src/HOL/Product_Type.thy
author haftmann
Mon Jun 06 21:28:45 2016 +0200 (2016-06-06)
changeset 63237 3e908f762817
parent 63007 aa894a49f77d
child 63399 d1742d1b7f0f
permissions -rw-r--r--
conventional syntax for unit abstractions
haftmann@58469
     1
(*  Title:      HOL/Product_Type.thy
nipkow@10213
     2
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
nipkow@10213
     3
    Copyright   1992  University of Cambridge
wenzelm@11777
     4
*)
nipkow@10213
     5
wenzelm@60758
     6
section \<open>Cartesian products\<close>
nipkow@10213
     7
nipkow@15131
     8
theory Product_Type
haftmann@33959
     9
imports Typedef Inductive Fun
wenzelm@46950
    10
keywords "inductive_set" "coinductive_set" :: thy_decl
nipkow@15131
    11
begin
wenzelm@11838
    12
wenzelm@60758
    13
subsection \<open>@{typ bool} is a datatype\<close>
haftmann@24699
    14
blanchet@62594
    15
free_constructors (discs_sels) case_bool for True | False
blanchet@58189
    16
  by auto
blanchet@55393
    17
wenzelm@61799
    18
text \<open>Avoid name clashes by prefixing the output of \<open>old_rep_datatype\<close> with \<open>old\<close>.\<close>
blanchet@55442
    19
wenzelm@60758
    20
setup \<open>Sign.mandatory_path "old"\<close>
blanchet@55393
    21
blanchet@58306
    22
old_rep_datatype True False by (auto intro: bool_induct)
haftmann@24699
    23
wenzelm@60758
    24
setup \<open>Sign.parent_path\<close>
blanchet@55393
    25
wenzelm@61799
    26
text \<open>But erase the prefix for properties that are not generated by \<open>free_constructors\<close>.\<close>
blanchet@55442
    27
wenzelm@60758
    28
setup \<open>Sign.mandatory_path "bool"\<close>
blanchet@55393
    29
blanchet@55393
    30
lemmas induct = old.bool.induct
blanchet@55393
    31
lemmas inducts = old.bool.inducts
blanchet@55642
    32
lemmas rec = old.bool.rec
blanchet@55642
    33
lemmas simps = bool.distinct bool.case bool.rec
blanchet@55393
    34
wenzelm@60758
    35
setup \<open>Sign.parent_path\<close>
blanchet@55393
    36
haftmann@24699
    37
declare case_split [cases type: bool]
wenzelm@61799
    38
  \<comment> "prefer plain propositional version"
haftmann@24699
    39
haftmann@28346
    40
lemma
haftmann@38857
    41
  shows [code]: "HOL.equal False P \<longleftrightarrow> \<not> P"
haftmann@38857
    42
    and [code]: "HOL.equal True P \<longleftrightarrow> P" 
haftmann@46630
    43
    and [code]: "HOL.equal P False \<longleftrightarrow> \<not> P"
haftmann@38857
    44
    and [code]: "HOL.equal P True \<longleftrightarrow> P"
haftmann@38857
    45
    and [code nbe]: "HOL.equal P P \<longleftrightarrow> True"
haftmann@38857
    46
  by (simp_all add: equal)
haftmann@25534
    47
haftmann@43654
    48
lemma If_case_cert:
haftmann@43654
    49
  assumes "CASE \<equiv> (\<lambda>b. If b f g)"
haftmann@43654
    50
  shows "(CASE True \<equiv> f) &&& (CASE False \<equiv> g)"
haftmann@43654
    51
  using assms by simp_all
haftmann@43654
    52
wenzelm@60758
    53
setup \<open>Code.add_case @{thm If_case_cert}\<close>
haftmann@43654
    54
haftmann@52435
    55
code_printing
haftmann@52435
    56
  constant "HOL.equal :: bool \<Rightarrow> bool \<Rightarrow> bool" \<rightharpoonup> (Haskell) infix 4 "=="
haftmann@52435
    57
| class_instance "bool" :: "equal" \<rightharpoonup> (Haskell) -
haftmann@24699
    58
haftmann@26358
    59
wenzelm@61799
    60
subsection \<open>The \<open>unit\<close> type\<close>
wenzelm@11838
    61
wenzelm@49834
    62
typedef unit = "{True}"
wenzelm@45694
    63
  by auto
wenzelm@11838
    64
wenzelm@45694
    65
definition Unity :: unit  ("'(')")
wenzelm@45694
    66
  where "() = Abs_unit True"
wenzelm@11838
    67
blanchet@35828
    68
lemma unit_eq [no_atp]: "u = ()"
huffman@40590
    69
  by (induct u) (simp add: Unity_def)
wenzelm@11838
    70
wenzelm@60758
    71
text \<open>
wenzelm@11838
    72
  Simplification procedure for @{thm [source] unit_eq}.  Cannot use
wenzelm@11838
    73
  this rule directly --- it loops!
wenzelm@60758
    74
\<close>
wenzelm@11838
    75
wenzelm@60758
    76
simproc_setup unit_eq ("x::unit") = \<open>
wenzelm@43594
    77
  fn _ => fn _ => fn ct =>
wenzelm@59582
    78
    if HOLogic.is_unit (Thm.term_of ct) then NONE
wenzelm@43594
    79
    else SOME (mk_meta_eq @{thm unit_eq})
wenzelm@60758
    80
\<close>
wenzelm@11838
    81
blanchet@55469
    82
free_constructors case_unit for "()"
blanchet@58189
    83
  by auto
blanchet@55393
    84
wenzelm@61799
    85
text \<open>Avoid name clashes by prefixing the output of \<open>old_rep_datatype\<close> with \<open>old\<close>.\<close>
blanchet@55442
    86
wenzelm@60758
    87
setup \<open>Sign.mandatory_path "old"\<close>
blanchet@55393
    88
blanchet@58306
    89
old_rep_datatype "()" by simp
haftmann@24699
    90
wenzelm@60758
    91
setup \<open>Sign.parent_path\<close>
blanchet@55393
    92
wenzelm@61799
    93
text \<open>But erase the prefix for properties that are not generated by \<open>free_constructors\<close>.\<close>
blanchet@55442
    94
wenzelm@60758
    95
setup \<open>Sign.mandatory_path "unit"\<close>
blanchet@55393
    96
blanchet@55393
    97
lemmas induct = old.unit.induct
blanchet@55393
    98
lemmas inducts = old.unit.inducts
blanchet@55642
    99
lemmas rec = old.unit.rec
blanchet@55642
   100
lemmas simps = unit.case unit.rec
blanchet@55393
   101
wenzelm@60758
   102
setup \<open>Sign.parent_path\<close>
blanchet@55393
   103
wenzelm@11838
   104
lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()"
wenzelm@11838
   105
  by simp
wenzelm@11838
   106
wenzelm@11838
   107
lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P"
wenzelm@11838
   108
  by (rule triv_forall_equality)
wenzelm@11838
   109
wenzelm@60758
   110
text \<open>
wenzelm@61799
   111
  This rewrite counters the effect of simproc \<open>unit_eq\<close> on @{term
wenzelm@11838
   112
  [source] "%u::unit. f u"}, replacing it by @{term [source]
wenzelm@11838
   113
  f} rather than by @{term [source] "%u. f ()"}.
wenzelm@60758
   114
\<close>
wenzelm@11838
   115
blanchet@54147
   116
lemma unit_abs_eta_conv [simp]: "(%u::unit. f ()) = f"
wenzelm@11838
   117
  by (rule ext) simp
nipkow@10213
   118
blanchet@54147
   119
lemma UNIV_unit:
haftmann@43866
   120
  "UNIV = {()}" by auto
haftmann@43866
   121
haftmann@30924
   122
instantiation unit :: default
haftmann@30924
   123
begin
haftmann@30924
   124
haftmann@30924
   125
definition "default = ()"
haftmann@30924
   126
haftmann@30924
   127
instance ..
haftmann@30924
   128
haftmann@30924
   129
end
nipkow@10213
   130
haftmann@57233
   131
instantiation unit :: "{complete_boolean_algebra, complete_linorder, wellorder}"
nipkow@57016
   132
begin
nipkow@57016
   133
haftmann@57233
   134
definition less_eq_unit :: "unit \<Rightarrow> unit \<Rightarrow> bool"
haftmann@57233
   135
where
haftmann@57233
   136
  "(_::unit) \<le> _ \<longleftrightarrow> True"
haftmann@57233
   137
haftmann@57233
   138
lemma less_eq_unit [iff]:
haftmann@57233
   139
  "(u::unit) \<le> v"
haftmann@57233
   140
  by (simp add: less_eq_unit_def)
haftmann@57233
   141
haftmann@57233
   142
definition less_unit :: "unit \<Rightarrow> unit \<Rightarrow> bool"
haftmann@57233
   143
where
haftmann@57233
   144
  "(_::unit) < _ \<longleftrightarrow> False"
haftmann@57233
   145
haftmann@57233
   146
lemma less_unit [iff]:
haftmann@57233
   147
  "\<not> (u::unit) < v"
haftmann@57233
   148
  by (simp_all add: less_eq_unit_def less_unit_def)
haftmann@57233
   149
haftmann@57233
   150
definition bot_unit :: unit
haftmann@57233
   151
where
haftmann@57233
   152
  [code_unfold]: "\<bottom> = ()"
haftmann@57233
   153
haftmann@57233
   154
definition top_unit :: unit
haftmann@57233
   155
where
haftmann@57233
   156
  [code_unfold]: "\<top> = ()"
nipkow@57016
   157
haftmann@57233
   158
definition inf_unit :: "unit \<Rightarrow> unit \<Rightarrow> unit"
haftmann@57233
   159
where
haftmann@57233
   160
  [simp]: "_ \<sqinter> _ = ()"
haftmann@57233
   161
haftmann@57233
   162
definition sup_unit :: "unit \<Rightarrow> unit \<Rightarrow> unit"
haftmann@57233
   163
where
haftmann@57233
   164
  [simp]: "_ \<squnion> _ = ()"
haftmann@57233
   165
haftmann@57233
   166
definition Inf_unit :: "unit set \<Rightarrow> unit"
haftmann@57233
   167
where
haftmann@57233
   168
  [simp]: "\<Sqinter>_ = ()"
nipkow@57016
   169
haftmann@57233
   170
definition Sup_unit :: "unit set \<Rightarrow> unit"
haftmann@57233
   171
where
haftmann@57233
   172
  [simp]: "\<Squnion>_ = ()"
haftmann@57233
   173
haftmann@57233
   174
definition uminus_unit :: "unit \<Rightarrow> unit"
haftmann@57233
   175
where
haftmann@57233
   176
  [simp]: "- _ = ()"
haftmann@57233
   177
haftmann@57233
   178
declare less_eq_unit_def [abs_def, code_unfold]
haftmann@57233
   179
  less_unit_def [abs_def, code_unfold]
haftmann@57233
   180
  inf_unit_def [abs_def, code_unfold]
haftmann@57233
   181
  sup_unit_def [abs_def, code_unfold]
haftmann@57233
   182
  Inf_unit_def [abs_def, code_unfold]
haftmann@57233
   183
  Sup_unit_def [abs_def, code_unfold]
haftmann@57233
   184
  uminus_unit_def [abs_def, code_unfold]
nipkow@57016
   185
nipkow@57016
   186
instance
haftmann@57233
   187
  by intro_classes auto
nipkow@57016
   188
nipkow@57016
   189
end
nipkow@57016
   190
haftmann@28562
   191
lemma [code]:
wenzelm@61076
   192
  "HOL.equal (u::unit) v \<longleftrightarrow> True" unfolding equal unit_eq [of u] unit_eq [of v] by rule+
haftmann@26358
   193
haftmann@52435
   194
code_printing
haftmann@52435
   195
  type_constructor unit \<rightharpoonup>
haftmann@52435
   196
    (SML) "unit"
haftmann@52435
   197
    and (OCaml) "unit"
haftmann@52435
   198
    and (Haskell) "()"
haftmann@52435
   199
    and (Scala) "Unit"
haftmann@52435
   200
| constant Unity \<rightharpoonup>
haftmann@52435
   201
    (SML) "()"
haftmann@52435
   202
    and (OCaml) "()"
haftmann@52435
   203
    and (Haskell) "()"
haftmann@52435
   204
    and (Scala) "()"
haftmann@52435
   205
| class_instance unit :: equal \<rightharpoonup>
haftmann@52435
   206
    (Haskell) -
haftmann@52435
   207
| constant "HOL.equal :: unit \<Rightarrow> unit \<Rightarrow> bool" \<rightharpoonup>
haftmann@52435
   208
    (Haskell) infix 4 "=="
haftmann@26358
   209
haftmann@26358
   210
code_reserved SML
haftmann@26358
   211
  unit
haftmann@26358
   212
haftmann@26358
   213
code_reserved OCaml
haftmann@26358
   214
  unit
haftmann@26358
   215
haftmann@34886
   216
code_reserved Scala
haftmann@34886
   217
  Unit
haftmann@34886
   218
haftmann@26358
   219
wenzelm@60758
   220
subsection \<open>The product type\<close>
nipkow@10213
   221
wenzelm@60758
   222
subsubsection \<open>Type definition\<close>
haftmann@37166
   223
haftmann@37166
   224
definition Pair_Rep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" where
haftmann@26358
   225
  "Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)"
nipkow@10213
   226
wenzelm@61076
   227
definition "prod = {f. \<exists>a b. f = Pair_Rep (a::'a) (b::'b)}"
wenzelm@45696
   228
wenzelm@61955
   229
typedef ('a, 'b) prod ("(_ \<times>/ _)" [21, 20] 20) = "prod :: ('a \<Rightarrow> 'b \<Rightarrow> bool) set"
wenzelm@45696
   230
  unfolding prod_def by auto
nipkow@10213
   231
wenzelm@61955
   232
type_notation (ASCII)
wenzelm@61955
   233
  prod  (infixr "*" 20)
nipkow@10213
   234
haftmann@37389
   235
definition Pair :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b" where
haftmann@37389
   236
  "Pair a b = Abs_prod (Pair_Rep a b)"
haftmann@37166
   237
blanchet@55393
   238
lemma prod_cases: "(\<And>a b. P (Pair a b)) \<Longrightarrow> P p"
blanchet@55393
   239
  by (cases p) (auto simp add: prod_def Pair_def Pair_Rep_def)
blanchet@55393
   240
haftmann@61424
   241
free_constructors case_prod for Pair fst snd
blanchet@55393
   242
proof -
blanchet@55393
   243
  fix P :: bool and p :: "'a \<times> 'b"
blanchet@55393
   244
  show "(\<And>x1 x2. p = Pair x1 x2 \<Longrightarrow> P) \<Longrightarrow> P"
blanchet@55393
   245
    by (cases p) (auto simp add: prod_def Pair_def Pair_Rep_def)
haftmann@37166
   246
next
haftmann@37166
   247
  fix a c :: 'a and b d :: 'b
haftmann@37166
   248
  have "Pair_Rep a b = Pair_Rep c d \<longleftrightarrow> a = c \<and> b = d"
nipkow@39302
   249
    by (auto simp add: Pair_Rep_def fun_eq_iff)
haftmann@37389
   250
  moreover have "Pair_Rep a b \<in> prod" and "Pair_Rep c d \<in> prod"
haftmann@37389
   251
    by (auto simp add: prod_def)
haftmann@37166
   252
  ultimately show "Pair a b = Pair c d \<longleftrightarrow> a = c \<and> b = d"
haftmann@37389
   253
    by (simp add: Pair_def Abs_prod_inject)
haftmann@37166
   254
qed
haftmann@37166
   255
wenzelm@61799
   256
text \<open>Avoid name clashes by prefixing the output of \<open>old_rep_datatype\<close> with \<open>old\<close>.\<close>
blanchet@55442
   257
wenzelm@60758
   258
setup \<open>Sign.mandatory_path "old"\<close>
blanchet@55393
   259
blanchet@58306
   260
old_rep_datatype Pair
blanchet@55403
   261
by (erule prod_cases) (rule prod.inject)
blanchet@55393
   262
wenzelm@60758
   263
setup \<open>Sign.parent_path\<close>
blanchet@37704
   264
wenzelm@61799
   265
text \<open>But erase the prefix for properties that are not generated by \<open>free_constructors\<close>.\<close>
blanchet@55442
   266
wenzelm@60758
   267
setup \<open>Sign.mandatory_path "prod"\<close>
blanchet@55393
   268
haftmann@61424
   269
declare old.prod.inject [iff del]
blanchet@55393
   270
blanchet@55393
   271
lemmas induct = old.prod.induct
blanchet@55393
   272
lemmas inducts = old.prod.inducts
blanchet@55642
   273
lemmas rec = old.prod.rec
blanchet@55642
   274
lemmas simps = prod.inject prod.case prod.rec
blanchet@55393
   275
wenzelm@60758
   276
setup \<open>Sign.parent_path\<close>
blanchet@55393
   277
blanchet@55393
   278
declare prod.case [nitpick_simp del]
blanchet@57983
   279
declare prod.case_cong_weak [cong del]
haftmann@61424
   280
declare prod.case_eq_if [mono]
haftmann@61424
   281
declare prod.split [no_atp]
haftmann@61424
   282
declare prod.split_asm [no_atp]
haftmann@61424
   283
haftmann@61424
   284
text \<open>
wenzelm@61799
   285
  @{thm [source] prod.split} could be declared as \<open>[split]\<close>
haftmann@61424
   286
  done after the Splitter has been speeded up significantly;
haftmann@61424
   287
  precompute the constants involved and don't do anything unless the
haftmann@61424
   288
  current goal contains one of those constants.
haftmann@61424
   289
\<close>
haftmann@37411
   290
haftmann@37166
   291
wenzelm@60758
   292
subsubsection \<open>Tuple syntax\<close>
haftmann@37166
   293
wenzelm@60758
   294
text \<open>
wenzelm@11777
   295
  Patterns -- extends pre-defined type @{typ pttrn} used in
wenzelm@11777
   296
  abstractions.
wenzelm@60758
   297
\<close>
nipkow@10213
   298
wenzelm@41229
   299
nonterminal tuple_args and patterns
nipkow@10213
   300
nipkow@10213
   301
syntax
nipkow@10213
   302
  "_tuple"      :: "'a => tuple_args => 'a * 'b"        ("(1'(_,/ _'))")
nipkow@10213
   303
  "_tuple_arg"  :: "'a => tuple_args"                   ("_")
nipkow@10213
   304
  "_tuple_args" :: "'a => tuple_args => tuple_args"     ("_,/ _")
oheimb@11025
   305
  "_pattern"    :: "[pttrn, patterns] => pttrn"         ("'(_,/ _')")
oheimb@11025
   306
  ""            :: "pttrn => patterns"                  ("_")
oheimb@11025
   307
  "_patterns"   :: "[pttrn, patterns] => patterns"      ("_,/ _")
haftmann@63237
   308
  "_unit"       :: pttrn                                ("'(')")
nipkow@10213
   309
nipkow@10213
   310
translations
haftmann@61124
   311
  "(x, y)" \<rightleftharpoons> "CONST Pair x y"
haftmann@61124
   312
  "_pattern x y" \<rightleftharpoons> "CONST Pair x y"
haftmann@61124
   313
  "_patterns x y" \<rightleftharpoons> "CONST Pair x y"
haftmann@61124
   314
  "_tuple x (_tuple_args y z)" \<rightleftharpoons> "_tuple x (_tuple_arg (_tuple y z))"
haftmann@61424
   315
  "\<lambda>(x, y, zs). b" \<rightleftharpoons> "CONST case_prod (\<lambda>x (y, zs). b)"
haftmann@61424
   316
  "\<lambda>(x, y). b" \<rightleftharpoons> "CONST case_prod (\<lambda>x y. b)"
haftmann@61124
   317
  "_abs (CONST Pair x y) t" \<rightharpoonup> "\<lambda>(x, y). t"
wenzelm@61799
   318
  \<comment> \<open>This rule accommodates tuples in \<open>case C \<dots> (x, y) \<dots> \<Rightarrow> \<dots>\<close>:
wenzelm@61799
   319
     The \<open>(x, y)\<close> is parsed as \<open>Pair x y\<close> because it is \<open>logic\<close>,
wenzelm@61799
   320
     not \<open>pttrn\<close>.\<close>
haftmann@63237
   321
  "\<lambda>(). b" \<rightleftharpoons> "CONST case_unit b"
haftmann@63237
   322
  "_abs (CONST Unity) t" \<rightharpoonup> "\<lambda>(). t"
nipkow@10213
   323
haftmann@61424
   324
text \<open>print @{term "case_prod f"} as @{term "\<lambda>(x, y). f x y"} and
haftmann@61424
   325
  @{term "case_prod (\<lambda>x. f x)"} as @{term "\<lambda>(x, y). f x y"}\<close>
haftmann@61226
   326
haftmann@61226
   327
typed_print_translation \<open>
haftmann@61226
   328
  let
haftmann@61424
   329
    fun case_prod_guess_names_tr' T [Abs (x, _, Abs _)] = raise Match
haftmann@61424
   330
      | case_prod_guess_names_tr' T [Abs (x, xT, t)] =
haftmann@61226
   331
          (case (head_of t) of
haftmann@61424
   332
            Const (@{const_syntax case_prod}, _) => raise Match
haftmann@61226
   333
          | _ =>
haftmann@61226
   334
            let 
haftmann@61226
   335
              val (_ :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match;
haftmann@61226
   336
              val (y, t') = Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 1 t $ Bound 0);
haftmann@61226
   337
              val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, xT, t');
haftmann@61226
   338
            in
haftmann@61226
   339
              Syntax.const @{syntax_const "_abs"} $
haftmann@61226
   340
                (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
haftmann@61226
   341
            end)
haftmann@61424
   342
      | case_prod_guess_names_tr' T [t] =
haftmann@61226
   343
          (case head_of t of
haftmann@61424
   344
            Const (@{const_syntax case_prod}, _) => raise Match
haftmann@61226
   345
          | _ =>
haftmann@61226
   346
            let
haftmann@61226
   347
              val (xT :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match;
haftmann@61226
   348
              val (y, t') =
haftmann@61226
   349
                Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 2 t $ Bound 1 $ Bound 0);
haftmann@61226
   350
              val (x', t'') = Syntax_Trans.atomic_abs_tr' ("x", xT, t');
haftmann@61226
   351
            in
haftmann@61226
   352
              Syntax.const @{syntax_const "_abs"} $
haftmann@61226
   353
                (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
haftmann@61226
   354
            end)
haftmann@61424
   355
      | case_prod_guess_names_tr' _ _ = raise Match;
haftmann@61424
   356
  in [(@{const_syntax case_prod}, K case_prod_guess_names_tr')] end
haftmann@61226
   357
\<close>
haftmann@61226
   358
haftmann@61425
   359
text \<open>reconstruct pattern from (nested) @{const case_prod}s,
haftmann@61425
   360
  avoiding eta-contraction of body; required for enclosing "let",
haftmann@61425
   361
  if "let" does not avoid eta-contraction, which has been observed to occur\<close>
haftmann@61425
   362
haftmann@61425
   363
print_translation \<open>
haftmann@61425
   364
  let
haftmann@61425
   365
    fun case_prod_tr' [Abs (x, T, t as (Abs abs))] =
haftmann@61425
   366
          (* case_prod (\<lambda>x y. t) \<Rightarrow> \<lambda>(x, y) t *)
haftmann@61425
   367
          let
haftmann@61425
   368
            val (y, t') = Syntax_Trans.atomic_abs_tr' abs;
haftmann@61425
   369
            val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t');
haftmann@61425
   370
          in
haftmann@61425
   371
            Syntax.const @{syntax_const "_abs"} $
haftmann@61425
   372
              (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
haftmann@61425
   373
          end
haftmann@61425
   374
      | case_prod_tr' [Abs (x, T, (s as Const (@{const_syntax case_prod}, _) $ t))] =
haftmann@61425
   375
          (* case_prod (\<lambda>x. (case_prod (\<lambda>y z. t))) \<Rightarrow> \<lambda>(x, y, z). t *)
haftmann@61425
   376
          let
haftmann@61425
   377
            val Const (@{syntax_const "_abs"}, _) $
haftmann@61425
   378
              (Const (@{syntax_const "_pattern"}, _) $ y $ z) $ t' =
haftmann@61425
   379
                case_prod_tr' [t];
haftmann@61425
   380
            val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t');
haftmann@61425
   381
          in
haftmann@61425
   382
            Syntax.const @{syntax_const "_abs"} $
haftmann@61425
   383
              (Syntax.const @{syntax_const "_pattern"} $ x' $
haftmann@61425
   384
                (Syntax.const @{syntax_const "_patterns"} $ y $ z)) $ t''
haftmann@61425
   385
          end
haftmann@61425
   386
      | case_prod_tr' [Const (@{const_syntax case_prod}, _) $ t] =
haftmann@61425
   387
          (* case_prod (case_prod (\<lambda>x y z. t)) \<Rightarrow> \<lambda>((x, y), z). t *)
haftmann@61425
   388
          case_prod_tr' [(case_prod_tr' [t])]
haftmann@61425
   389
            (* inner case_prod_tr' creates next pattern *)
haftmann@61425
   390
      | case_prod_tr' [Const (@{syntax_const "_abs"}, _) $ x_y $ Abs abs] =
haftmann@61425
   391
          (* case_prod (\<lambda>pttrn z. t) \<Rightarrow> \<lambda>(pttrn, z). t *)
haftmann@61425
   392
          let val (z, t) = Syntax_Trans.atomic_abs_tr' abs in
haftmann@61425
   393
            Syntax.const @{syntax_const "_abs"} $
haftmann@61425
   394
              (Syntax.const @{syntax_const "_pattern"} $ x_y $ z) $ t
haftmann@61425
   395
          end
haftmann@61425
   396
      | case_prod_tr' _ = raise Match;
haftmann@61425
   397
  in [(@{const_syntax case_prod}, K case_prod_tr')] end
haftmann@61425
   398
\<close>
haftmann@61425
   399
nipkow@10213
   400
wenzelm@60758
   401
subsubsection \<open>Code generator setup\<close>
haftmann@37166
   402
haftmann@52435
   403
code_printing
haftmann@52435
   404
  type_constructor prod \<rightharpoonup>
haftmann@52435
   405
    (SML) infix 2 "*"
haftmann@52435
   406
    and (OCaml) infix 2 "*"
haftmann@52435
   407
    and (Haskell) "!((_),/ (_))"
haftmann@52435
   408
    and (Scala) "((_),/ (_))"
haftmann@52435
   409
| constant Pair \<rightharpoonup>
haftmann@52435
   410
    (SML) "!((_),/ (_))"
haftmann@52435
   411
    and (OCaml) "!((_),/ (_))"
haftmann@52435
   412
    and (Haskell) "!((_),/ (_))"
haftmann@52435
   413
    and (Scala) "!((_),/ (_))"
haftmann@52435
   414
| class_instance  prod :: equal \<rightharpoonup>
haftmann@52435
   415
    (Haskell) -
haftmann@52435
   416
| constant "HOL.equal :: 'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool" \<rightharpoonup>
haftmann@52435
   417
    (Haskell) infix 4 "=="
haftmann@61424
   418
| constant fst \<rightharpoonup> (Haskell) "fst"
haftmann@61424
   419
| constant snd \<rightharpoonup> (Haskell) "snd"
haftmann@37166
   420
haftmann@37166
   421
wenzelm@60758
   422
subsubsection \<open>Fundamental operations and properties\<close>
wenzelm@11838
   423
bulwahn@49897
   424
lemma Pair_inject:
bulwahn@49897
   425
  assumes "(a, b) = (a', b')"
haftmann@61424
   426
    and "a = a' \<Longrightarrow> b = b' \<Longrightarrow> R"
bulwahn@49897
   427
  shows R
bulwahn@49897
   428
  using assms by simp
bulwahn@49897
   429
haftmann@26358
   430
lemma surj_pair [simp]: "EX x y. p = (x, y)"
haftmann@37166
   431
  by (cases p) simp
nipkow@10213
   432
wenzelm@11838
   433
lemma fst_eqD: "fst (x, y) = a ==> x = a"
wenzelm@11838
   434
  by simp
wenzelm@11838
   435
wenzelm@11838
   436
lemma snd_eqD: "snd (x, y) = a ==> y = a"
wenzelm@11838
   437
  by simp
wenzelm@11838
   438
haftmann@61424
   439
lemma case_prod_unfold [nitpick_unfold]: "case_prod = (\<lambda>c p. c (fst p) (snd p))"
haftmann@61424
   440
  by (simp add: fun_eq_iff split: prod.split)
haftmann@61424
   441
haftmann@61424
   442
lemma case_prod_conv [simp, code]: "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
haftmann@61424
   443
  by (fact prod.case)
haftmann@61424
   444
blanchet@55393
   445
lemmas surjective_pairing = prod.collapse [symmetric]
wenzelm@11838
   446
huffman@44066
   447
lemma prod_eq_iff: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t"
haftmann@37166
   448
  by (cases s, cases t) simp
haftmann@37166
   449
haftmann@37166
   450
lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q"
huffman@44066
   451
  by (simp add: prod_eq_iff)
haftmann@37166
   452
haftmann@61424
   453
lemma case_prodI: "f a b \<Longrightarrow> case (a, b) of (c, d) \<Rightarrow> f c d"
haftmann@61424
   454
  by (rule prod.case [THEN iffD2])
haftmann@37166
   455
haftmann@61424
   456
lemma case_prodD: "(case (a, b) of (c, d) \<Rightarrow> f c d) \<Longrightarrow> f a b"
haftmann@61424
   457
  by (rule prod.case [THEN iffD1])
haftmann@37166
   458
haftmann@61424
   459
lemma case_prod_Pair [simp]: "case_prod Pair = id"
nipkow@39302
   460
  by (simp add: fun_eq_iff split: prod.split)
haftmann@37166
   461
haftmann@61424
   462
lemma case_prod_eta: "(\<lambda>(x, y). f (x, y)) = f"
wenzelm@61799
   463
  \<comment> \<open>Subsumes the old \<open>split_Pair\<close> when @{term f} is the identity function.\<close>
nipkow@39302
   464
  by (simp add: fun_eq_iff split: prod.split)
haftmann@37166
   465
haftmann@61424
   466
lemma case_prod_comp: "(case x of (a, b) \<Rightarrow> (f \<circ> g) a b) = f (g (fst x)) (snd x)"
haftmann@37166
   467
  by (cases x) simp
haftmann@37166
   468
haftmann@61424
   469
lemma The_case_prod: "The (case_prod P) = (THE xy. P (fst xy) (snd xy))"
blanchet@55414
   470
  by (simp add: case_prod_unfold)
haftmann@37166
   471
haftmann@61424
   472
lemma cond_case_prod_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
haftmann@61424
   473
  by (simp add: case_prod_eta)
haftmann@37166
   474
blanchet@47740
   475
lemma split_paired_all [no_atp]: "(!!x. PROP P x) == (!!a b. PROP P (a, b))"
wenzelm@11820
   476
proof
wenzelm@11820
   477
  fix a b
wenzelm@11820
   478
  assume "!!x. PROP P x"
wenzelm@19535
   479
  then show "PROP P (a, b)" .
wenzelm@11820
   480
next
wenzelm@11820
   481
  fix x
wenzelm@11820
   482
  assume "!!a b. PROP P (a, b)"
wenzelm@60758
   483
  from \<open>PROP P (fst x, snd x)\<close> show "PROP P x" by simp
wenzelm@11820
   484
qed
wenzelm@11820
   485
wenzelm@60758
   486
text \<open>
wenzelm@11838
   487
  The rule @{thm [source] split_paired_all} does not work with the
wenzelm@11838
   488
  Simplifier because it also affects premises in congrence rules,
wenzelm@61799
   489
  where this can lead to premises of the form \<open>!!a b. ... =
wenzelm@61799
   490
  ?P(a, b)\<close> which cannot be solved by reflexivity.
wenzelm@60758
   491
\<close>
wenzelm@11838
   492
haftmann@26358
   493
lemmas split_tupled_all = split_paired_all unit_all_eq2
haftmann@26358
   494
wenzelm@60758
   495
ML \<open>
wenzelm@11838
   496
  (* replace parameters of product type by individual component parameters *)
wenzelm@11838
   497
  local (* filtering with exists_paired_all is an essential optimization *)
wenzelm@56245
   498
    fun exists_paired_all (Const (@{const_name Pure.all}, _) $ Abs (_, T, t)) =
wenzelm@11838
   499
          can HOLogic.dest_prodT T orelse exists_paired_all t
wenzelm@11838
   500
      | exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u
wenzelm@11838
   501
      | exists_paired_all (Abs (_, _, t)) = exists_paired_all t
wenzelm@11838
   502
      | exists_paired_all _ = false;
wenzelm@51717
   503
    val ss =
wenzelm@51717
   504
      simpset_of
wenzelm@51717
   505
       (put_simpset HOL_basic_ss @{context}
wenzelm@51717
   506
        addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}]
wenzelm@51717
   507
        addsimprocs [@{simproc unit_eq}]);
wenzelm@11838
   508
  in
wenzelm@51717
   509
    fun split_all_tac ctxt = SUBGOAL (fn (t, i) =>
wenzelm@51717
   510
      if exists_paired_all t then safe_full_simp_tac (put_simpset ss ctxt) i else no_tac);
wenzelm@51717
   511
wenzelm@51717
   512
    fun unsafe_split_all_tac ctxt = SUBGOAL (fn (t, i) =>
wenzelm@51717
   513
      if exists_paired_all t then full_simp_tac (put_simpset ss ctxt) i else no_tac);
wenzelm@51717
   514
wenzelm@51717
   515
    fun split_all ctxt th =
wenzelm@51717
   516
      if exists_paired_all (Thm.prop_of th)
wenzelm@51717
   517
      then full_simplify (put_simpset ss ctxt) th else th;
wenzelm@11838
   518
  end;
wenzelm@60758
   519
\<close>
wenzelm@11838
   520
wenzelm@60758
   521
setup \<open>map_theory_claset (fn ctxt => ctxt addSbefore ("split_all_tac", split_all_tac))\<close>
wenzelm@11838
   522
blanchet@47740
   523
lemma split_paired_All [simp, no_atp]: "(ALL x. P x) = (ALL a b. P (a, b))"
wenzelm@61799
   524
  \<comment> \<open>\<open>[iff]\<close> is not a good idea because it makes \<open>blast\<close> loop\<close>
wenzelm@11838
   525
  by fast
wenzelm@11838
   526
blanchet@47740
   527
lemma split_paired_Ex [simp, no_atp]: "(EX x. P x) = (EX a b. P (a, b))"
haftmann@26358
   528
  by fast
haftmann@26358
   529
blanchet@47740
   530
lemma split_paired_The [no_atp]: "(THE x. P x) = (THE (a, b). P (a, b))"
wenzelm@61799
   531
  \<comment> \<open>Can't be added to simpset: loops!\<close>
haftmann@61424
   532
  by (simp add: case_prod_eta)
wenzelm@11838
   533
wenzelm@60758
   534
text \<open>
haftmann@61424
   535
  Simplification procedure for @{thm [source] cond_case_prod_eta}.  Using
haftmann@61424
   536
  @{thm [source] case_prod_eta} as a rewrite rule is not general enough,
haftmann@61424
   537
  and using @{thm [source] cond_case_prod_eta} directly would render some
wenzelm@61799
   538
  existing proofs very inefficient; similarly for \<open>prod.case_eq_if\<close>.
wenzelm@60758
   539
\<close>
wenzelm@11838
   540
wenzelm@60758
   541
ML \<open>
wenzelm@11838
   542
local
haftmann@61424
   543
  val cond_case_prod_eta_ss =
haftmann@61424
   544
    simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms cond_case_prod_eta});
wenzelm@35364
   545
  fun Pair_pat k 0 (Bound m) = (m = k)
wenzelm@35364
   546
    | Pair_pat k i (Const (@{const_name Pair},  _) $ Bound m $ t) =
wenzelm@35364
   547
        i > 0 andalso m = k + i andalso Pair_pat k (i - 1) t
wenzelm@35364
   548
    | Pair_pat _ _ _ = false;
wenzelm@35364
   549
  fun no_args k i (Abs (_, _, t)) = no_args (k + 1) i t
wenzelm@35364
   550
    | no_args k i (t $ u) = no_args k i t andalso no_args k i u
wenzelm@35364
   551
    | no_args k i (Bound m) = m < k orelse m > k + i
wenzelm@35364
   552
    | no_args _ _ _ = true;
wenzelm@35364
   553
  fun split_pat tp i (Abs  (_, _, t)) = if tp 0 i t then SOME (i, t) else NONE
haftmann@61424
   554
    | split_pat tp i (Const (@{const_name case_prod}, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t
wenzelm@35364
   555
    | split_pat tp i _ = NONE;
wenzelm@51717
   556
  fun metaeq ctxt lhs rhs = mk_meta_eq (Goal.prove ctxt [] []
wenzelm@35364
   557
        (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)))
haftmann@61424
   558
        (K (simp_tac (put_simpset cond_case_prod_eta_ss ctxt) 1)));
wenzelm@11838
   559
wenzelm@35364
   560
  fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k + 1) i t
wenzelm@35364
   561
    | beta_term_pat k i (t $ u) =
wenzelm@35364
   562
        Pair_pat k i (t $ u) orelse (beta_term_pat k i t andalso beta_term_pat k i u)
wenzelm@35364
   563
    | beta_term_pat k i t = no_args k i t;
wenzelm@35364
   564
  fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
wenzelm@35364
   565
    | eta_term_pat _ _ _ = false;
wenzelm@11838
   566
  fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
wenzelm@35364
   567
    | subst arg k i (t $ u) =
wenzelm@35364
   568
        if Pair_pat k i (t $ u) then incr_boundvars k arg
wenzelm@35364
   569
        else (subst arg k i t $ subst arg k i u)
wenzelm@35364
   570
    | subst arg k i t = t;
wenzelm@43595
   571
in
haftmann@61424
   572
  fun beta_proc ctxt (s as Const (@{const_name case_prod}, _) $ Abs (_, _, t) $ arg) =
wenzelm@11838
   573
        (case split_pat beta_term_pat 1 t of
wenzelm@51717
   574
          SOME (i, f) => SOME (metaeq ctxt s (subst arg 0 i f))
skalberg@15531
   575
        | NONE => NONE)
wenzelm@35364
   576
    | beta_proc _ _ = NONE;
haftmann@61424
   577
  fun eta_proc ctxt (s as Const (@{const_name case_prod}, _) $ Abs (_, _, t)) =
wenzelm@11838
   578
        (case split_pat eta_term_pat 1 t of
haftmann@58468
   579
          SOME (_, ft) => SOME (metaeq ctxt s (let val f $ _ = ft in f end))
skalberg@15531
   580
        | NONE => NONE)
wenzelm@35364
   581
    | eta_proc _ _ = NONE;
wenzelm@11838
   582
end;
wenzelm@60758
   583
\<close>
haftmann@61424
   584
simproc_setup case_prod_beta ("case_prod f z") =
wenzelm@60758
   585
  \<open>fn _ => fn ctxt => fn ct => beta_proc ctxt (Thm.term_of ct)\<close>
haftmann@61424
   586
simproc_setup case_prod_eta ("case_prod f") =
wenzelm@60758
   587
  \<open>fn _ => fn ctxt => fn ct => eta_proc ctxt (Thm.term_of ct)\<close>
wenzelm@11838
   588
haftmann@61424
   589
lemma case_prod_beta': "(\<lambda>(x,y). f x y) = (\<lambda>x. f (fst x) (snd x))"
hoelzl@50104
   590
  by (auto simp: fun_eq_iff)
hoelzl@50104
   591
wenzelm@60758
   592
text \<open>
haftmann@61424
   593
  \medskip @{const case_prod} used as a logical connective or set former.
wenzelm@11838
   594
wenzelm@61799
   595
  \medskip These rules are for use with \<open>blast\<close>; could instead
wenzelm@61799
   596
  call \<open>simp\<close> using @{thm [source] prod.split} as rewrite.\<close>
wenzelm@11838
   597
haftmann@61424
   598
lemma case_prodI2:
haftmann@61424
   599
  "\<And>p. (\<And>a b. p = (a, b) \<Longrightarrow> c a b) \<Longrightarrow> case p of (a, b) \<Rightarrow> c a b"
haftmann@61424
   600
  by (simp add: split_tupled_all)
wenzelm@11838
   601
haftmann@61424
   602
lemma case_prodI2':
haftmann@61424
   603
  "\<And>p. (\<And>a b. (a, b) = p \<Longrightarrow> c a b x) \<Longrightarrow> (case p of (a, b) \<Rightarrow> c a b) x"
haftmann@61424
   604
  by (simp add: split_tupled_all)
wenzelm@11838
   605
haftmann@61424
   606
lemma case_prodE [elim!]:
haftmann@61424
   607
  "(case p of (a, b) \<Rightarrow> c a b) \<Longrightarrow> (\<And>x y. p = (x, y) \<Longrightarrow> c x y \<Longrightarrow> Q) \<Longrightarrow> Q"
haftmann@61424
   608
  by (induct p) simp
wenzelm@11838
   609
haftmann@61424
   610
lemma case_prodE' [elim!]:
haftmann@61424
   611
  "(case p of (a, b) \<Rightarrow> c a b) z \<Longrightarrow> (\<And>x y. p = (x, y) \<Longrightarrow> c x y z \<Longrightarrow> Q) \<Longrightarrow> Q"
haftmann@61424
   612
  by (induct p) simp
wenzelm@11838
   613
haftmann@61424
   614
lemma case_prodE2:
haftmann@61424
   615
  assumes q: "Q (case z of (a, b) \<Rightarrow> P a b)"
haftmann@61424
   616
    and r: "\<And>x y. z = (x, y) \<Longrightarrow> Q (P x y) \<Longrightarrow> R"
haftmann@61424
   617
  shows R
haftmann@61424
   618
proof (rule r)
haftmann@61424
   619
  show "z = (fst z, snd z)" by simp
haftmann@61424
   620
  then show "Q (P (fst z) (snd z))"
haftmann@61424
   621
    using q by (simp add: case_prod_unfold)
wenzelm@11838
   622
qed
wenzelm@11838
   623
haftmann@61424
   624
lemma case_prodD':
haftmann@61127
   625
  "(case (a, b) of (c, d) \<Rightarrow> R c d) c \<Longrightarrow> R a b c"
wenzelm@11838
   626
  by simp
wenzelm@11838
   627
haftmann@61424
   628
lemma mem_case_prodI:
haftmann@61127
   629
  "z \<in> c a b \<Longrightarrow> z \<in> (case (a, b) of (d, e) \<Rightarrow> c d e)"
wenzelm@11838
   630
  by simp
wenzelm@11838
   631
haftmann@61424
   632
lemma mem_case_prodI2 [intro!]:
haftmann@61127
   633
  "\<And>p. (\<And>a b. p = (a, b) \<Longrightarrow> z \<in> c a b) \<Longrightarrow> z \<in> (case p of (a, b) \<Rightarrow> c a b)"
haftmann@61127
   634
  by (simp only: split_tupled_all) simp
wenzelm@11838
   635
wenzelm@61799
   636
declare mem_case_prodI [intro!] \<comment> \<open>postponed to maintain traditional declaration order!\<close>
wenzelm@61799
   637
declare case_prodI2' [intro!] \<comment> \<open>postponed to maintain traditional declaration order!\<close>
wenzelm@61799
   638
declare case_prodI2 [intro!] \<comment> \<open>postponed to maintain traditional declaration order!\<close>
wenzelm@61799
   639
declare case_prodI [intro!] \<comment> \<open>postponed to maintain traditional declaration order!\<close>
haftmann@61424
   640
  
haftmann@61424
   641
lemma mem_case_prodE [elim!]:
haftmann@61424
   642
  assumes "z \<in> case_prod c p"
haftmann@58468
   643
  obtains x y where "p = (x, y)" and "z \<in> c x y"
haftmann@61424
   644
  using assms by (rule case_prodE2)
wenzelm@11838
   645
wenzelm@60758
   646
ML \<open>
wenzelm@11838
   647
local (* filtering with exists_p_split is an essential optimization *)
haftmann@61424
   648
  fun exists_p_split (Const (@{const_name case_prod},_) $ _ $ (Const (@{const_name Pair},_)$_$_)) = true
wenzelm@11838
   649
    | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
wenzelm@11838
   650
    | exists_p_split (Abs (_, _, t)) = exists_p_split t
wenzelm@11838
   651
    | exists_p_split _ = false;
wenzelm@11838
   652
in
wenzelm@51717
   653
fun split_conv_tac ctxt = SUBGOAL (fn (t, i) =>
wenzelm@51717
   654
  if exists_p_split t
haftmann@61424
   655
  then safe_full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps @{thms case_prod_conv}) i
wenzelm@51717
   656
  else no_tac);
wenzelm@11838
   657
end;
wenzelm@60758
   658
\<close>
wenzelm@26340
   659
wenzelm@11838
   660
(* This prevents applications of splitE for already splitted arguments leading
wenzelm@11838
   661
   to quite time-consuming computations (in particular for nested tuples) *)
wenzelm@60758
   662
setup \<open>map_theory_claset (fn ctxt => ctxt addSbefore ("split_conv_tac", split_conv_tac))\<close>
wenzelm@11838
   663
blanchet@54147
   664
lemma split_eta_SetCompr [simp, no_atp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
wenzelm@18372
   665
  by (rule ext) fast
wenzelm@11838
   666
haftmann@61424
   667
lemma split_eta_SetCompr2 [simp, no_atp]: "(%u. EX x y. u = (x, y) & P x y) = case_prod P"
wenzelm@18372
   668
  by (rule ext) fast
wenzelm@11838
   669
haftmann@61424
   670
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & case_prod Q ab)"
wenzelm@61799
   671
  \<comment> \<open>Allows simplifications of nested splits in case of independent predicates.\<close>
wenzelm@18372
   672
  by (rule ext) blast
wenzelm@11838
   673
nipkow@14337
   674
(* Do NOT make this a simp rule as it
nipkow@14337
   675
   a) only helps in special situations
nipkow@14337
   676
   b) can lead to nontermination in the presence of split_def
nipkow@14337
   677
*)
nipkow@14337
   678
lemma split_comp_eq: 
paulson@20415
   679
  fixes f :: "'a => 'b => 'c" and g :: "'d => 'a"
haftmann@61424
   680
  shows "(%u. f (g (fst u)) (snd u)) = (case_prod (%x. f (g x)))"
wenzelm@18372
   681
  by (rule ext) auto
oheimb@14101
   682
haftmann@26358
   683
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
haftmann@26358
   684
  apply (rule_tac x = "(a, b)" in image_eqI)
haftmann@26358
   685
   apply auto
haftmann@26358
   686
  done
haftmann@26358
   687
wenzelm@11838
   688
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
wenzelm@11838
   689
  by blast
wenzelm@11838
   690
wenzelm@11838
   691
(*
wenzelm@11838
   692
the following  would be slightly more general,
wenzelm@11838
   693
but cannot be used as rewrite rule:
wenzelm@11838
   694
### Cannot add premise as rewrite rule because it contains (type) unknowns:
wenzelm@11838
   695
### ?y = .x
wenzelm@11838
   696
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
paulson@14208
   697
by (rtac some_equality 1)
paulson@14208
   698
by ( Simp_tac 1)
paulson@14208
   699
by (split_all_tac 1)
paulson@14208
   700
by (Asm_full_simp_tac 1)
wenzelm@11838
   701
qed "The_split_eq";
wenzelm@11838
   702
*)
wenzelm@11838
   703
blanchet@55414
   704
lemma case_prod_beta:
haftmann@61424
   705
  "case_prod f p = f (fst p) (snd p)"
haftmann@61424
   706
  by (fact prod.case_eq_if)
bulwahn@26143
   707
blanchet@55417
   708
lemma prod_cases3 [cases type]:
haftmann@24699
   709
  obtains (fields) a b c where "y = (a, b, c)"
haftmann@24699
   710
  by (cases y, case_tac b) blast
haftmann@24699
   711
haftmann@24699
   712
lemma prod_induct3 [case_names fields, induct type]:
haftmann@24699
   713
    "(!!a b c. P (a, b, c)) ==> P x"
haftmann@24699
   714
  by (cases x) blast
haftmann@24699
   715
blanchet@55417
   716
lemma prod_cases4 [cases type]:
haftmann@24699
   717
  obtains (fields) a b c d where "y = (a, b, c, d)"
haftmann@24699
   718
  by (cases y, case_tac c) blast
haftmann@24699
   719
haftmann@24699
   720
lemma prod_induct4 [case_names fields, induct type]:
haftmann@24699
   721
    "(!!a b c d. P (a, b, c, d)) ==> P x"
haftmann@24699
   722
  by (cases x) blast
haftmann@24699
   723
blanchet@55417
   724
lemma prod_cases5 [cases type]:
haftmann@24699
   725
  obtains (fields) a b c d e where "y = (a, b, c, d, e)"
haftmann@24699
   726
  by (cases y, case_tac d) blast
haftmann@24699
   727
haftmann@24699
   728
lemma prod_induct5 [case_names fields, induct type]:
haftmann@24699
   729
    "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
haftmann@24699
   730
  by (cases x) blast
haftmann@24699
   731
blanchet@55417
   732
lemma prod_cases6 [cases type]:
haftmann@24699
   733
  obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
haftmann@24699
   734
  by (cases y, case_tac e) blast
haftmann@24699
   735
haftmann@24699
   736
lemma prod_induct6 [case_names fields, induct type]:
haftmann@24699
   737
    "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
haftmann@24699
   738
  by (cases x) blast
haftmann@24699
   739
blanchet@55417
   740
lemma prod_cases7 [cases type]:
haftmann@24699
   741
  obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
haftmann@24699
   742
  by (cases y, case_tac f) blast
haftmann@24699
   743
haftmann@24699
   744
lemma prod_induct7 [case_names fields, induct type]:
haftmann@24699
   745
    "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
haftmann@24699
   746
  by (cases x) blast
haftmann@24699
   747
haftmann@61424
   748
definition internal_case_prod :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where
haftmann@61424
   749
  "internal_case_prod == case_prod"
haftmann@37166
   750
haftmann@61424
   751
lemma internal_case_prod_conv: "internal_case_prod c (a, b) = c a b"
haftmann@61424
   752
  by (simp only: internal_case_prod_def case_prod_conv)
haftmann@37166
   753
wenzelm@48891
   754
ML_file "Tools/split_rule.ML"
haftmann@37166
   755
haftmann@61424
   756
hide_const internal_case_prod
haftmann@37166
   757
haftmann@24699
   758
wenzelm@60758
   759
subsubsection \<open>Derived operations\<close>
haftmann@26358
   760
haftmann@61424
   761
definition curry :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c" where
haftmann@37387
   762
  "curry = (\<lambda>c x y. c (x, y))"
haftmann@37166
   763
haftmann@37166
   764
lemma curry_conv [simp, code]: "curry f a b = f (a, b)"
haftmann@37166
   765
  by (simp add: curry_def)
haftmann@37166
   766
haftmann@37166
   767
lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b"
haftmann@37166
   768
  by (simp add: curry_def)
haftmann@37166
   769
haftmann@37166
   770
lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)"
haftmann@37166
   771
  by (simp add: curry_def)
haftmann@37166
   772
haftmann@37166
   773
lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q"
haftmann@37166
   774
  by (simp add: curry_def)
haftmann@37166
   775
haftmann@61424
   776
lemma curry_case_prod [simp]: "curry (case_prod f) = f"
haftmann@61032
   777
  by (simp add: curry_def case_prod_unfold)
haftmann@37166
   778
haftmann@61424
   779
lemma case_prod_curry [simp]: "case_prod (curry f) = f"
haftmann@61032
   780
  by (simp add: curry_def case_prod_unfold)
haftmann@37166
   781
Andreas@54630
   782
lemma curry_K: "curry (\<lambda>x. c) = (\<lambda>x y. c)"
Andreas@54630
   783
by(simp add: fun_eq_iff)
Andreas@54630
   784
wenzelm@60758
   785
text \<open>
haftmann@26358
   786
  The composition-uncurry combinator.
wenzelm@60758
   787
\<close>
haftmann@26358
   788
haftmann@37751
   789
notation fcomp (infixl "\<circ>>" 60)
haftmann@26358
   790
haftmann@37751
   791
definition scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "\<circ>\<rightarrow>" 60) where
haftmann@61424
   792
  "f \<circ>\<rightarrow> g = (\<lambda>x. case_prod g (f x))"
haftmann@26358
   793
haftmann@37678
   794
lemma scomp_unfold: "scomp = (\<lambda>f g x. g (fst (f x)) (snd (f x)))"
blanchet@55414
   795
  by (simp add: fun_eq_iff scomp_def case_prod_unfold)
haftmann@37678
   796
haftmann@61424
   797
lemma scomp_apply [simp]: "(f \<circ>\<rightarrow> g) x = case_prod g (f x)"
blanchet@55414
   798
  by (simp add: scomp_unfold case_prod_unfold)
haftmann@26358
   799
haftmann@37751
   800
lemma Pair_scomp: "Pair x \<circ>\<rightarrow> f = f x"
huffman@44921
   801
  by (simp add: fun_eq_iff)
haftmann@26358
   802
haftmann@37751
   803
lemma scomp_Pair: "x \<circ>\<rightarrow> Pair = x"
huffman@44921
   804
  by (simp add: fun_eq_iff)
haftmann@26358
   805
haftmann@37751
   806
lemma scomp_scomp: "(f \<circ>\<rightarrow> g) \<circ>\<rightarrow> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>\<rightarrow> h)"
nipkow@39302
   807
  by (simp add: fun_eq_iff scomp_unfold)
haftmann@26358
   808
haftmann@37751
   809
lemma scomp_fcomp: "(f \<circ>\<rightarrow> g) \<circ>> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>> h)"
nipkow@39302
   810
  by (simp add: fun_eq_iff scomp_unfold fcomp_def)
haftmann@26358
   811
haftmann@37751
   812
lemma fcomp_scomp: "(f \<circ>> g) \<circ>\<rightarrow> h = f \<circ>> (g \<circ>\<rightarrow> h)"
huffman@44921
   813
  by (simp add: fun_eq_iff scomp_unfold)
haftmann@26358
   814
haftmann@52435
   815
code_printing
haftmann@52435
   816
  constant scomp \<rightharpoonup> (Eval) infixl 3 "#->"
haftmann@31202
   817
haftmann@37751
   818
no_notation fcomp (infixl "\<circ>>" 60)
haftmann@37751
   819
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@26358
   820
wenzelm@60758
   821
text \<open>
blanchet@55932
   822
  @{term map_prod} --- action of the product functor upon
krauss@36664
   823
  functions.
wenzelm@60758
   824
\<close>
haftmann@21195
   825
blanchet@55932
   826
definition map_prod :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where
blanchet@55932
   827
  "map_prod f g = (\<lambda>(x, y). (f x, g y))"
haftmann@26358
   828
blanchet@55932
   829
lemma map_prod_simp [simp, code]:
blanchet@55932
   830
  "map_prod f g (a, b) = (f a, g b)"
blanchet@55932
   831
  by (simp add: map_prod_def)
haftmann@26358
   832
blanchet@55932
   833
functor map_prod: map_prod
huffman@44921
   834
  by (auto simp add: split_paired_all)
nipkow@37278
   835
blanchet@55932
   836
lemma fst_map_prod [simp]:
blanchet@55932
   837
  "fst (map_prod f g x) = f (fst x)"
haftmann@40607
   838
  by (cases x) simp_all
nipkow@37278
   839
traytel@58916
   840
lemma snd_map_prod [simp]:
blanchet@55932
   841
  "snd (map_prod f g x) = g (snd x)"
haftmann@40607
   842
  by (cases x) simp_all
nipkow@37278
   843
blanchet@55932
   844
lemma fst_comp_map_prod [simp]:
blanchet@55932
   845
  "fst \<circ> map_prod f g = f \<circ> fst"
haftmann@40607
   846
  by (rule ext) simp_all
nipkow@37278
   847
blanchet@55932
   848
lemma snd_comp_map_prod [simp]:
blanchet@55932
   849
  "snd \<circ> map_prod f g = g \<circ> snd"
haftmann@40607
   850
  by (rule ext) simp_all
haftmann@26358
   851
blanchet@55932
   852
lemma map_prod_compose:
blanchet@55932
   853
  "map_prod (f1 o f2) (g1 o g2) = (map_prod f1 g1 o map_prod f2 g2)"
blanchet@55932
   854
  by (rule ext) (simp add: map_prod.compositionality comp_def)
haftmann@26358
   855
blanchet@55932
   856
lemma map_prod_ident [simp]:
blanchet@55932
   857
  "map_prod (%x. x) (%y. y) = (%z. z)"
blanchet@55932
   858
  by (rule ext) (simp add: map_prod.identity)
haftmann@40607
   859
blanchet@55932
   860
lemma map_prod_imageI [intro]:
blanchet@55932
   861
  "(a, b) \<in> R \<Longrightarrow> (f a, g b) \<in> map_prod f g ` R"
haftmann@40607
   862
  by (rule image_eqI) simp_all
haftmann@21195
   863
haftmann@26358
   864
lemma prod_fun_imageE [elim!]:
blanchet@55932
   865
  assumes major: "c \<in> map_prod f g ` R"
haftmann@40607
   866
    and cases: "\<And>x y. c = (f x, g y) \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> P"
haftmann@26358
   867
  shows P
haftmann@26358
   868
  apply (rule major [THEN imageE])
haftmann@37166
   869
  apply (case_tac x)
haftmann@26358
   870
  apply (rule cases)
haftmann@40607
   871
  apply simp_all
haftmann@26358
   872
  done
haftmann@26358
   873
haftmann@37166
   874
definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" where
blanchet@55932
   875
  "apfst f = map_prod f id"
haftmann@26358
   876
haftmann@37166
   877
definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" where
blanchet@55932
   878
  "apsnd f = map_prod id f"
haftmann@26358
   879
haftmann@26358
   880
lemma apfst_conv [simp, code]:
haftmann@26358
   881
  "apfst f (x, y) = (f x, y)" 
haftmann@26358
   882
  by (simp add: apfst_def)
haftmann@26358
   883
hoelzl@33638
   884
lemma apsnd_conv [simp, code]:
haftmann@26358
   885
  "apsnd f (x, y) = (x, f y)" 
haftmann@26358
   886
  by (simp add: apsnd_def)
haftmann@21195
   887
haftmann@33594
   888
lemma fst_apfst [simp]:
haftmann@33594
   889
  "fst (apfst f x) = f (fst x)"
haftmann@33594
   890
  by (cases x) simp
haftmann@33594
   891
haftmann@51173
   892
lemma fst_comp_apfst [simp]:
haftmann@51173
   893
  "fst \<circ> apfst f = f \<circ> fst"
haftmann@51173
   894
  by (simp add: fun_eq_iff)
haftmann@51173
   895
haftmann@33594
   896
lemma fst_apsnd [simp]:
haftmann@33594
   897
  "fst (apsnd f x) = fst x"
haftmann@33594
   898
  by (cases x) simp
haftmann@33594
   899
haftmann@51173
   900
lemma fst_comp_apsnd [simp]:
haftmann@51173
   901
  "fst \<circ> apsnd f = fst"
haftmann@51173
   902
  by (simp add: fun_eq_iff)
haftmann@51173
   903
haftmann@33594
   904
lemma snd_apfst [simp]:
haftmann@33594
   905
  "snd (apfst f x) = snd x"
haftmann@33594
   906
  by (cases x) simp
haftmann@33594
   907
haftmann@51173
   908
lemma snd_comp_apfst [simp]:
haftmann@51173
   909
  "snd \<circ> apfst f = snd"
haftmann@51173
   910
  by (simp add: fun_eq_iff)
haftmann@51173
   911
haftmann@33594
   912
lemma snd_apsnd [simp]:
haftmann@33594
   913
  "snd (apsnd f x) = f (snd x)"
haftmann@33594
   914
  by (cases x) simp
haftmann@33594
   915
haftmann@51173
   916
lemma snd_comp_apsnd [simp]:
haftmann@51173
   917
  "snd \<circ> apsnd f = f \<circ> snd"
haftmann@51173
   918
  by (simp add: fun_eq_iff)
haftmann@51173
   919
haftmann@33594
   920
lemma apfst_compose:
haftmann@33594
   921
  "apfst f (apfst g x) = apfst (f \<circ> g) x"
haftmann@33594
   922
  by (cases x) simp
haftmann@33594
   923
haftmann@33594
   924
lemma apsnd_compose:
haftmann@33594
   925
  "apsnd f (apsnd g x) = apsnd (f \<circ> g) x"
haftmann@33594
   926
  by (cases x) simp
haftmann@33594
   927
haftmann@33594
   928
lemma apfst_apsnd [simp]:
haftmann@33594
   929
  "apfst f (apsnd g x) = (f (fst x), g (snd x))"
haftmann@33594
   930
  by (cases x) simp
haftmann@33594
   931
haftmann@33594
   932
lemma apsnd_apfst [simp]:
haftmann@33594
   933
  "apsnd f (apfst g x) = (g (fst x), f (snd x))"
haftmann@33594
   934
  by (cases x) simp
haftmann@33594
   935
haftmann@33594
   936
lemma apfst_id [simp] :
haftmann@33594
   937
  "apfst id = id"
nipkow@39302
   938
  by (simp add: fun_eq_iff)
haftmann@33594
   939
haftmann@33594
   940
lemma apsnd_id [simp] :
haftmann@33594
   941
  "apsnd id = id"
nipkow@39302
   942
  by (simp add: fun_eq_iff)
haftmann@33594
   943
haftmann@33594
   944
lemma apfst_eq_conv [simp]:
haftmann@33594
   945
  "apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)"
haftmann@33594
   946
  by (cases x) simp
haftmann@33594
   947
haftmann@33594
   948
lemma apsnd_eq_conv [simp]:
haftmann@33594
   949
  "apsnd f x = apsnd g x \<longleftrightarrow> f (snd x) = g (snd x)"
haftmann@33594
   950
  by (cases x) simp
haftmann@33594
   951
hoelzl@33638
   952
lemma apsnd_apfst_commute:
hoelzl@33638
   953
  "apsnd f (apfst g p) = apfst g (apsnd f p)"
hoelzl@33638
   954
  by simp
haftmann@21195
   955
haftmann@56626
   956
context
haftmann@56626
   957
begin
haftmann@56626
   958
wenzelm@60758
   959
local_setup \<open>Local_Theory.map_background_naming (Name_Space.mandatory_path "prod")\<close>
haftmann@56626
   960
haftmann@56545
   961
definition swap :: "'a \<times> 'b \<Rightarrow> 'b \<times> 'a"
haftmann@56545
   962
where
haftmann@56545
   963
  "swap p = (snd p, fst p)"
haftmann@56545
   964
haftmann@56626
   965
end
haftmann@56626
   966
haftmann@56545
   967
lemma swap_simp [simp]:
haftmann@56626
   968
  "prod.swap (x, y) = (y, x)"
haftmann@56626
   969
  by (simp add: prod.swap_def)
haftmann@56545
   970
haftmann@58195
   971
lemma swap_swap [simp]:
haftmann@58195
   972
  "prod.swap (prod.swap p) = p"
haftmann@58195
   973
  by (cases p) simp
haftmann@58195
   974
haftmann@58195
   975
lemma swap_comp_swap [simp]:
haftmann@58195
   976
  "prod.swap \<circ> prod.swap = id"
haftmann@58195
   977
  by (simp add: fun_eq_iff)
haftmann@58195
   978
haftmann@56545
   979
lemma pair_in_swap_image [simp]:
haftmann@56626
   980
  "(y, x) \<in> prod.swap ` A \<longleftrightarrow> (x, y) \<in> A"
haftmann@56545
   981
  by (auto intro!: image_eqI)
haftmann@56545
   982
haftmann@56545
   983
lemma inj_swap [simp]:
haftmann@56626
   984
  "inj_on prod.swap A"
haftmann@56626
   985
  by (rule inj_onI) auto
haftmann@56626
   986
haftmann@56626
   987
lemma swap_inj_on:
haftmann@56626
   988
  "inj_on (\<lambda>(i, j). (j, i)) A"
haftmann@56626
   989
  by (rule inj_onI) auto
haftmann@56545
   990
haftmann@58195
   991
lemma surj_swap [simp]:
haftmann@58195
   992
  "surj prod.swap"
haftmann@58195
   993
  by (rule surjI [of _ prod.swap]) simp
haftmann@58195
   994
haftmann@58195
   995
lemma bij_swap [simp]:
haftmann@58195
   996
  "bij prod.swap"
haftmann@58195
   997
  by (simp add: bij_def)
haftmann@58195
   998
haftmann@56545
   999
lemma case_swap [simp]:
haftmann@56626
  1000
  "(case prod.swap p of (y, x) \<Rightarrow> f x y) = (case p of (x, y) \<Rightarrow> f x y)"
haftmann@56545
  1001
  by (cases p) simp
haftmann@56545
  1002
Andreas@62139
  1003
lemma fst_swap [simp]: "fst (prod.swap x) = snd x"
Andreas@62139
  1004
by(cases x) simp
Andreas@62139
  1005
Andreas@62139
  1006
lemma snd_swap [simp]: "snd (prod.swap x) = fst x"
Andreas@62139
  1007
by(cases x) simp
Andreas@62139
  1008
wenzelm@60758
  1009
text \<open>
haftmann@26358
  1010
  Disjoint union of a family of sets -- Sigma.
wenzelm@60758
  1011
\<close>
haftmann@26358
  1012
haftmann@45986
  1013
definition Sigma :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<times> 'b) set" where
haftmann@26358
  1014
  Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}"
haftmann@26358
  1015
haftmann@26358
  1016
abbreviation
wenzelm@61943
  1017
  Times :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set"  (infixr "\<times>" 80) where
wenzelm@61943
  1018
  "A \<times> B == Sigma A (%_. B)"
berghofe@15394
  1019
nipkow@45662
  1020
hide_const (open) Times
nipkow@45662
  1021
haftmann@26358
  1022
syntax
wenzelm@35115
  1023
  "_Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set"  ("(3SIGMA _:_./ _)" [0, 0, 10] 10)
haftmann@26358
  1024
translations
wenzelm@35115
  1025
  "SIGMA x:A. B" == "CONST Sigma A (%x. B)"
haftmann@26358
  1026
haftmann@26358
  1027
lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
haftmann@26358
  1028
  by (unfold Sigma_def) blast
haftmann@26358
  1029
haftmann@26358
  1030
lemma SigmaE [elim!]:
haftmann@26358
  1031
    "[| c: Sigma A B;
haftmann@26358
  1032
        !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
haftmann@26358
  1033
     |] ==> P"
wenzelm@61799
  1034
  \<comment> \<open>The general elimination rule.\<close>
haftmann@26358
  1035
  by (unfold Sigma_def) blast
haftmann@20588
  1036
wenzelm@60758
  1037
text \<open>
haftmann@26358
  1038
  Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
haftmann@26358
  1039
  eigenvariables.
wenzelm@60758
  1040
\<close>
haftmann@26358
  1041
haftmann@26358
  1042
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
haftmann@26358
  1043
  by blast
haftmann@26358
  1044
haftmann@26358
  1045
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
haftmann@26358
  1046
  by blast
haftmann@26358
  1047
haftmann@26358
  1048
lemma SigmaE2:
haftmann@26358
  1049
    "[| (a, b) : Sigma A B;
haftmann@26358
  1050
        [| a:A;  b:B(a) |] ==> P
haftmann@26358
  1051
     |] ==> P"
haftmann@26358
  1052
  by blast
haftmann@20588
  1053
haftmann@26358
  1054
lemma Sigma_cong:
haftmann@26358
  1055
     "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>
haftmann@26358
  1056
      \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
haftmann@26358
  1057
  by auto
haftmann@26358
  1058
haftmann@26358
  1059
lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
haftmann@26358
  1060
  by blast
haftmann@26358
  1061
haftmann@26358
  1062
lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
haftmann@26358
  1063
  by blast
haftmann@26358
  1064
wenzelm@61943
  1065
lemma Sigma_empty2 [simp]: "A \<times> {} = {}"
haftmann@26358
  1066
  by blast
haftmann@26358
  1067
wenzelm@61943
  1068
lemma UNIV_Times_UNIV [simp]: "UNIV \<times> UNIV = UNIV"
haftmann@26358
  1069
  by auto
haftmann@21908
  1070
wenzelm@61943
  1071
lemma Compl_Times_UNIV1 [simp]: "- (UNIV \<times> A) = UNIV \<times> (-A)"
haftmann@26358
  1072
  by auto
haftmann@26358
  1073
wenzelm@61943
  1074
lemma Compl_Times_UNIV2 [simp]: "- (A \<times> UNIV) = (-A) \<times> UNIV"
haftmann@26358
  1075
  by auto
haftmann@26358
  1076
haftmann@26358
  1077
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
haftmann@26358
  1078
  by blast
haftmann@26358
  1079
hoelzl@62101
  1080
lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
hoelzl@62101
  1081
  by (induct x) simp
hoelzl@62101
  1082
hoelzl@59000
  1083
lemma Sigma_empty_iff: "(SIGMA i:I. X i) = {} \<longleftrightarrow> (\<forall>i\<in>I. X i = {})"
hoelzl@59000
  1084
  by auto
hoelzl@59000
  1085
wenzelm@61943
  1086
lemma Times_subset_cancel2: "x:C ==> (A \<times> C <= B \<times> C) = (A <= B)"
haftmann@26358
  1087
  by blast
haftmann@26358
  1088
wenzelm@61943
  1089
lemma Times_eq_cancel2: "x:C ==> (A \<times> C = B \<times> C) = (A = B)"
haftmann@26358
  1090
  by (blast elim: equalityE)
haftmann@20588
  1091
haftmann@61424
  1092
lemma Collect_case_prod_Sigma:
haftmann@61127
  1093
  "{(x, y). P x \<and> Q x y} = (SIGMA x:Collect P. Collect (Q x))"
haftmann@26358
  1094
  by blast
haftmann@26358
  1095
haftmann@61424
  1096
lemma Collect_case_prod [simp]:
haftmann@61127
  1097
  "{(a, b). P a \<and> Q b} = Collect P \<times> Collect Q "
haftmann@61424
  1098
  by (fact Collect_case_prod_Sigma)
haftmann@26358
  1099
haftmann@61424
  1100
lemma Collect_case_prodD:
haftmann@61424
  1101
  "x \<in> Collect (case_prod A) \<Longrightarrow> A (fst x) (snd x)"
haftmann@61422
  1102
  by auto
haftmann@61422
  1103
haftmann@61424
  1104
lemma Collect_case_prod_mono:
haftmann@61424
  1105
  "A \<le> B \<Longrightarrow> Collect (case_prod A) \<subseteq> Collect (case_prod B)"
haftmann@61422
  1106
  by auto (auto elim!: le_funE)
haftmann@61422
  1107
haftmann@61422
  1108
lemma Collect_split_mono_strong: 
haftmann@61422
  1109
  "X = fst ` A \<Longrightarrow> Y = snd ` A \<Longrightarrow> \<forall>a\<in>X. \<forall>b \<in> Y. P a b \<longrightarrow> Q a b
haftmann@61424
  1110
    \<Longrightarrow> A \<subseteq> Collect (case_prod P) \<Longrightarrow> A \<subseteq> Collect (case_prod Q)"
haftmann@61422
  1111
  by fastforce
haftmann@61422
  1112
  
haftmann@26358
  1113
lemma UN_Times_distrib:
haftmann@61127
  1114
  "(\<Union>(a, b)\<in>A \<times> B. E a \<times> F b) = UNION A E \<times> UNION B F"
wenzelm@61799
  1115
  \<comment> \<open>Suggested by Pierre Chartier\<close>
haftmann@26358
  1116
  by blast
haftmann@26358
  1117
blanchet@47740
  1118
lemma split_paired_Ball_Sigma [simp, no_atp]:
haftmann@61127
  1119
  "(\<forall>z\<in>Sigma A B. P z) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>B x. P (x, y))"
haftmann@26358
  1120
  by blast
haftmann@26358
  1121
blanchet@47740
  1122
lemma split_paired_Bex_Sigma [simp, no_atp]:
haftmann@61127
  1123
  "(\<exists>z\<in>Sigma A B. P z) \<longleftrightarrow> (\<exists>x\<in>A. \<exists>y\<in>B x. P (x, y))"
haftmann@61127
  1124
  by blast
haftmann@61127
  1125
haftmann@61127
  1126
lemma Sigma_Un_distrib1:
haftmann@61127
  1127
  "Sigma (I \<union> J) C = Sigma I C \<union> Sigma J C"
haftmann@26358
  1128
  by blast
haftmann@21908
  1129
haftmann@61127
  1130
lemma Sigma_Un_distrib2:
haftmann@61127
  1131
  "(SIGMA i:I. A i \<union> B i) = Sigma I A \<union> Sigma I B"
haftmann@26358
  1132
  by blast
haftmann@26358
  1133
haftmann@61127
  1134
lemma Sigma_Int_distrib1:
haftmann@61127
  1135
  "Sigma (I \<inter> J) C = Sigma I C \<inter> Sigma J C"
haftmann@26358
  1136
  by blast
haftmann@26358
  1137
haftmann@61127
  1138
lemma Sigma_Int_distrib2:
haftmann@61127
  1139
  "(SIGMA i:I. A i \<inter> B i) = Sigma I A \<inter> Sigma I B"
haftmann@26358
  1140
  by blast
haftmann@26358
  1141
haftmann@61127
  1142
lemma Sigma_Diff_distrib1:
haftmann@61127
  1143
  "Sigma (I - J) C = Sigma I C - Sigma J C"
haftmann@26358
  1144
  by blast
haftmann@26358
  1145
haftmann@61127
  1146
lemma Sigma_Diff_distrib2:
haftmann@61127
  1147
  "(SIGMA i:I. A i - B i) = Sigma I A - Sigma I B"
haftmann@26358
  1148
  by blast
haftmann@21908
  1149
haftmann@61127
  1150
lemma Sigma_Union:
haftmann@61127
  1151
  "Sigma (\<Union>X) B = (\<Union>A\<in>X. Sigma A B)"
haftmann@26358
  1152
  by blast
haftmann@26358
  1153
Andreas@61630
  1154
lemma Pair_vimage_Sigma: "Pair x -` Sigma A f = (if x \<in> A then f x else {})"
Andreas@61630
  1155
  by auto
Andreas@61630
  1156
wenzelm@60758
  1157
text \<open>
haftmann@26358
  1158
  Non-dependent versions are needed to avoid the need for higher-order
haftmann@26358
  1159
  matching, especially when the rules are re-oriented.
wenzelm@60758
  1160
\<close>
haftmann@21908
  1161
haftmann@61127
  1162
lemma Times_Un_distrib1:
haftmann@61127
  1163
  "(A \<union> B) \<times> C = A \<times> C \<union> B \<times> C "
haftmann@56545
  1164
  by (fact Sigma_Un_distrib1)
haftmann@26358
  1165
haftmann@61127
  1166
lemma Times_Int_distrib1:
haftmann@61127
  1167
  "(A \<inter> B) \<times> C = A \<times> C \<inter> B \<times> C "
haftmann@56545
  1168
  by (fact Sigma_Int_distrib1)
haftmann@26358
  1169
haftmann@61127
  1170
lemma Times_Diff_distrib1:
haftmann@61127
  1171
  "(A - B) \<times> C = A \<times> C - B \<times> C "
haftmann@56545
  1172
  by (fact Sigma_Diff_distrib1)
haftmann@26358
  1173
haftmann@61127
  1174
lemma Times_empty [simp]:
haftmann@61127
  1175
  "A \<times> B = {} \<longleftrightarrow> A = {} \<or> B = {}"
hoelzl@36622
  1176
  by auto
hoelzl@36622
  1177
haftmann@61127
  1178
lemma times_eq_iff:
haftmann@61127
  1179
  "A \<times> B = C \<times> D \<longleftrightarrow> A = C \<and> B = D \<or> (A = {} \<or> B = {}) \<and> (C = {} \<or> D = {})"
hoelzl@50104
  1180
  by auto
hoelzl@50104
  1181
haftmann@61127
  1182
lemma fst_image_times [simp]:
haftmann@61127
  1183
  "fst ` (A \<times> B) = (if B = {} then {} else A)"
huffman@44921
  1184
  by force
hoelzl@36622
  1185
haftmann@61127
  1186
lemma snd_image_times [simp]:
haftmann@61127
  1187
  "snd ` (A \<times> B) = (if A = {} then {} else B)"
huffman@44921
  1188
  by force
hoelzl@36622
  1189
lp15@62379
  1190
lemma fst_image_Sigma:
lp15@62379
  1191
  "fst ` (Sigma A B) = {x \<in> A. B(x) \<noteq> {}}"
lp15@62379
  1192
  by force
lp15@62379
  1193
lp15@62379
  1194
lemma snd_image_Sigma:
lp15@62379
  1195
  "snd ` (Sigma A B) = (\<Union> x \<in> A. B x)"
lp15@62379
  1196
  by force
lp15@62379
  1197
haftmann@56545
  1198
lemma vimage_fst:
haftmann@56545
  1199
  "fst -` A = A \<times> UNIV"
haftmann@56545
  1200
  by auto
haftmann@56545
  1201
haftmann@56545
  1202
lemma vimage_snd:
haftmann@56545
  1203
  "snd -` A = UNIV \<times> A"
haftmann@56545
  1204
  by auto
haftmann@56545
  1205
nipkow@28719
  1206
lemma insert_times_insert[simp]:
nipkow@28719
  1207
  "insert a A \<times> insert b B =
nipkow@28719
  1208
   insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)"
haftmann@61127
  1209
  by blast
haftmann@26358
  1210
haftmann@61127
  1211
lemma vimage_Times:
haftmann@61127
  1212
  "f -` (A \<times> B) = (fst \<circ> f) -` A \<inter> (snd \<circ> f) -` B"
haftmann@61127
  1213
proof (rule set_eqI)
haftmann@61127
  1214
  fix x
haftmann@61127
  1215
  show "x \<in> f -` (A \<times> B) \<longleftrightarrow> x \<in> (fst \<circ> f) -` A \<inter> (snd \<circ> f) -` B"
haftmann@61127
  1216
    by (cases "f x") (auto split: prod.split)
haftmann@61127
  1217
qed
paulson@33271
  1218
haftmann@61127
  1219
lemma times_Int_times:
haftmann@61127
  1220
  "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"
hoelzl@50104
  1221
  by auto
hoelzl@50104
  1222
haftmann@56626
  1223
lemma product_swap:
haftmann@56626
  1224
  "prod.swap ` (A \<times> B) = B \<times> A"
haftmann@56626
  1225
  by (auto simp add: set_eq_iff)
haftmann@35822
  1226
haftmann@35822
  1227
lemma swap_product:
haftmann@56626
  1228
  "(\<lambda>(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
haftmann@56626
  1229
  by (auto simp add: set_eq_iff)
haftmann@35822
  1230
hoelzl@36622
  1231
lemma image_split_eq_Sigma:
hoelzl@36622
  1232
  "(\<lambda>x. (f x, g x)) ` A = Sigma (f ` A) (\<lambda>x. g ` (f -` {x} \<inter> A))"
haftmann@46128
  1233
proof (safe intro!: imageI)
hoelzl@36622
  1234
  fix a b assume *: "a \<in> A" "b \<in> A" and eq: "f a = f b"
hoelzl@36622
  1235
  show "(f b, g a) \<in> (\<lambda>x. (f x, g x)) ` A"
hoelzl@36622
  1236
    using * eq[symmetric] by auto
hoelzl@36622
  1237
qed simp_all
haftmann@35822
  1238
lp15@63007
  1239
lemma subset_fst_snd: "A \<subseteq> (fst ` A \<times> snd ` A)"
lp15@63007
  1240
  by force
lp15@63007
  1241
Andreas@60057
  1242
lemma inj_on_apfst [simp]: "inj_on (apfst f) (A \<times> UNIV) \<longleftrightarrow> inj_on f A"
Andreas@60057
  1243
by(auto simp add: inj_on_def)
Andreas@60057
  1244
Andreas@60057
  1245
lemma inj_apfst [simp]: "inj (apfst f) \<longleftrightarrow> inj f"
Andreas@60057
  1246
using inj_on_apfst[of f UNIV] by simp
Andreas@60057
  1247
Andreas@60057
  1248
lemma inj_on_apsnd [simp]: "inj_on (apsnd f) (UNIV \<times> A) \<longleftrightarrow> inj_on f A"
Andreas@60057
  1249
by(auto simp add: inj_on_def)
Andreas@60057
  1250
Andreas@60057
  1251
lemma inj_apsnd [simp]: "inj (apsnd f) \<longleftrightarrow> inj f"
Andreas@60057
  1252
using inj_on_apsnd[of f UNIV] by simp
Andreas@60057
  1253
haftmann@61127
  1254
context
haftmann@61127
  1255
begin
haftmann@61127
  1256
haftmann@61127
  1257
qualified definition product :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" where
haftmann@46128
  1258
  [code_abbrev]: "product A B = A \<times> B"
haftmann@46128
  1259
haftmann@46128
  1260
lemma member_product:
haftmann@46128
  1261
  "x \<in> Product_Type.product A B \<longleftrightarrow> x \<in> A \<times> B"
haftmann@61127
  1262
  by (simp add: Product_Type.product_def)
haftmann@46128
  1263
haftmann@61127
  1264
end
haftmann@61127
  1265
  
wenzelm@60758
  1266
text \<open>The following @{const map_prod} lemmas are due to Joachim Breitner:\<close>
haftmann@40607
  1267
blanchet@55932
  1268
lemma map_prod_inj_on:
haftmann@40607
  1269
  assumes "inj_on f A" and "inj_on g B"
blanchet@55932
  1270
  shows "inj_on (map_prod f g) (A \<times> B)"
haftmann@40607
  1271
proof (rule inj_onI)
haftmann@40607
  1272
  fix x :: "'a \<times> 'c" and y :: "'a \<times> 'c"
haftmann@40607
  1273
  assume "x \<in> A \<times> B" hence "fst x \<in> A" and "snd x \<in> B" by auto
haftmann@40607
  1274
  assume "y \<in> A \<times> B" hence "fst y \<in> A" and "snd y \<in> B" by auto
blanchet@55932
  1275
  assume "map_prod f g x = map_prod f g y"
blanchet@55932
  1276
  hence "fst (map_prod f g x) = fst (map_prod f g y)" by (auto)
haftmann@40607
  1277
  hence "f (fst x) = f (fst y)" by (cases x,cases y,auto)
wenzelm@60758
  1278
  with \<open>inj_on f A\<close> and \<open>fst x \<in> A\<close> and \<open>fst y \<in> A\<close>
haftmann@40607
  1279
  have "fst x = fst y" by (auto dest:dest:inj_onD)
wenzelm@60758
  1280
  moreover from \<open>map_prod f g x = map_prod f g y\<close>
blanchet@55932
  1281
  have "snd (map_prod f g x) = snd (map_prod f g y)" by (auto)
haftmann@40607
  1282
  hence "g (snd x) = g (snd y)" by (cases x,cases y,auto)
wenzelm@60758
  1283
  with \<open>inj_on g B\<close> and \<open>snd x \<in> B\<close> and \<open>snd y \<in> B\<close>
haftmann@40607
  1284
  have "snd x = snd y" by (auto dest:dest:inj_onD)
haftmann@40607
  1285
  ultimately show "x = y" by(rule prod_eqI)
haftmann@40607
  1286
qed
haftmann@40607
  1287
blanchet@55932
  1288
lemma map_prod_surj:
hoelzl@40702
  1289
  fixes f :: "'a \<Rightarrow> 'b" and g :: "'c \<Rightarrow> 'd"
haftmann@40607
  1290
  assumes "surj f" and "surj g"
blanchet@55932
  1291
  shows "surj (map_prod f g)"
haftmann@40607
  1292
unfolding surj_def
haftmann@40607
  1293
proof
haftmann@40607
  1294
  fix y :: "'b \<times> 'd"
wenzelm@60758
  1295
  from \<open>surj f\<close> obtain a where "fst y = f a" by (auto elim:surjE)
haftmann@40607
  1296
  moreover
wenzelm@60758
  1297
  from \<open>surj g\<close> obtain b where "snd y = g b" by (auto elim:surjE)
blanchet@55932
  1298
  ultimately have "(fst y, snd y) = map_prod f g (a,b)" by auto
blanchet@55932
  1299
  thus "\<exists>x. y = map_prod f g x" by auto
haftmann@40607
  1300
qed
haftmann@40607
  1301
blanchet@55932
  1302
lemma map_prod_surj_on:
haftmann@40607
  1303
  assumes "f ` A = A'" and "g ` B = B'"
blanchet@55932
  1304
  shows "map_prod f g ` (A \<times> B) = A' \<times> B'"
haftmann@40607
  1305
unfolding image_def
haftmann@40607
  1306
proof(rule set_eqI,rule iffI)
haftmann@40607
  1307
  fix x :: "'a \<times> 'c"
wenzelm@61076
  1308
  assume "x \<in> {y::'a \<times> 'c. \<exists>x::'b \<times> 'd\<in>A \<times> B. y = map_prod f g x}"
blanchet@55932
  1309
  then obtain y where "y \<in> A \<times> B" and "x = map_prod f g y" by blast
wenzelm@60758
  1310
  from \<open>image f A = A'\<close> and \<open>y \<in> A \<times> B\<close> have "f (fst y) \<in> A'" by auto
wenzelm@60758
  1311
  moreover from \<open>image g B = B'\<close> and \<open>y \<in> A \<times> B\<close> have "g (snd y) \<in> B'" by auto
haftmann@40607
  1312
  ultimately have "(f (fst y), g (snd y)) \<in> (A' \<times> B')" by auto
wenzelm@60758
  1313
  with \<open>x = map_prod f g y\<close> show "x \<in> A' \<times> B'" by (cases y, auto)
haftmann@40607
  1314
next
haftmann@40607
  1315
  fix x :: "'a \<times> 'c"
haftmann@40607
  1316
  assume "x \<in> A' \<times> B'" hence "fst x \<in> A'" and "snd x \<in> B'" by auto
wenzelm@60758
  1317
  from \<open>image f A = A'\<close> and \<open>fst x \<in> A'\<close> have "fst x \<in> image f A" by auto
haftmann@40607
  1318
  then obtain a where "a \<in> A" and "fst x = f a" by (rule imageE)
wenzelm@60758
  1319
  moreover from \<open>image g B = B'\<close> and \<open>snd x \<in> B'\<close>
haftmann@40607
  1320
  obtain b where "b \<in> B" and "snd x = g b" by auto
blanchet@55932
  1321
  ultimately have "(fst x, snd x) = map_prod f g (a,b)" by auto
wenzelm@60758
  1322
  moreover from \<open>a \<in> A\<close> and  \<open>b \<in> B\<close> have "(a , b) \<in> A \<times> B" by auto
blanchet@55932
  1323
  ultimately have "\<exists>y \<in> A \<times> B. x = map_prod f g y" by auto
blanchet@55932
  1324
  thus "x \<in> {x. \<exists>y \<in> A \<times> B. x = map_prod f g y}" by auto
haftmann@40607
  1325
qed
haftmann@40607
  1326
haftmann@21908
  1327
wenzelm@60758
  1328
subsection \<open>Simproc for rewriting a set comprehension into a pointfree expression\<close>
bulwahn@49764
  1329
bulwahn@49764
  1330
ML_file "Tools/set_comprehension_pointfree.ML"
bulwahn@49764
  1331
wenzelm@60758
  1332
setup \<open>
wenzelm@51717
  1333
  Code_Preproc.map_pre (fn ctxt => ctxt addsimprocs
wenzelm@61144
  1334
    [Simplifier.make_simproc @{context} "set comprehension"
wenzelm@61144
  1335
      {lhss = [@{term "Collect P"}],
wenzelm@62913
  1336
       proc = K Set_Comprehension_Pointfree.code_simproc}])
wenzelm@60758
  1337
\<close>
bulwahn@49764
  1338
bulwahn@49764
  1339
wenzelm@60758
  1340
subsection \<open>Inductively defined sets\<close>
berghofe@15394
  1341
wenzelm@56512
  1342
(* simplify {(x1, ..., xn). (x1, ..., xn) : S} to S *)
wenzelm@60758
  1343
simproc_setup Collect_mem ("Collect t") = \<open>
wenzelm@56512
  1344
  fn _ => fn ctxt => fn ct =>
wenzelm@59582
  1345
    (case Thm.term_of ct of
wenzelm@56512
  1346
      S as Const (@{const_name Collect}, Type (@{type_name fun}, [_, T])) $ t =>
haftmann@61424
  1347
        let val (u, _, ps) = HOLogic.strip_ptupleabs t in
wenzelm@56512
  1348
          (case u of
wenzelm@56512
  1349
            (c as Const (@{const_name Set.member}, _)) $ q $ S' =>
wenzelm@56512
  1350
              (case try (HOLogic.strip_ptuple ps) q of
wenzelm@56512
  1351
                NONE => NONE
wenzelm@56512
  1352
              | SOME ts =>
wenzelm@56512
  1353
                  if not (Term.is_open S') andalso
wenzelm@56512
  1354
                    ts = map Bound (length ps downto 0)
wenzelm@56512
  1355
                  then
wenzelm@56512
  1356
                    let val simp =
wenzelm@56512
  1357
                      full_simp_tac (put_simpset HOL_basic_ss ctxt
haftmann@61424
  1358
                        addsimps [@{thm split_paired_all}, @{thm case_prod_conv}]) 1
wenzelm@56512
  1359
                    in
wenzelm@56512
  1360
                      SOME (Goal.prove ctxt [] []
wenzelm@56512
  1361
                        (Const (@{const_name Pure.eq}, T --> T --> propT) $ S $ S')
wenzelm@56512
  1362
                        (K (EVERY
wenzelm@59498
  1363
                          [resolve_tac ctxt [eq_reflection] 1,
wenzelm@59498
  1364
                           resolve_tac ctxt @{thms subset_antisym} 1,
wenzelm@59498
  1365
                           resolve_tac ctxt [subsetI] 1, dresolve_tac ctxt [CollectD] 1, simp,
wenzelm@59498
  1366
                           resolve_tac ctxt [subsetI] 1, resolve_tac ctxt [CollectI] 1, simp])))
wenzelm@56512
  1367
                    end
wenzelm@56512
  1368
                  else NONE)
wenzelm@56512
  1369
          | _ => NONE)
wenzelm@56512
  1370
        end
wenzelm@56512
  1371
    | _ => NONE)
wenzelm@60758
  1372
\<close>
blanchet@58389
  1373
wenzelm@48891
  1374
ML_file "Tools/inductive_set.ML"
haftmann@24699
  1375
haftmann@37166
  1376
wenzelm@60758
  1377
subsection \<open>Legacy theorem bindings and duplicates\<close>
haftmann@37166
  1378
blanchet@55393
  1379
lemmas fst_conv = prod.sel(1)
blanchet@55393
  1380
lemmas snd_conv = prod.sel(2)
haftmann@61032
  1381
lemmas split_def = case_prod_unfold
haftmann@61424
  1382
lemmas split_beta' = case_prod_beta'
haftmann@61424
  1383
lemmas split_beta = prod.case_eq_if
haftmann@61424
  1384
lemmas split_conv = case_prod_conv
haftmann@61424
  1385
lemmas split = case_prod_conv
huffman@44066
  1386
huffman@45204
  1387
hide_const (open) prod
huffman@45204
  1388
nipkow@10213
  1389
end