src/HOL/Product_Type.thy
author haftmann
Fri Jun 17 16:12:49 2005 +0200 (2005-06-17)
changeset 16417 9bc16273c2d4
parent 16121 a80aa66d2271
child 16634 f19d58cfb47a
permissions -rw-r--r--
migrated theory headers to new format
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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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imports Fun
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uses ("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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(* print "split f" as "\<lambda>(x,y). f x y" and "split (\<lambda>x. f x)" as "\<lambda>(x,y). f x y" *) 
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typed_print_translation {*
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let
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  fun split_guess_names_tr' _ T [Abs (x,_,Abs _)] = raise Match
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    | split_guess_names_tr' _ T  [Abs (x,xT,t)] =
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        (case (head_of t) of
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           Const ("split",_) => raise Match
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         | _ => let 
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                  val (_::yT::_) = binder_types (domain_type T) handle Bind => raise Match;
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                  val (y,t') = atomic_abs_tr' ("y",yT,(incr_boundvars 1 t)$Bound 0); 
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                  val (x',t'') = atomic_abs_tr' (x,xT,t');
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                in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end)
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    | split_guess_names_tr' _ T [t] =
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       (case (head_of t) of
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           Const ("split",_) => raise Match 
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         | _ => let 
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                  val (xT::yT::_) = binder_types (domain_type T) handle Bind => raise Match;
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                  val (y,t') = 
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                        atomic_abs_tr' ("y",yT,(incr_boundvars 2 t)$Bound 1$Bound 0); 
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                  val (x',t'') = atomic_abs_tr' ("x",xT,t');
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                in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end)
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    | split_guess_names_tr' _ _ _ = raise Match;
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in [("split", split_guess_names_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 {*
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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);
wenzelm@16121
   323
*}
wenzelm@11838
   324
wenzelm@11838
   325
lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))"
wenzelm@11838
   326
  -- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *}
wenzelm@11838
   327
  by fast
wenzelm@11838
   328
skalberg@14189
   329
lemma curry_split [simp]: "curry (split f) = f"
skalberg@14189
   330
  by (simp add: curry_def split_def)
skalberg@14189
   331
skalberg@14189
   332
lemma split_curry [simp]: "split (curry f) = f"
skalberg@14189
   333
  by (simp add: curry_def split_def)
skalberg@14189
   334
skalberg@14189
   335
lemma curryI [intro!]: "f (a,b) ==> curry f a b"
skalberg@14189
   336
  by (simp add: curry_def)
skalberg@14189
   337
skalberg@14190
   338
lemma curryD [dest!]: "curry f a b ==> f (a,b)"
skalberg@14189
   339
  by (simp add: curry_def)
skalberg@14189
   340
skalberg@14190
   341
lemma curryE: "[| curry f a b ; f (a,b) ==> Q |] ==> Q"
skalberg@14189
   342
  by (simp add: curry_def)
skalberg@14189
   343
skalberg@14189
   344
lemma curry_conv [simp]: "curry f a b = f (a,b)"
skalberg@14189
   345
  by (simp add: curry_def)
skalberg@14189
   346
wenzelm@11838
   347
lemma prod_induct [induct type: *]: "!!x. (!!a b. P (a, b)) ==> P x"
wenzelm@11838
   348
  by fast
wenzelm@11838
   349
wenzelm@11838
   350
lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))"
wenzelm@11838
   351
  by fast
wenzelm@11838
   352
wenzelm@11838
   353
lemma split_conv [simp]: "split c (a, b) = c a b"
wenzelm@11838
   354
  by (simp add: split_def)
wenzelm@11838
   355
wenzelm@11838
   356
lemmas split = split_conv  -- {* for backwards compatibility *}
wenzelm@11838
   357
wenzelm@11838
   358
lemmas splitI = split_conv [THEN iffD2, standard]
wenzelm@11838
   359
lemmas splitD = split_conv [THEN iffD1, standard]
wenzelm@11838
   360
wenzelm@11838
   361
lemma split_Pair_apply: "split (%x y. f (x, y)) = f"
wenzelm@11838
   362
  -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *}
wenzelm@11838
   363
  apply (rule ext)
paulson@14208
   364
  apply (tactic {* pair_tac "x" 1 *}, simp)
wenzelm@11838
   365
  done
wenzelm@11838
   366
wenzelm@11838
   367
lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))"
wenzelm@11838
   368
  -- {* Can't be added to simpset: loops! *}
wenzelm@11838
   369
  by (simp add: split_Pair_apply)
wenzelm@11838
   370
wenzelm@11838
   371
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
wenzelm@11838
   372
  by (simp add: split_def)
wenzelm@11838
   373
wenzelm@11838
   374
lemma Pair_fst_snd_eq: "!!s t. (s = t) = (fst s = fst t & snd s = snd t)"
paulson@14208
   375
by (simp only: split_tupled_all, simp)
wenzelm@11838
   376
wenzelm@11838
   377
lemma prod_eqI [intro?]: "fst p = fst q ==> snd p = snd q ==> p = q"
wenzelm@11838
   378
  by (simp add: Pair_fst_snd_eq)
wenzelm@11838
   379
wenzelm@11838
   380
lemma split_weak_cong: "p = q ==> split c p = split c q"
wenzelm@11838
   381
  -- {* Prevents simplification of @{term c}: much faster *}
wenzelm@11838
   382
  by (erule arg_cong)
wenzelm@11838
   383
wenzelm@11838
   384
lemma split_eta: "(%(x, y). f (x, y)) = f"
wenzelm@11838
   385
  apply (rule ext)
wenzelm@11838
   386
  apply (simp only: split_tupled_all)
wenzelm@11838
   387
  apply (rule split_conv)
wenzelm@11838
   388
  done
wenzelm@11838
   389
wenzelm@11838
   390
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
wenzelm@11838
   391
  by (simp add: split_eta)
wenzelm@11838
   392
wenzelm@11838
   393
text {*
wenzelm@11838
   394
  Simplification procedure for @{thm [source] cond_split_eta}.  Using
wenzelm@11838
   395
  @{thm [source] split_eta} as a rewrite rule is not general enough,
wenzelm@11838
   396
  and using @{thm [source] cond_split_eta} directly would render some
wenzelm@11838
   397
  existing proofs very inefficient; similarly for @{text
wenzelm@11838
   398
  split_beta}. *}
wenzelm@11838
   399
wenzelm@11838
   400
ML_setup {*
wenzelm@11838
   401
wenzelm@11838
   402
local
wenzelm@11838
   403
  val cond_split_eta = thm "cond_split_eta";
wenzelm@11838
   404
  fun  Pair_pat k 0 (Bound m) = (m = k)
wenzelm@11838
   405
  |    Pair_pat k i (Const ("Pair",  _) $ Bound m $ t) = i > 0 andalso
wenzelm@11838
   406
                        m = k+i andalso Pair_pat k (i-1) t
wenzelm@11838
   407
  |    Pair_pat _ _ _ = false;
wenzelm@11838
   408
  fun no_args k i (Abs (_, _, t)) = no_args (k+1) i t
wenzelm@11838
   409
  |   no_args k i (t $ u) = no_args k i t andalso no_args k i u
wenzelm@11838
   410
  |   no_args k i (Bound m) = m < k orelse m > k+i
wenzelm@11838
   411
  |   no_args _ _ _ = true;
skalberg@15531
   412
  fun split_pat tp i (Abs  (_,_,t)) = if tp 0 i t then SOME (i,t) else NONE
wenzelm@11838
   413
  |   split_pat tp i (Const ("split", _) $ Abs (_, _, t)) = split_pat tp (i+1) t
skalberg@15531
   414
  |   split_pat tp i _ = NONE;
wenzelm@13480
   415
  fun metaeq sg lhs rhs = mk_meta_eq (Tactic.prove sg [] []
wenzelm@13480
   416
        (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs,rhs)))
wenzelm@13480
   417
        (K (simp_tac (HOL_basic_ss addsimps [cond_split_eta]) 1)));
wenzelm@11838
   418
wenzelm@11838
   419
  fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k+1) i t
wenzelm@11838
   420
  |   beta_term_pat k i (t $ u) = Pair_pat k i (t $ u) orelse
wenzelm@11838
   421
                        (beta_term_pat k i t andalso beta_term_pat k i u)
wenzelm@11838
   422
  |   beta_term_pat k i t = no_args k i t;
wenzelm@11838
   423
  fun  eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
wenzelm@11838
   424
  |    eta_term_pat _ _ _ = false;
wenzelm@11838
   425
  fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
wenzelm@11838
   426
  |   subst arg k i (t $ u) = if Pair_pat k i (t $ u) then incr_boundvars k arg
wenzelm@11838
   427
                              else (subst arg k i t $ subst arg k i u)
wenzelm@11838
   428
  |   subst arg k i t = t;
wenzelm@11838
   429
  fun beta_proc sg _ (s as Const ("split", _) $ Abs (_, _, t) $ arg) =
wenzelm@11838
   430
        (case split_pat beta_term_pat 1 t of
skalberg@15531
   431
        SOME (i,f) => SOME (metaeq sg s (subst arg 0 i f))
skalberg@15531
   432
        | NONE => NONE)
skalberg@15531
   433
  |   beta_proc _ _ _ = NONE;
wenzelm@11838
   434
  fun eta_proc sg _ (s as Const ("split", _) $ Abs (_, _, t)) =
wenzelm@11838
   435
        (case split_pat eta_term_pat 1 t of
skalberg@15531
   436
          SOME (_,ft) => SOME (metaeq sg s (let val (f $ arg) = ft in f end))
skalberg@15531
   437
        | NONE => NONE)
skalberg@15531
   438
  |   eta_proc _ _ _ = NONE;
wenzelm@11838
   439
in
wenzelm@13462
   440
  val split_beta_proc = Simplifier.simproc (Theory.sign_of (the_context ()))
wenzelm@13462
   441
    "split_beta" ["split f z"] beta_proc;
wenzelm@13462
   442
  val split_eta_proc = Simplifier.simproc (Theory.sign_of (the_context ()))
wenzelm@13462
   443
    "split_eta" ["split f"] eta_proc;
wenzelm@11838
   444
end;
wenzelm@11838
   445
wenzelm@11838
   446
Addsimprocs [split_beta_proc, split_eta_proc];
wenzelm@11838
   447
*}
wenzelm@11838
   448
wenzelm@11838
   449
lemma split_beta: "(%(x, y). P x y) z = P (fst z) (snd z)"
wenzelm@11838
   450
  by (subst surjective_pairing, rule split_conv)
wenzelm@11838
   451
wenzelm@11838
   452
lemma split_split: "R (split c p) = (ALL x y. p = (x, y) --> R (c x y))"
wenzelm@11838
   453
  -- {* For use with @{text split} and the Simplifier. *}
paulson@15481
   454
  by (insert surj_pair [of p], clarify, simp)
wenzelm@11838
   455
wenzelm@11838
   456
text {*
wenzelm@11838
   457
  @{thm [source] split_split} could be declared as @{text "[split]"}
wenzelm@11838
   458
  done after the Splitter has been speeded up significantly;
wenzelm@11838
   459
  precompute the constants involved and don't do anything unless the
wenzelm@11838
   460
  current goal contains one of those constants.
wenzelm@11838
   461
*}
wenzelm@11838
   462
wenzelm@11838
   463
lemma split_split_asm: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))"
paulson@14208
   464
by (subst split_split, simp)
wenzelm@11838
   465
wenzelm@11838
   466
wenzelm@11838
   467
text {*
wenzelm@11838
   468
  \medskip @{term split} used as a logical connective or set former.
wenzelm@11838
   469
wenzelm@11838
   470
  \medskip These rules are for use with @{text blast}; could instead
wenzelm@11838
   471
  call @{text simp} using @{thm [source] split} as rewrite. *}
wenzelm@11838
   472
wenzelm@11838
   473
lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"
wenzelm@11838
   474
  apply (simp only: split_tupled_all)
wenzelm@11838
   475
  apply (simp (no_asm_simp))
wenzelm@11838
   476
  done
wenzelm@11838
   477
wenzelm@11838
   478
lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"
wenzelm@11838
   479
  apply (simp only: split_tupled_all)
wenzelm@11838
   480
  apply (simp (no_asm_simp))
wenzelm@11838
   481
  done
wenzelm@11838
   482
wenzelm@11838
   483
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
wenzelm@11838
   484
  by (induct p) (auto simp add: split_def)
wenzelm@11838
   485
wenzelm@11838
   486
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
wenzelm@11838
   487
  by (induct p) (auto simp add: split_def)
wenzelm@11838
   488
wenzelm@11838
   489
lemma splitE2:
wenzelm@11838
   490
  "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
wenzelm@11838
   491
proof -
wenzelm@11838
   492
  assume q: "Q (split P z)"
wenzelm@11838
   493
  assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
wenzelm@11838
   494
  show R
wenzelm@11838
   495
    apply (rule r surjective_pairing)+
wenzelm@11838
   496
    apply (rule split_beta [THEN subst], rule q)
wenzelm@11838
   497
    done
wenzelm@11838
   498
qed
wenzelm@11838
   499
wenzelm@11838
   500
lemma splitD': "split R (a,b) c ==> R a b c"
wenzelm@11838
   501
  by simp
wenzelm@11838
   502
wenzelm@11838
   503
lemma mem_splitI: "z: c a b ==> z: split c (a, b)"
wenzelm@11838
   504
  by simp
wenzelm@11838
   505
wenzelm@11838
   506
lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"
paulson@14208
   507
by (simp only: split_tupled_all, simp)
wenzelm@11838
   508
wenzelm@11838
   509
lemma mem_splitE: "[| z: split c p; !!x y. [| p = (x,y); z: c x y |] ==> Q |] ==> Q"
wenzelm@11838
   510
proof -
wenzelm@11838
   511
  case rule_context [unfolded split_def]
wenzelm@11838
   512
  show ?thesis by (rule rule_context surjective_pairing)+
wenzelm@11838
   513
qed
wenzelm@11838
   514
wenzelm@11838
   515
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
wenzelm@11838
   516
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]
wenzelm@11838
   517
wenzelm@16121
   518
ML_setup {*
wenzelm@11838
   519
local (* filtering with exists_p_split is an essential optimization *)
wenzelm@16121
   520
  fun exists_p_split (Const ("split",_) $ _ $ (Const ("Pair",_)$_$_)) = true
wenzelm@11838
   521
    | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
wenzelm@11838
   522
    | exists_p_split (Abs (_, _, t)) = exists_p_split t
wenzelm@11838
   523
    | exists_p_split _ = false;
wenzelm@16121
   524
  val ss = HOL_basic_ss addsimps [thm "split_conv"];
wenzelm@11838
   525
in
wenzelm@11838
   526
val split_conv_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838
   527
    if exists_p_split t then safe_full_simp_tac ss i else no_tac);
wenzelm@11838
   528
end;
wenzelm@11838
   529
(* This prevents applications of splitE for already splitted arguments leading
wenzelm@11838
   530
   to quite time-consuming computations (in particular for nested tuples) *)
wenzelm@16121
   531
claset_ref() := claset() addSbefore ("split_conv_tac", split_conv_tac);
wenzelm@16121
   532
*}
wenzelm@11838
   533
wenzelm@11838
   534
lemma split_eta_SetCompr [simp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
paulson@14208
   535
by (rule ext, fast)
wenzelm@11838
   536
wenzelm@11838
   537
lemma split_eta_SetCompr2 [simp]: "(%u. EX x y. u = (x, y) & P x y) = split P"
paulson@14208
   538
by (rule ext, fast)
wenzelm@11838
   539
wenzelm@11838
   540
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"
wenzelm@11838
   541
  -- {* Allows simplifications of nested splits in case of independent predicates. *}
paulson@14208
   542
  apply (rule ext, blast)
wenzelm@11838
   543
  done
wenzelm@11838
   544
nipkow@14337
   545
(* Do NOT make this a simp rule as it
nipkow@14337
   546
   a) only helps in special situations
nipkow@14337
   547
   b) can lead to nontermination in the presence of split_def
nipkow@14337
   548
*)
nipkow@14337
   549
lemma split_comp_eq: 
oheimb@14101
   550
"(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"
oheimb@14101
   551
by (rule ext, auto)
oheimb@14101
   552
wenzelm@11838
   553
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
wenzelm@11838
   554
  by blast
wenzelm@11838
   555
wenzelm@11838
   556
(*
wenzelm@11838
   557
the following  would be slightly more general,
wenzelm@11838
   558
but cannot be used as rewrite rule:
wenzelm@11838
   559
### Cannot add premise as rewrite rule because it contains (type) unknowns:
wenzelm@11838
   560
### ?y = .x
wenzelm@11838
   561
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
paulson@14208
   562
by (rtac some_equality 1)
paulson@14208
   563
by ( Simp_tac 1)
paulson@14208
   564
by (split_all_tac 1)
paulson@14208
   565
by (Asm_full_simp_tac 1)
wenzelm@11838
   566
qed "The_split_eq";
wenzelm@11838
   567
*)
wenzelm@11838
   568
wenzelm@11838
   569
lemma injective_fst_snd: "!!x y. [|fst x = fst y; snd x = snd y|] ==> x = y"
wenzelm@11838
   570
  by auto
wenzelm@11838
   571
wenzelm@11838
   572
wenzelm@11838
   573
text {*
wenzelm@11838
   574
  \bigskip @{term prod_fun} --- action of the product functor upon
wenzelm@11838
   575
  functions.
wenzelm@11838
   576
*}
wenzelm@11838
   577
wenzelm@11838
   578
lemma prod_fun [simp]: "prod_fun f g (a, b) = (f a, g b)"
wenzelm@11838
   579
  by (simp add: prod_fun_def)
wenzelm@11838
   580
wenzelm@11838
   581
lemma prod_fun_compose: "prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)"
wenzelm@11838
   582
  apply (rule ext)
paulson@14208
   583
  apply (tactic {* pair_tac "x" 1 *}, simp)
wenzelm@11838
   584
  done
wenzelm@11838
   585
wenzelm@11838
   586
lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)"
wenzelm@11838
   587
  apply (rule ext)
paulson@14208
   588
  apply (tactic {* pair_tac "z" 1 *}, simp)
wenzelm@11838
   589
  done
wenzelm@11838
   590
wenzelm@11838
   591
lemma prod_fun_imageI [intro]: "(a, b) : r ==> (f a, g b) : prod_fun f g ` r"
wenzelm@11838
   592
  apply (rule image_eqI)
paulson@14208
   593
  apply (rule prod_fun [symmetric], assumption)
wenzelm@11838
   594
  done
wenzelm@11838
   595
wenzelm@11838
   596
lemma prod_fun_imageE [elim!]:
wenzelm@11838
   597
  "[| c: (prod_fun f g)`r;  !!x y. [| c=(f(x),g(y));  (x,y):r |] ==> P
wenzelm@11838
   598
    |] ==> P"
wenzelm@11838
   599
proof -
wenzelm@11838
   600
  case rule_context
wenzelm@11838
   601
  assume major: "c: (prod_fun f g)`r"
wenzelm@11838
   602
  show ?thesis
wenzelm@11838
   603
    apply (rule major [THEN imageE])
wenzelm@11838
   604
    apply (rule_tac p = x in PairE)
wenzelm@11838
   605
    apply (rule rule_context)
wenzelm@11838
   606
     prefer 2
wenzelm@11838
   607
     apply blast
wenzelm@11838
   608
    apply (blast intro: prod_fun)
wenzelm@11838
   609
    done
wenzelm@11838
   610
qed
wenzelm@11838
   611
wenzelm@11838
   612
oheimb@14101
   613
constdefs
oheimb@14101
   614
  upd_fst :: "('a => 'c) => 'a * 'b => 'c * 'b"
oheimb@14101
   615
 "upd_fst f == prod_fun f id"
oheimb@14101
   616
oheimb@14101
   617
  upd_snd :: "('b => 'c) => 'a * 'b => 'a * 'c"
oheimb@14101
   618
 "upd_snd f == prod_fun id f"
oheimb@14101
   619
oheimb@14101
   620
lemma upd_fst_conv [simp]: "upd_fst f (x,y) = (f x,y)" 
oheimb@14101
   621
by (simp add: upd_fst_def)
oheimb@14101
   622
oheimb@14101
   623
lemma upd_snd_conv [simp]: "upd_snd f (x,y) = (x,f y)" 
oheimb@14101
   624
by (simp add: upd_snd_def)
oheimb@14101
   625
wenzelm@11838
   626
text {*
wenzelm@11838
   627
  \bigskip Disjoint union of a family of sets -- Sigma.
wenzelm@11838
   628
*}
wenzelm@11838
   629
wenzelm@11838
   630
lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
wenzelm@11838
   631
  by (unfold Sigma_def) blast
wenzelm@11838
   632
paulson@14952
   633
lemma SigmaE [elim!]:
wenzelm@11838
   634
    "[| c: Sigma A B;
wenzelm@11838
   635
        !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
wenzelm@11838
   636
     |] ==> P"
wenzelm@11838
   637
  -- {* The general elimination rule. *}
wenzelm@11838
   638
  by (unfold Sigma_def) blast
wenzelm@11838
   639
wenzelm@11838
   640
text {*
schirmer@15422
   641
  Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
wenzelm@11838
   642
  eigenvariables.
wenzelm@11838
   643
*}
wenzelm@11838
   644
wenzelm@11838
   645
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
paulson@14952
   646
by blast
wenzelm@11838
   647
wenzelm@11838
   648
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
paulson@14952
   649
by blast
wenzelm@11838
   650
wenzelm@11838
   651
lemma SigmaE2:
wenzelm@11838
   652
    "[| (a, b) : Sigma A B;
wenzelm@11838
   653
        [| a:A;  b:B(a) |] ==> P
wenzelm@11838
   654
     |] ==> P"
paulson@14952
   655
  by blast
wenzelm@11838
   656
paulson@14952
   657
lemma Sigma_cong:
schirmer@15422
   658
     "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>
schirmer@15422
   659
      \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
paulson@14952
   660
by auto
wenzelm@11838
   661
wenzelm@11838
   662
lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
wenzelm@11838
   663
  by blast
wenzelm@11838
   664
wenzelm@11838
   665
lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
wenzelm@11838
   666
  by blast
wenzelm@11838
   667
wenzelm@11838
   668
lemma Sigma_empty2 [simp]: "A <*> {} = {}"
wenzelm@11838
   669
  by blast
wenzelm@11838
   670
wenzelm@11838
   671
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
wenzelm@11838
   672
  by auto
wenzelm@11838
   673
wenzelm@11838
   674
lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
wenzelm@11838
   675
  by auto
wenzelm@11838
   676
wenzelm@11838
   677
lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
wenzelm@11838
   678
  by auto
wenzelm@11838
   679
wenzelm@11838
   680
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
wenzelm@11838
   681
  by blast
wenzelm@11838
   682
wenzelm@11838
   683
lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
wenzelm@11838
   684
  by blast
wenzelm@11838
   685
wenzelm@11838
   686
lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
wenzelm@11838
   687
  by (blast elim: equalityE)
wenzelm@11838
   688
wenzelm@11838
   689
lemma SetCompr_Sigma_eq:
wenzelm@11838
   690
    "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
wenzelm@11838
   691
  by blast
wenzelm@11838
   692
wenzelm@11838
   693
text {*
wenzelm@11838
   694
  \bigskip Complex rules for Sigma.
wenzelm@11838
   695
*}
wenzelm@11838
   696
wenzelm@11838
   697
lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"
wenzelm@11838
   698
  by blast
wenzelm@11838
   699
wenzelm@11838
   700
lemma UN_Times_distrib:
wenzelm@11838
   701
  "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"
wenzelm@11838
   702
  -- {* Suggested by Pierre Chartier *}
wenzelm@11838
   703
  by blast
wenzelm@11838
   704
wenzelm@11838
   705
lemma split_paired_Ball_Sigma [simp]:
wenzelm@11838
   706
    "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"
wenzelm@11838
   707
  by blast
wenzelm@11838
   708
wenzelm@11838
   709
lemma split_paired_Bex_Sigma [simp]:
wenzelm@11838
   710
    "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"
wenzelm@11838
   711
  by blast
wenzelm@11838
   712
wenzelm@11838
   713
lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"
wenzelm@11838
   714
  by blast
wenzelm@11838
   715
wenzelm@11838
   716
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
   717
  by blast
wenzelm@11838
   718
wenzelm@11838
   719
lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"
wenzelm@11838
   720
  by blast
wenzelm@11838
   721
wenzelm@11838
   722
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
   723
  by blast
wenzelm@11838
   724
wenzelm@11838
   725
lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"
wenzelm@11838
   726
  by blast
wenzelm@11838
   727
wenzelm@11838
   728
lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"
wenzelm@11838
   729
  by blast
wenzelm@11838
   730
wenzelm@11838
   731
lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"
wenzelm@11838
   732
  by blast
wenzelm@11838
   733
wenzelm@11838
   734
text {*
wenzelm@11838
   735
  Non-dependent versions are needed to avoid the need for higher-order
wenzelm@11838
   736
  matching, especially when the rules are re-oriented.
wenzelm@11838
   737
*}
wenzelm@11838
   738
wenzelm@11838
   739
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"
wenzelm@11838
   740
  by blast
wenzelm@11838
   741
wenzelm@11838
   742
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"
wenzelm@11838
   743
  by blast
wenzelm@11838
   744
wenzelm@11838
   745
lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
wenzelm@11838
   746
  by blast
wenzelm@11838
   747
wenzelm@11838
   748
oheimb@11493
   749
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
wenzelm@11777
   750
  apply (rule_tac x = "(a, b)" in image_eqI)
wenzelm@11777
   751
   apply auto
wenzelm@11777
   752
  done
wenzelm@11777
   753
oheimb@11493
   754
wenzelm@11838
   755
text {*
wenzelm@11838
   756
  Setup of internal @{text split_rule}.
wenzelm@11838
   757
*}
wenzelm@11838
   758
wenzelm@11032
   759
constdefs
wenzelm@11425
   760
  internal_split :: "('a => 'b => 'c) => 'a * 'b => 'c"
wenzelm@11032
   761
  "internal_split == split"
wenzelm@11032
   762
wenzelm@11032
   763
lemma internal_split_conv: "internal_split c (a, b) = c a b"
wenzelm@11032
   764
  by (simp only: internal_split_def split_conv)
wenzelm@11032
   765
wenzelm@11032
   766
hide const internal_split
wenzelm@11032
   767
oheimb@11025
   768
use "Tools/split_rule.ML"
wenzelm@11032
   769
setup SplitRule.setup
nipkow@10213
   770
berghofe@15394
   771
berghofe@15394
   772
subsection {* Code generator setup *}
berghofe@15394
   773
berghofe@15394
   774
types_code
berghofe@15394
   775
  "*"     ("(_ */ _)")
berghofe@15394
   776
berghofe@15394
   777
consts_code
berghofe@15394
   778
  "Pair"    ("(_,/ _)")
berghofe@15394
   779
  "fst"     ("fst")
berghofe@15394
   780
  "snd"     ("snd")
berghofe@15394
   781
berghofe@15394
   782
ML {*
berghofe@15394
   783
fun term_of_id_42 f T g U (x, y) = HOLogic.pair_const T U $ f x $ g y;
berghofe@15394
   784
berghofe@15394
   785
fun gen_id_42 aG bG i = (aG i, bG i);
berghofe@15394
   786
berghofe@15394
   787
local
berghofe@15394
   788
berghofe@15394
   789
fun strip_abs 0 t = ([], t)
berghofe@15394
   790
  | strip_abs i (Abs (s, T, t)) =
berghofe@15394
   791
    let
berghofe@15394
   792
      val s' = Codegen.new_name t s;
berghofe@15394
   793
      val v = Free (s', T)
berghofe@15394
   794
    in apfst (cons v) (strip_abs (i-1) (subst_bound (v, t))) end
berghofe@15394
   795
  | strip_abs i (u as Const ("split", _) $ t) = (case strip_abs (i+1) t of
berghofe@15394
   796
        (v :: v' :: vs, u) => (HOLogic.mk_prod (v, v') :: vs, u)
berghofe@15394
   797
      | _ => ([], u))
berghofe@15394
   798
  | strip_abs i t = ([], t);
berghofe@15394
   799
berghofe@15394
   800
fun let_codegen thy gr dep brack (t as Const ("Let", _) $ _ $ _) =
berghofe@15394
   801
    let
berghofe@15394
   802
      fun dest_let (l as Const ("Let", _) $ t $ u) =
berghofe@15394
   803
          (case strip_abs 1 u of
berghofe@15394
   804
             ([p], u') => apfst (cons (p, t)) (dest_let u')
berghofe@15394
   805
           | _ => ([], l))
berghofe@15394
   806
        | dest_let t = ([], t);
berghofe@15394
   807
      fun mk_code (gr, (l, r)) =
berghofe@15394
   808
        let
berghofe@15394
   809
          val (gr1, pl) = Codegen.invoke_codegen thy dep false (gr, l);
berghofe@15394
   810
          val (gr2, pr) = Codegen.invoke_codegen thy dep false (gr1, r);
berghofe@15394
   811
        in (gr2, (pl, pr)) end
berghofe@15394
   812
    in case dest_let t of
skalberg@15531
   813
        ([], _) => NONE
berghofe@15394
   814
      | (ps, u) =>
berghofe@15394
   815
          let
berghofe@15394
   816
            val (gr1, qs) = foldl_map mk_code (gr, ps);
berghofe@15394
   817
            val (gr2, pu) = Codegen.invoke_codegen thy dep false (gr1, u)
berghofe@15394
   818
          in
skalberg@15570
   819
            SOME (gr2, Pretty.blk (0, [Pretty.str "let ", Pretty.blk (0, List.concat
berghofe@15394
   820
                (separate [Pretty.str ";", Pretty.brk 1] (map (fn (pl, pr) =>
berghofe@15394
   821
                  [Pretty.block [Pretty.str "val ", pl, Pretty.str " =",
berghofe@15394
   822
                     Pretty.brk 1, pr]]) qs))),
berghofe@15394
   823
              Pretty.brk 1, Pretty.str "in ", pu,
berghofe@15394
   824
              Pretty.brk 1, Pretty.str "end"]))
berghofe@15394
   825
          end
berghofe@15394
   826
    end
skalberg@15531
   827
  | let_codegen thy gr dep brack t = NONE;
berghofe@15394
   828
berghofe@15394
   829
fun split_codegen thy gr dep brack (t as Const ("split", _) $ _) =
berghofe@15394
   830
    (case strip_abs 1 t of
berghofe@15394
   831
       ([p], u) =>
berghofe@15394
   832
         let
berghofe@15394
   833
           val (gr1, q) = Codegen.invoke_codegen thy dep false (gr, p);
berghofe@15394
   834
           val (gr2, pu) = Codegen.invoke_codegen thy dep false (gr1, u)
berghofe@15394
   835
         in
skalberg@15531
   836
           SOME (gr2, Pretty.block [Pretty.str "(fn ", q, Pretty.str " =>",
berghofe@15394
   837
             Pretty.brk 1, pu, Pretty.str ")"])
berghofe@15394
   838
         end
skalberg@15531
   839
     | _ => NONE)
skalberg@15531
   840
  | split_codegen thy gr dep brack t = NONE;
berghofe@15394
   841
berghofe@15394
   842
in
berghofe@15394
   843
berghofe@15394
   844
val prod_codegen_setup =
berghofe@15394
   845
  [Codegen.add_codegen "let_codegen" let_codegen,
berghofe@15394
   846
   Codegen.add_codegen "split_codegen" split_codegen];
berghofe@15394
   847
berghofe@15394
   848
end;
berghofe@15394
   849
*}
berghofe@15394
   850
berghofe@15394
   851
setup prod_codegen_setup
berghofe@15394
   852
paulson@15404
   853
ML
paulson@15404
   854
{*
paulson@15404
   855
val Collect_split = thm "Collect_split";
paulson@15404
   856
val Compl_Times_UNIV1 = thm "Compl_Times_UNIV1";
paulson@15404
   857
val Compl_Times_UNIV2 = thm "Compl_Times_UNIV2";
paulson@15404
   858
val PairE = thm "PairE";
paulson@15404
   859
val PairE_lemma = thm "PairE_lemma";
paulson@15404
   860
val Pair_Rep_inject = thm "Pair_Rep_inject";
paulson@15404
   861
val Pair_def = thm "Pair_def";
paulson@15404
   862
val Pair_eq = thm "Pair_eq";
paulson@15404
   863
val Pair_fst_snd_eq = thm "Pair_fst_snd_eq";
paulson@15404
   864
val Pair_inject = thm "Pair_inject";
paulson@15404
   865
val ProdI = thm "ProdI";
paulson@15404
   866
val SetCompr_Sigma_eq = thm "SetCompr_Sigma_eq";
paulson@15404
   867
val SigmaD1 = thm "SigmaD1";
paulson@15404
   868
val SigmaD2 = thm "SigmaD2";
paulson@15404
   869
val SigmaE = thm "SigmaE";
paulson@15404
   870
val SigmaE2 = thm "SigmaE2";
paulson@15404
   871
val SigmaI = thm "SigmaI";
paulson@15404
   872
val Sigma_Diff_distrib1 = thm "Sigma_Diff_distrib1";
paulson@15404
   873
val Sigma_Diff_distrib2 = thm "Sigma_Diff_distrib2";
paulson@15404
   874
val Sigma_Int_distrib1 = thm "Sigma_Int_distrib1";
paulson@15404
   875
val Sigma_Int_distrib2 = thm "Sigma_Int_distrib2";
paulson@15404
   876
val Sigma_Un_distrib1 = thm "Sigma_Un_distrib1";
paulson@15404
   877
val Sigma_Un_distrib2 = thm "Sigma_Un_distrib2";
paulson@15404
   878
val Sigma_Union = thm "Sigma_Union";
paulson@15404
   879
val Sigma_def = thm "Sigma_def";
paulson@15404
   880
val Sigma_empty1 = thm "Sigma_empty1";
paulson@15404
   881
val Sigma_empty2 = thm "Sigma_empty2";
paulson@15404
   882
val Sigma_mono = thm "Sigma_mono";
paulson@15404
   883
val The_split = thm "The_split";
paulson@15404
   884
val The_split_eq = thm "The_split_eq";
paulson@15404
   885
val The_split_eq = thm "The_split_eq";
paulson@15404
   886
val Times_Diff_distrib1 = thm "Times_Diff_distrib1";
paulson@15404
   887
val Times_Int_distrib1 = thm "Times_Int_distrib1";
paulson@15404
   888
val Times_Un_distrib1 = thm "Times_Un_distrib1";
paulson@15404
   889
val Times_eq_cancel2 = thm "Times_eq_cancel2";
paulson@15404
   890
val Times_subset_cancel2 = thm "Times_subset_cancel2";
paulson@15404
   891
val UNIV_Times_UNIV = thm "UNIV_Times_UNIV";
paulson@15404
   892
val UN_Times_distrib = thm "UN_Times_distrib";
paulson@15404
   893
val Unity_def = thm "Unity_def";
paulson@15404
   894
val cond_split_eta = thm "cond_split_eta";
paulson@15404
   895
val fst_conv = thm "fst_conv";
paulson@15404
   896
val fst_def = thm "fst_def";
paulson@15404
   897
val fst_eqD = thm "fst_eqD";
paulson@15404
   898
val inj_on_Abs_Prod = thm "inj_on_Abs_Prod";
paulson@15404
   899
val injective_fst_snd = thm "injective_fst_snd";
paulson@15404
   900
val mem_Sigma_iff = thm "mem_Sigma_iff";
paulson@15404
   901
val mem_splitE = thm "mem_splitE";
paulson@15404
   902
val mem_splitI = thm "mem_splitI";
paulson@15404
   903
val mem_splitI2 = thm "mem_splitI2";
paulson@15404
   904
val prod_eqI = thm "prod_eqI";
paulson@15404
   905
val prod_fun = thm "prod_fun";
paulson@15404
   906
val prod_fun_compose = thm "prod_fun_compose";
paulson@15404
   907
val prod_fun_def = thm "prod_fun_def";
paulson@15404
   908
val prod_fun_ident = thm "prod_fun_ident";
paulson@15404
   909
val prod_fun_imageE = thm "prod_fun_imageE";
paulson@15404
   910
val prod_fun_imageI = thm "prod_fun_imageI";
paulson@15404
   911
val prod_induct = thm "prod_induct";
paulson@15404
   912
val snd_conv = thm "snd_conv";
paulson@15404
   913
val snd_def = thm "snd_def";
paulson@15404
   914
val snd_eqD = thm "snd_eqD";
paulson@15404
   915
val split = thm "split";
paulson@15404
   916
val splitD = thm "splitD";
paulson@15404
   917
val splitD' = thm "splitD'";
paulson@15404
   918
val splitE = thm "splitE";
paulson@15404
   919
val splitE' = thm "splitE'";
paulson@15404
   920
val splitE2 = thm "splitE2";
paulson@15404
   921
val splitI = thm "splitI";
paulson@15404
   922
val splitI2 = thm "splitI2";
paulson@15404
   923
val splitI2' = thm "splitI2'";
paulson@15404
   924
val split_Pair_apply = thm "split_Pair_apply";
paulson@15404
   925
val split_beta = thm "split_beta";
paulson@15404
   926
val split_conv = thm "split_conv";
paulson@15404
   927
val split_def = thm "split_def";
paulson@15404
   928
val split_eta = thm "split_eta";
paulson@15404
   929
val split_eta_SetCompr = thm "split_eta_SetCompr";
paulson@15404
   930
val split_eta_SetCompr2 = thm "split_eta_SetCompr2";
paulson@15404
   931
val split_paired_All = thm "split_paired_All";
paulson@15404
   932
val split_paired_Ball_Sigma = thm "split_paired_Ball_Sigma";
paulson@15404
   933
val split_paired_Bex_Sigma = thm "split_paired_Bex_Sigma";
paulson@15404
   934
val split_paired_Ex = thm "split_paired_Ex";
paulson@15404
   935
val split_paired_The = thm "split_paired_The";
paulson@15404
   936
val split_paired_all = thm "split_paired_all";
paulson@15404
   937
val split_part = thm "split_part";
paulson@15404
   938
val split_split = thm "split_split";
paulson@15404
   939
val split_split_asm = thm "split_split_asm";
paulson@15404
   940
val split_tupled_all = thms "split_tupled_all";
paulson@15404
   941
val split_weak_cong = thm "split_weak_cong";
paulson@15404
   942
val surj_pair = thm "surj_pair";
paulson@15404
   943
val surjective_pairing = thm "surjective_pairing";
paulson@15404
   944
val unit_abs_eta_conv = thm "unit_abs_eta_conv";
paulson@15404
   945
val unit_all_eq1 = thm "unit_all_eq1";
paulson@15404
   946
val unit_all_eq2 = thm "unit_all_eq2";
paulson@15404
   947
val unit_eq = thm "unit_eq";
paulson@15404
   948
val unit_induct = thm "unit_induct";
paulson@15404
   949
*}
paulson@15404
   950
nipkow@10213
   951
end