src/HOL/Product_Type.thy
author nipkow
Fri Mar 06 17:38:47 2009 +0100 (2009-03-06)
changeset 30313 b2441b0c8d38
parent 28719 01e04e41cc7b
child 30604 2a9911f4b0a3
permissions -rw-r--r--
added lemmas
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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 True False by (auto intro: bool_induct)
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declare case_split [cases type: bool]
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  -- "prefer plain propositional version"
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lemma
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  shows [code]: "eq_class.eq False P \<longleftrightarrow> \<not> P"
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    and [code]: "eq_class.eq True P \<longleftrightarrow> P" 
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    and [code]: "eq_class.eq P False \<longleftrightarrow> \<not> P" 
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    and [code]: "eq_class.eq P True \<longleftrightarrow> P"
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    and [code nbe]: "eq_class.eq P P \<longleftrightarrow> True"
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  by (simp_all add: eq)
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code_const "eq_class.eq \<Colon> bool \<Rightarrow> bool \<Rightarrow> bool"
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  (Haskell infixl 4 "==")
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code_instance bool :: eq
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  (Haskell -)
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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 {*
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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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rep_datatype "()" by simp
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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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text {* code generator setup *}
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instance unit :: eq ..
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lemma [code]:
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  "eq_class.eq (u\<Colon>unit) v \<longleftrightarrow> True" unfolding eq unit_eq [of u] unit_eq [of v] by rule+
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code_type unit
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  (SML "unit")
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  (OCaml "unit")
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  (Haskell "()")
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code_instance unit :: eq
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  (Haskell -)
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code_const "eq_class.eq \<Colon> unit \<Rightarrow> unit \<Rightarrow> bool"
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  (Haskell infixl 4 "==")
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code_const Unity
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  (SML "()")
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  (OCaml "()")
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  (Haskell "()")
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code_reserved SML
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  unit
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code_reserved OCaml
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  unit
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subsection {* Pairs *}
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subsubsection {* Product type, basic operations and concrete syntax *}
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definition
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  Pair_Rep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
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where
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  "Pair_Rep a b = (\<lambda>x y. x = a \<and> 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. \<exists>a b. f = Pair_Rep (a\<Colon>'a) (b\<Colon>'b)}"
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proof
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  fix a b show "Pair_Rep a b \<in> ?Prod"
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    by rule+
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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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consts
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  Pair     :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b"
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  fst      :: "'a \<times> 'b \<Rightarrow> 'a"
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  snd      :: "'a \<times> 'b \<Rightarrow> 'b"
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  split    :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c"
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  curry    :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c"
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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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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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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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(* 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 {* Towards a datatype declaration *}
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lemma surj_pair [simp]: "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: *]:
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  obtains x y where "p = (x, y)"
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  using surj_pair [of p] by blast
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lemma ProdI: "Pair_Rep a b \<in> Prod"
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  unfolding Prod_def by rule+
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lemma Pair_Rep_inject: "Pair_Rep a b = Pair_Rep a' b' \<Longrightarrow> a = a' \<and> b = b'"
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  unfolding Pair_Rep_def by (drule fun_cong, drule fun_cong) blast
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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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rep_datatype (prod) Pair
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proof -
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  fix P p
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  assume "\<And>x y. P (x, y)"
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  then show "P p" by (cases p) simp
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qed (auto elim: Pair_inject)
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lemmas Pair_eq = prod.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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subsubsection {* Basic rules and proof tools *}
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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 pair_collapse [simp]: "(fst p, snd p) = p"
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  by (cases p) simp
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lemmas surjective_pairing = pair_collapse [symmetric]
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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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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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lemmas split_tupled_all = split_paired_all unit_all_eq2
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ML {*
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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
   335
    val unsafe_split_all_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838
   336
      if exists_paired_all t then full_simp_tac ss i else no_tac);
wenzelm@11838
   337
    fun split_all th =
wenzelm@26340
   338
   if exists_paired_all (Thm.prop_of th) then full_simplify ss th else th;
wenzelm@11838
   339
  end;
wenzelm@26340
   340
*}
wenzelm@11838
   341
wenzelm@26340
   342
declaration {* fn _ =>
wenzelm@26340
   343
  Classical.map_cs (fn cs => cs addSbefore ("split_all_tac", split_all_tac))
wenzelm@16121
   344
*}
wenzelm@11838
   345
wenzelm@11838
   346
lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))"
wenzelm@11838
   347
  -- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *}
wenzelm@11838
   348
  by fast
wenzelm@11838
   349
haftmann@26358
   350
lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))"
haftmann@26358
   351
  by fast
haftmann@26358
   352
haftmann@26358
   353
lemma Pair_fst_snd_eq: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t"
haftmann@26358
   354
  by (cases s, cases t) simp
haftmann@26358
   355
haftmann@26358
   356
lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q"
haftmann@26358
   357
  by (simp add: Pair_fst_snd_eq)
haftmann@26358
   358
haftmann@26358
   359
haftmann@26358
   360
subsubsection {* @{text split} and @{text curry} *}
haftmann@26358
   361
haftmann@28562
   362
lemma split_conv [simp, code]: "split f (a, b) = f a b"
haftmann@26358
   363
  by (simp add: split_def)
haftmann@26358
   364
haftmann@28562
   365
lemma curry_conv [simp, code]: "curry f a b = f (a, b)"
haftmann@26358
   366
  by (simp add: curry_def)
haftmann@26358
   367
haftmann@26358
   368
lemmas split = split_conv  -- {* for backwards compatibility *}
haftmann@26358
   369
haftmann@26358
   370
lemma splitI: "f a b \<Longrightarrow> split f (a, b)"
haftmann@26358
   371
  by (rule split_conv [THEN iffD2])
haftmann@26358
   372
haftmann@26358
   373
lemma splitD: "split f (a, b) \<Longrightarrow> f a b"
haftmann@26358
   374
  by (rule split_conv [THEN iffD1])
haftmann@26358
   375
haftmann@26358
   376
lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b"
haftmann@26358
   377
  by (simp add: curry_def)
haftmann@26358
   378
haftmann@26358
   379
lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)"
haftmann@26358
   380
  by (simp add: curry_def)
haftmann@26358
   381
haftmann@26358
   382
lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q"
haftmann@26358
   383
  by (simp add: curry_def)
haftmann@26358
   384
skalberg@14189
   385
lemma curry_split [simp]: "curry (split f) = f"
skalberg@14189
   386
  by (simp add: curry_def split_def)
skalberg@14189
   387
skalberg@14189
   388
lemma split_curry [simp]: "split (curry f) = f"
skalberg@14189
   389
  by (simp add: curry_def split_def)
skalberg@14189
   390
haftmann@26358
   391
lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id"
haftmann@26358
   392
  by (simp add: split_def id_def)
wenzelm@11838
   393
haftmann@26358
   394
lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f"
haftmann@26358
   395
  -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *}
haftmann@26358
   396
  by (rule ext) auto
wenzelm@11838
   397
haftmann@26358
   398
lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)"
haftmann@26358
   399
  by (cases x) simp
wenzelm@11838
   400
haftmann@26358
   401
lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p"
haftmann@26358
   402
  unfolding split_def ..
wenzelm@11838
   403
wenzelm@11838
   404
lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))"
wenzelm@11838
   405
  -- {* Can't be added to simpset: loops! *}
haftmann@26358
   406
  by (simp add: split_eta)
wenzelm@11838
   407
wenzelm@11838
   408
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
wenzelm@11838
   409
  by (simp add: split_def)
wenzelm@11838
   410
haftmann@26358
   411
lemma split_weak_cong: "p = q \<Longrightarrow> split c p = split c q"
wenzelm@11838
   412
  -- {* Prevents simplification of @{term c}: much faster *}
wenzelm@11838
   413
  by (erule arg_cong)
wenzelm@11838
   414
wenzelm@11838
   415
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
wenzelm@11838
   416
  by (simp add: split_eta)
wenzelm@11838
   417
wenzelm@11838
   418
text {*
wenzelm@11838
   419
  Simplification procedure for @{thm [source] cond_split_eta}.  Using
wenzelm@11838
   420
  @{thm [source] split_eta} as a rewrite rule is not general enough,
wenzelm@11838
   421
  and using @{thm [source] cond_split_eta} directly would render some
wenzelm@11838
   422
  existing proofs very inefficient; similarly for @{text
haftmann@26358
   423
  split_beta}.
haftmann@26358
   424
*}
wenzelm@11838
   425
wenzelm@26480
   426
ML {*
wenzelm@11838
   427
wenzelm@11838
   428
local
wenzelm@18328
   429
  val cond_split_eta_ss = HOL_basic_ss addsimps [thm "cond_split_eta"]
wenzelm@11838
   430
  fun  Pair_pat k 0 (Bound m) = (m = k)
wenzelm@11838
   431
  |    Pair_pat k i (Const ("Pair",  _) $ Bound m $ t) = i > 0 andalso
wenzelm@11838
   432
                        m = k+i andalso Pair_pat k (i-1) t
wenzelm@11838
   433
  |    Pair_pat _ _ _ = false;
wenzelm@11838
   434
  fun no_args k i (Abs (_, _, t)) = no_args (k+1) i t
wenzelm@11838
   435
  |   no_args k i (t $ u) = no_args k i t andalso no_args k i u
wenzelm@11838
   436
  |   no_args k i (Bound m) = m < k orelse m > k+i
wenzelm@11838
   437
  |   no_args _ _ _ = true;
skalberg@15531
   438
  fun split_pat tp i (Abs  (_,_,t)) = if tp 0 i t then SOME (i,t) else NONE
wenzelm@11838
   439
  |   split_pat tp i (Const ("split", _) $ Abs (_, _, t)) = split_pat tp (i+1) t
skalberg@15531
   440
  |   split_pat tp i _ = NONE;
wenzelm@20044
   441
  fun metaeq ss lhs rhs = mk_meta_eq (Goal.prove (Simplifier.the_context ss) [] []
wenzelm@13480
   442
        (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs,rhs)))
wenzelm@18328
   443
        (K (simp_tac (Simplifier.inherit_context ss cond_split_eta_ss) 1)));
wenzelm@11838
   444
wenzelm@11838
   445
  fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k+1) i t
wenzelm@11838
   446
  |   beta_term_pat k i (t $ u) = Pair_pat k i (t $ u) orelse
wenzelm@11838
   447
                        (beta_term_pat k i t andalso beta_term_pat k i u)
wenzelm@11838
   448
  |   beta_term_pat k i t = no_args k i t;
wenzelm@11838
   449
  fun  eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
wenzelm@11838
   450
  |    eta_term_pat _ _ _ = false;
wenzelm@11838
   451
  fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
wenzelm@11838
   452
  |   subst arg k i (t $ u) = if Pair_pat k i (t $ u) then incr_boundvars k arg
wenzelm@11838
   453
                              else (subst arg k i t $ subst arg k i u)
wenzelm@11838
   454
  |   subst arg k i t = t;
wenzelm@20044
   455
  fun beta_proc ss (s as Const ("split", _) $ Abs (_, _, t) $ arg) =
wenzelm@11838
   456
        (case split_pat beta_term_pat 1 t of
wenzelm@20044
   457
        SOME (i,f) => SOME (metaeq ss s (subst arg 0 i f))
skalberg@15531
   458
        | NONE => NONE)
wenzelm@20044
   459
  |   beta_proc _ _ = NONE;
wenzelm@20044
   460
  fun eta_proc ss (s as Const ("split", _) $ Abs (_, _, t)) =
wenzelm@11838
   461
        (case split_pat eta_term_pat 1 t of
wenzelm@20044
   462
          SOME (_,ft) => SOME (metaeq ss s (let val (f $ arg) = ft in f end))
skalberg@15531
   463
        | NONE => NONE)
wenzelm@20044
   464
  |   eta_proc _ _ = NONE;
wenzelm@11838
   465
in
wenzelm@28262
   466
  val split_beta_proc = Simplifier.simproc (the_context ()) "split_beta" ["split f z"] (K beta_proc);
wenzelm@28262
   467
  val split_eta_proc = Simplifier.simproc (the_context ()) "split_eta" ["split f"] (K eta_proc);
wenzelm@11838
   468
end;
wenzelm@11838
   469
wenzelm@11838
   470
Addsimprocs [split_beta_proc, split_eta_proc];
wenzelm@11838
   471
*}
wenzelm@11838
   472
berghofe@26798
   473
lemma split_beta [mono]: "(%(x, y). P x y) z = P (fst z) (snd z)"
wenzelm@11838
   474
  by (subst surjective_pairing, rule split_conv)
wenzelm@11838
   475
paulson@24286
   476
lemma split_split [noatp]: "R(split c p) = (ALL x y. p = (x, y) --> R(c x y))"
wenzelm@11838
   477
  -- {* For use with @{text split} and the Simplifier. *}
paulson@15481
   478
  by (insert surj_pair [of p], clarify, simp)
wenzelm@11838
   479
wenzelm@11838
   480
text {*
wenzelm@11838
   481
  @{thm [source] split_split} could be declared as @{text "[split]"}
wenzelm@11838
   482
  done after the Splitter has been speeded up significantly;
wenzelm@11838
   483
  precompute the constants involved and don't do anything unless the
wenzelm@11838
   484
  current goal contains one of those constants.
wenzelm@11838
   485
*}
wenzelm@11838
   486
paulson@24286
   487
lemma split_split_asm [noatp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))"
paulson@14208
   488
by (subst split_split, simp)
wenzelm@11838
   489
wenzelm@11838
   490
wenzelm@11838
   491
text {*
wenzelm@11838
   492
  \medskip @{term split} used as a logical connective or set former.
wenzelm@11838
   493
wenzelm@11838
   494
  \medskip These rules are for use with @{text blast}; could instead
wenzelm@11838
   495
  call @{text simp} using @{thm [source] split} as rewrite. *}
wenzelm@11838
   496
wenzelm@11838
   497
lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"
wenzelm@11838
   498
  apply (simp only: split_tupled_all)
wenzelm@11838
   499
  apply (simp (no_asm_simp))
wenzelm@11838
   500
  done
wenzelm@11838
   501
wenzelm@11838
   502
lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"
wenzelm@11838
   503
  apply (simp only: split_tupled_all)
wenzelm@11838
   504
  apply (simp (no_asm_simp))
wenzelm@11838
   505
  done
wenzelm@11838
   506
wenzelm@11838
   507
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
wenzelm@11838
   508
  by (induct p) (auto simp add: split_def)
wenzelm@11838
   509
wenzelm@11838
   510
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
wenzelm@11838
   511
  by (induct p) (auto simp add: split_def)
wenzelm@11838
   512
wenzelm@11838
   513
lemma splitE2:
wenzelm@11838
   514
  "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
wenzelm@11838
   515
proof -
wenzelm@11838
   516
  assume q: "Q (split P z)"
wenzelm@11838
   517
  assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
wenzelm@11838
   518
  show R
wenzelm@11838
   519
    apply (rule r surjective_pairing)+
wenzelm@11838
   520
    apply (rule split_beta [THEN subst], rule q)
wenzelm@11838
   521
    done
wenzelm@11838
   522
qed
wenzelm@11838
   523
wenzelm@11838
   524
lemma splitD': "split R (a,b) c ==> R a b c"
wenzelm@11838
   525
  by simp
wenzelm@11838
   526
wenzelm@11838
   527
lemma mem_splitI: "z: c a b ==> z: split c (a, b)"
wenzelm@11838
   528
  by simp
wenzelm@11838
   529
wenzelm@11838
   530
lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"
paulson@14208
   531
by (simp only: split_tupled_all, simp)
wenzelm@11838
   532
wenzelm@18372
   533
lemma mem_splitE:
wenzelm@18372
   534
  assumes major: "z: split c p"
wenzelm@18372
   535
    and cases: "!!x y. [| p = (x,y); z: c x y |] ==> Q"
wenzelm@18372
   536
  shows Q
wenzelm@18372
   537
  by (rule major [unfolded split_def] cases surjective_pairing)+
wenzelm@11838
   538
wenzelm@11838
   539
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
wenzelm@11838
   540
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]
wenzelm@11838
   541
wenzelm@26340
   542
ML {*
wenzelm@11838
   543
local (* filtering with exists_p_split is an essential optimization *)
wenzelm@16121
   544
  fun exists_p_split (Const ("split",_) $ _ $ (Const ("Pair",_)$_$_)) = true
wenzelm@11838
   545
    | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
wenzelm@11838
   546
    | exists_p_split (Abs (_, _, t)) = exists_p_split t
wenzelm@11838
   547
    | exists_p_split _ = false;
wenzelm@16121
   548
  val ss = HOL_basic_ss addsimps [thm "split_conv"];
wenzelm@11838
   549
in
wenzelm@11838
   550
val split_conv_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838
   551
    if exists_p_split t then safe_full_simp_tac ss i else no_tac);
wenzelm@11838
   552
end;
wenzelm@26340
   553
*}
wenzelm@26340
   554
wenzelm@11838
   555
(* This prevents applications of splitE for already splitted arguments leading
wenzelm@11838
   556
   to quite time-consuming computations (in particular for nested tuples) *)
wenzelm@26340
   557
declaration {* fn _ =>
wenzelm@26340
   558
  Classical.map_cs (fn cs => cs addSbefore ("split_conv_tac", split_conv_tac))
wenzelm@16121
   559
*}
wenzelm@11838
   560
paulson@24286
   561
lemma split_eta_SetCompr [simp,noatp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
wenzelm@18372
   562
  by (rule ext) fast
wenzelm@11838
   563
paulson@24286
   564
lemma split_eta_SetCompr2 [simp,noatp]: "(%u. EX x y. u = (x, y) & P x y) = split P"
wenzelm@18372
   565
  by (rule ext) fast
wenzelm@11838
   566
wenzelm@11838
   567
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"
wenzelm@11838
   568
  -- {* Allows simplifications of nested splits in case of independent predicates. *}
wenzelm@18372
   569
  by (rule ext) blast
wenzelm@11838
   570
nipkow@14337
   571
(* Do NOT make this a simp rule as it
nipkow@14337
   572
   a) only helps in special situations
nipkow@14337
   573
   b) can lead to nontermination in the presence of split_def
nipkow@14337
   574
*)
nipkow@14337
   575
lemma split_comp_eq: 
paulson@20415
   576
  fixes f :: "'a => 'b => 'c" and g :: "'d => 'a"
paulson@20415
   577
  shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"
wenzelm@18372
   578
  by (rule ext) auto
oheimb@14101
   579
haftmann@26358
   580
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
haftmann@26358
   581
  apply (rule_tac x = "(a, b)" in image_eqI)
haftmann@26358
   582
   apply auto
haftmann@26358
   583
  done
haftmann@26358
   584
wenzelm@11838
   585
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
wenzelm@11838
   586
  by blast
wenzelm@11838
   587
wenzelm@11838
   588
(*
wenzelm@11838
   589
the following  would be slightly more general,
wenzelm@11838
   590
but cannot be used as rewrite rule:
wenzelm@11838
   591
### Cannot add premise as rewrite rule because it contains (type) unknowns:
wenzelm@11838
   592
### ?y = .x
wenzelm@11838
   593
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
paulson@14208
   594
by (rtac some_equality 1)
paulson@14208
   595
by ( Simp_tac 1)
paulson@14208
   596
by (split_all_tac 1)
paulson@14208
   597
by (Asm_full_simp_tac 1)
wenzelm@11838
   598
qed "The_split_eq";
wenzelm@11838
   599
*)
wenzelm@11838
   600
wenzelm@11838
   601
text {*
wenzelm@11838
   602
  Setup of internal @{text split_rule}.
wenzelm@11838
   603
*}
wenzelm@11838
   604
haftmann@25511
   605
definition
haftmann@25511
   606
  internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c"
haftmann@25511
   607
where
wenzelm@11032
   608
  "internal_split == split"
wenzelm@11032
   609
wenzelm@11032
   610
lemma internal_split_conv: "internal_split c (a, b) = c a b"
wenzelm@11032
   611
  by (simp only: internal_split_def split_conv)
wenzelm@11032
   612
wenzelm@11032
   613
hide const internal_split
wenzelm@11032
   614
oheimb@11025
   615
use "Tools/split_rule.ML"
wenzelm@11032
   616
setup SplitRule.setup
nipkow@10213
   617
haftmann@24699
   618
lemmas prod_caseI = prod.cases [THEN iffD2, standard]
haftmann@24699
   619
haftmann@24699
   620
lemma prod_caseI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> prod_case c p"
haftmann@24699
   621
  by auto
haftmann@24699
   622
haftmann@24699
   623
lemma prod_caseI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> prod_case c p x"
haftmann@24699
   624
  by (auto simp: split_tupled_all)
haftmann@24699
   625
haftmann@24699
   626
lemma prod_caseE: "prod_case c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
haftmann@24699
   627
  by (induct p) auto
haftmann@24699
   628
haftmann@24699
   629
lemma prod_caseE': "prod_case c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
haftmann@24699
   630
  by (induct p) auto
haftmann@24699
   631
haftmann@24699
   632
lemma prod_case_unfold: "prod_case = (%c p. c (fst p) (snd p))"
haftmann@24699
   633
  by (simp add: expand_fun_eq)
haftmann@24699
   634
haftmann@24699
   635
declare prod_caseI2' [intro!] prod_caseI2 [intro!] prod_caseI [intro!]
haftmann@24699
   636
declare prod_caseE' [elim!] prod_caseE [elim!]
haftmann@24699
   637
haftmann@24844
   638
lemma prod_case_split:
haftmann@24699
   639
  "prod_case = split"
haftmann@24699
   640
  by (auto simp add: expand_fun_eq)
haftmann@24699
   641
bulwahn@26143
   642
lemma prod_case_beta:
bulwahn@26143
   643
  "prod_case f p = f (fst p) (snd p)"
bulwahn@26143
   644
  unfolding prod_case_split split_beta ..
bulwahn@26143
   645
haftmann@24699
   646
haftmann@24699
   647
subsection {* Further cases/induct rules for tuples *}
haftmann@24699
   648
haftmann@24699
   649
lemma prod_cases3 [cases type]:
haftmann@24699
   650
  obtains (fields) a b c where "y = (a, b, c)"
haftmann@24699
   651
  by (cases y, case_tac b) blast
haftmann@24699
   652
haftmann@24699
   653
lemma prod_induct3 [case_names fields, induct type]:
haftmann@24699
   654
    "(!!a b c. P (a, b, c)) ==> P x"
haftmann@24699
   655
  by (cases x) blast
haftmann@24699
   656
haftmann@24699
   657
lemma prod_cases4 [cases type]:
haftmann@24699
   658
  obtains (fields) a b c d where "y = (a, b, c, d)"
haftmann@24699
   659
  by (cases y, case_tac c) blast
haftmann@24699
   660
haftmann@24699
   661
lemma prod_induct4 [case_names fields, induct type]:
haftmann@24699
   662
    "(!!a b c d. P (a, b, c, d)) ==> P x"
haftmann@24699
   663
  by (cases x) blast
haftmann@24699
   664
haftmann@24699
   665
lemma prod_cases5 [cases type]:
haftmann@24699
   666
  obtains (fields) a b c d e where "y = (a, b, c, d, e)"
haftmann@24699
   667
  by (cases y, case_tac d) blast
haftmann@24699
   668
haftmann@24699
   669
lemma prod_induct5 [case_names fields, induct type]:
haftmann@24699
   670
    "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
haftmann@24699
   671
  by (cases x) blast
haftmann@24699
   672
haftmann@24699
   673
lemma prod_cases6 [cases type]:
haftmann@24699
   674
  obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
haftmann@24699
   675
  by (cases y, case_tac e) blast
haftmann@24699
   676
haftmann@24699
   677
lemma prod_induct6 [case_names fields, induct type]:
haftmann@24699
   678
    "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
haftmann@24699
   679
  by (cases x) blast
haftmann@24699
   680
haftmann@24699
   681
lemma prod_cases7 [cases type]:
haftmann@24699
   682
  obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
haftmann@24699
   683
  by (cases y, case_tac f) blast
haftmann@24699
   684
haftmann@24699
   685
lemma prod_induct7 [case_names fields, induct type]:
haftmann@24699
   686
    "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
haftmann@24699
   687
  by (cases x) blast
haftmann@24699
   688
haftmann@24699
   689
haftmann@26358
   690
subsubsection {* Derived operations *}
haftmann@26358
   691
haftmann@26358
   692
text {*
haftmann@26358
   693
  The composition-uncurry combinator.
haftmann@26358
   694
*}
haftmann@26358
   695
haftmann@26588
   696
notation fcomp (infixl "o>" 60)
haftmann@26358
   697
haftmann@26588
   698
definition
haftmann@26588
   699
  scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "o\<rightarrow>" 60)
haftmann@26588
   700
where
haftmann@26588
   701
  "f o\<rightarrow> g = (\<lambda>x. split g (f x))"
haftmann@26358
   702
haftmann@26588
   703
lemma scomp_apply:  "(f o\<rightarrow> g) x = split g (f x)"
haftmann@26588
   704
  by (simp add: scomp_def)
haftmann@26358
   705
haftmann@26588
   706
lemma Pair_scomp: "Pair x o\<rightarrow> f = f x"
haftmann@26588
   707
  by (simp add: expand_fun_eq scomp_apply)
haftmann@26358
   708
haftmann@26588
   709
lemma scomp_Pair: "x o\<rightarrow> Pair = x"
haftmann@26588
   710
  by (simp add: expand_fun_eq scomp_apply)
haftmann@26358
   711
haftmann@26588
   712
lemma scomp_scomp: "(f o\<rightarrow> g) o\<rightarrow> h = f o\<rightarrow> (\<lambda>x. g x o\<rightarrow> h)"
haftmann@26588
   713
  by (simp add: expand_fun_eq split_twice scomp_def)
haftmann@26358
   714
haftmann@26588
   715
lemma scomp_fcomp: "(f o\<rightarrow> g) o> h = f o\<rightarrow> (\<lambda>x. g x o> h)"
haftmann@26588
   716
  by (simp add: expand_fun_eq scomp_apply fcomp_def split_def)
haftmann@26358
   717
haftmann@26588
   718
lemma fcomp_scomp: "(f o> g) o\<rightarrow> h = f o> (g o\<rightarrow> h)"
haftmann@26588
   719
  by (simp add: expand_fun_eq scomp_apply fcomp_apply)
haftmann@26358
   720
haftmann@26588
   721
no_notation fcomp (infixl "o>" 60)
haftmann@26588
   722
no_notation scomp (infixl "o\<rightarrow>" 60)
haftmann@26358
   723
haftmann@26358
   724
haftmann@26358
   725
text {*
haftmann@26358
   726
  @{term prod_fun} --- action of the product functor upon
haftmann@26358
   727
  functions.
haftmann@26358
   728
*}
haftmann@21195
   729
haftmann@26358
   730
definition prod_fun :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where
haftmann@28562
   731
  [code del]: "prod_fun f g = (\<lambda>(x, y). (f x, g y))"
haftmann@26358
   732
haftmann@28562
   733
lemma prod_fun [simp, code]: "prod_fun f g (a, b) = (f a, g b)"
haftmann@26358
   734
  by (simp add: prod_fun_def)
haftmann@26358
   735
haftmann@26358
   736
lemma prod_fun_compose: "prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)"
haftmann@26358
   737
  by (rule ext) auto
haftmann@26358
   738
haftmann@26358
   739
lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)"
haftmann@26358
   740
  by (rule ext) auto
haftmann@26358
   741
haftmann@26358
   742
lemma prod_fun_imageI [intro]: "(a, b) : r ==> (f a, g b) : prod_fun f g ` r"
haftmann@26358
   743
  apply (rule image_eqI)
haftmann@26358
   744
  apply (rule prod_fun [symmetric], assumption)
haftmann@26358
   745
  done
haftmann@21195
   746
haftmann@26358
   747
lemma prod_fun_imageE [elim!]:
haftmann@26358
   748
  assumes major: "c: (prod_fun f g)`r"
haftmann@26358
   749
    and cases: "!!x y. [| c=(f(x),g(y));  (x,y):r |] ==> P"
haftmann@26358
   750
  shows P
haftmann@26358
   751
  apply (rule major [THEN imageE])
haftmann@26358
   752
  apply (rule_tac p = x in PairE)
haftmann@26358
   753
  apply (rule cases)
haftmann@26358
   754
   apply (blast intro: prod_fun)
haftmann@26358
   755
  apply blast
haftmann@26358
   756
  done
haftmann@26358
   757
haftmann@26358
   758
definition
haftmann@26358
   759
  apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b"
haftmann@26358
   760
where
haftmann@28562
   761
  [code del]: "apfst f = prod_fun f id"
haftmann@26358
   762
haftmann@26358
   763
definition
haftmann@26358
   764
  apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c"
haftmann@26358
   765
where
haftmann@28562
   766
  [code del]: "apsnd f = prod_fun id f"
haftmann@26358
   767
haftmann@26358
   768
lemma apfst_conv [simp, code]:
haftmann@26358
   769
  "apfst f (x, y) = (f x, y)" 
haftmann@26358
   770
  by (simp add: apfst_def)
haftmann@26358
   771
haftmann@26358
   772
lemma upd_snd_conv [simp, code]:
haftmann@26358
   773
  "apsnd f (x, y) = (x, f y)" 
haftmann@26358
   774
  by (simp add: apsnd_def)
haftmann@21195
   775
haftmann@21195
   776
haftmann@26358
   777
text {*
haftmann@26358
   778
  Disjoint union of a family of sets -- Sigma.
haftmann@26358
   779
*}
haftmann@26358
   780
haftmann@26358
   781
definition  Sigma :: "['a set, 'a => 'b set] => ('a \<times> 'b) set" where
haftmann@26358
   782
  Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}"
haftmann@26358
   783
haftmann@26358
   784
abbreviation
haftmann@26358
   785
  Times :: "['a set, 'b set] => ('a * 'b) set"
haftmann@26358
   786
    (infixr "<*>" 80) where
haftmann@26358
   787
  "A <*> B == Sigma A (%_. B)"
haftmann@26358
   788
haftmann@26358
   789
notation (xsymbols)
haftmann@26358
   790
  Times  (infixr "\<times>" 80)
berghofe@15394
   791
haftmann@26358
   792
notation (HTML output)
haftmann@26358
   793
  Times  (infixr "\<times>" 80)
haftmann@26358
   794
haftmann@26358
   795
syntax
haftmann@26358
   796
  "@Sigma" ::"[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" [0, 0, 10] 10)
haftmann@26358
   797
haftmann@26358
   798
translations
haftmann@26358
   799
  "SIGMA x:A. B" == "Product_Type.Sigma A (%x. B)"
haftmann@26358
   800
haftmann@26358
   801
lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
haftmann@26358
   802
  by (unfold Sigma_def) blast
haftmann@26358
   803
haftmann@26358
   804
lemma SigmaE [elim!]:
haftmann@26358
   805
    "[| c: Sigma A B;
haftmann@26358
   806
        !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
haftmann@26358
   807
     |] ==> P"
haftmann@26358
   808
  -- {* The general elimination rule. *}
haftmann@26358
   809
  by (unfold Sigma_def) blast
haftmann@20588
   810
haftmann@26358
   811
text {*
haftmann@26358
   812
  Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
haftmann@26358
   813
  eigenvariables.
haftmann@26358
   814
*}
haftmann@26358
   815
haftmann@26358
   816
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
haftmann@26358
   817
  by blast
haftmann@26358
   818
haftmann@26358
   819
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
haftmann@26358
   820
  by blast
haftmann@26358
   821
haftmann@26358
   822
lemma SigmaE2:
haftmann@26358
   823
    "[| (a, b) : Sigma A B;
haftmann@26358
   824
        [| a:A;  b:B(a) |] ==> P
haftmann@26358
   825
     |] ==> P"
haftmann@26358
   826
  by blast
haftmann@20588
   827
haftmann@26358
   828
lemma Sigma_cong:
haftmann@26358
   829
     "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>
haftmann@26358
   830
      \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
haftmann@26358
   831
  by auto
haftmann@26358
   832
haftmann@26358
   833
lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
haftmann@26358
   834
  by blast
haftmann@26358
   835
haftmann@26358
   836
lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
haftmann@26358
   837
  by blast
haftmann@26358
   838
haftmann@26358
   839
lemma Sigma_empty2 [simp]: "A <*> {} = {}"
haftmann@26358
   840
  by blast
haftmann@26358
   841
haftmann@26358
   842
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
haftmann@26358
   843
  by auto
haftmann@21908
   844
haftmann@26358
   845
lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
haftmann@26358
   846
  by auto
haftmann@26358
   847
haftmann@26358
   848
lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
haftmann@26358
   849
  by auto
haftmann@26358
   850
haftmann@26358
   851
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
haftmann@26358
   852
  by blast
haftmann@26358
   853
haftmann@26358
   854
lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
haftmann@26358
   855
  by blast
haftmann@26358
   856
haftmann@26358
   857
lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
haftmann@26358
   858
  by (blast elim: equalityE)
haftmann@20588
   859
haftmann@26358
   860
lemma SetCompr_Sigma_eq:
haftmann@26358
   861
    "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
haftmann@26358
   862
  by blast
haftmann@26358
   863
haftmann@26358
   864
lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"
haftmann@26358
   865
  by blast
haftmann@26358
   866
haftmann@26358
   867
lemma UN_Times_distrib:
haftmann@26358
   868
  "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"
haftmann@26358
   869
  -- {* Suggested by Pierre Chartier *}
haftmann@26358
   870
  by blast
haftmann@26358
   871
haftmann@26358
   872
lemma split_paired_Ball_Sigma [simp,noatp]:
haftmann@26358
   873
    "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"
haftmann@26358
   874
  by blast
haftmann@26358
   875
haftmann@26358
   876
lemma split_paired_Bex_Sigma [simp,noatp]:
haftmann@26358
   877
    "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"
haftmann@26358
   878
  by blast
haftmann@21908
   879
haftmann@26358
   880
lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"
haftmann@26358
   881
  by blast
haftmann@26358
   882
haftmann@26358
   883
lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))"
haftmann@26358
   884
  by blast
haftmann@26358
   885
haftmann@26358
   886
lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"
haftmann@26358
   887
  by blast
haftmann@26358
   888
haftmann@26358
   889
lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))"
haftmann@26358
   890
  by blast
haftmann@26358
   891
haftmann@26358
   892
lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"
haftmann@26358
   893
  by blast
haftmann@26358
   894
haftmann@26358
   895
lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"
haftmann@26358
   896
  by blast
haftmann@21908
   897
haftmann@26358
   898
lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"
haftmann@26358
   899
  by blast
haftmann@26358
   900
haftmann@26358
   901
text {*
haftmann@26358
   902
  Non-dependent versions are needed to avoid the need for higher-order
haftmann@26358
   903
  matching, especially when the rules are re-oriented.
haftmann@26358
   904
*}
haftmann@21908
   905
haftmann@26358
   906
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"
nipkow@28719
   907
by blast
haftmann@26358
   908
haftmann@26358
   909
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"
nipkow@28719
   910
by blast
haftmann@26358
   911
haftmann@26358
   912
lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
nipkow@28719
   913
by blast
haftmann@26358
   914
nipkow@28719
   915
lemma insert_times_insert[simp]:
nipkow@28719
   916
  "insert a A \<times> insert b B =
nipkow@28719
   917
   insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)"
nipkow@28719
   918
by blast
haftmann@26358
   919
haftmann@26358
   920
subsubsection {* Code generator setup *}
haftmann@21908
   921
haftmann@20588
   922
instance * :: (eq, eq) eq ..
haftmann@20588
   923
haftmann@28562
   924
lemma [code]:
haftmann@28346
   925
  "eq_class.eq (x1\<Colon>'a\<Colon>eq, y1\<Colon>'b\<Colon>eq) (x2, y2) \<longleftrightarrow> x1 = x2 \<and> y1 = y2" by (simp add: eq)
haftmann@20588
   926
haftmann@24844
   927
lemma split_case_cert:
haftmann@24844
   928
  assumes "CASE \<equiv> split f"
haftmann@24844
   929
  shows "CASE (a, b) \<equiv> f a b"
haftmann@24844
   930
  using assms by simp
haftmann@24844
   931
haftmann@24844
   932
setup {*
haftmann@24844
   933
  Code.add_case @{thm split_case_cert}
haftmann@24844
   934
*}
haftmann@24844
   935
haftmann@21908
   936
code_type *
haftmann@21908
   937
  (SML infix 2 "*")
haftmann@21908
   938
  (OCaml infix 2 "*")
haftmann@21908
   939
  (Haskell "!((_),/ (_))")
haftmann@21908
   940
haftmann@20588
   941
code_instance * :: eq
haftmann@20588
   942
  (Haskell -)
haftmann@20588
   943
haftmann@28346
   944
code_const "eq_class.eq \<Colon> 'a\<Colon>eq \<times> 'b\<Colon>eq \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"
haftmann@20588
   945
  (Haskell infixl 4 "==")
haftmann@20588
   946
haftmann@21908
   947
code_const Pair
haftmann@21908
   948
  (SML "!((_),/ (_))")
haftmann@21908
   949
  (OCaml "!((_),/ (_))")
haftmann@21908
   950
  (Haskell "!((_),/ (_))")
haftmann@20588
   951
haftmann@22389
   952
code_const fst and snd
haftmann@22389
   953
  (Haskell "fst" and "snd")
haftmann@22389
   954
berghofe@15394
   955
types_code
berghofe@15394
   956
  "*"     ("(_ */ _)")
berghofe@16770
   957
attach (term_of) {*
berghofe@25885
   958
fun term_of_id_42 aF aT bF bT (x, y) = HOLogic.pair_const aT bT $ aF x $ bF y;
berghofe@16770
   959
*}
berghofe@16770
   960
attach (test) {*
berghofe@25885
   961
fun gen_id_42 aG aT bG bT i =
berghofe@25885
   962
  let
berghofe@25885
   963
    val (x, t) = aG i;
berghofe@25885
   964
    val (y, u) = bG i
berghofe@25885
   965
  in ((x, y), fn () => HOLogic.pair_const aT bT $ t () $ u ()) end;
berghofe@16770
   966
*}
berghofe@15394
   967
berghofe@18706
   968
consts_code
berghofe@18706
   969
  "Pair"    ("(_,/ _)")
berghofe@18706
   970
haftmann@21908
   971
setup {*
haftmann@21908
   972
haftmann@21908
   973
let
haftmann@18013
   974
haftmann@19039
   975
fun strip_abs_split 0 t = ([], t)
haftmann@19039
   976
  | strip_abs_split i (Abs (s, T, t)) =
haftmann@18013
   977
      let
haftmann@18013
   978
        val s' = Codegen.new_name t s;
haftmann@18013
   979
        val v = Free (s', T)
haftmann@19039
   980
      in apfst (cons v) (strip_abs_split (i-1) (subst_bound (v, t))) end
haftmann@19039
   981
  | strip_abs_split i (u as Const ("split", _) $ t) = (case strip_abs_split (i+1) t of
berghofe@15394
   982
        (v :: v' :: vs, u) => (HOLogic.mk_prod (v, v') :: vs, u)
berghofe@15394
   983
      | _ => ([], u))
haftmann@19039
   984
  | strip_abs_split i t = ([], t);
haftmann@18013
   985
haftmann@28537
   986
fun let_codegen thy defs dep thyname brack t gr = (case strip_comb t of
berghofe@16634
   987
    (t1 as Const ("Let", _), t2 :: t3 :: ts) =>
berghofe@15394
   988
    let
berghofe@15394
   989
      fun dest_let (l as Const ("Let", _) $ t $ u) =
haftmann@19039
   990
          (case strip_abs_split 1 u of
berghofe@15394
   991
             ([p], u') => apfst (cons (p, t)) (dest_let u')
berghofe@15394
   992
           | _ => ([], l))
berghofe@15394
   993
        | dest_let t = ([], t);
haftmann@28537
   994
      fun mk_code (l, r) gr =
berghofe@15394
   995
        let
haftmann@28537
   996
          val (pl, gr1) = Codegen.invoke_codegen thy defs dep thyname false l gr;
haftmann@28537
   997
          val (pr, gr2) = Codegen.invoke_codegen thy defs dep thyname false r gr1;
haftmann@28537
   998
        in ((pl, pr), gr2) end
berghofe@16634
   999
    in case dest_let (t1 $ t2 $ t3) of
skalberg@15531
  1000
        ([], _) => NONE
berghofe@15394
  1001
      | (ps, u) =>
berghofe@15394
  1002
          let
haftmann@28537
  1003
            val (qs, gr1) = fold_map mk_code ps gr;
haftmann@28537
  1004
            val (pu, gr2) = Codegen.invoke_codegen thy defs dep thyname false u gr1;
haftmann@28537
  1005
            val (pargs, gr3) = fold_map
haftmann@28537
  1006
              (Codegen.invoke_codegen thy defs dep thyname true) ts gr2
berghofe@15394
  1007
          in
haftmann@28537
  1008
            SOME (Codegen.mk_app brack
berghofe@26975
  1009
              (Pretty.blk (0, [Codegen.str "let ", Pretty.blk (0, List.concat
berghofe@26975
  1010
                  (separate [Codegen.str ";", Pretty.brk 1] (map (fn (pl, pr) =>
berghofe@26975
  1011
                    [Pretty.block [Codegen.str "val ", pl, Codegen.str " =",
berghofe@16634
  1012
                       Pretty.brk 1, pr]]) qs))),
berghofe@26975
  1013
                Pretty.brk 1, Codegen.str "in ", pu,
haftmann@28537
  1014
                Pretty.brk 1, Codegen.str "end"])) pargs, gr3)
berghofe@15394
  1015
          end
berghofe@15394
  1016
    end
berghofe@16634
  1017
  | _ => NONE);
berghofe@15394
  1018
haftmann@28537
  1019
fun split_codegen thy defs dep thyname brack t gr = (case strip_comb t of
berghofe@16634
  1020
    (t1 as Const ("split", _), t2 :: ts) =>
haftmann@19039
  1021
      (case strip_abs_split 1 (t1 $ t2) of
berghofe@16634
  1022
         ([p], u) =>
berghofe@16634
  1023
           let
haftmann@28537
  1024
             val (q, gr1) = Codegen.invoke_codegen thy defs dep thyname false p gr;
haftmann@28537
  1025
             val (pu, gr2) = Codegen.invoke_codegen thy defs dep thyname false u gr1;
haftmann@28537
  1026
             val (pargs, gr3) = fold_map
haftmann@28537
  1027
               (Codegen.invoke_codegen thy defs dep thyname true) ts gr2
berghofe@16634
  1028
           in
haftmann@28537
  1029
             SOME (Codegen.mk_app brack
berghofe@26975
  1030
               (Pretty.block [Codegen.str "(fn ", q, Codegen.str " =>",
haftmann@28537
  1031
                 Pretty.brk 1, pu, Codegen.str ")"]) pargs, gr2)
berghofe@16634
  1032
           end
berghofe@16634
  1033
       | _ => NONE)
berghofe@16634
  1034
  | _ => NONE);
berghofe@15394
  1035
haftmann@21908
  1036
in
haftmann@21908
  1037
haftmann@20105
  1038
  Codegen.add_codegen "let_codegen" let_codegen
haftmann@20105
  1039
  #> Codegen.add_codegen "split_codegen" split_codegen
berghofe@15394
  1040
haftmann@21908
  1041
end
berghofe@15394
  1042
*}
berghofe@15394
  1043
haftmann@24699
  1044
haftmann@24699
  1045
subsection {* Legacy bindings *}
haftmann@24699
  1046
haftmann@21908
  1047
ML {*
paulson@15404
  1048
val Collect_split = thm "Collect_split";
paulson@15404
  1049
val Compl_Times_UNIV1 = thm "Compl_Times_UNIV1";
paulson@15404
  1050
val Compl_Times_UNIV2 = thm "Compl_Times_UNIV2";
paulson@15404
  1051
val PairE = thm "PairE";
paulson@15404
  1052
val Pair_Rep_inject = thm "Pair_Rep_inject";
paulson@15404
  1053
val Pair_def = thm "Pair_def";
haftmann@27104
  1054
val Pair_eq = @{thm "prod.inject"};
paulson@15404
  1055
val Pair_fst_snd_eq = thm "Pair_fst_snd_eq";
paulson@15404
  1056
val ProdI = thm "ProdI";
paulson@15404
  1057
val SetCompr_Sigma_eq = thm "SetCompr_Sigma_eq";
paulson@15404
  1058
val SigmaD1 = thm "SigmaD1";
paulson@15404
  1059
val SigmaD2 = thm "SigmaD2";
paulson@15404
  1060
val SigmaE = thm "SigmaE";
paulson@15404
  1061
val SigmaE2 = thm "SigmaE2";
paulson@15404
  1062
val SigmaI = thm "SigmaI";
paulson@15404
  1063
val Sigma_Diff_distrib1 = thm "Sigma_Diff_distrib1";
paulson@15404
  1064
val Sigma_Diff_distrib2 = thm "Sigma_Diff_distrib2";
paulson@15404
  1065
val Sigma_Int_distrib1 = thm "Sigma_Int_distrib1";
paulson@15404
  1066
val Sigma_Int_distrib2 = thm "Sigma_Int_distrib2";
paulson@15404
  1067
val Sigma_Un_distrib1 = thm "Sigma_Un_distrib1";
paulson@15404
  1068
val Sigma_Un_distrib2 = thm "Sigma_Un_distrib2";
paulson@15404
  1069
val Sigma_Union = thm "Sigma_Union";
paulson@15404
  1070
val Sigma_def = thm "Sigma_def";
paulson@15404
  1071
val Sigma_empty1 = thm "Sigma_empty1";
paulson@15404
  1072
val Sigma_empty2 = thm "Sigma_empty2";
paulson@15404
  1073
val Sigma_mono = thm "Sigma_mono";
paulson@15404
  1074
val The_split = thm "The_split";
paulson@15404
  1075
val The_split_eq = thm "The_split_eq";
paulson@15404
  1076
val The_split_eq = thm "The_split_eq";
paulson@15404
  1077
val Times_Diff_distrib1 = thm "Times_Diff_distrib1";
paulson@15404
  1078
val Times_Int_distrib1 = thm "Times_Int_distrib1";
paulson@15404
  1079
val Times_Un_distrib1 = thm "Times_Un_distrib1";
paulson@15404
  1080
val Times_eq_cancel2 = thm "Times_eq_cancel2";
paulson@15404
  1081
val Times_subset_cancel2 = thm "Times_subset_cancel2";
paulson@15404
  1082
val UNIV_Times_UNIV = thm "UNIV_Times_UNIV";
paulson@15404
  1083
val UN_Times_distrib = thm "UN_Times_distrib";
paulson@15404
  1084
val Unity_def = thm "Unity_def";
paulson@15404
  1085
val cond_split_eta = thm "cond_split_eta";
paulson@15404
  1086
val fst_conv = thm "fst_conv";
paulson@15404
  1087
val fst_def = thm "fst_def";
paulson@15404
  1088
val fst_eqD = thm "fst_eqD";
paulson@15404
  1089
val inj_on_Abs_Prod = thm "inj_on_Abs_Prod";
paulson@15404
  1090
val mem_Sigma_iff = thm "mem_Sigma_iff";
paulson@15404
  1091
val mem_splitE = thm "mem_splitE";
paulson@15404
  1092
val mem_splitI = thm "mem_splitI";
paulson@15404
  1093
val mem_splitI2 = thm "mem_splitI2";
paulson@15404
  1094
val prod_eqI = thm "prod_eqI";
paulson@15404
  1095
val prod_fun = thm "prod_fun";
paulson@15404
  1096
val prod_fun_compose = thm "prod_fun_compose";
paulson@15404
  1097
val prod_fun_def = thm "prod_fun_def";
paulson@15404
  1098
val prod_fun_ident = thm "prod_fun_ident";
paulson@15404
  1099
val prod_fun_imageE = thm "prod_fun_imageE";
paulson@15404
  1100
val prod_fun_imageI = thm "prod_fun_imageI";
haftmann@27104
  1101
val prod_induct = thm "prod.induct";
paulson@15404
  1102
val snd_conv = thm "snd_conv";
paulson@15404
  1103
val snd_def = thm "snd_def";
paulson@15404
  1104
val snd_eqD = thm "snd_eqD";
paulson@15404
  1105
val split = thm "split";
paulson@15404
  1106
val splitD = thm "splitD";
paulson@15404
  1107
val splitD' = thm "splitD'";
paulson@15404
  1108
val splitE = thm "splitE";
paulson@15404
  1109
val splitE' = thm "splitE'";
paulson@15404
  1110
val splitE2 = thm "splitE2";
paulson@15404
  1111
val splitI = thm "splitI";
paulson@15404
  1112
val splitI2 = thm "splitI2";
paulson@15404
  1113
val splitI2' = thm "splitI2'";
paulson@15404
  1114
val split_beta = thm "split_beta";
paulson@15404
  1115
val split_conv = thm "split_conv";
paulson@15404
  1116
val split_def = thm "split_def";
paulson@15404
  1117
val split_eta = thm "split_eta";
paulson@15404
  1118
val split_eta_SetCompr = thm "split_eta_SetCompr";
paulson@15404
  1119
val split_eta_SetCompr2 = thm "split_eta_SetCompr2";
paulson@15404
  1120
val split_paired_All = thm "split_paired_All";
paulson@15404
  1121
val split_paired_Ball_Sigma = thm "split_paired_Ball_Sigma";
paulson@15404
  1122
val split_paired_Bex_Sigma = thm "split_paired_Bex_Sigma";
paulson@15404
  1123
val split_paired_Ex = thm "split_paired_Ex";
paulson@15404
  1124
val split_paired_The = thm "split_paired_The";
paulson@15404
  1125
val split_paired_all = thm "split_paired_all";
paulson@15404
  1126
val split_part = thm "split_part";
paulson@15404
  1127
val split_split = thm "split_split";
paulson@15404
  1128
val split_split_asm = thm "split_split_asm";
paulson@15404
  1129
val split_tupled_all = thms "split_tupled_all";
paulson@15404
  1130
val split_weak_cong = thm "split_weak_cong";
paulson@15404
  1131
val surj_pair = thm "surj_pair";
paulson@15404
  1132
val surjective_pairing = thm "surjective_pairing";
paulson@15404
  1133
val unit_abs_eta_conv = thm "unit_abs_eta_conv";
paulson@15404
  1134
val unit_all_eq1 = thm "unit_all_eq1";
paulson@15404
  1135
val unit_all_eq2 = thm "unit_all_eq2";
paulson@15404
  1136
val unit_eq = thm "unit_eq";
paulson@15404
  1137
*}
paulson@15404
  1138
haftmann@24699
  1139
haftmann@24699
  1140
subsection {* Further inductive packages *}
haftmann@24699
  1141
haftmann@24699
  1142
use "Tools/inductive_realizer.ML"
haftmann@24699
  1143
setup InductiveRealizer.setup
haftmann@24699
  1144
haftmann@24699
  1145
use "Tools/inductive_set_package.ML"
haftmann@24699
  1146
setup InductiveSetPackage.setup
haftmann@24699
  1147
haftmann@24699
  1148
use "Tools/datatype_realizer.ML"
haftmann@24699
  1149
setup DatatypeRealizer.setup
haftmann@24699
  1150
nipkow@10213
  1151
end