src/HOL/Groups.thy
author wenzelm
Tue Oct 06 15:14:28 2015 +0200 (2015-10-06)
changeset 61337 4645502c3c64
parent 61169 4de9ff3ea29a
child 61378 3e04c9ca001a
permissions -rw-r--r--
fewer aliases for toplevel theorem statements;
haftmann@35050
     1
(*  Title:   HOL/Groups.thy
wenzelm@29269
     2
    Author:  Gertrud Bauer, Steven Obua, Lawrence C Paulson, Markus Wenzel, Jeremy Avigad
obua@14738
     3
*)
obua@14738
     4
wenzelm@60758
     5
section \<open>Groups, also combined with orderings\<close>
obua@14738
     6
haftmann@35050
     7
theory Groups
haftmann@35092
     8
imports Orderings
nipkow@15131
     9
begin
obua@14738
    10
wenzelm@60758
    11
subsection \<open>Dynamic facts\<close>
haftmann@35301
    12
wenzelm@57950
    13
named_theorems ac_simps "associativity and commutativity simplification rules"
haftmann@36343
    14
haftmann@36343
    15
wenzelm@60758
    16
text\<open>The rewrites accumulated in @{text algebra_simps} deal with the
haftmann@36348
    17
classical algebraic structures of groups, rings and family. They simplify
haftmann@36348
    18
terms by multiplying everything out (in case of a ring) and bringing sums and
haftmann@36348
    19
products into a canonical form (by ordered rewriting). As a result it decides
haftmann@36348
    20
group and ring equalities but also helps with inequalities.
haftmann@36348
    21
haftmann@36348
    22
Of course it also works for fields, but it knows nothing about multiplicative
wenzelm@60758
    23
inverses or division. This is catered for by @{text field_simps}.\<close>
haftmann@36348
    24
wenzelm@57950
    25
named_theorems algebra_simps "algebra simplification rules"
haftmann@35301
    26
haftmann@35301
    27
wenzelm@60758
    28
text\<open>Lemmas @{text field_simps} multiply with denominators in (in)equations
haftmann@36348
    29
if they can be proved to be non-zero (for equations) or positive/negative
haftmann@36348
    30
(for inequations). Can be too aggressive and is therefore separate from the
wenzelm@60758
    31
more benign @{text algebra_simps}.\<close>
haftmann@35301
    32
wenzelm@57950
    33
named_theorems field_simps "algebra simplification rules for fields"
haftmann@35301
    34
haftmann@35301
    35
wenzelm@60758
    36
subsection \<open>Abstract structures\<close>
haftmann@35301
    37
wenzelm@60758
    38
text \<open>
haftmann@35301
    39
  These locales provide basic structures for interpretation into
haftmann@35301
    40
  bigger structures;  extensions require careful thinking, otherwise
haftmann@35301
    41
  undesired effects may occur due to interpretation.
wenzelm@60758
    42
\<close>
haftmann@35301
    43
haftmann@35301
    44
locale semigroup =
haftmann@35301
    45
  fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
haftmann@35301
    46
  assumes assoc [ac_simps]: "a * b * c = a * (b * c)"
haftmann@35301
    47
haftmann@35301
    48
locale abel_semigroup = semigroup +
haftmann@35301
    49
  assumes commute [ac_simps]: "a * b = b * a"
haftmann@35301
    50
begin
haftmann@35301
    51
haftmann@35301
    52
lemma left_commute [ac_simps]:
haftmann@35301
    53
  "b * (a * c) = a * (b * c)"
haftmann@35301
    54
proof -
haftmann@35301
    55
  have "(b * a) * c = (a * b) * c"
haftmann@35301
    56
    by (simp only: commute)
haftmann@35301
    57
  then show ?thesis
haftmann@35301
    58
    by (simp only: assoc)
haftmann@35301
    59
qed
haftmann@35301
    60
haftmann@35301
    61
end
haftmann@35301
    62
haftmann@35720
    63
locale monoid = semigroup +
haftmann@35723
    64
  fixes z :: 'a ("1")
haftmann@35723
    65
  assumes left_neutral [simp]: "1 * a = a"
haftmann@35723
    66
  assumes right_neutral [simp]: "a * 1 = a"
haftmann@35720
    67
haftmann@35720
    68
locale comm_monoid = abel_semigroup +
haftmann@35723
    69
  fixes z :: 'a ("1")
haftmann@35723
    70
  assumes comm_neutral: "a * 1 = a"
haftmann@54868
    71
begin
haftmann@35720
    72
haftmann@54868
    73
sublocale monoid
wenzelm@61169
    74
  by standard (simp_all add: commute comm_neutral)
haftmann@35720
    75
haftmann@54868
    76
end
haftmann@54868
    77
haftmann@35301
    78
wenzelm@60758
    79
subsection \<open>Generic operations\<close>
haftmann@35267
    80
haftmann@35267
    81
class zero = 
haftmann@35267
    82
  fixes zero :: 'a  ("0")
haftmann@35267
    83
haftmann@35267
    84
class one =
haftmann@35267
    85
  fixes one  :: 'a  ("1")
haftmann@35267
    86
wenzelm@36176
    87
hide_const (open) zero one
haftmann@35267
    88
haftmann@35267
    89
lemma Let_0 [simp]: "Let 0 f = f 0"
haftmann@35267
    90
  unfolding Let_def ..
haftmann@35267
    91
haftmann@35267
    92
lemma Let_1 [simp]: "Let 1 f = f 1"
haftmann@35267
    93
  unfolding Let_def ..
haftmann@35267
    94
wenzelm@60758
    95
setup \<open>
haftmann@35267
    96
  Reorient_Proc.add
haftmann@35267
    97
    (fn Const(@{const_name Groups.zero}, _) => true
haftmann@35267
    98
      | Const(@{const_name Groups.one}, _) => true
haftmann@35267
    99
      | _ => false)
wenzelm@60758
   100
\<close>
haftmann@35267
   101
haftmann@35267
   102
simproc_setup reorient_zero ("0 = x") = Reorient_Proc.proc
haftmann@35267
   103
simproc_setup reorient_one ("1 = x") = Reorient_Proc.proc
haftmann@35267
   104
wenzelm@60758
   105
typed_print_translation \<open>
wenzelm@42247
   106
  let
wenzelm@42247
   107
    fun tr' c = (c, fn ctxt => fn T => fn ts =>
wenzelm@52210
   108
      if null ts andalso Printer.type_emphasis ctxt T then
wenzelm@42248
   109
        Syntax.const @{syntax_const "_constrain"} $ Syntax.const c $
wenzelm@52210
   110
          Syntax_Phases.term_of_typ ctxt T
wenzelm@52210
   111
      else raise Match);
wenzelm@42247
   112
  in map tr' [@{const_syntax Groups.one}, @{const_syntax Groups.zero}] end;
wenzelm@60758
   113
\<close> -- \<open>show types that are presumably too general\<close>
haftmann@35267
   114
haftmann@35267
   115
class plus =
haftmann@35267
   116
  fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "+" 65)
haftmann@35267
   117
haftmann@35267
   118
class minus =
haftmann@35267
   119
  fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "-" 65)
haftmann@35267
   120
haftmann@35267
   121
class uminus =
haftmann@35267
   122
  fixes uminus :: "'a \<Rightarrow> 'a"  ("- _" [81] 80)
haftmann@35267
   123
haftmann@35267
   124
class times =
haftmann@35267
   125
  fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "*" 70)
haftmann@35267
   126
haftmann@35092
   127
wenzelm@60758
   128
subsection \<open>Semigroups and Monoids\<close>
obua@14738
   129
haftmann@22390
   130
class semigroup_add = plus +
haftmann@36348
   131
  assumes add_assoc [algebra_simps, field_simps]: "(a + b) + c = a + (b + c)"
haftmann@54868
   132
begin
haftmann@34973
   133
haftmann@54868
   134
sublocale add!: semigroup plus
wenzelm@61169
   135
  by standard (fact add_assoc)
haftmann@22390
   136
haftmann@54868
   137
end
haftmann@54868
   138
haftmann@57512
   139
hide_fact add_assoc
haftmann@57512
   140
haftmann@22390
   141
class ab_semigroup_add = semigroup_add +
haftmann@36348
   142
  assumes add_commute [algebra_simps, field_simps]: "a + b = b + a"
haftmann@54868
   143
begin
haftmann@34973
   144
haftmann@54868
   145
sublocale add!: abel_semigroup plus
wenzelm@61169
   146
  by standard (fact add_commute)
haftmann@34973
   147
haftmann@57512
   148
declare add.left_commute [algebra_simps, field_simps]
haftmann@25062
   149
wenzelm@61337
   150
lemmas add_ac = add.assoc add.commute add.left_commute
nipkow@57571
   151
haftmann@25062
   152
end
obua@14738
   153
haftmann@57512
   154
hide_fact add_commute
haftmann@57512
   155
wenzelm@61337
   156
lemmas add_ac = add.assoc add.commute add.left_commute
nipkow@57571
   157
haftmann@22390
   158
class semigroup_mult = times +
haftmann@36348
   159
  assumes mult_assoc [algebra_simps, field_simps]: "(a * b) * c = a * (b * c)"
haftmann@54868
   160
begin
haftmann@34973
   161
haftmann@54868
   162
sublocale mult!: semigroup times
wenzelm@61169
   163
  by standard (fact mult_assoc)
obua@14738
   164
haftmann@54868
   165
end
haftmann@54868
   166
haftmann@57512
   167
hide_fact mult_assoc
haftmann@57512
   168
haftmann@22390
   169
class ab_semigroup_mult = semigroup_mult +
haftmann@36348
   170
  assumes mult_commute [algebra_simps, field_simps]: "a * b = b * a"
haftmann@54868
   171
begin
haftmann@34973
   172
haftmann@54868
   173
sublocale mult!: abel_semigroup times
wenzelm@61169
   174
  by standard (fact mult_commute)
haftmann@34973
   175
haftmann@57512
   176
declare mult.left_commute [algebra_simps, field_simps]
haftmann@25062
   177
wenzelm@61337
   178
lemmas mult_ac = mult.assoc mult.commute mult.left_commute
nipkow@57571
   179
haftmann@23181
   180
end
obua@14738
   181
haftmann@57512
   182
hide_fact mult_commute
haftmann@57512
   183
wenzelm@61337
   184
lemmas mult_ac = mult.assoc mult.commute mult.left_commute
nipkow@57571
   185
nipkow@23085
   186
class monoid_add = zero + semigroup_add +
haftmann@35720
   187
  assumes add_0_left: "0 + a = a"
haftmann@35720
   188
    and add_0_right: "a + 0 = a"
haftmann@54868
   189
begin
haftmann@35720
   190
haftmann@54868
   191
sublocale add!: monoid plus 0
wenzelm@61169
   192
  by standard (fact add_0_left add_0_right)+
nipkow@23085
   193
haftmann@54868
   194
end
haftmann@54868
   195
haftmann@26071
   196
lemma zero_reorient: "0 = x \<longleftrightarrow> x = 0"
haftmann@54868
   197
  by (fact eq_commute)
haftmann@26071
   198
haftmann@22390
   199
class comm_monoid_add = zero + ab_semigroup_add +
haftmann@25062
   200
  assumes add_0: "0 + a = a"
haftmann@54868
   201
begin
nipkow@23085
   202
haftmann@59559
   203
subclass monoid_add
wenzelm@61169
   204
  by standard (simp_all add: add_0 add.commute [of _ 0])
haftmann@25062
   205
haftmann@59559
   206
sublocale add!: comm_monoid plus 0
wenzelm@61169
   207
  by standard (simp add: ac_simps)
obua@14738
   208
haftmann@54868
   209
end
haftmann@54868
   210
haftmann@22390
   211
class monoid_mult = one + semigroup_mult +
haftmann@35720
   212
  assumes mult_1_left: "1 * a  = a"
haftmann@35720
   213
    and mult_1_right: "a * 1 = a"
haftmann@54868
   214
begin
haftmann@35720
   215
haftmann@54868
   216
sublocale mult!: monoid times 1
wenzelm@61169
   217
  by standard (fact mult_1_left mult_1_right)+
obua@14738
   218
haftmann@54868
   219
end
haftmann@54868
   220
haftmann@26071
   221
lemma one_reorient: "1 = x \<longleftrightarrow> x = 1"
haftmann@54868
   222
  by (fact eq_commute)
haftmann@26071
   223
haftmann@22390
   224
class comm_monoid_mult = one + ab_semigroup_mult +
haftmann@25062
   225
  assumes mult_1: "1 * a = a"
haftmann@54868
   226
begin
obua@14738
   227
haftmann@59559
   228
subclass monoid_mult
wenzelm@61169
   229
  by standard (simp_all add: mult_1 mult.commute [of _ 1])
haftmann@25062
   230
haftmann@59559
   231
sublocale mult!: comm_monoid times 1
wenzelm@61169
   232
  by standard (simp add: ac_simps)
obua@14738
   233
haftmann@54868
   234
end
haftmann@54868
   235
haftmann@22390
   236
class cancel_semigroup_add = semigroup_add +
haftmann@25062
   237
  assumes add_left_imp_eq: "a + b = a + c \<Longrightarrow> b = c"
haftmann@25062
   238
  assumes add_right_imp_eq: "b + a = c + a \<Longrightarrow> b = c"
huffman@27474
   239
begin
huffman@27474
   240
huffman@27474
   241
lemma add_left_cancel [simp]:
huffman@27474
   242
  "a + b = a + c \<longleftrightarrow> b = c"
nipkow@29667
   243
by (blast dest: add_left_imp_eq)
huffman@27474
   244
huffman@27474
   245
lemma add_right_cancel [simp]:
huffman@27474
   246
  "b + a = c + a \<longleftrightarrow> b = c"
nipkow@29667
   247
by (blast dest: add_right_imp_eq)
huffman@27474
   248
huffman@27474
   249
end
obua@14738
   250
haftmann@59815
   251
class cancel_ab_semigroup_add = ab_semigroup_add + minus +
haftmann@59815
   252
  assumes add_diff_cancel_left' [simp]: "(a + b) - a = b"
haftmann@59815
   253
  assumes diff_diff_add [algebra_simps, field_simps]: "a - b - c = a - (b + c)"
haftmann@25267
   254
begin
obua@14738
   255
haftmann@59815
   256
lemma add_diff_cancel_right' [simp]:
haftmann@59815
   257
  "(a + b) - b = a"
haftmann@59815
   258
  using add_diff_cancel_left' [of b a] by (simp add: ac_simps)
haftmann@59815
   259
haftmann@25267
   260
subclass cancel_semigroup_add
haftmann@28823
   261
proof
haftmann@22390
   262
  fix a b c :: 'a
haftmann@59815
   263
  assume "a + b = a + c"
haftmann@59815
   264
  then have "a + b - a = a + c - a"
haftmann@59815
   265
    by simp
haftmann@59815
   266
  then show "b = c"
haftmann@59815
   267
    by simp
haftmann@22390
   268
next
obua@14738
   269
  fix a b c :: 'a
obua@14738
   270
  assume "b + a = c + a"
haftmann@59815
   271
  then have "b + a - a = c + a - a"
haftmann@59815
   272
    by simp
haftmann@59815
   273
  then show "b = c"
haftmann@59815
   274
    by simp
obua@14738
   275
qed
obua@14738
   276
haftmann@59815
   277
lemma add_diff_cancel_left [simp]:
haftmann@59815
   278
  "(c + a) - (c + b) = a - b"
haftmann@59815
   279
  unfolding diff_diff_add [symmetric] by simp
haftmann@59815
   280
haftmann@59815
   281
lemma add_diff_cancel_right [simp]:
haftmann@59815
   282
  "(a + c) - (b + c) = a - b"
haftmann@59815
   283
  using add_diff_cancel_left [symmetric] by (simp add: ac_simps)
haftmann@59815
   284
haftmann@59815
   285
lemma diff_right_commute:
haftmann@59815
   286
  "a - c - b = a - b - c"
haftmann@59815
   287
  by (simp add: diff_diff_add add.commute)
haftmann@59815
   288
haftmann@25267
   289
end
haftmann@25267
   290
huffman@29904
   291
class cancel_comm_monoid_add = cancel_ab_semigroup_add + comm_monoid_add
haftmann@59322
   292
begin
haftmann@59322
   293
haftmann@59815
   294
lemma diff_zero [simp]:
haftmann@59815
   295
  "a - 0 = a"
haftmann@59815
   296
  using add_diff_cancel_right' [of a 0] by simp
haftmann@59322
   297
haftmann@59322
   298
lemma diff_cancel [simp]:
haftmann@59322
   299
  "a - a = 0"
haftmann@59322
   300
proof -
haftmann@59322
   301
  have "(a + 0) - (a + 0) = 0" by (simp only: add_diff_cancel_left diff_zero)
haftmann@59322
   302
  then show ?thesis by simp
haftmann@59322
   303
qed
haftmann@59322
   304
haftmann@59322
   305
lemma add_implies_diff:
haftmann@59322
   306
  assumes "c + b = a"
haftmann@59322
   307
  shows "c = a - b"
haftmann@59322
   308
proof -
haftmann@59322
   309
  from assms have "(b + c) - (b + 0) = a - b" by (simp add: add.commute)
haftmann@59322
   310
  then show "c = a - b" by simp
haftmann@59322
   311
qed
haftmann@59322
   312
haftmann@59815
   313
end  
haftmann@59815
   314
haftmann@59815
   315
class comm_monoid_diff = cancel_comm_monoid_add +
haftmann@59815
   316
  assumes zero_diff [simp]: "0 - a = 0"
haftmann@59815
   317
begin
haftmann@59815
   318
haftmann@59815
   319
lemma diff_add_zero [simp]:
haftmann@59815
   320
  "a - (a + b) = 0"
haftmann@59815
   321
proof -
haftmann@59815
   322
  have "a - (a + b) = (a + 0) - (a + b)" by simp
haftmann@59815
   323
  also have "\<dots> = 0" by (simp only: add_diff_cancel_left zero_diff)
haftmann@59815
   324
  finally show ?thesis .
haftmann@59815
   325
qed
haftmann@59815
   326
haftmann@59322
   327
end
haftmann@59322
   328
huffman@29904
   329
wenzelm@60758
   330
subsection \<open>Groups\<close>
nipkow@23085
   331
haftmann@25762
   332
class group_add = minus + uminus + monoid_add +
haftmann@25062
   333
  assumes left_minus [simp]: "- a + a = 0"
haftmann@54230
   334
  assumes add_uminus_conv_diff [simp]: "a + (- b) = a - b"
haftmann@25062
   335
begin
nipkow@23085
   336
haftmann@54230
   337
lemma diff_conv_add_uminus:
haftmann@54230
   338
  "a - b = a + (- b)"
haftmann@54230
   339
  by simp
haftmann@54230
   340
huffman@34147
   341
lemma minus_unique:
huffman@34147
   342
  assumes "a + b = 0" shows "- a = b"
huffman@34147
   343
proof -
huffman@34147
   344
  have "- a = - a + (a + b)" using assms by simp
haftmann@57512
   345
  also have "\<dots> = b" by (simp add: add.assoc [symmetric])
huffman@34147
   346
  finally show ?thesis .
huffman@34147
   347
qed
huffman@34147
   348
haftmann@25062
   349
lemma minus_zero [simp]: "- 0 = 0"
obua@14738
   350
proof -
huffman@34147
   351
  have "0 + 0 = 0" by (rule add_0_right)
huffman@34147
   352
  thus "- 0 = 0" by (rule minus_unique)
obua@14738
   353
qed
obua@14738
   354
haftmann@25062
   355
lemma minus_minus [simp]: "- (- a) = a"
nipkow@23085
   356
proof -
huffman@34147
   357
  have "- a + a = 0" by (rule left_minus)
huffman@34147
   358
  thus "- (- a) = a" by (rule minus_unique)
nipkow@23085
   359
qed
obua@14738
   360
haftmann@54230
   361
lemma right_minus: "a + - a = 0"
obua@14738
   362
proof -
haftmann@25062
   363
  have "a + - a = - (- a) + - a" by simp
haftmann@25062
   364
  also have "\<dots> = 0" by (rule left_minus)
obua@14738
   365
  finally show ?thesis .
obua@14738
   366
qed
obua@14738
   367
haftmann@54230
   368
lemma diff_self [simp]:
haftmann@54230
   369
  "a - a = 0"
haftmann@54230
   370
  using right_minus [of a] by simp
haftmann@54230
   371
haftmann@40368
   372
subclass cancel_semigroup_add
haftmann@40368
   373
proof
haftmann@40368
   374
  fix a b c :: 'a
haftmann@40368
   375
  assume "a + b = a + c"
haftmann@40368
   376
  then have "- a + a + b = - a + a + c"
haftmann@57512
   377
    unfolding add.assoc by simp
haftmann@40368
   378
  then show "b = c" by simp
haftmann@40368
   379
next
haftmann@40368
   380
  fix a b c :: 'a
haftmann@40368
   381
  assume "b + a = c + a"
haftmann@40368
   382
  then have "b + a + - a = c + a  + - a" by simp
haftmann@57512
   383
  then show "b = c" unfolding add.assoc by simp
haftmann@40368
   384
qed
haftmann@40368
   385
haftmann@54230
   386
lemma minus_add_cancel [simp]:
haftmann@54230
   387
  "- a + (a + b) = b"
haftmann@57512
   388
  by (simp add: add.assoc [symmetric])
haftmann@54230
   389
haftmann@54230
   390
lemma add_minus_cancel [simp]:
haftmann@54230
   391
  "a + (- a + b) = b"
haftmann@57512
   392
  by (simp add: add.assoc [symmetric])
huffman@34147
   393
haftmann@54230
   394
lemma diff_add_cancel [simp]:
haftmann@54230
   395
  "a - b + b = a"
haftmann@57512
   396
  by (simp only: diff_conv_add_uminus add.assoc) simp
huffman@34147
   397
haftmann@54230
   398
lemma add_diff_cancel [simp]:
haftmann@54230
   399
  "a + b - b = a"
haftmann@57512
   400
  by (simp only: diff_conv_add_uminus add.assoc) simp
haftmann@54230
   401
haftmann@54230
   402
lemma minus_add:
haftmann@54230
   403
  "- (a + b) = - b + - a"
huffman@34147
   404
proof -
huffman@34147
   405
  have "(a + b) + (- b + - a) = 0"
haftmann@57512
   406
    by (simp only: add.assoc add_minus_cancel) simp
haftmann@54230
   407
  then show "- (a + b) = - b + - a"
huffman@34147
   408
    by (rule minus_unique)
huffman@34147
   409
qed
huffman@34147
   410
haftmann@54230
   411
lemma right_minus_eq [simp]:
haftmann@54230
   412
  "a - b = 0 \<longleftrightarrow> a = b"
obua@14738
   413
proof
nipkow@23085
   414
  assume "a - b = 0"
haftmann@57512
   415
  have "a = (a - b) + b" by (simp add: add.assoc)
wenzelm@60758
   416
  also have "\<dots> = b" using \<open>a - b = 0\<close> by simp
nipkow@23085
   417
  finally show "a = b" .
obua@14738
   418
next
haftmann@54230
   419
  assume "a = b" thus "a - b = 0" by simp
obua@14738
   420
qed
obua@14738
   421
haftmann@54230
   422
lemma eq_iff_diff_eq_0:
haftmann@54230
   423
  "a = b \<longleftrightarrow> a - b = 0"
haftmann@54230
   424
  by (fact right_minus_eq [symmetric])
obua@14738
   425
haftmann@54230
   426
lemma diff_0 [simp]:
haftmann@54230
   427
  "0 - a = - a"
haftmann@54230
   428
  by (simp only: diff_conv_add_uminus add_0_left)
obua@14738
   429
haftmann@54230
   430
lemma diff_0_right [simp]:
haftmann@54230
   431
  "a - 0 = a" 
haftmann@54230
   432
  by (simp only: diff_conv_add_uminus minus_zero add_0_right)
obua@14738
   433
haftmann@54230
   434
lemma diff_minus_eq_add [simp]:
haftmann@54230
   435
  "a - - b = a + b"
haftmann@54230
   436
  by (simp only: diff_conv_add_uminus minus_minus)
obua@14738
   437
haftmann@25062
   438
lemma neg_equal_iff_equal [simp]:
haftmann@25062
   439
  "- a = - b \<longleftrightarrow> a = b" 
obua@14738
   440
proof 
obua@14738
   441
  assume "- a = - b"
nipkow@29667
   442
  hence "- (- a) = - (- b)" by simp
haftmann@25062
   443
  thus "a = b" by simp
obua@14738
   444
next
haftmann@25062
   445
  assume "a = b"
haftmann@25062
   446
  thus "- a = - b" by simp
obua@14738
   447
qed
obua@14738
   448
haftmann@25062
   449
lemma neg_equal_0_iff_equal [simp]:
haftmann@25062
   450
  "- a = 0 \<longleftrightarrow> a = 0"
haftmann@54230
   451
  by (subst neg_equal_iff_equal [symmetric]) simp
obua@14738
   452
haftmann@25062
   453
lemma neg_0_equal_iff_equal [simp]:
haftmann@25062
   454
  "0 = - a \<longleftrightarrow> 0 = a"
haftmann@54230
   455
  by (subst neg_equal_iff_equal [symmetric]) simp
obua@14738
   456
wenzelm@60758
   457
text\<open>The next two equations can make the simplifier loop!\<close>
obua@14738
   458
haftmann@25062
   459
lemma equation_minus_iff:
haftmann@25062
   460
  "a = - b \<longleftrightarrow> b = - a"
obua@14738
   461
proof -
haftmann@25062
   462
  have "- (- a) = - b \<longleftrightarrow> - a = b" by (rule neg_equal_iff_equal)
haftmann@25062
   463
  thus ?thesis by (simp add: eq_commute)
haftmann@25062
   464
qed
haftmann@25062
   465
haftmann@25062
   466
lemma minus_equation_iff:
haftmann@25062
   467
  "- a = b \<longleftrightarrow> - b = a"
haftmann@25062
   468
proof -
haftmann@25062
   469
  have "- a = - (- b) \<longleftrightarrow> a = -b" by (rule neg_equal_iff_equal)
obua@14738
   470
  thus ?thesis by (simp add: eq_commute)
obua@14738
   471
qed
obua@14738
   472
haftmann@54230
   473
lemma eq_neg_iff_add_eq_0:
haftmann@54230
   474
  "a = - b \<longleftrightarrow> a + b = 0"
huffman@29914
   475
proof
huffman@29914
   476
  assume "a = - b" then show "a + b = 0" by simp
huffman@29914
   477
next
huffman@29914
   478
  assume "a + b = 0"
huffman@29914
   479
  moreover have "a + (b + - b) = (a + b) + - b"
haftmann@57512
   480
    by (simp only: add.assoc)
huffman@29914
   481
  ultimately show "a = - b" by simp
huffman@29914
   482
qed
huffman@29914
   483
haftmann@54230
   484
lemma add_eq_0_iff2:
haftmann@54230
   485
  "a + b = 0 \<longleftrightarrow> a = - b"
haftmann@54230
   486
  by (fact eq_neg_iff_add_eq_0 [symmetric])
haftmann@54230
   487
haftmann@54230
   488
lemma neg_eq_iff_add_eq_0:
haftmann@54230
   489
  "- a = b \<longleftrightarrow> a + b = 0"
haftmann@54230
   490
  by (auto simp add: add_eq_0_iff2)
huffman@44348
   491
haftmann@54230
   492
lemma add_eq_0_iff:
haftmann@54230
   493
  "a + b = 0 \<longleftrightarrow> b = - a"
haftmann@54230
   494
  by (auto simp add: neg_eq_iff_add_eq_0 [symmetric])
huffman@45548
   495
haftmann@54230
   496
lemma minus_diff_eq [simp]:
haftmann@54230
   497
  "- (a - b) = b - a"
haftmann@57512
   498
  by (simp only: neg_eq_iff_add_eq_0 diff_conv_add_uminus add.assoc minus_add_cancel) simp
huffman@45548
   499
haftmann@54230
   500
lemma add_diff_eq [algebra_simps, field_simps]:
haftmann@54230
   501
  "a + (b - c) = (a + b) - c"
haftmann@57512
   502
  by (simp only: diff_conv_add_uminus add.assoc)
huffman@45548
   503
haftmann@54230
   504
lemma diff_add_eq_diff_diff_swap:
haftmann@54230
   505
  "a - (b + c) = a - c - b"
haftmann@57512
   506
  by (simp only: diff_conv_add_uminus add.assoc minus_add)
huffman@45548
   507
haftmann@54230
   508
lemma diff_eq_eq [algebra_simps, field_simps]:
haftmann@54230
   509
  "a - b = c \<longleftrightarrow> a = c + b"
haftmann@54230
   510
  by auto
huffman@45548
   511
haftmann@54230
   512
lemma eq_diff_eq [algebra_simps, field_simps]:
haftmann@54230
   513
  "a = c - b \<longleftrightarrow> a + b = c"
haftmann@54230
   514
  by auto
haftmann@54230
   515
haftmann@54230
   516
lemma diff_diff_eq2 [algebra_simps, field_simps]:
haftmann@54230
   517
  "a - (b - c) = (a + c) - b"
haftmann@57512
   518
  by (simp only: diff_conv_add_uminus add.assoc) simp
huffman@45548
   519
huffman@45548
   520
lemma diff_eq_diff_eq:
huffman@45548
   521
  "a - b = c - d \<Longrightarrow> a = b \<longleftrightarrow> c = d"
haftmann@54230
   522
  by (simp only: eq_iff_diff_eq_0 [of a b] eq_iff_diff_eq_0 [of c d])
huffman@45548
   523
haftmann@25062
   524
end
haftmann@25062
   525
haftmann@25762
   526
class ab_group_add = minus + uminus + comm_monoid_add +
haftmann@25062
   527
  assumes ab_left_minus: "- a + a = 0"
haftmann@59557
   528
  assumes ab_diff_conv_add_uminus: "a - b = a + (- b)"
haftmann@25267
   529
begin
haftmann@25062
   530
haftmann@25267
   531
subclass group_add
haftmann@59557
   532
  proof qed (simp_all add: ab_left_minus ab_diff_conv_add_uminus)
haftmann@25062
   533
huffman@29904
   534
subclass cancel_comm_monoid_add
haftmann@28823
   535
proof
haftmann@25062
   536
  fix a b c :: 'a
haftmann@59815
   537
  have "b + a - a = b"
haftmann@59815
   538
    by simp
haftmann@59815
   539
  then show "a + b - a = b"
haftmann@59815
   540
    by (simp add: ac_simps)
haftmann@59815
   541
  show "a - b - c = a - (b + c)"
haftmann@59815
   542
    by (simp add: algebra_simps)
haftmann@25062
   543
qed
haftmann@25062
   544
haftmann@54230
   545
lemma uminus_add_conv_diff [simp]:
haftmann@25062
   546
  "- a + b = b - a"
haftmann@57512
   547
  by (simp add: add.commute)
haftmann@25062
   548
haftmann@25062
   549
lemma minus_add_distrib [simp]:
haftmann@25062
   550
  "- (a + b) = - a + - b"
haftmann@54230
   551
  by (simp add: algebra_simps)
haftmann@25062
   552
haftmann@54230
   553
lemma diff_add_eq [algebra_simps, field_simps]:
haftmann@54230
   554
  "(a - b) + c = (a + c) - b"
haftmann@54230
   555
  by (simp add: algebra_simps)
haftmann@25077
   556
haftmann@25062
   557
end
obua@14738
   558
haftmann@37884
   559
wenzelm@60758
   560
subsection \<open>(Partially) Ordered Groups\<close> 
obua@14738
   561
wenzelm@60758
   562
text \<open>
haftmann@35301
   563
  The theory of partially ordered groups is taken from the books:
haftmann@35301
   564
  \begin{itemize}
haftmann@35301
   565
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
haftmann@35301
   566
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
haftmann@35301
   567
  \end{itemize}
haftmann@35301
   568
  Most of the used notions can also be looked up in 
haftmann@35301
   569
  \begin{itemize}
wenzelm@54703
   570
  \item @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
haftmann@35301
   571
  \item \emph{Algebra I} by van der Waerden, Springer.
haftmann@35301
   572
  \end{itemize}
wenzelm@60758
   573
\<close>
haftmann@35301
   574
haftmann@35028
   575
class ordered_ab_semigroup_add = order + ab_semigroup_add +
haftmann@25062
   576
  assumes add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b"
haftmann@25062
   577
begin
haftmann@24380
   578
haftmann@25062
   579
lemma add_right_mono:
haftmann@25062
   580
  "a \<le> b \<Longrightarrow> a + c \<le> b + c"
haftmann@57512
   581
by (simp add: add.commute [of _ c] add_left_mono)
obua@14738
   582
wenzelm@60758
   583
text \<open>non-strict, in both arguments\<close>
obua@14738
   584
lemma add_mono:
haftmann@25062
   585
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c \<le> b + d"
obua@14738
   586
  apply (erule add_right_mono [THEN order_trans])
haftmann@57512
   587
  apply (simp add: add.commute add_left_mono)
obua@14738
   588
  done
obua@14738
   589
haftmann@25062
   590
end
haftmann@25062
   591
haftmann@35028
   592
class ordered_cancel_ab_semigroup_add =
haftmann@35028
   593
  ordered_ab_semigroup_add + cancel_ab_semigroup_add
haftmann@25062
   594
begin
haftmann@25062
   595
obua@14738
   596
lemma add_strict_left_mono:
haftmann@25062
   597
  "a < b \<Longrightarrow> c + a < c + b"
nipkow@29667
   598
by (auto simp add: less_le add_left_mono)
obua@14738
   599
obua@14738
   600
lemma add_strict_right_mono:
haftmann@25062
   601
  "a < b \<Longrightarrow> a + c < b + c"
haftmann@57512
   602
by (simp add: add.commute [of _ c] add_strict_left_mono)
obua@14738
   603
wenzelm@60758
   604
text\<open>Strict monotonicity in both arguments\<close>
haftmann@25062
   605
lemma add_strict_mono:
haftmann@25062
   606
  "a < b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
haftmann@25062
   607
apply (erule add_strict_right_mono [THEN less_trans])
obua@14738
   608
apply (erule add_strict_left_mono)
obua@14738
   609
done
obua@14738
   610
obua@14738
   611
lemma add_less_le_mono:
haftmann@25062
   612
  "a < b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c < b + d"
haftmann@25062
   613
apply (erule add_strict_right_mono [THEN less_le_trans])
haftmann@25062
   614
apply (erule add_left_mono)
obua@14738
   615
done
obua@14738
   616
obua@14738
   617
lemma add_le_less_mono:
haftmann@25062
   618
  "a \<le> b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
haftmann@25062
   619
apply (erule add_right_mono [THEN le_less_trans])
obua@14738
   620
apply (erule add_strict_left_mono) 
obua@14738
   621
done
obua@14738
   622
haftmann@25062
   623
end
haftmann@25062
   624
haftmann@35028
   625
class ordered_ab_semigroup_add_imp_le =
haftmann@35028
   626
  ordered_cancel_ab_semigroup_add +
haftmann@25062
   627
  assumes add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b"
haftmann@25062
   628
begin
haftmann@25062
   629
obua@14738
   630
lemma add_less_imp_less_left:
nipkow@29667
   631
  assumes less: "c + a < c + b" shows "a < b"
obua@14738
   632
proof -
obua@14738
   633
  from less have le: "c + a <= c + b" by (simp add: order_le_less)
obua@14738
   634
  have "a <= b" 
obua@14738
   635
    apply (insert le)
obua@14738
   636
    apply (drule add_le_imp_le_left)
obua@14738
   637
    by (insert le, drule add_le_imp_le_left, assumption)
obua@14738
   638
  moreover have "a \<noteq> b"
obua@14738
   639
  proof (rule ccontr)
obua@14738
   640
    assume "~(a \<noteq> b)"
obua@14738
   641
    then have "a = b" by simp
obua@14738
   642
    then have "c + a = c + b" by simp
obua@14738
   643
    with less show "False"by simp
obua@14738
   644
  qed
obua@14738
   645
  ultimately show "a < b" by (simp add: order_le_less)
obua@14738
   646
qed
obua@14738
   647
obua@14738
   648
lemma add_less_imp_less_right:
haftmann@25062
   649
  "a + c < b + c \<Longrightarrow> a < b"
obua@14738
   650
apply (rule add_less_imp_less_left [of c])
haftmann@57512
   651
apply (simp add: add.commute)  
obua@14738
   652
done
obua@14738
   653
obua@14738
   654
lemma add_less_cancel_left [simp]:
haftmann@25062
   655
  "c + a < c + b \<longleftrightarrow> a < b"
haftmann@54230
   656
  by (blast intro: add_less_imp_less_left add_strict_left_mono) 
obua@14738
   657
obua@14738
   658
lemma add_less_cancel_right [simp]:
haftmann@25062
   659
  "a + c < b + c \<longleftrightarrow> a < b"
haftmann@54230
   660
  by (blast intro: add_less_imp_less_right add_strict_right_mono)
obua@14738
   661
obua@14738
   662
lemma add_le_cancel_left [simp]:
haftmann@25062
   663
  "c + a \<le> c + b \<longleftrightarrow> a \<le> b"
haftmann@54230
   664
  by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono) 
obua@14738
   665
obua@14738
   666
lemma add_le_cancel_right [simp]:
haftmann@25062
   667
  "a + c \<le> b + c \<longleftrightarrow> a \<le> b"
haftmann@57512
   668
  by (simp add: add.commute [of a c] add.commute [of b c])
obua@14738
   669
obua@14738
   670
lemma add_le_imp_le_right:
haftmann@25062
   671
  "a + c \<le> b + c \<Longrightarrow> a \<le> b"
nipkow@29667
   672
by simp
haftmann@25062
   673
haftmann@25077
   674
lemma max_add_distrib_left:
haftmann@25077
   675
  "max x y + z = max (x + z) (y + z)"
haftmann@25077
   676
  unfolding max_def by auto
haftmann@25077
   677
haftmann@25077
   678
lemma min_add_distrib_left:
haftmann@25077
   679
  "min x y + z = min (x + z) (y + z)"
haftmann@25077
   680
  unfolding min_def by auto
haftmann@25077
   681
huffman@44848
   682
lemma max_add_distrib_right:
huffman@44848
   683
  "x + max y z = max (x + y) (x + z)"
huffman@44848
   684
  unfolding max_def by auto
huffman@44848
   685
huffman@44848
   686
lemma min_add_distrib_right:
huffman@44848
   687
  "x + min y z = min (x + y) (x + z)"
huffman@44848
   688
  unfolding min_def by auto
huffman@44848
   689
haftmann@25062
   690
end
haftmann@25062
   691
haftmann@52289
   692
class ordered_cancel_comm_monoid_diff = comm_monoid_diff + ordered_ab_semigroup_add_imp_le +
haftmann@52289
   693
  assumes le_iff_add: "a \<le> b \<longleftrightarrow> (\<exists>c. b = a + c)"
haftmann@52289
   694
begin
haftmann@52289
   695
haftmann@52289
   696
context
haftmann@52289
   697
  fixes a b
haftmann@52289
   698
  assumes "a \<le> b"
haftmann@52289
   699
begin
haftmann@52289
   700
haftmann@52289
   701
lemma add_diff_inverse:
haftmann@52289
   702
  "a + (b - a) = b"
wenzelm@60758
   703
  using \<open>a \<le> b\<close> by (auto simp add: le_iff_add)
haftmann@52289
   704
haftmann@52289
   705
lemma add_diff_assoc:
haftmann@52289
   706
  "c + (b - a) = c + b - a"
wenzelm@60758
   707
  using \<open>a \<le> b\<close> by (auto simp add: le_iff_add add.left_commute [of c])
haftmann@52289
   708
haftmann@52289
   709
lemma add_diff_assoc2:
haftmann@52289
   710
  "b - a + c = b + c - a"
wenzelm@60758
   711
  using \<open>a \<le> b\<close> by (auto simp add: le_iff_add add.assoc)
haftmann@52289
   712
haftmann@52289
   713
lemma diff_add_assoc:
haftmann@52289
   714
  "c + b - a = c + (b - a)"
wenzelm@60758
   715
  using \<open>a \<le> b\<close> by (simp add: add.commute add_diff_assoc)
haftmann@52289
   716
haftmann@52289
   717
lemma diff_add_assoc2:
haftmann@52289
   718
  "b + c - a = b - a + c"
wenzelm@60758
   719
  using \<open>a \<le> b\<close>by (simp add: add.commute add_diff_assoc)
haftmann@52289
   720
haftmann@52289
   721
lemma diff_diff_right:
haftmann@52289
   722
  "c - (b - a) = c + a - b"
haftmann@57512
   723
  by (simp add: add_diff_inverse add_diff_cancel_left [of a c "b - a", symmetric] add.commute)
haftmann@52289
   724
haftmann@52289
   725
lemma diff_add:
haftmann@52289
   726
  "b - a + a = b"
haftmann@57512
   727
  by (simp add: add.commute add_diff_inverse)
haftmann@52289
   728
haftmann@52289
   729
lemma le_add_diff:
haftmann@52289
   730
  "c \<le> b + c - a"
haftmann@57512
   731
  by (auto simp add: add.commute diff_add_assoc2 le_iff_add)
haftmann@52289
   732
haftmann@52289
   733
lemma le_imp_diff_is_add:
haftmann@52289
   734
  "a \<le> b \<Longrightarrow> b - a = c \<longleftrightarrow> b = c + a"
haftmann@57512
   735
  by (auto simp add: add.commute add_diff_inverse)
haftmann@52289
   736
haftmann@52289
   737
lemma le_diff_conv2:
haftmann@52289
   738
  "c \<le> b - a \<longleftrightarrow> c + a \<le> b" (is "?P \<longleftrightarrow> ?Q")
haftmann@52289
   739
proof
haftmann@52289
   740
  assume ?P
haftmann@52289
   741
  then have "c + a \<le> b - a + a" by (rule add_right_mono)
haftmann@57512
   742
  then show ?Q by (simp add: add_diff_inverse add.commute)
haftmann@52289
   743
next
haftmann@52289
   744
  assume ?Q
haftmann@57512
   745
  then have "a + c \<le> a + (b - a)" by (simp add: add_diff_inverse add.commute)
haftmann@52289
   746
  then show ?P by simp
haftmann@52289
   747
qed
haftmann@52289
   748
haftmann@52289
   749
end
haftmann@52289
   750
haftmann@52289
   751
end
haftmann@52289
   752
haftmann@52289
   753
wenzelm@60758
   754
subsection \<open>Support for reasoning about signs\<close>
haftmann@25303
   755
haftmann@35028
   756
class ordered_comm_monoid_add =
haftmann@35028
   757
  ordered_cancel_ab_semigroup_add + comm_monoid_add
haftmann@25303
   758
begin
haftmann@25303
   759
haftmann@25303
   760
lemma add_pos_nonneg:
nipkow@29667
   761
  assumes "0 < a" and "0 \<le> b" shows "0 < a + b"
haftmann@25303
   762
proof -
haftmann@25303
   763
  have "0 + 0 < a + b" 
haftmann@25303
   764
    using assms by (rule add_less_le_mono)
haftmann@25303
   765
  then show ?thesis by simp
haftmann@25303
   766
qed
haftmann@25303
   767
haftmann@25303
   768
lemma add_pos_pos:
nipkow@29667
   769
  assumes "0 < a" and "0 < b" shows "0 < a + b"
nipkow@29667
   770
by (rule add_pos_nonneg) (insert assms, auto)
haftmann@25303
   771
haftmann@25303
   772
lemma add_nonneg_pos:
nipkow@29667
   773
  assumes "0 \<le> a" and "0 < b" shows "0 < a + b"
haftmann@25303
   774
proof -
haftmann@25303
   775
  have "0 + 0 < a + b" 
haftmann@25303
   776
    using assms by (rule add_le_less_mono)
haftmann@25303
   777
  then show ?thesis by simp
haftmann@25303
   778
qed
haftmann@25303
   779
huffman@36977
   780
lemma add_nonneg_nonneg [simp]:
nipkow@29667
   781
  assumes "0 \<le> a" and "0 \<le> b" shows "0 \<le> a + b"
haftmann@25303
   782
proof -
haftmann@25303
   783
  have "0 + 0 \<le> a + b" 
haftmann@25303
   784
    using assms by (rule add_mono)
haftmann@25303
   785
  then show ?thesis by simp
haftmann@25303
   786
qed
haftmann@25303
   787
huffman@30691
   788
lemma add_neg_nonpos:
nipkow@29667
   789
  assumes "a < 0" and "b \<le> 0" shows "a + b < 0"
haftmann@25303
   790
proof -
haftmann@25303
   791
  have "a + b < 0 + 0"
haftmann@25303
   792
    using assms by (rule add_less_le_mono)
haftmann@25303
   793
  then show ?thesis by simp
haftmann@25303
   794
qed
haftmann@25303
   795
haftmann@25303
   796
lemma add_neg_neg: 
nipkow@29667
   797
  assumes "a < 0" and "b < 0" shows "a + b < 0"
nipkow@29667
   798
by (rule add_neg_nonpos) (insert assms, auto)
haftmann@25303
   799
haftmann@25303
   800
lemma add_nonpos_neg:
nipkow@29667
   801
  assumes "a \<le> 0" and "b < 0" shows "a + b < 0"
haftmann@25303
   802
proof -
haftmann@25303
   803
  have "a + b < 0 + 0"
haftmann@25303
   804
    using assms by (rule add_le_less_mono)
haftmann@25303
   805
  then show ?thesis by simp
haftmann@25303
   806
qed
haftmann@25303
   807
haftmann@25303
   808
lemma add_nonpos_nonpos:
nipkow@29667
   809
  assumes "a \<le> 0" and "b \<le> 0" shows "a + b \<le> 0"
haftmann@25303
   810
proof -
haftmann@25303
   811
  have "a + b \<le> 0 + 0"
haftmann@25303
   812
    using assms by (rule add_mono)
haftmann@25303
   813
  then show ?thesis by simp
haftmann@25303
   814
qed
haftmann@25303
   815
huffman@30691
   816
lemmas add_sign_intros =
huffman@30691
   817
  add_pos_nonneg add_pos_pos add_nonneg_pos add_nonneg_nonneg
huffman@30691
   818
  add_neg_nonpos add_neg_neg add_nonpos_neg add_nonpos_nonpos
huffman@30691
   819
huffman@29886
   820
lemma add_nonneg_eq_0_iff:
huffman@29886
   821
  assumes x: "0 \<le> x" and y: "0 \<le> y"
huffman@29886
   822
  shows "x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@29886
   823
proof (intro iffI conjI)
huffman@29886
   824
  have "x = x + 0" by simp
huffman@29886
   825
  also have "x + 0 \<le> x + y" using y by (rule add_left_mono)
huffman@29886
   826
  also assume "x + y = 0"
huffman@29886
   827
  also have "0 \<le> x" using x .
huffman@29886
   828
  finally show "x = 0" .
huffman@29886
   829
next
huffman@29886
   830
  have "y = 0 + y" by simp
huffman@29886
   831
  also have "0 + y \<le> x + y" using x by (rule add_right_mono)
huffman@29886
   832
  also assume "x + y = 0"
huffman@29886
   833
  also have "0 \<le> y" using y .
huffman@29886
   834
  finally show "y = 0" .
huffman@29886
   835
next
huffman@29886
   836
  assume "x = 0 \<and> y = 0"
huffman@29886
   837
  then show "x + y = 0" by simp
huffman@29886
   838
qed
huffman@29886
   839
haftmann@54230
   840
lemma add_increasing:
haftmann@54230
   841
  "0 \<le> a \<Longrightarrow> b \<le> c \<Longrightarrow> b \<le> a + c"
haftmann@54230
   842
  by (insert add_mono [of 0 a b c], simp)
haftmann@54230
   843
haftmann@54230
   844
lemma add_increasing2:
haftmann@54230
   845
  "0 \<le> c \<Longrightarrow> b \<le> a \<Longrightarrow> b \<le> a + c"
haftmann@57512
   846
  by (simp add: add_increasing add.commute [of a])
haftmann@54230
   847
haftmann@54230
   848
lemma add_strict_increasing:
haftmann@54230
   849
  "0 < a \<Longrightarrow> b \<le> c \<Longrightarrow> b < a + c"
haftmann@54230
   850
  by (insert add_less_le_mono [of 0 a b c], simp)
haftmann@54230
   851
haftmann@54230
   852
lemma add_strict_increasing2:
haftmann@54230
   853
  "0 \<le> a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@54230
   854
  by (insert add_le_less_mono [of 0 a b c], simp)
haftmann@54230
   855
haftmann@25303
   856
end
haftmann@25303
   857
haftmann@35028
   858
class ordered_ab_group_add =
haftmann@35028
   859
  ab_group_add + ordered_ab_semigroup_add
haftmann@25062
   860
begin
haftmann@25062
   861
haftmann@35028
   862
subclass ordered_cancel_ab_semigroup_add ..
haftmann@25062
   863
haftmann@35028
   864
subclass ordered_ab_semigroup_add_imp_le
haftmann@28823
   865
proof
haftmann@25062
   866
  fix a b c :: 'a
haftmann@25062
   867
  assume "c + a \<le> c + b"
haftmann@25062
   868
  hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono)
haftmann@57512
   869
  hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add.assoc)
haftmann@25062
   870
  thus "a \<le> b" by simp
haftmann@25062
   871
qed
haftmann@25062
   872
haftmann@35028
   873
subclass ordered_comm_monoid_add ..
haftmann@25303
   874
haftmann@54230
   875
lemma add_less_same_cancel1 [simp]:
haftmann@54230
   876
  "b + a < b \<longleftrightarrow> a < 0"
haftmann@54230
   877
  using add_less_cancel_left [of _ _ 0] by simp
haftmann@54230
   878
haftmann@54230
   879
lemma add_less_same_cancel2 [simp]:
haftmann@54230
   880
  "a + b < b \<longleftrightarrow> a < 0"
haftmann@54230
   881
  using add_less_cancel_right [of _ _ 0] by simp
haftmann@54230
   882
haftmann@54230
   883
lemma less_add_same_cancel1 [simp]:
haftmann@54230
   884
  "a < a + b \<longleftrightarrow> 0 < b"
haftmann@54230
   885
  using add_less_cancel_left [of _ 0] by simp
haftmann@54230
   886
haftmann@54230
   887
lemma less_add_same_cancel2 [simp]:
haftmann@54230
   888
  "a < b + a \<longleftrightarrow> 0 < b"
haftmann@54230
   889
  using add_less_cancel_right [of 0] by simp
haftmann@54230
   890
haftmann@54230
   891
lemma add_le_same_cancel1 [simp]:
haftmann@54230
   892
  "b + a \<le> b \<longleftrightarrow> a \<le> 0"
haftmann@54230
   893
  using add_le_cancel_left [of _ _ 0] by simp
haftmann@54230
   894
haftmann@54230
   895
lemma add_le_same_cancel2 [simp]:
haftmann@54230
   896
  "a + b \<le> b \<longleftrightarrow> a \<le> 0"
haftmann@54230
   897
  using add_le_cancel_right [of _ _ 0] by simp
haftmann@54230
   898
haftmann@54230
   899
lemma le_add_same_cancel1 [simp]:
haftmann@54230
   900
  "a \<le> a + b \<longleftrightarrow> 0 \<le> b"
haftmann@54230
   901
  using add_le_cancel_left [of _ 0] by simp
haftmann@54230
   902
haftmann@54230
   903
lemma le_add_same_cancel2 [simp]:
haftmann@54230
   904
  "a \<le> b + a \<longleftrightarrow> 0 \<le> b"
haftmann@54230
   905
  using add_le_cancel_right [of 0] by simp
haftmann@54230
   906
haftmann@25077
   907
lemma max_diff_distrib_left:
haftmann@25077
   908
  shows "max x y - z = max (x - z) (y - z)"
haftmann@54230
   909
  using max_add_distrib_left [of x y "- z"] by simp
haftmann@25077
   910
haftmann@25077
   911
lemma min_diff_distrib_left:
haftmann@25077
   912
  shows "min x y - z = min (x - z) (y - z)"
haftmann@54230
   913
  using min_add_distrib_left [of x y "- z"] by simp
haftmann@25077
   914
haftmann@25077
   915
lemma le_imp_neg_le:
nipkow@29667
   916
  assumes "a \<le> b" shows "-b \<le> -a"
haftmann@25077
   917
proof -
wenzelm@60758
   918
  have "-a+a \<le> -a+b" using \<open>a \<le> b\<close> by (rule add_left_mono) 
haftmann@54230
   919
  then have "0 \<le> -a+b" by simp
haftmann@54230
   920
  then have "0 + (-b) \<le> (-a + b) + (-b)" by (rule add_right_mono) 
haftmann@54230
   921
  then show ?thesis by (simp add: algebra_simps)
haftmann@25077
   922
qed
haftmann@25077
   923
haftmann@25077
   924
lemma neg_le_iff_le [simp]: "- b \<le> - a \<longleftrightarrow> a \<le> b"
haftmann@25077
   925
proof 
haftmann@25077
   926
  assume "- b \<le> - a"
nipkow@29667
   927
  hence "- (- a) \<le> - (- b)" by (rule le_imp_neg_le)
haftmann@25077
   928
  thus "a\<le>b" by simp
haftmann@25077
   929
next
haftmann@25077
   930
  assume "a\<le>b"
haftmann@25077
   931
  thus "-b \<le> -a" by (rule le_imp_neg_le)
haftmann@25077
   932
qed
haftmann@25077
   933
haftmann@25077
   934
lemma neg_le_0_iff_le [simp]: "- a \<le> 0 \<longleftrightarrow> 0 \<le> a"
nipkow@29667
   935
by (subst neg_le_iff_le [symmetric], simp)
haftmann@25077
   936
haftmann@25077
   937
lemma neg_0_le_iff_le [simp]: "0 \<le> - a \<longleftrightarrow> a \<le> 0"
nipkow@29667
   938
by (subst neg_le_iff_le [symmetric], simp)
haftmann@25077
   939
haftmann@25077
   940
lemma neg_less_iff_less [simp]: "- b < - a \<longleftrightarrow> a < b"
nipkow@29667
   941
by (force simp add: less_le) 
haftmann@25077
   942
haftmann@25077
   943
lemma neg_less_0_iff_less [simp]: "- a < 0 \<longleftrightarrow> 0 < a"
nipkow@29667
   944
by (subst neg_less_iff_less [symmetric], simp)
haftmann@25077
   945
haftmann@25077
   946
lemma neg_0_less_iff_less [simp]: "0 < - a \<longleftrightarrow> a < 0"
nipkow@29667
   947
by (subst neg_less_iff_less [symmetric], simp)
haftmann@25077
   948
wenzelm@60758
   949
text\<open>The next several equations can make the simplifier loop!\<close>
haftmann@25077
   950
haftmann@25077
   951
lemma less_minus_iff: "a < - b \<longleftrightarrow> b < - a"
haftmann@25077
   952
proof -
haftmann@25077
   953
  have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less)
haftmann@25077
   954
  thus ?thesis by simp
haftmann@25077
   955
qed
haftmann@25077
   956
haftmann@25077
   957
lemma minus_less_iff: "- a < b \<longleftrightarrow> - b < a"
haftmann@25077
   958
proof -
haftmann@25077
   959
  have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less)
haftmann@25077
   960
  thus ?thesis by simp
haftmann@25077
   961
qed
haftmann@25077
   962
haftmann@25077
   963
lemma le_minus_iff: "a \<le> - b \<longleftrightarrow> b \<le> - a"
haftmann@25077
   964
proof -
haftmann@25077
   965
  have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff)
haftmann@25077
   966
  have "(- (- a) <= -b) = (b <= - a)" 
haftmann@25077
   967
    apply (auto simp only: le_less)
haftmann@25077
   968
    apply (drule mm)
haftmann@25077
   969
    apply (simp_all)
haftmann@25077
   970
    apply (drule mm[simplified], assumption)
haftmann@25077
   971
    done
haftmann@25077
   972
  then show ?thesis by simp
haftmann@25077
   973
qed
haftmann@25077
   974
haftmann@25077
   975
lemma minus_le_iff: "- a \<le> b \<longleftrightarrow> - b \<le> a"
nipkow@29667
   976
by (auto simp add: le_less minus_less_iff)
haftmann@25077
   977
blanchet@54148
   978
lemma diff_less_0_iff_less [simp]:
haftmann@37884
   979
  "a - b < 0 \<longleftrightarrow> a < b"
haftmann@25077
   980
proof -
haftmann@54230
   981
  have "a - b < 0 \<longleftrightarrow> a + (- b) < b + (- b)" by simp
haftmann@37884
   982
  also have "... \<longleftrightarrow> a < b" by (simp only: add_less_cancel_right)
haftmann@25077
   983
  finally show ?thesis .
haftmann@25077
   984
qed
haftmann@25077
   985
haftmann@37884
   986
lemmas less_iff_diff_less_0 = diff_less_0_iff_less [symmetric]
haftmann@37884
   987
haftmann@54230
   988
lemma diff_less_eq [algebra_simps, field_simps]:
haftmann@54230
   989
  "a - b < c \<longleftrightarrow> a < c + b"
haftmann@25077
   990
apply (subst less_iff_diff_less_0 [of a])
haftmann@25077
   991
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
haftmann@54230
   992
apply (simp add: algebra_simps)
haftmann@25077
   993
done
haftmann@25077
   994
haftmann@54230
   995
lemma less_diff_eq[algebra_simps, field_simps]:
haftmann@54230
   996
  "a < c - b \<longleftrightarrow> a + b < c"
haftmann@36302
   997
apply (subst less_iff_diff_less_0 [of "a + b"])
haftmann@25077
   998
apply (subst less_iff_diff_less_0 [of a])
haftmann@54230
   999
apply (simp add: algebra_simps)
haftmann@25077
  1000
done
haftmann@25077
  1001
haftmann@36348
  1002
lemma diff_le_eq[algebra_simps, field_simps]: "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
haftmann@54230
  1003
by (auto simp add: le_less diff_less_eq )
haftmann@25077
  1004
haftmann@36348
  1005
lemma le_diff_eq[algebra_simps, field_simps]: "a \<le> c - b \<longleftrightarrow> a + b \<le> c"
haftmann@54230
  1006
by (auto simp add: le_less less_diff_eq)
haftmann@25077
  1007
blanchet@54148
  1008
lemma diff_le_0_iff_le [simp]:
haftmann@37884
  1009
  "a - b \<le> 0 \<longleftrightarrow> a \<le> b"
haftmann@37884
  1010
  by (simp add: algebra_simps)
haftmann@37884
  1011
haftmann@37884
  1012
lemmas le_iff_diff_le_0 = diff_le_0_iff_le [symmetric]
haftmann@37884
  1013
haftmann@37884
  1014
lemma diff_eq_diff_less:
haftmann@37884
  1015
  "a - b = c - d \<Longrightarrow> a < b \<longleftrightarrow> c < d"
haftmann@37884
  1016
  by (auto simp only: less_iff_diff_less_0 [of a b] less_iff_diff_less_0 [of c d])
haftmann@37884
  1017
haftmann@37889
  1018
lemma diff_eq_diff_less_eq:
haftmann@37889
  1019
  "a - b = c - d \<Longrightarrow> a \<le> b \<longleftrightarrow> c \<le> d"
haftmann@37889
  1020
  by (auto simp only: le_iff_diff_le_0 [of a b] le_iff_diff_le_0 [of c d])
haftmann@25077
  1021
hoelzl@56950
  1022
lemma diff_mono: "a \<le> b \<Longrightarrow> d \<le> c \<Longrightarrow> a - c \<le> b - d"
hoelzl@56950
  1023
  by (simp add: field_simps add_mono)
hoelzl@56950
  1024
hoelzl@56950
  1025
lemma diff_left_mono: "b \<le> a \<Longrightarrow> c - a \<le> c - b"
hoelzl@56950
  1026
  by (simp add: field_simps)
hoelzl@56950
  1027
hoelzl@56950
  1028
lemma diff_right_mono: "a \<le> b \<Longrightarrow> a - c \<le> b - c"
hoelzl@56950
  1029
  by (simp add: field_simps)
hoelzl@56950
  1030
hoelzl@56950
  1031
lemma diff_strict_mono: "a < b \<Longrightarrow> d < c \<Longrightarrow> a - c < b - d"
hoelzl@56950
  1032
  by (simp add: field_simps add_strict_mono)
hoelzl@56950
  1033
hoelzl@56950
  1034
lemma diff_strict_left_mono: "b < a \<Longrightarrow> c - a < c - b"
hoelzl@56950
  1035
  by (simp add: field_simps)
hoelzl@56950
  1036
hoelzl@56950
  1037
lemma diff_strict_right_mono: "a < b \<Longrightarrow> a - c < b - c"
hoelzl@56950
  1038
  by (simp add: field_simps)
hoelzl@56950
  1039
haftmann@25077
  1040
end
haftmann@25077
  1041
wenzelm@48891
  1042
ML_file "Tools/group_cancel.ML"
huffman@48556
  1043
huffman@48556
  1044
simproc_setup group_cancel_add ("a + b::'a::ab_group_add") =
wenzelm@60758
  1045
  \<open>fn phi => fn ss => try Group_Cancel.cancel_add_conv\<close>
huffman@48556
  1046
huffman@48556
  1047
simproc_setup group_cancel_diff ("a - b::'a::ab_group_add") =
wenzelm@60758
  1048
  \<open>fn phi => fn ss => try Group_Cancel.cancel_diff_conv\<close>
haftmann@37884
  1049
huffman@48556
  1050
simproc_setup group_cancel_eq ("a = (b::'a::ab_group_add)") =
wenzelm@60758
  1051
  \<open>fn phi => fn ss => try Group_Cancel.cancel_eq_conv\<close>
haftmann@37889
  1052
huffman@48556
  1053
simproc_setup group_cancel_le ("a \<le> (b::'a::ordered_ab_group_add)") =
wenzelm@60758
  1054
  \<open>fn phi => fn ss => try Group_Cancel.cancel_le_conv\<close>
huffman@48556
  1055
huffman@48556
  1056
simproc_setup group_cancel_less ("a < (b::'a::ordered_ab_group_add)") =
wenzelm@60758
  1057
  \<open>fn phi => fn ss => try Group_Cancel.cancel_less_conv\<close>
haftmann@37884
  1058
haftmann@35028
  1059
class linordered_ab_semigroup_add =
haftmann@35028
  1060
  linorder + ordered_ab_semigroup_add
haftmann@25062
  1061
haftmann@35028
  1062
class linordered_cancel_ab_semigroup_add =
haftmann@35028
  1063
  linorder + ordered_cancel_ab_semigroup_add
haftmann@25267
  1064
begin
haftmann@25062
  1065
haftmann@35028
  1066
subclass linordered_ab_semigroup_add ..
haftmann@25062
  1067
haftmann@35028
  1068
subclass ordered_ab_semigroup_add_imp_le
haftmann@28823
  1069
proof
haftmann@25062
  1070
  fix a b c :: 'a
haftmann@25062
  1071
  assume le: "c + a <= c + b"  
haftmann@25062
  1072
  show "a <= b"
haftmann@25062
  1073
  proof (rule ccontr)
haftmann@25062
  1074
    assume w: "~ a \<le> b"
haftmann@25062
  1075
    hence "b <= a" by (simp add: linorder_not_le)
haftmann@25062
  1076
    hence le2: "c + b <= c + a" by (rule add_left_mono)
haftmann@25062
  1077
    have "a = b" 
haftmann@25062
  1078
      apply (insert le)
haftmann@25062
  1079
      apply (insert le2)
haftmann@25062
  1080
      apply (drule antisym, simp_all)
haftmann@25062
  1081
      done
haftmann@25062
  1082
    with w show False 
haftmann@25062
  1083
      by (simp add: linorder_not_le [symmetric])
haftmann@25062
  1084
  qed
haftmann@25062
  1085
qed
haftmann@25062
  1086
haftmann@25267
  1087
end
haftmann@25267
  1088
haftmann@35028
  1089
class linordered_ab_group_add = linorder + ordered_ab_group_add
haftmann@25267
  1090
begin
haftmann@25230
  1091
haftmann@35028
  1092
subclass linordered_cancel_ab_semigroup_add ..
haftmann@25230
  1093
haftmann@35036
  1094
lemma equal_neg_zero [simp]:
haftmann@25303
  1095
  "a = - a \<longleftrightarrow> a = 0"
haftmann@25303
  1096
proof
haftmann@25303
  1097
  assume "a = 0" then show "a = - a" by simp
haftmann@25303
  1098
next
haftmann@25303
  1099
  assume A: "a = - a" show "a = 0"
haftmann@25303
  1100
  proof (cases "0 \<le> a")
haftmann@25303
  1101
    case True with A have "0 \<le> - a" by auto
haftmann@25303
  1102
    with le_minus_iff have "a \<le> 0" by simp
haftmann@25303
  1103
    with True show ?thesis by (auto intro: order_trans)
haftmann@25303
  1104
  next
haftmann@25303
  1105
    case False then have B: "a \<le> 0" by auto
haftmann@25303
  1106
    with A have "- a \<le> 0" by auto
haftmann@25303
  1107
    with B show ?thesis by (auto intro: order_trans)
haftmann@25303
  1108
  qed
haftmann@25303
  1109
qed
haftmann@25303
  1110
haftmann@35036
  1111
lemma neg_equal_zero [simp]:
haftmann@25303
  1112
  "- a = a \<longleftrightarrow> a = 0"
haftmann@35036
  1113
  by (auto dest: sym)
haftmann@35036
  1114
haftmann@54250
  1115
lemma neg_less_eq_nonneg [simp]:
haftmann@54250
  1116
  "- a \<le> a \<longleftrightarrow> 0 \<le> a"
haftmann@54250
  1117
proof
haftmann@54250
  1118
  assume A: "- a \<le> a" show "0 \<le> a"
haftmann@54250
  1119
  proof (rule classical)
haftmann@54250
  1120
    assume "\<not> 0 \<le> a"
haftmann@54250
  1121
    then have "a < 0" by auto
haftmann@54250
  1122
    with A have "- a < 0" by (rule le_less_trans)
haftmann@54250
  1123
    then show ?thesis by auto
haftmann@54250
  1124
  qed
haftmann@54250
  1125
next
haftmann@54250
  1126
  assume A: "0 \<le> a" show "- a \<le> a"
haftmann@54250
  1127
  proof (rule order_trans)
haftmann@54250
  1128
    show "- a \<le> 0" using A by (simp add: minus_le_iff)
haftmann@54250
  1129
  next
haftmann@54250
  1130
    show "0 \<le> a" using A .
haftmann@54250
  1131
  qed
haftmann@54250
  1132
qed
haftmann@54250
  1133
haftmann@54250
  1134
lemma neg_less_pos [simp]:
haftmann@54250
  1135
  "- a < a \<longleftrightarrow> 0 < a"
haftmann@54250
  1136
  by (auto simp add: less_le)
haftmann@54250
  1137
haftmann@54250
  1138
lemma less_eq_neg_nonpos [simp]:
haftmann@54250
  1139
  "a \<le> - a \<longleftrightarrow> a \<le> 0"
haftmann@54250
  1140
  using neg_less_eq_nonneg [of "- a"] by simp
haftmann@54250
  1141
haftmann@54250
  1142
lemma less_neg_neg [simp]:
haftmann@54250
  1143
  "a < - a \<longleftrightarrow> a < 0"
haftmann@54250
  1144
  using neg_less_pos [of "- a"] by simp
haftmann@54250
  1145
haftmann@35036
  1146
lemma double_zero [simp]:
haftmann@35036
  1147
  "a + a = 0 \<longleftrightarrow> a = 0"
haftmann@35036
  1148
proof
haftmann@35036
  1149
  assume assm: "a + a = 0"
haftmann@35036
  1150
  then have a: "- a = a" by (rule minus_unique)
huffman@35216
  1151
  then show "a = 0" by (simp only: neg_equal_zero)
haftmann@35036
  1152
qed simp
haftmann@35036
  1153
haftmann@35036
  1154
lemma double_zero_sym [simp]:
haftmann@35036
  1155
  "0 = a + a \<longleftrightarrow> a = 0"
haftmann@35036
  1156
  by (rule, drule sym) simp_all
haftmann@35036
  1157
haftmann@35036
  1158
lemma zero_less_double_add_iff_zero_less_single_add [simp]:
haftmann@35036
  1159
  "0 < a + a \<longleftrightarrow> 0 < a"
haftmann@35036
  1160
proof
haftmann@35036
  1161
  assume "0 < a + a"
haftmann@35036
  1162
  then have "0 - a < a" by (simp only: diff_less_eq)
haftmann@35036
  1163
  then have "- a < a" by simp
haftmann@54250
  1164
  then show "0 < a" by simp
haftmann@35036
  1165
next
haftmann@35036
  1166
  assume "0 < a"
haftmann@35036
  1167
  with this have "0 + 0 < a + a"
haftmann@35036
  1168
    by (rule add_strict_mono)
haftmann@35036
  1169
  then show "0 < a + a" by simp
haftmann@35036
  1170
qed
haftmann@35036
  1171
haftmann@35036
  1172
lemma zero_le_double_add_iff_zero_le_single_add [simp]:
haftmann@35036
  1173
  "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
haftmann@35036
  1174
  by (auto simp add: le_less)
haftmann@35036
  1175
haftmann@35036
  1176
lemma double_add_less_zero_iff_single_add_less_zero [simp]:
haftmann@35036
  1177
  "a + a < 0 \<longleftrightarrow> a < 0"
haftmann@35036
  1178
proof -
haftmann@35036
  1179
  have "\<not> a + a < 0 \<longleftrightarrow> \<not> a < 0"
haftmann@35036
  1180
    by (simp add: not_less)
haftmann@35036
  1181
  then show ?thesis by simp
haftmann@35036
  1182
qed
haftmann@35036
  1183
haftmann@35036
  1184
lemma double_add_le_zero_iff_single_add_le_zero [simp]:
haftmann@35036
  1185
  "a + a \<le> 0 \<longleftrightarrow> a \<le> 0" 
haftmann@35036
  1186
proof -
haftmann@35036
  1187
  have "\<not> a + a \<le> 0 \<longleftrightarrow> \<not> a \<le> 0"
haftmann@35036
  1188
    by (simp add: not_le)
haftmann@35036
  1189
  then show ?thesis by simp
haftmann@35036
  1190
qed
haftmann@35036
  1191
haftmann@35036
  1192
lemma minus_max_eq_min:
haftmann@35036
  1193
  "- max x y = min (-x) (-y)"
haftmann@35036
  1194
  by (auto simp add: max_def min_def)
haftmann@35036
  1195
haftmann@35036
  1196
lemma minus_min_eq_max:
haftmann@35036
  1197
  "- min x y = max (-x) (-y)"
haftmann@35036
  1198
  by (auto simp add: max_def min_def)
haftmann@25303
  1199
haftmann@25267
  1200
end
haftmann@25267
  1201
haftmann@35092
  1202
class abs =
haftmann@35092
  1203
  fixes abs :: "'a \<Rightarrow> 'a"
haftmann@35092
  1204
begin
haftmann@35092
  1205
haftmann@35092
  1206
notation (xsymbols)
haftmann@35092
  1207
  abs  ("\<bar>_\<bar>")
haftmann@35092
  1208
haftmann@35092
  1209
notation (HTML output)
haftmann@35092
  1210
  abs  ("\<bar>_\<bar>")
haftmann@35092
  1211
haftmann@35092
  1212
end
haftmann@35092
  1213
haftmann@35092
  1214
class sgn =
haftmann@35092
  1215
  fixes sgn :: "'a \<Rightarrow> 'a"
haftmann@35092
  1216
haftmann@35092
  1217
class abs_if = minus + uminus + ord + zero + abs +
haftmann@35092
  1218
  assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)"
haftmann@35092
  1219
haftmann@35092
  1220
class sgn_if = minus + uminus + zero + one + ord + sgn +
haftmann@35092
  1221
  assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
haftmann@35092
  1222
begin
haftmann@35092
  1223
haftmann@35092
  1224
lemma sgn0 [simp]: "sgn 0 = 0"
haftmann@35092
  1225
  by (simp add:sgn_if)
haftmann@35092
  1226
haftmann@35092
  1227
end
obua@14738
  1228
haftmann@35028
  1229
class ordered_ab_group_add_abs = ordered_ab_group_add + abs +
haftmann@25303
  1230
  assumes abs_ge_zero [simp]: "\<bar>a\<bar> \<ge> 0"
haftmann@25303
  1231
    and abs_ge_self: "a \<le> \<bar>a\<bar>"
haftmann@25303
  1232
    and abs_leI: "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
haftmann@25303
  1233
    and abs_minus_cancel [simp]: "\<bar>-a\<bar> = \<bar>a\<bar>"
haftmann@25303
  1234
    and abs_triangle_ineq: "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@25303
  1235
begin
haftmann@25303
  1236
haftmann@25307
  1237
lemma abs_minus_le_zero: "- \<bar>a\<bar> \<le> 0"
haftmann@25307
  1238
  unfolding neg_le_0_iff_le by simp
haftmann@25307
  1239
haftmann@25307
  1240
lemma abs_of_nonneg [simp]:
nipkow@29667
  1241
  assumes nonneg: "0 \<le> a" shows "\<bar>a\<bar> = a"
haftmann@25307
  1242
proof (rule antisym)
haftmann@25307
  1243
  from nonneg le_imp_neg_le have "- a \<le> 0" by simp
haftmann@25307
  1244
  from this nonneg have "- a \<le> a" by (rule order_trans)
haftmann@25307
  1245
  then show "\<bar>a\<bar> \<le> a" by (auto intro: abs_leI)
haftmann@25307
  1246
qed (rule abs_ge_self)
haftmann@25307
  1247
haftmann@25307
  1248
lemma abs_idempotent [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"
nipkow@29667
  1249
by (rule antisym)
haftmann@36302
  1250
   (auto intro!: abs_ge_self abs_leI order_trans [of "- \<bar>a\<bar>" 0 "\<bar>a\<bar>"])
haftmann@25307
  1251
haftmann@25307
  1252
lemma abs_eq_0 [simp]: "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0"
haftmann@25307
  1253
proof -
haftmann@25307
  1254
  have "\<bar>a\<bar> = 0 \<Longrightarrow> a = 0"
haftmann@25307
  1255
  proof (rule antisym)
haftmann@25307
  1256
    assume zero: "\<bar>a\<bar> = 0"
haftmann@25307
  1257
    with abs_ge_self show "a \<le> 0" by auto
haftmann@25307
  1258
    from zero have "\<bar>-a\<bar> = 0" by simp
haftmann@36302
  1259
    with abs_ge_self [of "- a"] have "- a \<le> 0" by auto
haftmann@25307
  1260
    with neg_le_0_iff_le show "0 \<le> a" by auto
haftmann@25307
  1261
  qed
haftmann@25307
  1262
  then show ?thesis by auto
haftmann@25307
  1263
qed
haftmann@25307
  1264
haftmann@25303
  1265
lemma abs_zero [simp]: "\<bar>0\<bar> = 0"
nipkow@29667
  1266
by simp
avigad@16775
  1267
blanchet@54148
  1268
lemma abs_0_eq [simp]: "0 = \<bar>a\<bar> \<longleftrightarrow> a = 0"
haftmann@25303
  1269
proof -
haftmann@25303
  1270
  have "0 = \<bar>a\<bar> \<longleftrightarrow> \<bar>a\<bar> = 0" by (simp only: eq_ac)
haftmann@25303
  1271
  thus ?thesis by simp
haftmann@25303
  1272
qed
haftmann@25303
  1273
haftmann@25303
  1274
lemma abs_le_zero_iff [simp]: "\<bar>a\<bar> \<le> 0 \<longleftrightarrow> a = 0" 
haftmann@25303
  1275
proof
haftmann@25303
  1276
  assume "\<bar>a\<bar> \<le> 0"
haftmann@25303
  1277
  then have "\<bar>a\<bar> = 0" by (rule antisym) simp
haftmann@25303
  1278
  thus "a = 0" by simp
haftmann@25303
  1279
next
haftmann@25303
  1280
  assume "a = 0"
haftmann@25303
  1281
  thus "\<bar>a\<bar> \<le> 0" by simp
haftmann@25303
  1282
qed
haftmann@25303
  1283
haftmann@25303
  1284
lemma zero_less_abs_iff [simp]: "0 < \<bar>a\<bar> \<longleftrightarrow> a \<noteq> 0"
nipkow@29667
  1285
by (simp add: less_le)
haftmann@25303
  1286
haftmann@25303
  1287
lemma abs_not_less_zero [simp]: "\<not> \<bar>a\<bar> < 0"
haftmann@25303
  1288
proof -
haftmann@25303
  1289
  have a: "\<And>x y. x \<le> y \<Longrightarrow> \<not> y < x" by auto
haftmann@25303
  1290
  show ?thesis by (simp add: a)
haftmann@25303
  1291
qed
avigad@16775
  1292
haftmann@25303
  1293
lemma abs_ge_minus_self: "- a \<le> \<bar>a\<bar>"
haftmann@25303
  1294
proof -
haftmann@25303
  1295
  have "- a \<le> \<bar>-a\<bar>" by (rule abs_ge_self)
haftmann@25303
  1296
  then show ?thesis by simp
haftmann@25303
  1297
qed
haftmann@25303
  1298
haftmann@25303
  1299
lemma abs_minus_commute: 
haftmann@25303
  1300
  "\<bar>a - b\<bar> = \<bar>b - a\<bar>"
haftmann@25303
  1301
proof -
haftmann@25303
  1302
  have "\<bar>a - b\<bar> = \<bar>- (a - b)\<bar>" by (simp only: abs_minus_cancel)
haftmann@25303
  1303
  also have "... = \<bar>b - a\<bar>" by simp
haftmann@25303
  1304
  finally show ?thesis .
haftmann@25303
  1305
qed
haftmann@25303
  1306
haftmann@25303
  1307
lemma abs_of_pos: "0 < a \<Longrightarrow> \<bar>a\<bar> = a"
nipkow@29667
  1308
by (rule abs_of_nonneg, rule less_imp_le)
avigad@16775
  1309
haftmann@25303
  1310
lemma abs_of_nonpos [simp]:
nipkow@29667
  1311
  assumes "a \<le> 0" shows "\<bar>a\<bar> = - a"
haftmann@25303
  1312
proof -
haftmann@25303
  1313
  let ?b = "- a"
haftmann@25303
  1314
  have "- ?b \<le> 0 \<Longrightarrow> \<bar>- ?b\<bar> = - (- ?b)"
haftmann@25303
  1315
  unfolding abs_minus_cancel [of "?b"]
haftmann@25303
  1316
  unfolding neg_le_0_iff_le [of "?b"]
haftmann@25303
  1317
  unfolding minus_minus by (erule abs_of_nonneg)
haftmann@25303
  1318
  then show ?thesis using assms by auto
haftmann@25303
  1319
qed
haftmann@25303
  1320
  
haftmann@25303
  1321
lemma abs_of_neg: "a < 0 \<Longrightarrow> \<bar>a\<bar> = - a"
nipkow@29667
  1322
by (rule abs_of_nonpos, rule less_imp_le)
haftmann@25303
  1323
haftmann@25303
  1324
lemma abs_le_D1: "\<bar>a\<bar> \<le> b \<Longrightarrow> a \<le> b"
nipkow@29667
  1325
by (insert abs_ge_self, blast intro: order_trans)
haftmann@25303
  1326
haftmann@25303
  1327
lemma abs_le_D2: "\<bar>a\<bar> \<le> b \<Longrightarrow> - a \<le> b"
haftmann@36302
  1328
by (insert abs_le_D1 [of "- a"], simp)
haftmann@25303
  1329
haftmann@25303
  1330
lemma abs_le_iff: "\<bar>a\<bar> \<le> b \<longleftrightarrow> a \<le> b \<and> - a \<le> b"
nipkow@29667
  1331
by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)
haftmann@25303
  1332
haftmann@25303
  1333
lemma abs_triangle_ineq2: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>a - b\<bar>"
haftmann@36302
  1334
proof -
haftmann@36302
  1335
  have "\<bar>a\<bar> = \<bar>b + (a - b)\<bar>"
haftmann@54230
  1336
    by (simp add: algebra_simps)
haftmann@36302
  1337
  then have "\<bar>a\<bar> \<le> \<bar>b\<bar> + \<bar>a - b\<bar>"
haftmann@36302
  1338
    by (simp add: abs_triangle_ineq)
haftmann@36302
  1339
  then show ?thesis
haftmann@36302
  1340
    by (simp add: algebra_simps)
haftmann@36302
  1341
qed
haftmann@36302
  1342
haftmann@36302
  1343
lemma abs_triangle_ineq2_sym: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>b - a\<bar>"
haftmann@36302
  1344
  by (simp only: abs_minus_commute [of b] abs_triangle_ineq2)
avigad@16775
  1345
haftmann@25303
  1346
lemma abs_triangle_ineq3: "\<bar>\<bar>a\<bar> - \<bar>b\<bar>\<bar> \<le> \<bar>a - b\<bar>"
haftmann@36302
  1347
  by (simp add: abs_le_iff abs_triangle_ineq2 abs_triangle_ineq2_sym)
avigad@16775
  1348
haftmann@25303
  1349
lemma abs_triangle_ineq4: "\<bar>a - b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@25303
  1350
proof -
haftmann@54230
  1351
  have "\<bar>a - b\<bar> = \<bar>a + - b\<bar>" by (simp add: algebra_simps)
haftmann@36302
  1352
  also have "... \<le> \<bar>a\<bar> + \<bar>- b\<bar>" by (rule abs_triangle_ineq)
nipkow@29667
  1353
  finally show ?thesis by simp
haftmann@25303
  1354
qed
avigad@16775
  1355
haftmann@25303
  1356
lemma abs_diff_triangle_ineq: "\<bar>a + b - (c + d)\<bar> \<le> \<bar>a - c\<bar> + \<bar>b - d\<bar>"
haftmann@25303
  1357
proof -
haftmann@54230
  1358
  have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: algebra_simps)
haftmann@25303
  1359
  also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq)
haftmann@25303
  1360
  finally show ?thesis .
haftmann@25303
  1361
qed
avigad@16775
  1362
haftmann@25303
  1363
lemma abs_add_abs [simp]:
haftmann@25303
  1364
  "\<bar>\<bar>a\<bar> + \<bar>b\<bar>\<bar> = \<bar>a\<bar> + \<bar>b\<bar>" (is "?L = ?R")
haftmann@25303
  1365
proof (rule antisym)
haftmann@25303
  1366
  show "?L \<ge> ?R" by(rule abs_ge_self)
haftmann@25303
  1367
next
haftmann@25303
  1368
  have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by(rule abs_triangle_ineq)
haftmann@25303
  1369
  also have "\<dots> = ?R" by simp
haftmann@25303
  1370
  finally show "?L \<le> ?R" .
haftmann@25303
  1371
qed
haftmann@25303
  1372
haftmann@25303
  1373
end
obua@14738
  1374
paulson@60762
  1375
lemma dense_eq0_I:
paulson@60762
  1376
  fixes x::"'a::{dense_linorder,ordered_ab_group_add_abs}"
paulson@60762
  1377
  shows "(\<And>e. 0 < e \<Longrightarrow> \<bar>x\<bar> \<le> e) ==> x = 0"
paulson@60762
  1378
  apply (cases "abs x=0", simp)
paulson@60762
  1379
  apply (simp only: zero_less_abs_iff [symmetric])
paulson@60762
  1380
  apply (drule dense)
paulson@60762
  1381
  apply (auto simp add: not_less [symmetric])
paulson@60762
  1382
  done
paulson@60762
  1383
haftmann@59815
  1384
hide_fact (open) ab_diff_conv_add_uminus add_0 mult_1 ab_left_minus
haftmann@59815
  1385
haftmann@59815
  1386
lemmas add_0 = add_0_left -- \<open>FIXME duplicate\<close>
haftmann@59815
  1387
lemmas mult_1 = mult_1_left -- \<open>FIXME duplicate\<close>
haftmann@59815
  1388
lemmas ab_left_minus = left_minus -- \<open>FIXME duplicate\<close>
haftmann@59815
  1389
lemmas diff_diff_eq = diff_diff_add -- \<open>FIXME duplicate\<close>
haftmann@59815
  1390
obua@15178
  1391
wenzelm@60758
  1392
subsection \<open>Tools setup\<close>
haftmann@25090
  1393
blanchet@54147
  1394
lemma add_mono_thms_linordered_semiring:
wenzelm@61076
  1395
  fixes i j k :: "'a::ordered_ab_semigroup_add"
haftmann@25077
  1396
  shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077
  1397
    and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077
  1398
    and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077
  1399
    and "i = j \<and> k = l \<Longrightarrow> i + k = j + l"
haftmann@25077
  1400
by (rule add_mono, clarify+)+
haftmann@25077
  1401
blanchet@54147
  1402
lemma add_mono_thms_linordered_field:
wenzelm@61076
  1403
  fixes i j k :: "'a::ordered_cancel_ab_semigroup_add"
haftmann@25077
  1404
  shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1405
    and "i = j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1406
    and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1407
    and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1408
    and "i < j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1409
by (auto intro: add_strict_right_mono add_strict_left_mono
haftmann@25077
  1410
  add_less_le_mono add_le_less_mono add_strict_mono)
haftmann@25077
  1411
haftmann@52435
  1412
code_identifier
haftmann@52435
  1413
  code_module Groups \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
  1414
obua@14738
  1415
end
haftmann@49388
  1416