src/HOL/Product_Type.thy
author haftmann
Thu Mar 20 12:04:54 2008 +0100 (2008-03-20)
changeset 26358 d6a508c16908
parent 26340 a85fe32e7b2f
child 26480 544cef16045b
permissions -rw-r--r--
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
nipkow@10213
     1
(*  Title:      HOL/Product_Type.thy
nipkow@10213
     2
    ID:         $Id$
nipkow@10213
     3
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
nipkow@10213
     4
    Copyright   1992  University of Cambridge
wenzelm@11777
     5
*)
nipkow@10213
     6
wenzelm@11838
     7
header {* Cartesian products *}
nipkow@10213
     8
nipkow@15131
     9
theory Product_Type
haftmann@24699
    10
imports Inductive
haftmann@24699
    11
uses
haftmann@24699
    12
  ("Tools/split_rule.ML")
haftmann@24699
    13
  ("Tools/inductive_set_package.ML")
haftmann@24699
    14
  ("Tools/inductive_realizer.ML")
haftmann@24699
    15
  ("Tools/datatype_realizer.ML")
nipkow@15131
    16
begin
wenzelm@11838
    17
haftmann@24699
    18
subsection {* @{typ bool} is a datatype *}
haftmann@24699
    19
haftmann@24699
    20
rep_datatype bool
haftmann@24699
    21
  distinct True_not_False False_not_True
haftmann@24699
    22
  induction bool_induct
haftmann@24699
    23
haftmann@24699
    24
declare case_split [cases type: bool]
haftmann@24699
    25
  -- "prefer plain propositional version"
haftmann@24699
    26
haftmann@25534
    27
lemma [code func]:
haftmann@25534
    28
  shows "False = P \<longleftrightarrow> \<not> P"
haftmann@25534
    29
    and "True = P \<longleftrightarrow> P" 
haftmann@25534
    30
    and "P = False \<longleftrightarrow> \<not> P" 
haftmann@25534
    31
    and "P = True \<longleftrightarrow> P" by simp_all
haftmann@25534
    32
haftmann@25534
    33
code_const "op = \<Colon> bool \<Rightarrow> bool \<Rightarrow> bool"
haftmann@25534
    34
  (Haskell infixl 4 "==")
haftmann@25534
    35
haftmann@25534
    36
code_instance bool :: eq
haftmann@25534
    37
  (Haskell -)
haftmann@24699
    38
haftmann@26358
    39
wenzelm@11838
    40
subsection {* Unit *}
wenzelm@11838
    41
wenzelm@11838
    42
typedef unit = "{True}"
wenzelm@11838
    43
proof
haftmann@20588
    44
  show "True : ?unit" ..
wenzelm@11838
    45
qed
wenzelm@11838
    46
haftmann@24699
    47
definition
wenzelm@11838
    48
  Unity :: unit    ("'(')")
haftmann@24699
    49
where
haftmann@24699
    50
  "() = Abs_unit True"
wenzelm@11838
    51
paulson@24286
    52
lemma unit_eq [noatp]: "u = ()"
wenzelm@11838
    53
  by (induct u) (simp add: unit_def Unity_def)
wenzelm@11838
    54
wenzelm@11838
    55
text {*
wenzelm@11838
    56
  Simplification procedure for @{thm [source] unit_eq}.  Cannot use
wenzelm@11838
    57
  this rule directly --- it loops!
wenzelm@11838
    58
*}
wenzelm@11838
    59
wenzelm@11838
    60
ML_setup {*
wenzelm@13462
    61
  val unit_eq_proc =
haftmann@24699
    62
    let val unit_meta_eq = mk_meta_eq @{thm unit_eq} in
haftmann@24699
    63
      Simplifier.simproc @{theory} "unit_eq" ["x::unit"]
skalberg@15531
    64
      (fn _ => fn _ => fn t => if HOLogic.is_unit t then NONE else SOME unit_meta_eq)
wenzelm@13462
    65
    end;
wenzelm@11838
    66
wenzelm@11838
    67
  Addsimprocs [unit_eq_proc];
wenzelm@11838
    68
*}
wenzelm@11838
    69
haftmann@24699
    70
lemma unit_induct [noatp,induct type: unit]: "P () ==> P x"
haftmann@24699
    71
  by simp
haftmann@24699
    72
haftmann@24699
    73
rep_datatype unit
haftmann@24699
    74
  induction unit_induct
haftmann@24699
    75
wenzelm@11838
    76
lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()"
wenzelm@11838
    77
  by simp
wenzelm@11838
    78
wenzelm@11838
    79
lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P"
wenzelm@11838
    80
  by (rule triv_forall_equality)
wenzelm@11838
    81
wenzelm@11838
    82
text {*
wenzelm@11838
    83
  This rewrite counters the effect of @{text unit_eq_proc} on @{term
wenzelm@11838
    84
  [source] "%u::unit. f u"}, replacing it by @{term [source]
wenzelm@11838
    85
  f} rather than by @{term [source] "%u. f ()"}.
wenzelm@11838
    86
*}
wenzelm@11838
    87
paulson@24286
    88
lemma unit_abs_eta_conv [simp,noatp]: "(%u::unit. f ()) = f"
wenzelm@11838
    89
  by (rule ext) simp
nipkow@10213
    90
nipkow@10213
    91
haftmann@26358
    92
text {* code generator setup *}
haftmann@26358
    93
haftmann@26358
    94
instance unit :: eq ..
haftmann@26358
    95
haftmann@26358
    96
lemma [code func]:
haftmann@26358
    97
  "(u\<Colon>unit) = v \<longleftrightarrow> True" unfolding unit_eq [of u] unit_eq [of v] by rule+
haftmann@26358
    98
haftmann@26358
    99
code_type unit
haftmann@26358
   100
  (SML "unit")
haftmann@26358
   101
  (OCaml "unit")
haftmann@26358
   102
  (Haskell "()")
haftmann@26358
   103
haftmann@26358
   104
code_instance unit :: eq
haftmann@26358
   105
  (Haskell -)
haftmann@26358
   106
haftmann@26358
   107
code_const "op = \<Colon> unit \<Rightarrow> unit \<Rightarrow> bool"
haftmann@26358
   108
  (Haskell infixl 4 "==")
haftmann@26358
   109
haftmann@26358
   110
code_const Unity
haftmann@26358
   111
  (SML "()")
haftmann@26358
   112
  (OCaml "()")
haftmann@26358
   113
  (Haskell "()")
haftmann@26358
   114
haftmann@26358
   115
code_reserved SML
haftmann@26358
   116
  unit
haftmann@26358
   117
haftmann@26358
   118
code_reserved OCaml
haftmann@26358
   119
  unit
haftmann@26358
   120
haftmann@26358
   121
wenzelm@11838
   122
subsection {* Pairs *}
nipkow@10213
   123
haftmann@26358
   124
subsubsection {* Product type, basic operations and concrete syntax *}
nipkow@10213
   125
haftmann@26358
   126
definition
haftmann@26358
   127
  Pair_Rep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
haftmann@26358
   128
where
haftmann@26358
   129
  "Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)"
nipkow@10213
   130
nipkow@10213
   131
global
nipkow@10213
   132
nipkow@10213
   133
typedef (Prod)
haftmann@22838
   134
  ('a, 'b) "*"    (infixr "*" 20)
haftmann@26358
   135
    = "{f. \<exists>a b. f = Pair_Rep (a\<Colon>'a) (b\<Colon>'b)}"
oheimb@11025
   136
proof
haftmann@26358
   137
  fix a b show "Pair_Rep a b \<in> ?Prod"
haftmann@26358
   138
    by rule+
oheimb@11025
   139
qed
nipkow@10213
   140
wenzelm@12114
   141
syntax (xsymbols)
schirmer@15422
   142
  "*"      :: "[type, type] => type"         ("(_ \<times>/ _)" [21, 20] 20)
nipkow@10213
   143
syntax (HTML output)
schirmer@15422
   144
  "*"      :: "[type, type] => type"         ("(_ \<times>/ _)" [21, 20] 20)
nipkow@10213
   145
nipkow@10213
   146
consts
haftmann@26358
   147
  Pair     :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b"
haftmann@26358
   148
  fst      :: "'a \<times> 'b \<Rightarrow> 'a"
haftmann@26358
   149
  snd      :: "'a \<times> 'b \<Rightarrow> 'b"
haftmann@26358
   150
  split    :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c"
haftmann@26358
   151
  curry    :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c"
nipkow@10213
   152
wenzelm@11777
   153
local
nipkow@10213
   154
wenzelm@19535
   155
defs
wenzelm@19535
   156
  Pair_def:     "Pair a b == Abs_Prod (Pair_Rep a b)"
wenzelm@19535
   157
  fst_def:      "fst p == THE a. EX b. p = Pair a b"
wenzelm@19535
   158
  snd_def:      "snd p == THE b. EX a. p = Pair a b"
haftmann@24162
   159
  split_def:    "split == (%c p. c (fst p) (snd p))"
haftmann@24162
   160
  curry_def:    "curry == (%c x y. c (Pair x y))"
wenzelm@19535
   161
wenzelm@11777
   162
text {*
wenzelm@11777
   163
  Patterns -- extends pre-defined type @{typ pttrn} used in
wenzelm@11777
   164
  abstractions.
wenzelm@11777
   165
*}
nipkow@10213
   166
nipkow@10213
   167
nonterminals
nipkow@10213
   168
  tuple_args patterns
nipkow@10213
   169
nipkow@10213
   170
syntax
nipkow@10213
   171
  "_tuple"      :: "'a => tuple_args => 'a * 'b"        ("(1'(_,/ _'))")
nipkow@10213
   172
  "_tuple_arg"  :: "'a => tuple_args"                   ("_")
nipkow@10213
   173
  "_tuple_args" :: "'a => tuple_args => tuple_args"     ("_,/ _")
oheimb@11025
   174
  "_pattern"    :: "[pttrn, patterns] => pttrn"         ("'(_,/ _')")
oheimb@11025
   175
  ""            :: "pttrn => patterns"                  ("_")
oheimb@11025
   176
  "_patterns"   :: "[pttrn, patterns] => patterns"      ("_,/ _")
nipkow@10213
   177
nipkow@10213
   178
translations
nipkow@10213
   179
  "(x, y)"       == "Pair x y"
nipkow@10213
   180
  "_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))"
nipkow@10213
   181
  "%(x,y,zs).b"  == "split(%x (y,zs).b)"
nipkow@10213
   182
  "%(x,y).b"     == "split(%x y. b)"
nipkow@10213
   183
  "_abs (Pair x y) t" => "%(x,y).t"
nipkow@10213
   184
  (* The last rule accommodates tuples in `case C ... (x,y) ... => ...'
nipkow@10213
   185
     The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *)
nipkow@10213
   186
schirmer@14359
   187
(* reconstructs pattern from (nested) splits, avoiding eta-contraction of body*)
schirmer@14359
   188
(* works best with enclosing "let", if "let" does not avoid eta-contraction   *)
schirmer@14359
   189
print_translation {*
schirmer@14359
   190
let fun split_tr' [Abs (x,T,t as (Abs abs))] =
schirmer@14359
   191
      (* split (%x y. t) => %(x,y) t *)
schirmer@14359
   192
      let val (y,t') = atomic_abs_tr' abs;
schirmer@14359
   193
          val (x',t'') = atomic_abs_tr' (x,T,t');
schirmer@14359
   194
    
schirmer@14359
   195
      in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end
schirmer@14359
   196
    | split_tr' [Abs (x,T,(s as Const ("split",_)$t))] =
schirmer@14359
   197
       (* split (%x. (split (%y z. t))) => %(x,y,z). t *)
schirmer@14359
   198
       let val (Const ("_abs",_)$(Const ("_pattern",_)$y$z)$t') = split_tr' [t];
schirmer@14359
   199
           val (x',t'') = atomic_abs_tr' (x,T,t');
schirmer@14359
   200
       in Syntax.const "_abs"$ 
schirmer@14359
   201
           (Syntax.const "_pattern"$x'$(Syntax.const "_patterns"$y$z))$t'' end
schirmer@14359
   202
    | split_tr' [Const ("split",_)$t] =
schirmer@14359
   203
       (* split (split (%x y z. t)) => %((x,y),z). t *)   
schirmer@14359
   204
       split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *)
schirmer@14359
   205
    | split_tr' [Const ("_abs",_)$x_y$(Abs abs)] =
schirmer@14359
   206
       (* split (%pttrn z. t) => %(pttrn,z). t *)
schirmer@14359
   207
       let val (z,t) = atomic_abs_tr' abs;
schirmer@14359
   208
       in Syntax.const "_abs" $ (Syntax.const "_pattern" $x_y$z) $ t end
schirmer@14359
   209
    | split_tr' _ =  raise Match;
schirmer@14359
   210
in [("split", split_tr')]
schirmer@14359
   211
end
schirmer@14359
   212
*}
schirmer@14359
   213
schirmer@15422
   214
(* print "split f" as "\<lambda>(x,y). f x y" and "split (\<lambda>x. f x)" as "\<lambda>(x,y). f x y" *) 
schirmer@15422
   215
typed_print_translation {*
schirmer@15422
   216
let
schirmer@15422
   217
  fun split_guess_names_tr' _ T [Abs (x,_,Abs _)] = raise Match
schirmer@15422
   218
    | split_guess_names_tr' _ T  [Abs (x,xT,t)] =
schirmer@15422
   219
        (case (head_of t) of
schirmer@15422
   220
           Const ("split",_) => raise Match
schirmer@15422
   221
         | _ => let 
schirmer@15422
   222
                  val (_::yT::_) = binder_types (domain_type T) handle Bind => raise Match;
schirmer@15422
   223
                  val (y,t') = atomic_abs_tr' ("y",yT,(incr_boundvars 1 t)$Bound 0); 
schirmer@15422
   224
                  val (x',t'') = atomic_abs_tr' (x,xT,t');
schirmer@15422
   225
                in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end)
schirmer@15422
   226
    | split_guess_names_tr' _ T [t] =
schirmer@15422
   227
       (case (head_of t) of
schirmer@15422
   228
           Const ("split",_) => raise Match 
schirmer@15422
   229
         | _ => let 
schirmer@15422
   230
                  val (xT::yT::_) = binder_types (domain_type T) handle Bind => raise Match;
schirmer@15422
   231
                  val (y,t') = 
schirmer@15422
   232
                        atomic_abs_tr' ("y",yT,(incr_boundvars 2 t)$Bound 1$Bound 0); 
schirmer@15422
   233
                  val (x',t'') = atomic_abs_tr' ("x",xT,t');
schirmer@15422
   234
                in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end)
schirmer@15422
   235
    | split_guess_names_tr' _ _ _ = raise Match;
schirmer@15422
   236
in [("split", split_guess_names_tr')]
schirmer@15422
   237
end 
schirmer@15422
   238
*}
schirmer@15422
   239
nipkow@10213
   240
haftmann@26358
   241
text {* Towards a datatype declaration *}
wenzelm@11838
   242
haftmann@26358
   243
lemma surj_pair [simp]: "EX x y. p = (x, y)"
haftmann@26358
   244
  apply (unfold Pair_def)
haftmann@26358
   245
  apply (rule Rep_Prod [unfolded Prod_def, THEN CollectE])
haftmann@26358
   246
  apply (erule exE, erule exE, rule exI, rule exI)
haftmann@26358
   247
  apply (rule Rep_Prod_inverse [symmetric, THEN trans])
haftmann@26358
   248
  apply (erule arg_cong)
haftmann@26358
   249
  done
wenzelm@11838
   250
haftmann@26358
   251
lemma PairE [cases type: *]:
haftmann@26358
   252
  obtains x y where "p = (x, y)"
haftmann@26358
   253
  using surj_pair [of p] by blast
haftmann@26358
   254
haftmann@26358
   255
haftmann@26358
   256
lemma prod_induct [induct type: *]: "(\<And>a b. P (a, b)) \<Longrightarrow> P x"
haftmann@26358
   257
  by (cases x) simp
haftmann@26358
   258
haftmann@26358
   259
lemma ProdI: "Pair_Rep a b \<in> Prod"
haftmann@26358
   260
  unfolding Prod_def by rule+
haftmann@26358
   261
haftmann@26358
   262
lemma Pair_Rep_inject: "Pair_Rep a b = Pair_Rep a' b' \<Longrightarrow> a = a' \<and> b = b'"
haftmann@26358
   263
  unfolding Pair_Rep_def by (drule fun_cong, drule fun_cong) blast
nipkow@10213
   264
wenzelm@11838
   265
lemma inj_on_Abs_Prod: "inj_on Abs_Prod Prod"
wenzelm@11838
   266
  apply (rule inj_on_inverseI)
wenzelm@11838
   267
  apply (erule Abs_Prod_inverse)
wenzelm@11838
   268
  done
wenzelm@11838
   269
wenzelm@11838
   270
lemma Pair_inject:
wenzelm@18372
   271
  assumes "(a, b) = (a', b')"
wenzelm@18372
   272
    and "a = a' ==> b = b' ==> R"
wenzelm@18372
   273
  shows R
wenzelm@18372
   274
  apply (insert prems [unfolded Pair_def])
wenzelm@18372
   275
  apply (rule inj_on_Abs_Prod [THEN inj_onD, THEN Pair_Rep_inject, THEN conjE])
wenzelm@18372
   276
  apply (assumption | rule ProdI)+
wenzelm@18372
   277
  done
nipkow@10213
   278
wenzelm@11838
   279
lemma Pair_eq [iff]: "((a, b) = (a', b')) = (a = a' & b = b')"
wenzelm@11838
   280
  by (blast elim!: Pair_inject)
wenzelm@11838
   281
haftmann@22886
   282
lemma fst_conv [simp, code]: "fst (a, b) = a"
wenzelm@19535
   283
  unfolding fst_def by blast
wenzelm@11838
   284
haftmann@22886
   285
lemma snd_conv [simp, code]: "snd (a, b) = b"
wenzelm@19535
   286
  unfolding snd_def by blast
oheimb@11025
   287
haftmann@26358
   288
rep_datatype prod
haftmann@26358
   289
  inject Pair_eq
haftmann@26358
   290
  induction prod_induct
haftmann@26358
   291
haftmann@26358
   292
haftmann@26358
   293
subsubsection {* Basic rules and proof tools *}
haftmann@26358
   294
wenzelm@11838
   295
lemma fst_eqD: "fst (x, y) = a ==> x = a"
wenzelm@11838
   296
  by simp
wenzelm@11838
   297
wenzelm@11838
   298
lemma snd_eqD: "snd (x, y) = a ==> y = a"
wenzelm@11838
   299
  by simp
wenzelm@11838
   300
haftmann@26358
   301
lemma pair_collapse [simp]: "(fst p, snd p) = p"
wenzelm@11838
   302
  by (cases p) simp
wenzelm@11838
   303
haftmann@26358
   304
lemmas surjective_pairing = pair_collapse [symmetric]
wenzelm@11838
   305
wenzelm@11838
   306
lemma split_paired_all: "(!!x. PROP P x) == (!!a b. PROP P (a, b))"
wenzelm@11820
   307
proof
wenzelm@11820
   308
  fix a b
wenzelm@11820
   309
  assume "!!x. PROP P x"
wenzelm@19535
   310
  then show "PROP P (a, b)" .
wenzelm@11820
   311
next
wenzelm@11820
   312
  fix x
wenzelm@11820
   313
  assume "!!a b. PROP P (a, b)"
wenzelm@19535
   314
  from `PROP P (fst x, snd x)` show "PROP P x" by simp
wenzelm@11820
   315
qed
wenzelm@11820
   316
wenzelm@11838
   317
text {*
wenzelm@11838
   318
  The rule @{thm [source] split_paired_all} does not work with the
wenzelm@11838
   319
  Simplifier because it also affects premises in congrence rules,
wenzelm@11838
   320
  where this can lead to premises of the form @{text "!!a b. ... =
wenzelm@11838
   321
  ?P(a, b)"} which cannot be solved by reflexivity.
wenzelm@11838
   322
*}
wenzelm@11838
   323
haftmann@26358
   324
lemmas split_tupled_all = split_paired_all unit_all_eq2
haftmann@26358
   325
haftmann@26358
   326
ML_setup {*
wenzelm@11838
   327
  (* replace parameters of product type by individual component parameters *)
wenzelm@11838
   328
  val safe_full_simp_tac = generic_simp_tac true (true, false, false);
wenzelm@11838
   329
  local (* filtering with exists_paired_all is an essential optimization *)
wenzelm@16121
   330
    fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) =
wenzelm@11838
   331
          can HOLogic.dest_prodT T orelse exists_paired_all t
wenzelm@11838
   332
      | exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u
wenzelm@11838
   333
      | exists_paired_all (Abs (_, _, t)) = exists_paired_all t
wenzelm@11838
   334
      | exists_paired_all _ = false;
wenzelm@11838
   335
    val ss = HOL_basic_ss
wenzelm@26340
   336
      addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}]
wenzelm@11838
   337
      addsimprocs [unit_eq_proc];
wenzelm@11838
   338
  in
wenzelm@11838
   339
    val split_all_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838
   340
      if exists_paired_all t then safe_full_simp_tac ss i else no_tac);
wenzelm@11838
   341
    val unsafe_split_all_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838
   342
      if exists_paired_all t then full_simp_tac ss i else no_tac);
wenzelm@11838
   343
    fun split_all th =
wenzelm@26340
   344
   if exists_paired_all (Thm.prop_of th) then full_simplify ss th else th;
wenzelm@11838
   345
  end;
wenzelm@26340
   346
*}
wenzelm@11838
   347
wenzelm@26340
   348
declaration {* fn _ =>
wenzelm@26340
   349
  Classical.map_cs (fn cs => cs addSbefore ("split_all_tac", split_all_tac))
wenzelm@16121
   350
*}
wenzelm@11838
   351
wenzelm@11838
   352
lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))"
wenzelm@11838
   353
  -- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *}
wenzelm@11838
   354
  by fast
wenzelm@11838
   355
haftmann@26358
   356
lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))"
haftmann@26358
   357
  by fast
haftmann@26358
   358
haftmann@26358
   359
lemma Pair_fst_snd_eq: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t"
haftmann@26358
   360
  by (cases s, cases t) simp
haftmann@26358
   361
haftmann@26358
   362
lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q"
haftmann@26358
   363
  by (simp add: Pair_fst_snd_eq)
haftmann@26358
   364
haftmann@26358
   365
haftmann@26358
   366
subsubsection {* @{text split} and @{text curry} *}
haftmann@26358
   367
haftmann@26358
   368
lemma split_conv [simp, code func]: "split f (a, b) = f a b"
haftmann@26358
   369
  by (simp add: split_def)
haftmann@26358
   370
haftmann@26358
   371
lemma curry_conv [simp, code func]: "curry f a b = f (a, b)"
haftmann@26358
   372
  by (simp add: curry_def)
haftmann@26358
   373
haftmann@26358
   374
lemmas split = split_conv  -- {* for backwards compatibility *}
haftmann@26358
   375
haftmann@26358
   376
lemma splitI: "f a b \<Longrightarrow> split f (a, b)"
haftmann@26358
   377
  by (rule split_conv [THEN iffD2])
haftmann@26358
   378
haftmann@26358
   379
lemma splitD: "split f (a, b) \<Longrightarrow> f a b"
haftmann@26358
   380
  by (rule split_conv [THEN iffD1])
haftmann@26358
   381
haftmann@26358
   382
lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b"
haftmann@26358
   383
  by (simp add: curry_def)
haftmann@26358
   384
haftmann@26358
   385
lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)"
haftmann@26358
   386
  by (simp add: curry_def)
haftmann@26358
   387
haftmann@26358
   388
lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q"
haftmann@26358
   389
  by (simp add: curry_def)
haftmann@26358
   390
skalberg@14189
   391
lemma curry_split [simp]: "curry (split f) = f"
skalberg@14189
   392
  by (simp add: curry_def split_def)
skalberg@14189
   393
skalberg@14189
   394
lemma split_curry [simp]: "split (curry f) = f"
skalberg@14189
   395
  by (simp add: curry_def split_def)
skalberg@14189
   396
haftmann@26358
   397
lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id"
haftmann@26358
   398
  by (simp add: split_def id_def)
wenzelm@11838
   399
haftmann@26358
   400
lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f"
haftmann@26358
   401
  -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *}
haftmann@26358
   402
  by (rule ext) auto
wenzelm@11838
   403
haftmann@26358
   404
lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)"
haftmann@26358
   405
  by (cases x) simp
wenzelm@11838
   406
haftmann@26358
   407
lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p"
haftmann@26358
   408
  unfolding split_def ..
wenzelm@11838
   409
wenzelm@11838
   410
lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))"
wenzelm@11838
   411
  -- {* Can't be added to simpset: loops! *}
haftmann@26358
   412
  by (simp add: split_eta)
wenzelm@11838
   413
wenzelm@11838
   414
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
wenzelm@11838
   415
  by (simp add: split_def)
wenzelm@11838
   416
haftmann@26358
   417
lemma split_weak_cong: "p = q \<Longrightarrow> split c p = split c q"
wenzelm@11838
   418
  -- {* Prevents simplification of @{term c}: much faster *}
wenzelm@11838
   419
  by (erule arg_cong)
wenzelm@11838
   420
wenzelm@11838
   421
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
wenzelm@11838
   422
  by (simp add: split_eta)
wenzelm@11838
   423
wenzelm@11838
   424
text {*
wenzelm@11838
   425
  Simplification procedure for @{thm [source] cond_split_eta}.  Using
wenzelm@11838
   426
  @{thm [source] split_eta} as a rewrite rule is not general enough,
wenzelm@11838
   427
  and using @{thm [source] cond_split_eta} directly would render some
wenzelm@11838
   428
  existing proofs very inefficient; similarly for @{text
haftmann@26358
   429
  split_beta}.
haftmann@26358
   430
*}
wenzelm@11838
   431
wenzelm@11838
   432
ML_setup {*
wenzelm@11838
   433
wenzelm@11838
   434
local
wenzelm@18328
   435
  val cond_split_eta_ss = HOL_basic_ss addsimps [thm "cond_split_eta"]
wenzelm@11838
   436
  fun  Pair_pat k 0 (Bound m) = (m = k)
wenzelm@11838
   437
  |    Pair_pat k i (Const ("Pair",  _) $ Bound m $ t) = i > 0 andalso
wenzelm@11838
   438
                        m = k+i andalso Pair_pat k (i-1) t
wenzelm@11838
   439
  |    Pair_pat _ _ _ = false;
wenzelm@11838
   440
  fun no_args k i (Abs (_, _, t)) = no_args (k+1) i t
wenzelm@11838
   441
  |   no_args k i (t $ u) = no_args k i t andalso no_args k i u
wenzelm@11838
   442
  |   no_args k i (Bound m) = m < k orelse m > k+i
wenzelm@11838
   443
  |   no_args _ _ _ = true;
skalberg@15531
   444
  fun split_pat tp i (Abs  (_,_,t)) = if tp 0 i t then SOME (i,t) else NONE
wenzelm@11838
   445
  |   split_pat tp i (Const ("split", _) $ Abs (_, _, t)) = split_pat tp (i+1) t
skalberg@15531
   446
  |   split_pat tp i _ = NONE;
wenzelm@20044
   447
  fun metaeq ss lhs rhs = mk_meta_eq (Goal.prove (Simplifier.the_context ss) [] []
wenzelm@13480
   448
        (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs,rhs)))
wenzelm@18328
   449
        (K (simp_tac (Simplifier.inherit_context ss cond_split_eta_ss) 1)));
wenzelm@11838
   450
wenzelm@11838
   451
  fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k+1) i t
wenzelm@11838
   452
  |   beta_term_pat k i (t $ u) = Pair_pat k i (t $ u) orelse
wenzelm@11838
   453
                        (beta_term_pat k i t andalso beta_term_pat k i u)
wenzelm@11838
   454
  |   beta_term_pat k i t = no_args k i t;
wenzelm@11838
   455
  fun  eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
wenzelm@11838
   456
  |    eta_term_pat _ _ _ = false;
wenzelm@11838
   457
  fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
wenzelm@11838
   458
  |   subst arg k i (t $ u) = if Pair_pat k i (t $ u) then incr_boundvars k arg
wenzelm@11838
   459
                              else (subst arg k i t $ subst arg k i u)
wenzelm@11838
   460
  |   subst arg k i t = t;
wenzelm@20044
   461
  fun beta_proc ss (s as Const ("split", _) $ Abs (_, _, t) $ arg) =
wenzelm@11838
   462
        (case split_pat beta_term_pat 1 t of
wenzelm@20044
   463
        SOME (i,f) => SOME (metaeq ss s (subst arg 0 i f))
skalberg@15531
   464
        | NONE => NONE)
wenzelm@20044
   465
  |   beta_proc _ _ = NONE;
wenzelm@20044
   466
  fun eta_proc ss (s as Const ("split", _) $ Abs (_, _, t)) =
wenzelm@11838
   467
        (case split_pat eta_term_pat 1 t of
wenzelm@20044
   468
          SOME (_,ft) => SOME (metaeq ss s (let val (f $ arg) = ft in f end))
skalberg@15531
   469
        | NONE => NONE)
wenzelm@20044
   470
  |   eta_proc _ _ = NONE;
wenzelm@11838
   471
in
wenzelm@22577
   472
  val split_beta_proc = Simplifier.simproc @{theory} "split_beta" ["split f z"] (K beta_proc);
wenzelm@22577
   473
  val split_eta_proc = Simplifier.simproc @{theory} "split_eta" ["split f"] (K eta_proc);
wenzelm@11838
   474
end;
wenzelm@11838
   475
wenzelm@11838
   476
Addsimprocs [split_beta_proc, split_eta_proc];
wenzelm@11838
   477
*}
wenzelm@11838
   478
wenzelm@11838
   479
lemma split_beta: "(%(x, y). P x y) z = P (fst z) (snd z)"
wenzelm@11838
   480
  by (subst surjective_pairing, rule split_conv)
wenzelm@11838
   481
paulson@24286
   482
lemma split_split [noatp]: "R(split c p) = (ALL x y. p = (x, y) --> R(c x y))"
wenzelm@11838
   483
  -- {* For use with @{text split} and the Simplifier. *}
paulson@15481
   484
  by (insert surj_pair [of p], clarify, simp)
wenzelm@11838
   485
wenzelm@11838
   486
text {*
wenzelm@11838
   487
  @{thm [source] split_split} could be declared as @{text "[split]"}
wenzelm@11838
   488
  done after the Splitter has been speeded up significantly;
wenzelm@11838
   489
  precompute the constants involved and don't do anything unless the
wenzelm@11838
   490
  current goal contains one of those constants.
wenzelm@11838
   491
*}
wenzelm@11838
   492
paulson@24286
   493
lemma split_split_asm [noatp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))"
paulson@14208
   494
by (subst split_split, simp)
wenzelm@11838
   495
wenzelm@11838
   496
wenzelm@11838
   497
text {*
wenzelm@11838
   498
  \medskip @{term split} used as a logical connective or set former.
wenzelm@11838
   499
wenzelm@11838
   500
  \medskip These rules are for use with @{text blast}; could instead
wenzelm@11838
   501
  call @{text simp} using @{thm [source] split} as rewrite. *}
wenzelm@11838
   502
wenzelm@11838
   503
lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"
wenzelm@11838
   504
  apply (simp only: split_tupled_all)
wenzelm@11838
   505
  apply (simp (no_asm_simp))
wenzelm@11838
   506
  done
wenzelm@11838
   507
wenzelm@11838
   508
lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"
wenzelm@11838
   509
  apply (simp only: split_tupled_all)
wenzelm@11838
   510
  apply (simp (no_asm_simp))
wenzelm@11838
   511
  done
wenzelm@11838
   512
wenzelm@11838
   513
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
wenzelm@11838
   514
  by (induct p) (auto simp add: split_def)
wenzelm@11838
   515
wenzelm@11838
   516
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
wenzelm@11838
   517
  by (induct p) (auto simp add: split_def)
wenzelm@11838
   518
wenzelm@11838
   519
lemma splitE2:
wenzelm@11838
   520
  "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
wenzelm@11838
   521
proof -
wenzelm@11838
   522
  assume q: "Q (split P z)"
wenzelm@11838
   523
  assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
wenzelm@11838
   524
  show R
wenzelm@11838
   525
    apply (rule r surjective_pairing)+
wenzelm@11838
   526
    apply (rule split_beta [THEN subst], rule q)
wenzelm@11838
   527
    done
wenzelm@11838
   528
qed
wenzelm@11838
   529
wenzelm@11838
   530
lemma splitD': "split R (a,b) c ==> R a b c"
wenzelm@11838
   531
  by simp
wenzelm@11838
   532
wenzelm@11838
   533
lemma mem_splitI: "z: c a b ==> z: split c (a, b)"
wenzelm@11838
   534
  by simp
wenzelm@11838
   535
wenzelm@11838
   536
lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"
paulson@14208
   537
by (simp only: split_tupled_all, simp)
wenzelm@11838
   538
wenzelm@18372
   539
lemma mem_splitE:
wenzelm@18372
   540
  assumes major: "z: split c p"
wenzelm@18372
   541
    and cases: "!!x y. [| p = (x,y); z: c x y |] ==> Q"
wenzelm@18372
   542
  shows Q
wenzelm@18372
   543
  by (rule major [unfolded split_def] cases surjective_pairing)+
wenzelm@11838
   544
wenzelm@11838
   545
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
wenzelm@11838
   546
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]
wenzelm@11838
   547
wenzelm@26340
   548
ML {*
wenzelm@11838
   549
local (* filtering with exists_p_split is an essential optimization *)
wenzelm@16121
   550
  fun exists_p_split (Const ("split",_) $ _ $ (Const ("Pair",_)$_$_)) = true
wenzelm@11838
   551
    | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
wenzelm@11838
   552
    | exists_p_split (Abs (_, _, t)) = exists_p_split t
wenzelm@11838
   553
    | exists_p_split _ = false;
wenzelm@16121
   554
  val ss = HOL_basic_ss addsimps [thm "split_conv"];
wenzelm@11838
   555
in
wenzelm@11838
   556
val split_conv_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838
   557
    if exists_p_split t then safe_full_simp_tac ss i else no_tac);
wenzelm@11838
   558
end;
wenzelm@26340
   559
*}
wenzelm@26340
   560
wenzelm@11838
   561
(* This prevents applications of splitE for already splitted arguments leading
wenzelm@11838
   562
   to quite time-consuming computations (in particular for nested tuples) *)
wenzelm@26340
   563
declaration {* fn _ =>
wenzelm@26340
   564
  Classical.map_cs (fn cs => cs addSbefore ("split_conv_tac", split_conv_tac))
wenzelm@16121
   565
*}
wenzelm@11838
   566
paulson@24286
   567
lemma split_eta_SetCompr [simp,noatp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
wenzelm@18372
   568
  by (rule ext) fast
wenzelm@11838
   569
paulson@24286
   570
lemma split_eta_SetCompr2 [simp,noatp]: "(%u. EX x y. u = (x, y) & P x y) = split P"
wenzelm@18372
   571
  by (rule ext) fast
wenzelm@11838
   572
wenzelm@11838
   573
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"
wenzelm@11838
   574
  -- {* Allows simplifications of nested splits in case of independent predicates. *}
wenzelm@18372
   575
  by (rule ext) blast
wenzelm@11838
   576
nipkow@14337
   577
(* Do NOT make this a simp rule as it
nipkow@14337
   578
   a) only helps in special situations
nipkow@14337
   579
   b) can lead to nontermination in the presence of split_def
nipkow@14337
   580
*)
nipkow@14337
   581
lemma split_comp_eq: 
paulson@20415
   582
  fixes f :: "'a => 'b => 'c" and g :: "'d => 'a"
paulson@20415
   583
  shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"
wenzelm@18372
   584
  by (rule ext) auto
oheimb@14101
   585
haftmann@26358
   586
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
haftmann@26358
   587
  apply (rule_tac x = "(a, b)" in image_eqI)
haftmann@26358
   588
   apply auto
haftmann@26358
   589
  done
haftmann@26358
   590
wenzelm@11838
   591
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
wenzelm@11838
   592
  by blast
wenzelm@11838
   593
wenzelm@11838
   594
(*
wenzelm@11838
   595
the following  would be slightly more general,
wenzelm@11838
   596
but cannot be used as rewrite rule:
wenzelm@11838
   597
### Cannot add premise as rewrite rule because it contains (type) unknowns:
wenzelm@11838
   598
### ?y = .x
wenzelm@11838
   599
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
paulson@14208
   600
by (rtac some_equality 1)
paulson@14208
   601
by ( Simp_tac 1)
paulson@14208
   602
by (split_all_tac 1)
paulson@14208
   603
by (Asm_full_simp_tac 1)
wenzelm@11838
   604
qed "The_split_eq";
wenzelm@11838
   605
*)
wenzelm@11838
   606
wenzelm@11838
   607
text {*
wenzelm@11838
   608
  Setup of internal @{text split_rule}.
wenzelm@11838
   609
*}
wenzelm@11838
   610
haftmann@25511
   611
definition
haftmann@25511
   612
  internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c"
haftmann@25511
   613
where
wenzelm@11032
   614
  "internal_split == split"
wenzelm@11032
   615
wenzelm@11032
   616
lemma internal_split_conv: "internal_split c (a, b) = c a b"
wenzelm@11032
   617
  by (simp only: internal_split_def split_conv)
wenzelm@11032
   618
wenzelm@11032
   619
hide const internal_split
wenzelm@11032
   620
oheimb@11025
   621
use "Tools/split_rule.ML"
wenzelm@11032
   622
setup SplitRule.setup
nipkow@10213
   623
haftmann@24699
   624
lemmas prod_caseI = prod.cases [THEN iffD2, standard]
haftmann@24699
   625
haftmann@24699
   626
lemma prod_caseI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> prod_case c p"
haftmann@24699
   627
  by auto
haftmann@24699
   628
haftmann@24699
   629
lemma prod_caseI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> prod_case c p x"
haftmann@24699
   630
  by (auto simp: split_tupled_all)
haftmann@24699
   631
haftmann@24699
   632
lemma prod_caseE: "prod_case c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
haftmann@24699
   633
  by (induct p) auto
haftmann@24699
   634
haftmann@24699
   635
lemma prod_caseE': "prod_case c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
haftmann@24699
   636
  by (induct p) auto
haftmann@24699
   637
haftmann@24699
   638
lemma prod_case_unfold: "prod_case = (%c p. c (fst p) (snd p))"
haftmann@24699
   639
  by (simp add: expand_fun_eq)
haftmann@24699
   640
haftmann@24699
   641
declare prod_caseI2' [intro!] prod_caseI2 [intro!] prod_caseI [intro!]
haftmann@24699
   642
declare prod_caseE' [elim!] prod_caseE [elim!]
haftmann@24699
   643
haftmann@24844
   644
lemma prod_case_split:
haftmann@24699
   645
  "prod_case = split"
haftmann@24699
   646
  by (auto simp add: expand_fun_eq)
haftmann@24699
   647
bulwahn@26143
   648
lemma prod_case_beta:
bulwahn@26143
   649
  "prod_case f p = f (fst p) (snd p)"
bulwahn@26143
   650
  unfolding prod_case_split split_beta ..
bulwahn@26143
   651
haftmann@24699
   652
haftmann@24699
   653
subsection {* Further cases/induct rules for tuples *}
haftmann@24699
   654
haftmann@24699
   655
lemma prod_cases3 [cases type]:
haftmann@24699
   656
  obtains (fields) a b c where "y = (a, b, c)"
haftmann@24699
   657
  by (cases y, case_tac b) blast
haftmann@24699
   658
haftmann@24699
   659
lemma prod_induct3 [case_names fields, induct type]:
haftmann@24699
   660
    "(!!a b c. P (a, b, c)) ==> P x"
haftmann@24699
   661
  by (cases x) blast
haftmann@24699
   662
haftmann@24699
   663
lemma prod_cases4 [cases type]:
haftmann@24699
   664
  obtains (fields) a b c d where "y = (a, b, c, d)"
haftmann@24699
   665
  by (cases y, case_tac c) blast
haftmann@24699
   666
haftmann@24699
   667
lemma prod_induct4 [case_names fields, induct type]:
haftmann@24699
   668
    "(!!a b c d. P (a, b, c, d)) ==> P x"
haftmann@24699
   669
  by (cases x) blast
haftmann@24699
   670
haftmann@24699
   671
lemma prod_cases5 [cases type]:
haftmann@24699
   672
  obtains (fields) a b c d e where "y = (a, b, c, d, e)"
haftmann@24699
   673
  by (cases y, case_tac d) blast
haftmann@24699
   674
haftmann@24699
   675
lemma prod_induct5 [case_names fields, induct type]:
haftmann@24699
   676
    "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
haftmann@24699
   677
  by (cases x) blast
haftmann@24699
   678
haftmann@24699
   679
lemma prod_cases6 [cases type]:
haftmann@24699
   680
  obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
haftmann@24699
   681
  by (cases y, case_tac e) blast
haftmann@24699
   682
haftmann@24699
   683
lemma prod_induct6 [case_names fields, induct type]:
haftmann@24699
   684
    "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
haftmann@24699
   685
  by (cases x) blast
haftmann@24699
   686
haftmann@24699
   687
lemma prod_cases7 [cases type]:
haftmann@24699
   688
  obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
haftmann@24699
   689
  by (cases y, case_tac f) blast
haftmann@24699
   690
haftmann@24699
   691
lemma prod_induct7 [case_names fields, induct type]:
haftmann@24699
   692
    "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
haftmann@24699
   693
  by (cases x) blast
haftmann@24699
   694
haftmann@24699
   695
haftmann@26358
   696
subsubsection {* Derived operations *}
haftmann@26358
   697
haftmann@26358
   698
text {*
haftmann@26358
   699
  The composition-uncurry combinator.
haftmann@26358
   700
*}
haftmann@26358
   701
haftmann@26358
   702
definition
haftmann@26358
   703
  mbind :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "o->" 60)
haftmann@26358
   704
where
haftmann@26358
   705
  "f o-> g = (\<lambda>x. split g (f x))"
haftmann@26358
   706
haftmann@26358
   707
notation (xsymbols)
haftmann@26358
   708
  mbind  (infixl "\<circ>\<rightarrow>" 60)
haftmann@26358
   709
haftmann@26358
   710
notation (HTML output)
haftmann@26358
   711
  mbind  (infixl "\<circ>\<rightarrow>" 60)
haftmann@26358
   712
haftmann@26358
   713
lemma mbind_apply:  "(f \<circ>\<rightarrow> g) x = split g (f x)"
haftmann@26358
   714
  by (simp add: mbind_def)
haftmann@26358
   715
haftmann@26358
   716
lemma Pair_mbind: "Pair x \<circ>\<rightarrow> f = f x"
haftmann@26358
   717
  by (simp add: expand_fun_eq mbind_apply)
haftmann@26358
   718
haftmann@26358
   719
lemma mbind_Pair: "x \<circ>\<rightarrow> Pair = x"
haftmann@26358
   720
  by (simp add: expand_fun_eq mbind_apply)
haftmann@26358
   721
haftmann@26358
   722
lemma mbind_mbind: "(f \<circ>\<rightarrow> g) \<circ>\<rightarrow> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>\<rightarrow> h)"
haftmann@26358
   723
  by (simp add: expand_fun_eq split_twice mbind_def)
haftmann@26358
   724
haftmann@26358
   725
lemma mbind_fcomp: "(f \<circ>\<rightarrow> g) \<circ>> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>> h)"
haftmann@26358
   726
  by (simp add: expand_fun_eq mbind_apply fcomp_def split_def)
haftmann@26358
   727
haftmann@26358
   728
lemma fcomp_mbind: "(f \<circ>> g) \<circ>\<rightarrow> h = f \<circ>> (g \<circ>\<rightarrow> h)"
haftmann@26358
   729
  by (simp add: expand_fun_eq mbind_apply fcomp_apply)
haftmann@26358
   730
haftmann@26358
   731
haftmann@26358
   732
text {*
haftmann@26358
   733
  @{term prod_fun} --- action of the product functor upon
haftmann@26358
   734
  functions.
haftmann@26358
   735
*}
haftmann@21195
   736
haftmann@26358
   737
definition prod_fun :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where
haftmann@26358
   738
  [code func del]: "prod_fun f g = (\<lambda>(x, y). (f x, g y))"
haftmann@26358
   739
haftmann@26358
   740
lemma prod_fun [simp, code func]: "prod_fun f g (a, b) = (f a, g b)"
haftmann@26358
   741
  by (simp add: prod_fun_def)
haftmann@26358
   742
haftmann@26358
   743
lemma prod_fun_compose: "prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)"
haftmann@26358
   744
  by (rule ext) auto
haftmann@26358
   745
haftmann@26358
   746
lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)"
haftmann@26358
   747
  by (rule ext) auto
haftmann@26358
   748
haftmann@26358
   749
lemma prod_fun_imageI [intro]: "(a, b) : r ==> (f a, g b) : prod_fun f g ` r"
haftmann@26358
   750
  apply (rule image_eqI)
haftmann@26358
   751
  apply (rule prod_fun [symmetric], assumption)
haftmann@26358
   752
  done
haftmann@21195
   753
haftmann@26358
   754
lemma prod_fun_imageE [elim!]:
haftmann@26358
   755
  assumes major: "c: (prod_fun f g)`r"
haftmann@26358
   756
    and cases: "!!x y. [| c=(f(x),g(y));  (x,y):r |] ==> P"
haftmann@26358
   757
  shows P
haftmann@26358
   758
  apply (rule major [THEN imageE])
haftmann@26358
   759
  apply (rule_tac p = x in PairE)
haftmann@26358
   760
  apply (rule cases)
haftmann@26358
   761
   apply (blast intro: prod_fun)
haftmann@26358
   762
  apply blast
haftmann@26358
   763
  done
haftmann@26358
   764
haftmann@26358
   765
definition
haftmann@26358
   766
  apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b"
haftmann@26358
   767
where
haftmann@26358
   768
  [code func del]: "apfst f = prod_fun f id"
haftmann@26358
   769
haftmann@26358
   770
definition
haftmann@26358
   771
  apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c"
haftmann@26358
   772
where
haftmann@26358
   773
  [code func del]: "apsnd f = prod_fun id f"
haftmann@26358
   774
haftmann@26358
   775
lemma apfst_conv [simp, code]:
haftmann@26358
   776
  "apfst f (x, y) = (f x, y)" 
haftmann@26358
   777
  by (simp add: apfst_def)
haftmann@26358
   778
haftmann@26358
   779
lemma upd_snd_conv [simp, code]:
haftmann@26358
   780
  "apsnd f (x, y) = (x, f y)" 
haftmann@26358
   781
  by (simp add: apsnd_def)
haftmann@21195
   782
haftmann@21195
   783
haftmann@26358
   784
text {*
haftmann@26358
   785
  Disjoint union of a family of sets -- Sigma.
haftmann@26358
   786
*}
haftmann@26358
   787
haftmann@26358
   788
definition  Sigma :: "['a set, 'a => 'b set] => ('a \<times> 'b) set" where
haftmann@26358
   789
  Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}"
haftmann@26358
   790
haftmann@26358
   791
abbreviation
haftmann@26358
   792
  Times :: "['a set, 'b set] => ('a * 'b) set"
haftmann@26358
   793
    (infixr "<*>" 80) where
haftmann@26358
   794
  "A <*> B == Sigma A (%_. B)"
haftmann@26358
   795
haftmann@26358
   796
notation (xsymbols)
haftmann@26358
   797
  Times  (infixr "\<times>" 80)
berghofe@15394
   798
haftmann@26358
   799
notation (HTML output)
haftmann@26358
   800
  Times  (infixr "\<times>" 80)
haftmann@26358
   801
haftmann@26358
   802
syntax
haftmann@26358
   803
  "@Sigma" ::"[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" [0, 0, 10] 10)
haftmann@26358
   804
haftmann@26358
   805
translations
haftmann@26358
   806
  "SIGMA x:A. B" == "Product_Type.Sigma A (%x. B)"
haftmann@26358
   807
haftmann@26358
   808
lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
haftmann@26358
   809
  by (unfold Sigma_def) blast
haftmann@26358
   810
haftmann@26358
   811
lemma SigmaE [elim!]:
haftmann@26358
   812
    "[| c: Sigma A B;
haftmann@26358
   813
        !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
haftmann@26358
   814
     |] ==> P"
haftmann@26358
   815
  -- {* The general elimination rule. *}
haftmann@26358
   816
  by (unfold Sigma_def) blast
haftmann@20588
   817
haftmann@26358
   818
text {*
haftmann@26358
   819
  Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
haftmann@26358
   820
  eigenvariables.
haftmann@26358
   821
*}
haftmann@26358
   822
haftmann@26358
   823
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
haftmann@26358
   824
  by blast
haftmann@26358
   825
haftmann@26358
   826
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
haftmann@26358
   827
  by blast
haftmann@26358
   828
haftmann@26358
   829
lemma SigmaE2:
haftmann@26358
   830
    "[| (a, b) : Sigma A B;
haftmann@26358
   831
        [| a:A;  b:B(a) |] ==> P
haftmann@26358
   832
     |] ==> P"
haftmann@26358
   833
  by blast
haftmann@20588
   834
haftmann@26358
   835
lemma Sigma_cong:
haftmann@26358
   836
     "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>
haftmann@26358
   837
      \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
haftmann@26358
   838
  by auto
haftmann@26358
   839
haftmann@26358
   840
lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
haftmann@26358
   841
  by blast
haftmann@26358
   842
haftmann@26358
   843
lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
haftmann@26358
   844
  by blast
haftmann@26358
   845
haftmann@26358
   846
lemma Sigma_empty2 [simp]: "A <*> {} = {}"
haftmann@26358
   847
  by blast
haftmann@26358
   848
haftmann@26358
   849
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
haftmann@26358
   850
  by auto
haftmann@21908
   851
haftmann@26358
   852
lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
haftmann@26358
   853
  by auto
haftmann@26358
   854
haftmann@26358
   855
lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
haftmann@26358
   856
  by auto
haftmann@26358
   857
haftmann@26358
   858
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
haftmann@26358
   859
  by blast
haftmann@26358
   860
haftmann@26358
   861
lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
haftmann@26358
   862
  by blast
haftmann@26358
   863
haftmann@26358
   864
lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
haftmann@26358
   865
  by (blast elim: equalityE)
haftmann@20588
   866
haftmann@26358
   867
lemma SetCompr_Sigma_eq:
haftmann@26358
   868
    "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
haftmann@26358
   869
  by blast
haftmann@26358
   870
haftmann@26358
   871
lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"
haftmann@26358
   872
  by blast
haftmann@26358
   873
haftmann@26358
   874
lemma UN_Times_distrib:
haftmann@26358
   875
  "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"
haftmann@26358
   876
  -- {* Suggested by Pierre Chartier *}
haftmann@26358
   877
  by blast
haftmann@26358
   878
haftmann@26358
   879
lemma split_paired_Ball_Sigma [simp,noatp]:
haftmann@26358
   880
    "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"
haftmann@26358
   881
  by blast
haftmann@26358
   882
haftmann@26358
   883
lemma split_paired_Bex_Sigma [simp,noatp]:
haftmann@26358
   884
    "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"
haftmann@26358
   885
  by blast
haftmann@21908
   886
haftmann@26358
   887
lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"
haftmann@26358
   888
  by blast
haftmann@26358
   889
haftmann@26358
   890
lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))"
haftmann@26358
   891
  by blast
haftmann@26358
   892
haftmann@26358
   893
lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"
haftmann@26358
   894
  by blast
haftmann@26358
   895
haftmann@26358
   896
lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))"
haftmann@26358
   897
  by blast
haftmann@26358
   898
haftmann@26358
   899
lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"
haftmann@26358
   900
  by blast
haftmann@26358
   901
haftmann@26358
   902
lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"
haftmann@26358
   903
  by blast
haftmann@21908
   904
haftmann@26358
   905
lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"
haftmann@26358
   906
  by blast
haftmann@26358
   907
haftmann@26358
   908
text {*
haftmann@26358
   909
  Non-dependent versions are needed to avoid the need for higher-order
haftmann@26358
   910
  matching, especially when the rules are re-oriented.
haftmann@26358
   911
*}
haftmann@21908
   912
haftmann@26358
   913
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"
haftmann@26358
   914
  by blast
haftmann@26358
   915
haftmann@26358
   916
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"
haftmann@26358
   917
  by blast
haftmann@26358
   918
haftmann@26358
   919
lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
haftmann@26358
   920
  by blast
haftmann@26358
   921
haftmann@26358
   922
haftmann@26358
   923
subsubsection {* Code generator setup *}
haftmann@21908
   924
haftmann@20588
   925
instance * :: (eq, eq) eq ..
haftmann@20588
   926
haftmann@20588
   927
lemma [code func]:
haftmann@21454
   928
  "(x1\<Colon>'a\<Colon>eq, y1\<Colon>'b\<Colon>eq) = (x2, y2) \<longleftrightarrow> x1 = x2 \<and> y1 = y2" by auto
haftmann@20588
   929
haftmann@24844
   930
lemma split_case_cert:
haftmann@24844
   931
  assumes "CASE \<equiv> split f"
haftmann@24844
   932
  shows "CASE (a, b) \<equiv> f a b"
haftmann@24844
   933
  using assms by simp
haftmann@24844
   934
haftmann@24844
   935
setup {*
haftmann@24844
   936
  Code.add_case @{thm split_case_cert}
haftmann@24844
   937
*}
haftmann@24844
   938
haftmann@21908
   939
code_type *
haftmann@21908
   940
  (SML infix 2 "*")
haftmann@21908
   941
  (OCaml infix 2 "*")
haftmann@21908
   942
  (Haskell "!((_),/ (_))")
haftmann@21908
   943
haftmann@20588
   944
code_instance * :: eq
haftmann@20588
   945
  (Haskell -)
haftmann@20588
   946
haftmann@21908
   947
code_const "op = \<Colon> 'a\<Colon>eq \<times> 'b\<Colon>eq \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"
haftmann@20588
   948
  (Haskell infixl 4 "==")
haftmann@20588
   949
haftmann@21908
   950
code_const Pair
haftmann@21908
   951
  (SML "!((_),/ (_))")
haftmann@21908
   952
  (OCaml "!((_),/ (_))")
haftmann@21908
   953
  (Haskell "!((_),/ (_))")
haftmann@20588
   954
haftmann@22389
   955
code_const fst and snd
haftmann@22389
   956
  (Haskell "fst" and "snd")
haftmann@22389
   957
berghofe@15394
   958
types_code
berghofe@15394
   959
  "*"     ("(_ */ _)")
berghofe@16770
   960
attach (term_of) {*
berghofe@25885
   961
fun term_of_id_42 aF aT bF bT (x, y) = HOLogic.pair_const aT bT $ aF x $ bF y;
berghofe@16770
   962
*}
berghofe@16770
   963
attach (test) {*
berghofe@25885
   964
fun gen_id_42 aG aT bG bT i =
berghofe@25885
   965
  let
berghofe@25885
   966
    val (x, t) = aG i;
berghofe@25885
   967
    val (y, u) = bG i
berghofe@25885
   968
  in ((x, y), fn () => HOLogic.pair_const aT bT $ t () $ u ()) end;
berghofe@16770
   969
*}
berghofe@15394
   970
berghofe@18706
   971
consts_code
berghofe@18706
   972
  "Pair"    ("(_,/ _)")
berghofe@18706
   973
haftmann@21908
   974
setup {*
haftmann@21908
   975
haftmann@21908
   976
let
haftmann@18013
   977
haftmann@19039
   978
fun strip_abs_split 0 t = ([], t)
haftmann@19039
   979
  | strip_abs_split i (Abs (s, T, t)) =
haftmann@18013
   980
      let
haftmann@18013
   981
        val s' = Codegen.new_name t s;
haftmann@18013
   982
        val v = Free (s', T)
haftmann@19039
   983
      in apfst (cons v) (strip_abs_split (i-1) (subst_bound (v, t))) end
haftmann@19039
   984
  | strip_abs_split i (u as Const ("split", _) $ t) = (case strip_abs_split (i+1) t of
berghofe@15394
   985
        (v :: v' :: vs, u) => (HOLogic.mk_prod (v, v') :: vs, u)
berghofe@15394
   986
      | _ => ([], u))
haftmann@19039
   987
  | strip_abs_split i t = ([], t);
haftmann@18013
   988
berghofe@16634
   989
fun let_codegen thy defs gr dep thyname brack t = (case strip_comb t of
berghofe@16634
   990
    (t1 as Const ("Let", _), t2 :: t3 :: ts) =>
berghofe@15394
   991
    let
berghofe@15394
   992
      fun dest_let (l as Const ("Let", _) $ t $ u) =
haftmann@19039
   993
          (case strip_abs_split 1 u of
berghofe@15394
   994
             ([p], u') => apfst (cons (p, t)) (dest_let u')
berghofe@15394
   995
           | _ => ([], l))
berghofe@15394
   996
        | dest_let t = ([], t);
berghofe@15394
   997
      fun mk_code (gr, (l, r)) =
berghofe@15394
   998
        let
berghofe@16634
   999
          val (gr1, pl) = Codegen.invoke_codegen thy defs dep thyname false (gr, l);
berghofe@16634
  1000
          val (gr2, pr) = Codegen.invoke_codegen thy defs dep thyname false (gr1, r);
berghofe@15394
  1001
        in (gr2, (pl, pr)) end
berghofe@16634
  1002
    in case dest_let (t1 $ t2 $ t3) of
skalberg@15531
  1003
        ([], _) => NONE
berghofe@15394
  1004
      | (ps, u) =>
berghofe@15394
  1005
          let
berghofe@15394
  1006
            val (gr1, qs) = foldl_map mk_code (gr, ps);
berghofe@16634
  1007
            val (gr2, pu) = Codegen.invoke_codegen thy defs dep thyname false (gr1, u);
berghofe@16634
  1008
            val (gr3, pargs) = foldl_map
berghofe@17021
  1009
              (Codegen.invoke_codegen thy defs dep thyname true) (gr2, ts)
berghofe@15394
  1010
          in
berghofe@16634
  1011
            SOME (gr3, Codegen.mk_app brack
berghofe@16634
  1012
              (Pretty.blk (0, [Pretty.str "let ", Pretty.blk (0, List.concat
berghofe@16634
  1013
                  (separate [Pretty.str ";", Pretty.brk 1] (map (fn (pl, pr) =>
berghofe@16634
  1014
                    [Pretty.block [Pretty.str "val ", pl, Pretty.str " =",
berghofe@16634
  1015
                       Pretty.brk 1, pr]]) qs))),
berghofe@16634
  1016
                Pretty.brk 1, Pretty.str "in ", pu,
berghofe@16634
  1017
                Pretty.brk 1, Pretty.str "end"])) pargs)
berghofe@15394
  1018
          end
berghofe@15394
  1019
    end
berghofe@16634
  1020
  | _ => NONE);
berghofe@15394
  1021
berghofe@16634
  1022
fun split_codegen thy defs gr dep thyname brack t = (case strip_comb t of
berghofe@16634
  1023
    (t1 as Const ("split", _), t2 :: ts) =>
haftmann@19039
  1024
      (case strip_abs_split 1 (t1 $ t2) of
berghofe@16634
  1025
         ([p], u) =>
berghofe@16634
  1026
           let
berghofe@16634
  1027
             val (gr1, q) = Codegen.invoke_codegen thy defs dep thyname false (gr, p);
berghofe@16634
  1028
             val (gr2, pu) = Codegen.invoke_codegen thy defs dep thyname false (gr1, u);
berghofe@16634
  1029
             val (gr3, pargs) = foldl_map
berghofe@17021
  1030
               (Codegen.invoke_codegen thy defs dep thyname true) (gr2, ts)
berghofe@16634
  1031
           in
berghofe@16634
  1032
             SOME (gr2, Codegen.mk_app brack
berghofe@16634
  1033
               (Pretty.block [Pretty.str "(fn ", q, Pretty.str " =>",
berghofe@16634
  1034
                 Pretty.brk 1, pu, Pretty.str ")"]) pargs)
berghofe@16634
  1035
           end
berghofe@16634
  1036
       | _ => NONE)
berghofe@16634
  1037
  | _ => NONE);
berghofe@15394
  1038
haftmann@21908
  1039
in
haftmann@21908
  1040
haftmann@20105
  1041
  Codegen.add_codegen "let_codegen" let_codegen
haftmann@20105
  1042
  #> Codegen.add_codegen "split_codegen" split_codegen
berghofe@15394
  1043
haftmann@21908
  1044
end
berghofe@15394
  1045
*}
berghofe@15394
  1046
haftmann@24699
  1047
haftmann@24699
  1048
subsection {* Legacy bindings *}
haftmann@24699
  1049
haftmann@21908
  1050
ML {*
paulson@15404
  1051
val Collect_split = thm "Collect_split";
paulson@15404
  1052
val Compl_Times_UNIV1 = thm "Compl_Times_UNIV1";
paulson@15404
  1053
val Compl_Times_UNIV2 = thm "Compl_Times_UNIV2";
paulson@15404
  1054
val PairE = thm "PairE";
paulson@15404
  1055
val Pair_Rep_inject = thm "Pair_Rep_inject";
paulson@15404
  1056
val Pair_def = thm "Pair_def";
paulson@15404
  1057
val Pair_eq = thm "Pair_eq";
paulson@15404
  1058
val Pair_fst_snd_eq = thm "Pair_fst_snd_eq";
paulson@15404
  1059
val ProdI = thm "ProdI";
paulson@15404
  1060
val SetCompr_Sigma_eq = thm "SetCompr_Sigma_eq";
paulson@15404
  1061
val SigmaD1 = thm "SigmaD1";
paulson@15404
  1062
val SigmaD2 = thm "SigmaD2";
paulson@15404
  1063
val SigmaE = thm "SigmaE";
paulson@15404
  1064
val SigmaE2 = thm "SigmaE2";
paulson@15404
  1065
val SigmaI = thm "SigmaI";
paulson@15404
  1066
val Sigma_Diff_distrib1 = thm "Sigma_Diff_distrib1";
paulson@15404
  1067
val Sigma_Diff_distrib2 = thm "Sigma_Diff_distrib2";
paulson@15404
  1068
val Sigma_Int_distrib1 = thm "Sigma_Int_distrib1";
paulson@15404
  1069
val Sigma_Int_distrib2 = thm "Sigma_Int_distrib2";
paulson@15404
  1070
val Sigma_Un_distrib1 = thm "Sigma_Un_distrib1";
paulson@15404
  1071
val Sigma_Un_distrib2 = thm "Sigma_Un_distrib2";
paulson@15404
  1072
val Sigma_Union = thm "Sigma_Union";
paulson@15404
  1073
val Sigma_def = thm "Sigma_def";
paulson@15404
  1074
val Sigma_empty1 = thm "Sigma_empty1";
paulson@15404
  1075
val Sigma_empty2 = thm "Sigma_empty2";
paulson@15404
  1076
val Sigma_mono = thm "Sigma_mono";
paulson@15404
  1077
val The_split = thm "The_split";
paulson@15404
  1078
val The_split_eq = thm "The_split_eq";
paulson@15404
  1079
val The_split_eq = thm "The_split_eq";
paulson@15404
  1080
val Times_Diff_distrib1 = thm "Times_Diff_distrib1";
paulson@15404
  1081
val Times_Int_distrib1 = thm "Times_Int_distrib1";
paulson@15404
  1082
val Times_Un_distrib1 = thm "Times_Un_distrib1";
paulson@15404
  1083
val Times_eq_cancel2 = thm "Times_eq_cancel2";
paulson@15404
  1084
val Times_subset_cancel2 = thm "Times_subset_cancel2";
paulson@15404
  1085
val UNIV_Times_UNIV = thm "UNIV_Times_UNIV";
paulson@15404
  1086
val UN_Times_distrib = thm "UN_Times_distrib";
paulson@15404
  1087
val Unity_def = thm "Unity_def";
paulson@15404
  1088
val cond_split_eta = thm "cond_split_eta";
paulson@15404
  1089
val fst_conv = thm "fst_conv";
paulson@15404
  1090
val fst_def = thm "fst_def";
paulson@15404
  1091
val fst_eqD = thm "fst_eqD";
paulson@15404
  1092
val inj_on_Abs_Prod = thm "inj_on_Abs_Prod";
paulson@15404
  1093
val mem_Sigma_iff = thm "mem_Sigma_iff";
paulson@15404
  1094
val mem_splitE = thm "mem_splitE";
paulson@15404
  1095
val mem_splitI = thm "mem_splitI";
paulson@15404
  1096
val mem_splitI2 = thm "mem_splitI2";
paulson@15404
  1097
val prod_eqI = thm "prod_eqI";
paulson@15404
  1098
val prod_fun = thm "prod_fun";
paulson@15404
  1099
val prod_fun_compose = thm "prod_fun_compose";
paulson@15404
  1100
val prod_fun_def = thm "prod_fun_def";
paulson@15404
  1101
val prod_fun_ident = thm "prod_fun_ident";
paulson@15404
  1102
val prod_fun_imageE = thm "prod_fun_imageE";
paulson@15404
  1103
val prod_fun_imageI = thm "prod_fun_imageI";
paulson@15404
  1104
val prod_induct = thm "prod_induct";
paulson@15404
  1105
val snd_conv = thm "snd_conv";
paulson@15404
  1106
val snd_def = thm "snd_def";
paulson@15404
  1107
val snd_eqD = thm "snd_eqD";
paulson@15404
  1108
val split = thm "split";
paulson@15404
  1109
val splitD = thm "splitD";
paulson@15404
  1110
val splitD' = thm "splitD'";
paulson@15404
  1111
val splitE = thm "splitE";
paulson@15404
  1112
val splitE' = thm "splitE'";
paulson@15404
  1113
val splitE2 = thm "splitE2";
paulson@15404
  1114
val splitI = thm "splitI";
paulson@15404
  1115
val splitI2 = thm "splitI2";
paulson@15404
  1116
val splitI2' = thm "splitI2'";
paulson@15404
  1117
val split_beta = thm "split_beta";
paulson@15404
  1118
val split_conv = thm "split_conv";
paulson@15404
  1119
val split_def = thm "split_def";
paulson@15404
  1120
val split_eta = thm "split_eta";
paulson@15404
  1121
val split_eta_SetCompr = thm "split_eta_SetCompr";
paulson@15404
  1122
val split_eta_SetCompr2 = thm "split_eta_SetCompr2";
paulson@15404
  1123
val split_paired_All = thm "split_paired_All";
paulson@15404
  1124
val split_paired_Ball_Sigma = thm "split_paired_Ball_Sigma";
paulson@15404
  1125
val split_paired_Bex_Sigma = thm "split_paired_Bex_Sigma";
paulson@15404
  1126
val split_paired_Ex = thm "split_paired_Ex";
paulson@15404
  1127
val split_paired_The = thm "split_paired_The";
paulson@15404
  1128
val split_paired_all = thm "split_paired_all";
paulson@15404
  1129
val split_part = thm "split_part";
paulson@15404
  1130
val split_split = thm "split_split";
paulson@15404
  1131
val split_split_asm = thm "split_split_asm";
paulson@15404
  1132
val split_tupled_all = thms "split_tupled_all";
paulson@15404
  1133
val split_weak_cong = thm "split_weak_cong";
paulson@15404
  1134
val surj_pair = thm "surj_pair";
paulson@15404
  1135
val surjective_pairing = thm "surjective_pairing";
paulson@15404
  1136
val unit_abs_eta_conv = thm "unit_abs_eta_conv";
paulson@15404
  1137
val unit_all_eq1 = thm "unit_all_eq1";
paulson@15404
  1138
val unit_all_eq2 = thm "unit_all_eq2";
paulson@15404
  1139
val unit_eq = thm "unit_eq";
paulson@15404
  1140
*}
paulson@15404
  1141
haftmann@24699
  1142
haftmann@24699
  1143
subsection {* Further inductive packages *}
haftmann@24699
  1144
haftmann@24699
  1145
use "Tools/inductive_realizer.ML"
haftmann@24699
  1146
setup InductiveRealizer.setup
haftmann@24699
  1147
haftmann@24699
  1148
use "Tools/inductive_set_package.ML"
haftmann@24699
  1149
setup InductiveSetPackage.setup
haftmann@24699
  1150
haftmann@24699
  1151
use "Tools/datatype_realizer.ML"
haftmann@24699
  1152
setup DatatypeRealizer.setup
haftmann@24699
  1153
nipkow@10213
  1154
end