src/HOL/Rings.thy
author hoelzl
Fri Feb 19 13:40:50 2016 +0100 (2016-02-19)
changeset 62378 85ed00c1fe7c
parent 62377 ace69956d018
child 62390 842917225d56
permissions -rw-r--r--
generalize more theorems to support enat and ennreal
haftmann@35050
     1
(*  Title:      HOL/Rings.thy
wenzelm@32960
     2
    Author:     Gertrud Bauer
wenzelm@32960
     3
    Author:     Steven Obua
wenzelm@32960
     4
    Author:     Tobias Nipkow
wenzelm@32960
     5
    Author:     Lawrence C Paulson
wenzelm@32960
     6
    Author:     Markus Wenzel
wenzelm@32960
     7
    Author:     Jeremy Avigad
paulson@14265
     8
*)
paulson@14265
     9
wenzelm@60758
    10
section \<open>Rings\<close>
paulson@14265
    11
haftmann@35050
    12
theory Rings
haftmann@62366
    13
imports Groups Set
nipkow@15131
    14
begin
paulson@14504
    15
haftmann@22390
    16
class semiring = ab_semigroup_add + semigroup_mult +
hoelzl@58776
    17
  assumes distrib_right[algebra_simps]: "(a + b) * c = a * c + b * c"
hoelzl@58776
    18
  assumes distrib_left[algebra_simps]: "a * (b + c) = a * b + a * c"
haftmann@25152
    19
begin
haftmann@25152
    20
wenzelm@61799
    21
text\<open>For the \<open>combine_numerals\<close> simproc\<close>
haftmann@25152
    22
lemma combine_common_factor:
haftmann@25152
    23
  "a * e + (b * e + c) = (a + b) * e + c"
haftmann@57514
    24
by (simp add: distrib_right ac_simps)
haftmann@25152
    25
haftmann@25152
    26
end
paulson@14504
    27
haftmann@22390
    28
class mult_zero = times + zero +
haftmann@25062
    29
  assumes mult_zero_left [simp]: "0 * a = 0"
haftmann@25062
    30
  assumes mult_zero_right [simp]: "a * 0 = 0"
haftmann@58195
    31
begin
haftmann@58195
    32
haftmann@58195
    33
lemma mult_not_zero:
haftmann@58195
    34
  "a * b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<and> b \<noteq> 0"
haftmann@58195
    35
  by auto
haftmann@58195
    36
haftmann@58195
    37
end
krauss@21199
    38
haftmann@58198
    39
class semiring_0 = semiring + comm_monoid_add + mult_zero
haftmann@58198
    40
huffman@29904
    41
class semiring_0_cancel = semiring + cancel_comm_monoid_add
haftmann@25186
    42
begin
paulson@14504
    43
haftmann@25186
    44
subclass semiring_0
haftmann@28823
    45
proof
krauss@21199
    46
  fix a :: 'a
webertj@49962
    47
  have "0 * a + 0 * a = 0 * a + 0" by (simp add: distrib_right [symmetric])
nipkow@29667
    48
  thus "0 * a = 0" by (simp only: add_left_cancel)
haftmann@25152
    49
next
haftmann@25152
    50
  fix a :: 'a
webertj@49962
    51
  have "a * 0 + a * 0 = a * 0 + 0" by (simp add: distrib_left [symmetric])
nipkow@29667
    52
  thus "a * 0 = 0" by (simp only: add_left_cancel)
krauss@21199
    53
qed
obua@14940
    54
haftmann@25186
    55
end
haftmann@25152
    56
haftmann@22390
    57
class comm_semiring = ab_semigroup_add + ab_semigroup_mult +
haftmann@25062
    58
  assumes distrib: "(a + b) * c = a * c + b * c"
haftmann@25152
    59
begin
paulson@14504
    60
haftmann@25152
    61
subclass semiring
haftmann@28823
    62
proof
obua@14738
    63
  fix a b c :: 'a
obua@14738
    64
  show "(a + b) * c = a * c + b * c" by (simp add: distrib)
haftmann@57514
    65
  have "a * (b + c) = (b + c) * a" by (simp add: ac_simps)
obua@14738
    66
  also have "... = b * a + c * a" by (simp only: distrib)
haftmann@57514
    67
  also have "... = a * b + a * c" by (simp add: ac_simps)
obua@14738
    68
  finally show "a * (b + c) = a * b + a * c" by blast
paulson@14504
    69
qed
paulson@14504
    70
haftmann@25152
    71
end
paulson@14504
    72
haftmann@25152
    73
class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero
haftmann@25152
    74
begin
haftmann@25152
    75
huffman@27516
    76
subclass semiring_0 ..
haftmann@25152
    77
haftmann@25152
    78
end
paulson@14504
    79
huffman@29904
    80
class comm_semiring_0_cancel = comm_semiring + cancel_comm_monoid_add
haftmann@25186
    81
begin
obua@14940
    82
huffman@27516
    83
subclass semiring_0_cancel ..
obua@14940
    84
huffman@28141
    85
subclass comm_semiring_0 ..
huffman@28141
    86
haftmann@25186
    87
end
krauss@21199
    88
haftmann@22390
    89
class zero_neq_one = zero + one +
haftmann@25062
    90
  assumes zero_neq_one [simp]: "0 \<noteq> 1"
haftmann@26193
    91
begin
haftmann@26193
    92
haftmann@26193
    93
lemma one_neq_zero [simp]: "1 \<noteq> 0"
nipkow@29667
    94
by (rule not_sym) (rule zero_neq_one)
haftmann@26193
    95
haftmann@54225
    96
definition of_bool :: "bool \<Rightarrow> 'a"
haftmann@54225
    97
where
lp15@60562
    98
  "of_bool p = (if p then 1 else 0)"
haftmann@54225
    99
haftmann@54225
   100
lemma of_bool_eq [simp, code]:
haftmann@54225
   101
  "of_bool False = 0"
haftmann@54225
   102
  "of_bool True = 1"
haftmann@54225
   103
  by (simp_all add: of_bool_def)
haftmann@54225
   104
haftmann@54225
   105
lemma of_bool_eq_iff:
haftmann@54225
   106
  "of_bool p = of_bool q \<longleftrightarrow> p = q"
haftmann@54225
   107
  by (simp add: of_bool_def)
haftmann@54225
   108
haftmann@55187
   109
lemma split_of_bool [split]:
haftmann@55187
   110
  "P (of_bool p) \<longleftrightarrow> (p \<longrightarrow> P 1) \<and> (\<not> p \<longrightarrow> P 0)"
haftmann@55187
   111
  by (cases p) simp_all
haftmann@55187
   112
haftmann@55187
   113
lemma split_of_bool_asm:
haftmann@55187
   114
  "P (of_bool p) \<longleftrightarrow> \<not> (p \<and> \<not> P 1 \<or> \<not> p \<and> \<not> P 0)"
haftmann@55187
   115
  by (cases p) simp_all
lp15@60562
   116
lp15@60562
   117
end
paulson@14265
   118
haftmann@22390
   119
class semiring_1 = zero_neq_one + semiring_0 + monoid_mult
paulson@14504
   120
wenzelm@60758
   121
text \<open>Abstract divisibility\<close>
haftmann@27651
   122
haftmann@27651
   123
class dvd = times
haftmann@27651
   124
begin
haftmann@27651
   125
nipkow@50420
   126
definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "dvd" 50) where
haftmann@37767
   127
  "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"
haftmann@27651
   128
haftmann@27651
   129
lemma dvdI [intro?]: "a = b * k \<Longrightarrow> b dvd a"
haftmann@27651
   130
  unfolding dvd_def ..
haftmann@27651
   131
haftmann@27651
   132
lemma dvdE [elim?]: "b dvd a \<Longrightarrow> (\<And>k. a = b * k \<Longrightarrow> P) \<Longrightarrow> P"
lp15@60562
   133
  unfolding dvd_def by blast
haftmann@27651
   134
haftmann@27651
   135
end
haftmann@27651
   136
haftmann@59009
   137
context comm_monoid_mult
haftmann@25152
   138
begin
obua@14738
   139
haftmann@59009
   140
subclass dvd .
haftmann@25152
   141
haftmann@59009
   142
lemma dvd_refl [simp]:
haftmann@59009
   143
  "a dvd a"
haftmann@28559
   144
proof
haftmann@28559
   145
  show "a = a * 1" by simp
haftmann@27651
   146
qed
haftmann@27651
   147
haftmann@62349
   148
lemma dvd_trans [trans]:
haftmann@27651
   149
  assumes "a dvd b" and "b dvd c"
haftmann@27651
   150
  shows "a dvd c"
haftmann@27651
   151
proof -
haftmann@28559
   152
  from assms obtain v where "b = a * v" by (auto elim!: dvdE)
haftmann@28559
   153
  moreover from assms obtain w where "c = b * w" by (auto elim!: dvdE)
haftmann@57512
   154
  ultimately have "c = a * (v * w)" by (simp add: mult.assoc)
haftmann@28559
   155
  then show ?thesis ..
haftmann@27651
   156
qed
haftmann@27651
   157
haftmann@62366
   158
lemma subset_divisors_dvd:
haftmann@62366
   159
  "{c. c dvd a} \<subseteq> {c. c dvd b} \<longleftrightarrow> a dvd b"
haftmann@62366
   160
  by (auto simp add: subset_iff intro: dvd_trans)
haftmann@62366
   161
haftmann@62366
   162
lemma strict_subset_divisors_dvd:
haftmann@62366
   163
  "{c. c dvd a} \<subset> {c. c dvd b} \<longleftrightarrow> a dvd b \<and> \<not> b dvd a"
haftmann@62366
   164
  by (auto simp add: subset_iff intro: dvd_trans)
haftmann@62366
   165
haftmann@59009
   166
lemma one_dvd [simp]:
haftmann@59009
   167
  "1 dvd a"
haftmann@59009
   168
  by (auto intro!: dvdI)
haftmann@28559
   169
haftmann@59009
   170
lemma dvd_mult [simp]:
haftmann@59009
   171
  "a dvd c \<Longrightarrow> a dvd (b * c)"
haftmann@59009
   172
  by (auto intro!: mult.left_commute dvdI elim!: dvdE)
haftmann@27651
   173
haftmann@59009
   174
lemma dvd_mult2 [simp]:
haftmann@59009
   175
  "a dvd b \<Longrightarrow> a dvd (b * c)"
lp15@60562
   176
  using dvd_mult [of a b c] by (simp add: ac_simps)
haftmann@27651
   177
haftmann@59009
   178
lemma dvd_triv_right [simp]:
haftmann@59009
   179
  "a dvd b * a"
haftmann@59009
   180
  by (rule dvd_mult) (rule dvd_refl)
haftmann@27651
   181
haftmann@59009
   182
lemma dvd_triv_left [simp]:
haftmann@59009
   183
  "a dvd a * b"
haftmann@59009
   184
  by (rule dvd_mult2) (rule dvd_refl)
haftmann@27651
   185
haftmann@27651
   186
lemma mult_dvd_mono:
nipkow@30042
   187
  assumes "a dvd b"
nipkow@30042
   188
    and "c dvd d"
haftmann@27651
   189
  shows "a * c dvd b * d"
haftmann@27651
   190
proof -
wenzelm@60758
   191
  from \<open>a dvd b\<close> obtain b' where "b = a * b'" ..
wenzelm@60758
   192
  moreover from \<open>c dvd d\<close> obtain d' where "d = c * d'" ..
haftmann@57514
   193
  ultimately have "b * d = (a * c) * (b' * d')" by (simp add: ac_simps)
haftmann@27651
   194
  then show ?thesis ..
haftmann@27651
   195
qed
haftmann@27651
   196
haftmann@59009
   197
lemma dvd_mult_left:
haftmann@59009
   198
  "a * b dvd c \<Longrightarrow> a dvd c"
haftmann@59009
   199
  by (simp add: dvd_def mult.assoc) blast
haftmann@27651
   200
haftmann@59009
   201
lemma dvd_mult_right:
haftmann@59009
   202
  "a * b dvd c \<Longrightarrow> b dvd c"
haftmann@59009
   203
  using dvd_mult_left [of b a c] by (simp add: ac_simps)
lp15@60562
   204
haftmann@59009
   205
end
haftmann@59009
   206
haftmann@59009
   207
class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult
haftmann@59009
   208
begin
haftmann@59009
   209
haftmann@59009
   210
subclass semiring_1 ..
haftmann@27651
   211
haftmann@59009
   212
lemma dvd_0_left_iff [simp]:
haftmann@59009
   213
  "0 dvd a \<longleftrightarrow> a = 0"
haftmann@59009
   214
  by (auto intro: dvd_refl elim!: dvdE)
haftmann@27651
   215
haftmann@59009
   216
lemma dvd_0_right [iff]:
haftmann@59009
   217
  "a dvd 0"
haftmann@59009
   218
proof
haftmann@59009
   219
  show "0 = a * 0" by simp
haftmann@59009
   220
qed
haftmann@59009
   221
haftmann@59009
   222
lemma dvd_0_left:
haftmann@59009
   223
  "0 dvd a \<Longrightarrow> a = 0"
haftmann@59009
   224
  by simp
haftmann@59009
   225
haftmann@59009
   226
lemma dvd_add [simp]:
haftmann@59009
   227
  assumes "a dvd b" and "a dvd c"
haftmann@59009
   228
  shows "a dvd (b + c)"
haftmann@27651
   229
proof -
wenzelm@60758
   230
  from \<open>a dvd b\<close> obtain b' where "b = a * b'" ..
wenzelm@60758
   231
  moreover from \<open>a dvd c\<close> obtain c' where "c = a * c'" ..
webertj@49962
   232
  ultimately have "b + c = a * (b' + c')" by (simp add: distrib_left)
haftmann@27651
   233
  then show ?thesis ..
haftmann@27651
   234
qed
haftmann@27651
   235
haftmann@25152
   236
end
paulson@14421
   237
huffman@29904
   238
class semiring_1_cancel = semiring + cancel_comm_monoid_add
huffman@29904
   239
  + zero_neq_one + monoid_mult
haftmann@25267
   240
begin
obua@14940
   241
huffman@27516
   242
subclass semiring_0_cancel ..
haftmann@25512
   243
huffman@27516
   244
subclass semiring_1 ..
haftmann@25267
   245
haftmann@25267
   246
end
krauss@21199
   247
lp15@60562
   248
class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add +
lp15@60562
   249
                               zero_neq_one + comm_monoid_mult +
lp15@60562
   250
  assumes right_diff_distrib' [algebra_simps]: "a * (b - c) = a * b - a * c"
haftmann@25267
   251
begin
obua@14738
   252
huffman@27516
   253
subclass semiring_1_cancel ..
huffman@27516
   254
subclass comm_semiring_0_cancel ..
huffman@27516
   255
subclass comm_semiring_1 ..
haftmann@25267
   256
haftmann@59816
   257
lemma left_diff_distrib' [algebra_simps]:
haftmann@59816
   258
  "(b - c) * a = b * a - c * a"
haftmann@59816
   259
  by (simp add: algebra_simps)
haftmann@59816
   260
haftmann@59816
   261
lemma dvd_add_times_triv_left_iff [simp]:
haftmann@59816
   262
  "a dvd c * a + b \<longleftrightarrow> a dvd b"
haftmann@59816
   263
proof -
haftmann@59816
   264
  have "a dvd a * c + b \<longleftrightarrow> a dvd b" (is "?P \<longleftrightarrow> ?Q")
haftmann@59816
   265
  proof
haftmann@59816
   266
    assume ?Q then show ?P by simp
haftmann@59816
   267
  next
haftmann@59816
   268
    assume ?P
haftmann@59816
   269
    then obtain d where "a * c + b = a * d" ..
haftmann@59816
   270
    then have "a * c + b - a * c = a * d - a * c" by simp
haftmann@59816
   271
    then have "b = a * d - a * c" by simp
lp15@60562
   272
    then have "b = a * (d - c)" by (simp add: algebra_simps)
haftmann@59816
   273
    then show ?Q ..
haftmann@59816
   274
  qed
haftmann@59816
   275
  then show "a dvd c * a + b \<longleftrightarrow> a dvd b" by (simp add: ac_simps)
haftmann@59816
   276
qed
haftmann@59816
   277
haftmann@59816
   278
lemma dvd_add_times_triv_right_iff [simp]:
haftmann@59816
   279
  "a dvd b + c * a \<longleftrightarrow> a dvd b"
haftmann@59816
   280
  using dvd_add_times_triv_left_iff [of a c b] by (simp add: ac_simps)
haftmann@59816
   281
haftmann@59816
   282
lemma dvd_add_triv_left_iff [simp]:
haftmann@59816
   283
  "a dvd a + b \<longleftrightarrow> a dvd b"
haftmann@59816
   284
  using dvd_add_times_triv_left_iff [of a 1 b] by simp
haftmann@59816
   285
haftmann@59816
   286
lemma dvd_add_triv_right_iff [simp]:
haftmann@59816
   287
  "a dvd b + a \<longleftrightarrow> a dvd b"
haftmann@59816
   288
  using dvd_add_times_triv_right_iff [of a b 1] by simp
haftmann@59816
   289
haftmann@59816
   290
lemma dvd_add_right_iff:
haftmann@59816
   291
  assumes "a dvd b"
haftmann@59816
   292
  shows "a dvd b + c \<longleftrightarrow> a dvd c" (is "?P \<longleftrightarrow> ?Q")
haftmann@59816
   293
proof
haftmann@59816
   294
  assume ?P then obtain d where "b + c = a * d" ..
wenzelm@60758
   295
  moreover from \<open>a dvd b\<close> obtain e where "b = a * e" ..
haftmann@59816
   296
  ultimately have "a * e + c = a * d" by simp
haftmann@59816
   297
  then have "a * e + c - a * e = a * d - a * e" by simp
haftmann@59816
   298
  then have "c = a * d - a * e" by simp
haftmann@59816
   299
  then have "c = a * (d - e)" by (simp add: algebra_simps)
haftmann@59816
   300
  then show ?Q ..
haftmann@59816
   301
next
haftmann@59816
   302
  assume ?Q with assms show ?P by simp
haftmann@59816
   303
qed
haftmann@59816
   304
haftmann@59816
   305
lemma dvd_add_left_iff:
haftmann@59816
   306
  assumes "a dvd c"
haftmann@59816
   307
  shows "a dvd b + c \<longleftrightarrow> a dvd b"
haftmann@59816
   308
  using assms dvd_add_right_iff [of a c b] by (simp add: ac_simps)
haftmann@59816
   309
haftmann@59816
   310
end
haftmann@59816
   311
haftmann@22390
   312
class ring = semiring + ab_group_add
haftmann@25267
   313
begin
haftmann@25152
   314
huffman@27516
   315
subclass semiring_0_cancel ..
haftmann@25152
   316
wenzelm@60758
   317
text \<open>Distribution rules\<close>
haftmann@25152
   318
haftmann@25152
   319
lemma minus_mult_left: "- (a * b) = - a * b"
lp15@60562
   320
by (rule minus_unique) (simp add: distrib_right [symmetric])
haftmann@25152
   321
haftmann@25152
   322
lemma minus_mult_right: "- (a * b) = a * - b"
lp15@60562
   323
by (rule minus_unique) (simp add: distrib_left [symmetric])
haftmann@25152
   324
wenzelm@60758
   325
text\<open>Extract signs from products\<close>
blanchet@54147
   326
lemmas mult_minus_left [simp] = minus_mult_left [symmetric]
blanchet@54147
   327
lemmas mult_minus_right [simp] = minus_mult_right [symmetric]
huffman@29407
   328
haftmann@25152
   329
lemma minus_mult_minus [simp]: "- a * - b = a * b"
nipkow@29667
   330
by simp
haftmann@25152
   331
haftmann@25152
   332
lemma minus_mult_commute: "- a * b = a * - b"
nipkow@29667
   333
by simp
nipkow@29667
   334
hoelzl@58776
   335
lemma right_diff_distrib [algebra_simps]:
haftmann@54230
   336
  "a * (b - c) = a * b - a * c"
haftmann@54230
   337
  using distrib_left [of a b "-c "] by simp
nipkow@29667
   338
hoelzl@58776
   339
lemma left_diff_distrib [algebra_simps]:
haftmann@54230
   340
  "(a - b) * c = a * c - b * c"
haftmann@54230
   341
  using distrib_right [of a "- b" c] by simp
haftmann@25152
   342
blanchet@54147
   343
lemmas ring_distribs =
webertj@49962
   344
  distrib_left distrib_right left_diff_distrib right_diff_distrib
haftmann@25152
   345
haftmann@25230
   346
lemma eq_add_iff1:
haftmann@25230
   347
  "a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"
nipkow@29667
   348
by (simp add: algebra_simps)
haftmann@25230
   349
haftmann@25230
   350
lemma eq_add_iff2:
haftmann@25230
   351
  "a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"
nipkow@29667
   352
by (simp add: algebra_simps)
haftmann@25230
   353
haftmann@25152
   354
end
haftmann@25152
   355
blanchet@54147
   356
lemmas ring_distribs =
webertj@49962
   357
  distrib_left distrib_right left_diff_distrib right_diff_distrib
haftmann@25152
   358
haftmann@22390
   359
class comm_ring = comm_semiring + ab_group_add
haftmann@25267
   360
begin
obua@14738
   361
huffman@27516
   362
subclass ring ..
huffman@28141
   363
subclass comm_semiring_0_cancel ..
haftmann@25267
   364
huffman@44350
   365
lemma square_diff_square_factored:
huffman@44350
   366
  "x * x - y * y = (x + y) * (x - y)"
huffman@44350
   367
  by (simp add: algebra_simps)
huffman@44350
   368
haftmann@25267
   369
end
obua@14738
   370
haftmann@22390
   371
class ring_1 = ring + zero_neq_one + monoid_mult
haftmann@25267
   372
begin
paulson@14265
   373
huffman@27516
   374
subclass semiring_1_cancel ..
haftmann@25267
   375
huffman@44346
   376
lemma square_diff_one_factored:
huffman@44346
   377
  "x * x - 1 = (x + 1) * (x - 1)"
huffman@44346
   378
  by (simp add: algebra_simps)
huffman@44346
   379
haftmann@25267
   380
end
haftmann@25152
   381
haftmann@22390
   382
class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
haftmann@25267
   383
begin
obua@14738
   384
huffman@27516
   385
subclass ring_1 ..
lp15@60562
   386
subclass comm_semiring_1_cancel
haftmann@59816
   387
  by unfold_locales (simp add: algebra_simps)
haftmann@58647
   388
huffman@29465
   389
lemma dvd_minus_iff [simp]: "x dvd - y \<longleftrightarrow> x dvd y"
huffman@29408
   390
proof
huffman@29408
   391
  assume "x dvd - y"
huffman@29408
   392
  then have "x dvd - 1 * - y" by (rule dvd_mult)
huffman@29408
   393
  then show "x dvd y" by simp
huffman@29408
   394
next
huffman@29408
   395
  assume "x dvd y"
huffman@29408
   396
  then have "x dvd - 1 * y" by (rule dvd_mult)
huffman@29408
   397
  then show "x dvd - y" by simp
huffman@29408
   398
qed
huffman@29408
   399
huffman@29465
   400
lemma minus_dvd_iff [simp]: "- x dvd y \<longleftrightarrow> x dvd y"
huffman@29408
   401
proof
huffman@29408
   402
  assume "- x dvd y"
huffman@29408
   403
  then obtain k where "y = - x * k" ..
huffman@29408
   404
  then have "y = x * - k" by simp
huffman@29408
   405
  then show "x dvd y" ..
huffman@29408
   406
next
huffman@29408
   407
  assume "x dvd y"
huffman@29408
   408
  then obtain k where "y = x * k" ..
huffman@29408
   409
  then have "y = - x * - k" by simp
huffman@29408
   410
  then show "- x dvd y" ..
huffman@29408
   411
qed
huffman@29408
   412
haftmann@54230
   413
lemma dvd_diff [simp]:
haftmann@54230
   414
  "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
haftmann@54230
   415
  using dvd_add [of x y "- z"] by simp
huffman@29409
   416
haftmann@25267
   417
end
haftmann@25152
   418
haftmann@59833
   419
class semiring_no_zero_divisors = semiring_0 +
haftmann@59833
   420
  assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
haftmann@25230
   421
begin
haftmann@25230
   422
haftmann@59833
   423
lemma divisors_zero:
haftmann@59833
   424
  assumes "a * b = 0"
haftmann@59833
   425
  shows "a = 0 \<or> b = 0"
haftmann@59833
   426
proof (rule classical)
haftmann@59833
   427
  assume "\<not> (a = 0 \<or> b = 0)"
haftmann@59833
   428
  then have "a \<noteq> 0" and "b \<noteq> 0" by auto
haftmann@59833
   429
  with no_zero_divisors have "a * b \<noteq> 0" by blast
haftmann@59833
   430
  with assms show ?thesis by simp
haftmann@59833
   431
qed
haftmann@59833
   432
haftmann@25230
   433
lemma mult_eq_0_iff [simp]:
haftmann@58952
   434
  shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
haftmann@25230
   435
proof (cases "a = 0 \<or> b = 0")
haftmann@25230
   436
  case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
haftmann@25230
   437
    then show ?thesis using no_zero_divisors by simp
haftmann@25230
   438
next
haftmann@25230
   439
  case True then show ?thesis by auto
haftmann@25230
   440
qed
haftmann@25230
   441
haftmann@58952
   442
end
haftmann@58952
   443
haftmann@60516
   444
class semiring_no_zero_divisors_cancel = semiring_no_zero_divisors +
haftmann@60516
   445
  assumes mult_cancel_right [simp]: "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   446
    and mult_cancel_left [simp]: "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@58952
   447
begin
haftmann@58952
   448
haftmann@58952
   449
lemma mult_left_cancel:
haftmann@58952
   450
  "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b"
lp15@60562
   451
  by simp
lp15@56217
   452
haftmann@58952
   453
lemma mult_right_cancel:
haftmann@58952
   454
  "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b"
lp15@60562
   455
  by simp
lp15@56217
   456
haftmann@25230
   457
end
huffman@22990
   458
haftmann@60516
   459
class ring_no_zero_divisors = ring + semiring_no_zero_divisors
haftmann@60516
   460
begin
haftmann@60516
   461
haftmann@60516
   462
subclass semiring_no_zero_divisors_cancel
haftmann@60516
   463
proof
haftmann@60516
   464
  fix a b c
haftmann@60516
   465
  have "a * c = b * c \<longleftrightarrow> (a - b) * c = 0"
haftmann@60516
   466
    by (simp add: algebra_simps)
haftmann@60516
   467
  also have "\<dots> \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   468
    by auto
haftmann@60516
   469
  finally show "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" .
haftmann@60516
   470
  have "c * a = c * b \<longleftrightarrow> c * (a - b) = 0"
haftmann@60516
   471
    by (simp add: algebra_simps)
haftmann@60516
   472
  also have "\<dots> \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   473
    by auto
haftmann@60516
   474
  finally show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" .
haftmann@60516
   475
qed
haftmann@60516
   476
haftmann@60516
   477
end
haftmann@60516
   478
huffman@23544
   479
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
haftmann@26274
   480
begin
haftmann@26274
   481
huffman@36970
   482
lemma square_eq_1_iff:
huffman@36821
   483
  "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
huffman@36821
   484
proof -
huffman@36821
   485
  have "(x - 1) * (x + 1) = x * x - 1"
huffman@36821
   486
    by (simp add: algebra_simps)
huffman@36821
   487
  hence "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
huffman@36821
   488
    by simp
huffman@36821
   489
  thus ?thesis
huffman@36821
   490
    by (simp add: eq_neg_iff_add_eq_0)
huffman@36821
   491
qed
huffman@36821
   492
haftmann@26274
   493
lemma mult_cancel_right1 [simp]:
haftmann@26274
   494
  "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   495
by (insert mult_cancel_right [of 1 c b], force)
haftmann@26274
   496
haftmann@26274
   497
lemma mult_cancel_right2 [simp]:
haftmann@26274
   498
  "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   499
by (insert mult_cancel_right [of a c 1], simp)
lp15@60562
   500
haftmann@26274
   501
lemma mult_cancel_left1 [simp]:
haftmann@26274
   502
  "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   503
by (insert mult_cancel_left [of c 1 b], force)
haftmann@26274
   504
haftmann@26274
   505
lemma mult_cancel_left2 [simp]:
haftmann@26274
   506
  "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   507
by (insert mult_cancel_left [of c a 1], simp)
haftmann@26274
   508
haftmann@26274
   509
end
huffman@22990
   510
lp15@60562
   511
class semidom = comm_semiring_1_cancel + semiring_no_zero_divisors
haftmann@59833
   512
haftmann@59833
   513
class idom = comm_ring_1 + semiring_no_zero_divisors
haftmann@25186
   514
begin
paulson@14421
   515
haftmann@59833
   516
subclass semidom ..
haftmann@59833
   517
huffman@27516
   518
subclass ring_1_no_zero_divisors ..
huffman@22990
   519
huffman@29981
   520
lemma dvd_mult_cancel_right [simp]:
huffman@29981
   521
  "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   522
proof -
huffman@29981
   523
  have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
haftmann@57514
   524
    unfolding dvd_def by (simp add: ac_simps)
huffman@29981
   525
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   526
    unfolding dvd_def by simp
huffman@29981
   527
  finally show ?thesis .
huffman@29981
   528
qed
huffman@29981
   529
huffman@29981
   530
lemma dvd_mult_cancel_left [simp]:
huffman@29981
   531
  "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   532
proof -
huffman@29981
   533
  have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
haftmann@57514
   534
    unfolding dvd_def by (simp add: ac_simps)
huffman@29981
   535
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   536
    unfolding dvd_def by simp
huffman@29981
   537
  finally show ?thesis .
huffman@29981
   538
qed
huffman@29981
   539
haftmann@60516
   540
lemma square_eq_iff: "a * a = b * b \<longleftrightarrow> a = b \<or> a = - b"
haftmann@59833
   541
proof
haftmann@59833
   542
  assume "a * a = b * b"
haftmann@59833
   543
  then have "(a - b) * (a + b) = 0"
haftmann@59833
   544
    by (simp add: algebra_simps)
haftmann@59833
   545
  then show "a = b \<or> a = - b"
haftmann@59833
   546
    by (simp add: eq_neg_iff_add_eq_0)
haftmann@59833
   547
next
haftmann@59833
   548
  assume "a = b \<or> a = - b"
haftmann@59833
   549
  then show "a * a = b * b" by auto
haftmann@59833
   550
qed
haftmann@59833
   551
haftmann@25186
   552
end
haftmann@25152
   553
wenzelm@60758
   554
text \<open>
haftmann@35302
   555
  The theory of partially ordered rings is taken from the books:
haftmann@35302
   556
  \begin{itemize}
lp15@60562
   557
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979
haftmann@35302
   558
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
haftmann@35302
   559
  \end{itemize}
lp15@60562
   560
  Most of the used notions can also be looked up in
haftmann@35302
   561
  \begin{itemize}
wenzelm@54703
   562
  \item @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
haftmann@35302
   563
  \item \emph{Algebra I} by van der Waerden, Springer.
haftmann@35302
   564
  \end{itemize}
wenzelm@60758
   565
\<close>
haftmann@35302
   566
haftmann@60353
   567
class divide =
haftmann@60429
   568
  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "div" 70)
haftmann@60353
   569
wenzelm@60758
   570
setup \<open>Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a \<Rightarrow> 'a \<Rightarrow> 'a"})\<close>
haftmann@60353
   571
haftmann@60353
   572
context semiring
haftmann@60353
   573
begin
haftmann@60353
   574
haftmann@60353
   575
lemma [field_simps]:
haftmann@60429
   576
  shows distrib_left_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b + c) = a * b + a * c"
haftmann@60429
   577
    and distrib_right_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a + b) * c = a * c + b * c"
haftmann@60353
   578
  by (rule distrib_left distrib_right)+
haftmann@60353
   579
haftmann@60353
   580
end
haftmann@60353
   581
haftmann@60353
   582
context ring
haftmann@60353
   583
begin
haftmann@60353
   584
haftmann@60353
   585
lemma [field_simps]:
haftmann@60429
   586
  shows left_diff_distrib_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a - b) * c = a * c - b * c"
haftmann@60429
   587
    and right_diff_distrib_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b - c) = a * b - a * c"
haftmann@60353
   588
  by (rule left_diff_distrib right_diff_distrib)+
haftmann@60353
   589
haftmann@60353
   590
end
haftmann@60353
   591
wenzelm@60758
   592
setup \<open>Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a::divide \<Rightarrow> 'a \<Rightarrow> 'a"})\<close>
haftmann@60353
   593
haftmann@60353
   594
class semidom_divide = semidom + divide +
haftmann@60429
   595
  assumes nonzero_mult_divide_cancel_right [simp]: "b \<noteq> 0 \<Longrightarrow> (a * b) div b = a"
haftmann@60429
   596
  assumes divide_zero [simp]: "a div 0 = 0"
haftmann@60353
   597
begin
haftmann@60353
   598
haftmann@60353
   599
lemma nonzero_mult_divide_cancel_left [simp]:
haftmann@60429
   600
  "a \<noteq> 0 \<Longrightarrow> (a * b) div a = b"
haftmann@60353
   601
  using nonzero_mult_divide_cancel_right [of a b] by (simp add: ac_simps)
haftmann@60353
   602
haftmann@60516
   603
subclass semiring_no_zero_divisors_cancel
haftmann@60516
   604
proof
haftmann@60516
   605
  fix a b c
haftmann@60516
   606
  { fix a b c
haftmann@60516
   607
    show "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   608
    proof (cases "c = 0")
haftmann@60516
   609
      case True then show ?thesis by simp
haftmann@60516
   610
    next
haftmann@60516
   611
      case False
haftmann@60516
   612
      { assume "a * c = b * c"
haftmann@60516
   613
        then have "a * c div c = b * c div c"
haftmann@60516
   614
          by simp
haftmann@60516
   615
        with False have "a = b"
haftmann@60516
   616
          by simp
haftmann@60516
   617
      } then show ?thesis by auto
haftmann@60516
   618
    qed
haftmann@60516
   619
  }
haftmann@60516
   620
  from this [of a c b]
haftmann@60516
   621
  show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   622
    by (simp add: ac_simps)
haftmann@60516
   623
qed
haftmann@60516
   624
haftmann@60516
   625
lemma div_self [simp]:
haftmann@60516
   626
  assumes "a \<noteq> 0"
haftmann@60516
   627
  shows "a div a = 1"
haftmann@60516
   628
  using assms nonzero_mult_divide_cancel_left [of a 1] by simp
haftmann@60516
   629
haftmann@60570
   630
lemma divide_zero_left [simp]:
haftmann@60570
   631
  "0 div a = 0"
haftmann@60570
   632
proof (cases "a = 0")
haftmann@60570
   633
  case True then show ?thesis by simp
haftmann@60570
   634
next
haftmann@60570
   635
  case False then have "a * 0 div a = 0"
haftmann@60570
   636
    by (rule nonzero_mult_divide_cancel_left)
haftmann@60570
   637
  then show ?thesis by simp
hoelzl@62376
   638
qed
haftmann@60570
   639
haftmann@60690
   640
lemma divide_1 [simp]:
haftmann@60690
   641
  "a div 1 = a"
haftmann@60690
   642
  using nonzero_mult_divide_cancel_left [of 1 a] by simp
haftmann@60690
   643
haftmann@60867
   644
end
haftmann@60867
   645
haftmann@60867
   646
class idom_divide = idom + semidom_divide
haftmann@60867
   647
haftmann@60867
   648
class algebraic_semidom = semidom_divide
haftmann@60867
   649
begin
haftmann@60867
   650
haftmann@60867
   651
text \<open>
haftmann@60867
   652
  Class @{class algebraic_semidom} enriches a integral domain
haftmann@60867
   653
  by notions from algebra, like units in a ring.
haftmann@60867
   654
  It is a separate class to avoid spoiling fields with notions
haftmann@60867
   655
  which are degenerated there.
haftmann@60867
   656
\<close>
haftmann@60867
   657
haftmann@60690
   658
lemma dvd_times_left_cancel_iff [simp]:
haftmann@60690
   659
  assumes "a \<noteq> 0"
haftmann@60690
   660
  shows "a * b dvd a * c \<longleftrightarrow> b dvd c" (is "?P \<longleftrightarrow> ?Q")
haftmann@60690
   661
proof
haftmann@60690
   662
  assume ?P then obtain d where "a * c = a * b * d" ..
haftmann@60690
   663
  with assms have "c = b * d" by (simp add: ac_simps)
haftmann@60690
   664
  then show ?Q ..
haftmann@60690
   665
next
hoelzl@62376
   666
  assume ?Q then obtain d where "c = b * d" ..
haftmann@60690
   667
  then have "a * c = a * b * d" by (simp add: ac_simps)
haftmann@60690
   668
  then show ?P ..
haftmann@60690
   669
qed
hoelzl@62376
   670
haftmann@60690
   671
lemma dvd_times_right_cancel_iff [simp]:
haftmann@60690
   672
  assumes "a \<noteq> 0"
haftmann@60690
   673
  shows "b * a dvd c * a \<longleftrightarrow> b dvd c" (is "?P \<longleftrightarrow> ?Q")
haftmann@60690
   674
using dvd_times_left_cancel_iff [of a b c] assms by (simp add: ac_simps)
hoelzl@62376
   675
haftmann@60690
   676
lemma div_dvd_iff_mult:
haftmann@60690
   677
  assumes "b \<noteq> 0" and "b dvd a"
haftmann@60690
   678
  shows "a div b dvd c \<longleftrightarrow> a dvd c * b"
haftmann@60690
   679
proof -
haftmann@60690
   680
  from \<open>b dvd a\<close> obtain d where "a = b * d" ..
haftmann@60690
   681
  with \<open>b \<noteq> 0\<close> show ?thesis by (simp add: ac_simps)
haftmann@60690
   682
qed
haftmann@60690
   683
haftmann@60690
   684
lemma dvd_div_iff_mult:
haftmann@60690
   685
  assumes "c \<noteq> 0" and "c dvd b"
haftmann@60690
   686
  shows "a dvd b div c \<longleftrightarrow> a * c dvd b"
haftmann@60690
   687
proof -
haftmann@60690
   688
  from \<open>c dvd b\<close> obtain d where "b = c * d" ..
haftmann@60690
   689
  with \<open>c \<noteq> 0\<close> show ?thesis by (simp add: mult.commute [of a])
haftmann@60690
   690
qed
haftmann@60690
   691
haftmann@60867
   692
lemma div_dvd_div [simp]:
haftmann@60867
   693
  assumes "a dvd b" and "a dvd c"
haftmann@60867
   694
  shows "b div a dvd c div a \<longleftrightarrow> b dvd c"
haftmann@60867
   695
proof (cases "a = 0")
haftmann@60867
   696
  case True with assms show ?thesis by simp
haftmann@60867
   697
next
haftmann@60867
   698
  case False
haftmann@60867
   699
  moreover from assms obtain k l where "b = a * k" and "c = a * l"
haftmann@60867
   700
    by (auto elim!: dvdE)
haftmann@60867
   701
  ultimately show ?thesis by simp
haftmann@60867
   702
qed
haftmann@60353
   703
haftmann@60867
   704
lemma div_add [simp]:
haftmann@60867
   705
  assumes "c dvd a" and "c dvd b"
haftmann@60867
   706
  shows "(a + b) div c = a div c + b div c"
haftmann@60867
   707
proof (cases "c = 0")
haftmann@60867
   708
  case True then show ?thesis by simp
haftmann@60867
   709
next
haftmann@60867
   710
  case False
haftmann@60867
   711
  moreover from assms obtain k l where "a = c * k" and "b = c * l"
haftmann@60867
   712
    by (auto elim!: dvdE)
haftmann@60867
   713
  moreover have "c * k + c * l = c * (k + l)"
haftmann@60867
   714
    by (simp add: algebra_simps)
haftmann@60867
   715
  ultimately show ?thesis
haftmann@60867
   716
    by simp
haftmann@60867
   717
qed
haftmann@60517
   718
haftmann@60867
   719
lemma div_mult_div_if_dvd:
haftmann@60867
   720
  assumes "b dvd a" and "d dvd c"
haftmann@60867
   721
  shows "(a div b) * (c div d) = (a * c) div (b * d)"
haftmann@60867
   722
proof (cases "b = 0 \<or> c = 0")
haftmann@60867
   723
  case True with assms show ?thesis by auto
haftmann@60867
   724
next
haftmann@60867
   725
  case False
haftmann@60867
   726
  moreover from assms obtain k l where "a = b * k" and "c = d * l"
haftmann@60867
   727
    by (auto elim!: dvdE)
haftmann@60867
   728
  moreover have "b * k * (d * l) div (b * d) = (b * d) * (k * l) div (b * d)"
haftmann@60867
   729
    by (simp add: ac_simps)
haftmann@60867
   730
  ultimately show ?thesis by simp
haftmann@60867
   731
qed
haftmann@60867
   732
haftmann@60867
   733
lemma dvd_div_eq_mult:
haftmann@60867
   734
  assumes "a \<noteq> 0" and "a dvd b"
haftmann@60867
   735
  shows "b div a = c \<longleftrightarrow> b = c * a"
haftmann@60867
   736
proof
haftmann@60867
   737
  assume "b = c * a"
haftmann@60867
   738
  then show "b div a = c" by (simp add: assms)
haftmann@60867
   739
next
haftmann@60867
   740
  assume "b div a = c"
haftmann@60867
   741
  then have "b div a * a = c * a" by simp
haftmann@60867
   742
  moreover from \<open>a \<noteq> 0\<close> \<open>a dvd b\<close> have "b div a * a = b"
haftmann@60867
   743
    by (auto elim!: dvdE simp add: ac_simps)
haftmann@60867
   744
  ultimately show "b = c * a" by simp
haftmann@60867
   745
qed
haftmann@60688
   746
haftmann@60517
   747
lemma dvd_div_mult_self [simp]:
haftmann@60517
   748
  "a dvd b \<Longrightarrow> b div a * a = b"
haftmann@60517
   749
  by (cases "a = 0") (auto elim: dvdE simp add: ac_simps)
haftmann@60517
   750
haftmann@60517
   751
lemma dvd_mult_div_cancel [simp]:
haftmann@60517
   752
  "a dvd b \<Longrightarrow> a * (b div a) = b"
haftmann@60517
   753
  using dvd_div_mult_self [of a b] by (simp add: ac_simps)
lp15@60562
   754
haftmann@60517
   755
lemma div_mult_swap:
haftmann@60517
   756
  assumes "c dvd b"
haftmann@60517
   757
  shows "a * (b div c) = (a * b) div c"
haftmann@60517
   758
proof (cases "c = 0")
haftmann@60517
   759
  case True then show ?thesis by simp
haftmann@60517
   760
next
haftmann@60517
   761
  case False from assms obtain d where "b = c * d" ..
haftmann@60517
   762
  moreover from False have "a * divide (d * c) c = ((a * d) * c) div c"
haftmann@60517
   763
    by simp
haftmann@60517
   764
  ultimately show ?thesis by (simp add: ac_simps)
haftmann@60517
   765
qed
haftmann@60517
   766
haftmann@60517
   767
lemma dvd_div_mult:
haftmann@60517
   768
  assumes "c dvd b"
haftmann@60517
   769
  shows "b div c * a = (b * a) div c"
haftmann@60517
   770
  using assms div_mult_swap [of c b a] by (simp add: ac_simps)
haftmann@60517
   771
haftmann@60570
   772
lemma dvd_div_mult2_eq:
haftmann@60570
   773
  assumes "b * c dvd a"
haftmann@60570
   774
  shows "a div (b * c) = a div b div c"
haftmann@60570
   775
using assms proof
haftmann@60570
   776
  fix k
haftmann@60570
   777
  assume "a = b * c * k"
haftmann@60570
   778
  then show ?thesis
haftmann@60570
   779
    by (cases "b = 0 \<or> c = 0") (auto, simp add: ac_simps)
haftmann@60570
   780
qed
haftmann@60570
   781
haftmann@60867
   782
lemma dvd_div_div_eq_mult:
haftmann@60867
   783
  assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d"
haftmann@60867
   784
  shows "b div a = d div c \<longleftrightarrow> b * c = a * d" (is "?P \<longleftrightarrow> ?Q")
haftmann@60867
   785
proof -
haftmann@60867
   786
  from assms have "a * c \<noteq> 0" by simp
haftmann@60867
   787
  then have "?P \<longleftrightarrow> b div a * (a * c) = d div c * (a * c)"
haftmann@60867
   788
    by simp
haftmann@60867
   789
  also have "\<dots> \<longleftrightarrow> (a * (b div a)) * c = (c * (d div c)) * a"
haftmann@60867
   790
    by (simp add: ac_simps)
haftmann@60867
   791
  also have "\<dots> \<longleftrightarrow> (a * b div a) * c = (c * d div c) * a"
haftmann@60867
   792
    using assms by (simp add: div_mult_swap)
haftmann@60867
   793
  also have "\<dots> \<longleftrightarrow> ?Q"
haftmann@60867
   794
    using assms by (simp add: ac_simps)
haftmann@60867
   795
  finally show ?thesis .
haftmann@60867
   796
qed
haftmann@60867
   797
lp15@60562
   798
haftmann@60517
   799
text \<open>Units: invertible elements in a ring\<close>
haftmann@60517
   800
haftmann@60517
   801
abbreviation is_unit :: "'a \<Rightarrow> bool"
haftmann@60517
   802
where
haftmann@60517
   803
  "is_unit a \<equiv> a dvd 1"
haftmann@60517
   804
haftmann@60517
   805
lemma not_is_unit_0 [simp]:
haftmann@60517
   806
  "\<not> is_unit 0"
haftmann@60517
   807
  by simp
haftmann@60517
   808
lp15@60562
   809
lemma unit_imp_dvd [dest]:
haftmann@60517
   810
  "is_unit b \<Longrightarrow> b dvd a"
haftmann@60517
   811
  by (rule dvd_trans [of _ 1]) simp_all
haftmann@60517
   812
haftmann@60517
   813
lemma unit_dvdE:
haftmann@60517
   814
  assumes "is_unit a"
haftmann@60517
   815
  obtains c where "a \<noteq> 0" and "b = a * c"
haftmann@60517
   816
proof -
haftmann@60517
   817
  from assms have "a dvd b" by auto
haftmann@60517
   818
  then obtain c where "b = a * c" ..
haftmann@60517
   819
  moreover from assms have "a \<noteq> 0" by auto
haftmann@60517
   820
  ultimately show thesis using that by blast
haftmann@60517
   821
qed
haftmann@60517
   822
haftmann@60517
   823
lemma dvd_unit_imp_unit:
haftmann@60517
   824
  "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a"
haftmann@60517
   825
  by (rule dvd_trans)
haftmann@60517
   826
haftmann@60517
   827
lemma unit_div_1_unit [simp, intro]:
haftmann@60517
   828
  assumes "is_unit a"
haftmann@60517
   829
  shows "is_unit (1 div a)"
haftmann@60517
   830
proof -
haftmann@60517
   831
  from assms have "1 = 1 div a * a" by simp
haftmann@60517
   832
  then show "is_unit (1 div a)" by (rule dvdI)
haftmann@60517
   833
qed
haftmann@60517
   834
haftmann@60517
   835
lemma is_unitE [elim?]:
haftmann@60517
   836
  assumes "is_unit a"
haftmann@60517
   837
  obtains b where "a \<noteq> 0" and "b \<noteq> 0"
haftmann@60517
   838
    and "is_unit b" and "1 div a = b" and "1 div b = a"
haftmann@60517
   839
    and "a * b = 1" and "c div a = c * b"
haftmann@60517
   840
proof (rule that)
haftmann@60517
   841
  def b \<equiv> "1 div a"
haftmann@60517
   842
  then show "1 div a = b" by simp
wenzelm@60758
   843
  from b_def \<open>is_unit a\<close> show "is_unit b" by simp
wenzelm@60758
   844
  from \<open>is_unit a\<close> and \<open>is_unit b\<close> show "a \<noteq> 0" and "b \<noteq> 0" by auto
wenzelm@60758
   845
  from b_def \<open>is_unit a\<close> show "a * b = 1" by simp
haftmann@60517
   846
  then have "1 = a * b" ..
wenzelm@60758
   847
  with b_def \<open>b \<noteq> 0\<close> show "1 div b = a" by simp
wenzelm@60758
   848
  from \<open>is_unit a\<close> have "a dvd c" ..
haftmann@60517
   849
  then obtain d where "c = a * d" ..
wenzelm@60758
   850
  with \<open>a \<noteq> 0\<close> \<open>a * b = 1\<close> show "c div a = c * b"
haftmann@60517
   851
    by (simp add: mult.assoc mult.left_commute [of a])
haftmann@60517
   852
qed
haftmann@60517
   853
haftmann@60517
   854
lemma unit_prod [intro]:
haftmann@60517
   855
  "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)"
lp15@60562
   856
  by (subst mult_1_left [of 1, symmetric]) (rule mult_dvd_mono)
lp15@60562
   857
haftmann@62366
   858
lemma is_unit_mult_iff:
haftmann@62366
   859
  "is_unit (a * b) \<longleftrightarrow> is_unit a \<and> is_unit b" (is "?P \<longleftrightarrow> ?Q")
haftmann@62366
   860
  by (auto dest: dvd_mult_left dvd_mult_right)
haftmann@62366
   861
haftmann@60517
   862
lemma unit_div [intro]:
haftmann@60517
   863
  "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)"
haftmann@60517
   864
  by (erule is_unitE [of b a]) (simp add: ac_simps unit_prod)
haftmann@60517
   865
haftmann@60517
   866
lemma mult_unit_dvd_iff:
haftmann@60517
   867
  assumes "is_unit b"
haftmann@60517
   868
  shows "a * b dvd c \<longleftrightarrow> a dvd c"
haftmann@60517
   869
proof
haftmann@60517
   870
  assume "a * b dvd c"
haftmann@60517
   871
  with assms show "a dvd c"
haftmann@60517
   872
    by (simp add: dvd_mult_left)
haftmann@60517
   873
next
haftmann@60517
   874
  assume "a dvd c"
haftmann@60517
   875
  then obtain k where "c = a * k" ..
haftmann@60517
   876
  with assms have "c = (a * b) * (1 div b * k)"
haftmann@60517
   877
    by (simp add: mult_ac)
haftmann@60517
   878
  then show "a * b dvd c" by (rule dvdI)
haftmann@60517
   879
qed
haftmann@60517
   880
haftmann@60517
   881
lemma dvd_mult_unit_iff:
haftmann@60517
   882
  assumes "is_unit b"
haftmann@60517
   883
  shows "a dvd c * b \<longleftrightarrow> a dvd c"
haftmann@60517
   884
proof
haftmann@60517
   885
  assume "a dvd c * b"
haftmann@60517
   886
  with assms have "c * b dvd c * (b * (1 div b))"
haftmann@60517
   887
    by (subst mult_assoc [symmetric]) simp
wenzelm@60758
   888
  also from \<open>is_unit b\<close> have "b * (1 div b) = 1" by (rule is_unitE) simp
haftmann@60517
   889
  finally have "c * b dvd c" by simp
wenzelm@60758
   890
  with \<open>a dvd c * b\<close> show "a dvd c" by (rule dvd_trans)
haftmann@60517
   891
next
haftmann@60517
   892
  assume "a dvd c"
haftmann@60517
   893
  then show "a dvd c * b" by simp
haftmann@60517
   894
qed
haftmann@60517
   895
haftmann@60517
   896
lemma div_unit_dvd_iff:
haftmann@60517
   897
  "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c"
haftmann@60517
   898
  by (erule is_unitE [of _ a]) (auto simp add: mult_unit_dvd_iff)
haftmann@60517
   899
haftmann@60517
   900
lemma dvd_div_unit_iff:
haftmann@60517
   901
  "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c"
haftmann@60517
   902
  by (erule is_unitE [of _ c]) (simp add: dvd_mult_unit_iff)
haftmann@60517
   903
haftmann@60517
   904
lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff
wenzelm@61799
   905
  dvd_mult_unit_iff dvd_div_unit_iff \<comment> \<open>FIXME consider fact collection\<close>
haftmann@60517
   906
haftmann@60517
   907
lemma unit_mult_div_div [simp]:
haftmann@60517
   908
  "is_unit a \<Longrightarrow> b * (1 div a) = b div a"
haftmann@60517
   909
  by (erule is_unitE [of _ b]) simp
haftmann@60517
   910
haftmann@60517
   911
lemma unit_div_mult_self [simp]:
haftmann@60517
   912
  "is_unit a \<Longrightarrow> b div a * a = b"
haftmann@60517
   913
  by (rule dvd_div_mult_self) auto
haftmann@60517
   914
haftmann@60517
   915
lemma unit_div_1_div_1 [simp]:
haftmann@60517
   916
  "is_unit a \<Longrightarrow> 1 div (1 div a) = a"
haftmann@60517
   917
  by (erule is_unitE) simp
haftmann@60517
   918
haftmann@60517
   919
lemma unit_div_mult_swap:
haftmann@60517
   920
  "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c"
haftmann@60517
   921
  by (erule unit_dvdE [of _ b]) (simp add: mult.left_commute [of _ c])
haftmann@60517
   922
haftmann@60517
   923
lemma unit_div_commute:
haftmann@60517
   924
  "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b"
haftmann@60517
   925
  using unit_div_mult_swap [of b c a] by (simp add: ac_simps)
haftmann@60517
   926
haftmann@60517
   927
lemma unit_eq_div1:
haftmann@60517
   928
  "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b"
haftmann@60517
   929
  by (auto elim: is_unitE)
haftmann@60517
   930
haftmann@60517
   931
lemma unit_eq_div2:
haftmann@60517
   932
  "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c"
haftmann@60517
   933
  using unit_eq_div1 [of b c a] by auto
haftmann@60517
   934
haftmann@60517
   935
lemma unit_mult_left_cancel:
haftmann@60517
   936
  assumes "is_unit a"
haftmann@60517
   937
  shows "a * b = a * c \<longleftrightarrow> b = c" (is "?P \<longleftrightarrow> ?Q")
lp15@60562
   938
  using assms mult_cancel_left [of a b c] by auto
haftmann@60517
   939
haftmann@60517
   940
lemma unit_mult_right_cancel:
haftmann@60517
   941
  "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c"
haftmann@60517
   942
  using unit_mult_left_cancel [of a b c] by (auto simp add: ac_simps)
haftmann@60517
   943
haftmann@60517
   944
lemma unit_div_cancel:
haftmann@60517
   945
  assumes "is_unit a"
haftmann@60517
   946
  shows "b div a = c div a \<longleftrightarrow> b = c"
haftmann@60517
   947
proof -
haftmann@60517
   948
  from assms have "is_unit (1 div a)" by simp
haftmann@60517
   949
  then have "b * (1 div a) = c * (1 div a) \<longleftrightarrow> b = c"
haftmann@60517
   950
    by (rule unit_mult_right_cancel)
haftmann@60517
   951
  with assms show ?thesis by simp
haftmann@60517
   952
qed
lp15@60562
   953
haftmann@60570
   954
lemma is_unit_div_mult2_eq:
haftmann@60570
   955
  assumes "is_unit b" and "is_unit c"
haftmann@60570
   956
  shows "a div (b * c) = a div b div c"
haftmann@60570
   957
proof -
haftmann@60570
   958
  from assms have "is_unit (b * c)" by (simp add: unit_prod)
haftmann@60570
   959
  then have "b * c dvd a"
haftmann@60570
   960
    by (rule unit_imp_dvd)
haftmann@60570
   961
  then show ?thesis
haftmann@60570
   962
    by (rule dvd_div_mult2_eq)
haftmann@60570
   963
qed
haftmann@60570
   964
lp15@60562
   965
lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff
haftmann@60517
   966
  dvd_div_unit_iff unit_div_mult_swap unit_div_commute
lp15@60562
   967
  unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel
haftmann@60517
   968
  unit_eq_div1 unit_eq_div2
haftmann@60517
   969
haftmann@60685
   970
lemma is_unit_divide_mult_cancel_left:
haftmann@60685
   971
  assumes "a \<noteq> 0" and "is_unit b"
haftmann@60685
   972
  shows "a div (a * b) = 1 div b"
haftmann@60685
   973
proof -
haftmann@60685
   974
  from assms have "a div (a * b) = a div a div b"
haftmann@60685
   975
    by (simp add: mult_unit_dvd_iff dvd_div_mult2_eq)
haftmann@60685
   976
  with assms show ?thesis by simp
haftmann@60685
   977
qed
haftmann@60685
   978
haftmann@60685
   979
lemma is_unit_divide_mult_cancel_right:
haftmann@60685
   980
  assumes "a \<noteq> 0" and "is_unit b"
haftmann@60685
   981
  shows "a div (b * a) = 1 div b"
haftmann@60685
   982
  using assms is_unit_divide_mult_cancel_left [of a b] by (simp add: ac_simps)
haftmann@60685
   983
haftmann@60685
   984
end
haftmann@60685
   985
haftmann@60685
   986
class normalization_semidom = algebraic_semidom +
haftmann@60685
   987
  fixes normalize :: "'a \<Rightarrow> 'a"
haftmann@60685
   988
    and unit_factor :: "'a \<Rightarrow> 'a"
haftmann@60685
   989
  assumes unit_factor_mult_normalize [simp]: "unit_factor a * normalize a = a"
haftmann@60685
   990
  assumes normalize_0 [simp]: "normalize 0 = 0"
haftmann@60685
   991
    and unit_factor_0 [simp]: "unit_factor 0 = 0"
haftmann@60685
   992
  assumes is_unit_normalize:
haftmann@60685
   993
    "is_unit a  \<Longrightarrow> normalize a = 1"
hoelzl@62376
   994
  assumes unit_factor_is_unit [iff]:
haftmann@60685
   995
    "a \<noteq> 0 \<Longrightarrow> is_unit (unit_factor a)"
haftmann@60685
   996
  assumes unit_factor_mult: "unit_factor (a * b) = unit_factor a * unit_factor b"
haftmann@60685
   997
begin
haftmann@60685
   998
haftmann@60688
   999
text \<open>
haftmann@60688
  1000
  Class @{class normalization_semidom} cultivates the idea that
haftmann@60688
  1001
  each integral domain can be split into equivalence classes
haftmann@60688
  1002
  whose representants are associated, i.e. divide each other.
haftmann@60688
  1003
  @{const normalize} specifies a canonical representant for each equivalence
haftmann@60688
  1004
  class.  The rationale behind this is that it is easier to reason about equality
haftmann@60688
  1005
  than equivalences, hence we prefer to think about equality of normalized
haftmann@60688
  1006
  values rather than associated elements.
haftmann@60688
  1007
\<close>
haftmann@60688
  1008
haftmann@60685
  1009
lemma unit_factor_dvd [simp]:
haftmann@60685
  1010
  "a \<noteq> 0 \<Longrightarrow> unit_factor a dvd b"
haftmann@60685
  1011
  by (rule unit_imp_dvd) simp
haftmann@60685
  1012
haftmann@60685
  1013
lemma unit_factor_self [simp]:
haftmann@60685
  1014
  "unit_factor a dvd a"
hoelzl@62376
  1015
  by (cases "a = 0") simp_all
hoelzl@62376
  1016
haftmann@60685
  1017
lemma normalize_mult_unit_factor [simp]:
haftmann@60685
  1018
  "normalize a * unit_factor a = a"
haftmann@60685
  1019
  using unit_factor_mult_normalize [of a] by (simp add: ac_simps)
haftmann@60685
  1020
haftmann@60685
  1021
lemma normalize_eq_0_iff [simp]:
haftmann@60685
  1022
  "normalize a = 0 \<longleftrightarrow> a = 0" (is "?P \<longleftrightarrow> ?Q")
haftmann@60685
  1023
proof
haftmann@60685
  1024
  assume ?P
haftmann@60685
  1025
  moreover have "unit_factor a * normalize a = a" by simp
hoelzl@62376
  1026
  ultimately show ?Q by simp
haftmann@60685
  1027
next
haftmann@60685
  1028
  assume ?Q then show ?P by simp
haftmann@60685
  1029
qed
haftmann@60685
  1030
haftmann@60685
  1031
lemma unit_factor_eq_0_iff [simp]:
haftmann@60685
  1032
  "unit_factor a = 0 \<longleftrightarrow> a = 0" (is "?P \<longleftrightarrow> ?Q")
haftmann@60685
  1033
proof
haftmann@60685
  1034
  assume ?P
haftmann@60685
  1035
  moreover have "unit_factor a * normalize a = a" by simp
hoelzl@62376
  1036
  ultimately show ?Q by simp
haftmann@60685
  1037
next
haftmann@60685
  1038
  assume ?Q then show ?P by simp
haftmann@60685
  1039
qed
haftmann@60685
  1040
haftmann@60685
  1041
lemma is_unit_unit_factor:
haftmann@60685
  1042
  assumes "is_unit a" shows "unit_factor a = a"
hoelzl@62376
  1043
proof -
haftmann@60685
  1044
  from assms have "normalize a = 1" by (rule is_unit_normalize)
haftmann@60685
  1045
  moreover from unit_factor_mult_normalize have "unit_factor a * normalize a = a" .
haftmann@60685
  1046
  ultimately show ?thesis by simp
haftmann@60685
  1047
qed
haftmann@60685
  1048
haftmann@60685
  1049
lemma unit_factor_1 [simp]:
haftmann@60685
  1050
  "unit_factor 1 = 1"
haftmann@60685
  1051
  by (rule is_unit_unit_factor) simp
haftmann@60685
  1052
haftmann@60685
  1053
lemma normalize_1 [simp]:
haftmann@60685
  1054
  "normalize 1 = 1"
haftmann@60685
  1055
  by (rule is_unit_normalize) simp
haftmann@60685
  1056
haftmann@60685
  1057
lemma normalize_1_iff:
haftmann@60685
  1058
  "normalize a = 1 \<longleftrightarrow> is_unit a" (is "?P \<longleftrightarrow> ?Q")
haftmann@60685
  1059
proof
haftmann@60685
  1060
  assume ?Q then show ?P by (rule is_unit_normalize)
haftmann@60685
  1061
next
haftmann@60685
  1062
  assume ?P
haftmann@60685
  1063
  then have "a \<noteq> 0" by auto
haftmann@60685
  1064
  from \<open>?P\<close> have "unit_factor a * normalize a = unit_factor a * 1"
haftmann@60685
  1065
    by simp
haftmann@60685
  1066
  then have "unit_factor a = a"
haftmann@60685
  1067
    by simp
haftmann@60685
  1068
  moreover have "is_unit (unit_factor a)"
haftmann@60685
  1069
    using \<open>a \<noteq> 0\<close> by simp
haftmann@60685
  1070
  ultimately show ?Q by simp
haftmann@60685
  1071
qed
hoelzl@62376
  1072
haftmann@60685
  1073
lemma div_normalize [simp]:
haftmann@60685
  1074
  "a div normalize a = unit_factor a"
haftmann@60685
  1075
proof (cases "a = 0")
haftmann@60685
  1076
  case True then show ?thesis by simp
haftmann@60685
  1077
next
hoelzl@62376
  1078
  case False then have "normalize a \<noteq> 0" by simp
haftmann@60685
  1079
  with nonzero_mult_divide_cancel_right
haftmann@60685
  1080
  have "unit_factor a * normalize a div normalize a = unit_factor a" by blast
haftmann@60685
  1081
  then show ?thesis by simp
haftmann@60685
  1082
qed
haftmann@60685
  1083
haftmann@60685
  1084
lemma div_unit_factor [simp]:
haftmann@60685
  1085
  "a div unit_factor a = normalize a"
haftmann@60685
  1086
proof (cases "a = 0")
haftmann@60685
  1087
  case True then show ?thesis by simp
haftmann@60685
  1088
next
hoelzl@62376
  1089
  case False then have "unit_factor a \<noteq> 0" by simp
haftmann@60685
  1090
  with nonzero_mult_divide_cancel_left
haftmann@60685
  1091
  have "unit_factor a * normalize a div unit_factor a = normalize a" by blast
haftmann@60685
  1092
  then show ?thesis by simp
haftmann@60685
  1093
qed
haftmann@60685
  1094
haftmann@60685
  1095
lemma normalize_div [simp]:
haftmann@60685
  1096
  "normalize a div a = 1 div unit_factor a"
haftmann@60685
  1097
proof (cases "a = 0")
haftmann@60685
  1098
  case True then show ?thesis by simp
haftmann@60685
  1099
next
haftmann@60685
  1100
  case False
haftmann@60685
  1101
  have "normalize a div a = normalize a div (unit_factor a * normalize a)"
haftmann@60685
  1102
    by simp
haftmann@60685
  1103
  also have "\<dots> = 1 div unit_factor a"
haftmann@60685
  1104
    using False by (subst is_unit_divide_mult_cancel_right) simp_all
haftmann@60685
  1105
  finally show ?thesis .
haftmann@60685
  1106
qed
haftmann@60685
  1107
haftmann@60685
  1108
lemma mult_one_div_unit_factor [simp]:
haftmann@60685
  1109
  "a * (1 div unit_factor b) = a div unit_factor b"
haftmann@60685
  1110
  by (cases "b = 0") simp_all
haftmann@60685
  1111
haftmann@60685
  1112
lemma normalize_mult:
haftmann@60685
  1113
  "normalize (a * b) = normalize a * normalize b"
haftmann@60685
  1114
proof (cases "a = 0 \<or> b = 0")
haftmann@60685
  1115
  case True then show ?thesis by auto
haftmann@60685
  1116
next
haftmann@60685
  1117
  case False
haftmann@60685
  1118
  from unit_factor_mult_normalize have "unit_factor (a * b) * normalize (a * b) = a * b" .
haftmann@60685
  1119
  then have "normalize (a * b) = a * b div unit_factor (a * b)" by simp
haftmann@60685
  1120
  also have "\<dots> = a * b div unit_factor (b * a)" by (simp add: ac_simps)
haftmann@60685
  1121
  also have "\<dots> = a * b div unit_factor b div unit_factor a"
haftmann@60685
  1122
    using False by (simp add: unit_factor_mult is_unit_div_mult2_eq [symmetric])
haftmann@60685
  1123
  also have "\<dots> = a * (b div unit_factor b) div unit_factor a"
haftmann@60685
  1124
    using False by (subst unit_div_mult_swap) simp_all
haftmann@60685
  1125
  also have "\<dots> = normalize a * normalize b"
haftmann@60685
  1126
    using False by (simp add: mult.commute [of a] mult.commute [of "normalize a"] unit_div_mult_swap [symmetric])
haftmann@60685
  1127
  finally show ?thesis .
haftmann@60685
  1128
qed
hoelzl@62376
  1129
haftmann@60685
  1130
lemma unit_factor_idem [simp]:
haftmann@60685
  1131
  "unit_factor (unit_factor a) = unit_factor a"
haftmann@60685
  1132
  by (cases "a = 0") (auto intro: is_unit_unit_factor)
haftmann@60685
  1133
haftmann@60685
  1134
lemma normalize_unit_factor [simp]:
haftmann@60685
  1135
  "a \<noteq> 0 \<Longrightarrow> normalize (unit_factor a) = 1"
haftmann@60685
  1136
  by (rule is_unit_normalize) simp
hoelzl@62376
  1137
haftmann@60685
  1138
lemma normalize_idem [simp]:
haftmann@60685
  1139
  "normalize (normalize a) = normalize a"
haftmann@60685
  1140
proof (cases "a = 0")
haftmann@60685
  1141
  case True then show ?thesis by simp
haftmann@60685
  1142
next
haftmann@60685
  1143
  case False
haftmann@60685
  1144
  have "normalize a = normalize (unit_factor a * normalize a)" by simp
haftmann@60685
  1145
  also have "\<dots> = normalize (unit_factor a) * normalize (normalize a)"
haftmann@60685
  1146
    by (simp only: normalize_mult)
haftmann@60685
  1147
  finally show ?thesis using False by simp_all
haftmann@60685
  1148
qed
haftmann@60685
  1149
haftmann@60685
  1150
lemma unit_factor_normalize [simp]:
haftmann@60685
  1151
  assumes "a \<noteq> 0"
haftmann@60685
  1152
  shows "unit_factor (normalize a) = 1"
haftmann@60685
  1153
proof -
haftmann@60685
  1154
  from assms have "normalize a \<noteq> 0" by simp
haftmann@60685
  1155
  have "unit_factor (normalize a) * normalize (normalize a) = normalize a"
haftmann@60685
  1156
    by (simp only: unit_factor_mult_normalize)
haftmann@60685
  1157
  then have "unit_factor (normalize a) * normalize a = normalize a"
haftmann@60685
  1158
    by simp
haftmann@60685
  1159
  with \<open>normalize a \<noteq> 0\<close>
haftmann@60685
  1160
  have "unit_factor (normalize a) * normalize a div normalize a = normalize a div normalize a"
haftmann@60685
  1161
    by simp
haftmann@60685
  1162
  with \<open>normalize a \<noteq> 0\<close>
haftmann@60685
  1163
  show ?thesis by simp
haftmann@60685
  1164
qed
haftmann@60685
  1165
haftmann@60685
  1166
lemma dvd_unit_factor_div:
haftmann@60685
  1167
  assumes "b dvd a"
haftmann@60685
  1168
  shows "unit_factor (a div b) = unit_factor a div unit_factor b"
haftmann@60685
  1169
proof -
haftmann@60685
  1170
  from assms have "a = a div b * b"
haftmann@60685
  1171
    by simp
haftmann@60685
  1172
  then have "unit_factor a = unit_factor (a div b * b)"
haftmann@60685
  1173
    by simp
haftmann@60685
  1174
  then show ?thesis
haftmann@60685
  1175
    by (cases "b = 0") (simp_all add: unit_factor_mult)
haftmann@60685
  1176
qed
haftmann@60685
  1177
haftmann@60685
  1178
lemma dvd_normalize_div:
haftmann@60685
  1179
  assumes "b dvd a"
haftmann@60685
  1180
  shows "normalize (a div b) = normalize a div normalize b"
haftmann@60685
  1181
proof -
haftmann@60685
  1182
  from assms have "a = a div b * b"
haftmann@60685
  1183
    by simp
haftmann@60685
  1184
  then have "normalize a = normalize (a div b * b)"
haftmann@60685
  1185
    by simp
haftmann@60685
  1186
  then show ?thesis
haftmann@60685
  1187
    by (cases "b = 0") (simp_all add: normalize_mult)
haftmann@60685
  1188
qed
haftmann@60685
  1189
haftmann@60685
  1190
lemma normalize_dvd_iff [simp]:
haftmann@60685
  1191
  "normalize a dvd b \<longleftrightarrow> a dvd b"
haftmann@60685
  1192
proof -
haftmann@60685
  1193
  have "normalize a dvd b \<longleftrightarrow> unit_factor a * normalize a dvd b"
haftmann@60685
  1194
    using mult_unit_dvd_iff [of "unit_factor a" "normalize a" b]
haftmann@60685
  1195
      by (cases "a = 0") simp_all
haftmann@60685
  1196
  then show ?thesis by simp
haftmann@60685
  1197
qed
haftmann@60685
  1198
haftmann@60685
  1199
lemma dvd_normalize_iff [simp]:
haftmann@60685
  1200
  "a dvd normalize b \<longleftrightarrow> a dvd b"
haftmann@60685
  1201
proof -
haftmann@60685
  1202
  have "a dvd normalize  b \<longleftrightarrow> a dvd normalize b * unit_factor b"
haftmann@60685
  1203
    using dvd_mult_unit_iff [of "unit_factor b" a "normalize b"]
haftmann@60685
  1204
      by (cases "b = 0") simp_all
haftmann@60685
  1205
  then show ?thesis by simp
haftmann@60685
  1206
qed
haftmann@60685
  1207
haftmann@60688
  1208
text \<open>
haftmann@60688
  1209
  We avoid an explicit definition of associated elements but prefer
haftmann@60688
  1210
  explicit normalisation instead.  In theory we could define an abbreviation
haftmann@60688
  1211
  like @{prop "associated a b \<longleftrightarrow> normalize a = normalize b"} but this is
haftmann@60688
  1212
  counterproductive without suggestive infix syntax, which we do not want
haftmann@60688
  1213
  to sacrifice for this purpose here.
haftmann@60688
  1214
\<close>
haftmann@60685
  1215
haftmann@60688
  1216
lemma associatedI:
haftmann@60688
  1217
  assumes "a dvd b" and "b dvd a"
haftmann@60688
  1218
  shows "normalize a = normalize b"
haftmann@60685
  1219
proof (cases "a = 0 \<or> b = 0")
haftmann@60688
  1220
  case True with assms show ?thesis by auto
haftmann@60685
  1221
next
haftmann@60685
  1222
  case False
haftmann@60688
  1223
  from \<open>a dvd b\<close> obtain c where b: "b = a * c" ..
haftmann@60688
  1224
  moreover from \<open>b dvd a\<close> obtain d where a: "a = b * d" ..
haftmann@60688
  1225
  ultimately have "b * 1 = b * (c * d)" by (simp add: ac_simps)
haftmann@60688
  1226
  with False have "1 = c * d"
haftmann@60688
  1227
    unfolding mult_cancel_left by simp
haftmann@60688
  1228
  then have "is_unit c" and "is_unit d" by auto
haftmann@60688
  1229
  with a b show ?thesis by (simp add: normalize_mult is_unit_normalize)
haftmann@60688
  1230
qed
haftmann@60688
  1231
haftmann@60688
  1232
lemma associatedD1:
haftmann@60688
  1233
  "normalize a = normalize b \<Longrightarrow> a dvd b"
haftmann@60688
  1234
  using dvd_normalize_iff [of _ b, symmetric] normalize_dvd_iff [of a _, symmetric]
haftmann@60688
  1235
  by simp
haftmann@60688
  1236
haftmann@60688
  1237
lemma associatedD2:
haftmann@60688
  1238
  "normalize a = normalize b \<Longrightarrow> b dvd a"
haftmann@60688
  1239
  using dvd_normalize_iff [of _ a, symmetric] normalize_dvd_iff [of b _, symmetric]
haftmann@60688
  1240
  by simp
haftmann@60688
  1241
haftmann@60688
  1242
lemma associated_unit:
haftmann@60688
  1243
  "normalize a = normalize b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"
haftmann@60688
  1244
  using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2)
haftmann@60688
  1245
haftmann@60688
  1246
lemma associated_iff_dvd:
haftmann@60688
  1247
  "normalize a = normalize b \<longleftrightarrow> a dvd b \<and> b dvd a" (is "?P \<longleftrightarrow> ?Q")
haftmann@60688
  1248
proof
haftmann@60688
  1249
  assume ?Q then show ?P by (auto intro!: associatedI)
haftmann@60688
  1250
next
haftmann@60688
  1251
  assume ?P
haftmann@60688
  1252
  then have "unit_factor a * normalize a = unit_factor a * normalize b"
haftmann@60688
  1253
    by simp
haftmann@60688
  1254
  then have *: "normalize b * unit_factor a = a"
haftmann@60688
  1255
    by (simp add: ac_simps)
haftmann@60688
  1256
  show ?Q
haftmann@60688
  1257
  proof (cases "a = 0 \<or> b = 0")
haftmann@60688
  1258
    case True with \<open>?P\<close> show ?thesis by auto
haftmann@60685
  1259
  next
hoelzl@62376
  1260
    case False
haftmann@60688
  1261
    then have "b dvd normalize b * unit_factor a" and "normalize b * unit_factor a dvd b"
haftmann@60688
  1262
      by (simp_all add: mult_unit_dvd_iff dvd_mult_unit_iff)
haftmann@60688
  1263
    with * show ?thesis by simp
haftmann@60685
  1264
  qed
haftmann@60685
  1265
qed
haftmann@60685
  1266
haftmann@60685
  1267
lemma associated_eqI:
haftmann@60688
  1268
  assumes "a dvd b" and "b dvd a"
haftmann@60688
  1269
  assumes "normalize a = a" and "normalize b = b"
haftmann@60685
  1270
  shows "a = b"
haftmann@60688
  1271
proof -
haftmann@60688
  1272
  from assms have "normalize a = normalize b"
haftmann@60688
  1273
    unfolding associated_iff_dvd by simp
haftmann@60688
  1274
  with \<open>normalize a = a\<close> have "a = normalize b" by simp
haftmann@60688
  1275
  with \<open>normalize b = b\<close> show "a = b" by simp
haftmann@60685
  1276
qed
haftmann@60685
  1277
haftmann@60685
  1278
end
haftmann@60685
  1279
hoelzl@62376
  1280
class ordered_semiring = semiring + ordered_comm_monoid_add +
haftmann@38642
  1281
  assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@38642
  1282
  assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
haftmann@25230
  1283
begin
haftmann@25230
  1284
haftmann@25230
  1285
lemma mult_mono:
haftmann@38642
  1286
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
  1287
apply (erule mult_right_mono [THEN order_trans], assumption)
haftmann@25230
  1288
apply (erule mult_left_mono, assumption)
haftmann@25230
  1289
done
haftmann@25230
  1290
haftmann@25230
  1291
lemma mult_mono':
haftmann@38642
  1292
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
  1293
apply (rule mult_mono)
haftmann@25230
  1294
apply (fast intro: order_trans)+
haftmann@25230
  1295
done
haftmann@25230
  1296
haftmann@25230
  1297
end
krauss@21199
  1298
hoelzl@62377
  1299
class ordered_semiring_0 = semiring_0 + ordered_semiring
haftmann@25267
  1300
begin
paulson@14268
  1301
nipkow@56536
  1302
lemma mult_nonneg_nonneg[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
haftmann@36301
  1303
using mult_left_mono [of 0 b a] by simp
haftmann@25230
  1304
haftmann@25230
  1305
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
haftmann@36301
  1306
using mult_left_mono [of b 0 a] by simp
huffman@30692
  1307
huffman@30692
  1308
lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"
haftmann@36301
  1309
using mult_right_mono [of a 0 b] by simp
huffman@30692
  1310
wenzelm@61799
  1311
text \<open>Legacy - use \<open>mult_nonpos_nonneg\<close>\<close>
lp15@60562
  1312
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0"
haftmann@36301
  1313
by (drule mult_right_mono [of b 0], auto)
haftmann@25230
  1314
hoelzl@62378
  1315
lemma split_mult_neg_le: "(0 \<le> a \<and> b \<le> 0) \<or> (a \<le> 0 \<and> 0 \<le> b) \<Longrightarrow> a * b \<le> 0"
nipkow@29667
  1316
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
haftmann@25230
  1317
haftmann@25230
  1318
end
haftmann@25230
  1319
hoelzl@62377
  1320
class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add
hoelzl@62377
  1321
begin
hoelzl@62377
  1322
hoelzl@62377
  1323
subclass semiring_0_cancel ..
hoelzl@62377
  1324
subclass ordered_semiring_0 ..
hoelzl@62377
  1325
hoelzl@62377
  1326
end
hoelzl@62377
  1327
haftmann@38642
  1328
class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add
haftmann@25267
  1329
begin
haftmann@25230
  1330
haftmann@35028
  1331
subclass ordered_cancel_semiring ..
haftmann@35028
  1332
hoelzl@62376
  1333
subclass ordered_cancel_comm_monoid_add ..
haftmann@25304
  1334
haftmann@25230
  1335
lemma mult_left_less_imp_less:
haftmann@25230
  1336
  "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
  1337
by (force simp add: mult_left_mono not_le [symmetric])
lp15@60562
  1338
haftmann@25230
  1339
lemma mult_right_less_imp_less:
haftmann@25230
  1340
  "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
  1341
by (force simp add: mult_right_mono not_le [symmetric])
obua@23521
  1342
haftmann@25186
  1343
end
haftmann@25152
  1344
haftmann@35043
  1345
class linordered_semiring_1 = linordered_semiring + semiring_1
hoelzl@36622
  1346
begin
hoelzl@36622
  1347
hoelzl@36622
  1348
lemma convex_bound_le:
hoelzl@36622
  1349
  assumes "x \<le> a" "y \<le> a" "0 \<le> u" "0 \<le> v" "u + v = 1"
hoelzl@36622
  1350
  shows "u * x + v * y \<le> a"
hoelzl@36622
  1351
proof-
hoelzl@36622
  1352
  from assms have "u * x + v * y \<le> u * a + v * a"
hoelzl@36622
  1353
    by (simp add: add_mono mult_left_mono)
webertj@49962
  1354
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
hoelzl@36622
  1355
qed
hoelzl@36622
  1356
hoelzl@36622
  1357
end
haftmann@35043
  1358
haftmann@35043
  1359
class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +
haftmann@25062
  1360
  assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25062
  1361
  assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
haftmann@25267
  1362
begin
paulson@14341
  1363
huffman@27516
  1364
subclass semiring_0_cancel ..
obua@14940
  1365
haftmann@35028
  1366
subclass linordered_semiring
haftmann@28823
  1367
proof
huffman@23550
  1368
  fix a b c :: 'a
huffman@23550
  1369
  assume A: "a \<le> b" "0 \<le> c"
huffman@23550
  1370
  from A show "c * a \<le> c * b"
haftmann@25186
  1371
    unfolding le_less
haftmann@25186
  1372
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
  1373
  from A show "a * c \<le> b * c"
haftmann@25152
  1374
    unfolding le_less
haftmann@25186
  1375
    using mult_strict_right_mono by (cases "c = 0") auto
haftmann@25152
  1376
qed
haftmann@25152
  1377
haftmann@25230
  1378
lemma mult_left_le_imp_le:
haftmann@25230
  1379
  "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
  1380
by (force simp add: mult_strict_left_mono _not_less [symmetric])
lp15@60562
  1381
haftmann@25230
  1382
lemma mult_right_le_imp_le:
haftmann@25230
  1383
  "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
  1384
by (force simp add: mult_strict_right_mono not_less [symmetric])
haftmann@25230
  1385
nipkow@56544
  1386
lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
haftmann@36301
  1387
using mult_strict_left_mono [of 0 b a] by simp
huffman@30692
  1388
huffman@30692
  1389
lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
haftmann@36301
  1390
using mult_strict_left_mono [of b 0 a] by simp
huffman@30692
  1391
huffman@30692
  1392
lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"
haftmann@36301
  1393
using mult_strict_right_mono [of a 0 b] by simp
huffman@30692
  1394
wenzelm@61799
  1395
text \<open>Legacy - use \<open>mult_neg_pos\<close>\<close>
lp15@60562
  1396
lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0"
haftmann@36301
  1397
by (drule mult_strict_right_mono [of b 0], auto)
haftmann@25230
  1398
haftmann@25230
  1399
lemma zero_less_mult_pos:
haftmann@25230
  1400
  "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
  1401
apply (cases "b\<le>0")
haftmann@25230
  1402
 apply (auto simp add: le_less not_less)
huffman@30692
  1403
apply (drule_tac mult_pos_neg [of a b])
haftmann@25230
  1404
 apply (auto dest: less_not_sym)
haftmann@25230
  1405
done
haftmann@25230
  1406
haftmann@25230
  1407
lemma zero_less_mult_pos2:
haftmann@25230
  1408
  "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
  1409
apply (cases "b\<le>0")
haftmann@25230
  1410
 apply (auto simp add: le_less not_less)
huffman@30692
  1411
apply (drule_tac mult_pos_neg2 [of a b])
haftmann@25230
  1412
 apply (auto dest: less_not_sym)
haftmann@25230
  1413
done
haftmann@25230
  1414
wenzelm@60758
  1415
text\<open>Strict monotonicity in both arguments\<close>
haftmann@26193
  1416
lemma mult_strict_mono:
haftmann@26193
  1417
  assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
haftmann@26193
  1418
  shows "a * c < b * d"
haftmann@26193
  1419
  using assms apply (cases "c=0")
nipkow@56544
  1420
  apply (simp)
haftmann@26193
  1421
  apply (erule mult_strict_right_mono [THEN less_trans])
huffman@30692
  1422
  apply (force simp add: le_less)
haftmann@26193
  1423
  apply (erule mult_strict_left_mono, assumption)
haftmann@26193
  1424
  done
haftmann@26193
  1425
wenzelm@60758
  1426
text\<open>This weaker variant has more natural premises\<close>
haftmann@26193
  1427
lemma mult_strict_mono':
haftmann@26193
  1428
  assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
haftmann@26193
  1429
  shows "a * c < b * d"
nipkow@29667
  1430
by (rule mult_strict_mono) (insert assms, auto)
haftmann@26193
  1431
haftmann@26193
  1432
lemma mult_less_le_imp_less:
haftmann@26193
  1433
  assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
haftmann@26193
  1434
  shows "a * c < b * d"
haftmann@26193
  1435
  using assms apply (subgoal_tac "a * c < b * c")
haftmann@26193
  1436
  apply (erule less_le_trans)
haftmann@26193
  1437
  apply (erule mult_left_mono)
haftmann@26193
  1438
  apply simp
haftmann@26193
  1439
  apply (erule mult_strict_right_mono)
haftmann@26193
  1440
  apply assumption
haftmann@26193
  1441
  done
haftmann@26193
  1442
haftmann@26193
  1443
lemma mult_le_less_imp_less:
haftmann@26193
  1444
  assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
haftmann@26193
  1445
  shows "a * c < b * d"
haftmann@26193
  1446
  using assms apply (subgoal_tac "a * c \<le> b * c")
haftmann@26193
  1447
  apply (erule le_less_trans)
haftmann@26193
  1448
  apply (erule mult_strict_left_mono)
haftmann@26193
  1449
  apply simp
haftmann@26193
  1450
  apply (erule mult_right_mono)
haftmann@26193
  1451
  apply simp
haftmann@26193
  1452
  done
haftmann@26193
  1453
haftmann@25230
  1454
end
haftmann@25230
  1455
haftmann@35097
  1456
class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1
hoelzl@36622
  1457
begin
hoelzl@36622
  1458
hoelzl@36622
  1459
subclass linordered_semiring_1 ..
hoelzl@36622
  1460
hoelzl@36622
  1461
lemma convex_bound_lt:
hoelzl@36622
  1462
  assumes "x < a" "y < a" "0 \<le> u" "0 \<le> v" "u + v = 1"
hoelzl@36622
  1463
  shows "u * x + v * y < a"
hoelzl@36622
  1464
proof -
hoelzl@36622
  1465
  from assms have "u * x + v * y < u * a + v * a"
hoelzl@36622
  1466
    by (cases "u = 0")
hoelzl@36622
  1467
       (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
webertj@49962
  1468
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
hoelzl@36622
  1469
qed
hoelzl@36622
  1470
hoelzl@36622
  1471
end
haftmann@33319
  1472
lp15@60562
  1473
class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add +
haftmann@38642
  1474
  assumes comm_mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@25186
  1475
begin
haftmann@25152
  1476
haftmann@35028
  1477
subclass ordered_semiring
haftmann@28823
  1478
proof
krauss@21199
  1479
  fix a b c :: 'a
huffman@23550
  1480
  assume "a \<le> b" "0 \<le> c"
haftmann@38642
  1481
  thus "c * a \<le> c * b" by (rule comm_mult_left_mono)
haftmann@57512
  1482
  thus "a * c \<le> b * c" by (simp only: mult.commute)
krauss@21199
  1483
qed
paulson@14265
  1484
haftmann@25267
  1485
end
haftmann@25267
  1486
haftmann@38642
  1487
class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add
haftmann@25267
  1488
begin
paulson@14265
  1489
haftmann@38642
  1490
subclass comm_semiring_0_cancel ..
haftmann@35028
  1491
subclass ordered_comm_semiring ..
haftmann@35028
  1492
subclass ordered_cancel_semiring ..
haftmann@25267
  1493
haftmann@25267
  1494
end
haftmann@25267
  1495
haftmann@35028
  1496
class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +
haftmann@38642
  1497
  assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25267
  1498
begin
haftmann@25267
  1499
haftmann@35043
  1500
subclass linordered_semiring_strict
haftmann@28823
  1501
proof
huffman@23550
  1502
  fix a b c :: 'a
huffman@23550
  1503
  assume "a < b" "0 < c"
haftmann@38642
  1504
  thus "c * a < c * b" by (rule comm_mult_strict_left_mono)
haftmann@57512
  1505
  thus "a * c < b * c" by (simp only: mult.commute)
huffman@23550
  1506
qed
paulson@14272
  1507
haftmann@35028
  1508
subclass ordered_cancel_comm_semiring
haftmann@28823
  1509
proof
huffman@23550
  1510
  fix a b c :: 'a
huffman@23550
  1511
  assume "a \<le> b" "0 \<le> c"
huffman@23550
  1512
  thus "c * a \<le> c * b"
haftmann@25186
  1513
    unfolding le_less
haftmann@26193
  1514
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
  1515
qed
paulson@14272
  1516
haftmann@25267
  1517
end
haftmann@25230
  1518
lp15@60562
  1519
class ordered_ring = ring + ordered_cancel_semiring
haftmann@25267
  1520
begin
haftmann@25230
  1521
haftmann@35028
  1522
subclass ordered_ab_group_add ..
paulson@14270
  1523
haftmann@25230
  1524
lemma less_add_iff1:
haftmann@25230
  1525
  "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
nipkow@29667
  1526
by (simp add: algebra_simps)
haftmann@25230
  1527
haftmann@25230
  1528
lemma less_add_iff2:
haftmann@25230
  1529
  "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
nipkow@29667
  1530
by (simp add: algebra_simps)
haftmann@25230
  1531
haftmann@25230
  1532
lemma le_add_iff1:
haftmann@25230
  1533
  "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
nipkow@29667
  1534
by (simp add: algebra_simps)
haftmann@25230
  1535
haftmann@25230
  1536
lemma le_add_iff2:
haftmann@25230
  1537
  "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
nipkow@29667
  1538
by (simp add: algebra_simps)
haftmann@25230
  1539
haftmann@25230
  1540
lemma mult_left_mono_neg:
haftmann@25230
  1541
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
haftmann@36301
  1542
  apply (drule mult_left_mono [of _ _ "- c"])
huffman@35216
  1543
  apply simp_all
haftmann@25230
  1544
  done
haftmann@25230
  1545
haftmann@25230
  1546
lemma mult_right_mono_neg:
haftmann@25230
  1547
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
haftmann@36301
  1548
  apply (drule mult_right_mono [of _ _ "- c"])
huffman@35216
  1549
  apply simp_all
haftmann@25230
  1550
  done
haftmann@25230
  1551
huffman@30692
  1552
lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
haftmann@36301
  1553
using mult_right_mono_neg [of a 0 b] by simp
haftmann@25230
  1554
haftmann@25230
  1555
lemma split_mult_pos_le:
haftmann@25230
  1556
  "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
nipkow@56536
  1557
by (auto simp add: mult_nonpos_nonpos)
haftmann@25186
  1558
haftmann@25186
  1559
end
paulson@14270
  1560
haftmann@35028
  1561
class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if
haftmann@25304
  1562
begin
haftmann@25304
  1563
haftmann@35028
  1564
subclass ordered_ring ..
haftmann@35028
  1565
haftmann@35028
  1566
subclass ordered_ab_group_add_abs
haftmann@28823
  1567
proof
haftmann@25304
  1568
  fix a b
haftmann@25304
  1569
  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@54230
  1570
    by (auto simp add: abs_if not_le not_less algebra_simps simp del: add.commute dest: add_neg_neg add_nonneg_nonneg)
huffman@35216
  1571
qed (auto simp add: abs_if)
haftmann@25304
  1572
huffman@35631
  1573
lemma zero_le_square [simp]: "0 \<le> a * a"
huffman@35631
  1574
  using linear [of 0 a]
nipkow@56536
  1575
  by (auto simp add: mult_nonpos_nonpos)
huffman@35631
  1576
huffman@35631
  1577
lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
huffman@35631
  1578
  by (simp add: not_less)
huffman@35631
  1579
wenzelm@61944
  1580
proposition abs_eq_iff: "\<bar>x\<bar> = \<bar>y\<bar> \<longleftrightarrow> x = y \<or> x = -y"
lp15@61762
  1581
  by (auto simp add: abs_if split: split_if_asm)
lp15@61762
  1582
haftmann@62347
  1583
lemma sum_squares_ge_zero:
haftmann@62347
  1584
  "0 \<le> x * x + y * y"
haftmann@62347
  1585
  by (intro add_nonneg_nonneg zero_le_square)
haftmann@62347
  1586
haftmann@62347
  1587
lemma not_sum_squares_lt_zero:
haftmann@62347
  1588
  "\<not> x * x + y * y < 0"
haftmann@62347
  1589
  by (simp add: not_less sum_squares_ge_zero)
haftmann@62347
  1590
haftmann@25304
  1591
end
obua@23521
  1592
haftmann@35043
  1593
class linordered_ring_strict = ring + linordered_semiring_strict
haftmann@25304
  1594
  + ordered_ab_group_add + abs_if
haftmann@25230
  1595
begin
paulson@14348
  1596
haftmann@35028
  1597
subclass linordered_ring ..
haftmann@25304
  1598
huffman@30692
  1599
lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
huffman@30692
  1600
using mult_strict_left_mono [of b a "- c"] by simp
huffman@30692
  1601
huffman@30692
  1602
lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
huffman@30692
  1603
using mult_strict_right_mono [of b a "- c"] by simp
huffman@30692
  1604
huffman@30692
  1605
lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
haftmann@36301
  1606
using mult_strict_right_mono_neg [of a 0 b] by simp
obua@14738
  1607
haftmann@25917
  1608
subclass ring_no_zero_divisors
haftmann@28823
  1609
proof
haftmann@25917
  1610
  fix a b
haftmann@25917
  1611
  assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
haftmann@25917
  1612
  assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
haftmann@25917
  1613
  have "a * b < 0 \<or> 0 < a * b"
haftmann@25917
  1614
  proof (cases "a < 0")
haftmann@25917
  1615
    case True note A' = this
haftmann@25917
  1616
    show ?thesis proof (cases "b < 0")
haftmann@25917
  1617
      case True with A'
haftmann@25917
  1618
      show ?thesis by (auto dest: mult_neg_neg)
haftmann@25917
  1619
    next
haftmann@25917
  1620
      case False with B have "0 < b" by auto
haftmann@25917
  1621
      with A' show ?thesis by (auto dest: mult_strict_right_mono)
haftmann@25917
  1622
    qed
haftmann@25917
  1623
  next
haftmann@25917
  1624
    case False with A have A': "0 < a" by auto
haftmann@25917
  1625
    show ?thesis proof (cases "b < 0")
haftmann@25917
  1626
      case True with A'
haftmann@25917
  1627
      show ?thesis by (auto dest: mult_strict_right_mono_neg)
haftmann@25917
  1628
    next
haftmann@25917
  1629
      case False with B have "0 < b" by auto
nipkow@56544
  1630
      with A' show ?thesis by auto
haftmann@25917
  1631
    qed
haftmann@25917
  1632
  qed
haftmann@25917
  1633
  then show "a * b \<noteq> 0" by (simp add: neq_iff)
haftmann@25917
  1634
qed
haftmann@25304
  1635
hoelzl@56480
  1636
lemma zero_less_mult_iff: "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
hoelzl@56480
  1637
  by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
nipkow@56544
  1638
     (auto simp add: mult_neg_neg not_less le_less dest: zero_less_mult_pos zero_less_mult_pos2)
huffman@22990
  1639
hoelzl@56480
  1640
lemma zero_le_mult_iff: "0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
hoelzl@56480
  1641
  by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
paulson@14265
  1642
paulson@14265
  1643
lemma mult_less_0_iff:
haftmann@25917
  1644
  "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
huffman@35216
  1645
  apply (insert zero_less_mult_iff [of "-a" b])
huffman@35216
  1646
  apply force
haftmann@25917
  1647
  done
paulson@14265
  1648
paulson@14265
  1649
lemma mult_le_0_iff:
haftmann@25917
  1650
  "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
lp15@60562
  1651
  apply (insert zero_le_mult_iff [of "-a" b])
huffman@35216
  1652
  apply force
haftmann@25917
  1653
  done
haftmann@25917
  1654
wenzelm@60758
  1655
text\<open>Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
wenzelm@61799
  1656
   also with the relations \<open>\<le>\<close> and equality.\<close>
haftmann@26193
  1657
wenzelm@60758
  1658
text\<open>These ``disjunction'' versions produce two cases when the comparison is
wenzelm@60758
  1659
 an assumption, but effectively four when the comparison is a goal.\<close>
haftmann@26193
  1660
haftmann@26193
  1661
lemma mult_less_cancel_right_disj:
haftmann@26193
  1662
  "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
  1663
  apply (cases "c = 0")
lp15@60562
  1664
  apply (auto simp add: neq_iff mult_strict_right_mono
haftmann@26193
  1665
                      mult_strict_right_mono_neg)
lp15@60562
  1666
  apply (auto simp add: not_less
haftmann@26193
  1667
                      not_le [symmetric, of "a*c"]
haftmann@26193
  1668
                      not_le [symmetric, of a])
haftmann@26193
  1669
  apply (erule_tac [!] notE)
lp15@60562
  1670
  apply (auto simp add: less_imp_le mult_right_mono
haftmann@26193
  1671
                      mult_right_mono_neg)
haftmann@26193
  1672
  done
haftmann@26193
  1673
haftmann@26193
  1674
lemma mult_less_cancel_left_disj:
haftmann@26193
  1675
  "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
  1676
  apply (cases "c = 0")
lp15@60562
  1677
  apply (auto simp add: neq_iff mult_strict_left_mono
haftmann@26193
  1678
                      mult_strict_left_mono_neg)
lp15@60562
  1679
  apply (auto simp add: not_less
haftmann@26193
  1680
                      not_le [symmetric, of "c*a"]
haftmann@26193
  1681
                      not_le [symmetric, of a])
haftmann@26193
  1682
  apply (erule_tac [!] notE)
lp15@60562
  1683
  apply (auto simp add: less_imp_le mult_left_mono
haftmann@26193
  1684
                      mult_left_mono_neg)
haftmann@26193
  1685
  done
haftmann@26193
  1686
wenzelm@60758
  1687
text\<open>The ``conjunction of implication'' lemmas produce two cases when the
wenzelm@60758
  1688
comparison is a goal, but give four when the comparison is an assumption.\<close>
haftmann@26193
  1689
haftmann@26193
  1690
lemma mult_less_cancel_right:
haftmann@26193
  1691
  "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
  1692
  using mult_less_cancel_right_disj [of a c b] by auto
haftmann@26193
  1693
haftmann@26193
  1694
lemma mult_less_cancel_left:
haftmann@26193
  1695
  "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
  1696
  using mult_less_cancel_left_disj [of c a b] by auto
haftmann@26193
  1697
haftmann@26193
  1698
lemma mult_le_cancel_right:
haftmann@26193
  1699
   "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
  1700
by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
haftmann@26193
  1701
haftmann@26193
  1702
lemma mult_le_cancel_left:
haftmann@26193
  1703
  "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
  1704
by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
haftmann@26193
  1705
nipkow@30649
  1706
lemma mult_le_cancel_left_pos:
nipkow@30649
  1707
  "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
nipkow@30649
  1708
by (auto simp: mult_le_cancel_left)
nipkow@30649
  1709
nipkow@30649
  1710
lemma mult_le_cancel_left_neg:
nipkow@30649
  1711
  "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
nipkow@30649
  1712
by (auto simp: mult_le_cancel_left)
nipkow@30649
  1713
nipkow@30649
  1714
lemma mult_less_cancel_left_pos:
nipkow@30649
  1715
  "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
nipkow@30649
  1716
by (auto simp: mult_less_cancel_left)
nipkow@30649
  1717
nipkow@30649
  1718
lemma mult_less_cancel_left_neg:
nipkow@30649
  1719
  "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
nipkow@30649
  1720
by (auto simp: mult_less_cancel_left)
nipkow@30649
  1721
haftmann@25917
  1722
end
paulson@14265
  1723
huffman@30692
  1724
lemmas mult_sign_intros =
huffman@30692
  1725
  mult_nonneg_nonneg mult_nonneg_nonpos
huffman@30692
  1726
  mult_nonpos_nonneg mult_nonpos_nonpos
huffman@30692
  1727
  mult_pos_pos mult_pos_neg
huffman@30692
  1728
  mult_neg_pos mult_neg_neg
haftmann@25230
  1729
haftmann@35028
  1730
class ordered_comm_ring = comm_ring + ordered_comm_semiring
haftmann@25267
  1731
begin
haftmann@25230
  1732
haftmann@35028
  1733
subclass ordered_ring ..
haftmann@35028
  1734
subclass ordered_cancel_comm_semiring ..
haftmann@25230
  1735
haftmann@25267
  1736
end
haftmann@25230
  1737
hoelzl@62378
  1738
class zero_less_one = order + zero + one +
haftmann@25230
  1739
  assumes zero_less_one [simp]: "0 < 1"
hoelzl@62378
  1740
hoelzl@62378
  1741
class linordered_nonzero_semiring = ordered_comm_semiring + monoid_mult + linorder + zero_less_one
hoelzl@62378
  1742
begin
hoelzl@62378
  1743
hoelzl@62378
  1744
subclass zero_neq_one
hoelzl@62378
  1745
  proof qed (insert zero_less_one, blast)
hoelzl@62378
  1746
hoelzl@62378
  1747
subclass comm_semiring_1
hoelzl@62378
  1748
  proof qed (rule mult_1_left)
hoelzl@62378
  1749
hoelzl@62378
  1750
lemma zero_le_one [simp]: "0 \<le> 1"
hoelzl@62378
  1751
by (rule zero_less_one [THEN less_imp_le])
hoelzl@62378
  1752
hoelzl@62378
  1753
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
hoelzl@62378
  1754
by (simp add: not_le)
hoelzl@62378
  1755
hoelzl@62378
  1756
lemma not_one_less_zero [simp]: "\<not> 1 < 0"
hoelzl@62378
  1757
by (simp add: not_less)
hoelzl@62378
  1758
hoelzl@62378
  1759
lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a"
hoelzl@62378
  1760
  using mult_left_mono[of c 1 a] by simp
hoelzl@62378
  1761
hoelzl@62378
  1762
lemma mult_le_one: "a \<le> 1 \<Longrightarrow> 0 \<le> b \<Longrightarrow> b \<le> 1 \<Longrightarrow> a * b \<le> 1"
hoelzl@62378
  1763
  using mult_mono[of a 1 b 1] by simp
hoelzl@62378
  1764
hoelzl@62378
  1765
lemma zero_less_two: "0 < 1 + 1"
hoelzl@62378
  1766
  using add_pos_pos[OF zero_less_one zero_less_one] .
hoelzl@62378
  1767
hoelzl@62378
  1768
end
hoelzl@62378
  1769
hoelzl@62378
  1770
class linordered_semidom = semidom + linordered_comm_semiring_strict + zero_less_one +
lp15@60562
  1771
  assumes le_add_diff_inverse2 [simp]: "b \<le> a \<Longrightarrow> a - b + b = a"
haftmann@25230
  1772
begin
haftmann@25230
  1773
hoelzl@62378
  1774
subclass linordered_nonzero_semiring
hoelzl@62378
  1775
  proof qed
hoelzl@62378
  1776
wenzelm@60758
  1777
text \<open>Addition is the inverse of subtraction.\<close>
lp15@60562
  1778
lp15@60562
  1779
lemma le_add_diff_inverse [simp]: "b \<le> a \<Longrightarrow> b + (a - b) = a"
lp15@60562
  1780
  by (frule le_add_diff_inverse2) (simp add: add.commute)
lp15@60562
  1781
hoelzl@62378
  1782
lemma add_diff_inverse: "\<not> a < b \<Longrightarrow> b + (a - b) = a"
lp15@60562
  1783
  by simp
lp15@60615
  1784
hoelzl@62376
  1785
lemma add_le_imp_le_diff:
lp15@60615
  1786
  shows "i + k \<le> n \<Longrightarrow> i \<le> n - k"
lp15@60615
  1787
  apply (subst add_le_cancel_right [where c=k, symmetric])
lp15@60615
  1788
  apply (frule le_add_diff_inverse2)
lp15@60615
  1789
  apply (simp only: add.assoc [symmetric])
lp15@60615
  1790
  using add_implies_diff by fastforce
lp15@60615
  1791
hoelzl@62376
  1792
lemma add_le_add_imp_diff_le:
lp15@60615
  1793
  assumes a1: "i + k \<le> n"
lp15@60615
  1794
      and a2: "n \<le> j + k"
lp15@60615
  1795
  shows "\<lbrakk>i + k \<le> n; n \<le> j + k\<rbrakk> \<Longrightarrow> n - k \<le> j"
lp15@60615
  1796
proof -
lp15@60615
  1797
  have "n - (i + k) + (i + k) = n"
lp15@60615
  1798
    using a1 by simp
lp15@60615
  1799
  moreover have "n - k = n - k - i + i"
lp15@60615
  1800
    using a1 by (simp add: add_le_imp_le_diff)
lp15@60615
  1801
  ultimately show ?thesis
lp15@60615
  1802
    using a2
lp15@60615
  1803
    apply (simp add: add.assoc [symmetric])
lp15@60615
  1804
    apply (rule add_le_imp_le_diff [of _ k "j+k", simplified add_diff_cancel_right'])
lp15@60615
  1805
    by (simp add: add.commute diff_diff_add)
lp15@60615
  1806
qed
lp15@60615
  1807
haftmann@26193
  1808
lemma less_1_mult:
hoelzl@62378
  1809
  "1 < m \<Longrightarrow> 1 < n \<Longrightarrow> 1 < m * n"
hoelzl@62378
  1810
  using mult_strict_mono [of 1 m 1 n] by (simp add: less_trans [OF zero_less_one])
hoelzl@59000
  1811
haftmann@25230
  1812
end
haftmann@25230
  1813
hoelzl@62378
  1814
class linordered_idom =
hoelzl@62378
  1815
  comm_ring_1 + linordered_comm_semiring_strict + ordered_ab_group_add + abs_if + sgn_if
haftmann@25917
  1816
begin
haftmann@25917
  1817
hoelzl@36622
  1818
subclass linordered_semiring_1_strict ..
haftmann@35043
  1819
subclass linordered_ring_strict ..
haftmann@35028
  1820
subclass ordered_comm_ring ..
huffman@27516
  1821
subclass idom ..
haftmann@25917
  1822
haftmann@35028
  1823
subclass linordered_semidom
haftmann@28823
  1824
proof
haftmann@26193
  1825
  have "0 \<le> 1 * 1" by (rule zero_le_square)
haftmann@26193
  1826
  thus "0 < 1" by (simp add: le_less)
lp15@60562
  1827
  show "\<And>b a. b \<le> a \<Longrightarrow> a - b + b = a"
lp15@60562
  1828
    by simp
lp15@60562
  1829
qed
haftmann@25917
  1830
haftmann@35028
  1831
lemma linorder_neqE_linordered_idom:
haftmann@26193
  1832
  assumes "x \<noteq> y" obtains "x < y" | "y < x"
haftmann@26193
  1833
  using assms by (rule neqE)
haftmann@26193
  1834
wenzelm@60758
  1835
text \<open>These cancellation simprules also produce two cases when the comparison is a goal.\<close>
haftmann@26274
  1836
haftmann@26274
  1837
lemma mult_le_cancel_right1:
haftmann@26274
  1838
  "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1839
by (insert mult_le_cancel_right [of 1 c b], simp)
haftmann@26274
  1840
haftmann@26274
  1841
lemma mult_le_cancel_right2:
haftmann@26274
  1842
  "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1843
by (insert mult_le_cancel_right [of a c 1], simp)
haftmann@26274
  1844
haftmann@26274
  1845
lemma mult_le_cancel_left1:
haftmann@26274
  1846
  "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1847
by (insert mult_le_cancel_left [of c 1 b], simp)
haftmann@26274
  1848
haftmann@26274
  1849
lemma mult_le_cancel_left2:
haftmann@26274
  1850
  "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1851
by (insert mult_le_cancel_left [of c a 1], simp)
haftmann@26274
  1852
haftmann@26274
  1853
lemma mult_less_cancel_right1:
haftmann@26274
  1854
  "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1855
by (insert mult_less_cancel_right [of 1 c b], simp)
haftmann@26274
  1856
haftmann@26274
  1857
lemma mult_less_cancel_right2:
haftmann@26274
  1858
  "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1859
by (insert mult_less_cancel_right [of a c 1], simp)
haftmann@26274
  1860
haftmann@26274
  1861
lemma mult_less_cancel_left1:
haftmann@26274
  1862
  "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1863
by (insert mult_less_cancel_left [of c 1 b], simp)
haftmann@26274
  1864
haftmann@26274
  1865
lemma mult_less_cancel_left2:
haftmann@26274
  1866
  "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1867
by (insert mult_less_cancel_left [of c a 1], simp)
haftmann@26274
  1868
haftmann@27651
  1869
lemma sgn_sgn [simp]:
haftmann@27651
  1870
  "sgn (sgn a) = sgn a"
nipkow@29700
  1871
unfolding sgn_if by simp
haftmann@27651
  1872
haftmann@27651
  1873
lemma sgn_0_0:
haftmann@27651
  1874
  "sgn a = 0 \<longleftrightarrow> a = 0"
nipkow@29700
  1875
unfolding sgn_if by simp
haftmann@27651
  1876
haftmann@27651
  1877
lemma sgn_1_pos:
haftmann@27651
  1878
  "sgn a = 1 \<longleftrightarrow> a > 0"
huffman@35216
  1879
unfolding sgn_if by simp
haftmann@27651
  1880
haftmann@27651
  1881
lemma sgn_1_neg:
haftmann@27651
  1882
  "sgn a = - 1 \<longleftrightarrow> a < 0"
huffman@35216
  1883
unfolding sgn_if by auto
haftmann@27651
  1884
haftmann@29940
  1885
lemma sgn_pos [simp]:
haftmann@29940
  1886
  "0 < a \<Longrightarrow> sgn a = 1"
haftmann@29940
  1887
unfolding sgn_1_pos .
haftmann@29940
  1888
haftmann@29940
  1889
lemma sgn_neg [simp]:
haftmann@29940
  1890
  "a < 0 \<Longrightarrow> sgn a = - 1"
haftmann@29940
  1891
unfolding sgn_1_neg .
haftmann@29940
  1892
haftmann@27651
  1893
lemma sgn_times:
haftmann@27651
  1894
  "sgn (a * b) = sgn a * sgn b"
nipkow@29667
  1895
by (auto simp add: sgn_if zero_less_mult_iff)
haftmann@27651
  1896
haftmann@36301
  1897
lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"
nipkow@29700
  1898
unfolding sgn_if abs_if by auto
nipkow@29700
  1899
haftmann@29940
  1900
lemma sgn_greater [simp]:
haftmann@29940
  1901
  "0 < sgn a \<longleftrightarrow> 0 < a"
haftmann@29940
  1902
  unfolding sgn_if by auto
haftmann@29940
  1903
haftmann@29940
  1904
lemma sgn_less [simp]:
haftmann@29940
  1905
  "sgn a < 0 \<longleftrightarrow> a < 0"
haftmann@29940
  1906
  unfolding sgn_if by auto
haftmann@29940
  1907
haftmann@62347
  1908
lemma abs_sgn_eq:
haftmann@62347
  1909
  "\<bar>sgn a\<bar> = (if a = 0 then 0 else 1)"
haftmann@62347
  1910
  by (simp add: sgn_if)
haftmann@62347
  1911
haftmann@36301
  1912
lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"
huffman@29949
  1913
  by (simp add: abs_if)
huffman@29949
  1914
haftmann@36301
  1915
lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"
huffman@29949
  1916
  by (simp add: abs_if)
haftmann@29653
  1917
nipkow@33676
  1918
lemma dvd_if_abs_eq:
haftmann@36301
  1919
  "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
nipkow@33676
  1920
by(subst abs_dvd_iff[symmetric]) simp
nipkow@33676
  1921
wenzelm@60758
  1922
text \<open>The following lemmas can be proven in more general structures, but
lp15@60562
  1923
are dangerous as simp rules in absence of @{thm neg_equal_zero},
wenzelm@60758
  1924
@{thm neg_less_pos}, @{thm neg_less_eq_nonneg}.\<close>
haftmann@54489
  1925
haftmann@54489
  1926
lemma equation_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1927
  "1 = - a \<longleftrightarrow> a = - 1"
haftmann@54489
  1928
  by (fact equation_minus_iff)
haftmann@54489
  1929
haftmann@54489
  1930
lemma minus_equation_iff_1 [simp, no_atp]:
haftmann@54489
  1931
  "- a = 1 \<longleftrightarrow> a = - 1"
haftmann@54489
  1932
  by (subst minus_equation_iff, auto)
haftmann@54489
  1933
haftmann@54489
  1934
lemma le_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1935
  "1 \<le> - b \<longleftrightarrow> b \<le> - 1"
haftmann@54489
  1936
  by (fact le_minus_iff)
haftmann@54489
  1937
haftmann@54489
  1938
lemma minus_le_iff_1 [simp, no_atp]:
haftmann@54489
  1939
  "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"
haftmann@54489
  1940
  by (fact minus_le_iff)
haftmann@54489
  1941
haftmann@54489
  1942
lemma less_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1943
  "1 < - b \<longleftrightarrow> b < - 1"
haftmann@54489
  1944
  by (fact less_minus_iff)
haftmann@54489
  1945
haftmann@54489
  1946
lemma minus_less_iff_1 [simp, no_atp]:
haftmann@54489
  1947
  "- a < 1 \<longleftrightarrow> - 1 < a"
haftmann@54489
  1948
  by (fact minus_less_iff)
haftmann@54489
  1949
haftmann@25917
  1950
end
haftmann@25230
  1951
wenzelm@60758
  1952
text \<open>Simprules for comparisons where common factors can be cancelled.\<close>
paulson@15234
  1953
blanchet@54147
  1954
lemmas mult_compare_simps =
paulson@15234
  1955
    mult_le_cancel_right mult_le_cancel_left
paulson@15234
  1956
    mult_le_cancel_right1 mult_le_cancel_right2
paulson@15234
  1957
    mult_le_cancel_left1 mult_le_cancel_left2
paulson@15234
  1958
    mult_less_cancel_right mult_less_cancel_left
paulson@15234
  1959
    mult_less_cancel_right1 mult_less_cancel_right2
paulson@15234
  1960
    mult_less_cancel_left1 mult_less_cancel_left2
paulson@15234
  1961
    mult_cancel_right mult_cancel_left
paulson@15234
  1962
    mult_cancel_right1 mult_cancel_right2
paulson@15234
  1963
    mult_cancel_left1 mult_cancel_left2
paulson@15234
  1964
wenzelm@60758
  1965
text \<open>Reasoning about inequalities with division\<close>
avigad@16775
  1966
haftmann@35028
  1967
context linordered_semidom
haftmann@25193
  1968
begin
haftmann@25193
  1969
haftmann@25193
  1970
lemma less_add_one: "a < a + 1"
paulson@14293
  1971
proof -
haftmann@25193
  1972
  have "a + 0 < a + 1"
nipkow@23482
  1973
    by (blast intro: zero_less_one add_strict_left_mono)
paulson@14293
  1974
  thus ?thesis by simp
paulson@14293
  1975
qed
paulson@14293
  1976
haftmann@25193
  1977
end
paulson@14365
  1978
haftmann@36301
  1979
context linordered_idom
haftmann@36301
  1980
begin
paulson@15234
  1981
haftmann@36301
  1982
lemma mult_right_le_one_le:
haftmann@36301
  1983
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"
haftmann@59833
  1984
  by (rule mult_left_le)
haftmann@36301
  1985
haftmann@36301
  1986
lemma mult_left_le_one_le:
haftmann@36301
  1987
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"
haftmann@36301
  1988
  by (auto simp add: mult_le_cancel_right2)
haftmann@36301
  1989
haftmann@36301
  1990
end
haftmann@36301
  1991
wenzelm@60758
  1992
text \<open>Absolute Value\<close>
paulson@14293
  1993
haftmann@35028
  1994
context linordered_idom
haftmann@25304
  1995
begin
haftmann@25304
  1996
haftmann@36301
  1997
lemma mult_sgn_abs:
haftmann@36301
  1998
  "sgn x * \<bar>x\<bar> = x"
haftmann@25304
  1999
  unfolding abs_if sgn_if by auto
haftmann@25304
  2000
haftmann@36301
  2001
lemma abs_one [simp]:
haftmann@36301
  2002
  "\<bar>1\<bar> = 1"
huffman@44921
  2003
  by (simp add: abs_if)
haftmann@36301
  2004
haftmann@25304
  2005
end
nipkow@24491
  2006
haftmann@35028
  2007
class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +
haftmann@25304
  2008
  assumes abs_eq_mult:
haftmann@25304
  2009
    "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0) \<Longrightarrow> \<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
haftmann@25304
  2010
haftmann@35028
  2011
context linordered_idom
haftmann@30961
  2012
begin
haftmann@30961
  2013
haftmann@35028
  2014
subclass ordered_ring_abs proof
huffman@35216
  2015
qed (auto simp add: abs_if not_less mult_less_0_iff)
haftmann@30961
  2016
haftmann@30961
  2017
lemma abs_mult:
lp15@60562
  2018
  "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
haftmann@30961
  2019
  by (rule abs_eq_mult) auto
haftmann@30961
  2020
lp15@61649
  2021
lemma abs_mult_self [simp]:
haftmann@36301
  2022
  "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
lp15@60562
  2023
  by (simp add: abs_if)
haftmann@30961
  2024
paulson@14294
  2025
lemma abs_mult_less:
haftmann@36301
  2026
  "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"
paulson@14294
  2027
proof -
haftmann@36301
  2028
  assume ac: "\<bar>a\<bar> < c"
haftmann@36301
  2029
  hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)
haftmann@36301
  2030
  assume "\<bar>b\<bar> < d"
lp15@60562
  2031
  thus ?thesis by (simp add: ac cpos mult_strict_mono)
paulson@14294
  2032
qed
paulson@14293
  2033
haftmann@36301
  2034
lemma abs_less_iff:
lp15@60562
  2035
  "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b"
haftmann@36301
  2036
  by (simp add: less_le abs_le_iff) (auto simp add: abs_if)
obua@14738
  2037
haftmann@36301
  2038
lemma abs_mult_pos:
haftmann@36301
  2039
  "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"
haftmann@36301
  2040
  by (simp add: abs_mult)
haftmann@36301
  2041
hoelzl@51520
  2042
lemma abs_diff_less_iff:
hoelzl@51520
  2043
  "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"
hoelzl@51520
  2044
  by (auto simp add: diff_less_eq ac_simps abs_less_iff)
hoelzl@51520
  2045
lp15@59865
  2046
lemma abs_diff_le_iff:
lp15@59865
  2047
   "\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r"
lp15@59865
  2048
  by (auto simp add: diff_le_eq ac_simps abs_le_iff)
lp15@59865
  2049
haftmann@36301
  2050
end
avigad@16775
  2051
hoelzl@62376
  2052
subsection \<open>Dioids\<close>
hoelzl@62376
  2053
hoelzl@62376
  2054
text \<open>Dioids are the alternative extensions of semirings, a semiring can either be a ring or a dioid
hoelzl@62376
  2055
but never both.\<close>
hoelzl@62376
  2056
hoelzl@62376
  2057
class dioid = semiring_1 + canonically_ordered_monoid_add
hoelzl@62376
  2058
begin
hoelzl@62376
  2059
hoelzl@62376
  2060
subclass ordered_semiring
hoelzl@62376
  2061
  proof qed (auto simp: le_iff_add distrib_left distrib_right)
hoelzl@62376
  2062
hoelzl@62376
  2063
end
hoelzl@62376
  2064
hoelzl@62376
  2065
haftmann@59557
  2066
hide_fact (open) comm_mult_left_mono comm_mult_strict_left_mono distrib
haftmann@59557
  2067
haftmann@52435
  2068
code_identifier
haftmann@52435
  2069
  code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
  2070
paulson@14265
  2071
end