src/HOL/Product_Type.thy
author bulwahn
Tue Feb 26 11:18:43 2008 +0100 (2008-02-26)
changeset 26143 314c0bcb7df7
parent 25885 6fbc3f54f819
child 26340 a85fe32e7b2f
permissions -rw-r--r--
Added useful general lemmas from the work with the HeapMonad
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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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lemma [code func]:
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  shows "False = P \<longleftrightarrow> \<not> P"
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    and "True = P \<longleftrightarrow> P" 
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    and "P = False \<longleftrightarrow> \<not> P" 
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    and "P = True \<longleftrightarrow> P" by simp_all
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code_const "op = \<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_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
wenzelm@11838
   332
      | exists_paired_all (Abs (_, _, t)) = exists_paired_all t
wenzelm@11838
   333
      | exists_paired_all _ = false;
wenzelm@11838
   334
    val ss = HOL_basic_ss
wenzelm@16121
   335
      addsimps [thm "split_paired_all", thm "unit_all_eq2", thm "unit_abs_eta_conv"]
wenzelm@11838
   336
      addsimprocs [unit_eq_proc];
wenzelm@11838
   337
  in
wenzelm@11838
   338
    val split_all_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838
   339
      if exists_paired_all t then safe_full_simp_tac ss i else no_tac);
wenzelm@11838
   340
    val unsafe_split_all_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838
   341
      if exists_paired_all t then full_simp_tac ss i else no_tac);
wenzelm@11838
   342
    fun split_all th =
wenzelm@11838
   343
   if exists_paired_all (#prop (Thm.rep_thm th)) then full_simplify ss th else th;
wenzelm@11838
   344
  end;
wenzelm@11838
   345
wenzelm@17875
   346
change_claset (fn cs => cs addSbefore ("split_all_tac", split_all_tac));
wenzelm@16121
   347
*}
wenzelm@11838
   348
wenzelm@11838
   349
lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))"
wenzelm@11838
   350
  -- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *}
wenzelm@11838
   351
  by fast
wenzelm@11838
   352
skalberg@14189
   353
lemma curry_split [simp]: "curry (split f) = f"
skalberg@14189
   354
  by (simp add: curry_def split_def)
skalberg@14189
   355
skalberg@14189
   356
lemma split_curry [simp]: "split (curry f) = f"
skalberg@14189
   357
  by (simp add: curry_def split_def)
skalberg@14189
   358
skalberg@14189
   359
lemma curryI [intro!]: "f (a,b) ==> curry f a b"
skalberg@14189
   360
  by (simp add: curry_def)
skalberg@14189
   361
skalberg@14190
   362
lemma curryD [dest!]: "curry f a b ==> f (a,b)"
skalberg@14189
   363
  by (simp add: curry_def)
skalberg@14189
   364
skalberg@14190
   365
lemma curryE: "[| curry f a b ; f (a,b) ==> Q |] ==> Q"
skalberg@14189
   366
  by (simp add: curry_def)
skalberg@14189
   367
haftmann@24162
   368
lemma curry_conv [simp, code func]: "curry f a b = f (a,b)"
skalberg@14189
   369
  by (simp add: curry_def)
skalberg@14189
   370
wenzelm@11838
   371
lemma prod_induct [induct type: *]: "!!x. (!!a b. P (a, b)) ==> P x"
wenzelm@11838
   372
  by fast
wenzelm@11838
   373
haftmann@24699
   374
rep_datatype prod
haftmann@24699
   375
  inject Pair_eq
haftmann@24699
   376
  induction prod_induct
haftmann@24699
   377
wenzelm@11838
   378
lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))"
wenzelm@11838
   379
  by fast
wenzelm@11838
   380
haftmann@24162
   381
lemma split_conv [simp, code func]: "split c (a, b) = c a b"
wenzelm@11838
   382
  by (simp add: split_def)
wenzelm@11838
   383
wenzelm@11838
   384
lemmas split = split_conv  -- {* for backwards compatibility *}
wenzelm@11838
   385
wenzelm@11838
   386
lemmas splitI = split_conv [THEN iffD2, standard]
wenzelm@11838
   387
lemmas splitD = split_conv [THEN iffD1, standard]
wenzelm@11838
   388
wenzelm@11838
   389
lemma split_Pair_apply: "split (%x y. f (x, y)) = f"
wenzelm@11838
   390
  -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *}
wenzelm@11838
   391
  apply (rule ext)
paulson@14208
   392
  apply (tactic {* pair_tac "x" 1 *}, simp)
wenzelm@11838
   393
  done
wenzelm@11838
   394
wenzelm@11838
   395
lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))"
wenzelm@11838
   396
  -- {* Can't be added to simpset: loops! *}
wenzelm@11838
   397
  by (simp add: split_Pair_apply)
wenzelm@11838
   398
wenzelm@11838
   399
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
wenzelm@11838
   400
  by (simp add: split_def)
wenzelm@11838
   401
wenzelm@11838
   402
lemma Pair_fst_snd_eq: "!!s t. (s = t) = (fst s = fst t & snd s = snd t)"
paulson@14208
   403
by (simp only: split_tupled_all, simp)
wenzelm@11838
   404
wenzelm@11838
   405
lemma prod_eqI [intro?]: "fst p = fst q ==> snd p = snd q ==> p = q"
wenzelm@11838
   406
  by (simp add: Pair_fst_snd_eq)
wenzelm@11838
   407
wenzelm@11838
   408
lemma split_weak_cong: "p = q ==> split c p = split c q"
wenzelm@11838
   409
  -- {* Prevents simplification of @{term c}: much faster *}
wenzelm@11838
   410
  by (erule arg_cong)
wenzelm@11838
   411
wenzelm@11838
   412
lemma split_eta: "(%(x, y). f (x, y)) = f"
wenzelm@11838
   413
  apply (rule ext)
wenzelm@11838
   414
  apply (simp only: split_tupled_all)
wenzelm@11838
   415
  apply (rule split_conv)
wenzelm@11838
   416
  done
wenzelm@11838
   417
wenzelm@11838
   418
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
wenzelm@11838
   419
  by (simp add: split_eta)
wenzelm@11838
   420
wenzelm@11838
   421
text {*
wenzelm@11838
   422
  Simplification procedure for @{thm [source] cond_split_eta}.  Using
wenzelm@11838
   423
  @{thm [source] split_eta} as a rewrite rule is not general enough,
wenzelm@11838
   424
  and using @{thm [source] cond_split_eta} directly would render some
wenzelm@11838
   425
  existing proofs very inefficient; similarly for @{text
wenzelm@11838
   426
  split_beta}. *}
wenzelm@11838
   427
wenzelm@11838
   428
ML_setup {*
wenzelm@11838
   429
wenzelm@11838
   430
local
wenzelm@18328
   431
  val cond_split_eta_ss = HOL_basic_ss addsimps [thm "cond_split_eta"]
wenzelm@11838
   432
  fun  Pair_pat k 0 (Bound m) = (m = k)
wenzelm@11838
   433
  |    Pair_pat k i (Const ("Pair",  _) $ Bound m $ t) = i > 0 andalso
wenzelm@11838
   434
                        m = k+i andalso Pair_pat k (i-1) t
wenzelm@11838
   435
  |    Pair_pat _ _ _ = false;
wenzelm@11838
   436
  fun no_args k i (Abs (_, _, t)) = no_args (k+1) i t
wenzelm@11838
   437
  |   no_args k i (t $ u) = no_args k i t andalso no_args k i u
wenzelm@11838
   438
  |   no_args k i (Bound m) = m < k orelse m > k+i
wenzelm@11838
   439
  |   no_args _ _ _ = true;
skalberg@15531
   440
  fun split_pat tp i (Abs  (_,_,t)) = if tp 0 i t then SOME (i,t) else NONE
wenzelm@11838
   441
  |   split_pat tp i (Const ("split", _) $ Abs (_, _, t)) = split_pat tp (i+1) t
skalberg@15531
   442
  |   split_pat tp i _ = NONE;
wenzelm@20044
   443
  fun metaeq ss lhs rhs = mk_meta_eq (Goal.prove (Simplifier.the_context ss) [] []
wenzelm@13480
   444
        (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs,rhs)))
wenzelm@18328
   445
        (K (simp_tac (Simplifier.inherit_context ss cond_split_eta_ss) 1)));
wenzelm@11838
   446
wenzelm@11838
   447
  fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k+1) i t
wenzelm@11838
   448
  |   beta_term_pat k i (t $ u) = Pair_pat k i (t $ u) orelse
wenzelm@11838
   449
                        (beta_term_pat k i t andalso beta_term_pat k i u)
wenzelm@11838
   450
  |   beta_term_pat k i t = no_args k i t;
wenzelm@11838
   451
  fun  eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
wenzelm@11838
   452
  |    eta_term_pat _ _ _ = false;
wenzelm@11838
   453
  fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
wenzelm@11838
   454
  |   subst arg k i (t $ u) = if Pair_pat k i (t $ u) then incr_boundvars k arg
wenzelm@11838
   455
                              else (subst arg k i t $ subst arg k i u)
wenzelm@11838
   456
  |   subst arg k i t = t;
wenzelm@20044
   457
  fun beta_proc ss (s as Const ("split", _) $ Abs (_, _, t) $ arg) =
wenzelm@11838
   458
        (case split_pat beta_term_pat 1 t of
wenzelm@20044
   459
        SOME (i,f) => SOME (metaeq ss s (subst arg 0 i f))
skalberg@15531
   460
        | NONE => NONE)
wenzelm@20044
   461
  |   beta_proc _ _ = NONE;
wenzelm@20044
   462
  fun eta_proc ss (s as Const ("split", _) $ Abs (_, _, t)) =
wenzelm@11838
   463
        (case split_pat eta_term_pat 1 t of
wenzelm@20044
   464
          SOME (_,ft) => SOME (metaeq ss s (let val (f $ arg) = ft in f end))
skalberg@15531
   465
        | NONE => NONE)
wenzelm@20044
   466
  |   eta_proc _ _ = NONE;
wenzelm@11838
   467
in
wenzelm@22577
   468
  val split_beta_proc = Simplifier.simproc @{theory} "split_beta" ["split f z"] (K beta_proc);
wenzelm@22577
   469
  val split_eta_proc = Simplifier.simproc @{theory} "split_eta" ["split f"] (K eta_proc);
wenzelm@11838
   470
end;
wenzelm@11838
   471
wenzelm@11838
   472
Addsimprocs [split_beta_proc, split_eta_proc];
wenzelm@11838
   473
*}
wenzelm@11838
   474
wenzelm@11838
   475
lemma split_beta: "(%(x, y). P x y) z = P (fst z) (snd z)"
wenzelm@11838
   476
  by (subst surjective_pairing, rule split_conv)
wenzelm@11838
   477
paulson@24286
   478
lemma split_split [noatp]: "R(split c p) = (ALL x y. p = (x, y) --> R(c x y))"
wenzelm@11838
   479
  -- {* For use with @{text split} and the Simplifier. *}
paulson@15481
   480
  by (insert surj_pair [of p], clarify, simp)
wenzelm@11838
   481
wenzelm@11838
   482
text {*
wenzelm@11838
   483
  @{thm [source] split_split} could be declared as @{text "[split]"}
wenzelm@11838
   484
  done after the Splitter has been speeded up significantly;
wenzelm@11838
   485
  precompute the constants involved and don't do anything unless the
wenzelm@11838
   486
  current goal contains one of those constants.
wenzelm@11838
   487
*}
wenzelm@11838
   488
paulson@24286
   489
lemma split_split_asm [noatp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))"
paulson@14208
   490
by (subst split_split, simp)
wenzelm@11838
   491
wenzelm@11838
   492
wenzelm@11838
   493
text {*
wenzelm@11838
   494
  \medskip @{term split} used as a logical connective or set former.
wenzelm@11838
   495
wenzelm@11838
   496
  \medskip These rules are for use with @{text blast}; could instead
wenzelm@11838
   497
  call @{text simp} using @{thm [source] split} as rewrite. *}
wenzelm@11838
   498
wenzelm@11838
   499
lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"
wenzelm@11838
   500
  apply (simp only: split_tupled_all)
wenzelm@11838
   501
  apply (simp (no_asm_simp))
wenzelm@11838
   502
  done
wenzelm@11838
   503
wenzelm@11838
   504
lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"
wenzelm@11838
   505
  apply (simp only: split_tupled_all)
wenzelm@11838
   506
  apply (simp (no_asm_simp))
wenzelm@11838
   507
  done
wenzelm@11838
   508
wenzelm@11838
   509
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
wenzelm@11838
   510
  by (induct p) (auto simp add: split_def)
wenzelm@11838
   511
wenzelm@11838
   512
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
wenzelm@11838
   513
  by (induct p) (auto simp add: split_def)
wenzelm@11838
   514
wenzelm@11838
   515
lemma splitE2:
wenzelm@11838
   516
  "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
wenzelm@11838
   517
proof -
wenzelm@11838
   518
  assume q: "Q (split P z)"
wenzelm@11838
   519
  assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
wenzelm@11838
   520
  show R
wenzelm@11838
   521
    apply (rule r surjective_pairing)+
wenzelm@11838
   522
    apply (rule split_beta [THEN subst], rule q)
wenzelm@11838
   523
    done
wenzelm@11838
   524
qed
wenzelm@11838
   525
wenzelm@11838
   526
lemma splitD': "split R (a,b) c ==> R a b c"
wenzelm@11838
   527
  by simp
wenzelm@11838
   528
wenzelm@11838
   529
lemma mem_splitI: "z: c a b ==> z: split c (a, b)"
wenzelm@11838
   530
  by simp
wenzelm@11838
   531
wenzelm@11838
   532
lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"
paulson@14208
   533
by (simp only: split_tupled_all, simp)
wenzelm@11838
   534
wenzelm@18372
   535
lemma mem_splitE:
wenzelm@18372
   536
  assumes major: "z: split c p"
wenzelm@18372
   537
    and cases: "!!x y. [| p = (x,y); z: c x y |] ==> Q"
wenzelm@18372
   538
  shows Q
wenzelm@18372
   539
  by (rule major [unfolded split_def] cases surjective_pairing)+
wenzelm@11838
   540
wenzelm@11838
   541
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
wenzelm@11838
   542
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]
wenzelm@11838
   543
wenzelm@16121
   544
ML_setup {*
wenzelm@11838
   545
local (* filtering with exists_p_split is an essential optimization *)
wenzelm@16121
   546
  fun exists_p_split (Const ("split",_) $ _ $ (Const ("Pair",_)$_$_)) = true
wenzelm@11838
   547
    | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
wenzelm@11838
   548
    | exists_p_split (Abs (_, _, t)) = exists_p_split t
wenzelm@11838
   549
    | exists_p_split _ = false;
wenzelm@16121
   550
  val ss = HOL_basic_ss addsimps [thm "split_conv"];
wenzelm@11838
   551
in
wenzelm@11838
   552
val split_conv_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838
   553
    if exists_p_split t then safe_full_simp_tac ss i else no_tac);
wenzelm@11838
   554
end;
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@17875
   557
change_claset (fn cs => cs addSbefore ("split_conv_tac", split_conv_tac));
wenzelm@16121
   558
*}
wenzelm@11838
   559
paulson@24286
   560
lemma split_eta_SetCompr [simp,noatp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
wenzelm@18372
   561
  by (rule ext) fast
wenzelm@11838
   562
paulson@24286
   563
lemma split_eta_SetCompr2 [simp,noatp]: "(%u. EX x y. u = (x, y) & P x y) = split P"
wenzelm@18372
   564
  by (rule ext) fast
wenzelm@11838
   565
wenzelm@11838
   566
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"
wenzelm@11838
   567
  -- {* Allows simplifications of nested splits in case of independent predicates. *}
wenzelm@18372
   568
  by (rule ext) blast
wenzelm@11838
   569
nipkow@14337
   570
(* Do NOT make this a simp rule as it
nipkow@14337
   571
   a) only helps in special situations
nipkow@14337
   572
   b) can lead to nontermination in the presence of split_def
nipkow@14337
   573
*)
nipkow@14337
   574
lemma split_comp_eq: 
paulson@20415
   575
  fixes f :: "'a => 'b => 'c" and g :: "'d => 'a"
paulson@20415
   576
  shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"
wenzelm@18372
   577
  by (rule ext) auto
oheimb@14101
   578
wenzelm@11838
   579
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
wenzelm@11838
   580
  by blast
wenzelm@11838
   581
wenzelm@11838
   582
(*
wenzelm@11838
   583
the following  would be slightly more general,
wenzelm@11838
   584
but cannot be used as rewrite rule:
wenzelm@11838
   585
### Cannot add premise as rewrite rule because it contains (type) unknowns:
wenzelm@11838
   586
### ?y = .x
wenzelm@11838
   587
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
paulson@14208
   588
by (rtac some_equality 1)
paulson@14208
   589
by ( Simp_tac 1)
paulson@14208
   590
by (split_all_tac 1)
paulson@14208
   591
by (Asm_full_simp_tac 1)
wenzelm@11838
   592
qed "The_split_eq";
wenzelm@11838
   593
*)
wenzelm@11838
   594
wenzelm@11838
   595
lemma injective_fst_snd: "!!x y. [|fst x = fst y; snd x = snd y|] ==> x = y"
wenzelm@11838
   596
  by auto
wenzelm@11838
   597
wenzelm@11838
   598
wenzelm@11838
   599
text {*
wenzelm@11838
   600
  \bigskip @{term prod_fun} --- action of the product functor upon
wenzelm@11838
   601
  functions.
wenzelm@11838
   602
*}
wenzelm@11838
   603
haftmann@24162
   604
lemma prod_fun [simp, code func]: "prod_fun f g (a, b) = (f a, g b)"
wenzelm@11838
   605
  by (simp add: prod_fun_def)
wenzelm@11838
   606
wenzelm@11838
   607
lemma prod_fun_compose: "prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)"
wenzelm@11838
   608
  apply (rule ext)
paulson@14208
   609
  apply (tactic {* pair_tac "x" 1 *}, simp)
wenzelm@11838
   610
  done
wenzelm@11838
   611
wenzelm@11838
   612
lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)"
wenzelm@11838
   613
  apply (rule ext)
paulson@14208
   614
  apply (tactic {* pair_tac "z" 1 *}, simp)
wenzelm@11838
   615
  done
wenzelm@11838
   616
wenzelm@11838
   617
lemma prod_fun_imageI [intro]: "(a, b) : r ==> (f a, g b) : prod_fun f g ` r"
wenzelm@11838
   618
  apply (rule image_eqI)
paulson@14208
   619
  apply (rule prod_fun [symmetric], assumption)
wenzelm@11838
   620
  done
wenzelm@11838
   621
wenzelm@11838
   622
lemma prod_fun_imageE [elim!]:
wenzelm@18372
   623
  assumes major: "c: (prod_fun f g)`r"
wenzelm@18372
   624
    and cases: "!!x y. [| c=(f(x),g(y));  (x,y):r |] ==> P"
wenzelm@18372
   625
  shows P
wenzelm@18372
   626
  apply (rule major [THEN imageE])
wenzelm@18372
   627
  apply (rule_tac p = x in PairE)
wenzelm@18372
   628
  apply (rule cases)
wenzelm@18372
   629
   apply (blast intro: prod_fun)
wenzelm@18372
   630
  apply blast
wenzelm@18372
   631
  done
wenzelm@11838
   632
wenzelm@11838
   633
haftmann@22744
   634
definition
haftmann@22744
   635
  upd_fst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b"
haftmann@22744
   636
where
haftmann@22886
   637
  [code func del]: "upd_fst f = prod_fun f id"
oheimb@14101
   638
haftmann@22744
   639
definition
haftmann@22744
   640
  upd_snd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c"
haftmann@22744
   641
where
haftmann@22886
   642
  [code func del]: "upd_snd f = prod_fun id f"
oheimb@14101
   643
haftmann@22886
   644
lemma upd_fst_conv [simp, code]:
haftmann@22744
   645
  "upd_fst f (x, y) = (f x, y)" 
wenzelm@18372
   646
  by (simp add: upd_fst_def)
oheimb@14101
   647
haftmann@22886
   648
lemma upd_snd_conv [simp, code]:
haftmann@22744
   649
  "upd_snd f (x, y) = (x, f y)" 
wenzelm@18372
   650
  by (simp add: upd_snd_def)
oheimb@14101
   651
wenzelm@11838
   652
text {*
wenzelm@11838
   653
  \bigskip Disjoint union of a family of sets -- Sigma.
wenzelm@11838
   654
*}
wenzelm@11838
   655
wenzelm@11838
   656
lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
wenzelm@11838
   657
  by (unfold Sigma_def) blast
wenzelm@11838
   658
paulson@14952
   659
lemma SigmaE [elim!]:
wenzelm@11838
   660
    "[| c: Sigma A B;
wenzelm@11838
   661
        !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
wenzelm@11838
   662
     |] ==> P"
wenzelm@11838
   663
  -- {* The general elimination rule. *}
wenzelm@11838
   664
  by (unfold Sigma_def) blast
wenzelm@11838
   665
wenzelm@11838
   666
text {*
schirmer@15422
   667
  Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
wenzelm@11838
   668
  eigenvariables.
wenzelm@11838
   669
*}
wenzelm@11838
   670
wenzelm@11838
   671
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
wenzelm@18372
   672
  by blast
wenzelm@11838
   673
wenzelm@11838
   674
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
wenzelm@18372
   675
  by blast
wenzelm@11838
   676
wenzelm@11838
   677
lemma SigmaE2:
wenzelm@11838
   678
    "[| (a, b) : Sigma A B;
wenzelm@11838
   679
        [| a:A;  b:B(a) |] ==> P
wenzelm@11838
   680
     |] ==> P"
paulson@14952
   681
  by blast
wenzelm@11838
   682
paulson@14952
   683
lemma Sigma_cong:
schirmer@15422
   684
     "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>
schirmer@15422
   685
      \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
wenzelm@18372
   686
  by auto
wenzelm@11838
   687
wenzelm@11838
   688
lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
wenzelm@11838
   689
  by blast
wenzelm@11838
   690
wenzelm@11838
   691
lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
wenzelm@11838
   692
  by blast
wenzelm@11838
   693
wenzelm@11838
   694
lemma Sigma_empty2 [simp]: "A <*> {} = {}"
wenzelm@11838
   695
  by blast
wenzelm@11838
   696
wenzelm@11838
   697
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
wenzelm@11838
   698
  by auto
wenzelm@11838
   699
wenzelm@11838
   700
lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
wenzelm@11838
   701
  by auto
wenzelm@11838
   702
wenzelm@11838
   703
lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
wenzelm@11838
   704
  by auto
wenzelm@11838
   705
wenzelm@11838
   706
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
wenzelm@11838
   707
  by blast
wenzelm@11838
   708
wenzelm@11838
   709
lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
wenzelm@11838
   710
  by blast
wenzelm@11838
   711
wenzelm@11838
   712
lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
wenzelm@11838
   713
  by (blast elim: equalityE)
wenzelm@11838
   714
wenzelm@11838
   715
lemma SetCompr_Sigma_eq:
wenzelm@11838
   716
    "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
wenzelm@11838
   717
  by blast
wenzelm@11838
   718
wenzelm@11838
   719
text {*
wenzelm@11838
   720
  \bigskip Complex rules for Sigma.
wenzelm@11838
   721
*}
wenzelm@11838
   722
wenzelm@11838
   723
lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"
wenzelm@11838
   724
  by blast
wenzelm@11838
   725
wenzelm@11838
   726
lemma UN_Times_distrib:
wenzelm@11838
   727
  "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"
wenzelm@11838
   728
  -- {* Suggested by Pierre Chartier *}
wenzelm@11838
   729
  by blast
wenzelm@11838
   730
paulson@24286
   731
lemma split_paired_Ball_Sigma [simp,noatp]:
wenzelm@11838
   732
    "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"
wenzelm@11838
   733
  by blast
wenzelm@11838
   734
paulson@24286
   735
lemma split_paired_Bex_Sigma [simp,noatp]:
wenzelm@11838
   736
    "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"
wenzelm@11838
   737
  by blast
wenzelm@11838
   738
wenzelm@11838
   739
lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"
wenzelm@11838
   740
  by blast
wenzelm@11838
   741
wenzelm@11838
   742
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
   743
  by blast
wenzelm@11838
   744
wenzelm@11838
   745
lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"
wenzelm@11838
   746
  by blast
wenzelm@11838
   747
wenzelm@11838
   748
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
   749
  by blast
wenzelm@11838
   750
wenzelm@11838
   751
lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"
wenzelm@11838
   752
  by blast
wenzelm@11838
   753
wenzelm@11838
   754
lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"
wenzelm@11838
   755
  by blast
wenzelm@11838
   756
wenzelm@11838
   757
lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"
wenzelm@11838
   758
  by blast
wenzelm@11838
   759
wenzelm@11838
   760
text {*
wenzelm@11838
   761
  Non-dependent versions are needed to avoid the need for higher-order
wenzelm@11838
   762
  matching, especially when the rules are re-oriented.
wenzelm@11838
   763
*}
wenzelm@11838
   764
wenzelm@11838
   765
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"
wenzelm@11838
   766
  by blast
wenzelm@11838
   767
wenzelm@11838
   768
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"
wenzelm@11838
   769
  by blast
wenzelm@11838
   770
wenzelm@11838
   771
lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
wenzelm@11838
   772
  by blast
wenzelm@11838
   773
wenzelm@11838
   774
oheimb@11493
   775
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
wenzelm@11777
   776
  apply (rule_tac x = "(a, b)" in image_eqI)
wenzelm@11777
   777
   apply auto
wenzelm@11777
   778
  done
wenzelm@11777
   779
oheimb@11493
   780
wenzelm@11838
   781
text {*
wenzelm@11838
   782
  Setup of internal @{text split_rule}.
wenzelm@11838
   783
*}
wenzelm@11838
   784
haftmann@25511
   785
definition
haftmann@25511
   786
  internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c"
haftmann@25511
   787
where
wenzelm@11032
   788
  "internal_split == split"
wenzelm@11032
   789
wenzelm@11032
   790
lemma internal_split_conv: "internal_split c (a, b) = c a b"
wenzelm@11032
   791
  by (simp only: internal_split_def split_conv)
wenzelm@11032
   792
wenzelm@11032
   793
hide const internal_split
wenzelm@11032
   794
oheimb@11025
   795
use "Tools/split_rule.ML"
wenzelm@11032
   796
setup SplitRule.setup
nipkow@10213
   797
berghofe@15394
   798
haftmann@24699
   799
haftmann@24699
   800
lemmas prod_caseI = prod.cases [THEN iffD2, standard]
haftmann@24699
   801
haftmann@24699
   802
lemma prod_caseI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> prod_case c p"
haftmann@24699
   803
  by auto
haftmann@24699
   804
haftmann@24699
   805
lemma prod_caseI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> prod_case c p x"
haftmann@24699
   806
  by (auto simp: split_tupled_all)
haftmann@24699
   807
haftmann@24699
   808
lemma prod_caseE: "prod_case c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
haftmann@24699
   809
  by (induct p) auto
haftmann@24699
   810
haftmann@24699
   811
lemma prod_caseE': "prod_case c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
haftmann@24699
   812
  by (induct p) auto
haftmann@24699
   813
haftmann@24699
   814
lemma prod_case_unfold: "prod_case = (%c p. c (fst p) (snd p))"
haftmann@24699
   815
  by (simp add: expand_fun_eq)
haftmann@24699
   816
haftmann@24699
   817
declare prod_caseI2' [intro!] prod_caseI2 [intro!] prod_caseI [intro!]
haftmann@24699
   818
declare prod_caseE' [elim!] prod_caseE [elim!]
haftmann@24699
   819
haftmann@24844
   820
lemma prod_case_split:
haftmann@24699
   821
  "prod_case = split"
haftmann@24699
   822
  by (auto simp add: expand_fun_eq)
haftmann@24699
   823
bulwahn@26143
   824
lemma prod_case_beta:
bulwahn@26143
   825
  "prod_case f p = f (fst p) (snd p)"
bulwahn@26143
   826
  unfolding prod_case_split split_beta ..
bulwahn@26143
   827
haftmann@24699
   828
haftmann@24699
   829
subsection {* Further cases/induct rules for tuples *}
haftmann@24699
   830
haftmann@24699
   831
lemma prod_cases3 [cases type]:
haftmann@24699
   832
  obtains (fields) a b c where "y = (a, b, c)"
haftmann@24699
   833
  by (cases y, case_tac b) blast
haftmann@24699
   834
haftmann@24699
   835
lemma prod_induct3 [case_names fields, induct type]:
haftmann@24699
   836
    "(!!a b c. P (a, b, c)) ==> P x"
haftmann@24699
   837
  by (cases x) blast
haftmann@24699
   838
haftmann@24699
   839
lemma prod_cases4 [cases type]:
haftmann@24699
   840
  obtains (fields) a b c d where "y = (a, b, c, d)"
haftmann@24699
   841
  by (cases y, case_tac c) blast
haftmann@24699
   842
haftmann@24699
   843
lemma prod_induct4 [case_names fields, induct type]:
haftmann@24699
   844
    "(!!a b c d. P (a, b, c, d)) ==> P x"
haftmann@24699
   845
  by (cases x) blast
haftmann@24699
   846
haftmann@24699
   847
lemma prod_cases5 [cases type]:
haftmann@24699
   848
  obtains (fields) a b c d e where "y = (a, b, c, d, e)"
haftmann@24699
   849
  by (cases y, case_tac d) blast
haftmann@24699
   850
haftmann@24699
   851
lemma prod_induct5 [case_names fields, induct type]:
haftmann@24699
   852
    "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
haftmann@24699
   853
  by (cases x) blast
haftmann@24699
   854
haftmann@24699
   855
lemma prod_cases6 [cases type]:
haftmann@24699
   856
  obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
haftmann@24699
   857
  by (cases y, case_tac e) blast
haftmann@24699
   858
haftmann@24699
   859
lemma prod_induct6 [case_names fields, induct type]:
haftmann@24699
   860
    "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
haftmann@24699
   861
  by (cases x) blast
haftmann@24699
   862
haftmann@24699
   863
lemma prod_cases7 [cases type]:
haftmann@24699
   864
  obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
haftmann@24699
   865
  by (cases y, case_tac f) blast
haftmann@24699
   866
haftmann@24699
   867
lemma prod_induct7 [case_names fields, induct type]:
haftmann@24699
   868
    "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
haftmann@24699
   869
  by (cases x) blast
haftmann@24699
   870
haftmann@24699
   871
haftmann@21195
   872
subsection {* Further lemmas *}
haftmann@21195
   873
haftmann@21195
   874
lemma
haftmann@21195
   875
  split_Pair: "split Pair x = x"
haftmann@21195
   876
  unfolding split_def by auto
haftmann@21195
   877
haftmann@21195
   878
lemma
haftmann@21195
   879
  split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)"
haftmann@21195
   880
  by (cases x, simp)
haftmann@21195
   881
haftmann@21195
   882
berghofe@15394
   883
subsection {* Code generator setup *}
berghofe@15394
   884
haftmann@20588
   885
instance unit :: eq ..
haftmann@20588
   886
haftmann@20588
   887
lemma [code func]:
haftmann@21454
   888
  "(u\<Colon>unit) = v \<longleftrightarrow> True" unfolding unit_eq [of u] unit_eq [of v] by rule+
haftmann@20588
   889
haftmann@21908
   890
code_type unit
haftmann@21908
   891
  (SML "unit")
haftmann@21908
   892
  (OCaml "unit")
haftmann@21908
   893
  (Haskell "()")
haftmann@21908
   894
haftmann@20588
   895
code_instance unit :: eq
haftmann@20588
   896
  (Haskell -)
haftmann@20588
   897
haftmann@21908
   898
code_const "op = \<Colon> unit \<Rightarrow> unit \<Rightarrow> bool"
haftmann@21908
   899
  (Haskell infixl 4 "==")
haftmann@21908
   900
haftmann@21908
   901
code_const Unity
haftmann@21908
   902
  (SML "()")
haftmann@21908
   903
  (OCaml "()")
haftmann@21908
   904
  (Haskell "()")
haftmann@21908
   905
haftmann@21908
   906
code_reserved SML
haftmann@21908
   907
  unit
haftmann@21908
   908
haftmann@21908
   909
code_reserved OCaml
haftmann@21908
   910
  unit
haftmann@21908
   911
haftmann@20588
   912
instance * :: (eq, eq) eq ..
haftmann@20588
   913
haftmann@20588
   914
lemma [code func]:
haftmann@21454
   915
  "(x1\<Colon>'a\<Colon>eq, y1\<Colon>'b\<Colon>eq) = (x2, y2) \<longleftrightarrow> x1 = x2 \<and> y1 = y2" by auto
haftmann@20588
   916
haftmann@24844
   917
lemma split_case_cert:
haftmann@24844
   918
  assumes "CASE \<equiv> split f"
haftmann@24844
   919
  shows "CASE (a, b) \<equiv> f a b"
haftmann@24844
   920
  using assms by simp
haftmann@24844
   921
haftmann@24844
   922
setup {*
haftmann@24844
   923
  Code.add_case @{thm split_case_cert}
haftmann@24844
   924
*}
haftmann@24844
   925
haftmann@21908
   926
code_type *
haftmann@21908
   927
  (SML infix 2 "*")
haftmann@21908
   928
  (OCaml infix 2 "*")
haftmann@21908
   929
  (Haskell "!((_),/ (_))")
haftmann@21908
   930
haftmann@20588
   931
code_instance * :: eq
haftmann@20588
   932
  (Haskell -)
haftmann@20588
   933
haftmann@21908
   934
code_const "op = \<Colon> 'a\<Colon>eq \<times> 'b\<Colon>eq \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"
haftmann@20588
   935
  (Haskell infixl 4 "==")
haftmann@20588
   936
haftmann@21908
   937
code_const Pair
haftmann@21908
   938
  (SML "!((_),/ (_))")
haftmann@21908
   939
  (OCaml "!((_),/ (_))")
haftmann@21908
   940
  (Haskell "!((_),/ (_))")
haftmann@20588
   941
haftmann@22389
   942
code_const fst and snd
haftmann@22389
   943
  (Haskell "fst" and "snd")
haftmann@22389
   944
berghofe@15394
   945
types_code
berghofe@15394
   946
  "*"     ("(_ */ _)")
berghofe@16770
   947
attach (term_of) {*
berghofe@25885
   948
fun term_of_id_42 aF aT bF bT (x, y) = HOLogic.pair_const aT bT $ aF x $ bF y;
berghofe@16770
   949
*}
berghofe@16770
   950
attach (test) {*
berghofe@25885
   951
fun gen_id_42 aG aT bG bT i =
berghofe@25885
   952
  let
berghofe@25885
   953
    val (x, t) = aG i;
berghofe@25885
   954
    val (y, u) = bG i
berghofe@25885
   955
  in ((x, y), fn () => HOLogic.pair_const aT bT $ t () $ u ()) end;
berghofe@16770
   956
*}
berghofe@15394
   957
berghofe@18706
   958
consts_code
berghofe@18706
   959
  "Pair"    ("(_,/ _)")
berghofe@18706
   960
haftmann@21908
   961
setup {*
haftmann@21908
   962
haftmann@21908
   963
let
haftmann@18013
   964
haftmann@19039
   965
fun strip_abs_split 0 t = ([], t)
haftmann@19039
   966
  | strip_abs_split i (Abs (s, T, t)) =
haftmann@18013
   967
      let
haftmann@18013
   968
        val s' = Codegen.new_name t s;
haftmann@18013
   969
        val v = Free (s', T)
haftmann@19039
   970
      in apfst (cons v) (strip_abs_split (i-1) (subst_bound (v, t))) end
haftmann@19039
   971
  | strip_abs_split i (u as Const ("split", _) $ t) = (case strip_abs_split (i+1) t of
berghofe@15394
   972
        (v :: v' :: vs, u) => (HOLogic.mk_prod (v, v') :: vs, u)
berghofe@15394
   973
      | _ => ([], u))
haftmann@19039
   974
  | strip_abs_split i t = ([], t);
haftmann@18013
   975
berghofe@16634
   976
fun let_codegen thy defs gr dep thyname brack t = (case strip_comb t of
berghofe@16634
   977
    (t1 as Const ("Let", _), t2 :: t3 :: ts) =>
berghofe@15394
   978
    let
berghofe@15394
   979
      fun dest_let (l as Const ("Let", _) $ t $ u) =
haftmann@19039
   980
          (case strip_abs_split 1 u of
berghofe@15394
   981
             ([p], u') => apfst (cons (p, t)) (dest_let u')
berghofe@15394
   982
           | _ => ([], l))
berghofe@15394
   983
        | dest_let t = ([], t);
berghofe@15394
   984
      fun mk_code (gr, (l, r)) =
berghofe@15394
   985
        let
berghofe@16634
   986
          val (gr1, pl) = Codegen.invoke_codegen thy defs dep thyname false (gr, l);
berghofe@16634
   987
          val (gr2, pr) = Codegen.invoke_codegen thy defs dep thyname false (gr1, r);
berghofe@15394
   988
        in (gr2, (pl, pr)) end
berghofe@16634
   989
    in case dest_let (t1 $ t2 $ t3) of
skalberg@15531
   990
        ([], _) => NONE
berghofe@15394
   991
      | (ps, u) =>
berghofe@15394
   992
          let
berghofe@15394
   993
            val (gr1, qs) = foldl_map mk_code (gr, ps);
berghofe@16634
   994
            val (gr2, pu) = Codegen.invoke_codegen thy defs dep thyname false (gr1, u);
berghofe@16634
   995
            val (gr3, pargs) = foldl_map
berghofe@17021
   996
              (Codegen.invoke_codegen thy defs dep thyname true) (gr2, ts)
berghofe@15394
   997
          in
berghofe@16634
   998
            SOME (gr3, Codegen.mk_app brack
berghofe@16634
   999
              (Pretty.blk (0, [Pretty.str "let ", Pretty.blk (0, List.concat
berghofe@16634
  1000
                  (separate [Pretty.str ";", Pretty.brk 1] (map (fn (pl, pr) =>
berghofe@16634
  1001
                    [Pretty.block [Pretty.str "val ", pl, Pretty.str " =",
berghofe@16634
  1002
                       Pretty.brk 1, pr]]) qs))),
berghofe@16634
  1003
                Pretty.brk 1, Pretty.str "in ", pu,
berghofe@16634
  1004
                Pretty.brk 1, Pretty.str "end"])) pargs)
berghofe@15394
  1005
          end
berghofe@15394
  1006
    end
berghofe@16634
  1007
  | _ => NONE);
berghofe@15394
  1008
berghofe@16634
  1009
fun split_codegen thy defs gr dep thyname brack t = (case strip_comb t of
berghofe@16634
  1010
    (t1 as Const ("split", _), t2 :: ts) =>
haftmann@19039
  1011
      (case strip_abs_split 1 (t1 $ t2) of
berghofe@16634
  1012
         ([p], u) =>
berghofe@16634
  1013
           let
berghofe@16634
  1014
             val (gr1, q) = Codegen.invoke_codegen thy defs dep thyname false (gr, p);
berghofe@16634
  1015
             val (gr2, pu) = Codegen.invoke_codegen thy defs dep thyname false (gr1, u);
berghofe@16634
  1016
             val (gr3, pargs) = foldl_map
berghofe@17021
  1017
               (Codegen.invoke_codegen thy defs dep thyname true) (gr2, ts)
berghofe@16634
  1018
           in
berghofe@16634
  1019
             SOME (gr2, Codegen.mk_app brack
berghofe@16634
  1020
               (Pretty.block [Pretty.str "(fn ", q, Pretty.str " =>",
berghofe@16634
  1021
                 Pretty.brk 1, pu, Pretty.str ")"]) pargs)
berghofe@16634
  1022
           end
berghofe@16634
  1023
       | _ => NONE)
berghofe@16634
  1024
  | _ => NONE);
berghofe@15394
  1025
haftmann@21908
  1026
in
haftmann@21908
  1027
haftmann@20105
  1028
  Codegen.add_codegen "let_codegen" let_codegen
haftmann@20105
  1029
  #> Codegen.add_codegen "split_codegen" split_codegen
berghofe@15394
  1030
haftmann@21908
  1031
end
berghofe@15394
  1032
*}
berghofe@15394
  1033
haftmann@24699
  1034
haftmann@24699
  1035
subsection {* Legacy bindings *}
haftmann@24699
  1036
haftmann@21908
  1037
ML {*
paulson@15404
  1038
val Collect_split = thm "Collect_split";
paulson@15404
  1039
val Compl_Times_UNIV1 = thm "Compl_Times_UNIV1";
paulson@15404
  1040
val Compl_Times_UNIV2 = thm "Compl_Times_UNIV2";
paulson@15404
  1041
val PairE = thm "PairE";
paulson@15404
  1042
val PairE_lemma = thm "PairE_lemma";
paulson@15404
  1043
val Pair_Rep_inject = thm "Pair_Rep_inject";
paulson@15404
  1044
val Pair_def = thm "Pair_def";
paulson@15404
  1045
val Pair_eq = thm "Pair_eq";
paulson@15404
  1046
val Pair_fst_snd_eq = thm "Pair_fst_snd_eq";
paulson@15404
  1047
val Pair_inject = thm "Pair_inject";
paulson@15404
  1048
val ProdI = thm "ProdI";
paulson@15404
  1049
val SetCompr_Sigma_eq = thm "SetCompr_Sigma_eq";
paulson@15404
  1050
val SigmaD1 = thm "SigmaD1";
paulson@15404
  1051
val SigmaD2 = thm "SigmaD2";
paulson@15404
  1052
val SigmaE = thm "SigmaE";
paulson@15404
  1053
val SigmaE2 = thm "SigmaE2";
paulson@15404
  1054
val SigmaI = thm "SigmaI";
paulson@15404
  1055
val Sigma_Diff_distrib1 = thm "Sigma_Diff_distrib1";
paulson@15404
  1056
val Sigma_Diff_distrib2 = thm "Sigma_Diff_distrib2";
paulson@15404
  1057
val Sigma_Int_distrib1 = thm "Sigma_Int_distrib1";
paulson@15404
  1058
val Sigma_Int_distrib2 = thm "Sigma_Int_distrib2";
paulson@15404
  1059
val Sigma_Un_distrib1 = thm "Sigma_Un_distrib1";
paulson@15404
  1060
val Sigma_Un_distrib2 = thm "Sigma_Un_distrib2";
paulson@15404
  1061
val Sigma_Union = thm "Sigma_Union";
paulson@15404
  1062
val Sigma_def = thm "Sigma_def";
paulson@15404
  1063
val Sigma_empty1 = thm "Sigma_empty1";
paulson@15404
  1064
val Sigma_empty2 = thm "Sigma_empty2";
paulson@15404
  1065
val Sigma_mono = thm "Sigma_mono";
paulson@15404
  1066
val The_split = thm "The_split";
paulson@15404
  1067
val The_split_eq = thm "The_split_eq";
paulson@15404
  1068
val The_split_eq = thm "The_split_eq";
paulson@15404
  1069
val Times_Diff_distrib1 = thm "Times_Diff_distrib1";
paulson@15404
  1070
val Times_Int_distrib1 = thm "Times_Int_distrib1";
paulson@15404
  1071
val Times_Un_distrib1 = thm "Times_Un_distrib1";
paulson@15404
  1072
val Times_eq_cancel2 = thm "Times_eq_cancel2";
paulson@15404
  1073
val Times_subset_cancel2 = thm "Times_subset_cancel2";
paulson@15404
  1074
val UNIV_Times_UNIV = thm "UNIV_Times_UNIV";
paulson@15404
  1075
val UN_Times_distrib = thm "UN_Times_distrib";
paulson@15404
  1076
val Unity_def = thm "Unity_def";
paulson@15404
  1077
val cond_split_eta = thm "cond_split_eta";
paulson@15404
  1078
val fst_conv = thm "fst_conv";
paulson@15404
  1079
val fst_def = thm "fst_def";
paulson@15404
  1080
val fst_eqD = thm "fst_eqD";
paulson@15404
  1081
val inj_on_Abs_Prod = thm "inj_on_Abs_Prod";
paulson@15404
  1082
val injective_fst_snd = thm "injective_fst_snd";
paulson@15404
  1083
val mem_Sigma_iff = thm "mem_Sigma_iff";
paulson@15404
  1084
val mem_splitE = thm "mem_splitE";
paulson@15404
  1085
val mem_splitI = thm "mem_splitI";
paulson@15404
  1086
val mem_splitI2 = thm "mem_splitI2";
paulson@15404
  1087
val prod_eqI = thm "prod_eqI";
paulson@15404
  1088
val prod_fun = thm "prod_fun";
paulson@15404
  1089
val prod_fun_compose = thm "prod_fun_compose";
paulson@15404
  1090
val prod_fun_def = thm "prod_fun_def";
paulson@15404
  1091
val prod_fun_ident = thm "prod_fun_ident";
paulson@15404
  1092
val prod_fun_imageE = thm "prod_fun_imageE";
paulson@15404
  1093
val prod_fun_imageI = thm "prod_fun_imageI";
paulson@15404
  1094
val prod_induct = thm "prod_induct";
paulson@15404
  1095
val snd_conv = thm "snd_conv";
paulson@15404
  1096
val snd_def = thm "snd_def";
paulson@15404
  1097
val snd_eqD = thm "snd_eqD";
paulson@15404
  1098
val split = thm "split";
paulson@15404
  1099
val splitD = thm "splitD";
paulson@15404
  1100
val splitD' = thm "splitD'";
paulson@15404
  1101
val splitE = thm "splitE";
paulson@15404
  1102
val splitE' = thm "splitE'";
paulson@15404
  1103
val splitE2 = thm "splitE2";
paulson@15404
  1104
val splitI = thm "splitI";
paulson@15404
  1105
val splitI2 = thm "splitI2";
paulson@15404
  1106
val splitI2' = thm "splitI2'";
paulson@15404
  1107
val split_Pair_apply = thm "split_Pair_apply";
paulson@15404
  1108
val split_beta = thm "split_beta";
paulson@15404
  1109
val split_conv = thm "split_conv";
paulson@15404
  1110
val split_def = thm "split_def";
paulson@15404
  1111
val split_eta = thm "split_eta";
paulson@15404
  1112
val split_eta_SetCompr = thm "split_eta_SetCompr";
paulson@15404
  1113
val split_eta_SetCompr2 = thm "split_eta_SetCompr2";
paulson@15404
  1114
val split_paired_All = thm "split_paired_All";
paulson@15404
  1115
val split_paired_Ball_Sigma = thm "split_paired_Ball_Sigma";
paulson@15404
  1116
val split_paired_Bex_Sigma = thm "split_paired_Bex_Sigma";
paulson@15404
  1117
val split_paired_Ex = thm "split_paired_Ex";
paulson@15404
  1118
val split_paired_The = thm "split_paired_The";
paulson@15404
  1119
val split_paired_all = thm "split_paired_all";
paulson@15404
  1120
val split_part = thm "split_part";
paulson@15404
  1121
val split_split = thm "split_split";
paulson@15404
  1122
val split_split_asm = thm "split_split_asm";
paulson@15404
  1123
val split_tupled_all = thms "split_tupled_all";
paulson@15404
  1124
val split_weak_cong = thm "split_weak_cong";
paulson@15404
  1125
val surj_pair = thm "surj_pair";
paulson@15404
  1126
val surjective_pairing = thm "surjective_pairing";
paulson@15404
  1127
val unit_abs_eta_conv = thm "unit_abs_eta_conv";
paulson@15404
  1128
val unit_all_eq1 = thm "unit_all_eq1";
paulson@15404
  1129
val unit_all_eq2 = thm "unit_all_eq2";
paulson@15404
  1130
val unit_eq = thm "unit_eq";
paulson@15404
  1131
val unit_induct = thm "unit_induct";
paulson@15404
  1132
*}
paulson@15404
  1133
haftmann@24699
  1134
haftmann@24699
  1135
subsection {* Further inductive packages *}
haftmann@24699
  1136
haftmann@24699
  1137
use "Tools/inductive_realizer.ML"
haftmann@24699
  1138
setup InductiveRealizer.setup
haftmann@24699
  1139
haftmann@24699
  1140
use "Tools/inductive_set_package.ML"
haftmann@24699
  1141
setup InductiveSetPackage.setup
haftmann@24699
  1142
haftmann@24699
  1143
use "Tools/datatype_realizer.ML"
haftmann@24699
  1144
setup DatatypeRealizer.setup
haftmann@24699
  1145
nipkow@10213
  1146
end