src/HOL/Rings.thy
author haftmann
Wed Jul 08 14:01:41 2015 +0200 (2015-07-08)
changeset 60688 01488b559910
parent 60685 cb21b7022b00
child 60690 a9e45c9588c3
permissions -rw-r--r--
avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
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@58889
    10
section {* Rings *}
paulson@14265
    11
haftmann@35050
    12
theory Rings
haftmann@35050
    13
imports Groups
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
haftmann@25152
    21
text{*For the @{text combine_numerals} simproc*}
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
haftmann@27651
   121
text {* Abstract divisibility *}
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@27651
   148
lemma dvd_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@59009
   158
lemma one_dvd [simp]:
haftmann@59009
   159
  "1 dvd a"
haftmann@59009
   160
  by (auto intro!: dvdI)
haftmann@28559
   161
haftmann@59009
   162
lemma dvd_mult [simp]:
haftmann@59009
   163
  "a dvd c \<Longrightarrow> a dvd (b * c)"
haftmann@59009
   164
  by (auto intro!: mult.left_commute dvdI elim!: dvdE)
haftmann@27651
   165
haftmann@59009
   166
lemma dvd_mult2 [simp]:
haftmann@59009
   167
  "a dvd b \<Longrightarrow> a dvd (b * c)"
lp15@60562
   168
  using dvd_mult [of a b c] by (simp add: ac_simps)
haftmann@27651
   169
haftmann@59009
   170
lemma dvd_triv_right [simp]:
haftmann@59009
   171
  "a dvd b * a"
haftmann@59009
   172
  by (rule dvd_mult) (rule dvd_refl)
haftmann@27651
   173
haftmann@59009
   174
lemma dvd_triv_left [simp]:
haftmann@59009
   175
  "a dvd a * b"
haftmann@59009
   176
  by (rule dvd_mult2) (rule dvd_refl)
haftmann@27651
   177
haftmann@27651
   178
lemma mult_dvd_mono:
nipkow@30042
   179
  assumes "a dvd b"
nipkow@30042
   180
    and "c dvd d"
haftmann@27651
   181
  shows "a * c dvd b * d"
haftmann@27651
   182
proof -
nipkow@30042
   183
  from `a dvd b` obtain b' where "b = a * b'" ..
nipkow@30042
   184
  moreover from `c dvd d` obtain d' where "d = c * d'" ..
haftmann@57514
   185
  ultimately have "b * d = (a * c) * (b' * d')" by (simp add: ac_simps)
haftmann@27651
   186
  then show ?thesis ..
haftmann@27651
   187
qed
haftmann@27651
   188
haftmann@59009
   189
lemma dvd_mult_left:
haftmann@59009
   190
  "a * b dvd c \<Longrightarrow> a dvd c"
haftmann@59009
   191
  by (simp add: dvd_def mult.assoc) blast
haftmann@27651
   192
haftmann@59009
   193
lemma dvd_mult_right:
haftmann@59009
   194
  "a * b dvd c \<Longrightarrow> b dvd c"
haftmann@59009
   195
  using dvd_mult_left [of b a c] by (simp add: ac_simps)
lp15@60562
   196
haftmann@59009
   197
end
haftmann@59009
   198
haftmann@59009
   199
class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult
haftmann@59009
   200
begin
haftmann@59009
   201
haftmann@59009
   202
subclass semiring_1 ..
haftmann@27651
   203
haftmann@59009
   204
lemma dvd_0_left_iff [simp]:
haftmann@59009
   205
  "0 dvd a \<longleftrightarrow> a = 0"
haftmann@59009
   206
  by (auto intro: dvd_refl elim!: dvdE)
haftmann@27651
   207
haftmann@59009
   208
lemma dvd_0_right [iff]:
haftmann@59009
   209
  "a dvd 0"
haftmann@59009
   210
proof
haftmann@59009
   211
  show "0 = a * 0" by simp
haftmann@59009
   212
qed
haftmann@59009
   213
haftmann@59009
   214
lemma dvd_0_left:
haftmann@59009
   215
  "0 dvd a \<Longrightarrow> a = 0"
haftmann@59009
   216
  by simp
haftmann@59009
   217
haftmann@59009
   218
lemma dvd_add [simp]:
haftmann@59009
   219
  assumes "a dvd b" and "a dvd c"
haftmann@59009
   220
  shows "a dvd (b + c)"
haftmann@27651
   221
proof -
nipkow@29925
   222
  from `a dvd b` obtain b' where "b = a * b'" ..
nipkow@29925
   223
  moreover from `a dvd c` obtain c' where "c = a * c'" ..
webertj@49962
   224
  ultimately have "b + c = a * (b' + c')" by (simp add: distrib_left)
haftmann@27651
   225
  then show ?thesis ..
haftmann@27651
   226
qed
haftmann@27651
   227
haftmann@25152
   228
end
paulson@14421
   229
huffman@29904
   230
class semiring_1_cancel = semiring + cancel_comm_monoid_add
huffman@29904
   231
  + zero_neq_one + monoid_mult
haftmann@25267
   232
begin
obua@14940
   233
huffman@27516
   234
subclass semiring_0_cancel ..
haftmann@25512
   235
huffman@27516
   236
subclass semiring_1 ..
haftmann@25267
   237
haftmann@25267
   238
end
krauss@21199
   239
lp15@60562
   240
class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add +
lp15@60562
   241
                               zero_neq_one + comm_monoid_mult +
lp15@60562
   242
  assumes right_diff_distrib' [algebra_simps]: "a * (b - c) = a * b - a * c"
haftmann@25267
   243
begin
obua@14738
   244
huffman@27516
   245
subclass semiring_1_cancel ..
huffman@27516
   246
subclass comm_semiring_0_cancel ..
huffman@27516
   247
subclass comm_semiring_1 ..
haftmann@25267
   248
haftmann@59816
   249
lemma left_diff_distrib' [algebra_simps]:
haftmann@59816
   250
  "(b - c) * a = b * a - c * a"
haftmann@59816
   251
  by (simp add: algebra_simps)
haftmann@59816
   252
haftmann@59816
   253
lemma dvd_add_times_triv_left_iff [simp]:
haftmann@59816
   254
  "a dvd c * a + b \<longleftrightarrow> a dvd b"
haftmann@59816
   255
proof -
haftmann@59816
   256
  have "a dvd a * c + b \<longleftrightarrow> a dvd b" (is "?P \<longleftrightarrow> ?Q")
haftmann@59816
   257
  proof
haftmann@59816
   258
    assume ?Q then show ?P by simp
haftmann@59816
   259
  next
haftmann@59816
   260
    assume ?P
haftmann@59816
   261
    then obtain d where "a * c + b = a * d" ..
haftmann@59816
   262
    then have "a * c + b - a * c = a * d - a * c" by simp
haftmann@59816
   263
    then have "b = a * d - a * c" by simp
lp15@60562
   264
    then have "b = a * (d - c)" by (simp add: algebra_simps)
haftmann@59816
   265
    then show ?Q ..
haftmann@59816
   266
  qed
haftmann@59816
   267
  then show "a dvd c * a + b \<longleftrightarrow> a dvd b" by (simp add: ac_simps)
haftmann@59816
   268
qed
haftmann@59816
   269
haftmann@59816
   270
lemma dvd_add_times_triv_right_iff [simp]:
haftmann@59816
   271
  "a dvd b + c * a \<longleftrightarrow> a dvd b"
haftmann@59816
   272
  using dvd_add_times_triv_left_iff [of a c b] by (simp add: ac_simps)
haftmann@59816
   273
haftmann@59816
   274
lemma dvd_add_triv_left_iff [simp]:
haftmann@59816
   275
  "a dvd a + b \<longleftrightarrow> a dvd b"
haftmann@59816
   276
  using dvd_add_times_triv_left_iff [of a 1 b] by simp
haftmann@59816
   277
haftmann@59816
   278
lemma dvd_add_triv_right_iff [simp]:
haftmann@59816
   279
  "a dvd b + a \<longleftrightarrow> a dvd b"
haftmann@59816
   280
  using dvd_add_times_triv_right_iff [of a b 1] by simp
haftmann@59816
   281
haftmann@59816
   282
lemma dvd_add_right_iff:
haftmann@59816
   283
  assumes "a dvd b"
haftmann@59816
   284
  shows "a dvd b + c \<longleftrightarrow> a dvd c" (is "?P \<longleftrightarrow> ?Q")
haftmann@59816
   285
proof
haftmann@59816
   286
  assume ?P then obtain d where "b + c = a * d" ..
haftmann@59816
   287
  moreover from `a dvd b` obtain e where "b = a * e" ..
haftmann@59816
   288
  ultimately have "a * e + c = a * d" by simp
haftmann@59816
   289
  then have "a * e + c - a * e = a * d - a * e" by simp
haftmann@59816
   290
  then have "c = a * d - a * e" by simp
haftmann@59816
   291
  then have "c = a * (d - e)" by (simp add: algebra_simps)
haftmann@59816
   292
  then show ?Q ..
haftmann@59816
   293
next
haftmann@59816
   294
  assume ?Q with assms show ?P by simp
haftmann@59816
   295
qed
haftmann@59816
   296
haftmann@59816
   297
lemma dvd_add_left_iff:
haftmann@59816
   298
  assumes "a dvd c"
haftmann@59816
   299
  shows "a dvd b + c \<longleftrightarrow> a dvd b"
haftmann@59816
   300
  using assms dvd_add_right_iff [of a c b] by (simp add: ac_simps)
haftmann@59816
   301
haftmann@59816
   302
end
haftmann@59816
   303
haftmann@22390
   304
class ring = semiring + ab_group_add
haftmann@25267
   305
begin
haftmann@25152
   306
huffman@27516
   307
subclass semiring_0_cancel ..
haftmann@25152
   308
haftmann@25152
   309
text {* Distribution rules *}
haftmann@25152
   310
haftmann@25152
   311
lemma minus_mult_left: "- (a * b) = - a * b"
lp15@60562
   312
by (rule minus_unique) (simp add: distrib_right [symmetric])
haftmann@25152
   313
haftmann@25152
   314
lemma minus_mult_right: "- (a * b) = a * - b"
lp15@60562
   315
by (rule minus_unique) (simp add: distrib_left [symmetric])
haftmann@25152
   316
huffman@29407
   317
text{*Extract signs from products*}
blanchet@54147
   318
lemmas mult_minus_left [simp] = minus_mult_left [symmetric]
blanchet@54147
   319
lemmas mult_minus_right [simp] = minus_mult_right [symmetric]
huffman@29407
   320
haftmann@25152
   321
lemma minus_mult_minus [simp]: "- a * - b = a * b"
nipkow@29667
   322
by simp
haftmann@25152
   323
haftmann@25152
   324
lemma minus_mult_commute: "- a * b = a * - b"
nipkow@29667
   325
by simp
nipkow@29667
   326
hoelzl@58776
   327
lemma right_diff_distrib [algebra_simps]:
haftmann@54230
   328
  "a * (b - c) = a * b - a * c"
haftmann@54230
   329
  using distrib_left [of a b "-c "] by simp
nipkow@29667
   330
hoelzl@58776
   331
lemma left_diff_distrib [algebra_simps]:
haftmann@54230
   332
  "(a - b) * c = a * c - b * c"
haftmann@54230
   333
  using distrib_right [of a "- b" c] by simp
haftmann@25152
   334
blanchet@54147
   335
lemmas ring_distribs =
webertj@49962
   336
  distrib_left distrib_right left_diff_distrib right_diff_distrib
haftmann@25152
   337
haftmann@25230
   338
lemma eq_add_iff1:
haftmann@25230
   339
  "a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"
nipkow@29667
   340
by (simp add: algebra_simps)
haftmann@25230
   341
haftmann@25230
   342
lemma eq_add_iff2:
haftmann@25230
   343
  "a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"
nipkow@29667
   344
by (simp add: algebra_simps)
haftmann@25230
   345
haftmann@25152
   346
end
haftmann@25152
   347
blanchet@54147
   348
lemmas ring_distribs =
webertj@49962
   349
  distrib_left distrib_right left_diff_distrib right_diff_distrib
haftmann@25152
   350
haftmann@22390
   351
class comm_ring = comm_semiring + ab_group_add
haftmann@25267
   352
begin
obua@14738
   353
huffman@27516
   354
subclass ring ..
huffman@28141
   355
subclass comm_semiring_0_cancel ..
haftmann@25267
   356
huffman@44350
   357
lemma square_diff_square_factored:
huffman@44350
   358
  "x * x - y * y = (x + y) * (x - y)"
huffman@44350
   359
  by (simp add: algebra_simps)
huffman@44350
   360
haftmann@25267
   361
end
obua@14738
   362
haftmann@22390
   363
class ring_1 = ring + zero_neq_one + monoid_mult
haftmann@25267
   364
begin
paulson@14265
   365
huffman@27516
   366
subclass semiring_1_cancel ..
haftmann@25267
   367
huffman@44346
   368
lemma square_diff_one_factored:
huffman@44346
   369
  "x * x - 1 = (x + 1) * (x - 1)"
huffman@44346
   370
  by (simp add: algebra_simps)
huffman@44346
   371
haftmann@25267
   372
end
haftmann@25152
   373
haftmann@22390
   374
class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
haftmann@25267
   375
begin
obua@14738
   376
huffman@27516
   377
subclass ring_1 ..
lp15@60562
   378
subclass comm_semiring_1_cancel
haftmann@59816
   379
  by unfold_locales (simp add: algebra_simps)
haftmann@58647
   380
huffman@29465
   381
lemma dvd_minus_iff [simp]: "x dvd - y \<longleftrightarrow> x dvd y"
huffman@29408
   382
proof
huffman@29408
   383
  assume "x dvd - y"
huffman@29408
   384
  then have "x dvd - 1 * - y" by (rule dvd_mult)
huffman@29408
   385
  then show "x dvd y" by simp
huffman@29408
   386
next
huffman@29408
   387
  assume "x dvd y"
huffman@29408
   388
  then have "x dvd - 1 * y" by (rule dvd_mult)
huffman@29408
   389
  then show "x dvd - y" by simp
huffman@29408
   390
qed
huffman@29408
   391
huffman@29465
   392
lemma minus_dvd_iff [simp]: "- x dvd y \<longleftrightarrow> x dvd y"
huffman@29408
   393
proof
huffman@29408
   394
  assume "- x dvd y"
huffman@29408
   395
  then obtain k where "y = - x * k" ..
huffman@29408
   396
  then have "y = x * - k" by simp
huffman@29408
   397
  then show "x dvd y" ..
huffman@29408
   398
next
huffman@29408
   399
  assume "x dvd y"
huffman@29408
   400
  then obtain k where "y = x * k" ..
huffman@29408
   401
  then have "y = - x * - k" by simp
huffman@29408
   402
  then show "- x dvd y" ..
huffman@29408
   403
qed
huffman@29408
   404
haftmann@54230
   405
lemma dvd_diff [simp]:
haftmann@54230
   406
  "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
haftmann@54230
   407
  using dvd_add [of x y "- z"] by simp
huffman@29409
   408
haftmann@25267
   409
end
haftmann@25152
   410
haftmann@59833
   411
class semiring_no_zero_divisors = semiring_0 +
haftmann@59833
   412
  assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
haftmann@25230
   413
begin
haftmann@25230
   414
haftmann@59833
   415
lemma divisors_zero:
haftmann@59833
   416
  assumes "a * b = 0"
haftmann@59833
   417
  shows "a = 0 \<or> b = 0"
haftmann@59833
   418
proof (rule classical)
haftmann@59833
   419
  assume "\<not> (a = 0 \<or> b = 0)"
haftmann@59833
   420
  then have "a \<noteq> 0" and "b \<noteq> 0" by auto
haftmann@59833
   421
  with no_zero_divisors have "a * b \<noteq> 0" by blast
haftmann@59833
   422
  with assms show ?thesis by simp
haftmann@59833
   423
qed
haftmann@59833
   424
haftmann@25230
   425
lemma mult_eq_0_iff [simp]:
haftmann@58952
   426
  shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
haftmann@25230
   427
proof (cases "a = 0 \<or> b = 0")
haftmann@25230
   428
  case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
haftmann@25230
   429
    then show ?thesis using no_zero_divisors by simp
haftmann@25230
   430
next
haftmann@25230
   431
  case True then show ?thesis by auto
haftmann@25230
   432
qed
haftmann@25230
   433
haftmann@58952
   434
end
haftmann@58952
   435
haftmann@60516
   436
class semiring_no_zero_divisors_cancel = semiring_no_zero_divisors +
haftmann@60516
   437
  assumes mult_cancel_right [simp]: "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   438
    and mult_cancel_left [simp]: "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@58952
   439
begin
haftmann@58952
   440
haftmann@58952
   441
lemma mult_left_cancel:
haftmann@58952
   442
  "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b"
lp15@60562
   443
  by simp
lp15@56217
   444
haftmann@58952
   445
lemma mult_right_cancel:
haftmann@58952
   446
  "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b"
lp15@60562
   447
  by simp
lp15@56217
   448
haftmann@25230
   449
end
huffman@22990
   450
haftmann@60516
   451
class ring_no_zero_divisors = ring + semiring_no_zero_divisors
haftmann@60516
   452
begin
haftmann@60516
   453
haftmann@60516
   454
subclass semiring_no_zero_divisors_cancel
haftmann@60516
   455
proof
haftmann@60516
   456
  fix a b c
haftmann@60516
   457
  have "a * c = b * c \<longleftrightarrow> (a - b) * c = 0"
haftmann@60516
   458
    by (simp add: algebra_simps)
haftmann@60516
   459
  also have "\<dots> \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   460
    by auto
haftmann@60516
   461
  finally show "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" .
haftmann@60516
   462
  have "c * a = c * b \<longleftrightarrow> c * (a - b) = 0"
haftmann@60516
   463
    by (simp add: algebra_simps)
haftmann@60516
   464
  also have "\<dots> \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   465
    by auto
haftmann@60516
   466
  finally show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" .
haftmann@60516
   467
qed
haftmann@60516
   468
haftmann@60516
   469
end
haftmann@60516
   470
huffman@23544
   471
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
haftmann@26274
   472
begin
haftmann@26274
   473
huffman@36970
   474
lemma square_eq_1_iff:
huffman@36821
   475
  "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
huffman@36821
   476
proof -
huffman@36821
   477
  have "(x - 1) * (x + 1) = x * x - 1"
huffman@36821
   478
    by (simp add: algebra_simps)
huffman@36821
   479
  hence "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
huffman@36821
   480
    by simp
huffman@36821
   481
  thus ?thesis
huffman@36821
   482
    by (simp add: eq_neg_iff_add_eq_0)
huffman@36821
   483
qed
huffman@36821
   484
haftmann@26274
   485
lemma mult_cancel_right1 [simp]:
haftmann@26274
   486
  "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   487
by (insert mult_cancel_right [of 1 c b], force)
haftmann@26274
   488
haftmann@26274
   489
lemma mult_cancel_right2 [simp]:
haftmann@26274
   490
  "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   491
by (insert mult_cancel_right [of a c 1], simp)
lp15@60562
   492
haftmann@26274
   493
lemma mult_cancel_left1 [simp]:
haftmann@26274
   494
  "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   495
by (insert mult_cancel_left [of c 1 b], force)
haftmann@26274
   496
haftmann@26274
   497
lemma mult_cancel_left2 [simp]:
haftmann@26274
   498
  "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   499
by (insert mult_cancel_left [of c a 1], simp)
haftmann@26274
   500
haftmann@26274
   501
end
huffman@22990
   502
lp15@60562
   503
class semidom = comm_semiring_1_cancel + semiring_no_zero_divisors
haftmann@59833
   504
haftmann@59833
   505
class idom = comm_ring_1 + semiring_no_zero_divisors
haftmann@25186
   506
begin
paulson@14421
   507
haftmann@59833
   508
subclass semidom ..
haftmann@59833
   509
huffman@27516
   510
subclass ring_1_no_zero_divisors ..
huffman@22990
   511
huffman@29981
   512
lemma dvd_mult_cancel_right [simp]:
huffman@29981
   513
  "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   514
proof -
huffman@29981
   515
  have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
haftmann@57514
   516
    unfolding dvd_def by (simp add: ac_simps)
huffman@29981
   517
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   518
    unfolding dvd_def by simp
huffman@29981
   519
  finally show ?thesis .
huffman@29981
   520
qed
huffman@29981
   521
huffman@29981
   522
lemma dvd_mult_cancel_left [simp]:
huffman@29981
   523
  "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   524
proof -
huffman@29981
   525
  have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
haftmann@57514
   526
    unfolding dvd_def by (simp add: ac_simps)
huffman@29981
   527
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   528
    unfolding dvd_def by simp
huffman@29981
   529
  finally show ?thesis .
huffman@29981
   530
qed
huffman@29981
   531
haftmann@60516
   532
lemma square_eq_iff: "a * a = b * b \<longleftrightarrow> a = b \<or> a = - b"
haftmann@59833
   533
proof
haftmann@59833
   534
  assume "a * a = b * b"
haftmann@59833
   535
  then have "(a - b) * (a + b) = 0"
haftmann@59833
   536
    by (simp add: algebra_simps)
haftmann@59833
   537
  then show "a = b \<or> a = - b"
haftmann@59833
   538
    by (simp add: eq_neg_iff_add_eq_0)
haftmann@59833
   539
next
haftmann@59833
   540
  assume "a = b \<or> a = - b"
haftmann@59833
   541
  then show "a * a = b * b" by auto
haftmann@59833
   542
qed
haftmann@59833
   543
haftmann@25186
   544
end
haftmann@25152
   545
haftmann@35302
   546
text {*
haftmann@35302
   547
  The theory of partially ordered rings is taken from the books:
haftmann@35302
   548
  \begin{itemize}
lp15@60562
   549
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979
haftmann@35302
   550
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
haftmann@35302
   551
  \end{itemize}
lp15@60562
   552
  Most of the used notions can also be looked up in
haftmann@35302
   553
  \begin{itemize}
wenzelm@54703
   554
  \item @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
haftmann@35302
   555
  \item \emph{Algebra I} by van der Waerden, Springer.
haftmann@35302
   556
  \end{itemize}
haftmann@35302
   557
*}
haftmann@35302
   558
haftmann@60353
   559
class divide =
haftmann@60429
   560
  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "div" 70)
haftmann@60353
   561
haftmann@60353
   562
setup {* Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a \<Rightarrow> 'a \<Rightarrow> 'a"}) *}
haftmann@60353
   563
haftmann@60353
   564
context semiring
haftmann@60353
   565
begin
haftmann@60353
   566
haftmann@60353
   567
lemma [field_simps]:
haftmann@60429
   568
  shows distrib_left_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b + c) = a * b + a * c"
haftmann@60429
   569
    and distrib_right_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a + b) * c = a * c + b * c"
haftmann@60353
   570
  by (rule distrib_left distrib_right)+
haftmann@60353
   571
haftmann@60353
   572
end
haftmann@60353
   573
haftmann@60353
   574
context ring
haftmann@60353
   575
begin
haftmann@60353
   576
haftmann@60353
   577
lemma [field_simps]:
haftmann@60429
   578
  shows left_diff_distrib_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a - b) * c = a * c - b * c"
haftmann@60429
   579
    and right_diff_distrib_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b - c) = a * b - a * c"
haftmann@60353
   580
  by (rule left_diff_distrib right_diff_distrib)+
haftmann@60353
   581
haftmann@60353
   582
end
haftmann@60353
   583
haftmann@60353
   584
setup {* Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a::divide \<Rightarrow> 'a \<Rightarrow> 'a"}) *}
haftmann@60353
   585
haftmann@60353
   586
class semidom_divide = semidom + divide +
haftmann@60429
   587
  assumes nonzero_mult_divide_cancel_right [simp]: "b \<noteq> 0 \<Longrightarrow> (a * b) div b = a"
haftmann@60429
   588
  assumes divide_zero [simp]: "a div 0 = 0"
haftmann@60353
   589
begin
haftmann@60353
   590
haftmann@60353
   591
lemma nonzero_mult_divide_cancel_left [simp]:
haftmann@60429
   592
  "a \<noteq> 0 \<Longrightarrow> (a * b) div a = b"
haftmann@60353
   593
  using nonzero_mult_divide_cancel_right [of a b] by (simp add: ac_simps)
haftmann@60353
   594
haftmann@60516
   595
subclass semiring_no_zero_divisors_cancel
haftmann@60516
   596
proof
haftmann@60516
   597
  fix a b c
haftmann@60516
   598
  { fix a b c
haftmann@60516
   599
    show "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   600
    proof (cases "c = 0")
haftmann@60516
   601
      case True then show ?thesis by simp
haftmann@60516
   602
    next
haftmann@60516
   603
      case False
haftmann@60516
   604
      { assume "a * c = b * c"
haftmann@60516
   605
        then have "a * c div c = b * c div c"
haftmann@60516
   606
          by simp
haftmann@60516
   607
        with False have "a = b"
haftmann@60516
   608
          by simp
haftmann@60516
   609
      } then show ?thesis by auto
haftmann@60516
   610
    qed
haftmann@60516
   611
  }
haftmann@60516
   612
  from this [of a c b]
haftmann@60516
   613
  show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   614
    by (simp add: ac_simps)
haftmann@60516
   615
qed
haftmann@60516
   616
haftmann@60516
   617
lemma div_self [simp]:
haftmann@60516
   618
  assumes "a \<noteq> 0"
haftmann@60516
   619
  shows "a div a = 1"
haftmann@60516
   620
  using assms nonzero_mult_divide_cancel_left [of a 1] by simp
haftmann@60516
   621
haftmann@60570
   622
lemma divide_zero_left [simp]:
haftmann@60570
   623
  "0 div a = 0"
haftmann@60570
   624
proof (cases "a = 0")
haftmann@60570
   625
  case True then show ?thesis by simp
haftmann@60570
   626
next
haftmann@60570
   627
  case False then have "a * 0 div a = 0"
haftmann@60570
   628
    by (rule nonzero_mult_divide_cancel_left)
haftmann@60570
   629
  then show ?thesis by simp
haftmann@60570
   630
qed 
haftmann@60570
   631
haftmann@60353
   632
end
haftmann@60353
   633
haftmann@60353
   634
class idom_divide = idom + semidom_divide
haftmann@60353
   635
haftmann@60517
   636
class algebraic_semidom = semidom_divide
haftmann@60517
   637
begin
haftmann@60517
   638
haftmann@60688
   639
text \<open>
haftmann@60688
   640
  Class @{class algebraic_semidom} enriches a integral domain
haftmann@60688
   641
  by notions from algebra, like units in a ring.
haftmann@60688
   642
  It is a separate class to avoid spoiling fields with notions
haftmann@60688
   643
  which are degenerated there.
haftmann@60688
   644
\<close>
haftmann@60688
   645
haftmann@60517
   646
lemma dvd_div_mult_self [simp]:
haftmann@60517
   647
  "a dvd b \<Longrightarrow> b div a * a = b"
haftmann@60517
   648
  by (cases "a = 0") (auto elim: dvdE simp add: ac_simps)
haftmann@60517
   649
haftmann@60517
   650
lemma dvd_mult_div_cancel [simp]:
haftmann@60517
   651
  "a dvd b \<Longrightarrow> a * (b div a) = b"
haftmann@60517
   652
  using dvd_div_mult_self [of a b] by (simp add: ac_simps)
lp15@60562
   653
haftmann@60517
   654
lemma div_mult_swap:
haftmann@60517
   655
  assumes "c dvd b"
haftmann@60517
   656
  shows "a * (b div c) = (a * b) div c"
haftmann@60517
   657
proof (cases "c = 0")
haftmann@60517
   658
  case True then show ?thesis by simp
haftmann@60517
   659
next
haftmann@60517
   660
  case False from assms obtain d where "b = c * d" ..
haftmann@60517
   661
  moreover from False have "a * divide (d * c) c = ((a * d) * c) div c"
haftmann@60517
   662
    by simp
haftmann@60517
   663
  ultimately show ?thesis by (simp add: ac_simps)
haftmann@60517
   664
qed
haftmann@60517
   665
haftmann@60517
   666
lemma dvd_div_mult:
haftmann@60517
   667
  assumes "c dvd b"
haftmann@60517
   668
  shows "b div c * a = (b * a) div c"
haftmann@60517
   669
  using assms div_mult_swap [of c b a] by (simp add: ac_simps)
haftmann@60517
   670
haftmann@60570
   671
lemma dvd_div_mult2_eq:
haftmann@60570
   672
  assumes "b * c dvd a"
haftmann@60570
   673
  shows "a div (b * c) = a div b div c"
haftmann@60570
   674
using assms proof
haftmann@60570
   675
  fix k
haftmann@60570
   676
  assume "a = b * c * k"
haftmann@60570
   677
  then show ?thesis
haftmann@60570
   678
    by (cases "b = 0 \<or> c = 0") (auto, simp add: ac_simps)
haftmann@60570
   679
qed
haftmann@60570
   680
lp15@60562
   681
haftmann@60517
   682
text \<open>Units: invertible elements in a ring\<close>
haftmann@60517
   683
haftmann@60517
   684
abbreviation is_unit :: "'a \<Rightarrow> bool"
haftmann@60517
   685
where
haftmann@60517
   686
  "is_unit a \<equiv> a dvd 1"
haftmann@60517
   687
haftmann@60517
   688
lemma not_is_unit_0 [simp]:
haftmann@60517
   689
  "\<not> is_unit 0"
haftmann@60517
   690
  by simp
haftmann@60517
   691
lp15@60562
   692
lemma unit_imp_dvd [dest]:
haftmann@60517
   693
  "is_unit b \<Longrightarrow> b dvd a"
haftmann@60517
   694
  by (rule dvd_trans [of _ 1]) simp_all
haftmann@60517
   695
haftmann@60517
   696
lemma unit_dvdE:
haftmann@60517
   697
  assumes "is_unit a"
haftmann@60517
   698
  obtains c where "a \<noteq> 0" and "b = a * c"
haftmann@60517
   699
proof -
haftmann@60517
   700
  from assms have "a dvd b" by auto
haftmann@60517
   701
  then obtain c where "b = a * c" ..
haftmann@60517
   702
  moreover from assms have "a \<noteq> 0" by auto
haftmann@60517
   703
  ultimately show thesis using that by blast
haftmann@60517
   704
qed
haftmann@60517
   705
haftmann@60517
   706
lemma dvd_unit_imp_unit:
haftmann@60517
   707
  "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a"
haftmann@60517
   708
  by (rule dvd_trans)
haftmann@60517
   709
haftmann@60517
   710
lemma unit_div_1_unit [simp, intro]:
haftmann@60517
   711
  assumes "is_unit a"
haftmann@60517
   712
  shows "is_unit (1 div a)"
haftmann@60517
   713
proof -
haftmann@60517
   714
  from assms have "1 = 1 div a * a" by simp
haftmann@60517
   715
  then show "is_unit (1 div a)" by (rule dvdI)
haftmann@60517
   716
qed
haftmann@60517
   717
haftmann@60517
   718
lemma is_unitE [elim?]:
haftmann@60517
   719
  assumes "is_unit a"
haftmann@60517
   720
  obtains b where "a \<noteq> 0" and "b \<noteq> 0"
haftmann@60517
   721
    and "is_unit b" and "1 div a = b" and "1 div b = a"
haftmann@60517
   722
    and "a * b = 1" and "c div a = c * b"
haftmann@60517
   723
proof (rule that)
haftmann@60517
   724
  def b \<equiv> "1 div a"
haftmann@60517
   725
  then show "1 div a = b" by simp
haftmann@60517
   726
  from b_def `is_unit a` show "is_unit b" by simp
haftmann@60517
   727
  from `is_unit a` and `is_unit b` show "a \<noteq> 0" and "b \<noteq> 0" by auto
haftmann@60517
   728
  from b_def `is_unit a` show "a * b = 1" by simp
haftmann@60517
   729
  then have "1 = a * b" ..
haftmann@60517
   730
  with b_def `b \<noteq> 0` show "1 div b = a" by simp
haftmann@60517
   731
  from `is_unit a` have "a dvd c" ..
haftmann@60517
   732
  then obtain d where "c = a * d" ..
haftmann@60517
   733
  with `a \<noteq> 0` `a * b = 1` show "c div a = c * b"
haftmann@60517
   734
    by (simp add: mult.assoc mult.left_commute [of a])
haftmann@60517
   735
qed
haftmann@60517
   736
haftmann@60517
   737
lemma unit_prod [intro]:
haftmann@60517
   738
  "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)"
lp15@60562
   739
  by (subst mult_1_left [of 1, symmetric]) (rule mult_dvd_mono)
lp15@60562
   740
haftmann@60517
   741
lemma unit_div [intro]:
haftmann@60517
   742
  "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)"
haftmann@60517
   743
  by (erule is_unitE [of b a]) (simp add: ac_simps unit_prod)
haftmann@60517
   744
haftmann@60517
   745
lemma mult_unit_dvd_iff:
haftmann@60517
   746
  assumes "is_unit b"
haftmann@60517
   747
  shows "a * b dvd c \<longleftrightarrow> a dvd c"
haftmann@60517
   748
proof
haftmann@60517
   749
  assume "a * b dvd c"
haftmann@60517
   750
  with assms show "a dvd c"
haftmann@60517
   751
    by (simp add: dvd_mult_left)
haftmann@60517
   752
next
haftmann@60517
   753
  assume "a dvd c"
haftmann@60517
   754
  then obtain k where "c = a * k" ..
haftmann@60517
   755
  with assms have "c = (a * b) * (1 div b * k)"
haftmann@60517
   756
    by (simp add: mult_ac)
haftmann@60517
   757
  then show "a * b dvd c" by (rule dvdI)
haftmann@60517
   758
qed
haftmann@60517
   759
haftmann@60517
   760
lemma dvd_mult_unit_iff:
haftmann@60517
   761
  assumes "is_unit b"
haftmann@60517
   762
  shows "a dvd c * b \<longleftrightarrow> a dvd c"
haftmann@60517
   763
proof
haftmann@60517
   764
  assume "a dvd c * b"
haftmann@60517
   765
  with assms have "c * b dvd c * (b * (1 div b))"
haftmann@60517
   766
    by (subst mult_assoc [symmetric]) simp
haftmann@60517
   767
  also from `is_unit b` have "b * (1 div b) = 1" by (rule is_unitE) simp
haftmann@60517
   768
  finally have "c * b dvd c" by simp
haftmann@60517
   769
  with `a dvd c * b` show "a dvd c" by (rule dvd_trans)
haftmann@60517
   770
next
haftmann@60517
   771
  assume "a dvd c"
haftmann@60517
   772
  then show "a dvd c * b" by simp
haftmann@60517
   773
qed
haftmann@60517
   774
haftmann@60517
   775
lemma div_unit_dvd_iff:
haftmann@60517
   776
  "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c"
haftmann@60517
   777
  by (erule is_unitE [of _ a]) (auto simp add: mult_unit_dvd_iff)
haftmann@60517
   778
haftmann@60517
   779
lemma dvd_div_unit_iff:
haftmann@60517
   780
  "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c"
haftmann@60517
   781
  by (erule is_unitE [of _ c]) (simp add: dvd_mult_unit_iff)
haftmann@60517
   782
haftmann@60517
   783
lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff
haftmann@60517
   784
  dvd_mult_unit_iff dvd_div_unit_iff -- \<open>FIXME consider fact collection\<close>
haftmann@60517
   785
haftmann@60517
   786
lemma unit_mult_div_div [simp]:
haftmann@60517
   787
  "is_unit a \<Longrightarrow> b * (1 div a) = b div a"
haftmann@60517
   788
  by (erule is_unitE [of _ b]) simp
haftmann@60517
   789
haftmann@60517
   790
lemma unit_div_mult_self [simp]:
haftmann@60517
   791
  "is_unit a \<Longrightarrow> b div a * a = b"
haftmann@60517
   792
  by (rule dvd_div_mult_self) auto
haftmann@60517
   793
haftmann@60517
   794
lemma unit_div_1_div_1 [simp]:
haftmann@60517
   795
  "is_unit a \<Longrightarrow> 1 div (1 div a) = a"
haftmann@60517
   796
  by (erule is_unitE) simp
haftmann@60517
   797
haftmann@60517
   798
lemma unit_div_mult_swap:
haftmann@60517
   799
  "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c"
haftmann@60517
   800
  by (erule unit_dvdE [of _ b]) (simp add: mult.left_commute [of _ c])
haftmann@60517
   801
haftmann@60517
   802
lemma unit_div_commute:
haftmann@60517
   803
  "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b"
haftmann@60517
   804
  using unit_div_mult_swap [of b c a] by (simp add: ac_simps)
haftmann@60517
   805
haftmann@60517
   806
lemma unit_eq_div1:
haftmann@60517
   807
  "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b"
haftmann@60517
   808
  by (auto elim: is_unitE)
haftmann@60517
   809
haftmann@60517
   810
lemma unit_eq_div2:
haftmann@60517
   811
  "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c"
haftmann@60517
   812
  using unit_eq_div1 [of b c a] by auto
haftmann@60517
   813
haftmann@60517
   814
lemma unit_mult_left_cancel:
haftmann@60517
   815
  assumes "is_unit a"
haftmann@60517
   816
  shows "a * b = a * c \<longleftrightarrow> b = c" (is "?P \<longleftrightarrow> ?Q")
lp15@60562
   817
  using assms mult_cancel_left [of a b c] by auto
haftmann@60517
   818
haftmann@60517
   819
lemma unit_mult_right_cancel:
haftmann@60517
   820
  "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c"
haftmann@60517
   821
  using unit_mult_left_cancel [of a b c] by (auto simp add: ac_simps)
haftmann@60517
   822
haftmann@60517
   823
lemma unit_div_cancel:
haftmann@60517
   824
  assumes "is_unit a"
haftmann@60517
   825
  shows "b div a = c div a \<longleftrightarrow> b = c"
haftmann@60517
   826
proof -
haftmann@60517
   827
  from assms have "is_unit (1 div a)" by simp
haftmann@60517
   828
  then have "b * (1 div a) = c * (1 div a) \<longleftrightarrow> b = c"
haftmann@60517
   829
    by (rule unit_mult_right_cancel)
haftmann@60517
   830
  with assms show ?thesis by simp
haftmann@60517
   831
qed
lp15@60562
   832
haftmann@60570
   833
lemma is_unit_div_mult2_eq:
haftmann@60570
   834
  assumes "is_unit b" and "is_unit c"
haftmann@60570
   835
  shows "a div (b * c) = a div b div c"
haftmann@60570
   836
proof -
haftmann@60570
   837
  from assms have "is_unit (b * c)" by (simp add: unit_prod)
haftmann@60570
   838
  then have "b * c dvd a"
haftmann@60570
   839
    by (rule unit_imp_dvd)
haftmann@60570
   840
  then show ?thesis
haftmann@60570
   841
    by (rule dvd_div_mult2_eq)
haftmann@60570
   842
qed
haftmann@60570
   843
lp15@60562
   844
lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff
haftmann@60517
   845
  dvd_div_unit_iff unit_div_mult_swap unit_div_commute
lp15@60562
   846
  unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel
haftmann@60517
   847
  unit_eq_div1 unit_eq_div2
haftmann@60517
   848
haftmann@60685
   849
lemma is_unit_divide_mult_cancel_left:
haftmann@60685
   850
  assumes "a \<noteq> 0" and "is_unit b"
haftmann@60685
   851
  shows "a div (a * b) = 1 div b"
haftmann@60685
   852
proof -
haftmann@60685
   853
  from assms have "a div (a * b) = a div a div b"
haftmann@60685
   854
    by (simp add: mult_unit_dvd_iff dvd_div_mult2_eq)
haftmann@60685
   855
  with assms show ?thesis by simp
haftmann@60685
   856
qed
haftmann@60685
   857
haftmann@60685
   858
lemma is_unit_divide_mult_cancel_right:
haftmann@60685
   859
  assumes "a \<noteq> 0" and "is_unit b"
haftmann@60685
   860
  shows "a div (b * a) = 1 div b"
haftmann@60685
   861
  using assms is_unit_divide_mult_cancel_left [of a b] by (simp add: ac_simps)
haftmann@60685
   862
haftmann@60685
   863
end
haftmann@60685
   864
haftmann@60685
   865
class normalization_semidom = algebraic_semidom +
haftmann@60685
   866
  fixes normalize :: "'a \<Rightarrow> 'a"
haftmann@60685
   867
    and unit_factor :: "'a \<Rightarrow> 'a"
haftmann@60685
   868
  assumes unit_factor_mult_normalize [simp]: "unit_factor a * normalize a = a"
haftmann@60685
   869
  assumes normalize_0 [simp]: "normalize 0 = 0"
haftmann@60685
   870
    and unit_factor_0 [simp]: "unit_factor 0 = 0"
haftmann@60685
   871
  assumes is_unit_normalize:
haftmann@60685
   872
    "is_unit a  \<Longrightarrow> normalize a = 1"
haftmann@60685
   873
  assumes unit_factor_is_unit [iff]: 
haftmann@60685
   874
    "a \<noteq> 0 \<Longrightarrow> is_unit (unit_factor a)"
haftmann@60685
   875
  assumes unit_factor_mult: "unit_factor (a * b) = unit_factor a * unit_factor b"
haftmann@60685
   876
begin
haftmann@60685
   877
haftmann@60688
   878
text \<open>
haftmann@60688
   879
  Class @{class normalization_semidom} cultivates the idea that
haftmann@60688
   880
  each integral domain can be split into equivalence classes
haftmann@60688
   881
  whose representants are associated, i.e. divide each other.
haftmann@60688
   882
  @{const normalize} specifies a canonical representant for each equivalence
haftmann@60688
   883
  class.  The rationale behind this is that it is easier to reason about equality
haftmann@60688
   884
  than equivalences, hence we prefer to think about equality of normalized
haftmann@60688
   885
  values rather than associated elements.
haftmann@60688
   886
\<close>
haftmann@60688
   887
haftmann@60685
   888
lemma unit_factor_dvd [simp]:
haftmann@60685
   889
  "a \<noteq> 0 \<Longrightarrow> unit_factor a dvd b"
haftmann@60685
   890
  by (rule unit_imp_dvd) simp
haftmann@60685
   891
haftmann@60685
   892
lemma unit_factor_self [simp]:
haftmann@60685
   893
  "unit_factor a dvd a"
haftmann@60685
   894
  by (cases "a = 0") simp_all 
haftmann@60685
   895
  
haftmann@60685
   896
lemma normalize_mult_unit_factor [simp]:
haftmann@60685
   897
  "normalize a * unit_factor a = a"
haftmann@60685
   898
  using unit_factor_mult_normalize [of a] by (simp add: ac_simps)
haftmann@60685
   899
haftmann@60685
   900
lemma normalize_eq_0_iff [simp]:
haftmann@60685
   901
  "normalize a = 0 \<longleftrightarrow> a = 0" (is "?P \<longleftrightarrow> ?Q")
haftmann@60685
   902
proof
haftmann@60685
   903
  assume ?P
haftmann@60685
   904
  moreover have "unit_factor a * normalize a = a" by simp
haftmann@60685
   905
  ultimately show ?Q by simp 
haftmann@60685
   906
next
haftmann@60685
   907
  assume ?Q then show ?P by simp
haftmann@60685
   908
qed
haftmann@60685
   909
haftmann@60685
   910
lemma unit_factor_eq_0_iff [simp]:
haftmann@60685
   911
  "unit_factor a = 0 \<longleftrightarrow> a = 0" (is "?P \<longleftrightarrow> ?Q")
haftmann@60685
   912
proof
haftmann@60685
   913
  assume ?P
haftmann@60685
   914
  moreover have "unit_factor a * normalize a = a" by simp
haftmann@60685
   915
  ultimately show ?Q by simp 
haftmann@60685
   916
next
haftmann@60685
   917
  assume ?Q then show ?P by simp
haftmann@60685
   918
qed
haftmann@60685
   919
haftmann@60685
   920
lemma is_unit_unit_factor:
haftmann@60685
   921
  assumes "is_unit a" shows "unit_factor a = a"
haftmann@60685
   922
proof - 
haftmann@60685
   923
  from assms have "normalize a = 1" by (rule is_unit_normalize)
haftmann@60685
   924
  moreover from unit_factor_mult_normalize have "unit_factor a * normalize a = a" .
haftmann@60685
   925
  ultimately show ?thesis by simp
haftmann@60685
   926
qed
haftmann@60685
   927
haftmann@60685
   928
lemma unit_factor_1 [simp]:
haftmann@60685
   929
  "unit_factor 1 = 1"
haftmann@60685
   930
  by (rule is_unit_unit_factor) simp
haftmann@60685
   931
haftmann@60685
   932
lemma normalize_1 [simp]:
haftmann@60685
   933
  "normalize 1 = 1"
haftmann@60685
   934
  by (rule is_unit_normalize) simp
haftmann@60685
   935
haftmann@60685
   936
lemma normalize_1_iff:
haftmann@60685
   937
  "normalize a = 1 \<longleftrightarrow> is_unit a" (is "?P \<longleftrightarrow> ?Q")
haftmann@60685
   938
proof
haftmann@60685
   939
  assume ?Q then show ?P by (rule is_unit_normalize)
haftmann@60685
   940
next
haftmann@60685
   941
  assume ?P
haftmann@60685
   942
  then have "a \<noteq> 0" by auto
haftmann@60685
   943
  from \<open>?P\<close> have "unit_factor a * normalize a = unit_factor a * 1"
haftmann@60685
   944
    by simp
haftmann@60685
   945
  then have "unit_factor a = a"
haftmann@60685
   946
    by simp
haftmann@60685
   947
  moreover have "is_unit (unit_factor a)"
haftmann@60685
   948
    using \<open>a \<noteq> 0\<close> by simp
haftmann@60685
   949
  ultimately show ?Q by simp
haftmann@60685
   950
qed
haftmann@60685
   951
  
haftmann@60685
   952
lemma div_normalize [simp]:
haftmann@60685
   953
  "a div normalize a = unit_factor a"
haftmann@60685
   954
proof (cases "a = 0")
haftmann@60685
   955
  case True then show ?thesis by simp
haftmann@60685
   956
next
haftmann@60685
   957
  case False then have "normalize a \<noteq> 0" by simp 
haftmann@60685
   958
  with nonzero_mult_divide_cancel_right
haftmann@60685
   959
  have "unit_factor a * normalize a div normalize a = unit_factor a" by blast
haftmann@60685
   960
  then show ?thesis by simp
haftmann@60685
   961
qed
haftmann@60685
   962
haftmann@60685
   963
lemma div_unit_factor [simp]:
haftmann@60685
   964
  "a div unit_factor a = normalize a"
haftmann@60685
   965
proof (cases "a = 0")
haftmann@60685
   966
  case True then show ?thesis by simp
haftmann@60685
   967
next
haftmann@60685
   968
  case False then have "unit_factor a \<noteq> 0" by simp 
haftmann@60685
   969
  with nonzero_mult_divide_cancel_left
haftmann@60685
   970
  have "unit_factor a * normalize a div unit_factor a = normalize a" by blast
haftmann@60685
   971
  then show ?thesis by simp
haftmann@60685
   972
qed
haftmann@60685
   973
haftmann@60685
   974
lemma normalize_div [simp]:
haftmann@60685
   975
  "normalize a div a = 1 div unit_factor a"
haftmann@60685
   976
proof (cases "a = 0")
haftmann@60685
   977
  case True then show ?thesis by simp
haftmann@60685
   978
next
haftmann@60685
   979
  case False
haftmann@60685
   980
  have "normalize a div a = normalize a div (unit_factor a * normalize a)"
haftmann@60685
   981
    by simp
haftmann@60685
   982
  also have "\<dots> = 1 div unit_factor a"
haftmann@60685
   983
    using False by (subst is_unit_divide_mult_cancel_right) simp_all
haftmann@60685
   984
  finally show ?thesis .
haftmann@60685
   985
qed
haftmann@60685
   986
haftmann@60685
   987
lemma mult_one_div_unit_factor [simp]:
haftmann@60685
   988
  "a * (1 div unit_factor b) = a div unit_factor b"
haftmann@60685
   989
  by (cases "b = 0") simp_all
haftmann@60685
   990
haftmann@60685
   991
lemma normalize_mult:
haftmann@60685
   992
  "normalize (a * b) = normalize a * normalize b"
haftmann@60685
   993
proof (cases "a = 0 \<or> b = 0")
haftmann@60685
   994
  case True then show ?thesis by auto
haftmann@60685
   995
next
haftmann@60685
   996
  case False
haftmann@60685
   997
  from unit_factor_mult_normalize have "unit_factor (a * b) * normalize (a * b) = a * b" .
haftmann@60685
   998
  then have "normalize (a * b) = a * b div unit_factor (a * b)" by simp
haftmann@60685
   999
  also have "\<dots> = a * b div unit_factor (b * a)" by (simp add: ac_simps)
haftmann@60685
  1000
  also have "\<dots> = a * b div unit_factor b div unit_factor a"
haftmann@60685
  1001
    using False by (simp add: unit_factor_mult is_unit_div_mult2_eq [symmetric])
haftmann@60685
  1002
  also have "\<dots> = a * (b div unit_factor b) div unit_factor a"
haftmann@60685
  1003
    using False by (subst unit_div_mult_swap) simp_all
haftmann@60685
  1004
  also have "\<dots> = normalize a * normalize b"
haftmann@60685
  1005
    using False by (simp add: mult.commute [of a] mult.commute [of "normalize a"] unit_div_mult_swap [symmetric])
haftmann@60685
  1006
  finally show ?thesis .
haftmann@60685
  1007
qed
haftmann@60685
  1008
 
haftmann@60685
  1009
lemma unit_factor_idem [simp]:
haftmann@60685
  1010
  "unit_factor (unit_factor a) = unit_factor a"
haftmann@60685
  1011
  by (cases "a = 0") (auto intro: is_unit_unit_factor)
haftmann@60685
  1012
haftmann@60685
  1013
lemma normalize_unit_factor [simp]:
haftmann@60685
  1014
  "a \<noteq> 0 \<Longrightarrow> normalize (unit_factor a) = 1"
haftmann@60685
  1015
  by (rule is_unit_normalize) simp
haftmann@60685
  1016
  
haftmann@60685
  1017
lemma normalize_idem [simp]:
haftmann@60685
  1018
  "normalize (normalize a) = normalize a"
haftmann@60685
  1019
proof (cases "a = 0")
haftmann@60685
  1020
  case True then show ?thesis by simp
haftmann@60685
  1021
next
haftmann@60685
  1022
  case False
haftmann@60685
  1023
  have "normalize a = normalize (unit_factor a * normalize a)" by simp
haftmann@60685
  1024
  also have "\<dots> = normalize (unit_factor a) * normalize (normalize a)"
haftmann@60685
  1025
    by (simp only: normalize_mult)
haftmann@60685
  1026
  finally show ?thesis using False by simp_all
haftmann@60685
  1027
qed
haftmann@60685
  1028
haftmann@60685
  1029
lemma unit_factor_normalize [simp]:
haftmann@60685
  1030
  assumes "a \<noteq> 0"
haftmann@60685
  1031
  shows "unit_factor (normalize a) = 1"
haftmann@60685
  1032
proof -
haftmann@60685
  1033
  from assms have "normalize a \<noteq> 0" by simp
haftmann@60685
  1034
  have "unit_factor (normalize a) * normalize (normalize a) = normalize a"
haftmann@60685
  1035
    by (simp only: unit_factor_mult_normalize)
haftmann@60685
  1036
  then have "unit_factor (normalize a) * normalize a = normalize a"
haftmann@60685
  1037
    by simp
haftmann@60685
  1038
  with \<open>normalize a \<noteq> 0\<close>
haftmann@60685
  1039
  have "unit_factor (normalize a) * normalize a div normalize a = normalize a div normalize a"
haftmann@60685
  1040
    by simp
haftmann@60685
  1041
  with \<open>normalize a \<noteq> 0\<close>
haftmann@60685
  1042
  show ?thesis by simp
haftmann@60685
  1043
qed
haftmann@60685
  1044
haftmann@60685
  1045
lemma dvd_unit_factor_div:
haftmann@60685
  1046
  assumes "b dvd a"
haftmann@60685
  1047
  shows "unit_factor (a div b) = unit_factor a div unit_factor b"
haftmann@60685
  1048
proof -
haftmann@60685
  1049
  from assms have "a = a div b * b"
haftmann@60685
  1050
    by simp
haftmann@60685
  1051
  then have "unit_factor a = unit_factor (a div b * b)"
haftmann@60685
  1052
    by simp
haftmann@60685
  1053
  then show ?thesis
haftmann@60685
  1054
    by (cases "b = 0") (simp_all add: unit_factor_mult)
haftmann@60685
  1055
qed
haftmann@60685
  1056
haftmann@60685
  1057
lemma dvd_normalize_div:
haftmann@60685
  1058
  assumes "b dvd a"
haftmann@60685
  1059
  shows "normalize (a div b) = normalize a div normalize b"
haftmann@60685
  1060
proof -
haftmann@60685
  1061
  from assms have "a = a div b * b"
haftmann@60685
  1062
    by simp
haftmann@60685
  1063
  then have "normalize a = normalize (a div b * b)"
haftmann@60685
  1064
    by simp
haftmann@60685
  1065
  then show ?thesis
haftmann@60685
  1066
    by (cases "b = 0") (simp_all add: normalize_mult)
haftmann@60685
  1067
qed
haftmann@60685
  1068
haftmann@60685
  1069
lemma normalize_dvd_iff [simp]:
haftmann@60685
  1070
  "normalize a dvd b \<longleftrightarrow> a dvd b"
haftmann@60685
  1071
proof -
haftmann@60685
  1072
  have "normalize a dvd b \<longleftrightarrow> unit_factor a * normalize a dvd b"
haftmann@60685
  1073
    using mult_unit_dvd_iff [of "unit_factor a" "normalize a" b]
haftmann@60685
  1074
      by (cases "a = 0") simp_all
haftmann@60685
  1075
  then show ?thesis by simp
haftmann@60685
  1076
qed
haftmann@60685
  1077
haftmann@60685
  1078
lemma dvd_normalize_iff [simp]:
haftmann@60685
  1079
  "a dvd normalize b \<longleftrightarrow> a dvd b"
haftmann@60685
  1080
proof -
haftmann@60685
  1081
  have "a dvd normalize  b \<longleftrightarrow> a dvd normalize b * unit_factor b"
haftmann@60685
  1082
    using dvd_mult_unit_iff [of "unit_factor b" a "normalize b"]
haftmann@60685
  1083
      by (cases "b = 0") simp_all
haftmann@60685
  1084
  then show ?thesis by simp
haftmann@60685
  1085
qed
haftmann@60685
  1086
haftmann@60688
  1087
text \<open>
haftmann@60688
  1088
  We avoid an explicit definition of associated elements but prefer
haftmann@60688
  1089
  explicit normalisation instead.  In theory we could define an abbreviation
haftmann@60688
  1090
  like @{prop "associated a b \<longleftrightarrow> normalize a = normalize b"} but this is
haftmann@60688
  1091
  counterproductive without suggestive infix syntax, which we do not want
haftmann@60688
  1092
  to sacrifice for this purpose here.
haftmann@60688
  1093
\<close>
haftmann@60685
  1094
haftmann@60688
  1095
lemma associatedI:
haftmann@60688
  1096
  assumes "a dvd b" and "b dvd a"
haftmann@60688
  1097
  shows "normalize a = normalize b"
haftmann@60685
  1098
proof (cases "a = 0 \<or> b = 0")
haftmann@60688
  1099
  case True with assms show ?thesis by auto
haftmann@60685
  1100
next
haftmann@60685
  1101
  case False
haftmann@60688
  1102
  from \<open>a dvd b\<close> obtain c where b: "b = a * c" ..
haftmann@60688
  1103
  moreover from \<open>b dvd a\<close> obtain d where a: "a = b * d" ..
haftmann@60688
  1104
  ultimately have "b * 1 = b * (c * d)" by (simp add: ac_simps)
haftmann@60688
  1105
  with False have "1 = c * d"
haftmann@60688
  1106
    unfolding mult_cancel_left by simp
haftmann@60688
  1107
  then have "is_unit c" and "is_unit d" by auto
haftmann@60688
  1108
  with a b show ?thesis by (simp add: normalize_mult is_unit_normalize)
haftmann@60688
  1109
qed
haftmann@60688
  1110
haftmann@60688
  1111
lemma associatedD1:
haftmann@60688
  1112
  "normalize a = normalize b \<Longrightarrow> a dvd b"
haftmann@60688
  1113
  using dvd_normalize_iff [of _ b, symmetric] normalize_dvd_iff [of a _, symmetric]
haftmann@60688
  1114
  by simp
haftmann@60688
  1115
haftmann@60688
  1116
lemma associatedD2:
haftmann@60688
  1117
  "normalize a = normalize b \<Longrightarrow> b dvd a"
haftmann@60688
  1118
  using dvd_normalize_iff [of _ a, symmetric] normalize_dvd_iff [of b _, symmetric]
haftmann@60688
  1119
  by simp
haftmann@60688
  1120
haftmann@60688
  1121
lemma associated_unit:
haftmann@60688
  1122
  "normalize a = normalize b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"
haftmann@60688
  1123
  using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2)
haftmann@60688
  1124
haftmann@60688
  1125
lemma associated_iff_dvd:
haftmann@60688
  1126
  "normalize a = normalize b \<longleftrightarrow> a dvd b \<and> b dvd a" (is "?P \<longleftrightarrow> ?Q")
haftmann@60688
  1127
proof
haftmann@60688
  1128
  assume ?Q then show ?P by (auto intro!: associatedI)
haftmann@60688
  1129
next
haftmann@60688
  1130
  assume ?P
haftmann@60688
  1131
  then have "unit_factor a * normalize a = unit_factor a * normalize b"
haftmann@60688
  1132
    by simp
haftmann@60688
  1133
  then have *: "normalize b * unit_factor a = a"
haftmann@60688
  1134
    by (simp add: ac_simps)
haftmann@60688
  1135
  show ?Q
haftmann@60688
  1136
  proof (cases "a = 0 \<or> b = 0")
haftmann@60688
  1137
    case True with \<open>?P\<close> show ?thesis by auto
haftmann@60685
  1138
  next
haftmann@60688
  1139
    case False 
haftmann@60688
  1140
    then have "b dvd normalize b * unit_factor a" and "normalize b * unit_factor a dvd b"
haftmann@60688
  1141
      by (simp_all add: mult_unit_dvd_iff dvd_mult_unit_iff)
haftmann@60688
  1142
    with * show ?thesis by simp
haftmann@60685
  1143
  qed
haftmann@60685
  1144
qed
haftmann@60685
  1145
haftmann@60685
  1146
lemma associated_eqI:
haftmann@60688
  1147
  assumes "a dvd b" and "b dvd a"
haftmann@60688
  1148
  assumes "normalize a = a" and "normalize b = b"
haftmann@60685
  1149
  shows "a = b"
haftmann@60688
  1150
proof -
haftmann@60688
  1151
  from assms have "normalize a = normalize b"
haftmann@60688
  1152
    unfolding associated_iff_dvd by simp
haftmann@60688
  1153
  with \<open>normalize a = a\<close> have "a = normalize b" by simp
haftmann@60688
  1154
  with \<open>normalize b = b\<close> show "a = b" by simp
haftmann@60685
  1155
qed
haftmann@60685
  1156
haftmann@60685
  1157
end
haftmann@60685
  1158
haftmann@38642
  1159
class ordered_semiring = semiring + comm_monoid_add + ordered_ab_semigroup_add +
haftmann@38642
  1160
  assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@38642
  1161
  assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
haftmann@25230
  1162
begin
haftmann@25230
  1163
haftmann@25230
  1164
lemma mult_mono:
haftmann@38642
  1165
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
  1166
apply (erule mult_right_mono [THEN order_trans], assumption)
haftmann@25230
  1167
apply (erule mult_left_mono, assumption)
haftmann@25230
  1168
done
haftmann@25230
  1169
haftmann@25230
  1170
lemma mult_mono':
haftmann@38642
  1171
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
  1172
apply (rule mult_mono)
haftmann@25230
  1173
apply (fast intro: order_trans)+
haftmann@25230
  1174
done
haftmann@25230
  1175
haftmann@25230
  1176
end
krauss@21199
  1177
haftmann@38642
  1178
class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add
haftmann@25267
  1179
begin
paulson@14268
  1180
huffman@27516
  1181
subclass semiring_0_cancel ..
obua@23521
  1182
nipkow@56536
  1183
lemma mult_nonneg_nonneg[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
haftmann@36301
  1184
using mult_left_mono [of 0 b a] by simp
haftmann@25230
  1185
haftmann@25230
  1186
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
haftmann@36301
  1187
using mult_left_mono [of b 0 a] by simp
huffman@30692
  1188
huffman@30692
  1189
lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"
haftmann@36301
  1190
using mult_right_mono [of a 0 b] by simp
huffman@30692
  1191
huffman@30692
  1192
text {* Legacy - use @{text mult_nonpos_nonneg} *}
lp15@60562
  1193
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0"
haftmann@36301
  1194
by (drule mult_right_mono [of b 0], auto)
haftmann@25230
  1195
lp15@60562
  1196
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0"
nipkow@29667
  1197
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
haftmann@25230
  1198
haftmann@25230
  1199
end
haftmann@25230
  1200
haftmann@38642
  1201
class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add
haftmann@25267
  1202
begin
haftmann@25230
  1203
haftmann@35028
  1204
subclass ordered_cancel_semiring ..
haftmann@35028
  1205
haftmann@35028
  1206
subclass ordered_comm_monoid_add ..
haftmann@25304
  1207
haftmann@25230
  1208
lemma mult_left_less_imp_less:
haftmann@25230
  1209
  "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
  1210
by (force simp add: mult_left_mono not_le [symmetric])
lp15@60562
  1211
haftmann@25230
  1212
lemma mult_right_less_imp_less:
haftmann@25230
  1213
  "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
  1214
by (force simp add: mult_right_mono not_le [symmetric])
obua@23521
  1215
haftmann@25186
  1216
end
haftmann@25152
  1217
haftmann@35043
  1218
class linordered_semiring_1 = linordered_semiring + semiring_1
hoelzl@36622
  1219
begin
hoelzl@36622
  1220
hoelzl@36622
  1221
lemma convex_bound_le:
hoelzl@36622
  1222
  assumes "x \<le> a" "y \<le> a" "0 \<le> u" "0 \<le> v" "u + v = 1"
hoelzl@36622
  1223
  shows "u * x + v * y \<le> a"
hoelzl@36622
  1224
proof-
hoelzl@36622
  1225
  from assms have "u * x + v * y \<le> u * a + v * a"
hoelzl@36622
  1226
    by (simp add: add_mono mult_left_mono)
webertj@49962
  1227
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
hoelzl@36622
  1228
qed
hoelzl@36622
  1229
hoelzl@36622
  1230
end
haftmann@35043
  1231
haftmann@35043
  1232
class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +
haftmann@25062
  1233
  assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25062
  1234
  assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
haftmann@25267
  1235
begin
paulson@14341
  1236
huffman@27516
  1237
subclass semiring_0_cancel ..
obua@14940
  1238
haftmann@35028
  1239
subclass linordered_semiring
haftmann@28823
  1240
proof
huffman@23550
  1241
  fix a b c :: 'a
huffman@23550
  1242
  assume A: "a \<le> b" "0 \<le> c"
huffman@23550
  1243
  from A show "c * a \<le> c * b"
haftmann@25186
  1244
    unfolding le_less
haftmann@25186
  1245
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
  1246
  from A show "a * c \<le> b * c"
haftmann@25152
  1247
    unfolding le_less
haftmann@25186
  1248
    using mult_strict_right_mono by (cases "c = 0") auto
haftmann@25152
  1249
qed
haftmann@25152
  1250
haftmann@25230
  1251
lemma mult_left_le_imp_le:
haftmann@25230
  1252
  "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
  1253
by (force simp add: mult_strict_left_mono _not_less [symmetric])
lp15@60562
  1254
haftmann@25230
  1255
lemma mult_right_le_imp_le:
haftmann@25230
  1256
  "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
  1257
by (force simp add: mult_strict_right_mono not_less [symmetric])
haftmann@25230
  1258
nipkow@56544
  1259
lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
haftmann@36301
  1260
using mult_strict_left_mono [of 0 b a] by simp
huffman@30692
  1261
huffman@30692
  1262
lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
haftmann@36301
  1263
using mult_strict_left_mono [of b 0 a] by simp
huffman@30692
  1264
huffman@30692
  1265
lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"
haftmann@36301
  1266
using mult_strict_right_mono [of a 0 b] by simp
huffman@30692
  1267
huffman@30692
  1268
text {* Legacy - use @{text mult_neg_pos} *}
lp15@60562
  1269
lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0"
haftmann@36301
  1270
by (drule mult_strict_right_mono [of b 0], auto)
haftmann@25230
  1271
haftmann@25230
  1272
lemma zero_less_mult_pos:
haftmann@25230
  1273
  "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
  1274
apply (cases "b\<le>0")
haftmann@25230
  1275
 apply (auto simp add: le_less not_less)
huffman@30692
  1276
apply (drule_tac mult_pos_neg [of a b])
haftmann@25230
  1277
 apply (auto dest: less_not_sym)
haftmann@25230
  1278
done
haftmann@25230
  1279
haftmann@25230
  1280
lemma zero_less_mult_pos2:
haftmann@25230
  1281
  "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
  1282
apply (cases "b\<le>0")
haftmann@25230
  1283
 apply (auto simp add: le_less not_less)
huffman@30692
  1284
apply (drule_tac mult_pos_neg2 [of a b])
haftmann@25230
  1285
 apply (auto dest: less_not_sym)
haftmann@25230
  1286
done
haftmann@25230
  1287
haftmann@26193
  1288
text{*Strict monotonicity in both arguments*}
haftmann@26193
  1289
lemma mult_strict_mono:
haftmann@26193
  1290
  assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
haftmann@26193
  1291
  shows "a * c < b * d"
haftmann@26193
  1292
  using assms apply (cases "c=0")
nipkow@56544
  1293
  apply (simp)
haftmann@26193
  1294
  apply (erule mult_strict_right_mono [THEN less_trans])
huffman@30692
  1295
  apply (force simp add: le_less)
haftmann@26193
  1296
  apply (erule mult_strict_left_mono, assumption)
haftmann@26193
  1297
  done
haftmann@26193
  1298
haftmann@26193
  1299
text{*This weaker variant has more natural premises*}
haftmann@26193
  1300
lemma mult_strict_mono':
haftmann@26193
  1301
  assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
haftmann@26193
  1302
  shows "a * c < b * d"
nipkow@29667
  1303
by (rule mult_strict_mono) (insert assms, auto)
haftmann@26193
  1304
haftmann@26193
  1305
lemma mult_less_le_imp_less:
haftmann@26193
  1306
  assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
haftmann@26193
  1307
  shows "a * c < b * d"
haftmann@26193
  1308
  using assms apply (subgoal_tac "a * c < b * c")
haftmann@26193
  1309
  apply (erule less_le_trans)
haftmann@26193
  1310
  apply (erule mult_left_mono)
haftmann@26193
  1311
  apply simp
haftmann@26193
  1312
  apply (erule mult_strict_right_mono)
haftmann@26193
  1313
  apply assumption
haftmann@26193
  1314
  done
haftmann@26193
  1315
haftmann@26193
  1316
lemma mult_le_less_imp_less:
haftmann@26193
  1317
  assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
haftmann@26193
  1318
  shows "a * c < b * d"
haftmann@26193
  1319
  using assms apply (subgoal_tac "a * c \<le> b * c")
haftmann@26193
  1320
  apply (erule le_less_trans)
haftmann@26193
  1321
  apply (erule mult_strict_left_mono)
haftmann@26193
  1322
  apply simp
haftmann@26193
  1323
  apply (erule mult_right_mono)
haftmann@26193
  1324
  apply simp
haftmann@26193
  1325
  done
haftmann@26193
  1326
haftmann@25230
  1327
end
haftmann@25230
  1328
haftmann@35097
  1329
class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1
hoelzl@36622
  1330
begin
hoelzl@36622
  1331
hoelzl@36622
  1332
subclass linordered_semiring_1 ..
hoelzl@36622
  1333
hoelzl@36622
  1334
lemma convex_bound_lt:
hoelzl@36622
  1335
  assumes "x < a" "y < a" "0 \<le> u" "0 \<le> v" "u + v = 1"
hoelzl@36622
  1336
  shows "u * x + v * y < a"
hoelzl@36622
  1337
proof -
hoelzl@36622
  1338
  from assms have "u * x + v * y < u * a + v * a"
hoelzl@36622
  1339
    by (cases "u = 0")
hoelzl@36622
  1340
       (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
webertj@49962
  1341
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
hoelzl@36622
  1342
qed
hoelzl@36622
  1343
hoelzl@36622
  1344
end
haftmann@33319
  1345
lp15@60562
  1346
class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add +
haftmann@38642
  1347
  assumes comm_mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@25186
  1348
begin
haftmann@25152
  1349
haftmann@35028
  1350
subclass ordered_semiring
haftmann@28823
  1351
proof
krauss@21199
  1352
  fix a b c :: 'a
huffman@23550
  1353
  assume "a \<le> b" "0 \<le> c"
haftmann@38642
  1354
  thus "c * a \<le> c * b" by (rule comm_mult_left_mono)
haftmann@57512
  1355
  thus "a * c \<le> b * c" by (simp only: mult.commute)
krauss@21199
  1356
qed
paulson@14265
  1357
haftmann@25267
  1358
end
haftmann@25267
  1359
haftmann@38642
  1360
class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add
haftmann@25267
  1361
begin
paulson@14265
  1362
haftmann@38642
  1363
subclass comm_semiring_0_cancel ..
haftmann@35028
  1364
subclass ordered_comm_semiring ..
haftmann@35028
  1365
subclass ordered_cancel_semiring ..
haftmann@25267
  1366
haftmann@25267
  1367
end
haftmann@25267
  1368
haftmann@35028
  1369
class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +
haftmann@38642
  1370
  assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25267
  1371
begin
haftmann@25267
  1372
haftmann@35043
  1373
subclass linordered_semiring_strict
haftmann@28823
  1374
proof
huffman@23550
  1375
  fix a b c :: 'a
huffman@23550
  1376
  assume "a < b" "0 < c"
haftmann@38642
  1377
  thus "c * a < c * b" by (rule comm_mult_strict_left_mono)
haftmann@57512
  1378
  thus "a * c < b * c" by (simp only: mult.commute)
huffman@23550
  1379
qed
paulson@14272
  1380
haftmann@35028
  1381
subclass ordered_cancel_comm_semiring
haftmann@28823
  1382
proof
huffman@23550
  1383
  fix a b c :: 'a
huffman@23550
  1384
  assume "a \<le> b" "0 \<le> c"
huffman@23550
  1385
  thus "c * a \<le> c * b"
haftmann@25186
  1386
    unfolding le_less
haftmann@26193
  1387
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
  1388
qed
paulson@14272
  1389
haftmann@25267
  1390
end
haftmann@25230
  1391
lp15@60562
  1392
class ordered_ring = ring + ordered_cancel_semiring
haftmann@25267
  1393
begin
haftmann@25230
  1394
haftmann@35028
  1395
subclass ordered_ab_group_add ..
paulson@14270
  1396
haftmann@25230
  1397
lemma less_add_iff1:
haftmann@25230
  1398
  "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
nipkow@29667
  1399
by (simp add: algebra_simps)
haftmann@25230
  1400
haftmann@25230
  1401
lemma less_add_iff2:
haftmann@25230
  1402
  "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
nipkow@29667
  1403
by (simp add: algebra_simps)
haftmann@25230
  1404
haftmann@25230
  1405
lemma le_add_iff1:
haftmann@25230
  1406
  "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
nipkow@29667
  1407
by (simp add: algebra_simps)
haftmann@25230
  1408
haftmann@25230
  1409
lemma le_add_iff2:
haftmann@25230
  1410
  "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
nipkow@29667
  1411
by (simp add: algebra_simps)
haftmann@25230
  1412
haftmann@25230
  1413
lemma mult_left_mono_neg:
haftmann@25230
  1414
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
haftmann@36301
  1415
  apply (drule mult_left_mono [of _ _ "- c"])
huffman@35216
  1416
  apply simp_all
haftmann@25230
  1417
  done
haftmann@25230
  1418
haftmann@25230
  1419
lemma mult_right_mono_neg:
haftmann@25230
  1420
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
haftmann@36301
  1421
  apply (drule mult_right_mono [of _ _ "- c"])
huffman@35216
  1422
  apply simp_all
haftmann@25230
  1423
  done
haftmann@25230
  1424
huffman@30692
  1425
lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
haftmann@36301
  1426
using mult_right_mono_neg [of a 0 b] by simp
haftmann@25230
  1427
haftmann@25230
  1428
lemma split_mult_pos_le:
haftmann@25230
  1429
  "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
nipkow@56536
  1430
by (auto simp add: mult_nonpos_nonpos)
haftmann@25186
  1431
haftmann@25186
  1432
end
paulson@14270
  1433
haftmann@35028
  1434
class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if
haftmann@25304
  1435
begin
haftmann@25304
  1436
haftmann@35028
  1437
subclass ordered_ring ..
haftmann@35028
  1438
haftmann@35028
  1439
subclass ordered_ab_group_add_abs
haftmann@28823
  1440
proof
haftmann@25304
  1441
  fix a b
haftmann@25304
  1442
  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@54230
  1443
    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
  1444
qed (auto simp add: abs_if)
haftmann@25304
  1445
huffman@35631
  1446
lemma zero_le_square [simp]: "0 \<le> a * a"
huffman@35631
  1447
  using linear [of 0 a]
nipkow@56536
  1448
  by (auto simp add: mult_nonpos_nonpos)
huffman@35631
  1449
huffman@35631
  1450
lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
huffman@35631
  1451
  by (simp add: not_less)
huffman@35631
  1452
haftmann@25304
  1453
end
obua@23521
  1454
haftmann@35043
  1455
class linordered_ring_strict = ring + linordered_semiring_strict
haftmann@25304
  1456
  + ordered_ab_group_add + abs_if
haftmann@25230
  1457
begin
paulson@14348
  1458
haftmann@35028
  1459
subclass linordered_ring ..
haftmann@25304
  1460
huffman@30692
  1461
lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
huffman@30692
  1462
using mult_strict_left_mono [of b a "- c"] by simp
huffman@30692
  1463
huffman@30692
  1464
lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
huffman@30692
  1465
using mult_strict_right_mono [of b a "- c"] by simp
huffman@30692
  1466
huffman@30692
  1467
lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
haftmann@36301
  1468
using mult_strict_right_mono_neg [of a 0 b] by simp
obua@14738
  1469
haftmann@25917
  1470
subclass ring_no_zero_divisors
haftmann@28823
  1471
proof
haftmann@25917
  1472
  fix a b
haftmann@25917
  1473
  assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
haftmann@25917
  1474
  assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
haftmann@25917
  1475
  have "a * b < 0 \<or> 0 < a * b"
haftmann@25917
  1476
  proof (cases "a < 0")
haftmann@25917
  1477
    case True note A' = this
haftmann@25917
  1478
    show ?thesis proof (cases "b < 0")
haftmann@25917
  1479
      case True with A'
haftmann@25917
  1480
      show ?thesis by (auto dest: mult_neg_neg)
haftmann@25917
  1481
    next
haftmann@25917
  1482
      case False with B have "0 < b" by auto
haftmann@25917
  1483
      with A' show ?thesis by (auto dest: mult_strict_right_mono)
haftmann@25917
  1484
    qed
haftmann@25917
  1485
  next
haftmann@25917
  1486
    case False with A have A': "0 < a" by auto
haftmann@25917
  1487
    show ?thesis proof (cases "b < 0")
haftmann@25917
  1488
      case True with A'
haftmann@25917
  1489
      show ?thesis by (auto dest: mult_strict_right_mono_neg)
haftmann@25917
  1490
    next
haftmann@25917
  1491
      case False with B have "0 < b" by auto
nipkow@56544
  1492
      with A' show ?thesis by auto
haftmann@25917
  1493
    qed
haftmann@25917
  1494
  qed
haftmann@25917
  1495
  then show "a * b \<noteq> 0" by (simp add: neq_iff)
haftmann@25917
  1496
qed
haftmann@25304
  1497
hoelzl@56480
  1498
lemma zero_less_mult_iff: "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
hoelzl@56480
  1499
  by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
nipkow@56544
  1500
     (auto simp add: mult_neg_neg not_less le_less dest: zero_less_mult_pos zero_less_mult_pos2)
huffman@22990
  1501
hoelzl@56480
  1502
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
  1503
  by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
paulson@14265
  1504
paulson@14265
  1505
lemma mult_less_0_iff:
haftmann@25917
  1506
  "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
huffman@35216
  1507
  apply (insert zero_less_mult_iff [of "-a" b])
huffman@35216
  1508
  apply force
haftmann@25917
  1509
  done
paulson@14265
  1510
paulson@14265
  1511
lemma mult_le_0_iff:
haftmann@25917
  1512
  "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
lp15@60562
  1513
  apply (insert zero_le_mult_iff [of "-a" b])
huffman@35216
  1514
  apply force
haftmann@25917
  1515
  done
haftmann@25917
  1516
haftmann@26193
  1517
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
haftmann@26193
  1518
   also with the relations @{text "\<le>"} and equality.*}
haftmann@26193
  1519
haftmann@26193
  1520
text{*These ``disjunction'' versions produce two cases when the comparison is
haftmann@26193
  1521
 an assumption, but effectively four when the comparison is a goal.*}
haftmann@26193
  1522
haftmann@26193
  1523
lemma mult_less_cancel_right_disj:
haftmann@26193
  1524
  "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
  1525
  apply (cases "c = 0")
lp15@60562
  1526
  apply (auto simp add: neq_iff mult_strict_right_mono
haftmann@26193
  1527
                      mult_strict_right_mono_neg)
lp15@60562
  1528
  apply (auto simp add: not_less
haftmann@26193
  1529
                      not_le [symmetric, of "a*c"]
haftmann@26193
  1530
                      not_le [symmetric, of a])
haftmann@26193
  1531
  apply (erule_tac [!] notE)
lp15@60562
  1532
  apply (auto simp add: less_imp_le mult_right_mono
haftmann@26193
  1533
                      mult_right_mono_neg)
haftmann@26193
  1534
  done
haftmann@26193
  1535
haftmann@26193
  1536
lemma mult_less_cancel_left_disj:
haftmann@26193
  1537
  "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
  1538
  apply (cases "c = 0")
lp15@60562
  1539
  apply (auto simp add: neq_iff mult_strict_left_mono
haftmann@26193
  1540
                      mult_strict_left_mono_neg)
lp15@60562
  1541
  apply (auto simp add: not_less
haftmann@26193
  1542
                      not_le [symmetric, of "c*a"]
haftmann@26193
  1543
                      not_le [symmetric, of a])
haftmann@26193
  1544
  apply (erule_tac [!] notE)
lp15@60562
  1545
  apply (auto simp add: less_imp_le mult_left_mono
haftmann@26193
  1546
                      mult_left_mono_neg)
haftmann@26193
  1547
  done
haftmann@26193
  1548
haftmann@26193
  1549
text{*The ``conjunction of implication'' lemmas produce two cases when the
haftmann@26193
  1550
comparison is a goal, but give four when the comparison is an assumption.*}
haftmann@26193
  1551
haftmann@26193
  1552
lemma mult_less_cancel_right:
haftmann@26193
  1553
  "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
  1554
  using mult_less_cancel_right_disj [of a c b] by auto
haftmann@26193
  1555
haftmann@26193
  1556
lemma mult_less_cancel_left:
haftmann@26193
  1557
  "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
  1558
  using mult_less_cancel_left_disj [of c a b] by auto
haftmann@26193
  1559
haftmann@26193
  1560
lemma mult_le_cancel_right:
haftmann@26193
  1561
   "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
  1562
by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
haftmann@26193
  1563
haftmann@26193
  1564
lemma mult_le_cancel_left:
haftmann@26193
  1565
  "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
  1566
by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
haftmann@26193
  1567
nipkow@30649
  1568
lemma mult_le_cancel_left_pos:
nipkow@30649
  1569
  "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
nipkow@30649
  1570
by (auto simp: mult_le_cancel_left)
nipkow@30649
  1571
nipkow@30649
  1572
lemma mult_le_cancel_left_neg:
nipkow@30649
  1573
  "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
nipkow@30649
  1574
by (auto simp: mult_le_cancel_left)
nipkow@30649
  1575
nipkow@30649
  1576
lemma mult_less_cancel_left_pos:
nipkow@30649
  1577
  "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
nipkow@30649
  1578
by (auto simp: mult_less_cancel_left)
nipkow@30649
  1579
nipkow@30649
  1580
lemma mult_less_cancel_left_neg:
nipkow@30649
  1581
  "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
nipkow@30649
  1582
by (auto simp: mult_less_cancel_left)
nipkow@30649
  1583
haftmann@25917
  1584
end
paulson@14265
  1585
huffman@30692
  1586
lemmas mult_sign_intros =
huffman@30692
  1587
  mult_nonneg_nonneg mult_nonneg_nonpos
huffman@30692
  1588
  mult_nonpos_nonneg mult_nonpos_nonpos
huffman@30692
  1589
  mult_pos_pos mult_pos_neg
huffman@30692
  1590
  mult_neg_pos mult_neg_neg
haftmann@25230
  1591
haftmann@35028
  1592
class ordered_comm_ring = comm_ring + ordered_comm_semiring
haftmann@25267
  1593
begin
haftmann@25230
  1594
haftmann@35028
  1595
subclass ordered_ring ..
haftmann@35028
  1596
subclass ordered_cancel_comm_semiring ..
haftmann@25230
  1597
haftmann@25267
  1598
end
haftmann@25230
  1599
haftmann@59833
  1600
class linordered_semidom = semidom + linordered_comm_semiring_strict +
haftmann@25230
  1601
  assumes zero_less_one [simp]: "0 < 1"
lp15@60562
  1602
  assumes le_add_diff_inverse2 [simp]: "b \<le> a \<Longrightarrow> a - b + b = a"
haftmann@25230
  1603
begin
haftmann@25230
  1604
lp15@60562
  1605
text {* Addition is the inverse of subtraction. *}
lp15@60562
  1606
lp15@60562
  1607
lemma le_add_diff_inverse [simp]: "b \<le> a \<Longrightarrow> b + (a - b) = a"
lp15@60562
  1608
  by (frule le_add_diff_inverse2) (simp add: add.commute)
lp15@60562
  1609
lp15@60562
  1610
lemma add_diff_inverse: "~ a<b \<Longrightarrow> b + (a - b) = a"
lp15@60562
  1611
  by simp
lp15@60615
  1612
lp15@60615
  1613
lemma add_le_imp_le_diff: 
lp15@60615
  1614
  shows "i + k \<le> n \<Longrightarrow> i \<le> n - k"
lp15@60615
  1615
  apply (subst add_le_cancel_right [where c=k, symmetric])
lp15@60615
  1616
  apply (frule le_add_diff_inverse2)
lp15@60615
  1617
  apply (simp only: add.assoc [symmetric])
lp15@60615
  1618
  using add_implies_diff by fastforce
lp15@60615
  1619
lp15@60615
  1620
lemma add_le_add_imp_diff_le: 
lp15@60615
  1621
  assumes a1: "i + k \<le> n"
lp15@60615
  1622
      and a2: "n \<le> j + k"
lp15@60615
  1623
  shows "\<lbrakk>i + k \<le> n; n \<le> j + k\<rbrakk> \<Longrightarrow> n - k \<le> j"
lp15@60615
  1624
proof -
lp15@60615
  1625
  have "n - (i + k) + (i + k) = n"
lp15@60615
  1626
    using a1 by simp
lp15@60615
  1627
  moreover have "n - k = n - k - i + i"
lp15@60615
  1628
    using a1 by (simp add: add_le_imp_le_diff)
lp15@60615
  1629
  ultimately show ?thesis
lp15@60615
  1630
    using a2
lp15@60615
  1631
    apply (simp add: add.assoc [symmetric])
lp15@60615
  1632
    apply (rule add_le_imp_le_diff [of _ k "j+k", simplified add_diff_cancel_right'])
lp15@60615
  1633
    by (simp add: add.commute diff_diff_add)
lp15@60615
  1634
qed
lp15@60615
  1635
haftmann@25230
  1636
lemma pos_add_strict:
haftmann@25230
  1637
  shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@36301
  1638
  using add_strict_mono [of 0 a b c] by simp
haftmann@25230
  1639
haftmann@26193
  1640
lemma zero_le_one [simp]: "0 \<le> 1"
lp15@60562
  1641
by (rule zero_less_one [THEN less_imp_le])
haftmann@26193
  1642
haftmann@26193
  1643
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
lp15@60562
  1644
by (simp add: not_le)
haftmann@26193
  1645
haftmann@26193
  1646
lemma not_one_less_zero [simp]: "\<not> 1 < 0"
lp15@60562
  1647
by (simp add: not_less)
haftmann@26193
  1648
haftmann@26193
  1649
lemma less_1_mult:
haftmann@26193
  1650
  assumes "1 < m" and "1 < n"
haftmann@26193
  1651
  shows "1 < m * n"
haftmann@26193
  1652
  using assms mult_strict_mono [of 1 m 1 n]
lp15@60562
  1653
    by (simp add:  less_trans [OF zero_less_one])
haftmann@26193
  1654
hoelzl@59000
  1655
lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a"
hoelzl@59000
  1656
  using mult_left_mono[of c 1 a] by simp
hoelzl@59000
  1657
hoelzl@59000
  1658
lemma mult_le_one: "a \<le> 1 \<Longrightarrow> 0 \<le> b \<Longrightarrow> b \<le> 1 \<Longrightarrow> a * b \<le> 1"
hoelzl@59000
  1659
  using mult_mono[of a 1 b 1] by simp
hoelzl@59000
  1660
haftmann@25230
  1661
end
haftmann@25230
  1662
haftmann@35028
  1663
class linordered_idom = comm_ring_1 +
haftmann@35028
  1664
  linordered_comm_semiring_strict + ordered_ab_group_add +
haftmann@25230
  1665
  abs_if + sgn_if
haftmann@25917
  1666
begin
haftmann@25917
  1667
hoelzl@36622
  1668
subclass linordered_semiring_1_strict ..
haftmann@35043
  1669
subclass linordered_ring_strict ..
haftmann@35028
  1670
subclass ordered_comm_ring ..
huffman@27516
  1671
subclass idom ..
haftmann@25917
  1672
haftmann@35028
  1673
subclass linordered_semidom
haftmann@28823
  1674
proof
haftmann@26193
  1675
  have "0 \<le> 1 * 1" by (rule zero_le_square)
haftmann@26193
  1676
  thus "0 < 1" by (simp add: le_less)
lp15@60562
  1677
  show "\<And>b a. b \<le> a \<Longrightarrow> a - b + b = a"
lp15@60562
  1678
    by simp
lp15@60562
  1679
qed
haftmann@25917
  1680
haftmann@35028
  1681
lemma linorder_neqE_linordered_idom:
haftmann@26193
  1682
  assumes "x \<noteq> y" obtains "x < y" | "y < x"
haftmann@26193
  1683
  using assms by (rule neqE)
haftmann@26193
  1684
haftmann@26274
  1685
text {* These cancellation simprules also produce two cases when the comparison is a goal. *}
haftmann@26274
  1686
haftmann@26274
  1687
lemma mult_le_cancel_right1:
haftmann@26274
  1688
  "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1689
by (insert mult_le_cancel_right [of 1 c b], simp)
haftmann@26274
  1690
haftmann@26274
  1691
lemma mult_le_cancel_right2:
haftmann@26274
  1692
  "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1693
by (insert mult_le_cancel_right [of a c 1], simp)
haftmann@26274
  1694
haftmann@26274
  1695
lemma mult_le_cancel_left1:
haftmann@26274
  1696
  "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1697
by (insert mult_le_cancel_left [of c 1 b], simp)
haftmann@26274
  1698
haftmann@26274
  1699
lemma mult_le_cancel_left2:
haftmann@26274
  1700
  "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1701
by (insert mult_le_cancel_left [of c a 1], simp)
haftmann@26274
  1702
haftmann@26274
  1703
lemma mult_less_cancel_right1:
haftmann@26274
  1704
  "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1705
by (insert mult_less_cancel_right [of 1 c b], simp)
haftmann@26274
  1706
haftmann@26274
  1707
lemma mult_less_cancel_right2:
haftmann@26274
  1708
  "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1709
by (insert mult_less_cancel_right [of a c 1], simp)
haftmann@26274
  1710
haftmann@26274
  1711
lemma mult_less_cancel_left1:
haftmann@26274
  1712
  "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1713
by (insert mult_less_cancel_left [of c 1 b], simp)
haftmann@26274
  1714
haftmann@26274
  1715
lemma mult_less_cancel_left2:
haftmann@26274
  1716
  "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1717
by (insert mult_less_cancel_left [of c a 1], simp)
haftmann@26274
  1718
haftmann@27651
  1719
lemma sgn_sgn [simp]:
haftmann@27651
  1720
  "sgn (sgn a) = sgn a"
nipkow@29700
  1721
unfolding sgn_if by simp
haftmann@27651
  1722
haftmann@27651
  1723
lemma sgn_0_0:
haftmann@27651
  1724
  "sgn a = 0 \<longleftrightarrow> a = 0"
nipkow@29700
  1725
unfolding sgn_if by simp
haftmann@27651
  1726
haftmann@27651
  1727
lemma sgn_1_pos:
haftmann@27651
  1728
  "sgn a = 1 \<longleftrightarrow> a > 0"
huffman@35216
  1729
unfolding sgn_if by simp
haftmann@27651
  1730
haftmann@27651
  1731
lemma sgn_1_neg:
haftmann@27651
  1732
  "sgn a = - 1 \<longleftrightarrow> a < 0"
huffman@35216
  1733
unfolding sgn_if by auto
haftmann@27651
  1734
haftmann@29940
  1735
lemma sgn_pos [simp]:
haftmann@29940
  1736
  "0 < a \<Longrightarrow> sgn a = 1"
haftmann@29940
  1737
unfolding sgn_1_pos .
haftmann@29940
  1738
haftmann@29940
  1739
lemma sgn_neg [simp]:
haftmann@29940
  1740
  "a < 0 \<Longrightarrow> sgn a = - 1"
haftmann@29940
  1741
unfolding sgn_1_neg .
haftmann@29940
  1742
haftmann@27651
  1743
lemma sgn_times:
haftmann@27651
  1744
  "sgn (a * b) = sgn a * sgn b"
nipkow@29667
  1745
by (auto simp add: sgn_if zero_less_mult_iff)
haftmann@27651
  1746
haftmann@36301
  1747
lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"
nipkow@29700
  1748
unfolding sgn_if abs_if by auto
nipkow@29700
  1749
haftmann@29940
  1750
lemma sgn_greater [simp]:
haftmann@29940
  1751
  "0 < sgn a \<longleftrightarrow> 0 < a"
haftmann@29940
  1752
  unfolding sgn_if by auto
haftmann@29940
  1753
haftmann@29940
  1754
lemma sgn_less [simp]:
haftmann@29940
  1755
  "sgn a < 0 \<longleftrightarrow> a < 0"
haftmann@29940
  1756
  unfolding sgn_if by auto
haftmann@29940
  1757
haftmann@36301
  1758
lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"
huffman@29949
  1759
  by (simp add: abs_if)
huffman@29949
  1760
haftmann@36301
  1761
lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"
huffman@29949
  1762
  by (simp add: abs_if)
haftmann@29653
  1763
nipkow@33676
  1764
lemma dvd_if_abs_eq:
haftmann@36301
  1765
  "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
nipkow@33676
  1766
by(subst abs_dvd_iff[symmetric]) simp
nipkow@33676
  1767
huffman@55912
  1768
text {* The following lemmas can be proven in more general structures, but
lp15@60562
  1769
are dangerous as simp rules in absence of @{thm neg_equal_zero},
haftmann@54489
  1770
@{thm neg_less_pos}, @{thm neg_less_eq_nonneg}. *}
haftmann@54489
  1771
haftmann@54489
  1772
lemma equation_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1773
  "1 = - a \<longleftrightarrow> a = - 1"
haftmann@54489
  1774
  by (fact equation_minus_iff)
haftmann@54489
  1775
haftmann@54489
  1776
lemma minus_equation_iff_1 [simp, no_atp]:
haftmann@54489
  1777
  "- a = 1 \<longleftrightarrow> a = - 1"
haftmann@54489
  1778
  by (subst minus_equation_iff, auto)
haftmann@54489
  1779
haftmann@54489
  1780
lemma le_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1781
  "1 \<le> - b \<longleftrightarrow> b \<le> - 1"
haftmann@54489
  1782
  by (fact le_minus_iff)
haftmann@54489
  1783
haftmann@54489
  1784
lemma minus_le_iff_1 [simp, no_atp]:
haftmann@54489
  1785
  "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"
haftmann@54489
  1786
  by (fact minus_le_iff)
haftmann@54489
  1787
haftmann@54489
  1788
lemma less_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1789
  "1 < - b \<longleftrightarrow> b < - 1"
haftmann@54489
  1790
  by (fact less_minus_iff)
haftmann@54489
  1791
haftmann@54489
  1792
lemma minus_less_iff_1 [simp, no_atp]:
haftmann@54489
  1793
  "- a < 1 \<longleftrightarrow> - 1 < a"
haftmann@54489
  1794
  by (fact minus_less_iff)
haftmann@54489
  1795
haftmann@25917
  1796
end
haftmann@25230
  1797
haftmann@26274
  1798
text {* Simprules for comparisons where common factors can be cancelled. *}
paulson@15234
  1799
blanchet@54147
  1800
lemmas mult_compare_simps =
paulson@15234
  1801
    mult_le_cancel_right mult_le_cancel_left
paulson@15234
  1802
    mult_le_cancel_right1 mult_le_cancel_right2
paulson@15234
  1803
    mult_le_cancel_left1 mult_le_cancel_left2
paulson@15234
  1804
    mult_less_cancel_right mult_less_cancel_left
paulson@15234
  1805
    mult_less_cancel_right1 mult_less_cancel_right2
paulson@15234
  1806
    mult_less_cancel_left1 mult_less_cancel_left2
paulson@15234
  1807
    mult_cancel_right mult_cancel_left
paulson@15234
  1808
    mult_cancel_right1 mult_cancel_right2
paulson@15234
  1809
    mult_cancel_left1 mult_cancel_left2
paulson@15234
  1810
haftmann@36301
  1811
text {* Reasoning about inequalities with division *}
avigad@16775
  1812
haftmann@35028
  1813
context linordered_semidom
haftmann@25193
  1814
begin
haftmann@25193
  1815
haftmann@25193
  1816
lemma less_add_one: "a < a + 1"
paulson@14293
  1817
proof -
haftmann@25193
  1818
  have "a + 0 < a + 1"
nipkow@23482
  1819
    by (blast intro: zero_less_one add_strict_left_mono)
paulson@14293
  1820
  thus ?thesis by simp
paulson@14293
  1821
qed
paulson@14293
  1822
haftmann@25193
  1823
lemma zero_less_two: "0 < 1 + 1"
nipkow@29667
  1824
by (blast intro: less_trans zero_less_one less_add_one)
haftmann@25193
  1825
haftmann@25193
  1826
end
paulson@14365
  1827
haftmann@36301
  1828
context linordered_idom
haftmann@36301
  1829
begin
paulson@15234
  1830
haftmann@36301
  1831
lemma mult_right_le_one_le:
haftmann@36301
  1832
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"
haftmann@59833
  1833
  by (rule mult_left_le)
haftmann@36301
  1834
haftmann@36301
  1835
lemma mult_left_le_one_le:
haftmann@36301
  1836
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"
haftmann@36301
  1837
  by (auto simp add: mult_le_cancel_right2)
haftmann@36301
  1838
haftmann@36301
  1839
end
haftmann@36301
  1840
haftmann@36301
  1841
text {* Absolute Value *}
paulson@14293
  1842
haftmann@35028
  1843
context linordered_idom
haftmann@25304
  1844
begin
haftmann@25304
  1845
haftmann@36301
  1846
lemma mult_sgn_abs:
haftmann@36301
  1847
  "sgn x * \<bar>x\<bar> = x"
haftmann@25304
  1848
  unfolding abs_if sgn_if by auto
haftmann@25304
  1849
haftmann@36301
  1850
lemma abs_one [simp]:
haftmann@36301
  1851
  "\<bar>1\<bar> = 1"
huffman@44921
  1852
  by (simp add: abs_if)
haftmann@36301
  1853
haftmann@25304
  1854
end
nipkow@24491
  1855
haftmann@35028
  1856
class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +
haftmann@25304
  1857
  assumes abs_eq_mult:
haftmann@25304
  1858
    "(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
  1859
haftmann@35028
  1860
context linordered_idom
haftmann@30961
  1861
begin
haftmann@30961
  1862
haftmann@35028
  1863
subclass ordered_ring_abs proof
huffman@35216
  1864
qed (auto simp add: abs_if not_less mult_less_0_iff)
haftmann@30961
  1865
haftmann@30961
  1866
lemma abs_mult:
lp15@60562
  1867
  "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
haftmann@30961
  1868
  by (rule abs_eq_mult) auto
haftmann@30961
  1869
haftmann@30961
  1870
lemma abs_mult_self:
haftmann@36301
  1871
  "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
lp15@60562
  1872
  by (simp add: abs_if)
haftmann@30961
  1873
paulson@14294
  1874
lemma abs_mult_less:
haftmann@36301
  1875
  "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"
paulson@14294
  1876
proof -
haftmann@36301
  1877
  assume ac: "\<bar>a\<bar> < c"
haftmann@36301
  1878
  hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)
haftmann@36301
  1879
  assume "\<bar>b\<bar> < d"
lp15@60562
  1880
  thus ?thesis by (simp add: ac cpos mult_strict_mono)
paulson@14294
  1881
qed
paulson@14293
  1882
haftmann@36301
  1883
lemma abs_less_iff:
lp15@60562
  1884
  "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b"
haftmann@36301
  1885
  by (simp add: less_le abs_le_iff) (auto simp add: abs_if)
obua@14738
  1886
haftmann@36301
  1887
lemma abs_mult_pos:
haftmann@36301
  1888
  "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"
haftmann@36301
  1889
  by (simp add: abs_mult)
haftmann@36301
  1890
hoelzl@51520
  1891
lemma abs_diff_less_iff:
hoelzl@51520
  1892
  "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"
hoelzl@51520
  1893
  by (auto simp add: diff_less_eq ac_simps abs_less_iff)
hoelzl@51520
  1894
lp15@59865
  1895
lemma abs_diff_le_iff:
lp15@59865
  1896
   "\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r"
lp15@59865
  1897
  by (auto simp add: diff_le_eq ac_simps abs_le_iff)
lp15@59865
  1898
haftmann@36301
  1899
end
avigad@16775
  1900
haftmann@59557
  1901
hide_fact (open) comm_mult_left_mono comm_mult_strict_left_mono distrib
haftmann@59557
  1902
haftmann@52435
  1903
code_identifier
haftmann@52435
  1904
  code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
  1905
paulson@14265
  1906
end