src/HOL/Product_Type.thy

   (Haskell infixl 4 "==")
 
 code_instance bool :: eq
   (Haskell -)
 
+
 subsection {* Unit *}
 
 typedef unit = "{True}"
 proof
   show "True : ?unit" ..

 
 lemma unit_abs_eta_conv [simp,noatp]: "(%u::unit. f ()) = f"
   by (rule ext) simp
 
 
+text {* code generator setup *}
+
+instance unit :: eq ..
+
+lemma [code func]:
+  "(u\<Colon>unit) = v \<longleftrightarrow> True" unfolding unit_eq [of u] unit_eq [of v] by rule+
+
+code_type unit
+  (SML "unit")
+  (OCaml "unit")
+  (Haskell "()")
+
+code_instance unit :: eq
+  (Haskell -)
+
+code_const "op = \<Colon> unit \<Rightarrow> unit \<Rightarrow> bool"
+  (Haskell infixl 4 "==")
+
+code_const Unity
+  (SML "()")
+  (OCaml "()")
+  (Haskell "()")
+
+code_reserved SML
+  unit
+
+code_reserved OCaml
+  unit
+
+
 subsection {* Pairs *}
 
-subsubsection {* Type definition *}
-
-constdefs
-  Pair_Rep :: "['a, 'b] => ['a, 'b] => bool"
-  "Pair_Rep == (%a b. %x y. x=a & y=b)"
+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. EX a b. f = Pair_Rep (a::'a) (b::'b)}"
+    = "{f. \<exists>a b. f = Pair_Rep (a\<Colon>'a) (b\<Colon>'b)}"
 proof
-  fix a b show "Pair_Rep a b : ?Prod"
-    by blast
+  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)
 
-local
-
-subsubsection {* Definitions *}
-
-global
-
 consts
-  fst      :: "'a * 'b => 'a"
-  snd      :: "'a * 'b => 'b"
-  split    :: "[['a, 'b] => 'c, 'a * 'b] => 'c"
-  curry    :: "['a * 'b => 'c, 'a, 'b] => 'c"
-  prod_fun :: "['a => 'b, 'c => 'd, 'a * 'c] => 'b * 'd"
-  Pair     :: "['a, 'b] => 'a * 'b"
-  Sigma    :: "['a set, 'a => 'b set] => ('a * 'b) set"
+  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))"
-  prod_fun_def: "prod_fun f g == split (%x y. Pair (f x) (g y))"
-  Sigma_def [code func]:    "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)
-
-
-subsubsection {* Concrete syntax *}
 
 text {*
   Patterns -- extends pre-defined type @{typ pttrn} used in
   abstractions.
 *}

   "_tuple_arg"  :: "'a => tuple_args"                   ("_")
   "_tuple_args" :: "'a => tuple_args => tuple_args"     ("_,/ _")
   "_pattern"    :: "[pttrn, patterns] => pttrn"         ("'(_,/ _')")
   ""            :: "pttrn => patterns"                  ("_")
   "_patterns"   :: "[pttrn, patterns] => patterns"      ("_,/ _")
-  "@Sigma" ::"[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" [0, 0, 10] 10)
 
 translations
   "(x, y)"       == "Pair x y"
   "_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))"
   "%(x,y,zs).b"  == "split(%x (y,zs).b)"
   "%(x,y).b"     == "split(%x y. b)"
   "_abs (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 *)
-  "SIGMA x:A. B" == "Sigma A (%x. B)"
 
 (* reconstructs 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))] =

 in [("split", split_guess_names_tr')]
 end 
 *}
 
 
-subsubsection {* Lemmas and proof tool setup *}
-
-lemma ProdI: "Pair_Rep a b : Prod"
-  unfolding Prod_def by blast
-
-lemma Pair_Rep_inject: "Pair_Rep a b = Pair_Rep a' b' ==> a = a' & b = b'"
-  apply (unfold Pair_Rep_def)
-  apply (drule fun_cong [THEN fun_cong], blast)
+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 prod_induct [induct type: *]: "(\<And>a b. P (a, b)) \<Longrightarrow> P x"
+  by (cases x) simp
+
+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

   unfolding fst_def by blast
 
 lemma snd_conv [simp, code]: "snd (a, b) = b"
   unfolding snd_def by blast
 
+rep_datatype prod
+  inject Pair_eq
+  induction prod_induct
+
+
+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 PairE_lemma: "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: *]: "(!!x y. p = (x, y) ==> Q) ==> Q"
-  using PairE_lemma [of p] by blast
-
-ML {*
-  local val PairE = thm "PairE" in
-    fun pair_tac s =
-      EVERY' [res_inst_tac [("p", s)] PairE, hyp_subst_tac, K prune_params_tac];
-  end;
-*}
-
-lemma surjective_pairing: "p = (fst p, snd p)"
-  -- {* Do not add as rewrite rule: invalidates some proofs in IMP *}
+lemma pair_collapse [simp]: "(fst p, snd p) = p"
   by (cases p) simp
 
-lemmas pair_collapse = surjective_pairing [symmetric]
-declare pair_collapse [simp]
-
-lemma surj_pair [simp]: "EX x y. z = (x, y)"
-  apply (rule exI)
-  apply (rule exI)
-  apply (rule surjective_pairing)
-  done
+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"

   fix x
   assume "!!a b. PROP P (a, b)"
   from `PROP P (fst x, snd x)` show "PROP P x" by simp
 qed
 
-lemmas split_tupled_all = split_paired_all unit_all_eq2
-
 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.
 *}
 
-ML {*
+lemmas split_tupled_all = split_paired_all unit_all_eq2
+
+ML_setup {*
   (* 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

 
 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 func]: "split f (a, b) = f a b"
+  by (simp add: split_def)
+
+lemma curry_conv [simp, code func]: "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 curryI [intro!]: "f (a,b) ==> curry f a b"
-  by (simp add: curry_def)
-
-lemma curryD [dest!]: "curry f a b ==> f (a,b)"
-  by (simp add: curry_def)
-
-lemma curryE: "[| curry f a b ; f (a,b) ==> Q |] ==> Q"
-  by (simp add: curry_def)
-
-lemma curry_conv [simp, code func]: "curry f a b = f (a,b)"
-  by (simp add: curry_def)
-
-lemma prod_induct [induct type: *]: "!!x. (!!a b. P (a, b)) ==> P x"
-  by fast
-
-rep_datatype prod
-  inject Pair_eq
-  induction prod_induct
-
-lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))"
-  by fast
-
-lemma split_conv [simp, code func]: "split c (a, b) = c a b"
-  by (simp add: split_def)
-
-lemmas split = split_conv  -- {* for backwards compatibility *}
-
-lemmas splitI = split_conv [THEN iffD2, standard]
-lemmas splitD = split_conv [THEN iffD1, standard]
-
-lemma split_Pair_apply: "split (%x y. f (x, y)) = f"
+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 function. *}
-  apply (rule ext)
-  apply (tactic {* pair_tac "x" 1 *}, simp)
-  done
+  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_Pair_apply)
+  by (simp add: split_eta)
 
 lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
   by (simp add: split_def)
 
-lemma Pair_fst_snd_eq: "!!s t. (s = t) = (fst s = fst t & snd s = snd t)"
-by (simp only: split_tupled_all, simp)
-
-lemma prod_eqI [intro?]: "fst p = fst q ==> snd p = snd q ==> p = q"
-  by (simp add: Pair_fst_snd_eq)
-
-lemma split_weak_cong: "p = q ==> split c p = split c q"
+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 split_eta: "(%(x, y). f (x, y)) = f"
-  apply (rule ext)
-  apply (simp only: split_tupled_all)
-  apply (rule split_conv)
-  done
 
 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}. *}
+  split_beta}.
+*}
 
 ML_setup {*
 
 local
   val cond_split_eta_ss = HOL_basic_ss addsimps [thm "cond_split_eta"]

 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,

 by (split_all_tac 1)
 by (Asm_full_simp_tac 1)
 qed "The_split_eq";
 *)
 
-lemma injective_fst_snd: "!!x y. [|fst x = fst y; snd x = snd y|] ==> x = y"
+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
 
-
-text {*
-  \bigskip @{term prod_fun} --- action of the product functor upon
+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.
+*}
+
+definition
+  mbind :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "o->" 60)
+where
+  "f o-> g = (\<lambda>x. split g (f x))"
+
+notation (xsymbols)
+  mbind  (infixl "\<circ>\<rightarrow>" 60)
+
+notation (HTML output)
+  mbind  (infixl "\<circ>\<rightarrow>" 60)
+
+lemma mbind_apply:  "(f \<circ>\<rightarrow> g) x = split g (f x)"
+  by (simp add: mbind_def)
+
+lemma Pair_mbind: "Pair x \<circ>\<rightarrow> f = f x"
+  by (simp add: expand_fun_eq mbind_apply)
+
+lemma mbind_Pair: "x \<circ>\<rightarrow> Pair = x"
+  by (simp add: expand_fun_eq mbind_apply)
+
+lemma mbind_mbind: "(f \<circ>\<rightarrow> g) \<circ>\<rightarrow> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>\<rightarrow> h)"
+  by (simp add: expand_fun_eq split_twice mbind_def)
+
+lemma mbind_fcomp: "(f \<circ>\<rightarrow> g) \<circ>> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>> h)"
+  by (simp add: expand_fun_eq mbind_apply fcomp_def split_def)
+
+lemma fcomp_mbind: "(f \<circ>> g) \<circ>\<rightarrow> h = f \<circ>> (g \<circ>\<rightarrow> h)"
+  by (simp add: expand_fun_eq mbind_apply fcomp_apply)
+
+
+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
+  [code func del]: "prod_fun f g = (\<lambda>(x, y). (f x, g y))"
 
 lemma prod_fun [simp, code func]: "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)"
-  apply (rule ext)
-  apply (tactic {* pair_tac "x" 1 *}, simp)
-  done
+  by (rule ext) auto
 
 lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)"
-  apply (rule ext)
-  apply (tactic {* pair_tac "z" 1 *}, simp)
-  done
+  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

   apply (rule cases)
    apply (blast intro: prod_fun)
   apply blast
   done
 
-
 definition
-  upd_fst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b"
+  apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b"
 where
-  [code func del]: "upd_fst f = prod_fun f id"
+  [code func del]: "apfst f = prod_fun f id"
 
 definition
-  upd_snd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c"
+  apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c"
 where
-  [code func del]: "upd_snd f = prod_fun id f"
-
-lemma upd_fst_conv [simp, code]:
-  "upd_fst f (x, y) = (f x, y)" 
-  by (simp add: upd_fst_def)
+  [code func 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 upd_snd_conv [simp, code]:
-  "upd_snd f (x, y) = (x, f y)" 
-  by (simp add: upd_snd_def)
-
-text {*
-  \bigskip Disjoint union of a family of sets -- Sigma.
-*}
+  "apsnd f (x, y) = (x, f y)" 
+  by (simp add: apsnd_def)
+
+
+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" == "Product_Type.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!]:

 
 lemma SetCompr_Sigma_eq:
     "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
   by blast
 
-text {*
-  \bigskip Complex rules for Sigma.
-*}
-
 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)"

 
 lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
   by blast
 
 
-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
-
-
-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
-
-
-subsection {* Further lemmas *}
-
-lemma
-  split_Pair: "split Pair x = x"
-  unfolding split_def by auto
-
-lemma
-  split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)"
-  by (cases x, simp)
-
-
-subsection {* Code generator setup *}
-
-instance unit :: eq ..
-
-lemma [code func]:
-  "(u\<Colon>unit) = v \<longleftrightarrow> True" unfolding unit_eq [of u] unit_eq [of v] by rule+
-
-code_type unit
-  (SML "unit")
-  (OCaml "unit")
-  (Haskell "()")
-
-code_instance unit :: eq
-  (Haskell -)
-
-code_const "op = \<Colon> unit \<Rightarrow> unit \<Rightarrow> bool"
-  (Haskell infixl 4 "==")
-
-code_const Unity
-  (SML "()")
-  (OCaml "()")
-  (Haskell "()")
-
-code_reserved SML
-  unit
-
-code_reserved OCaml
-  unit
+subsubsection {* Code generator setup *}
 
 instance * :: (eq, eq) eq ..
 
 lemma [code func]:
   "(x1\<Colon>'a\<Colon>eq, y1\<Colon>'b\<Colon>eq) = (x2, y2) \<longleftrightarrow> x1 = x2 \<and> y1 = y2" by auto

 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 PairE_lemma = thm "PairE_lemma";
 val Pair_Rep_inject = thm "Pair_Rep_inject";
 val Pair_def = thm "Pair_def";
 val Pair_eq = thm "Pair_eq";
 val Pair_fst_snd_eq = thm "Pair_fst_snd_eq";
-val Pair_inject = thm "Pair_inject";
 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 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 injective_fst_snd = thm "injective_fst_snd";
 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 splitE' = thm "splitE'";
 val splitE2 = thm "splitE2";
 val splitI = thm "splitI";
 val splitI2 = thm "splitI2";
 val splitI2' = thm "splitI2'";
-val split_Pair_apply = thm "split_Pair_apply";
 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 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";
-val unit_induct = thm "unit_induct";
 *}
 
 
 subsection {* Further inductive packages *}