src/HOL/Product_Type.thy
author haftmann
Sun Jun 23 21:16:07 2013 +0200 (2013-06-23)
changeset 52435 6646bb548c6b
parent 52143 36ffe23b25f8
child 54147 97a8ff4e4ac9
permissions -rw-r--r--
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
     1 (*  Title:      HOL/Product_Type.thy
     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     3     Copyright   1992  University of Cambridge
     4 *)
     5 
     6 header {* Cartesian products *}
     7 
     8 theory Product_Type
     9 imports Typedef Inductive Fun
    10 keywords "inductive_set" "coinductive_set" :: thy_decl
    11 begin
    12 
    13 subsection {* @{typ bool} is a datatype *}
    14 
    15 rep_datatype True False by (auto intro: bool_induct)
    16 
    17 declare case_split [cases type: bool]
    18   -- "prefer plain propositional version"
    19 
    20 lemma
    21   shows [code]: "HOL.equal False P \<longleftrightarrow> \<not> P"
    22     and [code]: "HOL.equal True P \<longleftrightarrow> P" 
    23     and [code]: "HOL.equal P False \<longleftrightarrow> \<not> P"
    24     and [code]: "HOL.equal P True \<longleftrightarrow> P"
    25     and [code nbe]: "HOL.equal P P \<longleftrightarrow> True"
    26   by (simp_all add: equal)
    27 
    28 lemma If_case_cert:
    29   assumes "CASE \<equiv> (\<lambda>b. If b f g)"
    30   shows "(CASE True \<equiv> f) &&& (CASE False \<equiv> g)"
    31   using assms by simp_all
    32 
    33 setup {*
    34   Code.add_case @{thm If_case_cert}
    35 *}
    36 
    37 code_printing
    38   constant "HOL.equal :: bool \<Rightarrow> bool \<Rightarrow> bool" \<rightharpoonup> (Haskell) infix 4 "=="
    39 | class_instance "bool" :: "equal" \<rightharpoonup> (Haskell) -
    40 
    41 
    42 subsection {* The @{text unit} type *}
    43 
    44 typedef unit = "{True}"
    45   by auto
    46 
    47 definition Unity :: unit  ("'(')")
    48   where "() = Abs_unit True"
    49 
    50 lemma unit_eq [no_atp]: "u = ()"
    51   by (induct u) (simp add: Unity_def)
    52 
    53 text {*
    54   Simplification procedure for @{thm [source] unit_eq}.  Cannot use
    55   this rule directly --- it loops!
    56 *}
    57 
    58 simproc_setup unit_eq ("x::unit") = {*
    59   fn _ => fn _ => fn ct =>
    60     if HOLogic.is_unit (term_of ct) then NONE
    61     else SOME (mk_meta_eq @{thm unit_eq})
    62 *}
    63 
    64 rep_datatype "()" by simp
    65 
    66 lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()"
    67   by simp
    68 
    69 lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P"
    70   by (rule triv_forall_equality)
    71 
    72 text {*
    73   This rewrite counters the effect of simproc @{text unit_eq} on @{term
    74   [source] "%u::unit. f u"}, replacing it by @{term [source]
    75   f} rather than by @{term [source] "%u. f ()"}.
    76 *}
    77 
    78 lemma unit_abs_eta_conv [simp, no_atp]: "(%u::unit. f ()) = f"
    79   by (rule ext) simp
    80 
    81 lemma UNIV_unit [no_atp]:
    82   "UNIV = {()}" by auto
    83 
    84 instantiation unit :: default
    85 begin
    86 
    87 definition "default = ()"
    88 
    89 instance ..
    90 
    91 end
    92 
    93 lemma [code]:
    94   "HOL.equal (u\<Colon>unit) v \<longleftrightarrow> True" unfolding equal unit_eq [of u] unit_eq [of v] by rule+
    95 
    96 code_printing
    97   type_constructor unit \<rightharpoonup>
    98     (SML) "unit"
    99     and (OCaml) "unit"
   100     and (Haskell) "()"
   101     and (Scala) "Unit"
   102 | constant Unity \<rightharpoonup>
   103     (SML) "()"
   104     and (OCaml) "()"
   105     and (Haskell) "()"
   106     and (Scala) "()"
   107 | class_instance unit :: equal \<rightharpoonup>
   108     (Haskell) -
   109 | constant "HOL.equal :: unit \<Rightarrow> unit \<Rightarrow> bool" \<rightharpoonup>
   110     (Haskell) infix 4 "=="
   111 
   112 code_reserved SML
   113   unit
   114 
   115 code_reserved OCaml
   116   unit
   117 
   118 code_reserved Scala
   119   Unit
   120 
   121 
   122 subsection {* The product type *}
   123 
   124 subsubsection {* Type definition *}
   125 
   126 definition Pair_Rep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" where
   127   "Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)"
   128 
   129 definition "prod = {f. \<exists>a b. f = Pair_Rep (a\<Colon>'a) (b\<Colon>'b)}"
   130 
   131 typedef ('a, 'b) prod (infixr "*" 20) = "prod :: ('a \<Rightarrow> 'b \<Rightarrow> bool) set"
   132   unfolding prod_def by auto
   133 
   134 type_notation (xsymbols)
   135   "prod"  ("(_ \<times>/ _)" [21, 20] 20)
   136 type_notation (HTML output)
   137   "prod"  ("(_ \<times>/ _)" [21, 20] 20)
   138 
   139 definition Pair :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b" where
   140   "Pair a b = Abs_prod (Pair_Rep a b)"
   141 
   142 rep_datatype Pair proof -
   143   fix P :: "'a \<times> 'b \<Rightarrow> bool" and p
   144   assume "\<And>a b. P (Pair a b)"
   145   then show "P p" by (cases p) (auto simp add: prod_def Pair_def Pair_Rep_def)
   146 next
   147   fix a c :: 'a and b d :: 'b
   148   have "Pair_Rep a b = Pair_Rep c d \<longleftrightarrow> a = c \<and> b = d"
   149     by (auto simp add: Pair_Rep_def fun_eq_iff)
   150   moreover have "Pair_Rep a b \<in> prod" and "Pair_Rep c d \<in> prod"
   151     by (auto simp add: prod_def)
   152   ultimately show "Pair a b = Pair c d \<longleftrightarrow> a = c \<and> b = d"
   153     by (simp add: Pair_def Abs_prod_inject)
   154 qed
   155 
   156 declare prod.simps(2) [nitpick_simp del]
   157 
   158 declare prod.weak_case_cong [cong del]
   159 
   160 
   161 subsubsection {* Tuple syntax *}
   162 
   163 abbreviation (input) split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where
   164   "split \<equiv> prod_case"
   165 
   166 text {*
   167   Patterns -- extends pre-defined type @{typ pttrn} used in
   168   abstractions.
   169 *}
   170 
   171 nonterminal tuple_args and patterns
   172 
   173 syntax
   174   "_tuple"      :: "'a => tuple_args => 'a * 'b"        ("(1'(_,/ _'))")
   175   "_tuple_arg"  :: "'a => tuple_args"                   ("_")
   176   "_tuple_args" :: "'a => tuple_args => tuple_args"     ("_,/ _")
   177   "_pattern"    :: "[pttrn, patterns] => pttrn"         ("'(_,/ _')")
   178   ""            :: "pttrn => patterns"                  ("_")
   179   "_patterns"   :: "[pttrn, patterns] => patterns"      ("_,/ _")
   180 
   181 translations
   182   "(x, y)" == "CONST Pair x y"
   183   "_pattern x y" => "CONST Pair x y"
   184   "_patterns x y" => "CONST Pair x y"
   185   "_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))"
   186   "%(x, y, zs). b" == "CONST prod_case (%x (y, zs). b)"
   187   "%(x, y). b" == "CONST prod_case (%x y. b)"
   188   "_abs (CONST Pair x y) t" => "%(x, y). t"
   189   -- {* The last rule accommodates tuples in `case C ... (x,y) ... => ...'
   190      The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *}
   191 
   192 (*reconstruct pattern from (nested) splits, avoiding eta-contraction of body;
   193   works best with enclosing "let", if "let" does not avoid eta-contraction*)
   194 print_translation {*
   195   let
   196     fun split_tr' [Abs (x, T, t as (Abs abs))] =
   197           (* split (%x y. t) => %(x,y) t *)
   198           let
   199             val (y, t') = Syntax_Trans.atomic_abs_tr' abs;
   200             val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t');
   201           in
   202             Syntax.const @{syntax_const "_abs"} $
   203               (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
   204           end
   205       | split_tr' [Abs (x, T, (s as Const (@{const_syntax prod_case}, _) $ t))] =
   206           (* split (%x. (split (%y z. t))) => %(x,y,z). t *)
   207           let
   208             val Const (@{syntax_const "_abs"}, _) $
   209               (Const (@{syntax_const "_pattern"}, _) $ y $ z) $ t' = split_tr' [t];
   210             val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t');
   211           in
   212             Syntax.const @{syntax_const "_abs"} $
   213               (Syntax.const @{syntax_const "_pattern"} $ x' $
   214                 (Syntax.const @{syntax_const "_patterns"} $ y $ z)) $ t''
   215           end
   216       | split_tr' [Const (@{const_syntax prod_case}, _) $ t] =
   217           (* split (split (%x y z. t)) => %((x, y), z). t *)
   218           split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *)
   219       | split_tr' [Const (@{syntax_const "_abs"}, _) $ x_y $ Abs abs] =
   220           (* split (%pttrn z. t) => %(pttrn,z). t *)
   221           let val (z, t) = Syntax_Trans.atomic_abs_tr' abs in
   222             Syntax.const @{syntax_const "_abs"} $
   223               (Syntax.const @{syntax_const "_pattern"} $ x_y $ z) $ t
   224           end
   225       | split_tr' _ = raise Match;
   226   in [(@{const_syntax prod_case}, K split_tr')] end
   227 *}
   228 
   229 (* print "split f" as "\<lambda>(x,y). f x y" and "split (\<lambda>x. f x)" as "\<lambda>(x,y). f x y" *) 
   230 typed_print_translation {*
   231   let
   232     fun split_guess_names_tr' T [Abs (x, _, Abs _)] = raise Match
   233       | split_guess_names_tr' T [Abs (x, xT, t)] =
   234           (case (head_of t) of
   235             Const (@{const_syntax prod_case}, _) => raise Match
   236           | _ =>
   237             let 
   238               val (_ :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match;
   239               val (y, t') = Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 1 t $ Bound 0);
   240               val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, xT, t');
   241             in
   242               Syntax.const @{syntax_const "_abs"} $
   243                 (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
   244             end)
   245       | split_guess_names_tr' T [t] =
   246           (case head_of t of
   247             Const (@{const_syntax prod_case}, _) => raise Match
   248           | _ =>
   249             let
   250               val (xT :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match;
   251               val (y, t') =
   252                 Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 2 t $ Bound 1 $ Bound 0);
   253               val (x', t'') = Syntax_Trans.atomic_abs_tr' ("x", xT, t');
   254             in
   255               Syntax.const @{syntax_const "_abs"} $
   256                 (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
   257             end)
   258       | split_guess_names_tr' _ _ = raise Match;
   259   in [(@{const_syntax prod_case}, K split_guess_names_tr')] end
   260 *}
   261 
   262 (* Force eta-contraction for terms of the form "Q A (%p. prod_case P p)"
   263    where Q is some bounded quantifier or set operator.
   264    Reason: the above prints as "Q p : A. case p of (x,y) \<Rightarrow> P x y"
   265    whereas we want "Q (x,y):A. P x y".
   266    Otherwise prevent eta-contraction.
   267 *)
   268 print_translation {*
   269   let
   270     fun contract Q tr ctxt ts =
   271       (case ts of
   272         [A, Abs (_, _, (s as Const (@{const_syntax prod_case},_) $ t) $ Bound 0)] =>
   273           if Term.is_dependent t then tr ctxt ts
   274           else Syntax.const Q $ A $ s
   275       | _ => tr ctxt ts);
   276   in
   277     [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax Ball} @{syntax_const "_Ball"},
   278      Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax Bex} @{syntax_const "_Bex"},
   279      Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"},
   280      Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}]
   281     |> map (fn (Q, tr) => (Q, contract Q tr))
   282   end
   283 *}
   284 
   285 subsubsection {* Code generator setup *}
   286 
   287 code_printing
   288   type_constructor prod \<rightharpoonup>
   289     (SML) infix 2 "*"
   290     and (OCaml) infix 2 "*"
   291     and (Haskell) "!((_),/ (_))"
   292     and (Scala) "((_),/ (_))"
   293 | constant Pair \<rightharpoonup>
   294     (SML) "!((_),/ (_))"
   295     and (OCaml) "!((_),/ (_))"
   296     and (Haskell) "!((_),/ (_))"
   297     and (Scala) "!((_),/ (_))"
   298 | class_instance  prod :: equal \<rightharpoonup>
   299     (Haskell) -
   300 | constant "HOL.equal :: 'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool" \<rightharpoonup>
   301     (Haskell) infix 4 "=="
   302 
   303 
   304 subsubsection {* Fundamental operations and properties *}
   305 
   306 lemma Pair_inject:
   307   assumes "(a, b) = (a', b')"
   308     and "a = a' ==> b = b' ==> R"
   309   shows R
   310   using assms by simp
   311 
   312 lemma surj_pair [simp]: "EX x y. p = (x, y)"
   313   by (cases p) simp
   314 
   315 definition fst :: "'a \<times> 'b \<Rightarrow> 'a" where
   316   "fst p = (case p of (a, b) \<Rightarrow> a)"
   317 
   318 definition snd :: "'a \<times> 'b \<Rightarrow> 'b" where
   319   "snd p = (case p of (a, b) \<Rightarrow> b)"
   320 
   321 lemma fst_conv [simp, code]: "fst (a, b) = a"
   322   unfolding fst_def by simp
   323 
   324 lemma snd_conv [simp, code]: "snd (a, b) = b"
   325   unfolding snd_def by simp
   326 
   327 code_printing
   328   constant fst \<rightharpoonup> (Haskell) "fst"
   329 | constant snd \<rightharpoonup> (Haskell) "snd"
   330 
   331 lemma prod_case_unfold [nitpick_unfold]: "prod_case = (%c p. c (fst p) (snd p))"
   332   by (simp add: fun_eq_iff split: prod.split)
   333 
   334 lemma fst_eqD: "fst (x, y) = a ==> x = a"
   335   by simp
   336 
   337 lemma snd_eqD: "snd (x, y) = a ==> y = a"
   338   by simp
   339 
   340 lemma pair_collapse [simp]: "(fst p, snd p) = p"
   341   by (cases p) simp
   342 
   343 lemmas surjective_pairing = pair_collapse [symmetric]
   344 
   345 lemma prod_eq_iff: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t"
   346   by (cases s, cases t) simp
   347 
   348 lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q"
   349   by (simp add: prod_eq_iff)
   350 
   351 lemma split_conv [simp, code]: "split f (a, b) = f a b"
   352   by (fact prod.cases)
   353 
   354 lemma splitI: "f a b \<Longrightarrow> split f (a, b)"
   355   by (rule split_conv [THEN iffD2])
   356 
   357 lemma splitD: "split f (a, b) \<Longrightarrow> f a b"
   358   by (rule split_conv [THEN iffD1])
   359 
   360 lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id"
   361   by (simp add: fun_eq_iff split: prod.split)
   362 
   363 lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f"
   364   -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *}
   365   by (simp add: fun_eq_iff split: prod.split)
   366 
   367 lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)"
   368   by (cases x) simp
   369 
   370 lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p"
   371   by (cases p) simp
   372 
   373 lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
   374   by (simp add: prod_case_unfold)
   375 
   376 lemma split_weak_cong: "p = q \<Longrightarrow> split c p = split c q"
   377   -- {* Prevents simplification of @{term c}: much faster *}
   378   by (fact prod.weak_case_cong)
   379 
   380 lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
   381   by (simp add: split_eta)
   382 
   383 lemma split_paired_all [no_atp]: "(!!x. PROP P x) == (!!a b. PROP P (a, b))"
   384 proof
   385   fix a b
   386   assume "!!x. PROP P x"
   387   then show "PROP P (a, b)" .
   388 next
   389   fix x
   390   assume "!!a b. PROP P (a, b)"
   391   from `PROP P (fst x, snd x)` show "PROP P x" by simp
   392 qed
   393 
   394 lemma case_prod_distrib: "f (case x of (x, y) \<Rightarrow> g x y) = (case x of (x, y) \<Rightarrow> f (g x y))"
   395   by (cases x) simp
   396 
   397 text {*
   398   The rule @{thm [source] split_paired_all} does not work with the
   399   Simplifier because it also affects premises in congrence rules,
   400   where this can lead to premises of the form @{text "!!a b. ... =
   401   ?P(a, b)"} which cannot be solved by reflexivity.
   402 *}
   403 
   404 lemmas split_tupled_all = split_paired_all unit_all_eq2
   405 
   406 ML {*
   407   (* replace parameters of product type by individual component parameters *)
   408   local (* filtering with exists_paired_all is an essential optimization *)
   409     fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) =
   410           can HOLogic.dest_prodT T orelse exists_paired_all t
   411       | exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u
   412       | exists_paired_all (Abs (_, _, t)) = exists_paired_all t
   413       | exists_paired_all _ = false;
   414     val ss =
   415       simpset_of
   416        (put_simpset HOL_basic_ss @{context}
   417         addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}]
   418         addsimprocs [@{simproc unit_eq}]);
   419   in
   420     fun split_all_tac ctxt = SUBGOAL (fn (t, i) =>
   421       if exists_paired_all t then safe_full_simp_tac (put_simpset ss ctxt) i else no_tac);
   422 
   423     fun unsafe_split_all_tac ctxt = SUBGOAL (fn (t, i) =>
   424       if exists_paired_all t then full_simp_tac (put_simpset ss ctxt) i else no_tac);
   425 
   426     fun split_all ctxt th =
   427       if exists_paired_all (Thm.prop_of th)
   428       then full_simplify (put_simpset ss ctxt) th else th;
   429   end;
   430 *}
   431 
   432 setup {* map_theory_claset (fn ctxt => ctxt addSbefore ("split_all_tac", split_all_tac)) *}
   433 
   434 lemma split_paired_All [simp, no_atp]: "(ALL x. P x) = (ALL a b. P (a, b))"
   435   -- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *}
   436   by fast
   437 
   438 lemma split_paired_Ex [simp, no_atp]: "(EX x. P x) = (EX a b. P (a, b))"
   439   by fast
   440 
   441 lemma split_paired_The [no_atp]: "(THE x. P x) = (THE (a, b). P (a, b))"
   442   -- {* Can't be added to simpset: loops! *}
   443   by (simp add: split_eta)
   444 
   445 text {*
   446   Simplification procedure for @{thm [source] cond_split_eta}.  Using
   447   @{thm [source] split_eta} as a rewrite rule is not general enough,
   448   and using @{thm [source] cond_split_eta} directly would render some
   449   existing proofs very inefficient; similarly for @{text
   450   split_beta}.
   451 *}
   452 
   453 ML {*
   454 local
   455   val cond_split_eta_ss =
   456     simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms cond_split_eta});
   457   fun Pair_pat k 0 (Bound m) = (m = k)
   458     | Pair_pat k i (Const (@{const_name Pair},  _) $ Bound m $ t) =
   459         i > 0 andalso m = k + i andalso Pair_pat k (i - 1) t
   460     | Pair_pat _ _ _ = false;
   461   fun no_args k i (Abs (_, _, t)) = no_args (k + 1) i t
   462     | no_args k i (t $ u) = no_args k i t andalso no_args k i u
   463     | no_args k i (Bound m) = m < k orelse m > k + i
   464     | no_args _ _ _ = true;
   465   fun split_pat tp i (Abs  (_, _, t)) = if tp 0 i t then SOME (i, t) else NONE
   466     | split_pat tp i (Const (@{const_name prod_case}, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t
   467     | split_pat tp i _ = NONE;
   468   fun metaeq ctxt lhs rhs = mk_meta_eq (Goal.prove ctxt [] []
   469         (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)))
   470         (K (simp_tac (put_simpset cond_split_eta_ss ctxt) 1)));
   471 
   472   fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k + 1) i t
   473     | beta_term_pat k i (t $ u) =
   474         Pair_pat k i (t $ u) orelse (beta_term_pat k i t andalso beta_term_pat k i u)
   475     | beta_term_pat k i t = no_args k i t;
   476   fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
   477     | eta_term_pat _ _ _ = false;
   478   fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
   479     | subst arg k i (t $ u) =
   480         if Pair_pat k i (t $ u) then incr_boundvars k arg
   481         else (subst arg k i t $ subst arg k i u)
   482     | subst arg k i t = t;
   483 in
   484   fun beta_proc ctxt (s as Const (@{const_name prod_case}, _) $ Abs (_, _, t) $ arg) =
   485         (case split_pat beta_term_pat 1 t of
   486           SOME (i, f) => SOME (metaeq ctxt s (subst arg 0 i f))
   487         | NONE => NONE)
   488     | beta_proc _ _ = NONE;
   489   fun eta_proc ctxt (s as Const (@{const_name prod_case}, _) $ Abs (_, _, t)) =
   490         (case split_pat eta_term_pat 1 t of
   491           SOME (_, ft) => SOME (metaeq ctxt s (let val (f $ arg) = ft in f end))
   492         | NONE => NONE)
   493     | eta_proc _ _ = NONE;
   494 end;
   495 *}
   496 simproc_setup split_beta ("split f z") = {* fn _ => fn ctxt => fn ct => beta_proc ctxt (term_of ct) *}
   497 simproc_setup split_eta ("split f") = {* fn _ => fn ctxt => fn ct => eta_proc ctxt (term_of ct) *}
   498 
   499 lemma split_beta [mono]: "(%(x, y). P x y) z = P (fst z) (snd z)"
   500   by (subst surjective_pairing, rule split_conv)
   501 
   502 lemma split_beta': "(\<lambda>(x,y). f x y) = (\<lambda>x. f (fst x) (snd x))"
   503   by (auto simp: fun_eq_iff)
   504 
   505 
   506 lemma split_split [no_atp]: "R(split c p) = (ALL x y. p = (x, y) --> R(c x y))"
   507   -- {* For use with @{text split} and the Simplifier. *}
   508   by (insert surj_pair [of p], clarify, simp)
   509 
   510 text {*
   511   @{thm [source] split_split} could be declared as @{text "[split]"}
   512   done after the Splitter has been speeded up significantly;
   513   precompute the constants involved and don't do anything unless the
   514   current goal contains one of those constants.
   515 *}
   516 
   517 lemma split_split_asm [no_atp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))"
   518 by (subst split_split, simp)
   519 
   520 text {*
   521   \medskip @{term split} used as a logical connective or set former.
   522 
   523   \medskip These rules are for use with @{text blast}; could instead
   524   call @{text simp} using @{thm [source] prod.split} as rewrite. *}
   525 
   526 lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"
   527   apply (simp only: split_tupled_all)
   528   apply (simp (no_asm_simp))
   529   done
   530 
   531 lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"
   532   apply (simp only: split_tupled_all)
   533   apply (simp (no_asm_simp))
   534   done
   535 
   536 lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
   537   by (induct p) auto
   538 
   539 lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
   540   by (induct p) auto
   541 
   542 lemma splitE2:
   543   "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
   544 proof -
   545   assume q: "Q (split P z)"
   546   assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
   547   show R
   548     apply (rule r surjective_pairing)+
   549     apply (rule split_beta [THEN subst], rule q)
   550     done
   551 qed
   552 
   553 lemma splitD': "split R (a,b) c ==> R a b c"
   554   by simp
   555 
   556 lemma mem_splitI: "z: c a b ==> z: split c (a, b)"
   557   by simp
   558 
   559 lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"
   560 by (simp only: split_tupled_all, simp)
   561 
   562 lemma mem_splitE:
   563   assumes major: "z \<in> split c p"
   564     and cases: "\<And>x y. p = (x, y) \<Longrightarrow> z \<in> c x y \<Longrightarrow> Q"
   565   shows Q
   566   by (rule major [unfolded prod_case_unfold] cases surjective_pairing)+
   567 
   568 declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
   569 declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]
   570 
   571 ML {*
   572 local (* filtering with exists_p_split is an essential optimization *)
   573   fun exists_p_split (Const (@{const_name prod_case},_) $ _ $ (Const (@{const_name Pair},_)$_$_)) = true
   574     | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
   575     | exists_p_split (Abs (_, _, t)) = exists_p_split t
   576     | exists_p_split _ = false;
   577 in
   578 fun split_conv_tac ctxt = SUBGOAL (fn (t, i) =>
   579   if exists_p_split t
   580   then safe_full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps @{thms split_conv}) i
   581   else no_tac);
   582 end;
   583 *}
   584 
   585 (* This prevents applications of splitE for already splitted arguments leading
   586    to quite time-consuming computations (in particular for nested tuples) *)
   587 setup {* map_theory_claset (fn ctxt => ctxt addSbefore ("split_conv_tac", split_conv_tac)) *}
   588 
   589 lemma split_eta_SetCompr [simp,no_atp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
   590   by (rule ext) fast
   591 
   592 lemma split_eta_SetCompr2 [simp,no_atp]: "(%u. EX x y. u = (x, y) & P x y) = split P"
   593   by (rule ext) fast
   594 
   595 lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"
   596   -- {* Allows simplifications of nested splits in case of independent predicates. *}
   597   by (rule ext) blast
   598 
   599 (* Do NOT make this a simp rule as it
   600    a) only helps in special situations
   601    b) can lead to nontermination in the presence of split_def
   602 *)
   603 lemma split_comp_eq: 
   604   fixes f :: "'a => 'b => 'c" and g :: "'d => 'a"
   605   shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"
   606   by (rule ext) auto
   607 
   608 lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
   609   apply (rule_tac x = "(a, b)" in image_eqI)
   610    apply auto
   611   done
   612 
   613 lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
   614   by blast
   615 
   616 (*
   617 the following  would be slightly more general,
   618 but cannot be used as rewrite rule:
   619 ### Cannot add premise as rewrite rule because it contains (type) unknowns:
   620 ### ?y = .x
   621 Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
   622 by (rtac some_equality 1)
   623 by ( Simp_tac 1)
   624 by (split_all_tac 1)
   625 by (Asm_full_simp_tac 1)
   626 qed "The_split_eq";
   627 *)
   628 
   629 text {*
   630   Setup of internal @{text split_rule}.
   631 *}
   632 
   633 lemmas prod_caseI = prod.cases [THEN iffD2]
   634 
   635 lemma prod_caseI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> prod_case c p"
   636   by (fact splitI2)
   637 
   638 lemma prod_caseI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> prod_case c p x"
   639   by (fact splitI2')
   640 
   641 lemma prod_caseE: "prod_case c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
   642   by (fact splitE)
   643 
   644 lemma prod_caseE': "prod_case c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
   645   by (fact splitE')
   646 
   647 declare prod_caseI [intro!]
   648 
   649 lemma prod_case_beta:
   650   "prod_case f p = f (fst p) (snd p)"
   651   by (fact split_beta)
   652 
   653 lemma prod_cases3 [cases type]:
   654   obtains (fields) a b c where "y = (a, b, c)"
   655   by (cases y, case_tac b) blast
   656 
   657 lemma prod_induct3 [case_names fields, induct type]:
   658     "(!!a b c. P (a, b, c)) ==> P x"
   659   by (cases x) blast
   660 
   661 lemma prod_cases4 [cases type]:
   662   obtains (fields) a b c d where "y = (a, b, c, d)"
   663   by (cases y, case_tac c) blast
   664 
   665 lemma prod_induct4 [case_names fields, induct type]:
   666     "(!!a b c d. P (a, b, c, d)) ==> P x"
   667   by (cases x) blast
   668 
   669 lemma prod_cases5 [cases type]:
   670   obtains (fields) a b c d e where "y = (a, b, c, d, e)"
   671   by (cases y, case_tac d) blast
   672 
   673 lemma prod_induct5 [case_names fields, induct type]:
   674     "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
   675   by (cases x) blast
   676 
   677 lemma prod_cases6 [cases type]:
   678   obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
   679   by (cases y, case_tac e) blast
   680 
   681 lemma prod_induct6 [case_names fields, induct type]:
   682     "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
   683   by (cases x) blast
   684 
   685 lemma prod_cases7 [cases type]:
   686   obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
   687   by (cases y, case_tac f) blast
   688 
   689 lemma prod_induct7 [case_names fields, induct type]:
   690     "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
   691   by (cases x) blast
   692 
   693 lemma split_def:
   694   "split = (\<lambda>c p. c (fst p) (snd p))"
   695   by (fact prod_case_unfold)
   696 
   697 definition internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where
   698   "internal_split == split"
   699 
   700 lemma internal_split_conv: "internal_split c (a, b) = c a b"
   701   by (simp only: internal_split_def split_conv)
   702 
   703 ML_file "Tools/split_rule.ML"
   704 setup Split_Rule.setup
   705 
   706 hide_const internal_split
   707 
   708 
   709 subsubsection {* Derived operations *}
   710 
   711 definition curry    :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c" where
   712   "curry = (\<lambda>c x y. c (x, y))"
   713 
   714 lemma curry_conv [simp, code]: "curry f a b = f (a, b)"
   715   by (simp add: curry_def)
   716 
   717 lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b"
   718   by (simp add: curry_def)
   719 
   720 lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)"
   721   by (simp add: curry_def)
   722 
   723 lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q"
   724   by (simp add: curry_def)
   725 
   726 lemma curry_split [simp]: "curry (split f) = f"
   727   by (simp add: curry_def split_def)
   728 
   729 lemma split_curry [simp]: "split (curry f) = f"
   730   by (simp add: curry_def split_def)
   731 
   732 text {*
   733   The composition-uncurry combinator.
   734 *}
   735 
   736 notation fcomp (infixl "\<circ>>" 60)
   737 
   738 definition scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "\<circ>\<rightarrow>" 60) where
   739   "f \<circ>\<rightarrow> g = (\<lambda>x. prod_case g (f x))"
   740 
   741 lemma scomp_unfold: "scomp = (\<lambda>f g x. g (fst (f x)) (snd (f x)))"
   742   by (simp add: fun_eq_iff scomp_def prod_case_unfold)
   743 
   744 lemma scomp_apply [simp]: "(f \<circ>\<rightarrow> g) x = prod_case g (f x)"
   745   by (simp add: scomp_unfold prod_case_unfold)
   746 
   747 lemma Pair_scomp: "Pair x \<circ>\<rightarrow> f = f x"
   748   by (simp add: fun_eq_iff)
   749 
   750 lemma scomp_Pair: "x \<circ>\<rightarrow> Pair = x"
   751   by (simp add: fun_eq_iff)
   752 
   753 lemma scomp_scomp: "(f \<circ>\<rightarrow> g) \<circ>\<rightarrow> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>\<rightarrow> h)"
   754   by (simp add: fun_eq_iff scomp_unfold)
   755 
   756 lemma scomp_fcomp: "(f \<circ>\<rightarrow> g) \<circ>> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>> h)"
   757   by (simp add: fun_eq_iff scomp_unfold fcomp_def)
   758 
   759 lemma fcomp_scomp: "(f \<circ>> g) \<circ>\<rightarrow> h = f \<circ>> (g \<circ>\<rightarrow> h)"
   760   by (simp add: fun_eq_iff scomp_unfold)
   761 
   762 code_printing
   763   constant scomp \<rightharpoonup> (Eval) infixl 3 "#->"
   764 
   765 no_notation fcomp (infixl "\<circ>>" 60)
   766 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
   767 
   768 text {*
   769   @{term map_pair} --- action of the product functor upon
   770   functions.
   771 *}
   772 
   773 definition map_pair :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where
   774   "map_pair f g = (\<lambda>(x, y). (f x, g y))"
   775 
   776 lemma map_pair_simp [simp, code]:
   777   "map_pair f g (a, b) = (f a, g b)"
   778   by (simp add: map_pair_def)
   779 
   780 enriched_type map_pair: map_pair
   781   by (auto simp add: split_paired_all)
   782 
   783 lemma fst_map_pair [simp]:
   784   "fst (map_pair f g x) = f (fst x)"
   785   by (cases x) simp_all
   786 
   787 lemma snd_prod_fun [simp]:
   788   "snd (map_pair f g x) = g (snd x)"
   789   by (cases x) simp_all
   790 
   791 lemma fst_comp_map_pair [simp]:
   792   "fst \<circ> map_pair f g = f \<circ> fst"
   793   by (rule ext) simp_all
   794 
   795 lemma snd_comp_map_pair [simp]:
   796   "snd \<circ> map_pair f g = g \<circ> snd"
   797   by (rule ext) simp_all
   798 
   799 lemma map_pair_compose:
   800   "map_pair (f1 o f2) (g1 o g2) = (map_pair f1 g1 o map_pair f2 g2)"
   801   by (rule ext) (simp add: map_pair.compositionality comp_def)
   802 
   803 lemma map_pair_ident [simp]:
   804   "map_pair (%x. x) (%y. y) = (%z. z)"
   805   by (rule ext) (simp add: map_pair.identity)
   806 
   807 lemma map_pair_imageI [intro]:
   808   "(a, b) \<in> R \<Longrightarrow> (f a, g b) \<in> map_pair f g ` R"
   809   by (rule image_eqI) simp_all
   810 
   811 lemma prod_fun_imageE [elim!]:
   812   assumes major: "c \<in> map_pair f g ` R"
   813     and cases: "\<And>x y. c = (f x, g y) \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> P"
   814   shows P
   815   apply (rule major [THEN imageE])
   816   apply (case_tac x)
   817   apply (rule cases)
   818   apply simp_all
   819   done
   820 
   821 definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" where
   822   "apfst f = map_pair f id"
   823 
   824 definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" where
   825   "apsnd f = map_pair id f"
   826 
   827 lemma apfst_conv [simp, code]:
   828   "apfst f (x, y) = (f x, y)" 
   829   by (simp add: apfst_def)
   830 
   831 lemma apsnd_conv [simp, code]:
   832   "apsnd f (x, y) = (x, f y)" 
   833   by (simp add: apsnd_def)
   834 
   835 lemma fst_apfst [simp]:
   836   "fst (apfst f x) = f (fst x)"
   837   by (cases x) simp
   838 
   839 lemma fst_comp_apfst [simp]:
   840   "fst \<circ> apfst f = f \<circ> fst"
   841   by (simp add: fun_eq_iff)
   842 
   843 lemma fst_apsnd [simp]:
   844   "fst (apsnd f x) = fst x"
   845   by (cases x) simp
   846 
   847 lemma fst_comp_apsnd [simp]:
   848   "fst \<circ> apsnd f = fst"
   849   by (simp add: fun_eq_iff)
   850 
   851 lemma snd_apfst [simp]:
   852   "snd (apfst f x) = snd x"
   853   by (cases x) simp
   854 
   855 lemma snd_comp_apfst [simp]:
   856   "snd \<circ> apfst f = snd"
   857   by (simp add: fun_eq_iff)
   858 
   859 lemma snd_apsnd [simp]:
   860   "snd (apsnd f x) = f (snd x)"
   861   by (cases x) simp
   862 
   863 lemma snd_comp_apsnd [simp]:
   864   "snd \<circ> apsnd f = f \<circ> snd"
   865   by (simp add: fun_eq_iff)
   866 
   867 lemma apfst_compose:
   868   "apfst f (apfst g x) = apfst (f \<circ> g) x"
   869   by (cases x) simp
   870 
   871 lemma apsnd_compose:
   872   "apsnd f (apsnd g x) = apsnd (f \<circ> g) x"
   873   by (cases x) simp
   874 
   875 lemma apfst_apsnd [simp]:
   876   "apfst f (apsnd g x) = (f (fst x), g (snd x))"
   877   by (cases x) simp
   878 
   879 lemma apsnd_apfst [simp]:
   880   "apsnd f (apfst g x) = (g (fst x), f (snd x))"
   881   by (cases x) simp
   882 
   883 lemma apfst_id [simp] :
   884   "apfst id = id"
   885   by (simp add: fun_eq_iff)
   886 
   887 lemma apsnd_id [simp] :
   888   "apsnd id = id"
   889   by (simp add: fun_eq_iff)
   890 
   891 lemma apfst_eq_conv [simp]:
   892   "apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)"
   893   by (cases x) simp
   894 
   895 lemma apsnd_eq_conv [simp]:
   896   "apsnd f x = apsnd g x \<longleftrightarrow> f (snd x) = g (snd x)"
   897   by (cases x) simp
   898 
   899 lemma apsnd_apfst_commute:
   900   "apsnd f (apfst g p) = apfst g (apsnd f p)"
   901   by simp
   902 
   903 text {*
   904   Disjoint union of a family of sets -- Sigma.
   905 *}
   906 
   907 definition Sigma :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<times> 'b) set" where
   908   Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}"
   909 
   910 abbreviation
   911   Times :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set"
   912     (infixr "<*>" 80) where
   913   "A <*> B == Sigma A (%_. B)"
   914 
   915 notation (xsymbols)
   916   Times  (infixr "\<times>" 80)
   917 
   918 notation (HTML output)
   919   Times  (infixr "\<times>" 80)
   920 
   921 hide_const (open) Times
   922 
   923 syntax
   924   "_Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set"  ("(3SIGMA _:_./ _)" [0, 0, 10] 10)
   925 translations
   926   "SIGMA x:A. B" == "CONST Sigma A (%x. B)"
   927 
   928 lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
   929   by (unfold Sigma_def) blast
   930 
   931 lemma SigmaE [elim!]:
   932     "[| c: Sigma A B;
   933         !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
   934      |] ==> P"
   935   -- {* The general elimination rule. *}
   936   by (unfold Sigma_def) blast
   937 
   938 text {*
   939   Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
   940   eigenvariables.
   941 *}
   942 
   943 lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
   944   by blast
   945 
   946 lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
   947   by blast
   948 
   949 lemma SigmaE2:
   950     "[| (a, b) : Sigma A B;
   951         [| a:A;  b:B(a) |] ==> P
   952      |] ==> P"
   953   by blast
   954 
   955 lemma Sigma_cong:
   956      "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>
   957       \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
   958   by auto
   959 
   960 lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
   961   by blast
   962 
   963 lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
   964   by blast
   965 
   966 lemma Sigma_empty2 [simp]: "A <*> {} = {}"
   967   by blast
   968 
   969 lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
   970   by auto
   971 
   972 lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
   973   by auto
   974 
   975 lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
   976   by auto
   977 
   978 lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
   979   by blast
   980 
   981 lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
   982   by blast
   983 
   984 lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
   985   by (blast elim: equalityE)
   986 
   987 lemma SetCompr_Sigma_eq:
   988     "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
   989   by blast
   990 
   991 lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"
   992   by blast
   993 
   994 lemma UN_Times_distrib:
   995   "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"
   996   -- {* Suggested by Pierre Chartier *}
   997   by blast
   998 
   999 lemma split_paired_Ball_Sigma [simp, no_atp]:
  1000     "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"
  1001   by blast
  1002 
  1003 lemma split_paired_Bex_Sigma [simp, no_atp]:
  1004     "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"
  1005   by blast
  1006 
  1007 lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"
  1008   by blast
  1009 
  1010 lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))"
  1011   by blast
  1012 
  1013 lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"
  1014   by blast
  1015 
  1016 lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))"
  1017   by blast
  1018 
  1019 lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"
  1020   by blast
  1021 
  1022 lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"
  1023   by blast
  1024 
  1025 lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"
  1026   by blast
  1027 
  1028 text {*
  1029   Non-dependent versions are needed to avoid the need for higher-order
  1030   matching, especially when the rules are re-oriented.
  1031 *}
  1032 
  1033 lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"
  1034 by blast
  1035 
  1036 lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"
  1037 by blast
  1038 
  1039 lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
  1040 by blast
  1041 
  1042 lemma Times_empty[simp]: "A \<times> B = {} \<longleftrightarrow> A = {} \<or> B = {}"
  1043   by auto
  1044 
  1045 lemma times_eq_iff: "A \<times> B = C \<times> D \<longleftrightarrow> A = C \<and> B = D \<or> ((A = {} \<or> B = {}) \<and> (C = {} \<or> D = {}))"
  1046   by auto
  1047 
  1048 lemma fst_image_times[simp]: "fst ` (A \<times> B) = (if B = {} then {} else A)"
  1049   by force
  1050 
  1051 lemma snd_image_times[simp]: "snd ` (A \<times> B) = (if A = {} then {} else B)"
  1052   by force
  1053 
  1054 lemma insert_times_insert[simp]:
  1055   "insert a A \<times> insert b B =
  1056    insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)"
  1057 by blast
  1058 
  1059 lemma vimage_Times: "f -` (A \<times> B) = ((fst \<circ> f) -` A) \<inter> ((snd \<circ> f) -` B)"
  1060   apply auto
  1061   apply (case_tac "f x")
  1062   apply auto
  1063   done
  1064 
  1065 lemma times_Int_times: "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"
  1066   by auto
  1067 
  1068 lemma swap_inj_on:
  1069   "inj_on (\<lambda>(i, j). (j, i)) A"
  1070   by (auto intro!: inj_onI)
  1071 
  1072 lemma swap_product:
  1073   "(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
  1074   by (simp add: split_def image_def) blast
  1075 
  1076 lemma image_split_eq_Sigma:
  1077   "(\<lambda>x. (f x, g x)) ` A = Sigma (f ` A) (\<lambda>x. g ` (f -` {x} \<inter> A))"
  1078 proof (safe intro!: imageI)
  1079   fix a b assume *: "a \<in> A" "b \<in> A" and eq: "f a = f b"
  1080   show "(f b, g a) \<in> (\<lambda>x. (f x, g x)) ` A"
  1081     using * eq[symmetric] by auto
  1082 qed simp_all
  1083 
  1084 definition product :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" where
  1085   [code_abbrev]: "product A B = A \<times> B"
  1086 
  1087 hide_const (open) product
  1088 
  1089 lemma member_product:
  1090   "x \<in> Product_Type.product A B \<longleftrightarrow> x \<in> A \<times> B"
  1091   by (simp add: product_def)
  1092 
  1093 text {* The following @{const map_pair} lemmas are due to Joachim Breitner: *}
  1094 
  1095 lemma map_pair_inj_on:
  1096   assumes "inj_on f A" and "inj_on g B"
  1097   shows "inj_on (map_pair f g) (A \<times> B)"
  1098 proof (rule inj_onI)
  1099   fix x :: "'a \<times> 'c" and y :: "'a \<times> 'c"
  1100   assume "x \<in> A \<times> B" hence "fst x \<in> A" and "snd x \<in> B" by auto
  1101   assume "y \<in> A \<times> B" hence "fst y \<in> A" and "snd y \<in> B" by auto
  1102   assume "map_pair f g x = map_pair f g y"
  1103   hence "fst (map_pair f g x) = fst (map_pair f g y)" by (auto)
  1104   hence "f (fst x) = f (fst y)" by (cases x,cases y,auto)
  1105   with `inj_on f A` and `fst x \<in> A` and `fst y \<in> A`
  1106   have "fst x = fst y" by (auto dest:dest:inj_onD)
  1107   moreover from `map_pair f g x = map_pair f g y`
  1108   have "snd (map_pair f g x) = snd (map_pair f g y)" by (auto)
  1109   hence "g (snd x) = g (snd y)" by (cases x,cases y,auto)
  1110   with `inj_on g B` and `snd x \<in> B` and `snd y \<in> B`
  1111   have "snd x = snd y" by (auto dest:dest:inj_onD)
  1112   ultimately show "x = y" by(rule prod_eqI)
  1113 qed
  1114 
  1115 lemma map_pair_surj:
  1116   fixes f :: "'a \<Rightarrow> 'b" and g :: "'c \<Rightarrow> 'd"
  1117   assumes "surj f" and "surj g"
  1118   shows "surj (map_pair f g)"
  1119 unfolding surj_def
  1120 proof
  1121   fix y :: "'b \<times> 'd"
  1122   from `surj f` obtain a where "fst y = f a" by (auto elim:surjE)
  1123   moreover
  1124   from `surj g` obtain b where "snd y = g b" by (auto elim:surjE)
  1125   ultimately have "(fst y, snd y) = map_pair f g (a,b)" by auto
  1126   thus "\<exists>x. y = map_pair f g x" by auto
  1127 qed
  1128 
  1129 lemma map_pair_surj_on:
  1130   assumes "f ` A = A'" and "g ` B = B'"
  1131   shows "map_pair f g ` (A \<times> B) = A' \<times> B'"
  1132 unfolding image_def
  1133 proof(rule set_eqI,rule iffI)
  1134   fix x :: "'a \<times> 'c"
  1135   assume "x \<in> {y\<Colon>'a \<times> 'c. \<exists>x\<Colon>'b \<times> 'd\<in>A \<times> B. y = map_pair f g x}"
  1136   then obtain y where "y \<in> A \<times> B" and "x = map_pair f g y" by blast
  1137   from `image f A = A'` and `y \<in> A \<times> B` have "f (fst y) \<in> A'" by auto
  1138   moreover from `image g B = B'` and `y \<in> A \<times> B` have "g (snd y) \<in> B'" by auto
  1139   ultimately have "(f (fst y), g (snd y)) \<in> (A' \<times> B')" by auto
  1140   with `x = map_pair f g y` show "x \<in> A' \<times> B'" by (cases y, auto)
  1141 next
  1142   fix x :: "'a \<times> 'c"
  1143   assume "x \<in> A' \<times> B'" hence "fst x \<in> A'" and "snd x \<in> B'" by auto
  1144   from `image f A = A'` and `fst x \<in> A'` have "fst x \<in> image f A" by auto
  1145   then obtain a where "a \<in> A" and "fst x = f a" by (rule imageE)
  1146   moreover from `image g B = B'` and `snd x \<in> B'`
  1147   obtain b where "b \<in> B" and "snd x = g b" by auto
  1148   ultimately have "(fst x, snd x) = map_pair f g (a,b)" by auto
  1149   moreover from `a \<in> A` and  `b \<in> B` have "(a , b) \<in> A \<times> B" by auto
  1150   ultimately have "\<exists>y \<in> A \<times> B. x = map_pair f g y" by auto
  1151   thus "x \<in> {x. \<exists>y \<in> A \<times> B. x = map_pair f g y}" by auto
  1152 qed
  1153 
  1154 
  1155 subsection {* Simproc for rewriting a set comprehension into a pointfree expression *}
  1156 
  1157 ML_file "Tools/set_comprehension_pointfree.ML"
  1158 
  1159 setup {*
  1160   Code_Preproc.map_pre (fn ctxt => ctxt addsimprocs
  1161     [Raw_Simplifier.make_simproc {name = "set comprehension", lhss = [@{cpat "Collect ?P"}],
  1162     proc = K Set_Comprehension_Pointfree.code_simproc, identifier = []}])
  1163 *}
  1164 
  1165 
  1166 subsection {* Inductively defined sets *}
  1167 
  1168 ML_file "Tools/inductive_set.ML"
  1169 setup Inductive_Set.setup
  1170 
  1171 
  1172 subsection {* Legacy theorem bindings and duplicates *}
  1173 
  1174 lemma PairE:
  1175   obtains x y where "p = (x, y)"
  1176   by (fact prod.exhaust)
  1177 
  1178 lemmas Pair_eq = prod.inject
  1179 
  1180 lemmas split = split_conv  -- {* for backwards compatibility *}
  1181 
  1182 lemmas Pair_fst_snd_eq = prod_eq_iff
  1183 
  1184 hide_const (open) prod
  1185 
  1186 end