src/HOL/Library/Old_Datatype.thy
author wenzelm
Thu Feb 15 12:11:00 2018 +0100 (20 months ago)
changeset 67613 ce654b0e6d69
parent 67319 07176d5b81d5
child 69605 a96320074298
permissions -rw-r--r--
more symbols;
     1 (*  Title:      HOL/Library/Old_Datatype.thy
     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     3     Author:     Stefan Berghofer and Markus Wenzel, TU Muenchen
     4 *)
     5 
     6 section \<open>Old Datatype package: constructing datatypes from Cartesian Products and Disjoint Sums\<close>
     7 
     8 theory Old_Datatype
     9 imports Main
    10 begin
    11 
    12 
    13 subsection \<open>The datatype universe\<close>
    14 
    15 definition "Node = {p. \<exists>f x k. p = (f :: nat => 'b + nat, x ::'a + nat) \<and> f k = Inr 0}"
    16 
    17 typedef ('a, 'b) node = "Node :: ((nat => 'b + nat) * ('a + nat)) set"
    18   morphisms Rep_Node Abs_Node
    19   unfolding Node_def by auto
    20 
    21 text\<open>Datatypes will be represented by sets of type \<open>node\<close>\<close>
    22 
    23 type_synonym 'a item        = "('a, unit) node set"
    24 type_synonym ('a, 'b) dtree = "('a, 'b) node set"
    25 
    26 definition Push :: "[('b + nat), nat => ('b + nat)] => (nat => ('b + nat))"
    27   (*crude "lists" of nats -- needed for the constructions*)
    28   where "Push == (%b h. case_nat b h)"
    29 
    30 definition Push_Node :: "[('b + nat), ('a, 'b) node] => ('a, 'b) node"
    31   where "Push_Node == (%n x. Abs_Node (apfst (Push n) (Rep_Node x)))"
    32 
    33 
    34 (** operations on S-expressions -- sets of nodes **)
    35 
    36 (*S-expression constructors*)
    37 definition Atom :: "('a + nat) => ('a, 'b) dtree"
    38   where "Atom == (%x. {Abs_Node((%k. Inr 0, x))})"
    39 definition Scons :: "[('a, 'b) dtree, ('a, 'b) dtree] => ('a, 'b) dtree"
    40   where "Scons M N == (Push_Node (Inr 1) ` M) Un (Push_Node (Inr (Suc 1)) ` N)"
    41 
    42 (*Leaf nodes, with arbitrary or nat labels*)
    43 definition Leaf :: "'a => ('a, 'b) dtree"
    44   where "Leaf == Atom \<circ> Inl"
    45 definition Numb :: "nat => ('a, 'b) dtree"
    46   where "Numb == Atom \<circ> Inr"
    47 
    48 (*Injections of the "disjoint sum"*)
    49 definition In0 :: "('a, 'b) dtree => ('a, 'b) dtree"
    50   where "In0(M) == Scons (Numb 0) M"
    51 definition In1 :: "('a, 'b) dtree => ('a, 'b) dtree"
    52   where "In1(M) == Scons (Numb 1) M"
    53 
    54 (*Function spaces*)
    55 definition Lim :: "('b => ('a, 'b) dtree) => ('a, 'b) dtree"
    56   where "Lim f == \<Union>{z. \<exists>x. z = Push_Node (Inl x) ` (f x)}"
    57 
    58 (*the set of nodes with depth less than k*)
    59 definition ndepth :: "('a, 'b) node => nat"
    60   where "ndepth(n) == (%(f,x). LEAST k. f k = Inr 0) (Rep_Node n)"
    61 definition ntrunc :: "[nat, ('a, 'b) dtree] => ('a, 'b) dtree"
    62   where "ntrunc k N == {n. n\<in>N \<and> ndepth(n)<k}"
    63 
    64 (*products and sums for the "universe"*)
    65 definition uprod :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set"
    66   where "uprod A B == UN x:A. UN y:B. { Scons x y }"
    67 definition usum :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set"
    68   where "usum A B == In0`A Un In1`B"
    69 
    70 (*the corresponding eliminators*)
    71 definition Split :: "[[('a, 'b) dtree, ('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c"
    72   where "Split c M == THE u. \<exists>x y. M = Scons x y \<and> u = c x y"
    73 
    74 definition Case :: "[[('a, 'b) dtree]=>'c, [('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c"
    75   where "Case c d M == THE u. (\<exists>x . M = In0(x) \<and> u = c(x)) \<or> (\<exists>y . M = In1(y) \<and> u = d(y))"
    76 
    77 
    78 (** equality for the "universe" **)
    79 
    80 definition dprod :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]
    81       => (('a, 'b) dtree * ('a, 'b) dtree)set"
    82   where "dprod r s == UN (x,x'):r. UN (y,y'):s. {(Scons x y, Scons x' y')}"
    83 
    84 definition dsum :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]
    85       => (('a, 'b) dtree * ('a, 'b) dtree)set"
    86   where "dsum r s == (UN (x,x'):r. {(In0(x),In0(x'))}) Un (UN (y,y'):s. {(In1(y),In1(y'))})"
    87 
    88 
    89 lemma apfst_convE: 
    90     "[| q = apfst f p;  !!x y. [| p = (x,y);  q = (f(x),y) |] ==> R  
    91      |] ==> R"
    92 by (force simp add: apfst_def)
    93 
    94 (** Push -- an injection, analogous to Cons on lists **)
    95 
    96 lemma Push_inject1: "Push i f = Push j g  ==> i=j"
    97 apply (simp add: Push_def fun_eq_iff) 
    98 apply (drule_tac x=0 in spec, simp) 
    99 done
   100 
   101 lemma Push_inject2: "Push i f = Push j g  ==> f=g"
   102 apply (auto simp add: Push_def fun_eq_iff) 
   103 apply (drule_tac x="Suc x" in spec, simp) 
   104 done
   105 
   106 lemma Push_inject:
   107     "[| Push i f =Push j g;  [| i=j;  f=g |] ==> P |] ==> P"
   108 by (blast dest: Push_inject1 Push_inject2) 
   109 
   110 lemma Push_neq_K0: "Push (Inr (Suc k)) f = (%z. Inr 0) ==> P"
   111 by (auto simp add: Push_def fun_eq_iff split: nat.split_asm)
   112 
   113 lemmas Abs_Node_inj = Abs_Node_inject [THEN [2] rev_iffD1]
   114 
   115 
   116 (*** Introduction rules for Node ***)
   117 
   118 lemma Node_K0_I: "(\<lambda>k. Inr 0, a) \<in> Node"
   119 by (simp add: Node_def)
   120 
   121 lemma Node_Push_I: "p \<in> Node \<Longrightarrow> apfst (Push i) p \<in> Node"
   122 apply (simp add: Node_def Push_def) 
   123 apply (fast intro!: apfst_conv nat.case(2)[THEN trans])
   124 done
   125 
   126 
   127 subsection\<open>Freeness: Distinctness of Constructors\<close>
   128 
   129 (** Scons vs Atom **)
   130 
   131 lemma Scons_not_Atom [iff]: "Scons M N \<noteq> Atom(a)"
   132 unfolding Atom_def Scons_def Push_Node_def One_nat_def
   133 by (blast intro: Node_K0_I Rep_Node [THEN Node_Push_I] 
   134          dest!: Abs_Node_inj 
   135          elim!: apfst_convE sym [THEN Push_neq_K0])  
   136 
   137 lemmas Atom_not_Scons [iff] = Scons_not_Atom [THEN not_sym]
   138 
   139 
   140 (*** Injectiveness ***)
   141 
   142 (** Atomic nodes **)
   143 
   144 lemma inj_Atom: "inj(Atom)"
   145 apply (simp add: Atom_def)
   146 apply (blast intro!: inj_onI Node_K0_I dest!: Abs_Node_inj)
   147 done
   148 lemmas Atom_inject = inj_Atom [THEN injD]
   149 
   150 lemma Atom_Atom_eq [iff]: "(Atom(a)=Atom(b)) = (a=b)"
   151 by (blast dest!: Atom_inject)
   152 
   153 lemma inj_Leaf: "inj(Leaf)"
   154 apply (simp add: Leaf_def o_def)
   155 apply (rule inj_onI)
   156 apply (erule Atom_inject [THEN Inl_inject])
   157 done
   158 
   159 lemmas Leaf_inject [dest!] = inj_Leaf [THEN injD]
   160 
   161 lemma inj_Numb: "inj(Numb)"
   162 apply (simp add: Numb_def o_def)
   163 apply (rule inj_onI)
   164 apply (erule Atom_inject [THEN Inr_inject])
   165 done
   166 
   167 lemmas Numb_inject [dest!] = inj_Numb [THEN injD]
   168 
   169 
   170 (** Injectiveness of Push_Node **)
   171 
   172 lemma Push_Node_inject:
   173     "[| Push_Node i m =Push_Node j n;  [| i=j;  m=n |] ==> P  
   174      |] ==> P"
   175 apply (simp add: Push_Node_def)
   176 apply (erule Abs_Node_inj [THEN apfst_convE])
   177 apply (rule Rep_Node [THEN Node_Push_I])+
   178 apply (erule sym [THEN apfst_convE]) 
   179 apply (blast intro: Rep_Node_inject [THEN iffD1] trans sym elim!: Push_inject)
   180 done
   181 
   182 
   183 (** Injectiveness of Scons **)
   184 
   185 lemma Scons_inject_lemma1: "Scons M N <= Scons M' N' ==> M<=M'"
   186 unfolding Scons_def One_nat_def
   187 by (blast dest!: Push_Node_inject)
   188 
   189 lemma Scons_inject_lemma2: "Scons M N <= Scons M' N' ==> N<=N'"
   190 unfolding Scons_def One_nat_def
   191 by (blast dest!: Push_Node_inject)
   192 
   193 lemma Scons_inject1: "Scons M N = Scons M' N' ==> M=M'"
   194 apply (erule equalityE)
   195 apply (iprover intro: equalityI Scons_inject_lemma1)
   196 done
   197 
   198 lemma Scons_inject2: "Scons M N = Scons M' N' ==> N=N'"
   199 apply (erule equalityE)
   200 apply (iprover intro: equalityI Scons_inject_lemma2)
   201 done
   202 
   203 lemma Scons_inject:
   204     "[| Scons M N = Scons M' N';  [| M=M';  N=N' |] ==> P |] ==> P"
   205 by (iprover dest: Scons_inject1 Scons_inject2)
   206 
   207 lemma Scons_Scons_eq [iff]: "(Scons M N = Scons M' N') = (M=M' \<and> N=N')"
   208 by (blast elim!: Scons_inject)
   209 
   210 (*** Distinctness involving Leaf and Numb ***)
   211 
   212 (** Scons vs Leaf **)
   213 
   214 lemma Scons_not_Leaf [iff]: "Scons M N \<noteq> Leaf(a)"
   215 unfolding Leaf_def o_def by (rule Scons_not_Atom)
   216 
   217 lemmas Leaf_not_Scons  [iff] = Scons_not_Leaf [THEN not_sym]
   218 
   219 (** Scons vs Numb **)
   220 
   221 lemma Scons_not_Numb [iff]: "Scons M N \<noteq> Numb(k)"
   222 unfolding Numb_def o_def by (rule Scons_not_Atom)
   223 
   224 lemmas Numb_not_Scons [iff] = Scons_not_Numb [THEN not_sym]
   225 
   226 
   227 (** Leaf vs Numb **)
   228 
   229 lemma Leaf_not_Numb [iff]: "Leaf(a) \<noteq> Numb(k)"
   230 by (simp add: Leaf_def Numb_def)
   231 
   232 lemmas Numb_not_Leaf [iff] = Leaf_not_Numb [THEN not_sym]
   233 
   234 
   235 (*** ndepth -- the depth of a node ***)
   236 
   237 lemma ndepth_K0: "ndepth (Abs_Node(%k. Inr 0, x)) = 0"
   238 by (simp add: ndepth_def  Node_K0_I [THEN Abs_Node_inverse] Least_equality)
   239 
   240 lemma ndepth_Push_Node_aux:
   241      "case_nat (Inr (Suc i)) f k = Inr 0 \<longrightarrow> Suc(LEAST x. f x = Inr 0) \<le> k"
   242 apply (induct_tac "k", auto)
   243 apply (erule Least_le)
   244 done
   245 
   246 lemma ndepth_Push_Node: 
   247     "ndepth (Push_Node (Inr (Suc i)) n) = Suc(ndepth(n))"
   248 apply (insert Rep_Node [of n, unfolded Node_def])
   249 apply (auto simp add: ndepth_def Push_Node_def
   250                  Rep_Node [THEN Node_Push_I, THEN Abs_Node_inverse])
   251 apply (rule Least_equality)
   252 apply (auto simp add: Push_def ndepth_Push_Node_aux)
   253 apply (erule LeastI)
   254 done
   255 
   256 
   257 (*** ntrunc applied to the various node sets ***)
   258 
   259 lemma ntrunc_0 [simp]: "ntrunc 0 M = {}"
   260 by (simp add: ntrunc_def)
   261 
   262 lemma ntrunc_Atom [simp]: "ntrunc (Suc k) (Atom a) = Atom(a)"
   263 by (auto simp add: Atom_def ntrunc_def ndepth_K0)
   264 
   265 lemma ntrunc_Leaf [simp]: "ntrunc (Suc k) (Leaf a) = Leaf(a)"
   266 unfolding Leaf_def o_def by (rule ntrunc_Atom)
   267 
   268 lemma ntrunc_Numb [simp]: "ntrunc (Suc k) (Numb i) = Numb(i)"
   269 unfolding Numb_def o_def by (rule ntrunc_Atom)
   270 
   271 lemma ntrunc_Scons [simp]: 
   272     "ntrunc (Suc k) (Scons M N) = Scons (ntrunc k M) (ntrunc k N)"
   273 unfolding Scons_def ntrunc_def One_nat_def
   274 by (auto simp add: ndepth_Push_Node)
   275 
   276 
   277 
   278 (** Injection nodes **)
   279 
   280 lemma ntrunc_one_In0 [simp]: "ntrunc (Suc 0) (In0 M) = {}"
   281 apply (simp add: In0_def)
   282 apply (simp add: Scons_def)
   283 done
   284 
   285 lemma ntrunc_In0 [simp]: "ntrunc (Suc(Suc k)) (In0 M) = In0 (ntrunc (Suc k) M)"
   286 by (simp add: In0_def)
   287 
   288 lemma ntrunc_one_In1 [simp]: "ntrunc (Suc 0) (In1 M) = {}"
   289 apply (simp add: In1_def)
   290 apply (simp add: Scons_def)
   291 done
   292 
   293 lemma ntrunc_In1 [simp]: "ntrunc (Suc(Suc k)) (In1 M) = In1 (ntrunc (Suc k) M)"
   294 by (simp add: In1_def)
   295 
   296 
   297 subsection\<open>Set Constructions\<close>
   298 
   299 
   300 (*** Cartesian Product ***)
   301 
   302 lemma uprodI [intro!]: "\<lbrakk>M\<in>A; N\<in>B\<rbrakk> \<Longrightarrow> Scons M N \<in> uprod A B"
   303 by (simp add: uprod_def)
   304 
   305 (*The general elimination rule*)
   306 lemma uprodE [elim!]:
   307     "\<lbrakk>c \<in> uprod A B;   
   308         \<And>x y. \<lbrakk>x \<in> A; y \<in> B; c = Scons x y\<rbrakk> \<Longrightarrow> P  
   309      \<rbrakk> \<Longrightarrow> P"
   310 by (auto simp add: uprod_def) 
   311 
   312 
   313 (*Elimination of a pair -- introduces no eigenvariables*)
   314 lemma uprodE2: "\<lbrakk>Scons M N \<in> uprod A B; \<lbrakk>M \<in> A; N \<in> B\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
   315 by (auto simp add: uprod_def)
   316 
   317 
   318 (*** Disjoint Sum ***)
   319 
   320 lemma usum_In0I [intro]: "M \<in> A \<Longrightarrow> In0(M) \<in> usum A B"
   321 by (simp add: usum_def)
   322 
   323 lemma usum_In1I [intro]: "N \<in> B \<Longrightarrow> In1(N) \<in> usum A B"
   324 by (simp add: usum_def)
   325 
   326 lemma usumE [elim!]: 
   327     "\<lbrakk>u \<in> usum A B;   
   328         \<And>x. \<lbrakk>x \<in> A; u=In0(x)\<rbrakk> \<Longrightarrow> P;  
   329         \<And>y. \<lbrakk>y \<in> B; u=In1(y)\<rbrakk> \<Longrightarrow> P  
   330      \<rbrakk> \<Longrightarrow> P"
   331 by (auto simp add: usum_def)
   332 
   333 
   334 (** Injection **)
   335 
   336 lemma In0_not_In1 [iff]: "In0(M) \<noteq> In1(N)"
   337 unfolding In0_def In1_def One_nat_def by auto
   338 
   339 lemmas In1_not_In0 [iff] = In0_not_In1 [THEN not_sym]
   340 
   341 lemma In0_inject: "In0(M) = In0(N) ==>  M=N"
   342 by (simp add: In0_def)
   343 
   344 lemma In1_inject: "In1(M) = In1(N) ==>  M=N"
   345 by (simp add: In1_def)
   346 
   347 lemma In0_eq [iff]: "(In0 M = In0 N) = (M=N)"
   348 by (blast dest!: In0_inject)
   349 
   350 lemma In1_eq [iff]: "(In1 M = In1 N) = (M=N)"
   351 by (blast dest!: In1_inject)
   352 
   353 lemma inj_In0: "inj In0"
   354 by (blast intro!: inj_onI)
   355 
   356 lemma inj_In1: "inj In1"
   357 by (blast intro!: inj_onI)
   358 
   359 
   360 (*** Function spaces ***)
   361 
   362 lemma Lim_inject: "Lim f = Lim g ==> f = g"
   363 apply (simp add: Lim_def)
   364 apply (rule ext)
   365 apply (blast elim!: Push_Node_inject)
   366 done
   367 
   368 
   369 (*** proving equality of sets and functions using ntrunc ***)
   370 
   371 lemma ntrunc_subsetI: "ntrunc k M <= M"
   372 by (auto simp add: ntrunc_def)
   373 
   374 lemma ntrunc_subsetD: "(!!k. ntrunc k M <= N) ==> M<=N"
   375 by (auto simp add: ntrunc_def)
   376 
   377 (*A generalized form of the take-lemma*)
   378 lemma ntrunc_equality: "(!!k. ntrunc k M = ntrunc k N) ==> M=N"
   379 apply (rule equalityI)
   380 apply (rule_tac [!] ntrunc_subsetD)
   381 apply (rule_tac [!] ntrunc_subsetI [THEN [2] subset_trans], auto) 
   382 done
   383 
   384 lemma ntrunc_o_equality: 
   385     "[| !!k. (ntrunc(k) \<circ> h1) = (ntrunc(k) \<circ> h2) |] ==> h1=h2"
   386 apply (rule ntrunc_equality [THEN ext])
   387 apply (simp add: fun_eq_iff) 
   388 done
   389 
   390 
   391 (*** Monotonicity ***)
   392 
   393 lemma uprod_mono: "[| A<=A';  B<=B' |] ==> uprod A B <= uprod A' B'"
   394 by (simp add: uprod_def, blast)
   395 
   396 lemma usum_mono: "[| A<=A';  B<=B' |] ==> usum A B <= usum A' B'"
   397 by (simp add: usum_def, blast)
   398 
   399 lemma Scons_mono: "[| M<=M';  N<=N' |] ==> Scons M N <= Scons M' N'"
   400 by (simp add: Scons_def, blast)
   401 
   402 lemma In0_mono: "M<=N ==> In0(M) <= In0(N)"
   403 by (simp add: In0_def Scons_mono)
   404 
   405 lemma In1_mono: "M<=N ==> In1(M) <= In1(N)"
   406 by (simp add: In1_def Scons_mono)
   407 
   408 
   409 (*** Split and Case ***)
   410 
   411 lemma Split [simp]: "Split c (Scons M N) = c M N"
   412 by (simp add: Split_def)
   413 
   414 lemma Case_In0 [simp]: "Case c d (In0 M) = c(M)"
   415 by (simp add: Case_def)
   416 
   417 lemma Case_In1 [simp]: "Case c d (In1 N) = d(N)"
   418 by (simp add: Case_def)
   419 
   420 
   421 
   422 (**** UN x. B(x) rules ****)
   423 
   424 lemma ntrunc_UN1: "ntrunc k (UN x. f(x)) = (UN x. ntrunc k (f x))"
   425 by (simp add: ntrunc_def, blast)
   426 
   427 lemma Scons_UN1_x: "Scons (UN x. f x) M = (UN x. Scons (f x) M)"
   428 by (simp add: Scons_def, blast)
   429 
   430 lemma Scons_UN1_y: "Scons M (UN x. f x) = (UN x. Scons M (f x))"
   431 by (simp add: Scons_def, blast)
   432 
   433 lemma In0_UN1: "In0(UN x. f(x)) = (UN x. In0(f(x)))"
   434 by (simp add: In0_def Scons_UN1_y)
   435 
   436 lemma In1_UN1: "In1(UN x. f(x)) = (UN x. In1(f(x)))"
   437 by (simp add: In1_def Scons_UN1_y)
   438 
   439 
   440 (*** Equality for Cartesian Product ***)
   441 
   442 lemma dprodI [intro!]: 
   443     "\<lbrakk>(M,M') \<in> r; (N,N') \<in> s\<rbrakk> \<Longrightarrow> (Scons M N, Scons M' N') \<in> dprod r s"
   444 by (auto simp add: dprod_def)
   445 
   446 (*The general elimination rule*)
   447 lemma dprodE [elim!]: 
   448     "\<lbrakk>c \<in> dprod r s;   
   449         \<And>x y x' y'. \<lbrakk>(x,x') \<in> r; (y,y') \<in> s;  
   450                         c = (Scons x y, Scons x' y')\<rbrakk> \<Longrightarrow> P  
   451      \<rbrakk> \<Longrightarrow> P"
   452 by (auto simp add: dprod_def)
   453 
   454 
   455 (*** Equality for Disjoint Sum ***)
   456 
   457 lemma dsum_In0I [intro]: "(M,M') \<in> r \<Longrightarrow> (In0(M), In0(M')) \<in> dsum r s"
   458 by (auto simp add: dsum_def)
   459 
   460 lemma dsum_In1I [intro]: "(N,N') \<in> s \<Longrightarrow> (In1(N), In1(N')) \<in> dsum r s"
   461 by (auto simp add: dsum_def)
   462 
   463 lemma dsumE [elim!]: 
   464     "\<lbrakk>w \<in> dsum r s;   
   465         \<And>x x'. \<lbrakk> (x,x') \<in> r;  w = (In0(x), In0(x')) \<rbrakk> \<Longrightarrow> P;  
   466         \<And>y y'. \<lbrakk> (y,y') \<in> s;  w = (In1(y), In1(y')) \<rbrakk> \<Longrightarrow> P  
   467      \<rbrakk> \<Longrightarrow> P"
   468 by (auto simp add: dsum_def)
   469 
   470 
   471 (*** Monotonicity ***)
   472 
   473 lemma dprod_mono: "[| r<=r';  s<=s' |] ==> dprod r s <= dprod r' s'"
   474 by blast
   475 
   476 lemma dsum_mono: "[| r<=r';  s<=s' |] ==> dsum r s <= dsum r' s'"
   477 by blast
   478 
   479 
   480 (*** Bounding theorems ***)
   481 
   482 lemma dprod_Sigma: "(dprod (A \<times> B) (C \<times> D)) <= (uprod A C) \<times> (uprod B D)"
   483 by blast
   484 
   485 lemmas dprod_subset_Sigma = subset_trans [OF dprod_mono dprod_Sigma]
   486 
   487 (*Dependent version*)
   488 lemma dprod_subset_Sigma2:
   489     "(dprod (Sigma A B) (Sigma C D)) <= Sigma (uprod A C) (Split (%x y. uprod (B x) (D y)))"
   490 by auto
   491 
   492 lemma dsum_Sigma: "(dsum (A \<times> B) (C \<times> D)) <= (usum A C) \<times> (usum B D)"
   493 by blast
   494 
   495 lemmas dsum_subset_Sigma = subset_trans [OF dsum_mono dsum_Sigma]
   496 
   497 
   498 (*** Domain theorems ***)
   499 
   500 lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)"
   501   by auto
   502 
   503 lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)"
   504   by auto
   505 
   506 
   507 text \<open>hides popular names\<close>
   508 hide_type (open) node item
   509 hide_const (open) Push Node Atom Leaf Numb Lim Split Case
   510 
   511 ML_file "~~/src/HOL/Tools/Old_Datatype/old_datatype.ML"
   512 
   513 end