src/HOL/Rings.thy
author huffman
Mon Aug 08 09:52:09 2011 -0700 (2011-08-08)
changeset 44064 5bce8ff0d9ae
parent 38642 8fa437809c67
child 44346 00dd3c4dabe0
permissions -rw-r--r--
moved division ring stuff from Rings.thy to Fields.thy
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
haftmann@35050
    10
header {* 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 +
haftmann@36348
    17
  assumes left_distrib[algebra_simps, field_simps]: "(a + b) * c = a * c + b * c"
haftmann@36348
    18
  assumes right_distrib[algebra_simps, field_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"
nipkow@29667
    24
by (simp add: left_distrib add_ac)
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"
krauss@21199
    31
haftmann@22390
    32
class semiring_0 = semiring + comm_monoid_add + mult_zero
krauss@21199
    33
huffman@29904
    34
class semiring_0_cancel = semiring + cancel_comm_monoid_add
haftmann@25186
    35
begin
paulson@14504
    36
haftmann@25186
    37
subclass semiring_0
haftmann@28823
    38
proof
krauss@21199
    39
  fix a :: 'a
nipkow@29667
    40
  have "0 * a + 0 * a = 0 * a + 0" by (simp add: left_distrib [symmetric])
nipkow@29667
    41
  thus "0 * a = 0" by (simp only: add_left_cancel)
haftmann@25152
    42
next
haftmann@25152
    43
  fix a :: 'a
nipkow@29667
    44
  have "a * 0 + a * 0 = a * 0 + 0" by (simp add: right_distrib [symmetric])
nipkow@29667
    45
  thus "a * 0 = 0" by (simp only: add_left_cancel)
krauss@21199
    46
qed
obua@14940
    47
haftmann@25186
    48
end
haftmann@25152
    49
haftmann@22390
    50
class comm_semiring = ab_semigroup_add + ab_semigroup_mult +
haftmann@25062
    51
  assumes distrib: "(a + b) * c = a * c + b * c"
haftmann@25152
    52
begin
paulson@14504
    53
haftmann@25152
    54
subclass semiring
haftmann@28823
    55
proof
obua@14738
    56
  fix a b c :: 'a
obua@14738
    57
  show "(a + b) * c = a * c + b * c" by (simp add: distrib)
obua@14738
    58
  have "a * (b + c) = (b + c) * a" by (simp add: mult_ac)
obua@14738
    59
  also have "... = b * a + c * a" by (simp only: distrib)
obua@14738
    60
  also have "... = a * b + a * c" by (simp add: mult_ac)
obua@14738
    61
  finally show "a * (b + c) = a * b + a * c" by blast
paulson@14504
    62
qed
paulson@14504
    63
haftmann@25152
    64
end
paulson@14504
    65
haftmann@25152
    66
class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero
haftmann@25152
    67
begin
haftmann@25152
    68
huffman@27516
    69
subclass semiring_0 ..
haftmann@25152
    70
haftmann@25152
    71
end
paulson@14504
    72
huffman@29904
    73
class comm_semiring_0_cancel = comm_semiring + cancel_comm_monoid_add
haftmann@25186
    74
begin
obua@14940
    75
huffman@27516
    76
subclass semiring_0_cancel ..
obua@14940
    77
huffman@28141
    78
subclass comm_semiring_0 ..
huffman@28141
    79
haftmann@25186
    80
end
krauss@21199
    81
haftmann@22390
    82
class zero_neq_one = zero + one +
haftmann@25062
    83
  assumes zero_neq_one [simp]: "0 \<noteq> 1"
haftmann@26193
    84
begin
haftmann@26193
    85
haftmann@26193
    86
lemma one_neq_zero [simp]: "1 \<noteq> 0"
nipkow@29667
    87
by (rule not_sym) (rule zero_neq_one)
haftmann@26193
    88
haftmann@26193
    89
end
paulson@14265
    90
haftmann@22390
    91
class semiring_1 = zero_neq_one + semiring_0 + monoid_mult
paulson@14504
    92
haftmann@27651
    93
text {* Abstract divisibility *}
haftmann@27651
    94
haftmann@27651
    95
class dvd = times
haftmann@27651
    96
begin
haftmann@27651
    97
haftmann@28559
    98
definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "dvd" 50) where
haftmann@37767
    99
  "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"
haftmann@27651
   100
haftmann@27651
   101
lemma dvdI [intro?]: "a = b * k \<Longrightarrow> b dvd a"
haftmann@27651
   102
  unfolding dvd_def ..
haftmann@27651
   103
haftmann@27651
   104
lemma dvdE [elim?]: "b dvd a \<Longrightarrow> (\<And>k. a = b * k \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@27651
   105
  unfolding dvd_def by blast 
haftmann@27651
   106
haftmann@27651
   107
end
haftmann@27651
   108
haftmann@27651
   109
class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult + dvd
haftmann@22390
   110
  (*previously almost_semiring*)
haftmann@25152
   111
begin
obua@14738
   112
huffman@27516
   113
subclass semiring_1 ..
haftmann@25152
   114
nipkow@29925
   115
lemma dvd_refl[simp]: "a dvd a"
haftmann@28559
   116
proof
haftmann@28559
   117
  show "a = a * 1" by simp
haftmann@27651
   118
qed
haftmann@27651
   119
haftmann@27651
   120
lemma dvd_trans:
haftmann@27651
   121
  assumes "a dvd b" and "b dvd c"
haftmann@27651
   122
  shows "a dvd c"
haftmann@27651
   123
proof -
haftmann@28559
   124
  from assms obtain v where "b = a * v" by (auto elim!: dvdE)
haftmann@28559
   125
  moreover from assms obtain w where "c = b * w" by (auto elim!: dvdE)
haftmann@27651
   126
  ultimately have "c = a * (v * w)" by (simp add: mult_assoc)
haftmann@28559
   127
  then show ?thesis ..
haftmann@27651
   128
qed
haftmann@27651
   129
blanchet@35828
   130
lemma dvd_0_left_iff [no_atp, simp]: "0 dvd a \<longleftrightarrow> a = 0"
nipkow@29667
   131
by (auto intro: dvd_refl elim!: dvdE)
haftmann@28559
   132
haftmann@28559
   133
lemma dvd_0_right [iff]: "a dvd 0"
haftmann@28559
   134
proof
haftmann@27651
   135
  show "0 = a * 0" by simp
haftmann@27651
   136
qed
haftmann@27651
   137
haftmann@27651
   138
lemma one_dvd [simp]: "1 dvd a"
nipkow@29667
   139
by (auto intro!: dvdI)
haftmann@27651
   140
nipkow@30042
   141
lemma dvd_mult[simp]: "a dvd c \<Longrightarrow> a dvd (b * c)"
nipkow@29667
   142
by (auto intro!: mult_left_commute dvdI elim!: dvdE)
haftmann@27651
   143
nipkow@30042
   144
lemma dvd_mult2[simp]: "a dvd b \<Longrightarrow> a dvd (b * c)"
haftmann@27651
   145
  apply (subst mult_commute)
haftmann@27651
   146
  apply (erule dvd_mult)
haftmann@27651
   147
  done
haftmann@27651
   148
haftmann@27651
   149
lemma dvd_triv_right [simp]: "a dvd b * a"
nipkow@29667
   150
by (rule dvd_mult) (rule dvd_refl)
haftmann@27651
   151
haftmann@27651
   152
lemma dvd_triv_left [simp]: "a dvd a * b"
nipkow@29667
   153
by (rule dvd_mult2) (rule dvd_refl)
haftmann@27651
   154
haftmann@27651
   155
lemma mult_dvd_mono:
nipkow@30042
   156
  assumes "a dvd b"
nipkow@30042
   157
    and "c dvd d"
haftmann@27651
   158
  shows "a * c dvd b * d"
haftmann@27651
   159
proof -
nipkow@30042
   160
  from `a dvd b` obtain b' where "b = a * b'" ..
nipkow@30042
   161
  moreover from `c dvd d` obtain d' where "d = c * d'" ..
haftmann@27651
   162
  ultimately have "b * d = (a * c) * (b' * d')" by (simp add: mult_ac)
haftmann@27651
   163
  then show ?thesis ..
haftmann@27651
   164
qed
haftmann@27651
   165
haftmann@27651
   166
lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c"
nipkow@29667
   167
by (simp add: dvd_def mult_assoc, blast)
haftmann@27651
   168
haftmann@27651
   169
lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c"
haftmann@27651
   170
  unfolding mult_ac [of a] by (rule dvd_mult_left)
haftmann@27651
   171
haftmann@27651
   172
lemma dvd_0_left: "0 dvd a \<Longrightarrow> a = 0"
nipkow@29667
   173
by simp
haftmann@27651
   174
nipkow@29925
   175
lemma dvd_add[simp]:
nipkow@29925
   176
  assumes "a dvd b" and "a dvd c" shows "a dvd (b + c)"
haftmann@27651
   177
proof -
nipkow@29925
   178
  from `a dvd b` obtain b' where "b = a * b'" ..
nipkow@29925
   179
  moreover from `a dvd c` obtain c' where "c = a * c'" ..
haftmann@27651
   180
  ultimately have "b + c = a * (b' + c')" by (simp add: right_distrib)
haftmann@27651
   181
  then show ?thesis ..
haftmann@27651
   182
qed
haftmann@27651
   183
haftmann@25152
   184
end
paulson@14421
   185
haftmann@22390
   186
class no_zero_divisors = zero + times +
haftmann@25062
   187
  assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
haftmann@36719
   188
begin
haftmann@36719
   189
haftmann@36719
   190
lemma divisors_zero:
haftmann@36719
   191
  assumes "a * b = 0"
haftmann@36719
   192
  shows "a = 0 \<or> b = 0"
haftmann@36719
   193
proof (rule classical)
haftmann@36719
   194
  assume "\<not> (a = 0 \<or> b = 0)"
haftmann@36719
   195
  then have "a \<noteq> 0" and "b \<noteq> 0" by auto
haftmann@36719
   196
  with no_zero_divisors have "a * b \<noteq> 0" by blast
haftmann@36719
   197
  with assms show ?thesis by simp
haftmann@36719
   198
qed
haftmann@36719
   199
haftmann@36719
   200
end
paulson@14504
   201
huffman@29904
   202
class semiring_1_cancel = semiring + cancel_comm_monoid_add
huffman@29904
   203
  + zero_neq_one + monoid_mult
haftmann@25267
   204
begin
obua@14940
   205
huffman@27516
   206
subclass semiring_0_cancel ..
haftmann@25512
   207
huffman@27516
   208
subclass semiring_1 ..
haftmann@25267
   209
haftmann@25267
   210
end
krauss@21199
   211
huffman@29904
   212
class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add
huffman@29904
   213
  + zero_neq_one + comm_monoid_mult
haftmann@25267
   214
begin
obua@14738
   215
huffman@27516
   216
subclass semiring_1_cancel ..
huffman@27516
   217
subclass comm_semiring_0_cancel ..
huffman@27516
   218
subclass comm_semiring_1 ..
haftmann@25267
   219
haftmann@25267
   220
end
haftmann@25152
   221
haftmann@22390
   222
class ring = semiring + ab_group_add
haftmann@25267
   223
begin
haftmann@25152
   224
huffman@27516
   225
subclass semiring_0_cancel ..
haftmann@25152
   226
haftmann@25152
   227
text {* Distribution rules *}
haftmann@25152
   228
haftmann@25152
   229
lemma minus_mult_left: "- (a * b) = - a * b"
huffman@34146
   230
by (rule minus_unique) (simp add: left_distrib [symmetric]) 
haftmann@25152
   231
haftmann@25152
   232
lemma minus_mult_right: "- (a * b) = a * - b"
huffman@34146
   233
by (rule minus_unique) (simp add: right_distrib [symmetric]) 
haftmann@25152
   234
huffman@29407
   235
text{*Extract signs from products*}
blanchet@35828
   236
lemmas mult_minus_left [simp, no_atp] = minus_mult_left [symmetric]
blanchet@35828
   237
lemmas mult_minus_right [simp,no_atp] = minus_mult_right [symmetric]
huffman@29407
   238
haftmann@25152
   239
lemma minus_mult_minus [simp]: "- a * - b = a * b"
nipkow@29667
   240
by simp
haftmann@25152
   241
haftmann@25152
   242
lemma minus_mult_commute: "- a * b = a * - b"
nipkow@29667
   243
by simp
nipkow@29667
   244
haftmann@36348
   245
lemma right_diff_distrib[algebra_simps, field_simps]: "a * (b - c) = a * b - a * c"
nipkow@29667
   246
by (simp add: right_distrib diff_minus)
nipkow@29667
   247
haftmann@36348
   248
lemma left_diff_distrib[algebra_simps, field_simps]: "(a - b) * c = a * c - b * c"
nipkow@29667
   249
by (simp add: left_distrib diff_minus)
haftmann@25152
   250
blanchet@35828
   251
lemmas ring_distribs[no_atp] =
haftmann@25152
   252
  right_distrib left_distrib left_diff_distrib right_diff_distrib
haftmann@25152
   253
haftmann@25230
   254
lemma eq_add_iff1:
haftmann@25230
   255
  "a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"
nipkow@29667
   256
by (simp add: algebra_simps)
haftmann@25230
   257
haftmann@25230
   258
lemma eq_add_iff2:
haftmann@25230
   259
  "a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"
nipkow@29667
   260
by (simp add: algebra_simps)
haftmann@25230
   261
haftmann@25152
   262
end
haftmann@25152
   263
blanchet@35828
   264
lemmas ring_distribs[no_atp] =
haftmann@25152
   265
  right_distrib left_distrib left_diff_distrib right_diff_distrib
haftmann@25152
   266
haftmann@22390
   267
class comm_ring = comm_semiring + ab_group_add
haftmann@25267
   268
begin
obua@14738
   269
huffman@27516
   270
subclass ring ..
huffman@28141
   271
subclass comm_semiring_0_cancel ..
haftmann@25267
   272
haftmann@25267
   273
end
obua@14738
   274
haftmann@22390
   275
class ring_1 = ring + zero_neq_one + monoid_mult
haftmann@25267
   276
begin
paulson@14265
   277
huffman@27516
   278
subclass semiring_1_cancel ..
haftmann@25267
   279
haftmann@25267
   280
end
haftmann@25152
   281
haftmann@22390
   282
class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
haftmann@22390
   283
  (*previously ring*)
haftmann@25267
   284
begin
obua@14738
   285
huffman@27516
   286
subclass ring_1 ..
huffman@27516
   287
subclass comm_semiring_1_cancel ..
haftmann@25267
   288
huffman@29465
   289
lemma dvd_minus_iff [simp]: "x dvd - y \<longleftrightarrow> x dvd y"
huffman@29408
   290
proof
huffman@29408
   291
  assume "x dvd - y"
huffman@29408
   292
  then have "x dvd - 1 * - y" by (rule dvd_mult)
huffman@29408
   293
  then show "x dvd y" by simp
huffman@29408
   294
next
huffman@29408
   295
  assume "x dvd y"
huffman@29408
   296
  then have "x dvd - 1 * y" by (rule dvd_mult)
huffman@29408
   297
  then show "x dvd - y" by simp
huffman@29408
   298
qed
huffman@29408
   299
huffman@29465
   300
lemma minus_dvd_iff [simp]: "- x dvd y \<longleftrightarrow> x dvd y"
huffman@29408
   301
proof
huffman@29408
   302
  assume "- x dvd y"
huffman@29408
   303
  then obtain k where "y = - x * k" ..
huffman@29408
   304
  then have "y = x * - k" by simp
huffman@29408
   305
  then show "x dvd y" ..
huffman@29408
   306
next
huffman@29408
   307
  assume "x dvd y"
huffman@29408
   308
  then obtain k where "y = x * k" ..
huffman@29408
   309
  then have "y = - x * - k" by simp
huffman@29408
   310
  then show "- x dvd y" ..
huffman@29408
   311
qed
huffman@29408
   312
nipkow@30042
   313
lemma dvd_diff[simp]: "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
huffman@35216
   314
by (simp only: diff_minus dvd_add dvd_minus_iff)
huffman@29409
   315
haftmann@25267
   316
end
haftmann@25152
   317
huffman@22990
   318
class ring_no_zero_divisors = ring + no_zero_divisors
haftmann@25230
   319
begin
haftmann@25230
   320
haftmann@25230
   321
lemma mult_eq_0_iff [simp]:
haftmann@25230
   322
  shows "a * b = 0 \<longleftrightarrow> (a = 0 \<or> b = 0)"
haftmann@25230
   323
proof (cases "a = 0 \<or> b = 0")
haftmann@25230
   324
  case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
haftmann@25230
   325
    then show ?thesis using no_zero_divisors by simp
haftmann@25230
   326
next
haftmann@25230
   327
  case True then show ?thesis by auto
haftmann@25230
   328
qed
haftmann@25230
   329
haftmann@26193
   330
text{*Cancellation of equalities with a common factor*}
blanchet@35828
   331
lemma mult_cancel_right [simp, no_atp]:
haftmann@26193
   332
  "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@26193
   333
proof -
haftmann@26193
   334
  have "(a * c = b * c) = ((a - b) * c = 0)"
huffman@35216
   335
    by (simp add: algebra_simps)
huffman@35216
   336
  thus ?thesis by (simp add: disj_commute)
haftmann@26193
   337
qed
haftmann@26193
   338
blanchet@35828
   339
lemma mult_cancel_left [simp, no_atp]:
haftmann@26193
   340
  "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@26193
   341
proof -
haftmann@26193
   342
  have "(c * a = c * b) = (c * (a - b) = 0)"
huffman@35216
   343
    by (simp add: algebra_simps)
huffman@35216
   344
  thus ?thesis by simp
haftmann@26193
   345
qed
haftmann@26193
   346
haftmann@25230
   347
end
huffman@22990
   348
huffman@23544
   349
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
haftmann@26274
   350
begin
haftmann@26274
   351
huffman@36970
   352
lemma square_eq_1_iff:
huffman@36821
   353
  "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
huffman@36821
   354
proof -
huffman@36821
   355
  have "(x - 1) * (x + 1) = x * x - 1"
huffman@36821
   356
    by (simp add: algebra_simps)
huffman@36821
   357
  hence "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
huffman@36821
   358
    by simp
huffman@36821
   359
  thus ?thesis
huffman@36821
   360
    by (simp add: eq_neg_iff_add_eq_0)
huffman@36821
   361
qed
huffman@36821
   362
haftmann@26274
   363
lemma mult_cancel_right1 [simp]:
haftmann@26274
   364
  "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   365
by (insert mult_cancel_right [of 1 c b], force)
haftmann@26274
   366
haftmann@26274
   367
lemma mult_cancel_right2 [simp]:
haftmann@26274
   368
  "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   369
by (insert mult_cancel_right [of a c 1], simp)
haftmann@26274
   370
 
haftmann@26274
   371
lemma mult_cancel_left1 [simp]:
haftmann@26274
   372
  "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   373
by (insert mult_cancel_left [of c 1 b], force)
haftmann@26274
   374
haftmann@26274
   375
lemma mult_cancel_left2 [simp]:
haftmann@26274
   376
  "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   377
by (insert mult_cancel_left [of c a 1], simp)
haftmann@26274
   378
haftmann@26274
   379
end
huffman@22990
   380
haftmann@22390
   381
class idom = comm_ring_1 + no_zero_divisors
haftmann@25186
   382
begin
paulson@14421
   383
huffman@27516
   384
subclass ring_1_no_zero_divisors ..
huffman@22990
   385
huffman@29915
   386
lemma square_eq_iff: "a * a = b * b \<longleftrightarrow> (a = b \<or> a = - b)"
huffman@29915
   387
proof
huffman@29915
   388
  assume "a * a = b * b"
huffman@29915
   389
  then have "(a - b) * (a + b) = 0"
huffman@29915
   390
    by (simp add: algebra_simps)
huffman@29915
   391
  then show "a = b \<or> a = - b"
huffman@35216
   392
    by (simp add: eq_neg_iff_add_eq_0)
huffman@29915
   393
next
huffman@29915
   394
  assume "a = b \<or> a = - b"
huffman@29915
   395
  then show "a * a = b * b" by auto
huffman@29915
   396
qed
huffman@29915
   397
huffman@29981
   398
lemma dvd_mult_cancel_right [simp]:
huffman@29981
   399
  "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   400
proof -
huffman@29981
   401
  have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
huffman@29981
   402
    unfolding dvd_def by (simp add: mult_ac)
huffman@29981
   403
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   404
    unfolding dvd_def by simp
huffman@29981
   405
  finally show ?thesis .
huffman@29981
   406
qed
huffman@29981
   407
huffman@29981
   408
lemma dvd_mult_cancel_left [simp]:
huffman@29981
   409
  "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   410
proof -
huffman@29981
   411
  have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
huffman@29981
   412
    unfolding dvd_def by (simp add: mult_ac)
huffman@29981
   413
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   414
    unfolding dvd_def by simp
huffman@29981
   415
  finally show ?thesis .
huffman@29981
   416
qed
huffman@29981
   417
haftmann@25186
   418
end
haftmann@25152
   419
haftmann@35302
   420
text {*
haftmann@35302
   421
  The theory of partially ordered rings is taken from the books:
haftmann@35302
   422
  \begin{itemize}
haftmann@35302
   423
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
haftmann@35302
   424
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
haftmann@35302
   425
  \end{itemize}
haftmann@35302
   426
  Most of the used notions can also be looked up in 
haftmann@35302
   427
  \begin{itemize}
haftmann@35302
   428
  \item \url{http://www.mathworld.com} by Eric Weisstein et. al.
haftmann@35302
   429
  \item \emph{Algebra I} by van der Waerden, Springer.
haftmann@35302
   430
  \end{itemize}
haftmann@35302
   431
*}
haftmann@35302
   432
haftmann@38642
   433
class ordered_semiring = semiring + comm_monoid_add + ordered_ab_semigroup_add +
haftmann@38642
   434
  assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@38642
   435
  assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
haftmann@25230
   436
begin
haftmann@25230
   437
haftmann@25230
   438
lemma mult_mono:
haftmann@38642
   439
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   440
apply (erule mult_right_mono [THEN order_trans], assumption)
haftmann@25230
   441
apply (erule mult_left_mono, assumption)
haftmann@25230
   442
done
haftmann@25230
   443
haftmann@25230
   444
lemma mult_mono':
haftmann@38642
   445
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   446
apply (rule mult_mono)
haftmann@25230
   447
apply (fast intro: order_trans)+
haftmann@25230
   448
done
haftmann@25230
   449
haftmann@25230
   450
end
krauss@21199
   451
haftmann@38642
   452
class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add
haftmann@25267
   453
begin
paulson@14268
   454
huffman@27516
   455
subclass semiring_0_cancel ..
obua@23521
   456
haftmann@25230
   457
lemma mult_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
haftmann@36301
   458
using mult_left_mono [of 0 b a] by simp
haftmann@25230
   459
haftmann@25230
   460
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
haftmann@36301
   461
using mult_left_mono [of b 0 a] by simp
huffman@30692
   462
huffman@30692
   463
lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"
haftmann@36301
   464
using mult_right_mono [of a 0 b] by simp
huffman@30692
   465
huffman@30692
   466
text {* Legacy - use @{text mult_nonpos_nonneg} *}
haftmann@25230
   467
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0" 
haftmann@36301
   468
by (drule mult_right_mono [of b 0], auto)
haftmann@25230
   469
haftmann@26234
   470
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0" 
nipkow@29667
   471
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
haftmann@25230
   472
haftmann@25230
   473
end
haftmann@25230
   474
haftmann@38642
   475
class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add
haftmann@25267
   476
begin
haftmann@25230
   477
haftmann@35028
   478
subclass ordered_cancel_semiring ..
haftmann@35028
   479
haftmann@35028
   480
subclass ordered_comm_monoid_add ..
haftmann@25304
   481
haftmann@25230
   482
lemma mult_left_less_imp_less:
haftmann@25230
   483
  "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
   484
by (force simp add: mult_left_mono not_le [symmetric])
haftmann@25230
   485
 
haftmann@25230
   486
lemma mult_right_less_imp_less:
haftmann@25230
   487
  "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
   488
by (force simp add: mult_right_mono not_le [symmetric])
obua@23521
   489
haftmann@25186
   490
end
haftmann@25152
   491
haftmann@35043
   492
class linordered_semiring_1 = linordered_semiring + semiring_1
hoelzl@36622
   493
begin
hoelzl@36622
   494
hoelzl@36622
   495
lemma convex_bound_le:
hoelzl@36622
   496
  assumes "x \<le> a" "y \<le> a" "0 \<le> u" "0 \<le> v" "u + v = 1"
hoelzl@36622
   497
  shows "u * x + v * y \<le> a"
hoelzl@36622
   498
proof-
hoelzl@36622
   499
  from assms have "u * x + v * y \<le> u * a + v * a"
hoelzl@36622
   500
    by (simp add: add_mono mult_left_mono)
hoelzl@36622
   501
  thus ?thesis using assms unfolding left_distrib[symmetric] by simp
hoelzl@36622
   502
qed
hoelzl@36622
   503
hoelzl@36622
   504
end
haftmann@35043
   505
haftmann@35043
   506
class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +
haftmann@25062
   507
  assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25062
   508
  assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
haftmann@25267
   509
begin
paulson@14341
   510
huffman@27516
   511
subclass semiring_0_cancel ..
obua@14940
   512
haftmann@35028
   513
subclass linordered_semiring
haftmann@28823
   514
proof
huffman@23550
   515
  fix a b c :: 'a
huffman@23550
   516
  assume A: "a \<le> b" "0 \<le> c"
huffman@23550
   517
  from A show "c * a \<le> c * b"
haftmann@25186
   518
    unfolding le_less
haftmann@25186
   519
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
   520
  from A show "a * c \<le> b * c"
haftmann@25152
   521
    unfolding le_less
haftmann@25186
   522
    using mult_strict_right_mono by (cases "c = 0") auto
haftmann@25152
   523
qed
haftmann@25152
   524
haftmann@25230
   525
lemma mult_left_le_imp_le:
haftmann@25230
   526
  "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
   527
by (force simp add: mult_strict_left_mono _not_less [symmetric])
haftmann@25230
   528
 
haftmann@25230
   529
lemma mult_right_le_imp_le:
haftmann@25230
   530
  "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
   531
by (force simp add: mult_strict_right_mono not_less [symmetric])
haftmann@25230
   532
huffman@30692
   533
lemma mult_pos_pos: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
haftmann@36301
   534
using mult_strict_left_mono [of 0 b a] by simp
huffman@30692
   535
huffman@30692
   536
lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
haftmann@36301
   537
using mult_strict_left_mono [of b 0 a] by simp
huffman@30692
   538
huffman@30692
   539
lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"
haftmann@36301
   540
using mult_strict_right_mono [of a 0 b] by simp
huffman@30692
   541
huffman@30692
   542
text {* Legacy - use @{text mult_neg_pos} *}
huffman@30692
   543
lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0" 
haftmann@36301
   544
by (drule mult_strict_right_mono [of b 0], auto)
haftmann@25230
   545
haftmann@25230
   546
lemma zero_less_mult_pos:
haftmann@25230
   547
  "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
   548
apply (cases "b\<le>0")
haftmann@25230
   549
 apply (auto simp add: le_less not_less)
huffman@30692
   550
apply (drule_tac mult_pos_neg [of a b])
haftmann@25230
   551
 apply (auto dest: less_not_sym)
haftmann@25230
   552
done
haftmann@25230
   553
haftmann@25230
   554
lemma zero_less_mult_pos2:
haftmann@25230
   555
  "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
   556
apply (cases "b\<le>0")
haftmann@25230
   557
 apply (auto simp add: le_less not_less)
huffman@30692
   558
apply (drule_tac mult_pos_neg2 [of a b])
haftmann@25230
   559
 apply (auto dest: less_not_sym)
haftmann@25230
   560
done
haftmann@25230
   561
haftmann@26193
   562
text{*Strict monotonicity in both arguments*}
haftmann@26193
   563
lemma mult_strict_mono:
haftmann@26193
   564
  assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
haftmann@26193
   565
  shows "a * c < b * d"
haftmann@26193
   566
  using assms apply (cases "c=0")
huffman@30692
   567
  apply (simp add: mult_pos_pos)
haftmann@26193
   568
  apply (erule mult_strict_right_mono [THEN less_trans])
huffman@30692
   569
  apply (force simp add: le_less)
haftmann@26193
   570
  apply (erule mult_strict_left_mono, assumption)
haftmann@26193
   571
  done
haftmann@26193
   572
haftmann@26193
   573
text{*This weaker variant has more natural premises*}
haftmann@26193
   574
lemma mult_strict_mono':
haftmann@26193
   575
  assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
haftmann@26193
   576
  shows "a * c < b * d"
nipkow@29667
   577
by (rule mult_strict_mono) (insert assms, auto)
haftmann@26193
   578
haftmann@26193
   579
lemma mult_less_le_imp_less:
haftmann@26193
   580
  assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
haftmann@26193
   581
  shows "a * c < b * d"
haftmann@26193
   582
  using assms apply (subgoal_tac "a * c < b * c")
haftmann@26193
   583
  apply (erule less_le_trans)
haftmann@26193
   584
  apply (erule mult_left_mono)
haftmann@26193
   585
  apply simp
haftmann@26193
   586
  apply (erule mult_strict_right_mono)
haftmann@26193
   587
  apply assumption
haftmann@26193
   588
  done
haftmann@26193
   589
haftmann@26193
   590
lemma mult_le_less_imp_less:
haftmann@26193
   591
  assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
haftmann@26193
   592
  shows "a * c < b * d"
haftmann@26193
   593
  using assms apply (subgoal_tac "a * c \<le> b * c")
haftmann@26193
   594
  apply (erule le_less_trans)
haftmann@26193
   595
  apply (erule mult_strict_left_mono)
haftmann@26193
   596
  apply simp
haftmann@26193
   597
  apply (erule mult_right_mono)
haftmann@26193
   598
  apply simp
haftmann@26193
   599
  done
haftmann@26193
   600
haftmann@26193
   601
lemma mult_less_imp_less_left:
haftmann@26193
   602
  assumes less: "c * a < c * b" and nonneg: "0 \<le> c"
haftmann@26193
   603
  shows "a < b"
haftmann@26193
   604
proof (rule ccontr)
haftmann@26193
   605
  assume "\<not>  a < b"
haftmann@26193
   606
  hence "b \<le> a" by (simp add: linorder_not_less)
haftmann@26193
   607
  hence "c * b \<le> c * a" using nonneg by (rule mult_left_mono)
nipkow@29667
   608
  with this and less show False by (simp add: not_less [symmetric])
haftmann@26193
   609
qed
haftmann@26193
   610
haftmann@26193
   611
lemma mult_less_imp_less_right:
haftmann@26193
   612
  assumes less: "a * c < b * c" and nonneg: "0 \<le> c"
haftmann@26193
   613
  shows "a < b"
haftmann@26193
   614
proof (rule ccontr)
haftmann@26193
   615
  assume "\<not> a < b"
haftmann@26193
   616
  hence "b \<le> a" by (simp add: linorder_not_less)
haftmann@26193
   617
  hence "b * c \<le> a * c" using nonneg by (rule mult_right_mono)
nipkow@29667
   618
  with this and less show False by (simp add: not_less [symmetric])
haftmann@26193
   619
qed  
haftmann@26193
   620
haftmann@25230
   621
end
haftmann@25230
   622
haftmann@35097
   623
class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1
hoelzl@36622
   624
begin
hoelzl@36622
   625
hoelzl@36622
   626
subclass linordered_semiring_1 ..
hoelzl@36622
   627
hoelzl@36622
   628
lemma convex_bound_lt:
hoelzl@36622
   629
  assumes "x < a" "y < a" "0 \<le> u" "0 \<le> v" "u + v = 1"
hoelzl@36622
   630
  shows "u * x + v * y < a"
hoelzl@36622
   631
proof -
hoelzl@36622
   632
  from assms have "u * x + v * y < u * a + v * a"
hoelzl@36622
   633
    by (cases "u = 0")
hoelzl@36622
   634
       (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
hoelzl@36622
   635
  thus ?thesis using assms unfolding left_distrib[symmetric] by simp
hoelzl@36622
   636
qed
hoelzl@36622
   637
hoelzl@36622
   638
end
haftmann@33319
   639
haftmann@38642
   640
class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add + 
haftmann@38642
   641
  assumes comm_mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@25186
   642
begin
haftmann@25152
   643
haftmann@35028
   644
subclass ordered_semiring
haftmann@28823
   645
proof
krauss@21199
   646
  fix a b c :: 'a
huffman@23550
   647
  assume "a \<le> b" "0 \<le> c"
haftmann@38642
   648
  thus "c * a \<le> c * b" by (rule comm_mult_left_mono)
huffman@23550
   649
  thus "a * c \<le> b * c" by (simp only: mult_commute)
krauss@21199
   650
qed
paulson@14265
   651
haftmann@25267
   652
end
haftmann@25267
   653
haftmann@38642
   654
class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add
haftmann@25267
   655
begin
paulson@14265
   656
haftmann@38642
   657
subclass comm_semiring_0_cancel ..
haftmann@35028
   658
subclass ordered_comm_semiring ..
haftmann@35028
   659
subclass ordered_cancel_semiring ..
haftmann@25267
   660
haftmann@25267
   661
end
haftmann@25267
   662
haftmann@35028
   663
class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +
haftmann@38642
   664
  assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25267
   665
begin
haftmann@25267
   666
haftmann@35043
   667
subclass linordered_semiring_strict
haftmann@28823
   668
proof
huffman@23550
   669
  fix a b c :: 'a
huffman@23550
   670
  assume "a < b" "0 < c"
haftmann@38642
   671
  thus "c * a < c * b" by (rule comm_mult_strict_left_mono)
huffman@23550
   672
  thus "a * c < b * c" by (simp only: mult_commute)
huffman@23550
   673
qed
paulson@14272
   674
haftmann@35028
   675
subclass ordered_cancel_comm_semiring
haftmann@28823
   676
proof
huffman@23550
   677
  fix a b c :: 'a
huffman@23550
   678
  assume "a \<le> b" "0 \<le> c"
huffman@23550
   679
  thus "c * a \<le> c * b"
haftmann@25186
   680
    unfolding le_less
haftmann@26193
   681
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
   682
qed
paulson@14272
   683
haftmann@25267
   684
end
haftmann@25230
   685
haftmann@35028
   686
class ordered_ring = ring + ordered_cancel_semiring 
haftmann@25267
   687
begin
haftmann@25230
   688
haftmann@35028
   689
subclass ordered_ab_group_add ..
paulson@14270
   690
haftmann@25230
   691
lemma less_add_iff1:
haftmann@25230
   692
  "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
nipkow@29667
   693
by (simp add: algebra_simps)
haftmann@25230
   694
haftmann@25230
   695
lemma less_add_iff2:
haftmann@25230
   696
  "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
nipkow@29667
   697
by (simp add: algebra_simps)
haftmann@25230
   698
haftmann@25230
   699
lemma le_add_iff1:
haftmann@25230
   700
  "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
nipkow@29667
   701
by (simp add: algebra_simps)
haftmann@25230
   702
haftmann@25230
   703
lemma le_add_iff2:
haftmann@25230
   704
  "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
nipkow@29667
   705
by (simp add: algebra_simps)
haftmann@25230
   706
haftmann@25230
   707
lemma mult_left_mono_neg:
haftmann@25230
   708
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
haftmann@36301
   709
  apply (drule mult_left_mono [of _ _ "- c"])
huffman@35216
   710
  apply simp_all
haftmann@25230
   711
  done
haftmann@25230
   712
haftmann@25230
   713
lemma mult_right_mono_neg:
haftmann@25230
   714
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
haftmann@36301
   715
  apply (drule mult_right_mono [of _ _ "- c"])
huffman@35216
   716
  apply simp_all
haftmann@25230
   717
  done
haftmann@25230
   718
huffman@30692
   719
lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
haftmann@36301
   720
using mult_right_mono_neg [of a 0 b] by simp
haftmann@25230
   721
haftmann@25230
   722
lemma split_mult_pos_le:
haftmann@25230
   723
  "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
nipkow@29667
   724
by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)
haftmann@25186
   725
haftmann@25186
   726
end
paulson@14270
   727
haftmann@35028
   728
class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if
haftmann@25304
   729
begin
haftmann@25304
   730
haftmann@35028
   731
subclass ordered_ring ..
haftmann@35028
   732
haftmann@35028
   733
subclass ordered_ab_group_add_abs
haftmann@28823
   734
proof
haftmann@25304
   735
  fix a b
haftmann@25304
   736
  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
huffman@35216
   737
    by (auto simp add: abs_if not_less)
huffman@35216
   738
    (auto simp del: minus_add_distrib simp add: minus_add_distrib [symmetric],
huffman@36977
   739
     auto intro!: less_imp_le add_neg_neg)
huffman@35216
   740
qed (auto simp add: abs_if)
haftmann@25304
   741
huffman@35631
   742
lemma zero_le_square [simp]: "0 \<le> a * a"
huffman@35631
   743
  using linear [of 0 a]
huffman@35631
   744
  by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)
huffman@35631
   745
huffman@35631
   746
lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
huffman@35631
   747
  by (simp add: not_less)
huffman@35631
   748
haftmann@25304
   749
end
obua@23521
   750
haftmann@35028
   751
(* The "strict" suffix can be seen as describing the combination of linordered_ring and no_zero_divisors.
haftmann@35043
   752
   Basically, linordered_ring + no_zero_divisors = linordered_ring_strict.
haftmann@25230
   753
 *)
haftmann@35043
   754
class linordered_ring_strict = ring + linordered_semiring_strict
haftmann@25304
   755
  + ordered_ab_group_add + abs_if
haftmann@25230
   756
begin
paulson@14348
   757
haftmann@35028
   758
subclass linordered_ring ..
haftmann@25304
   759
huffman@30692
   760
lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
huffman@30692
   761
using mult_strict_left_mono [of b a "- c"] by simp
huffman@30692
   762
huffman@30692
   763
lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
huffman@30692
   764
using mult_strict_right_mono [of b a "- c"] by simp
huffman@30692
   765
huffman@30692
   766
lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
haftmann@36301
   767
using mult_strict_right_mono_neg [of a 0 b] by simp
obua@14738
   768
haftmann@25917
   769
subclass ring_no_zero_divisors
haftmann@28823
   770
proof
haftmann@25917
   771
  fix a b
haftmann@25917
   772
  assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
haftmann@25917
   773
  assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
haftmann@25917
   774
  have "a * b < 0 \<or> 0 < a * b"
haftmann@25917
   775
  proof (cases "a < 0")
haftmann@25917
   776
    case True note A' = this
haftmann@25917
   777
    show ?thesis proof (cases "b < 0")
haftmann@25917
   778
      case True with A'
haftmann@25917
   779
      show ?thesis by (auto dest: mult_neg_neg)
haftmann@25917
   780
    next
haftmann@25917
   781
      case False with B have "0 < b" by auto
haftmann@25917
   782
      with A' show ?thesis by (auto dest: mult_strict_right_mono)
haftmann@25917
   783
    qed
haftmann@25917
   784
  next
haftmann@25917
   785
    case False with A have A': "0 < a" by auto
haftmann@25917
   786
    show ?thesis proof (cases "b < 0")
haftmann@25917
   787
      case True with A'
haftmann@25917
   788
      show ?thesis by (auto dest: mult_strict_right_mono_neg)
haftmann@25917
   789
    next
haftmann@25917
   790
      case False with B have "0 < b" by auto
haftmann@25917
   791
      with A' show ?thesis by (auto dest: mult_pos_pos)
haftmann@25917
   792
    qed
haftmann@25917
   793
  qed
haftmann@25917
   794
  then show "a * b \<noteq> 0" by (simp add: neq_iff)
haftmann@25917
   795
qed
haftmann@25304
   796
paulson@14265
   797
lemma zero_less_mult_iff:
haftmann@25917
   798
  "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
haftmann@25917
   799
  apply (auto simp add: mult_pos_pos mult_neg_neg)
haftmann@25917
   800
  apply (simp_all add: not_less le_less)
haftmann@25917
   801
  apply (erule disjE) apply assumption defer
haftmann@25917
   802
  apply (erule disjE) defer apply (drule sym) apply simp
haftmann@25917
   803
  apply (erule disjE) defer apply (drule sym) apply simp
haftmann@25917
   804
  apply (erule disjE) apply assumption apply (drule sym) apply simp
haftmann@25917
   805
  apply (drule sym) apply simp
haftmann@25917
   806
  apply (blast dest: zero_less_mult_pos)
haftmann@25230
   807
  apply (blast dest: zero_less_mult_pos2)
haftmann@25230
   808
  done
huffman@22990
   809
paulson@14265
   810
lemma zero_le_mult_iff:
haftmann@25917
   811
  "0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
nipkow@29667
   812
by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
paulson@14265
   813
paulson@14265
   814
lemma mult_less_0_iff:
haftmann@25917
   815
  "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
huffman@35216
   816
  apply (insert zero_less_mult_iff [of "-a" b])
huffman@35216
   817
  apply force
haftmann@25917
   818
  done
paulson@14265
   819
paulson@14265
   820
lemma mult_le_0_iff:
haftmann@25917
   821
  "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
haftmann@25917
   822
  apply (insert zero_le_mult_iff [of "-a" b]) 
huffman@35216
   823
  apply force
haftmann@25917
   824
  done
haftmann@25917
   825
haftmann@26193
   826
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
haftmann@26193
   827
   also with the relations @{text "\<le>"} and equality.*}
haftmann@26193
   828
haftmann@26193
   829
text{*These ``disjunction'' versions produce two cases when the comparison is
haftmann@26193
   830
 an assumption, but effectively four when the comparison is a goal.*}
haftmann@26193
   831
haftmann@26193
   832
lemma mult_less_cancel_right_disj:
haftmann@26193
   833
  "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
   834
  apply (cases "c = 0")
haftmann@26193
   835
  apply (auto simp add: neq_iff mult_strict_right_mono 
haftmann@26193
   836
                      mult_strict_right_mono_neg)
haftmann@26193
   837
  apply (auto simp add: not_less 
haftmann@26193
   838
                      not_le [symmetric, of "a*c"]
haftmann@26193
   839
                      not_le [symmetric, of a])
haftmann@26193
   840
  apply (erule_tac [!] notE)
haftmann@26193
   841
  apply (auto simp add: less_imp_le mult_right_mono 
haftmann@26193
   842
                      mult_right_mono_neg)
haftmann@26193
   843
  done
haftmann@26193
   844
haftmann@26193
   845
lemma mult_less_cancel_left_disj:
haftmann@26193
   846
  "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
   847
  apply (cases "c = 0")
haftmann@26193
   848
  apply (auto simp add: neq_iff mult_strict_left_mono 
haftmann@26193
   849
                      mult_strict_left_mono_neg)
haftmann@26193
   850
  apply (auto simp add: not_less 
haftmann@26193
   851
                      not_le [symmetric, of "c*a"]
haftmann@26193
   852
                      not_le [symmetric, of a])
haftmann@26193
   853
  apply (erule_tac [!] notE)
haftmann@26193
   854
  apply (auto simp add: less_imp_le mult_left_mono 
haftmann@26193
   855
                      mult_left_mono_neg)
haftmann@26193
   856
  done
haftmann@26193
   857
haftmann@26193
   858
text{*The ``conjunction of implication'' lemmas produce two cases when the
haftmann@26193
   859
comparison is a goal, but give four when the comparison is an assumption.*}
haftmann@26193
   860
haftmann@26193
   861
lemma mult_less_cancel_right:
haftmann@26193
   862
  "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
   863
  using mult_less_cancel_right_disj [of a c b] by auto
haftmann@26193
   864
haftmann@26193
   865
lemma mult_less_cancel_left:
haftmann@26193
   866
  "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
   867
  using mult_less_cancel_left_disj [of c a b] by auto
haftmann@26193
   868
haftmann@26193
   869
lemma mult_le_cancel_right:
haftmann@26193
   870
   "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
   871
by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
haftmann@26193
   872
haftmann@26193
   873
lemma mult_le_cancel_left:
haftmann@26193
   874
  "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
   875
by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
haftmann@26193
   876
nipkow@30649
   877
lemma mult_le_cancel_left_pos:
nipkow@30649
   878
  "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
nipkow@30649
   879
by (auto simp: mult_le_cancel_left)
nipkow@30649
   880
nipkow@30649
   881
lemma mult_le_cancel_left_neg:
nipkow@30649
   882
  "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
nipkow@30649
   883
by (auto simp: mult_le_cancel_left)
nipkow@30649
   884
nipkow@30649
   885
lemma mult_less_cancel_left_pos:
nipkow@30649
   886
  "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
nipkow@30649
   887
by (auto simp: mult_less_cancel_left)
nipkow@30649
   888
nipkow@30649
   889
lemma mult_less_cancel_left_neg:
nipkow@30649
   890
  "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
nipkow@30649
   891
by (auto simp: mult_less_cancel_left)
nipkow@30649
   892
haftmann@25917
   893
end
paulson@14265
   894
huffman@30692
   895
lemmas mult_sign_intros =
huffman@30692
   896
  mult_nonneg_nonneg mult_nonneg_nonpos
huffman@30692
   897
  mult_nonpos_nonneg mult_nonpos_nonpos
huffman@30692
   898
  mult_pos_pos mult_pos_neg
huffman@30692
   899
  mult_neg_pos mult_neg_neg
haftmann@25230
   900
haftmann@35028
   901
class ordered_comm_ring = comm_ring + ordered_comm_semiring
haftmann@25267
   902
begin
haftmann@25230
   903
haftmann@35028
   904
subclass ordered_ring ..
haftmann@35028
   905
subclass ordered_cancel_comm_semiring ..
haftmann@25230
   906
haftmann@25267
   907
end
haftmann@25230
   908
haftmann@35028
   909
class linordered_semidom = comm_semiring_1_cancel + linordered_comm_semiring_strict +
haftmann@35028
   910
  (*previously linordered_semiring*)
haftmann@25230
   911
  assumes zero_less_one [simp]: "0 < 1"
haftmann@25230
   912
begin
haftmann@25230
   913
haftmann@25230
   914
lemma pos_add_strict:
haftmann@25230
   915
  shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@36301
   916
  using add_strict_mono [of 0 a b c] by simp
haftmann@25230
   917
haftmann@26193
   918
lemma zero_le_one [simp]: "0 \<le> 1"
nipkow@29667
   919
by (rule zero_less_one [THEN less_imp_le]) 
haftmann@26193
   920
haftmann@26193
   921
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
nipkow@29667
   922
by (simp add: not_le) 
haftmann@26193
   923
haftmann@26193
   924
lemma not_one_less_zero [simp]: "\<not> 1 < 0"
nipkow@29667
   925
by (simp add: not_less) 
haftmann@26193
   926
haftmann@26193
   927
lemma less_1_mult:
haftmann@26193
   928
  assumes "1 < m" and "1 < n"
haftmann@26193
   929
  shows "1 < m * n"
haftmann@26193
   930
  using assms mult_strict_mono [of 1 m 1 n]
haftmann@26193
   931
    by (simp add:  less_trans [OF zero_less_one]) 
haftmann@26193
   932
haftmann@25230
   933
end
haftmann@25230
   934
haftmann@35028
   935
class linordered_idom = comm_ring_1 +
haftmann@35028
   936
  linordered_comm_semiring_strict + ordered_ab_group_add +
haftmann@25230
   937
  abs_if + sgn_if
haftmann@35028
   938
  (*previously linordered_ring*)
haftmann@25917
   939
begin
haftmann@25917
   940
hoelzl@36622
   941
subclass linordered_semiring_1_strict ..
haftmann@35043
   942
subclass linordered_ring_strict ..
haftmann@35028
   943
subclass ordered_comm_ring ..
huffman@27516
   944
subclass idom ..
haftmann@25917
   945
haftmann@35028
   946
subclass linordered_semidom
haftmann@28823
   947
proof
haftmann@26193
   948
  have "0 \<le> 1 * 1" by (rule zero_le_square)
haftmann@26193
   949
  thus "0 < 1" by (simp add: le_less)
haftmann@25917
   950
qed 
haftmann@25917
   951
haftmann@35028
   952
lemma linorder_neqE_linordered_idom:
haftmann@26193
   953
  assumes "x \<noteq> y" obtains "x < y" | "y < x"
haftmann@26193
   954
  using assms by (rule neqE)
haftmann@26193
   955
haftmann@26274
   956
text {* These cancellation simprules also produce two cases when the comparison is a goal. *}
haftmann@26274
   957
haftmann@26274
   958
lemma mult_le_cancel_right1:
haftmann@26274
   959
  "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
   960
by (insert mult_le_cancel_right [of 1 c b], simp)
haftmann@26274
   961
haftmann@26274
   962
lemma mult_le_cancel_right2:
haftmann@26274
   963
  "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
   964
by (insert mult_le_cancel_right [of a c 1], simp)
haftmann@26274
   965
haftmann@26274
   966
lemma mult_le_cancel_left1:
haftmann@26274
   967
  "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
   968
by (insert mult_le_cancel_left [of c 1 b], simp)
haftmann@26274
   969
haftmann@26274
   970
lemma mult_le_cancel_left2:
haftmann@26274
   971
  "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
   972
by (insert mult_le_cancel_left [of c a 1], simp)
haftmann@26274
   973
haftmann@26274
   974
lemma mult_less_cancel_right1:
haftmann@26274
   975
  "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
   976
by (insert mult_less_cancel_right [of 1 c b], simp)
haftmann@26274
   977
haftmann@26274
   978
lemma mult_less_cancel_right2:
haftmann@26274
   979
  "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
   980
by (insert mult_less_cancel_right [of a c 1], simp)
haftmann@26274
   981
haftmann@26274
   982
lemma mult_less_cancel_left1:
haftmann@26274
   983
  "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
   984
by (insert mult_less_cancel_left [of c 1 b], simp)
haftmann@26274
   985
haftmann@26274
   986
lemma mult_less_cancel_left2:
haftmann@26274
   987
  "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
   988
by (insert mult_less_cancel_left [of c a 1], simp)
haftmann@26274
   989
haftmann@27651
   990
lemma sgn_sgn [simp]:
haftmann@27651
   991
  "sgn (sgn a) = sgn a"
nipkow@29700
   992
unfolding sgn_if by simp
haftmann@27651
   993
haftmann@27651
   994
lemma sgn_0_0:
haftmann@27651
   995
  "sgn a = 0 \<longleftrightarrow> a = 0"
nipkow@29700
   996
unfolding sgn_if by simp
haftmann@27651
   997
haftmann@27651
   998
lemma sgn_1_pos:
haftmann@27651
   999
  "sgn a = 1 \<longleftrightarrow> a > 0"
huffman@35216
  1000
unfolding sgn_if by simp
haftmann@27651
  1001
haftmann@27651
  1002
lemma sgn_1_neg:
haftmann@27651
  1003
  "sgn a = - 1 \<longleftrightarrow> a < 0"
huffman@35216
  1004
unfolding sgn_if by auto
haftmann@27651
  1005
haftmann@29940
  1006
lemma sgn_pos [simp]:
haftmann@29940
  1007
  "0 < a \<Longrightarrow> sgn a = 1"
haftmann@29940
  1008
unfolding sgn_1_pos .
haftmann@29940
  1009
haftmann@29940
  1010
lemma sgn_neg [simp]:
haftmann@29940
  1011
  "a < 0 \<Longrightarrow> sgn a = - 1"
haftmann@29940
  1012
unfolding sgn_1_neg .
haftmann@29940
  1013
haftmann@27651
  1014
lemma sgn_times:
haftmann@27651
  1015
  "sgn (a * b) = sgn a * sgn b"
nipkow@29667
  1016
by (auto simp add: sgn_if zero_less_mult_iff)
haftmann@27651
  1017
haftmann@36301
  1018
lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"
nipkow@29700
  1019
unfolding sgn_if abs_if by auto
nipkow@29700
  1020
haftmann@29940
  1021
lemma sgn_greater [simp]:
haftmann@29940
  1022
  "0 < sgn a \<longleftrightarrow> 0 < a"
haftmann@29940
  1023
  unfolding sgn_if by auto
haftmann@29940
  1024
haftmann@29940
  1025
lemma sgn_less [simp]:
haftmann@29940
  1026
  "sgn a < 0 \<longleftrightarrow> a < 0"
haftmann@29940
  1027
  unfolding sgn_if by auto
haftmann@29940
  1028
haftmann@36301
  1029
lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"
huffman@29949
  1030
  by (simp add: abs_if)
huffman@29949
  1031
haftmann@36301
  1032
lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"
huffman@29949
  1033
  by (simp add: abs_if)
haftmann@29653
  1034
nipkow@33676
  1035
lemma dvd_if_abs_eq:
haftmann@36301
  1036
  "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
nipkow@33676
  1037
by(subst abs_dvd_iff[symmetric]) simp
nipkow@33676
  1038
haftmann@25917
  1039
end
haftmann@25230
  1040
haftmann@26274
  1041
text {* Simprules for comparisons where common factors can be cancelled. *}
paulson@15234
  1042
blanchet@35828
  1043
lemmas mult_compare_simps[no_atp] =
paulson@15234
  1044
    mult_le_cancel_right mult_le_cancel_left
paulson@15234
  1045
    mult_le_cancel_right1 mult_le_cancel_right2
paulson@15234
  1046
    mult_le_cancel_left1 mult_le_cancel_left2
paulson@15234
  1047
    mult_less_cancel_right mult_less_cancel_left
paulson@15234
  1048
    mult_less_cancel_right1 mult_less_cancel_right2
paulson@15234
  1049
    mult_less_cancel_left1 mult_less_cancel_left2
paulson@15234
  1050
    mult_cancel_right mult_cancel_left
paulson@15234
  1051
    mult_cancel_right1 mult_cancel_right2
paulson@15234
  1052
    mult_cancel_left1 mult_cancel_left2
paulson@15234
  1053
haftmann@36301
  1054
text {* Reasoning about inequalities with division *}
avigad@16775
  1055
haftmann@35028
  1056
context linordered_semidom
haftmann@25193
  1057
begin
haftmann@25193
  1058
haftmann@25193
  1059
lemma less_add_one: "a < a + 1"
paulson@14293
  1060
proof -
haftmann@25193
  1061
  have "a + 0 < a + 1"
nipkow@23482
  1062
    by (blast intro: zero_less_one add_strict_left_mono)
paulson@14293
  1063
  thus ?thesis by simp
paulson@14293
  1064
qed
paulson@14293
  1065
haftmann@25193
  1066
lemma zero_less_two: "0 < 1 + 1"
nipkow@29667
  1067
by (blast intro: less_trans zero_less_one less_add_one)
haftmann@25193
  1068
haftmann@25193
  1069
end
paulson@14365
  1070
haftmann@36301
  1071
context linordered_idom
haftmann@36301
  1072
begin
paulson@15234
  1073
haftmann@36301
  1074
lemma mult_right_le_one_le:
haftmann@36301
  1075
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"
haftmann@36301
  1076
  by (auto simp add: mult_le_cancel_left2)
haftmann@36301
  1077
haftmann@36301
  1078
lemma mult_left_le_one_le:
haftmann@36301
  1079
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"
haftmann@36301
  1080
  by (auto simp add: mult_le_cancel_right2)
haftmann@36301
  1081
haftmann@36301
  1082
end
haftmann@36301
  1083
haftmann@36301
  1084
text {* Absolute Value *}
paulson@14293
  1085
haftmann@35028
  1086
context linordered_idom
haftmann@25304
  1087
begin
haftmann@25304
  1088
haftmann@36301
  1089
lemma mult_sgn_abs:
haftmann@36301
  1090
  "sgn x * \<bar>x\<bar> = x"
haftmann@25304
  1091
  unfolding abs_if sgn_if by auto
haftmann@25304
  1092
haftmann@36301
  1093
lemma abs_one [simp]:
haftmann@36301
  1094
  "\<bar>1\<bar> = 1"
haftmann@36301
  1095
  by (simp add: abs_if zero_less_one [THEN less_not_sym])
haftmann@36301
  1096
haftmann@25304
  1097
end
nipkow@24491
  1098
haftmann@35028
  1099
class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +
haftmann@25304
  1100
  assumes abs_eq_mult:
haftmann@25304
  1101
    "(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
  1102
haftmann@35028
  1103
context linordered_idom
haftmann@30961
  1104
begin
haftmann@30961
  1105
haftmann@35028
  1106
subclass ordered_ring_abs proof
huffman@35216
  1107
qed (auto simp add: abs_if not_less mult_less_0_iff)
haftmann@30961
  1108
haftmann@30961
  1109
lemma abs_mult:
haftmann@36301
  1110
  "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>" 
haftmann@30961
  1111
  by (rule abs_eq_mult) auto
haftmann@30961
  1112
haftmann@30961
  1113
lemma abs_mult_self:
haftmann@36301
  1114
  "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
haftmann@30961
  1115
  by (simp add: abs_if) 
haftmann@30961
  1116
paulson@14294
  1117
lemma abs_mult_less:
haftmann@36301
  1118
  "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"
paulson@14294
  1119
proof -
haftmann@36301
  1120
  assume ac: "\<bar>a\<bar> < c"
haftmann@36301
  1121
  hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)
haftmann@36301
  1122
  assume "\<bar>b\<bar> < d"
paulson@14294
  1123
  thus ?thesis by (simp add: ac cpos mult_strict_mono) 
paulson@14294
  1124
qed
paulson@14293
  1125
haftmann@36301
  1126
lemma less_minus_self_iff:
haftmann@36301
  1127
  "a < - a \<longleftrightarrow> a < 0"
haftmann@36301
  1128
  by (simp only: less_le less_eq_neg_nonpos equal_neg_zero)
obua@14738
  1129
haftmann@36301
  1130
lemma abs_less_iff:
haftmann@36301
  1131
  "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b" 
haftmann@36301
  1132
  by (simp add: less_le abs_le_iff) (auto simp add: abs_if)
obua@14738
  1133
haftmann@36301
  1134
lemma abs_mult_pos:
haftmann@36301
  1135
  "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"
haftmann@36301
  1136
  by (simp add: abs_mult)
haftmann@36301
  1137
haftmann@36301
  1138
end
avigad@16775
  1139
haftmann@33364
  1140
code_modulename SML
haftmann@35050
  1141
  Rings Arith
haftmann@33364
  1142
haftmann@33364
  1143
code_modulename OCaml
haftmann@35050
  1144
  Rings Arith
haftmann@33364
  1145
haftmann@33364
  1146
code_modulename Haskell
haftmann@35050
  1147
  Rings Arith
haftmann@33364
  1148
paulson@14265
  1149
end