src/HOL/Product_Type.thy
author wenzelm
Wed Sep 17 21:27:14 2008 +0200 (2008-09-17)
changeset 28263 69eaa97e7e96
parent 28262 aa7ca36d67fd
child 28346 b8390cd56b8f
permissions -rw-r--r--
moved global ML bindings to global place;
     1 (*  Title:      HOL/Product_Type.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1992  University of Cambridge
     5 *)
     6 
     7 header {* Cartesian products *}
     8 
     9 theory Product_Type
    10 imports Inductive
    11 uses
    12   ("Tools/split_rule.ML")
    13   ("Tools/inductive_set_package.ML")
    14   ("Tools/inductive_realizer.ML")
    15   ("Tools/datatype_realizer.ML")
    16 begin
    17 
    18 subsection {* @{typ bool} is a datatype *}
    19 
    20 rep_datatype True False by (auto intro: bool_induct)
    21 
    22 declare case_split [cases type: bool]
    23   -- "prefer plain propositional version"
    24 
    25 lemma [code func]:
    26   shows "False = P \<longleftrightarrow> \<not> P"
    27     and "True = P \<longleftrightarrow> P" 
    28     and "P = False \<longleftrightarrow> \<not> P" 
    29     and "P = True \<longleftrightarrow> P" by simp_all
    30 
    31 code_const "op = \<Colon> bool \<Rightarrow> bool \<Rightarrow> bool"
    32   (Haskell infixl 4 "==")
    33 
    34 code_instance bool :: eq
    35   (Haskell -)
    36 
    37 
    38 subsection {* Unit *}
    39 
    40 typedef unit = "{True}"
    41 proof
    42   show "True : ?unit" ..
    43 qed
    44 
    45 definition
    46   Unity :: unit    ("'(')")
    47 where
    48   "() = Abs_unit True"
    49 
    50 lemma unit_eq [noatp]: "u = ()"
    51   by (induct u) (simp add: unit_def Unity_def)
    52 
    53 text {*
    54   Simplification procedure for @{thm [source] unit_eq}.  Cannot use
    55   this rule directly --- it loops!
    56 *}
    57 
    58 ML {*
    59   val unit_eq_proc =
    60     let val unit_meta_eq = mk_meta_eq @{thm unit_eq} in
    61       Simplifier.simproc @{theory} "unit_eq" ["x::unit"]
    62       (fn _ => fn _ => fn t => if HOLogic.is_unit t then NONE else SOME unit_meta_eq)
    63     end;
    64 
    65   Addsimprocs [unit_eq_proc];
    66 *}
    67 
    68 rep_datatype "()" by simp
    69 
    70 lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()"
    71   by simp
    72 
    73 lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P"
    74   by (rule triv_forall_equality)
    75 
    76 text {*
    77   This rewrite counters the effect of @{text unit_eq_proc} on @{term
    78   [source] "%u::unit. f u"}, replacing it by @{term [source]
    79   f} rather than by @{term [source] "%u. f ()"}.
    80 *}
    81 
    82 lemma unit_abs_eta_conv [simp,noatp]: "(%u::unit. f ()) = f"
    83   by (rule ext) simp
    84 
    85 
    86 text {* code generator setup *}
    87 
    88 instance unit :: eq ..
    89 
    90 lemma [code func]:
    91   "(u\<Colon>unit) = v \<longleftrightarrow> True" unfolding unit_eq [of u] unit_eq [of v] by rule+
    92 
    93 code_type unit
    94   (SML "unit")
    95   (OCaml "unit")
    96   (Haskell "()")
    97 
    98 code_instance unit :: eq
    99   (Haskell -)
   100 
   101 code_const "op = \<Colon> unit \<Rightarrow> unit \<Rightarrow> bool"
   102   (Haskell infixl 4 "==")
   103 
   104 code_const Unity
   105   (SML "()")
   106   (OCaml "()")
   107   (Haskell "()")
   108 
   109 code_reserved SML
   110   unit
   111 
   112 code_reserved OCaml
   113   unit
   114 
   115 
   116 subsection {* Pairs *}
   117 
   118 subsubsection {* Product type, basic operations and concrete syntax *}
   119 
   120 definition
   121   Pair_Rep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
   122 where
   123   "Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)"
   124 
   125 global
   126 
   127 typedef (Prod)
   128   ('a, 'b) "*"    (infixr "*" 20)
   129     = "{f. \<exists>a b. f = Pair_Rep (a\<Colon>'a) (b\<Colon>'b)}"
   130 proof
   131   fix a b show "Pair_Rep a b \<in> ?Prod"
   132     by rule+
   133 qed
   134 
   135 syntax (xsymbols)
   136   "*"      :: "[type, type] => type"         ("(_ \<times>/ _)" [21, 20] 20)
   137 syntax (HTML output)
   138   "*"      :: "[type, type] => type"         ("(_ \<times>/ _)" [21, 20] 20)
   139 
   140 consts
   141   Pair     :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b"
   142   fst      :: "'a \<times> 'b \<Rightarrow> 'a"
   143   snd      :: "'a \<times> 'b \<Rightarrow> 'b"
   144   split    :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c"
   145   curry    :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c"
   146 
   147 local
   148 
   149 defs
   150   Pair_def:     "Pair a b == Abs_Prod (Pair_Rep a b)"
   151   fst_def:      "fst p == THE a. EX b. p = Pair a b"
   152   snd_def:      "snd p == THE b. EX a. p = Pair a b"
   153   split_def:    "split == (%c p. c (fst p) (snd p))"
   154   curry_def:    "curry == (%c x y. c (Pair x y))"
   155 
   156 text {*
   157   Patterns -- extends pre-defined type @{typ pttrn} used in
   158   abstractions.
   159 *}
   160 
   161 nonterminals
   162   tuple_args patterns
   163 
   164 syntax
   165   "_tuple"      :: "'a => tuple_args => 'a * 'b"        ("(1'(_,/ _'))")
   166   "_tuple_arg"  :: "'a => tuple_args"                   ("_")
   167   "_tuple_args" :: "'a => tuple_args => tuple_args"     ("_,/ _")
   168   "_pattern"    :: "[pttrn, patterns] => pttrn"         ("'(_,/ _')")
   169   ""            :: "pttrn => patterns"                  ("_")
   170   "_patterns"   :: "[pttrn, patterns] => patterns"      ("_,/ _")
   171 
   172 translations
   173   "(x, y)"       == "Pair x y"
   174   "_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))"
   175   "%(x,y,zs).b"  == "split(%x (y,zs).b)"
   176   "%(x,y).b"     == "split(%x y. b)"
   177   "_abs (Pair x y) t" => "%(x,y).t"
   178   (* The last rule accommodates tuples in `case C ... (x,y) ... => ...'
   179      The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *)
   180 
   181 (* reconstructs pattern from (nested) splits, avoiding eta-contraction of body*)
   182 (* works best with enclosing "let", if "let" does not avoid eta-contraction   *)
   183 print_translation {*
   184 let fun split_tr' [Abs (x,T,t as (Abs abs))] =
   185       (* split (%x y. t) => %(x,y) t *)
   186       let val (y,t') = atomic_abs_tr' abs;
   187           val (x',t'') = atomic_abs_tr' (x,T,t');
   188     
   189       in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end
   190     | split_tr' [Abs (x,T,(s as Const ("split",_)$t))] =
   191        (* split (%x. (split (%y z. t))) => %(x,y,z). t *)
   192        let val (Const ("_abs",_)$(Const ("_pattern",_)$y$z)$t') = split_tr' [t];
   193            val (x',t'') = atomic_abs_tr' (x,T,t');
   194        in Syntax.const "_abs"$ 
   195            (Syntax.const "_pattern"$x'$(Syntax.const "_patterns"$y$z))$t'' end
   196     | split_tr' [Const ("split",_)$t] =
   197        (* split (split (%x y z. t)) => %((x,y),z). t *)   
   198        split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *)
   199     | split_tr' [Const ("_abs",_)$x_y$(Abs abs)] =
   200        (* split (%pttrn z. t) => %(pttrn,z). t *)
   201        let val (z,t) = atomic_abs_tr' abs;
   202        in Syntax.const "_abs" $ (Syntax.const "_pattern" $x_y$z) $ t end
   203     | split_tr' _ =  raise Match;
   204 in [("split", split_tr')]
   205 end
   206 *}
   207 
   208 (* print "split f" as "\<lambda>(x,y). f x y" and "split (\<lambda>x. f x)" as "\<lambda>(x,y). f x y" *) 
   209 typed_print_translation {*
   210 let
   211   fun split_guess_names_tr' _ T [Abs (x,_,Abs _)] = raise Match
   212     | split_guess_names_tr' _ T  [Abs (x,xT,t)] =
   213         (case (head_of t) of
   214            Const ("split",_) => raise Match
   215          | _ => let 
   216                   val (_::yT::_) = binder_types (domain_type T) handle Bind => raise Match;
   217                   val (y,t') = atomic_abs_tr' ("y",yT,(incr_boundvars 1 t)$Bound 0); 
   218                   val (x',t'') = atomic_abs_tr' (x,xT,t');
   219                 in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end)
   220     | split_guess_names_tr' _ T [t] =
   221        (case (head_of t) of
   222            Const ("split",_) => raise Match 
   223          | _ => let 
   224                   val (xT::yT::_) = binder_types (domain_type T) handle Bind => raise Match;
   225                   val (y,t') = 
   226                         atomic_abs_tr' ("y",yT,(incr_boundvars 2 t)$Bound 1$Bound 0); 
   227                   val (x',t'') = atomic_abs_tr' ("x",xT,t');
   228                 in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end)
   229     | split_guess_names_tr' _ _ _ = raise Match;
   230 in [("split", split_guess_names_tr')]
   231 end 
   232 *}
   233 
   234 
   235 text {* Towards a datatype declaration *}
   236 
   237 lemma surj_pair [simp]: "EX x y. p = (x, y)"
   238   apply (unfold Pair_def)
   239   apply (rule Rep_Prod [unfolded Prod_def, THEN CollectE])
   240   apply (erule exE, erule exE, rule exI, rule exI)
   241   apply (rule Rep_Prod_inverse [symmetric, THEN trans])
   242   apply (erule arg_cong)
   243   done
   244 
   245 lemma PairE [cases type: *]:
   246   obtains x y where "p = (x, y)"
   247   using surj_pair [of p] by blast
   248 
   249 lemma ProdI: "Pair_Rep a b \<in> Prod"
   250   unfolding Prod_def by rule+
   251 
   252 lemma Pair_Rep_inject: "Pair_Rep a b = Pair_Rep a' b' \<Longrightarrow> a = a' \<and> b = b'"
   253   unfolding Pair_Rep_def by (drule fun_cong, drule fun_cong) blast
   254 
   255 lemma inj_on_Abs_Prod: "inj_on Abs_Prod Prod"
   256   apply (rule inj_on_inverseI)
   257   apply (erule Abs_Prod_inverse)
   258   done
   259 
   260 lemma Pair_inject:
   261   assumes "(a, b) = (a', b')"
   262     and "a = a' ==> b = b' ==> R"
   263   shows R
   264   apply (insert prems [unfolded Pair_def])
   265   apply (rule inj_on_Abs_Prod [THEN inj_onD, THEN Pair_Rep_inject, THEN conjE])
   266   apply (assumption | rule ProdI)+
   267   done
   268 
   269 rep_datatype (prod) Pair
   270 proof -
   271   fix P p
   272   assume "\<And>x y. P (x, y)"
   273   then show "P p" by (cases p) simp
   274 qed (auto elim: Pair_inject)
   275 
   276 lemmas Pair_eq = prod.inject
   277 
   278 lemma fst_conv [simp, code]: "fst (a, b) = a"
   279   unfolding fst_def by blast
   280 
   281 lemma snd_conv [simp, code]: "snd (a, b) = b"
   282   unfolding snd_def by blast
   283 
   284 
   285 subsubsection {* Basic rules and proof tools *}
   286 
   287 lemma fst_eqD: "fst (x, y) = a ==> x = a"
   288   by simp
   289 
   290 lemma snd_eqD: "snd (x, y) = a ==> y = a"
   291   by simp
   292 
   293 lemma pair_collapse [simp]: "(fst p, snd p) = p"
   294   by (cases p) simp
   295 
   296 lemmas surjective_pairing = pair_collapse [symmetric]
   297 
   298 lemma split_paired_all: "(!!x. PROP P x) == (!!a b. PROP P (a, b))"
   299 proof
   300   fix a b
   301   assume "!!x. PROP P x"
   302   then show "PROP P (a, b)" .
   303 next
   304   fix x
   305   assume "!!a b. PROP P (a, b)"
   306   from `PROP P (fst x, snd x)` show "PROP P x" by simp
   307 qed
   308 
   309 text {*
   310   The rule @{thm [source] split_paired_all} does not work with the
   311   Simplifier because it also affects premises in congrence rules,
   312   where this can lead to premises of the form @{text "!!a b. ... =
   313   ?P(a, b)"} which cannot be solved by reflexivity.
   314 *}
   315 
   316 lemmas split_tupled_all = split_paired_all unit_all_eq2
   317 
   318 ML {*
   319   (* replace parameters of product type by individual component parameters *)
   320   val safe_full_simp_tac = generic_simp_tac true (true, false, false);
   321   local (* filtering with exists_paired_all is an essential optimization *)
   322     fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) =
   323           can HOLogic.dest_prodT T orelse exists_paired_all t
   324       | exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u
   325       | exists_paired_all (Abs (_, _, t)) = exists_paired_all t
   326       | exists_paired_all _ = false;
   327     val ss = HOL_basic_ss
   328       addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}]
   329       addsimprocs [unit_eq_proc];
   330   in
   331     val split_all_tac = SUBGOAL (fn (t, i) =>
   332       if exists_paired_all t then safe_full_simp_tac ss i else no_tac);
   333     val unsafe_split_all_tac = SUBGOAL (fn (t, i) =>
   334       if exists_paired_all t then full_simp_tac ss i else no_tac);
   335     fun split_all th =
   336    if exists_paired_all (Thm.prop_of th) then full_simplify ss th else th;
   337   end;
   338 *}
   339 
   340 declaration {* fn _ =>
   341   Classical.map_cs (fn cs => cs addSbefore ("split_all_tac", split_all_tac))
   342 *}
   343 
   344 lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))"
   345   -- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *}
   346   by fast
   347 
   348 lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))"
   349   by fast
   350 
   351 lemma Pair_fst_snd_eq: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t"
   352   by (cases s, cases t) simp
   353 
   354 lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q"
   355   by (simp add: Pair_fst_snd_eq)
   356 
   357 
   358 subsubsection {* @{text split} and @{text curry} *}
   359 
   360 lemma split_conv [simp, code func]: "split f (a, b) = f a b"
   361   by (simp add: split_def)
   362 
   363 lemma curry_conv [simp, code func]: "curry f a b = f (a, b)"
   364   by (simp add: curry_def)
   365 
   366 lemmas split = split_conv  -- {* for backwards compatibility *}
   367 
   368 lemma splitI: "f a b \<Longrightarrow> split f (a, b)"
   369   by (rule split_conv [THEN iffD2])
   370 
   371 lemma splitD: "split f (a, b) \<Longrightarrow> f a b"
   372   by (rule split_conv [THEN iffD1])
   373 
   374 lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b"
   375   by (simp add: curry_def)
   376 
   377 lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)"
   378   by (simp add: curry_def)
   379 
   380 lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q"
   381   by (simp add: curry_def)
   382 
   383 lemma curry_split [simp]: "curry (split f) = f"
   384   by (simp add: curry_def split_def)
   385 
   386 lemma split_curry [simp]: "split (curry f) = f"
   387   by (simp add: curry_def split_def)
   388 
   389 lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id"
   390   by (simp add: split_def id_def)
   391 
   392 lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f"
   393   -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *}
   394   by (rule ext) auto
   395 
   396 lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)"
   397   by (cases x) simp
   398 
   399 lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p"
   400   unfolding split_def ..
   401 
   402 lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))"
   403   -- {* Can't be added to simpset: loops! *}
   404   by (simp add: split_eta)
   405 
   406 lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
   407   by (simp add: split_def)
   408 
   409 lemma split_weak_cong: "p = q \<Longrightarrow> split c p = split c q"
   410   -- {* Prevents simplification of @{term c}: much faster *}
   411   by (erule arg_cong)
   412 
   413 lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
   414   by (simp add: split_eta)
   415 
   416 text {*
   417   Simplification procedure for @{thm [source] cond_split_eta}.  Using
   418   @{thm [source] split_eta} as a rewrite rule is not general enough,
   419   and using @{thm [source] cond_split_eta} directly would render some
   420   existing proofs very inefficient; similarly for @{text
   421   split_beta}.
   422 *}
   423 
   424 ML {*
   425 
   426 local
   427   val cond_split_eta_ss = HOL_basic_ss addsimps [thm "cond_split_eta"]
   428   fun  Pair_pat k 0 (Bound m) = (m = k)
   429   |    Pair_pat k i (Const ("Pair",  _) $ Bound m $ t) = i > 0 andalso
   430                         m = k+i andalso Pair_pat k (i-1) t
   431   |    Pair_pat _ _ _ = false;
   432   fun no_args k i (Abs (_, _, t)) = no_args (k+1) i t
   433   |   no_args k i (t $ u) = no_args k i t andalso no_args k i u
   434   |   no_args k i (Bound m) = m < k orelse m > k+i
   435   |   no_args _ _ _ = true;
   436   fun split_pat tp i (Abs  (_,_,t)) = if tp 0 i t then SOME (i,t) else NONE
   437   |   split_pat tp i (Const ("split", _) $ Abs (_, _, t)) = split_pat tp (i+1) t
   438   |   split_pat tp i _ = NONE;
   439   fun metaeq ss lhs rhs = mk_meta_eq (Goal.prove (Simplifier.the_context ss) [] []
   440         (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs,rhs)))
   441         (K (simp_tac (Simplifier.inherit_context ss cond_split_eta_ss) 1)));
   442 
   443   fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k+1) i t
   444   |   beta_term_pat k i (t $ u) = Pair_pat k i (t $ u) orelse
   445                         (beta_term_pat k i t andalso beta_term_pat k i u)
   446   |   beta_term_pat k i t = no_args k i t;
   447   fun  eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
   448   |    eta_term_pat _ _ _ = false;
   449   fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
   450   |   subst arg k i (t $ u) = if Pair_pat k i (t $ u) then incr_boundvars k arg
   451                               else (subst arg k i t $ subst arg k i u)
   452   |   subst arg k i t = t;
   453   fun beta_proc ss (s as Const ("split", _) $ Abs (_, _, t) $ arg) =
   454         (case split_pat beta_term_pat 1 t of
   455         SOME (i,f) => SOME (metaeq ss s (subst arg 0 i f))
   456         | NONE => NONE)
   457   |   beta_proc _ _ = NONE;
   458   fun eta_proc ss (s as Const ("split", _) $ Abs (_, _, t)) =
   459         (case split_pat eta_term_pat 1 t of
   460           SOME (_,ft) => SOME (metaeq ss s (let val (f $ arg) = ft in f end))
   461         | NONE => NONE)
   462   |   eta_proc _ _ = NONE;
   463 in
   464   val split_beta_proc = Simplifier.simproc (the_context ()) "split_beta" ["split f z"] (K beta_proc);
   465   val split_eta_proc = Simplifier.simproc (the_context ()) "split_eta" ["split f"] (K eta_proc);
   466 end;
   467 
   468 Addsimprocs [split_beta_proc, split_eta_proc];
   469 *}
   470 
   471 lemma split_beta [mono]: "(%(x, y). P x y) z = P (fst z) (snd z)"
   472   by (subst surjective_pairing, rule split_conv)
   473 
   474 lemma split_split [noatp]: "R(split c p) = (ALL x y. p = (x, y) --> R(c x y))"
   475   -- {* For use with @{text split} and the Simplifier. *}
   476   by (insert surj_pair [of p], clarify, simp)
   477 
   478 text {*
   479   @{thm [source] split_split} could be declared as @{text "[split]"}
   480   done after the Splitter has been speeded up significantly;
   481   precompute the constants involved and don't do anything unless the
   482   current goal contains one of those constants.
   483 *}
   484 
   485 lemma split_split_asm [noatp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))"
   486 by (subst split_split, simp)
   487 
   488 
   489 text {*
   490   \medskip @{term split} used as a logical connective or set former.
   491 
   492   \medskip These rules are for use with @{text blast}; could instead
   493   call @{text simp} using @{thm [source] split} as rewrite. *}
   494 
   495 lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"
   496   apply (simp only: split_tupled_all)
   497   apply (simp (no_asm_simp))
   498   done
   499 
   500 lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"
   501   apply (simp only: split_tupled_all)
   502   apply (simp (no_asm_simp))
   503   done
   504 
   505 lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
   506   by (induct p) (auto simp add: split_def)
   507 
   508 lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
   509   by (induct p) (auto simp add: split_def)
   510 
   511 lemma splitE2:
   512   "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
   513 proof -
   514   assume q: "Q (split P z)"
   515   assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
   516   show R
   517     apply (rule r surjective_pairing)+
   518     apply (rule split_beta [THEN subst], rule q)
   519     done
   520 qed
   521 
   522 lemma splitD': "split R (a,b) c ==> R a b c"
   523   by simp
   524 
   525 lemma mem_splitI: "z: c a b ==> z: split c (a, b)"
   526   by simp
   527 
   528 lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"
   529 by (simp only: split_tupled_all, simp)
   530 
   531 lemma mem_splitE:
   532   assumes major: "z: split c p"
   533     and cases: "!!x y. [| p = (x,y); z: c x y |] ==> Q"
   534   shows Q
   535   by (rule major [unfolded split_def] cases surjective_pairing)+
   536 
   537 declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
   538 declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]
   539 
   540 ML {*
   541 local (* filtering with exists_p_split is an essential optimization *)
   542   fun exists_p_split (Const ("split",_) $ _ $ (Const ("Pair",_)$_$_)) = true
   543     | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
   544     | exists_p_split (Abs (_, _, t)) = exists_p_split t
   545     | exists_p_split _ = false;
   546   val ss = HOL_basic_ss addsimps [thm "split_conv"];
   547 in
   548 val split_conv_tac = SUBGOAL (fn (t, i) =>
   549     if exists_p_split t then safe_full_simp_tac ss i else no_tac);
   550 end;
   551 *}
   552 
   553 (* This prevents applications of splitE for already splitted arguments leading
   554    to quite time-consuming computations (in particular for nested tuples) *)
   555 declaration {* fn _ =>
   556   Classical.map_cs (fn cs => cs addSbefore ("split_conv_tac", split_conv_tac))
   557 *}
   558 
   559 lemma split_eta_SetCompr [simp,noatp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
   560   by (rule ext) fast
   561 
   562 lemma split_eta_SetCompr2 [simp,noatp]: "(%u. EX x y. u = (x, y) & P x y) = split P"
   563   by (rule ext) fast
   564 
   565 lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"
   566   -- {* Allows simplifications of nested splits in case of independent predicates. *}
   567   by (rule ext) blast
   568 
   569 (* Do NOT make this a simp rule as it
   570    a) only helps in special situations
   571    b) can lead to nontermination in the presence of split_def
   572 *)
   573 lemma split_comp_eq: 
   574   fixes f :: "'a => 'b => 'c" and g :: "'d => 'a"
   575   shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"
   576   by (rule ext) auto
   577 
   578 lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
   579   apply (rule_tac x = "(a, b)" in image_eqI)
   580    apply auto
   581   done
   582 
   583 lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
   584   by blast
   585 
   586 (*
   587 the following  would be slightly more general,
   588 but cannot be used as rewrite rule:
   589 ### Cannot add premise as rewrite rule because it contains (type) unknowns:
   590 ### ?y = .x
   591 Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
   592 by (rtac some_equality 1)
   593 by ( Simp_tac 1)
   594 by (split_all_tac 1)
   595 by (Asm_full_simp_tac 1)
   596 qed "The_split_eq";
   597 *)
   598 
   599 text {*
   600   Setup of internal @{text split_rule}.
   601 *}
   602 
   603 definition
   604   internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c"
   605 where
   606   "internal_split == split"
   607 
   608 lemma internal_split_conv: "internal_split c (a, b) = c a b"
   609   by (simp only: internal_split_def split_conv)
   610 
   611 hide const internal_split
   612 
   613 use "Tools/split_rule.ML"
   614 setup SplitRule.setup
   615 
   616 lemmas prod_caseI = prod.cases [THEN iffD2, standard]
   617 
   618 lemma prod_caseI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> prod_case c p"
   619   by auto
   620 
   621 lemma prod_caseI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> prod_case c p x"
   622   by (auto simp: split_tupled_all)
   623 
   624 lemma prod_caseE: "prod_case c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
   625   by (induct p) auto
   626 
   627 lemma prod_caseE': "prod_case c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
   628   by (induct p) auto
   629 
   630 lemma prod_case_unfold: "prod_case = (%c p. c (fst p) (snd p))"
   631   by (simp add: expand_fun_eq)
   632 
   633 declare prod_caseI2' [intro!] prod_caseI2 [intro!] prod_caseI [intro!]
   634 declare prod_caseE' [elim!] prod_caseE [elim!]
   635 
   636 lemma prod_case_split:
   637   "prod_case = split"
   638   by (auto simp add: expand_fun_eq)
   639 
   640 lemma prod_case_beta:
   641   "prod_case f p = f (fst p) (snd p)"
   642   unfolding prod_case_split split_beta ..
   643 
   644 
   645 subsection {* Further cases/induct rules for tuples *}
   646 
   647 lemma prod_cases3 [cases type]:
   648   obtains (fields) a b c where "y = (a, b, c)"
   649   by (cases y, case_tac b) blast
   650 
   651 lemma prod_induct3 [case_names fields, induct type]:
   652     "(!!a b c. P (a, b, c)) ==> P x"
   653   by (cases x) blast
   654 
   655 lemma prod_cases4 [cases type]:
   656   obtains (fields) a b c d where "y = (a, b, c, d)"
   657   by (cases y, case_tac c) blast
   658 
   659 lemma prod_induct4 [case_names fields, induct type]:
   660     "(!!a b c d. P (a, b, c, d)) ==> P x"
   661   by (cases x) blast
   662 
   663 lemma prod_cases5 [cases type]:
   664   obtains (fields) a b c d e where "y = (a, b, c, d, e)"
   665   by (cases y, case_tac d) blast
   666 
   667 lemma prod_induct5 [case_names fields, induct type]:
   668     "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
   669   by (cases x) blast
   670 
   671 lemma prod_cases6 [cases type]:
   672   obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
   673   by (cases y, case_tac e) blast
   674 
   675 lemma prod_induct6 [case_names fields, induct type]:
   676     "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
   677   by (cases x) blast
   678 
   679 lemma prod_cases7 [cases type]:
   680   obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
   681   by (cases y, case_tac f) blast
   682 
   683 lemma prod_induct7 [case_names fields, induct type]:
   684     "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
   685   by (cases x) blast
   686 
   687 
   688 subsubsection {* Derived operations *}
   689 
   690 text {*
   691   The composition-uncurry combinator.
   692 *}
   693 
   694 notation fcomp (infixl "o>" 60)
   695 
   696 definition
   697   scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "o\<rightarrow>" 60)
   698 where
   699   "f o\<rightarrow> g = (\<lambda>x. split g (f x))"
   700 
   701 lemma scomp_apply:  "(f o\<rightarrow> g) x = split g (f x)"
   702   by (simp add: scomp_def)
   703 
   704 lemma Pair_scomp: "Pair x o\<rightarrow> f = f x"
   705   by (simp add: expand_fun_eq scomp_apply)
   706 
   707 lemma scomp_Pair: "x o\<rightarrow> Pair = x"
   708   by (simp add: expand_fun_eq scomp_apply)
   709 
   710 lemma scomp_scomp: "(f o\<rightarrow> g) o\<rightarrow> h = f o\<rightarrow> (\<lambda>x. g x o\<rightarrow> h)"
   711   by (simp add: expand_fun_eq split_twice scomp_def)
   712 
   713 lemma scomp_fcomp: "(f o\<rightarrow> g) o> h = f o\<rightarrow> (\<lambda>x. g x o> h)"
   714   by (simp add: expand_fun_eq scomp_apply fcomp_def split_def)
   715 
   716 lemma fcomp_scomp: "(f o> g) o\<rightarrow> h = f o> (g o\<rightarrow> h)"
   717   by (simp add: expand_fun_eq scomp_apply fcomp_apply)
   718 
   719 no_notation fcomp (infixl "o>" 60)
   720 no_notation scomp (infixl "o\<rightarrow>" 60)
   721 
   722 
   723 text {*
   724   @{term prod_fun} --- action of the product functor upon
   725   functions.
   726 *}
   727 
   728 definition prod_fun :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where
   729   [code func del]: "prod_fun f g = (\<lambda>(x, y). (f x, g y))"
   730 
   731 lemma prod_fun [simp, code func]: "prod_fun f g (a, b) = (f a, g b)"
   732   by (simp add: prod_fun_def)
   733 
   734 lemma prod_fun_compose: "prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)"
   735   by (rule ext) auto
   736 
   737 lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)"
   738   by (rule ext) auto
   739 
   740 lemma prod_fun_imageI [intro]: "(a, b) : r ==> (f a, g b) : prod_fun f g ` r"
   741   apply (rule image_eqI)
   742   apply (rule prod_fun [symmetric], assumption)
   743   done
   744 
   745 lemma prod_fun_imageE [elim!]:
   746   assumes major: "c: (prod_fun f g)`r"
   747     and cases: "!!x y. [| c=(f(x),g(y));  (x,y):r |] ==> P"
   748   shows P
   749   apply (rule major [THEN imageE])
   750   apply (rule_tac p = x in PairE)
   751   apply (rule cases)
   752    apply (blast intro: prod_fun)
   753   apply blast
   754   done
   755 
   756 definition
   757   apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b"
   758 where
   759   [code func del]: "apfst f = prod_fun f id"
   760 
   761 definition
   762   apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c"
   763 where
   764   [code func del]: "apsnd f = prod_fun id f"
   765 
   766 lemma apfst_conv [simp, code]:
   767   "apfst f (x, y) = (f x, y)" 
   768   by (simp add: apfst_def)
   769 
   770 lemma upd_snd_conv [simp, code]:
   771   "apsnd f (x, y) = (x, f y)" 
   772   by (simp add: apsnd_def)
   773 
   774 
   775 text {*
   776   Disjoint union of a family of sets -- Sigma.
   777 *}
   778 
   779 definition  Sigma :: "['a set, 'a => 'b set] => ('a \<times> 'b) set" where
   780   Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}"
   781 
   782 abbreviation
   783   Times :: "['a set, 'b set] => ('a * 'b) set"
   784     (infixr "<*>" 80) where
   785   "A <*> B == Sigma A (%_. B)"
   786 
   787 notation (xsymbols)
   788   Times  (infixr "\<times>" 80)
   789 
   790 notation (HTML output)
   791   Times  (infixr "\<times>" 80)
   792 
   793 syntax
   794   "@Sigma" ::"[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" [0, 0, 10] 10)
   795 
   796 translations
   797   "SIGMA x:A. B" == "Product_Type.Sigma A (%x. B)"
   798 
   799 lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
   800   by (unfold Sigma_def) blast
   801 
   802 lemma SigmaE [elim!]:
   803     "[| c: Sigma A B;
   804         !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
   805      |] ==> P"
   806   -- {* The general elimination rule. *}
   807   by (unfold Sigma_def) blast
   808 
   809 text {*
   810   Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
   811   eigenvariables.
   812 *}
   813 
   814 lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
   815   by blast
   816 
   817 lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
   818   by blast
   819 
   820 lemma SigmaE2:
   821     "[| (a, b) : Sigma A B;
   822         [| a:A;  b:B(a) |] ==> P
   823      |] ==> P"
   824   by blast
   825 
   826 lemma Sigma_cong:
   827      "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>
   828       \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
   829   by auto
   830 
   831 lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
   832   by blast
   833 
   834 lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
   835   by blast
   836 
   837 lemma Sigma_empty2 [simp]: "A <*> {} = {}"
   838   by blast
   839 
   840 lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
   841   by auto
   842 
   843 lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
   844   by auto
   845 
   846 lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
   847   by auto
   848 
   849 lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
   850   by blast
   851 
   852 lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
   853   by blast
   854 
   855 lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
   856   by (blast elim: equalityE)
   857 
   858 lemma SetCompr_Sigma_eq:
   859     "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
   860   by blast
   861 
   862 lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"
   863   by blast
   864 
   865 lemma UN_Times_distrib:
   866   "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"
   867   -- {* Suggested by Pierre Chartier *}
   868   by blast
   869 
   870 lemma split_paired_Ball_Sigma [simp,noatp]:
   871     "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"
   872   by blast
   873 
   874 lemma split_paired_Bex_Sigma [simp,noatp]:
   875     "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"
   876   by blast
   877 
   878 lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"
   879   by blast
   880 
   881 lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))"
   882   by blast
   883 
   884 lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"
   885   by blast
   886 
   887 lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))"
   888   by blast
   889 
   890 lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"
   891   by blast
   892 
   893 lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"
   894   by blast
   895 
   896 lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"
   897   by blast
   898 
   899 text {*
   900   Non-dependent versions are needed to avoid the need for higher-order
   901   matching, especially when the rules are re-oriented.
   902 *}
   903 
   904 lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"
   905   by blast
   906 
   907 lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"
   908   by blast
   909 
   910 lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
   911   by blast
   912 
   913 
   914 subsubsection {* Code generator setup *}
   915 
   916 instance * :: (eq, eq) eq ..
   917 
   918 lemma [code func]:
   919   "(x1\<Colon>'a\<Colon>eq, y1\<Colon>'b\<Colon>eq) = (x2, y2) \<longleftrightarrow> x1 = x2 \<and> y1 = y2" by auto
   920 
   921 lemma split_case_cert:
   922   assumes "CASE \<equiv> split f"
   923   shows "CASE (a, b) \<equiv> f a b"
   924   using assms by simp
   925 
   926 setup {*
   927   Code.add_case @{thm split_case_cert}
   928 *}
   929 
   930 code_type *
   931   (SML infix 2 "*")
   932   (OCaml infix 2 "*")
   933   (Haskell "!((_),/ (_))")
   934 
   935 code_instance * :: eq
   936   (Haskell -)
   937 
   938 code_const "op = \<Colon> 'a\<Colon>eq \<times> 'b\<Colon>eq \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"
   939   (Haskell infixl 4 "==")
   940 
   941 code_const Pair
   942   (SML "!((_),/ (_))")
   943   (OCaml "!((_),/ (_))")
   944   (Haskell "!((_),/ (_))")
   945 
   946 code_const fst and snd
   947   (Haskell "fst" and "snd")
   948 
   949 types_code
   950   "*"     ("(_ */ _)")
   951 attach (term_of) {*
   952 fun term_of_id_42 aF aT bF bT (x, y) = HOLogic.pair_const aT bT $ aF x $ bF y;
   953 *}
   954 attach (test) {*
   955 fun gen_id_42 aG aT bG bT i =
   956   let
   957     val (x, t) = aG i;
   958     val (y, u) = bG i
   959   in ((x, y), fn () => HOLogic.pair_const aT bT $ t () $ u ()) end;
   960 *}
   961 
   962 consts_code
   963   "Pair"    ("(_,/ _)")
   964 
   965 setup {*
   966 
   967 let
   968 
   969 fun strip_abs_split 0 t = ([], t)
   970   | strip_abs_split i (Abs (s, T, t)) =
   971       let
   972         val s' = Codegen.new_name t s;
   973         val v = Free (s', T)
   974       in apfst (cons v) (strip_abs_split (i-1) (subst_bound (v, t))) end
   975   | strip_abs_split i (u as Const ("split", _) $ t) = (case strip_abs_split (i+1) t of
   976         (v :: v' :: vs, u) => (HOLogic.mk_prod (v, v') :: vs, u)
   977       | _ => ([], u))
   978   | strip_abs_split i t = ([], t);
   979 
   980 fun let_codegen thy defs gr dep thyname brack t = (case strip_comb t of
   981     (t1 as Const ("Let", _), t2 :: t3 :: ts) =>
   982     let
   983       fun dest_let (l as Const ("Let", _) $ t $ u) =
   984           (case strip_abs_split 1 u of
   985              ([p], u') => apfst (cons (p, t)) (dest_let u')
   986            | _ => ([], l))
   987         | dest_let t = ([], t);
   988       fun mk_code (gr, (l, r)) =
   989         let
   990           val (gr1, pl) = Codegen.invoke_codegen thy defs dep thyname false (gr, l);
   991           val (gr2, pr) = Codegen.invoke_codegen thy defs dep thyname false (gr1, r);
   992         in (gr2, (pl, pr)) end
   993     in case dest_let (t1 $ t2 $ t3) of
   994         ([], _) => NONE
   995       | (ps, u) =>
   996           let
   997             val (gr1, qs) = foldl_map mk_code (gr, ps);
   998             val (gr2, pu) = Codegen.invoke_codegen thy defs dep thyname false (gr1, u);
   999             val (gr3, pargs) = foldl_map
  1000               (Codegen.invoke_codegen thy defs dep thyname true) (gr2, ts)
  1001           in
  1002             SOME (gr3, Codegen.mk_app brack
  1003               (Pretty.blk (0, [Codegen.str "let ", Pretty.blk (0, List.concat
  1004                   (separate [Codegen.str ";", Pretty.brk 1] (map (fn (pl, pr) =>
  1005                     [Pretty.block [Codegen.str "val ", pl, Codegen.str " =",
  1006                        Pretty.brk 1, pr]]) qs))),
  1007                 Pretty.brk 1, Codegen.str "in ", pu,
  1008                 Pretty.brk 1, Codegen.str "end"])) pargs)
  1009           end
  1010     end
  1011   | _ => NONE);
  1012 
  1013 fun split_codegen thy defs gr dep thyname brack t = (case strip_comb t of
  1014     (t1 as Const ("split", _), t2 :: ts) =>
  1015       (case strip_abs_split 1 (t1 $ t2) of
  1016          ([p], u) =>
  1017            let
  1018              val (gr1, q) = Codegen.invoke_codegen thy defs dep thyname false (gr, p);
  1019              val (gr2, pu) = Codegen.invoke_codegen thy defs dep thyname false (gr1, u);
  1020              val (gr3, pargs) = foldl_map
  1021                (Codegen.invoke_codegen thy defs dep thyname true) (gr2, ts)
  1022            in
  1023              SOME (gr2, Codegen.mk_app brack
  1024                (Pretty.block [Codegen.str "(fn ", q, Codegen.str " =>",
  1025                  Pretty.brk 1, pu, Codegen.str ")"]) pargs)
  1026            end
  1027        | _ => NONE)
  1028   | _ => NONE);
  1029 
  1030 in
  1031 
  1032   Codegen.add_codegen "let_codegen" let_codegen
  1033   #> Codegen.add_codegen "split_codegen" split_codegen
  1034 
  1035 end
  1036 *}
  1037 
  1038 
  1039 subsection {* Legacy bindings *}
  1040 
  1041 ML {*
  1042 val Collect_split = thm "Collect_split";
  1043 val Compl_Times_UNIV1 = thm "Compl_Times_UNIV1";
  1044 val Compl_Times_UNIV2 = thm "Compl_Times_UNIV2";
  1045 val PairE = thm "PairE";
  1046 val Pair_Rep_inject = thm "Pair_Rep_inject";
  1047 val Pair_def = thm "Pair_def";
  1048 val Pair_eq = @{thm "prod.inject"};
  1049 val Pair_fst_snd_eq = thm "Pair_fst_snd_eq";
  1050 val ProdI = thm "ProdI";
  1051 val SetCompr_Sigma_eq = thm "SetCompr_Sigma_eq";
  1052 val SigmaD1 = thm "SigmaD1";
  1053 val SigmaD2 = thm "SigmaD2";
  1054 val SigmaE = thm "SigmaE";
  1055 val SigmaE2 = thm "SigmaE2";
  1056 val SigmaI = thm "SigmaI";
  1057 val Sigma_Diff_distrib1 = thm "Sigma_Diff_distrib1";
  1058 val Sigma_Diff_distrib2 = thm "Sigma_Diff_distrib2";
  1059 val Sigma_Int_distrib1 = thm "Sigma_Int_distrib1";
  1060 val Sigma_Int_distrib2 = thm "Sigma_Int_distrib2";
  1061 val Sigma_Un_distrib1 = thm "Sigma_Un_distrib1";
  1062 val Sigma_Un_distrib2 = thm "Sigma_Un_distrib2";
  1063 val Sigma_Union = thm "Sigma_Union";
  1064 val Sigma_def = thm "Sigma_def";
  1065 val Sigma_empty1 = thm "Sigma_empty1";
  1066 val Sigma_empty2 = thm "Sigma_empty2";
  1067 val Sigma_mono = thm "Sigma_mono";
  1068 val The_split = thm "The_split";
  1069 val The_split_eq = thm "The_split_eq";
  1070 val The_split_eq = thm "The_split_eq";
  1071 val Times_Diff_distrib1 = thm "Times_Diff_distrib1";
  1072 val Times_Int_distrib1 = thm "Times_Int_distrib1";
  1073 val Times_Un_distrib1 = thm "Times_Un_distrib1";
  1074 val Times_eq_cancel2 = thm "Times_eq_cancel2";
  1075 val Times_subset_cancel2 = thm "Times_subset_cancel2";
  1076 val UNIV_Times_UNIV = thm "UNIV_Times_UNIV";
  1077 val UN_Times_distrib = thm "UN_Times_distrib";
  1078 val Unity_def = thm "Unity_def";
  1079 val cond_split_eta = thm "cond_split_eta";
  1080 val fst_conv = thm "fst_conv";
  1081 val fst_def = thm "fst_def";
  1082 val fst_eqD = thm "fst_eqD";
  1083 val inj_on_Abs_Prod = thm "inj_on_Abs_Prod";
  1084 val mem_Sigma_iff = thm "mem_Sigma_iff";
  1085 val mem_splitE = thm "mem_splitE";
  1086 val mem_splitI = thm "mem_splitI";
  1087 val mem_splitI2 = thm "mem_splitI2";
  1088 val prod_eqI = thm "prod_eqI";
  1089 val prod_fun = thm "prod_fun";
  1090 val prod_fun_compose = thm "prod_fun_compose";
  1091 val prod_fun_def = thm "prod_fun_def";
  1092 val prod_fun_ident = thm "prod_fun_ident";
  1093 val prod_fun_imageE = thm "prod_fun_imageE";
  1094 val prod_fun_imageI = thm "prod_fun_imageI";
  1095 val prod_induct = thm "prod.induct";
  1096 val snd_conv = thm "snd_conv";
  1097 val snd_def = thm "snd_def";
  1098 val snd_eqD = thm "snd_eqD";
  1099 val split = thm "split";
  1100 val splitD = thm "splitD";
  1101 val splitD' = thm "splitD'";
  1102 val splitE = thm "splitE";
  1103 val splitE' = thm "splitE'";
  1104 val splitE2 = thm "splitE2";
  1105 val splitI = thm "splitI";
  1106 val splitI2 = thm "splitI2";
  1107 val splitI2' = thm "splitI2'";
  1108 val split_beta = thm "split_beta";
  1109 val split_conv = thm "split_conv";
  1110 val split_def = thm "split_def";
  1111 val split_eta = thm "split_eta";
  1112 val split_eta_SetCompr = thm "split_eta_SetCompr";
  1113 val split_eta_SetCompr2 = thm "split_eta_SetCompr2";
  1114 val split_paired_All = thm "split_paired_All";
  1115 val split_paired_Ball_Sigma = thm "split_paired_Ball_Sigma";
  1116 val split_paired_Bex_Sigma = thm "split_paired_Bex_Sigma";
  1117 val split_paired_Ex = thm "split_paired_Ex";
  1118 val split_paired_The = thm "split_paired_The";
  1119 val split_paired_all = thm "split_paired_all";
  1120 val split_part = thm "split_part";
  1121 val split_split = thm "split_split";
  1122 val split_split_asm = thm "split_split_asm";
  1123 val split_tupled_all = thms "split_tupled_all";
  1124 val split_weak_cong = thm "split_weak_cong";
  1125 val surj_pair = thm "surj_pair";
  1126 val surjective_pairing = thm "surjective_pairing";
  1127 val unit_abs_eta_conv = thm "unit_abs_eta_conv";
  1128 val unit_all_eq1 = thm "unit_all_eq1";
  1129 val unit_all_eq2 = thm "unit_all_eq2";
  1130 val unit_eq = thm "unit_eq";
  1131 *}
  1132 
  1133 
  1134 subsection {* Further inductive packages *}
  1135 
  1136 use "Tools/inductive_realizer.ML"
  1137 setup InductiveRealizer.setup
  1138 
  1139 use "Tools/inductive_set_package.ML"
  1140 setup InductiveSetPackage.setup
  1141 
  1142 use "Tools/datatype_realizer.ML"
  1143 setup DatatypeRealizer.setup
  1144 
  1145 end