src/HOL/Rings.thy
author paulson <lp15@cam.ac.uk>
Tue Jun 30 13:56:16 2015 +0100 (2015-06-30)
changeset 60615 e5fa1d5d3952
parent 60570 7ed2cde6806d
child 60685 cb21b7022b00
permissions -rw-r--r--
Useful lemmas. The theorem concerning swapping the variables in a double integral.
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@60517
   639
lemma dvd_div_mult_self [simp]:
haftmann@60517
   640
  "a dvd b \<Longrightarrow> b div a * a = b"
haftmann@60517
   641
  by (cases "a = 0") (auto elim: dvdE simp add: ac_simps)
haftmann@60517
   642
haftmann@60517
   643
lemma dvd_mult_div_cancel [simp]:
haftmann@60517
   644
  "a dvd b \<Longrightarrow> a * (b div a) = b"
haftmann@60517
   645
  using dvd_div_mult_self [of a b] by (simp add: ac_simps)
lp15@60562
   646
haftmann@60517
   647
lemma div_mult_swap:
haftmann@60517
   648
  assumes "c dvd b"
haftmann@60517
   649
  shows "a * (b div c) = (a * b) div c"
haftmann@60517
   650
proof (cases "c = 0")
haftmann@60517
   651
  case True then show ?thesis by simp
haftmann@60517
   652
next
haftmann@60517
   653
  case False from assms obtain d where "b = c * d" ..
haftmann@60517
   654
  moreover from False have "a * divide (d * c) c = ((a * d) * c) div c"
haftmann@60517
   655
    by simp
haftmann@60517
   656
  ultimately show ?thesis by (simp add: ac_simps)
haftmann@60517
   657
qed
haftmann@60517
   658
haftmann@60517
   659
lemma dvd_div_mult:
haftmann@60517
   660
  assumes "c dvd b"
haftmann@60517
   661
  shows "b div c * a = (b * a) div c"
haftmann@60517
   662
  using assms div_mult_swap [of c b a] by (simp add: ac_simps)
haftmann@60517
   663
haftmann@60570
   664
lemma dvd_div_mult2_eq:
haftmann@60570
   665
  assumes "b * c dvd a"
haftmann@60570
   666
  shows "a div (b * c) = a div b div c"
haftmann@60570
   667
using assms proof
haftmann@60570
   668
  fix k
haftmann@60570
   669
  assume "a = b * c * k"
haftmann@60570
   670
  then show ?thesis
haftmann@60570
   671
    by (cases "b = 0 \<or> c = 0") (auto, simp add: ac_simps)
haftmann@60570
   672
qed
haftmann@60570
   673
lp15@60562
   674
haftmann@60517
   675
text \<open>Units: invertible elements in a ring\<close>
haftmann@60517
   676
haftmann@60517
   677
abbreviation is_unit :: "'a \<Rightarrow> bool"
haftmann@60517
   678
where
haftmann@60517
   679
  "is_unit a \<equiv> a dvd 1"
haftmann@60517
   680
haftmann@60517
   681
lemma not_is_unit_0 [simp]:
haftmann@60517
   682
  "\<not> is_unit 0"
haftmann@60517
   683
  by simp
haftmann@60517
   684
lp15@60562
   685
lemma unit_imp_dvd [dest]:
haftmann@60517
   686
  "is_unit b \<Longrightarrow> b dvd a"
haftmann@60517
   687
  by (rule dvd_trans [of _ 1]) simp_all
haftmann@60517
   688
haftmann@60517
   689
lemma unit_dvdE:
haftmann@60517
   690
  assumes "is_unit a"
haftmann@60517
   691
  obtains c where "a \<noteq> 0" and "b = a * c"
haftmann@60517
   692
proof -
haftmann@60517
   693
  from assms have "a dvd b" by auto
haftmann@60517
   694
  then obtain c where "b = a * c" ..
haftmann@60517
   695
  moreover from assms have "a \<noteq> 0" by auto
haftmann@60517
   696
  ultimately show thesis using that by blast
haftmann@60517
   697
qed
haftmann@60517
   698
haftmann@60517
   699
lemma dvd_unit_imp_unit:
haftmann@60517
   700
  "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a"
haftmann@60517
   701
  by (rule dvd_trans)
haftmann@60517
   702
haftmann@60517
   703
lemma unit_div_1_unit [simp, intro]:
haftmann@60517
   704
  assumes "is_unit a"
haftmann@60517
   705
  shows "is_unit (1 div a)"
haftmann@60517
   706
proof -
haftmann@60517
   707
  from assms have "1 = 1 div a * a" by simp
haftmann@60517
   708
  then show "is_unit (1 div a)" by (rule dvdI)
haftmann@60517
   709
qed
haftmann@60517
   710
haftmann@60517
   711
lemma is_unitE [elim?]:
haftmann@60517
   712
  assumes "is_unit a"
haftmann@60517
   713
  obtains b where "a \<noteq> 0" and "b \<noteq> 0"
haftmann@60517
   714
    and "is_unit b" and "1 div a = b" and "1 div b = a"
haftmann@60517
   715
    and "a * b = 1" and "c div a = c * b"
haftmann@60517
   716
proof (rule that)
haftmann@60517
   717
  def b \<equiv> "1 div a"
haftmann@60517
   718
  then show "1 div a = b" by simp
haftmann@60517
   719
  from b_def `is_unit a` show "is_unit b" by simp
haftmann@60517
   720
  from `is_unit a` and `is_unit b` show "a \<noteq> 0" and "b \<noteq> 0" by auto
haftmann@60517
   721
  from b_def `is_unit a` show "a * b = 1" by simp
haftmann@60517
   722
  then have "1 = a * b" ..
haftmann@60517
   723
  with b_def `b \<noteq> 0` show "1 div b = a" by simp
haftmann@60517
   724
  from `is_unit a` have "a dvd c" ..
haftmann@60517
   725
  then obtain d where "c = a * d" ..
haftmann@60517
   726
  with `a \<noteq> 0` `a * b = 1` show "c div a = c * b"
haftmann@60517
   727
    by (simp add: mult.assoc mult.left_commute [of a])
haftmann@60517
   728
qed
haftmann@60517
   729
haftmann@60517
   730
lemma unit_prod [intro]:
haftmann@60517
   731
  "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)"
lp15@60562
   732
  by (subst mult_1_left [of 1, symmetric]) (rule mult_dvd_mono)
lp15@60562
   733
haftmann@60517
   734
lemma unit_div [intro]:
haftmann@60517
   735
  "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)"
haftmann@60517
   736
  by (erule is_unitE [of b a]) (simp add: ac_simps unit_prod)
haftmann@60517
   737
haftmann@60517
   738
lemma mult_unit_dvd_iff:
haftmann@60517
   739
  assumes "is_unit b"
haftmann@60517
   740
  shows "a * b dvd c \<longleftrightarrow> a dvd c"
haftmann@60517
   741
proof
haftmann@60517
   742
  assume "a * b dvd c"
haftmann@60517
   743
  with assms show "a dvd c"
haftmann@60517
   744
    by (simp add: dvd_mult_left)
haftmann@60517
   745
next
haftmann@60517
   746
  assume "a dvd c"
haftmann@60517
   747
  then obtain k where "c = a * k" ..
haftmann@60517
   748
  with assms have "c = (a * b) * (1 div b * k)"
haftmann@60517
   749
    by (simp add: mult_ac)
haftmann@60517
   750
  then show "a * b dvd c" by (rule dvdI)
haftmann@60517
   751
qed
haftmann@60517
   752
haftmann@60517
   753
lemma dvd_mult_unit_iff:
haftmann@60517
   754
  assumes "is_unit b"
haftmann@60517
   755
  shows "a dvd c * b \<longleftrightarrow> a dvd c"
haftmann@60517
   756
proof
haftmann@60517
   757
  assume "a dvd c * b"
haftmann@60517
   758
  with assms have "c * b dvd c * (b * (1 div b))"
haftmann@60517
   759
    by (subst mult_assoc [symmetric]) simp
haftmann@60517
   760
  also from `is_unit b` have "b * (1 div b) = 1" by (rule is_unitE) simp
haftmann@60517
   761
  finally have "c * b dvd c" by simp
haftmann@60517
   762
  with `a dvd c * b` show "a dvd c" by (rule dvd_trans)
haftmann@60517
   763
next
haftmann@60517
   764
  assume "a dvd c"
haftmann@60517
   765
  then show "a dvd c * b" by simp
haftmann@60517
   766
qed
haftmann@60517
   767
haftmann@60517
   768
lemma div_unit_dvd_iff:
haftmann@60517
   769
  "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c"
haftmann@60517
   770
  by (erule is_unitE [of _ a]) (auto simp add: mult_unit_dvd_iff)
haftmann@60517
   771
haftmann@60517
   772
lemma dvd_div_unit_iff:
haftmann@60517
   773
  "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c"
haftmann@60517
   774
  by (erule is_unitE [of _ c]) (simp add: dvd_mult_unit_iff)
haftmann@60517
   775
haftmann@60517
   776
lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff
haftmann@60517
   777
  dvd_mult_unit_iff dvd_div_unit_iff -- \<open>FIXME consider fact collection\<close>
haftmann@60517
   778
haftmann@60517
   779
lemma unit_mult_div_div [simp]:
haftmann@60517
   780
  "is_unit a \<Longrightarrow> b * (1 div a) = b div a"
haftmann@60517
   781
  by (erule is_unitE [of _ b]) simp
haftmann@60517
   782
haftmann@60517
   783
lemma unit_div_mult_self [simp]:
haftmann@60517
   784
  "is_unit a \<Longrightarrow> b div a * a = b"
haftmann@60517
   785
  by (rule dvd_div_mult_self) auto
haftmann@60517
   786
haftmann@60517
   787
lemma unit_div_1_div_1 [simp]:
haftmann@60517
   788
  "is_unit a \<Longrightarrow> 1 div (1 div a) = a"
haftmann@60517
   789
  by (erule is_unitE) simp
haftmann@60517
   790
haftmann@60517
   791
lemma unit_div_mult_swap:
haftmann@60517
   792
  "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c"
haftmann@60517
   793
  by (erule unit_dvdE [of _ b]) (simp add: mult.left_commute [of _ c])
haftmann@60517
   794
haftmann@60517
   795
lemma unit_div_commute:
haftmann@60517
   796
  "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b"
haftmann@60517
   797
  using unit_div_mult_swap [of b c a] by (simp add: ac_simps)
haftmann@60517
   798
haftmann@60517
   799
lemma unit_eq_div1:
haftmann@60517
   800
  "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b"
haftmann@60517
   801
  by (auto elim: is_unitE)
haftmann@60517
   802
haftmann@60517
   803
lemma unit_eq_div2:
haftmann@60517
   804
  "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c"
haftmann@60517
   805
  using unit_eq_div1 [of b c a] by auto
haftmann@60517
   806
haftmann@60517
   807
lemma unit_mult_left_cancel:
haftmann@60517
   808
  assumes "is_unit a"
haftmann@60517
   809
  shows "a * b = a * c \<longleftrightarrow> b = c" (is "?P \<longleftrightarrow> ?Q")
lp15@60562
   810
  using assms mult_cancel_left [of a b c] by auto
haftmann@60517
   811
haftmann@60517
   812
lemma unit_mult_right_cancel:
haftmann@60517
   813
  "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c"
haftmann@60517
   814
  using unit_mult_left_cancel [of a b c] by (auto simp add: ac_simps)
haftmann@60517
   815
haftmann@60517
   816
lemma unit_div_cancel:
haftmann@60517
   817
  assumes "is_unit a"
haftmann@60517
   818
  shows "b div a = c div a \<longleftrightarrow> b = c"
haftmann@60517
   819
proof -
haftmann@60517
   820
  from assms have "is_unit (1 div a)" by simp
haftmann@60517
   821
  then have "b * (1 div a) = c * (1 div a) \<longleftrightarrow> b = c"
haftmann@60517
   822
    by (rule unit_mult_right_cancel)
haftmann@60517
   823
  with assms show ?thesis by simp
haftmann@60517
   824
qed
lp15@60562
   825
haftmann@60570
   826
lemma is_unit_div_mult2_eq:
haftmann@60570
   827
  assumes "is_unit b" and "is_unit c"
haftmann@60570
   828
  shows "a div (b * c) = a div b div c"
haftmann@60570
   829
proof -
haftmann@60570
   830
  from assms have "is_unit (b * c)" by (simp add: unit_prod)
haftmann@60570
   831
  then have "b * c dvd a"
haftmann@60570
   832
    by (rule unit_imp_dvd)
haftmann@60570
   833
  then show ?thesis
haftmann@60570
   834
    by (rule dvd_div_mult2_eq)
haftmann@60570
   835
qed
haftmann@60570
   836
haftmann@60517
   837
wenzelm@60529
   838
text \<open>Associated elements in a ring --- an equivalence relation induced
wenzelm@60529
   839
  by the quasi-order divisibility.\<close>
haftmann@60517
   840
lp15@60562
   841
definition associated :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
haftmann@60517
   842
where
haftmann@60517
   843
  "associated a b \<longleftrightarrow> a dvd b \<and> b dvd a"
haftmann@60517
   844
haftmann@60517
   845
lemma associatedI:
haftmann@60517
   846
  "a dvd b \<Longrightarrow> b dvd a \<Longrightarrow> associated a b"
haftmann@60517
   847
  by (simp add: associated_def)
haftmann@60517
   848
haftmann@60517
   849
lemma associatedD1:
haftmann@60517
   850
  "associated a b \<Longrightarrow> a dvd b"
haftmann@60517
   851
  by (simp add: associated_def)
haftmann@60517
   852
haftmann@60517
   853
lemma associatedD2:
haftmann@60517
   854
  "associated a b \<Longrightarrow> b dvd a"
haftmann@60517
   855
  by (simp add: associated_def)
haftmann@60517
   856
haftmann@60517
   857
lemma associated_refl [simp]:
haftmann@60517
   858
  "associated a a"
haftmann@60517
   859
  by (auto intro: associatedI)
haftmann@60517
   860
haftmann@60517
   861
lemma associated_sym:
haftmann@60517
   862
  "associated b a \<longleftrightarrow> associated a b"
haftmann@60517
   863
  by (auto intro: associatedI dest: associatedD1 associatedD2)
haftmann@60517
   864
haftmann@60517
   865
lemma associated_trans:
haftmann@60517
   866
  "associated a b \<Longrightarrow> associated b c \<Longrightarrow> associated a c"
haftmann@60517
   867
  by (auto intro: associatedI dvd_trans dest: associatedD1 associatedD2)
haftmann@60517
   868
haftmann@60517
   869
lemma associated_0 [simp]:
haftmann@60517
   870
  "associated 0 b \<longleftrightarrow> b = 0"
haftmann@60517
   871
  "associated a 0 \<longleftrightarrow> a = 0"
haftmann@60517
   872
  by (auto dest: associatedD1 associatedD2)
haftmann@60517
   873
haftmann@60517
   874
lemma associated_unit:
haftmann@60517
   875
  "associated a b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"
haftmann@60517
   876
  using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2)
haftmann@60517
   877
haftmann@60517
   878
lemma is_unit_associatedI:
haftmann@60517
   879
  assumes "is_unit c" and "a = c * b"
haftmann@60517
   880
  shows "associated a b"
haftmann@60517
   881
proof (rule associatedI)
haftmann@60517
   882
  from `a = c * b` show "b dvd a" by auto
haftmann@60517
   883
  from `is_unit c` obtain d where "c * d = 1" by (rule is_unitE)
haftmann@60517
   884
  moreover from `a = c * b` have "d * a = d * (c * b)" by simp
haftmann@60517
   885
  ultimately have "b = a * d" by (simp add: ac_simps)
haftmann@60517
   886
  then show "a dvd b" ..
haftmann@60517
   887
qed
haftmann@60517
   888
haftmann@60517
   889
lemma associated_is_unitE:
haftmann@60517
   890
  assumes "associated a b"
haftmann@60517
   891
  obtains c where "is_unit c" and "a = c * b"
haftmann@60517
   892
proof (cases "b = 0")
haftmann@60517
   893
  case True with assms have "is_unit 1" and "a = 1 * b" by simp_all
haftmann@60517
   894
  with that show thesis .
haftmann@60517
   895
next
haftmann@60517
   896
  case False
haftmann@60517
   897
  from assms have "a dvd b" and "b dvd a" by (auto dest: associatedD1 associatedD2)
haftmann@60517
   898
  then obtain c d where "b = a * d" and "a = b * c" by (blast elim: dvdE)
haftmann@60517
   899
  then have "a = c * b" and "(c * d) * b = 1 * b" by (simp_all add: ac_simps)
haftmann@60517
   900
  with False have "c * d = 1" using mult_cancel_right [of "c * d" b 1] by simp
haftmann@60517
   901
  then have "is_unit c" by auto
haftmann@60517
   902
  with `a = c * b` that show thesis by blast
haftmann@60517
   903
qed
lp15@60562
   904
lp15@60562
   905
lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff
haftmann@60517
   906
  dvd_div_unit_iff unit_div_mult_swap unit_div_commute
lp15@60562
   907
  unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel
haftmann@60517
   908
  unit_eq_div1 unit_eq_div2
haftmann@60517
   909
haftmann@60517
   910
end
haftmann@60517
   911
haftmann@38642
   912
class ordered_semiring = semiring + comm_monoid_add + ordered_ab_semigroup_add +
haftmann@38642
   913
  assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@38642
   914
  assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
haftmann@25230
   915
begin
haftmann@25230
   916
haftmann@25230
   917
lemma mult_mono:
haftmann@38642
   918
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   919
apply (erule mult_right_mono [THEN order_trans], assumption)
haftmann@25230
   920
apply (erule mult_left_mono, assumption)
haftmann@25230
   921
done
haftmann@25230
   922
haftmann@25230
   923
lemma mult_mono':
haftmann@38642
   924
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   925
apply (rule mult_mono)
haftmann@25230
   926
apply (fast intro: order_trans)+
haftmann@25230
   927
done
haftmann@25230
   928
haftmann@25230
   929
end
krauss@21199
   930
haftmann@38642
   931
class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add
haftmann@25267
   932
begin
paulson@14268
   933
huffman@27516
   934
subclass semiring_0_cancel ..
obua@23521
   935
nipkow@56536
   936
lemma mult_nonneg_nonneg[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
haftmann@36301
   937
using mult_left_mono [of 0 b a] by simp
haftmann@25230
   938
haftmann@25230
   939
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
haftmann@36301
   940
using mult_left_mono [of b 0 a] by simp
huffman@30692
   941
huffman@30692
   942
lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"
haftmann@36301
   943
using mult_right_mono [of a 0 b] by simp
huffman@30692
   944
huffman@30692
   945
text {* Legacy - use @{text mult_nonpos_nonneg} *}
lp15@60562
   946
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0"
haftmann@36301
   947
by (drule mult_right_mono [of b 0], auto)
haftmann@25230
   948
lp15@60562
   949
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0"
nipkow@29667
   950
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
haftmann@25230
   951
haftmann@25230
   952
end
haftmann@25230
   953
haftmann@38642
   954
class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add
haftmann@25267
   955
begin
haftmann@25230
   956
haftmann@35028
   957
subclass ordered_cancel_semiring ..
haftmann@35028
   958
haftmann@35028
   959
subclass ordered_comm_monoid_add ..
haftmann@25304
   960
haftmann@25230
   961
lemma mult_left_less_imp_less:
haftmann@25230
   962
  "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
   963
by (force simp add: mult_left_mono not_le [symmetric])
lp15@60562
   964
haftmann@25230
   965
lemma mult_right_less_imp_less:
haftmann@25230
   966
  "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
   967
by (force simp add: mult_right_mono not_le [symmetric])
obua@23521
   968
haftmann@25186
   969
end
haftmann@25152
   970
haftmann@35043
   971
class linordered_semiring_1 = linordered_semiring + semiring_1
hoelzl@36622
   972
begin
hoelzl@36622
   973
hoelzl@36622
   974
lemma convex_bound_le:
hoelzl@36622
   975
  assumes "x \<le> a" "y \<le> a" "0 \<le> u" "0 \<le> v" "u + v = 1"
hoelzl@36622
   976
  shows "u * x + v * y \<le> a"
hoelzl@36622
   977
proof-
hoelzl@36622
   978
  from assms have "u * x + v * y \<le> u * a + v * a"
hoelzl@36622
   979
    by (simp add: add_mono mult_left_mono)
webertj@49962
   980
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
hoelzl@36622
   981
qed
hoelzl@36622
   982
hoelzl@36622
   983
end
haftmann@35043
   984
haftmann@35043
   985
class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +
haftmann@25062
   986
  assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25062
   987
  assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
haftmann@25267
   988
begin
paulson@14341
   989
huffman@27516
   990
subclass semiring_0_cancel ..
obua@14940
   991
haftmann@35028
   992
subclass linordered_semiring
haftmann@28823
   993
proof
huffman@23550
   994
  fix a b c :: 'a
huffman@23550
   995
  assume A: "a \<le> b" "0 \<le> c"
huffman@23550
   996
  from A show "c * a \<le> c * b"
haftmann@25186
   997
    unfolding le_less
haftmann@25186
   998
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
   999
  from A show "a * c \<le> b * c"
haftmann@25152
  1000
    unfolding le_less
haftmann@25186
  1001
    using mult_strict_right_mono by (cases "c = 0") auto
haftmann@25152
  1002
qed
haftmann@25152
  1003
haftmann@25230
  1004
lemma mult_left_le_imp_le:
haftmann@25230
  1005
  "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
  1006
by (force simp add: mult_strict_left_mono _not_less [symmetric])
lp15@60562
  1007
haftmann@25230
  1008
lemma mult_right_le_imp_le:
haftmann@25230
  1009
  "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
  1010
by (force simp add: mult_strict_right_mono not_less [symmetric])
haftmann@25230
  1011
nipkow@56544
  1012
lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
haftmann@36301
  1013
using mult_strict_left_mono [of 0 b a] by simp
huffman@30692
  1014
huffman@30692
  1015
lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
haftmann@36301
  1016
using mult_strict_left_mono [of b 0 a] by simp
huffman@30692
  1017
huffman@30692
  1018
lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"
haftmann@36301
  1019
using mult_strict_right_mono [of a 0 b] by simp
huffman@30692
  1020
huffman@30692
  1021
text {* Legacy - use @{text mult_neg_pos} *}
lp15@60562
  1022
lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0"
haftmann@36301
  1023
by (drule mult_strict_right_mono [of b 0], auto)
haftmann@25230
  1024
haftmann@25230
  1025
lemma zero_less_mult_pos:
haftmann@25230
  1026
  "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
  1027
apply (cases "b\<le>0")
haftmann@25230
  1028
 apply (auto simp add: le_less not_less)
huffman@30692
  1029
apply (drule_tac mult_pos_neg [of a b])
haftmann@25230
  1030
 apply (auto dest: less_not_sym)
haftmann@25230
  1031
done
haftmann@25230
  1032
haftmann@25230
  1033
lemma zero_less_mult_pos2:
haftmann@25230
  1034
  "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
  1035
apply (cases "b\<le>0")
haftmann@25230
  1036
 apply (auto simp add: le_less not_less)
huffman@30692
  1037
apply (drule_tac mult_pos_neg2 [of a b])
haftmann@25230
  1038
 apply (auto dest: less_not_sym)
haftmann@25230
  1039
done
haftmann@25230
  1040
haftmann@26193
  1041
text{*Strict monotonicity in both arguments*}
haftmann@26193
  1042
lemma mult_strict_mono:
haftmann@26193
  1043
  assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
haftmann@26193
  1044
  shows "a * c < b * d"
haftmann@26193
  1045
  using assms apply (cases "c=0")
nipkow@56544
  1046
  apply (simp)
haftmann@26193
  1047
  apply (erule mult_strict_right_mono [THEN less_trans])
huffman@30692
  1048
  apply (force simp add: le_less)
haftmann@26193
  1049
  apply (erule mult_strict_left_mono, assumption)
haftmann@26193
  1050
  done
haftmann@26193
  1051
haftmann@26193
  1052
text{*This weaker variant has more natural premises*}
haftmann@26193
  1053
lemma mult_strict_mono':
haftmann@26193
  1054
  assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
haftmann@26193
  1055
  shows "a * c < b * d"
nipkow@29667
  1056
by (rule mult_strict_mono) (insert assms, auto)
haftmann@26193
  1057
haftmann@26193
  1058
lemma mult_less_le_imp_less:
haftmann@26193
  1059
  assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
haftmann@26193
  1060
  shows "a * c < b * d"
haftmann@26193
  1061
  using assms apply (subgoal_tac "a * c < b * c")
haftmann@26193
  1062
  apply (erule less_le_trans)
haftmann@26193
  1063
  apply (erule mult_left_mono)
haftmann@26193
  1064
  apply simp
haftmann@26193
  1065
  apply (erule mult_strict_right_mono)
haftmann@26193
  1066
  apply assumption
haftmann@26193
  1067
  done
haftmann@26193
  1068
haftmann@26193
  1069
lemma mult_le_less_imp_less:
haftmann@26193
  1070
  assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
haftmann@26193
  1071
  shows "a * c < b * d"
haftmann@26193
  1072
  using assms apply (subgoal_tac "a * c \<le> b * c")
haftmann@26193
  1073
  apply (erule le_less_trans)
haftmann@26193
  1074
  apply (erule mult_strict_left_mono)
haftmann@26193
  1075
  apply simp
haftmann@26193
  1076
  apply (erule mult_right_mono)
haftmann@26193
  1077
  apply simp
haftmann@26193
  1078
  done
haftmann@26193
  1079
haftmann@25230
  1080
end
haftmann@25230
  1081
haftmann@35097
  1082
class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1
hoelzl@36622
  1083
begin
hoelzl@36622
  1084
hoelzl@36622
  1085
subclass linordered_semiring_1 ..
hoelzl@36622
  1086
hoelzl@36622
  1087
lemma convex_bound_lt:
hoelzl@36622
  1088
  assumes "x < a" "y < a" "0 \<le> u" "0 \<le> v" "u + v = 1"
hoelzl@36622
  1089
  shows "u * x + v * y < a"
hoelzl@36622
  1090
proof -
hoelzl@36622
  1091
  from assms have "u * x + v * y < u * a + v * a"
hoelzl@36622
  1092
    by (cases "u = 0")
hoelzl@36622
  1093
       (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
webertj@49962
  1094
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
hoelzl@36622
  1095
qed
hoelzl@36622
  1096
hoelzl@36622
  1097
end
haftmann@33319
  1098
lp15@60562
  1099
class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add +
haftmann@38642
  1100
  assumes comm_mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@25186
  1101
begin
haftmann@25152
  1102
haftmann@35028
  1103
subclass ordered_semiring
haftmann@28823
  1104
proof
krauss@21199
  1105
  fix a b c :: 'a
huffman@23550
  1106
  assume "a \<le> b" "0 \<le> c"
haftmann@38642
  1107
  thus "c * a \<le> c * b" by (rule comm_mult_left_mono)
haftmann@57512
  1108
  thus "a * c \<le> b * c" by (simp only: mult.commute)
krauss@21199
  1109
qed
paulson@14265
  1110
haftmann@25267
  1111
end
haftmann@25267
  1112
haftmann@38642
  1113
class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add
haftmann@25267
  1114
begin
paulson@14265
  1115
haftmann@38642
  1116
subclass comm_semiring_0_cancel ..
haftmann@35028
  1117
subclass ordered_comm_semiring ..
haftmann@35028
  1118
subclass ordered_cancel_semiring ..
haftmann@25267
  1119
haftmann@25267
  1120
end
haftmann@25267
  1121
haftmann@35028
  1122
class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +
haftmann@38642
  1123
  assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25267
  1124
begin
haftmann@25267
  1125
haftmann@35043
  1126
subclass linordered_semiring_strict
haftmann@28823
  1127
proof
huffman@23550
  1128
  fix a b c :: 'a
huffman@23550
  1129
  assume "a < b" "0 < c"
haftmann@38642
  1130
  thus "c * a < c * b" by (rule comm_mult_strict_left_mono)
haftmann@57512
  1131
  thus "a * c < b * c" by (simp only: mult.commute)
huffman@23550
  1132
qed
paulson@14272
  1133
haftmann@35028
  1134
subclass ordered_cancel_comm_semiring
haftmann@28823
  1135
proof
huffman@23550
  1136
  fix a b c :: 'a
huffman@23550
  1137
  assume "a \<le> b" "0 \<le> c"
huffman@23550
  1138
  thus "c * a \<le> c * b"
haftmann@25186
  1139
    unfolding le_less
haftmann@26193
  1140
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
  1141
qed
paulson@14272
  1142
haftmann@25267
  1143
end
haftmann@25230
  1144
lp15@60562
  1145
class ordered_ring = ring + ordered_cancel_semiring
haftmann@25267
  1146
begin
haftmann@25230
  1147
haftmann@35028
  1148
subclass ordered_ab_group_add ..
paulson@14270
  1149
haftmann@25230
  1150
lemma less_add_iff1:
haftmann@25230
  1151
  "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
nipkow@29667
  1152
by (simp add: algebra_simps)
haftmann@25230
  1153
haftmann@25230
  1154
lemma less_add_iff2:
haftmann@25230
  1155
  "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
nipkow@29667
  1156
by (simp add: algebra_simps)
haftmann@25230
  1157
haftmann@25230
  1158
lemma le_add_iff1:
haftmann@25230
  1159
  "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
nipkow@29667
  1160
by (simp add: algebra_simps)
haftmann@25230
  1161
haftmann@25230
  1162
lemma le_add_iff2:
haftmann@25230
  1163
  "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
nipkow@29667
  1164
by (simp add: algebra_simps)
haftmann@25230
  1165
haftmann@25230
  1166
lemma mult_left_mono_neg:
haftmann@25230
  1167
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
haftmann@36301
  1168
  apply (drule mult_left_mono [of _ _ "- c"])
huffman@35216
  1169
  apply simp_all
haftmann@25230
  1170
  done
haftmann@25230
  1171
haftmann@25230
  1172
lemma mult_right_mono_neg:
haftmann@25230
  1173
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
haftmann@36301
  1174
  apply (drule mult_right_mono [of _ _ "- c"])
huffman@35216
  1175
  apply simp_all
haftmann@25230
  1176
  done
haftmann@25230
  1177
huffman@30692
  1178
lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
haftmann@36301
  1179
using mult_right_mono_neg [of a 0 b] by simp
haftmann@25230
  1180
haftmann@25230
  1181
lemma split_mult_pos_le:
haftmann@25230
  1182
  "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
nipkow@56536
  1183
by (auto simp add: mult_nonpos_nonpos)
haftmann@25186
  1184
haftmann@25186
  1185
end
paulson@14270
  1186
haftmann@35028
  1187
class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if
haftmann@25304
  1188
begin
haftmann@25304
  1189
haftmann@35028
  1190
subclass ordered_ring ..
haftmann@35028
  1191
haftmann@35028
  1192
subclass ordered_ab_group_add_abs
haftmann@28823
  1193
proof
haftmann@25304
  1194
  fix a b
haftmann@25304
  1195
  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@54230
  1196
    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
  1197
qed (auto simp add: abs_if)
haftmann@25304
  1198
huffman@35631
  1199
lemma zero_le_square [simp]: "0 \<le> a * a"
huffman@35631
  1200
  using linear [of 0 a]
nipkow@56536
  1201
  by (auto simp add: mult_nonpos_nonpos)
huffman@35631
  1202
huffman@35631
  1203
lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
huffman@35631
  1204
  by (simp add: not_less)
huffman@35631
  1205
haftmann@25304
  1206
end
obua@23521
  1207
haftmann@35043
  1208
class linordered_ring_strict = ring + linordered_semiring_strict
haftmann@25304
  1209
  + ordered_ab_group_add + abs_if
haftmann@25230
  1210
begin
paulson@14348
  1211
haftmann@35028
  1212
subclass linordered_ring ..
haftmann@25304
  1213
huffman@30692
  1214
lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
huffman@30692
  1215
using mult_strict_left_mono [of b a "- c"] by simp
huffman@30692
  1216
huffman@30692
  1217
lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
huffman@30692
  1218
using mult_strict_right_mono [of b a "- c"] by simp
huffman@30692
  1219
huffman@30692
  1220
lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
haftmann@36301
  1221
using mult_strict_right_mono_neg [of a 0 b] by simp
obua@14738
  1222
haftmann@25917
  1223
subclass ring_no_zero_divisors
haftmann@28823
  1224
proof
haftmann@25917
  1225
  fix a b
haftmann@25917
  1226
  assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
haftmann@25917
  1227
  assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
haftmann@25917
  1228
  have "a * b < 0 \<or> 0 < a * b"
haftmann@25917
  1229
  proof (cases "a < 0")
haftmann@25917
  1230
    case True note A' = this
haftmann@25917
  1231
    show ?thesis proof (cases "b < 0")
haftmann@25917
  1232
      case True with A'
haftmann@25917
  1233
      show ?thesis by (auto dest: mult_neg_neg)
haftmann@25917
  1234
    next
haftmann@25917
  1235
      case False with B have "0 < b" by auto
haftmann@25917
  1236
      with A' show ?thesis by (auto dest: mult_strict_right_mono)
haftmann@25917
  1237
    qed
haftmann@25917
  1238
  next
haftmann@25917
  1239
    case False with A have A': "0 < a" by auto
haftmann@25917
  1240
    show ?thesis proof (cases "b < 0")
haftmann@25917
  1241
      case True with A'
haftmann@25917
  1242
      show ?thesis by (auto dest: mult_strict_right_mono_neg)
haftmann@25917
  1243
    next
haftmann@25917
  1244
      case False with B have "0 < b" by auto
nipkow@56544
  1245
      with A' show ?thesis by auto
haftmann@25917
  1246
    qed
haftmann@25917
  1247
  qed
haftmann@25917
  1248
  then show "a * b \<noteq> 0" by (simp add: neq_iff)
haftmann@25917
  1249
qed
haftmann@25304
  1250
hoelzl@56480
  1251
lemma zero_less_mult_iff: "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
hoelzl@56480
  1252
  by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
nipkow@56544
  1253
     (auto simp add: mult_neg_neg not_less le_less dest: zero_less_mult_pos zero_less_mult_pos2)
huffman@22990
  1254
hoelzl@56480
  1255
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
  1256
  by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
paulson@14265
  1257
paulson@14265
  1258
lemma mult_less_0_iff:
haftmann@25917
  1259
  "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
huffman@35216
  1260
  apply (insert zero_less_mult_iff [of "-a" b])
huffman@35216
  1261
  apply force
haftmann@25917
  1262
  done
paulson@14265
  1263
paulson@14265
  1264
lemma mult_le_0_iff:
haftmann@25917
  1265
  "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
lp15@60562
  1266
  apply (insert zero_le_mult_iff [of "-a" b])
huffman@35216
  1267
  apply force
haftmann@25917
  1268
  done
haftmann@25917
  1269
haftmann@26193
  1270
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
haftmann@26193
  1271
   also with the relations @{text "\<le>"} and equality.*}
haftmann@26193
  1272
haftmann@26193
  1273
text{*These ``disjunction'' versions produce two cases when the comparison is
haftmann@26193
  1274
 an assumption, but effectively four when the comparison is a goal.*}
haftmann@26193
  1275
haftmann@26193
  1276
lemma mult_less_cancel_right_disj:
haftmann@26193
  1277
  "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
  1278
  apply (cases "c = 0")
lp15@60562
  1279
  apply (auto simp add: neq_iff mult_strict_right_mono
haftmann@26193
  1280
                      mult_strict_right_mono_neg)
lp15@60562
  1281
  apply (auto simp add: not_less
haftmann@26193
  1282
                      not_le [symmetric, of "a*c"]
haftmann@26193
  1283
                      not_le [symmetric, of a])
haftmann@26193
  1284
  apply (erule_tac [!] notE)
lp15@60562
  1285
  apply (auto simp add: less_imp_le mult_right_mono
haftmann@26193
  1286
                      mult_right_mono_neg)
haftmann@26193
  1287
  done
haftmann@26193
  1288
haftmann@26193
  1289
lemma mult_less_cancel_left_disj:
haftmann@26193
  1290
  "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
  1291
  apply (cases "c = 0")
lp15@60562
  1292
  apply (auto simp add: neq_iff mult_strict_left_mono
haftmann@26193
  1293
                      mult_strict_left_mono_neg)
lp15@60562
  1294
  apply (auto simp add: not_less
haftmann@26193
  1295
                      not_le [symmetric, of "c*a"]
haftmann@26193
  1296
                      not_le [symmetric, of a])
haftmann@26193
  1297
  apply (erule_tac [!] notE)
lp15@60562
  1298
  apply (auto simp add: less_imp_le mult_left_mono
haftmann@26193
  1299
                      mult_left_mono_neg)
haftmann@26193
  1300
  done
haftmann@26193
  1301
haftmann@26193
  1302
text{*The ``conjunction of implication'' lemmas produce two cases when the
haftmann@26193
  1303
comparison is a goal, but give four when the comparison is an assumption.*}
haftmann@26193
  1304
haftmann@26193
  1305
lemma mult_less_cancel_right:
haftmann@26193
  1306
  "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
  1307
  using mult_less_cancel_right_disj [of a c b] by auto
haftmann@26193
  1308
haftmann@26193
  1309
lemma mult_less_cancel_left:
haftmann@26193
  1310
  "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
  1311
  using mult_less_cancel_left_disj [of c a b] by auto
haftmann@26193
  1312
haftmann@26193
  1313
lemma mult_le_cancel_right:
haftmann@26193
  1314
   "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
  1315
by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
haftmann@26193
  1316
haftmann@26193
  1317
lemma mult_le_cancel_left:
haftmann@26193
  1318
  "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
  1319
by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
haftmann@26193
  1320
nipkow@30649
  1321
lemma mult_le_cancel_left_pos:
nipkow@30649
  1322
  "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
nipkow@30649
  1323
by (auto simp: mult_le_cancel_left)
nipkow@30649
  1324
nipkow@30649
  1325
lemma mult_le_cancel_left_neg:
nipkow@30649
  1326
  "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
nipkow@30649
  1327
by (auto simp: mult_le_cancel_left)
nipkow@30649
  1328
nipkow@30649
  1329
lemma mult_less_cancel_left_pos:
nipkow@30649
  1330
  "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
nipkow@30649
  1331
by (auto simp: mult_less_cancel_left)
nipkow@30649
  1332
nipkow@30649
  1333
lemma mult_less_cancel_left_neg:
nipkow@30649
  1334
  "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
nipkow@30649
  1335
by (auto simp: mult_less_cancel_left)
nipkow@30649
  1336
haftmann@25917
  1337
end
paulson@14265
  1338
huffman@30692
  1339
lemmas mult_sign_intros =
huffman@30692
  1340
  mult_nonneg_nonneg mult_nonneg_nonpos
huffman@30692
  1341
  mult_nonpos_nonneg mult_nonpos_nonpos
huffman@30692
  1342
  mult_pos_pos mult_pos_neg
huffman@30692
  1343
  mult_neg_pos mult_neg_neg
haftmann@25230
  1344
haftmann@35028
  1345
class ordered_comm_ring = comm_ring + ordered_comm_semiring
haftmann@25267
  1346
begin
haftmann@25230
  1347
haftmann@35028
  1348
subclass ordered_ring ..
haftmann@35028
  1349
subclass ordered_cancel_comm_semiring ..
haftmann@25230
  1350
haftmann@25267
  1351
end
haftmann@25230
  1352
haftmann@59833
  1353
class linordered_semidom = semidom + linordered_comm_semiring_strict +
haftmann@25230
  1354
  assumes zero_less_one [simp]: "0 < 1"
lp15@60562
  1355
  assumes le_add_diff_inverse2 [simp]: "b \<le> a \<Longrightarrow> a - b + b = a"
haftmann@25230
  1356
begin
haftmann@25230
  1357
lp15@60562
  1358
text {* Addition is the inverse of subtraction. *}
lp15@60562
  1359
lp15@60562
  1360
lemma le_add_diff_inverse [simp]: "b \<le> a \<Longrightarrow> b + (a - b) = a"
lp15@60562
  1361
  by (frule le_add_diff_inverse2) (simp add: add.commute)
lp15@60562
  1362
lp15@60562
  1363
lemma add_diff_inverse: "~ a<b \<Longrightarrow> b + (a - b) = a"
lp15@60562
  1364
  by simp
lp15@60615
  1365
lp15@60615
  1366
lemma add_le_imp_le_diff: 
lp15@60615
  1367
  shows "i + k \<le> n \<Longrightarrow> i \<le> n - k"
lp15@60615
  1368
  apply (subst add_le_cancel_right [where c=k, symmetric])
lp15@60615
  1369
  apply (frule le_add_diff_inverse2)
lp15@60615
  1370
  apply (simp only: add.assoc [symmetric])
lp15@60615
  1371
  using add_implies_diff by fastforce
lp15@60615
  1372
lp15@60615
  1373
lemma add_le_add_imp_diff_le: 
lp15@60615
  1374
  assumes a1: "i + k \<le> n"
lp15@60615
  1375
      and a2: "n \<le> j + k"
lp15@60615
  1376
  shows "\<lbrakk>i + k \<le> n; n \<le> j + k\<rbrakk> \<Longrightarrow> n - k \<le> j"
lp15@60615
  1377
proof -
lp15@60615
  1378
  have "n - (i + k) + (i + k) = n"
lp15@60615
  1379
    using a1 by simp
lp15@60615
  1380
  moreover have "n - k = n - k - i + i"
lp15@60615
  1381
    using a1 by (simp add: add_le_imp_le_diff)
lp15@60615
  1382
  ultimately show ?thesis
lp15@60615
  1383
    using a2
lp15@60615
  1384
    apply (simp add: add.assoc [symmetric])
lp15@60615
  1385
    apply (rule add_le_imp_le_diff [of _ k "j+k", simplified add_diff_cancel_right'])
lp15@60615
  1386
    by (simp add: add.commute diff_diff_add)
lp15@60615
  1387
qed
lp15@60615
  1388
haftmann@25230
  1389
lemma pos_add_strict:
haftmann@25230
  1390
  shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@36301
  1391
  using add_strict_mono [of 0 a b c] by simp
haftmann@25230
  1392
haftmann@26193
  1393
lemma zero_le_one [simp]: "0 \<le> 1"
lp15@60562
  1394
by (rule zero_less_one [THEN less_imp_le])
haftmann@26193
  1395
haftmann@26193
  1396
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
lp15@60562
  1397
by (simp add: not_le)
haftmann@26193
  1398
haftmann@26193
  1399
lemma not_one_less_zero [simp]: "\<not> 1 < 0"
lp15@60562
  1400
by (simp add: not_less)
haftmann@26193
  1401
haftmann@26193
  1402
lemma less_1_mult:
haftmann@26193
  1403
  assumes "1 < m" and "1 < n"
haftmann@26193
  1404
  shows "1 < m * n"
haftmann@26193
  1405
  using assms mult_strict_mono [of 1 m 1 n]
lp15@60562
  1406
    by (simp add:  less_trans [OF zero_less_one])
haftmann@26193
  1407
hoelzl@59000
  1408
lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a"
hoelzl@59000
  1409
  using mult_left_mono[of c 1 a] by simp
hoelzl@59000
  1410
hoelzl@59000
  1411
lemma mult_le_one: "a \<le> 1 \<Longrightarrow> 0 \<le> b \<Longrightarrow> b \<le> 1 \<Longrightarrow> a * b \<le> 1"
hoelzl@59000
  1412
  using mult_mono[of a 1 b 1] by simp
hoelzl@59000
  1413
haftmann@25230
  1414
end
haftmann@25230
  1415
haftmann@35028
  1416
class linordered_idom = comm_ring_1 +
haftmann@35028
  1417
  linordered_comm_semiring_strict + ordered_ab_group_add +
haftmann@25230
  1418
  abs_if + sgn_if
haftmann@25917
  1419
begin
haftmann@25917
  1420
hoelzl@36622
  1421
subclass linordered_semiring_1_strict ..
haftmann@35043
  1422
subclass linordered_ring_strict ..
haftmann@35028
  1423
subclass ordered_comm_ring ..
huffman@27516
  1424
subclass idom ..
haftmann@25917
  1425
haftmann@35028
  1426
subclass linordered_semidom
haftmann@28823
  1427
proof
haftmann@26193
  1428
  have "0 \<le> 1 * 1" by (rule zero_le_square)
haftmann@26193
  1429
  thus "0 < 1" by (simp add: le_less)
lp15@60562
  1430
  show "\<And>b a. b \<le> a \<Longrightarrow> a - b + b = a"
lp15@60562
  1431
    by simp
lp15@60562
  1432
qed
haftmann@25917
  1433
haftmann@35028
  1434
lemma linorder_neqE_linordered_idom:
haftmann@26193
  1435
  assumes "x \<noteq> y" obtains "x < y" | "y < x"
haftmann@26193
  1436
  using assms by (rule neqE)
haftmann@26193
  1437
haftmann@26274
  1438
text {* These cancellation simprules also produce two cases when the comparison is a goal. *}
haftmann@26274
  1439
haftmann@26274
  1440
lemma mult_le_cancel_right1:
haftmann@26274
  1441
  "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1442
by (insert mult_le_cancel_right [of 1 c b], simp)
haftmann@26274
  1443
haftmann@26274
  1444
lemma mult_le_cancel_right2:
haftmann@26274
  1445
  "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1446
by (insert mult_le_cancel_right [of a c 1], simp)
haftmann@26274
  1447
haftmann@26274
  1448
lemma mult_le_cancel_left1:
haftmann@26274
  1449
  "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1450
by (insert mult_le_cancel_left [of c 1 b], simp)
haftmann@26274
  1451
haftmann@26274
  1452
lemma mult_le_cancel_left2:
haftmann@26274
  1453
  "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1454
by (insert mult_le_cancel_left [of c a 1], simp)
haftmann@26274
  1455
haftmann@26274
  1456
lemma mult_less_cancel_right1:
haftmann@26274
  1457
  "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1458
by (insert mult_less_cancel_right [of 1 c b], simp)
haftmann@26274
  1459
haftmann@26274
  1460
lemma mult_less_cancel_right2:
haftmann@26274
  1461
  "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1462
by (insert mult_less_cancel_right [of a c 1], simp)
haftmann@26274
  1463
haftmann@26274
  1464
lemma mult_less_cancel_left1:
haftmann@26274
  1465
  "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1466
by (insert mult_less_cancel_left [of c 1 b], simp)
haftmann@26274
  1467
haftmann@26274
  1468
lemma mult_less_cancel_left2:
haftmann@26274
  1469
  "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1470
by (insert mult_less_cancel_left [of c a 1], simp)
haftmann@26274
  1471
haftmann@27651
  1472
lemma sgn_sgn [simp]:
haftmann@27651
  1473
  "sgn (sgn a) = sgn a"
nipkow@29700
  1474
unfolding sgn_if by simp
haftmann@27651
  1475
haftmann@27651
  1476
lemma sgn_0_0:
haftmann@27651
  1477
  "sgn a = 0 \<longleftrightarrow> a = 0"
nipkow@29700
  1478
unfolding sgn_if by simp
haftmann@27651
  1479
haftmann@27651
  1480
lemma sgn_1_pos:
haftmann@27651
  1481
  "sgn a = 1 \<longleftrightarrow> a > 0"
huffman@35216
  1482
unfolding sgn_if by simp
haftmann@27651
  1483
haftmann@27651
  1484
lemma sgn_1_neg:
haftmann@27651
  1485
  "sgn a = - 1 \<longleftrightarrow> a < 0"
huffman@35216
  1486
unfolding sgn_if by auto
haftmann@27651
  1487
haftmann@29940
  1488
lemma sgn_pos [simp]:
haftmann@29940
  1489
  "0 < a \<Longrightarrow> sgn a = 1"
haftmann@29940
  1490
unfolding sgn_1_pos .
haftmann@29940
  1491
haftmann@29940
  1492
lemma sgn_neg [simp]:
haftmann@29940
  1493
  "a < 0 \<Longrightarrow> sgn a = - 1"
haftmann@29940
  1494
unfolding sgn_1_neg .
haftmann@29940
  1495
haftmann@27651
  1496
lemma sgn_times:
haftmann@27651
  1497
  "sgn (a * b) = sgn a * sgn b"
nipkow@29667
  1498
by (auto simp add: sgn_if zero_less_mult_iff)
haftmann@27651
  1499
haftmann@36301
  1500
lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"
nipkow@29700
  1501
unfolding sgn_if abs_if by auto
nipkow@29700
  1502
haftmann@29940
  1503
lemma sgn_greater [simp]:
haftmann@29940
  1504
  "0 < sgn a \<longleftrightarrow> 0 < a"
haftmann@29940
  1505
  unfolding sgn_if by auto
haftmann@29940
  1506
haftmann@29940
  1507
lemma sgn_less [simp]:
haftmann@29940
  1508
  "sgn a < 0 \<longleftrightarrow> a < 0"
haftmann@29940
  1509
  unfolding sgn_if by auto
haftmann@29940
  1510
haftmann@36301
  1511
lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"
huffman@29949
  1512
  by (simp add: abs_if)
huffman@29949
  1513
haftmann@36301
  1514
lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"
huffman@29949
  1515
  by (simp add: abs_if)
haftmann@29653
  1516
nipkow@33676
  1517
lemma dvd_if_abs_eq:
haftmann@36301
  1518
  "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
nipkow@33676
  1519
by(subst abs_dvd_iff[symmetric]) simp
nipkow@33676
  1520
huffman@55912
  1521
text {* The following lemmas can be proven in more general structures, but
lp15@60562
  1522
are dangerous as simp rules in absence of @{thm neg_equal_zero},
haftmann@54489
  1523
@{thm neg_less_pos}, @{thm neg_less_eq_nonneg}. *}
haftmann@54489
  1524
haftmann@54489
  1525
lemma equation_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1526
  "1 = - a \<longleftrightarrow> a = - 1"
haftmann@54489
  1527
  by (fact equation_minus_iff)
haftmann@54489
  1528
haftmann@54489
  1529
lemma minus_equation_iff_1 [simp, no_atp]:
haftmann@54489
  1530
  "- a = 1 \<longleftrightarrow> a = - 1"
haftmann@54489
  1531
  by (subst minus_equation_iff, auto)
haftmann@54489
  1532
haftmann@54489
  1533
lemma le_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1534
  "1 \<le> - b \<longleftrightarrow> b \<le> - 1"
haftmann@54489
  1535
  by (fact le_minus_iff)
haftmann@54489
  1536
haftmann@54489
  1537
lemma minus_le_iff_1 [simp, no_atp]:
haftmann@54489
  1538
  "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"
haftmann@54489
  1539
  by (fact minus_le_iff)
haftmann@54489
  1540
haftmann@54489
  1541
lemma less_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1542
  "1 < - b \<longleftrightarrow> b < - 1"
haftmann@54489
  1543
  by (fact less_minus_iff)
haftmann@54489
  1544
haftmann@54489
  1545
lemma minus_less_iff_1 [simp, no_atp]:
haftmann@54489
  1546
  "- a < 1 \<longleftrightarrow> - 1 < a"
haftmann@54489
  1547
  by (fact minus_less_iff)
haftmann@54489
  1548
haftmann@25917
  1549
end
haftmann@25230
  1550
haftmann@26274
  1551
text {* Simprules for comparisons where common factors can be cancelled. *}
paulson@15234
  1552
blanchet@54147
  1553
lemmas mult_compare_simps =
paulson@15234
  1554
    mult_le_cancel_right mult_le_cancel_left
paulson@15234
  1555
    mult_le_cancel_right1 mult_le_cancel_right2
paulson@15234
  1556
    mult_le_cancel_left1 mult_le_cancel_left2
paulson@15234
  1557
    mult_less_cancel_right mult_less_cancel_left
paulson@15234
  1558
    mult_less_cancel_right1 mult_less_cancel_right2
paulson@15234
  1559
    mult_less_cancel_left1 mult_less_cancel_left2
paulson@15234
  1560
    mult_cancel_right mult_cancel_left
paulson@15234
  1561
    mult_cancel_right1 mult_cancel_right2
paulson@15234
  1562
    mult_cancel_left1 mult_cancel_left2
paulson@15234
  1563
haftmann@36301
  1564
text {* Reasoning about inequalities with division *}
avigad@16775
  1565
haftmann@35028
  1566
context linordered_semidom
haftmann@25193
  1567
begin
haftmann@25193
  1568
haftmann@25193
  1569
lemma less_add_one: "a < a + 1"
paulson@14293
  1570
proof -
haftmann@25193
  1571
  have "a + 0 < a + 1"
nipkow@23482
  1572
    by (blast intro: zero_less_one add_strict_left_mono)
paulson@14293
  1573
  thus ?thesis by simp
paulson@14293
  1574
qed
paulson@14293
  1575
haftmann@25193
  1576
lemma zero_less_two: "0 < 1 + 1"
nipkow@29667
  1577
by (blast intro: less_trans zero_less_one less_add_one)
haftmann@25193
  1578
haftmann@25193
  1579
end
paulson@14365
  1580
haftmann@36301
  1581
context linordered_idom
haftmann@36301
  1582
begin
paulson@15234
  1583
haftmann@36301
  1584
lemma mult_right_le_one_le:
haftmann@36301
  1585
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"
haftmann@59833
  1586
  by (rule mult_left_le)
haftmann@36301
  1587
haftmann@36301
  1588
lemma mult_left_le_one_le:
haftmann@36301
  1589
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"
haftmann@36301
  1590
  by (auto simp add: mult_le_cancel_right2)
haftmann@36301
  1591
haftmann@36301
  1592
end
haftmann@36301
  1593
haftmann@36301
  1594
text {* Absolute Value *}
paulson@14293
  1595
haftmann@35028
  1596
context linordered_idom
haftmann@25304
  1597
begin
haftmann@25304
  1598
haftmann@36301
  1599
lemma mult_sgn_abs:
haftmann@36301
  1600
  "sgn x * \<bar>x\<bar> = x"
haftmann@25304
  1601
  unfolding abs_if sgn_if by auto
haftmann@25304
  1602
haftmann@36301
  1603
lemma abs_one [simp]:
haftmann@36301
  1604
  "\<bar>1\<bar> = 1"
huffman@44921
  1605
  by (simp add: abs_if)
haftmann@36301
  1606
haftmann@25304
  1607
end
nipkow@24491
  1608
haftmann@35028
  1609
class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +
haftmann@25304
  1610
  assumes abs_eq_mult:
haftmann@25304
  1611
    "(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
  1612
haftmann@35028
  1613
context linordered_idom
haftmann@30961
  1614
begin
haftmann@30961
  1615
haftmann@35028
  1616
subclass ordered_ring_abs proof
huffman@35216
  1617
qed (auto simp add: abs_if not_less mult_less_0_iff)
haftmann@30961
  1618
haftmann@30961
  1619
lemma abs_mult:
lp15@60562
  1620
  "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
haftmann@30961
  1621
  by (rule abs_eq_mult) auto
haftmann@30961
  1622
haftmann@30961
  1623
lemma abs_mult_self:
haftmann@36301
  1624
  "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
lp15@60562
  1625
  by (simp add: abs_if)
haftmann@30961
  1626
paulson@14294
  1627
lemma abs_mult_less:
haftmann@36301
  1628
  "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"
paulson@14294
  1629
proof -
haftmann@36301
  1630
  assume ac: "\<bar>a\<bar> < c"
haftmann@36301
  1631
  hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)
haftmann@36301
  1632
  assume "\<bar>b\<bar> < d"
lp15@60562
  1633
  thus ?thesis by (simp add: ac cpos mult_strict_mono)
paulson@14294
  1634
qed
paulson@14293
  1635
haftmann@36301
  1636
lemma abs_less_iff:
lp15@60562
  1637
  "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b"
haftmann@36301
  1638
  by (simp add: less_le abs_le_iff) (auto simp add: abs_if)
obua@14738
  1639
haftmann@36301
  1640
lemma abs_mult_pos:
haftmann@36301
  1641
  "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"
haftmann@36301
  1642
  by (simp add: abs_mult)
haftmann@36301
  1643
hoelzl@51520
  1644
lemma abs_diff_less_iff:
hoelzl@51520
  1645
  "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"
hoelzl@51520
  1646
  by (auto simp add: diff_less_eq ac_simps abs_less_iff)
hoelzl@51520
  1647
lp15@59865
  1648
lemma abs_diff_le_iff:
lp15@59865
  1649
   "\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r"
lp15@59865
  1650
  by (auto simp add: diff_le_eq ac_simps abs_le_iff)
lp15@59865
  1651
haftmann@36301
  1652
end
avigad@16775
  1653
haftmann@59557
  1654
hide_fact (open) comm_mult_left_mono comm_mult_strict_left_mono distrib
haftmann@59557
  1655
haftmann@52435
  1656
code_identifier
haftmann@52435
  1657
  code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
  1658
paulson@14265
  1659
end