(*  Title:      HOL/Product_Type.thy

     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

     Copyright   1992  University of Cambridge

 *)

 header {* Cartesian products *}

 theory Product_Type

 imports Typedef Inductive Fun

 uses

   ("Tools/split_rule.ML")

   ("Tools/inductive_codegen.ML")

   ("Tools/inductive_set.ML")

 begin

 subsection {* @{typ bool} is a datatype *}

 rep_datatype True False by (auto intro: bool_induct)

 declare case_split [cases type: bool]

   -- "prefer plain propositional version"

 lemma

   shows [code]: "HOL.equal False P \<longleftrightarrow> \<not> P"

     and [code]: "HOL.equal True P \<longleftrightarrow> P" 

     and [code]: "HOL.equal P False \<longleftrightarrow> \<not> P" 

     and [code]: "HOL.equal P True \<longleftrightarrow> P"

     and [code nbe]: "HOL.equal P P \<longleftrightarrow> True"

   by (simp_all add: equal)

 code_const "HOL.equal \<Colon> bool \<Rightarrow> bool \<Rightarrow> bool"

   (Haskell infixl 4 "==")

 code_instance bool :: equal

   (Haskell -)

 subsection {* The @{text unit} type *}

 typedef unit = "{True}"

 proof

   show "True : ?unit" ..

 qed

 definition

   Unity :: unit    ("'(')")

 where

   "() = Abs_unit True"

 lemma unit_eq [no_atp]: "u = ()"

   by (induct u) (simp add: unit_def Unity_def)

 text {*

   Simplification procedure for @{thm [source] unit_eq}.  Cannot use

   this rule directly --- it loops!

 *}

 ML {*

   val unit_eq_proc =

     let val unit_meta_eq = mk_meta_eq @{thm unit_eq} in

       Simplifier.simproc_global @{theory} "unit_eq" ["x::unit"]

       (fn _ => fn _ => fn t => if HOLogic.is_unit t then NONE else SOME unit_meta_eq)

     end;

   Addsimprocs [unit_eq_proc];

 *}

 rep_datatype "()" by simp

 lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()"

   by simp

 lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P"

   by (rule triv_forall_equality)

 text {*

   This rewrite counters the effect of @{text unit_eq_proc} on @{term

   [source] "%u::unit. f u"}, replacing it by @{term [source]

   f} rather than by @{term [source] "%u. f ()"}.

 *}

 lemma unit_abs_eta_conv [simp,no_atp]: "(%u::unit. f ()) = f"

   by (rule ext) simp

 instantiation unit :: default

 begin

 definition "default = ()"

 instance ..

 end

 lemma [code]:

   "HOL.equal (u\<Colon>unit) v \<longleftrightarrow> True" unfolding equal unit_eq [of u] unit_eq [of v] by rule+

 code_type unit

   (SML "unit")

   (OCaml "unit")

   (Haskell "()")

   (Scala "Unit")

 code_const Unity

   (SML "()")

   (OCaml "()")

   (Haskell "()")

   (Scala "()")

 code_instance unit :: equal

   (Haskell -)

 code_const "HOL.equal \<Colon> unit \<Rightarrow> unit \<Rightarrow> bool"

   (Haskell infixl 4 "==")

 code_reserved SML

   unit

 code_reserved OCaml

   unit

 code_reserved Scala

   Unit

 subsection {* The product type *}

 subsubsection {* Type definition *}

 definition Pair_Rep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" where

   "Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)"

 typedef ('a, 'b) prod (infixr "*" 20)

   = "{f. \<exists>a b. f = Pair_Rep (a\<Colon>'a) (b\<Colon>'b)}"

 proof

   fix a b show "Pair_Rep a b \<in> ?prod"

     by rule+

 qed

 type_notation (xsymbols)

   "prod"  ("(_ \<times>/ _)" [21, 20] 20)

 type_notation (HTML output)

   "prod"  ("(_ \<times>/ _)" [21, 20] 20)

 definition Pair :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b" where

   "Pair a b = Abs_prod (Pair_Rep a b)"

 rep_datatype Pair proof -

   fix P :: "'a \<times> 'b \<Rightarrow> bool" and p

   assume "\<And>a b. P (Pair a b)"

   then show "P p" by (cases p) (auto simp add: prod_def Pair_def Pair_Rep_def)

 next

   fix a c :: 'a and b d :: 'b

   have "Pair_Rep a b = Pair_Rep c d \<longleftrightarrow> a = c \<and> b = d"

     by (auto simp add: Pair_Rep_def expand_fun_eq)

   moreover have "Pair_Rep a b \<in> prod" and "Pair_Rep c d \<in> prod"

     by (auto simp add: prod_def)

   ultimately show "Pair a b = Pair c d \<longleftrightarrow> a = c \<and> b = d"

     by (simp add: Pair_def Abs_prod_inject)

 qed

 declare prod.simps(2) [nitpick_simp del]

 declare weak_case_cong [cong del]

 subsubsection {* Tuple syntax *}

 abbreviation (input) split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where

   "split \<equiv> prod_case"

 text {*

   Patterns -- extends pre-defined type @{typ pttrn} used in

   abstractions.

 *}

 nonterminals

   tuple_args patterns

 syntax

   "_tuple"      :: "'a => tuple_args => 'a * 'b"        ("(1'(_,/ _'))")

   "_tuple_arg"  :: "'a => tuple_args"                   ("_")

   "_tuple_args" :: "'a => tuple_args => tuple_args"     ("_,/ _")

   "_pattern"    :: "[pttrn, patterns] => pttrn"         ("'(_,/ _')")

   ""            :: "pttrn => patterns"                  ("_")

   "_patterns"   :: "[pttrn, patterns] => patterns"      ("_,/ _")

 translations

   "(x, y)" == "CONST Pair x y"

   "_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))"

   "%(x, y, zs). b" == "CONST prod_case (%x (y, zs). b)"

   "%(x, y). b" == "CONST prod_case (%x y. b)"

   "_abs (CONST Pair x y) t" => "%(x, y). t"

   -- {* The last rule accommodates tuples in `case C ... (x,y) ... => ...'

      The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *}

 (*reconstruct pattern from (nested) splits, avoiding eta-contraction of body;

   works best with enclosing "let", if "let" does not avoid eta-contraction*)

 print_translation {*

 let

   fun split_tr' [Abs (x, T, t as (Abs abs))] =

         (* split (%x y. t) => %(x,y) t *)

         let

           val (y, t') = atomic_abs_tr' abs;

           val (x', t'') = atomic_abs_tr' (x, T, t');

         in

           Syntax.const @{syntax_const "_abs"} $

             (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''

         end

     | split_tr' [Abs (x, T, (s as Const (@{const_syntax prod_case}, _) $ t))] =

         (* split (%x. (split (%y z. t))) => %(x,y,z). t *)

         let

           val Const (@{syntax_const "_abs"}, _) $

             (Const (@{syntax_const "_pattern"}, _) $ y $ z) $ t' = split_tr' [t];

           val (x', t'') = atomic_abs_tr' (x, T, t');

         in

           Syntax.const @{syntax_const "_abs"} $

             (Syntax.const @{syntax_const "_pattern"} $ x' $

               (Syntax.const @{syntax_const "_patterns"} $ y $ z)) $ t''

         end

     | split_tr' [Const (@{const_syntax prod_case}, _) $ t] =

         (* split (split (%x y z. t)) => %((x, y), z). t *)

         split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *)

     | split_tr' [Const (@{syntax_const "_abs"}, _) $ x_y $ Abs abs] =

         (* split (%pttrn z. t) => %(pttrn,z). t *)

         let val (z, t) = atomic_abs_tr' abs in

           Syntax.const @{syntax_const "_abs"} $

             (Syntax.const @{syntax_const "_pattern"} $ x_y $ z) $ t

         end

     | split_tr' _ = raise Match;

 in [(@{const_syntax prod_case}, split_tr')] end

 *}

 (* print "split f" as "\<lambda>(x,y). f x y" and "split (\<lambda>x. f x)" as "\<lambda>(x,y). f x y" *) 

 typed_print_translation {*

 let

   fun split_guess_names_tr' _ T [Abs (x, _, Abs _)] = raise Match

     | split_guess_names_tr' _ T [Abs (x, xT, t)] =

         (case (head_of t) of

           Const (@{const_syntax prod_case}, _) => raise Match

         | _ =>

           let 

             val (_ :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match;

             val (y, t') = atomic_abs_tr' ("y", yT, incr_boundvars 1 t $ Bound 0);

             val (x', t'') = atomic_abs_tr' (x, xT, t');

           in

             Syntax.const @{syntax_const "_abs"} $

               (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''

           end)

     | split_guess_names_tr' _ T [t] =

         (case head_of t of

           Const (@{const_syntax prod_case}, _) => raise Match

         | _ =>

           let

             val (xT :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match;

             val (y, t') = atomic_abs_tr' ("y", yT, incr_boundvars 2 t $ Bound 1 $ Bound 0);

             val (x', t'') = atomic_abs_tr' ("x", xT, t');

           in

             Syntax.const @{syntax_const "_abs"} $

               (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''

           end)

     | split_guess_names_tr' _ _ _ = raise Match;

 in [(@{const_syntax prod_case}, split_guess_names_tr')] end

 *}

 subsubsection {* Code generator setup *}

 code_type prod

   (SML infix 2 "*")

   (OCaml infix 2 "*")

   (Haskell "!((_),/ (_))")

   (Scala "((_),/ (_))")

 code_const Pair

   (SML "!((_),/ (_))")

   (OCaml "!((_),/ (_))")

   (Haskell "!((_),/ (_))")

   (Scala "!((_),/ (_))")

 code_instance prod :: equal

   (Haskell -)

 code_const "HOL.equal \<Colon> 'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"

   (Haskell infixl 4 "==")

 types_code

   "prod"     ("(_ */ _)")

 attach (term_of) {*

 fun term_of_prod aF aT bF bT (x, y) = HOLogic.pair_const aT bT $ aF x $ bF y;

 *}

 attach (test) {*

 fun gen_prod aG aT bG bT i =

   let

     val (x, t) = aG i;

     val (y, u) = bG i

   in ((x, y), fn () => HOLogic.pair_const aT bT $ t () $ u ()) end;

 *}

 consts_code

   "Pair"    ("(_,/ _)")

 setup {*

 let

 fun strip_abs_split 0 t = ([], t)

   | strip_abs_split i (Abs (s, T, t)) =

       let

         val s' = Codegen.new_name t s;

         val v = Free (s', T)

       in apfst (cons v) (strip_abs_split (i-1) (subst_bound (v, t))) end

   | strip_abs_split i (u as Const (@{const_name prod_case}, _) $ t) =

       (case strip_abs_split (i+1) t of

         (v :: v' :: vs, u) => (HOLogic.mk_prod (v, v') :: vs, u)

       | _ => ([], u))

   | strip_abs_split i t =

       strip_abs_split i (Abs ("x", hd (binder_types (fastype_of t)), t $ Bound 0));

 fun let_codegen thy defs dep thyname brack t gr =

   (case strip_comb t of

     (t1 as Const (@{const_name Let}, _), t2 :: t3 :: ts) =>

     let

       fun dest_let (l as Const (@{const_name Let}, _) $ t $ u) =

           (case strip_abs_split 1 u of

              ([p], u') => apfst (cons (p, t)) (dest_let u')

            | _ => ([], l))

         | dest_let t = ([], t);

       fun mk_code (l, r) gr =

         let

           val (pl, gr1) = Codegen.invoke_codegen thy defs dep thyname false l gr;

           val (pr, gr2) = Codegen.invoke_codegen thy defs dep thyname false r gr1;

         in ((pl, pr), gr2) end

     in case dest_let (t1 $ t2 $ t3) of

         ([], _) => NONE

       | (ps, u) =>

           let

             val (qs, gr1) = fold_map mk_code ps gr;

             val (pu, gr2) = Codegen.invoke_codegen thy defs dep thyname false u gr1;

             val (pargs, gr3) = fold_map

               (Codegen.invoke_codegen thy defs dep thyname true) ts gr2

           in

             SOME (Codegen.mk_app brack

               (Pretty.blk (0, [Codegen.str "let ", Pretty.blk (0, flat

                   (separate [Codegen.str ";", Pretty.brk 1] (map (fn (pl, pr) =>

                     [Pretty.block [Codegen.str "val ", pl, Codegen.str " =",

                        Pretty.brk 1, pr]]) qs))),

                 Pretty.brk 1, Codegen.str "in ", pu,

                 Pretty.brk 1, Codegen.str "end"])) pargs, gr3)

           end

     end

   | _ => NONE);

 fun split_codegen thy defs dep thyname brack t gr = (case strip_comb t of

     (t1 as Const (@{const_name prod_case}, _), t2 :: ts) =>

       let

         val ([p], u) = strip_abs_split 1 (t1 $ t2);

         val (q, gr1) = Codegen.invoke_codegen thy defs dep thyname false p gr;

         val (pu, gr2) = Codegen.invoke_codegen thy defs dep thyname false u gr1;

         val (pargs, gr3) = fold_map

           (Codegen.invoke_codegen thy defs dep thyname true) ts gr2

       in

         SOME (Codegen.mk_app brack

           (Pretty.block [Codegen.str "(fn ", q, Codegen.str " =>",

             Pretty.brk 1, pu, Codegen.str ")"]) pargs, gr2)

       end

   | _ => NONE);

 in

   Codegen.add_codegen "let_codegen" let_codegen

   #> Codegen.add_codegen "split_codegen" split_codegen

 end

 *}

 subsubsection {* Fundamental operations and properties *}

 lemma surj_pair [simp]: "EX x y. p = (x, y)"

   by (cases p) simp

 definition fst :: "'a \<times> 'b \<Rightarrow> 'a" where

   "fst p = (case p of (a, b) \<Rightarrow> a)"

 definition snd :: "'a \<times> 'b \<Rightarrow> 'b" where

   "snd p = (case p of (a, b) \<Rightarrow> b)"

 lemma fst_conv [simp, code]: "fst (a, b) = a"

   unfolding fst_def by simp

 lemma snd_conv [simp, code]: "snd (a, b) = b"

   unfolding snd_def by simp

 code_const fst and snd

   (Haskell "fst" and "snd")

 lemma prod_case_unfold [nitpick_def]: "prod_case = (%c p. c (fst p) (snd p))"

   by (simp add: expand_fun_eq split: prod.split)

 lemma fst_eqD: "fst (x, y) = a ==> x = a"

   by simp

 lemma snd_eqD: "snd (x, y) = a ==> y = a"

   by simp

 lemma pair_collapse [simp]: "(fst p, snd p) = p"

   by (cases p) simp

 lemmas surjective_pairing = pair_collapse [symmetric]

 lemma Pair_fst_snd_eq: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t"

   by (cases s, cases t) simp

 lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q"

   by (simp add: Pair_fst_snd_eq)

 lemma split_conv [simp, code]: "split f (a, b) = f a b"

   by (fact prod.cases)

 lemma splitI: "f a b \<Longrightarrow> split f (a, b)"

   by (rule split_conv [THEN iffD2])

 lemma splitD: "split f (a, b) \<Longrightarrow> f a b"

   by (rule split_conv [THEN iffD1])

 lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id"

   by (simp add: expand_fun_eq split: prod.split)

 lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f"

   -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *}

   by (simp add: expand_fun_eq split: prod.split)

 lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)"

   by (cases x) simp

 lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p"

   by (cases p) simp

 lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"

   by (simp add: prod_case_unfold)

 lemma split_weak_cong: "p = q \<Longrightarrow> split c p = split c q"

   -- {* Prevents simplification of @{term c}: much faster *}

   by (fact weak_case_cong)

 lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"

   by (simp add: split_eta)

 lemma split_paired_all: "(!!x. PROP P x) == (!!a b. PROP P (a, b))"

 proof

   fix a b

   assume "!!x. PROP P x"

   then show "PROP P (a, b)" .

 next

   fix x

   assume "!!a b. PROP P (a, b)"

   from `PROP P (fst x, snd x)` show "PROP P x" by simp

 qed

 text {*

   The rule @{thm [source] split_paired_all} does not work with the

   Simplifier because it also affects premises in congrence rules,

   where this can lead to premises of the form @{text "!!a b. ... =

   ?P(a, b)"} which cannot be solved by reflexivity.

 *}

 lemmas split_tupled_all = split_paired_all unit_all_eq2

 ML {*

   (* replace parameters of product type by individual component parameters *)

   val safe_full_simp_tac = generic_simp_tac true (true, false, false);

   local (* filtering with exists_paired_all is an essential optimization *)

     fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) =

           can HOLogic.dest_prodT T orelse exists_paired_all t

       | exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u

       | exists_paired_all (Abs (_, _, t)) = exists_paired_all t

       | exists_paired_all _ = false;

     val ss = HOL_basic_ss

       addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}]

       addsimprocs [unit_eq_proc];

   in

     val split_all_tac = SUBGOAL (fn (t, i) =>

       if exists_paired_all t then safe_full_simp_tac ss i else no_tac);

     val unsafe_split_all_tac = SUBGOAL (fn (t, i) =>

       if exists_paired_all t then full_simp_tac ss i else no_tac);

     fun split_all th =

    if exists_paired_all (Thm.prop_of th) then full_simplify ss th else th;

   end;

 *}

 declaration {* fn _ =>

   Classical.map_cs (fn cs => cs addSbefore ("split_all_tac", split_all_tac))

 *}

 lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))"

   -- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *}

   by fast

 lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))"

   by fast

 lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))"

   -- {* Can't be added to simpset: loops! *}

   by (simp add: split_eta)

 text {*

   Simplification procedure for @{thm [source] cond_split_eta}.  Using

   @{thm [source] split_eta} as a rewrite rule is not general enough,

   and using @{thm [source] cond_split_eta} directly would render some

   existing proofs very inefficient; similarly for @{text

   split_beta}.

 *}

 ML {*

 local

   val cond_split_eta_ss = HOL_basic_ss addsimps @{thms cond_split_eta};

   fun Pair_pat k 0 (Bound m) = (m = k)

     | Pair_pat k i (Const (@{const_name Pair},  _) $ Bound m $ t) =

         i > 0 andalso m = k + i andalso Pair_pat k (i - 1) t

     | Pair_pat _ _ _ = false;

   fun no_args k i (Abs (_, _, t)) = no_args (k + 1) i t

     | no_args k i (t $ u) = no_args k i t andalso no_args k i u

     | no_args k i (Bound m) = m < k orelse m > k + i

     | no_args _ _ _ = true;

   fun split_pat tp i (Abs  (_, _, t)) = if tp 0 i t then SOME (i, t) else NONE

     | split_pat tp i (Const (@{const_name prod_case}, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t

     | split_pat tp i _ = NONE;

   fun metaeq ss lhs rhs = mk_meta_eq (Goal.prove (Simplifier.the_context ss) [] []

         (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)))

         (K (simp_tac (Simplifier.inherit_context ss cond_split_eta_ss) 1)));

   fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k + 1) i t

     | beta_term_pat k i (t $ u) =

         Pair_pat k i (t $ u) orelse (beta_term_pat k i t andalso beta_term_pat k i u)

     | beta_term_pat k i t = no_args k i t;

   fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg

     | eta_term_pat _ _ _ = false;

   fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)

     | subst arg k i (t $ u) =

         if Pair_pat k i (t $ u) then incr_boundvars k arg

         else (subst arg k i t $ subst arg k i u)

     | subst arg k i t = t;

   fun beta_proc ss (s as Const (@{const_name prod_case}, _) $ Abs (_, _, t) $ arg) =

         (case split_pat beta_term_pat 1 t of

           SOME (i, f) => SOME (metaeq ss s (subst arg 0 i f))

         | NONE => NONE)

     | beta_proc _ _ = NONE;

   fun eta_proc ss (s as Const (@{const_name prod_case}, _) $ Abs (_, _, t)) =

         (case split_pat eta_term_pat 1 t of

           SOME (_, ft) => SOME (metaeq ss s (let val (f $ arg) = ft in f end))

         | NONE => NONE)

     | eta_proc _ _ = NONE;

 in

   val split_beta_proc = Simplifier.simproc_global @{theory} "split_beta" ["split f z"] (K beta_proc);

   val split_eta_proc = Simplifier.simproc_global @{theory} "split_eta" ["split f"] (K eta_proc);

 end;

 Addsimprocs [split_beta_proc, split_eta_proc];

 *}

 lemma split_beta [mono]: "(%(x, y). P x y) z = P (fst z) (snd z)"

   by (subst surjective_pairing, rule split_conv)

 lemma split_split [no_atp]: "R(split c p) = (ALL x y. p = (x, y) --> R(c x y))"

   -- {* For use with @{text split} and the Simplifier. *}

   by (insert surj_pair [of p], clarify, simp)

 text {*

   @{thm [source] split_split} could be declared as @{text "[split]"}

   done after the Splitter has been speeded up significantly;

   precompute the constants involved and don't do anything unless the

   current goal contains one of those constants.

 *}

 lemma split_split_asm [no_atp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))"

 by (subst split_split, simp)

 text {*

   \medskip @{term split} used as a logical connective or set former.

   \medskip These rules are for use with @{text blast}; could instead

   call @{text simp} using @{thm [source] split} as rewrite. *}

 lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"

   apply (simp only: split_tupled_all)

   apply (simp (no_asm_simp))

   done

 lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"

   apply (simp only: split_tupled_all)

   apply (simp (no_asm_simp))

   done

 lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"

   by (induct p) auto

 lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"

   by (induct p) auto

 lemma splitE2:

   "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"

 proof -

   assume q: "Q (split P z)"

   assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"

   show R

     apply (rule r surjective_pairing)+

     apply (rule split_beta [THEN subst], rule q)

     done

 qed

 lemma splitD': "split R (a,b) c ==> R a b c"

   by simp

 lemma mem_splitI: "z: c a b ==> z: split c (a, b)"

   by simp

 lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"

 by (simp only: split_tupled_all, simp)

 lemma mem_splitE:

   assumes major: "z \<in> split c p"

     and cases: "\<And>x y. p = (x, y) \<Longrightarrow> z \<in> c x y \<Longrightarrow> Q"

   shows Q

   by (rule major [unfolded prod_case_unfold] cases surjective_pairing)+

 declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]

 declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]

 ML {*

 local (* filtering with exists_p_split is an essential optimization *)

   fun exists_p_split (Const (@{const_name prod_case},_) $ _ $ (Const (@{const_name Pair},_)$_$_)) = true

     | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u

     | exists_p_split (Abs (_, _, t)) = exists_p_split t

     | exists_p_split _ = false;

   val ss = HOL_basic_ss addsimps @{thms split_conv};

 in

 val split_conv_tac = SUBGOAL (fn (t, i) =>

     if exists_p_split t then safe_full_simp_tac ss i else no_tac);

 end;

 *}

 (* This prevents applications of splitE for already splitted arguments leading

    to quite time-consuming computations (in particular for nested tuples) *)

 declaration {* fn _ =>

   Classical.map_cs (fn cs => cs addSbefore ("split_conv_tac", split_conv_tac))

 *}

 lemma split_eta_SetCompr [simp,no_atp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"

   by (rule ext) fast

 lemma split_eta_SetCompr2 [simp,no_atp]: "(%u. EX x y. u = (x, y) & P x y) = split P"

   by (rule ext) fast

 lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"

   -- {* Allows simplifications of nested splits in case of independent predicates. *}

   by (rule ext) blast

 (* Do NOT make this a simp rule as it

    a) only helps in special situations

    b) can lead to nontermination in the presence of split_def

 *)

 lemma split_comp_eq: 

   fixes f :: "'a => 'b => 'c" and g :: "'d => 'a"

   shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"

   by (rule ext) auto

 lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"

   apply (rule_tac x = "(a, b)" in image_eqI)

    apply auto

   done

 lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"

   by blast

 (*

 the following  would be slightly more general,

 but cannot be used as rewrite rule:

 ### Cannot add premise as rewrite rule because it contains (type) unknowns:

 ### ?y = .x

 Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"

 by (rtac some_equality 1)

 by ( Simp_tac 1)

 by (split_all_tac 1)

 by (Asm_full_simp_tac 1)

 qed "The_split_eq";

 *)

 text {*

   Setup of internal @{text split_rule}.

 *}

 lemmas prod_caseI = prod.cases [THEN iffD2, standard]

 lemma prod_caseI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> prod_case c p"

   by (fact splitI2)

 lemma prod_caseI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> prod_case c p x"

   by (fact splitI2')

 lemma prod_caseE: "prod_case c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"

   by (fact splitE)

 lemma prod_caseE': "prod_case c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"

   by (fact splitE')

 declare prod_caseI [intro!]

 lemma prod_case_beta:

   "prod_case f p = f (fst p) (snd p)"

   by (fact split_beta)

 lemma prod_cases3 [cases type]:

   obtains (fields) a b c where "y = (a, b, c)"

   by (cases y, case_tac b) blast

 lemma prod_induct3 [case_names fields, induct type]:

     "(!!a b c. P (a, b, c)) ==> P x"

   by (cases x) blast

 lemma prod_cases4 [cases type]:

   obtains (fields) a b c d where "y = (a, b, c, d)"

   by (cases y, case_tac c) blast

 lemma prod_induct4 [case_names fields, induct type]:

     "(!!a b c d. P (a, b, c, d)) ==> P x"

   by (cases x) blast

 lemma prod_cases5 [cases type]:

   obtains (fields) a b c d e where "y = (a, b, c, d, e)"

   by (cases y, case_tac d) blast

 lemma prod_induct5 [case_names fields, induct type]:

     "(!!a b c d e. P (a, b, c, d, e)) ==> P x"

   by (cases x) blast

 lemma prod_cases6 [cases type]:

   obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"

   by (cases y, case_tac e) blast

 lemma prod_induct6 [case_names fields, induct type]:

     "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"

   by (cases x) blast

 lemma prod_cases7 [cases type]:

   obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"

   by (cases y, case_tac f) blast

 lemma prod_induct7 [case_names fields, induct type]:

     "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"

   by (cases x) blast

 lemma split_def:

   "split = (\<lambda>c p. c (fst p) (snd p))"

   by (fact prod_case_unfold)

 definition internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where

   "internal_split == split"

 lemma internal_split_conv: "internal_split c (a, b) = c a b"

   by (simp only: internal_split_def split_conv)

 use "Tools/split_rule.ML"

 setup Split_Rule.setup

 hide_const internal_split

 subsubsection {* Derived operations *}

 definition curry    :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c" where

   "curry = (\<lambda>c x y. c (x, y))"

 lemma curry_conv [simp, code]: "curry f a b = f (a, b)"

   by (simp add: curry_def)

 lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b"

   by (simp add: curry_def)

 lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)"

   by (simp add: curry_def)

 lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q"

   by (simp add: curry_def)

 lemma curry_split [simp]: "curry (split f) = f"

   by (simp add: curry_def split_def)

 lemma split_curry [simp]: "split (curry f) = f"

   by (simp add: curry_def split_def)

 text {*

   The composition-uncurry combinator.

 *}

 notation fcomp (infixl "\<circ>>" 60)

 definition scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "\<circ>\<rightarrow>" 60) where

   "f \<circ>\<rightarrow> g = (\<lambda>x. prod_case g (f x))"

 lemma scomp_unfold: "scomp = (\<lambda>f g x. g (fst (f x)) (snd (f x)))"

   by (simp add: expand_fun_eq scomp_def prod_case_unfold)

 lemma scomp_apply [simp]: "(f \<circ>\<rightarrow> g) x = prod_case g (f x)"

   by (simp add: scomp_unfold prod_case_unfold)

 lemma Pair_scomp: "Pair x \<circ>\<rightarrow> f = f x"

   by (simp add: expand_fun_eq scomp_apply)

 lemma scomp_Pair: "x \<circ>\<rightarrow> Pair = x"

   by (simp add: expand_fun_eq scomp_apply)

 lemma scomp_scomp: "(f \<circ>\<rightarrow> g) \<circ>\<rightarrow> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>\<rightarrow> h)"

   by (simp add: expand_fun_eq scomp_unfold)

 lemma scomp_fcomp: "(f \<circ>\<rightarrow> g) \<circ>> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>> h)"

   by (simp add: expand_fun_eq scomp_unfold fcomp_def)

 lemma fcomp_scomp: "(f \<circ>> g) \<circ>\<rightarrow> h = f \<circ>> (g \<circ>\<rightarrow> h)"

   by (simp add: expand_fun_eq scomp_unfold fcomp_apply)

 code_const scomp

   (Eval infixl 3 "#->")

 no_notation fcomp (infixl "\<circ>>" 60)

 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)

 text {*

   @{term prod_fun} --- action of the product functor upon

   functions.

 *}

 definition prod_fun :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where

   "prod_fun f g = (\<lambda>(x, y). (f x, g y))"

 lemma prod_fun [simp, code]: "prod_fun f g (a, b) = (f a, g b)"

   by (simp add: prod_fun_def)

 lemma fst_prod_fun[simp]: "fst (prod_fun f g x) = f (fst x)"

 by (cases x, auto)

 lemma snd_prod_fun[simp]: "snd (prod_fun f g x) = g (snd x)"

 by (cases x, auto)

 lemma fst_comp_prod_fun[simp]: "fst \<circ> prod_fun f g = f \<circ> fst"

 by (rule ext) auto

 lemma snd_comp_prod_fun[simp]: "snd \<circ> prod_fun f g = g \<circ> snd"

 by (rule ext) auto

 lemma prod_fun_compose:

   "prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)"

 by (rule ext) auto

 lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)"

   by (rule ext) auto

 lemma prod_fun_imageI [intro]: "(a, b) : r ==> (f a, g b) : prod_fun f g ` r"

   apply (rule image_eqI)

   apply (rule prod_fun [symmetric], assumption)

   done

 lemma prod_fun_imageE [elim!]:

   assumes major: "c: (prod_fun f g)`r"

     and cases: "!!x y. [| c=(f(x),g(y));  (x,y):r |] ==> P"

   shows P

   apply (rule major [THEN imageE])

   apply (case_tac x)

   apply (rule cases)

    apply (blast intro: prod_fun)

   apply blast

   done

 definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" where

   "apfst f = prod_fun f id"

 definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" where

   "apsnd f = prod_fun id f"

 lemma apfst_conv [simp, code]:

   "apfst f (x, y) = (f x, y)" 

   by (simp add: apfst_def)

 lemma apsnd_conv [simp, code]:

   "apsnd f (x, y) = (x, f y)" 

   by (simp add: apsnd_def)

 lemma fst_apfst [simp]:

   "fst (apfst f x) = f (fst x)"

   by (cases x) simp

 lemma fst_apsnd [simp]:

   "fst (apsnd f x) = fst x"

   by (cases x) simp

 lemma snd_apfst [simp]:

   "snd (apfst f x) = snd x"

   by (cases x) simp

 lemma snd_apsnd [simp]:

   "snd (apsnd f x) = f (snd x)"

   by (cases x) simp

 lemma apfst_compose:

   "apfst f (apfst g x) = apfst (f \<circ> g) x"

   by (cases x) simp

 lemma apsnd_compose:

   "apsnd f (apsnd g x) = apsnd (f \<circ> g) x"

   by (cases x) simp

 lemma apfst_apsnd [simp]:

   "apfst f (apsnd g x) = (f (fst x), g (snd x))"

   by (cases x) simp

 lemma apsnd_apfst [simp]:

   "apsnd f (apfst g x) = (g (fst x), f (snd x))"

   by (cases x) simp

 lemma apfst_id [simp] :

   "apfst id = id"

   by (simp add: expand_fun_eq)

 lemma apsnd_id [simp] :

   "apsnd id = id"

   by (simp add: expand_fun_eq)

 lemma apfst_eq_conv [simp]:

   "apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)"

   by (cases x) simp

 lemma apsnd_eq_conv [simp]:

   "apsnd f x = apsnd g x \<longleftrightarrow> f (snd x) = g (snd x)"

   by (cases x) simp

 lemma apsnd_apfst_commute:

   "apsnd f (apfst g p) = apfst g (apsnd f p)"

   by simp

 text {*

   Disjoint union of a family of sets -- Sigma.

 *}

 definition  Sigma :: "['a set, 'a => 'b set] => ('a \<times> 'b) set" where

   Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}"

 abbreviation

   Times :: "['a set, 'b set] => ('a * 'b) set"

     (infixr "<*>" 80) where

   "A <*> B == Sigma A (%_. B)"

 notation (xsymbols)

   Times  (infixr "\<times>" 80)

 notation (HTML output)

   Times  (infixr "\<times>" 80)

 syntax

   "_Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set"  ("(3SIGMA _:_./ _)" [0, 0, 10] 10)

 translations

   "SIGMA x:A. B" == "CONST Sigma A (%x. B)"

 lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"

   by (unfold Sigma_def) blast

 lemma SigmaE [elim!]:

     "[| c: Sigma A B;

         !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P

      |] ==> P"

   -- {* The general elimination rule. *}

   by (unfold Sigma_def) blast

 text {*

   Elimination of @{term "(a, b) : A \<times> B"} -- introduces no

   eigenvariables.

 *}

 lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"

   by blast

 lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"

   by blast

 lemma SigmaE2:

     "[| (a, b) : Sigma A B;

         [| a:A;  b:B(a) |] ==> P

      |] ==> P"

   by blast

 lemma Sigma_cong:

      "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>

       \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"

   by auto

 lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"

   by blast

 lemma Sigma_empty1 [simp]: "Sigma {} B = {}"

   by blast

 lemma Sigma_empty2 [simp]: "A <*> {} = {}"

   by blast

 lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"

   by auto

 lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"

   by auto

 lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"

   by auto

 lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"

   by blast

 lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"

   by blast

 lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"

   by (blast elim: equalityE)

 lemma SetCompr_Sigma_eq:

     "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"

   by blast

 lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"

   by blast

 lemma UN_Times_distrib:

   "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"

   -- {* Suggested by Pierre Chartier *}

   by blast

 lemma split_paired_Ball_Sigma [simp,no_atp]:

     "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"

   by blast

 lemma split_paired_Bex_Sigma [simp,no_atp]:

     "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"

   by blast

 lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"

   by blast

 lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))"

   by blast

 lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"

   by blast

 lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))"

   by blast

 lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"

   by blast

 lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"

   by blast

 lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"

   by blast

 text {*

   Non-dependent versions are needed to avoid the need for higher-order

   matching, especially when the rules are re-oriented.

 *}

 lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"

 by blast

 lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"

 by blast

 lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"

 by blast

 lemma Times_empty[simp]: "A \<times> B = {} \<longleftrightarrow> A = {} \<or> B = {}"

   by auto

 lemma fst_image_times[simp]: "fst ` (A \<times> B) = (if B = {} then {} else A)"

   by (auto intro!: image_eqI)

 lemma snd_image_times[simp]: "snd ` (A \<times> B) = (if A = {} then {} else B)"

   by (auto intro!: image_eqI)

 lemma insert_times_insert[simp]:

   "insert a A \<times> insert b B =

    insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)"

 by blast

 lemma vimage_Times: "f -` (A \<times> B) = ((fst \<circ> f) -` A) \<inter> ((snd \<circ> f) -` B)"

   by (auto, case_tac "f x", auto)

 text{* The following @{const prod_fun} lemmas are due to Joachim Breitner: *}

 lemma prod_fun_inj_on:

   assumes "inj_on f A" and "inj_on g B"

   shows "inj_on (prod_fun f g) (A \<times> B)"

 proof (rule inj_onI)

   fix x :: "'a \<times> 'c" and y :: "'a \<times> 'c"

   assume "x \<in> A \<times> B" hence "fst x \<in> A" and "snd x \<in> B" by auto

   assume "y \<in> A \<times> B" hence "fst y \<in> A" and "snd y \<in> B" by auto

   assume "prod_fun f g x = prod_fun f g y"

   hence "fst (prod_fun f g x) = fst (prod_fun f g y)" by (auto)

   hence "f (fst x) = f (fst y)" by (cases x,cases y,auto)

   with `inj_on f A` and `fst x \<in> A` and `fst y \<in> A`

   have "fst x = fst y" by (auto dest:dest:inj_onD)

   moreover from `prod_fun f g x = prod_fun f g y`

   have "snd (prod_fun f g x) = snd (prod_fun f g y)" by (auto)

   hence "g (snd x) = g (snd y)" by (cases x,cases y,auto)

   with `inj_on g B` and `snd x \<in> B` and `snd y \<in> B`

   have "snd x = snd y" by (auto dest:dest:inj_onD)

   ultimately show "x = y" by(rule prod_eqI)

 qed

 lemma prod_fun_surj:

   assumes "surj f" and "surj g"

   shows "surj (prod_fun f g)"

 unfolding surj_def

 proof

   fix y :: "'b \<times> 'd"

   from `surj f` obtain a where "fst y = f a" by (auto elim:surjE)

   moreover

   from `surj g` obtain b where "snd y = g b" by (auto elim:surjE)

   ultimately have "(fst y, snd y) = prod_fun f g (a,b)" by auto

   thus "\<exists>x. y = prod_fun f g x" by auto

 qed

 lemma prod_fun_surj_on:

   assumes "f ` A = A'" and "g ` B = B'"

   shows "prod_fun f g ` (A \<times> B) = A' \<times> B'"

 unfolding image_def

 proof(rule set_ext,rule iffI)

   fix x :: "'a \<times> 'c"

   assume "x \<in> {y\<Colon>'a \<times> 'c. \<exists>x\<Colon>'b \<times> 'd\<in>A \<times> B. y = prod_fun f g x}"

   then obtain y where "y \<in> A \<times> B" and "x = prod_fun f g y" by blast

   from `image f A = A'` and `y \<in> A \<times> B` have "f (fst y) \<in> A'" by auto

   moreover from `image g B = B'` and `y \<in> A \<times> B` have "g (snd y) \<in> B'" by auto

   ultimately have "(f (fst y), g (snd y)) \<in> (A' \<times> B')" by auto

   with `x = prod_fun f g y` show "x \<in> A' \<times> B'" by (cases y, auto)

 next

   fix x :: "'a \<times> 'c"

   assume "x \<in> A' \<times> B'" hence "fst x \<in> A'" and "snd x \<in> B'" by auto

   from `image f A = A'` and `fst x \<in> A'` have "fst x \<in> image f A" by auto

   then obtain a where "a \<in> A" and "fst x = f a" by (rule imageE)

   moreover from `image g B = B'` and `snd x \<in> B'`

   obtain b where "b \<in> B" and "snd x = g b" by auto

   ultimately have "(fst x, snd x) = prod_fun f g (a,b)" by auto

   moreover from `a \<in> A` and  `b \<in> B` have "(a , b) \<in> A \<times> B" by auto

   ultimately have "\<exists>y \<in> A \<times> B. x = prod_fun f g y" by auto

   thus "x \<in> {x. \<exists>y \<in> A \<times> B. x = prod_fun f g y}" by auto

 qed

 lemma swap_inj_on:

   "inj_on (\<lambda>(i, j). (j, i)) A"

   by (auto intro!: inj_onI)

 lemma swap_product:

   "(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A"

   by (simp add: split_def image_def) blast

 lemma image_split_eq_Sigma:

   "(\<lambda>x. (f x, g x)) ` A = Sigma (f ` A) (\<lambda>x. g ` (f -` {x} \<inter> A))"

 proof (safe intro!: imageI vimageI)

   fix a b assume *: "a \<in> A" "b \<in> A" and eq: "f a = f b"

   show "(f b, g a) \<in> (\<lambda>x. (f x, g x)) ` A"

     using * eq[symmetric] by auto

 qed simp_all

 subsection {* Inductively defined sets *}

 use "Tools/inductive_codegen.ML"

 setup Inductive_Codegen.setup

 use "Tools/inductive_set.ML"

 setup Inductive_Set.setup

 subsection {* Legacy theorem bindings and duplicates *}

 lemma PairE:

   obtains x y where "p = (x, y)"

   by (fact prod.exhaust)

 lemma Pair_inject:

   assumes "(a, b) = (a', b')"

     and "a = a' ==> b = b' ==> R"

   shows R

   using assms by simp

 lemmas Pair_eq = prod.inject

 lemmas split = split_conv  -- {* for backwards compatibility *}

 end