src/HOL/Set.ML
author paulson
Thu Nov 20 10:54:04 1997 +0100 (1997-11-20)
changeset 4240 8ba60a4cd380
parent 4231 a73f5a63f197
child 4423 a129b817b58a
permissions -rw-r--r--
Renamed "overload" to "overloaded" for sml/nj compatibility
     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 open Set;
    10 
    11 section "Relating predicates and sets";
    12 
    13 Addsimps [Collect_mem_eq];
    14 AddIffs  [mem_Collect_eq];
    15 
    16 goal Set.thy "!!a. P(a) ==> a : {x. P(x)}";
    17 by (Asm_simp_tac 1);
    18 qed "CollectI";
    19 
    20 val prems = goal Set.thy "!!a. a : {x. P(x)} ==> P(a)";
    21 by (Asm_full_simp_tac 1);
    22 qed "CollectD";
    23 
    24 val [prem] = goal Set.thy "[| !!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 Set.thy "[| !!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 Set.thy [Ball_def]
    43     "[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)";
    44 by (REPEAT (ares_tac (prems @ [allI,impI]) 1));
    45 qed "ballI";
    46 
    47 val [major,minor] = goalw Set.thy [Ball_def]
    48     "[| ! x:A. P(x);  x:A |] ==> P(x)";
    49 by (rtac (minor RS (major RS spec RS mp)) 1);
    50 qed "bspec";
    51 
    52 val major::prems = goalw Set.thy [Ball_def]
    53     "[| ! x:A. P(x);  P(x) ==> Q;  x~:A ==> Q |] ==> Q";
    54 by (rtac (major RS spec RS impCE) 1);
    55 by (REPEAT (eresolve_tac prems 1));
    56 qed "ballE";
    57 
    58 (*Takes assumptions ! x:A.P(x) and a:A; creates assumption P(a)*)
    59 fun ball_tac i = etac ballE i THEN contr_tac (i+1);
    60 
    61 AddSIs [ballI];
    62 AddEs  [ballE];
    63 
    64 val prems = goalw Set.thy [Bex_def]
    65     "[| P(x);  x:A |] ==> ? x:A. P(x)";
    66 by (REPEAT (ares_tac (prems @ [exI,conjI]) 1));
    67 qed "bexI";
    68 
    69 qed_goal "bexCI" Set.thy 
    70    "[| ! x:A. ~P(x) ==> P(a);  a:A |] ==> ? x:A. P(x)"
    71  (fn prems=>
    72   [ (rtac classical 1),
    73     (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1))  ]);
    74 
    75 val major::prems = goalw Set.thy [Bex_def]
    76     "[| ? x:A. P(x);  !!x. [| x:A; P(x) |] ==> Q  |] ==> Q";
    77 by (rtac (major RS exE) 1);
    78 by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1));
    79 qed "bexE";
    80 
    81 AddIs  [bexI];
    82 AddSEs [bexE];
    83 
    84 (*Trival rewrite rule*)
    85 goal Set.thy "(! x:A. P) = ((? x. x:A) --> P)";
    86 by (simp_tac (simpset() addsimps [Ball_def]) 1);
    87 qed "ball_triv";
    88 
    89 (*Dual form for existentials*)
    90 goal Set.thy "(? x:A. P) = ((? x. x:A) & P)";
    91 by (simp_tac (simpset() addsimps [Bex_def]) 1);
    92 qed "bex_triv";
    93 
    94 Addsimps [ball_triv, bex_triv];
    95 
    96 (** Congruence rules **)
    97 
    98 val prems = goal Set.thy
    99     "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
   100 \    (! x:A. P(x)) = (! x:B. Q(x))";
   101 by (resolve_tac (prems RL [ssubst]) 1);
   102 by (REPEAT (ares_tac [ballI,iffI] 1
   103      ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1));
   104 qed "ball_cong";
   105 
   106 val prems = goal Set.thy
   107     "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
   108 \    (? x:A. P(x)) = (? x:B. Q(x))";
   109 by (resolve_tac (prems RL [ssubst]) 1);
   110 by (REPEAT (etac bexE 1
   111      ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1));
   112 qed "bex_cong";
   113 
   114 section "Subsets";
   115 
   116 val prems = goalw Set.thy [subset_def] "(!!x. x:A ==> x:B) ==> A <= B";
   117 by (REPEAT (ares_tac (prems @ [ballI]) 1));
   118 qed "subsetI";
   119 
   120 Blast.overloaded ("op <=", domain_type); (*The <= relation is overloaded*)
   121 
   122 (*While (:) is not, its type must be kept
   123   for overloading of = to work.*)
   124 Blast.overloaded ("op :", domain_type);
   125 seq (fn a => Blast.overloaded (a, HOLogic.dest_setT o domain_type))
   126     ["Ball", "Bex"];
   127 (*need UNION, INTER also?*)
   128 
   129 
   130 (*Rule in Modus Ponens style*)
   131 val major::prems = goalw Set.thy [subset_def] "[| A <= B;  c:A |] ==> c:B";
   132 by (rtac (major RS bspec) 1);
   133 by (resolve_tac prems 1);
   134 qed "subsetD";
   135 
   136 (*The same, with reversed premises for use with etac -- cf rev_mp*)
   137 qed_goal "rev_subsetD" Set.thy "[| c:A;  A <= B |] ==> c:B"
   138  (fn prems=>  [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]);
   139 
   140 (*Converts A<=B to x:A ==> x:B*)
   141 fun impOfSubs th = th RSN (2, rev_subsetD);
   142 
   143 qed_goal "contra_subsetD" Set.thy "!!c. [| A <= B; c ~: B |] ==> c ~: A"
   144  (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
   145 
   146 qed_goal "rev_contra_subsetD" Set.thy "!!c. [| c ~: B;  A <= B |] ==> c ~: A"
   147  (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
   148 
   149 (*Classical elimination rule*)
   150 val major::prems = goalw Set.thy [subset_def] 
   151     "[| A <= B;  c~:A ==> P;  c:B ==> P |] ==> P";
   152 by (rtac (major RS ballE) 1);
   153 by (REPEAT (eresolve_tac prems 1));
   154 qed "subsetCE";
   155 
   156 (*Takes assumptions A<=B; c:A and creates the assumption c:B *)
   157 fun set_mp_tac i = etac subsetCE i  THEN  mp_tac i;
   158 
   159 AddSIs [subsetI];
   160 AddEs  [subsetD, subsetCE];
   161 
   162 qed_goal "subset_refl" Set.thy "A <= (A::'a set)"
   163  (fn _=> [Fast_tac 1]);		(*Blast_tac would try order_refl and fail*)
   164 
   165 val prems = goal Set.thy "!!B. [| A<=B;  B<=C |] ==> A<=(C::'a set)";
   166 by (Blast_tac 1);
   167 qed "subset_trans";
   168 
   169 
   170 section "Equality";
   171 
   172 (*Anti-symmetry of the subset relation*)
   173 val prems = goal Set.thy "[| A <= B;  B <= A |] ==> A = (B::'a set)";
   174 by (rtac (iffI RS set_ext) 1);
   175 by (REPEAT (ares_tac (prems RL [subsetD]) 1));
   176 qed "subset_antisym";
   177 val equalityI = subset_antisym;
   178 
   179 AddSIs [equalityI];
   180 
   181 (* Equality rules from ZF set theory -- are they appropriate here? *)
   182 val prems = goal Set.thy "A = B ==> A<=(B::'a set)";
   183 by (resolve_tac (prems RL [subst]) 1);
   184 by (rtac subset_refl 1);
   185 qed "equalityD1";
   186 
   187 val prems = goal Set.thy "A = B ==> B<=(A::'a set)";
   188 by (resolve_tac (prems RL [subst]) 1);
   189 by (rtac subset_refl 1);
   190 qed "equalityD2";
   191 
   192 val prems = goal Set.thy
   193     "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";
   194 by (resolve_tac prems 1);
   195 by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
   196 qed "equalityE";
   197 
   198 val major::prems = goal Set.thy
   199     "[| A = B;  [| c:A; c:B |] ==> P;  [| c~:A; c~:B |] ==> P |]  ==>  P";
   200 by (rtac (major RS equalityE) 1);
   201 by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
   202 qed "equalityCE";
   203 
   204 (*Lemma for creating induction formulae -- for "pattern matching" on p
   205   To make the induction hypotheses usable, apply "spec" or "bspec" to
   206   put universal quantifiers over the free variables in p. *)
   207 val prems = goal Set.thy 
   208     "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
   209 by (rtac mp 1);
   210 by (REPEAT (resolve_tac (refl::prems) 1));
   211 qed "setup_induction";
   212 
   213 
   214 section "The universal set -- UNIV";
   215 
   216 qed_goalw "UNIV_I" Set.thy [UNIV_def] "x : UNIV"
   217   (fn _ => [rtac CollectI 1, rtac TrueI 1]);
   218 
   219 AddIffs [UNIV_I];
   220 
   221 qed_goal "subset_UNIV" Set.thy "A <= UNIV"
   222   (fn _ => [rtac subsetI 1, rtac UNIV_I 1]);
   223 
   224 (** Eta-contracting these two rules (to remove P) causes them to be ignored
   225     because of their interaction with congruence rules. **)
   226 
   227 goalw Set.thy [Ball_def] "Ball UNIV P = All P";
   228 by (Simp_tac 1);
   229 qed "ball_UNIV";
   230 
   231 goalw Set.thy [Bex_def] "Bex UNIV P = Ex P";
   232 by (Simp_tac 1);
   233 qed "bex_UNIV";
   234 Addsimps [ball_UNIV, bex_UNIV];
   235 
   236 
   237 section "The empty set -- {}";
   238 
   239 qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False"
   240  (fn _ => [ (Blast_tac 1) ]);
   241 
   242 Addsimps [empty_iff];
   243 
   244 qed_goal "emptyE" Set.thy "!!a. a:{} ==> P"
   245  (fn _ => [Full_simp_tac 1]);
   246 
   247 AddSEs [emptyE];
   248 
   249 qed_goal "empty_subsetI" Set.thy "{} <= A"
   250  (fn _ => [ (Blast_tac 1) ]);
   251 
   252 qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}"
   253  (fn [prem]=>
   254   [ (blast_tac (claset() addIs [prem RS FalseE]) 1) ]);
   255 
   256 qed_goal "equals0D" Set.thy "!!a. [| A={};  a:A |] ==> P"
   257  (fn _ => [ (Blast_tac 1) ]);
   258 
   259 goalw Set.thy [Ball_def] "Ball {} P = True";
   260 by (Simp_tac 1);
   261 qed "ball_empty";
   262 
   263 goalw Set.thy [Bex_def] "Bex {} P = False";
   264 by (Simp_tac 1);
   265 qed "bex_empty";
   266 Addsimps [ball_empty, bex_empty];
   267 
   268 goal thy "UNIV ~= {}";
   269 by (blast_tac (claset() addEs [equalityE]) 1);
   270 qed "UNIV_not_empty";
   271 AddIffs [UNIV_not_empty];
   272 
   273 
   274 
   275 section "The Powerset operator -- Pow";
   276 
   277 qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)"
   278  (fn _ => [ (Asm_simp_tac 1) ]);
   279 
   280 AddIffs [Pow_iff]; 
   281 
   282 qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
   283  (fn _ => [ (etac CollectI 1) ]);
   284 
   285 qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B)  ==>  A<=B"
   286  (fn _=> [ (etac CollectD 1) ]);
   287 
   288 val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)
   289 val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)
   290 
   291 
   292 section "Set complement -- Compl";
   293 
   294 qed_goalw "Compl_iff" Set.thy [Compl_def] "(c : Compl(A)) = (c~:A)"
   295  (fn _ => [ (Blast_tac 1) ]);
   296 
   297 Addsimps [Compl_iff];
   298 
   299 val prems = goalw Set.thy [Compl_def]
   300     "[| c:A ==> False |] ==> c : Compl(A)";
   301 by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
   302 qed "ComplI";
   303 
   304 (*This form, with negated conclusion, works well with the Classical prover.
   305   Negated assumptions behave like formulae on the right side of the notional
   306   turnstile...*)
   307 val major::prems = goalw Set.thy [Compl_def]
   308     "c : Compl(A) ==> c~:A";
   309 by (rtac (major RS CollectD) 1);
   310 qed "ComplD";
   311 
   312 val ComplE = make_elim ComplD;
   313 
   314 AddSIs [ComplI];
   315 AddSEs [ComplE];
   316 
   317 
   318 section "Binary union -- Un";
   319 
   320 qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)"
   321  (fn _ => [ Blast_tac 1 ]);
   322 
   323 Addsimps [Un_iff];
   324 
   325 goal Set.thy "!!c. c:A ==> c : A Un B";
   326 by (Asm_simp_tac 1);
   327 qed "UnI1";
   328 
   329 goal Set.thy "!!c. c:B ==> c : A Un B";
   330 by (Asm_simp_tac 1);
   331 qed "UnI2";
   332 
   333 (*Classical introduction rule: no commitment to A vs B*)
   334 qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B"
   335  (fn prems=>
   336   [ (Simp_tac 1),
   337     (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
   338 
   339 val major::prems = goalw Set.thy [Un_def]
   340     "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
   341 by (rtac (major RS CollectD RS disjE) 1);
   342 by (REPEAT (eresolve_tac prems 1));
   343 qed "UnE";
   344 
   345 AddSIs [UnCI];
   346 AddSEs [UnE];
   347 
   348 
   349 section "Binary intersection -- Int";
   350 
   351 qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)"
   352  (fn _ => [ (Blast_tac 1) ]);
   353 
   354 Addsimps [Int_iff];
   355 
   356 goal Set.thy "!!c. [| c:A;  c:B |] ==> c : A Int B";
   357 by (Asm_simp_tac 1);
   358 qed "IntI";
   359 
   360 goal Set.thy "!!c. c : A Int B ==> c:A";
   361 by (Asm_full_simp_tac 1);
   362 qed "IntD1";
   363 
   364 goal Set.thy "!!c. c : A Int B ==> c:B";
   365 by (Asm_full_simp_tac 1);
   366 qed "IntD2";
   367 
   368 val [major,minor] = goal Set.thy
   369     "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
   370 by (rtac minor 1);
   371 by (rtac (major RS IntD1) 1);
   372 by (rtac (major RS IntD2) 1);
   373 qed "IntE";
   374 
   375 AddSIs [IntI];
   376 AddSEs [IntE];
   377 
   378 section "Set difference";
   379 
   380 qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)"
   381  (fn _ => [ (Blast_tac 1) ]);
   382 
   383 Addsimps [Diff_iff];
   384 
   385 qed_goal "DiffI" Set.thy "!!c. [| c : A;  c ~: B |] ==> c : A - B"
   386  (fn _=> [ Asm_simp_tac 1 ]);
   387 
   388 qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A"
   389  (fn _=> [ (Asm_full_simp_tac 1) ]);
   390 
   391 qed_goal "DiffD2" Set.thy "!!c. [| c : A - B;  c : B |] ==> P"
   392  (fn _=> [ (Asm_full_simp_tac 1) ]);
   393 
   394 qed_goal "DiffE" Set.thy "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P"
   395  (fn prems=>
   396   [ (resolve_tac prems 1),
   397     (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]);
   398 
   399 AddSIs [DiffI];
   400 AddSEs [DiffE];
   401 
   402 
   403 section "Augmenting a set -- insert";
   404 
   405 qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)"
   406  (fn _ => [Blast_tac 1]);
   407 
   408 Addsimps [insert_iff];
   409 
   410 qed_goal "insertI1" Set.thy "a : insert a B"
   411  (fn _ => [Simp_tac 1]);
   412 
   413 qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B"
   414  (fn _=> [Asm_simp_tac 1]);
   415 
   416 qed_goalw "insertE" Set.thy [insert_def]
   417     "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P"
   418  (fn major::prems=>
   419   [ (rtac (major RS UnE) 1),
   420     (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
   421 
   422 (*Classical introduction rule*)
   423 qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
   424  (fn prems=>
   425   [ (Simp_tac 1),
   426     (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
   427 
   428 AddSIs [insertCI]; 
   429 AddSEs [insertE];
   430 
   431 section "Singletons, using insert";
   432 
   433 qed_goal "singletonI" Set.thy "a : {a}"
   434  (fn _=> [ (rtac insertI1 1) ]);
   435 
   436 goal Set.thy "!!a. b : {a} ==> b=a";
   437 by (Blast_tac 1);
   438 qed "singletonD";
   439 
   440 bind_thm ("singletonE", make_elim singletonD);
   441 
   442 qed_goal "singleton_iff" thy "(b : {a}) = (b=a)" 
   443 (fn _ => [Blast_tac 1]);
   444 
   445 goal Set.thy "!!a b. {a}={b} ==> a=b";
   446 by (blast_tac (claset() addEs [equalityE]) 1);
   447 qed "singleton_inject";
   448 
   449 (*Redundant? But unlike insertCI, it proves the subgoal immediately!*)
   450 AddSIs [singletonI];   
   451 AddSDs [singleton_inject];
   452 AddSEs [singletonE];
   453 
   454 goal Set.thy "{x. x=a} = {a}";
   455 by(Blast_tac 1);
   456 qed "singleton_conv";
   457 Addsimps [singleton_conv];
   458 
   459 
   460 section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
   461 
   462 goalw Set.thy [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
   463 by (Blast_tac 1);
   464 qed "UN_iff";
   465 
   466 Addsimps [UN_iff];
   467 
   468 (*The order of the premises presupposes that A is rigid; b may be flexible*)
   469 goal Set.thy "!!b. [| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
   470 by (Auto_tac());
   471 qed "UN_I";
   472 
   473 val major::prems = goalw Set.thy [UNION_def]
   474     "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
   475 by (rtac (major RS CollectD RS bexE) 1);
   476 by (REPEAT (ares_tac prems 1));
   477 qed "UN_E";
   478 
   479 AddIs  [UN_I];
   480 AddSEs [UN_E];
   481 
   482 val prems = goal Set.thy
   483     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
   484 \    (UN x:A. C(x)) = (UN x:B. D(x))";
   485 by (REPEAT (etac UN_E 1
   486      ORELSE ares_tac ([UN_I,equalityI,subsetI] @ 
   487                       (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
   488 qed "UN_cong";
   489 
   490 
   491 section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
   492 
   493 goalw Set.thy [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
   494 by (Auto_tac());
   495 qed "INT_iff";
   496 
   497 Addsimps [INT_iff];
   498 
   499 val prems = goalw Set.thy [INTER_def]
   500     "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
   501 by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
   502 qed "INT_I";
   503 
   504 goal Set.thy "!!b. [| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
   505 by (Auto_tac());
   506 qed "INT_D";
   507 
   508 (*"Classical" elimination -- by the Excluded Middle on a:A *)
   509 val major::prems = goalw Set.thy [INTER_def]
   510     "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
   511 by (rtac (major RS CollectD RS ballE) 1);
   512 by (REPEAT (eresolve_tac prems 1));
   513 qed "INT_E";
   514 
   515 AddSIs [INT_I];
   516 AddEs  [INT_D, INT_E];
   517 
   518 val prems = goal Set.thy
   519     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
   520 \    (INT x:A. C(x)) = (INT x:B. D(x))";
   521 by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
   522 by (REPEAT (dtac INT_D 1
   523      ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
   524 qed "INT_cong";
   525 
   526 
   527 section "Union";
   528 
   529 goalw Set.thy [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
   530 by (Blast_tac 1);
   531 qed "Union_iff";
   532 
   533 Addsimps [Union_iff];
   534 
   535 (*The order of the premises presupposes that C is rigid; A may be flexible*)
   536 goal Set.thy "!!X. [| X:C;  A:X |] ==> A : Union(C)";
   537 by (Auto_tac());
   538 qed "UnionI";
   539 
   540 val major::prems = goalw Set.thy [Union_def]
   541     "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
   542 by (rtac (major RS UN_E) 1);
   543 by (REPEAT (ares_tac prems 1));
   544 qed "UnionE";
   545 
   546 AddIs  [UnionI];
   547 AddSEs [UnionE];
   548 
   549 
   550 section "Inter";
   551 
   552 goalw Set.thy [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
   553 by (Blast_tac 1);
   554 qed "Inter_iff";
   555 
   556 Addsimps [Inter_iff];
   557 
   558 val prems = goalw Set.thy [Inter_def]
   559     "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
   560 by (REPEAT (ares_tac ([INT_I] @ prems) 1));
   561 qed "InterI";
   562 
   563 (*A "destruct" rule -- every X in C contains A as an element, but
   564   A:X can hold when X:C does not!  This rule is analogous to "spec". *)
   565 goal Set.thy "!!X. [| A : Inter(C);  X:C |] ==> A:X";
   566 by (Auto_tac());
   567 qed "InterD";
   568 
   569 (*"Classical" elimination rule -- does not require proving X:C *)
   570 val major::prems = goalw Set.thy [Inter_def]
   571     "[| A : Inter(C);  X~:C ==> R;  A:X ==> R |] ==> R";
   572 by (rtac (major RS INT_E) 1);
   573 by (REPEAT (eresolve_tac prems 1));
   574 qed "InterE";
   575 
   576 AddSIs [InterI];
   577 AddEs  [InterD, InterE];
   578 
   579 
   580 (*** Image of a set under a function ***)
   581 
   582 (*Frequently b does not have the syntactic form of f(x).*)
   583 val prems = goalw thy [image_def] "[| b=f(x);  x:A |] ==> b : f``A";
   584 by (REPEAT (resolve_tac (prems @ [CollectI,bexI,prem]) 1));
   585 qed "image_eqI";
   586 Addsimps [image_eqI];
   587 
   588 bind_thm ("imageI", refl RS image_eqI);
   589 
   590 (*The eta-expansion gives variable-name preservation.*)
   591 val major::prems = goalw thy [image_def]
   592     "[| b : (%x. f(x))``A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P"; 
   593 by (rtac (major RS CollectD RS bexE) 1);
   594 by (REPEAT (ares_tac prems 1));
   595 qed "imageE";
   596 
   597 AddIs  [image_eqI];
   598 AddSEs [imageE]; 
   599 
   600 goalw thy [o_def] "(f o g)``r = f``(g``r)";
   601 by (Blast_tac 1);
   602 qed "image_compose";
   603 
   604 goal thy "f``(A Un B) = f``A Un f``B";
   605 by (Blast_tac 1);
   606 qed "image_Un";
   607 
   608 goal Set.thy "(z : f``A) = (EX x:A. z = f x)";
   609 by (Blast_tac 1);
   610 qed "image_iff";
   611 
   612 
   613 (*** Range of a function -- just a translation for image! ***)
   614 
   615 goal thy "!!b. b=f(x) ==> b : range(f)";
   616 by (EVERY1 [etac image_eqI, rtac UNIV_I]);
   617 bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI));
   618 
   619 bind_thm ("rangeI", UNIV_I RS imageI);
   620 
   621 val [major,minor] = goal thy 
   622     "[| b : range(%x. f(x));  !!x. b=f(x) ==> P |] ==> P"; 
   623 by (rtac (major RS imageE) 1);
   624 by (etac minor 1);
   625 qed "rangeE";
   626 
   627 
   628 (*** Set reasoning tools ***)
   629 
   630 
   631 (** Rewrite rules for boolean case-splitting: faster than 
   632 	addsplits[expand_if]
   633 **)
   634 
   635 bind_thm ("expand_if_eq1", read_instantiate [("P", "%x. x = ?b")] expand_if);
   636 bind_thm ("expand_if_eq2", read_instantiate [("P", "%x. ?a = x")] expand_if);
   637 
   638 bind_thm ("expand_if_mem1", 
   639     read_instantiate_sg (sign_of Set.thy) [("P", "%x. x : ?b")] expand_if);
   640 bind_thm ("expand_if_mem2", 
   641     read_instantiate_sg (sign_of Set.thy) [("P", "%x. ?a : x")] expand_if);
   642 
   643 val expand_ifs = [if_bool_eq, expand_if_eq1, expand_if_eq2,
   644 		  expand_if_mem1, expand_if_mem2];
   645 
   646 
   647 (*Each of these has ALREADY been added to simpset() above.*)
   648 val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff, 
   649                  mem_Collect_eq, UN_iff, Union_iff, INT_iff, Inter_iff];
   650 
   651 (*Not for Addsimps -- it can cause goals to blow up!*)
   652 goal Set.thy "(a : (if Q then x else y)) = ((Q --> a:x) & (~Q --> a : y))";
   653 by (simp_tac (simpset() addsplits [expand_if]) 1);
   654 qed "mem_if";
   655 
   656 val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs;
   657 
   658 simpset_ref() := simpset() addcongs [ball_cong,bex_cong]
   659                     setmksimps (mksimps mksimps_pairs);
   660 
   661 Addsimps[subset_UNIV, empty_subsetI, subset_refl];
   662 
   663 
   664 (*** < ***)
   665 
   666 goalw Set.thy [psubset_def] "!!A::'a set. [| A <= B; A ~= B |] ==> A<B";
   667 by (Blast_tac 1);
   668 qed "psubsetI";
   669 
   670 goalw Set.thy [psubset_def]
   671     "!!x. A < insert x B ==> (x ~: A) & A<=B | x:A & A-{x}<B";
   672 by (Auto_tac());
   673 qed "psubset_insertD";
   674 
   675 bind_thm ("psubset_eq", psubset_def RS meta_eq_to_obj_eq);