
nipkow@10213: (*  Title:      HOL/Product_Type.thy
nipkow@10213:     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
nipkow@10213:     Copyright   1992  University of Cambridge
wenzelm@11777: *)
nipkow@10213: 
wenzelm@11838: header {* Cartesian products *}
nipkow@10213: 
nipkow@15131: theory Product_Type
haftmann@33959: imports Typedef Inductive Fun
wenzelm@46950: keywords "inductive_set" "coinductive_set" :: thy_decl
nipkow@15131: begin
wenzelm@11838: 
haftmann@24699: subsection {* @{typ bool} is a datatype *}
haftmann@24699: 
blanchet@55414: wrap_free_constructors [True, False] case_bool [=]
blanchet@55393: by auto
blanchet@55393: 
blanchet@55393: -- {* Avoid name clashes by prefixing the output of @{text rep_datatype} with @{text old}. *}
blanchet@55393: setup {* Sign.mandatory_path "old" *}
blanchet@55393: 
haftmann@27104: rep_datatype True False by (auto intro: bool_induct)
haftmann@24699: 
blanchet@55393: setup {* Sign.parent_path *}
blanchet@55393: 
blanchet@55393: -- {* But erase the prefix for properties that are not generated by @{text wrap_free_constructors}. *}
blanchet@55393: setup {* Sign.mandatory_path "bool" *}
blanchet@55393: 
blanchet@55393: lemmas induct = old.bool.induct
blanchet@55393: lemmas inducts = old.bool.inducts
blanchet@55393: lemmas recs = old.bool.recs
blanchet@55393: lemmas cases = bool.case
blanchet@55393: lemmas simps = bool.distinct bool.case old.bool.recs
blanchet@55393: 
blanchet@55393: setup {* Sign.parent_path *}
blanchet@55393: 
haftmann@24699: declare case_split [cases type: bool]
haftmann@24699:   -- "prefer plain propositional version"
haftmann@24699: 
haftmann@28346: lemma
haftmann@38857:   shows [code]: "HOL.equal False P \<longleftrightarrow> \<not> P"
haftmann@38857:     and [code]: "HOL.equal True P \<longleftrightarrow> P" 
haftmann@46630:     and [code]: "HOL.equal P False \<longleftrightarrow> \<not> P"
haftmann@38857:     and [code]: "HOL.equal P True \<longleftrightarrow> P"
haftmann@38857:     and [code nbe]: "HOL.equal P P \<longleftrightarrow> True"
haftmann@38857:   by (simp_all add: equal)
haftmann@25534: 
haftmann@43654: lemma If_case_cert:
haftmann@43654:   assumes "CASE \<equiv> (\<lambda>b. If b f g)"
haftmann@43654:   shows "(CASE True \<equiv> f) &&& (CASE False \<equiv> g)"
haftmann@43654:   using assms by simp_all
haftmann@43654: 
haftmann@43654: setup {*
haftmann@43654:   Code.add_case @{thm If_case_cert}
haftmann@43654: *}
haftmann@43654: 
haftmann@52435: code_printing
haftmann@52435:   constant "HOL.equal :: bool \<Rightarrow> bool \<Rightarrow> bool" \<rightharpoonup> (Haskell) infix 4 "=="
haftmann@52435: | class_instance "bool" :: "equal" \<rightharpoonup> (Haskell) -
haftmann@24699: 
haftmann@26358: 
haftmann@37166: subsection {* The @{text unit} type *}
wenzelm@11838: 
wenzelm@49834: typedef unit = "{True}"
wenzelm@45694:   by auto
wenzelm@11838: 
wenzelm@45694: definition Unity :: unit  ("'(')")
wenzelm@45694:   where "() = Abs_unit True"
wenzelm@11838: 
blanchet@35828: lemma unit_eq [no_atp]: "u = ()"
huffman@40590:   by (induct u) (simp add: Unity_def)
wenzelm@11838: 
wenzelm@11838: text {*
wenzelm@11838:   Simplification procedure for @{thm [source] unit_eq}.  Cannot use
wenzelm@11838:   this rule directly --- it loops!
wenzelm@11838: *}
wenzelm@11838: 
wenzelm@43594: simproc_setup unit_eq ("x::unit") = {*
wenzelm@43594:   fn _ => fn _ => fn ct =>
wenzelm@43594:     if HOLogic.is_unit (term_of ct) then NONE
wenzelm@43594:     else SOME (mk_meta_eq @{thm unit_eq})
wenzelm@11838: *}
wenzelm@11838: 
blanchet@55414: wrap_free_constructors ["()"] case_unit
blanchet@55393: by auto
blanchet@55393: 
blanchet@55393: -- {* Avoid name clashes by prefixing the output of @{text rep_datatype} with @{text old}. *}
blanchet@55393: setup {* Sign.mandatory_path "old" *}
blanchet@55393: 
haftmann@27104: rep_datatype "()" by simp
haftmann@24699: 
blanchet@55393: setup {* Sign.parent_path *}
blanchet@55393: 
blanchet@55393: -- {* But erase the prefix for properties that are not generated by @{text wrap_free_constructors}. *}
blanchet@55393: setup {* Sign.mandatory_path "unit" *}
blanchet@55393: 
blanchet@55393: lemmas induct = old.unit.induct
blanchet@55393: lemmas inducts = old.unit.inducts
blanchet@55393: lemmas recs = old.unit.recs
blanchet@55393: lemmas cases = unit.case
blanchet@55393: lemmas simps = unit.case old.unit.recs
blanchet@55393: 
blanchet@55393: setup {* Sign.parent_path *}
blanchet@55393: 
wenzelm@11838: lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()"
wenzelm@11838:   by simp
wenzelm@11838: 
wenzelm@11838: lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P"
wenzelm@11838:   by (rule triv_forall_equality)
wenzelm@11838: 
wenzelm@11838: text {*
wenzelm@43594:   This rewrite counters the effect of simproc @{text unit_eq} on @{term
wenzelm@11838:   [source] "%u::unit. f u"}, replacing it by @{term [source]
wenzelm@11838:   f} rather than by @{term [source] "%u. f ()"}.
wenzelm@11838: *}
wenzelm@11838: 
blanchet@54147: lemma unit_abs_eta_conv [simp]: "(%u::unit. f ()) = f"
wenzelm@11838:   by (rule ext) simp
nipkow@10213: 
blanchet@54147: lemma UNIV_unit:
haftmann@43866:   "UNIV = {()}" by auto
haftmann@43866: 
haftmann@30924: instantiation unit :: default
haftmann@30924: begin
haftmann@30924: 
haftmann@30924: definition "default = ()"
haftmann@30924: 
haftmann@30924: instance ..
haftmann@30924: 
haftmann@30924: end
nipkow@10213: 
haftmann@28562: lemma [code]:
haftmann@38857:   "HOL.equal (u\<Colon>unit) v \<longleftrightarrow> True" unfolding equal unit_eq [of u] unit_eq [of v] by rule+
haftmann@26358: 
haftmann@52435: code_printing
haftmann@52435:   type_constructor unit \<rightharpoonup>
haftmann@52435:     (SML) "unit"
haftmann@52435:     and (OCaml) "unit"
haftmann@52435:     and (Haskell) "()"
haftmann@52435:     and (Scala) "Unit"
haftmann@52435: | constant Unity \<rightharpoonup>
haftmann@52435:     (SML) "()"
haftmann@52435:     and (OCaml) "()"
haftmann@52435:     and (Haskell) "()"
haftmann@52435:     and (Scala) "()"
haftmann@52435: | class_instance unit :: equal \<rightharpoonup>
haftmann@52435:     (Haskell) -
haftmann@52435: | constant "HOL.equal :: unit \<Rightarrow> unit \<Rightarrow> bool" \<rightharpoonup>
haftmann@52435:     (Haskell) infix 4 "=="
haftmann@26358: 
haftmann@26358: code_reserved SML
haftmann@26358:   unit
haftmann@26358: 
haftmann@26358: code_reserved OCaml
haftmann@26358:   unit
haftmann@26358: 
haftmann@34886: code_reserved Scala
haftmann@34886:   Unit
haftmann@34886: 
haftmann@26358: 
haftmann@37166: subsection {* The product type *}
nipkow@10213: 
haftmann@37166: subsubsection {* Type definition *}
haftmann@37166: 
haftmann@37166: definition Pair_Rep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" where
haftmann@26358:   "Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)"
nipkow@10213: 
wenzelm@45696: definition "prod = {f. \<exists>a b. f = Pair_Rep (a\<Colon>'a) (b\<Colon>'b)}"
wenzelm@45696: 
wenzelm@49834: typedef ('a, 'b) prod (infixr "*" 20) = "prod :: ('a \<Rightarrow> 'b \<Rightarrow> bool) set"
wenzelm@45696:   unfolding prod_def by auto
nipkow@10213: 
wenzelm@35427: type_notation (xsymbols)
haftmann@37678:   "prod"  ("(_ \<times>/ _)" [21, 20] 20)
wenzelm@35427: type_notation (HTML output)
haftmann@37678:   "prod"  ("(_ \<times>/ _)" [21, 20] 20)
nipkow@10213: 
haftmann@37389: definition Pair :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b" where
haftmann@37389:   "Pair a b = Abs_prod (Pair_Rep a b)"
haftmann@37166: 
blanchet@55393: lemma prod_cases: "(\<And>a b. P (Pair a b)) \<Longrightarrow> P p"
blanchet@55393:   by (cases p) (auto simp add: prod_def Pair_def Pair_Rep_def)
blanchet@55393: 
blanchet@55414: wrap_free_constructors [Pair] case_prod [] [[fst, snd]]
blanchet@55393: proof -
blanchet@55393:   fix P :: bool and p :: "'a \<times> 'b"
blanchet@55393:   show "(\<And>x1 x2. p = Pair x1 x2 \<Longrightarrow> P) \<Longrightarrow> P"
blanchet@55393:     by (cases p) (auto simp add: prod_def Pair_def Pair_Rep_def)
haftmann@37166: next
haftmann@37166:   fix a c :: 'a and b d :: 'b
haftmann@37166:   have "Pair_Rep a b = Pair_Rep c d \<longleftrightarrow> a = c \<and> b = d"
nipkow@39302:     by (auto simp add: Pair_Rep_def fun_eq_iff)
haftmann@37389:   moreover have "Pair_Rep a b \<in> prod" and "Pair_Rep c d \<in> prod"
haftmann@37389:     by (auto simp add: prod_def)
haftmann@37166:   ultimately show "Pair a b = Pair c d \<longleftrightarrow> a = c \<and> b = d"
haftmann@37389:     by (simp add: Pair_def Abs_prod_inject)
haftmann@37166: qed
haftmann@37166: 
blanchet@55393: -- {* Avoid name clashes by prefixing the output of @{text rep_datatype} with @{text old}. *}
blanchet@55393: setup {* Sign.mandatory_path "old" *}
blanchet@55393: 
blanchet@55393: rep_datatype Pair
blanchet@55403: by (erule prod_cases) (rule prod.inject)
blanchet@55393: 
blanchet@55393: setup {* Sign.parent_path *}
blanchet@37704: 
blanchet@55393: -- {* But erase the prefix for properties that are not generated by @{text wrap_free_constructors}. *}
blanchet@55393: setup {* Sign.mandatory_path "prod" *}
blanchet@55393: 
blanchet@55393: declare
blanchet@55393:   old.prod.inject[iff del]
blanchet@55393: 
blanchet@55393: lemmas induct = old.prod.induct
blanchet@55393: lemmas inducts = old.prod.inducts
blanchet@55393: lemmas recs = old.prod.recs
blanchet@55393: lemmas cases = prod.case
blanchet@55393: lemmas simps = prod.inject prod.case old.prod.recs
blanchet@55393: 
blanchet@55393: setup {* Sign.parent_path *}
blanchet@55393: 
blanchet@55393: declare prod.case [nitpick_simp del]
huffman@40929: declare prod.weak_case_cong [cong del]
haftmann@37411: 
haftmann@37166: 
haftmann@37166: subsubsection {* Tuple syntax *}
haftmann@37166: 
haftmann@37591: abbreviation (input) split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where
blanchet@55414:   "split \<equiv> case_prod"
wenzelm@19535: 
wenzelm@11777: text {*
wenzelm@11777:   Patterns -- extends pre-defined type @{typ pttrn} used in
wenzelm@11777:   abstractions.
wenzelm@11777: *}
nipkow@10213: 
wenzelm@41229: nonterminal tuple_args and patterns
nipkow@10213: 
nipkow@10213: syntax
nipkow@10213:   "_tuple"      :: "'a => tuple_args => 'a * 'b"        ("(1'(_,/ _'))")
nipkow@10213:   "_tuple_arg"  :: "'a => tuple_args"                   ("_")
nipkow@10213:   "_tuple_args" :: "'a => tuple_args => tuple_args"     ("_,/ _")
oheimb@11025:   "_pattern"    :: "[pttrn, patterns] => pttrn"         ("'(_,/ _')")
oheimb@11025:   ""            :: "pttrn => patterns"                  ("_")
oheimb@11025:   "_patterns"   :: "[pttrn, patterns] => patterns"      ("_,/ _")
nipkow@10213: 
nipkow@10213: translations
wenzelm@35115:   "(x, y)" == "CONST Pair x y"
nipkow@51392:   "_pattern x y" => "CONST Pair x y"
nipkow@51392:   "_patterns x y" => "CONST Pair x y"
nipkow@10213:   "_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))"
blanchet@55414:   "%(x, y, zs). b" == "CONST case_prod (%x (y, zs). b)"
blanchet@55414:   "%(x, y). b" == "CONST case_prod (%x y. b)"
wenzelm@35115:   "_abs (CONST Pair x y) t" => "%(x, y). t"
haftmann@37166:   -- {* The last rule accommodates tuples in `case C ... (x,y) ... => ...'
haftmann@37166:      The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *}
nipkow@10213: 
wenzelm@35115: (*reconstruct pattern from (nested) splits, avoiding eta-contraction of body;
wenzelm@35115:   works best with enclosing "let", if "let" does not avoid eta-contraction*)
schirmer@14359: print_translation {*
wenzelm@52143:   let
wenzelm@52143:     fun split_tr' [Abs (x, T, t as (Abs abs))] =
wenzelm@52143:           (* split (%x y. t) => %(x,y) t *)
wenzelm@52143:           let
wenzelm@52143:             val (y, t') = Syntax_Trans.atomic_abs_tr' abs;
wenzelm@52143:             val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t');
wenzelm@52143:           in
wenzelm@52143:             Syntax.const @{syntax_const "_abs"} $
wenzelm@52143:               (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
wenzelm@52143:           end
blanchet@55414:       | split_tr' [Abs (x, T, (s as Const (@{const_syntax case_prod}, _) $ t))] =
wenzelm@52143:           (* split (%x. (split (%y z. t))) => %(x,y,z). t *)
wenzelm@52143:           let
wenzelm@52143:             val Const (@{syntax_const "_abs"}, _) $
wenzelm@52143:               (Const (@{syntax_const "_pattern"}, _) $ y $ z) $ t' = split_tr' [t];
wenzelm@52143:             val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t');
wenzelm@52143:           in
wenzelm@52143:             Syntax.const @{syntax_const "_abs"} $
wenzelm@52143:               (Syntax.const @{syntax_const "_pattern"} $ x' $
wenzelm@52143:                 (Syntax.const @{syntax_const "_patterns"} $ y $ z)) $ t''
wenzelm@52143:           end
blanchet@55414:       | split_tr' [Const (@{const_syntax case_prod}, _) $ t] =
wenzelm@52143:           (* split (split (%x y z. t)) => %((x, y), z). t *)
wenzelm@52143:           split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *)
wenzelm@52143:       | split_tr' [Const (@{syntax_const "_abs"}, _) $ x_y $ Abs abs] =
wenzelm@52143:           (* split (%pttrn z. t) => %(pttrn,z). t *)
wenzelm@52143:           let val (z, t) = Syntax_Trans.atomic_abs_tr' abs in
wenzelm@52143:             Syntax.const @{syntax_const "_abs"} $
wenzelm@52143:               (Syntax.const @{syntax_const "_pattern"} $ x_y $ z) $ t
wenzelm@52143:           end
wenzelm@52143:       | split_tr' _ = raise Match;
blanchet@55414:   in [(@{const_syntax case_prod}, K split_tr')] end
schirmer@14359: *}
schirmer@14359: 
schirmer@15422: (* print "split f" as "\<lambda>(x,y). f x y" and "split (\<lambda>x. f x)" as "\<lambda>(x,y). f x y" *) 
schirmer@15422: typed_print_translation {*
wenzelm@52143:   let
wenzelm@52143:     fun split_guess_names_tr' T [Abs (x, _, Abs _)] = raise Match
wenzelm@52143:       | split_guess_names_tr' T [Abs (x, xT, t)] =
wenzelm@52143:           (case (head_of t) of
blanchet@55414:             Const (@{const_syntax case_prod}, _) => raise Match
wenzelm@52143:           | _ =>
wenzelm@52143:             let 
wenzelm@52143:               val (_ :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match;
wenzelm@52143:               val (y, t') = Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 1 t $ Bound 0);
wenzelm@52143:               val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, xT, t');
wenzelm@52143:             in
wenzelm@52143:               Syntax.const @{syntax_const "_abs"} $
wenzelm@52143:                 (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
wenzelm@52143:             end)
wenzelm@52143:       | split_guess_names_tr' T [t] =
wenzelm@52143:           (case head_of t of
blanchet@55414:             Const (@{const_syntax case_prod}, _) => raise Match
wenzelm@52143:           | _ =>
wenzelm@52143:             let
wenzelm@52143:               val (xT :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match;
wenzelm@52143:               val (y, t') =
wenzelm@52143:                 Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 2 t $ Bound 1 $ Bound 0);
wenzelm@52143:               val (x', t'') = Syntax_Trans.atomic_abs_tr' ("x", xT, t');
wenzelm@52143:             in
wenzelm@52143:               Syntax.const @{syntax_const "_abs"} $
wenzelm@52143:                 (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
wenzelm@52143:             end)
wenzelm@52143:       | split_guess_names_tr' _ _ = raise Match;
blanchet@55414:   in [(@{const_syntax case_prod}, K split_guess_names_tr')] end
schirmer@15422: *}
schirmer@15422: 
blanchet@55414: (* Force eta-contraction for terms of the form "Q A (%p. case_prod P p)"
nipkow@42059:    where Q is some bounded quantifier or set operator.
nipkow@42059:    Reason: the above prints as "Q p : A. case p of (x,y) \<Rightarrow> P x y"
nipkow@42059:    whereas we want "Q (x,y):A. P x y".
nipkow@42059:    Otherwise prevent eta-contraction.
nipkow@42059: *)
nipkow@42059: print_translation {*
wenzelm@52143:   let
wenzelm@52143:     fun contract Q tr ctxt ts =
wenzelm@52143:       (case ts of
blanchet@55414:         [A, Abs (_, _, (s as Const (@{const_syntax case_prod},_) $ t) $ Bound 0)] =>
wenzelm@52143:           if Term.is_dependent t then tr ctxt ts
wenzelm@52143:           else Syntax.const Q $ A $ s
wenzelm@52143:       | _ => tr ctxt ts);
wenzelm@52143:   in
wenzelm@42284:     [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax Ball} @{syntax_const "_Ball"},
wenzelm@42284:      Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax Bex} @{syntax_const "_Bex"},
wenzelm@42284:      Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"},
wenzelm@42284:      Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}]
wenzelm@52143:     |> map (fn (Q, tr) => (Q, contract Q tr))
wenzelm@52143:   end
nipkow@42059: *}
nipkow@10213: 
haftmann@37166: subsubsection {* Code generator setup *}
haftmann@37166: 
haftmann@52435: code_printing
haftmann@52435:   type_constructor prod \<rightharpoonup>
haftmann@52435:     (SML) infix 2 "*"
haftmann@52435:     and (OCaml) infix 2 "*"
haftmann@52435:     and (Haskell) "!((_),/ (_))"
haftmann@52435:     and (Scala) "((_),/ (_))"
haftmann@52435: | constant Pair \<rightharpoonup>
haftmann@52435:     (SML) "!((_),/ (_))"
haftmann@52435:     and (OCaml) "!((_),/ (_))"
haftmann@52435:     and (Haskell) "!((_),/ (_))"
haftmann@52435:     and (Scala) "!((_),/ (_))"
haftmann@52435: | class_instance  prod :: equal \<rightharpoonup>
haftmann@52435:     (Haskell) -
haftmann@52435: | constant "HOL.equal :: 'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool" \<rightharpoonup>
haftmann@52435:     (Haskell) infix 4 "=="
haftmann@37166: 
haftmann@37166: 
haftmann@37166: subsubsection {* Fundamental operations and properties *}
wenzelm@11838: 
bulwahn@49897: lemma Pair_inject:
bulwahn@49897:   assumes "(a, b) = (a', b')"
bulwahn@49897:     and "a = a' ==> b = b' ==> R"
bulwahn@49897:   shows R
bulwahn@49897:   using assms by simp
bulwahn@49897: 
haftmann@26358: lemma surj_pair [simp]: "EX x y. p = (x, y)"
haftmann@37166:   by (cases p) simp
nipkow@10213: 
haftmann@52435: code_printing
haftmann@52435:   constant fst \<rightharpoonup> (Haskell) "fst"
haftmann@52435: | constant snd \<rightharpoonup> (Haskell) "snd"
haftmann@26358: 
blanchet@55414: lemma case_prod_unfold [nitpick_unfold]: "case_prod = (%c p. c (fst p) (snd p))"
nipkow@39302:   by (simp add: fun_eq_iff split: prod.split)
haftmann@26358: 
wenzelm@11838: lemma fst_eqD: "fst (x, y) = a ==> x = a"
wenzelm@11838:   by simp
wenzelm@11838: 
wenzelm@11838: lemma snd_eqD: "snd (x, y) = a ==> y = a"
wenzelm@11838:   by simp
wenzelm@11838: 
blanchet@55393: lemmas surjective_pairing = prod.collapse [symmetric]
wenzelm@11838: 
huffman@44066: lemma prod_eq_iff: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t"
haftmann@37166:   by (cases s, cases t) simp
haftmann@37166: 
haftmann@37166: lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q"
huffman@44066:   by (simp add: prod_eq_iff)
haftmann@37166: 
haftmann@37166: lemma split_conv [simp, code]: "split f (a, b) = f a b"
haftmann@37591:   by (fact prod.cases)
haftmann@37166: 
haftmann@37166: lemma splitI: "f a b \<Longrightarrow> split f (a, b)"
haftmann@37166:   by (rule split_conv [THEN iffD2])
haftmann@37166: 
haftmann@37166: lemma splitD: "split f (a, b) \<Longrightarrow> f a b"
haftmann@37166:   by (rule split_conv [THEN iffD1])
haftmann@37166: 
haftmann@37166: lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id"
nipkow@39302:   by (simp add: fun_eq_iff split: prod.split)
haftmann@37166: 
haftmann@37166: lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f"
haftmann@37166:   -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *}
nipkow@39302:   by (simp add: fun_eq_iff split: prod.split)
haftmann@37166: 
haftmann@37166: lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)"
haftmann@37166:   by (cases x) simp
haftmann@37166: 
haftmann@37166: lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p"
haftmann@37166:   by (cases p) simp
haftmann@37166: 
haftmann@37166: lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
blanchet@55414:   by (simp add: case_prod_unfold)
haftmann@37166: 
haftmann@37166: lemma split_weak_cong: "p = q \<Longrightarrow> split c p = split c q"
haftmann@37166:   -- {* Prevents simplification of @{term c}: much faster *}
huffman@40929:   by (fact prod.weak_case_cong)
haftmann@37166: 
haftmann@37166: lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
haftmann@37166:   by (simp add: split_eta)
haftmann@37166: 
blanchet@47740: lemma split_paired_all [no_atp]: "(!!x. PROP P x) == (!!a b. PROP P (a, b))"
wenzelm@11820: proof
wenzelm@11820:   fix a b
wenzelm@11820:   assume "!!x. PROP P x"
wenzelm@19535:   then show "PROP P (a, b)" .
wenzelm@11820: next
wenzelm@11820:   fix x
wenzelm@11820:   assume "!!a b. PROP P (a, b)"
wenzelm@19535:   from `PROP P (fst x, snd x)` show "PROP P x" by simp
wenzelm@11820: qed
wenzelm@11820: 
hoelzl@50104: lemma case_prod_distrib: "f (case x of (x, y) \<Rightarrow> g x y) = (case x of (x, y) \<Rightarrow> f (g x y))"
hoelzl@50104:   by (cases x) simp
hoelzl@50104: 
wenzelm@11838: text {*
wenzelm@11838:   The rule @{thm [source] split_paired_all} does not work with the
wenzelm@11838:   Simplifier because it also affects premises in congrence rules,
wenzelm@11838:   where this can lead to premises of the form @{text "!!a b. ... =
wenzelm@11838:   ?P(a, b)"} which cannot be solved by reflexivity.
wenzelm@11838: *}
wenzelm@11838: 
haftmann@26358: lemmas split_tupled_all = split_paired_all unit_all_eq2
haftmann@26358: 
wenzelm@26480: ML {*
wenzelm@11838:   (* replace parameters of product type by individual component parameters *)
wenzelm@11838:   local (* filtering with exists_paired_all is an essential optimization *)
wenzelm@16121:     fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) =
wenzelm@11838:           can HOLogic.dest_prodT T orelse exists_paired_all t
wenzelm@11838:       | exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u
wenzelm@11838:       | exists_paired_all (Abs (_, _, t)) = exists_paired_all t
wenzelm@11838:       | exists_paired_all _ = false;
wenzelm@51717:     val ss =
wenzelm@51717:       simpset_of
wenzelm@51717:        (put_simpset HOL_basic_ss @{context}
wenzelm@51717:         addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}]
wenzelm@51717:         addsimprocs [@{simproc unit_eq}]);
wenzelm@11838:   in
wenzelm@51717:     fun split_all_tac ctxt = SUBGOAL (fn (t, i) =>
wenzelm@51717:       if exists_paired_all t then safe_full_simp_tac (put_simpset ss ctxt) i else no_tac);
wenzelm@51717: 
wenzelm@51717:     fun unsafe_split_all_tac ctxt = SUBGOAL (fn (t, i) =>
wenzelm@51717:       if exists_paired_all t then full_simp_tac (put_simpset ss ctxt) i else no_tac);
wenzelm@51717: 
wenzelm@51717:     fun split_all ctxt th =
wenzelm@51717:       if exists_paired_all (Thm.prop_of th)
wenzelm@51717:       then full_simplify (put_simpset ss ctxt) th else th;
wenzelm@11838:   end;
wenzelm@26340: *}
wenzelm@11838: 
wenzelm@51703: setup {* map_theory_claset (fn ctxt => ctxt addSbefore ("split_all_tac", split_all_tac)) *}
wenzelm@11838: 
blanchet@47740: lemma split_paired_All [simp, no_atp]: "(ALL x. P x) = (ALL a b. P (a, b))"
wenzelm@11838:   -- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *}
wenzelm@11838:   by fast
wenzelm@11838: 
blanchet@47740: lemma split_paired_Ex [simp, no_atp]: "(EX x. P x) = (EX a b. P (a, b))"
haftmann@26358:   by fast
haftmann@26358: 
blanchet@47740: lemma split_paired_The [no_atp]: "(THE x. P x) = (THE (a, b). P (a, b))"
wenzelm@11838:   -- {* Can't be added to simpset: loops! *}
haftmann@26358:   by (simp add: split_eta)
wenzelm@11838: 
wenzelm@11838: text {*
wenzelm@11838:   Simplification procedure for @{thm [source] cond_split_eta}.  Using
wenzelm@11838:   @{thm [source] split_eta} as a rewrite rule is not general enough,
wenzelm@11838:   and using @{thm [source] cond_split_eta} directly would render some
wenzelm@11838:   existing proofs very inefficient; similarly for @{text
haftmann@26358:   split_beta}.
haftmann@26358: *}
wenzelm@11838: 
wenzelm@26480: ML {*
wenzelm@11838: local
wenzelm@51717:   val cond_split_eta_ss =
wenzelm@51717:     simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms cond_split_eta});
wenzelm@35364:   fun Pair_pat k 0 (Bound m) = (m = k)
wenzelm@35364:     | Pair_pat k i (Const (@{const_name Pair},  _) $ Bound m $ t) =
wenzelm@35364:         i > 0 andalso m = k + i andalso Pair_pat k (i - 1) t
wenzelm@35364:     | Pair_pat _ _ _ = false;
wenzelm@35364:   fun no_args k i (Abs (_, _, t)) = no_args (k + 1) i t
wenzelm@35364:     | no_args k i (t $ u) = no_args k i t andalso no_args k i u
wenzelm@35364:     | no_args k i (Bound m) = m < k orelse m > k + i
wenzelm@35364:     | no_args _ _ _ = true;
wenzelm@35364:   fun split_pat tp i (Abs  (_, _, t)) = if tp 0 i t then SOME (i, t) else NONE
blanchet@55414:     | split_pat tp i (Const (@{const_name case_prod}, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t
wenzelm@35364:     | split_pat tp i _ = NONE;
wenzelm@51717:   fun metaeq ctxt lhs rhs = mk_meta_eq (Goal.prove ctxt [] []
wenzelm@35364:         (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)))
wenzelm@51717:         (K (simp_tac (put_simpset cond_split_eta_ss ctxt) 1)));
wenzelm@11838: 
wenzelm@35364:   fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k + 1) i t
wenzelm@35364:     | beta_term_pat k i (t $ u) =
wenzelm@35364:         Pair_pat k i (t $ u) orelse (beta_term_pat k i t andalso beta_term_pat k i u)
wenzelm@35364:     | beta_term_pat k i t = no_args k i t;
wenzelm@35364:   fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
wenzelm@35364:     | eta_term_pat _ _ _ = false;
wenzelm@11838:   fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
wenzelm@35364:     | subst arg k i (t $ u) =
wenzelm@35364:         if Pair_pat k i (t $ u) then incr_boundvars k arg
wenzelm@35364:         else (subst arg k i t $ subst arg k i u)
wenzelm@35364:     | subst arg k i t = t;
wenzelm@43595: in
blanchet@55414:   fun beta_proc ctxt (s as Const (@{const_name case_prod}, _) $ Abs (_, _, t) $ arg) =
wenzelm@11838:         (case split_pat beta_term_pat 1 t of
wenzelm@51717:           SOME (i, f) => SOME (metaeq ctxt s (subst arg 0 i f))
skalberg@15531:         | NONE => NONE)
wenzelm@35364:     | beta_proc _ _ = NONE;
blanchet@55414:   fun eta_proc ctxt (s as Const (@{const_name case_prod}, _) $ Abs (_, _, t)) =
wenzelm@11838:         (case split_pat eta_term_pat 1 t of
wenzelm@51717:           SOME (_, ft) => SOME (metaeq ctxt s (let val (f $ arg) = ft in f end))
skalberg@15531:         | NONE => NONE)
wenzelm@35364:     | eta_proc _ _ = NONE;
wenzelm@11838: end;
wenzelm@11838: *}
wenzelm@51717: simproc_setup split_beta ("split f z") = {* fn _ => fn ctxt => fn ct => beta_proc ctxt (term_of ct) *}
wenzelm@51717: simproc_setup split_eta ("split f") = {* fn _ => fn ctxt => fn ct => eta_proc ctxt (term_of ct) *}
wenzelm@11838: 
berghofe@26798: lemma split_beta [mono]: "(%(x, y). P x y) z = P (fst z) (snd z)"
wenzelm@11838:   by (subst surjective_pairing, rule split_conv)
wenzelm@11838: 
hoelzl@50104: lemma split_beta': "(\<lambda>(x,y). f x y) = (\<lambda>x. f (fst x) (snd x))"
hoelzl@50104:   by (auto simp: fun_eq_iff)
hoelzl@50104: 
hoelzl@50104: 
blanchet@35828: lemma split_split [no_atp]: "R(split c p) = (ALL x y. p = (x, y) --> R(c x y))"
wenzelm@11838:   -- {* For use with @{text split} and the Simplifier. *}
paulson@15481:   by (insert surj_pair [of p], clarify, simp)
wenzelm@11838: 
wenzelm@11838: text {*
wenzelm@11838:   @{thm [source] split_split} could be declared as @{text "[split]"}
wenzelm@11838:   done after the Splitter has been speeded up significantly;
wenzelm@11838:   precompute the constants involved and don't do anything unless the
wenzelm@11838:   current goal contains one of those constants.
wenzelm@11838: *}
wenzelm@11838: 
blanchet@35828: lemma split_split_asm [no_atp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))"
paulson@14208: by (subst split_split, simp)
wenzelm@11838: 
wenzelm@11838: text {*
wenzelm@11838:   \medskip @{term split} used as a logical connective or set former.
wenzelm@11838: 
wenzelm@11838:   \medskip These rules are for use with @{text blast}; could instead
huffman@40929:   call @{text simp} using @{thm [source] prod.split} as rewrite. *}
wenzelm@11838: 
wenzelm@11838: lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"
wenzelm@11838:   apply (simp only: split_tupled_all)
wenzelm@11838:   apply (simp (no_asm_simp))
wenzelm@11838:   done
wenzelm@11838: 
wenzelm@11838: lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"
wenzelm@11838:   apply (simp only: split_tupled_all)
wenzelm@11838:   apply (simp (no_asm_simp))
wenzelm@11838:   done
wenzelm@11838: 
wenzelm@11838: lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
haftmann@37591:   by (induct p) auto
wenzelm@11838: 
wenzelm@11838: lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
haftmann@37591:   by (induct p) auto
wenzelm@11838: 
wenzelm@11838: lemma splitE2:
wenzelm@11838:   "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
wenzelm@11838: proof -
wenzelm@11838:   assume q: "Q (split P z)"
wenzelm@11838:   assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
wenzelm@11838:   show R
wenzelm@11838:     apply (rule r surjective_pairing)+
wenzelm@11838:     apply (rule split_beta [THEN subst], rule q)
wenzelm@11838:     done
wenzelm@11838: qed
wenzelm@11838: 
wenzelm@11838: lemma splitD': "split R (a,b) c ==> R a b c"
wenzelm@11838:   by simp
wenzelm@11838: 
wenzelm@11838: lemma mem_splitI: "z: c a b ==> z: split c (a, b)"
wenzelm@11838:   by simp
wenzelm@11838: 
wenzelm@11838: lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"
paulson@14208: by (simp only: split_tupled_all, simp)
wenzelm@11838: 
wenzelm@18372: lemma mem_splitE:
haftmann@37166:   assumes major: "z \<in> split c p"
haftmann@37166:     and cases: "\<And>x y. p = (x, y) \<Longrightarrow> z \<in> c x y \<Longrightarrow> Q"
wenzelm@18372:   shows Q
blanchet@55414:   by (rule major [unfolded case_prod_unfold] cases surjective_pairing)+
wenzelm@11838: 
wenzelm@11838: declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
wenzelm@11838: declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]
wenzelm@11838: 
wenzelm@26340: ML {*
wenzelm@11838: local (* filtering with exists_p_split is an essential optimization *)
blanchet@55414:   fun exists_p_split (Const (@{const_name case_prod},_) $ _ $ (Const (@{const_name Pair},_)$_$_)) = true
wenzelm@11838:     | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
wenzelm@11838:     | exists_p_split (Abs (_, _, t)) = exists_p_split t
wenzelm@11838:     | exists_p_split _ = false;
wenzelm@11838: in
wenzelm@51717: fun split_conv_tac ctxt = SUBGOAL (fn (t, i) =>
wenzelm@51717:   if exists_p_split t
wenzelm@51717:   then safe_full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps @{thms split_conv}) i
wenzelm@51717:   else no_tac);
wenzelm@11838: end;
wenzelm@26340: *}
wenzelm@26340: 
wenzelm@11838: (* This prevents applications of splitE for already splitted arguments leading
wenzelm@11838:    to quite time-consuming computations (in particular for nested tuples) *)
wenzelm@51703: setup {* map_theory_claset (fn ctxt => ctxt addSbefore ("split_conv_tac", split_conv_tac)) *}
wenzelm@11838: 
blanchet@54147: lemma split_eta_SetCompr [simp, no_atp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
wenzelm@18372:   by (rule ext) fast
wenzelm@11838: 
blanchet@54147: lemma split_eta_SetCompr2 [simp, no_atp]: "(%u. EX x y. u = (x, y) & P x y) = split P"
wenzelm@18372:   by (rule ext) fast
wenzelm@11838: 
wenzelm@11838: lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"
wenzelm@11838:   -- {* Allows simplifications of nested splits in case of independent predicates. *}
wenzelm@18372:   by (rule ext) blast
wenzelm@11838: 
nipkow@14337: (* Do NOT make this a simp rule as it
nipkow@14337:    a) only helps in special situations
nipkow@14337:    b) can lead to nontermination in the presence of split_def
nipkow@14337: *)
nipkow@14337: lemma split_comp_eq: 
paulson@20415:   fixes f :: "'a => 'b => 'c" and g :: "'d => 'a"
paulson@20415:   shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"
wenzelm@18372:   by (rule ext) auto
oheimb@14101: 
haftmann@26358: lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
haftmann@26358:   apply (rule_tac x = "(a, b)" in image_eqI)
haftmann@26358:    apply auto
haftmann@26358:   done
haftmann@26358: 
wenzelm@11838: lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
wenzelm@11838:   by blast
wenzelm@11838: 
wenzelm@11838: (*
wenzelm@11838: the following  would be slightly more general,
wenzelm@11838: but cannot be used as rewrite rule:
wenzelm@11838: ### Cannot add premise as rewrite rule because it contains (type) unknowns:
wenzelm@11838: ### ?y = .x
wenzelm@11838: Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
paulson@14208: by (rtac some_equality 1)
paulson@14208: by ( Simp_tac 1)
paulson@14208: by (split_all_tac 1)
paulson@14208: by (Asm_full_simp_tac 1)
wenzelm@11838: qed "The_split_eq";
wenzelm@11838: *)
wenzelm@11838: 
wenzelm@11838: text {*
wenzelm@11838:   Setup of internal @{text split_rule}.
wenzelm@11838: *}
wenzelm@11838: 
blanchet@55414: lemmas case_prodI = prod.cases [THEN iffD2]
haftmann@24699: 
blanchet@55414: lemma case_prodI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> case_prod c p"
haftmann@37678:   by (fact splitI2)
haftmann@24699: 
blanchet@55414: lemma case_prodI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> case_prod c p x"
haftmann@37678:   by (fact splitI2')
haftmann@24699: 
blanchet@55414: lemma case_prodE: "case_prod c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
haftmann@37678:   by (fact splitE)
haftmann@24699: 
blanchet@55414: lemma case_prodE': "case_prod c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
haftmann@37678:   by (fact splitE')
haftmann@24699: 
blanchet@55414: declare case_prodI [intro!]
haftmann@24699: 
blanchet@55414: lemma case_prod_beta:
blanchet@55414:   "case_prod f p = f (fst p) (snd p)"
haftmann@37591:   by (fact split_beta)
bulwahn@26143: 
blanchet@55417: lemma prod_cases3 [cases type]:
haftmann@24699:   obtains (fields) a b c where "y = (a, b, c)"
haftmann@24699:   by (cases y, case_tac b) blast
haftmann@24699: 
haftmann@24699: lemma prod_induct3 [case_names fields, induct type]:
haftmann@24699:     "(!!a b c. P (a, b, c)) ==> P x"
haftmann@24699:   by (cases x) blast
haftmann@24699: 
blanchet@55417: lemma prod_cases4 [cases type]:
haftmann@24699:   obtains (fields) a b c d where "y = (a, b, c, d)"
haftmann@24699:   by (cases y, case_tac c) blast
haftmann@24699: 
haftmann@24699: lemma prod_induct4 [case_names fields, induct type]:
haftmann@24699:     "(!!a b c d. P (a, b, c, d)) ==> P x"
haftmann@24699:   by (cases x) blast
haftmann@24699: 
blanchet@55417: lemma prod_cases5 [cases type]:
haftmann@24699:   obtains (fields) a b c d e where "y = (a, b, c, d, e)"
haftmann@24699:   by (cases y, case_tac d) blast
haftmann@24699: 
haftmann@24699: lemma prod_induct5 [case_names fields, induct type]:
haftmann@24699:     "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
haftmann@24699:   by (cases x) blast
haftmann@24699: 
blanchet@55417: lemma prod_cases6 [cases type]:
haftmann@24699:   obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
haftmann@24699:   by (cases y, case_tac e) blast
haftmann@24699: 
haftmann@24699: lemma prod_induct6 [case_names fields, induct type]:
haftmann@24699:     "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
haftmann@24699:   by (cases x) blast
haftmann@24699: 
blanchet@55417: lemma prod_cases7 [cases type]:
haftmann@24699:   obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
haftmann@24699:   by (cases y, case_tac f) blast
haftmann@24699: 
haftmann@24699: lemma prod_induct7 [case_names fields, induct type]:
haftmann@24699:     "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
haftmann@24699:   by (cases x) blast
haftmann@24699: 
haftmann@37166: lemma split_def:
haftmann@37166:   "split = (\<lambda>c p. c (fst p) (snd p))"
blanchet@55414:   by (fact case_prod_unfold)
haftmann@37166: 
haftmann@37166: definition internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where
haftmann@37166:   "internal_split == split"
haftmann@37166: 
haftmann@37166: lemma internal_split_conv: "internal_split c (a, b) = c a b"
haftmann@37166:   by (simp only: internal_split_def split_conv)
haftmann@37166: 
wenzelm@48891: ML_file "Tools/split_rule.ML"
haftmann@37166: setup Split_Rule.setup
haftmann@37166: 
haftmann@37166: hide_const internal_split
haftmann@37166: 
haftmann@24699: 
haftmann@26358: subsubsection {* Derived operations *}
haftmann@26358: 
haftmann@37387: definition curry    :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c" where
haftmann@37387:   "curry = (\<lambda>c x y. c (x, y))"
haftmann@37166: 
haftmann@37166: lemma curry_conv [simp, code]: "curry f a b = f (a, b)"
haftmann@37166:   by (simp add: curry_def)
haftmann@37166: 
haftmann@37166: lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b"
haftmann@37166:   by (simp add: curry_def)
haftmann@37166: 
haftmann@37166: lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)"
haftmann@37166:   by (simp add: curry_def)
haftmann@37166: 
haftmann@37166: lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q"
haftmann@37166:   by (simp add: curry_def)
haftmann@37166: 
haftmann@37166: lemma curry_split [simp]: "curry (split f) = f"
haftmann@37166:   by (simp add: curry_def split_def)
haftmann@37166: 
haftmann@37166: lemma split_curry [simp]: "split (curry f) = f"
haftmann@37166:   by (simp add: curry_def split_def)
haftmann@37166: 
Andreas@54630: lemma curry_K: "curry (\<lambda>x. c) = (\<lambda>x y. c)"
Andreas@54630: by(simp add: fun_eq_iff)
Andreas@54630: 
haftmann@26358: text {*
haftmann@26358:   The composition-uncurry combinator.
haftmann@26358: *}
haftmann@26358: 
haftmann@37751: notation fcomp (infixl "\<circ>>" 60)
haftmann@26358: 
haftmann@37751: definition scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "\<circ>\<rightarrow>" 60) where
blanchet@55414:   "f \<circ>\<rightarrow> g = (\<lambda>x. case_prod g (f x))"
haftmann@26358: 
haftmann@37678: lemma scomp_unfold: "scomp = (\<lambda>f g x. g (fst (f x)) (snd (f x)))"
blanchet@55414:   by (simp add: fun_eq_iff scomp_def case_prod_unfold)
haftmann@37678: 
blanchet@55414: lemma scomp_apply [simp]: "(f \<circ>\<rightarrow> g) x = case_prod g (f x)"
blanchet@55414:   by (simp add: scomp_unfold case_prod_unfold)
haftmann@26358: 
haftmann@37751: lemma Pair_scomp: "Pair x \<circ>\<rightarrow> f = f x"
huffman@44921:   by (simp add: fun_eq_iff)
haftmann@26358: 
haftmann@37751: lemma scomp_Pair: "x \<circ>\<rightarrow> Pair = x"
huffman@44921:   by (simp add: fun_eq_iff)
haftmann@26358: 
haftmann@37751: lemma scomp_scomp: "(f \<circ>\<rightarrow> g) \<circ>\<rightarrow> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>\<rightarrow> h)"
nipkow@39302:   by (simp add: fun_eq_iff scomp_unfold)
haftmann@26358: 
haftmann@37751: lemma scomp_fcomp: "(f \<circ>\<rightarrow> g) \<circ>> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>> h)"
nipkow@39302:   by (simp add: fun_eq_iff scomp_unfold fcomp_def)
haftmann@26358: 
haftmann@37751: lemma fcomp_scomp: "(f \<circ>> g) \<circ>\<rightarrow> h = f \<circ>> (g \<circ>\<rightarrow> h)"
huffman@44921:   by (simp add: fun_eq_iff scomp_unfold)
haftmann@26358: 
haftmann@52435: code_printing
haftmann@52435:   constant scomp \<rightharpoonup> (Eval) infixl 3 "#->"
haftmann@31202: 
haftmann@37751: no_notation fcomp (infixl "\<circ>>" 60)
haftmann@37751: no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@26358: 
haftmann@26358: text {*
haftmann@40607:   @{term map_pair} --- action of the product functor upon
krauss@36664:   functions.
haftmann@26358: *}
haftmann@21195: 
haftmann@40607: definition map_pair :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where
haftmann@40607:   "map_pair f g = (\<lambda>(x, y). (f x, g y))"
haftmann@26358: 
haftmann@40607: lemma map_pair_simp [simp, code]:
haftmann@40607:   "map_pair f g (a, b) = (f a, g b)"
haftmann@40607:   by (simp add: map_pair_def)
haftmann@26358: 
haftmann@41505: enriched_type map_pair: map_pair
huffman@44921:   by (auto simp add: split_paired_all)
nipkow@37278: 
haftmann@40607: lemma fst_map_pair [simp]:
haftmann@40607:   "fst (map_pair f g x) = f (fst x)"
haftmann@40607:   by (cases x) simp_all
nipkow@37278: 
haftmann@40607: lemma snd_prod_fun [simp]:
haftmann@40607:   "snd (map_pair f g x) = g (snd x)"
haftmann@40607:   by (cases x) simp_all
nipkow@37278: 
haftmann@40607: lemma fst_comp_map_pair [simp]:
haftmann@40607:   "fst \<circ> map_pair f g = f \<circ> fst"
haftmann@40607:   by (rule ext) simp_all
nipkow@37278: 
haftmann@40607: lemma snd_comp_map_pair [simp]:
haftmann@40607:   "snd \<circ> map_pair f g = g \<circ> snd"
haftmann@40607:   by (rule ext) simp_all
haftmann@26358: 
haftmann@40607: lemma map_pair_compose:
haftmann@40607:   "map_pair (f1 o f2) (g1 o g2) = (map_pair f1 g1 o map_pair f2 g2)"
haftmann@40607:   by (rule ext) (simp add: map_pair.compositionality comp_def)
haftmann@26358: 
haftmann@40607: lemma map_pair_ident [simp]:
haftmann@40607:   "map_pair (%x. x) (%y. y) = (%z. z)"
haftmann@40607:   by (rule ext) (simp add: map_pair.identity)
haftmann@40607: 
haftmann@40607: lemma map_pair_imageI [intro]:
haftmann@40607:   "(a, b) \<in> R \<Longrightarrow> (f a, g b) \<in> map_pair f g ` R"
haftmann@40607:   by (rule image_eqI) simp_all
haftmann@21195: 
haftmann@26358: lemma prod_fun_imageE [elim!]:
haftmann@40607:   assumes major: "c \<in> map_pair f g ` R"
haftmann@40607:     and cases: "\<And>x y. c = (f x, g y) \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> P"
haftmann@26358:   shows P
haftmann@26358:   apply (rule major [THEN imageE])
haftmann@37166:   apply (case_tac x)
haftmann@26358:   apply (rule cases)
haftmann@40607:   apply simp_all
haftmann@26358:   done
haftmann@26358: 
haftmann@37166: definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" where
haftmann@40607:   "apfst f = map_pair f id"
haftmann@26358: 
haftmann@37166: definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" where
haftmann@40607:   "apsnd f = map_pair id f"
haftmann@26358: 
haftmann@26358: lemma apfst_conv [simp, code]:
haftmann@26358:   "apfst f (x, y) = (f x, y)" 
haftmann@26358:   by (simp add: apfst_def)
haftmann@26358: 
hoelzl@33638: lemma apsnd_conv [simp, code]:
haftmann@26358:   "apsnd f (x, y) = (x, f y)" 
haftmann@26358:   by (simp add: apsnd_def)
haftmann@21195: 
haftmann@33594: lemma fst_apfst [simp]:
haftmann@33594:   "fst (apfst f x) = f (fst x)"
haftmann@33594:   by (cases x) simp
haftmann@33594: 
haftmann@51173: lemma fst_comp_apfst [simp]:
haftmann@51173:   "fst \<circ> apfst f = f \<circ> fst"
haftmann@51173:   by (simp add: fun_eq_iff)
haftmann@51173: 
haftmann@33594: lemma fst_apsnd [simp]:
haftmann@33594:   "fst (apsnd f x) = fst x"
haftmann@33594:   by (cases x) simp
haftmann@33594: 
haftmann@51173: lemma fst_comp_apsnd [simp]:
haftmann@51173:   "fst \<circ> apsnd f = fst"
haftmann@51173:   by (simp add: fun_eq_iff)
haftmann@51173: 
haftmann@33594: lemma snd_apfst [simp]:
haftmann@33594:   "snd (apfst f x) = snd x"
haftmann@33594:   by (cases x) simp
haftmann@33594: 
haftmann@51173: lemma snd_comp_apfst [simp]:
haftmann@51173:   "snd \<circ> apfst f = snd"
haftmann@51173:   by (simp add: fun_eq_iff)
haftmann@51173: 
haftmann@33594: lemma snd_apsnd [simp]:
haftmann@33594:   "snd (apsnd f x) = f (snd x)"
haftmann@33594:   by (cases x) simp
haftmann@33594: 
haftmann@51173: lemma snd_comp_apsnd [simp]:
haftmann@51173:   "snd \<circ> apsnd f = f \<circ> snd"
haftmann@51173:   by (simp add: fun_eq_iff)
haftmann@51173: 
haftmann@33594: lemma apfst_compose:
haftmann@33594:   "apfst f (apfst g x) = apfst (f \<circ> g) x"
haftmann@33594:   by (cases x) simp
haftmann@33594: 
haftmann@33594: lemma apsnd_compose:
haftmann@33594:   "apsnd f (apsnd g x) = apsnd (f \<circ> g) x"
haftmann@33594:   by (cases x) simp
haftmann@33594: 
haftmann@33594: lemma apfst_apsnd [simp]:
haftmann@33594:   "apfst f (apsnd g x) = (f (fst x), g (snd x))"
haftmann@33594:   by (cases x) simp
haftmann@33594: 
haftmann@33594: lemma apsnd_apfst [simp]:
haftmann@33594:   "apsnd f (apfst g x) = (g (fst x), f (snd x))"
haftmann@33594:   by (cases x) simp
haftmann@33594: 
haftmann@33594: lemma apfst_id [simp] :
haftmann@33594:   "apfst id = id"
nipkow@39302:   by (simp add: fun_eq_iff)
haftmann@33594: 
haftmann@33594: lemma apsnd_id [simp] :
haftmann@33594:   "apsnd id = id"
nipkow@39302:   by (simp add: fun_eq_iff)
haftmann@33594: 
haftmann@33594: lemma apfst_eq_conv [simp]:
haftmann@33594:   "apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)"
haftmann@33594:   by (cases x) simp
haftmann@33594: 
haftmann@33594: lemma apsnd_eq_conv [simp]:
haftmann@33594:   "apsnd f x = apsnd g x \<longleftrightarrow> f (snd x) = g (snd x)"
haftmann@33594:   by (cases x) simp
haftmann@33594: 
hoelzl@33638: lemma apsnd_apfst_commute:
hoelzl@33638:   "apsnd f (apfst g p) = apfst g (apsnd f p)"
hoelzl@33638:   by simp
haftmann@21195: 
haftmann@26358: text {*
haftmann@26358:   Disjoint union of a family of sets -- Sigma.
haftmann@26358: *}
haftmann@26358: 
haftmann@45986: definition Sigma :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<times> 'b) set" where
haftmann@26358:   Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}"
haftmann@26358: 
haftmann@26358: abbreviation
haftmann@45986:   Times :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set"
haftmann@26358:     (infixr "<*>" 80) where
haftmann@26358:   "A <*> B == Sigma A (%_. B)"
haftmann@26358: 
haftmann@26358: notation (xsymbols)
haftmann@26358:   Times  (infixr "\<times>" 80)
berghofe@15394: 
haftmann@26358: notation (HTML output)
haftmann@26358:   Times  (infixr "\<times>" 80)
haftmann@26358: 
nipkow@45662: hide_const (open) Times
nipkow@45662: 
haftmann@26358: syntax
wenzelm@35115:   "_Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set"  ("(3SIGMA _:_./ _)" [0, 0, 10] 10)
haftmann@26358: translations
wenzelm@35115:   "SIGMA x:A. B" == "CONST Sigma A (%x. B)"
haftmann@26358: 
haftmann@26358: lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
haftmann@26358:   by (unfold Sigma_def) blast
haftmann@26358: 
haftmann@26358: lemma SigmaE [elim!]:
haftmann@26358:     "[| c: Sigma A B;
haftmann@26358:         !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
haftmann@26358:      |] ==> P"
haftmann@26358:   -- {* The general elimination rule. *}
haftmann@26358:   by (unfold Sigma_def) blast
haftmann@20588: 
haftmann@26358: text {*
haftmann@26358:   Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
haftmann@26358:   eigenvariables.
haftmann@26358: *}
haftmann@26358: 
haftmann@26358: lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma SigmaE2:
haftmann@26358:     "[| (a, b) : Sigma A B;
haftmann@26358:         [| a:A;  b:B(a) |] ==> P
haftmann@26358:      |] ==> P"
haftmann@26358:   by blast
haftmann@20588: 
haftmann@26358: lemma Sigma_cong:
haftmann@26358:      "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>
haftmann@26358:       \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
haftmann@26358:   by auto
haftmann@26358: 
haftmann@26358: lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma Sigma_empty2 [simp]: "A <*> {} = {}"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
haftmann@26358:   by auto
haftmann@21908: 
haftmann@26358: lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
haftmann@26358:   by auto
haftmann@26358: 
haftmann@26358: lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
haftmann@26358:   by auto
haftmann@26358: 
haftmann@26358: lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
haftmann@26358:   by (blast elim: equalityE)
haftmann@20588: 
haftmann@26358: lemma SetCompr_Sigma_eq:
haftmann@26358:     "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma UN_Times_distrib:
haftmann@26358:   "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"
haftmann@26358:   -- {* Suggested by Pierre Chartier *}
haftmann@26358:   by blast
haftmann@26358: 
blanchet@47740: lemma split_paired_Ball_Sigma [simp, no_atp]:
haftmann@26358:     "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"
haftmann@26358:   by blast
haftmann@26358: 
blanchet@47740: lemma split_paired_Bex_Sigma [simp, no_atp]:
haftmann@26358:     "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"
haftmann@26358:   by blast
haftmann@21908: 
haftmann@26358: lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"
haftmann@26358:   by blast
haftmann@21908: 
haftmann@26358: lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: text {*
haftmann@26358:   Non-dependent versions are needed to avoid the need for higher-order
haftmann@26358:   matching, especially when the rules are re-oriented.
haftmann@26358: *}
haftmann@21908: 
haftmann@26358: lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"
nipkow@28719: by blast
haftmann@26358: 
haftmann@26358: lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"
nipkow@28719: by blast
haftmann@26358: 
haftmann@26358: lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
nipkow@28719: by blast
haftmann@26358: 
hoelzl@36622: lemma Times_empty[simp]: "A \<times> B = {} \<longleftrightarrow> A = {} \<or> B = {}"
hoelzl@36622:   by auto
hoelzl@36622: 
hoelzl@50104: lemma times_eq_iff: "A \<times> B = C \<times> D \<longleftrightarrow> A = C \<and> B = D \<or> ((A = {} \<or> B = {}) \<and> (C = {} \<or> D = {}))"
hoelzl@50104:   by auto
hoelzl@50104: 
hoelzl@36622: lemma fst_image_times[simp]: "fst ` (A \<times> B) = (if B = {} then {} else A)"
huffman@44921:   by force
hoelzl@36622: 
hoelzl@36622: lemma snd_image_times[simp]: "snd ` (A \<times> B) = (if A = {} then {} else B)"
huffman@44921:   by force
hoelzl@36622: 
nipkow@28719: lemma insert_times_insert[simp]:
nipkow@28719:   "insert a A \<times> insert b B =
nipkow@28719:    insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)"
nipkow@28719: by blast
haftmann@26358: 
paulson@33271: lemma vimage_Times: "f -` (A \<times> B) = ((fst \<circ> f) -` A) \<inter> ((snd \<circ> f) -` B)"
wenzelm@47988:   apply auto
wenzelm@47988:   apply (case_tac "f x")
wenzelm@47988:   apply auto
wenzelm@47988:   done
paulson@33271: 
hoelzl@50104: lemma times_Int_times: "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"
hoelzl@50104:   by auto
hoelzl@50104: 
haftmann@35822: lemma swap_inj_on:
hoelzl@36622:   "inj_on (\<lambda>(i, j). (j, i)) A"
hoelzl@36622:   by (auto intro!: inj_onI)
haftmann@35822: 
haftmann@35822: lemma swap_product:
haftmann@35822:   "(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
haftmann@35822:   by (simp add: split_def image_def) blast
haftmann@35822: 
hoelzl@36622: lemma image_split_eq_Sigma:
hoelzl@36622:   "(\<lambda>x. (f x, g x)) ` A = Sigma (f ` A) (\<lambda>x. g ` (f -` {x} \<inter> A))"
haftmann@46128: proof (safe intro!: imageI)
hoelzl@36622:   fix a b assume *: "a \<in> A" "b \<in> A" and eq: "f a = f b"
hoelzl@36622:   show "(f b, g a) \<in> (\<lambda>x. (f x, g x)) ` A"
hoelzl@36622:     using * eq[symmetric] by auto
hoelzl@36622: qed simp_all
haftmann@35822: 
haftmann@46128: definition product :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" where
haftmann@46128:   [code_abbrev]: "product A B = A \<times> B"
haftmann@46128: 
haftmann@46128: hide_const (open) product
haftmann@46128: 
haftmann@46128: lemma member_product:
haftmann@46128:   "x \<in> Product_Type.product A B \<longleftrightarrow> x \<in> A \<times> B"
haftmann@46128:   by (simp add: product_def)
haftmann@46128: 
haftmann@40607: text {* The following @{const map_pair} lemmas are due to Joachim Breitner: *}
haftmann@40607: 
haftmann@40607: lemma map_pair_inj_on:
haftmann@40607:   assumes "inj_on f A" and "inj_on g B"
haftmann@40607:   shows "inj_on (map_pair f g) (A \<times> B)"
haftmann@40607: proof (rule inj_onI)
haftmann@40607:   fix x :: "'a \<times> 'c" and y :: "'a \<times> 'c"
haftmann@40607:   assume "x \<in> A \<times> B" hence "fst x \<in> A" and "snd x \<in> B" by auto
haftmann@40607:   assume "y \<in> A \<times> B" hence "fst y \<in> A" and "snd y \<in> B" by auto
haftmann@40607:   assume "map_pair f g x = map_pair f g y"
haftmann@40607:   hence "fst (map_pair f g x) = fst (map_pair f g y)" by (auto)
haftmann@40607:   hence "f (fst x) = f (fst y)" by (cases x,cases y,auto)
haftmann@40607:   with `inj_on f A` and `fst x \<in> A` and `fst y \<in> A`
haftmann@40607:   have "fst x = fst y" by (auto dest:dest:inj_onD)
haftmann@40607:   moreover from `map_pair f g x = map_pair f g y`
haftmann@40607:   have "snd (map_pair f g x) = snd (map_pair f g y)" by (auto)
haftmann@40607:   hence "g (snd x) = g (snd y)" by (cases x,cases y,auto)
haftmann@40607:   with `inj_on g B` and `snd x \<in> B` and `snd y \<in> B`
haftmann@40607:   have "snd x = snd y" by (auto dest:dest:inj_onD)
haftmann@40607:   ultimately show "x = y" by(rule prod_eqI)
haftmann@40607: qed
haftmann@40607: 
haftmann@40607: lemma map_pair_surj:
hoelzl@40702:   fixes f :: "'a \<Rightarrow> 'b" and g :: "'c \<Rightarrow> 'd"
haftmann@40607:   assumes "surj f" and "surj g"
haftmann@40607:   shows "surj (map_pair f g)"
haftmann@40607: unfolding surj_def
haftmann@40607: proof
haftmann@40607:   fix y :: "'b \<times> 'd"
haftmann@40607:   from `surj f` obtain a where "fst y = f a" by (auto elim:surjE)
haftmann@40607:   moreover
haftmann@40607:   from `surj g` obtain b where "snd y = g b" by (auto elim:surjE)
haftmann@40607:   ultimately have "(fst y, snd y) = map_pair f g (a,b)" by auto
haftmann@40607:   thus "\<exists>x. y = map_pair f g x" by auto
haftmann@40607: qed
haftmann@40607: 
haftmann@40607: lemma map_pair_surj_on:
haftmann@40607:   assumes "f ` A = A'" and "g ` B = B'"
haftmann@40607:   shows "map_pair f g ` (A \<times> B) = A' \<times> B'"
haftmann@40607: unfolding image_def
haftmann@40607: proof(rule set_eqI,rule iffI)
haftmann@40607:   fix x :: "'a \<times> 'c"
haftmann@40607:   assume "x \<in> {y\<Colon>'a \<times> 'c. \<exists>x\<Colon>'b \<times> 'd\<in>A \<times> B. y = map_pair f g x}"
haftmann@40607:   then obtain y where "y \<in> A \<times> B" and "x = map_pair f g y" by blast
haftmann@40607:   from `image f A = A'` and `y \<in> A \<times> B` have "f (fst y) \<in> A'" by auto
haftmann@40607:   moreover from `image g B = B'` and `y \<in> A \<times> B` have "g (snd y) \<in> B'" by auto
haftmann@40607:   ultimately have "(f (fst y), g (snd y)) \<in> (A' \<times> B')" by auto
haftmann@40607:   with `x = map_pair f g y` show "x \<in> A' \<times> B'" by (cases y, auto)
haftmann@40607: next
haftmann@40607:   fix x :: "'a \<times> 'c"
haftmann@40607:   assume "x \<in> A' \<times> B'" hence "fst x \<in> A'" and "snd x \<in> B'" by auto
haftmann@40607:   from `image f A = A'` and `fst x \<in> A'` have "fst x \<in> image f A" by auto
haftmann@40607:   then obtain a where "a \<in> A" and "fst x = f a" by (rule imageE)
haftmann@40607:   moreover from `image g B = B'` and `snd x \<in> B'`
haftmann@40607:   obtain b where "b \<in> B" and "snd x = g b" by auto
haftmann@40607:   ultimately have "(fst x, snd x) = map_pair f g (a,b)" by auto
haftmann@40607:   moreover from `a \<in> A` and  `b \<in> B` have "(a , b) \<in> A \<times> B" by auto
haftmann@40607:   ultimately have "\<exists>y \<in> A \<times> B. x = map_pair f g y" by auto
haftmann@40607:   thus "x \<in> {x. \<exists>y \<in> A \<times> B. x = map_pair f g y}" by auto
haftmann@40607: qed
haftmann@40607: 
haftmann@21908: 
bulwahn@49764: subsection {* Simproc for rewriting a set comprehension into a pointfree expression *}
bulwahn@49764: 
bulwahn@49764: ML_file "Tools/set_comprehension_pointfree.ML"
bulwahn@49764: 
bulwahn@49764: setup {*
wenzelm@51717:   Code_Preproc.map_pre (fn ctxt => ctxt addsimprocs
bulwahn@49764:     [Raw_Simplifier.make_simproc {name = "set comprehension", lhss = [@{cpat "Collect ?P"}],
bulwahn@49764:     proc = K Set_Comprehension_Pointfree.code_simproc, identifier = []}])
bulwahn@49764: *}
bulwahn@49764: 
bulwahn@49764: 
haftmann@37166: subsection {* Inductively defined sets *}
berghofe@15394: 
wenzelm@48891: ML_file "Tools/inductive_set.ML"
haftmann@31723: setup Inductive_Set.setup
haftmann@24699: 
haftmann@37166: 
haftmann@37166: subsection {* Legacy theorem bindings and duplicates *}
haftmann@37166: 
haftmann@37166: lemma PairE:
haftmann@37166:   obtains x y where "p = (x, y)"
haftmann@37166:   by (fact prod.exhaust)
haftmann@37166: 
haftmann@37166: lemmas Pair_eq = prod.inject
blanchet@55393: lemmas fst_conv = prod.sel(1)
blanchet@55393: lemmas snd_conv = prod.sel(2)
blanchet@55393: lemmas pair_collapse = prod.collapse
blanchet@55393: lemmas split = split_conv
huffman@44066: lemmas Pair_fst_snd_eq = prod_eq_iff
huffman@44066: 
huffman@45204: hide_const (open) prod
huffman@45204: 
nipkow@10213: end




