src/HOL/Product_Type.thy

modernized translations;
formal markup of @{syntax_const} and @{const_syntax};
minor tuning;

(*  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_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]: "eq_class.eq False P \<longleftrightarrow> \<not> P"
    and [code]: "eq_class.eq True P \<longleftrightarrow> P" 
    and [code]: "eq_class.eq P False \<longleftrightarrow> \<not> P" 
    and [code]: "eq_class.eq P True \<longleftrightarrow> P"
    and [code nbe]: "eq_class.eq P P \<longleftrightarrow> True"
  by (simp_all add: eq)

code_const "eq_class.eq \<Colon> bool \<Rightarrow> bool \<Rightarrow> bool"
  (Haskell infixl 4 "==")

code_instance bool :: eq
  (Haskell -)


subsection {* Unit *}

typedef unit = "{True}"
proof
  show "True : ?unit" ..
qed

definition
  Unity :: unit    ("'(')")
where
  "() = Abs_unit True"

lemma unit_eq [noatp]: "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 @{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,noatp]: "(%u::unit. f ()) = f"
  by (rule ext) simp

instantiation unit :: default
begin

definition "default = ()"

instance ..

end

text {* code generator setup *}

lemma [code]:
  "eq_class.eq (u\<Colon>unit) v \<longleftrightarrow> True" unfolding eq unit_eq [of u] unit_eq [of v] by rule+

code_type unit
  (SML "unit")
  (OCaml "unit")
  (Haskell "()")
  (Scala "Unit")

code_instance unit :: eq
  (Haskell -)

code_const "eq_class.eq \<Colon> unit \<Rightarrow> unit \<Rightarrow> bool"
  (Haskell infixl 4 "==")

code_const Unity
  (SML "()")
  (OCaml "()")
  (Haskell "()")
  (Scala "()")

code_reserved SML
  unit

code_reserved OCaml
  unit

code_reserved Scala
  Unit


subsection {* Pairs *}

subsubsection {* Product type, basic operations and concrete syntax *}

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)"

global

typedef (Prod)
  ('a, 'b) "*"    (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

syntax (xsymbols)
  "*"      :: "[type, type] => type"         ("(_ \<times>/ _)" [21, 20] 20)
syntax (HTML output)
  "*"      :: "[type, type] => type"         ("(_ \<times>/ _)" [21, 20] 20)

consts
  Pair     :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b"
  fst      :: "'a \<times> 'b \<Rightarrow> 'a"
  snd      :: "'a \<times> 'b \<Rightarrow> 'b"
  split    :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c"
  curry    :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c"

local

defs
  Pair_def:     "Pair a b == Abs_Prod (Pair_Rep a b)"
  fst_def:      "fst p == THE a. EX b. p = Pair a b"
  snd_def:      "snd p == THE b. EX a. p = Pair a b"
  split_def:    "split == (%c p. c (fst p) (snd p))"
  curry_def:    "curry == (%c x y. c (Pair x y))"

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 split (%x (y, zs). b)"
  "%(x, y). b" == "CONST split (%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 split}, _) $ 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 split}, _) $ 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 split}, 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 split}, _) => 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 split}, _) => 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 split}, split_guess_names_tr')] end
*}


text {* Towards a datatype declaration *}

lemma surj_pair [simp]: "EX x y. p = (x, y)"
  apply (unfold Pair_def)
  apply (rule Rep_Prod [unfolded Prod_def, THEN CollectE])
  apply (erule exE, erule exE, rule exI, rule exI)
  apply (rule Rep_Prod_inverse [symmetric, THEN trans])
  apply (erule arg_cong)
  done

lemma PairE [cases type: *]:
  obtains x y where "p = (x, y)"
  using surj_pair [of p] by blast

lemma ProdI: "Pair_Rep a b \<in> Prod"
  unfolding Prod_def by rule+

lemma Pair_Rep_inject: "Pair_Rep a b = Pair_Rep a' b' \<Longrightarrow> a = a' \<and> b = b'"
  unfolding Pair_Rep_def by (drule fun_cong, drule fun_cong) blast

lemma inj_on_Abs_Prod: "inj_on Abs_Prod Prod"
  apply (rule inj_on_inverseI)
  apply (erule Abs_Prod_inverse)
  done

lemma Pair_inject:
  assumes "(a, b) = (a', b')"
    and "a = a' ==> b = b' ==> R"
  shows R
  apply (insert prems [unfolded Pair_def])
  apply (rule inj_on_Abs_Prod [THEN inj_onD, THEN Pair_Rep_inject, THEN conjE])
  apply (assumption | rule ProdI)+
  done

rep_datatype (prod) Pair
proof -
  fix P p
  assume "\<And>x y. P (x, y)"
  then show "P p" by (cases p) simp
qed (auto elim: Pair_inject)

lemmas Pair_eq = prod.inject

lemma fst_conv [simp, code]: "fst (a, b) = a"
  unfolding fst_def by blast

lemma snd_conv [simp, code]: "snd (a, b) = b"
  unfolding snd_def by blast


subsubsection {* Basic rules and proof tools *}

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 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 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)


subsubsection {* @{text split} and @{text curry} *}

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

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

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

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 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)

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

lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f"
  -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity Datatype. *}
  by (rule ext) auto

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"
  unfolding split_def ..

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)

lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
  by (simp add: split_def)

lemma split_weak_cong: "p = q \<Longrightarrow> split c p = split c q"
  -- {* Prevents simplification of @{term c}: much faster *}
  by (erule arg_cong)

lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
  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 [thm "cond_split_eta"]
  fun  Pair_pat k 0 (Bound m) = (m = k)
  |    Pair_pat k i (Const ("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 ("split", _) $ 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 ("split", _) $ 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 ("split", _) $ 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 @{theory} "split_beta" ["split f z"] (K beta_proc);
  val split_eta_proc = Simplifier.simproc @{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 [noatp]: "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 [noatp]: "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 simp add: split_def)

lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
  by (induct p) (auto simp add: split_def)

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: split c p"
    and cases: "!!x y. [| p = (x,y); z: c x y |] ==> Q"
  shows Q
  by (rule major [unfolded split_def] 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 ("split",_) $ _ $ (Const ("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 [thm "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,noatp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
  by (rule ext) fast

lemma split_eta_SetCompr2 [simp,noatp]: "(%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}.
*}

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)

hide const internal_split

use "Tools/split_rule.ML"
setup SplitRule.setup

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 auto

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

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

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

lemma prod_case_unfold: "prod_case = (%c p. c (fst p) (snd p))"
  by (simp add: expand_fun_eq)

declare prod_caseI2' [intro!] prod_caseI2 [intro!] prod_caseI [intro!]
declare prod_caseE' [elim!] prod_caseE [elim!]

lemma prod_case_split:
  "prod_case = split"
  by (auto simp add: expand_fun_eq)

lemma prod_case_beta:
  "prod_case f p = f (fst p) (snd p)"
  unfolding prod_case_split split_beta ..


subsection {* Further cases/induct rules for tuples *}

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


subsubsection {* Derived operations *}

text {*
  The composition-uncurry combinator.
*}

notation fcomp (infixl "o>" 60)

definition
  scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "o\<rightarrow>" 60)
where
  "f o\<rightarrow> g = (\<lambda>x. split g (f x))"

lemma scomp_apply:  "(f o\<rightarrow> g) x = split g (f x)"
  by (simp add: scomp_def)

lemma Pair_scomp: "Pair x o\<rightarrow> f = f x"
  by (simp add: expand_fun_eq scomp_apply)

lemma scomp_Pair: "x o\<rightarrow> Pair = x"
  by (simp add: expand_fun_eq scomp_apply)

lemma scomp_scomp: "(f o\<rightarrow> g) o\<rightarrow> h = f o\<rightarrow> (\<lambda>x. g x o\<rightarrow> h)"
  by (simp add: expand_fun_eq split_twice scomp_def)

lemma scomp_fcomp: "(f o\<rightarrow> g) o> h = f o\<rightarrow> (\<lambda>x. g x o> h)"
  by (simp add: expand_fun_eq scomp_apply fcomp_def split_def)

lemma fcomp_scomp: "(f o> g) o\<rightarrow> h = f o> (g o\<rightarrow> h)"
  by (simp add: expand_fun_eq scomp_apply fcomp_apply)

code_const scomp
  (Eval infixl 3 "#->")

no_notation fcomp (infixl "o>" 60)
no_notation scomp (infixl "o\<rightarrow>" 60)


text {*
  @{term prod_fun} --- action of the product functor upon
  Datatypes.
*}

definition prod_fun :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where
  [code del]: "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 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 (rule_tac p = x in PairE)
  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
  [code del]: "apfst f = prod_fun f id"

definition
  apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c"
where
  [code del]: "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,noatp]:
    "(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,noatp]:
    "(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 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, rule_tac p = "f x" in PairE, auto)

subsubsection {* Code generator setup *}

lemma [code]:
  "eq_class.eq (x1\<Colon>'a\<Colon>eq, y1\<Colon>'b\<Colon>eq) (x2, y2) \<longleftrightarrow> x1 = x2 \<and> y1 = y2" by (simp add: eq)

lemma split_case_cert:
  assumes "CASE \<equiv> split f"
  shows "CASE (a, b) \<equiv> f a b"
  using assms by simp

setup {*
  Code.add_case @{thm split_case_cert}
*}

code_type *
  (SML infix 2 "*")
  (OCaml infix 2 "*")
  (Haskell "!((_),/ (_))")
  (Scala "((_),/ (_))")

code_instance * :: eq
  (Haskell -)

code_const "eq_class.eq \<Colon> 'a\<Colon>eq \<times> 'b\<Colon>eq \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"
  (Haskell infixl 4 "==")

code_const Pair
  (SML "!((_),/ (_))")
  (OCaml "!((_),/ (_))")
  (Haskell "!((_),/ (_))")
  (Scala "!((_),/ (_))")

code_const fst and snd
  (Haskell "fst" and "snd")

types_code
  "*"     ("(_ */ _)")
attach (term_of) {*
fun term_of_id_42 aF aT bF bT (x, y) = HOLogic.pair_const aT bT $ aF x $ bF y;
*}
attach (test) {*
fun gen_id_42 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 ("split", _) $ 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 ("Let", _), t2 :: t3 :: ts) =>
    let
      fun dest_let (l as Const ("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 ("split", _), 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
*}


subsection {* Legacy bindings *}

ML {*
val Collect_split = thm "Collect_split";
val Compl_Times_UNIV1 = thm "Compl_Times_UNIV1";
val Compl_Times_UNIV2 = thm "Compl_Times_UNIV2";
val PairE = thm "PairE";
val Pair_Rep_inject = thm "Pair_Rep_inject";
val Pair_def = thm "Pair_def";
val Pair_eq = @{thm "prod.inject"};
val Pair_fst_snd_eq = thm "Pair_fst_snd_eq";
val ProdI = thm "ProdI";
val SetCompr_Sigma_eq = thm "SetCompr_Sigma_eq";
val SigmaD1 = thm "SigmaD1";
val SigmaD2 = thm "SigmaD2";
val SigmaE = thm "SigmaE";
val SigmaE2 = thm "SigmaE2";
val SigmaI = thm "SigmaI";
val Sigma_Diff_distrib1 = thm "Sigma_Diff_distrib1";
val Sigma_Diff_distrib2 = thm "Sigma_Diff_distrib2";
val Sigma_Int_distrib1 = thm "Sigma_Int_distrib1";
val Sigma_Int_distrib2 = thm "Sigma_Int_distrib2";
val Sigma_Un_distrib1 = thm "Sigma_Un_distrib1";
val Sigma_Un_distrib2 = thm "Sigma_Un_distrib2";
val Sigma_Union = thm "Sigma_Union";
val Sigma_def = thm "Sigma_def";
val Sigma_empty1 = thm "Sigma_empty1";
val Sigma_empty2 = thm "Sigma_empty2";
val Sigma_mono = thm "Sigma_mono";
val The_split = thm "The_split";
val The_split_eq = thm "The_split_eq";
val The_split_eq = thm "The_split_eq";
val Times_Diff_distrib1 = thm "Times_Diff_distrib1";
val Times_Int_distrib1 = thm "Times_Int_distrib1";
val Times_Un_distrib1 = thm "Times_Un_distrib1";
val Times_eq_cancel2 = thm "Times_eq_cancel2";
val Times_subset_cancel2 = thm "Times_subset_cancel2";
val UNIV_Times_UNIV = thm "UNIV_Times_UNIV";
val UN_Times_distrib = thm "UN_Times_distrib";
val Unity_def = thm "Unity_def";
val cond_split_eta = thm "cond_split_eta";
val fst_conv = thm "fst_conv";
val fst_def = thm "fst_def";
val fst_eqD = thm "fst_eqD";
val inj_on_Abs_Prod = thm "inj_on_Abs_Prod";
val mem_Sigma_iff = thm "mem_Sigma_iff";
val mem_splitE = thm "mem_splitE";
val mem_splitI = thm "mem_splitI";
val mem_splitI2 = thm "mem_splitI2";
val prod_eqI = thm "prod_eqI";
val prod_fun = thm "prod_fun";
val prod_fun_compose = thm "prod_fun_compose";
val prod_fun_def = thm "prod_fun_def";
val prod_fun_ident = thm "prod_fun_ident";
val prod_fun_imageE = thm "prod_fun_imageE";
val prod_fun_imageI = thm "prod_fun_imageI";
val prod_induct = thm "prod.induct";
val snd_conv = thm "snd_conv";
val snd_def = thm "snd_def";
val snd_eqD = thm "snd_eqD";
val split = thm "split";
val splitD = thm "splitD";
val splitD' = thm "splitD'";
val splitE = thm "splitE";
val splitE' = thm "splitE'";
val splitE2 = thm "splitE2";
val splitI = thm "splitI";
val splitI2 = thm "splitI2";
val splitI2' = thm "splitI2'";
val split_beta = thm "split_beta";
val split_conv = thm "split_conv";
val split_def = thm "split_def";
val split_eta = thm "split_eta";
val split_eta_SetCompr = thm "split_eta_SetCompr";
val split_eta_SetCompr2 = thm "split_eta_SetCompr2";
val split_paired_All = thm "split_paired_All";
val split_paired_Ball_Sigma = thm "split_paired_Ball_Sigma";
val split_paired_Bex_Sigma = thm "split_paired_Bex_Sigma";
val split_paired_Ex = thm "split_paired_Ex";
val split_paired_The = thm "split_paired_The";
val split_paired_all = thm "split_paired_all";
val split_part = thm "split_part";
val split_split = thm "split_split";
val split_split_asm = thm "split_split_asm";
val split_tupled_all = thms "split_tupled_all";
val split_weak_cong = thm "split_weak_cong";
val surj_pair = thm "surj_pair";
val surjective_pairing = thm "surjective_pairing";
val unit_abs_eta_conv = thm "unit_abs_eta_conv";
val unit_all_eq1 = thm "unit_all_eq1";
val unit_all_eq2 = thm "unit_all_eq2";
val unit_eq = thm "unit_eq";
*}

use "Tools/inductive_set.ML"
setup Inductive_Set.setup

end