src/HOL/Product_Type.thy
author nipkow
Mon Aug 16 14:22:27 2004 +0200 (2004-08-16)
changeset 15131 c69542757a4d
parent 14952 47455995693d
child 15140 322485b816ac
permissions -rw-r--r--
New theory header syntax.
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(*  Title:      HOL/Product_Type.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1992  University of Cambridge
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*)
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header {* Cartesian products *}
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theory Product_Type
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import Fun
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files ("Tools/split_rule.ML")
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begin
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subsection {* Unit *}
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typedef unit = "{True}"
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proof
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  show "True : ?unit" by blast
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qed
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constdefs
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  Unity :: unit    ("'(')")
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  "() == Abs_unit True"
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lemma unit_eq: "u = ()"
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  by (induct u) (simp add: unit_def Unity_def)
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text {*
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  Simplification procedure for @{thm [source] unit_eq}.  Cannot use
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  this rule directly --- it loops!
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*}
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ML_setup {*
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  val unit_eq_proc =
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    let val unit_meta_eq = mk_meta_eq (thm "unit_eq") in
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      Simplifier.simproc (Theory.sign_of (the_context ())) "unit_eq" ["x::unit"]
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      (fn _ => fn _ => fn t => if HOLogic.is_unit t then None else Some unit_meta_eq)
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    end;
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  Addsimprocs [unit_eq_proc];
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*}
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lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()"
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  by simp
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lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P"
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  by (rule triv_forall_equality)
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lemma unit_induct [induct type: unit]: "P () ==> P x"
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  by simp
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text {*
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  This rewrite counters the effect of @{text unit_eq_proc} on @{term
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  [source] "%u::unit. f u"}, replacing it by @{term [source]
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  f} rather than by @{term [source] "%u. f ()"}.
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*}
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lemma unit_abs_eta_conv [simp]: "(%u::unit. f ()) = f"
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  by (rule ext) simp
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subsection {* Pairs *}
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subsubsection {* Type definition *}
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constdefs
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  Pair_Rep :: "['a, 'b] => ['a, 'b] => bool"
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  "Pair_Rep == (%a b. %x y. x=a & y=b)"
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global
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typedef (Prod)
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  ('a, 'b) "*"    (infixr 20)
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    = "{f. EX a b. f = Pair_Rep (a::'a) (b::'b)}"
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proof
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  fix a b show "Pair_Rep a b : ?Prod"
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    by blast
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qed
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syntax (xsymbols)
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  "*"      :: "[type, type] => type"         ("(_ \<times>/ _)" [21, 20] 20)
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syntax (HTML output)
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  "*"      :: "[type, type] => type"         ("(_ \<times>/ _)" [21, 20] 20)
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local
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subsubsection {* Abstract constants and syntax *}
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global
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consts
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  fst      :: "'a * 'b => 'a"
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  snd      :: "'a * 'b => 'b"
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  split    :: "[['a, 'b] => 'c, 'a * 'b] => 'c"
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  curry    :: "['a * 'b => 'c, 'a, 'b] => 'c"
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  prod_fun :: "['a => 'b, 'c => 'd, 'a * 'c] => 'b * 'd"
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  Pair     :: "['a, 'b] => 'a * 'b"
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  Sigma    :: "['a set, 'a => 'b set] => ('a * 'b) set"
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local
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text {*
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  Patterns -- extends pre-defined type @{typ pttrn} used in
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  abstractions.
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*}
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nonterminals
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  tuple_args patterns
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syntax
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  "_tuple"      :: "'a => tuple_args => 'a * 'b"        ("(1'(_,/ _'))")
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  "_tuple_arg"  :: "'a => tuple_args"                   ("_")
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  "_tuple_args" :: "'a => tuple_args => tuple_args"     ("_,/ _")
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  "_pattern"    :: "[pttrn, patterns] => pttrn"         ("'(_,/ _')")
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  ""            :: "pttrn => patterns"                  ("_")
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  "_patterns"   :: "[pttrn, patterns] => patterns"      ("_,/ _")
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  "@Sigma" ::"[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" 10)
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  "@Times" ::"['a set,  'a => 'b set] => ('a * 'b) set" (infixr "<*>" 80)
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translations
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  "(x, y)"       == "Pair x y"
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  "_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))"
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  "%(x,y,zs).b"  == "split(%x (y,zs).b)"
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  "%(x,y).b"     == "split(%x y. b)"
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  "_abs (Pair x y) t" => "%(x,y).t"
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  (* The last rule accommodates tuples in `case C ... (x,y) ... => ...'
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     The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *)
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  "SIGMA x:A. B" => "Sigma A (%x. B)"
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  "A <*> B"      => "Sigma A (_K B)"
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(* reconstructs pattern from (nested) splits, avoiding eta-contraction of body*)
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(* works best with enclosing "let", if "let" does not avoid eta-contraction   *)
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print_translation {*
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let fun split_tr' [Abs (x,T,t as (Abs abs))] =
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      (* split (%x y. t) => %(x,y) t *)
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      let val (y,t') = atomic_abs_tr' abs;
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          val (x',t'') = atomic_abs_tr' (x,T,t');
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      in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end
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    | split_tr' [Abs (x,T,(s as Const ("split",_)$t))] =
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       (* split (%x. (split (%y z. t))) => %(x,y,z). t *)
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       let val (Const ("_abs",_)$(Const ("_pattern",_)$y$z)$t') = split_tr' [t];
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           val (x',t'') = atomic_abs_tr' (x,T,t');
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       in Syntax.const "_abs"$ 
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           (Syntax.const "_pattern"$x'$(Syntax.const "_patterns"$y$z))$t'' end
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    | split_tr' [Const ("split",_)$t] =
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       (* split (split (%x y z. t)) => %((x,y),z). t *)   
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       split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *)
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    | split_tr' [Const ("_abs",_)$x_y$(Abs abs)] =
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       (* split (%pttrn z. t) => %(pttrn,z). t *)
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       let val (z,t) = atomic_abs_tr' abs;
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       in Syntax.const "_abs" $ (Syntax.const "_pattern" $x_y$z) $ t end
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    | split_tr' _ =  raise Match;
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in [("split", split_tr')]
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end
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*}
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text{*Deleted x-symbol and html support using @{text"\<Sigma>"} (Sigma) because of the danger of confusion with Sum.*}
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syntax (xsymbols)
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  "@Times" :: "['a set,  'a => 'b set] => ('a * 'b) set"  ("_ \<times> _" [81, 80] 80)
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syntax (HTML output)
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  "@Times" :: "['a set,  'a => 'b set] => ('a * 'b) set"  ("_ \<times> _" [81, 80] 80)
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print_translation {* [("Sigma", dependent_tr' ("@Sigma", "@Times"))] *}
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subsubsection {* Definitions *}
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defs
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  Pair_def:     "Pair a b == Abs_Prod(Pair_Rep a b)"
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  fst_def:      "fst p == THE a. EX b. p = (a, b)"
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  snd_def:      "snd p == THE b. EX a. p = (a, b)"
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  split_def:    "split == (%c p. c (fst p) (snd p))"
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  curry_def:    "curry == (%c x y. c (x,y))"
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  prod_fun_def: "prod_fun f g == split(%x y.(f(x), g(y)))"
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  Sigma_def:    "Sigma A B == UN x:A. UN y:B(x). {(x, y)}"
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subsubsection {* Lemmas and proof tool setup *}
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lemma ProdI: "Pair_Rep a b : Prod"
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  by (unfold Prod_def) blast
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lemma Pair_Rep_inject: "Pair_Rep a b = Pair_Rep a' b' ==> a = a' & b = b'"
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  apply (unfold Pair_Rep_def)
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  apply (drule fun_cong [THEN fun_cong], blast)
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  done
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lemma inj_on_Abs_Prod: "inj_on Abs_Prod Prod"
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  apply (rule inj_on_inverseI)
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  apply (erule Abs_Prod_inverse)
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  done
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lemma Pair_inject:
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  "(a, b) = (a', b') ==> (a = a' ==> b = b' ==> R) ==> R"
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proof -
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  case rule_context [unfolded Pair_def]
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  show ?thesis
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    apply (rule inj_on_Abs_Prod [THEN inj_onD, THEN Pair_Rep_inject, THEN conjE])
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    apply (rule rule_context ProdI)+
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    .
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qed
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lemma Pair_eq [iff]: "((a, b) = (a', b')) = (a = a' & b = b')"
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  by (blast elim!: Pair_inject)
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lemma fst_conv [simp]: "fst (a, b) = a"
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  by (unfold fst_def) blast
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lemma snd_conv [simp]: "snd (a, b) = b"
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  by (unfold snd_def) blast
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lemma fst_eqD: "fst (x, y) = a ==> x = a"
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  by simp
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lemma snd_eqD: "snd (x, y) = a ==> y = a"
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  by simp
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lemma PairE_lemma: "EX x y. p = (x, y)"
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  apply (unfold Pair_def)
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  apply (rule Rep_Prod [unfolded Prod_def, THEN CollectE])
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  apply (erule exE, erule exE, rule exI, rule exI)
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  apply (rule Rep_Prod_inverse [symmetric, THEN trans])
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  apply (erule arg_cong)
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  done
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lemma PairE [cases type: *]: "(!!x y. p = (x, y) ==> Q) ==> Q"
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  by (insert PairE_lemma [of p]) blast
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ML_setup {*
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  local val PairE = thm "PairE" in
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    fun pair_tac s =
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      EVERY' [res_inst_tac [("p", s)] PairE, hyp_subst_tac, K prune_params_tac];
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  end;
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*}
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lemma surjective_pairing: "p = (fst p, snd p)"
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  -- {* Do not add as rewrite rule: invalidates some proofs in IMP *}
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  by (cases p) simp
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declare surjective_pairing [symmetric, simp]
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lemma surj_pair [simp]: "EX x y. z = (x, y)"
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  apply (rule exI)
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  apply (rule exI)
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  apply (rule surjective_pairing)
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  done
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lemma split_paired_all: "(!!x. PROP P x) == (!!a b. PROP P (a, b))"
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proof
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  fix a b
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  assume "!!x. PROP P x"
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  thus "PROP P (a, b)" .
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next
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  fix x
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  assume "!!a b. PROP P (a, b)"
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  hence "PROP P (fst x, snd x)" .
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  thus "PROP P x" by simp
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qed
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lemmas split_tupled_all = split_paired_all unit_all_eq2
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text {*
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  The rule @{thm [source] split_paired_all} does not work with the
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  Simplifier because it also affects premises in congrence rules,
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  where this can lead to premises of the form @{text "!!a b. ... =
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  ?P(a, b)"} which cannot be solved by reflexivity.
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*}
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ML_setup "
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  (* replace parameters of product type by individual component parameters *)
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  val safe_full_simp_tac = generic_simp_tac true (true, false, false);
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  local (* filtering with exists_paired_all is an essential optimization *)
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    fun exists_paired_all (Const (\"all\", _) $ Abs (_, T, t)) =
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          can HOLogic.dest_prodT T orelse exists_paired_all t
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      | exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u
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      | exists_paired_all (Abs (_, _, t)) = exists_paired_all t
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      | exists_paired_all _ = false;
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    val ss = HOL_basic_ss
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      addsimps [thm \"split_paired_all\", thm \"unit_all_eq2\", thm \"unit_abs_eta_conv\"]
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      addsimprocs [unit_eq_proc];
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  in
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    val split_all_tac = SUBGOAL (fn (t, i) =>
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      if exists_paired_all t then safe_full_simp_tac ss i else no_tac);
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    val unsafe_split_all_tac = SUBGOAL (fn (t, i) =>
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      if exists_paired_all t then full_simp_tac ss i else no_tac);
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    fun split_all th =
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   if exists_paired_all (#prop (Thm.rep_thm th)) then full_simplify ss th else th;
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  end;
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claset_ref() := claset() addSbefore (\"split_all_tac\", split_all_tac);
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"
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lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))"
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  -- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *}
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  by fast
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lemma curry_split [simp]: "curry (split f) = f"
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  by (simp add: curry_def split_def)
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lemma split_curry [simp]: "split (curry f) = f"
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  by (simp add: curry_def split_def)
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lemma curryI [intro!]: "f (a,b) ==> curry f a b"
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  by (simp add: curry_def)
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lemma curryD [dest!]: "curry f a b ==> f (a,b)"
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  by (simp add: curry_def)
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lemma curryE: "[| curry f a b ; f (a,b) ==> Q |] ==> Q"
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  by (simp add: curry_def)
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lemma curry_conv [simp]: "curry f a b = f (a,b)"
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  by (simp add: curry_def)
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lemma prod_induct [induct type: *]: "!!x. (!!a b. P (a, b)) ==> P x"
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  by fast
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lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))"
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  by fast
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lemma split_conv [simp]: "split c (a, b) = c a b"
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  by (simp add: split_def)
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   327
wenzelm@11838
   328
lemmas split = split_conv  -- {* for backwards compatibility *}
wenzelm@11838
   329
wenzelm@11838
   330
lemmas splitI = split_conv [THEN iffD2, standard]
wenzelm@11838
   331
lemmas splitD = split_conv [THEN iffD1, standard]
wenzelm@11838
   332
wenzelm@11838
   333
lemma split_Pair_apply: "split (%x y. f (x, y)) = f"
wenzelm@11838
   334
  -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *}
wenzelm@11838
   335
  apply (rule ext)
paulson@14208
   336
  apply (tactic {* pair_tac "x" 1 *}, simp)
wenzelm@11838
   337
  done
wenzelm@11838
   338
wenzelm@11838
   339
lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))"
wenzelm@11838
   340
  -- {* Can't be added to simpset: loops! *}
wenzelm@11838
   341
  by (simp add: split_Pair_apply)
wenzelm@11838
   342
wenzelm@11838
   343
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
wenzelm@11838
   344
  by (simp add: split_def)
wenzelm@11838
   345
wenzelm@11838
   346
lemma Pair_fst_snd_eq: "!!s t. (s = t) = (fst s = fst t & snd s = snd t)"
paulson@14208
   347
by (simp only: split_tupled_all, simp)
wenzelm@11838
   348
wenzelm@11838
   349
lemma prod_eqI [intro?]: "fst p = fst q ==> snd p = snd q ==> p = q"
wenzelm@11838
   350
  by (simp add: Pair_fst_snd_eq)
wenzelm@11838
   351
wenzelm@11838
   352
lemma split_weak_cong: "p = q ==> split c p = split c q"
wenzelm@11838
   353
  -- {* Prevents simplification of @{term c}: much faster *}
wenzelm@11838
   354
  by (erule arg_cong)
wenzelm@11838
   355
wenzelm@11838
   356
lemma split_eta: "(%(x, y). f (x, y)) = f"
wenzelm@11838
   357
  apply (rule ext)
wenzelm@11838
   358
  apply (simp only: split_tupled_all)
wenzelm@11838
   359
  apply (rule split_conv)
wenzelm@11838
   360
  done
wenzelm@11838
   361
wenzelm@11838
   362
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
wenzelm@11838
   363
  by (simp add: split_eta)
wenzelm@11838
   364
wenzelm@11838
   365
text {*
wenzelm@11838
   366
  Simplification procedure for @{thm [source] cond_split_eta}.  Using
wenzelm@11838
   367
  @{thm [source] split_eta} as a rewrite rule is not general enough,
wenzelm@11838
   368
  and using @{thm [source] cond_split_eta} directly would render some
wenzelm@11838
   369
  existing proofs very inefficient; similarly for @{text
wenzelm@11838
   370
  split_beta}. *}
wenzelm@11838
   371
wenzelm@11838
   372
ML_setup {*
wenzelm@11838
   373
wenzelm@11838
   374
local
wenzelm@11838
   375
  val cond_split_eta = thm "cond_split_eta";
wenzelm@11838
   376
  fun  Pair_pat k 0 (Bound m) = (m = k)
wenzelm@11838
   377
  |    Pair_pat k i (Const ("Pair",  _) $ Bound m $ t) = i > 0 andalso
wenzelm@11838
   378
                        m = k+i andalso Pair_pat k (i-1) t
wenzelm@11838
   379
  |    Pair_pat _ _ _ = false;
wenzelm@11838
   380
  fun no_args k i (Abs (_, _, t)) = no_args (k+1) i t
wenzelm@11838
   381
  |   no_args k i (t $ u) = no_args k i t andalso no_args k i u
wenzelm@11838
   382
  |   no_args k i (Bound m) = m < k orelse m > k+i
wenzelm@11838
   383
  |   no_args _ _ _ = true;
wenzelm@11838
   384
  fun split_pat tp i (Abs  (_,_,t)) = if tp 0 i t then Some (i,t) else None
wenzelm@11838
   385
  |   split_pat tp i (Const ("split", _) $ Abs (_, _, t)) = split_pat tp (i+1) t
wenzelm@11838
   386
  |   split_pat tp i _ = None;
wenzelm@13480
   387
  fun metaeq sg lhs rhs = mk_meta_eq (Tactic.prove sg [] []
wenzelm@13480
   388
        (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs,rhs)))
wenzelm@13480
   389
        (K (simp_tac (HOL_basic_ss addsimps [cond_split_eta]) 1)));
wenzelm@11838
   390
wenzelm@11838
   391
  fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k+1) i t
wenzelm@11838
   392
  |   beta_term_pat k i (t $ u) = Pair_pat k i (t $ u) orelse
wenzelm@11838
   393
                        (beta_term_pat k i t andalso beta_term_pat k i u)
wenzelm@11838
   394
  |   beta_term_pat k i t = no_args k i t;
wenzelm@11838
   395
  fun  eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
wenzelm@11838
   396
  |    eta_term_pat _ _ _ = false;
wenzelm@11838
   397
  fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
wenzelm@11838
   398
  |   subst arg k i (t $ u) = if Pair_pat k i (t $ u) then incr_boundvars k arg
wenzelm@11838
   399
                              else (subst arg k i t $ subst arg k i u)
wenzelm@11838
   400
  |   subst arg k i t = t;
wenzelm@11838
   401
  fun beta_proc sg _ (s as Const ("split", _) $ Abs (_, _, t) $ arg) =
wenzelm@11838
   402
        (case split_pat beta_term_pat 1 t of
wenzelm@11838
   403
        Some (i,f) => Some (metaeq sg s (subst arg 0 i f))
wenzelm@11838
   404
        | None => None)
wenzelm@11838
   405
  |   beta_proc _ _ _ = None;
wenzelm@11838
   406
  fun eta_proc sg _ (s as Const ("split", _) $ Abs (_, _, t)) =
wenzelm@11838
   407
        (case split_pat eta_term_pat 1 t of
wenzelm@11838
   408
          Some (_,ft) => Some (metaeq sg s (let val (f $ arg) = ft in f end))
wenzelm@11838
   409
        | None => None)
wenzelm@11838
   410
  |   eta_proc _ _ _ = None;
wenzelm@11838
   411
in
wenzelm@13462
   412
  val split_beta_proc = Simplifier.simproc (Theory.sign_of (the_context ()))
wenzelm@13462
   413
    "split_beta" ["split f z"] beta_proc;
wenzelm@13462
   414
  val split_eta_proc = Simplifier.simproc (Theory.sign_of (the_context ()))
wenzelm@13462
   415
    "split_eta" ["split f"] eta_proc;
wenzelm@11838
   416
end;
wenzelm@11838
   417
wenzelm@11838
   418
Addsimprocs [split_beta_proc, split_eta_proc];
wenzelm@11838
   419
*}
wenzelm@11838
   420
wenzelm@11838
   421
lemma split_beta: "(%(x, y). P x y) z = P (fst z) (snd z)"
wenzelm@11838
   422
  by (subst surjective_pairing, rule split_conv)
wenzelm@11838
   423
wenzelm@11838
   424
lemma split_split: "R (split c p) = (ALL x y. p = (x, y) --> R (c x y))"
wenzelm@11838
   425
  -- {* For use with @{text split} and the Simplifier. *}
wenzelm@11838
   426
  apply (subst surjective_pairing)
paulson@14208
   427
  apply (subst split_conv, blast)
wenzelm@11838
   428
  done
wenzelm@11838
   429
wenzelm@11838
   430
text {*
wenzelm@11838
   431
  @{thm [source] split_split} could be declared as @{text "[split]"}
wenzelm@11838
   432
  done after the Splitter has been speeded up significantly;
wenzelm@11838
   433
  precompute the constants involved and don't do anything unless the
wenzelm@11838
   434
  current goal contains one of those constants.
wenzelm@11838
   435
*}
wenzelm@11838
   436
wenzelm@11838
   437
lemma split_split_asm: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))"
paulson@14208
   438
by (subst split_split, simp)
wenzelm@11838
   439
wenzelm@11838
   440
wenzelm@11838
   441
text {*
wenzelm@11838
   442
  \medskip @{term split} used as a logical connective or set former.
wenzelm@11838
   443
wenzelm@11838
   444
  \medskip These rules are for use with @{text blast}; could instead
wenzelm@11838
   445
  call @{text simp} using @{thm [source] split} as rewrite. *}
wenzelm@11838
   446
wenzelm@11838
   447
lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"
wenzelm@11838
   448
  apply (simp only: split_tupled_all)
wenzelm@11838
   449
  apply (simp (no_asm_simp))
wenzelm@11838
   450
  done
wenzelm@11838
   451
wenzelm@11838
   452
lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"
wenzelm@11838
   453
  apply (simp only: split_tupled_all)
wenzelm@11838
   454
  apply (simp (no_asm_simp))
wenzelm@11838
   455
  done
wenzelm@11838
   456
wenzelm@11838
   457
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
wenzelm@11838
   458
  by (induct p) (auto simp add: split_def)
wenzelm@11838
   459
wenzelm@11838
   460
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
wenzelm@11838
   461
  by (induct p) (auto simp add: split_def)
wenzelm@11838
   462
wenzelm@11838
   463
lemma splitE2:
wenzelm@11838
   464
  "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
wenzelm@11838
   465
proof -
wenzelm@11838
   466
  assume q: "Q (split P z)"
wenzelm@11838
   467
  assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
wenzelm@11838
   468
  show R
wenzelm@11838
   469
    apply (rule r surjective_pairing)+
wenzelm@11838
   470
    apply (rule split_beta [THEN subst], rule q)
wenzelm@11838
   471
    done
wenzelm@11838
   472
qed
wenzelm@11838
   473
wenzelm@11838
   474
lemma splitD': "split R (a,b) c ==> R a b c"
wenzelm@11838
   475
  by simp
wenzelm@11838
   476
wenzelm@11838
   477
lemma mem_splitI: "z: c a b ==> z: split c (a, b)"
wenzelm@11838
   478
  by simp
wenzelm@11838
   479
wenzelm@11838
   480
lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"
paulson@14208
   481
by (simp only: split_tupled_all, simp)
wenzelm@11838
   482
wenzelm@11838
   483
lemma mem_splitE: "[| z: split c p; !!x y. [| p = (x,y); z: c x y |] ==> Q |] ==> Q"
wenzelm@11838
   484
proof -
wenzelm@11838
   485
  case rule_context [unfolded split_def]
wenzelm@11838
   486
  show ?thesis by (rule rule_context surjective_pairing)+
wenzelm@11838
   487
qed
wenzelm@11838
   488
wenzelm@11838
   489
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
wenzelm@11838
   490
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]
wenzelm@11838
   491
wenzelm@11838
   492
ML_setup "
wenzelm@11838
   493
local (* filtering with exists_p_split is an essential optimization *)
wenzelm@11838
   494
  fun exists_p_split (Const (\"split\",_) $ _ $ (Const (\"Pair\",_)$_$_)) = true
wenzelm@11838
   495
    | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
wenzelm@11838
   496
    | exists_p_split (Abs (_, _, t)) = exists_p_split t
wenzelm@11838
   497
    | exists_p_split _ = false;
wenzelm@11838
   498
  val ss = HOL_basic_ss addsimps [thm \"split_conv\"];
wenzelm@11838
   499
in
wenzelm@11838
   500
val split_conv_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838
   501
    if exists_p_split t then safe_full_simp_tac ss i else no_tac);
wenzelm@11838
   502
end;
wenzelm@11838
   503
(* This prevents applications of splitE for already splitted arguments leading
wenzelm@11838
   504
   to quite time-consuming computations (in particular for nested tuples) *)
wenzelm@11838
   505
claset_ref() := claset() addSbefore (\"split_conv_tac\", split_conv_tac);
wenzelm@11838
   506
"
wenzelm@11838
   507
wenzelm@11838
   508
lemma split_eta_SetCompr [simp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
paulson@14208
   509
by (rule ext, fast)
wenzelm@11838
   510
wenzelm@11838
   511
lemma split_eta_SetCompr2 [simp]: "(%u. EX x y. u = (x, y) & P x y) = split P"
paulson@14208
   512
by (rule ext, fast)
wenzelm@11838
   513
wenzelm@11838
   514
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"
wenzelm@11838
   515
  -- {* Allows simplifications of nested splits in case of independent predicates. *}
paulson@14208
   516
  apply (rule ext, blast)
wenzelm@11838
   517
  done
wenzelm@11838
   518
nipkow@14337
   519
(* Do NOT make this a simp rule as it
nipkow@14337
   520
   a) only helps in special situations
nipkow@14337
   521
   b) can lead to nontermination in the presence of split_def
nipkow@14337
   522
*)
nipkow@14337
   523
lemma split_comp_eq: 
oheimb@14101
   524
"(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"
oheimb@14101
   525
by (rule ext, auto)
oheimb@14101
   526
wenzelm@11838
   527
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
wenzelm@11838
   528
  by blast
wenzelm@11838
   529
wenzelm@11838
   530
(*
wenzelm@11838
   531
the following  would be slightly more general,
wenzelm@11838
   532
but cannot be used as rewrite rule:
wenzelm@11838
   533
### Cannot add premise as rewrite rule because it contains (type) unknowns:
wenzelm@11838
   534
### ?y = .x
wenzelm@11838
   535
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
paulson@14208
   536
by (rtac some_equality 1)
paulson@14208
   537
by ( Simp_tac 1)
paulson@14208
   538
by (split_all_tac 1)
paulson@14208
   539
by (Asm_full_simp_tac 1)
wenzelm@11838
   540
qed "The_split_eq";
wenzelm@11838
   541
*)
wenzelm@11838
   542
wenzelm@11838
   543
lemma injective_fst_snd: "!!x y. [|fst x = fst y; snd x = snd y|] ==> x = y"
wenzelm@11838
   544
  by auto
wenzelm@11838
   545
wenzelm@11838
   546
wenzelm@11838
   547
text {*
wenzelm@11838
   548
  \bigskip @{term prod_fun} --- action of the product functor upon
wenzelm@11838
   549
  functions.
wenzelm@11838
   550
*}
wenzelm@11838
   551
wenzelm@11838
   552
lemma prod_fun [simp]: "prod_fun f g (a, b) = (f a, g b)"
wenzelm@11838
   553
  by (simp add: prod_fun_def)
wenzelm@11838
   554
wenzelm@11838
   555
lemma prod_fun_compose: "prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)"
wenzelm@11838
   556
  apply (rule ext)
paulson@14208
   557
  apply (tactic {* pair_tac "x" 1 *}, simp)
wenzelm@11838
   558
  done
wenzelm@11838
   559
wenzelm@11838
   560
lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)"
wenzelm@11838
   561
  apply (rule ext)
paulson@14208
   562
  apply (tactic {* pair_tac "z" 1 *}, simp)
wenzelm@11838
   563
  done
wenzelm@11838
   564
wenzelm@11838
   565
lemma prod_fun_imageI [intro]: "(a, b) : r ==> (f a, g b) : prod_fun f g ` r"
wenzelm@11838
   566
  apply (rule image_eqI)
paulson@14208
   567
  apply (rule prod_fun [symmetric], assumption)
wenzelm@11838
   568
  done
wenzelm@11838
   569
wenzelm@11838
   570
lemma prod_fun_imageE [elim!]:
wenzelm@11838
   571
  "[| c: (prod_fun f g)`r;  !!x y. [| c=(f(x),g(y));  (x,y):r |] ==> P
wenzelm@11838
   572
    |] ==> P"
wenzelm@11838
   573
proof -
wenzelm@11838
   574
  case rule_context
wenzelm@11838
   575
  assume major: "c: (prod_fun f g)`r"
wenzelm@11838
   576
  show ?thesis
wenzelm@11838
   577
    apply (rule major [THEN imageE])
wenzelm@11838
   578
    apply (rule_tac p = x in PairE)
wenzelm@11838
   579
    apply (rule rule_context)
wenzelm@11838
   580
     prefer 2
wenzelm@11838
   581
     apply blast
wenzelm@11838
   582
    apply (blast intro: prod_fun)
wenzelm@11838
   583
    done
wenzelm@11838
   584
qed
wenzelm@11838
   585
wenzelm@11838
   586
oheimb@14101
   587
constdefs
oheimb@14101
   588
  upd_fst :: "('a => 'c) => 'a * 'b => 'c * 'b"
oheimb@14101
   589
 "upd_fst f == prod_fun f id"
oheimb@14101
   590
oheimb@14101
   591
  upd_snd :: "('b => 'c) => 'a * 'b => 'a * 'c"
oheimb@14101
   592
 "upd_snd f == prod_fun id f"
oheimb@14101
   593
oheimb@14101
   594
lemma upd_fst_conv [simp]: "upd_fst f (x,y) = (f x,y)" 
oheimb@14101
   595
by (simp add: upd_fst_def)
oheimb@14101
   596
oheimb@14101
   597
lemma upd_snd_conv [simp]: "upd_snd f (x,y) = (x,f y)" 
oheimb@14101
   598
by (simp add: upd_snd_def)
oheimb@14101
   599
wenzelm@11838
   600
text {*
wenzelm@11838
   601
  \bigskip Disjoint union of a family of sets -- Sigma.
wenzelm@11838
   602
*}
wenzelm@11838
   603
wenzelm@11838
   604
lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
wenzelm@11838
   605
  by (unfold Sigma_def) blast
wenzelm@11838
   606
paulson@14952
   607
lemma SigmaE [elim!]:
wenzelm@11838
   608
    "[| c: Sigma A B;
wenzelm@11838
   609
        !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
wenzelm@11838
   610
     |] ==> P"
wenzelm@11838
   611
  -- {* The general elimination rule. *}
wenzelm@11838
   612
  by (unfold Sigma_def) blast
wenzelm@11838
   613
wenzelm@11838
   614
text {*
wenzelm@11838
   615
  Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
wenzelm@11838
   616
  eigenvariables.
wenzelm@11838
   617
*}
wenzelm@11838
   618
wenzelm@11838
   619
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
paulson@14952
   620
by blast
wenzelm@11838
   621
wenzelm@11838
   622
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
paulson@14952
   623
by blast
wenzelm@11838
   624
wenzelm@11838
   625
lemma SigmaE2:
wenzelm@11838
   626
    "[| (a, b) : Sigma A B;
wenzelm@11838
   627
        [| a:A;  b:B(a) |] ==> P
wenzelm@11838
   628
     |] ==> P"
paulson@14952
   629
  by blast
wenzelm@11838
   630
paulson@14952
   631
lemma Sigma_cong:
paulson@14952
   632
     "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>
paulson@14952
   633
      \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
paulson@14952
   634
by auto
wenzelm@11838
   635
wenzelm@11838
   636
lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
wenzelm@11838
   637
  by blast
wenzelm@11838
   638
wenzelm@11838
   639
lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
wenzelm@11838
   640
  by blast
wenzelm@11838
   641
wenzelm@11838
   642
lemma Sigma_empty2 [simp]: "A <*> {} = {}"
wenzelm@11838
   643
  by blast
wenzelm@11838
   644
wenzelm@11838
   645
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
wenzelm@11838
   646
  by auto
wenzelm@11838
   647
wenzelm@11838
   648
lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
wenzelm@11838
   649
  by auto
wenzelm@11838
   650
wenzelm@11838
   651
lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
wenzelm@11838
   652
  by auto
wenzelm@11838
   653
wenzelm@11838
   654
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
wenzelm@11838
   655
  by blast
wenzelm@11838
   656
wenzelm@11838
   657
lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
wenzelm@11838
   658
  by blast
wenzelm@11838
   659
wenzelm@11838
   660
lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
wenzelm@11838
   661
  by (blast elim: equalityE)
wenzelm@11838
   662
wenzelm@11838
   663
lemma SetCompr_Sigma_eq:
wenzelm@11838
   664
    "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
wenzelm@11838
   665
  by blast
wenzelm@11838
   666
wenzelm@11838
   667
text {*
wenzelm@11838
   668
  \bigskip Complex rules for Sigma.
wenzelm@11838
   669
*}
wenzelm@11838
   670
wenzelm@11838
   671
lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"
wenzelm@11838
   672
  by blast
wenzelm@11838
   673
wenzelm@11838
   674
lemma UN_Times_distrib:
wenzelm@11838
   675
  "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"
wenzelm@11838
   676
  -- {* Suggested by Pierre Chartier *}
wenzelm@11838
   677
  by blast
wenzelm@11838
   678
wenzelm@11838
   679
lemma split_paired_Ball_Sigma [simp]:
wenzelm@11838
   680
    "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"
wenzelm@11838
   681
  by blast
wenzelm@11838
   682
wenzelm@11838
   683
lemma split_paired_Bex_Sigma [simp]:
wenzelm@11838
   684
    "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"
wenzelm@11838
   685
  by blast
wenzelm@11838
   686
wenzelm@11838
   687
lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"
wenzelm@11838
   688
  by blast
wenzelm@11838
   689
wenzelm@11838
   690
lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))"
wenzelm@11838
   691
  by blast
wenzelm@11838
   692
wenzelm@11838
   693
lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"
wenzelm@11838
   694
  by blast
wenzelm@11838
   695
wenzelm@11838
   696
lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))"
wenzelm@11838
   697
  by blast
wenzelm@11838
   698
wenzelm@11838
   699
lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"
wenzelm@11838
   700
  by blast
wenzelm@11838
   701
wenzelm@11838
   702
lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"
wenzelm@11838
   703
  by blast
wenzelm@11838
   704
wenzelm@11838
   705
lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"
wenzelm@11838
   706
  by blast
wenzelm@11838
   707
wenzelm@11838
   708
text {*
wenzelm@11838
   709
  Non-dependent versions are needed to avoid the need for higher-order
wenzelm@11838
   710
  matching, especially when the rules are re-oriented.
wenzelm@11838
   711
*}
wenzelm@11838
   712
wenzelm@11838
   713
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"
wenzelm@11838
   714
  by blast
wenzelm@11838
   715
wenzelm@11838
   716
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"
wenzelm@11838
   717
  by blast
wenzelm@11838
   718
wenzelm@11838
   719
lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
wenzelm@11838
   720
  by blast
wenzelm@11838
   721
wenzelm@11838
   722
oheimb@11493
   723
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
wenzelm@11777
   724
  apply (rule_tac x = "(a, b)" in image_eqI)
wenzelm@11777
   725
   apply auto
wenzelm@11777
   726
  done
wenzelm@11777
   727
oheimb@11493
   728
wenzelm@11838
   729
text {*
wenzelm@11838
   730
  Setup of internal @{text split_rule}.
wenzelm@11838
   731
*}
wenzelm@11838
   732
wenzelm@11032
   733
constdefs
wenzelm@11425
   734
  internal_split :: "('a => 'b => 'c) => 'a * 'b => 'c"
wenzelm@11032
   735
  "internal_split == split"
wenzelm@11032
   736
wenzelm@11032
   737
lemma internal_split_conv: "internal_split c (a, b) = c a b"
wenzelm@11032
   738
  by (simp only: internal_split_def split_conv)
wenzelm@11032
   739
wenzelm@11032
   740
hide const internal_split
wenzelm@11032
   741
oheimb@11025
   742
use "Tools/split_rule.ML"
wenzelm@11032
   743
setup SplitRule.setup
nipkow@10213
   744
nipkow@10213
   745
end