src/HOL/Int.thy
author wenzelm
Tue Oct 10 19:23:03 2017 +0200 (23 months ago)
changeset 66831 29ea2b900a05
parent 66816 212a3334e7da
child 66836 4eb431c3f974
permissions -rw-r--r--
tuned: each session has at most one defining entry;
wenzelm@41959
     1
(*  Title:      HOL/Int.thy
haftmann@25919
     2
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
wenzelm@41959
     3
    Author:     Tobias Nipkow, Florian Haftmann, TU Muenchen
haftmann@25919
     4
*)
haftmann@25919
     5
wenzelm@60758
     6
section \<open>The Integers as Equivalence Classes over Pairs of Natural Numbers\<close>
haftmann@25919
     7
haftmann@25919
     8
theory Int
wenzelm@63652
     9
  imports Equiv_Relations Power Quotient Fun_Def
haftmann@25919
    10
begin
haftmann@25919
    11
wenzelm@60758
    12
subsection \<open>Definition of integers as a quotient type\<close>
haftmann@25919
    13
wenzelm@63652
    14
definition intrel :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool"
wenzelm@63652
    15
  where "intrel = (\<lambda>(x, y) (u, v). x + v = u + y)"
huffman@48045
    16
huffman@48045
    17
lemma intrel_iff [simp]: "intrel (x, y) (u, v) \<longleftrightarrow> x + v = u + y"
huffman@48045
    18
  by (simp add: intrel_def)
haftmann@25919
    19
huffman@48045
    20
quotient_type int = "nat \<times> nat" / "intrel"
wenzelm@45694
    21
  morphisms Rep_Integ Abs_Integ
huffman@48045
    22
proof (rule equivpI)
wenzelm@63652
    23
  show "reflp intrel" by (auto simp: reflp_def)
wenzelm@63652
    24
  show "symp intrel" by (auto simp: symp_def)
wenzelm@63652
    25
  show "transp intrel" by (auto simp: transp_def)
huffman@48045
    26
qed
haftmann@25919
    27
huffman@48045
    28
lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
wenzelm@63652
    29
  "(\<And>x y. z = Abs_Integ (x, y) \<Longrightarrow> P) \<Longrightarrow> P"
wenzelm@63652
    30
  by (induct z) auto
wenzelm@63652
    31
huffman@48045
    32
wenzelm@60758
    33
subsection \<open>Integers form a commutative ring\<close>
huffman@48045
    34
huffman@48045
    35
instantiation int :: comm_ring_1
haftmann@25919
    36
begin
haftmann@25919
    37
kuncar@51994
    38
lift_definition zero_int :: "int" is "(0, 0)" .
haftmann@25919
    39
kuncar@51994
    40
lift_definition one_int :: "int" is "(1, 0)" .
haftmann@25919
    41
huffman@48045
    42
lift_definition plus_int :: "int \<Rightarrow> int \<Rightarrow> int"
huffman@48045
    43
  is "\<lambda>(x, y) (u, v). (x + u, y + v)"
huffman@48045
    44
  by clarsimp
haftmann@25919
    45
huffman@48045
    46
lift_definition uminus_int :: "int \<Rightarrow> int"
huffman@48045
    47
  is "\<lambda>(x, y). (y, x)"
huffman@48045
    48
  by clarsimp
haftmann@25919
    49
huffman@48045
    50
lift_definition minus_int :: "int \<Rightarrow> int \<Rightarrow> int"
huffman@48045
    51
  is "\<lambda>(x, y) (u, v). (x + v, y + u)"
huffman@48045
    52
  by clarsimp
haftmann@25919
    53
huffman@48045
    54
lift_definition times_int :: "int \<Rightarrow> int \<Rightarrow> int"
huffman@48045
    55
  is "\<lambda>(x, y) (u, v). (x*u + y*v, x*v + y*u)"
huffman@48045
    56
proof (clarsimp)
huffman@48045
    57
  fix s t u v w x y z :: nat
huffman@48045
    58
  assume "s + v = u + t" and "w + z = y + x"
wenzelm@63652
    59
  then have "(s + v) * w + (u + t) * x + u * (w + z) + v * (y + x) =
wenzelm@63652
    60
    (u + t) * w + (s + v) * x + u * (y + x) + v * (w + z)"
huffman@48045
    61
    by simp
wenzelm@63652
    62
  then show "(s * w + t * x) + (u * z + v * y) = (u * y + v * z) + (s * x + t * w)"
huffman@48045
    63
    by (simp add: algebra_simps)
huffman@48045
    64
qed
haftmann@25919
    65
huffman@48045
    66
instance
wenzelm@63652
    67
  by standard (transfer; clarsimp simp: algebra_simps)+
haftmann@25919
    68
haftmann@25919
    69
end
haftmann@25919
    70
wenzelm@63652
    71
abbreviation int :: "nat \<Rightarrow> int"
wenzelm@63652
    72
  where "int \<equiv> of_nat"
huffman@44709
    73
huffman@48045
    74
lemma int_def: "int n = Abs_Integ (n, 0)"
wenzelm@63652
    75
  by (induct n) (simp add: zero_int.abs_eq, simp add: one_int.abs_eq plus_int.abs_eq)
haftmann@25919
    76
wenzelm@63652
    77
lemma int_transfer [transfer_rule]: "(rel_fun (op =) pcr_int) (\<lambda>n. (n, 0)) int"
wenzelm@63652
    78
  by (simp add: rel_fun_def int.pcr_cr_eq cr_int_def int_def)
haftmann@25919
    79
wenzelm@63652
    80
lemma int_diff_cases: obtains (diff) m n where "z = int m - int n"
huffman@48045
    81
  by transfer clarsimp
huffman@48045
    82
wenzelm@63652
    83
wenzelm@60758
    84
subsection \<open>Integers are totally ordered\<close>
haftmann@25919
    85
huffman@48045
    86
instantiation int :: linorder
huffman@48045
    87
begin
huffman@48045
    88
huffman@48045
    89
lift_definition less_eq_int :: "int \<Rightarrow> int \<Rightarrow> bool"
huffman@48045
    90
  is "\<lambda>(x, y) (u, v). x + v \<le> u + y"
huffman@48045
    91
  by auto
huffman@48045
    92
huffman@48045
    93
lift_definition less_int :: "int \<Rightarrow> int \<Rightarrow> bool"
huffman@48045
    94
  is "\<lambda>(x, y) (u, v). x + v < u + y"
huffman@48045
    95
  by auto
huffman@48045
    96
huffman@48045
    97
instance
wenzelm@61169
    98
  by standard (transfer, force)+
huffman@48045
    99
huffman@48045
   100
end
haftmann@25919
   101
haftmann@25919
   102
instantiation int :: distrib_lattice
haftmann@25919
   103
begin
haftmann@25919
   104
wenzelm@63652
   105
definition "(inf :: int \<Rightarrow> int \<Rightarrow> int) = min"
haftmann@25919
   106
wenzelm@63652
   107
definition "(sup :: int \<Rightarrow> int \<Rightarrow> int) = max"
haftmann@25919
   108
haftmann@25919
   109
instance
wenzelm@63652
   110
  by standard (auto simp add: inf_int_def sup_int_def max_min_distrib2)
haftmann@25919
   111
haftmann@25919
   112
end
haftmann@25919
   113
wenzelm@63652
   114
wenzelm@60758
   115
subsection \<open>Ordering properties of arithmetic operations\<close>
huffman@48045
   116
haftmann@35028
   117
instance int :: ordered_cancel_ab_semigroup_add
haftmann@25919
   118
proof
haftmann@25919
   119
  fix i j k :: int
haftmann@25919
   120
  show "i \<le> j \<Longrightarrow> k + i \<le> k + j"
huffman@48045
   121
    by transfer clarsimp
haftmann@25919
   122
qed
haftmann@25919
   123
wenzelm@63652
   124
text \<open>Strict Monotonicity of Multiplication.\<close>
haftmann@25919
   125
wenzelm@63652
   126
text \<open>Strict, in 1st argument; proof is by induction on \<open>k > 0\<close>.\<close>
wenzelm@63652
   127
lemma zmult_zless_mono2_lemma: "i < j \<Longrightarrow> 0 < k \<Longrightarrow> int k * i < int k * j"
wenzelm@63652
   128
  for i j :: int
wenzelm@63652
   129
proof (induct k)
wenzelm@63652
   130
  case 0
wenzelm@63652
   131
  then show ?case by simp
wenzelm@63652
   132
next
wenzelm@63652
   133
  case (Suc k)
wenzelm@63652
   134
  then show ?case
wenzelm@63652
   135
    by (cases "k = 0") (simp_all add: distrib_right add_strict_mono)
wenzelm@63652
   136
qed
haftmann@25919
   137
wenzelm@63652
   138
lemma zero_le_imp_eq_int: "0 \<le> k \<Longrightarrow> \<exists>n. k = int n"
wenzelm@63652
   139
  for k :: int
wenzelm@63652
   140
  apply transfer
wenzelm@63652
   141
  apply clarsimp
wenzelm@63652
   142
  apply (rule_tac x="a - b" in exI)
wenzelm@63652
   143
  apply simp
wenzelm@63652
   144
  done
haftmann@25919
   145
wenzelm@63652
   146
lemma zero_less_imp_eq_int: "0 < k \<Longrightarrow> \<exists>n>0. k = int n"
wenzelm@63652
   147
  for k :: int
wenzelm@63652
   148
  apply transfer
wenzelm@63652
   149
  apply clarsimp
wenzelm@63652
   150
  apply (rule_tac x="a - b" in exI)
wenzelm@63652
   151
  apply simp
wenzelm@63652
   152
  done
haftmann@25919
   153
wenzelm@63652
   154
lemma zmult_zless_mono2: "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
wenzelm@63652
   155
  for i j k :: int
wenzelm@63652
   156
  by (drule zero_less_imp_eq_int) (auto simp add: zmult_zless_mono2_lemma)
haftmann@25919
   157
wenzelm@63652
   158
wenzelm@63652
   159
text \<open>The integers form an ordered integral domain.\<close>
wenzelm@63652
   160
huffman@48045
   161
instantiation int :: linordered_idom
huffman@48045
   162
begin
huffman@48045
   163
wenzelm@63652
   164
definition zabs_def: "\<bar>i::int\<bar> = (if i < 0 then - i else i)"
huffman@48045
   165
wenzelm@63652
   166
definition zsgn_def: "sgn (i::int) = (if i = 0 then 0 else if 0 < i then 1 else - 1)"
huffman@48045
   167
wenzelm@63652
   168
instance
wenzelm@63652
   169
proof
haftmann@25919
   170
  fix i j k :: int
haftmann@25919
   171
  show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
haftmann@25919
   172
    by (rule zmult_zless_mono2)
haftmann@25919
   173
  show "\<bar>i\<bar> = (if i < 0 then -i else i)"
haftmann@25919
   174
    by (simp only: zabs_def)
wenzelm@61076
   175
  show "sgn (i::int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
haftmann@25919
   176
    by (simp only: zsgn_def)
haftmann@25919
   177
qed
haftmann@25919
   178
huffman@48045
   179
end
huffman@48045
   180
wenzelm@63652
   181
lemma zless_imp_add1_zle: "w < z \<Longrightarrow> w + 1 \<le> z"
wenzelm@63652
   182
  for w z :: int
huffman@48045
   183
  by transfer clarsimp
haftmann@25919
   184
wenzelm@63652
   185
lemma zless_iff_Suc_zadd: "w < z \<longleftrightarrow> (\<exists>n. z = w + int (Suc n))"
wenzelm@63652
   186
  for w z :: int
wenzelm@63652
   187
  apply transfer
wenzelm@63652
   188
  apply auto
wenzelm@63652
   189
  apply (rename_tac a b c d)
wenzelm@63652
   190
  apply (rule_tac x="c+b - Suc(a+d)" in exI)
wenzelm@63652
   191
  apply arith
wenzelm@63652
   192
  done
haftmann@25919
   193
wenzelm@63652
   194
lemma zabs_less_one_iff [simp]: "\<bar>z\<bar> < 1 \<longleftrightarrow> z = 0" (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@63652
   195
  for z :: int
haftmann@62347
   196
proof
wenzelm@63652
   197
  assume ?rhs
wenzelm@63652
   198
  then show ?lhs by simp
haftmann@62347
   199
next
wenzelm@63652
   200
  assume ?lhs
wenzelm@63652
   201
  with zless_imp_add1_zle [of "\<bar>z\<bar>" 1] have "\<bar>z\<bar> + 1 \<le> 1" by simp
wenzelm@63652
   202
  then have "\<bar>z\<bar> \<le> 0" by simp
wenzelm@63652
   203
  then show ?rhs by simp
haftmann@62347
   204
qed
haftmann@62347
   205
haftmann@25919
   206
lemmas int_distrib =
webertj@49962
   207
  distrib_right [of z1 z2 w]
webertj@49962
   208
  distrib_left [of w z1 z2]
wenzelm@45607
   209
  left_diff_distrib [of z1 z2 w]
wenzelm@45607
   210
  right_diff_distrib [of w z1 z2]
wenzelm@45607
   211
  for z1 z2 w :: int
haftmann@25919
   212
haftmann@25919
   213
wenzelm@61799
   214
subsection \<open>Embedding of the Integers into any \<open>ring_1\<close>: \<open>of_int\<close>\<close>
haftmann@25919
   215
haftmann@25919
   216
context ring_1
haftmann@25919
   217
begin
haftmann@25919
   218
wenzelm@63652
   219
lift_definition of_int :: "int \<Rightarrow> 'a"
wenzelm@63652
   220
  is "\<lambda>(i, j). of_nat i - of_nat j"
huffman@48045
   221
  by (clarsimp simp add: diff_eq_eq eq_diff_eq diff_add_eq
wenzelm@63652
   222
      of_nat_add [symmetric] simp del: of_nat_add)
haftmann@25919
   223
haftmann@25919
   224
lemma of_int_0 [simp]: "of_int 0 = 0"
huffman@48066
   225
  by transfer simp
haftmann@25919
   226
haftmann@25919
   227
lemma of_int_1 [simp]: "of_int 1 = 1"
huffman@48066
   228
  by transfer simp
haftmann@25919
   229
wenzelm@63652
   230
lemma of_int_add [simp]: "of_int (w + z) = of_int w + of_int z"
huffman@48066
   231
  by transfer (clarsimp simp add: algebra_simps)
haftmann@25919
   232
wenzelm@63652
   233
lemma of_int_minus [simp]: "of_int (- z) = - (of_int z)"
huffman@48066
   234
  by (transfer fixing: uminus) clarsimp
haftmann@25919
   235
haftmann@25919
   236
lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z"
haftmann@54230
   237
  using of_int_add [of w "- z"] by simp
haftmann@25919
   238
haftmann@25919
   239
lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
wenzelm@63652
   240
  by (transfer fixing: times) (clarsimp simp add: algebra_simps)
haftmann@25919
   241
eberlm@61531
   242
lemma mult_of_int_commute: "of_int x * y = y * of_int x"
eberlm@61531
   243
  by (transfer fixing: times) (auto simp: algebra_simps mult_of_nat_commute)
eberlm@61531
   244
wenzelm@63652
   245
text \<open>Collapse nested embeddings.\<close>
huffman@44709
   246
lemma of_int_of_nat_eq [simp]: "of_int (int n) = of_nat n"
wenzelm@63652
   247
  by (induct n) auto
haftmann@25919
   248
huffman@47108
   249
lemma of_int_numeral [simp, code_post]: "of_int (numeral k) = numeral k"
huffman@47108
   250
  by (simp add: of_nat_numeral [symmetric] of_int_of_nat_eq [symmetric])
huffman@47108
   251
haftmann@54489
   252
lemma of_int_neg_numeral [code_post]: "of_int (- numeral k) = - numeral k"
haftmann@54489
   253
  by simp
huffman@47108
   254
wenzelm@63652
   255
lemma of_int_power [simp]: "of_int (z ^ n) = of_int z ^ n"
haftmann@31015
   256
  by (induct n) simp_all
haftmann@31015
   257
haftmann@66816
   258
lemma of_int_of_bool [simp]:
haftmann@66816
   259
  "of_int (of_bool P) = of_bool P"
haftmann@66816
   260
  by auto
haftmann@66816
   261
haftmann@25919
   262
end
haftmann@25919
   263
huffman@47108
   264
context ring_char_0
haftmann@25919
   265
begin
haftmann@25919
   266
wenzelm@63652
   267
lemma of_int_eq_iff [simp]: "of_int w = of_int z \<longleftrightarrow> w = z"
wenzelm@63652
   268
  by transfer (clarsimp simp add: algebra_simps of_nat_add [symmetric] simp del: of_nat_add)
haftmann@25919
   269
wenzelm@63652
   270
text \<open>Special cases where either operand is zero.\<close>
wenzelm@63652
   271
lemma of_int_eq_0_iff [simp]: "of_int z = 0 \<longleftrightarrow> z = 0"
haftmann@36424
   272
  using of_int_eq_iff [of z 0] by simp
haftmann@36424
   273
wenzelm@63652
   274
lemma of_int_0_eq_iff [simp]: "0 = of_int z \<longleftrightarrow> z = 0"
haftmann@36424
   275
  using of_int_eq_iff [of 0 z] by simp
haftmann@25919
   276
wenzelm@63652
   277
lemma of_int_eq_1_iff [iff]: "of_int z = 1 \<longleftrightarrow> z = 1"
lp15@61234
   278
  using of_int_eq_iff [of z 1] by simp
lp15@61234
   279
haftmann@25919
   280
end
haftmann@25919
   281
haftmann@36424
   282
context linordered_idom
haftmann@36424
   283
begin
haftmann@36424
   284
wenzelm@63652
   285
text \<open>Every \<open>linordered_idom\<close> has characteristic zero.\<close>
haftmann@36424
   286
subclass ring_char_0 ..
haftmann@36424
   287
wenzelm@63652
   288
lemma of_int_le_iff [simp]: "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z"
wenzelm@63652
   289
  by (transfer fixing: less_eq)
wenzelm@63652
   290
    (clarsimp simp add: algebra_simps of_nat_add [symmetric] simp del: of_nat_add)
haftmann@36424
   291
wenzelm@63652
   292
lemma of_int_less_iff [simp]: "of_int w < of_int z \<longleftrightarrow> w < z"
haftmann@36424
   293
  by (simp add: less_le order_less_le)
haftmann@36424
   294
wenzelm@63652
   295
lemma of_int_0_le_iff [simp]: "0 \<le> of_int z \<longleftrightarrow> 0 \<le> z"
haftmann@36424
   296
  using of_int_le_iff [of 0 z] by simp
haftmann@36424
   297
wenzelm@63652
   298
lemma of_int_le_0_iff [simp]: "of_int z \<le> 0 \<longleftrightarrow> z \<le> 0"
haftmann@36424
   299
  using of_int_le_iff [of z 0] by simp
haftmann@36424
   300
wenzelm@63652
   301
lemma of_int_0_less_iff [simp]: "0 < of_int z \<longleftrightarrow> 0 < z"
haftmann@36424
   302
  using of_int_less_iff [of 0 z] by simp
haftmann@36424
   303
wenzelm@63652
   304
lemma of_int_less_0_iff [simp]: "of_int z < 0 \<longleftrightarrow> z < 0"
haftmann@36424
   305
  using of_int_less_iff [of z 0] by simp
haftmann@36424
   306
wenzelm@63652
   307
lemma of_int_1_le_iff [simp]: "1 \<le> of_int z \<longleftrightarrow> 1 \<le> z"
lp15@61234
   308
  using of_int_le_iff [of 1 z] by simp
lp15@61234
   309
wenzelm@63652
   310
lemma of_int_le_1_iff [simp]: "of_int z \<le> 1 \<longleftrightarrow> z \<le> 1"
lp15@61234
   311
  using of_int_le_iff [of z 1] by simp
lp15@61234
   312
wenzelm@63652
   313
lemma of_int_1_less_iff [simp]: "1 < of_int z \<longleftrightarrow> 1 < z"
lp15@61234
   314
  using of_int_less_iff [of 1 z] by simp
lp15@61234
   315
wenzelm@63652
   316
lemma of_int_less_1_iff [simp]: "of_int z < 1 \<longleftrightarrow> z < 1"
lp15@61234
   317
  using of_int_less_iff [of z 1] by simp
lp15@61234
   318
eberlm@62128
   319
lemma of_int_pos: "z > 0 \<Longrightarrow> of_int z > 0"
eberlm@62128
   320
  by simp
eberlm@62128
   321
eberlm@62128
   322
lemma of_int_nonneg: "z \<ge> 0 \<Longrightarrow> of_int z \<ge> 0"
eberlm@62128
   323
  by simp
eberlm@62128
   324
wenzelm@63652
   325
lemma of_int_abs [simp]: "of_int \<bar>x\<bar> = \<bar>of_int x\<bar>"
haftmann@62347
   326
  by (auto simp add: abs_if)
haftmann@62347
   327
haftmann@62347
   328
lemma of_int_lessD:
haftmann@62347
   329
  assumes "\<bar>of_int n\<bar> < x"
haftmann@62347
   330
  shows "n = 0 \<or> x > 1"
haftmann@62347
   331
proof (cases "n = 0")
wenzelm@63652
   332
  case True
wenzelm@63652
   333
  then show ?thesis by simp
haftmann@62347
   334
next
haftmann@62347
   335
  case False
haftmann@62347
   336
  then have "\<bar>n\<bar> \<noteq> 0" by simp
haftmann@62347
   337
  then have "\<bar>n\<bar> > 0" by simp
haftmann@62347
   338
  then have "\<bar>n\<bar> \<ge> 1"
haftmann@62347
   339
    using zless_imp_add1_zle [of 0 "\<bar>n\<bar>"] by simp
haftmann@62347
   340
  then have "\<bar>of_int n\<bar> \<ge> 1"
haftmann@62347
   341
    unfolding of_int_1_le_iff [of "\<bar>n\<bar>", symmetric] by simp
haftmann@62347
   342
  then have "1 < x" using assms by (rule le_less_trans)
haftmann@62347
   343
  then show ?thesis ..
haftmann@62347
   344
qed
haftmann@62347
   345
haftmann@62347
   346
lemma of_int_leD:
haftmann@62347
   347
  assumes "\<bar>of_int n\<bar> \<le> x"
haftmann@62347
   348
  shows "n = 0 \<or> 1 \<le> x"
haftmann@62347
   349
proof (cases "n = 0")
wenzelm@63652
   350
  case True
wenzelm@63652
   351
  then show ?thesis by simp
haftmann@62347
   352
next
haftmann@62347
   353
  case False
haftmann@62347
   354
  then have "\<bar>n\<bar> \<noteq> 0" by simp
haftmann@62347
   355
  then have "\<bar>n\<bar> > 0" by simp
haftmann@62347
   356
  then have "\<bar>n\<bar> \<ge> 1"
haftmann@62347
   357
    using zless_imp_add1_zle [of 0 "\<bar>n\<bar>"] by simp
haftmann@62347
   358
  then have "\<bar>of_int n\<bar> \<ge> 1"
haftmann@62347
   359
    unfolding of_int_1_le_iff [of "\<bar>n\<bar>", symmetric] by simp
haftmann@62347
   360
  then have "1 \<le> x" using assms by (rule order_trans)
haftmann@62347
   361
  then show ?thesis ..
haftmann@62347
   362
qed
haftmann@62347
   363
haftmann@36424
   364
end
haftmann@25919
   365
lp15@61234
   366
text \<open>Comparisons involving @{term of_int}.\<close>
lp15@61234
   367
wenzelm@63652
   368
lemma of_int_eq_numeral_iff [iff]: "of_int z = (numeral n :: 'a::ring_char_0) \<longleftrightarrow> z = numeral n"
lp15@61234
   369
  using of_int_eq_iff by fastforce
lp15@61234
   370
lp15@61649
   371
lemma of_int_le_numeral_iff [simp]:
wenzelm@63652
   372
  "of_int z \<le> (numeral n :: 'a::linordered_idom) \<longleftrightarrow> z \<le> numeral n"
lp15@61234
   373
  using of_int_le_iff [of z "numeral n"] by simp
lp15@61234
   374
lp15@61649
   375
lemma of_int_numeral_le_iff [simp]:
wenzelm@63652
   376
  "(numeral n :: 'a::linordered_idom) \<le> of_int z \<longleftrightarrow> numeral n \<le> z"
lp15@61234
   377
  using of_int_le_iff [of "numeral n"] by simp
lp15@61234
   378
lp15@61649
   379
lemma of_int_less_numeral_iff [simp]:
wenzelm@63652
   380
  "of_int z < (numeral n :: 'a::linordered_idom) \<longleftrightarrow> z < numeral n"
lp15@61234
   381
  using of_int_less_iff [of z "numeral n"] by simp
lp15@61234
   382
lp15@61649
   383
lemma of_int_numeral_less_iff [simp]:
wenzelm@63652
   384
  "(numeral n :: 'a::linordered_idom) < of_int z \<longleftrightarrow> numeral n < z"
lp15@61234
   385
  using of_int_less_iff [of "numeral n" z] by simp
lp15@61234
   386
wenzelm@63652
   387
lemma of_nat_less_of_int_iff: "(of_nat n::'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"
hoelzl@56889
   388
  by (metis of_int_of_nat_eq of_int_less_iff)
hoelzl@56889
   389
haftmann@25919
   390
lemma of_int_eq_id [simp]: "of_int = id"
haftmann@25919
   391
proof
wenzelm@63652
   392
  show "of_int z = id z" for z
wenzelm@63652
   393
    by (cases z rule: int_diff_cases) simp
haftmann@25919
   394
qed
haftmann@25919
   395
hoelzl@51329
   396
instance int :: no_top
wenzelm@61169
   397
  apply standard
hoelzl@51329
   398
  apply (rule_tac x="x + 1" in exI)
hoelzl@51329
   399
  apply simp
hoelzl@51329
   400
  done
hoelzl@51329
   401
hoelzl@51329
   402
instance int :: no_bot
wenzelm@61169
   403
  apply standard
hoelzl@51329
   404
  apply (rule_tac x="x - 1" in exI)
hoelzl@51329
   405
  apply simp
hoelzl@51329
   406
  done
hoelzl@51329
   407
wenzelm@63652
   408
wenzelm@61799
   409
subsection \<open>Magnitude of an Integer, as a Natural Number: \<open>nat\<close>\<close>
haftmann@25919
   410
huffman@48045
   411
lift_definition nat :: "int \<Rightarrow> nat" is "\<lambda>(x, y). x - y"
huffman@48045
   412
  by auto
haftmann@25919
   413
huffman@44709
   414
lemma nat_int [simp]: "nat (int n) = n"
huffman@48045
   415
  by transfer simp
haftmann@25919
   416
huffman@44709
   417
lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
huffman@48045
   418
  by transfer clarsimp
haftmann@25919
   419
wenzelm@63652
   420
lemma nat_0_le: "0 \<le> z \<Longrightarrow> int (nat z) = z"
wenzelm@63652
   421
  by simp
haftmann@25919
   422
wenzelm@63652
   423
lemma nat_le_0 [simp]: "z \<le> 0 \<Longrightarrow> nat z = 0"
huffman@48045
   424
  by transfer clarsimp
haftmann@25919
   425
wenzelm@63652
   426
lemma nat_le_eq_zle: "0 < w \<or> 0 \<le> z \<Longrightarrow> nat w \<le> nat z \<longleftrightarrow> w \<le> z"
huffman@48045
   427
  by transfer (clarsimp, arith)
haftmann@25919
   428
wenzelm@63652
   429
text \<open>An alternative condition is @{term "0 \<le> w"}.\<close>
wenzelm@63652
   430
lemma nat_mono_iff: "0 < z \<Longrightarrow> nat w < nat z \<longleftrightarrow> w < z"
wenzelm@63652
   431
  by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
haftmann@25919
   432
wenzelm@63652
   433
lemma nat_less_eq_zless: "0 \<le> w \<Longrightarrow> nat w < nat z \<longleftrightarrow> w < z"
wenzelm@63652
   434
  by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
haftmann@25919
   435
wenzelm@63652
   436
lemma zless_nat_conj [simp]: "nat w < nat z \<longleftrightarrow> 0 < z \<and> w < z"
huffman@48045
   437
  by transfer (clarsimp, arith)
haftmann@25919
   438
haftmann@64714
   439
lemma nonneg_int_cases:
haftmann@64714
   440
  assumes "0 \<le> k"
haftmann@64714
   441
  obtains n where "k = int n"
haftmann@64714
   442
proof -
haftmann@64714
   443
  from assms have "k = int (nat k)"
haftmann@64714
   444
    by simp
haftmann@64714
   445
  then show thesis
haftmann@64714
   446
    by (rule that)
haftmann@64714
   447
qed
haftmann@64714
   448
haftmann@64714
   449
lemma pos_int_cases:
haftmann@64714
   450
  assumes "0 < k"
haftmann@64714
   451
  obtains n where "k = int n" and "n > 0"
haftmann@64714
   452
proof -
haftmann@64714
   453
  from assms have "0 \<le> k"
haftmann@64714
   454
    by simp
haftmann@64714
   455
  then obtain n where "k = int n"
haftmann@64714
   456
    by (rule nonneg_int_cases)
haftmann@64714
   457
  moreover have "n > 0"
haftmann@64714
   458
    using \<open>k = int n\<close> assms by simp
haftmann@64714
   459
  ultimately show thesis
haftmann@64714
   460
    by (rule that)
haftmann@64714
   461
qed
haftmann@64714
   462
haftmann@64714
   463
lemma nonpos_int_cases:
haftmann@64714
   464
  assumes "k \<le> 0"
haftmann@64714
   465
  obtains n where "k = - int n"
haftmann@64714
   466
proof -
haftmann@64714
   467
  from assms have "- k \<ge> 0"
haftmann@64714
   468
    by simp
haftmann@64714
   469
  then obtain n where "- k = int n"
haftmann@64714
   470
    by (rule nonneg_int_cases)
haftmann@64714
   471
  then have "k = - int n"
haftmann@64714
   472
    by simp
haftmann@64714
   473
  then show thesis
haftmann@64714
   474
    by (rule that)
haftmann@64714
   475
qed
haftmann@64714
   476
haftmann@64714
   477
lemma neg_int_cases:
haftmann@64714
   478
  assumes "k < 0"
haftmann@64714
   479
  obtains n where "k = - int n" and "n > 0"
haftmann@64714
   480
proof -
haftmann@64714
   481
  from assms have "- k > 0"
haftmann@64714
   482
    by simp
haftmann@64714
   483
  then obtain n where "- k = int n" and "- k > 0"
haftmann@64714
   484
    by (blast elim: pos_int_cases)
haftmann@64714
   485
  then have "k = - int n" and "n > 0"
haftmann@64714
   486
    by simp_all
haftmann@64714
   487
  then show thesis
haftmann@64714
   488
    by (rule that)
haftmann@64714
   489
qed
haftmann@25919
   490
wenzelm@63652
   491
lemma nat_eq_iff: "nat w = m \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
huffman@48045
   492
  by transfer (clarsimp simp add: le_imp_diff_is_add)
lp15@60162
   493
wenzelm@63652
   494
lemma nat_eq_iff2: "m = nat w \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
haftmann@54223
   495
  using nat_eq_iff [of w m] by auto
haftmann@54223
   496
wenzelm@63652
   497
lemma nat_0 [simp]: "nat 0 = 0"
haftmann@54223
   498
  by (simp add: nat_eq_iff)
haftmann@25919
   499
wenzelm@63652
   500
lemma nat_1 [simp]: "nat 1 = Suc 0"
haftmann@54223
   501
  by (simp add: nat_eq_iff)
haftmann@54223
   502
wenzelm@63652
   503
lemma nat_numeral [simp]: "nat (numeral k) = numeral k"
haftmann@54223
   504
  by (simp add: nat_eq_iff)
haftmann@25919
   505
wenzelm@63652
   506
lemma nat_neg_numeral [simp]: "nat (- numeral k) = 0"
haftmann@54223
   507
  by simp
haftmann@54223
   508
haftmann@54223
   509
lemma nat_2: "nat 2 = Suc (Suc 0)"
haftmann@54223
   510
  by simp
lp15@60162
   511
wenzelm@63652
   512
lemma nat_less_iff: "0 \<le> w \<Longrightarrow> nat w < m \<longleftrightarrow> w < of_nat m"
huffman@48045
   513
  by transfer (clarsimp, arith)
haftmann@25919
   514
huffman@44709
   515
lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> int n"
huffman@48045
   516
  by transfer (clarsimp simp add: le_diff_conv)
huffman@44707
   517
huffman@44707
   518
lemma nat_mono: "x \<le> y \<Longrightarrow> nat x \<le> nat y"
huffman@48045
   519
  by transfer auto
huffman@44707
   520
wenzelm@63652
   521
lemma nat_0_iff[simp]: "nat i = 0 \<longleftrightarrow> i \<le> 0"
wenzelm@63652
   522
  for i :: int
huffman@48045
   523
  by transfer clarsimp
nipkow@29700
   524
wenzelm@63652
   525
lemma int_eq_iff: "of_nat m = z \<longleftrightarrow> m = nat z \<and> 0 \<le> z"
wenzelm@63652
   526
  by (auto simp add: nat_eq_iff2)
haftmann@25919
   527
wenzelm@63652
   528
lemma zero_less_nat_eq [simp]: "0 < nat z \<longleftrightarrow> 0 < z"
wenzelm@63652
   529
  using zless_nat_conj [of 0] by auto
haftmann@25919
   530
wenzelm@63652
   531
lemma nat_add_distrib: "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat (z + z') = nat z + nat z'"
huffman@48045
   532
  by transfer clarsimp
haftmann@25919
   533
wenzelm@63652
   534
lemma nat_diff_distrib': "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> nat (x - y) = nat x - nat y"
haftmann@54223
   535
  by transfer clarsimp
lp15@60162
   536
wenzelm@63652
   537
lemma nat_diff_distrib: "0 \<le> z' \<Longrightarrow> z' \<le> z \<Longrightarrow> nat (z - z') = nat z - nat z'"
haftmann@54223
   538
  by (rule nat_diff_distrib') auto
haftmann@25919
   539
huffman@44709
   540
lemma nat_zminus_int [simp]: "nat (- int n) = 0"
huffman@48045
   541
  by transfer simp
haftmann@25919
   542
wenzelm@63652
   543
lemma le_nat_iff: "k \<ge> 0 \<Longrightarrow> n \<le> nat k \<longleftrightarrow> int n \<le> k"
haftmann@53065
   544
  by transfer auto
lp15@60162
   545
wenzelm@63652
   546
lemma zless_nat_eq_int_zless: "m < nat z \<longleftrightarrow> int m < z"
huffman@48045
   547
  by transfer (clarsimp simp add: less_diff_conv)
haftmann@25919
   548
wenzelm@63652
   549
lemma (in ring_1) of_nat_nat [simp]: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
huffman@48066
   550
  by transfer (clarsimp simp add: of_nat_diff)
haftmann@25919
   551
wenzelm@63652
   552
lemma diff_nat_numeral [simp]: "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')"
haftmann@54249
   553
  by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral)
haftmann@54249
   554
haftmann@66816
   555
lemma nat_of_bool [simp]:
haftmann@66816
   556
  "nat (of_bool P) = of_bool P"
haftmann@66816
   557
  by auto
haftmann@66816
   558
haftmann@54249
   559
wenzelm@60758
   560
text \<open>For termination proofs:\<close>
wenzelm@63652
   561
lemma measure_function_int[measure_function]: "is_measure (nat \<circ> abs)" ..
krauss@29779
   562
haftmann@25919
   563
wenzelm@63652
   564
subsection \<open>Lemmas about the Function @{term of_nat} and Orderings\<close>
haftmann@25919
   565
wenzelm@61076
   566
lemma negative_zless_0: "- (int (Suc n)) < (0 :: int)"
wenzelm@63652
   567
  by (simp add: order_less_le del: of_nat_Suc)
haftmann@25919
   568
huffman@44709
   569
lemma negative_zless [iff]: "- (int (Suc n)) < int m"
wenzelm@63652
   570
  by (rule negative_zless_0 [THEN order_less_le_trans], simp)
haftmann@25919
   571
huffman@44709
   572
lemma negative_zle_0: "- int n \<le> 0"
wenzelm@63652
   573
  by (simp add: minus_le_iff)
haftmann@25919
   574
huffman@44709
   575
lemma negative_zle [iff]: "- int n \<le> int m"
wenzelm@63652
   576
  by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
haftmann@25919
   577
wenzelm@63652
   578
lemma not_zle_0_negative [simp]: "\<not> 0 \<le> - int (Suc n)"
wenzelm@63652
   579
  by (subst le_minus_iff) (simp del: of_nat_Suc)
haftmann@25919
   580
wenzelm@63652
   581
lemma int_zle_neg: "int n \<le> - int m \<longleftrightarrow> n = 0 \<and> m = 0"
huffman@48045
   582
  by transfer simp
haftmann@25919
   583
wenzelm@63652
   584
lemma not_int_zless_negative [simp]: "\<not> int n < - int m"
wenzelm@63652
   585
  by (simp add: linorder_not_less)
haftmann@25919
   586
wenzelm@63652
   587
lemma negative_eq_positive [simp]: "- int n = of_nat m \<longleftrightarrow> n = 0 \<and> m = 0"
wenzelm@63652
   588
  by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
haftmann@25919
   589
wenzelm@63652
   590
lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)"
wenzelm@63652
   591
  (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@62348
   592
proof
wenzelm@63652
   593
  assume ?rhs
wenzelm@63652
   594
  then show ?lhs by auto
haftmann@62348
   595
next
wenzelm@63652
   596
  assume ?lhs
haftmann@62348
   597
  then have "0 \<le> z - w" by simp
haftmann@62348
   598
  then obtain n where "z - w = int n"
haftmann@62348
   599
    using zero_le_imp_eq_int [of "z - w"] by blast
wenzelm@63652
   600
  then have "z = w + int n" by simp
wenzelm@63652
   601
  then show ?rhs ..
haftmann@25919
   602
qed
haftmann@25919
   603
huffman@44709
   604
lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
wenzelm@63652
   605
  by simp
haftmann@25919
   606
wenzelm@63652
   607
text \<open>
wenzelm@63652
   608
  This version is proved for all ordered rings, not just integers!
wenzelm@63652
   609
  It is proved here because attribute \<open>arith_split\<close> is not available
wenzelm@63652
   610
  in theory \<open>Rings\<close>.
wenzelm@63652
   611
  But is it really better than just rewriting with \<open>abs_if\<close>?
wenzelm@63652
   612
\<close>
wenzelm@63652
   613
lemma abs_split [arith_split, no_atp]: "P \<bar>a\<bar> \<longleftrightarrow> (0 \<le> a \<longrightarrow> P a) \<and> (a < 0 \<longrightarrow> P (- a))"
wenzelm@63652
   614
  for a :: "'a::linordered_idom"
wenzelm@63652
   615
  by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
haftmann@25919
   616
huffman@44709
   617
lemma negD: "x < 0 \<Longrightarrow> \<exists>n. x = - (int (Suc n))"
wenzelm@63652
   618
  apply transfer
wenzelm@63652
   619
  apply clarsimp
wenzelm@63652
   620
  apply (rule_tac x="b - Suc a" in exI)
wenzelm@63652
   621
  apply arith
wenzelm@63652
   622
  done
wenzelm@63652
   623
haftmann@25919
   624
wenzelm@60758
   625
subsection \<open>Cases and induction\<close>
haftmann@25919
   626
wenzelm@63652
   627
text \<open>
wenzelm@63652
   628
  Now we replace the case analysis rule by a more conventional one:
wenzelm@63652
   629
  whether an integer is negative or not.
wenzelm@63652
   630
\<close>
haftmann@25919
   631
wenzelm@63652
   632
text \<open>This version is symmetric in the two subgoals.\<close>
wenzelm@63652
   633
lemma int_cases2 [case_names nonneg nonpos, cases type: int]:
wenzelm@63652
   634
  "(\<And>n. z = int n \<Longrightarrow> P) \<Longrightarrow> (\<And>n. z = - (int n) \<Longrightarrow> P) \<Longrightarrow> P"
wenzelm@63652
   635
  by (cases "z < 0") (auto simp add: linorder_not_less dest!: negD nat_0_le [THEN sym])
lp15@59613
   636
wenzelm@63652
   637
text \<open>This is the default, with a negative case.\<close>
wenzelm@63652
   638
lemma int_cases [case_names nonneg neg, cases type: int]:
wenzelm@63652
   639
  "(\<And>n. z = int n \<Longrightarrow> P) \<Longrightarrow> (\<And>n. z = - (int (Suc n)) \<Longrightarrow> P) \<Longrightarrow> P"
wenzelm@63652
   640
  apply (cases "z < 0")
wenzelm@63652
   641
   apply (blast dest!: negD)
wenzelm@63652
   642
  apply (simp add: linorder_not_less del: of_nat_Suc)
wenzelm@63652
   643
  apply auto
wenzelm@63652
   644
  apply (blast dest: nat_0_le [THEN sym])
wenzelm@63652
   645
  done
haftmann@25919
   646
haftmann@60868
   647
lemma int_cases3 [case_names zero pos neg]:
haftmann@60868
   648
  fixes k :: int
haftmann@60868
   649
  assumes "k = 0 \<Longrightarrow> P" and "\<And>n. k = int n \<Longrightarrow> n > 0 \<Longrightarrow> P"
paulson@61204
   650
    and "\<And>n. k = - int n \<Longrightarrow> n > 0 \<Longrightarrow> P"
haftmann@60868
   651
  shows "P"
haftmann@60868
   652
proof (cases k "0::int" rule: linorder_cases)
wenzelm@63652
   653
  case equal
wenzelm@63652
   654
  with assms(1) show P by simp
haftmann@60868
   655
next
haftmann@60868
   656
  case greater
wenzelm@63539
   657
  then have *: "nat k > 0" by simp
wenzelm@63539
   658
  moreover from * have "k = int (nat k)" by auto
haftmann@60868
   659
  ultimately show P using assms(2) by blast
haftmann@60868
   660
next
haftmann@60868
   661
  case less
wenzelm@63539
   662
  then have *: "nat (- k) > 0" by simp
wenzelm@63539
   663
  moreover from * have "k = - int (nat (- k))" by auto
haftmann@60868
   664
  ultimately show P using assms(3) by blast
haftmann@60868
   665
qed
haftmann@60868
   666
wenzelm@63652
   667
lemma int_of_nat_induct [case_names nonneg neg, induct type: int]:
wenzelm@63652
   668
  "(\<And>n. P (int n)) \<Longrightarrow> (\<And>n. P (- (int (Suc n)))) \<Longrightarrow> P z"
wenzelm@42676
   669
  by (cases z) auto
haftmann@25919
   670
huffman@47108
   671
lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
wenzelm@61799
   672
  \<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
wenzelm@61799
   673
  by (fact Let_numeral) \<comment> \<open>FIXME drop\<close>
haftmann@37767
   674
haftmann@54489
   675
lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
wenzelm@61799
   676
  \<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
wenzelm@61799
   677
  by (fact Let_neg_numeral) \<comment> \<open>FIXME drop\<close>
haftmann@25919
   678
haftmann@66816
   679
lemma sgn_mult_dvd_iff [simp]:
haftmann@66816
   680
  "sgn r * l dvd k \<longleftrightarrow> l dvd k \<and> (r = 0 \<longrightarrow> k = 0)" for k l r :: int
haftmann@66816
   681
  by (cases r rule: int_cases3) auto
haftmann@66816
   682
haftmann@66816
   683
lemma mult_sgn_dvd_iff [simp]:
haftmann@66816
   684
  "l * sgn r dvd k \<longleftrightarrow> l dvd k \<and> (r = 0 \<longrightarrow> k = 0)" for k l r :: int
haftmann@66816
   685
  using sgn_mult_dvd_iff [of r l k] by (simp add: ac_simps)
haftmann@66816
   686
haftmann@66816
   687
lemma dvd_sgn_mult_iff [simp]:
haftmann@66816
   688
  "l dvd sgn r * k \<longleftrightarrow> l dvd k \<or> r = 0" for k l r :: int
haftmann@66816
   689
  by (cases r rule: int_cases3) simp_all
haftmann@66816
   690
haftmann@66816
   691
lemma dvd_mult_sgn_iff [simp]:
haftmann@66816
   692
  "l dvd k * sgn r \<longleftrightarrow> l dvd k \<or> r = 0" for k l r :: int
haftmann@66816
   693
  using dvd_sgn_mult_iff [of l r k] by (simp add: ac_simps)
haftmann@66816
   694
haftmann@66816
   695
lemma int_sgnE:
haftmann@66816
   696
  fixes k :: int
haftmann@66816
   697
  obtains n and l where "k = sgn l * int n"
haftmann@66816
   698
proof -
haftmann@66816
   699
  have "k = sgn k * int (nat \<bar>k\<bar>)"
haftmann@66816
   700
    by (simp add: sgn_mult_abs)
haftmann@66816
   701
  then show ?thesis ..
haftmann@66816
   702
qed
haftmann@66816
   703
wenzelm@61799
   704
text \<open>Unfold \<open>min\<close> and \<open>max\<close> on numerals.\<close>
huffman@28958
   705
huffman@47108
   706
lemmas max_number_of [simp] =
huffman@47108
   707
  max_def [of "numeral u" "numeral v"]
haftmann@54489
   708
  max_def [of "numeral u" "- numeral v"]
haftmann@54489
   709
  max_def [of "- numeral u" "numeral v"]
haftmann@54489
   710
  max_def [of "- numeral u" "- numeral v"] for u v
huffman@28958
   711
huffman@47108
   712
lemmas min_number_of [simp] =
huffman@47108
   713
  min_def [of "numeral u" "numeral v"]
haftmann@54489
   714
  min_def [of "numeral u" "- numeral v"]
haftmann@54489
   715
  min_def [of "- numeral u" "numeral v"]
haftmann@54489
   716
  min_def [of "- numeral u" "- numeral v"] for u v
huffman@26075
   717
haftmann@25919
   718
wenzelm@60758
   719
subsubsection \<open>Binary comparisons\<close>
huffman@28958
   720
wenzelm@60758
   721
text \<open>Preliminaries\<close>
huffman@28958
   722
lp15@60162
   723
lemma le_imp_0_less:
wenzelm@63652
   724
  fixes z :: int
huffman@28958
   725
  assumes le: "0 \<le> z"
wenzelm@63652
   726
  shows "0 < 1 + z"
huffman@28958
   727
proof -
huffman@28958
   728
  have "0 \<le> z" by fact
wenzelm@63652
   729
  also have "\<dots> < z + 1" by (rule less_add_one)
wenzelm@63652
   730
  also have "\<dots> = 1 + z" by (simp add: ac_simps)
huffman@28958
   731
  finally show "0 < 1 + z" .
huffman@28958
   732
qed
huffman@28958
   733
wenzelm@63652
   734
lemma odd_less_0_iff: "1 + z + z < 0 \<longleftrightarrow> z < 0"
wenzelm@63652
   735
  for z :: int
wenzelm@42676
   736
proof (cases z)
huffman@28958
   737
  case (nonneg n)
wenzelm@63652
   738
  then show ?thesis
wenzelm@63652
   739
    by (simp add: linorder_not_less add.assoc add_increasing le_imp_0_less [THEN order_less_imp_le])
huffman@28958
   740
next
huffman@28958
   741
  case (neg n)
wenzelm@63652
   742
  then show ?thesis
wenzelm@63652
   743
    by (simp del: of_nat_Suc of_nat_add of_nat_1
wenzelm@63652
   744
        add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
huffman@28958
   745
qed
huffman@28958
   746
wenzelm@63652
   747
wenzelm@60758
   748
subsubsection \<open>Comparisons, for Ordered Rings\<close>
haftmann@25919
   749
haftmann@25919
   750
lemmas double_eq_0_iff = double_zero
haftmann@25919
   751
wenzelm@63652
   752
lemma odd_nonzero: "1 + z + z \<noteq> 0"
wenzelm@63652
   753
  for z :: int
wenzelm@42676
   754
proof (cases z)
haftmann@25919
   755
  case (nonneg n)
wenzelm@63652
   756
  have le: "0 \<le> z + z"
wenzelm@63652
   757
    by (simp add: nonneg add_increasing)
wenzelm@63652
   758
  then show ?thesis
wenzelm@63652
   759
    using  le_imp_0_less [OF le] by (auto simp: add.assoc)
haftmann@25919
   760
next
haftmann@25919
   761
  case (neg n)
haftmann@25919
   762
  show ?thesis
haftmann@25919
   763
  proof
haftmann@25919
   764
    assume eq: "1 + z + z = 0"
wenzelm@63652
   765
    have "0 < 1 + (int n + int n)"
lp15@60162
   766
      by (simp add: le_imp_0_less add_increasing)
wenzelm@63652
   767
    also have "\<dots> = - (1 + z + z)"
lp15@60162
   768
      by (simp add: neg add.assoc [symmetric])
wenzelm@63652
   769
    also have "\<dots> = 0" by (simp add: eq)
haftmann@25919
   770
    finally have "0<0" ..
wenzelm@63652
   771
    then show False by blast
haftmann@25919
   772
  qed
haftmann@25919
   773
qed
haftmann@25919
   774
haftmann@30652
   775
wenzelm@60758
   776
subsection \<open>The Set of Integers\<close>
haftmann@25919
   777
haftmann@25919
   778
context ring_1
haftmann@25919
   779
begin
haftmann@25919
   780
wenzelm@61070
   781
definition Ints :: "'a set"  ("\<int>")
wenzelm@61070
   782
  where "\<int> = range of_int"
haftmann@25919
   783
huffman@35634
   784
lemma Ints_of_int [simp]: "of_int z \<in> \<int>"
huffman@35634
   785
  by (simp add: Ints_def)
huffman@35634
   786
huffman@35634
   787
lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>"
huffman@45533
   788
  using Ints_of_int [of "of_nat n"] by simp
huffman@35634
   789
haftmann@25919
   790
lemma Ints_0 [simp]: "0 \<in> \<int>"
huffman@45533
   791
  using Ints_of_int [of "0"] by simp
haftmann@25919
   792
haftmann@25919
   793
lemma Ints_1 [simp]: "1 \<in> \<int>"
huffman@45533
   794
  using Ints_of_int [of "1"] by simp
haftmann@25919
   795
eberlm@61552
   796
lemma Ints_numeral [simp]: "numeral n \<in> \<int>"
eberlm@61552
   797
  by (subst of_nat_numeral [symmetric], rule Ints_of_nat)
eberlm@61552
   798
haftmann@25919
   799
lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
wenzelm@63652
   800
  apply (auto simp add: Ints_def)
wenzelm@63652
   801
  apply (rule range_eqI)
wenzelm@63652
   802
  apply (rule of_int_add [symmetric])
wenzelm@63652
   803
  done
haftmann@25919
   804
haftmann@25919
   805
lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
wenzelm@63652
   806
  apply (auto simp add: Ints_def)
wenzelm@63652
   807
  apply (rule range_eqI)
wenzelm@63652
   808
  apply (rule of_int_minus [symmetric])
wenzelm@63652
   809
  done
haftmann@25919
   810
huffman@35634
   811
lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>"
wenzelm@63652
   812
  apply (auto simp add: Ints_def)
wenzelm@63652
   813
  apply (rule range_eqI)
wenzelm@63652
   814
  apply (rule of_int_diff [symmetric])
wenzelm@63652
   815
  done
huffman@35634
   816
haftmann@25919
   817
lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
wenzelm@63652
   818
  apply (auto simp add: Ints_def)
wenzelm@63652
   819
  apply (rule range_eqI)
wenzelm@63652
   820
  apply (rule of_int_mult [symmetric])
wenzelm@63652
   821
  done
haftmann@25919
   822
huffman@35634
   823
lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"
wenzelm@63652
   824
  by (induct n) simp_all
huffman@35634
   825
haftmann@25919
   826
lemma Ints_cases [cases set: Ints]:
haftmann@25919
   827
  assumes "q \<in> \<int>"
haftmann@25919
   828
  obtains (of_int) z where "q = of_int z"
haftmann@25919
   829
  unfolding Ints_def
haftmann@25919
   830
proof -
wenzelm@60758
   831
  from \<open>q \<in> \<int>\<close> have "q \<in> range of_int" unfolding Ints_def .
haftmann@25919
   832
  then obtain z where "q = of_int z" ..
haftmann@25919
   833
  then show thesis ..
haftmann@25919
   834
qed
haftmann@25919
   835
haftmann@25919
   836
lemma Ints_induct [case_names of_int, induct set: Ints]:
haftmann@25919
   837
  "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
haftmann@25919
   838
  by (rule Ints_cases) auto
haftmann@25919
   839
eberlm@61524
   840
lemma Nats_subset_Ints: "\<nat> \<subseteq> \<int>"
eberlm@61524
   841
  unfolding Nats_def Ints_def
eberlm@61524
   842
  by (rule subsetI, elim imageE, hypsubst, subst of_int_of_nat_eq[symmetric], rule imageI) simp_all
eberlm@61524
   843
eberlm@61524
   844
lemma Nats_altdef1: "\<nat> = {of_int n |n. n \<ge> 0}"
eberlm@61524
   845
proof (intro subsetI equalityI)
wenzelm@63652
   846
  fix x :: 'a
wenzelm@63652
   847
  assume "x \<in> {of_int n |n. n \<ge> 0}"
wenzelm@63652
   848
  then obtain n where "x = of_int n" "n \<ge> 0"
wenzelm@63652
   849
    by (auto elim!: Ints_cases)
wenzelm@63652
   850
  then have "x = of_nat (nat n)"
wenzelm@63652
   851
    by (subst of_nat_nat) simp_all
wenzelm@63652
   852
  then show "x \<in> \<nat>"
wenzelm@63652
   853
    by simp
eberlm@61524
   854
next
wenzelm@63652
   855
  fix x :: 'a
wenzelm@63652
   856
  assume "x \<in> \<nat>"
wenzelm@63652
   857
  then obtain n where "x = of_nat n"
wenzelm@63652
   858
    by (auto elim!: Nats_cases)
wenzelm@63652
   859
  then have "x = of_int (int n)" by simp
eberlm@61524
   860
  also have "int n \<ge> 0" by simp
wenzelm@63652
   861
  then have "of_int (int n) \<in> {of_int n |n. n \<ge> 0}" by blast
eberlm@61524
   862
  finally show "x \<in> {of_int n |n. n \<ge> 0}" .
eberlm@61524
   863
qed
eberlm@61524
   864
haftmann@25919
   865
end
haftmann@25919
   866
lp15@64758
   867
lemma (in linordered_idom) Ints_abs [simp]:
lp15@64758
   868
  shows "a \<in> \<int> \<Longrightarrow> abs a \<in> \<int>"
lp15@64758
   869
  by (auto simp: abs_if)
lp15@64758
   870
eberlm@61524
   871
lemma (in linordered_idom) Nats_altdef2: "\<nat> = {n \<in> \<int>. n \<ge> 0}"
eberlm@61524
   872
proof (intro subsetI equalityI)
wenzelm@63652
   873
  fix x :: 'a
wenzelm@63652
   874
  assume "x \<in> {n \<in> \<int>. n \<ge> 0}"
wenzelm@63652
   875
  then obtain n where "x = of_int n" "n \<ge> 0"
wenzelm@63652
   876
    by (auto elim!: Ints_cases)
wenzelm@63652
   877
  then have "x = of_nat (nat n)"
wenzelm@63652
   878
    by (subst of_nat_nat) simp_all
wenzelm@63652
   879
  then show "x \<in> \<nat>"
wenzelm@63652
   880
    by simp
eberlm@61524
   881
qed (auto elim!: Nats_cases)
eberlm@61524
   882
haftmann@64849
   883
lemma (in idom_divide) of_int_divide_in_Ints: 
haftmann@64849
   884
  "of_int a div of_int b \<in> \<int>" if "b dvd a"
haftmann@64849
   885
proof -
haftmann@64849
   886
  from that obtain c where "a = b * c" ..
haftmann@64849
   887
  then show ?thesis
haftmann@64849
   888
    by (cases "of_int b = 0") simp_all
haftmann@64849
   889
qed
eberlm@61524
   890
wenzelm@60758
   891
text \<open>The premise involving @{term Ints} prevents @{term "a = 1/2"}.\<close>
haftmann@25919
   892
haftmann@25919
   893
lemma Ints_double_eq_0_iff:
wenzelm@63652
   894
  fixes a :: "'a::ring_char_0"
wenzelm@61070
   895
  assumes in_Ints: "a \<in> \<int>"
wenzelm@63652
   896
  shows "a + a = 0 \<longleftrightarrow> a = 0"
wenzelm@63652
   897
    (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@25919
   898
proof -
wenzelm@63652
   899
  from in_Ints have "a \<in> range of_int"
wenzelm@63652
   900
    unfolding Ints_def [symmetric] .
haftmann@25919
   901
  then obtain z where a: "a = of_int z" ..
haftmann@25919
   902
  show ?thesis
haftmann@25919
   903
  proof
wenzelm@63652
   904
    assume ?rhs
wenzelm@63652
   905
    then show ?lhs by simp
haftmann@25919
   906
  next
wenzelm@63652
   907
    assume ?lhs
wenzelm@63652
   908
    with a have "of_int (z + z) = (of_int 0 :: 'a)" by simp
wenzelm@63652
   909
    then have "z + z = 0" by (simp only: of_int_eq_iff)
wenzelm@63652
   910
    then have "z = 0" by (simp only: double_eq_0_iff)
wenzelm@63652
   911
    with a show ?rhs by simp
haftmann@25919
   912
  qed
haftmann@25919
   913
qed
haftmann@25919
   914
haftmann@25919
   915
lemma Ints_odd_nonzero:
wenzelm@63652
   916
  fixes a :: "'a::ring_char_0"
wenzelm@61070
   917
  assumes in_Ints: "a \<in> \<int>"
wenzelm@63652
   918
  shows "1 + a + a \<noteq> 0"
haftmann@25919
   919
proof -
wenzelm@63652
   920
  from in_Ints have "a \<in> range of_int"
wenzelm@63652
   921
    unfolding Ints_def [symmetric] .
haftmann@25919
   922
  then obtain z where a: "a = of_int z" ..
haftmann@25919
   923
  show ?thesis
haftmann@25919
   924
  proof
wenzelm@63652
   925
    assume "1 + a + a = 0"
wenzelm@63652
   926
    with a have "of_int (1 + z + z) = (of_int 0 :: 'a)" by simp
wenzelm@63652
   927
    then have "1 + z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919
   928
    with odd_nonzero show False by blast
haftmann@25919
   929
  qed
lp15@60162
   930
qed
haftmann@25919
   931
wenzelm@61070
   932
lemma Nats_numeral [simp]: "numeral w \<in> \<nat>"
huffman@47108
   933
  using of_nat_in_Nats [of "numeral w"] by simp
huffman@35634
   934
lp15@60162
   935
lemma Ints_odd_less_0:
wenzelm@63652
   936
  fixes a :: "'a::linordered_idom"
wenzelm@61070
   937
  assumes in_Ints: "a \<in> \<int>"
wenzelm@63652
   938
  shows "1 + a + a < 0 \<longleftrightarrow> a < 0"
haftmann@25919
   939
proof -
wenzelm@63652
   940
  from in_Ints have "a \<in> range of_int"
wenzelm@63652
   941
    unfolding Ints_def [symmetric] .
haftmann@25919
   942
  then obtain z where a: "a = of_int z" ..
wenzelm@63652
   943
  with a have "1 + a + a < 0 \<longleftrightarrow> of_int (1 + z + z) < (of_int 0 :: 'a)"
wenzelm@63652
   944
    by simp
wenzelm@63652
   945
  also have "\<dots> \<longleftrightarrow> z < 0"
wenzelm@63652
   946
    by (simp only: of_int_less_iff odd_less_0_iff)
wenzelm@63652
   947
  also have "\<dots> \<longleftrightarrow> a < 0"
haftmann@25919
   948
    by (simp add: a)
haftmann@25919
   949
  finally show ?thesis .
haftmann@25919
   950
qed
haftmann@25919
   951
haftmann@25919
   952
nipkow@64272
   953
subsection \<open>@{term sum} and @{term prod}\<close>
haftmann@25919
   954
nipkow@64267
   955
lemma of_nat_sum [simp]: "of_nat (sum f A) = (\<Sum>x\<in>A. of_nat(f x))"
wenzelm@63652
   956
  by (induct A rule: infinite_finite_induct) auto
haftmann@25919
   957
nipkow@64267
   958
lemma of_int_sum [simp]: "of_int (sum f A) = (\<Sum>x\<in>A. of_int(f x))"
wenzelm@63652
   959
  by (induct A rule: infinite_finite_induct) auto
haftmann@25919
   960
nipkow@64272
   961
lemma of_nat_prod [simp]: "of_nat (prod f A) = (\<Prod>x\<in>A. of_nat(f x))"
wenzelm@63652
   962
  by (induct A rule: infinite_finite_induct) auto
haftmann@25919
   963
nipkow@64272
   964
lemma of_int_prod [simp]: "of_int (prod f A) = (\<Prod>x\<in>A. of_int(f x))"
wenzelm@63652
   965
  by (induct A rule: infinite_finite_induct) auto
haftmann@25919
   966
haftmann@25919
   967
wenzelm@60758
   968
text \<open>Legacy theorems\<close>
haftmann@25919
   969
haftmann@64714
   970
lemmas int_sum = of_nat_sum [where 'a=int]
haftmann@64714
   971
lemmas int_prod = of_nat_prod [where 'a=int]
haftmann@25919
   972
lemmas zle_int = of_nat_le_iff [where 'a=int]
haftmann@25919
   973
lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
haftmann@64714
   974
lemmas nonneg_eq_int = nonneg_int_cases
haftmann@25919
   975
wenzelm@63652
   976
wenzelm@60758
   977
subsection \<open>Setting up simplification procedures\<close>
huffman@30802
   978
haftmann@54249
   979
lemmas of_int_simps =
haftmann@54249
   980
  of_int_0 of_int_1 of_int_add of_int_mult
haftmann@54249
   981
wenzelm@48891
   982
ML_file "Tools/int_arith.ML"
wenzelm@60758
   983
declaration \<open>K Int_Arith.setup\<close>
haftmann@25919
   984
wenzelm@63652
   985
simproc_setup fast_arith
wenzelm@63652
   986
  ("(m::'a::linordered_idom) < n" |
wenzelm@63652
   987
    "(m::'a::linordered_idom) \<le> n" |
wenzelm@63652
   988
    "(m::'a::linordered_idom) = n") =
wenzelm@61144
   989
  \<open>K Lin_Arith.simproc\<close>
wenzelm@43595
   990
haftmann@25919
   991
wenzelm@60758
   992
subsection\<open>More Inequality Reasoning\<close>
haftmann@25919
   993
wenzelm@63652
   994
lemma zless_add1_eq: "w < z + 1 \<longleftrightarrow> w < z \<or> w = z"
wenzelm@63652
   995
  for w z :: int
wenzelm@63652
   996
  by arith
haftmann@25919
   997
wenzelm@63652
   998
lemma add1_zle_eq: "w + 1 \<le> z \<longleftrightarrow> w < z"
wenzelm@63652
   999
  for w z :: int
wenzelm@63652
  1000
  by arith
haftmann@25919
  1001
wenzelm@63652
  1002
lemma zle_diff1_eq [simp]: "w \<le> z - 1 \<longleftrightarrow> w < z"
wenzelm@63652
  1003
  for w z :: int
wenzelm@63652
  1004
  by arith
haftmann@25919
  1005
wenzelm@63652
  1006
lemma zle_add1_eq_le [simp]: "w < z + 1 \<longleftrightarrow> w \<le> z"
wenzelm@63652
  1007
  for w z :: int
wenzelm@63652
  1008
  by arith
haftmann@25919
  1009
wenzelm@63652
  1010
lemma int_one_le_iff_zero_less: "1 \<le> z \<longleftrightarrow> 0 < z"
wenzelm@63652
  1011
  for z :: int
wenzelm@63652
  1012
  by arith
haftmann@25919
  1013
lp15@64758
  1014
lemma Ints_nonzero_abs_ge1:
lp15@64758
  1015
  fixes x:: "'a :: linordered_idom"
lp15@64758
  1016
    assumes "x \<in> Ints" "x \<noteq> 0"
lp15@64758
  1017
    shows "1 \<le> abs x"
lp15@64758
  1018
proof (rule Ints_cases [OF \<open>x \<in> Ints\<close>])
lp15@64758
  1019
  fix z::int
lp15@64758
  1020
  assume "x = of_int z"
lp15@64758
  1021
    with \<open>x \<noteq> 0\<close> 
lp15@64758
  1022
  show "1 \<le> \<bar>x\<bar>"
lp15@64758
  1023
    apply (auto simp add: abs_if)
lp15@64758
  1024
    by (metis diff_0 of_int_1 of_int_le_iff of_int_minus zle_diff1_eq)
lp15@64758
  1025
qed
lp15@64758
  1026
  
lp15@64758
  1027
lemma Ints_nonzero_abs_less1:
lp15@64758
  1028
  fixes x:: "'a :: linordered_idom"
lp15@64758
  1029
  shows "\<lbrakk>x \<in> Ints; abs x < 1\<rbrakk> \<Longrightarrow> x = 0"
lp15@64758
  1030
    using Ints_nonzero_abs_ge1 [of x] by auto
lp15@64758
  1031
    
haftmann@25919
  1032
wenzelm@63652
  1033
subsection \<open>The functions @{term nat} and @{term int}\<close>
haftmann@25919
  1034
wenzelm@63652
  1035
text \<open>Simplify the term @{term "w + - z"}.\<close>
haftmann@25919
  1036
wenzelm@63652
  1037
lemma one_less_nat_eq [simp]: "Suc 0 < nat z \<longleftrightarrow> 1 < z"
lp15@60162
  1038
  using zless_nat_conj [of 1 z] by auto
haftmann@25919
  1039
wenzelm@63652
  1040
text \<open>
wenzelm@63652
  1041
  This simplifies expressions of the form @{term "int n = z"} where
wenzelm@63652
  1042
  \<open>z\<close> is an integer literal.
wenzelm@63652
  1043
\<close>
huffman@47108
  1044
lemmas int_eq_iff_numeral [simp] = int_eq_iff [of _ "numeral v"] for v
haftmann@25919
  1045
wenzelm@63652
  1046
lemma split_nat [arith_split]: "P (nat i) = ((\<forall>n. i = int n \<longrightarrow> P n) \<and> (i < 0 \<longrightarrow> P 0))"
wenzelm@63652
  1047
  (is "?P = (?L \<and> ?R)")
wenzelm@63652
  1048
  for i :: int
haftmann@25919
  1049
proof (cases "i < 0")
wenzelm@63652
  1050
  case True
wenzelm@63652
  1051
  then show ?thesis by auto
haftmann@25919
  1052
next
haftmann@25919
  1053
  case False
haftmann@25919
  1054
  have "?P = ?L"
haftmann@25919
  1055
  proof
wenzelm@63652
  1056
    assume ?P
wenzelm@63652
  1057
    then show ?L using False by auto
haftmann@25919
  1058
  next
wenzelm@63652
  1059
    assume ?L
wenzelm@63652
  1060
    then show ?P using False by simp
haftmann@25919
  1061
  qed
haftmann@25919
  1062
  with False show ?thesis by simp
haftmann@25919
  1063
qed
haftmann@25919
  1064
hoelzl@59000
  1065
lemma nat_abs_int_diff: "nat \<bar>int a - int b\<bar> = (if a \<le> b then b - a else a - b)"
hoelzl@59000
  1066
  by auto
hoelzl@59000
  1067
hoelzl@59000
  1068
lemma nat_int_add: "nat (int a + int b) = a + b"
hoelzl@59000
  1069
  by auto
hoelzl@59000
  1070
haftmann@25919
  1071
context ring_1
haftmann@25919
  1072
begin
haftmann@25919
  1073
blanchet@33056
  1074
lemma of_int_of_nat [nitpick_simp]:
haftmann@25919
  1075
  "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
haftmann@25919
  1076
proof (cases "k < 0")
wenzelm@63652
  1077
  case True
wenzelm@63652
  1078
  then have "0 \<le> - k" by simp
haftmann@25919
  1079
  then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
haftmann@25919
  1080
  with True show ?thesis by simp
haftmann@25919
  1081
next
wenzelm@63652
  1082
  case False
wenzelm@63652
  1083
  then show ?thesis by (simp add: not_less)
haftmann@25919
  1084
qed
haftmann@25919
  1085
haftmann@25919
  1086
end
haftmann@25919
  1087
haftmann@64014
  1088
lemma transfer_rule_of_int:
haftmann@64014
  1089
  fixes R :: "'a::ring_1 \<Rightarrow> 'b::ring_1 \<Rightarrow> bool"
haftmann@64014
  1090
  assumes [transfer_rule]: "R 0 0" "R 1 1"
haftmann@64014
  1091
    "rel_fun R (rel_fun R R) plus plus"
haftmann@64014
  1092
    "rel_fun R R uminus uminus"
haftmann@64014
  1093
  shows "rel_fun HOL.eq R of_int of_int"
haftmann@64014
  1094
proof -
haftmann@64014
  1095
  note transfer_rule_of_nat [transfer_rule]
haftmann@64014
  1096
  have [transfer_rule]: "rel_fun HOL.eq R of_nat of_nat"
haftmann@64014
  1097
    by transfer_prover
haftmann@64014
  1098
  show ?thesis
haftmann@64014
  1099
    by (unfold of_int_of_nat [abs_def]) transfer_prover
haftmann@64014
  1100
qed
haftmann@64014
  1101
haftmann@25919
  1102
lemma nat_mult_distrib:
haftmann@25919
  1103
  fixes z z' :: int
haftmann@25919
  1104
  assumes "0 \<le> z"
haftmann@25919
  1105
  shows "nat (z * z') = nat z * nat z'"
haftmann@25919
  1106
proof (cases "0 \<le> z'")
wenzelm@63652
  1107
  case False
wenzelm@63652
  1108
  with assms have "z * z' \<le> 0"
haftmann@25919
  1109
    by (simp add: not_le mult_le_0_iff)
haftmann@25919
  1110
  then have "nat (z * z') = 0" by simp
haftmann@25919
  1111
  moreover from False have "nat z' = 0" by simp
haftmann@25919
  1112
  ultimately show ?thesis by simp
haftmann@25919
  1113
next
wenzelm@63652
  1114
  case True
wenzelm@63652
  1115
  with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
haftmann@25919
  1116
  show ?thesis
haftmann@25919
  1117
    by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
haftmann@25919
  1118
      (simp only: of_nat_mult of_nat_nat [OF True]
haftmann@25919
  1119
         of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
haftmann@25919
  1120
qed
haftmann@25919
  1121
wenzelm@63652
  1122
lemma nat_mult_distrib_neg: "z \<le> 0 \<Longrightarrow> nat (z * z') = nat (- z) * nat (- z')"
wenzelm@63652
  1123
  for z z' :: int
wenzelm@63652
  1124
  apply (rule trans)
wenzelm@63652
  1125
   apply (rule_tac [2] nat_mult_distrib)
wenzelm@63652
  1126
   apply auto
wenzelm@63652
  1127
  done
haftmann@25919
  1128
wenzelm@61944
  1129
lemma nat_abs_mult_distrib: "nat \<bar>w * z\<bar> = nat \<bar>w\<bar> * nat \<bar>z\<bar>"
wenzelm@63652
  1130
  by (cases "z = 0 \<or> w = 0")
wenzelm@63652
  1131
    (auto simp add: abs_if nat_mult_distrib [symmetric]
wenzelm@63652
  1132
      nat_mult_distrib_neg [symmetric] mult_less_0_iff)
haftmann@25919
  1133
wenzelm@63652
  1134
lemma int_in_range_abs [simp]: "int n \<in> range abs"
haftmann@60570
  1135
proof (rule range_eqI)
wenzelm@63652
  1136
  show "int n = \<bar>int n\<bar>" by simp
haftmann@60570
  1137
qed
haftmann@60570
  1138
wenzelm@63652
  1139
lemma range_abs_Nats [simp]: "range abs = (\<nat> :: int set)"
haftmann@60570
  1140
proof -
haftmann@60570
  1141
  have "\<bar>k\<bar> \<in> \<nat>" for k :: int
haftmann@60570
  1142
    by (cases k) simp_all
haftmann@60570
  1143
  moreover have "k \<in> range abs" if "k \<in> \<nat>" for k :: int
haftmann@60570
  1144
    using that by induct simp
haftmann@60570
  1145
  ultimately show ?thesis by blast
paulson@61204
  1146
qed
haftmann@60570
  1147
wenzelm@63652
  1148
lemma Suc_nat_eq_nat_zadd1: "0 \<le> z \<Longrightarrow> Suc (nat z) = nat (1 + z)"
wenzelm@63652
  1149
  for z :: int
wenzelm@63652
  1150
  by (rule sym) (simp add: nat_eq_iff)
huffman@47207
  1151
huffman@47207
  1152
lemma diff_nat_eq_if:
wenzelm@63652
  1153
  "nat z - nat z' =
wenzelm@63652
  1154
    (if z' < 0 then nat z
wenzelm@63652
  1155
     else
wenzelm@63652
  1156
      let d = z - z'
wenzelm@63652
  1157
      in if d < 0 then 0 else nat d)"
wenzelm@63652
  1158
  by (simp add: Let_def nat_diff_distrib [symmetric])
huffman@47207
  1159
wenzelm@63652
  1160
lemma nat_numeral_diff_1 [simp]: "numeral v - (1::nat) = nat (numeral v - 1)"
huffman@47207
  1161
  using diff_nat_numeral [of v Num.One] by simp
huffman@47207
  1162
haftmann@25919
  1163
wenzelm@63652
  1164
subsection \<open>Induction principles for int\<close>
haftmann@25919
  1165
wenzelm@63652
  1166
text \<open>Well-founded segments of the integers.\<close>
haftmann@25919
  1167
wenzelm@63652
  1168
definition int_ge_less_than :: "int \<Rightarrow> (int \<times> int) set"
wenzelm@63652
  1169
  where "int_ge_less_than d = {(z', z). d \<le> z' \<and> z' < z}"
haftmann@25919
  1170
wenzelm@63652
  1171
lemma wf_int_ge_less_than: "wf (int_ge_less_than d)"
haftmann@25919
  1172
proof -
wenzelm@63652
  1173
  have "int_ge_less_than d \<subseteq> measure (\<lambda>z. nat (z - d))"
haftmann@25919
  1174
    by (auto simp add: int_ge_less_than_def)
wenzelm@63652
  1175
  then show ?thesis
lp15@60162
  1176
    by (rule wf_subset [OF wf_measure])
haftmann@25919
  1177
qed
haftmann@25919
  1178
wenzelm@63652
  1179
text \<open>
wenzelm@63652
  1180
  This variant looks odd, but is typical of the relations suggested
wenzelm@63652
  1181
  by RankFinder.\<close>
haftmann@25919
  1182
wenzelm@63652
  1183
definition int_ge_less_than2 :: "int \<Rightarrow> (int \<times> int) set"
wenzelm@63652
  1184
  where "int_ge_less_than2 d = {(z',z). d \<le> z \<and> z' < z}"
haftmann@25919
  1185
wenzelm@63652
  1186
lemma wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
haftmann@25919
  1187
proof -
wenzelm@63652
  1188
  have "int_ge_less_than2 d \<subseteq> measure (\<lambda>z. nat (1 + z - d))"
haftmann@25919
  1189
    by (auto simp add: int_ge_less_than2_def)
wenzelm@63652
  1190
  then show ?thesis
lp15@60162
  1191
    by (rule wf_subset [OF wf_measure])
haftmann@25919
  1192
qed
haftmann@25919
  1193
haftmann@25919
  1194
(* `set:int': dummy construction *)
haftmann@25919
  1195
theorem int_ge_induct [case_names base step, induct set: int]:
haftmann@25919
  1196
  fixes i :: int
wenzelm@63652
  1197
  assumes ge: "k \<le> i"
wenzelm@63652
  1198
    and base: "P k"
wenzelm@63652
  1199
    and step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@25919
  1200
  shows "P i"
haftmann@25919
  1201
proof -
wenzelm@63652
  1202
  have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i" for n
wenzelm@63652
  1203
  proof (induct n)
wenzelm@63652
  1204
    case 0
wenzelm@63652
  1205
    then have "i = k" by arith
wenzelm@63652
  1206
    with base show "P i" by simp
wenzelm@63652
  1207
  next
wenzelm@63652
  1208
    case (Suc n)
wenzelm@63652
  1209
    then have "n = nat ((i - 1) - k)" by arith
wenzelm@63652
  1210
    moreover have k: "k \<le> i - 1" using Suc.prems by arith
wenzelm@63652
  1211
    ultimately have "P (i - 1)" by (rule Suc.hyps)
wenzelm@63652
  1212
    from step [OF k this] show ?case by simp
wenzelm@63652
  1213
  qed
haftmann@25919
  1214
  with ge show ?thesis by fast
haftmann@25919
  1215
qed
haftmann@25919
  1216
haftmann@25928
  1217
(* `set:int': dummy construction *)
haftmann@25928
  1218
theorem int_gr_induct [case_names base step, induct set: int]:
wenzelm@63652
  1219
  fixes i k :: int
wenzelm@63652
  1220
  assumes gr: "k < i"
wenzelm@63652
  1221
    and base: "P (k + 1)"
wenzelm@63652
  1222
    and step: "\<And>i. k < i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@25919
  1223
  shows "P i"
wenzelm@63652
  1224
  apply (rule int_ge_induct[of "k + 1"])
haftmann@25919
  1225
  using gr apply arith
wenzelm@63652
  1226
   apply (rule base)
wenzelm@63652
  1227
  apply (rule step)
wenzelm@63652
  1228
   apply simp_all
wenzelm@63652
  1229
  done
haftmann@25919
  1230
wenzelm@42676
  1231
theorem int_le_induct [consumes 1, case_names base step]:
wenzelm@63652
  1232
  fixes i k :: int
wenzelm@63652
  1233
  assumes le: "i \<le> k"
wenzelm@63652
  1234
    and base: "P k"
wenzelm@63652
  1235
    and step: "\<And>i. i \<le> k \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
haftmann@25919
  1236
  shows "P i"
haftmann@25919
  1237
proof -
wenzelm@63652
  1238
  have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i" for n
wenzelm@63652
  1239
  proof (induct n)
wenzelm@63652
  1240
    case 0
wenzelm@63652
  1241
    then have "i = k" by arith
wenzelm@63652
  1242
    with base show "P i" by simp
wenzelm@63652
  1243
  next
wenzelm@63652
  1244
    case (Suc n)
wenzelm@63652
  1245
    then have "n = nat (k - (i + 1))" by arith
wenzelm@63652
  1246
    moreover have k: "i + 1 \<le> k" using Suc.prems by arith
wenzelm@63652
  1247
    ultimately have "P (i + 1)" by (rule Suc.hyps)
wenzelm@63652
  1248
    from step[OF k this] show ?case by simp
wenzelm@63652
  1249
  qed
haftmann@25919
  1250
  with le show ?thesis by fast
haftmann@25919
  1251
qed
haftmann@25919
  1252
wenzelm@42676
  1253
theorem int_less_induct [consumes 1, case_names base step]:
wenzelm@63652
  1254
  fixes i k :: int
wenzelm@63652
  1255
  assumes less: "i < k"
wenzelm@63652
  1256
    and base: "P (k - 1)"
wenzelm@63652
  1257
    and step: "\<And>i. i < k \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
haftmann@25919
  1258
  shows "P i"
wenzelm@63652
  1259
  apply (rule int_le_induct[of _ "k - 1"])
haftmann@25919
  1260
  using less apply arith
wenzelm@63652
  1261
   apply (rule base)
wenzelm@63652
  1262
  apply (rule step)
wenzelm@63652
  1263
   apply simp_all
wenzelm@63652
  1264
  done
haftmann@25919
  1265
haftmann@36811
  1266
theorem int_induct [case_names base step1 step2]:
haftmann@36801
  1267
  fixes k :: int
haftmann@36801
  1268
  assumes base: "P k"
haftmann@36801
  1269
    and step1: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@36801
  1270
    and step2: "\<And>i. k \<ge> i \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
haftmann@36801
  1271
  shows "P i"
haftmann@36801
  1272
proof -
haftmann@36801
  1273
  have "i \<le> k \<or> i \<ge> k" by arith
wenzelm@42676
  1274
  then show ?thesis
wenzelm@42676
  1275
  proof
wenzelm@42676
  1276
    assume "i \<ge> k"
wenzelm@63652
  1277
    then show ?thesis
wenzelm@63652
  1278
      using base by (rule int_ge_induct) (fact step1)
haftmann@36801
  1279
  next
wenzelm@42676
  1280
    assume "i \<le> k"
wenzelm@63652
  1281
    then show ?thesis
wenzelm@63652
  1282
      using base by (rule int_le_induct) (fact step2)
haftmann@36801
  1283
  qed
haftmann@36801
  1284
qed
haftmann@36801
  1285
wenzelm@63652
  1286
wenzelm@63652
  1287
subsection \<open>Intermediate value theorems\<close>
haftmann@25919
  1288
wenzelm@63652
  1289
lemma int_val_lemma: "(\<forall>i<n. \<bar>f (i + 1) - f i\<bar> \<le> 1) \<longrightarrow> f 0 \<le> k \<longrightarrow> k \<le> f n \<longrightarrow> (\<exists>i \<le> n. f i = k)"
wenzelm@63652
  1290
  for n :: nat and k :: int
wenzelm@63652
  1291
  unfolding One_nat_def
wenzelm@63652
  1292
  apply (induct n)
wenzelm@63652
  1293
   apply simp
wenzelm@63652
  1294
  apply (intro strip)
wenzelm@63652
  1295
  apply (erule impE)
wenzelm@63652
  1296
   apply simp
wenzelm@63652
  1297
  apply (erule_tac x = n in allE)
wenzelm@63652
  1298
  apply simp
wenzelm@63652
  1299
  apply (case_tac "k = f (Suc n)")
wenzelm@63652
  1300
   apply force
wenzelm@63652
  1301
  apply (erule impE)
wenzelm@63652
  1302
   apply (simp add: abs_if split: if_split_asm)
wenzelm@63652
  1303
  apply (blast intro: le_SucI)
wenzelm@63652
  1304
  done
haftmann@25919
  1305
haftmann@25919
  1306
lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
haftmann@25919
  1307
haftmann@25919
  1308
lemma nat_intermed_int_val:
wenzelm@63652
  1309
  "\<forall>i. m \<le> i \<and> i < n \<longrightarrow> \<bar>f (i + 1) - f i\<bar> \<le> 1 \<Longrightarrow> m < n \<Longrightarrow>
wenzelm@63652
  1310
    f m \<le> k \<Longrightarrow> k \<le> f n \<Longrightarrow> \<exists>i. m \<le> i \<and> i \<le> n \<and> f i = k"
wenzelm@63652
  1311
    for f :: "nat \<Rightarrow> int" and k :: int
wenzelm@63652
  1312
  apply (cut_tac n = "n-m" and f = "\<lambda>i. f (i + m)" and k = k in int_val_lemma)
wenzelm@63652
  1313
  unfolding One_nat_def
wenzelm@63652
  1314
  apply simp
wenzelm@63652
  1315
  apply (erule exE)
wenzelm@63652
  1316
  apply (rule_tac x = "i+m" in exI)
wenzelm@63652
  1317
  apply arith
wenzelm@63652
  1318
  done
haftmann@25919
  1319
haftmann@25919
  1320
wenzelm@63652
  1321
subsection \<open>Products and 1, by T. M. Rasmussen\<close>
haftmann@25919
  1322
paulson@34055
  1323
lemma abs_zmult_eq_1:
wenzelm@63652
  1324
  fixes m n :: int
paulson@34055
  1325
  assumes mn: "\<bar>m * n\<bar> = 1"
wenzelm@63652
  1326
  shows "\<bar>m\<bar> = 1"
paulson@34055
  1327
proof -
wenzelm@63652
  1328
  from mn have 0: "m \<noteq> 0" "n \<noteq> 0" by auto
wenzelm@63652
  1329
  have "\<not> 2 \<le> \<bar>m\<bar>"
paulson@34055
  1330
  proof
paulson@34055
  1331
    assume "2 \<le> \<bar>m\<bar>"
wenzelm@63652
  1332
    then have "2 * \<bar>n\<bar> \<le> \<bar>m\<bar> * \<bar>n\<bar>" by (simp add: mult_mono 0)
wenzelm@63652
  1333
    also have "\<dots> = \<bar>m * n\<bar>" by (simp add: abs_mult)
wenzelm@63652
  1334
    also from mn have "\<dots> = 1" by simp
wenzelm@63652
  1335
    finally have "2 * \<bar>n\<bar> \<le> 1" .
wenzelm@63652
  1336
    with 0 show "False" by arith
paulson@34055
  1337
  qed
wenzelm@63652
  1338
  with 0 show ?thesis by auto
paulson@34055
  1339
qed
haftmann@25919
  1340
wenzelm@63652
  1341
lemma pos_zmult_eq_1_iff_lemma: "m * n = 1 \<Longrightarrow> m = 1 \<or> m = - 1"
wenzelm@63652
  1342
  for m n :: int
wenzelm@63652
  1343
  using abs_zmult_eq_1 [of m n] by arith
haftmann@25919
  1344
boehmes@35815
  1345
lemma pos_zmult_eq_1_iff:
wenzelm@63652
  1346
  fixes m n :: int
wenzelm@63652
  1347
  assumes "0 < m"
wenzelm@63652
  1348
  shows "m * n = 1 \<longleftrightarrow> m = 1 \<and> n = 1"
boehmes@35815
  1349
proof -
wenzelm@63652
  1350
  from assms have "m * n = 1 \<Longrightarrow> m = 1"
wenzelm@63652
  1351
    by (auto dest: pos_zmult_eq_1_iff_lemma)
wenzelm@63652
  1352
  then show ?thesis
wenzelm@63652
  1353
    by (auto dest: pos_zmult_eq_1_iff_lemma)
boehmes@35815
  1354
qed
haftmann@25919
  1355
wenzelm@63652
  1356
lemma zmult_eq_1_iff: "m * n = 1 \<longleftrightarrow> (m = 1 \<and> n = 1) \<or> (m = - 1 \<and> n = - 1)"
wenzelm@63652
  1357
  for m n :: int
wenzelm@63652
  1358
  apply (rule iffI)
wenzelm@63652
  1359
   apply (frule pos_zmult_eq_1_iff_lemma)
wenzelm@63652
  1360
   apply (simp add: mult.commute [of m])
wenzelm@63652
  1361
   apply (frule pos_zmult_eq_1_iff_lemma)
wenzelm@63652
  1362
   apply auto
wenzelm@63652
  1363
  done
haftmann@25919
  1364
haftmann@33296
  1365
lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)"
haftmann@25919
  1366
proof
haftmann@33296
  1367
  assume "finite (UNIV::int set)"
wenzelm@61076
  1368
  moreover have "inj (\<lambda>i::int. 2 * i)"
haftmann@33296
  1369
    by (rule injI) simp
wenzelm@61076
  1370
  ultimately have "surj (\<lambda>i::int. 2 * i)"
haftmann@33296
  1371
    by (rule finite_UNIV_inj_surj)
haftmann@33296
  1372
  then obtain i :: int where "1 = 2 * i" by (rule surjE)
haftmann@33296
  1373
  then show False by (simp add: pos_zmult_eq_1_iff)
haftmann@25919
  1374
qed
haftmann@25919
  1375
haftmann@25919
  1376
wenzelm@60758
  1377
subsection \<open>Further theorems on numerals\<close>
haftmann@30652
  1378
wenzelm@63652
  1379
subsubsection \<open>Special Simplification for Constants\<close>
haftmann@30652
  1380
wenzelm@63652
  1381
text \<open>These distributive laws move literals inside sums and differences.\<close>
haftmann@30652
  1382
webertj@49962
  1383
lemmas distrib_right_numeral [simp] = distrib_right [of _ _ "numeral v"] for v
webertj@49962
  1384
lemmas distrib_left_numeral [simp] = distrib_left [of "numeral v"] for v
huffman@47108
  1385
lemmas left_diff_distrib_numeral [simp] = left_diff_distrib [of _ _ "numeral v"] for v
huffman@47108
  1386
lemmas right_diff_distrib_numeral [simp] = right_diff_distrib [of "numeral v"] for v
haftmann@30652
  1387
wenzelm@63652
  1388
text \<open>These are actually for fields, like real: but where else to put them?\<close>
haftmann@30652
  1389
huffman@47108
  1390
lemmas zero_less_divide_iff_numeral [simp, no_atp] = zero_less_divide_iff [of "numeral w"] for w
huffman@47108
  1391
lemmas divide_less_0_iff_numeral [simp, no_atp] = divide_less_0_iff [of "numeral w"] for w
huffman@47108
  1392
lemmas zero_le_divide_iff_numeral [simp, no_atp] = zero_le_divide_iff [of "numeral w"] for w
huffman@47108
  1393
lemmas divide_le_0_iff_numeral [simp, no_atp] = divide_le_0_iff [of "numeral w"] for w
haftmann@30652
  1394
haftmann@30652
  1395
wenzelm@61799
  1396
text \<open>Replaces \<open>inverse #nn\<close> by \<open>1/#nn\<close>.  It looks
wenzelm@60758
  1397
  strange, but then other simprocs simplify the quotient.\<close>
haftmann@30652
  1398
huffman@47108
  1399
lemmas inverse_eq_divide_numeral [simp] =
huffman@47108
  1400
  inverse_eq_divide [of "numeral w"] for w
huffman@47108
  1401
huffman@47108
  1402
lemmas inverse_eq_divide_neg_numeral [simp] =
haftmann@54489
  1403
  inverse_eq_divide [of "- numeral w"] for w
haftmann@30652
  1404
wenzelm@60758
  1405
text \<open>These laws simplify inequalities, moving unary minus from a term
wenzelm@63652
  1406
  into the literal.\<close>
haftmann@30652
  1407
haftmann@54489
  1408
lemmas equation_minus_iff_numeral [no_atp] =
haftmann@54489
  1409
  equation_minus_iff [of "numeral v"] for v
huffman@47108
  1410
haftmann@54489
  1411
lemmas minus_equation_iff_numeral [no_atp] =
haftmann@54489
  1412
  minus_equation_iff [of _ "numeral v"] for v
huffman@47108
  1413
haftmann@54489
  1414
lemmas le_minus_iff_numeral [no_atp] =
haftmann@54489
  1415
  le_minus_iff [of "numeral v"] for v
haftmann@30652
  1416
haftmann@54489
  1417
lemmas minus_le_iff_numeral [no_atp] =
haftmann@54489
  1418
  minus_le_iff [of _ "numeral v"] for v
haftmann@30652
  1419
haftmann@54489
  1420
lemmas less_minus_iff_numeral [no_atp] =
haftmann@54489
  1421
  less_minus_iff [of "numeral v"] for v
haftmann@30652
  1422
haftmann@54489
  1423
lemmas minus_less_iff_numeral [no_atp] =
haftmann@54489
  1424
  minus_less_iff [of _ "numeral v"] for v
haftmann@30652
  1425
wenzelm@63652
  1426
(* FIXME maybe simproc *)
haftmann@30652
  1427
haftmann@30652
  1428
wenzelm@61799
  1429
text \<open>Cancellation of constant factors in comparisons (\<open><\<close> and \<open>\<le>\<close>)\<close>
haftmann@30652
  1430
huffman@47108
  1431
lemmas mult_less_cancel_left_numeral [simp, no_atp] = mult_less_cancel_left [of "numeral v"] for v
huffman@47108
  1432
lemmas mult_less_cancel_right_numeral [simp, no_atp] = mult_less_cancel_right [of _ "numeral v"] for v
huffman@47108
  1433
lemmas mult_le_cancel_left_numeral [simp, no_atp] = mult_le_cancel_left [of "numeral v"] for v
huffman@47108
  1434
lemmas mult_le_cancel_right_numeral [simp, no_atp] = mult_le_cancel_right [of _ "numeral v"] for v
haftmann@30652
  1435
haftmann@30652
  1436
wenzelm@61799
  1437
text \<open>Multiplying out constant divisors in comparisons (\<open><\<close>, \<open>\<le>\<close> and \<open>=\<close>)\<close>
haftmann@30652
  1438
lp15@61738
  1439
named_theorems divide_const_simps "simplification rules to simplify comparisons involving constant divisors"
lp15@61738
  1440
lp15@61738
  1441
lemmas le_divide_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1442
  pos_le_divide_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1443
  neg_le_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1444
lp15@61738
  1445
lemmas divide_le_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1446
  pos_divide_le_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1447
  neg_divide_le_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1448
lp15@61738
  1449
lemmas less_divide_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1450
  pos_less_divide_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1451
  neg_less_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
haftmann@30652
  1452
lp15@61738
  1453
lemmas divide_less_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1454
  pos_divide_less_eq [of "numeral w", OF zero_less_numeral]
haftmann@54489
  1455
  neg_divide_less_eq [of "- numeral w", OF neg_numeral_less_zero] for w
huffman@47108
  1456
lp15@61738
  1457
lemmas eq_divide_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1458
  eq_divide_eq [of _ _ "numeral w"]
haftmann@54489
  1459
  eq_divide_eq [of _ _ "- numeral w"] for w
huffman@47108
  1460
lp15@61738
  1461
lemmas divide_eq_eq_numeral1 [simp,divide_const_simps] =
huffman@47108
  1462
  divide_eq_eq [of _ "numeral w"]
haftmann@54489
  1463
  divide_eq_eq [of _ "- numeral w"] for w
haftmann@54489
  1464
haftmann@30652
  1465
wenzelm@63652
  1466
subsubsection \<open>Optional Simplification Rules Involving Constants\<close>
haftmann@30652
  1467
wenzelm@63652
  1468
text \<open>Simplify quotients that are compared with a literal constant.\<close>
haftmann@30652
  1469
lp15@61738
  1470
lemmas le_divide_eq_numeral [divide_const_simps] =
huffman@47108
  1471
  le_divide_eq [of "numeral w"]
haftmann@54489
  1472
  le_divide_eq [of "- numeral w"] for w
huffman@47108
  1473
lp15@61738
  1474
lemmas divide_le_eq_numeral [divide_const_simps] =
huffman@47108
  1475
  divide_le_eq [of _ _ "numeral w"]
haftmann@54489
  1476
  divide_le_eq [of _ _ "- numeral w"] for w
huffman@47108
  1477
lp15@61738
  1478
lemmas less_divide_eq_numeral [divide_const_simps] =
huffman@47108
  1479
  less_divide_eq [of "numeral w"]
haftmann@54489
  1480
  less_divide_eq [of "- numeral w"] for w
huffman@47108
  1481
lp15@61738
  1482
lemmas divide_less_eq_numeral [divide_const_simps] =
huffman@47108
  1483
  divide_less_eq [of _ _ "numeral w"]
haftmann@54489
  1484
  divide_less_eq [of _ _ "- numeral w"] for w
huffman@47108
  1485
lp15@61738
  1486
lemmas eq_divide_eq_numeral [divide_const_simps] =
huffman@47108
  1487
  eq_divide_eq [of "numeral w"]
haftmann@54489
  1488
  eq_divide_eq [of "- numeral w"] for w
huffman@47108
  1489
lp15@61738
  1490
lemmas divide_eq_eq_numeral [divide_const_simps] =
huffman@47108
  1491
  divide_eq_eq [of _ _ "numeral w"]
haftmann@54489
  1492
  divide_eq_eq [of _ _ "- numeral w"] for w
haftmann@30652
  1493
haftmann@30652
  1494
wenzelm@63652
  1495
text \<open>Not good as automatic simprules because they cause case splits.\<close>
wenzelm@63652
  1496
lemmas [divide_const_simps] =
wenzelm@63652
  1497
  le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
haftmann@30652
  1498
haftmann@30652
  1499
wenzelm@60758
  1500
subsection \<open>The divides relation\<close>
haftmann@33320
  1501
wenzelm@63652
  1502
lemma zdvd_antisym_nonneg: "0 \<le> m \<Longrightarrow> 0 \<le> n \<Longrightarrow> m dvd n \<Longrightarrow> n dvd m \<Longrightarrow> m = n"
wenzelm@63652
  1503
  for m n :: int
wenzelm@63652
  1504
  by (auto simp add: dvd_def mult.assoc zero_le_mult_iff zmult_eq_1_iff)
haftmann@33320
  1505
wenzelm@63652
  1506
lemma zdvd_antisym_abs:
wenzelm@63652
  1507
  fixes a b :: int
wenzelm@63652
  1508
  assumes "a dvd b" and "b dvd a"
haftmann@33320
  1509
  shows "\<bar>a\<bar> = \<bar>b\<bar>"
wenzelm@63652
  1510
proof (cases "a = 0")
wenzelm@63652
  1511
  case True
wenzelm@63652
  1512
  with assms show ?thesis by simp
nipkow@33657
  1513
next
wenzelm@63652
  1514
  case False
wenzelm@63652
  1515
  from \<open>a dvd b\<close> obtain k where k: "b = a * k"
wenzelm@63652
  1516
    unfolding dvd_def by blast
wenzelm@63652
  1517
  from \<open>b dvd a\<close> obtain k' where k': "a = b * k'"
wenzelm@63652
  1518
    unfolding dvd_def by blast
wenzelm@63652
  1519
  from k k' have "a = a * k * k'" by simp
wenzelm@63652
  1520
  with mult_cancel_left1[where c="a" and b="k*k'"] have kk': "k * k' = 1"
wenzelm@63652
  1521
    using \<open>a \<noteq> 0\<close> by (simp add: mult.assoc)
wenzelm@63652
  1522
  then have "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1"
wenzelm@63652
  1523
    by (simp add: zmult_eq_1_iff)
wenzelm@63652
  1524
  with k k' show ?thesis by auto
haftmann@33320
  1525
qed
haftmann@33320
  1526
wenzelm@63652
  1527
lemma zdvd_zdiffD: "k dvd m - n \<Longrightarrow> k dvd n \<Longrightarrow> k dvd m"
wenzelm@63652
  1528
  for k m n :: int
lp15@60162
  1529
  using dvd_add_right_iff [of k "- n" m] by simp
haftmann@33320
  1530
wenzelm@63652
  1531
lemma zdvd_reduce: "k dvd n + k * m \<longleftrightarrow> k dvd n"
wenzelm@63652
  1532
  for k m n :: int
haftmann@58649
  1533
  using dvd_add_times_triv_right_iff [of k n m] by (simp add: ac_simps)
haftmann@33320
  1534
haftmann@33320
  1535
lemma dvd_imp_le_int:
haftmann@33320
  1536
  fixes d i :: int
haftmann@33320
  1537
  assumes "i \<noteq> 0" and "d dvd i"
haftmann@33320
  1538
  shows "\<bar>d\<bar> \<le> \<bar>i\<bar>"
haftmann@33320
  1539
proof -
wenzelm@60758
  1540
  from \<open>d dvd i\<close> obtain k where "i = d * k" ..
wenzelm@60758
  1541
  with \<open>i \<noteq> 0\<close> have "k \<noteq> 0" by auto
haftmann@33320
  1542
  then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto
haftmann@33320
  1543
  then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono)
wenzelm@60758
  1544
  with \<open>i = d * k\<close> show ?thesis by (simp add: abs_mult)
haftmann@33320
  1545
qed
haftmann@33320
  1546
haftmann@33320
  1547
lemma zdvd_not_zless:
haftmann@33320
  1548
  fixes m n :: int
haftmann@33320
  1549
  assumes "0 < m" and "m < n"
haftmann@33320
  1550
  shows "\<not> n dvd m"
haftmann@33320
  1551
proof
haftmann@33320
  1552
  from assms have "0 < n" by auto
haftmann@33320
  1553
  assume "n dvd m" then obtain k where k: "m = n * k" ..
wenzelm@60758
  1554
  with \<open>0 < m\<close> have "0 < n * k" by auto
wenzelm@60758
  1555
  with \<open>0 < n\<close> have "0 < k" by (simp add: zero_less_mult_iff)
wenzelm@60758
  1556
  with k \<open>0 < n\<close> \<open>m < n\<close> have "n * k < n * 1" by simp
wenzelm@60758
  1557
  with \<open>0 < n\<close> \<open>0 < k\<close> show False unfolding mult_less_cancel_left by auto
haftmann@33320
  1558
qed
haftmann@33320
  1559
wenzelm@63652
  1560
lemma zdvd_mult_cancel:
wenzelm@63652
  1561
  fixes k m n :: int
wenzelm@63652
  1562
  assumes d: "k * m dvd k * n"
wenzelm@63652
  1563
    and "k \<noteq> 0"
haftmann@33320
  1564
  shows "m dvd n"
wenzelm@63652
  1565
proof -
wenzelm@63652
  1566
  from d obtain h where h: "k * n = k * m * h"
wenzelm@63652
  1567
    unfolding dvd_def by blast
wenzelm@63652
  1568
  have "n = m * h"
wenzelm@63652
  1569
  proof (rule ccontr)
wenzelm@63652
  1570
    assume "\<not> ?thesis"
wenzelm@63652
  1571
    with \<open>k \<noteq> 0\<close> have "k * n \<noteq> k * (m * h)" by simp
wenzelm@63652
  1572
    with h show False
wenzelm@63652
  1573
      by (simp add: mult.assoc)
wenzelm@63652
  1574
  qed
wenzelm@63652
  1575
  then show ?thesis by simp
haftmann@33320
  1576
qed
haftmann@33320
  1577
wenzelm@63652
  1578
theorem zdvd_int: "x dvd y \<longleftrightarrow> int x dvd int y"
haftmann@33320
  1579
proof -
wenzelm@63652
  1580
  have "x dvd y" if "int y = int x * k" for k
wenzelm@63652
  1581
  proof (cases k)
wenzelm@63652
  1582
    case (nonneg n)
wenzelm@63652
  1583
    with that have "y = x * n"
wenzelm@63652
  1584
      by (simp del: of_nat_mult add: of_nat_mult [symmetric])
wenzelm@63652
  1585
    then show ?thesis ..
wenzelm@63652
  1586
  next
wenzelm@63652
  1587
    case (neg n)
wenzelm@63652
  1588
    with that have "int y = int x * (- int (Suc n))"
wenzelm@63652
  1589
      by simp
wenzelm@63652
  1590
    also have "\<dots> = - (int x * int (Suc n))"
wenzelm@63652
  1591
      by (simp only: mult_minus_right)
wenzelm@63652
  1592
    also have "\<dots> = - int (x * Suc n)"
wenzelm@63652
  1593
      by (simp only: of_nat_mult [symmetric])
wenzelm@63652
  1594
    finally have "- int (x * Suc n) = int y" ..
wenzelm@63652
  1595
    then show ?thesis
wenzelm@63652
  1596
      by (simp only: negative_eq_positive) auto
haftmann@33320
  1597
  qed
wenzelm@63652
  1598
  then show ?thesis
wenzelm@63652
  1599
    by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult)
haftmann@33320
  1600
qed
haftmann@33320
  1601
wenzelm@63652
  1602
lemma zdvd1_eq[simp]: "x dvd 1 \<longleftrightarrow> \<bar>x\<bar> = 1"
wenzelm@63652
  1603
  (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@63652
  1604
  for x :: int
haftmann@33320
  1605
proof
wenzelm@63652
  1606
  assume ?lhs
wenzelm@63652
  1607
  then have "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
wenzelm@63652
  1608
  then have "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
wenzelm@63652
  1609
  then have "nat \<bar>x\<bar> = 1" by simp
wenzelm@63652
  1610
  then show ?rhs by (cases "x < 0") auto
haftmann@33320
  1611
next
wenzelm@63652
  1612
  assume ?rhs
wenzelm@63652
  1613
  then have "x = 1 \<or> x = - 1" by auto
wenzelm@63652
  1614
  then show ?lhs by (auto intro: dvdI)
haftmann@33320
  1615
qed
haftmann@33320
  1616
lp15@60162
  1617
lemma zdvd_mult_cancel1:
wenzelm@63652
  1618
  fixes m :: int
wenzelm@63652
  1619
  assumes mp: "m \<noteq> 0"
wenzelm@63652
  1620
  shows "m * n dvd m \<longleftrightarrow> \<bar>n\<bar> = 1"
wenzelm@63652
  1621
    (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@33320
  1622
proof
wenzelm@63652
  1623
  assume ?rhs
wenzelm@63652
  1624
  then show ?lhs
wenzelm@63652
  1625
    by (cases "n > 0") (auto simp add: minus_equation_iff)
haftmann@33320
  1626
next
wenzelm@63652
  1627
  assume ?lhs
wenzelm@63652
  1628
  then have "m * n dvd m * 1" by simp
wenzelm@63652
  1629
  from zdvd_mult_cancel[OF this mp] show ?rhs
wenzelm@63652
  1630
    by (simp only: zdvd1_eq)
haftmann@33320
  1631
qed
haftmann@33320
  1632
wenzelm@63652
  1633
lemma int_dvd_iff: "int m dvd z \<longleftrightarrow> m dvd nat \<bar>z\<bar>"
wenzelm@63652
  1634
  by (cases "z \<ge> 0") (simp_all add: zdvd_int)
haftmann@33320
  1635
wenzelm@63652
  1636
lemma dvd_int_iff: "z dvd int m \<longleftrightarrow> nat \<bar>z\<bar> dvd m"
wenzelm@63652
  1637
  by (cases "z \<ge> 0") (simp_all add: zdvd_int)
haftmann@58650
  1638
wenzelm@63652
  1639
lemma dvd_int_unfold_dvd_nat: "k dvd l \<longleftrightarrow> nat \<bar>k\<bar> dvd nat \<bar>l\<bar>"
wenzelm@63652
  1640
  by (simp add: dvd_int_iff [symmetric])
wenzelm@63652
  1641
wenzelm@63652
  1642
lemma nat_dvd_iff: "nat z dvd m \<longleftrightarrow> (if 0 \<le> z then z dvd int m else m = 0)"
haftmann@33320
  1643
  by (auto simp add: dvd_int_iff)
haftmann@33320
  1644
wenzelm@63652
  1645
lemma eq_nat_nat_iff: "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'"
haftmann@33341
  1646
  by (auto elim!: nonneg_eq_int)
haftmann@33341
  1647
wenzelm@63652
  1648
lemma nat_power_eq: "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n"
haftmann@33341
  1649
  by (induct n) (simp_all add: nat_mult_distrib)
haftmann@33341
  1650
wenzelm@63652
  1651
lemma zdvd_imp_le: "z dvd n \<Longrightarrow> 0 < n \<Longrightarrow> z \<le> n"
wenzelm@63652
  1652
  for n z :: int
wenzelm@42676
  1653
  apply (cases n)
wenzelm@63652
  1654
   apply (auto simp add: dvd_int_iff)
wenzelm@42676
  1655
  apply (cases z)
wenzelm@63652
  1656
   apply (auto simp add: dvd_imp_le)
haftmann@33320
  1657
  done
haftmann@33320
  1658
haftmann@36749
  1659
lemma zdvd_period:
haftmann@36749
  1660
  fixes a d :: int
haftmann@36749
  1661
  assumes "a dvd d"
haftmann@36749
  1662
  shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"
wenzelm@63652
  1663
    (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@36749
  1664
proof -
haftmann@66816
  1665
  from assms have "a dvd (x + t) \<longleftrightarrow> a dvd ((x + t) + c * d)"
haftmann@66816
  1666
    by (simp add: dvd_add_left_iff)
haftmann@66816
  1667
  then show ?thesis
haftmann@66816
  1668
    by (simp add: ac_simps)
haftmann@36749
  1669
qed
haftmann@36749
  1670
haftmann@33320
  1671
wenzelm@60758
  1672
subsection \<open>Finiteness of intervals\<close>
bulwahn@46756
  1673
wenzelm@63652
  1674
lemma finite_interval_int1 [iff]: "finite {i :: int. a \<le> i \<and> i \<le> b}"
wenzelm@63652
  1675
proof (cases "a \<le> b")
bulwahn@46756
  1676
  case True
wenzelm@63652
  1677
  then show ?thesis
bulwahn@46756
  1678
  proof (induct b rule: int_ge_induct)
bulwahn@46756
  1679
    case base
wenzelm@63652
  1680
    have "{i. a \<le> i \<and> i \<le> a} = {a}" by auto
wenzelm@63652
  1681
    then show ?case by simp
bulwahn@46756
  1682
  next
bulwahn@46756
  1683
    case (step b)
wenzelm@63652
  1684
    then have "{i. a \<le> i \<and> i \<le> b + 1} = {i. a \<le> i \<and> i \<le> b} \<union> {b + 1}" by auto
wenzelm@63652
  1685
    with step show ?case by simp
bulwahn@46756
  1686
  qed
bulwahn@46756
  1687
next
wenzelm@63652
  1688
  case False
wenzelm@63652
  1689
  then show ?thesis
bulwahn@46756
  1690
    by (metis (lifting, no_types) Collect_empty_eq finite.emptyI order_trans)
bulwahn@46756
  1691
qed
bulwahn@46756
  1692
wenzelm@63652
  1693
lemma finite_interval_int2 [iff]: "finite {i :: int. a \<le> i \<and> i < b}"
wenzelm@63652
  1694
  by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1695
wenzelm@63652
  1696
lemma finite_interval_int3 [iff]: "finite {i :: int. a < i \<and> i \<le> b}"
wenzelm@63652
  1697
  by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1698
wenzelm@63652
  1699
lemma finite_interval_int4 [iff]: "finite {i :: int. a < i \<and> i < b}"
wenzelm@63652
  1700
  by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
bulwahn@46756
  1701
bulwahn@46756
  1702
wenzelm@60758
  1703
subsection \<open>Configuration of the code generator\<close>
haftmann@25919
  1704
wenzelm@60758
  1705
text \<open>Constructors\<close>
huffman@47108
  1706
wenzelm@63652
  1707
definition Pos :: "num \<Rightarrow> int"
wenzelm@63652
  1708
  where [simp, code_abbrev]: "Pos = numeral"
huffman@47108
  1709
wenzelm@63652
  1710
definition Neg :: "num \<Rightarrow> int"
wenzelm@63652
  1711
  where [simp, code_abbrev]: "Neg n = - (Pos n)"
huffman@47108
  1712
huffman@47108
  1713
code_datatype "0::int" Pos Neg
huffman@47108
  1714
huffman@47108
  1715
wenzelm@63652
  1716
text \<open>Auxiliary operations.\<close>
huffman@47108
  1717
wenzelm@63652
  1718
definition dup :: "int \<Rightarrow> int"
wenzelm@63652
  1719
  where [simp]: "dup k = k + k"
haftmann@26507
  1720
huffman@47108
  1721
lemma dup_code [code]:
huffman@47108
  1722
  "dup 0 = 0"
huffman@47108
  1723
  "dup (Pos n) = Pos (Num.Bit0 n)"
huffman@47108
  1724
  "dup (Neg n) = Neg (Num.Bit0 n)"
huffman@47108
  1725
  by (simp_all add: numeral_Bit0)
huffman@47108
  1726
wenzelm@63652
  1727
definition sub :: "num \<Rightarrow> num \<Rightarrow> int"
wenzelm@63652
  1728
  where [simp]: "sub m n = numeral m - numeral n"
haftmann@26507
  1729
huffman@47108
  1730
lemma sub_code [code]:
huffman@47108
  1731
  "sub Num.One Num.One = 0"
huffman@47108
  1732
  "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
huffman@47108
  1733
  "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
huffman@47108
  1734
  "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
huffman@47108
  1735
  "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
huffman@47108
  1736
  "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
huffman@47108
  1737
  "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
huffman@47108
  1738
  "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
huffman@47108
  1739
  "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
boehmes@66035
  1740
  by (simp_all only: sub_def dup_def numeral.simps Pos_def Neg_def numeral_BitM)
huffman@47108
  1741
wenzelm@63652
  1742
text \<open>Implementations.\<close>
huffman@47108
  1743
haftmann@64996
  1744
lemma one_int_code [code]: "1 = Pos Num.One"
huffman@47108
  1745
  by simp
huffman@47108
  1746
huffman@47108
  1747
lemma plus_int_code [code]:
wenzelm@63652
  1748
  "k + 0 = k"
wenzelm@63652
  1749
  "0 + l = l"
huffman@47108
  1750
  "Pos m + Pos n = Pos (m + n)"
huffman@47108
  1751
  "Pos m + Neg n = sub m n"
huffman@47108
  1752
  "Neg m + Pos n = sub n m"
huffman@47108
  1753
  "Neg m + Neg n = Neg (m + n)"
wenzelm@63652
  1754
  for k l :: int
huffman@47108
  1755
  by simp_all
haftmann@26507
  1756
huffman@47108
  1757
lemma uminus_int_code [code]:
huffman@47108
  1758
  "uminus 0 = (0::int)"
huffman@47108
  1759
  "uminus (Pos m) = Neg m"
huffman@47108
  1760
  "uminus (Neg m) = Pos m"
huffman@47108
  1761
  by simp_all
huffman@47108
  1762
huffman@47108
  1763
lemma minus_int_code [code]:
wenzelm@63652
  1764
  "k - 0 = k"
wenzelm@63652
  1765
  "0 - l = uminus l"
huffman@47108
  1766
  "Pos m - Pos n = sub m n"
huffman@47108
  1767
  "Pos m - Neg n = Pos (m + n)"
huffman@47108
  1768
  "Neg m - Pos n = Neg (m + n)"
huffman@47108
  1769
  "Neg m - Neg n = sub n m"
wenzelm@63652
  1770
  for k l :: int
huffman@47108
  1771
  by simp_all
huffman@47108
  1772
huffman@47108
  1773
lemma times_int_code [code]:
wenzelm@63652
  1774
  "k * 0 = 0"
wenzelm@63652
  1775
  "0 * l = 0"
huffman@47108
  1776
  "Pos m * Pos n = Pos (m * n)"
huffman@47108
  1777
  "Pos m * Neg n = Neg (m * n)"
huffman@47108
  1778
  "Neg m * Pos n = Neg (m * n)"
huffman@47108
  1779
  "Neg m * Neg n = Pos (m * n)"
wenzelm@63652
  1780
  for k l :: int
huffman@47108
  1781
  by simp_all
haftmann@26507
  1782
haftmann@38857
  1783
instantiation int :: equal
haftmann@26507
  1784
begin
haftmann@26507
  1785
wenzelm@63652
  1786
definition "HOL.equal k l \<longleftrightarrow> k = (l::int)"
haftmann@38857
  1787
wenzelm@61169
  1788
instance
wenzelm@61169
  1789
  by standard (rule equal_int_def)
haftmann@26507
  1790
haftmann@26507
  1791
end
haftmann@26507
  1792
huffman@47108
  1793
lemma equal_int_code [code]:
huffman@47108
  1794
  "HOL.equal 0 (0::int) \<longleftrightarrow> True"
huffman@47108
  1795
  "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
huffman@47108
  1796
  "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
huffman@47108
  1797
  "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
huffman@47108
  1798
  "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
huffman@47108
  1799
  "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
huffman@47108
  1800
  "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
huffman@47108
  1801
  "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
huffman@47108
  1802
  "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
huffman@47108
  1803
  by (auto simp add: equal)
haftmann@26507
  1804
wenzelm@63652
  1805
lemma equal_int_refl [code nbe]: "HOL.equal k k \<longleftrightarrow> True"
wenzelm@63652
  1806
  for k :: int
huffman@47108
  1807
  by (fact equal_refl)
haftmann@26507
  1808
haftmann@28562
  1809
lemma less_eq_int_code [code]:
huffman@47108
  1810
  "0 \<le> (0::int) \<longleftrightarrow> True"
huffman@47108
  1811
  "0 \<le> Pos l \<longleftrightarrow> True"
huffman@47108
  1812
  "0 \<le> Neg l \<longleftrightarrow> False"
huffman@47108
  1813
  "Pos k \<le> 0 \<longleftrightarrow> False"
huffman@47108
  1814
  "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
huffman@47108
  1815
  "Pos k \<le> Neg l \<longleftrightarrow> False"
huffman@47108
  1816
  "Neg k \<le> 0 \<longleftrightarrow> True"
huffman@47108
  1817
  "Neg k \<le> Pos l \<longleftrightarrow> True"
huffman@47108
  1818
  "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
huffman@28958
  1819
  by simp_all
haftmann@26507
  1820
haftmann@28562
  1821
lemma less_int_code [code]:
huffman@47108
  1822
  "0 < (0::int) \<longleftrightarrow> False"
huffman@47108
  1823
  "0 < Pos l \<longleftrightarrow> True"
huffman@47108
  1824
  "0 < Neg l \<longleftrightarrow> False"
huffman@47108
  1825
  "Pos k < 0 \<longleftrightarrow> False"
huffman@47108
  1826
  "Pos k < Pos l \<longleftrightarrow> k < l"
huffman@47108
  1827
  "Pos k < Neg l \<longleftrightarrow> False"
huffman@47108
  1828
  "Neg k < 0 \<longleftrightarrow> True"
huffman@47108
  1829
  "Neg k < Pos l \<longleftrightarrow> True"
huffman@47108
  1830
  "Neg k < Neg l \<longleftrightarrow> l < k"
huffman@28958
  1831
  by simp_all
haftmann@25919
  1832
huffman@47108
  1833
lemma nat_code [code]:
huffman@47108
  1834
  "nat (Int.Neg k) = 0"
huffman@47108
  1835
  "nat 0 = 0"
huffman@47108
  1836
  "nat (Int.Pos k) = nat_of_num k"
haftmann@54489
  1837
  by (simp_all add: nat_of_num_numeral)
haftmann@25928
  1838
huffman@47108
  1839
lemma (in ring_1) of_int_code [code]:
haftmann@54489
  1840
  "of_int (Int.Neg k) = - numeral k"
huffman@47108
  1841
  "of_int 0 = 0"
huffman@47108
  1842
  "of_int (Int.Pos k) = numeral k"
huffman@47108
  1843
  by simp_all
haftmann@25919
  1844
huffman@47108
  1845
wenzelm@63652
  1846
text \<open>Serializer setup.\<close>
haftmann@25919
  1847
haftmann@52435
  1848
code_identifier
haftmann@52435
  1849
  code_module Int \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@25919
  1850
haftmann@25919
  1851
quickcheck_params [default_type = int]
haftmann@25919
  1852
huffman@47108
  1853
hide_const (open) Pos Neg sub dup
haftmann@25919
  1854
haftmann@25919
  1855
wenzelm@61799
  1856
text \<open>De-register \<open>int\<close> as a quotient type:\<close>
huffman@48045
  1857
kuncar@53652
  1858
lifting_update int.lifting
kuncar@53652
  1859
lifting_forget int.lifting
huffman@48045
  1860
haftmann@25919
  1861
end