src/HOL/Datatype.thy
author haftmann
Mon Nov 30 11:42:49 2009 +0100 (2009-11-30)
changeset 33968 f94fb13ecbb3
parent 33963 977b94b64905
child 35216 7641e8d831d2
permissions -rw-r--r--
modernized structures and tuned headers of datatype package modules; joined former datatype.ML and datatype_rep_proofs.ML
     1 (*  Title:      HOL/Datatype.thy
     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     3     Author:     Stefan Berghofer and Markus Wenzel, TU Muenchen
     4 *)
     5 
     6 header {* Datatype package: constructing datatypes from Cartesian Products and Disjoint Sums *}
     7 
     8 theory Datatype
     9 imports Product_Type Sum_Type Nat
    10 uses
    11   ("Tools/Datatype/datatype.ML")
    12   ("Tools/inductive_realizer.ML")
    13   ("Tools/Datatype/datatype_realizer.ML")
    14 begin
    15 
    16 subsection {* The datatype universe *}
    17 
    18 typedef (Node)
    19   ('a,'b) node = "{p. EX f x k. p = (f::nat=>'b+nat, x::'a+nat) & f k = Inr 0}"
    20     --{*it is a subtype of @{text "(nat=>'b+nat) * ('a+nat)"}*}
    21   by auto
    22 
    23 text{*Datatypes will be represented by sets of type @{text node}*}
    24 
    25 types 'a item        = "('a, unit) node set"
    26       ('a, 'b) dtree = "('a, 'b) node set"
    27 
    28 consts
    29   Push      :: "[('b + nat), nat => ('b + nat)] => (nat => ('b + nat))"
    30 
    31   Push_Node :: "[('b + nat), ('a, 'b) node] => ('a, 'b) node"
    32   ndepth    :: "('a, 'b) node => nat"
    33 
    34   Atom      :: "('a + nat) => ('a, 'b) dtree"
    35   Leaf      :: "'a => ('a, 'b) dtree"
    36   Numb      :: "nat => ('a, 'b) dtree"
    37   Scons     :: "[('a, 'b) dtree, ('a, 'b) dtree] => ('a, 'b) dtree"
    38   In0       :: "('a, 'b) dtree => ('a, 'b) dtree"
    39   In1       :: "('a, 'b) dtree => ('a, 'b) dtree"
    40   Lim       :: "('b => ('a, 'b) dtree) => ('a, 'b) dtree"
    41 
    42   ntrunc    :: "[nat, ('a, 'b) dtree] => ('a, 'b) dtree"
    43 
    44   uprod     :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set"
    45   usum      :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set"
    46 
    47   Split     :: "[[('a, 'b) dtree, ('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c"
    48   Case      :: "[[('a, 'b) dtree]=>'c, [('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c"
    49 
    50   dprod     :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]
    51                 => (('a, 'b) dtree * ('a, 'b) dtree)set"
    52   dsum      :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]
    53                 => (('a, 'b) dtree * ('a, 'b) dtree)set"
    54 
    55 
    56 defs
    57 
    58   Push_Node_def:  "Push_Node == (%n x. Abs_Node (apfst (Push n) (Rep_Node x)))"
    59 
    60   (*crude "lists" of nats -- needed for the constructions*)
    61   Push_def:   "Push == (%b h. nat_case b h)"
    62 
    63   (** operations on S-expressions -- sets of nodes **)
    64 
    65   (*S-expression constructors*)
    66   Atom_def:   "Atom == (%x. {Abs_Node((%k. Inr 0, x))})"
    67   Scons_def:  "Scons M N == (Push_Node (Inr 1) ` M) Un (Push_Node (Inr (Suc 1)) ` N)"
    68 
    69   (*Leaf nodes, with arbitrary or nat labels*)
    70   Leaf_def:   "Leaf == Atom o Inl"
    71   Numb_def:   "Numb == Atom o Inr"
    72 
    73   (*Injections of the "disjoint sum"*)
    74   In0_def:    "In0(M) == Scons (Numb 0) M"
    75   In1_def:    "In1(M) == Scons (Numb 1) M"
    76 
    77   (*Function spaces*)
    78   Lim_def: "Lim f == Union {z. ? x. z = Push_Node (Inl x) ` (f x)}"
    79 
    80   (*the set of nodes with depth less than k*)
    81   ndepth_def: "ndepth(n) == (%(f,x). LEAST k. f k = Inr 0) (Rep_Node n)"
    82   ntrunc_def: "ntrunc k N == {n. n:N & ndepth(n)<k}"
    83 
    84   (*products and sums for the "universe"*)
    85   uprod_def:  "uprod A B == UN x:A. UN y:B. { Scons x y }"
    86   usum_def:   "usum A B == In0`A Un In1`B"
    87 
    88   (*the corresponding eliminators*)
    89   Split_def:  "Split c M == THE u. EX x y. M = Scons x y & u = c x y"
    90 
    91   Case_def:   "Case c d M == THE u.  (EX x . M = In0(x) & u = c(x))
    92                                   | (EX y . M = In1(y) & u = d(y))"
    93 
    94 
    95   (** equality for the "universe" **)
    96 
    97   dprod_def:  "dprod r s == UN (x,x'):r. UN (y,y'):s. {(Scons x y, Scons x' y')}"
    98 
    99   dsum_def:   "dsum r s == (UN (x,x'):r. {(In0(x),In0(x'))}) Un
   100                           (UN (y,y'):s. {(In1(y),In1(y'))})"
   101 
   102 
   103 
   104 lemma apfst_convE: 
   105     "[| q = apfst f p;  !!x y. [| p = (x,y);  q = (f(x),y) |] ==> R  
   106      |] ==> R"
   107 by (force simp add: apfst_def)
   108 
   109 (** Push -- an injection, analogous to Cons on lists **)
   110 
   111 lemma Push_inject1: "Push i f = Push j g  ==> i=j"
   112 apply (simp add: Push_def expand_fun_eq) 
   113 apply (drule_tac x=0 in spec, simp) 
   114 done
   115 
   116 lemma Push_inject2: "Push i f = Push j g  ==> f=g"
   117 apply (auto simp add: Push_def expand_fun_eq) 
   118 apply (drule_tac x="Suc x" in spec, simp) 
   119 done
   120 
   121 lemma Push_inject:
   122     "[| Push i f =Push j g;  [| i=j;  f=g |] ==> P |] ==> P"
   123 by (blast dest: Push_inject1 Push_inject2) 
   124 
   125 lemma Push_neq_K0: "Push (Inr (Suc k)) f = (%z. Inr 0) ==> P"
   126 by (auto simp add: Push_def expand_fun_eq split: nat.split_asm)
   127 
   128 lemmas Abs_Node_inj = Abs_Node_inject [THEN [2] rev_iffD1, standard]
   129 
   130 
   131 (*** Introduction rules for Node ***)
   132 
   133 lemma Node_K0_I: "(%k. Inr 0, a) : Node"
   134 by (simp add: Node_def)
   135 
   136 lemma Node_Push_I: "p: Node ==> apfst (Push i) p : Node"
   137 apply (simp add: Node_def Push_def) 
   138 apply (fast intro!: apfst_conv nat_case_Suc [THEN trans])
   139 done
   140 
   141 
   142 subsection{*Freeness: Distinctness of Constructors*}
   143 
   144 (** Scons vs Atom **)
   145 
   146 lemma Scons_not_Atom [iff]: "Scons M N \<noteq> Atom(a)"
   147 apply (simp add: Atom_def Scons_def Push_Node_def One_nat_def)
   148 apply (blast intro: Node_K0_I Rep_Node [THEN Node_Push_I] 
   149          dest!: Abs_Node_inj 
   150          elim!: apfst_convE sym [THEN Push_neq_K0])  
   151 done
   152 
   153 lemmas Atom_not_Scons [iff] = Scons_not_Atom [THEN not_sym, standard]
   154 
   155 
   156 (*** Injectiveness ***)
   157 
   158 (** Atomic nodes **)
   159 
   160 lemma inj_Atom: "inj(Atom)"
   161 apply (simp add: Atom_def)
   162 apply (blast intro!: inj_onI Node_K0_I dest!: Abs_Node_inj)
   163 done
   164 lemmas Atom_inject = inj_Atom [THEN injD, standard]
   165 
   166 lemma Atom_Atom_eq [iff]: "(Atom(a)=Atom(b)) = (a=b)"
   167 by (blast dest!: Atom_inject)
   168 
   169 lemma inj_Leaf: "inj(Leaf)"
   170 apply (simp add: Leaf_def o_def)
   171 apply (rule inj_onI)
   172 apply (erule Atom_inject [THEN Inl_inject])
   173 done
   174 
   175 lemmas Leaf_inject [dest!] = inj_Leaf [THEN injD, standard]
   176 
   177 lemma inj_Numb: "inj(Numb)"
   178 apply (simp add: Numb_def o_def)
   179 apply (rule inj_onI)
   180 apply (erule Atom_inject [THEN Inr_inject])
   181 done
   182 
   183 lemmas Numb_inject [dest!] = inj_Numb [THEN injD, standard]
   184 
   185 
   186 (** Injectiveness of Push_Node **)
   187 
   188 lemma Push_Node_inject:
   189     "[| Push_Node i m =Push_Node j n;  [| i=j;  m=n |] ==> P  
   190      |] ==> P"
   191 apply (simp add: Push_Node_def)
   192 apply (erule Abs_Node_inj [THEN apfst_convE])
   193 apply (rule Rep_Node [THEN Node_Push_I])+
   194 apply (erule sym [THEN apfst_convE]) 
   195 apply (blast intro: Rep_Node_inject [THEN iffD1] trans sym elim!: Push_inject)
   196 done
   197 
   198 
   199 (** Injectiveness of Scons **)
   200 
   201 lemma Scons_inject_lemma1: "Scons M N <= Scons M' N' ==> M<=M'"
   202 apply (simp add: Scons_def One_nat_def)
   203 apply (blast dest!: Push_Node_inject)
   204 done
   205 
   206 lemma Scons_inject_lemma2: "Scons M N <= Scons M' N' ==> N<=N'"
   207 apply (simp add: Scons_def One_nat_def)
   208 apply (blast dest!: Push_Node_inject)
   209 done
   210 
   211 lemma Scons_inject1: "Scons M N = Scons M' N' ==> M=M'"
   212 apply (erule equalityE)
   213 apply (iprover intro: equalityI Scons_inject_lemma1)
   214 done
   215 
   216 lemma Scons_inject2: "Scons M N = Scons M' N' ==> N=N'"
   217 apply (erule equalityE)
   218 apply (iprover intro: equalityI Scons_inject_lemma2)
   219 done
   220 
   221 lemma Scons_inject:
   222     "[| Scons M N = Scons M' N';  [| M=M';  N=N' |] ==> P |] ==> P"
   223 by (iprover dest: Scons_inject1 Scons_inject2)
   224 
   225 lemma Scons_Scons_eq [iff]: "(Scons M N = Scons M' N') = (M=M' & N=N')"
   226 by (blast elim!: Scons_inject)
   227 
   228 (*** Distinctness involving Leaf and Numb ***)
   229 
   230 (** Scons vs Leaf **)
   231 
   232 lemma Scons_not_Leaf [iff]: "Scons M N \<noteq> Leaf(a)"
   233 by (simp add: Leaf_def o_def Scons_not_Atom)
   234 
   235 lemmas Leaf_not_Scons  [iff] = Scons_not_Leaf [THEN not_sym, standard]
   236 
   237 (** Scons vs Numb **)
   238 
   239 lemma Scons_not_Numb [iff]: "Scons M N \<noteq> Numb(k)"
   240 by (simp add: Numb_def o_def Scons_not_Atom)
   241 
   242 lemmas Numb_not_Scons [iff] = Scons_not_Numb [THEN not_sym, standard]
   243 
   244 
   245 (** Leaf vs Numb **)
   246 
   247 lemma Leaf_not_Numb [iff]: "Leaf(a) \<noteq> Numb(k)"
   248 by (simp add: Leaf_def Numb_def)
   249 
   250 lemmas Numb_not_Leaf [iff] = Leaf_not_Numb [THEN not_sym, standard]
   251 
   252 
   253 (*** ndepth -- the depth of a node ***)
   254 
   255 lemma ndepth_K0: "ndepth (Abs_Node(%k. Inr 0, x)) = 0"
   256 by (simp add: ndepth_def  Node_K0_I [THEN Abs_Node_inverse] Least_equality)
   257 
   258 lemma ndepth_Push_Node_aux:
   259      "nat_case (Inr (Suc i)) f k = Inr 0 --> Suc(LEAST x. f x = Inr 0) <= k"
   260 apply (induct_tac "k", auto)
   261 apply (erule Least_le)
   262 done
   263 
   264 lemma ndepth_Push_Node: 
   265     "ndepth (Push_Node (Inr (Suc i)) n) = Suc(ndepth(n))"
   266 apply (insert Rep_Node [of n, unfolded Node_def])
   267 apply (auto simp add: ndepth_def Push_Node_def
   268                  Rep_Node [THEN Node_Push_I, THEN Abs_Node_inverse])
   269 apply (rule Least_equality)
   270 apply (auto simp add: Push_def ndepth_Push_Node_aux)
   271 apply (erule LeastI)
   272 done
   273 
   274 
   275 (*** ntrunc applied to the various node sets ***)
   276 
   277 lemma ntrunc_0 [simp]: "ntrunc 0 M = {}"
   278 by (simp add: ntrunc_def)
   279 
   280 lemma ntrunc_Atom [simp]: "ntrunc (Suc k) (Atom a) = Atom(a)"
   281 by (auto simp add: Atom_def ntrunc_def ndepth_K0)
   282 
   283 lemma ntrunc_Leaf [simp]: "ntrunc (Suc k) (Leaf a) = Leaf(a)"
   284 by (simp add: Leaf_def o_def ntrunc_Atom)
   285 
   286 lemma ntrunc_Numb [simp]: "ntrunc (Suc k) (Numb i) = Numb(i)"
   287 by (simp add: Numb_def o_def ntrunc_Atom)
   288 
   289 lemma ntrunc_Scons [simp]: 
   290     "ntrunc (Suc k) (Scons M N) = Scons (ntrunc k M) (ntrunc k N)"
   291 by (auto simp add: Scons_def ntrunc_def One_nat_def ndepth_Push_Node) 
   292 
   293 
   294 
   295 (** Injection nodes **)
   296 
   297 lemma ntrunc_one_In0 [simp]: "ntrunc (Suc 0) (In0 M) = {}"
   298 apply (simp add: In0_def)
   299 apply (simp add: Scons_def)
   300 done
   301 
   302 lemma ntrunc_In0 [simp]: "ntrunc (Suc(Suc k)) (In0 M) = In0 (ntrunc (Suc k) M)"
   303 by (simp add: In0_def)
   304 
   305 lemma ntrunc_one_In1 [simp]: "ntrunc (Suc 0) (In1 M) = {}"
   306 apply (simp add: In1_def)
   307 apply (simp add: Scons_def)
   308 done
   309 
   310 lemma ntrunc_In1 [simp]: "ntrunc (Suc(Suc k)) (In1 M) = In1 (ntrunc (Suc k) M)"
   311 by (simp add: In1_def)
   312 
   313 
   314 subsection{*Set Constructions*}
   315 
   316 
   317 (*** Cartesian Product ***)
   318 
   319 lemma uprodI [intro!]: "[| M:A;  N:B |] ==> Scons M N : uprod A B"
   320 by (simp add: uprod_def)
   321 
   322 (*The general elimination rule*)
   323 lemma uprodE [elim!]:
   324     "[| c : uprod A B;   
   325         !!x y. [| x:A;  y:B;  c = Scons x y |] ==> P  
   326      |] ==> P"
   327 by (auto simp add: uprod_def) 
   328 
   329 
   330 (*Elimination of a pair -- introduces no eigenvariables*)
   331 lemma uprodE2: "[| Scons M N : uprod A B;  [| M:A;  N:B |] ==> P |] ==> P"
   332 by (auto simp add: uprod_def)
   333 
   334 
   335 (*** Disjoint Sum ***)
   336 
   337 lemma usum_In0I [intro]: "M:A ==> In0(M) : usum A B"
   338 by (simp add: usum_def)
   339 
   340 lemma usum_In1I [intro]: "N:B ==> In1(N) : usum A B"
   341 by (simp add: usum_def)
   342 
   343 lemma usumE [elim!]: 
   344     "[| u : usum A B;   
   345         !!x. [| x:A;  u=In0(x) |] ==> P;  
   346         !!y. [| y:B;  u=In1(y) |] ==> P  
   347      |] ==> P"
   348 by (auto simp add: usum_def)
   349 
   350 
   351 (** Injection **)
   352 
   353 lemma In0_not_In1 [iff]: "In0(M) \<noteq> In1(N)"
   354 by (auto simp add: In0_def In1_def One_nat_def)
   355 
   356 lemmas In1_not_In0 [iff] = In0_not_In1 [THEN not_sym, standard]
   357 
   358 lemma In0_inject: "In0(M) = In0(N) ==>  M=N"
   359 by (simp add: In0_def)
   360 
   361 lemma In1_inject: "In1(M) = In1(N) ==>  M=N"
   362 by (simp add: In1_def)
   363 
   364 lemma In0_eq [iff]: "(In0 M = In0 N) = (M=N)"
   365 by (blast dest!: In0_inject)
   366 
   367 lemma In1_eq [iff]: "(In1 M = In1 N) = (M=N)"
   368 by (blast dest!: In1_inject)
   369 
   370 lemma inj_In0: "inj In0"
   371 by (blast intro!: inj_onI)
   372 
   373 lemma inj_In1: "inj In1"
   374 by (blast intro!: inj_onI)
   375 
   376 
   377 (*** Function spaces ***)
   378 
   379 lemma Lim_inject: "Lim f = Lim g ==> f = g"
   380 apply (simp add: Lim_def)
   381 apply (rule ext)
   382 apply (blast elim!: Push_Node_inject)
   383 done
   384 
   385 
   386 (*** proving equality of sets and functions using ntrunc ***)
   387 
   388 lemma ntrunc_subsetI: "ntrunc k M <= M"
   389 by (auto simp add: ntrunc_def)
   390 
   391 lemma ntrunc_subsetD: "(!!k. ntrunc k M <= N) ==> M<=N"
   392 by (auto simp add: ntrunc_def)
   393 
   394 (*A generalized form of the take-lemma*)
   395 lemma ntrunc_equality: "(!!k. ntrunc k M = ntrunc k N) ==> M=N"
   396 apply (rule equalityI)
   397 apply (rule_tac [!] ntrunc_subsetD)
   398 apply (rule_tac [!] ntrunc_subsetI [THEN [2] subset_trans], auto) 
   399 done
   400 
   401 lemma ntrunc_o_equality: 
   402     "[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2"
   403 apply (rule ntrunc_equality [THEN ext])
   404 apply (simp add: expand_fun_eq) 
   405 done
   406 
   407 
   408 (*** Monotonicity ***)
   409 
   410 lemma uprod_mono: "[| A<=A';  B<=B' |] ==> uprod A B <= uprod A' B'"
   411 by (simp add: uprod_def, blast)
   412 
   413 lemma usum_mono: "[| A<=A';  B<=B' |] ==> usum A B <= usum A' B'"
   414 by (simp add: usum_def, blast)
   415 
   416 lemma Scons_mono: "[| M<=M';  N<=N' |] ==> Scons M N <= Scons M' N'"
   417 by (simp add: Scons_def, blast)
   418 
   419 lemma In0_mono: "M<=N ==> In0(M) <= In0(N)"
   420 by (simp add: In0_def subset_refl Scons_mono)
   421 
   422 lemma In1_mono: "M<=N ==> In1(M) <= In1(N)"
   423 by (simp add: In1_def subset_refl Scons_mono)
   424 
   425 
   426 (*** Split and Case ***)
   427 
   428 lemma Split [simp]: "Split c (Scons M N) = c M N"
   429 by (simp add: Split_def)
   430 
   431 lemma Case_In0 [simp]: "Case c d (In0 M) = c(M)"
   432 by (simp add: Case_def)
   433 
   434 lemma Case_In1 [simp]: "Case c d (In1 N) = d(N)"
   435 by (simp add: Case_def)
   436 
   437 
   438 
   439 (**** UN x. B(x) rules ****)
   440 
   441 lemma ntrunc_UN1: "ntrunc k (UN x. f(x)) = (UN x. ntrunc k (f x))"
   442 by (simp add: ntrunc_def, blast)
   443 
   444 lemma Scons_UN1_x: "Scons (UN x. f x) M = (UN x. Scons (f x) M)"
   445 by (simp add: Scons_def, blast)
   446 
   447 lemma Scons_UN1_y: "Scons M (UN x. f x) = (UN x. Scons M (f x))"
   448 by (simp add: Scons_def, blast)
   449 
   450 lemma In0_UN1: "In0(UN x. f(x)) = (UN x. In0(f(x)))"
   451 by (simp add: In0_def Scons_UN1_y)
   452 
   453 lemma In1_UN1: "In1(UN x. f(x)) = (UN x. In1(f(x)))"
   454 by (simp add: In1_def Scons_UN1_y)
   455 
   456 
   457 (*** Equality for Cartesian Product ***)
   458 
   459 lemma dprodI [intro!]: 
   460     "[| (M,M'):r;  (N,N'):s |] ==> (Scons M N, Scons M' N') : dprod r s"
   461 by (auto simp add: dprod_def)
   462 
   463 (*The general elimination rule*)
   464 lemma dprodE [elim!]: 
   465     "[| c : dprod r s;   
   466         !!x y x' y'. [| (x,x') : r;  (y,y') : s;  
   467                         c = (Scons x y, Scons x' y') |] ==> P  
   468      |] ==> P"
   469 by (auto simp add: dprod_def)
   470 
   471 
   472 (*** Equality for Disjoint Sum ***)
   473 
   474 lemma dsum_In0I [intro]: "(M,M'):r ==> (In0(M), In0(M')) : dsum r s"
   475 by (auto simp add: dsum_def)
   476 
   477 lemma dsum_In1I [intro]: "(N,N'):s ==> (In1(N), In1(N')) : dsum r s"
   478 by (auto simp add: dsum_def)
   479 
   480 lemma dsumE [elim!]: 
   481     "[| w : dsum r s;   
   482         !!x x'. [| (x,x') : r;  w = (In0(x), In0(x')) |] ==> P;  
   483         !!y y'. [| (y,y') : s;  w = (In1(y), In1(y')) |] ==> P  
   484      |] ==> P"
   485 by (auto simp add: dsum_def)
   486 
   487 
   488 (*** Monotonicity ***)
   489 
   490 lemma dprod_mono: "[| r<=r';  s<=s' |] ==> dprod r s <= dprod r' s'"
   491 by blast
   492 
   493 lemma dsum_mono: "[| r<=r';  s<=s' |] ==> dsum r s <= dsum r' s'"
   494 by blast
   495 
   496 
   497 (*** Bounding theorems ***)
   498 
   499 lemma dprod_Sigma: "(dprod (A <*> B) (C <*> D)) <= (uprod A C) <*> (uprod B D)"
   500 by blast
   501 
   502 lemmas dprod_subset_Sigma = subset_trans [OF dprod_mono dprod_Sigma, standard]
   503 
   504 (*Dependent version*)
   505 lemma dprod_subset_Sigma2:
   506      "(dprod (Sigma A B) (Sigma C D)) <= 
   507       Sigma (uprod A C) (Split (%x y. uprod (B x) (D y)))"
   508 by auto
   509 
   510 lemma dsum_Sigma: "(dsum (A <*> B) (C <*> D)) <= (usum A C) <*> (usum B D)"
   511 by blast
   512 
   513 lemmas dsum_subset_Sigma = subset_trans [OF dsum_mono dsum_Sigma, standard]
   514 
   515 
   516 text {* hides popular names *}
   517 hide (open) type node item
   518 hide (open) const Push Node Atom Leaf Numb Lim Split Case
   519 
   520 use "Tools/Datatype/datatype.ML"
   521 
   522 use "Tools/inductive_realizer.ML"
   523 setup InductiveRealizer.setup
   524 
   525 use "Tools/Datatype/datatype_realizer.ML"
   526 setup Datatype_Realizer.setup
   527 
   528 end