src/HOL/Set.ML
author paulson
Wed Jun 07 12:06:36 2000 +0200 (2000-06-07)
changeset 9041 3730ae0f513a
parent 8913 0bc13d5e60b8
child 9075 e8521ed7f35b
permissions -rw-r--r--
tidied
     1 (*  Title:      HOL/set
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1991  University of Cambridge
     5 
     6 Set theory for higher-order logic.  A set is simply a predicate.
     7 *)
     8 
     9 section "Relating predicates and sets";
    10 
    11 Addsimps [Collect_mem_eq];
    12 AddIffs  [mem_Collect_eq];
    13 
    14 Goal "P(a) ==> a : {x. P(x)}";
    15 by (Asm_simp_tac 1);
    16 qed "CollectI";
    17 
    18 Goal "a : {x. P(x)} ==> P(a)";
    19 by (Asm_full_simp_tac 1);
    20 qed "CollectD";
    21 
    22 bind_thm ("CollectE", make_elim CollectD);
    23 
    24 val [prem] = Goal "[| !!x. (x:A) = (x:B) |] ==> A = B";
    25 by (rtac (prem RS ext RS arg_cong RS box_equals) 1);
    26 by (rtac Collect_mem_eq 1);
    27 by (rtac Collect_mem_eq 1);
    28 qed "set_ext";
    29 
    30 val [prem] = Goal "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}";
    31 by (rtac (prem RS ext RS arg_cong) 1);
    32 qed "Collect_cong";
    33 
    34 val CollectE = make_elim CollectD;
    35 
    36 AddSIs [CollectI];
    37 AddSEs [CollectE];
    38 
    39 
    40 section "Bounded quantifiers";
    41 
    42 val prems = Goalw [Ball_def]
    43     "[| !!x. x:A ==> P(x) |] ==> ALL x:A. P(x)";
    44 by (REPEAT (ares_tac (prems @ [allI,impI]) 1));
    45 qed "ballI";
    46 
    47 bind_thms ("strip", [impI, allI, ballI]);
    48 
    49 Goalw [Ball_def] "[| ALL x:A. P(x);  x:A |] ==> P(x)";
    50 by (Blast_tac 1);
    51 qed "bspec";
    52 
    53 val major::prems = Goalw [Ball_def]
    54     "[| ALL x:A. P(x);  P(x) ==> Q;  x~:A ==> Q |] ==> Q";
    55 by (rtac (major RS spec RS impCE) 1);
    56 by (REPEAT (eresolve_tac prems 1));
    57 qed "ballE";
    58 
    59 (*Takes assumptions ALL x:A.P(x) and a:A; creates assumption P(a)*)
    60 fun ball_tac i = etac ballE i THEN contr_tac (i+1);
    61 
    62 AddSIs [ballI];
    63 AddEs  [ballE];
    64 AddXDs [bspec];
    65 (* gives better instantiation for bound: *)
    66 claset_ref() := claset() addWrapper ("bspec", fn tac2 =>
    67 			 (dtac bspec THEN' atac) APPEND' tac2);
    68 
    69 (*Normally the best argument order: P(x) constrains the choice of x:A*)
    70 Goalw [Bex_def] "[| P(x);  x:A |] ==> EX x:A. P(x)";
    71 by (Blast_tac 1);
    72 qed "bexI";
    73 
    74 (*The best argument order when there is only one x:A*)
    75 Goalw [Bex_def] "[| x:A;  P(x) |] ==> EX x:A. P(x)";
    76 by (Blast_tac 1);
    77 qed "rev_bexI";
    78 
    79 val prems = Goal 
    80    "[| ALL x:A. ~P(x) ==> P(a);  a:A |] ==> EX x:A. P(x)";
    81 by (rtac classical 1);
    82 by (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1))  ;
    83 qed "bexCI";
    84 
    85 val major::prems = Goalw [Bex_def]
    86     "[| EX x:A. P(x);  !!x. [| x:A; P(x) |] ==> Q  |] ==> Q";
    87 by (rtac (major RS exE) 1);
    88 by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1));
    89 qed "bexE";
    90 
    91 AddIs  [bexI];
    92 AddSEs [bexE];
    93 
    94 (*Trival rewrite rule*)
    95 Goal "(ALL x:A. P) = ((EX x. x:A) --> P)";
    96 by (simp_tac (simpset() addsimps [Ball_def]) 1);
    97 qed "ball_triv";
    98 
    99 (*Dual form for existentials*)
   100 Goal "(EX x:A. P) = ((EX x. x:A) & P)";
   101 by (simp_tac (simpset() addsimps [Bex_def]) 1);
   102 qed "bex_triv";
   103 
   104 Addsimps [ball_triv, bex_triv];
   105 
   106 (** Congruence rules **)
   107 
   108 val prems = Goalw [Ball_def]
   109     "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
   110 \    (ALL x:A. P(x)) = (ALL x:B. Q(x))";
   111 by (asm_simp_tac (simpset() addsimps prems) 1);
   112 qed "ball_cong";
   113 
   114 val prems = Goalw [Bex_def]
   115     "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
   116 \    (EX x:A. P(x)) = (EX x:B. Q(x))";
   117 by (asm_simp_tac (simpset() addcongs [conj_cong] addsimps prems) 1);
   118 qed "bex_cong";
   119 
   120 Addcongs [ball_cong,bex_cong];
   121 
   122 section "Subsets";
   123 
   124 val prems = Goalw [subset_def] "(!!x. x:A ==> x:B) ==> A <= B";
   125 by (REPEAT (ares_tac (prems @ [ballI]) 1));
   126 qed "subsetI";
   127 
   128 (*Map the type ('a set => anything) to just 'a.
   129   For overloading constants whose first argument has type "'a set" *)
   130 fun overload_1st_set s = Blast.overloaded (s, HOLogic.dest_setT o domain_type);
   131 
   132 (*While (:) is not, its type must be kept
   133   for overloading of = to work.*)
   134 Blast.overloaded ("op :", domain_type);
   135 
   136 overload_1st_set "Ball";		(*need UNION, INTER also?*)
   137 overload_1st_set "Bex";
   138 
   139 (*Image: retain the type of the set being expressed*)
   140 Blast.overloaded ("image", domain_type);
   141 
   142 (*Rule in Modus Ponens style*)
   143 Goalw [subset_def] "[| A <= B;  c:A |] ==> c:B";
   144 by (Blast_tac 1);
   145 qed "subsetD";
   146 AddXIs [subsetD];
   147 
   148 (*The same, with reversed premises for use with etac -- cf rev_mp*)
   149 Goal "[| c:A;  A <= B |] ==> c:B";
   150 by (REPEAT (ares_tac [subsetD] 1)) ;
   151 qed "rev_subsetD";
   152 AddXIs [rev_subsetD];
   153 
   154 (*Converts A<=B to x:A ==> x:B*)
   155 fun impOfSubs th = th RSN (2, rev_subsetD);
   156 
   157 Goal "[| A <= B; c ~: B |] ==> c ~: A";
   158 by (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ;
   159 qed "contra_subsetD";
   160 
   161 Goal "[| c ~: B;  A <= B |] ==> c ~: A";
   162 by (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ;
   163 qed "rev_contra_subsetD";
   164 
   165 (*Classical elimination rule*)
   166 val major::prems = Goalw [subset_def] 
   167     "[| A <= B;  c~:A ==> P;  c:B ==> P |] ==> P";
   168 by (rtac (major RS ballE) 1);
   169 by (REPEAT (eresolve_tac prems 1));
   170 qed "subsetCE";
   171 
   172 (*Takes assumptions A<=B; c:A and creates the assumption c:B *)
   173 fun set_mp_tac i = etac subsetCE i  THEN  mp_tac i;
   174 
   175 AddSIs [subsetI];
   176 AddEs  [subsetD, subsetCE];
   177 
   178 Goal "A <= (A::'a set)";
   179 by (Fast_tac 1);
   180 qed "subset_refl";		(*Blast_tac would try order_refl and fail*)
   181 
   182 Goal "[| A<=B;  B<=C |] ==> A<=(C::'a set)";
   183 by (Blast_tac 1);
   184 qed "subset_trans";
   185 
   186 
   187 section "Equality";
   188 
   189 (*Anti-symmetry of the subset relation*)
   190 Goal "[| A <= B;  B <= A |] ==> A = (B::'a set)";
   191 by (rtac set_ext 1);
   192 by (blast_tac (claset() addIs [subsetD]) 1);
   193 qed "subset_antisym";
   194 val equalityI = subset_antisym;
   195 
   196 AddSIs [equalityI];
   197 
   198 (* Equality rules from ZF set theory -- are they appropriate here? *)
   199 Goal "A = B ==> A<=(B::'a set)";
   200 by (etac ssubst 1);
   201 by (rtac subset_refl 1);
   202 qed "equalityD1";
   203 
   204 Goal "A = B ==> B<=(A::'a set)";
   205 by (etac ssubst 1);
   206 by (rtac subset_refl 1);
   207 qed "equalityD2";
   208 
   209 val prems = Goal
   210     "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";
   211 by (resolve_tac prems 1);
   212 by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
   213 qed "equalityE";
   214 
   215 (*This could be tried.  Most things build fine with it.  However, some proofs
   216   become very slow or even fail.
   217   AddEs [equalityE];
   218 *)
   219 
   220 val major::prems = Goal
   221     "[| A = B;  [| c:A; c:B |] ==> P;  [| c~:A; c~:B |] ==> P |]  ==>  P";
   222 by (rtac (major RS equalityE) 1);
   223 by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
   224 qed "equalityCE";
   225 
   226 (*Lemma for creating induction formulae -- for "pattern matching" on p
   227   To make the induction hypotheses usable, apply "spec" or "bspec" to
   228   put universal quantifiers over the free variables in p. *)
   229 val prems = Goal 
   230     "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
   231 by (rtac mp 1);
   232 by (REPEAT (resolve_tac (refl::prems) 1));
   233 qed "setup_induction";
   234 
   235 Goal "A = B ==> (x : A) = (x : B)";
   236 by (Asm_simp_tac 1);
   237 qed "eqset_imp_iff";
   238 
   239 
   240 section "The universal set -- UNIV";
   241 
   242 Goalw [UNIV_def] "x : UNIV";
   243 by (rtac CollectI 1);
   244 by (rtac TrueI 1);
   245 qed "UNIV_I";
   246 
   247 Addsimps [UNIV_I];
   248 AddIs    [UNIV_I];  (*unsafe makes it less likely to cause problems*)
   249 
   250 Goal "A <= UNIV";
   251 by (rtac subsetI 1);
   252 by (rtac UNIV_I 1);
   253 qed "subset_UNIV";
   254 
   255 (** Eta-contracting these two rules (to remove P) causes them to be ignored
   256     because of their interaction with congruence rules. **)
   257 
   258 Goalw [Ball_def] "Ball UNIV P = All P";
   259 by (Simp_tac 1);
   260 qed "ball_UNIV";
   261 
   262 Goalw [Bex_def] "Bex UNIV P = Ex P";
   263 by (Simp_tac 1);
   264 qed "bex_UNIV";
   265 Addsimps [ball_UNIV, bex_UNIV];
   266 
   267 
   268 section "The empty set -- {}";
   269 
   270 Goalw [empty_def] "(c : {}) = False";
   271 by (Blast_tac 1) ;
   272 qed "empty_iff";
   273 
   274 Addsimps [empty_iff];
   275 
   276 Goal "a:{} ==> P";
   277 by (Full_simp_tac 1);
   278 qed "emptyE";
   279 
   280 AddSEs [emptyE];
   281 
   282 Goal "{} <= A";
   283 by (Blast_tac 1) ;
   284 qed "empty_subsetI";
   285 
   286 (*One effect is to delete the ASSUMPTION {} <= A*)
   287 AddIffs [empty_subsetI];
   288 
   289 val [prem]= Goal "[| !!y. y:A ==> False |] ==> A={}";
   290 by (blast_tac (claset() addIs [prem RS FalseE]) 1) ;
   291 qed "equals0I";
   292 
   293 (*Use for reasoning about disjointness: A Int B = {} *)
   294 Goal "A={} ==> a ~: A";
   295 by (Blast_tac 1) ;
   296 qed "equals0D";
   297 
   298 (* [| A = {};  a : A |] ==> R *)
   299 AddDs [equals0D, sym RS equals0D];
   300 
   301 Goalw [Ball_def] "Ball {} P = True";
   302 by (Simp_tac 1);
   303 qed "ball_empty";
   304 
   305 Goalw [Bex_def] "Bex {} P = False";
   306 by (Simp_tac 1);
   307 qed "bex_empty";
   308 Addsimps [ball_empty, bex_empty];
   309 
   310 Goal "UNIV ~= {}";
   311 by (blast_tac (claset() addEs [equalityE]) 1);
   312 qed "UNIV_not_empty";
   313 AddIffs [UNIV_not_empty];
   314 
   315 
   316 
   317 section "The Powerset operator -- Pow";
   318 
   319 Goalw [Pow_def] "(A : Pow(B)) = (A <= B)";
   320 by (Asm_simp_tac 1);
   321 qed "Pow_iff";
   322 
   323 AddIffs [Pow_iff]; 
   324 
   325 Goalw [Pow_def] "A <= B ==> A : Pow(B)";
   326 by (etac CollectI 1);
   327 qed "PowI";
   328 
   329 Goalw [Pow_def] "A : Pow(B)  ==>  A<=B";
   330 by (etac CollectD 1);
   331 qed "PowD";
   332 
   333 
   334 val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)
   335 val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)
   336 
   337 
   338 section "Set complement";
   339 
   340 Goalw [Compl_def] "(c : -A) = (c~:A)";
   341 by (Blast_tac 1);
   342 qed "Compl_iff";
   343 
   344 Addsimps [Compl_iff];
   345 
   346 val prems = Goalw [Compl_def] "[| c:A ==> False |] ==> c : -A";
   347 by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
   348 qed "ComplI";
   349 
   350 (*This form, with negated conclusion, works well with the Classical prover.
   351   Negated assumptions behave like formulae on the right side of the notional
   352   turnstile...*)
   353 Goalw [Compl_def] "c : -A ==> c~:A";
   354 by (etac CollectD 1);
   355 qed "ComplD";
   356 
   357 val ComplE = make_elim ComplD;
   358 
   359 AddSIs [ComplI];
   360 AddSEs [ComplE];
   361 
   362 
   363 section "Binary union -- Un";
   364 
   365 Goalw [Un_def] "(c : A Un B) = (c:A | c:B)";
   366 by (Blast_tac 1);
   367 qed "Un_iff";
   368 Addsimps [Un_iff];
   369 
   370 Goal "c:A ==> c : A Un B";
   371 by (Asm_simp_tac 1);
   372 qed "UnI1";
   373 
   374 Goal "c:B ==> c : A Un B";
   375 by (Asm_simp_tac 1);
   376 qed "UnI2";
   377 
   378 (*Classical introduction rule: no commitment to A vs B*)
   379 
   380 val prems = Goal "(c~:B ==> c:A) ==> c : A Un B";
   381 by (Simp_tac 1);
   382 by (REPEAT (ares_tac (prems@[disjCI]) 1)) ;
   383 qed "UnCI";
   384 
   385 val major::prems = Goalw [Un_def]
   386     "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
   387 by (rtac (major RS CollectD RS disjE) 1);
   388 by (REPEAT (eresolve_tac prems 1));
   389 qed "UnE";
   390 
   391 AddSIs [UnCI];
   392 AddSEs [UnE];
   393 
   394 
   395 section "Binary intersection -- Int";
   396 
   397 Goalw [Int_def] "(c : A Int B) = (c:A & c:B)";
   398 by (Blast_tac 1);
   399 qed "Int_iff";
   400 Addsimps [Int_iff];
   401 
   402 Goal "[| c:A;  c:B |] ==> c : A Int B";
   403 by (Asm_simp_tac 1);
   404 qed "IntI";
   405 
   406 Goal "c : A Int B ==> c:A";
   407 by (Asm_full_simp_tac 1);
   408 qed "IntD1";
   409 
   410 Goal "c : A Int B ==> c:B";
   411 by (Asm_full_simp_tac 1);
   412 qed "IntD2";
   413 
   414 val [major,minor] = Goal
   415     "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
   416 by (rtac minor 1);
   417 by (rtac (major RS IntD1) 1);
   418 by (rtac (major RS IntD2) 1);
   419 qed "IntE";
   420 
   421 AddSIs [IntI];
   422 AddSEs [IntE];
   423 
   424 section "Set difference";
   425 
   426 Goalw [set_diff_def] "(c : A-B) = (c:A & c~:B)";
   427 by (Blast_tac 1);
   428 qed "Diff_iff";
   429 Addsimps [Diff_iff];
   430 
   431 Goal "[| c : A;  c ~: B |] ==> c : A - B";
   432 by (Asm_simp_tac 1) ;
   433 qed "DiffI";
   434 
   435 Goal "c : A - B ==> c : A";
   436 by (Asm_full_simp_tac 1) ;
   437 qed "DiffD1";
   438 
   439 Goal "[| c : A - B;  c : B |] ==> P";
   440 by (Asm_full_simp_tac 1) ;
   441 qed "DiffD2";
   442 
   443 val prems = Goal "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P";
   444 by (resolve_tac prems 1);
   445 by (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ;
   446 qed "DiffE";
   447 
   448 AddSIs [DiffI];
   449 AddSEs [DiffE];
   450 
   451 
   452 section "Augmenting a set -- insert";
   453 
   454 Goalw [insert_def] "a : insert b A = (a=b | a:A)";
   455 by (Blast_tac 1);
   456 qed "insert_iff";
   457 Addsimps [insert_iff];
   458 
   459 Goal "a : insert a B";
   460 by (Simp_tac 1);
   461 qed "insertI1";
   462 
   463 Goal "!!a. a : B ==> a : insert b B";
   464 by (Asm_simp_tac 1);
   465 qed "insertI2";
   466 
   467 val major::prems = Goalw [insert_def]
   468     "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P";
   469 by (rtac (major RS UnE) 1);
   470 by (REPEAT (eresolve_tac (prems @ [CollectE]) 1));
   471 qed "insertE";
   472 
   473 (*Classical introduction rule*)
   474 val prems = Goal "(a~:B ==> a=b) ==> a: insert b B";
   475 by (Simp_tac 1);
   476 by (REPEAT (ares_tac (prems@[disjCI]) 1)) ;
   477 qed "insertCI";
   478 
   479 AddSIs [insertCI]; 
   480 AddSEs [insertE];
   481 
   482 Goal "A <= insert x B ==> A <= B & x ~: A | (EX B'. A = insert x B' & B' <= B)";
   483 by (case_tac "x:A" 1);
   484 by  (Fast_tac 2);
   485 by (rtac disjI2 1);
   486 by (res_inst_tac [("x","A-{x}")] exI 1);
   487 by (Fast_tac 1);
   488 qed "subset_insertD";
   489 
   490 section "Singletons, using insert";
   491 
   492 Goal "a : {a}";
   493 by (rtac insertI1 1) ;
   494 qed "singletonI";
   495 
   496 Goal "b : {a} ==> b=a";
   497 by (Blast_tac 1);
   498 qed "singletonD";
   499 
   500 bind_thm ("singletonE", make_elim singletonD);
   501 
   502 Goal "(b : {a}) = (b=a)";
   503 by (Blast_tac 1);
   504 qed "singleton_iff";
   505 
   506 Goal "{a}={b} ==> a=b";
   507 by (blast_tac (claset() addEs [equalityE]) 1);
   508 qed "singleton_inject";
   509 
   510 (*Redundant? But unlike insertCI, it proves the subgoal immediately!*)
   511 AddSIs [singletonI];   
   512 AddSDs [singleton_inject];
   513 AddSEs [singletonE];
   514 
   515 Goal "{b} = insert a A = (a = b & A <= {b})";
   516 by (blast_tac (claset() addSEs [equalityE]) 1);
   517 qed "singleton_insert_inj_eq";
   518 
   519 Goal "(insert a A = {b}) = (a = b & A <= {b})";
   520 by (blast_tac (claset() addSEs [equalityE]) 1);
   521 qed "singleton_insert_inj_eq'";
   522 
   523 AddIffs [singleton_insert_inj_eq, singleton_insert_inj_eq'];
   524 
   525 Goal "A <= {x} ==> A={} | A = {x}";
   526 by (Fast_tac 1);
   527 qed "subset_singletonD";
   528 
   529 Goal "{x. x=a} = {a}";
   530 by (Blast_tac 1);
   531 qed "singleton_conv";
   532 Addsimps [singleton_conv];
   533 
   534 Goal "{x. a=x} = {a}";
   535 by (Blast_tac 1);
   536 qed "singleton_conv2";
   537 Addsimps [singleton_conv2];
   538 
   539 
   540 section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
   541 
   542 Goalw [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
   543 by (Blast_tac 1);
   544 qed "UN_iff";
   545 
   546 Addsimps [UN_iff];
   547 
   548 (*The order of the premises presupposes that A is rigid; b may be flexible*)
   549 Goal "[| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
   550 by Auto_tac;
   551 qed "UN_I";
   552 
   553 val major::prems = Goalw [UNION_def]
   554     "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
   555 by (rtac (major RS CollectD RS bexE) 1);
   556 by (REPEAT (ares_tac prems 1));
   557 qed "UN_E";
   558 
   559 AddIs  [UN_I];
   560 AddSEs [UN_E];
   561 
   562 val prems = Goalw [UNION_def]
   563     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
   564 \    (UN x:A. C(x)) = (UN x:B. D(x))";
   565 by (asm_simp_tac (simpset() addsimps prems) 1);
   566 qed "UN_cong";
   567 
   568 
   569 section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
   570 
   571 Goalw [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
   572 by Auto_tac;
   573 qed "INT_iff";
   574 
   575 Addsimps [INT_iff];
   576 
   577 val prems = Goalw [INTER_def]
   578     "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
   579 by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
   580 qed "INT_I";
   581 
   582 Goal "[| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
   583 by Auto_tac;
   584 qed "INT_D";
   585 
   586 (*"Classical" elimination -- by the Excluded Middle on a:A *)
   587 val major::prems = Goalw [INTER_def]
   588     "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
   589 by (rtac (major RS CollectD RS ballE) 1);
   590 by (REPEAT (eresolve_tac prems 1));
   591 qed "INT_E";
   592 
   593 AddSIs [INT_I];
   594 AddEs  [INT_D, INT_E];
   595 
   596 val prems = Goalw [INTER_def]
   597     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
   598 \    (INT x:A. C(x)) = (INT x:B. D(x))";
   599 by (asm_simp_tac (simpset() addsimps prems) 1);
   600 qed "INT_cong";
   601 
   602 
   603 section "Union";
   604 
   605 Goalw [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
   606 by (Blast_tac 1);
   607 qed "Union_iff";
   608 
   609 Addsimps [Union_iff];
   610 
   611 (*The order of the premises presupposes that C is rigid; A may be flexible*)
   612 Goal "[| X:C;  A:X |] ==> A : Union(C)";
   613 by Auto_tac;
   614 qed "UnionI";
   615 
   616 val major::prems = Goalw [Union_def]
   617     "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
   618 by (rtac (major RS UN_E) 1);
   619 by (REPEAT (ares_tac prems 1));
   620 qed "UnionE";
   621 
   622 AddIs  [UnionI];
   623 AddSEs [UnionE];
   624 
   625 
   626 section "Inter";
   627 
   628 Goalw [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
   629 by (Blast_tac 1);
   630 qed "Inter_iff";
   631 
   632 Addsimps [Inter_iff];
   633 
   634 val prems = Goalw [Inter_def]
   635     "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
   636 by (REPEAT (ares_tac ([INT_I] @ prems) 1));
   637 qed "InterI";
   638 
   639 (*A "destruct" rule -- every X in C contains A as an element, but
   640   A:X can hold when X:C does not!  This rule is analogous to "spec". *)
   641 Goal "[| A : Inter(C);  X:C |] ==> A:X";
   642 by Auto_tac;
   643 qed "InterD";
   644 
   645 (*"Classical" elimination rule -- does not require proving X:C *)
   646 val major::prems = Goalw [Inter_def]
   647     "[| A : Inter(C);  X~:C ==> R;  A:X ==> R |] ==> R";
   648 by (rtac (major RS INT_E) 1);
   649 by (REPEAT (eresolve_tac prems 1));
   650 qed "InterE";
   651 
   652 AddSIs [InterI];
   653 AddEs  [InterD, InterE];
   654 
   655 
   656 (*** Image of a set under a function ***)
   657 
   658 (*Frequently b does not have the syntactic form of f(x).*)
   659 Goalw [image_def] "[| b=f(x);  x:A |] ==> b : f``A";
   660 by (Blast_tac 1);
   661 qed "image_eqI";
   662 Addsimps [image_eqI];
   663 
   664 bind_thm ("imageI", refl RS image_eqI);
   665 
   666 (*This version's more effective when we already have the required x*)
   667 Goalw [image_def] "[| x:A;  b=f(x) |] ==> b : f``A";
   668 by (Blast_tac 1);
   669 qed "rev_image_eqI";
   670 
   671 (*The eta-expansion gives variable-name preservation.*)
   672 val major::prems = Goalw [image_def]
   673     "[| b : (%x. f(x))``A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P"; 
   674 by (rtac (major RS CollectD RS bexE) 1);
   675 by (REPEAT (ares_tac prems 1));
   676 qed "imageE";
   677 
   678 AddIs  [image_eqI];
   679 AddSEs [imageE]; 
   680 
   681 Goal "f``(A Un B) = f``A Un f``B";
   682 by (Blast_tac 1);
   683 qed "image_Un";
   684 
   685 Goal "(z : f``A) = (EX x:A. z = f x)";
   686 by (Blast_tac 1);
   687 qed "image_iff";
   688 
   689 (*This rewrite rule would confuse users if made default.*)
   690 Goal "(f``A <= B) = (ALL x:A. f(x): B)";
   691 by (Blast_tac 1);
   692 qed "image_subset_iff";
   693 
   694 (*Replaces the three steps subsetI, imageE, hyp_subst_tac, but breaks too
   695   many existing proofs.*)
   696 val prems = Goal "(!!x. x:A ==> f(x) : B) ==> f``A <= B";
   697 by (blast_tac (claset() addIs prems) 1);
   698 qed "image_subsetI";
   699 
   700 
   701 (*** Range of a function -- just a translation for image! ***)
   702 
   703 Goal "b=f(x) ==> b : range(f)";
   704 by (EVERY1 [etac image_eqI, rtac UNIV_I]);
   705 bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI));
   706 
   707 bind_thm ("rangeI", UNIV_I RS imageI);
   708 
   709 val [major,minor] = Goal 
   710     "[| b : range(%x. f(x));  !!x. b=f(x) ==> P |] ==> P"; 
   711 by (rtac (major RS imageE) 1);
   712 by (etac minor 1);
   713 qed "rangeE";
   714 
   715 
   716 (*** Set reasoning tools ***)
   717 
   718 
   719 (** Rewrite rules for boolean case-splitting: faster than 
   720 	addsplits[split_if]
   721 **)
   722 
   723 bind_thm ("split_if_eq1", read_instantiate [("P", "%x. x = ?b")] split_if);
   724 bind_thm ("split_if_eq2", read_instantiate [("P", "%x. ?a = x")] split_if);
   725 
   726 (*Split ifs on either side of the membership relation.
   727 	Not for Addsimps -- can cause goals to blow up!*)
   728 bind_thm ("split_if_mem1", 
   729     read_instantiate_sg (Theory.sign_of Set.thy) [("P", "%x. x : ?b")] split_if);
   730 bind_thm ("split_if_mem2", 
   731     read_instantiate_sg (Theory.sign_of Set.thy) [("P", "%x. ?a : x")] split_if);
   732 
   733 val split_ifs = [if_bool_eq_conj, split_if_eq1, split_if_eq2,
   734 		  split_if_mem1, split_if_mem2];
   735 
   736 
   737 (*Each of these has ALREADY been added to simpset() above.*)
   738 val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff, 
   739                  mem_Collect_eq, UN_iff, Union_iff, INT_iff, Inter_iff];
   740 
   741 (*Would like to add these, but the existing code only searches for the 
   742   outer-level constant, which in this case is just "op :"; we instead need
   743   to use term-nets to associate patterns with rules.  Also, if a rule fails to
   744   apply, then the formula should be kept.
   745   [("uminus", Compl_iff RS iffD1), ("op -", [Diff_iff RS iffD1]), 
   746    ("op Int", [IntD1,IntD2]),
   747    ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
   748  *)
   749 val mksimps_pairs =
   750   [("Ball",[bspec])] @ mksimps_pairs;
   751 
   752 simpset_ref() := simpset() setmksimps (mksimps mksimps_pairs);
   753 
   754 Addsimps[subset_UNIV, subset_refl];
   755 
   756 
   757 (*** The 'proper subset' relation (<) ***)
   758 
   759 Goalw [psubset_def] "!!A::'a set. [| A <= B; A ~= B |] ==> A<B";
   760 by (Blast_tac 1);
   761 qed "psubsetI";
   762 AddSIs [psubsetI];
   763 
   764 Goalw [psubset_def] "A < insert x B ==> (x ~: A) & A<=B | x:A & A-{x}<B";
   765 by Auto_tac;
   766 qed "psubset_insertD";
   767 
   768 bind_thm ("psubset_eq", psubset_def RS meta_eq_to_obj_eq);
   769 
   770 bind_thm ("psubset_imp_subset", psubset_eq RS iffD1 RS conjunct1);
   771 
   772 Goal"[| (A::'a set) < B; B <= C |] ==> A < C";
   773 by (auto_tac (claset(), simpset() addsimps [psubset_eq]));
   774 qed "psubset_subset_trans";
   775 
   776 Goal"[| (A::'a set) <= B; B < C|] ==> A < C";
   777 by (auto_tac (claset(), simpset() addsimps [psubset_eq]));
   778 qed "subset_psubset_trans";
   779 
   780 Goalw [psubset_def] "A < B ==> EX b. b : (B - A)";
   781 by (Blast_tac 1);
   782 qed "psubset_imp_ex_mem";
   783 
   784 
   785 (* attributes *)
   786 
   787 local
   788 
   789 fun gen_rulify_prems x =
   790   Attrib.no_args (Drule.rule_attribute (fn _ => (standard o
   791     rule_by_tactic (REPEAT (ALLGOALS (resolve_tac [allI, ballI, impI])))))) x;
   792 
   793 in
   794 
   795 val rulify_prems_attrib_setup =
   796  [Attrib.add_attributes
   797   [("rulify_prems", (gen_rulify_prems, gen_rulify_prems), "put theorem into standard rule form")]];
   798 
   799 end;