src/HOL/Product_Type.thy
author haftmann
Wed Jun 10 15:04:33 2009 +0200 (2009-06-10)
changeset 31604 eb2f9d709296
parent 31202 52d332f8f909
child 31667 cc969090c204
permissions -rw-r--r--
separate directory for datatype package
     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_package/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
    26   shows [code]: "eq_class.eq False P \<longleftrightarrow> \<not> P"
    27     and [code]: "eq_class.eq True P \<longleftrightarrow> P" 
    28     and [code]: "eq_class.eq P False \<longleftrightarrow> \<not> P" 
    29     and [code]: "eq_class.eq P True \<longleftrightarrow> P"
    30     and [code nbe]: "eq_class.eq P P \<longleftrightarrow> True"
    31   by (simp_all add: eq)
    32 
    33 code_const "eq_class.eq \<Colon> bool \<Rightarrow> bool \<Rightarrow> bool"
    34   (Haskell infixl 4 "==")
    35 
    36 code_instance bool :: eq
    37   (Haskell -)
    38 
    39 
    40 subsection {* Unit *}
    41 
    42 typedef unit = "{True}"
    43 proof
    44   show "True : ?unit" ..
    45 qed
    46 
    47 definition
    48   Unity :: unit    ("'(')")
    49 where
    50   "() = Abs_unit True"
    51 
    52 lemma unit_eq [noatp]: "u = ()"
    53   by (induct u) (simp add: unit_def Unity_def)
    54 
    55 text {*
    56   Simplification procedure for @{thm [source] unit_eq}.  Cannot use
    57   this rule directly --- it loops!
    58 *}
    59 
    60 ML {*
    61   val unit_eq_proc =
    62     let val unit_meta_eq = mk_meta_eq @{thm unit_eq} in
    63       Simplifier.simproc @{theory} "unit_eq" ["x::unit"]
    64       (fn _ => fn _ => fn t => if HOLogic.is_unit t then NONE else SOME unit_meta_eq)
    65     end;
    66 
    67   Addsimprocs [unit_eq_proc];
    68 *}
    69 
    70 rep_datatype "()" by simp
    71 
    72 lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()"
    73   by simp
    74 
    75 lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P"
    76   by (rule triv_forall_equality)
    77 
    78 text {*
    79   This rewrite counters the effect of @{text unit_eq_proc} on @{term
    80   [source] "%u::unit. f u"}, replacing it by @{term [source]
    81   f} rather than by @{term [source] "%u. f ()"}.
    82 *}
    83 
    84 lemma unit_abs_eta_conv [simp,noatp]: "(%u::unit. f ()) = f"
    85   by (rule ext) simp
    86 
    87 instantiation unit :: default
    88 begin
    89 
    90 definition "default = ()"
    91 
    92 instance ..
    93 
    94 end
    95 
    96 text {* code generator setup *}
    97 
    98 instance unit :: eq ..
    99 
   100 lemma [code]:
   101   "eq_class.eq (u\<Colon>unit) v \<longleftrightarrow> True" unfolding eq unit_eq [of u] unit_eq [of v] by rule+
   102 
   103 code_type unit
   104   (SML "unit")
   105   (OCaml "unit")
   106   (Haskell "()")
   107 
   108 code_instance unit :: eq
   109   (Haskell -)
   110 
   111 code_const "eq_class.eq \<Colon> unit \<Rightarrow> unit \<Rightarrow> bool"
   112   (Haskell infixl 4 "==")
   113 
   114 code_const Unity
   115   (SML "()")
   116   (OCaml "()")
   117   (Haskell "()")
   118 
   119 code_reserved SML
   120   unit
   121 
   122 code_reserved OCaml
   123   unit
   124 
   125 
   126 subsection {* Pairs *}
   127 
   128 subsubsection {* Product type, basic operations and concrete syntax *}
   129 
   130 definition
   131   Pair_Rep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
   132 where
   133   "Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)"
   134 
   135 global
   136 
   137 typedef (Prod)
   138   ('a, 'b) "*"    (infixr "*" 20)
   139     = "{f. \<exists>a b. f = Pair_Rep (a\<Colon>'a) (b\<Colon>'b)}"
   140 proof
   141   fix a b show "Pair_Rep a b \<in> ?Prod"
   142     by rule+
   143 qed
   144 
   145 syntax (xsymbols)
   146   "*"      :: "[type, type] => type"         ("(_ \<times>/ _)" [21, 20] 20)
   147 syntax (HTML output)
   148   "*"      :: "[type, type] => type"         ("(_ \<times>/ _)" [21, 20] 20)
   149 
   150 consts
   151   Pair     :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b"
   152   fst      :: "'a \<times> 'b \<Rightarrow> 'a"
   153   snd      :: "'a \<times> 'b \<Rightarrow> 'b"
   154   split    :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c"
   155   curry    :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c"
   156 
   157 local
   158 
   159 defs
   160   Pair_def:     "Pair a b == Abs_Prod (Pair_Rep a b)"
   161   fst_def:      "fst p == THE a. EX b. p = Pair a b"
   162   snd_def:      "snd p == THE b. EX a. p = Pair a b"
   163   split_def:    "split == (%c p. c (fst p) (snd p))"
   164   curry_def:    "curry == (%c x y. c (Pair x y))"
   165 
   166 text {*
   167   Patterns -- extends pre-defined type @{typ pttrn} used in
   168   abstractions.
   169 *}
   170 
   171 nonterminals
   172   tuple_args patterns
   173 
   174 syntax
   175   "_tuple"      :: "'a => tuple_args => 'a * 'b"        ("(1'(_,/ _'))")
   176   "_tuple_arg"  :: "'a => tuple_args"                   ("_")
   177   "_tuple_args" :: "'a => tuple_args => tuple_args"     ("_,/ _")
   178   "_pattern"    :: "[pttrn, patterns] => pttrn"         ("'(_,/ _')")
   179   ""            :: "pttrn => patterns"                  ("_")
   180   "_patterns"   :: "[pttrn, patterns] => patterns"      ("_,/ _")
   181 
   182 translations
   183   "(x, y)"       == "Pair x y"
   184   "_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))"
   185   "%(x,y,zs).b"  == "split(%x (y,zs).b)"
   186   "%(x,y).b"     == "split(%x y. b)"
   187   "_abs (Pair x y) t" => "%(x,y).t"
   188   (* The last rule accommodates tuples in `case C ... (x,y) ... => ...'
   189      The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *)
   190 
   191 (* reconstructs pattern from (nested) splits, avoiding eta-contraction of body*)
   192 (* works best with enclosing "let", if "let" does not avoid eta-contraction   *)
   193 print_translation {*
   194 let fun split_tr' [Abs (x,T,t as (Abs abs))] =
   195       (* split (%x y. t) => %(x,y) t *)
   196       let val (y,t') = atomic_abs_tr' abs;
   197           val (x',t'') = atomic_abs_tr' (x,T,t');
   198     
   199       in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end
   200     | split_tr' [Abs (x,T,(s as Const ("split",_)$t))] =
   201        (* split (%x. (split (%y z. t))) => %(x,y,z). t *)
   202        let val (Const ("_abs",_)$(Const ("_pattern",_)$y$z)$t') = split_tr' [t];
   203            val (x',t'') = atomic_abs_tr' (x,T,t');
   204        in Syntax.const "_abs"$ 
   205            (Syntax.const "_pattern"$x'$(Syntax.const "_patterns"$y$z))$t'' end
   206     | split_tr' [Const ("split",_)$t] =
   207        (* split (split (%x y z. t)) => %((x,y),z). t *)   
   208        split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *)
   209     | split_tr' [Const ("_abs",_)$x_y$(Abs abs)] =
   210        (* split (%pttrn z. t) => %(pttrn,z). t *)
   211        let val (z,t) = atomic_abs_tr' abs;
   212        in Syntax.const "_abs" $ (Syntax.const "_pattern" $x_y$z) $ t end
   213     | split_tr' _ =  raise Match;
   214 in [("split", split_tr')]
   215 end
   216 *}
   217 
   218 (* print "split f" as "\<lambda>(x,y). f x y" and "split (\<lambda>x. f x)" as "\<lambda>(x,y). f x y" *) 
   219 typed_print_translation {*
   220 let
   221   fun split_guess_names_tr' _ T [Abs (x,_,Abs _)] = raise Match
   222     | split_guess_names_tr' _ T  [Abs (x,xT,t)] =
   223         (case (head_of t) of
   224            Const ("split",_) => raise Match
   225          | _ => let 
   226                   val (_::yT::_) = binder_types (domain_type T) handle Bind => raise Match;
   227                   val (y,t') = atomic_abs_tr' ("y",yT,(incr_boundvars 1 t)$Bound 0); 
   228                   val (x',t'') = atomic_abs_tr' (x,xT,t');
   229                 in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end)
   230     | split_guess_names_tr' _ T [t] =
   231        (case (head_of t) of
   232            Const ("split",_) => raise Match 
   233          | _ => let 
   234                   val (xT::yT::_) = binder_types (domain_type T) handle Bind => raise Match;
   235                   val (y,t') = 
   236                         atomic_abs_tr' ("y",yT,(incr_boundvars 2 t)$Bound 1$Bound 0); 
   237                   val (x',t'') = atomic_abs_tr' ("x",xT,t');
   238                 in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end)
   239     | split_guess_names_tr' _ _ _ = raise Match;
   240 in [("split", split_guess_names_tr')]
   241 end 
   242 *}
   243 
   244 
   245 text {* Towards a datatype declaration *}
   246 
   247 lemma surj_pair [simp]: "EX x y. p = (x, y)"
   248   apply (unfold Pair_def)
   249   apply (rule Rep_Prod [unfolded Prod_def, THEN CollectE])
   250   apply (erule exE, erule exE, rule exI, rule exI)
   251   apply (rule Rep_Prod_inverse [symmetric, THEN trans])
   252   apply (erule arg_cong)
   253   done
   254 
   255 lemma PairE [cases type: *]:
   256   obtains x y where "p = (x, y)"
   257   using surj_pair [of p] by blast
   258 
   259 lemma ProdI: "Pair_Rep a b \<in> Prod"
   260   unfolding Prod_def by rule+
   261 
   262 lemma Pair_Rep_inject: "Pair_Rep a b = Pair_Rep a' b' \<Longrightarrow> a = a' \<and> b = b'"
   263   unfolding Pair_Rep_def by (drule fun_cong, drule fun_cong) blast
   264 
   265 lemma inj_on_Abs_Prod: "inj_on Abs_Prod Prod"
   266   apply (rule inj_on_inverseI)
   267   apply (erule Abs_Prod_inverse)
   268   done
   269 
   270 lemma Pair_inject:
   271   assumes "(a, b) = (a', b')"
   272     and "a = a' ==> b = b' ==> R"
   273   shows R
   274   apply (insert prems [unfolded Pair_def])
   275   apply (rule inj_on_Abs_Prod [THEN inj_onD, THEN Pair_Rep_inject, THEN conjE])
   276   apply (assumption | rule ProdI)+
   277   done
   278 
   279 rep_datatype (prod) Pair
   280 proof -
   281   fix P p
   282   assume "\<And>x y. P (x, y)"
   283   then show "P p" by (cases p) simp
   284 qed (auto elim: Pair_inject)
   285 
   286 lemmas Pair_eq = prod.inject
   287 
   288 lemma fst_conv [simp, code]: "fst (a, b) = a"
   289   unfolding fst_def by blast
   290 
   291 lemma snd_conv [simp, code]: "snd (a, b) = b"
   292   unfolding snd_def by blast
   293 
   294 
   295 subsubsection {* Basic rules and proof tools *}
   296 
   297 lemma fst_eqD: "fst (x, y) = a ==> x = a"
   298   by simp
   299 
   300 lemma snd_eqD: "snd (x, y) = a ==> y = a"
   301   by simp
   302 
   303 lemma pair_collapse [simp]: "(fst p, snd p) = p"
   304   by (cases p) simp
   305 
   306 lemmas surjective_pairing = pair_collapse [symmetric]
   307 
   308 lemma split_paired_all: "(!!x. PROP P x) == (!!a b. PROP P (a, b))"
   309 proof
   310   fix a b
   311   assume "!!x. PROP P x"
   312   then show "PROP P (a, b)" .
   313 next
   314   fix x
   315   assume "!!a b. PROP P (a, b)"
   316   from `PROP P (fst x, snd x)` show "PROP P x" by simp
   317 qed
   318 
   319 text {*
   320   The rule @{thm [source] split_paired_all} does not work with the
   321   Simplifier because it also affects premises in congrence rules,
   322   where this can lead to premises of the form @{text "!!a b. ... =
   323   ?P(a, b)"} which cannot be solved by reflexivity.
   324 *}
   325 
   326 lemmas split_tupled_all = split_paired_all unit_all_eq2
   327 
   328 ML {*
   329   (* replace parameters of product type by individual component parameters *)
   330   val safe_full_simp_tac = generic_simp_tac true (true, false, false);
   331   local (* filtering with exists_paired_all is an essential optimization *)
   332     fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) =
   333           can HOLogic.dest_prodT T orelse exists_paired_all t
   334       | exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u
   335       | exists_paired_all (Abs (_, _, t)) = exists_paired_all t
   336       | exists_paired_all _ = false;
   337     val ss = HOL_basic_ss
   338       addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}]
   339       addsimprocs [unit_eq_proc];
   340   in
   341     val split_all_tac = SUBGOAL (fn (t, i) =>
   342       if exists_paired_all t then safe_full_simp_tac ss i else no_tac);
   343     val unsafe_split_all_tac = SUBGOAL (fn (t, i) =>
   344       if exists_paired_all t then full_simp_tac ss i else no_tac);
   345     fun split_all th =
   346    if exists_paired_all (Thm.prop_of th) then full_simplify ss th else th;
   347   end;
   348 *}
   349 
   350 declaration {* fn _ =>
   351   Classical.map_cs (fn cs => cs addSbefore ("split_all_tac", split_all_tac))
   352 *}
   353 
   354 lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))"
   355   -- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *}
   356   by fast
   357 
   358 lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))"
   359   by fast
   360 
   361 lemma Pair_fst_snd_eq: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t"
   362   by (cases s, cases t) simp
   363 
   364 lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q"
   365   by (simp add: Pair_fst_snd_eq)
   366 
   367 
   368 subsubsection {* @{text split} and @{text curry} *}
   369 
   370 lemma split_conv [simp, code]: "split f (a, b) = f a b"
   371   by (simp add: split_def)
   372 
   373 lemma curry_conv [simp, code]: "curry f a b = f (a, b)"
   374   by (simp add: curry_def)
   375 
   376 lemmas split = split_conv  -- {* for backwards compatibility *}
   377 
   378 lemma splitI: "f a b \<Longrightarrow> split f (a, b)"
   379   by (rule split_conv [THEN iffD2])
   380 
   381 lemma splitD: "split f (a, b) \<Longrightarrow> f a b"
   382   by (rule split_conv [THEN iffD1])
   383 
   384 lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b"
   385   by (simp add: curry_def)
   386 
   387 lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)"
   388   by (simp add: curry_def)
   389 
   390 lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q"
   391   by (simp add: curry_def)
   392 
   393 lemma curry_split [simp]: "curry (split f) = f"
   394   by (simp add: curry_def split_def)
   395 
   396 lemma split_curry [simp]: "split (curry f) = f"
   397   by (simp add: curry_def split_def)
   398 
   399 lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id"
   400   by (simp add: split_def id_def)
   401 
   402 lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f"
   403   -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *}
   404   by (rule ext) auto
   405 
   406 lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)"
   407   by (cases x) simp
   408 
   409 lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p"
   410   unfolding split_def ..
   411 
   412 lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))"
   413   -- {* Can't be added to simpset: loops! *}
   414   by (simp add: split_eta)
   415 
   416 lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
   417   by (simp add: split_def)
   418 
   419 lemma split_weak_cong: "p = q \<Longrightarrow> split c p = split c q"
   420   -- {* Prevents simplification of @{term c}: much faster *}
   421   by (erule arg_cong)
   422 
   423 lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
   424   by (simp add: split_eta)
   425 
   426 text {*
   427   Simplification procedure for @{thm [source] cond_split_eta}.  Using
   428   @{thm [source] split_eta} as a rewrite rule is not general enough,
   429   and using @{thm [source] cond_split_eta} directly would render some
   430   existing proofs very inefficient; similarly for @{text
   431   split_beta}.
   432 *}
   433 
   434 ML {*
   435 
   436 local
   437   val cond_split_eta_ss = HOL_basic_ss addsimps [thm "cond_split_eta"]
   438   fun  Pair_pat k 0 (Bound m) = (m = k)
   439   |    Pair_pat k i (Const ("Pair",  _) $ Bound m $ t) = i > 0 andalso
   440                         m = k+i andalso Pair_pat k (i-1) t
   441   |    Pair_pat _ _ _ = false;
   442   fun no_args k i (Abs (_, _, t)) = no_args (k+1) i t
   443   |   no_args k i (t $ u) = no_args k i t andalso no_args k i u
   444   |   no_args k i (Bound m) = m < k orelse m > k+i
   445   |   no_args _ _ _ = true;
   446   fun split_pat tp i (Abs  (_,_,t)) = if tp 0 i t then SOME (i,t) else NONE
   447   |   split_pat tp i (Const ("split", _) $ Abs (_, _, t)) = split_pat tp (i+1) t
   448   |   split_pat tp i _ = NONE;
   449   fun metaeq ss lhs rhs = mk_meta_eq (Goal.prove (Simplifier.the_context ss) [] []
   450         (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs,rhs)))
   451         (K (simp_tac (Simplifier.inherit_context ss cond_split_eta_ss) 1)));
   452 
   453   fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k+1) i t
   454   |   beta_term_pat k i (t $ u) = Pair_pat k i (t $ u) orelse
   455                         (beta_term_pat k i t andalso beta_term_pat k i u)
   456   |   beta_term_pat k i t = no_args k i t;
   457   fun  eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
   458   |    eta_term_pat _ _ _ = false;
   459   fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
   460   |   subst arg k i (t $ u) = if Pair_pat k i (t $ u) then incr_boundvars k arg
   461                               else (subst arg k i t $ subst arg k i u)
   462   |   subst arg k i t = t;
   463   fun beta_proc ss (s as Const ("split", _) $ Abs (_, _, t) $ arg) =
   464         (case split_pat beta_term_pat 1 t of
   465         SOME (i,f) => SOME (metaeq ss s (subst arg 0 i f))
   466         | NONE => NONE)
   467   |   beta_proc _ _ = NONE;
   468   fun eta_proc ss (s as Const ("split", _) $ Abs (_, _, t)) =
   469         (case split_pat eta_term_pat 1 t of
   470           SOME (_,ft) => SOME (metaeq ss s (let val (f $ arg) = ft in f end))
   471         | NONE => NONE)
   472   |   eta_proc _ _ = NONE;
   473 in
   474   val split_beta_proc = Simplifier.simproc (the_context ()) "split_beta" ["split f z"] (K beta_proc);
   475   val split_eta_proc = Simplifier.simproc (the_context ()) "split_eta" ["split f"] (K eta_proc);
   476 end;
   477 
   478 Addsimprocs [split_beta_proc, split_eta_proc];
   479 *}
   480 
   481 lemma split_beta [mono]: "(%(x, y). P x y) z = P (fst z) (snd z)"
   482   by (subst surjective_pairing, rule split_conv)
   483 
   484 lemma split_split [noatp]: "R(split c p) = (ALL x y. p = (x, y) --> R(c x y))"
   485   -- {* For use with @{text split} and the Simplifier. *}
   486   by (insert surj_pair [of p], clarify, simp)
   487 
   488 text {*
   489   @{thm [source] split_split} could be declared as @{text "[split]"}
   490   done after the Splitter has been speeded up significantly;
   491   precompute the constants involved and don't do anything unless the
   492   current goal contains one of those constants.
   493 *}
   494 
   495 lemma split_split_asm [noatp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))"
   496 by (subst split_split, simp)
   497 
   498 
   499 text {*
   500   \medskip @{term split} used as a logical connective or set former.
   501 
   502   \medskip These rules are for use with @{text blast}; could instead
   503   call @{text simp} using @{thm [source] split} as rewrite. *}
   504 
   505 lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"
   506   apply (simp only: split_tupled_all)
   507   apply (simp (no_asm_simp))
   508   done
   509 
   510 lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"
   511   apply (simp only: split_tupled_all)
   512   apply (simp (no_asm_simp))
   513   done
   514 
   515 lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
   516   by (induct p) (auto simp add: split_def)
   517 
   518 lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
   519   by (induct p) (auto simp add: split_def)
   520 
   521 lemma splitE2:
   522   "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
   523 proof -
   524   assume q: "Q (split P z)"
   525   assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
   526   show R
   527     apply (rule r surjective_pairing)+
   528     apply (rule split_beta [THEN subst], rule q)
   529     done
   530 qed
   531 
   532 lemma splitD': "split R (a,b) c ==> R a b c"
   533   by simp
   534 
   535 lemma mem_splitI: "z: c a b ==> z: split c (a, b)"
   536   by simp
   537 
   538 lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"
   539 by (simp only: split_tupled_all, simp)
   540 
   541 lemma mem_splitE:
   542   assumes major: "z: split c p"
   543     and cases: "!!x y. [| p = (x,y); z: c x y |] ==> Q"
   544   shows Q
   545   by (rule major [unfolded split_def] cases surjective_pairing)+
   546 
   547 declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
   548 declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]
   549 
   550 ML {*
   551 local (* filtering with exists_p_split is an essential optimization *)
   552   fun exists_p_split (Const ("split",_) $ _ $ (Const ("Pair",_)$_$_)) = true
   553     | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
   554     | exists_p_split (Abs (_, _, t)) = exists_p_split t
   555     | exists_p_split _ = false;
   556   val ss = HOL_basic_ss addsimps [thm "split_conv"];
   557 in
   558 val split_conv_tac = SUBGOAL (fn (t, i) =>
   559     if exists_p_split t then safe_full_simp_tac ss i else no_tac);
   560 end;
   561 *}
   562 
   563 (* This prevents applications of splitE for already splitted arguments leading
   564    to quite time-consuming computations (in particular for nested tuples) *)
   565 declaration {* fn _ =>
   566   Classical.map_cs (fn cs => cs addSbefore ("split_conv_tac", split_conv_tac))
   567 *}
   568 
   569 lemma split_eta_SetCompr [simp,noatp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
   570   by (rule ext) fast
   571 
   572 lemma split_eta_SetCompr2 [simp,noatp]: "(%u. EX x y. u = (x, y) & P x y) = split P"
   573   by (rule ext) fast
   574 
   575 lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"
   576   -- {* Allows simplifications of nested splits in case of independent predicates. *}
   577   by (rule ext) blast
   578 
   579 (* Do NOT make this a simp rule as it
   580    a) only helps in special situations
   581    b) can lead to nontermination in the presence of split_def
   582 *)
   583 lemma split_comp_eq: 
   584   fixes f :: "'a => 'b => 'c" and g :: "'d => 'a"
   585   shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"
   586   by (rule ext) auto
   587 
   588 lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
   589   apply (rule_tac x = "(a, b)" in image_eqI)
   590    apply auto
   591   done
   592 
   593 lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
   594   by blast
   595 
   596 (*
   597 the following  would be slightly more general,
   598 but cannot be used as rewrite rule:
   599 ### Cannot add premise as rewrite rule because it contains (type) unknowns:
   600 ### ?y = .x
   601 Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
   602 by (rtac some_equality 1)
   603 by ( Simp_tac 1)
   604 by (split_all_tac 1)
   605 by (Asm_full_simp_tac 1)
   606 qed "The_split_eq";
   607 *)
   608 
   609 text {*
   610   Setup of internal @{text split_rule}.
   611 *}
   612 
   613 definition
   614   internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c"
   615 where
   616   "internal_split == split"
   617 
   618 lemma internal_split_conv: "internal_split c (a, b) = c a b"
   619   by (simp only: internal_split_def split_conv)
   620 
   621 hide const internal_split
   622 
   623 use "Tools/split_rule.ML"
   624 setup SplitRule.setup
   625 
   626 lemmas prod_caseI = prod.cases [THEN iffD2, standard]
   627 
   628 lemma prod_caseI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> prod_case c p"
   629   by auto
   630 
   631 lemma prod_caseI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> prod_case c p x"
   632   by (auto simp: split_tupled_all)
   633 
   634 lemma prod_caseE: "prod_case c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
   635   by (induct p) auto
   636 
   637 lemma prod_caseE': "prod_case c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
   638   by (induct p) auto
   639 
   640 lemma prod_case_unfold: "prod_case = (%c p. c (fst p) (snd p))"
   641   by (simp add: expand_fun_eq)
   642 
   643 declare prod_caseI2' [intro!] prod_caseI2 [intro!] prod_caseI [intro!]
   644 declare prod_caseE' [elim!] prod_caseE [elim!]
   645 
   646 lemma prod_case_split:
   647   "prod_case = split"
   648   by (auto simp add: expand_fun_eq)
   649 
   650 lemma prod_case_beta:
   651   "prod_case f p = f (fst p) (snd p)"
   652   unfolding prod_case_split split_beta ..
   653 
   654 
   655 subsection {* Further cases/induct rules for tuples *}
   656 
   657 lemma prod_cases3 [cases type]:
   658   obtains (fields) a b c where "y = (a, b, c)"
   659   by (cases y, case_tac b) blast
   660 
   661 lemma prod_induct3 [case_names fields, induct type]:
   662     "(!!a b c. P (a, b, c)) ==> P x"
   663   by (cases x) blast
   664 
   665 lemma prod_cases4 [cases type]:
   666   obtains (fields) a b c d where "y = (a, b, c, d)"
   667   by (cases y, case_tac c) blast
   668 
   669 lemma prod_induct4 [case_names fields, induct type]:
   670     "(!!a b c d. P (a, b, c, d)) ==> P x"
   671   by (cases x) blast
   672 
   673 lemma prod_cases5 [cases type]:
   674   obtains (fields) a b c d e where "y = (a, b, c, d, e)"
   675   by (cases y, case_tac d) blast
   676 
   677 lemma prod_induct5 [case_names fields, induct type]:
   678     "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
   679   by (cases x) blast
   680 
   681 lemma prod_cases6 [cases type]:
   682   obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
   683   by (cases y, case_tac e) blast
   684 
   685 lemma prod_induct6 [case_names fields, induct type]:
   686     "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
   687   by (cases x) blast
   688 
   689 lemma prod_cases7 [cases type]:
   690   obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
   691   by (cases y, case_tac f) blast
   692 
   693 lemma prod_induct7 [case_names fields, induct type]:
   694     "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
   695   by (cases x) blast
   696 
   697 
   698 subsubsection {* Derived operations *}
   699 
   700 text {*
   701   The composition-uncurry combinator.
   702 *}
   703 
   704 notation fcomp (infixl "o>" 60)
   705 
   706 definition
   707   scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "o\<rightarrow>" 60)
   708 where
   709   "f o\<rightarrow> g = (\<lambda>x. split g (f x))"
   710 
   711 lemma scomp_apply:  "(f o\<rightarrow> g) x = split g (f x)"
   712   by (simp add: scomp_def)
   713 
   714 lemma Pair_scomp: "Pair x o\<rightarrow> f = f x"
   715   by (simp add: expand_fun_eq scomp_apply)
   716 
   717 lemma scomp_Pair: "x o\<rightarrow> Pair = x"
   718   by (simp add: expand_fun_eq scomp_apply)
   719 
   720 lemma scomp_scomp: "(f o\<rightarrow> g) o\<rightarrow> h = f o\<rightarrow> (\<lambda>x. g x o\<rightarrow> h)"
   721   by (simp add: expand_fun_eq split_twice scomp_def)
   722 
   723 lemma scomp_fcomp: "(f o\<rightarrow> g) o> h = f o\<rightarrow> (\<lambda>x. g x o> h)"
   724   by (simp add: expand_fun_eq scomp_apply fcomp_def split_def)
   725 
   726 lemma fcomp_scomp: "(f o> g) o\<rightarrow> h = f o> (g o\<rightarrow> h)"
   727   by (simp add: expand_fun_eq scomp_apply fcomp_apply)
   728 
   729 code_const scomp
   730   (Eval infixl 3 "#->")
   731 
   732 no_notation fcomp (infixl "o>" 60)
   733 no_notation scomp (infixl "o\<rightarrow>" 60)
   734 
   735 
   736 text {*
   737   @{term prod_fun} --- action of the product functor upon
   738   functions.
   739 *}
   740 
   741 definition prod_fun :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where
   742   [code del]: "prod_fun f g = (\<lambda>(x, y). (f x, g y))"
   743 
   744 lemma prod_fun [simp, code]: "prod_fun f g (a, b) = (f a, g b)"
   745   by (simp add: prod_fun_def)
   746 
   747 lemma prod_fun_compose: "prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)"
   748   by (rule ext) auto
   749 
   750 lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)"
   751   by (rule ext) auto
   752 
   753 lemma prod_fun_imageI [intro]: "(a, b) : r ==> (f a, g b) : prod_fun f g ` r"
   754   apply (rule image_eqI)
   755   apply (rule prod_fun [symmetric], assumption)
   756   done
   757 
   758 lemma prod_fun_imageE [elim!]:
   759   assumes major: "c: (prod_fun f g)`r"
   760     and cases: "!!x y. [| c=(f(x),g(y));  (x,y):r |] ==> P"
   761   shows P
   762   apply (rule major [THEN imageE])
   763   apply (rule_tac p = x in PairE)
   764   apply (rule cases)
   765    apply (blast intro: prod_fun)
   766   apply blast
   767   done
   768 
   769 definition
   770   apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b"
   771 where
   772   [code del]: "apfst f = prod_fun f id"
   773 
   774 definition
   775   apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c"
   776 where
   777   [code del]: "apsnd f = prod_fun id f"
   778 
   779 lemma apfst_conv [simp, code]:
   780   "apfst f (x, y) = (f x, y)" 
   781   by (simp add: apfst_def)
   782 
   783 lemma upd_snd_conv [simp, code]:
   784   "apsnd f (x, y) = (x, f y)" 
   785   by (simp add: apsnd_def)
   786 
   787 
   788 text {*
   789   Disjoint union of a family of sets -- Sigma.
   790 *}
   791 
   792 definition  Sigma :: "['a set, 'a => 'b set] => ('a \<times> 'b) set" where
   793   Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}"
   794 
   795 abbreviation
   796   Times :: "['a set, 'b set] => ('a * 'b) set"
   797     (infixr "<*>" 80) where
   798   "A <*> B == Sigma A (%_. B)"
   799 
   800 notation (xsymbols)
   801   Times  (infixr "\<times>" 80)
   802 
   803 notation (HTML output)
   804   Times  (infixr "\<times>" 80)
   805 
   806 syntax
   807   "@Sigma" ::"[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" [0, 0, 10] 10)
   808 
   809 translations
   810   "SIGMA x:A. B" == "Product_Type.Sigma A (%x. B)"
   811 
   812 lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
   813   by (unfold Sigma_def) blast
   814 
   815 lemma SigmaE [elim!]:
   816     "[| c: Sigma A B;
   817         !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
   818      |] ==> P"
   819   -- {* The general elimination rule. *}
   820   by (unfold Sigma_def) blast
   821 
   822 text {*
   823   Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
   824   eigenvariables.
   825 *}
   826 
   827 lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
   828   by blast
   829 
   830 lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
   831   by blast
   832 
   833 lemma SigmaE2:
   834     "[| (a, b) : Sigma A B;
   835         [| a:A;  b:B(a) |] ==> P
   836      |] ==> P"
   837   by blast
   838 
   839 lemma Sigma_cong:
   840      "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>
   841       \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
   842   by auto
   843 
   844 lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
   845   by blast
   846 
   847 lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
   848   by blast
   849 
   850 lemma Sigma_empty2 [simp]: "A <*> {} = {}"
   851   by blast
   852 
   853 lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
   854   by auto
   855 
   856 lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
   857   by auto
   858 
   859 lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
   860   by auto
   861 
   862 lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
   863   by blast
   864 
   865 lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
   866   by blast
   867 
   868 lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
   869   by (blast elim: equalityE)
   870 
   871 lemma SetCompr_Sigma_eq:
   872     "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
   873   by blast
   874 
   875 lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"
   876   by blast
   877 
   878 lemma UN_Times_distrib:
   879   "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"
   880   -- {* Suggested by Pierre Chartier *}
   881   by blast
   882 
   883 lemma split_paired_Ball_Sigma [simp,noatp]:
   884     "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"
   885   by blast
   886 
   887 lemma split_paired_Bex_Sigma [simp,noatp]:
   888     "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"
   889   by blast
   890 
   891 lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"
   892   by blast
   893 
   894 lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))"
   895   by blast
   896 
   897 lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"
   898   by blast
   899 
   900 lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))"
   901   by blast
   902 
   903 lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"
   904   by blast
   905 
   906 lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"
   907   by blast
   908 
   909 lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"
   910   by blast
   911 
   912 text {*
   913   Non-dependent versions are needed to avoid the need for higher-order
   914   matching, especially when the rules are re-oriented.
   915 *}
   916 
   917 lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"
   918 by blast
   919 
   920 lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"
   921 by blast
   922 
   923 lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
   924 by blast
   925 
   926 lemma insert_times_insert[simp]:
   927   "insert a A \<times> insert b B =
   928    insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)"
   929 by blast
   930 
   931 subsubsection {* Code generator setup *}
   932 
   933 instance * :: (eq, eq) eq ..
   934 
   935 lemma [code]:
   936   "eq_class.eq (x1\<Colon>'a\<Colon>eq, y1\<Colon>'b\<Colon>eq) (x2, y2) \<longleftrightarrow> x1 = x2 \<and> y1 = y2" by (simp add: eq)
   937 
   938 lemma split_case_cert:
   939   assumes "CASE \<equiv> split f"
   940   shows "CASE (a, b) \<equiv> f a b"
   941   using assms by simp
   942 
   943 setup {*
   944   Code.add_case @{thm split_case_cert}
   945 *}
   946 
   947 code_type *
   948   (SML infix 2 "*")
   949   (OCaml infix 2 "*")
   950   (Haskell "!((_),/ (_))")
   951 
   952 code_instance * :: eq
   953   (Haskell -)
   954 
   955 code_const "eq_class.eq \<Colon> 'a\<Colon>eq \<times> 'b\<Colon>eq \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"
   956   (Haskell infixl 4 "==")
   957 
   958 code_const Pair
   959   (SML "!((_),/ (_))")
   960   (OCaml "!((_),/ (_))")
   961   (Haskell "!((_),/ (_))")
   962 
   963 code_const fst and snd
   964   (Haskell "fst" and "snd")
   965 
   966 types_code
   967   "*"     ("(_ */ _)")
   968 attach (term_of) {*
   969 fun term_of_id_42 aF aT bF bT (x, y) = HOLogic.pair_const aT bT $ aF x $ bF y;
   970 *}
   971 attach (test) {*
   972 fun gen_id_42 aG aT bG bT i =
   973   let
   974     val (x, t) = aG i;
   975     val (y, u) = bG i
   976   in ((x, y), fn () => HOLogic.pair_const aT bT $ t () $ u ()) end;
   977 *}
   978 
   979 consts_code
   980   "Pair"    ("(_,/ _)")
   981 
   982 setup {*
   983 
   984 let
   985 
   986 fun strip_abs_split 0 t = ([], t)
   987   | strip_abs_split i (Abs (s, T, t)) =
   988       let
   989         val s' = Codegen.new_name t s;
   990         val v = Free (s', T)
   991       in apfst (cons v) (strip_abs_split (i-1) (subst_bound (v, t))) end
   992   | strip_abs_split i (u as Const ("split", _) $ t) = (case strip_abs_split (i+1) t of
   993         (v :: v' :: vs, u) => (HOLogic.mk_prod (v, v') :: vs, u)
   994       | _ => ([], u))
   995   | strip_abs_split i t =
   996       strip_abs_split i (Abs ("x", hd (binder_types (fastype_of t)), t $ Bound 0));
   997 
   998 fun let_codegen thy defs dep thyname brack t gr = (case strip_comb t of
   999     (t1 as Const ("Let", _), t2 :: t3 :: ts) =>
  1000     let
  1001       fun dest_let (l as Const ("Let", _) $ t $ u) =
  1002           (case strip_abs_split 1 u of
  1003              ([p], u') => apfst (cons (p, t)) (dest_let u')
  1004            | _ => ([], l))
  1005         | dest_let t = ([], t);
  1006       fun mk_code (l, r) gr =
  1007         let
  1008           val (pl, gr1) = Codegen.invoke_codegen thy defs dep thyname false l gr;
  1009           val (pr, gr2) = Codegen.invoke_codegen thy defs dep thyname false r gr1;
  1010         in ((pl, pr), gr2) end
  1011     in case dest_let (t1 $ t2 $ t3) of
  1012         ([], _) => NONE
  1013       | (ps, u) =>
  1014           let
  1015             val (qs, gr1) = fold_map mk_code ps gr;
  1016             val (pu, gr2) = Codegen.invoke_codegen thy defs dep thyname false u gr1;
  1017             val (pargs, gr3) = fold_map
  1018               (Codegen.invoke_codegen thy defs dep thyname true) ts gr2
  1019           in
  1020             SOME (Codegen.mk_app brack
  1021               (Pretty.blk (0, [Codegen.str "let ", Pretty.blk (0, List.concat
  1022                   (separate [Codegen.str ";", Pretty.brk 1] (map (fn (pl, pr) =>
  1023                     [Pretty.block [Codegen.str "val ", pl, Codegen.str " =",
  1024                        Pretty.brk 1, pr]]) qs))),
  1025                 Pretty.brk 1, Codegen.str "in ", pu,
  1026                 Pretty.brk 1, Codegen.str "end"])) pargs, gr3)
  1027           end
  1028     end
  1029   | _ => NONE);
  1030 
  1031 fun split_codegen thy defs dep thyname brack t gr = (case strip_comb t of
  1032     (t1 as Const ("split", _), t2 :: ts) =>
  1033       let
  1034         val ([p], u) = strip_abs_split 1 (t1 $ t2);
  1035         val (q, gr1) = Codegen.invoke_codegen thy defs dep thyname false p gr;
  1036         val (pu, gr2) = Codegen.invoke_codegen thy defs dep thyname false u gr1;
  1037         val (pargs, gr3) = fold_map
  1038           (Codegen.invoke_codegen thy defs dep thyname true) ts gr2
  1039       in
  1040         SOME (Codegen.mk_app brack
  1041           (Pretty.block [Codegen.str "(fn ", q, Codegen.str " =>",
  1042             Pretty.brk 1, pu, Codegen.str ")"]) pargs, gr2)
  1043       end
  1044   | _ => NONE);
  1045 
  1046 in
  1047 
  1048   Codegen.add_codegen "let_codegen" let_codegen
  1049   #> Codegen.add_codegen "split_codegen" split_codegen
  1050 
  1051 end
  1052 *}
  1053 
  1054 
  1055 subsection {* Legacy bindings *}
  1056 
  1057 ML {*
  1058 val Collect_split = thm "Collect_split";
  1059 val Compl_Times_UNIV1 = thm "Compl_Times_UNIV1";
  1060 val Compl_Times_UNIV2 = thm "Compl_Times_UNIV2";
  1061 val PairE = thm "PairE";
  1062 val Pair_Rep_inject = thm "Pair_Rep_inject";
  1063 val Pair_def = thm "Pair_def";
  1064 val Pair_eq = @{thm "prod.inject"};
  1065 val Pair_fst_snd_eq = thm "Pair_fst_snd_eq";
  1066 val ProdI = thm "ProdI";
  1067 val SetCompr_Sigma_eq = thm "SetCompr_Sigma_eq";
  1068 val SigmaD1 = thm "SigmaD1";
  1069 val SigmaD2 = thm "SigmaD2";
  1070 val SigmaE = thm "SigmaE";
  1071 val SigmaE2 = thm "SigmaE2";
  1072 val SigmaI = thm "SigmaI";
  1073 val Sigma_Diff_distrib1 = thm "Sigma_Diff_distrib1";
  1074 val Sigma_Diff_distrib2 = thm "Sigma_Diff_distrib2";
  1075 val Sigma_Int_distrib1 = thm "Sigma_Int_distrib1";
  1076 val Sigma_Int_distrib2 = thm "Sigma_Int_distrib2";
  1077 val Sigma_Un_distrib1 = thm "Sigma_Un_distrib1";
  1078 val Sigma_Un_distrib2 = thm "Sigma_Un_distrib2";
  1079 val Sigma_Union = thm "Sigma_Union";
  1080 val Sigma_def = thm "Sigma_def";
  1081 val Sigma_empty1 = thm "Sigma_empty1";
  1082 val Sigma_empty2 = thm "Sigma_empty2";
  1083 val Sigma_mono = thm "Sigma_mono";
  1084 val The_split = thm "The_split";
  1085 val The_split_eq = thm "The_split_eq";
  1086 val The_split_eq = thm "The_split_eq";
  1087 val Times_Diff_distrib1 = thm "Times_Diff_distrib1";
  1088 val Times_Int_distrib1 = thm "Times_Int_distrib1";
  1089 val Times_Un_distrib1 = thm "Times_Un_distrib1";
  1090 val Times_eq_cancel2 = thm "Times_eq_cancel2";
  1091 val Times_subset_cancel2 = thm "Times_subset_cancel2";
  1092 val UNIV_Times_UNIV = thm "UNIV_Times_UNIV";
  1093 val UN_Times_distrib = thm "UN_Times_distrib";
  1094 val Unity_def = thm "Unity_def";
  1095 val cond_split_eta = thm "cond_split_eta";
  1096 val fst_conv = thm "fst_conv";
  1097 val fst_def = thm "fst_def";
  1098 val fst_eqD = thm "fst_eqD";
  1099 val inj_on_Abs_Prod = thm "inj_on_Abs_Prod";
  1100 val mem_Sigma_iff = thm "mem_Sigma_iff";
  1101 val mem_splitE = thm "mem_splitE";
  1102 val mem_splitI = thm "mem_splitI";
  1103 val mem_splitI2 = thm "mem_splitI2";
  1104 val prod_eqI = thm "prod_eqI";
  1105 val prod_fun = thm "prod_fun";
  1106 val prod_fun_compose = thm "prod_fun_compose";
  1107 val prod_fun_def = thm "prod_fun_def";
  1108 val prod_fun_ident = thm "prod_fun_ident";
  1109 val prod_fun_imageE = thm "prod_fun_imageE";
  1110 val prod_fun_imageI = thm "prod_fun_imageI";
  1111 val prod_induct = thm "prod.induct";
  1112 val snd_conv = thm "snd_conv";
  1113 val snd_def = thm "snd_def";
  1114 val snd_eqD = thm "snd_eqD";
  1115 val split = thm "split";
  1116 val splitD = thm "splitD";
  1117 val splitD' = thm "splitD'";
  1118 val splitE = thm "splitE";
  1119 val splitE' = thm "splitE'";
  1120 val splitE2 = thm "splitE2";
  1121 val splitI = thm "splitI";
  1122 val splitI2 = thm "splitI2";
  1123 val splitI2' = thm "splitI2'";
  1124 val split_beta = thm "split_beta";
  1125 val split_conv = thm "split_conv";
  1126 val split_def = thm "split_def";
  1127 val split_eta = thm "split_eta";
  1128 val split_eta_SetCompr = thm "split_eta_SetCompr";
  1129 val split_eta_SetCompr2 = thm "split_eta_SetCompr2";
  1130 val split_paired_All = thm "split_paired_All";
  1131 val split_paired_Ball_Sigma = thm "split_paired_Ball_Sigma";
  1132 val split_paired_Bex_Sigma = thm "split_paired_Bex_Sigma";
  1133 val split_paired_Ex = thm "split_paired_Ex";
  1134 val split_paired_The = thm "split_paired_The";
  1135 val split_paired_all = thm "split_paired_all";
  1136 val split_part = thm "split_part";
  1137 val split_split = thm "split_split";
  1138 val split_split_asm = thm "split_split_asm";
  1139 val split_tupled_all = thms "split_tupled_all";
  1140 val split_weak_cong = thm "split_weak_cong";
  1141 val surj_pair = thm "surj_pair";
  1142 val surjective_pairing = thm "surjective_pairing";
  1143 val unit_abs_eta_conv = thm "unit_abs_eta_conv";
  1144 val unit_all_eq1 = thm "unit_all_eq1";
  1145 val unit_all_eq2 = thm "unit_all_eq2";
  1146 val unit_eq = thm "unit_eq";
  1147 *}
  1148 
  1149 
  1150 subsection {* Further inductive packages *}
  1151 
  1152 use "Tools/inductive_realizer.ML"
  1153 setup InductiveRealizer.setup
  1154 
  1155 use "Tools/inductive_set_package.ML"
  1156 setup InductiveSetPackage.setup
  1157 
  1158 use "Tools/datatype_package/datatype_realizer.ML"
  1159 setup DatatypeRealizer.setup
  1160 
  1161 end