  by (auto intro: nat_coprime_dvd_mult)
huffman@31706
   449
huffman@31706
   450
lemma int_coprime_dvd_mult_iff: "coprime (k::int) n \<Longrightarrow>
huffman@31706
   451
    (k dvd m * n) = (k dvd m)"
huffman@31706
   452
  by (auto intro: int_coprime_dvd_mult)
huffman@31706
   453
huffman@31706
   454
lemma nat_gcd_mult_cancel: "coprime k n \<Longrightarrow> gcd ((k::nat) * m) n = gcd m n"
wenzelm@21256
   455
  apply (rule dvd_anti_sym)
huffman@31706
   456
  apply (rule nat_gcd_greatest)
huffman@31706
   457
  apply (rule_tac n = k in nat_coprime_dvd_mult)
huffman@31706
   458
  apply (simp add: nat_gcd_assoc)
huffman@31706
   459
  apply (simp add: nat_gcd_commute)
huffman@31706
   460
  apply (simp_all add: mult_commute)
huffman@31706
   461
done
wenzelm@21256
   462
huffman@31706
   463
lemma int_gcd_mult_cancel:
nipkow@31813
   464
  "coprime (k::int) n \<Longrightarrow> gcd (k * m) n = gcd m n"
nipkow@31813
   465
apply (subst (1 2) int_gcd_abs)
nipkow@31813
   466
apply (subst abs_mult)
nipkow@31813
   467
apply (rule nat_gcd_mult_cancel [transferred], auto)
huffman@31706
   468
done
wenzelm@21256
   469
wenzelm@21256
   470
text {* \medskip Addition laws *}
wenzelm@21256
   471
huffman@31706
   472
lemma nat_gcd_add1 [simp]: "gcd ((m::nat) + n) n = gcd m n"
huffman@31706
   473
  apply (case_tac "n = 0")
huffman@31706
   474
  apply (simp_all add: nat_gcd_non_0)
huffman@31706
   475
done
huffman@31706
   476
huffman@31706
   477
lemma nat_gcd_add2 [simp]: "gcd (m::nat) (m + n) = gcd m n"
huffman@31706
   478
  apply (subst (1 2) nat_gcd_commute)
huffman@31706
   479
  apply (subst add_commute)
huffman@31706
   480
  apply simp
huffman@31706
   481
done
huffman@31706
   482
huffman@31706
   483
(* to do: add the other variations? *)
huffman@31706
   484
huffman@31706
   485
lemma nat_gcd_diff1: "(m::nat) >= n \<Longrightarrow> gcd (m - n) n = gcd m n"
huffman@31706
   486
  by (subst nat_gcd_add1 [symmetric], auto)
huffman@31706
   487
huffman@31706
   488
lemma nat_gcd_diff2: "(n::nat) >= m \<Longrightarrow> gcd (n - m) n = gcd m n"
huffman@31706
   489
  apply (subst nat_gcd_commute)
huffman@31706
   490
  apply (subst nat_gcd_diff1 [symmetric])
huffman@31706
   491
  apply auto
huffman@31706
   492
  apply (subst nat_gcd_commute)
huffman@31706
   493
  apply (subst nat_gcd_diff1)
huffman@31706
   494
  apply assumption
huffman@31706
   495
  apply (rule nat_gcd_commute)
huffman@31706
   496
done
huffman@31706
   497
huffman@31706
   498
lemma int_gcd_non_0: "(y::int) > 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
huffman@31706
   499
  apply (frule_tac b = y and a = x in pos_mod_sign)
huffman@31706
   500
  apply (simp del: pos_mod_sign add: gcd_int_def abs_if nat_mod_distrib)
huffman@31706
   501
  apply (auto simp add: nat_gcd_non_0 nat_mod_distrib [symmetric]
huffman@31706
   502
    zmod_zminus1_eq_if)
huffman@31706
   503
  apply (frule_tac a = x in pos_mod_bound)
huffman@31706
   504
  apply (subst (1 2) nat_gcd_commute)
huffman@31706
   505
  apply (simp del: pos_mod_bound add: nat_diff_distrib nat_gcd_diff2
huffman@31706
   506
    nat_le_eq_zle)
huffman@31706
   507
done
wenzelm@21256
   508
huffman@31706
   509
lemma int_gcd_red: "gcd (x::int) y = gcd y (x mod y)"
huffman@31706
   510
  apply (case_tac "y = 0")
huffman@31706
   511
  apply force
huffman@31706
   512
  apply (case_tac "y > 0")
huffman@31706
   513
  apply (subst int_gcd_non_0, auto)
huffman@31706
   514
  apply (insert int_gcd_non_0 [of "-y" "-x"])
huffman@31706
   515
  apply (auto simp add: int_gcd_neg1 int_gcd_neg2)
huffman@31706
   516
done
huffman@31706
   517
huffman@31706
   518
lemma int_gcd_add1 [simp]: "gcd ((m::int) + n) n = gcd m n"
nipkow@31798
   519
by (metis int_gcd_red mod_add_self1 zadd_commute)
huffman@31706
   520
huffman@31706
   521
lemma int_gcd_add2 [simp]: "gcd m ((m::int) + n) = gcd m n"
nipkow@31798
   522
by (metis int_gcd_add1 int_gcd_commute zadd_commute)
wenzelm@21256
   523
huffman@31706
   524
lemma nat_gcd_add_mult: "gcd (m::nat) (k * m + n) = gcd m n"
nipkow@31798
   525
by (metis mod_mult_self3 nat_gcd_commute nat_gcd_red)
wenzelm@21256
   526
huffman@31706
   527
lemma int_gcd_add_mult: "gcd (m::int) (k * m + n) = gcd m n"
nipkow@31798
   528
by (metis int_gcd_commute int_gcd_red mod_mult_self1 zadd_commute)
nipkow@31798
   529
wenzelm@21256
   530
huffman@31706
   531
(* to do: differences, and all variations of addition rules
huffman@31706
   532
    as simplification rules for nat and int *)
huffman@31706
   533
nipkow@31798
   534
(* FIXME remove iff *)
huffman@31706
   535
lemma nat_gcd_dvd_prod [iff]: "gcd (m::nat) n dvd k * n"
haftmann@23687
   536
  using mult_dvd_mono [of 1] by auto
chaieb@22027
   537
huffman@31706
   538
(* to do: add the three variations of these, and for ints? *)
huffman@31706
   539
nipkow@31734
   540
lemma finite_divisors_nat:
nipkow@31734
   541
  assumes "(m::nat)~= 0" shows "finite{d. d dvd m}"
nipkow@31734
   542
proof-
nipkow@31734
   543
  have "finite{d. d <= m}" by(blast intro: bounded_nat_set_is_finite)
nipkow@31734
   544
  from finite_subset[OF _ this] show ?thesis using assms
nipkow@31734
   545
    by(bestsimp intro!:dvd_imp_le)
nipkow@31734
   546
qed
nipkow@31734
   547
nipkow@31734
   548
lemma finite_divisors_int:
nipkow@31734
   549
  assumes "(i::int) ~= 0" shows "finite{d. d dvd i}"
nipkow@31734
   550
proof-
nipkow@31734
   551
  have "{d. abs d <= abs i} = {- abs i .. abs i}" by(auto simp:abs_if)
nipkow@31734
   552
  hence "finite{d. abs d <= abs i}" by simp
nipkow@31734
   553
  from finite_subset[OF _ this] show ?thesis using assms
nipkow@31734
   554
    by(bestsimp intro!:dvd_imp_le_int)
nipkow@31734
   555
qed
nipkow@31734
   556
nipkow@31734
   557
lemma gcd_is_Max_divisors_nat:
nipkow@31734
   558
  "m ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> gcd (m::nat) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
   559
apply(rule Max_eqI[THEN sym])
nipkow@31734
   560
  apply (metis dvd.eq_iff finite_Collect_conjI finite_divisors_nat)
nipkow@31734
   561
 apply simp
nipkow@31734
   562
 apply(metis Suc_diff_1 Suc_neq_Zero dvd_imp_le nat_gcd_greatest_iff nat_gcd_pos)
nipkow@31734
   563
apply simp
nipkow@31734
   564
done
nipkow@31734
   565
nipkow@31734
   566
lemma gcd_is_Max_divisors_int:
nipkow@31734
   567
  "m ~= 0 ==> n ~= 0 ==> gcd (m::int) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
   568
apply(rule Max_eqI[THEN sym])
nipkow@31734
   569
  apply (metis dvd.eq_iff finite_Collect_conjI finite_divisors_int)
nipkow@31734
   570
 apply simp
nipkow@31734
   571
 apply (metis int_gcd_greatest_iff int_gcd_pos zdvd_imp_le)
nipkow@31734
   572
apply simp
nipkow@31734
   573
done
nipkow@31734
   574
chaieb@22027
   575
huffman@31706
   576
subsection {* Coprimality *}
huffman@31706
   577
nipkow@31813
   578
lemma nat_div_gcd_coprime:
huffman@31706
   579
  assumes nz: "(a::nat) \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
   580
  shows "coprime (a div gcd a b) (b div gcd a b)"
wenzelm@22367
   581
proof -
haftmann@27556
   582
  let ?g = "gcd a b"
chaieb@22027
   583
  let ?a' = "a div ?g"
chaieb@22027
   584
  let ?b' = "b div ?g"
haftmann@27556
   585
  let ?g' = "gcd ?a' ?b'"
chaieb@22027
   586
  have dvdg: "?g dvd a" "?g dvd b" by simp_all
chaieb@22027
   587
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
wenzelm@22367
   588
  from dvdg dvdg' obtain ka kb ka' kb' where
wenzelm@22367
   589
      kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
chaieb@22027
   590
    unfolding dvd_def by blast
huffman@31706
   591
  then have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
huffman@31706
   592
    by simp_all
wenzelm@22367
   593
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
wenzelm@22367
   594
    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
wenzelm@22367
   595
      dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
huffman@31706
   596
  have "?g \<noteq> 0" using nz by (simp add: nat_gcd_zero)
huffman@31706
   597
  then have gp: "?g > 0" by arith
huffman@31706
   598
  from nat_gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
wenzelm@22367
   599
  with dvd_mult_cancel1 [OF gp] show "?g' = 1" by simp
chaieb@22027
   600
qed
chaieb@22027
   601
nipkow@31813
   602
lemma int_div_gcd_coprime:
huffman@31706
   603
  assumes nz: "(a::int) \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
   604
  shows "coprime (a div gcd a b) (b div gcd a b)"
nipkow@31813
   605
apply (subst (1 2 3) int_gcd_abs)
nipkow@31813
   606
apply (subst (1 2) abs_div)
nipkow@31813
   607
  apply simp
nipkow@31813
   608
 apply simp
nipkow@31813
   609
apply(subst (1 2) abs_gcd_int)
nipkow@31813
   610
apply (rule nat_div_gcd_coprime [transferred])
nipkow@31813
   611
using nz apply (auto simp add: int_gcd_abs [symmetric])
huffman@31706
   612
done
huffman@31706
   613
huffman@31706
   614
lemma nat_coprime: "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
huffman@31706
   615
  using nat_gcd_unique[of 1 a b, simplified] by auto
huffman@31706
   616
huffman@31706
   617
lemma nat_coprime_Suc_0:
huffman@31706
   618
    "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = Suc 0)"
huffman@31706
   619
  using nat_coprime by (simp add: One_nat_def)
huffman@31706
   620
huffman@31706
   621
lemma int_coprime: "coprime (a::int) b \<longleftrightarrow>
huffman@31706
   622
    (\<forall>d. d >= 0 \<and> d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
huffman@31706
   623
  using int_gcd_unique [of 1 a b]
huffman@31706
   624
  apply clarsimp
huffman@31706
   625
  apply (erule subst)
huffman@31706
   626
  apply (rule iffI)
huffman@31706
   627
  apply force
huffman@31706
   628
  apply (drule_tac x = "abs e" in exI)
huffman@31706
   629
  apply (case_tac "e >= 0")
huffman@31706
   630
  apply force
huffman@31706
   631
  apply force
huffman@31706
   632
done
huffman@31706
   633
huffman@31706
   634
lemma nat_gcd_coprime:
huffman@31706
   635
  assumes z: "gcd (a::nat) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
   636
    b: "b = b' * gcd a b"
huffman@31706
   637
  shows    "coprime a' b'"
huffman@31706
   638
huffman@31706
   639
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
   640
  apply (erule ssubst)
huffman@31706
   641
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
   642
  apply (erule ssubst)
huffman@31706
   643
  apply (rule nat_div_gcd_coprime)
huffman@31706
   644
  using prems
huffman@31706
   645
  apply force
huffman@31706
   646
  apply (subst (1) b)
huffman@31706
   647
  using z apply force
huffman@31706
   648
  apply (subst (1) a)
huffman@31706
   649
  using z apply force
huffman@31706
   650
done
huffman@31706
   651
huffman@31706
   652
lemma int_gcd_coprime:
huffman@31706
   653
  assumes z: "gcd (a::int) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
   654
    b: "b = b' * gcd a b"
huffman@31706
   655
  shows    "coprime a' b'"
huffman@31706
   656
huffman@31706
   657
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
   658
  apply (erule ssubst)
huffman@31706
   659
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
   660
  apply (erule ssubst)
huffman@31706
   661
  apply (rule int_div_gcd_coprime)
huffman@31706
   662
  using prems
huffman@31706
   663
  apply force
huffman@31706
   664
  apply (subst (1) b)
huffman@31706
   665
  using z apply force
huffman@31706
   666
  apply (subst (1) a)
huffman@31706
   667
  using z apply force
huffman@31706
   668
done
huffman@31706
   669
huffman@31706
   670
lemma nat_coprime_mult: assumes da: "coprime (d::nat) a" and db: "coprime d b"
huffman@31706
   671
    shows "coprime d (a * b)"
huffman@31706
   672
  apply (subst nat_gcd_commute)
huffman@31706
   673
  using da apply (subst nat_gcd_mult_cancel)
huffman@31706
   674
  apply (subst nat_gcd_commute, assumption)
huffman@31706
   675
  apply (subst nat_gcd_commute, rule db)
huffman@31706
   676
done
huffman@31706
   677
huffman@31706
   678
lemma int_coprime_mult: assumes da: "coprime (d::int) a" and db: "coprime d b"
huffman@31706
   679
    shows "coprime d (a * b)"
huffman@31706
   680
  apply (subst int_gcd_commute)
huffman@31706
   681
  using da apply (subst int_gcd_mult_cancel)
huffman@31706
   682
  apply (subst int_gcd_commute, assumption)
huffman@31706
   683
  apply (subst int_gcd_commute, rule db)
huffman@31706
   684
done
huffman@31706
   685
huffman@31706
   686
lemma nat_coprime_lmult:
huffman@31706
   687
  assumes dab: "coprime (d::nat) (a * b)" shows "coprime d a"
huffman@31706
   688
proof -
huffman@31706
   689
  have "gcd d a dvd gcd d (a * b)"
huffman@31706
   690
    by (rule nat_gcd_greatest, auto)
huffman@31706
   691
  with dab show ?thesis
huffman@31706
   692
    by auto
huffman@31706
   693
qed
huffman@31706
   694
huffman@31706
   695
lemma int_coprime_lmult:
nipkow@31798
   696
  assumes "coprime (d::int) (a * b)" shows "coprime d a"
huffman@31706
   697
proof -
huffman@31706
   698
  have "gcd d a dvd gcd d (a * b)"
huffman@31706
   699
    by (rule int_gcd_greatest, auto)
nipkow@31798
   700
  with assms show ?thesis
huffman@31706
   701
    by auto
huffman@31706
   702
qed
huffman@31706
   703
huffman@31706
   704
lemma nat_coprime_rmult:
nipkow@31798
   705
  assumes "coprime (d::nat) (a * b)" shows "coprime d b"
huffman@31706
   706
proof -
huffman@31706
   707
  have "gcd d b dvd gcd d (a * b)"
huffman@31706
   708
    by (rule nat_gcd_greatest, auto intro: dvd_mult)
nipkow@31798
   709
  with assms show ?thesis
huffman@31706
   710
    by auto
huffman@31706
   711
qed
huffman@31706
   712
huffman@31706
   713
lemma int_coprime_rmult:
huffman@31706
   714
  assumes dab: "coprime (d::int) (a * b)" shows "coprime d b"
huffman@31706
   715
proof -
huffman@31706
   716
  have "gcd d b dvd gcd d (a * b)"
huffman@31706
   717
    by (rule int_gcd_greatest, auto intro: dvd_mult)
huffman@31706
   718
  with dab show ?thesis
huffman@31706
   719
    by auto
huffman@31706
   720
qed
huffman@31706
   721
huffman@31706
   722
lemma nat_coprime_mul_eq: "coprime (d::nat) (a * b) \<longleftrightarrow>
huffman@31706
   723
    coprime d a \<and>  coprime d b"
huffman@31706
   724
  using nat_coprime_rmult[of d a b] nat_coprime_lmult[of d a b]
huffman@31706
   725
    nat_coprime_mult[of d a b]
huffman@31706
   726
  by blast
huffman@31706
   727
huffman@31706
   728
lemma int_coprime_mul_eq: "coprime (d::int) (a * b) \<longleftrightarrow>
huffman@31706
   729
    coprime d a \<and>  coprime d b"
huffman@31706
   730
  using int_coprime_rmult[of d a b] int_coprime_lmult[of d a b]
huffman@31706
   731
    int_coprime_mult[of d a b]
huffman@31706
   732
  by blast
huffman@31706
   733
huffman@31706
   734
lemma nat_gcd_coprime_exists:
huffman@31706
   735
    assumes nz: "gcd (a::nat) b \<noteq> 0"
huffman@31706
   736
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
   737
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
   738
  apply (rule_tac x = "b div gcd a b" in exI)
huffman@31706
   739
  using nz apply (auto simp add: nat_div_gcd_coprime dvd_div_mult)
huffman@31706
   740
done
huffman@31706
   741
huffman@31706
   742
lemma int_gcd_coprime_exists:
huffman@31706
   743
    assumes nz: "gcd (a::int) b \<noteq> 0"
huffman@31706
   744
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
   745
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
   746
  apply (rule_tac x = "b div gcd a b" in exI)
huffman@31706
   747
  using nz apply (auto simp add: int_div_gcd_coprime dvd_div_mult_self)
huffman@31706
   748
done
huffman@31706
   749
huffman@31706
   750
lemma nat_coprime_exp: "coprime (d::nat) a \<Longrightarrow> coprime d (a^n)"
huffman@31706
   751
  by (induct n, simp_all add: nat_coprime_mult)
huffman@31706
   752
huffman@31706
   753
lemma int_coprime_exp: "coprime (d::int) a \<Longrightarrow> coprime d (a^n)"
huffman@31706
   754
  by (induct n, simp_all add: int_coprime_mult)
huffman@31706
   755
huffman@31706
   756
lemma nat_coprime_exp2 [intro]: "coprime (a::nat) b \<Longrightarrow> coprime (a^n) (b^m)"
huffman@31706
   757
  apply (rule nat_coprime_exp)
huffman@31706
   758
  apply (subst nat_gcd_commute)
huffman@31706
   759
  apply (rule nat_coprime_exp)
huffman@31706
   760
  apply (subst nat_gcd_commute, assumption)
huffman@31706
   761
done
huffman@31706
   762
huffman@31706
   763
lemma int_coprime_exp2 [intro]: "coprime (a::int) b \<Longrightarrow> coprime (a^n) (b^m)"
huffman@31706
   764
  apply (rule int_coprime_exp)
huffman@31706
   765
  apply (subst int_gcd_commute)
huffman@31706
   766
  apply (rule int_coprime_exp)
huffman@31706
   767
  apply (subst int_gcd_commute, assumption)
huffman@31706
   768
done
huffman@31706
   769
huffman@31706
   770
lemma nat_gcd_exp: "gcd ((a::nat)^n) (b^n) = (gcd a b)^n"
huffman@31706
   771
proof (cases)
huffman@31706
   772
  assume "a = 0 & b = 0"
huffman@31706
   773
  thus ?thesis by simp
huffman@31706
   774
  next assume "~(a = 0 & b = 0)"
huffman@31706
   775
  hence "coprime ((a div gcd a b)^n) ((b div gcd a b)^n)"
nipkow@31813
   776
    by (auto simp:nat_div_gcd_coprime)
huffman@31706
   777
  hence "gcd ((a div gcd a b)^n * (gcd a b)^n)
huffman@31706
   778
      ((b div gcd a b)^n * (gcd a b)^n) = (gcd a b)^n"
huffman@31706
   779
    apply (subst (1 2) mult_commute)
huffman@31706
   780
    apply (subst nat_gcd_mult_distrib [symmetric])
huffman@31706
   781
    apply simp
huffman@31706
   782
    done
huffman@31706
   783
  also have "(a div gcd a b)^n * (gcd a b)^n = a^n"
huffman@31706
   784
    apply (subst div_power)
huffman@31706
   785
    apply auto
huffman@31706
   786
    apply (rule dvd_div_mult_self)
huffman@31706
   787
    apply (rule dvd_power_same)
huffman@31706
   788
    apply auto
huffman@31706
   789
    done
huffman@31706
   790
  also have "(b div gcd a b)^n * (gcd a b)^n = b^n"
huffman@31706
   791
    apply (subst div_power)
huffman@31706
   792
    apply auto
huffman@31706
   793
    apply (rule dvd_div_mult_self)
huffman@31706
   794
    apply (rule dvd_power_same)
huffman@31706
   795
    apply auto
huffman@31706
   796
    done
huffman@31706
   797
  finally show ?thesis .
huffman@31706
   798
qed
huffman@31706
   799
huffman@31706
   800
lemma int_gcd_exp: "gcd ((a::int)^n) (b^n) = (gcd a b)^n"
huffman@31706
   801
  apply (subst (1 2) int_gcd_abs)
huffman@31706
   802
  apply (subst (1 2) power_abs)
huffman@31706
   803
  apply (rule nat_gcd_exp [where n = n, transferred])
huffman@31706
   804
  apply auto
huffman@31706
   805
done
huffman@31706
   806
huffman@31706
   807
lemma nat_coprime_divprod: "(d::nat) dvd a * b  \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
huffman@31706
   808
  using nat_coprime_dvd_mult_iff[of d a b]
huffman@31706
   809
  by (auto simp add: mult_commute)
huffman@31706
   810
huffman@31706
   811
lemma int_coprime_divprod: "(d::int) dvd a * b  \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
huffman@31706
   812
  using int_coprime_dvd_mult_iff[of d a b]
huffman@31706
   813
  by (auto simp add: mult_commute)
huffman@31706
   814
huffman@31706
   815
lemma nat_division_decomp: assumes dc: "(a::nat) dvd b * c"
huffman@31706
   816
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
   817
proof-
huffman@31706
   818
  let ?g = "gcd a b"
huffman@31706
   819
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
   820
  moreover
huffman@31706
   821
  {assume z: "?g \<noteq> 0"
huffman@31706
   822
    from nat_gcd_coprime_exists[OF z]
huffman@31706
   823
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   824
      by blast
huffman@31706
   825
    have thb: "?g dvd b" by auto
huffman@31706
   826
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
   827
    with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
huffman@31706
   828
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
huffman@31706
   829
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
huffman@31706
   830
    with z have th_1: "a' dvd b' * c" by auto
huffman@31706
   831
    from nat_coprime_dvd_mult[OF ab'(3)] th_1
huffman@31706
   832
    have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
huffman@31706
   833
    from ab' have "a = ?g*a'" by algebra
huffman@31706
   834
    with thb thc have ?thesis by blast }
huffman@31706
   835
  ultimately show ?thesis by blast
huffman@31706
   836
qed
huffman@31706
   837
huffman@31706
   838
lemma int_division_decomp: assumes dc: "(a::int) dvd b * c"
huffman@31706
   839
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
   840
proof-
huffman@31706
   841
  let ?g = "gcd a b"
huffman@31706
   842
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
   843
  moreover
huffman@31706
   844
  {assume z: "?g \<noteq> 0"
huffman@31706
   845
    from int_gcd_coprime_exists[OF z]
huffman@31706
   846
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   847
      by blast
huffman@31706
   848
    have thb: "?g dvd b" by auto
huffman@31706
   849
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
   850
    with dc have th0: "a' dvd b*c"
huffman@31706
   851
      using dvd_trans[of a' a "b*c"] by simp
huffman@31706
   852
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
huffman@31706
   853
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
huffman@31706
   854
    with z have th_1: "a' dvd b' * c" by auto
huffman@31706
   855
    from int_coprime_dvd_mult[OF ab'(3)] th_1
huffman@31706
   856
    have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
huffman@31706
   857
    from ab' have "a = ?g*a'" by algebra
huffman@31706
   858
    with thb thc have ?thesis by blast }
huffman@31706
   859
  ultimately show ?thesis by blast
chaieb@27669
   860
qed
chaieb@27669
   861
huffman@31706
   862
lemma nat_pow_divides_pow:
huffman@31706
   863
  assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
   864
  shows "a dvd b"
huffman@31706
   865
proof-
huffman@31706
   866
  let ?g = "gcd a b"
huffman@31706
   867
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
   868
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
   869
  moreover
huffman@31706
   870
  {assume z: "?g \<noteq> 0"
huffman@31706
   871
    hence zn: "?g ^ n \<noteq> 0" using n by (simp add: neq0_conv)
huffman@31706
   872
    from nat_gcd_coprime_exists[OF z]
huffman@31706
   873
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   874
      by blast
huffman@31706
   875
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
   876
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
   877
    hence "?g^n*a'^n dvd ?g^n *b'^n"
huffman@31706
   878
      by (simp only: power_mult_distrib mult_commute)
huffman@31706
   879
    with zn z n have th0:"a'^n dvd b'^n" by auto
huffman@31706
   880
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
   881
    with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
huffman@31706
   882
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
huffman@31706
   883
    from nat_coprime_dvd_mult[OF nat_coprime_exp [OF ab'(3), of m]] th1
huffman@31706
   884
    have "a' dvd b'" by (subst (asm) mult_commute, blast)
huffman@31706
   885
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
   886
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
   887
  ultimately show ?thesis by blast
huffman@31706
   888
qed
huffman@31706
   889
huffman@31706
   890
lemma int_pow_divides_pow:
huffman@31706
   891
  assumes ab: "(a::int) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
   892
  shows "a dvd b"
chaieb@27669
   893
proof-
huffman@31706
   894
  let ?g = "gcd a b"
huffman@31706
   895
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
   896
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
   897
  moreover
huffman@31706
   898
  {assume z: "?g \<noteq> 0"
huffman@31706
   899
    hence zn: "?g ^ n \<noteq> 0" using n by (simp add: neq0_conv)
huffman@31706
   900
    from int_gcd_coprime_exists[OF z]
huffman@31706
   901
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   902
      by blast
huffman@31706
   903
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
   904
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
   905
    hence "?g^n*a'^n dvd ?g^n *b'^n"
huffman@31706
   906
      by (simp only: power_mult_distrib mult_commute)
huffman@31706
   907
    with zn z n have th0:"a'^n dvd b'^n" by auto
huffman@31706
   908
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
   909
    with th0 have "a' dvd b'^n"
huffman@31706
   910
      using dvd_trans[of a' "a'^n" "b'^n"] by simp
huffman@31706
   911
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
huffman@31706
   912
    from int_coprime_dvd_mult[OF int_coprime_exp [OF ab'(3), of m]] th1
huffman@31706
   913
    have "a' dvd b'" by (subst (asm) mult_commute, blast)
huffman@31706
   914
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
   915
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
   916
  ultimately show ?thesis by blast
huffman@31706
   917
qed
huffman@31706
   918
nipkow@31798
   919
(* FIXME move to Divides(?) *)
huffman@31706
   920
lemma nat_pow_divides_eq [simp]: "n ~= 0 \<Longrightarrow> ((a::nat)^n dvd b^n) = (a dvd b)"
huffman@31706
   921
  by (auto intro: nat_pow_divides_pow dvd_power_same)
huffman@31706
   922
huffman@31706
   923
lemma int_pow_divides_eq [simp]: "n ~= 0 \<Longrightarrow> ((a::int)^n dvd b^n) = (a dvd b)"
huffman@31706
   924
  by (auto intro: int_pow_divides_pow dvd_power_same)
huffman@31706
   925
huffman@31706
   926
lemma nat_divides_mult:
huffman@31706
   927
  assumes mr: "(m::nat) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
   928
  shows "m * n dvd r"
huffman@31706
   929
proof-
huffman@31706
   930
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
   931
    unfolding dvd_def by blast
huffman@31706
   932
  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
huffman@31706
   933
  hence "m dvd n'" using nat_coprime_dvd_mult_iff[OF mn] by simp
huffman@31706
   934
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
   935
  from n' k show ?thesis unfolding dvd_def by auto
huffman@31706
   936
qed
huffman@31706
   937
huffman@31706
   938
lemma int_divides_mult:
huffman@31706
   939
  assumes mr: "(m::int) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
   940
  shows "m * n dvd r"
huffman@31706
   941
proof-
huffman@31706
   942
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
   943
    unfolding dvd_def by blast
huffman@31706
   944
  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
huffman@31706
   945
  hence "m dvd n'" using int_coprime_dvd_mult_iff[OF mn] by simp
huffman@31706
   946
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
   947
  from n' k show ?thesis unfolding dvd_def by auto
chaieb@27669
   948
qed
chaieb@27669
   949
huffman@31706
   950
lemma nat_coprime_plus_one [simp]: "coprime ((n::nat) + 1) n"
huffman@31706
   951
  apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
huffman@31706
   952
  apply force
huffman@31706
   953
  apply (rule nat_dvd_diff)
huffman@31706
   954
  apply auto
huffman@31706
   955
done
huffman@31706
   956
huffman@31706
   957
lemma nat_coprime_Suc [simp]: "coprime (Suc n) n"
huffman@31706
   958
  using nat_coprime_plus_one by (simp add: One_nat_def)
huffman@31706
   959
huffman@31706
   960
lemma int_coprime_plus_one [simp]: "coprime ((n::int) + 1) n"
huffman@31706
   961
  apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
huffman@31706
   962
  apply force
huffman@31706
   963
  apply (rule dvd_diff)
huffman@31706
   964
  apply auto
huffman@31706
   965
done
huffman@31706
   966
huffman@31706
   967
lemma nat_coprime_minus_one: "(n::nat) \<noteq> 0 \<Longrightarrow> coprime (n - 1) n"
huffman@31706
   968
  using nat_coprime_plus_one [of "n - 1"]
huffman@31706
   969
    nat_gcd_commute [of "n - 1" n] by auto
huffman@31706
   970
huffman@31706
   971
lemma int_coprime_minus_one: "coprime ((n::int) - 1) n"
huffman@31706
   972
  using int_coprime_plus_one [of "n - 1"]
huffman@31706
   973
    int_gcd_commute [of "n - 1" n] by auto
huffman@31706
   974
huffman@31706
   975
lemma nat_setprod_coprime [rule_format]:
huffman@31706
   976
    "(ALL i: A. coprime (f i) (x::nat)) --> coprime (PROD i:A. f i) x"
huffman@31706
   977
  apply (case_tac "finite A")
huffman@31706
   978
  apply (induct set: finite)
huffman@31706
   979
  apply (auto simp add: nat_gcd_mult_cancel)
huffman@31706
   980
done
huffman@31706
   981
huffman@31706
   982
lemma int_setprod_coprime [rule_format]:
huffman@31706
   983
    "(ALL i: A. coprime (f i) (x::int)) --> coprime (PROD i:A. f i) x"
huffman@31706
   984
  apply (case_tac "finite A")
huffman@31706
   985
  apply (induct set: finite)
huffman@31706
   986
  apply (auto simp add: int_gcd_mult_cancel)
huffman@31706
   987
done
huffman@31706
   988
huffman@31706
   989
lemma nat_prime_odd: "prime (p::nat) \<Longrightarrow> p > 2 \<Longrightarrow> odd p"
huffman@31706
   990
  unfolding prime_nat_def
huffman@31706
   991
  apply (subst even_mult_two_ex)
huffman@31706
   992
  apply clarify
huffman@31706
   993
  apply (drule_tac x = 2 in spec)
huffman@31706
   994
  apply auto
huffman@31706
   995
done
huffman@31706
   996
huffman@31706
   997
lemma int_prime_odd: "prime (p::int) \<Longrightarrow> p > 2 \<Longrightarrow> odd p"
huffman@31706
   998
  unfolding prime_int_def
huffman@31706
   999
  apply (frule nat_prime_odd)
huffman@31706
  1000
  apply (auto simp add: even_nat_def)
huffman@31706
  1001
done
huffman@31706
  1002
huffman@31706
  1003
lemma nat_coprime_common_divisor: "coprime (a::nat) b \<Longrightarrow> x dvd a \<Longrightarrow>
huffman@31706
  1004
    x dvd b \<Longrightarrow> x = 1"
huffman@31706
  1005
  apply (subgoal_tac "x dvd gcd a b")
huffman@31706
  1006
  apply simp
huffman@31706
  1007
  apply (erule (1) nat_gcd_greatest)
huffman@31706
  1008
done
huffman@31706
  1009
huffman@31706
  1010
lemma int_coprime_common_divisor: "coprime (a::int) b \<Longrightarrow> x dvd a \<Longrightarrow>
huffman@31706
  1011
    x dvd b \<Longrightarrow> abs x = 1"
huffman@31706
  1012
  apply (subgoal_tac "x dvd gcd a b")
huffman@31706
  1013
  apply simp
huffman@31706
  1014
  apply (erule (1) int_gcd_greatest)
huffman@31706
  1015
done
huffman@31706
  1016
huffman@31706
  1017
lemma nat_coprime_divisors: "(d::int) dvd a \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow>
huffman@31706
  1018
    coprime d e"
huffman@31706
  1019
  apply (auto simp add: dvd_def)
huffman@31706
  1020
  apply (frule int_coprime_lmult)
huffman@31706
  1021
  apply (subst int_gcd_commute)
huffman@31706
  1022
  apply (subst (asm) (2) int_gcd_commute)
huffman@31706
  1023
  apply (erule int_coprime_lmult)
huffman@31706
  1024
done
huffman@31706
  1025
huffman@31706
  1026
lemma nat_invertible_coprime: "(x::nat) * y mod m = 1 \<Longrightarrow> coprime x m"
huffman@31706
  1027
apply (metis nat_coprime_lmult nat_gcd_1 nat_gcd_commute nat_gcd_red)
huffman@31706
  1028
done
huffman@31706
  1029
huffman@31706
  1030
lemma int_invertible_coprime: "(x::int) * y mod m = 1 \<Longrightarrow> coprime x m"
huffman@31706
  1031
apply (metis int_coprime_lmult int_gcd_1 int_gcd_commute int_gcd_red)
huffman@31706
  1032
done
huffman@31706
  1033
huffman@31706
  1034
huffman@31706
  1035
subsection {* Bezout's theorem *}
huffman@31706
  1036
huffman@31706
  1037
(* Function bezw returns a pair of witnesses to Bezout's theorem --
huffman@31706
  1038
   see the theorems that follow the definition. *)
huffman@31706
  1039
fun
huffman@31706
  1040
  bezw  :: "nat \<Rightarrow> nat \<Rightarrow> int * int"
huffman@31706
  1041
where
huffman@31706
  1042
  "bezw x y =
huffman@31706
  1043
  (if y = 0 then (1, 0) else
huffman@31706
  1044
      (snd (bezw y (x mod y)),
huffman@31706
  1045
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y)))"
huffman@31706
  1046
huffman@31706
  1047
lemma bezw_0 [simp]: "bezw x 0 = (1, 0)" by simp
huffman@31706
  1048
huffman@31706
  1049
lemma bezw_non_0: "y > 0 \<Longrightarrow> bezw x y = (snd (bezw y (x mod y)),
huffman@31706
  1050
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y))"
huffman@31706
  1051
  by simp
huffman@31706
  1052
huffman@31706
  1053
declare bezw.simps [simp del]
huffman@31706
  1054
huffman@31706
  1055
lemma bezw_aux [rule_format]:
huffman@31706
  1056
    "fst (bezw x y) * int x + snd (bezw x y) * int y = int (gcd x y)"
huffman@31706
  1057
proof (induct x y rule: nat_gcd_induct)
huffman@31706
  1058
  fix m :: nat
huffman@31706
  1059
  show "fst (bezw m 0) * int m + snd (bezw m 0) * int 0 = int (gcd m 0)"
huffman@31706
  1060
    by auto
huffman@31706
  1061
  next fix m :: nat and n
huffman@31706
  1062
    assume ngt0: "n > 0" and
huffman@31706
  1063
      ih: "fst (bezw n (m mod n)) * int n +
huffman@31706
  1064
        snd (bezw n (m mod n)) * int (m mod n) =
huffman@31706
  1065
        int (gcd n (m mod n))"
huffman@31706
  1066
    thus "fst (bezw m n) * int m + snd (bezw m n) * int n = int (gcd m n)"
huffman@31706
  1067
      apply (simp add: bezw_non_0 nat_gcd_non_0)
huffman@31706
  1068
      apply (erule subst)
huffman@31706
  1069
      apply (simp add: ring_simps)
huffman@31706
  1070
      apply (subst mod_div_equality [of m n, symmetric])
huffman@31706
  1071
      (* applying simp here undoes the last substitution!
huffman@31706
  1072
         what is procedure cancel_div_mod? *)
huffman@31706
  1073
      apply (simp only: ring_simps zadd_int [symmetric]
huffman@31706
  1074
        zmult_int [symmetric])
huffman@31706
  1075
      done
huffman@31706
  1076
qed
huffman@31706
  1077
huffman@31706
  1078
lemma int_bezout:
huffman@31706
  1079
  fixes x y
huffman@31706
  1080
  shows "EX u v. u * (x::int) + v * y = gcd x y"
huffman@31706
  1081
proof -
huffman@31706
  1082
  have bezout_aux: "!!x y. x \<ge> (0::int) \<Longrightarrow> y \<ge> 0 \<Longrightarrow>
huffman@31706
  1083
      EX u v. u * x + v * y = gcd x y"
huffman@31706
  1084
    apply (rule_tac x = "fst (bezw (nat x) (nat y))" in exI)
huffman@31706
  1085
    apply (rule_tac x = "snd (bezw (nat x) (nat y))" in exI)
huffman@31706
  1086
    apply (unfold gcd_int_def)
huffman@31706
  1087
    apply simp
huffman@31706
  1088
    apply (subst bezw_aux [symmetric])
huffman@31706
  1089
    apply auto
huffman@31706
  1090
    done
huffman@31706
  1091
  have "(x \<ge> 0 \<and> y \<ge> 0) | (x \<ge> 0 \<and> y \<le> 0) | (x \<le> 0 \<and> y \<ge> 0) |
huffman@31706
  1092
      (x \<le> 0 \<and> y \<le> 0)"
huffman@31706
  1093
    by auto
huffman@31706
  1094
  moreover have "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> ?thesis"
huffman@31706
  1095
    by (erule (1) bezout_aux)
huffman@31706
  1096
  moreover have "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1097
    apply (insert bezout_aux [of x "-y"])
huffman@31706
  1098
    apply auto
huffman@31706
  1099
    apply (rule_tac x = u in exI)
huffman@31706
  1100
    apply (rule_tac x = "-v" in exI)
huffman@31706
  1101
    apply (subst int_gcd_neg2 [symmetric])
huffman@31706
  1102
    apply auto
huffman@31706
  1103
    done
huffman@31706
  1104
  moreover have "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1105
    apply (insert bezout_aux [of "-x" y])
huffman@31706
  1106
    apply auto
huffman@31706
  1107
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1108
    apply (rule_tac x = v in exI)
huffman@31706
  1109
    apply (subst int_gcd_neg1 [symmetric])
huffman@31706
  1110
    apply auto
huffman@31706
  1111
    done
huffman@31706
  1112
  moreover have "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1113
    apply (insert bezout_aux [of "-x" "-y"])
huffman@31706
  1114
    apply auto
huffman@31706
  1115
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1116
    apply (rule_tac x = "-v" in exI)
huffman@31706
  1117
    apply (subst int_gcd_neg1 [symmetric])
huffman@31706
  1118
    apply (subst int_gcd_neg2 [symmetric])
huffman@31706
  1119
    apply auto
huffman@31706
  1120
    done
huffman@31706
  1121
  ultimately show ?thesis by blast
huffman@31706
  1122
qed
huffman@31706
  1123
huffman@31706
  1124
text {* versions of Bezout for nat, by Amine Chaieb *}
huffman@31706
  1125
huffman@31706
  1126
lemma ind_euclid:
huffman@31706
  1127
  assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
huffman@31706
  1128
  and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
chaieb@27669
  1129
  shows "P a b"
chaieb@27669
  1130
proof(induct n\<equiv>"a+b" arbitrary: a b rule: nat_less_induct)
chaieb@27669
  1131
  fix n a b
chaieb@27669
  1132
  assume H: "\<forall>m < n. \<forall>a b. m = a + b \<longrightarrow> P a b" "n = a + b"
chaieb@27669
  1133
  have "a = b \<or> a < b \<or> b < a" by arith
chaieb@27669
  1134
  moreover {assume eq: "a= b"
huffman@31706
  1135
    from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq
huffman@31706
  1136
    by simp}
chaieb@27669
  1137
  moreover
chaieb@27669
  1138
  {assume lt: "a < b"
chaieb@27669
  1139
    hence "a + b - a < n \<or> a = 0"  using H(2) by arith
chaieb@27669
  1140
    moreover
chaieb@27669
  1141
    {assume "a =0" with z c have "P a b" by blast }
chaieb@27669
  1142
    moreover
chaieb@27669
  1143
    {assume ab: "a + b - a < n"
chaieb@27669
  1144
      have th0: "a + b - a = a + (b - a)" using lt by arith
chaieb@27669
  1145
      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
chaieb@27669
  1146
      have "P a b" by (simp add: th0[symmetric])}
chaieb@27669
  1147
    ultimately have "P a b" by blast}
chaieb@27669
  1148
  moreover
chaieb@27669
  1149
  {assume lt: "a > b"
chaieb@27669
  1150
    hence "b + a - b < n \<or> b = 0"  using H(2) by arith
chaieb@27669
  1151
    moreover
chaieb@27669
  1152
    {assume "b =0" with z c have "P a b" by blast }
chaieb@27669
  1153
    moreover
chaieb@27669
  1154
    {assume ab: "b + a - b < n"
chaieb@27669
  1155
      have th0: "b + a - b = b + (a - b)" using lt by arith
chaieb@27669
  1156
      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
chaieb@27669
  1157
      have "P b a" by (simp add: th0[symmetric])
chaieb@27669
  1158
      hence "P a b" using c by blast }
chaieb@27669
  1159
    ultimately have "P a b" by blast}
chaieb@27669
  1160
ultimately  show "P a b" by blast
chaieb@27669
  1161
qed
chaieb@27669
  1162
huffman@31706
  1163
lemma nat_bezout_lemma:
huffman@31706
  1164
  assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1165
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1166
  shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and>
huffman@31706
  1167
    (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
huffman@31706
  1168
  using ex
huffman@31706
  1169
  apply clarsimp
huffman@31706
  1170
  apply (rule_tac x="d" in exI, simp add: dvd_add)
huffman@31706
  1171
  apply (case_tac "a * x = b * y + d" , simp_all)
huffman@31706
  1172
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1173
  apply (rule_tac x="y" in exI)
huffman@31706
  1174
  apply algebra
huffman@31706
  1175
  apply (rule_tac x="x" in exI)
huffman@31706
  1176
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1177
  apply algebra
chaieb@27669
  1178
done
chaieb@27669
  1179
huffman@31706
  1180
lemma nat_bezout_add: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1181
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1182
  apply(induct a b rule: ind_euclid)
huffman@31706
  1183
  apply blast
huffman@31706
  1184
  apply clarify
huffman@31706
  1185
  apply (rule_tac x="a" in exI, simp add: dvd_add)
huffman@31706
  1186
  apply clarsimp
huffman@31706
  1187
  apply (rule_tac x="d" in exI)
huffman@31706
  1188
  apply (case_tac "a * x = b * y + d", simp_all add: dvd_add)
huffman@31706
  1189
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1190
  apply (rule_tac x="y" in exI)
huffman@31706
  1191
  apply algebra
huffman@31706
  1192
  apply (rule_tac x="x" in exI)
huffman@31706
  1193
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1194
  apply algebra
chaieb@27669
  1195
done
chaieb@27669
  1196
huffman@31706
  1197
lemma nat_bezout1: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1198
    (a * x - b * y = d \<or> b * x - a * y = d)"
huffman@31706
  1199
  using nat_bezout_add[of a b]
huffman@31706
  1200
  apply clarsimp
huffman@31706
  1201
  apply (rule_tac x="d" in exI, simp)
huffman@31706
  1202
  apply (rule_tac x="x" in exI)
huffman@31706
  1203
  apply (rule_tac x="y" in exI)
huffman@31706
  1204
  apply auto
chaieb@27669
  1205
done
chaieb@27669
  1206
huffman@31706
  1207
lemma nat_bezout_add_strong: assumes nz: "a \<noteq> (0::nat)"
chaieb@27669
  1208
  shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
chaieb@27669
  1209
proof-
huffman@31706
  1210
 from nz have ap: "a > 0" by simp
huffman@31706
  1211
 from nat_bezout_add[of a b]
huffman@31706
  1212
 have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or>
huffman@31706
  1213
   (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
chaieb@27669
  1214
 moreover
huffman@31706
  1215
    {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
huffman@31706
  1216
     from H have ?thesis by blast }
chaieb@27669
  1217
 moreover
chaieb@27669
  1218
 {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
chaieb@27669
  1219
   {assume b0: "b = 0" with H  have ?thesis by simp}
huffman@31706
  1220
   moreover
chaieb@27669
  1221
   {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
huffman@31706
  1222
     from b dvd_imp_le [OF H(2)] have "d < b \<or> d = b"
huffman@31706
  1223
       by auto
chaieb@27669
  1224
     moreover
chaieb@27669
  1225
     {assume db: "d=b"
chaieb@27669
  1226
       from prems have ?thesis apply simp
chaieb@27669
  1227
	 apply (rule exI[where x = b], simp)
chaieb@27669
  1228
	 apply (rule exI[where x = b])
chaieb@27669
  1229
	by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
chaieb@27669
  1230
    moreover
huffman@31706
  1231
    {assume db: "d < b"
chaieb@27669
  1232
	{assume "x=0" hence ?thesis  using prems by simp }
chaieb@27669
  1233
	moreover
chaieb@27669
  1234
	{assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
chaieb@27669
  1235
	  from db have "d \<le> b - 1" by simp
chaieb@27669
  1236
	  hence "d*b \<le> b*(b - 1)" by simp
chaieb@27669
  1237
	  with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
chaieb@27669
  1238
	  have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
huffman@31706
  1239
	  from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)"
huffman@31706
  1240
            by simp
huffman@31706
  1241
	  hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
huffman@31706
  1242
	    by (simp only: mult_assoc right_distrib)
huffman@31706
  1243
	  hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)"
huffman@31706
  1244
            by algebra
chaieb@27669
  1245
	  hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
huffman@31706
  1246
	  hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
chaieb@27669
  1247
	    by (simp only: diff_add_assoc[OF dble, of d, symmetric])
chaieb@27669
  1248
	  hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
chaieb@27669
  1249
	    by (simp only: diff_mult_distrib2 add_commute mult_ac)
chaieb@27669
  1250
	  hence ?thesis using H(1,2)
chaieb@27669
  1251
	    apply -
chaieb@27669
  1252
	    apply (rule exI[where x=d], simp)
chaieb@27669
  1253
	    apply (rule exI[where x="(b - 1) * y"])
chaieb@27669
  1254
	    by (rule exI[where x="x*(b - 1) - d"], simp)}
chaieb@27669
  1255
	ultimately have ?thesis by blast}
chaieb@27669
  1256
    ultimately have ?thesis by blast}
chaieb@27669
  1257
  ultimately have ?thesis by blast}
chaieb@27669
  1258
 ultimately show ?thesis by blast
chaieb@27669
  1259
qed
chaieb@27669
  1260
huffman@31706
  1261
lemma nat_bezout: assumes a: "(a::nat) \<noteq> 0"
chaieb@27669
  1262
  shows "\<exists>x y. a * x = b * y + gcd a b"
chaieb@27669
  1263
proof-
chaieb@27669
  1264
  let ?g = "gcd a b"
huffman@31706
  1265
  from nat_bezout_add_strong[OF a, of b]
chaieb@27669
  1266
  obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
chaieb@27669
  1267
  from d(1,2) have "d dvd ?g" by simp
chaieb@27669
  1268
  then obtain k where k: "?g = d*k" unfolding dvd_def by blast
huffman@31706
  1269
  from d(3) have "a * x * k = (b * y + d) *k " by auto
chaieb@27669
  1270
  hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
chaieb@27669
  1271
  thus ?thesis by blast
chaieb@27669
  1272
qed
chaieb@27669
  1273
huffman@31706
  1274
huffman@31706
  1275
subsection {* LCM *}
huffman@31706
  1276
huffman@31706
  1277
lemma int_lcm_altdef: "lcm (a::int) b = (abs a) * (abs b) div gcd a b"
huffman@31706
  1278
  by (simp add: lcm_int_def lcm_nat_def zdiv_int
huffman@31706
  1279
    zmult_int [symmetric] gcd_int_def)
huffman@31706
  1280
huffman@31706
  1281
lemma nat_prod_gcd_lcm: "(m::nat) * n = gcd m n * lcm m n"
huffman@31706
  1282
  unfolding lcm_nat_def
huffman@31706
  1283
  by (simp add: dvd_mult_div_cancel [OF nat_gcd_dvd_prod])
huffman@31706
  1284
huffman@31706
  1285
lemma int_prod_gcd_lcm: "abs(m::int) * abs n = gcd m n * lcm m n"
huffman@31706
  1286
  unfolding lcm_int_def gcd_int_def
huffman@31706
  1287
  apply (subst int_mult [symmetric])
huffman@31706
  1288
  apply (subst nat_prod_gcd_lcm [symmetric])
huffman@31706
  1289
  apply (subst nat_abs_mult_distrib [symmetric])
huffman@31706
  1290
  apply (simp, simp add: abs_mult)
huffman@31706
  1291
done
huffman@31706
  1292
huffman@31706
  1293
lemma nat_lcm_0 [simp]: "lcm (m::nat) 0 = 0"
huffman@31706
  1294
  unfolding lcm_nat_def by simp
huffman@31706
  1295
huffman@31706
  1296
lemma int_lcm_0 [simp]: "lcm (m::int) 0 = 0"
huffman@31706
  1297
  unfolding lcm_int_def by simp
huffman@31706
  1298
huffman@31706
  1299
lemma nat_lcm_0_left [simp]: "lcm (0::nat) n = 0"
huffman@31706
  1300
  unfolding lcm_nat_def by simp
chaieb@27669
  1301
huffman@31706
  1302
lemma int_lcm_0_left [simp]: "lcm (0::int) n = 0"
huffman@31706
  1303
  unfolding lcm_int_def by simp
huffman@31706
  1304
huffman@31706
  1305
lemma nat_lcm_commute: "lcm (m::nat) n = lcm n m"
huffman@31706
  1306
  unfolding lcm_nat_def by (simp add: nat_gcd_commute ring_simps)
huffman@31706
  1307
huffman@31706
  1308
lemma int_lcm_commute: "lcm (m::int) n = lcm n m"
huffman@31706
  1309
  unfolding lcm_int_def by (subst nat_lcm_commute, rule refl)
huffman@31706
  1310
huffman@31706
  1311
huffman@31706
  1312
lemma nat_lcm_pos:
nipkow@31798
  1313
  "(m::nat) > 0 \<Longrightarrow> n>0 \<Longrightarrow> lcm m n > 0"
nipkow@31798
  1314
by (metis gr0I mult_is_0 nat_prod_gcd_lcm)
chaieb@27669
  1315
huffman@31706
  1316
lemma int_lcm_pos:
nipkow@31798
  1317
  "(m::int) ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> lcm m n > 0"
huffman@31706
  1318
  apply (subst int_lcm_abs)
huffman@31706
  1319
  apply (rule nat_lcm_pos [transferred])
nipkow@31798
  1320
  apply auto
huffman@31706
  1321
done
haftmann@23687
  1322
huffman@31706
  1323
lemma nat_dvd_pos:
haftmann@23687
  1324
  fixes n m :: nat
haftmann@23687
  1325
  assumes "n > 0" and "m dvd n"
haftmann@23687
  1326
  shows "m > 0"
haftmann@23687
  1327
using assms by (cases m) auto
haftmann@23687
  1328
huffman@31706
  1329
lemma nat_lcm_least:
huffman@31706
  1330
  assumes "(m::nat) dvd k" and "n dvd k"
haftmann@27556
  1331
  shows "lcm m n dvd k"
haftmann@23687
  1332
proof (cases k)
haftmann@23687
  1333
  case 0 then show ?thesis by auto
haftmann@23687
  1334
next
haftmann@23687
  1335
  case (Suc _) then have pos_k: "k > 0" by auto
huffman@31706
  1336
  from assms nat_dvd_pos [OF this] have pos_mn: "m > 0" "n > 0" by auto
huffman@31706
  1337
  with nat_gcd_zero [of m n] have pos_gcd: "gcd m n > 0" by simp
haftmann@23687
  1338
  from assms obtain p where k_m: "k = m * p" using dvd_def by blast
haftmann@23687
  1339
  from assms obtain q where k_n: "k = n * q" using dvd_def by blast
haftmann@23687
  1340
  from pos_k k_m have pos_p: "p > 0" by auto
haftmann@23687
  1341
  from pos_k k_n have pos_q: "q > 0" by auto
haftmann@27556
  1342
  have "k * k * gcd q p = k * gcd (k * q) (k * p)"
huffman@31706
  1343
    by (simp add: mult_ac nat_gcd_mult_distrib)
haftmann@27556
  1344
  also have "\<dots> = k * gcd (m * p * q) (n * q * p)"
haftmann@23687
  1345
    by (simp add: k_m [symmetric] k_n [symmetric])
haftmann@27556
  1346
  also have "\<dots> = k * p * q * gcd m n"
huffman@31706
  1347
    by (simp add: mult_ac nat_gcd_mult_distrib)
haftmann@27556
  1348
  finally have "(m * p) * (n * q) * gcd q p = k * p * q * gcd m n"
haftmann@23687
  1349
    by (simp only: k_m [symmetric] k_n [symmetric])
haftmann@27556
  1350
  then have "p * q * m * n * gcd q p = p * q * k * gcd m n"
haftmann@23687
  1351
    by (simp add: mult_ac)
haftmann@27556
  1352
  with pos_p pos_q have "m * n * gcd q p = k * gcd m n"
haftmann@23687
  1353
    by simp
huffman@31706
  1354
  with nat_prod_gcd_lcm [of m n]
haftmann@27556
  1355
  have "lcm m n * gcd q p * gcd m n = k * gcd m n"
haftmann@23687
  1356
    by (simp add: mult_ac)
huffman@31706
  1357
  with pos_gcd have "lcm m n * gcd q p = k" by auto
haftmann@23687
  1358
  then show ?thesis using dvd_def by auto
haftmann@23687
  1359
qed
haftmann@23687
  1360
huffman@31706
  1361
lemma int_lcm_least:
nipkow@31798
  1362
  "(m::int) dvd k \<Longrightarrow> n dvd k \<Longrightarrow> lcm m n dvd k"
nipkow@31798
  1363
apply (subst int_lcm_abs)
nipkow@31798
  1364
apply (rule dvd_trans)
nipkow@31798
  1365
apply (rule nat_lcm_least [transferred, of _ "abs k" _])
nipkow@31798
  1366
apply auto
huffman@31706
  1367
done
huffman@31706
  1368
nipkow@31730
  1369
lemma nat_lcm_dvd1: "(m::nat) dvd lcm m n"
haftmann@23687
  1370
proof (cases m)
haftmann@23687
  1371
  case 0 then show ?thesis by simp
haftmann@23687
  1372
next
haftmann@23687
  1373
  case (Suc _)
haftmann@23687
  1374
  then have mpos: "m > 0" by simp
haftmann@23687
  1375
  show ?thesis
haftmann@23687
  1376
  proof (cases n)
haftmann@23687
  1377
    case 0 then show ?thesis by simp
haftmann@23687
  1378
  next
haftmann@23687
  1379
    case (Suc _)
haftmann@23687
  1380
    then have npos: "n > 0" by simp
haftmann@27556
  1381
    have "gcd m n dvd n" by simp
haftmann@27556
  1382
    then obtain k where "n = gcd m n * k" using dvd_def by auto
huffman@31706
  1383
    then have "m * n div gcd m n = m * (gcd m n * k) div gcd m n"
huffman@31706
  1384
      by (simp add: mult_ac)
huffman@31706
  1385
    also have "\<dots> = m * k" using mpos npos nat_gcd_zero by simp
huffman@31706
  1386
    finally show ?thesis by (simp add: lcm_nat_def)
haftmann@23687
  1387
  qed
haftmann@23687
  1388
qed
haftmann@23687
  1389
nipkow@31730
  1390
lemma int_lcm_dvd1: "(m::int) dvd lcm m n"
huffman@31706
  1391
  apply (subst int_lcm_abs)
huffman@31706
  1392
  apply (rule dvd_trans)
huffman@31706
  1393
  prefer 2
huffman@31706
  1394
  apply (rule nat_lcm_dvd1 [transferred])
huffman@31706
  1395
  apply auto
huffman@31706
  1396
done
huffman@31706
  1397
nipkow@31730
  1398
lemma nat_lcm_dvd2: "(n::nat) dvd lcm m n"
huffman@31706
  1399
  by (subst nat_lcm_commute, rule nat_lcm_dvd1)
huffman@31706
  1400
nipkow@31730
  1401
lemma int_lcm_dvd2: "(n::int) dvd lcm m n"
huffman@31706
  1402
  by (subst int_lcm_commute, rule int_lcm_dvd1)
huffman@31706
  1403
nipkow@31730
  1404
lemma dvd_lcm_I1_nat[simp]: "(k::nat) dvd m \<Longrightarrow> k dvd lcm m n"
nipkow@31729
  1405
by(metis nat_lcm_dvd1 dvd_trans)
nipkow@31729
  1406
nipkow@31730
  1407
lemma dvd_lcm_I2_nat[simp]: "(k::nat) dvd n \<Longrightarrow> k dvd lcm m n"
nipkow@31729
  1408
by(metis nat_lcm_dvd2 dvd_trans)
nipkow@31729
  1409
nipkow@31730
  1410
lemma dvd_lcm_I1_int[simp]: "(i::int) dvd m \<Longrightarrow> i dvd lcm m n"
nipkow@31729
  1411
by(metis int_lcm_dvd1 dvd_trans)
nipkow@31729
  1412
nipkow@31730
  1413
lemma dvd_lcm_I2_int[simp]: "(i::int) dvd n \<Longrightarrow> i dvd lcm m n"
nipkow@31729
  1414
by(metis int_lcm_dvd2 dvd_trans)
nipkow@31729
  1415
huffman@31706
  1416
lemma nat_lcm_unique: "(a::nat) dvd d \<and> b dvd d \<and>
huffman@31706
  1417
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
nipkow@31730
  1418
  by (auto intro: dvd_anti_sym nat_lcm_least nat_lcm_dvd1 nat_lcm_dvd2)
chaieb@27568
  1419
huffman@31706
  1420
lemma int_lcm_unique: "d >= 0 \<and> (a::int) dvd d \<and> b dvd d \<and>
huffman@31706
  1421
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
huffman@31706
  1422
  by (auto intro: dvd_anti_sym [transferred] int_lcm_least)
huffman@31706
  1423
nipkow@31798
  1424
lemma lcm_proj2_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm x y = y"
huffman@31706
  1425
  apply (rule sym)
huffman@31706
  1426
  apply (subst nat_lcm_unique [symmetric])
huffman@31706
  1427
  apply auto
huffman@31706
  1428
done
huffman@31706
  1429
nipkow@31798
  1430
lemma lcm_proj2_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm x y = abs y"
huffman@31706
  1431
  apply (rule sym)
huffman@31706
  1432
  apply (subst int_lcm_unique [symmetric])
huffman@31706
  1433
  apply auto
huffman@31706
  1434
done
huffman@31706
  1435
nipkow@31798
  1436
lemma lcm_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm y x = y"
nipkow@31798
  1437
by (subst nat_lcm_commute, erule lcm_proj2_if_dvd_nat)
huffman@31706
  1438
nipkow@31798
  1439
lemma lcm_proj1_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm y x = abs y"
nipkow@31798
  1440
by (subst int_lcm_commute, erule lcm_proj2_if_dvd_int)
huffman@31706
  1441
chaieb@27568
  1442
nipkow@31766
  1443
lemma lcm_assoc_nat: "lcm (lcm n m) (p::nat) = lcm n (lcm m p)"
nipkow@31766
  1444
apply(rule nat_lcm_unique[THEN iffD1])
nipkow@31766
  1445
apply (metis dvd.order_trans nat_lcm_unique)
nipkow@31766
  1446
done
nipkow@31766
  1447
nipkow@31766
  1448
lemma lcm_assoc_int: "lcm (lcm n m) (p::int) = lcm n (lcm m p)"
nipkow@31766
  1449
apply(rule int_lcm_unique[THEN iffD1])
nipkow@31766
  1450
apply (metis dvd_trans int_lcm_unique)
nipkow@31766
  1451
done
nipkow@31766
  1452
nipkow@31766
  1453
lemmas lcm_left_commute_nat =
nipkow@31766
  1454
  mk_left_commute[of lcm, OF lcm_assoc_nat nat_lcm_commute]
nipkow@31766
  1455
nipkow@31766
  1456
lemmas lcm_left_commute_int =
nipkow@31766
  1457
  mk_left_commute[of lcm, OF lcm_assoc_int int_lcm_commute]
nipkow@31766
  1458
nipkow@31766
  1459
lemmas lcm_ac_nat = lcm_assoc_nat nat_lcm_commute lcm_left_commute_nat
nipkow@31766
  1460
lemmas lcm_ac_int = lcm_assoc_int int_lcm_commute lcm_left_commute_int
nipkow@31766
  1461
haftmann@23687
  1462
huffman@31706
  1463
subsection {* Primes *}
wenzelm@22367
  1464
huffman@31706
  1465
(* Is there a better way to handle these, rather than making them
huffman@31706
  1466
   elim rules? *)
chaieb@22027
  1467
huffman@31706
  1468
lemma nat_prime_ge_0 [elim]: "prime (p::nat) \<Longrightarrow> p >= 0"
huffman@31706
  1469
  by (unfold prime_nat_def, auto)
chaieb@22027
  1470
huffman@31706
  1471
lemma nat_prime_gt_0 [elim]: "prime (p::nat) \<Longrightarrow> p > 0"
huffman@31706
  1472
  by (unfold prime_nat_def, auto)
wenzelm@22367
  1473
huffman@31706
  1474
lemma nat_prime_ge_1 [elim]: "prime (p::nat) \<Longrightarrow> p >= 1"
huffman@31706
  1475
  by (unfold prime_nat_def, auto)
chaieb@22027
  1476
huffman@31706
  1477
lemma nat_prime_gt_1 [elim]: "prime (p::nat) \<Longrightarrow> p > 1"
huffman@31706
  1478
  by (unfold prime_nat_def, auto)
wenzelm@22367
  1479
huffman@31706
  1480
lemma nat_prime_ge_Suc_0 [elim]: "prime (p::nat) \<Longrightarrow> p >= Suc 0"
huffman@31706
  1481
  by (unfold prime_nat_def, auto)
wenzelm@22367
  1482
huffman@31706
  1483
lemma nat_prime_gt_Suc_0 [elim]: "prime (p::nat) \<Longrightarrow> p > Suc 0"
huffman@31706
  1484
  by (unfold prime_nat_def, auto)
huffman@31706
  1485
huffman@31706
  1486
lemma nat_prime_ge_2 [elim]: "prime (p::nat) \<Longrightarrow> p >= 2"
huffman@31706
  1487
  by (unfold prime_nat_def, auto)
huffman@31706
  1488
huffman@31706
  1489
lemma int_prime_ge_0 [elim]: "prime (p::int) \<Longrightarrow> p >= 0"
huffman@31706
  1490
  by (unfold prime_int_def prime_nat_def, auto)
wenzelm@22367
  1491
huffman@31706
  1492
lemma int_prime_gt_0 [elim]: "prime (p::int) \<Longrightarrow> p > 0"
huffman@31706
  1493
  by (unfold prime_int_def prime_nat_def, auto)
huffman@31706
  1494
huffman@31706
  1495
lemma int_prime_ge_1 [elim]: "prime (p::int) \<Longrightarrow> p >= 1"
huffman@31706
  1496
  by (unfold prime_int_def prime_nat_def, auto)
chaieb@22027
  1497
huffman@31706
  1498
lemma int_prime_gt_1 [elim]: "prime (p::int) \<Longrightarrow> p > 1"
huffman@31706
  1499
  by (unfold prime_int_def prime_nat_def, auto)
huffman@31706
  1500
huffman@31706
  1501
lemma int_prime_ge_2 [elim]: "prime (p::int) \<Longrightarrow> p >= 2"
huffman@31706
  1502
  by (unfold prime_int_def prime_nat_def, auto)
wenzelm@22367
  1503
huffman@31706
  1504
thm prime_nat_def;
huffman@31706
  1505
thm prime_nat_def [transferred];
huffman@31706
  1506
huffman@31706
  1507
lemma prime_int_altdef: "prime (p::int) = (1 < p \<and> (\<forall>m \<ge> 0. m dvd p \<longrightarrow>
huffman@31706
  1508
    m = 1 \<or> m = p))"
huffman@31706
  1509
  using prime_nat_def [transferred]
huffman@31706
  1510
    apply (case_tac "p >= 0")
huffman@31706
  1511
    by (blast, auto simp add: int_prime_ge_0)
huffman@31706
  1512
huffman@31706
  1513
(* To do: determine primality of any numeral *)
huffman@31706
  1514
huffman@31706
  1515
lemma nat_zero_not_prime [simp]: "~prime (0::nat)"
huffman@31706
  1516
  by (simp add: prime_nat_def)
huffman@31706
  1517
huffman@31706
  1518
lemma int_zero_not_prime [simp]: "~prime (0::int)"
huffman@31706
  1519
  by (simp add: prime_int_def)
huffman@31706
  1520
huffman@31706
  1521
lemma nat_one_not_prime [simp]: "~prime (1::nat)"
huffman@31706
  1522
  by (simp add: prime_nat_def)
chaieb@22027
  1523
huffman@31706
  1524
lemma nat_Suc_0_not_prime [simp]: "~prime (Suc 0)"
huffman@31706
  1525
  by (simp add: prime_nat_def One_nat_def)
huffman@31706
  1526
huffman@31706
  1527
lemma int_one_not_prime [simp]: "~prime (1::int)"
huffman@31706
  1528
  by (simp add: prime_int_def)
huffman@31706
  1529
huffman@31706
  1530
lemma nat_two_is_prime [simp]: "prime (2::nat)"
huffman@31706
  1531
  apply (auto simp add: prime_nat_def)
huffman@31706
  1532
  apply (case_tac m)
huffman@31706
  1533
  apply (auto dest!: dvd_imp_le)
huffman@31706
  1534
  done
chaieb@22027
  1535
huffman@31706
  1536
lemma int_two_is_prime [simp]: "prime (2::int)"
huffman@31706
  1537
  by (rule nat_two_is_prime [transferred direction: nat "op <= (0::int)"])
chaieb@27568
  1538
huffman@31706
  1539
lemma nat_prime_imp_coprime: "prime (p::nat) \<Longrightarrow> \<not> p dvd n \<Longrightarrow> coprime p n"
huffman@31706
  1540
  apply (unfold prime_nat_def)
huffman@31706
  1541
  apply (metis nat_gcd_dvd1 nat_gcd_dvd2)
huffman@31706
  1542
  done
huffman@31706
  1543
huffman@31706
  1544
lemma int_prime_imp_coprime: "prime (p::int) \<Longrightarrow> \<not> p dvd n \<Longrightarrow> coprime p n"
huffman@31706
  1545
  apply (unfold prime_int_altdef)
huffman@31706
  1546
  apply (metis int_gcd_dvd1 int_gcd_dvd2 int_gcd_ge_0)
chaieb@27568
  1547
  done
chaieb@27568
  1548
huffman@31706
  1549
lemma nat_prime_dvd_mult: "prime (p::nat) \<Longrightarrow> p dvd m * n \<Longrightarrow> p dvd m \<or> p dvd n"
huffman@31706
  1550
  by (blast intro: nat_coprime_dvd_mult nat_prime_imp_coprime)
huffman@31706
  1551
huffman@31706
  1552
lemma int_prime_dvd_mult: "prime (p::int) \<Longrightarrow> p dvd m * n \<Longrightarrow> p dvd m \<or> p dvd n"
huffman@31706
  1553
  by (blast intro: int_coprime_dvd_mult int_prime_imp_coprime)
huffman@31706
  1554
huffman@31706
  1555
lemma nat_prime_dvd_mult_eq [simp]: "prime (p::nat) \<Longrightarrow>
huffman@31706
  1556
    p dvd m * n = (p dvd m \<or> p dvd n)"
huffman@31706
  1557
  by (rule iffI, rule nat_prime_dvd_mult, auto)
chaieb@27568
  1558
huffman@31706
  1559
lemma int_prime_dvd_mult_eq [simp]: "prime (p::int) \<Longrightarrow>
huffman@31706
  1560
    p dvd m * n = (p dvd m \<or> p dvd n)"
huffman@31706
  1561
  by (rule iffI, rule int_prime_dvd_mult, auto)
chaieb@27568
  1562
huffman@31706
  1563
lemma nat_not_prime_eq_prod: "(n::nat) > 1 \<Longrightarrow> ~ prime n \<Longrightarrow>
huffman@31706
  1564
    EX m k. n = m * k & 1 < m & m < n & 1 < k & k < n"
huffman@31706
  1565
  unfolding prime_nat_def dvd_def apply auto
huffman@31706
  1566
  apply (subgoal_tac "k > 1")
huffman@31706
  1567
  apply force
huffman@31706
  1568
  apply (subgoal_tac "k ~= 0")
huffman@31706
  1569
  apply force
huffman@31706
  1570
  apply (rule notI)
huffman@31706
  1571
  apply force
huffman@31706
  1572
done
chaieb@27568
  1573
huffman@31706
  1574
(* there's a lot of messing around with signs of products here --
huffman@31706
  1575
   could this be made more automatic? *)
huffman@31706
  1576
lemma int_not_prime_eq_prod: "(n::int) > 1 \<Longrightarrow> ~ prime n \<Longrightarrow>
huffman@31706
  1577
    EX m k. n = m * k & 1 < m & m < n & 1 < k & k < n"
huffman@31706
  1578
  unfolding prime_int_altdef dvd_def
huffman@31706
  1579
  apply auto
huffman@31706
  1580
  apply (rule_tac x = m in exI)
huffman@31706
  1581
  apply (rule_tac x = k in exI)
huffman@31706
  1582
  apply (auto simp add: mult_compare_simps)
huffman@31706
  1583
  apply (subgoal_tac "k > 0")
huffman@31706
  1584
  apply arith
huffman@31706
  1585
  apply (case_tac "k <= 0")
huffman@31706
  1586
  apply (subgoal_tac "m * k <= 0")
huffman@31706
  1587
  apply force
huffman@31706
  1588
  apply (subst zero_compare_simps(8))
huffman@31706
  1589
  apply auto
huffman@31706
  1590
  apply (subgoal_tac "m * k <= 0")
huffman@31706
  1591
  apply force
huffman@31706
  1592
  apply (subst zero_compare_simps(8))
huffman@31706
  1593
  apply auto
huffman@31706
  1594
done
chaieb@27568
  1595
huffman@31706
  1596
lemma nat_prime_dvd_power [rule_format]: "prime (p::nat) -->
huffman@31706
  1597
    n > 0 --> (p dvd x^n --> p dvd x)"
huffman@31706
  1598
  by (induct n rule: nat_induct, auto)
chaieb@27568
  1599
huffman@31706
  1600
lemma int_prime_dvd_power [rule_format]: "prime (p::int) -->
huffman@31706
  1601
    n > 0 --> (p dvd x^n --> p dvd x)"
huffman@31706
  1602
  apply (induct n rule: nat_induct, auto)
huffman@31706
  1603
  apply (frule int_prime_ge_0)
huffman@31706
  1604
  apply auto
huffman@31706
  1605
done
huffman@31706
  1606
huffman@31706
  1607
lemma nat_prime_imp_power_coprime: "prime (p::nat) \<Longrightarrow> ~ p dvd a \<Longrightarrow>
huffman@31706
  1608
    coprime a (p^m)"
huffman@31706
  1609
  apply (rule nat_coprime_exp)
huffman@31706
  1610
  apply (subst nat_gcd_commute)
huffman@31706
  1611
  apply (erule (1) nat_prime_imp_coprime)
huffman@31706
  1612
done
chaieb@27568
  1613
huffman@31706
  1614
lemma int_prime_imp_power_coprime: "prime (p::int) \<Longrightarrow> ~ p dvd a \<Longrightarrow>
huffman@31706
  1615
    coprime a (p^m)"
huffman@31706
  1616
  apply (rule int_coprime_exp)
huffman@31706
  1617
  apply (subst int_gcd_commute)
huffman@31706
  1618
  apply (erule (1) int_prime_imp_coprime)
huffman@31706
  1619
done
chaieb@27568
  1620
huffman@31706
  1621
lemma nat_primes_coprime: "prime (p::nat) \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
huffman@31706
  1622
  apply (rule nat_prime_imp_coprime, assumption)
huffman@31706
  1623
  apply (unfold prime_nat_def, auto)
huffman@31706
  1624
done
chaieb@27568
  1625
huffman@31706
  1626
lemma int_primes_coprime: "prime (p::int) \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
huffman@31706
  1627
  apply (rule int_prime_imp_coprime, assumption)
huffman@31706
  1628
  apply (unfold prime_int_altdef, clarify)
huffman@31706
  1629
  apply (drule_tac x = q in spec)
huffman@31706
  1630
  apply (drule_tac x = p in spec)
huffman@31706
  1631
  apply auto
huffman@31706
  1632
done
chaieb@27568
  1633
huffman@31706
  1634
lemma nat_primes_imp_powers_coprime: "prime (p::nat) \<Longrightarrow> prime q \<Longrightarrow> p ~= q \<Longrightarrow>
huffman@31706
  1635
    coprime (p^m) (q^n)"
huffman@31706
  1636
  by (rule nat_coprime_exp2, rule nat_primes_coprime)
chaieb@27568
  1637
huffman@31706
  1638
lemma int_primes_imp_powers_coprime: "prime (p::int) \<Longrightarrow> prime q \<Longrightarrow> p ~= q \<Longrightarrow>
huffman@31706
  1639
    coprime (p^m) (q^n)"
huffman@31706
  1640
  by (rule int_coprime_exp2, rule int_primes_coprime)
chaieb@27568
  1641
huffman@31706
  1642
lemma nat_prime_factor: "n \<noteq> (1::nat) \<Longrightarrow> \<exists> p. prime p \<and> p dvd n"
huffman@31706
  1643
  apply (induct n rule: nat_less_induct)
huffman@31706
  1644
  apply (case_tac "n = 0")
huffman@31706
  1645
  using nat_two_is_prime apply blast
huffman@31706
  1646
  apply (case_tac "prime n")
huffman@31706
  1647
  apply blast
huffman@31706
  1648
  apply (subgoal_tac "n > 1")
huffman@31706
  1649
  apply (frule (1) nat_not_prime_eq_prod)
huffman@31706
  1650
  apply (auto intro: dvd_mult dvd_mult2)
huffman@31706
  1651
done
chaieb@23244
  1652
huffman@31706
  1653
(* An Isar version:
huffman@31706
  1654
huffman@31706
  1655
lemma nat_prime_factor_b:
huffman@31706
  1656
  fixes n :: nat
huffman@31706
  1657
  assumes "n \<noteq> 1"
huffman@31706
  1658
  shows "\<exists>p. prime p \<and> p dvd n"
nipkow@23983
  1659
huffman@31706
  1660
using `n ~= 1`
huffman@31706
  1661
proof (induct n rule: nat_less_induct)
huffman@31706
  1662
  fix n :: nat
huffman@31706
  1663
  assume "n ~= 1" and
huffman@31706
  1664
    ih: "\<forall>m<n. m \<noteq> 1 \<longrightarrow> (\<exists>p. prime p \<and> p dvd m)"
huffman@31706
  1665
  thus "\<exists>p. prime p \<and> p dvd n"
huffman@31706
  1666
  proof -
huffman@31706
  1667
  {
huffman@31706
  1668
    assume "n = 0"
huffman@31706
  1669
    moreover note nat_two_is_prime
huffman@31706
  1670
    ultimately have ?thesis
huffman@31706
  1671
      by (auto simp del: nat_two_is_prime)
huffman@31706
  1672
  }
huffman@31706
  1673
  moreover
huffman@31706
  1674
  {
huffman@31706
  1675
    assume "prime n"
huffman@31706
  1676
    hence ?thesis by auto
huffman@31706
  1677
  }
huffman@31706
  1678
  moreover
huffman@31706
  1679
  {
huffman@31706
  1680
    assume "n ~= 0" and "~ prime n"
huffman@31706
  1681
    with `n ~= 1` have "n > 1" by auto
huffman@31706
  1682
    with `~ prime n` and nat_not_prime_eq_prod obtain m k where
huffman@31706
  1683
      "n = m * k" and "1 < m" and "m < n" by blast
huffman@31706
  1684
    with ih obtain p where "prime p" and "p dvd m" by blast
huffman@31706
  1685
    with `n = m * k` have ?thesis by auto
huffman@31706
  1686
  }
huffman@31706
  1687
  ultimately show ?thesis by blast
huffman@31706
  1688
  qed
nipkow@23983
  1689
qed
nipkow@23983
  1690
huffman@31706
  1691
*)
huffman@31706
  1692
huffman@31706
  1693
text {* One property of coprimality is easier to prove via prime factors. *}
huffman@31706
  1694
huffman@31706
  1695
lemma nat_prime_divprod_pow:
huffman@31706
  1696
  assumes p: "prime (p::nat)" and ab: "coprime a b" and pab: "p^n dvd a * b"
huffman@31706
  1697
  shows "p^n dvd a \<or> p^n dvd b"
huffman@31706
  1698
proof-
huffman@31706
  1699
  {assume "n = 0 \<or> a = 1 \<or> b = 1" with pab have ?thesis
huffman@31706
  1700
      apply (cases "n=0", simp_all)
huffman@31706
  1701
      apply (cases "a=1", simp_all) done}
huffman@31706
  1702
  moreover
huffman@31706
  1703
  {assume n: "n \<noteq> 0" and a: "a\<noteq>1" and b: "b\<noteq>1"
huffman@31706
  1704
    then obtain m where m: "n = Suc m" by (cases n, auto)
huffman@31706
  1705
    from n have "p dvd p^n" by (intro dvd_power, auto)
huffman@31706
  1706
    also note pab
huffman@31706
  1707
    finally have pab': "p dvd a * b".
huffman@31706
  1708
    from nat_prime_dvd_mult[OF p pab']
huffman@31706
  1709
    have "p dvd a \<or> p dvd b" .
huffman@31706
  1710
    moreover
huffman@31706
  1711
    {assume pa: "p dvd a"
huffman@31706
  1712
      have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
huffman@31706
  1713
      from nat_coprime_common_divisor [OF ab, OF pa] p have "\<not> p dvd b" by auto
huffman@31706
  1714
      with p have "coprime b p"
huffman@31706
  1715
        by (subst nat_gcd_commute, intro nat_prime_imp_coprime)
huffman@31706
  1716
      hence pnb: "coprime (p^n) b"
huffman@31706
  1717
        by (subst nat_gcd_commute, rule nat_coprime_exp)
huffman@31706
  1718
      from nat_coprime_divprod[OF pnba pnb] have ?thesis by blast }
huffman@31706
  1719
    moreover
huffman@31706
  1720
    {assume pb: "p dvd b"
huffman@31706
  1721
      have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
huffman@31706
  1722
      from nat_coprime_common_divisor [OF ab, of p] pb p have "\<not> p dvd a"
huffman@31706
  1723
        by auto
huffman@31706
  1724
      with p have "coprime a p"
huffman@31706
  1725
        by (subst nat_gcd_commute, intro nat_prime_imp_coprime)
huffman@31706
  1726
      hence pna: "coprime (p^n) a"
huffman@31706
  1727
        by (subst nat_gcd_commute, rule nat_coprime_exp)
huffman@31706
  1728
      from nat_coprime_divprod[OF pab pna] have ?thesis by blast }
huffman@31706
  1729
    ultimately have ?thesis by blast}
huffman@31706
  1730
  ultimately show ?thesis by blast
nipkow@23983
  1731
qed
nipkow@23983
  1732
wenzelm@21256
  1733
end