src/HOL/Product_Type.thy
author haftmann
Tue Sep 25 12:16:08 2007 +0200 (2007-09-25)
changeset 24699 c6674504103f
parent 24286 7619080e49f0
child 24844 98c006a30218
permissions -rw-r--r--
datatype interpretators for size and datatype_realizer
     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 bool
    21   distinct True_not_False False_not_True
    22   induction bool_induct
    23 
    24 declare case_split [cases type: bool]
    25   -- "prefer plain propositional version"
    26 
    27 
    28 subsection {* Unit *}
    29 
    30 typedef unit = "{True}"
    31 proof
    32   show "True : ?unit" ..
    33 qed
    34 
    35 definition
    36   Unity :: unit    ("'(')")
    37 where
    38   "() = Abs_unit True"
    39 
    40 lemma unit_eq [noatp]: "u = ()"
    41   by (induct u) (simp add: unit_def Unity_def)
    42 
    43 text {*
    44   Simplification procedure for @{thm [source] unit_eq}.  Cannot use
    45   this rule directly --- it loops!
    46 *}
    47 
    48 ML_setup {*
    49   val unit_eq_proc =
    50     let val unit_meta_eq = mk_meta_eq @{thm unit_eq} in
    51       Simplifier.simproc @{theory} "unit_eq" ["x::unit"]
    52       (fn _ => fn _ => fn t => if HOLogic.is_unit t then NONE else SOME unit_meta_eq)
    53     end;
    54 
    55   Addsimprocs [unit_eq_proc];
    56 *}
    57 
    58 lemma unit_induct [noatp,induct type: unit]: "P () ==> P x"
    59   by simp
    60 
    61 rep_datatype unit
    62   induction unit_induct
    63 
    64 lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()"
    65   by simp
    66 
    67 lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P"
    68   by (rule triv_forall_equality)
    69 
    70 text {*
    71   This rewrite counters the effect of @{text unit_eq_proc} on @{term
    72   [source] "%u::unit. f u"}, replacing it by @{term [source]
    73   f} rather than by @{term [source] "%u. f ()"}.
    74 *}
    75 
    76 lemma unit_abs_eta_conv [simp,noatp]: "(%u::unit. f ()) = f"
    77   by (rule ext) simp
    78 
    79 
    80 subsection {* Pairs *}
    81 
    82 subsubsection {* Type definition *}
    83 
    84 constdefs
    85   Pair_Rep :: "['a, 'b] => ['a, 'b] => bool"
    86   "Pair_Rep == (%a b. %x y. x=a & y=b)"
    87 
    88 global
    89 
    90 typedef (Prod)
    91   ('a, 'b) "*"    (infixr "*" 20)
    92     = "{f. EX a b. f = Pair_Rep (a::'a) (b::'b)}"
    93 proof
    94   fix a b show "Pair_Rep a b : ?Prod"
    95     by blast
    96 qed
    97 
    98 syntax (xsymbols)
    99   "*"      :: "[type, type] => type"         ("(_ \<times>/ _)" [21, 20] 20)
   100 syntax (HTML output)
   101   "*"      :: "[type, type] => type"         ("(_ \<times>/ _)" [21, 20] 20)
   102 
   103 local
   104 
   105 subsubsection {* Definitions *}
   106 
   107 global
   108 
   109 consts
   110   fst      :: "'a * 'b => 'a"
   111   snd      :: "'a * 'b => 'b"
   112   split    :: "[['a, 'b] => 'c, 'a * 'b] => 'c"
   113   curry    :: "['a * 'b => 'c, 'a, 'b] => 'c"
   114   prod_fun :: "['a => 'b, 'c => 'd, 'a * 'c] => 'b * 'd"
   115   Pair     :: "['a, 'b] => 'a * 'b"
   116   Sigma    :: "['a set, 'a => 'b set] => ('a * 'b) set"
   117 
   118 local
   119 
   120 defs
   121   Pair_def:     "Pair a b == Abs_Prod (Pair_Rep a b)"
   122   fst_def:      "fst p == THE a. EX b. p = Pair a b"
   123   snd_def:      "snd p == THE b. EX a. p = Pair a b"
   124   split_def:    "split == (%c p. c (fst p) (snd p))"
   125   curry_def:    "curry == (%c x y. c (Pair x y))"
   126   prod_fun_def: "prod_fun f g == split (%x y. Pair (f x) (g y))"
   127   Sigma_def [code func]:    "Sigma A B == UN x:A. UN y:B x. {Pair x y}"
   128 
   129 abbreviation
   130   Times :: "['a set, 'b set] => ('a * 'b) set"
   131     (infixr "<*>" 80) where
   132   "A <*> B == Sigma A (%_. B)"
   133 
   134 notation (xsymbols)
   135   Times  (infixr "\<times>" 80)
   136 
   137 notation (HTML output)
   138   Times  (infixr "\<times>" 80)
   139 
   140 
   141 subsubsection {* Concrete syntax *}
   142 
   143 text {*
   144   Patterns -- extends pre-defined type @{typ pttrn} used in
   145   abstractions.
   146 *}
   147 
   148 nonterminals
   149   tuple_args patterns
   150 
   151 syntax
   152   "_tuple"      :: "'a => tuple_args => 'a * 'b"        ("(1'(_,/ _'))")
   153   "_tuple_arg"  :: "'a => tuple_args"                   ("_")
   154   "_tuple_args" :: "'a => tuple_args => tuple_args"     ("_,/ _")
   155   "_pattern"    :: "[pttrn, patterns] => pttrn"         ("'(_,/ _')")
   156   ""            :: "pttrn => patterns"                  ("_")
   157   "_patterns"   :: "[pttrn, patterns] => patterns"      ("_,/ _")
   158   "@Sigma" ::"[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" [0, 0, 10] 10)
   159 
   160 translations
   161   "(x, y)"       == "Pair x y"
   162   "_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))"
   163   "%(x,y,zs).b"  == "split(%x (y,zs).b)"
   164   "%(x,y).b"     == "split(%x y. b)"
   165   "_abs (Pair x y) t" => "%(x,y).t"
   166   (* The last rule accommodates tuples in `case C ... (x,y) ... => ...'
   167      The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *)
   168   "SIGMA x:A. B" == "Sigma A (%x. B)"
   169 
   170 (* reconstructs pattern from (nested) splits, avoiding eta-contraction of body*)
   171 (* works best with enclosing "let", if "let" does not avoid eta-contraction   *)
   172 print_translation {*
   173 let fun split_tr' [Abs (x,T,t as (Abs abs))] =
   174       (* split (%x y. t) => %(x,y) t *)
   175       let val (y,t') = atomic_abs_tr' abs;
   176           val (x',t'') = atomic_abs_tr' (x,T,t');
   177     
   178       in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end
   179     | split_tr' [Abs (x,T,(s as Const ("split",_)$t))] =
   180        (* split (%x. (split (%y z. t))) => %(x,y,z). t *)
   181        let val (Const ("_abs",_)$(Const ("_pattern",_)$y$z)$t') = split_tr' [t];
   182            val (x',t'') = atomic_abs_tr' (x,T,t');
   183        in Syntax.const "_abs"$ 
   184            (Syntax.const "_pattern"$x'$(Syntax.const "_patterns"$y$z))$t'' end
   185     | split_tr' [Const ("split",_)$t] =
   186        (* split (split (%x y z. t)) => %((x,y),z). t *)   
   187        split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *)
   188     | split_tr' [Const ("_abs",_)$x_y$(Abs abs)] =
   189        (* split (%pttrn z. t) => %(pttrn,z). t *)
   190        let val (z,t) = atomic_abs_tr' abs;
   191        in Syntax.const "_abs" $ (Syntax.const "_pattern" $x_y$z) $ t end
   192     | split_tr' _ =  raise Match;
   193 in [("split", split_tr')]
   194 end
   195 *}
   196 
   197 (* print "split f" as "\<lambda>(x,y). f x y" and "split (\<lambda>x. f x)" as "\<lambda>(x,y). f x y" *) 
   198 typed_print_translation {*
   199 let
   200   fun split_guess_names_tr' _ T [Abs (x,_,Abs _)] = raise Match
   201     | split_guess_names_tr' _ T  [Abs (x,xT,t)] =
   202         (case (head_of t) of
   203            Const ("split",_) => raise Match
   204          | _ => let 
   205                   val (_::yT::_) = binder_types (domain_type T) handle Bind => raise Match;
   206                   val (y,t') = atomic_abs_tr' ("y",yT,(incr_boundvars 1 t)$Bound 0); 
   207                   val (x',t'') = atomic_abs_tr' (x,xT,t');
   208                 in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end)
   209     | split_guess_names_tr' _ T [t] =
   210        (case (head_of t) of
   211            Const ("split",_) => raise Match 
   212          | _ => let 
   213                   val (xT::yT::_) = binder_types (domain_type T) handle Bind => raise Match;
   214                   val (y,t') = 
   215                         atomic_abs_tr' ("y",yT,(incr_boundvars 2 t)$Bound 1$Bound 0); 
   216                   val (x',t'') = atomic_abs_tr' ("x",xT,t');
   217                 in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end)
   218     | split_guess_names_tr' _ _ _ = raise Match;
   219 in [("split", split_guess_names_tr')]
   220 end 
   221 *}
   222 
   223 
   224 subsubsection {* Lemmas and proof tool setup *}
   225 
   226 lemma ProdI: "Pair_Rep a b : Prod"
   227   unfolding Prod_def by blast
   228 
   229 lemma Pair_Rep_inject: "Pair_Rep a b = Pair_Rep a' b' ==> a = a' & b = b'"
   230   apply (unfold Pair_Rep_def)
   231   apply (drule fun_cong [THEN fun_cong], blast)
   232   done
   233 
   234 lemma inj_on_Abs_Prod: "inj_on Abs_Prod Prod"
   235   apply (rule inj_on_inverseI)
   236   apply (erule Abs_Prod_inverse)
   237   done
   238 
   239 lemma Pair_inject:
   240   assumes "(a, b) = (a', b')"
   241     and "a = a' ==> b = b' ==> R"
   242   shows R
   243   apply (insert prems [unfolded Pair_def])
   244   apply (rule inj_on_Abs_Prod [THEN inj_onD, THEN Pair_Rep_inject, THEN conjE])
   245   apply (assumption | rule ProdI)+
   246   done
   247 
   248 lemma Pair_eq [iff]: "((a, b) = (a', b')) = (a = a' & b = b')"
   249   by (blast elim!: Pair_inject)
   250 
   251 lemma fst_conv [simp, code]: "fst (a, b) = a"
   252   unfolding fst_def by blast
   253 
   254 lemma snd_conv [simp, code]: "snd (a, b) = b"
   255   unfolding snd_def by blast
   256 
   257 lemma fst_eqD: "fst (x, y) = a ==> x = a"
   258   by simp
   259 
   260 lemma snd_eqD: "snd (x, y) = a ==> y = a"
   261   by simp
   262 
   263 lemma PairE_lemma: "EX x y. p = (x, y)"
   264   apply (unfold Pair_def)
   265   apply (rule Rep_Prod [unfolded Prod_def, THEN CollectE])
   266   apply (erule exE, erule exE, rule exI, rule exI)
   267   apply (rule Rep_Prod_inverse [symmetric, THEN trans])
   268   apply (erule arg_cong)
   269   done
   270 
   271 lemma PairE [cases type: *]: "(!!x y. p = (x, y) ==> Q) ==> Q"
   272   using PairE_lemma [of p] by blast
   273 
   274 ML {*
   275   local val PairE = thm "PairE" in
   276     fun pair_tac s =
   277       EVERY' [res_inst_tac [("p", s)] PairE, hyp_subst_tac, K prune_params_tac];
   278   end;
   279 *}
   280 
   281 lemma surjective_pairing: "p = (fst p, snd p)"
   282   -- {* Do not add as rewrite rule: invalidates some proofs in IMP *}
   283   by (cases p) simp
   284 
   285 lemmas pair_collapse = surjective_pairing [symmetric]
   286 declare pair_collapse [simp]
   287 
   288 lemma surj_pair [simp]: "EX x y. z = (x, y)"
   289   apply (rule exI)
   290   apply (rule exI)
   291   apply (rule surjective_pairing)
   292   done
   293 
   294 lemma split_paired_all: "(!!x. PROP P x) == (!!a b. PROP P (a, b))"
   295 proof
   296   fix a b
   297   assume "!!x. PROP P x"
   298   then show "PROP P (a, b)" .
   299 next
   300   fix x
   301   assume "!!a b. PROP P (a, b)"
   302   from `PROP P (fst x, snd x)` show "PROP P x" by simp
   303 qed
   304 
   305 lemmas split_tupled_all = split_paired_all unit_all_eq2
   306 
   307 text {*
   308   The rule @{thm [source] split_paired_all} does not work with the
   309   Simplifier because it also affects premises in congrence rules,
   310   where this can lead to premises of the form @{text "!!a b. ... =
   311   ?P(a, b)"} which cannot be solved by reflexivity.
   312 *}
   313 
   314 ML_setup {*
   315   (* replace parameters of product type by individual component parameters *)
   316   val safe_full_simp_tac = generic_simp_tac true (true, false, false);
   317   local (* filtering with exists_paired_all is an essential optimization *)
   318     fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) =
   319           can HOLogic.dest_prodT T orelse exists_paired_all t
   320       | exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u
   321       | exists_paired_all (Abs (_, _, t)) = exists_paired_all t
   322       | exists_paired_all _ = false;
   323     val ss = HOL_basic_ss
   324       addsimps [thm "split_paired_all", thm "unit_all_eq2", thm "unit_abs_eta_conv"]
   325       addsimprocs [unit_eq_proc];
   326   in
   327     val split_all_tac = SUBGOAL (fn (t, i) =>
   328       if exists_paired_all t then safe_full_simp_tac ss i else no_tac);
   329     val unsafe_split_all_tac = SUBGOAL (fn (t, i) =>
   330       if exists_paired_all t then full_simp_tac ss i else no_tac);
   331     fun split_all th =
   332    if exists_paired_all (#prop (Thm.rep_thm th)) then full_simplify ss th else th;
   333   end;
   334 
   335 change_claset (fn cs => cs addSbefore ("split_all_tac", split_all_tac));
   336 *}
   337 
   338 lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))"
   339   -- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *}
   340   by fast
   341 
   342 lemma curry_split [simp]: "curry (split f) = f"
   343   by (simp add: curry_def split_def)
   344 
   345 lemma split_curry [simp]: "split (curry f) = f"
   346   by (simp add: curry_def split_def)
   347 
   348 lemma curryI [intro!]: "f (a,b) ==> curry f a b"
   349   by (simp add: curry_def)
   350 
   351 lemma curryD [dest!]: "curry f a b ==> f (a,b)"
   352   by (simp add: curry_def)
   353 
   354 lemma curryE: "[| curry f a b ; f (a,b) ==> Q |] ==> Q"
   355   by (simp add: curry_def)
   356 
   357 lemma curry_conv [simp, code func]: "curry f a b = f (a,b)"
   358   by (simp add: curry_def)
   359 
   360 lemma prod_induct [induct type: *]: "!!x. (!!a b. P (a, b)) ==> P x"
   361   by fast
   362 
   363 rep_datatype prod
   364   inject Pair_eq
   365   induction prod_induct
   366 
   367 lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))"
   368   by fast
   369 
   370 lemma split_conv [simp, code func]: "split c (a, b) = c a b"
   371   by (simp add: split_def)
   372 
   373 lemmas split = split_conv  -- {* for backwards compatibility *}
   374 
   375 lemmas splitI = split_conv [THEN iffD2, standard]
   376 lemmas splitD = split_conv [THEN iffD1, standard]
   377 
   378 lemma split_Pair_apply: "split (%x y. f (x, y)) = f"
   379   -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *}
   380   apply (rule ext)
   381   apply (tactic {* pair_tac "x" 1 *}, simp)
   382   done
   383 
   384 lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))"
   385   -- {* Can't be added to simpset: loops! *}
   386   by (simp add: split_Pair_apply)
   387 
   388 lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
   389   by (simp add: split_def)
   390 
   391 lemma Pair_fst_snd_eq: "!!s t. (s = t) = (fst s = fst t & snd s = snd t)"
   392 by (simp only: split_tupled_all, simp)
   393 
   394 lemma prod_eqI [intro?]: "fst p = fst q ==> snd p = snd q ==> p = q"
   395   by (simp add: Pair_fst_snd_eq)
   396 
   397 lemma split_weak_cong: "p = q ==> split c p = split c q"
   398   -- {* Prevents simplification of @{term c}: much faster *}
   399   by (erule arg_cong)
   400 
   401 lemma split_eta: "(%(x, y). f (x, y)) = f"
   402   apply (rule ext)
   403   apply (simp only: split_tupled_all)
   404   apply (rule split_conv)
   405   done
   406 
   407 lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
   408   by (simp add: split_eta)
   409 
   410 text {*
   411   Simplification procedure for @{thm [source] cond_split_eta}.  Using
   412   @{thm [source] split_eta} as a rewrite rule is not general enough,
   413   and using @{thm [source] cond_split_eta} directly would render some
   414   existing proofs very inefficient; similarly for @{text
   415   split_beta}. *}
   416 
   417 ML_setup {*
   418 
   419 local
   420   val cond_split_eta_ss = HOL_basic_ss addsimps [thm "cond_split_eta"]
   421   fun  Pair_pat k 0 (Bound m) = (m = k)
   422   |    Pair_pat k i (Const ("Pair",  _) $ Bound m $ t) = i > 0 andalso
   423                         m = k+i andalso Pair_pat k (i-1) t
   424   |    Pair_pat _ _ _ = false;
   425   fun no_args k i (Abs (_, _, t)) = no_args (k+1) i t
   426   |   no_args k i (t $ u) = no_args k i t andalso no_args k i u
   427   |   no_args k i (Bound m) = m < k orelse m > k+i
   428   |   no_args _ _ _ = true;
   429   fun split_pat tp i (Abs  (_,_,t)) = if tp 0 i t then SOME (i,t) else NONE
   430   |   split_pat tp i (Const ("split", _) $ Abs (_, _, t)) = split_pat tp (i+1) t
   431   |   split_pat tp i _ = NONE;
   432   fun metaeq ss lhs rhs = mk_meta_eq (Goal.prove (Simplifier.the_context ss) [] []
   433         (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs,rhs)))
   434         (K (simp_tac (Simplifier.inherit_context ss cond_split_eta_ss) 1)));
   435 
   436   fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k+1) i t
   437   |   beta_term_pat k i (t $ u) = Pair_pat k i (t $ u) orelse
   438                         (beta_term_pat k i t andalso beta_term_pat k i u)
   439   |   beta_term_pat k i t = no_args k i t;
   440   fun  eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
   441   |    eta_term_pat _ _ _ = false;
   442   fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
   443   |   subst arg k i (t $ u) = if Pair_pat k i (t $ u) then incr_boundvars k arg
   444                               else (subst arg k i t $ subst arg k i u)
   445   |   subst arg k i t = t;
   446   fun beta_proc ss (s as Const ("split", _) $ Abs (_, _, t) $ arg) =
   447         (case split_pat beta_term_pat 1 t of
   448         SOME (i,f) => SOME (metaeq ss s (subst arg 0 i f))
   449         | NONE => NONE)
   450   |   beta_proc _ _ = NONE;
   451   fun eta_proc ss (s as Const ("split", _) $ Abs (_, _, t)) =
   452         (case split_pat eta_term_pat 1 t of
   453           SOME (_,ft) => SOME (metaeq ss s (let val (f $ arg) = ft in f end))
   454         | NONE => NONE)
   455   |   eta_proc _ _ = NONE;
   456 in
   457   val split_beta_proc = Simplifier.simproc @{theory} "split_beta" ["split f z"] (K beta_proc);
   458   val split_eta_proc = Simplifier.simproc @{theory} "split_eta" ["split f"] (K eta_proc);
   459 end;
   460 
   461 Addsimprocs [split_beta_proc, split_eta_proc];
   462 *}
   463 
   464 lemma split_beta: "(%(x, y). P x y) z = P (fst z) (snd z)"
   465   by (subst surjective_pairing, rule split_conv)
   466 
   467 lemma split_split [noatp]: "R(split c p) = (ALL x y. p = (x, y) --> R(c x y))"
   468   -- {* For use with @{text split} and the Simplifier. *}
   469   by (insert surj_pair [of p], clarify, simp)
   470 
   471 text {*
   472   @{thm [source] split_split} could be declared as @{text "[split]"}
   473   done after the Splitter has been speeded up significantly;
   474   precompute the constants involved and don't do anything unless the
   475   current goal contains one of those constants.
   476 *}
   477 
   478 lemma split_split_asm [noatp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))"
   479 by (subst split_split, simp)
   480 
   481 
   482 text {*
   483   \medskip @{term split} used as a logical connective or set former.
   484 
   485   \medskip These rules are for use with @{text blast}; could instead
   486   call @{text simp} using @{thm [source] split} as rewrite. *}
   487 
   488 lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"
   489   apply (simp only: split_tupled_all)
   490   apply (simp (no_asm_simp))
   491   done
   492 
   493 lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"
   494   apply (simp only: split_tupled_all)
   495   apply (simp (no_asm_simp))
   496   done
   497 
   498 lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
   499   by (induct p) (auto simp add: split_def)
   500 
   501 lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
   502   by (induct p) (auto simp add: split_def)
   503 
   504 lemma splitE2:
   505   "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
   506 proof -
   507   assume q: "Q (split P z)"
   508   assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
   509   show R
   510     apply (rule r surjective_pairing)+
   511     apply (rule split_beta [THEN subst], rule q)
   512     done
   513 qed
   514 
   515 lemma splitD': "split R (a,b) c ==> R a b c"
   516   by simp
   517 
   518 lemma mem_splitI: "z: c a b ==> z: split c (a, b)"
   519   by simp
   520 
   521 lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"
   522 by (simp only: split_tupled_all, simp)
   523 
   524 lemma mem_splitE:
   525   assumes major: "z: split c p"
   526     and cases: "!!x y. [| p = (x,y); z: c x y |] ==> Q"
   527   shows Q
   528   by (rule major [unfolded split_def] cases surjective_pairing)+
   529 
   530 declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
   531 declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]
   532 
   533 ML_setup {*
   534 local (* filtering with exists_p_split is an essential optimization *)
   535   fun exists_p_split (Const ("split",_) $ _ $ (Const ("Pair",_)$_$_)) = true
   536     | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
   537     | exists_p_split (Abs (_, _, t)) = exists_p_split t
   538     | exists_p_split _ = false;
   539   val ss = HOL_basic_ss addsimps [thm "split_conv"];
   540 in
   541 val split_conv_tac = SUBGOAL (fn (t, i) =>
   542     if exists_p_split t then safe_full_simp_tac ss i else no_tac);
   543 end;
   544 (* This prevents applications of splitE for already splitted arguments leading
   545    to quite time-consuming computations (in particular for nested tuples) *)
   546 change_claset (fn cs => cs addSbefore ("split_conv_tac", split_conv_tac));
   547 *}
   548 
   549 lemma split_eta_SetCompr [simp,noatp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
   550   by (rule ext) fast
   551 
   552 lemma split_eta_SetCompr2 [simp,noatp]: "(%u. EX x y. u = (x, y) & P x y) = split P"
   553   by (rule ext) fast
   554 
   555 lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"
   556   -- {* Allows simplifications of nested splits in case of independent predicates. *}
   557   by (rule ext) blast
   558 
   559 (* Do NOT make this a simp rule as it
   560    a) only helps in special situations
   561    b) can lead to nontermination in the presence of split_def
   562 *)
   563 lemma split_comp_eq: 
   564   fixes f :: "'a => 'b => 'c" and g :: "'d => 'a"
   565   shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"
   566   by (rule ext) auto
   567 
   568 lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
   569   by blast
   570 
   571 (*
   572 the following  would be slightly more general,
   573 but cannot be used as rewrite rule:
   574 ### Cannot add premise as rewrite rule because it contains (type) unknowns:
   575 ### ?y = .x
   576 Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
   577 by (rtac some_equality 1)
   578 by ( Simp_tac 1)
   579 by (split_all_tac 1)
   580 by (Asm_full_simp_tac 1)
   581 qed "The_split_eq";
   582 *)
   583 
   584 lemma injective_fst_snd: "!!x y. [|fst x = fst y; snd x = snd y|] ==> x = y"
   585   by auto
   586 
   587 
   588 text {*
   589   \bigskip @{term prod_fun} --- action of the product functor upon
   590   functions.
   591 *}
   592 
   593 lemma prod_fun [simp, code func]: "prod_fun f g (a, b) = (f a, g b)"
   594   by (simp add: prod_fun_def)
   595 
   596 lemma prod_fun_compose: "prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)"
   597   apply (rule ext)
   598   apply (tactic {* pair_tac "x" 1 *}, simp)
   599   done
   600 
   601 lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)"
   602   apply (rule ext)
   603   apply (tactic {* pair_tac "z" 1 *}, simp)
   604   done
   605 
   606 lemma prod_fun_imageI [intro]: "(a, b) : r ==> (f a, g b) : prod_fun f g ` r"
   607   apply (rule image_eqI)
   608   apply (rule prod_fun [symmetric], assumption)
   609   done
   610 
   611 lemma prod_fun_imageE [elim!]:
   612   assumes major: "c: (prod_fun f g)`r"
   613     and cases: "!!x y. [| c=(f(x),g(y));  (x,y):r |] ==> P"
   614   shows P
   615   apply (rule major [THEN imageE])
   616   apply (rule_tac p = x in PairE)
   617   apply (rule cases)
   618    apply (blast intro: prod_fun)
   619   apply blast
   620   done
   621 
   622 
   623 definition
   624   upd_fst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b"
   625 where
   626   [code func del]: "upd_fst f = prod_fun f id"
   627 
   628 definition
   629   upd_snd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c"
   630 where
   631   [code func del]: "upd_snd f = prod_fun id f"
   632 
   633 lemma upd_fst_conv [simp, code]:
   634   "upd_fst f (x, y) = (f x, y)" 
   635   by (simp add: upd_fst_def)
   636 
   637 lemma upd_snd_conv [simp, code]:
   638   "upd_snd f (x, y) = (x, f y)" 
   639   by (simp add: upd_snd_def)
   640 
   641 text {*
   642   \bigskip Disjoint union of a family of sets -- Sigma.
   643 *}
   644 
   645 lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
   646   by (unfold Sigma_def) blast
   647 
   648 lemma SigmaE [elim!]:
   649     "[| c: Sigma A B;
   650         !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
   651      |] ==> P"
   652   -- {* The general elimination rule. *}
   653   by (unfold Sigma_def) blast
   654 
   655 text {*
   656   Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
   657   eigenvariables.
   658 *}
   659 
   660 lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
   661   by blast
   662 
   663 lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
   664   by blast
   665 
   666 lemma SigmaE2:
   667     "[| (a, b) : Sigma A B;
   668         [| a:A;  b:B(a) |] ==> P
   669      |] ==> P"
   670   by blast
   671 
   672 lemma Sigma_cong:
   673      "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>
   674       \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
   675   by auto
   676 
   677 lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
   678   by blast
   679 
   680 lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
   681   by blast
   682 
   683 lemma Sigma_empty2 [simp]: "A <*> {} = {}"
   684   by blast
   685 
   686 lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
   687   by auto
   688 
   689 lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
   690   by auto
   691 
   692 lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
   693   by auto
   694 
   695 lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
   696   by blast
   697 
   698 lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
   699   by blast
   700 
   701 lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
   702   by (blast elim: equalityE)
   703 
   704 lemma SetCompr_Sigma_eq:
   705     "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
   706   by blast
   707 
   708 text {*
   709   \bigskip Complex rules for Sigma.
   710 *}
   711 
   712 lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"
   713   by blast
   714 
   715 lemma UN_Times_distrib:
   716   "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"
   717   -- {* Suggested by Pierre Chartier *}
   718   by blast
   719 
   720 lemma split_paired_Ball_Sigma [simp,noatp]:
   721     "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"
   722   by blast
   723 
   724 lemma split_paired_Bex_Sigma [simp,noatp]:
   725     "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"
   726   by blast
   727 
   728 lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"
   729   by blast
   730 
   731 lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))"
   732   by blast
   733 
   734 lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"
   735   by blast
   736 
   737 lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))"
   738   by blast
   739 
   740 lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"
   741   by blast
   742 
   743 lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"
   744   by blast
   745 
   746 lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"
   747   by blast
   748 
   749 text {*
   750   Non-dependent versions are needed to avoid the need for higher-order
   751   matching, especially when the rules are re-oriented.
   752 *}
   753 
   754 lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"
   755   by blast
   756 
   757 lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"
   758   by blast
   759 
   760 lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
   761   by blast
   762 
   763 
   764 lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
   765   apply (rule_tac x = "(a, b)" in image_eqI)
   766    apply auto
   767   done
   768 
   769 
   770 text {*
   771   Setup of internal @{text split_rule}.
   772 *}
   773 
   774 constdefs
   775   internal_split :: "('a => 'b => 'c) => 'a * 'b => 'c"
   776   "internal_split == split"
   777 
   778 lemmas [code func del] = internal_split_def
   779 
   780 lemma internal_split_conv: "internal_split c (a, b) = c a b"
   781   by (simp only: internal_split_def split_conv)
   782 
   783 hide const internal_split
   784 
   785 use "Tools/split_rule.ML"
   786 setup SplitRule.setup
   787 
   788 
   789 
   790 lemmas prod_caseI = prod.cases [THEN iffD2, standard]
   791 
   792 lemma prod_caseI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> prod_case c p"
   793   by auto
   794 
   795 lemma prod_caseI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> prod_case c p x"
   796   by (auto simp: split_tupled_all)
   797 
   798 lemma prod_caseE: "prod_case c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
   799   by (induct p) auto
   800 
   801 lemma prod_caseE': "prod_case c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
   802   by (induct p) auto
   803 
   804 lemma prod_case_unfold: "prod_case = (%c p. c (fst p) (snd p))"
   805   by (simp add: expand_fun_eq)
   806 
   807 declare prod_caseI2' [intro!] prod_caseI2 [intro!] prod_caseI [intro!]
   808 declare prod_caseE' [elim!] prod_caseE [elim!]
   809 
   810 lemma prod_case_split [code post]:
   811   "prod_case = split"
   812   by (auto simp add: expand_fun_eq)
   813 
   814 lemmas [code inline] = prod_case_split [symmetric]
   815 
   816 
   817 subsection {* Further cases/induct rules for tuples *}
   818 
   819 lemma prod_cases3 [cases type]:
   820   obtains (fields) a b c where "y = (a, b, c)"
   821   by (cases y, case_tac b) blast
   822 
   823 lemma prod_induct3 [case_names fields, induct type]:
   824     "(!!a b c. P (a, b, c)) ==> P x"
   825   by (cases x) blast
   826 
   827 lemma prod_cases4 [cases type]:
   828   obtains (fields) a b c d where "y = (a, b, c, d)"
   829   by (cases y, case_tac c) blast
   830 
   831 lemma prod_induct4 [case_names fields, induct type]:
   832     "(!!a b c d. P (a, b, c, d)) ==> P x"
   833   by (cases x) blast
   834 
   835 lemma prod_cases5 [cases type]:
   836   obtains (fields) a b c d e where "y = (a, b, c, d, e)"
   837   by (cases y, case_tac d) blast
   838 
   839 lemma prod_induct5 [case_names fields, induct type]:
   840     "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
   841   by (cases x) blast
   842 
   843 lemma prod_cases6 [cases type]:
   844   obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
   845   by (cases y, case_tac e) blast
   846 
   847 lemma prod_induct6 [case_names fields, induct type]:
   848     "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
   849   by (cases x) blast
   850 
   851 lemma prod_cases7 [cases type]:
   852   obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
   853   by (cases y, case_tac f) blast
   854 
   855 lemma prod_induct7 [case_names fields, induct type]:
   856     "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
   857   by (cases x) blast
   858 
   859 
   860 subsection {* Further lemmas *}
   861 
   862 lemma
   863   split_Pair: "split Pair x = x"
   864   unfolding split_def by auto
   865 
   866 lemma
   867   split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)"
   868   by (cases x, simp)
   869 
   870 
   871 subsection {* Code generator setup *}
   872 
   873 instance unit :: eq ..
   874 
   875 lemma [code func]:
   876   "(u\<Colon>unit) = v \<longleftrightarrow> True" unfolding unit_eq [of u] unit_eq [of v] by rule+
   877 
   878 code_type unit
   879   (SML "unit")
   880   (OCaml "unit")
   881   (Haskell "()")
   882 
   883 code_instance unit :: eq
   884   (Haskell -)
   885 
   886 code_const "op = \<Colon> unit \<Rightarrow> unit \<Rightarrow> bool"
   887   (Haskell infixl 4 "==")
   888 
   889 code_const Unity
   890   (SML "()")
   891   (OCaml "()")
   892   (Haskell "()")
   893 
   894 code_reserved SML
   895   unit
   896 
   897 code_reserved OCaml
   898   unit
   899 
   900 instance * :: (eq, eq) eq ..
   901 
   902 lemma [code func]:
   903   "(x1\<Colon>'a\<Colon>eq, y1\<Colon>'b\<Colon>eq) = (x2, y2) \<longleftrightarrow> x1 = x2 \<and> y1 = y2" by auto
   904 
   905 code_type *
   906   (SML infix 2 "*")
   907   (OCaml infix 2 "*")
   908   (Haskell "!((_),/ (_))")
   909 
   910 code_instance * :: eq
   911   (Haskell -)
   912 
   913 code_const "op = \<Colon> 'a\<Colon>eq \<times> 'b\<Colon>eq \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"
   914   (Haskell infixl 4 "==")
   915 
   916 code_const Pair
   917   (SML "!((_),/ (_))")
   918   (OCaml "!((_),/ (_))")
   919   (Haskell "!((_),/ (_))")
   920 
   921 code_const fst and snd
   922   (Haskell "fst" and "snd")
   923 
   924 types_code
   925   "*"     ("(_ */ _)")
   926 attach (term_of) {*
   927 fun term_of_id_42 f T g U (x, y) = HOLogic.pair_const T U $ f x $ g y;
   928 *}
   929 attach (test) {*
   930 fun gen_id_42 aG bG i = (aG i, bG i);
   931 *}
   932 
   933 consts_code
   934   "Pair"    ("(_,/ _)")
   935 
   936 setup {*
   937 
   938 let
   939 
   940 fun strip_abs_split 0 t = ([], t)
   941   | strip_abs_split i (Abs (s, T, t)) =
   942       let
   943         val s' = Codegen.new_name t s;
   944         val v = Free (s', T)
   945       in apfst (cons v) (strip_abs_split (i-1) (subst_bound (v, t))) end
   946   | strip_abs_split i (u as Const ("split", _) $ t) = (case strip_abs_split (i+1) t of
   947         (v :: v' :: vs, u) => (HOLogic.mk_prod (v, v') :: vs, u)
   948       | _ => ([], u))
   949   | strip_abs_split i t = ([], t);
   950 
   951 fun let_codegen thy defs gr dep thyname brack t = (case strip_comb t of
   952     (t1 as Const ("Let", _), t2 :: t3 :: ts) =>
   953     let
   954       fun dest_let (l as Const ("Let", _) $ t $ u) =
   955           (case strip_abs_split 1 u of
   956              ([p], u') => apfst (cons (p, t)) (dest_let u')
   957            | _ => ([], l))
   958         | dest_let t = ([], t);
   959       fun mk_code (gr, (l, r)) =
   960         let
   961           val (gr1, pl) = Codegen.invoke_codegen thy defs dep thyname false (gr, l);
   962           val (gr2, pr) = Codegen.invoke_codegen thy defs dep thyname false (gr1, r);
   963         in (gr2, (pl, pr)) end
   964     in case dest_let (t1 $ t2 $ t3) of
   965         ([], _) => NONE
   966       | (ps, u) =>
   967           let
   968             val (gr1, qs) = foldl_map mk_code (gr, ps);
   969             val (gr2, pu) = Codegen.invoke_codegen thy defs dep thyname false (gr1, u);
   970             val (gr3, pargs) = foldl_map
   971               (Codegen.invoke_codegen thy defs dep thyname true) (gr2, ts)
   972           in
   973             SOME (gr3, Codegen.mk_app brack
   974               (Pretty.blk (0, [Pretty.str "let ", Pretty.blk (0, List.concat
   975                   (separate [Pretty.str ";", Pretty.brk 1] (map (fn (pl, pr) =>
   976                     [Pretty.block [Pretty.str "val ", pl, Pretty.str " =",
   977                        Pretty.brk 1, pr]]) qs))),
   978                 Pretty.brk 1, Pretty.str "in ", pu,
   979                 Pretty.brk 1, Pretty.str "end"])) pargs)
   980           end
   981     end
   982   | _ => NONE);
   983 
   984 fun split_codegen thy defs gr dep thyname brack t = (case strip_comb t of
   985     (t1 as Const ("split", _), t2 :: ts) =>
   986       (case strip_abs_split 1 (t1 $ t2) of
   987          ([p], u) =>
   988            let
   989              val (gr1, q) = Codegen.invoke_codegen thy defs dep thyname false (gr, p);
   990              val (gr2, pu) = Codegen.invoke_codegen thy defs dep thyname false (gr1, u);
   991              val (gr3, pargs) = foldl_map
   992                (Codegen.invoke_codegen thy defs dep thyname true) (gr2, ts)
   993            in
   994              SOME (gr2, Codegen.mk_app brack
   995                (Pretty.block [Pretty.str "(fn ", q, Pretty.str " =>",
   996                  Pretty.brk 1, pu, Pretty.str ")"]) pargs)
   997            end
   998        | _ => NONE)
   999   | _ => NONE);
  1000 
  1001 in
  1002 
  1003   Codegen.add_codegen "let_codegen" let_codegen
  1004   #> Codegen.add_codegen "split_codegen" split_codegen
  1005 
  1006 end
  1007 *}
  1008 
  1009 
  1010 subsection {* Legacy bindings *}
  1011 
  1012 ML {*
  1013 val Collect_split = thm "Collect_split";
  1014 val Compl_Times_UNIV1 = thm "Compl_Times_UNIV1";
  1015 val Compl_Times_UNIV2 = thm "Compl_Times_UNIV2";
  1016 val PairE = thm "PairE";
  1017 val PairE_lemma = thm "PairE_lemma";
  1018 val Pair_Rep_inject = thm "Pair_Rep_inject";
  1019 val Pair_def = thm "Pair_def";
  1020 val Pair_eq = thm "Pair_eq";
  1021 val Pair_fst_snd_eq = thm "Pair_fst_snd_eq";
  1022 val Pair_inject = thm "Pair_inject";
  1023 val ProdI = thm "ProdI";
  1024 val SetCompr_Sigma_eq = thm "SetCompr_Sigma_eq";
  1025 val SigmaD1 = thm "SigmaD1";
  1026 val SigmaD2 = thm "SigmaD2";
  1027 val SigmaE = thm "SigmaE";
  1028 val SigmaE2 = thm "SigmaE2";
  1029 val SigmaI = thm "SigmaI";
  1030 val Sigma_Diff_distrib1 = thm "Sigma_Diff_distrib1";
  1031 val Sigma_Diff_distrib2 = thm "Sigma_Diff_distrib2";
  1032 val Sigma_Int_distrib1 = thm "Sigma_Int_distrib1";
  1033 val Sigma_Int_distrib2 = thm "Sigma_Int_distrib2";
  1034 val Sigma_Un_distrib1 = thm "Sigma_Un_distrib1";
  1035 val Sigma_Un_distrib2 = thm "Sigma_Un_distrib2";
  1036 val Sigma_Union = thm "Sigma_Union";
  1037 val Sigma_def = thm "Sigma_def";
  1038 val Sigma_empty1 = thm "Sigma_empty1";
  1039 val Sigma_empty2 = thm "Sigma_empty2";
  1040 val Sigma_mono = thm "Sigma_mono";
  1041 val The_split = thm "The_split";
  1042 val The_split_eq = thm "The_split_eq";
  1043 val The_split_eq = thm "The_split_eq";
  1044 val Times_Diff_distrib1 = thm "Times_Diff_distrib1";
  1045 val Times_Int_distrib1 = thm "Times_Int_distrib1";
  1046 val Times_Un_distrib1 = thm "Times_Un_distrib1";
  1047 val Times_eq_cancel2 = thm "Times_eq_cancel2";
  1048 val Times_subset_cancel2 = thm "Times_subset_cancel2";
  1049 val UNIV_Times_UNIV = thm "UNIV_Times_UNIV";
  1050 val UN_Times_distrib = thm "UN_Times_distrib";
  1051 val Unity_def = thm "Unity_def";
  1052 val cond_split_eta = thm "cond_split_eta";
  1053 val fst_conv = thm "fst_conv";
  1054 val fst_def = thm "fst_def";
  1055 val fst_eqD = thm "fst_eqD";
  1056 val inj_on_Abs_Prod = thm "inj_on_Abs_Prod";
  1057 val injective_fst_snd = thm "injective_fst_snd";
  1058 val mem_Sigma_iff = thm "mem_Sigma_iff";
  1059 val mem_splitE = thm "mem_splitE";
  1060 val mem_splitI = thm "mem_splitI";
  1061 val mem_splitI2 = thm "mem_splitI2";
  1062 val prod_eqI = thm "prod_eqI";
  1063 val prod_fun = thm "prod_fun";
  1064 val prod_fun_compose = thm "prod_fun_compose";
  1065 val prod_fun_def = thm "prod_fun_def";
  1066 val prod_fun_ident = thm "prod_fun_ident";
  1067 val prod_fun_imageE = thm "prod_fun_imageE";
  1068 val prod_fun_imageI = thm "prod_fun_imageI";
  1069 val prod_induct = thm "prod_induct";
  1070 val snd_conv = thm "snd_conv";
  1071 val snd_def = thm "snd_def";
  1072 val snd_eqD = thm "snd_eqD";
  1073 val split = thm "split";
  1074 val splitD = thm "splitD";
  1075 val splitD' = thm "splitD'";
  1076 val splitE = thm "splitE";
  1077 val splitE' = thm "splitE'";
  1078 val splitE2 = thm "splitE2";
  1079 val splitI = thm "splitI";
  1080 val splitI2 = thm "splitI2";
  1081 val splitI2' = thm "splitI2'";
  1082 val split_Pair_apply = thm "split_Pair_apply";
  1083 val split_beta = thm "split_beta";
  1084 val split_conv = thm "split_conv";
  1085 val split_def = thm "split_def";
  1086 val split_eta = thm "split_eta";
  1087 val split_eta_SetCompr = thm "split_eta_SetCompr";
  1088 val split_eta_SetCompr2 = thm "split_eta_SetCompr2";
  1089 val split_paired_All = thm "split_paired_All";
  1090 val split_paired_Ball_Sigma = thm "split_paired_Ball_Sigma";
  1091 val split_paired_Bex_Sigma = thm "split_paired_Bex_Sigma";
  1092 val split_paired_Ex = thm "split_paired_Ex";
  1093 val split_paired_The = thm "split_paired_The";
  1094 val split_paired_all = thm "split_paired_all";
  1095 val split_part = thm "split_part";
  1096 val split_split = thm "split_split";
  1097 val split_split_asm = thm "split_split_asm";
  1098 val split_tupled_all = thms "split_tupled_all";
  1099 val split_weak_cong = thm "split_weak_cong";
  1100 val surj_pair = thm "surj_pair";
  1101 val surjective_pairing = thm "surjective_pairing";
  1102 val unit_abs_eta_conv = thm "unit_abs_eta_conv";
  1103 val unit_all_eq1 = thm "unit_all_eq1";
  1104 val unit_all_eq2 = thm "unit_all_eq2";
  1105 val unit_eq = thm "unit_eq";
  1106 val unit_induct = thm "unit_induct";
  1107 *}
  1108 
  1109 
  1110 subsection {* Further inductive packages *}
  1111 
  1112 use "Tools/inductive_realizer.ML"
  1113 setup InductiveRealizer.setup
  1114 
  1115 use "Tools/inductive_set_package.ML"
  1116 setup InductiveSetPackage.setup
  1117 
  1118 use "Tools/datatype_realizer.ML"
  1119 setup DatatypeRealizer.setup
  1120 
  1121 end