src/HOL/Product_Type.thy
author haftmann
Tue Sep 25 12:16:08 2007 +0200 (2007-09-25)
changeset 24699 c6674504103f
parent 24286 7619080e49f0
child 24844 98c006a30218
permissions -rw-r--r--
datatype interpretators for size and datatype_realizer
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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 Inductive
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uses
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  ("Tools/split_rule.ML")
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  ("Tools/inductive_set_package.ML")
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  ("Tools/inductive_realizer.ML")
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  ("Tools/datatype_realizer.ML")
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begin
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subsection {* @{typ bool} is a datatype *}
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rep_datatype bool
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  distinct True_not_False False_not_True
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  induction bool_induct
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declare case_split [cases type: bool]
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  -- "prefer plain propositional version"
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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" ..
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qed
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definition
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  Unity :: unit    ("'(')")
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where
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  "() = Abs_unit True"
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lemma unit_eq [noatp]: "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} "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_induct [noatp,induct type: unit]: "P () ==> P x"
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  by simp
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rep_datatype unit
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  induction unit_induct
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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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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,noatp]: "(%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 {* Definitions *}
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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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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 = Pair a b"
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  snd_def:      "snd p == THE b. EX a. p = Pair 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 (Pair x y))"
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  prod_fun_def: "prod_fun f g == split (%x y. Pair (f x) (g y))"
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  Sigma_def [code func]:    "Sigma A B == UN x:A. UN y:B x. {Pair x y}"
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abbreviation
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  Times :: "['a set, 'b set] => ('a * 'b) set"
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    (infixr "<*>" 80) where
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  "A <*> B == Sigma A (%_. B)"
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notation (xsymbols)
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  Times  (infixr "\<times>" 80)
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notation (HTML output)
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  Times  (infixr "\<times>" 80)
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subsubsection {* Concrete syntax *}
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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 _:_./ _)" [0, 0, 10] 10)
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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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(* 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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subsubsection {* Lemmas and proof tool setup *}
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lemma ProdI: "Pair_Rep a b : Prod"
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  unfolding Prod_def by 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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  assumes "(a, b) = (a', b')"
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    and "a = a' ==> b = b' ==> R"
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  shows R
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  apply (insert prems [unfolded Pair_def])
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  apply (rule inj_on_Abs_Prod [THEN inj_onD, THEN Pair_Rep_inject, THEN conjE])
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  apply (assumption | rule ProdI)+
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  done
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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, code]: "fst (a, b) = a"
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  unfolding fst_def by blast
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lemma snd_conv [simp, code]: "snd (a, b) = b"
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  unfolding snd_def by 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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  using PairE_lemma [of p] by 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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lemmas pair_collapse = surjective_pairing [symmetric]
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declare pair_collapse [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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  then show "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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  from `PROP P (fst x, snd x)` show "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);
wenzelm@11838
   329
    val unsafe_split_all_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838
   330
      if exists_paired_all t then full_simp_tac ss i else no_tac);
wenzelm@11838
   331
    fun split_all th =
wenzelm@11838
   332
   if exists_paired_all (#prop (Thm.rep_thm th)) then full_simplify ss th else th;
wenzelm@11838
   333
  end;
wenzelm@11838
   334
wenzelm@17875
   335
change_claset (fn cs => cs addSbefore ("split_all_tac", split_all_tac));
wenzelm@16121
   336
*}
wenzelm@11838
   337
wenzelm@11838
   338
lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))"
wenzelm@11838
   339
  -- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *}
wenzelm@11838
   340
  by fast
wenzelm@11838
   341
skalberg@14189
   342
lemma curry_split [simp]: "curry (split f) = f"
skalberg@14189
   343
  by (simp add: curry_def split_def)
skalberg@14189
   344
skalberg@14189
   345
lemma split_curry [simp]: "split (curry f) = f"
skalberg@14189
   346
  by (simp add: curry_def split_def)
skalberg@14189
   347
skalberg@14189
   348
lemma curryI [intro!]: "f (a,b) ==> curry f a b"
skalberg@14189
   349
  by (simp add: curry_def)
skalberg@14189
   350
skalberg@14190
   351
lemma curryD [dest!]: "curry f a b ==> f (a,b)"
skalberg@14189
   352
  by (simp add: curry_def)
skalberg@14189
   353
skalberg@14190
   354
lemma curryE: "[| curry f a b ; f (a,b) ==> Q |] ==> Q"
skalberg@14189
   355
  by (simp add: curry_def)
skalberg@14189
   356
haftmann@24162
   357
lemma curry_conv [simp, code func]: "curry f a b = f (a,b)"
skalberg@14189
   358
  by (simp add: curry_def)
skalberg@14189
   359
wenzelm@11838
   360
lemma prod_induct [induct type: *]: "!!x. (!!a b. P (a, b)) ==> P x"
wenzelm@11838
   361
  by fast
wenzelm@11838
   362
haftmann@24699
   363
rep_datatype prod
haftmann@24699
   364
  inject Pair_eq
haftmann@24699
   365
  induction prod_induct
haftmann@24699
   366
wenzelm@11838
   367
lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))"
wenzelm@11838
   368
  by fast
wenzelm@11838
   369
haftmann@24162
   370
lemma split_conv [simp, code func]: "split c (a, b) = c a b"
wenzelm@11838
   371
  by (simp add: split_def)
wenzelm@11838
   372
wenzelm@11838
   373
lemmas split = split_conv  -- {* for backwards compatibility *}
wenzelm@11838
   374
wenzelm@11838
   375
lemmas splitI = split_conv [THEN iffD2, standard]
wenzelm@11838
   376
lemmas splitD = split_conv [THEN iffD1, standard]
wenzelm@11838
   377
wenzelm@11838
   378
lemma split_Pair_apply: "split (%x y. f (x, y)) = f"
wenzelm@11838
   379
  -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *}
wenzelm@11838
   380
  apply (rule ext)
paulson@14208
   381
  apply (tactic {* pair_tac "x" 1 *}, simp)
wenzelm@11838
   382
  done
wenzelm@11838
   383
wenzelm@11838
   384
lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))"
wenzelm@11838
   385
  -- {* Can't be added to simpset: loops! *}
wenzelm@11838
   386
  by (simp add: split_Pair_apply)
wenzelm@11838
   387
wenzelm@11838
   388
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
wenzelm@11838
   389
  by (simp add: split_def)
wenzelm@11838
   390
wenzelm@11838
   391
lemma Pair_fst_snd_eq: "!!s t. (s = t) = (fst s = fst t & snd s = snd t)"
paulson@14208
   392
by (simp only: split_tupled_all, simp)
wenzelm@11838
   393
wenzelm@11838
   394
lemma prod_eqI [intro?]: "fst p = fst q ==> snd p = snd q ==> p = q"
wenzelm@11838
   395
  by (simp add: Pair_fst_snd_eq)
wenzelm@11838
   396
wenzelm@11838
   397
lemma split_weak_cong: "p = q ==> split c p = split c q"
wenzelm@11838
   398
  -- {* Prevents simplification of @{term c}: much faster *}
wenzelm@11838
   399
  by (erule arg_cong)
wenzelm@11838
   400
wenzelm@11838
   401
lemma split_eta: "(%(x, y). f (x, y)) = f"
wenzelm@11838
   402
  apply (rule ext)
wenzelm@11838
   403
  apply (simp only: split_tupled_all)
wenzelm@11838
   404
  apply (rule split_conv)
wenzelm@11838
   405
  done
wenzelm@11838
   406
wenzelm@11838
   407
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
wenzelm@11838
   408
  by (simp add: split_eta)
wenzelm@11838
   409
wenzelm@11838
   410
text {*
wenzelm@11838
   411
  Simplification procedure for @{thm [source] cond_split_eta}.  Using
wenzelm@11838
   412
  @{thm [source] split_eta} as a rewrite rule is not general enough,
wenzelm@11838
   413
  and using @{thm [source] cond_split_eta} directly would render some
wenzelm@11838
   414
  existing proofs very inefficient; similarly for @{text
wenzelm@11838
   415
  split_beta}. *}
wenzelm@11838
   416
wenzelm@11838
   417
ML_setup {*
wenzelm@11838
   418
wenzelm@11838
   419
local
wenzelm@18328
   420
  val cond_split_eta_ss = HOL_basic_ss addsimps [thm "cond_split_eta"]
wenzelm@11838
   421
  fun  Pair_pat k 0 (Bound m) = (m = k)
wenzelm@11838
   422
  |    Pair_pat k i (Const ("Pair",  _) $ Bound m $ t) = i > 0 andalso
wenzelm@11838
   423
                        m = k+i andalso Pair_pat k (i-1) t
wenzelm@11838
   424
  |    Pair_pat _ _ _ = false;
wenzelm@11838
   425
  fun no_args k i (Abs (_, _, t)) = no_args (k+1) i t
wenzelm@11838
   426
  |   no_args k i (t $ u) = no_args k i t andalso no_args k i u
wenzelm@11838
   427
  |   no_args k i (Bound m) = m < k orelse m > k+i
wenzelm@11838
   428
  |   no_args _ _ _ = true;
skalberg@15531
   429
  fun split_pat tp i (Abs  (_,_,t)) = if tp 0 i t then SOME (i,t) else NONE
wenzelm@11838
   430
  |   split_pat tp i (Const ("split", _) $ Abs (_, _, t)) = split_pat tp (i+1) t
skalberg@15531
   431
  |   split_pat tp i _ = NONE;
wenzelm@20044
   432
  fun metaeq ss lhs rhs = mk_meta_eq (Goal.prove (Simplifier.the_context ss) [] []
wenzelm@13480
   433
        (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs,rhs)))
wenzelm@18328
   434
        (K (simp_tac (Simplifier.inherit_context ss cond_split_eta_ss) 1)));
wenzelm@11838
   435
wenzelm@11838
   436
  fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k+1) i t
wenzelm@11838
   437
  |   beta_term_pat k i (t $ u) = Pair_pat k i (t $ u) orelse
wenzelm@11838
   438
                        (beta_term_pat k i t andalso beta_term_pat k i u)
wenzelm@11838
   439
  |   beta_term_pat k i t = no_args k i t;
wenzelm@11838
   440
  fun  eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
wenzelm@11838
   441
  |    eta_term_pat _ _ _ = false;
wenzelm@11838
   442
  fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
wenzelm@11838
   443
  |   subst arg k i (t $ u) = if Pair_pat k i (t $ u) then incr_boundvars k arg
wenzelm@11838
   444
                              else (subst arg k i t $ subst arg k i u)
wenzelm@11838
   445
  |   subst arg k i t = t;
wenzelm@20044
   446
  fun beta_proc ss (s as Const ("split", _) $ Abs (_, _, t) $ arg) =
wenzelm@11838
   447
        (case split_pat beta_term_pat 1 t of
wenzelm@20044
   448
        SOME (i,f) => SOME (metaeq ss s (subst arg 0 i f))
skalberg@15531
   449
        | NONE => NONE)
wenzelm@20044
   450
  |   beta_proc _ _ = NONE;
wenzelm@20044
   451
  fun eta_proc ss (s as Const ("split", _) $ Abs (_, _, t)) =
wenzelm@11838
   452
        (case split_pat eta_term_pat 1 t of
wenzelm@20044
   453
          SOME (_,ft) => SOME (metaeq ss s (let val (f $ arg) = ft in f end))
skalberg@15531
   454
        | NONE => NONE)
wenzelm@20044
   455
  |   eta_proc _ _ = NONE;
wenzelm@11838
   456
in
wenzelm@22577
   457
  val split_beta_proc = Simplifier.simproc @{theory} "split_beta" ["split f z"] (K beta_proc);
wenzelm@22577
   458
  val split_eta_proc = Simplifier.simproc @{theory} "split_eta" ["split f"] (K eta_proc);
wenzelm@11838
   459
end;
wenzelm@11838
   460
wenzelm@11838
   461
Addsimprocs [split_beta_proc, split_eta_proc];
wenzelm@11838
   462
*}
wenzelm@11838
   463
wenzelm@11838
   464
lemma split_beta: "(%(x, y). P x y) z = P (fst z) (snd z)"
wenzelm@11838
   465
  by (subst surjective_pairing, rule split_conv)
wenzelm@11838
   466
paulson@24286
   467
lemma split_split [noatp]: "R(split c p) = (ALL x y. p = (x, y) --> R(c x y))"
wenzelm@11838
   468
  -- {* For use with @{text split} and the Simplifier. *}
paulson@15481
   469
  by (insert surj_pair [of p], clarify, simp)
wenzelm@11838
   470
wenzelm@11838
   471
text {*
wenzelm@11838
   472
  @{thm [source] split_split} could be declared as @{text "[split]"}
wenzelm@11838
   473
  done after the Splitter has been speeded up significantly;
wenzelm@11838
   474
  precompute the constants involved and don't do anything unless the
wenzelm@11838
   475
  current goal contains one of those constants.
wenzelm@11838
   476
*}
wenzelm@11838
   477
paulson@24286
   478
lemma split_split_asm [noatp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))"
paulson@14208
   479
by (subst split_split, simp)
wenzelm@11838
   480
wenzelm@11838
   481
wenzelm@11838
   482
text {*
wenzelm@11838
   483
  \medskip @{term split} used as a logical connective or set former.
wenzelm@11838
   484
wenzelm@11838
   485
  \medskip These rules are for use with @{text blast}; could instead
wenzelm@11838
   486
  call @{text simp} using @{thm [source] split} as rewrite. *}
wenzelm@11838
   487
wenzelm@11838
   488
lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"
wenzelm@11838
   489
  apply (simp only: split_tupled_all)
wenzelm@11838
   490
  apply (simp (no_asm_simp))
wenzelm@11838
   491
  done
wenzelm@11838
   492
wenzelm@11838
   493
lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"
wenzelm@11838
   494
  apply (simp only: split_tupled_all)
wenzelm@11838
   495
  apply (simp (no_asm_simp))
wenzelm@11838
   496
  done
wenzelm@11838
   497
wenzelm@11838
   498
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
wenzelm@11838
   499
  by (induct p) (auto simp add: split_def)
wenzelm@11838
   500
wenzelm@11838
   501
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
wenzelm@11838
   502
  by (induct p) (auto simp add: split_def)
wenzelm@11838
   503
wenzelm@11838
   504
lemma splitE2:
wenzelm@11838
   505
  "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
wenzelm@11838
   506
proof -
wenzelm@11838
   507
  assume q: "Q (split P z)"
wenzelm@11838
   508
  assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
wenzelm@11838
   509
  show R
wenzelm@11838
   510
    apply (rule r surjective_pairing)+
wenzelm@11838
   511
    apply (rule split_beta [THEN subst], rule q)
wenzelm@11838
   512
    done
wenzelm@11838
   513
qed
wenzelm@11838
   514
wenzelm@11838
   515
lemma splitD': "split R (a,b) c ==> R a b c"
wenzelm@11838
   516
  by simp
wenzelm@11838
   517
wenzelm@11838
   518
lemma mem_splitI: "z: c a b ==> z: split c (a, b)"
wenzelm@11838
   519
  by simp
wenzelm@11838
   520
wenzelm@11838
   521
lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"
paulson@14208
   522
by (simp only: split_tupled_all, simp)
wenzelm@11838
   523
wenzelm@18372
   524
lemma mem_splitE:
wenzelm@18372
   525
  assumes major: "z: split c p"
wenzelm@18372
   526
    and cases: "!!x y. [| p = (x,y); z: c x y |] ==> Q"
wenzelm@18372
   527
  shows Q
wenzelm@18372
   528
  by (rule major [unfolded split_def] cases surjective_pairing)+
wenzelm@11838
   529
wenzelm@11838
   530
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
wenzelm@11838
   531
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]
wenzelm@11838
   532
wenzelm@16121
   533
ML_setup {*
wenzelm@11838
   534
local (* filtering with exists_p_split is an essential optimization *)
wenzelm@16121
   535
  fun exists_p_split (Const ("split",_) $ _ $ (Const ("Pair",_)$_$_)) = true
wenzelm@11838
   536
    | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
wenzelm@11838
   537
    | exists_p_split (Abs (_, _, t)) = exists_p_split t
wenzelm@11838
   538
    | exists_p_split _ = false;
wenzelm@16121
   539
  val ss = HOL_basic_ss addsimps [thm "split_conv"];
wenzelm@11838
   540
in
wenzelm@11838
   541
val split_conv_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838
   542
    if exists_p_split t then safe_full_simp_tac ss i else no_tac);
wenzelm@11838
   543
end;
wenzelm@11838
   544
(* This prevents applications of splitE for already splitted arguments leading
wenzelm@11838
   545
   to quite time-consuming computations (in particular for nested tuples) *)
wenzelm@17875
   546
change_claset (fn cs => cs addSbefore ("split_conv_tac", split_conv_tac));
wenzelm@16121
   547
*}
wenzelm@11838
   548
paulson@24286
   549
lemma split_eta_SetCompr [simp,noatp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
wenzelm@18372
   550
  by (rule ext) fast
wenzelm@11838
   551
paulson@24286
   552
lemma split_eta_SetCompr2 [simp,noatp]: "(%u. EX x y. u = (x, y) & P x y) = split P"
wenzelm@18372
   553
  by (rule ext) fast
wenzelm@11838
   554
wenzelm@11838
   555
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"
wenzelm@11838
   556
  -- {* Allows simplifications of nested splits in case of independent predicates. *}
wenzelm@18372
   557
  by (rule ext) blast
wenzelm@11838
   558
nipkow@14337
   559
(* Do NOT make this a simp rule as it
nipkow@14337
   560
   a) only helps in special situations
nipkow@14337
   561
   b) can lead to nontermination in the presence of split_def
nipkow@14337
   562
*)
nipkow@14337
   563
lemma split_comp_eq: 
paulson@20415
   564
  fixes f :: "'a => 'b => 'c" and g :: "'d => 'a"
paulson@20415
   565
  shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"
wenzelm@18372
   566
  by (rule ext) auto
oheimb@14101
   567
wenzelm@11838
   568
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
wenzelm@11838
   569
  by blast
wenzelm@11838
   570
wenzelm@11838
   571
(*
wenzelm@11838
   572
the following  would be slightly more general,
wenzelm@11838
   573
but cannot be used as rewrite rule:
wenzelm@11838
   574
### Cannot add premise as rewrite rule because it contains (type) unknowns:
wenzelm@11838
   575
### ?y = .x
wenzelm@11838
   576
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
paulson@14208
   577
by (rtac some_equality 1)
paulson@14208
   578
by ( Simp_tac 1)
paulson@14208
   579
by (split_all_tac 1)
paulson@14208
   580
by (Asm_full_simp_tac 1)
wenzelm@11838
   581
qed "The_split_eq";
wenzelm@11838
   582
*)
wenzelm@11838
   583
wenzelm@11838
   584
lemma injective_fst_snd: "!!x y. [|fst x = fst y; snd x = snd y|] ==> x = y"
wenzelm@11838
   585
  by auto
wenzelm@11838
   586
wenzelm@11838
   587
wenzelm@11838
   588
text {*
wenzelm@11838
   589
  \bigskip @{term prod_fun} --- action of the product functor upon
wenzelm@11838
   590
  functions.
wenzelm@11838
   591
*}
wenzelm@11838
   592
haftmann@24162
   593
lemma prod_fun [simp, code func]: "prod_fun f g (a, b) = (f a, g b)"
wenzelm@11838
   594
  by (simp add: prod_fun_def)
wenzelm@11838
   595
wenzelm@11838
   596
lemma prod_fun_compose: "prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)"
wenzelm@11838
   597
  apply (rule ext)
paulson@14208
   598
  apply (tactic {* pair_tac "x" 1 *}, simp)
wenzelm@11838
   599
  done
wenzelm@11838
   600
wenzelm@11838
   601
lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)"
wenzelm@11838
   602
  apply (rule ext)
paulson@14208
   603
  apply (tactic {* pair_tac "z" 1 *}, simp)
wenzelm@11838
   604
  done
wenzelm@11838
   605
wenzelm@11838
   606
lemma prod_fun_imageI [intro]: "(a, b) : r ==> (f a, g b) : prod_fun f g ` r"
wenzelm@11838
   607
  apply (rule image_eqI)
paulson@14208
   608
  apply (rule prod_fun [symmetric], assumption)
wenzelm@11838
   609
  done
wenzelm@11838
   610
wenzelm@11838
   611
lemma prod_fun_imageE [elim!]:
wenzelm@18372
   612
  assumes major: "c: (prod_fun f g)`r"
wenzelm@18372
   613
    and cases: "!!x y. [| c=(f(x),g(y));  (x,y):r |] ==> P"
wenzelm@18372
   614
  shows P
wenzelm@18372
   615
  apply (rule major [THEN imageE])
wenzelm@18372
   616
  apply (rule_tac p = x in PairE)
wenzelm@18372
   617
  apply (rule cases)
wenzelm@18372
   618
   apply (blast intro: prod_fun)
wenzelm@18372
   619
  apply blast
wenzelm@18372
   620
  done
wenzelm@11838
   621
wenzelm@11838
   622
haftmann@22744
   623
definition
haftmann@22744
   624
  upd_fst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b"
haftmann@22744
   625
where
haftmann@22886
   626
  [code func del]: "upd_fst f = prod_fun f id"
oheimb@14101
   627
haftmann@22744
   628
definition
haftmann@22744
   629
  upd_snd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c"
haftmann@22744
   630
where
haftmann@22886
   631
  [code func del]: "upd_snd f = prod_fun id f"
oheimb@14101
   632
haftmann@22886
   633
lemma upd_fst_conv [simp, code]:
haftmann@22744
   634
  "upd_fst f (x, y) = (f x, y)" 
wenzelm@18372
   635
  by (simp add: upd_fst_def)
oheimb@14101
   636
haftmann@22886
   637
lemma upd_snd_conv [simp, code]:
haftmann@22744
   638
  "upd_snd f (x, y) = (x, f y)" 
wenzelm@18372
   639
  by (simp add: upd_snd_def)
oheimb@14101
   640
wenzelm@11838
   641
text {*
wenzelm@11838
   642
  \bigskip Disjoint union of a family of sets -- Sigma.
wenzelm@11838
   643
*}
wenzelm@11838
   644
wenzelm@11838
   645
lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
wenzelm@11838
   646
  by (unfold Sigma_def) blast
wenzelm@11838
   647
paulson@14952
   648
lemma SigmaE [elim!]:
wenzelm@11838
   649
    "[| c: Sigma A B;
wenzelm@11838
   650
        !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
wenzelm@11838
   651
     |] ==> P"
wenzelm@11838
   652
  -- {* The general elimination rule. *}
wenzelm@11838
   653
  by (unfold Sigma_def) blast
wenzelm@11838
   654
wenzelm@11838
   655
text {*
schirmer@15422
   656
  Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
wenzelm@11838
   657
  eigenvariables.
wenzelm@11838
   658
*}
wenzelm@11838
   659
wenzelm@11838
   660
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
wenzelm@18372
   661
  by blast
wenzelm@11838
   662
wenzelm@11838
   663
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
wenzelm@18372
   664
  by blast
wenzelm@11838
   665
wenzelm@11838
   666
lemma SigmaE2:
wenzelm@11838
   667
    "[| (a, b) : Sigma A B;
wenzelm@11838
   668
        [| a:A;  b:B(a) |] ==> P
wenzelm@11838
   669
     |] ==> P"
paulson@14952
   670
  by blast
wenzelm@11838
   671
paulson@14952
   672
lemma Sigma_cong:
schirmer@15422
   673
     "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>
schirmer@15422
   674
      \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
wenzelm@18372
   675
  by auto
wenzelm@11838
   676
wenzelm@11838
   677
lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
wenzelm@11838
   678
  by blast
wenzelm@11838
   679
wenzelm@11838
   680
lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
wenzelm@11838
   681
  by blast
wenzelm@11838
   682
wenzelm@11838
   683
lemma Sigma_empty2 [simp]: "A <*> {} = {}"
wenzelm@11838
   684
  by blast
wenzelm@11838
   685
wenzelm@11838
   686
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
wenzelm@11838
   687
  by auto
wenzelm@11838
   688
wenzelm@11838
   689
lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
wenzelm@11838
   690
  by auto
wenzelm@11838
   691
wenzelm@11838
   692
lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
wenzelm@11838
   693
  by auto
wenzelm@11838
   694
wenzelm@11838
   695
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
wenzelm@11838
   696
  by blast
wenzelm@11838
   697
wenzelm@11838
   698
lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
wenzelm@11838
   699
  by blast
wenzelm@11838
   700
wenzelm@11838
   701
lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
wenzelm@11838
   702
  by (blast elim: equalityE)
wenzelm@11838
   703
wenzelm@11838
   704
lemma SetCompr_Sigma_eq:
wenzelm@11838
   705
    "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
wenzelm@11838
   706
  by blast
wenzelm@11838
   707
wenzelm@11838
   708
text {*
wenzelm@11838
   709
  \bigskip Complex rules for Sigma.
wenzelm@11838
   710
*}
wenzelm@11838
   711
wenzelm@11838
   712
lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"
wenzelm@11838
   713
  by blast
wenzelm@11838
   714
wenzelm@11838
   715
lemma UN_Times_distrib:
wenzelm@11838
   716
  "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"
wenzelm@11838
   717
  -- {* Suggested by Pierre Chartier *}
wenzelm@11838
   718
  by blast
wenzelm@11838
   719
paulson@24286
   720
lemma split_paired_Ball_Sigma [simp,noatp]:
wenzelm@11838
   721
    "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"
wenzelm@11838
   722
  by blast
wenzelm@11838
   723
paulson@24286
   724
lemma split_paired_Bex_Sigma [simp,noatp]:
wenzelm@11838
   725
    "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"
wenzelm@11838
   726
  by blast
wenzelm@11838
   727
wenzelm@11838
   728
lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"
wenzelm@11838
   729
  by blast
wenzelm@11838
   730
wenzelm@11838
   731
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
   732
  by blast
wenzelm@11838
   733
wenzelm@11838
   734
lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"
wenzelm@11838
   735
  by blast
wenzelm@11838
   736
wenzelm@11838
   737
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
   738
  by blast
wenzelm@11838
   739
wenzelm@11838
   740
lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"
wenzelm@11838
   741
  by blast
wenzelm@11838
   742
wenzelm@11838
   743
lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"
wenzelm@11838
   744
  by blast
wenzelm@11838
   745
wenzelm@11838
   746
lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"
wenzelm@11838
   747
  by blast
wenzelm@11838
   748
wenzelm@11838
   749
text {*
wenzelm@11838
   750
  Non-dependent versions are needed to avoid the need for higher-order
wenzelm@11838
   751
  matching, especially when the rules are re-oriented.
wenzelm@11838
   752
*}
wenzelm@11838
   753
wenzelm@11838
   754
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"
wenzelm@11838
   755
  by blast
wenzelm@11838
   756
wenzelm@11838
   757
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"
wenzelm@11838
   758
  by blast
wenzelm@11838
   759
wenzelm@11838
   760
lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
wenzelm@11838
   761
  by blast
wenzelm@11838
   762
wenzelm@11838
   763
oheimb@11493
   764
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
wenzelm@11777
   765
  apply (rule_tac x = "(a, b)" in image_eqI)
wenzelm@11777
   766
   apply auto
wenzelm@11777
   767
  done
wenzelm@11777
   768
oheimb@11493
   769
wenzelm@11838
   770
text {*
wenzelm@11838
   771
  Setup of internal @{text split_rule}.
wenzelm@11838
   772
*}
wenzelm@11838
   773
wenzelm@11032
   774
constdefs
wenzelm@11425
   775
  internal_split :: "('a => 'b => 'c) => 'a * 'b => 'c"
wenzelm@11032
   776
  "internal_split == split"
wenzelm@11032
   777
haftmann@24162
   778
lemmas [code func del] = internal_split_def
haftmann@24162
   779
wenzelm@11032
   780
lemma internal_split_conv: "internal_split c (a, b) = c a b"
wenzelm@11032
   781
  by (simp only: internal_split_def split_conv)
wenzelm@11032
   782
wenzelm@11032
   783
hide const internal_split
wenzelm@11032
   784
oheimb@11025
   785
use "Tools/split_rule.ML"
wenzelm@11032
   786
setup SplitRule.setup
nipkow@10213
   787
berghofe@15394
   788
haftmann@24699
   789
haftmann@24699
   790
lemmas prod_caseI = prod.cases [THEN iffD2, standard]
haftmann@24699
   791
haftmann@24699
   792
lemma prod_caseI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> prod_case c p"
haftmann@24699
   793
  by auto
haftmann@24699
   794
haftmann@24699
   795
lemma prod_caseI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> prod_case c p x"
haftmann@24699
   796
  by (auto simp: split_tupled_all)
haftmann@24699
   797
haftmann@24699
   798
lemma prod_caseE: "prod_case c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
haftmann@24699
   799
  by (induct p) auto
haftmann@24699
   800
haftmann@24699
   801
lemma prod_caseE': "prod_case c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
haftmann@24699
   802
  by (induct p) auto
haftmann@24699
   803
haftmann@24699
   804
lemma prod_case_unfold: "prod_case = (%c p. c (fst p) (snd p))"
haftmann@24699
   805
  by (simp add: expand_fun_eq)
haftmann@24699
   806
haftmann@24699
   807
declare prod_caseI2' [intro!] prod_caseI2 [intro!] prod_caseI [intro!]
haftmann@24699
   808
declare prod_caseE' [elim!] prod_caseE [elim!]
haftmann@24699
   809
haftmann@24699
   810
lemma prod_case_split [code post]:
haftmann@24699
   811
  "prod_case = split"
haftmann@24699
   812
  by (auto simp add: expand_fun_eq)
haftmann@24699
   813
haftmann@24699
   814
lemmas [code inline] = prod_case_split [symmetric]
haftmann@24699
   815
haftmann@24699
   816
haftmann@24699
   817
subsection {* Further cases/induct rules for tuples *}
haftmann@24699
   818
haftmann@24699
   819
lemma prod_cases3 [cases type]:
haftmann@24699
   820
  obtains (fields) a b c where "y = (a, b, c)"
haftmann@24699
   821
  by (cases y, case_tac b) blast
haftmann@24699
   822
haftmann@24699
   823
lemma prod_induct3 [case_names fields, induct type]:
haftmann@24699
   824
    "(!!a b c. P (a, b, c)) ==> P x"
haftmann@24699
   825
  by (cases x) blast
haftmann@24699
   826
haftmann@24699
   827
lemma prod_cases4 [cases type]:
haftmann@24699
   828
  obtains (fields) a b c d where "y = (a, b, c, d)"
haftmann@24699
   829
  by (cases y, case_tac c) blast
haftmann@24699
   830
haftmann@24699
   831
lemma prod_induct4 [case_names fields, induct type]:
haftmann@24699
   832
    "(!!a b c d. P (a, b, c, d)) ==> P x"
haftmann@24699
   833
  by (cases x) blast
haftmann@24699
   834
haftmann@24699
   835
lemma prod_cases5 [cases type]:
haftmann@24699
   836
  obtains (fields) a b c d e where "y = (a, b, c, d, e)"
haftmann@24699
   837
  by (cases y, case_tac d) blast
haftmann@24699
   838
haftmann@24699
   839
lemma prod_induct5 [case_names fields, induct type]:
haftmann@24699
   840
    "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
haftmann@24699
   841
  by (cases x) blast
haftmann@24699
   842
haftmann@24699
   843
lemma prod_cases6 [cases type]:
haftmann@24699
   844
  obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
haftmann@24699
   845
  by (cases y, case_tac e) blast
haftmann@24699
   846
haftmann@24699
   847
lemma prod_induct6 [case_names fields, induct type]:
haftmann@24699
   848
    "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
haftmann@24699
   849
  by (cases x) blast
haftmann@24699
   850
haftmann@24699
   851
lemma prod_cases7 [cases type]:
haftmann@24699
   852
  obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
haftmann@24699
   853
  by (cases y, case_tac f) blast
haftmann@24699
   854
haftmann@24699
   855
lemma prod_induct7 [case_names fields, induct type]:
haftmann@24699
   856
    "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
haftmann@24699
   857
  by (cases x) blast
haftmann@24699
   858
haftmann@24699
   859
haftmann@21195
   860
subsection {* Further lemmas *}
haftmann@21195
   861
haftmann@21195
   862
lemma
haftmann@21195
   863
  split_Pair: "split Pair x = x"
haftmann@21195
   864
  unfolding split_def by auto
haftmann@21195
   865
haftmann@21195
   866
lemma
haftmann@21195
   867
  split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)"
haftmann@21195
   868
  by (cases x, simp)
haftmann@21195
   869
haftmann@21195
   870
berghofe@15394
   871
subsection {* Code generator setup *}
berghofe@15394
   872
haftmann@20588
   873
instance unit :: eq ..
haftmann@20588
   874
haftmann@20588
   875
lemma [code func]:
haftmann@21454
   876
  "(u\<Colon>unit) = v \<longleftrightarrow> True" unfolding unit_eq [of u] unit_eq [of v] by rule+
haftmann@20588
   877
haftmann@21908
   878
code_type unit
haftmann@21908
   879
  (SML "unit")
haftmann@21908
   880
  (OCaml "unit")
haftmann@21908
   881
  (Haskell "()")
haftmann@21908
   882
haftmann@20588
   883
code_instance unit :: eq
haftmann@20588
   884
  (Haskell -)
haftmann@20588
   885
haftmann@21908
   886
code_const "op = \<Colon> unit \<Rightarrow> unit \<Rightarrow> bool"
haftmann@21908
   887
  (Haskell infixl 4 "==")
haftmann@21908
   888
haftmann@21908
   889
code_const Unity
haftmann@21908
   890
  (SML "()")
haftmann@21908
   891
  (OCaml "()")
haftmann@21908
   892
  (Haskell "()")
haftmann@21908
   893
haftmann@21908
   894
code_reserved SML
haftmann@21908
   895
  unit
haftmann@21908
   896
haftmann@21908
   897
code_reserved OCaml
haftmann@21908
   898
  unit
haftmann@21908
   899
haftmann@20588
   900
instance * :: (eq, eq) eq ..
haftmann@20588
   901
haftmann@20588
   902
lemma [code func]:
haftmann@21454
   903
  "(x1\<Colon>'a\<Colon>eq, y1\<Colon>'b\<Colon>eq) = (x2, y2) \<longleftrightarrow> x1 = x2 \<and> y1 = y2" by auto
haftmann@20588
   904
haftmann@21908
   905
code_type *
haftmann@21908
   906
  (SML infix 2 "*")
haftmann@21908
   907
  (OCaml infix 2 "*")
haftmann@21908
   908
  (Haskell "!((_),/ (_))")
haftmann@21908
   909
haftmann@20588
   910
code_instance * :: eq
haftmann@20588
   911
  (Haskell -)
haftmann@20588
   912
haftmann@21908
   913
code_const "op = \<Colon> 'a\<Colon>eq \<times> 'b\<Colon>eq \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"
haftmann@20588
   914
  (Haskell infixl 4 "==")
haftmann@20588
   915
haftmann@21908
   916
code_const Pair
haftmann@21908
   917
  (SML "!((_),/ (_))")
haftmann@21908
   918
  (OCaml "!((_),/ (_))")
haftmann@21908
   919
  (Haskell "!((_),/ (_))")
haftmann@20588
   920
haftmann@22389
   921
code_const fst and snd
haftmann@22389
   922
  (Haskell "fst" and "snd")
haftmann@22389
   923
berghofe@15394
   924
types_code
berghofe@15394
   925
  "*"     ("(_ */ _)")
berghofe@16770
   926
attach (term_of) {*
berghofe@16770
   927
fun term_of_id_42 f T g U (x, y) = HOLogic.pair_const T U $ f x $ g y;
berghofe@16770
   928
*}
berghofe@16770
   929
attach (test) {*
berghofe@16770
   930
fun gen_id_42 aG bG i = (aG i, bG i);
berghofe@16770
   931
*}
berghofe@15394
   932
berghofe@18706
   933
consts_code
berghofe@18706
   934
  "Pair"    ("(_,/ _)")
berghofe@18706
   935
haftmann@21908
   936
setup {*
haftmann@21908
   937
haftmann@21908
   938
let
haftmann@18013
   939
haftmann@19039
   940
fun strip_abs_split 0 t = ([], t)
haftmann@19039
   941
  | strip_abs_split i (Abs (s, T, t)) =
haftmann@18013
   942
      let
haftmann@18013
   943
        val s' = Codegen.new_name t s;
haftmann@18013
   944
        val v = Free (s', T)
haftmann@19039
   945
      in apfst (cons v) (strip_abs_split (i-1) (subst_bound (v, t))) end
haftmann@19039
   946
  | strip_abs_split i (u as Const ("split", _) $ t) = (case strip_abs_split (i+1) t of
berghofe@15394
   947
        (v :: v' :: vs, u) => (HOLogic.mk_prod (v, v') :: vs, u)
berghofe@15394
   948
      | _ => ([], u))
haftmann@19039
   949
  | strip_abs_split i t = ([], t);
haftmann@18013
   950
berghofe@16634
   951
fun let_codegen thy defs gr dep thyname brack t = (case strip_comb t of
berghofe@16634
   952
    (t1 as Const ("Let", _), t2 :: t3 :: ts) =>
berghofe@15394
   953
    let
berghofe@15394
   954
      fun dest_let (l as Const ("Let", _) $ t $ u) =
haftmann@19039
   955
          (case strip_abs_split 1 u of
berghofe@15394
   956
             ([p], u') => apfst (cons (p, t)) (dest_let u')
berghofe@15394
   957
           | _ => ([], l))
berghofe@15394
   958
        | dest_let t = ([], t);
berghofe@15394
   959
      fun mk_code (gr, (l, r)) =
berghofe@15394
   960
        let
berghofe@16634
   961
          val (gr1, pl) = Codegen.invoke_codegen thy defs dep thyname false (gr, l);
berghofe@16634
   962
          val (gr2, pr) = Codegen.invoke_codegen thy defs dep thyname false (gr1, r);
berghofe@15394
   963
        in (gr2, (pl, pr)) end
berghofe@16634
   964
    in case dest_let (t1 $ t2 $ t3) of
skalberg@15531
   965
        ([], _) => NONE
berghofe@15394
   966
      | (ps, u) =>
berghofe@15394
   967
          let
berghofe@15394
   968
            val (gr1, qs) = foldl_map mk_code (gr, ps);
berghofe@16634
   969
            val (gr2, pu) = Codegen.invoke_codegen thy defs dep thyname false (gr1, u);
berghofe@16634
   970
            val (gr3, pargs) = foldl_map
berghofe@17021
   971
              (Codegen.invoke_codegen thy defs dep thyname true) (gr2, ts)
berghofe@15394
   972
          in
berghofe@16634
   973
            SOME (gr3, Codegen.mk_app brack
berghofe@16634
   974
              (Pretty.blk (0, [Pretty.str "let ", Pretty.blk (0, List.concat
berghofe@16634
   975
                  (separate [Pretty.str ";", Pretty.brk 1] (map (fn (pl, pr) =>
berghofe@16634
   976
                    [Pretty.block [Pretty.str "val ", pl, Pretty.str " =",
berghofe@16634
   977
                       Pretty.brk 1, pr]]) qs))),
berghofe@16634
   978
                Pretty.brk 1, Pretty.str "in ", pu,
berghofe@16634
   979
                Pretty.brk 1, Pretty.str "end"])) pargs)
berghofe@15394
   980
          end
berghofe@15394
   981
    end
berghofe@16634
   982
  | _ => NONE);
berghofe@15394
   983
berghofe@16634
   984
fun split_codegen thy defs gr dep thyname brack t = (case strip_comb t of
berghofe@16634
   985
    (t1 as Const ("split", _), t2 :: ts) =>
haftmann@19039
   986
      (case strip_abs_split 1 (t1 $ t2) of
berghofe@16634
   987
         ([p], u) =>
berghofe@16634
   988
           let
berghofe@16634
   989
             val (gr1, q) = Codegen.invoke_codegen thy defs dep thyname false (gr, p);
berghofe@16634
   990
             val (gr2, pu) = Codegen.invoke_codegen thy defs dep thyname false (gr1, u);
berghofe@16634
   991
             val (gr3, pargs) = foldl_map
berghofe@17021
   992
               (Codegen.invoke_codegen thy defs dep thyname true) (gr2, ts)
berghofe@16634
   993
           in
berghofe@16634
   994
             SOME (gr2, Codegen.mk_app brack
berghofe@16634
   995
               (Pretty.block [Pretty.str "(fn ", q, Pretty.str " =>",
berghofe@16634
   996
                 Pretty.brk 1, pu, Pretty.str ")"]) pargs)
berghofe@16634
   997
           end
berghofe@16634
   998
       | _ => NONE)
berghofe@16634
   999
  | _ => NONE);
berghofe@15394
  1000
haftmann@21908
  1001
in
haftmann@21908
  1002
haftmann@20105
  1003
  Codegen.add_codegen "let_codegen" let_codegen
haftmann@20105
  1004
  #> Codegen.add_codegen "split_codegen" split_codegen
berghofe@15394
  1005
haftmann@21908
  1006
end
berghofe@15394
  1007
*}
berghofe@15394
  1008
haftmann@24699
  1009
haftmann@24699
  1010
subsection {* Legacy bindings *}
haftmann@24699
  1011
haftmann@21908
  1012
ML {*
paulson@15404
  1013
val Collect_split = thm "Collect_split";
paulson@15404
  1014
val Compl_Times_UNIV1 = thm "Compl_Times_UNIV1";
paulson@15404
  1015
val Compl_Times_UNIV2 = thm "Compl_Times_UNIV2";
paulson@15404
  1016
val PairE = thm "PairE";
paulson@15404
  1017
val PairE_lemma = thm "PairE_lemma";
paulson@15404
  1018
val Pair_Rep_inject = thm "Pair_Rep_inject";
paulson@15404
  1019
val Pair_def = thm "Pair_def";
paulson@15404
  1020
val Pair_eq = thm "Pair_eq";
paulson@15404
  1021
val Pair_fst_snd_eq = thm "Pair_fst_snd_eq";
paulson@15404
  1022
val Pair_inject = thm "Pair_inject";
paulson@15404
  1023
val ProdI = thm "ProdI";
paulson@15404
  1024
val SetCompr_Sigma_eq = thm "SetCompr_Sigma_eq";
paulson@15404
  1025
val SigmaD1 = thm "SigmaD1";
paulson@15404
  1026
val SigmaD2 = thm "SigmaD2";
paulson@15404
  1027
val SigmaE = thm "SigmaE";
paulson@15404
  1028
val SigmaE2 = thm "SigmaE2";
paulson@15404
  1029
val SigmaI = thm "SigmaI";
paulson@15404
  1030
val Sigma_Diff_distrib1 = thm "Sigma_Diff_distrib1";
paulson@15404
  1031
val Sigma_Diff_distrib2 = thm "Sigma_Diff_distrib2";
paulson@15404
  1032
val Sigma_Int_distrib1 = thm "Sigma_Int_distrib1";
paulson@15404
  1033
val Sigma_Int_distrib2 = thm "Sigma_Int_distrib2";
paulson@15404
  1034
val Sigma_Un_distrib1 = thm "Sigma_Un_distrib1";
paulson@15404
  1035
val Sigma_Un_distrib2 = thm "Sigma_Un_distrib2";
paulson@15404
  1036
val Sigma_Union = thm "Sigma_Union";
paulson@15404
  1037
val Sigma_def = thm "Sigma_def";
paulson@15404
  1038
val Sigma_empty1 = thm "Sigma_empty1";
paulson@15404
  1039
val Sigma_empty2 = thm "Sigma_empty2";
paulson@15404
  1040
val Sigma_mono = thm "Sigma_mono";
paulson@15404
  1041
val The_split = thm "The_split";
paulson@15404
  1042
val The_split_eq = thm "The_split_eq";
paulson@15404
  1043
val The_split_eq = thm "The_split_eq";
paulson@15404
  1044
val Times_Diff_distrib1 = thm "Times_Diff_distrib1";
paulson@15404
  1045
val Times_Int_distrib1 = thm "Times_Int_distrib1";
paulson@15404
  1046
val Times_Un_distrib1 = thm "Times_Un_distrib1";
paulson@15404
  1047
val Times_eq_cancel2 = thm "Times_eq_cancel2";
paulson@15404
  1048
val Times_subset_cancel2 = thm "Times_subset_cancel2";
paulson@15404
  1049
val UNIV_Times_UNIV = thm "UNIV_Times_UNIV";
paulson@15404
  1050
val UN_Times_distrib = thm "UN_Times_distrib";
paulson@15404
  1051
val Unity_def = thm "Unity_def";
paulson@15404
  1052
val cond_split_eta = thm "cond_split_eta";
paulson@15404
  1053
val fst_conv = thm "fst_conv";
paulson@15404
  1054
val fst_def = thm "fst_def";
paulson@15404
  1055
val fst_eqD = thm "fst_eqD";
paulson@15404
  1056
val inj_on_Abs_Prod = thm "inj_on_Abs_Prod";
paulson@15404
  1057
val injective_fst_snd = thm "injective_fst_snd";
paulson@15404
  1058
val mem_Sigma_iff = thm "mem_Sigma_iff";
paulson@15404
  1059
val mem_splitE = thm "mem_splitE";
paulson@15404
  1060
val mem_splitI = thm "mem_splitI";
paulson@15404
  1061
val mem_splitI2 = thm "mem_splitI2";
paulson@15404
  1062
val prod_eqI = thm "prod_eqI";
paulson@15404
  1063
val prod_fun = thm "prod_fun";
paulson@15404
  1064
val prod_fun_compose = thm "prod_fun_compose";
paulson@15404
  1065
val prod_fun_def = thm "prod_fun_def";
paulson@15404
  1066
val prod_fun_ident = thm "prod_fun_ident";
paulson@15404
  1067
val prod_fun_imageE = thm "prod_fun_imageE";
paulson@15404
  1068
val prod_fun_imageI = thm "prod_fun_imageI";
paulson@15404
  1069
val prod_induct = thm "prod_induct";
paulson@15404
  1070
val snd_conv = thm "snd_conv";
paulson@15404
  1071
val snd_def = thm "snd_def";
paulson@15404
  1072
val snd_eqD = thm "snd_eqD";
paulson@15404
  1073
val split = thm "split";
paulson@15404
  1074
val splitD = thm "splitD";
paulson@15404
  1075
val splitD' = thm "splitD'";
paulson@15404
  1076
val splitE = thm "splitE";
paulson@15404
  1077
val splitE' = thm "splitE'";
paulson@15404
  1078
val splitE2 = thm "splitE2";
paulson@15404
  1079
val splitI = thm "splitI";
paulson@15404
  1080
val splitI2 = thm "splitI2";
paulson@15404
  1081
val splitI2' = thm "splitI2'";
paulson@15404
  1082
val split_Pair_apply = thm "split_Pair_apply";
paulson@15404
  1083
val split_beta = thm "split_beta";
paulson@15404
  1084
val split_conv = thm "split_conv";
paulson@15404
  1085
val split_def = thm "split_def";
paulson@15404
  1086
val split_eta = thm "split_eta";
paulson@15404
  1087
val split_eta_SetCompr = thm "split_eta_SetCompr";
paulson@15404
  1088
val split_eta_SetCompr2 = thm "split_eta_SetCompr2";
paulson@15404
  1089
val split_paired_All = thm "split_paired_All";
paulson@15404
  1090
val split_paired_Ball_Sigma = thm "split_paired_Ball_Sigma";
paulson@15404
  1091
val split_paired_Bex_Sigma = thm "split_paired_Bex_Sigma";
paulson@15404
  1092
val split_paired_Ex = thm "split_paired_Ex";
paulson@15404
  1093
val split_paired_The = thm "split_paired_The";
paulson@15404
  1094
val split_paired_all = thm "split_paired_all";
paulson@15404
  1095
val split_part = thm "split_part";
paulson@15404
  1096
val split_split = thm "split_split";
paulson@15404
  1097
val split_split_asm = thm "split_split_asm";
paulson@15404
  1098
val split_tupled_all = thms "split_tupled_all";
paulson@15404
  1099
val split_weak_cong = thm "split_weak_cong";
paulson@15404
  1100
val surj_pair = thm "surj_pair";
paulson@15404
  1101
val surjective_pairing = thm "surjective_pairing";
paulson@15404
  1102
val unit_abs_eta_conv = thm "unit_abs_eta_conv";
paulson@15404
  1103
val unit_all_eq1 = thm "unit_all_eq1";
paulson@15404
  1104
val unit_all_eq2 = thm "unit_all_eq2";
paulson@15404
  1105
val unit_eq = thm "unit_eq";
paulson@15404
  1106
val unit_induct = thm "unit_induct";
paulson@15404
  1107
*}
paulson@15404
  1108
haftmann@24699
  1109
haftmann@24699
  1110
subsection {* Further inductive packages *}
haftmann@24699
  1111
haftmann@24699
  1112
use "Tools/inductive_realizer.ML"
haftmann@24699
  1113
setup InductiveRealizer.setup
haftmann@24699
  1114
haftmann@24699
  1115
use "Tools/inductive_set_package.ML"
haftmann@24699
  1116
setup InductiveSetPackage.setup
haftmann@24699
  1117
haftmann@24699
  1118
use "Tools/datatype_realizer.ML"
haftmann@24699
  1119
setup DatatypeRealizer.setup
haftmann@24699
  1120
nipkow@10213
  1121
end