src/HOL/Set.ML
 author paulson Wed Mar 05 09:59:55 1997 +0100 (1997-03-05) changeset 2721 f08042e18c7d parent 2608 450c9b682a92 child 2858 1f3f5c44e159 permissions -rw-r--r--
New version of InterE, like its ZF counterpart
```     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 AddIffs [mem_Collect_eq];
```
```    14
```
```    15 goal Set.thy "!!a. P(a) ==> a : {x.P(x)}";
```
```    16 by (Asm_simp_tac 1);
```
```    17 qed "CollectI";
```
```    18
```
```    19 val prems = goal Set.thy "!!a. a : {x.P(x)} ==> P(a)";
```
```    20 by (Asm_full_simp_tac 1);
```
```    21 qed "CollectD";
```
```    22
```
```    23 val [prem] = goal Set.thy "[| !!x. (x:A) = (x:B) |] ==> A = B";
```
```    24 by (rtac (prem RS ext RS arg_cong RS box_equals) 1);
```
```    25 by (rtac Collect_mem_eq 1);
```
```    26 by (rtac Collect_mem_eq 1);
```
```    27 qed "set_ext";
```
```    28
```
```    29 val [prem] = goal Set.thy "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}";
```
```    30 by (rtac (prem RS ext RS arg_cong) 1);
```
```    31 qed "Collect_cong";
```
```    32
```
```    33 val CollectE = make_elim CollectD;
```
```    34
```
```    35 AddSIs [CollectI];
```
```    36 AddSEs [CollectE];
```
```    37
```
```    38
```
```    39 section "Bounded quantifiers";
```
```    40
```
```    41 val prems = goalw Set.thy [Ball_def]
```
```    42     "[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)";
```
```    43 by (REPEAT (ares_tac (prems @ [allI,impI]) 1));
```
```    44 qed "ballI";
```
```    45
```
```    46 val [major,minor] = goalw Set.thy [Ball_def]
```
```    47     "[| ! x:A. P(x);  x:A |] ==> P(x)";
```
```    48 by (rtac (minor RS (major RS spec RS mp)) 1);
```
```    49 qed "bspec";
```
```    50
```
```    51 val major::prems = goalw Set.thy [Ball_def]
```
```    52     "[| ! x:A. P(x);  P(x) ==> Q;  x~:A ==> Q |] ==> Q";
```
```    53 by (rtac (major RS spec RS impCE) 1);
```
```    54 by (REPEAT (eresolve_tac prems 1));
```
```    55 qed "ballE";
```
```    56
```
```    57 (*Takes assumptions ! x:A.P(x) and a:A; creates assumption P(a)*)
```
```    58 fun ball_tac i = etac ballE i THEN contr_tac (i+1);
```
```    59
```
```    60 AddSIs [ballI];
```
```    61 AddEs  [ballE];
```
```    62
```
```    63 val prems = goalw Set.thy [Bex_def]
```
```    64     "[| P(x);  x:A |] ==> ? x:A. P(x)";
```
```    65 by (REPEAT (ares_tac (prems @ [exI,conjI]) 1));
```
```    66 qed "bexI";
```
```    67
```
```    68 qed_goal "bexCI" Set.thy
```
```    69    "[| ! x:A. ~P(x) ==> P(a);  a:A |] ==> ? x:A.P(x)"
```
```    70  (fn prems=>
```
```    71   [ (rtac classical 1),
```
```    72     (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1))  ]);
```
```    73
```
```    74 val major::prems = goalw Set.thy [Bex_def]
```
```    75     "[| ? x:A. P(x);  !!x. [| x:A; P(x) |] ==> Q  |] ==> Q";
```
```    76 by (rtac (major RS exE) 1);
```
```    77 by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1));
```
```    78 qed "bexE";
```
```    79
```
```    80 AddIs  [bexI];
```
```    81 AddSEs [bexE];
```
```    82
```
```    83 (*Trival rewrite rule;   (! x:A.P)=P holds only if A is nonempty!*)
```
```    84 goalw Set.thy [Ball_def] "(! x:A. True) = True";
```
```    85 by (Simp_tac 1);
```
```    86 qed "ball_True";
```
```    87
```
```    88 (*Dual form for existentials*)
```
```    89 goalw Set.thy [Bex_def] "(? x:A. False) = False";
```
```    90 by (Simp_tac 1);
```
```    91 qed "bex_False";
```
```    92
```
```    93 Addsimps [ball_True, bex_False];
```
```    94
```
```    95 (** Congruence rules **)
```
```    96
```
```    97 val prems = goal Set.thy
```
```    98     "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
```
```    99 \    (! x:A. P(x)) = (! x:B. Q(x))";
```
```   100 by (resolve_tac (prems RL [ssubst]) 1);
```
```   101 by (REPEAT (ares_tac [ballI,iffI] 1
```
```   102      ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1));
```
```   103 qed "ball_cong";
```
```   104
```
```   105 val prems = goal Set.thy
```
```   106     "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
```
```   107 \    (? x:A. P(x)) = (? x:B. Q(x))";
```
```   108 by (resolve_tac (prems RL [ssubst]) 1);
```
```   109 by (REPEAT (etac bexE 1
```
```   110      ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1));
```
```   111 qed "bex_cong";
```
```   112
```
```   113 section "Subsets";
```
```   114
```
```   115 val prems = goalw Set.thy [subset_def] "(!!x.x:A ==> x:B) ==> A <= B";
```
```   116 by (REPEAT (ares_tac (prems @ [ballI]) 1));
```
```   117 qed "subsetI";
```
```   118
```
```   119 (*Rule in Modus Ponens style*)
```
```   120 val major::prems = goalw Set.thy [subset_def] "[| A <= B;  c:A |] ==> c:B";
```
```   121 by (rtac (major RS bspec) 1);
```
```   122 by (resolve_tac prems 1);
```
```   123 qed "subsetD";
```
```   124
```
```   125 (*The same, with reversed premises for use with etac -- cf rev_mp*)
```
```   126 qed_goal "rev_subsetD" Set.thy "[| c:A;  A <= B |] ==> c:B"
```
```   127  (fn prems=>  [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]);
```
```   128
```
```   129 (*Converts A<=B to x:A ==> x:B*)
```
```   130 fun impOfSubs th = th RSN (2, rev_subsetD);
```
```   131
```
```   132 qed_goal "contra_subsetD" Set.thy "!!c. [| A <= B; c ~: B |] ==> c ~: A"
```
```   133  (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
```
```   134
```
```   135 qed_goal "rev_contra_subsetD" Set.thy "!!c. [| c ~: B;  A <= B |] ==> c ~: A"
```
```   136  (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
```
```   137
```
```   138 (*Classical elimination rule*)
```
```   139 val major::prems = goalw Set.thy [subset_def]
```
```   140     "[| A <= B;  c~:A ==> P;  c:B ==> P |] ==> P";
```
```   141 by (rtac (major RS ballE) 1);
```
```   142 by (REPEAT (eresolve_tac prems 1));
```
```   143 qed "subsetCE";
```
```   144
```
```   145 (*Takes assumptions A<=B; c:A and creates the assumption c:B *)
```
```   146 fun set_mp_tac i = etac subsetCE i  THEN  mp_tac i;
```
```   147
```
```   148 AddSIs [subsetI];
```
```   149 AddEs  [subsetD, subsetCE];
```
```   150
```
```   151 qed_goal "subset_refl" Set.thy "A <= (A::'a set)"
```
```   152  (fn _=> [Fast_tac 1]);
```
```   153
```
```   154 val prems = goal Set.thy "!!B. [| A<=B;  B<=C |] ==> A<=(C::'a set)";
```
```   155 by (Fast_tac 1);
```
```   156 qed "subset_trans";
```
```   157
```
```   158
```
```   159 section "Equality";
```
```   160
```
```   161 (*Anti-symmetry of the subset relation*)
```
```   162 val prems = goal Set.thy "[| A <= B;  B <= A |] ==> A = (B::'a set)";
```
```   163 by (rtac (iffI RS set_ext) 1);
```
```   164 by (REPEAT (ares_tac (prems RL [subsetD]) 1));
```
```   165 qed "subset_antisym";
```
```   166 val equalityI = subset_antisym;
```
```   167
```
```   168 AddSIs [equalityI];
```
```   169
```
```   170 (* Equality rules from ZF set theory -- are they appropriate here? *)
```
```   171 val prems = goal Set.thy "A = B ==> A<=(B::'a set)";
```
```   172 by (resolve_tac (prems RL [subst]) 1);
```
```   173 by (rtac subset_refl 1);
```
```   174 qed "equalityD1";
```
```   175
```
```   176 val prems = goal Set.thy "A = B ==> B<=(A::'a set)";
```
```   177 by (resolve_tac (prems RL [subst]) 1);
```
```   178 by (rtac subset_refl 1);
```
```   179 qed "equalityD2";
```
```   180
```
```   181 val prems = goal Set.thy
```
```   182     "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";
```
```   183 by (resolve_tac prems 1);
```
```   184 by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
```
```   185 qed "equalityE";
```
```   186
```
```   187 val major::prems = goal Set.thy
```
```   188     "[| A = B;  [| c:A; c:B |] ==> P;  [| c~:A; c~:B |] ==> P |]  ==>  P";
```
```   189 by (rtac (major RS equalityE) 1);
```
```   190 by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
```
```   191 qed "equalityCE";
```
```   192
```
```   193 (*Lemma for creating induction formulae -- for "pattern matching" on p
```
```   194   To make the induction hypotheses usable, apply "spec" or "bspec" to
```
```   195   put universal quantifiers over the free variables in p. *)
```
```   196 val prems = goal Set.thy
```
```   197     "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
```
```   198 by (rtac mp 1);
```
```   199 by (REPEAT (resolve_tac (refl::prems) 1));
```
```   200 qed "setup_induction";
```
```   201
```
```   202
```
```   203 section "Set complement -- Compl";
```
```   204
```
```   205 qed_goalw "Compl_iff" Set.thy [Compl_def] "(c : Compl(A)) = (c~:A)"
```
```   206  (fn _ => [ (Fast_tac 1) ]);
```
```   207
```
```   208 Addsimps [Compl_iff];
```
```   209
```
```   210 val prems = goalw Set.thy [Compl_def]
```
```   211     "[| c:A ==> False |] ==> c : Compl(A)";
```
```   212 by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
```
```   213 qed "ComplI";
```
```   214
```
```   215 (*This form, with negated conclusion, works well with the Classical prover.
```
```   216   Negated assumptions behave like formulae on the right side of the notional
```
```   217   turnstile...*)
```
```   218 val major::prems = goalw Set.thy [Compl_def]
```
```   219     "c : Compl(A) ==> c~:A";
```
```   220 by (rtac (major RS CollectD) 1);
```
```   221 qed "ComplD";
```
```   222
```
```   223 val ComplE = make_elim ComplD;
```
```   224
```
```   225 AddSIs [ComplI];
```
```   226 AddSEs [ComplE];
```
```   227
```
```   228
```
```   229 section "Binary union -- Un";
```
```   230
```
```   231 qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)"
```
```   232  (fn _ => [ Fast_tac 1 ]);
```
```   233
```
```   234 Addsimps [Un_iff];
```
```   235
```
```   236 goal Set.thy "!!c. c:A ==> c : A Un B";
```
```   237 by (Asm_simp_tac 1);
```
```   238 qed "UnI1";
```
```   239
```
```   240 goal Set.thy "!!c. c:B ==> c : A Un B";
```
```   241 by (Asm_simp_tac 1);
```
```   242 qed "UnI2";
```
```   243
```
```   244 (*Classical introduction rule: no commitment to A vs B*)
```
```   245 qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B"
```
```   246  (fn prems=>
```
```   247   [ (Simp_tac 1),
```
```   248     (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
```
```   249
```
```   250 val major::prems = goalw Set.thy [Un_def]
```
```   251     "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
```
```   252 by (rtac (major RS CollectD RS disjE) 1);
```
```   253 by (REPEAT (eresolve_tac prems 1));
```
```   254 qed "UnE";
```
```   255
```
```   256 AddSIs [UnCI];
```
```   257 AddSEs [UnE];
```
```   258
```
```   259
```
```   260 section "Binary intersection -- Int";
```
```   261
```
```   262 qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)"
```
```   263  (fn _ => [ (Fast_tac 1) ]);
```
```   264
```
```   265 Addsimps [Int_iff];
```
```   266
```
```   267 goal Set.thy "!!c. [| c:A;  c:B |] ==> c : A Int B";
```
```   268 by (Asm_simp_tac 1);
```
```   269 qed "IntI";
```
```   270
```
```   271 goal Set.thy "!!c. c : A Int B ==> c:A";
```
```   272 by (Asm_full_simp_tac 1);
```
```   273 qed "IntD1";
```
```   274
```
```   275 goal Set.thy "!!c. c : A Int B ==> c:B";
```
```   276 by (Asm_full_simp_tac 1);
```
```   277 qed "IntD2";
```
```   278
```
```   279 val [major,minor] = goal Set.thy
```
```   280     "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
```
```   281 by (rtac minor 1);
```
```   282 by (rtac (major RS IntD1) 1);
```
```   283 by (rtac (major RS IntD2) 1);
```
```   284 qed "IntE";
```
```   285
```
```   286 AddSIs [IntI];
```
```   287 AddSEs [IntE];
```
```   288
```
```   289 section "Set difference";
```
```   290
```
```   291 qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)"
```
```   292  (fn _ => [ (Fast_tac 1) ]);
```
```   293
```
```   294 Addsimps [Diff_iff];
```
```   295
```
```   296 qed_goal "DiffI" Set.thy "!!c. [| c : A;  c ~: B |] ==> c : A - B"
```
```   297  (fn _=> [ Asm_simp_tac 1 ]);
```
```   298
```
```   299 qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A"
```
```   300  (fn _=> [ (Asm_full_simp_tac 1) ]);
```
```   301
```
```   302 qed_goal "DiffD2" Set.thy "!!c. [| c : A - B;  c : B |] ==> P"
```
```   303  (fn _=> [ (Asm_full_simp_tac 1) ]);
```
```   304
```
```   305 qed_goal "DiffE" Set.thy "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P"
```
```   306  (fn prems=>
```
```   307   [ (resolve_tac prems 1),
```
```   308     (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]);
```
```   309
```
```   310 AddSIs [DiffI];
```
```   311 AddSEs [DiffE];
```
```   312
```
```   313 section "The empty set -- {}";
```
```   314
```
```   315 qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False"
```
```   316  (fn _ => [ (Fast_tac 1) ]);
```
```   317
```
```   318 Addsimps [empty_iff];
```
```   319
```
```   320 qed_goal "emptyE" Set.thy "!!a. a:{} ==> P"
```
```   321  (fn _ => [Full_simp_tac 1]);
```
```   322
```
```   323 AddSEs [emptyE];
```
```   324
```
```   325 qed_goal "empty_subsetI" Set.thy "{} <= A"
```
```   326  (fn _ => [ (Fast_tac 1) ]);
```
```   327
```
```   328 qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}"
```
```   329  (fn [prem]=>
```
```   330   [ (fast_tac (!claset addIs [prem RS FalseE]) 1) ]);
```
```   331
```
```   332 qed_goal "equals0D" Set.thy "!!a. [| A={};  a:A |] ==> P"
```
```   333  (fn _ => [ (Fast_tac 1) ]);
```
```   334
```
```   335 goal Set.thy "Ball {} P = True";
```
```   336 by (simp_tac (HOL_ss addsimps [mem_Collect_eq, Ball_def, empty_def]) 1);
```
```   337 qed "ball_empty";
```
```   338
```
```   339 goal Set.thy "Bex {} P = False";
```
```   340 by (simp_tac (HOL_ss addsimps [mem_Collect_eq, Bex_def, empty_def]) 1);
```
```   341 qed "bex_empty";
```
```   342 Addsimps [ball_empty, bex_empty];
```
```   343
```
```   344 goalw Set.thy [Ball_def] "(!x:A.False) = (A = {})";
```
```   345 by(Fast_tac 1);
```
```   346 qed "ball_False";
```
```   347 Addsimps [ball_False];
```
```   348
```
```   349 (* The dual is probably not helpful:
```
```   350 goal Set.thy "(? x:A.True) = (A ~= {})";
```
```   351 by(Fast_tac 1);
```
```   352 qed "bex_True";
```
```   353 Addsimps [bex_True];
```
```   354 *)
```
```   355
```
```   356
```
```   357 section "Augmenting a set -- insert";
```
```   358
```
```   359 qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)"
```
```   360  (fn _ => [Fast_tac 1]);
```
```   361
```
```   362 Addsimps [insert_iff];
```
```   363
```
```   364 qed_goal "insertI1" Set.thy "a : insert a B"
```
```   365  (fn _ => [Simp_tac 1]);
```
```   366
```
```   367 qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B"
```
```   368  (fn _=> [Asm_simp_tac 1]);
```
```   369
```
```   370 qed_goalw "insertE" Set.thy [insert_def]
```
```   371     "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P"
```
```   372  (fn major::prems=>
```
```   373   [ (rtac (major RS UnE) 1),
```
```   374     (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
```
```   375
```
```   376 (*Classical introduction rule*)
```
```   377 qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
```
```   378  (fn prems=>
```
```   379   [ (Simp_tac 1),
```
```   380     (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
```
```   381
```
```   382 AddSIs [insertCI];
```
```   383 AddSEs [insertE];
```
```   384
```
```   385 section "Singletons, using insert";
```
```   386
```
```   387 qed_goal "singletonI" Set.thy "a : {a}"
```
```   388  (fn _=> [ (rtac insertI1 1) ]);
```
```   389
```
```   390 goal Set.thy "!!a. b : {a} ==> b=a";
```
```   391 by (Fast_tac 1);
```
```   392 qed "singletonD";
```
```   393
```
```   394 bind_thm ("singletonE", make_elim singletonD);
```
```   395
```
```   396 qed_goal "singleton_iff" thy "(b : {a}) = (b=a)"
```
```   397 (fn _ => [Fast_tac 1]);
```
```   398
```
```   399 goal Set.thy "!!a b. {a}={b} ==> a=b";
```
```   400 by (fast_tac (!claset addEs [equalityE]) 1);
```
```   401 qed "singleton_inject";
```
```   402
```
```   403 AddSDs [singleton_inject];
```
```   404
```
```   405
```
```   406 section "The universal set -- UNIV";
```
```   407
```
```   408 qed_goal "UNIV_I" Set.thy "x : UNIV"
```
```   409   (fn _ => [rtac ComplI 1, etac emptyE 1]);
```
```   410
```
```   411 qed_goal "subset_UNIV" Set.thy "A <= UNIV"
```
```   412   (fn _ => [rtac subsetI 1, rtac UNIV_I 1]);
```
```   413
```
```   414
```
```   415 section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
```
```   416
```
```   417 goalw Set.thy [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
```
```   418 by (Fast_tac 1);
```
```   419 qed "UN_iff";
```
```   420
```
```   421 Addsimps [UN_iff];
```
```   422
```
```   423 (*The order of the premises presupposes that A is rigid; b may be flexible*)
```
```   424 goal Set.thy "!!b. [| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
```
```   425 by (Auto_tac());
```
```   426 qed "UN_I";
```
```   427
```
```   428 val major::prems = goalw Set.thy [UNION_def]
```
```   429     "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
```
```   430 by (rtac (major RS CollectD RS bexE) 1);
```
```   431 by (REPEAT (ares_tac prems 1));
```
```   432 qed "UN_E";
```
```   433
```
```   434 AddIs  [UN_I];
```
```   435 AddSEs [UN_E];
```
```   436
```
```   437 val prems = goal Set.thy
```
```   438     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
```
```   439 \    (UN x:A. C(x)) = (UN x:B. D(x))";
```
```   440 by (REPEAT (etac UN_E 1
```
```   441      ORELSE ares_tac ([UN_I,equalityI,subsetI] @
```
```   442                       (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
```
```   443 qed "UN_cong";
```
```   444
```
```   445
```
```   446 section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
```
```   447
```
```   448 goalw Set.thy [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
```
```   449 by (Auto_tac());
```
```   450 qed "INT_iff";
```
```   451
```
```   452 Addsimps [INT_iff];
```
```   453
```
```   454 val prems = goalw Set.thy [INTER_def]
```
```   455     "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
```
```   456 by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
```
```   457 qed "INT_I";
```
```   458
```
```   459 goal Set.thy "!!b. [| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
```
```   460 by (Auto_tac());
```
```   461 qed "INT_D";
```
```   462
```
```   463 (*"Classical" elimination -- by the Excluded Middle on a:A *)
```
```   464 val major::prems = goalw Set.thy [INTER_def]
```
```   465     "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
```
```   466 by (rtac (major RS CollectD RS ballE) 1);
```
```   467 by (REPEAT (eresolve_tac prems 1));
```
```   468 qed "INT_E";
```
```   469
```
```   470 AddSIs [INT_I];
```
```   471 AddEs  [INT_D, INT_E];
```
```   472
```
```   473 val prems = goal Set.thy
```
```   474     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
```
```   475 \    (INT x:A. C(x)) = (INT x:B. D(x))";
```
```   476 by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
```
```   477 by (REPEAT (dtac INT_D 1
```
```   478      ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
```
```   479 qed "INT_cong";
```
```   480
```
```   481
```
```   482 section "Unions over a type; UNION1(B) = Union(range(B))";
```
```   483
```
```   484 goalw Set.thy [UNION1_def] "(b: (UN x. B(x))) = (EX x. b: B(x))";
```
```   485 by (Simp_tac 1);
```
```   486 by (Fast_tac 1);
```
```   487 qed "UN1_iff";
```
```   488
```
```   489 Addsimps [UN1_iff];
```
```   490
```
```   491 (*The order of the premises presupposes that A is rigid; b may be flexible*)
```
```   492 goal Set.thy "!!b. b: B(x) ==> b: (UN x. B(x))";
```
```   493 by (Auto_tac());
```
```   494 qed "UN1_I";
```
```   495
```
```   496 val major::prems = goalw Set.thy [UNION1_def]
```
```   497     "[| b : (UN x. B(x));  !!x. b: B(x) ==> R |] ==> R";
```
```   498 by (rtac (major RS UN_E) 1);
```
```   499 by (REPEAT (ares_tac prems 1));
```
```   500 qed "UN1_E";
```
```   501
```
```   502 AddIs  [UN1_I];
```
```   503 AddSEs [UN1_E];
```
```   504
```
```   505
```
```   506 section "Intersections over a type; INTER1(B) = Inter(range(B))";
```
```   507
```
```   508 goalw Set.thy [INTER1_def] "(b: (INT x. B(x))) = (ALL x. b: B(x))";
```
```   509 by (Simp_tac 1);
```
```   510 by (Fast_tac 1);
```
```   511 qed "INT1_iff";
```
```   512
```
```   513 Addsimps [INT1_iff];
```
```   514
```
```   515 val prems = goalw Set.thy [INTER1_def]
```
```   516     "(!!x. b: B(x)) ==> b : (INT x. B(x))";
```
```   517 by (REPEAT (ares_tac (INT_I::prems) 1));
```
```   518 qed "INT1_I";
```
```   519
```
```   520 goal Set.thy "!!b. b : (INT x. B(x)) ==> b: B(a)";
```
```   521 by (Asm_full_simp_tac 1);
```
```   522 qed "INT1_D";
```
```   523
```
```   524 AddSIs [INT1_I];
```
```   525 AddDs  [INT1_D];
```
```   526
```
```   527
```
```   528 section "Union";
```
```   529
```
```   530 goalw Set.thy [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
```
```   531 by (Fast_tac 1);
```
```   532 qed "Union_iff";
```
```   533
```
```   534 Addsimps [Union_iff];
```
```   535
```
```   536 (*The order of the premises presupposes that C is rigid; A may be flexible*)
```
```   537 goal Set.thy "!!X. [| X:C;  A:X |] ==> A : Union(C)";
```
```   538 by (Auto_tac());
```
```   539 qed "UnionI";
```
```   540
```
```   541 val major::prems = goalw Set.thy [Union_def]
```
```   542     "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
```
```   543 by (rtac (major RS UN_E) 1);
```
```   544 by (REPEAT (ares_tac prems 1));
```
```   545 qed "UnionE";
```
```   546
```
```   547 AddIs  [UnionI];
```
```   548 AddSEs [UnionE];
```
```   549
```
```   550
```
```   551 section "Inter";
```
```   552
```
```   553 goalw Set.thy [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
```
```   554 by (Fast_tac 1);
```
```   555 qed "Inter_iff";
```
```   556
```
```   557 Addsimps [Inter_iff];
```
```   558
```
```   559 val prems = goalw Set.thy [Inter_def]
```
```   560     "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
```
```   561 by (REPEAT (ares_tac ([INT_I] @ prems) 1));
```
```   562 qed "InterI";
```
```   563
```
```   564 (*A "destruct" rule -- every X in C contains A as an element, but
```
```   565   A:X can hold when X:C does not!  This rule is analogous to "spec". *)
```
```   566 goal Set.thy "!!X. [| A : Inter(C);  X:C |] ==> A:X";
```
```   567 by (Auto_tac());
```
```   568 qed "InterD";
```
```   569
```
```   570 (*"Classical" elimination rule -- does not require proving X:C *)
```
```   571 val major::prems = goalw Set.thy [Inter_def]
```
```   572     "[| A : Inter(C);  X~:C ==> R;  A:X ==> R |] ==> R";
```
```   573 by (rtac (major RS INT_E) 1);
```
```   574 by (REPEAT (eresolve_tac prems 1));
```
```   575 qed "InterE";
```
```   576
```
```   577 AddSIs [InterI];
```
```   578 AddEs  [InterD, InterE];
```
```   579
```
```   580
```
```   581 section "The Powerset operator -- Pow";
```
```   582
```
```   583 qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)"
```
```   584  (fn _ => [ (Asm_simp_tac 1) ]);
```
```   585
```
```   586 AddIffs [Pow_iff];
```
```   587
```
```   588 qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
```
```   589  (fn _ => [ (etac CollectI 1) ]);
```
```   590
```
```   591 qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B)  ==>  A<=B"
```
```   592  (fn _=> [ (etac CollectD 1) ]);
```
```   593
```
```   594 val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)
```
```   595 val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)
```
```   596
```
```   597
```
```   598
```
```   599 (*** Set reasoning tools ***)
```
```   600
```
```   601
```
```   602 (*Each of these has ALREADY been added to !simpset above.*)
```
```   603 val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff,
```
```   604                  mem_Collect_eq,
```
```   605 		 UN_iff, UN1_iff, Union_iff,
```
```   606 		 INT_iff, INT1_iff, Inter_iff];
```
```   607
```
```   608 (*Not for Addsimps -- it can cause goals to blow up!*)
```
```   609 goal Set.thy "(a : (if Q then x else y)) = ((Q --> a:x) & (~Q --> a : y))";
```
```   610 by (simp_tac (!simpset setloop split_tac [expand_if]) 1);
```
```   611 qed "mem_if";
```
```   612
```
```   613 val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs;
```
```   614
```
```   615 simpset := !simpset addcongs [ball_cong,bex_cong]
```
```   616                     setmksimps (mksimps mksimps_pairs);
```