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
haftmann@24699: uses
haftmann@24699:   ("Tools/split_rule.ML")
haftmann@31723:   ("Tools/inductive_set.ML")
nipkow@15131: begin
wenzelm@11838: 
haftmann@24699: subsection {* @{typ bool} is a datatype *}
haftmann@24699: 
haftmann@27104: rep_datatype True False by (auto intro: bool_induct)
haftmann@24699: 
haftmann@24699: declare case_split [cases type: bool]
haftmann@24699:   -- "prefer plain propositional version"
haftmann@24699: 
haftmann@28346: lemma
haftmann@28562:   shows [code]: "eq_class.eq False P \<longleftrightarrow> \<not> P"
haftmann@28562:     and [code]: "eq_class.eq True P \<longleftrightarrow> P" 
haftmann@28562:     and [code]: "eq_class.eq P False \<longleftrightarrow> \<not> P" 
haftmann@28562:     and [code]: "eq_class.eq P True \<longleftrightarrow> P"
haftmann@28346:     and [code nbe]: "eq_class.eq P P \<longleftrightarrow> True"
haftmann@28346:   by (simp_all add: eq)
haftmann@25534: 
haftmann@28346: code_const "eq_class.eq \<Colon> bool \<Rightarrow> bool \<Rightarrow> bool"
haftmann@25534:   (Haskell infixl 4 "==")
haftmann@25534: 
haftmann@25534: code_instance bool :: eq
haftmann@25534:   (Haskell -)
haftmann@24699: 
haftmann@26358: 
haftmann@37166: subsection {* The @{text unit} type *}
wenzelm@11838: 
wenzelm@11838: typedef unit = "{True}"
wenzelm@11838: proof
haftmann@20588:   show "True : ?unit" ..
wenzelm@11838: qed
wenzelm@11838: 
haftmann@24699: definition
wenzelm@11838:   Unity :: unit    ("'(')")
haftmann@24699: where
haftmann@24699:   "() = Abs_unit True"
wenzelm@11838: 
blanchet@35828: lemma unit_eq [no_atp]: "u = ()"
wenzelm@11838:   by (induct u) (simp add: unit_def 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@26480: ML {*
wenzelm@13462:   val unit_eq_proc =
haftmann@24699:     let val unit_meta_eq = mk_meta_eq @{thm unit_eq} in
haftmann@24699:       Simplifier.simproc @{theory} "unit_eq" ["x::unit"]
skalberg@15531:       (fn _ => fn _ => fn t => if HOLogic.is_unit t then NONE else SOME unit_meta_eq)
wenzelm@13462:     end;
wenzelm@11838: 
wenzelm@11838:   Addsimprocs [unit_eq_proc];
wenzelm@11838: *}
wenzelm@11838: 
haftmann@27104: rep_datatype "()" by simp
haftmann@24699: 
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@11838:   This rewrite counters the effect of @{text unit_eq_proc} 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@35828: lemma unit_abs_eta_conv [simp,no_atp]: "(%u::unit. f ()) = f"
wenzelm@11838:   by (rule ext) simp
nipkow@10213: 
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@28346:   "eq_class.eq (u\<Colon>unit) v \<longleftrightarrow> True" unfolding eq unit_eq [of u] unit_eq [of v] by rule+
haftmann@26358: 
haftmann@26358: code_type unit
haftmann@26358:   (SML "unit")
haftmann@26358:   (OCaml "unit")
haftmann@26358:   (Haskell "()")
haftmann@34886:   (Scala "Unit")
haftmann@26358: 
haftmann@37166: code_const Unity
haftmann@37166:   (SML "()")
haftmann@37166:   (OCaml "()")
haftmann@37166:   (Haskell "()")
haftmann@37166:   (Scala "()")
haftmann@37166: 
haftmann@26358: code_instance unit :: eq
haftmann@26358:   (Haskell -)
haftmann@26358: 
haftmann@28346: code_const "eq_class.eq \<Colon> unit \<Rightarrow> unit \<Rightarrow> bool"
haftmann@26358:   (Haskell infixl 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: 
nipkow@10213: global
nipkow@10213: 
nipkow@10213: typedef (Prod)
haftmann@22838:   ('a, 'b) "*"    (infixr "*" 20)
haftmann@26358:     = "{f. \<exists>a b. f = Pair_Rep (a\<Colon>'a) (b\<Colon>'b)}"
oheimb@11025: proof
haftmann@26358:   fix a b show "Pair_Rep a b \<in> ?Prod"
haftmann@26358:     by rule+
oheimb@11025: qed
nipkow@10213: 
wenzelm@35427: type_notation (xsymbols)
wenzelm@35427:   "*"  ("(_ \<times>/ _)" [21, 20] 20)
wenzelm@35427: type_notation (HTML output)
wenzelm@35427:   "*"  ("(_ \<times>/ _)" [21, 20] 20)
nipkow@10213: 
nipkow@10213: consts
haftmann@26358:   Pair     :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b"
nipkow@10213: 
wenzelm@11777: local
nipkow@10213: 
wenzelm@19535: defs
wenzelm@19535:   Pair_def:     "Pair a b == Abs_Prod (Pair_Rep a b)"
haftmann@37166: 
haftmann@37166: rep_datatype (prod) Pair proof -
haftmann@37166:   fix P :: "'a \<times> 'b \<Rightarrow> bool" and p
haftmann@37166:   assume "\<And>a b. P (Pair a b)"
haftmann@37166:   then show "P p" 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"
haftmann@37166:     by (auto simp add: Pair_Rep_def expand_fun_eq)
haftmann@37166:   moreover have "Pair_Rep a b \<in> Prod" and "Pair_Rep c d \<in> Prod"
haftmann@37166:     by (auto simp add: Prod_def)
haftmann@37166:   ultimately show "Pair a b = Pair c d \<longleftrightarrow> a = c \<and> b = d"
haftmann@37166:     by (simp add: Pair_def Abs_Prod_inject)
haftmann@37166: qed
haftmann@37166: 
haftmann@37166: 
haftmann@37166: subsubsection {* Tuple syntax *}
haftmann@37166: 
haftmann@37166: global consts
haftmann@37166:   split    :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c"
haftmann@37166: 
haftmann@37166: local defs
haftmann@37166:   split_prod_case: "split == prod_case"
wenzelm@19535: 
wenzelm@11777: text {*
wenzelm@11777:   Patterns -- extends pre-defined type @{typ pttrn} used in
wenzelm@11777:   abstractions.
wenzelm@11777: *}
nipkow@10213: 
nipkow@10213: nonterminals
nipkow@10213:   tuple_args 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@10213:   "_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))"
wenzelm@35115:   "%(x, y, zs). b" == "CONST split (%x (y, zs). b)"
wenzelm@35115:   "%(x, y). b" == "CONST split (%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@35115: let
wenzelm@35115:   fun split_tr' [Abs (x, T, t as (Abs abs))] =
wenzelm@35115:         (* split (%x y. t) => %(x,y) t *)
wenzelm@35115:         let
wenzelm@35115:           val (y, t') = atomic_abs_tr' abs;
wenzelm@35115:           val (x', t'') = atomic_abs_tr' (x, T, t');
wenzelm@35115:         in
wenzelm@35115:           Syntax.const @{syntax_const "_abs"} $
wenzelm@35115:             (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
wenzelm@35115:         end
wenzelm@35115:     | split_tr' [Abs (x, T, (s as Const (@{const_syntax split}, _) $ t))] =
wenzelm@35115:         (* split (%x. (split (%y z. t))) => %(x,y,z). t *)
wenzelm@35115:         let
wenzelm@35115:           val Const (@{syntax_const "_abs"}, _) $
wenzelm@35115:             (Const (@{syntax_const "_pattern"}, _) $ y $ z) $ t' = split_tr' [t];
wenzelm@35115:           val (x', t'') = atomic_abs_tr' (x, T, t');
wenzelm@35115:         in
wenzelm@35115:           Syntax.const @{syntax_const "_abs"} $
wenzelm@35115:             (Syntax.const @{syntax_const "_pattern"} $ x' $
wenzelm@35115:               (Syntax.const @{syntax_const "_patterns"} $ y $ z)) $ t''
wenzelm@35115:         end
wenzelm@35115:     | split_tr' [Const (@{const_syntax split}, _) $ t] =
wenzelm@35115:         (* split (split (%x y z. t)) => %((x, y), z). t *)
wenzelm@35115:         split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *)
wenzelm@35115:     | split_tr' [Const (@{syntax_const "_abs"}, _) $ x_y $ Abs abs] =
wenzelm@35115:         (* split (%pttrn z. t) => %(pttrn,z). t *)
wenzelm@35115:         let val (z, t) = atomic_abs_tr' abs in
wenzelm@35115:           Syntax.const @{syntax_const "_abs"} $
wenzelm@35115:             (Syntax.const @{syntax_const "_pattern"} $ x_y $ z) $ t
wenzelm@35115:         end
wenzelm@35115:     | split_tr' _ = raise Match;
wenzelm@35115: in [(@{const_syntax split}, 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 {*
schirmer@15422: let
wenzelm@35115:   fun split_guess_names_tr' _ T [Abs (x, _, Abs _)] = raise Match
wenzelm@35115:     | split_guess_names_tr' _ T [Abs (x, xT, t)] =
schirmer@15422:         (case (head_of t) of
wenzelm@35115:           Const (@{const_syntax split}, _) => raise Match
wenzelm@35115:         | _ =>
wenzelm@35115:           let 
wenzelm@35115:             val (_ :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match;
wenzelm@35115:             val (y, t') = atomic_abs_tr' ("y", yT, incr_boundvars 1 t $ Bound 0);
wenzelm@35115:             val (x', t'') = atomic_abs_tr' (x, xT, t');
wenzelm@35115:           in
wenzelm@35115:             Syntax.const @{syntax_const "_abs"} $
wenzelm@35115:               (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
wenzelm@35115:           end)
schirmer@15422:     | split_guess_names_tr' _ T [t] =
wenzelm@35115:         (case head_of t of
wenzelm@35115:           Const (@{const_syntax split}, _) => raise Match
wenzelm@35115:         | _ =>
wenzelm@35115:           let
wenzelm@35115:             val (xT :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match;
wenzelm@35115:             val (y, t') = atomic_abs_tr' ("y", yT, incr_boundvars 2 t $ Bound 1 $ Bound 0);
wenzelm@35115:             val (x', t'') = atomic_abs_tr' ("x", xT, t');
wenzelm@35115:           in
wenzelm@35115:             Syntax.const @{syntax_const "_abs"} $
wenzelm@35115:               (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
wenzelm@35115:           end)
schirmer@15422:     | split_guess_names_tr' _ _ _ = raise Match;
wenzelm@35115: in [(@{const_syntax split}, split_guess_names_tr')] end
schirmer@15422: *}
schirmer@15422: 
nipkow@10213: 
haftmann@37166: subsubsection {* Code generator setup *}
haftmann@37166: 
haftmann@37166: lemma split_case_cert:
haftmann@37166:   assumes "CASE \<equiv> split f"
haftmann@37166:   shows "CASE (a, b) \<equiv> f a b"
haftmann@37166:   using assms by (simp add: split_prod_case)
haftmann@37166: 
haftmann@37166: setup {*
haftmann@37166:   Code.add_case @{thm split_case_cert}
haftmann@37166: *}
haftmann@37166: 
haftmann@37166: code_type *
haftmann@37166:   (SML infix 2 "*")
haftmann@37166:   (OCaml infix 2 "*")
haftmann@37166:   (Haskell "!((_),/ (_))")
haftmann@37166:   (Scala "((_),/ (_))")
haftmann@37166: 
haftmann@37166: code_const Pair
haftmann@37166:   (SML "!((_),/ (_))")
haftmann@37166:   (OCaml "!((_),/ (_))")
haftmann@37166:   (Haskell "!((_),/ (_))")
haftmann@37166:   (Scala "!((_),/ (_))")
haftmann@37166: 
haftmann@37166: code_instance * :: eq
haftmann@37166:   (Haskell -)
haftmann@37166: 
haftmann@37166: code_const "eq_class.eq \<Colon> 'a\<Colon>eq \<times> 'b\<Colon>eq \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"
haftmann@37166:   (Haskell infixl 4 "==")
haftmann@37166: 
haftmann@37166: types_code
haftmann@37166:   "*"     ("(_ */ _)")
haftmann@37166: attach (term_of) {*
haftmann@37166: fun term_of_id_42 aF aT bF bT (x, y) = HOLogic.pair_const aT bT $ aF x $ bF y;
haftmann@37166: *}
haftmann@37166: attach (test) {*
haftmann@37166: fun gen_id_42 aG aT bG bT i =
haftmann@37166:   let
haftmann@37166:     val (x, t) = aG i;
haftmann@37166:     val (y, u) = bG i
haftmann@37166:   in ((x, y), fn () => HOLogic.pair_const aT bT $ t () $ u ()) end;
haftmann@37166: *}
haftmann@37166: 
haftmann@37166: consts_code
haftmann@37166:   "Pair"    ("(_,/ _)")
haftmann@37166: 
haftmann@37166: setup {*
haftmann@37166: let
haftmann@37166: 
haftmann@37166: fun strip_abs_split 0 t = ([], t)
haftmann@37166:   | strip_abs_split i (Abs (s, T, t)) =
haftmann@37166:       let
haftmann@37166:         val s' = Codegen.new_name t s;
haftmann@37166:         val v = Free (s', T)
haftmann@37166:       in apfst (cons v) (strip_abs_split (i-1) (subst_bound (v, t))) end
haftmann@37166:   | strip_abs_split i (u as Const (@{const_name split}, _) $ t) =
haftmann@37166:       (case strip_abs_split (i+1) t of
haftmann@37166:         (v :: v' :: vs, u) => (HOLogic.mk_prod (v, v') :: vs, u)
haftmann@37166:       | _ => ([], u))
haftmann@37166:   | strip_abs_split i t =
haftmann@37166:       strip_abs_split i (Abs ("x", hd (binder_types (fastype_of t)), t $ Bound 0));
haftmann@37166: 
haftmann@37166: fun let_codegen thy defs dep thyname brack t gr =
haftmann@37166:   (case strip_comb t of
haftmann@37166:     (t1 as Const (@{const_name Let}, _), t2 :: t3 :: ts) =>
haftmann@37166:     let
haftmann@37166:       fun dest_let (l as Const (@{const_name Let}, _) $ t $ u) =
haftmann@37166:           (case strip_abs_split 1 u of
haftmann@37166:              ([p], u') => apfst (cons (p, t)) (dest_let u')
haftmann@37166:            | _ => ([], l))
haftmann@37166:         | dest_let t = ([], t);
haftmann@37166:       fun mk_code (l, r) gr =
haftmann@37166:         let
haftmann@37166:           val (pl, gr1) = Codegen.invoke_codegen thy defs dep thyname false l gr;
haftmann@37166:           val (pr, gr2) = Codegen.invoke_codegen thy defs dep thyname false r gr1;
haftmann@37166:         in ((pl, pr), gr2) end
haftmann@37166:     in case dest_let (t1 $ t2 $ t3) of
haftmann@37166:         ([], _) => NONE
haftmann@37166:       | (ps, u) =>
haftmann@37166:           let
haftmann@37166:             val (qs, gr1) = fold_map mk_code ps gr;
haftmann@37166:             val (pu, gr2) = Codegen.invoke_codegen thy defs dep thyname false u gr1;
haftmann@37166:             val (pargs, gr3) = fold_map
haftmann@37166:               (Codegen.invoke_codegen thy defs dep thyname true) ts gr2
haftmann@37166:           in
haftmann@37166:             SOME (Codegen.mk_app brack
haftmann@37166:               (Pretty.blk (0, [Codegen.str "let ", Pretty.blk (0, flat
haftmann@37166:                   (separate [Codegen.str ";", Pretty.brk 1] (map (fn (pl, pr) =>
haftmann@37166:                     [Pretty.block [Codegen.str "val ", pl, Codegen.str " =",
haftmann@37166:                        Pretty.brk 1, pr]]) qs))),
haftmann@37166:                 Pretty.brk 1, Codegen.str "in ", pu,
haftmann@37166:                 Pretty.brk 1, Codegen.str "end"])) pargs, gr3)
haftmann@37166:           end
haftmann@37166:     end
haftmann@37166:   | _ => NONE);
haftmann@37166: 
haftmann@37166: fun split_codegen thy defs dep thyname brack t gr = (case strip_comb t of
haftmann@37166:     (t1 as Const (@{const_name split}, _), t2 :: ts) =>
haftmann@37166:       let
haftmann@37166:         val ([p], u) = strip_abs_split 1 (t1 $ t2);
haftmann@37166:         val (q, gr1) = Codegen.invoke_codegen thy defs dep thyname false p gr;
haftmann@37166:         val (pu, gr2) = Codegen.invoke_codegen thy defs dep thyname false u gr1;
haftmann@37166:         val (pargs, gr3) = fold_map
haftmann@37166:           (Codegen.invoke_codegen thy defs dep thyname true) ts gr2
haftmann@37166:       in
haftmann@37166:         SOME (Codegen.mk_app brack
haftmann@37166:           (Pretty.block [Codegen.str "(fn ", q, Codegen.str " =>",
haftmann@37166:             Pretty.brk 1, pu, Codegen.str ")"]) pargs, gr2)
haftmann@37166:       end
haftmann@37166:   | _ => NONE);
haftmann@37166: 
haftmann@37166: in
haftmann@37166: 
haftmann@37166:   Codegen.add_codegen "let_codegen" let_codegen
haftmann@37166:   #> Codegen.add_codegen "split_codegen" split_codegen
haftmann@37166: 
haftmann@37166: end
haftmann@37166: *}
haftmann@37166: 
haftmann@37166: 
haftmann@37166: subsubsection {* Fundamental operations and properties *}
wenzelm@11838: 
haftmann@26358: lemma surj_pair [simp]: "EX x y. p = (x, y)"
haftmann@37166:   by (cases p) simp
nipkow@10213: 
haftmann@37166: global consts
haftmann@37166:   fst      :: "'a \<times> 'b \<Rightarrow> 'a"
haftmann@37166:   snd      :: "'a \<times> 'b \<Rightarrow> 'b"
wenzelm@11838: 
haftmann@37166: local defs
haftmann@37166:   fst_def:      "fst p == case p of (a, b) \<Rightarrow> a"
haftmann@37166:   snd_def:      "snd p == case p of (a, b) \<Rightarrow> b"
wenzelm@11838: 
haftmann@22886: lemma fst_conv [simp, code]: "fst (a, b) = a"
haftmann@37166:   unfolding fst_def by simp
wenzelm@11838: 
haftmann@22886: lemma snd_conv [simp, code]: "snd (a, b) = b"
haftmann@37166:   unfolding snd_def by simp
oheimb@11025: 
haftmann@37166: code_const fst and snd
haftmann@37166:   (Haskell "fst" and "snd")
haftmann@26358: 
haftmann@37166: lemma prod_case_unfold: "prod_case = (%c p. c (fst p) (snd p))"
haftmann@37166:   by (simp add: expand_fun_eq 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: 
haftmann@26358: lemma pair_collapse [simp]: "(fst p, snd p) = p"
wenzelm@11838:   by (cases p) simp
wenzelm@11838: 
haftmann@26358: lemmas surjective_pairing = pair_collapse [symmetric]
wenzelm@11838: 
haftmann@37166: lemma Pair_fst_snd_eq: "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"
haftmann@37166:   by (simp add: Pair_fst_snd_eq)
haftmann@37166: 
haftmann@37166: lemma split_conv [simp, code]: "split f (a, b) = f a b"
haftmann@37166:   by (simp add: split_prod_case)
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"
haftmann@37166:   by (simp add: split_prod_case expand_fun_eq 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. *}
haftmann@37166:   by (simp add: split_prod_case expand_fun_eq 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))"
haftmann@37166:   by (simp add: split_prod_case prod_case_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 *}
haftmann@37166:   by (erule arg_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: 
wenzelm@11838: lemma split_paired_all: "(!!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: 
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:   val safe_full_simp_tac = generic_simp_tac true (true, false, false);
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@11838:     val ss = HOL_basic_ss
wenzelm@26340:       addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}]
wenzelm@11838:       addsimprocs [unit_eq_proc];
wenzelm@11838:   in
wenzelm@11838:     val split_all_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838:       if exists_paired_all t then safe_full_simp_tac ss i else no_tac);
wenzelm@11838:     val unsafe_split_all_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838:       if exists_paired_all t then full_simp_tac ss i else no_tac);
wenzelm@11838:     fun split_all th =
wenzelm@26340:    if exists_paired_all (Thm.prop_of th) then full_simplify ss th else th;
wenzelm@11838:   end;
wenzelm@26340: *}
wenzelm@11838: 
wenzelm@26340: declaration {* fn _ =>
wenzelm@26340:   Classical.map_cs (fn cs => cs addSbefore ("split_all_tac", split_all_tac))
wenzelm@16121: *}
wenzelm@11838: 
wenzelm@11838: lemma split_paired_All [simp]: "(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: 
haftmann@26358: lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))"
haftmann@26358:   by fast
haftmann@26358: 
wenzelm@11838: lemma split_paired_The: "(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@35364:   val cond_split_eta_ss = HOL_basic_ss 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
wenzelm@35364:     | split_pat tp i (Const (@{const_name split}, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t
wenzelm@35364:     | split_pat tp i _ = NONE;
wenzelm@20044:   fun metaeq ss lhs rhs = mk_meta_eq (Goal.prove (Simplifier.the_context ss) [] []
wenzelm@35364:         (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)))
wenzelm@18328:         (K (simp_tac (Simplifier.inherit_context ss cond_split_eta_ss) 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@35364:   fun beta_proc ss (s as Const (@{const_name split}, _) $ Abs (_, _, t) $ arg) =
wenzelm@11838:         (case split_pat beta_term_pat 1 t of
wenzelm@35364:           SOME (i, f) => SOME (metaeq ss s (subst arg 0 i f))
skalberg@15531:         | NONE => NONE)
wenzelm@35364:     | beta_proc _ _ = NONE;
wenzelm@35364:   fun eta_proc ss (s as Const (@{const_name split}, _) $ Abs (_, _, t)) =
wenzelm@11838:         (case split_pat eta_term_pat 1 t of
wenzelm@35364:           SOME (_, ft) => SOME (metaeq ss s (let val (f $ arg) = ft in f end))
skalberg@15531:         | NONE => NONE)
wenzelm@35364:     | eta_proc _ _ = NONE;
wenzelm@11838: in
wenzelm@32010:   val split_beta_proc = Simplifier.simproc @{theory} "split_beta" ["split f z"] (K beta_proc);
wenzelm@32010:   val split_eta_proc = Simplifier.simproc @{theory} "split_eta" ["split f"] (K eta_proc);
wenzelm@11838: end;
wenzelm@11838: 
wenzelm@11838: Addsimprocs [split_beta_proc, split_eta_proc];
wenzelm@11838: *}
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: 
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
wenzelm@11838:   call @{text simp} using @{thm [source] 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@37166:   by (induct p) (auto simp add: split_prod_case)
wenzelm@11838: 
wenzelm@11838: lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
haftmann@37166:   by (induct p) (auto simp add: split_prod_case)
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
haftmann@37166:   by (rule major [unfolded split_prod_case prod_case_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 *)
wenzelm@35364:   fun exists_p_split (Const (@{const_name split},_) $ _ $ (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@35364:   val ss = HOL_basic_ss addsimps @{thms split_conv};
wenzelm@11838: in
wenzelm@11838: val split_conv_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838:     if exists_p_split t then safe_full_simp_tac ss i 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@26340: declaration {* fn _ =>
wenzelm@26340:   Classical.map_cs (fn cs => cs addSbefore ("split_conv_tac", split_conv_tac))
wenzelm@16121: *}
wenzelm@11838: 
blanchet@35828: 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@35828: 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: 
haftmann@24699: lemmas prod_caseI = prod.cases [THEN iffD2, standard]
haftmann@24699: 
haftmann@24699: lemma prod_caseI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> prod_case c p"
haftmann@24699:   by auto
haftmann@24699: 
haftmann@24699: lemma prod_caseI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> prod_case c p x"
haftmann@24699:   by (auto simp: split_tupled_all)
haftmann@24699: 
haftmann@24699: lemma prod_caseE: "prod_case c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
haftmann@24699:   by (induct p) auto
haftmann@24699: 
haftmann@24699: lemma prod_caseE': "prod_case c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
haftmann@24699:   by (induct p) auto
haftmann@24699: 
haftmann@24699: declare prod_caseI2' [intro!] prod_caseI2 [intro!] prod_caseI [intro!]
haftmann@24699: declare prod_caseE' [elim!] prod_caseE [elim!]
haftmann@24699: 
haftmann@24844: lemma prod_case_split:
haftmann@24699:   "prod_case = split"
haftmann@24699:   by (auto simp add: expand_fun_eq)
haftmann@24699: 
bulwahn@26143: lemma prod_case_beta:
bulwahn@26143:   "prod_case f p = f (fst p) (snd p)"
bulwahn@26143:   unfolding prod_case_split split_beta ..
bulwahn@26143: 
haftmann@24699: 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: 
haftmann@24699: 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: 
haftmann@24699: 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: 
haftmann@24699: 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: 
haftmann@24699: 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))"
haftmann@37166:   unfolding split_prod_case prod_case_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: 
haftmann@37166: use "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@37166: global consts
haftmann@37166:   curry    :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c"
haftmann@37166: 
haftmann@37166: local defs
haftmann@37166:   curry_def:    "curry == (%c x y. c (Pair 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: 
haftmann@26358: text {*
haftmann@26358:   The composition-uncurry combinator.
haftmann@26358: *}
haftmann@26358: 
haftmann@26588: notation fcomp (infixl "o>" 60)
haftmann@26358: 
haftmann@37166: definition scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "o\<rightarrow>" 60) where
haftmann@26588:   "f o\<rightarrow> g = (\<lambda>x. split g (f x))"
haftmann@26358: 
haftmann@26588: lemma scomp_apply:  "(f o\<rightarrow> g) x = split g (f x)"
haftmann@26588:   by (simp add: scomp_def)
haftmann@26358: 
haftmann@26588: lemma Pair_scomp: "Pair x o\<rightarrow> f = f x"
haftmann@26588:   by (simp add: expand_fun_eq scomp_apply)
haftmann@26358: 
haftmann@26588: lemma scomp_Pair: "x o\<rightarrow> Pair = x"
haftmann@26588:   by (simp add: expand_fun_eq scomp_apply)
haftmann@26358: 
haftmann@26588: lemma scomp_scomp: "(f o\<rightarrow> g) o\<rightarrow> h = f o\<rightarrow> (\<lambda>x. g x o\<rightarrow> h)"
haftmann@26588:   by (simp add: expand_fun_eq split_twice scomp_def)
haftmann@26358: 
haftmann@26588: lemma scomp_fcomp: "(f o\<rightarrow> g) o> h = f o\<rightarrow> (\<lambda>x. g x o> h)"
haftmann@26588:   by (simp add: expand_fun_eq scomp_apply fcomp_def split_def)
haftmann@26358: 
haftmann@26588: lemma fcomp_scomp: "(f o> g) o\<rightarrow> h = f o> (g o\<rightarrow> h)"
haftmann@26588:   by (simp add: expand_fun_eq scomp_apply fcomp_apply)
haftmann@26358: 
haftmann@31202: code_const scomp
haftmann@31202:   (Eval infixl 3 "#->")
haftmann@31202: 
haftmann@26588: no_notation fcomp (infixl "o>" 60)
haftmann@26588: no_notation scomp (infixl "o\<rightarrow>" 60)
haftmann@26358: 
haftmann@26358: text {*
haftmann@26358:   @{term prod_fun} --- action of the product functor upon
krauss@36664:   functions.
haftmann@26358: *}
haftmann@21195: 
haftmann@26358: definition prod_fun :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where
haftmann@28562:   [code del]: "prod_fun f g = (\<lambda>(x, y). (f x, g y))"
haftmann@26358: 
haftmann@28562: lemma prod_fun [simp, code]: "prod_fun f g (a, b) = (f a, g b)"
haftmann@26358:   by (simp add: prod_fun_def)
haftmann@26358: 
haftmann@26358: lemma prod_fun_compose: "prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)"
haftmann@26358:   by (rule ext) auto
haftmann@26358: 
haftmann@26358: lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)"
haftmann@26358:   by (rule ext) auto
haftmann@26358: 
haftmann@26358: lemma prod_fun_imageI [intro]: "(a, b) : r ==> (f a, g b) : prod_fun f g ` r"
haftmann@26358:   apply (rule image_eqI)
haftmann@26358:   apply (rule prod_fun [symmetric], assumption)
haftmann@26358:   done
haftmann@21195: 
haftmann@26358: lemma prod_fun_imageE [elim!]:
haftmann@26358:   assumes major: "c: (prod_fun f g)`r"
haftmann@26358:     and cases: "!!x y. [| c=(f(x),g(y));  (x,y):r |] ==> P"
haftmann@26358:   shows P
haftmann@26358:   apply (rule major [THEN imageE])
haftmann@37166:   apply (case_tac x)
haftmann@26358:   apply (rule cases)
haftmann@26358:    apply (blast intro: prod_fun)
haftmann@26358:   apply blast
haftmann@26358:   done
haftmann@26358: 
haftmann@37166: definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" where
haftmann@37166:   "apfst f = prod_fun f id"
haftmann@26358: 
haftmann@37166: definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" where
haftmann@37166:   "apsnd f = prod_fun 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@33594: lemma fst_apsnd [simp]:
haftmann@33594:   "fst (apsnd f x) = fst x"
haftmann@33594:   by (cases x) simp
haftmann@33594: 
haftmann@33594: lemma snd_apfst [simp]:
haftmann@33594:   "snd (apfst f x) = snd x"
haftmann@33594:   by (cases x) simp
haftmann@33594: 
haftmann@33594: lemma snd_apsnd [simp]:
haftmann@33594:   "snd (apsnd f x) = f (snd x)"
haftmann@33594:   by (cases x) simp
haftmann@33594: 
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"
haftmann@33594:   by (simp add: expand_fun_eq)
haftmann@33594: 
haftmann@33594: lemma apsnd_id [simp] :
haftmann@33594:   "apsnd id = id"
haftmann@33594:   by (simp add: expand_fun_eq)
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@26358: definition  Sigma :: "['a set, 'a => 'b set] => ('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@26358:   Times :: "['a set, 'b set] => ('a * '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: 
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@35828: 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@35828: 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@36622: lemma fst_image_times[simp]: "fst ` (A \<times> B) = (if B = {} then {} else A)"
hoelzl@36622:   by (auto intro!: image_eqI)
hoelzl@36622: 
hoelzl@36622: lemma snd_image_times[simp]: "snd ` (A \<times> B) = (if A = {} then {} else B)"
hoelzl@36622:   by (auto intro!: image_eqI)
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)"
haftmann@37166:   by (auto, case_tac "f x", auto)
paulson@33271: 
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))"
hoelzl@36622: proof (safe intro!: imageI vimageI)
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@21908: 
haftmann@37166: subsection {* Inductively defined sets *}
berghofe@15394: 
haftmann@31723: use "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: lemma Pair_inject:
haftmann@37166:   assumes "(a, b) = (a', b')"
haftmann@37166:     and "a = a' ==> b = b' ==> R"
haftmann@37166:   shows R
haftmann@37166:   using assms by simp
haftmann@37166: 
haftmann@37166: lemmas Pair_eq = prod.inject
haftmann@37166: 
haftmann@37166: lemmas split = split_conv  -- {* for backwards compatibility *}
haftmann@37166: 
nipkow@10213: end