src/HOL/Datatype_Universe.ML
author oheimb
Wed Jan 31 10:15:55 2001 +0100 (2001-01-31)
changeset 11008 f7333f055ef6
parent 10908 a7cfffb5d7dc
child 11464 ddea204de5bc
permissions -rw-r--r--
improved theory reference in comment
     1 (*  Title:      HOL/Datatype_Universe.ML
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1991  University of Cambridge
     5 *)
     6 
     7 (** apfst -- can be used in similar type definitions **)
     8 
     9 Goalw [apfst_def] "apfst f (a,b) = (f(a),b)";
    10 by (rtac split_conv 1);
    11 qed "apfst_conv";
    12 
    13 val [major,minor] = Goal
    14     "[| q = apfst f p;  !!x y. [| p = (x,y);  q = (f(x),y) |] ==> R \
    15 \    |] ==> R";
    16 by (rtac PairE 1);
    17 by (rtac minor 1);
    18 by (assume_tac 1);
    19 by (rtac (major RS trans) 1);
    20 by (etac ssubst 1);
    21 by (rtac apfst_conv 1);
    22 qed "apfst_convE";
    23 
    24 (** Push -- an injection, analogous to Cons on lists **)
    25 
    26 Goalw [Push_def] "Push i f = Push j g  ==> i=j";
    27 by (etac (fun_cong RS box_equals) 1);
    28 by (rtac nat_case_0 1);
    29 by (rtac nat_case_0 1);
    30 qed "Push_inject1";
    31 
    32 Goalw [Push_def] "Push i f = Push j g  ==> f=g";
    33 by (rtac (ext RS box_equals) 1);
    34 by (etac fun_cong 1);
    35 by (rtac (nat_case_Suc RS ext) 1);
    36 by (rtac (nat_case_Suc RS ext) 1);
    37 qed "Push_inject2";
    38 
    39 val [major,minor] = Goal
    40     "[| Push i f =Push j g;  [| i=j;  f=g |] ==> P \
    41 \    |] ==> P";
    42 by (rtac ((major RS Push_inject2) RS ((major RS Push_inject1) RS minor)) 1);
    43 qed "Push_inject";
    44 
    45 Goalw [Push_def] "Push (Inr (Suc k)) f = (%z. Inr 0) ==> P";
    46 by (rtac Suc_neq_Zero 1);
    47 by (etac (fun_cong RS box_equals RS Inr_inject) 1);
    48 by (rtac nat_case_0 1);
    49 by (rtac refl 1);
    50 qed "Push_neq_K0";
    51 
    52 (*** Isomorphisms ***)
    53 
    54 Goal "inj(Rep_Node)";
    55 by (rtac inj_inverseI 1);       (*cannot combine by RS: multiple unifiers*)
    56 by (rtac Rep_Node_inverse 1);
    57 qed "inj_Rep_Node";
    58 
    59 Goal "inj_on Abs_Node Node";
    60 by (rtac inj_on_inverseI 1);
    61 by (etac Abs_Node_inverse 1);
    62 qed "inj_on_Abs_Node";
    63 
    64 bind_thm ("Abs_Node_inj", inj_on_Abs_Node RS inj_onD);
    65 
    66 
    67 (*** Introduction rules for Node ***)
    68 
    69 Goalw [Node_def] "(%k. Inr 0, a) : Node";
    70 by (Blast_tac 1);
    71 qed "Node_K0_I";
    72 
    73 Goalw [Node_def,Push_def]
    74     "p: Node ==> apfst (Push i) p : Node";
    75 by (fast_tac (claset() addSIs [apfst_conv, nat_case_Suc RS trans]) 1);
    76 qed "Node_Push_I";
    77 
    78 
    79 (*** Distinctness of constructors ***)
    80 
    81 (** Scons vs Atom **)
    82 
    83 Goalw [Atom_def,Scons_def,Push_Node_def] "Scons M N ~= Atom(a)";
    84 by (rtac notI 1);
    85 by (etac (equalityD2 RS subsetD RS UnE) 1);
    86 by (rtac singletonI 1);
    87 by (REPEAT (eresolve_tac [imageE, Abs_Node_inj RS apfst_convE, 
    88                           Pair_inject, sym RS Push_neq_K0] 1
    89      ORELSE resolve_tac [Node_K0_I, Rep_Node RS Node_Push_I] 1));
    90 qed "Scons_not_Atom";
    91 bind_thm ("Atom_not_Scons", Scons_not_Atom RS not_sym);
    92 
    93 
    94 (*** Injectiveness ***)
    95 
    96 (** Atomic nodes **)
    97 
    98 Goalw [Atom_def] "inj(Atom)";
    99 by (blast_tac (claset() addSIs [injI, Node_K0_I] addSDs [Abs_Node_inj]) 1);
   100 qed "inj_Atom";
   101 bind_thm ("Atom_inject", inj_Atom RS injD);
   102 
   103 Goal "(Atom(a)=Atom(b)) = (a=b)";
   104 by (blast_tac (claset() addSDs [Atom_inject]) 1);
   105 qed "Atom_Atom_eq";
   106 AddIffs [Atom_Atom_eq];
   107 
   108 Goalw [Leaf_def,o_def] "inj(Leaf)";
   109 by (rtac injI 1);
   110 by (etac (Atom_inject RS Inl_inject) 1);
   111 qed "inj_Leaf";
   112 
   113 bind_thm ("Leaf_inject", inj_Leaf RS injD);
   114 AddSDs [Leaf_inject];
   115 
   116 Goalw [Numb_def,o_def] "inj(Numb)";
   117 by (rtac injI 1);
   118 by (etac (Atom_inject RS Inr_inject) 1);
   119 qed "inj_Numb";
   120 
   121 bind_thm ("Numb_inject", inj_Numb RS injD);
   122 AddSDs [Numb_inject];
   123 
   124 (** Injectiveness of Push_Node **)
   125 
   126 val [major,minor] = Goalw [Push_Node_def]
   127     "[| Push_Node i m =Push_Node j n;  [| i=j;  m=n |] ==> P \
   128 \    |] ==> P";
   129 by (rtac (major RS Abs_Node_inj RS apfst_convE) 1);
   130 by (REPEAT (resolve_tac [Rep_Node RS Node_Push_I] 1));
   131 by (etac (sym RS apfst_convE) 1);
   132 by (rtac minor 1);
   133 by (etac Pair_inject 1);
   134 by (etac (Push_inject1 RS sym) 1);
   135 by (rtac (inj_Rep_Node RS injD) 1);
   136 by (etac trans 1);
   137 by (safe_tac (claset() addSEs [Push_inject,sym]));
   138 qed "Push_Node_inject";
   139 
   140 
   141 (** Injectiveness of Scons **)
   142 
   143 Goalw [Scons_def] "Scons M N <= Scons M' N' ==> M<=M'";
   144 by (blast_tac (claset() addSDs [Push_Node_inject]) 1);
   145 qed "Scons_inject_lemma1";
   146 
   147 Goalw [Scons_def] "Scons M N <= Scons M' N' ==> N<=N'";
   148 by (blast_tac (claset() addSDs [Push_Node_inject]) 1);
   149 qed "Scons_inject_lemma2";
   150 
   151 Goal "Scons M N = Scons M' N' ==> M=M'";
   152 by (etac equalityE 1);
   153 by (REPEAT (ares_tac [equalityI, Scons_inject_lemma1] 1));
   154 qed "Scons_inject1";
   155 
   156 Goal "Scons M N = Scons M' N' ==> N=N'";
   157 by (etac equalityE 1);
   158 by (REPEAT (ares_tac [equalityI, Scons_inject_lemma2] 1));
   159 qed "Scons_inject2";
   160 
   161 val [major,minor] = Goal
   162     "[| Scons M N = Scons M' N';  [| M=M';  N=N' |] ==> P \
   163 \    |] ==> P";
   164 by (rtac ((major RS Scons_inject2) RS ((major RS Scons_inject1) RS minor)) 1);
   165 qed "Scons_inject";
   166 
   167 Goal "(Scons M N = Scons M' N') = (M=M' & N=N')";
   168 by (blast_tac (claset() addSEs [Scons_inject]) 1);
   169 qed "Scons_Scons_eq";
   170 
   171 (*** Distinctness involving Leaf and Numb ***)
   172 
   173 (** Scons vs Leaf **)
   174 
   175 Goalw [Leaf_def,o_def] "Scons M N ~= Leaf(a)";
   176 by (rtac Scons_not_Atom 1);
   177 qed "Scons_not_Leaf";
   178 bind_thm ("Leaf_not_Scons", Scons_not_Leaf RS not_sym);
   179 
   180 AddIffs [Scons_not_Leaf, Leaf_not_Scons];
   181 
   182 
   183 (** Scons vs Numb **)
   184 
   185 Goalw [Numb_def,o_def] "Scons M N ~= Numb(k)";
   186 by (rtac Scons_not_Atom 1);
   187 qed "Scons_not_Numb";
   188 bind_thm ("Numb_not_Scons", Scons_not_Numb RS not_sym);
   189 
   190 AddIffs [Scons_not_Numb, Numb_not_Scons];
   191 
   192 
   193 (** Leaf vs Numb **)
   194 
   195 Goalw [Leaf_def,Numb_def] "Leaf(a) ~= Numb(k)";
   196 by (simp_tac (simpset() addsimps [Inl_not_Inr]) 1);
   197 qed "Leaf_not_Numb";
   198 bind_thm ("Numb_not_Leaf", Leaf_not_Numb RS not_sym);
   199 
   200 AddIffs [Leaf_not_Numb, Numb_not_Leaf];
   201 
   202 
   203 (*** ndepth -- the depth of a node ***)
   204 
   205 Addsimps [apfst_conv];
   206 AddIffs  [Scons_not_Atom, Atom_not_Scons, Scons_Scons_eq];
   207 
   208 
   209 Goalw [ndepth_def] "ndepth (Abs_Node(%k. Inr 0, x)) = 0";
   210 by (EVERY1[stac (Node_K0_I RS Abs_Node_inverse), stac split_conv]);
   211 by (rtac Least_equality 1);
   212 by Auto_tac;  
   213 qed "ndepth_K0";
   214 
   215 Goal "nat_case (Inr (Suc i)) f k = Inr 0 --> Suc(LEAST x. f x = Inr 0) <= k";
   216 by (induct_tac "k" 1);
   217 by (ALLGOALS Simp_tac);
   218 by (rtac impI 1); 
   219 by (etac Least_le 1);
   220 val lemma = result();
   221 
   222 Goalw [ndepth_def,Push_Node_def]
   223     "ndepth (Push_Node (Inr (Suc i)) n) = Suc(ndepth(n))";
   224 by (stac (Rep_Node RS Node_Push_I RS Abs_Node_inverse) 1);
   225 by (cut_facts_tac [rewrite_rule [Node_def] Rep_Node] 1);
   226 by Safe_tac;
   227 by (etac ssubst 1);  (*instantiates type variables!*)
   228 by (Simp_tac 1);
   229 by (rtac Least_equality 1);
   230 by (rewtac Push_def);
   231 by (auto_tac (claset(), simpset() addsimps [lemma]));  
   232 by (etac LeastI 1);
   233 qed "ndepth_Push_Node";
   234 
   235 
   236 (*** ntrunc applied to the various node sets ***)
   237 
   238 Goalw [ntrunc_def] "ntrunc 0 M = {}";
   239 by (Blast_tac 1);
   240 qed "ntrunc_0";
   241 
   242 Goalw [Atom_def,ntrunc_def] "ntrunc (Suc k) (Atom a) = Atom(a)";
   243 by (fast_tac (claset() addss (simpset() addsimps [ndepth_K0])) 1);
   244 qed "ntrunc_Atom";
   245 
   246 Goalw [Leaf_def,o_def] "ntrunc (Suc k) (Leaf a) = Leaf(a)";
   247 by (rtac ntrunc_Atom 1);
   248 qed "ntrunc_Leaf";
   249 
   250 Goalw [Numb_def,o_def] "ntrunc (Suc k) (Numb i) = Numb(i)";
   251 by (rtac ntrunc_Atom 1);
   252 qed "ntrunc_Numb";
   253 
   254 Goalw [Scons_def,ntrunc_def]
   255     "ntrunc (Suc k) (Scons M N) = Scons (ntrunc k M) (ntrunc k N)";
   256 by (safe_tac (claset() addSIs [imageI]));
   257 by (REPEAT (stac ndepth_Push_Node 3 THEN etac Suc_mono 3));
   258 by (REPEAT (rtac Suc_less_SucD 1 THEN 
   259             rtac (ndepth_Push_Node RS subst) 1 THEN 
   260             assume_tac 1));
   261 qed "ntrunc_Scons";
   262 
   263 Addsimps [ntrunc_0, ntrunc_Atom, ntrunc_Leaf, ntrunc_Numb, ntrunc_Scons];
   264 
   265 
   266 (** Injection nodes **)
   267 
   268 Goalw [In0_def] "ntrunc 1 (In0 M) = {}";
   269 by (Simp_tac 1);
   270 by (rewtac Scons_def);
   271 by (Blast_tac 1);
   272 qed "ntrunc_one_In0";
   273 
   274 Goalw [In0_def]
   275     "ntrunc (Suc (Suc k)) (In0 M) = In0 (ntrunc (Suc k) M)";
   276 by (Simp_tac 1);
   277 qed "ntrunc_In0";
   278 
   279 Goalw [In1_def] "ntrunc 1 (In1 M) = {}";
   280 by (Simp_tac 1);
   281 by (rewtac Scons_def);
   282 by (Blast_tac 1);
   283 qed "ntrunc_one_In1";
   284 
   285 Goalw [In1_def]
   286     "ntrunc (Suc (Suc k)) (In1 M) = In1 (ntrunc (Suc k) M)";
   287 by (Simp_tac 1);
   288 qed "ntrunc_In1";
   289 
   290 Addsimps [ntrunc_one_In0, ntrunc_In0, ntrunc_one_In1, ntrunc_In1];
   291 
   292 
   293 (*** Cartesian Product ***)
   294 
   295 Goalw [uprod_def] "[| M:A;  N:B |] ==> Scons M N : uprod A B";
   296 by (REPEAT (ares_tac [singletonI,UN_I] 1));
   297 qed "uprodI";
   298 
   299 (*The general elimination rule*)
   300 val major::prems = Goalw [uprod_def]
   301     "[| c : uprod A B;  \
   302 \       !!x y. [| x:A;  y:B;  c = Scons x y |] ==> P \
   303 \    |] ==> P";
   304 by (cut_facts_tac [major] 1);
   305 by (REPEAT (eresolve_tac [asm_rl,singletonE,UN_E] 1
   306      ORELSE resolve_tac prems 1));
   307 qed "uprodE";
   308 
   309 (*Elimination of a pair -- introduces no eigenvariables*)
   310 val prems = Goal
   311     "[| Scons M N : uprod A B;      [| M:A;  N:B |] ==> P   \
   312 \    |] ==> P";
   313 by (rtac uprodE 1);
   314 by (REPEAT (ares_tac prems 1 ORELSE eresolve_tac [Scons_inject,ssubst] 1));
   315 qed "uprodE2";
   316 
   317 
   318 (*** Disjoint Sum ***)
   319 
   320 Goalw [usum_def] "M:A ==> In0(M) : usum A B";
   321 by (Blast_tac 1);
   322 qed "usum_In0I";
   323 
   324 Goalw [usum_def] "N:B ==> In1(N) : usum A B";
   325 by (Blast_tac 1);
   326 qed "usum_In1I";
   327 
   328 val major::prems = Goalw [usum_def]
   329     "[| u : usum A B;  \
   330 \       !!x. [| x:A;  u=In0(x) |] ==> P; \
   331 \       !!y. [| y:B;  u=In1(y) |] ==> P \
   332 \    |] ==> P";
   333 by (rtac (major RS UnE) 1);
   334 by (REPEAT (rtac refl 1 
   335      ORELSE eresolve_tac (prems@[imageE,ssubst]) 1));
   336 qed "usumE";
   337 
   338 
   339 (** Injection **)
   340 
   341 Goalw [In0_def,In1_def] "In0(M) ~= In1(N)";
   342 by (rtac notI 1);
   343 by (etac (Scons_inject1 RS Numb_inject RS Zero_neq_Suc) 1);
   344 qed "In0_not_In1";
   345 
   346 bind_thm ("In1_not_In0", In0_not_In1 RS not_sym);
   347 
   348 AddIffs [In0_not_In1, In1_not_In0];
   349 
   350 Goalw [In0_def] "In0(M) = In0(N) ==>  M=N";
   351 by (etac (Scons_inject2) 1);
   352 qed "In0_inject";
   353 
   354 Goalw [In1_def] "In1(M) = In1(N) ==>  M=N";
   355 by (etac (Scons_inject2) 1);
   356 qed "In1_inject";
   357 
   358 Goal "(In0 M = In0 N) = (M=N)";
   359 by (blast_tac (claset() addSDs [In0_inject]) 1);
   360 qed "In0_eq";
   361 
   362 Goal "(In1 M = In1 N) = (M=N)";
   363 by (blast_tac (claset() addSDs [In1_inject]) 1);
   364 qed "In1_eq";
   365 
   366 AddIffs [In0_eq, In1_eq];
   367 
   368 Goal "inj In0";
   369 by (blast_tac (claset() addSIs [injI]) 1);
   370 qed "inj_In0";
   371 
   372 Goal "inj In1";
   373 by (blast_tac (claset() addSIs [injI]) 1);
   374 qed "inj_In1";
   375 
   376 
   377 (*** Function spaces ***)
   378 
   379 Goalw [Lim_def] "Lim f = Lim g ==> f = g";
   380 by (rtac ext 1);
   381 by (blast_tac (claset() addSEs [Push_Node_inject]) 1);
   382 qed "Lim_inject";
   383 
   384 Goalw [Funs_def] "S <= T ==> Funs S <= Funs T";
   385 by (Blast_tac 1);
   386 qed "Funs_mono";
   387 
   388 val [prem] = Goalw [Funs_def] "(!!x. f x : S) ==> f : Funs S";
   389 by (blast_tac (claset() addIs [prem]) 1);
   390 qed "FunsI";
   391 
   392 Goalw [Funs_def] "f : Funs S ==> f x : S";
   393 by (etac CollectE 1);
   394 by (etac subsetD 1);
   395 by (rtac rangeI 1);
   396 qed "FunsD";
   397 
   398 val [p1, p2] = Goalw [o_def]
   399    "[| f : Funs R; !!x. x : R ==> r (a x) = x |] ==> r o (a o f) = f";
   400 by (rtac (p2 RS ext) 1);
   401 by (rtac (p1 RS FunsD) 1);
   402 qed "Funs_inv";
   403 
   404 val [p1, p2] = Goalw [o_def]
   405      "[| f : Funs (range g); !!h. f = g o h ==> P |] ==> P";
   406 by (res_inst_tac [("h", "%x. @y. (f::'a=>'b) x = g y")] p2 1);
   407 by (rtac ext 1);
   408 by (rtac (p1 RS FunsD RS rangeE) 1);
   409 by (etac (exI RS (some_eq_ex RS iffD2)) 1);
   410 qed "Funs_rangeE";
   411 
   412 Goal "a : S ==> (%x. a) : Funs S";
   413 by (rtac FunsI 1);
   414 by (assume_tac 1);
   415 qed "Funs_nonempty";
   416 
   417 
   418 (*** proving equality of sets and functions using ntrunc ***)
   419 
   420 Goalw [ntrunc_def] "ntrunc k M <= M";
   421 by (Blast_tac 1);
   422 qed "ntrunc_subsetI";
   423 
   424 val [major] = Goalw [ntrunc_def] "(!!k. ntrunc k M <= N) ==> M<=N";
   425 by (blast_tac (claset() addIs [less_add_Suc1, less_add_Suc2, 
   426 			       major RS subsetD]) 1);
   427 qed "ntrunc_subsetD";
   428 
   429 (*A generalized form of the take-lemma*)
   430 val [major] = Goal "(!!k. ntrunc k M = ntrunc k N) ==> M=N";
   431 by (rtac equalityI 1);
   432 by (ALLGOALS (rtac ntrunc_subsetD));
   433 by (ALLGOALS (rtac (ntrunc_subsetI RSN (2, subset_trans))));
   434 by (rtac (major RS equalityD1) 1);
   435 by (rtac (major RS equalityD2) 1);
   436 qed "ntrunc_equality";
   437 
   438 val [major] = Goalw [o_def]
   439     "[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2";
   440 by (rtac (ntrunc_equality RS ext) 1);
   441 by (rtac (major RS fun_cong) 1);
   442 qed "ntrunc_o_equality";
   443 
   444 (*** Monotonicity ***)
   445 
   446 Goalw [uprod_def] "[| A<=A';  B<=B' |] ==> uprod A B <= uprod A' B'";
   447 by (Blast_tac 1);
   448 qed "uprod_mono";
   449 
   450 Goalw [usum_def] "[| A<=A';  B<=B' |] ==> usum A B <= usum A' B'";
   451 by (Blast_tac 1);
   452 qed "usum_mono";
   453 
   454 Goalw [Scons_def] "[| M<=M';  N<=N' |] ==> Scons M N <= Scons M' N'";
   455 by (Blast_tac 1);
   456 qed "Scons_mono";
   457 
   458 Goalw [In0_def] "M<=N ==> In0(M) <= In0(N)";
   459 by (REPEAT (ares_tac [subset_refl,Scons_mono] 1));
   460 qed "In0_mono";
   461 
   462 Goalw [In1_def] "M<=N ==> In1(M) <= In1(N)";
   463 by (REPEAT (ares_tac [subset_refl,Scons_mono] 1));
   464 qed "In1_mono";
   465 
   466 
   467 (*** Split and Case ***)
   468 
   469 Goalw [Split_def] "Split c (Scons M N) = c M N";
   470 by (Blast_tac  1);
   471 qed "Split";
   472 
   473 Goalw [Case_def] "Case c d (In0 M) = c(M)";
   474 by (Blast_tac 1);
   475 qed "Case_In0";
   476 
   477 Goalw [Case_def] "Case c d (In1 N) = d(N)";
   478 by (Blast_tac 1);
   479 qed "Case_In1";
   480 
   481 Addsimps [Split, Case_In0, Case_In1];
   482 
   483 
   484 (**** UN x. B(x) rules ****)
   485 
   486 Goalw [ntrunc_def] "ntrunc k (UN x. f(x)) = (UN x. ntrunc k (f x))";
   487 by (Blast_tac 1);
   488 qed "ntrunc_UN1";
   489 
   490 Goalw [Scons_def] "Scons (UN x. f x) M = (UN x. Scons (f x) M)";
   491 by (Blast_tac 1);
   492 qed "Scons_UN1_x";
   493 
   494 Goalw [Scons_def] "Scons M (UN x. f x) = (UN x. Scons M (f x))";
   495 by (Blast_tac 1);
   496 qed "Scons_UN1_y";
   497 
   498 Goalw [In0_def] "In0(UN x. f(x)) = (UN x. In0(f(x)))";
   499 by (rtac Scons_UN1_y 1);
   500 qed "In0_UN1";
   501 
   502 Goalw [In1_def] "In1(UN x. f(x)) = (UN x. In1(f(x)))";
   503 by (rtac Scons_UN1_y 1);
   504 qed "In1_UN1";
   505 
   506 
   507 (*** Equality for Cartesian Product ***)
   508 
   509 Goalw [dprod_def]
   510     "[| (M,M'):r;  (N,N'):s |] ==> (Scons M N, Scons M' N') : dprod r s";
   511 by (Blast_tac 1);
   512 qed "dprodI";
   513 
   514 (*The general elimination rule*)
   515 val major::prems = Goalw [dprod_def]
   516     "[| c : dprod r s;  \
   517 \       !!x y x' y'. [| (x,x') : r;  (y,y') : s;  c = (Scons x y, Scons x' y') |] ==> P \
   518 \    |] ==> P";
   519 by (cut_facts_tac [major] 1);
   520 by (REPEAT_FIRST (eresolve_tac [asm_rl, UN_E, mem_splitE, singletonE]));
   521 by (REPEAT (ares_tac prems 1 ORELSE hyp_subst_tac 1));
   522 qed "dprodE";
   523 
   524 
   525 (*** Equality for Disjoint Sum ***)
   526 
   527 Goalw [dsum_def]  "(M,M'):r ==> (In0(M), In0(M')) : dsum r s";
   528 by (Blast_tac 1);
   529 qed "dsum_In0I";
   530 
   531 Goalw [dsum_def]  "(N,N'):s ==> (In1(N), In1(N')) : dsum r s";
   532 by (Blast_tac 1);
   533 qed "dsum_In1I";
   534 
   535 val major::prems = Goalw [dsum_def]
   536     "[| w : dsum r s;  \
   537 \       !!x x'. [| (x,x') : r;  w = (In0(x), In0(x')) |] ==> P; \
   538 \       !!y y'. [| (y,y') : s;  w = (In1(y), In1(y')) |] ==> P \
   539 \    |] ==> P";
   540 by (cut_facts_tac [major] 1);
   541 by (REPEAT_FIRST (eresolve_tac [asm_rl, UN_E, UnE, mem_splitE, singletonE]));
   542 by (DEPTH_SOLVE (ares_tac prems 1 ORELSE hyp_subst_tac 1));
   543 qed "dsumE";
   544 
   545 AddSIs [uprodI, dprodI];
   546 AddIs  [usum_In0I, usum_In1I, dsum_In0I, dsum_In1I];
   547 AddSEs [uprodE, dprodE, usumE, dsumE];
   548 
   549 
   550 (*** Monotonicity ***)
   551 
   552 Goal "[| r<=r';  s<=s' |] ==> dprod r s <= dprod r' s'";
   553 by (Blast_tac 1);
   554 qed "dprod_mono";
   555 
   556 Goal "[| r<=r';  s<=s' |] ==> dsum r s <= dsum r' s'";
   557 by (Blast_tac 1);
   558 qed "dsum_mono";
   559 
   560 
   561 (*** Bounding theorems ***)
   562 
   563 Goal "(dprod (A <*> B) (C <*> D)) <= (uprod A C) <*> (uprod B D)";
   564 by (Blast_tac 1);
   565 qed "dprod_Sigma";
   566 
   567 bind_thm ("dprod_subset_Sigma", [dprod_mono, dprod_Sigma] MRS subset_trans |> standard);
   568 
   569 (*Dependent version*)
   570 Goal "(dprod (Sigma A B) (Sigma C D)) <= Sigma (uprod A C) (Split (%x y. uprod (B x) (D y)))";
   571 by Safe_tac;
   572 by (stac Split 1);
   573 by (Blast_tac 1);
   574 qed "dprod_subset_Sigma2";
   575 
   576 Goal "(dsum (A <*> B) (C <*> D)) <= (usum A C) <*> (usum B D)";
   577 by (Blast_tac 1);
   578 qed "dsum_Sigma";
   579 
   580 bind_thm ("dsum_subset_Sigma", [dsum_mono, dsum_Sigma] MRS subset_trans |> standard);
   581 
   582 
   583 (*** Domain ***)
   584 
   585 Goal "Domain (dprod r s) = uprod (Domain r) (Domain s)";
   586 by Auto_tac;
   587 qed "Domain_dprod";
   588 
   589 Goal "Domain (dsum r s) = usum (Domain r) (Domain s)";
   590 by Auto_tac;
   591 qed "Domain_dsum";
   592 
   593 Addsimps [Domain_dprod, Domain_dsum];