src/HOL/Set.ML
 author paulson Wed Apr 02 11:25:04 1997 +0200 (1997-04-02) changeset 2858 1f3f5c44e159 parent 2721 f08042e18c7d child 2881 62ecde1015ae permissions -rw-r--r--
Re-ordering of rules to assist blast_tac
Powerset rules must not be the most recent
```     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 "The empty set -- {}";
```
```   204
```
```   205 qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False"
```
```   206  (fn _ => [ (Fast_tac 1) ]);
```
```   207
```
```   208 Addsimps [empty_iff];
```
```   209
```
```   210 qed_goal "emptyE" Set.thy "!!a. a:{} ==> P"
```
```   211  (fn _ => [Full_simp_tac 1]);
```
```   212
```
```   213 AddSEs [emptyE];
```
```   214
```
```   215 qed_goal "empty_subsetI" Set.thy "{} <= A"
```
```   216  (fn _ => [ (Fast_tac 1) ]);
```
```   217
```
```   218 qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}"
```
```   219  (fn [prem]=>
```
```   220   [ (fast_tac (!claset addIs [prem RS FalseE]) 1) ]);
```
```   221
```
```   222 qed_goal "equals0D" Set.thy "!!a. [| A={};  a:A |] ==> P"
```
```   223  (fn _ => [ (Fast_tac 1) ]);
```
```   224
```
```   225 goal Set.thy "Ball {} P = True";
```
```   226 by (simp_tac (HOL_ss addsimps [mem_Collect_eq, Ball_def, empty_def]) 1);
```
```   227 qed "ball_empty";
```
```   228
```
```   229 goal Set.thy "Bex {} P = False";
```
```   230 by (simp_tac (HOL_ss addsimps [mem_Collect_eq, Bex_def, empty_def]) 1);
```
```   231 qed "bex_empty";
```
```   232 Addsimps [ball_empty, bex_empty];
```
```   233
```
```   234 goalw Set.thy [Ball_def] "(!x:A.False) = (A = {})";
```
```   235 by(Fast_tac 1);
```
```   236 qed "ball_False";
```
```   237 Addsimps [ball_False];
```
```   238
```
```   239 (* The dual is probably not helpful:
```
```   240 goal Set.thy "(? x:A.True) = (A ~= {})";
```
```   241 by(Fast_tac 1);
```
```   242 qed "bex_True";
```
```   243 Addsimps [bex_True];
```
```   244 *)
```
```   245
```
```   246
```
```   247 section "The Powerset operator -- Pow";
```
```   248
```
```   249 qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)"
```
```   250  (fn _ => [ (Asm_simp_tac 1) ]);
```
```   251
```
```   252 AddIffs [Pow_iff];
```
```   253
```
```   254 qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
```
```   255  (fn _ => [ (etac CollectI 1) ]);
```
```   256
```
```   257 qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B)  ==>  A<=B"
```
```   258  (fn _=> [ (etac CollectD 1) ]);
```
```   259
```
```   260 val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)
```
```   261 val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)
```
```   262
```
```   263
```
```   264 section "Set complement -- Compl";
```
```   265
```
```   266 qed_goalw "Compl_iff" Set.thy [Compl_def] "(c : Compl(A)) = (c~:A)"
```
```   267  (fn _ => [ (Fast_tac 1) ]);
```
```   268
```
```   269 Addsimps [Compl_iff];
```
```   270
```
```   271 val prems = goalw Set.thy [Compl_def]
```
```   272     "[| c:A ==> False |] ==> c : Compl(A)";
```
```   273 by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
```
```   274 qed "ComplI";
```
```   275
```
```   276 (*This form, with negated conclusion, works well with the Classical prover.
```
```   277   Negated assumptions behave like formulae on the right side of the notional
```
```   278   turnstile...*)
```
```   279 val major::prems = goalw Set.thy [Compl_def]
```
```   280     "c : Compl(A) ==> c~:A";
```
```   281 by (rtac (major RS CollectD) 1);
```
```   282 qed "ComplD";
```
```   283
```
```   284 val ComplE = make_elim ComplD;
```
```   285
```
```   286 AddSIs [ComplI];
```
```   287 AddSEs [ComplE];
```
```   288
```
```   289
```
```   290 section "Binary union -- Un";
```
```   291
```
```   292 qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)"
```
```   293  (fn _ => [ Fast_tac 1 ]);
```
```   294
```
```   295 Addsimps [Un_iff];
```
```   296
```
```   297 goal Set.thy "!!c. c:A ==> c : A Un B";
```
```   298 by (Asm_simp_tac 1);
```
```   299 qed "UnI1";
```
```   300
```
```   301 goal Set.thy "!!c. c:B ==> c : A Un B";
```
```   302 by (Asm_simp_tac 1);
```
```   303 qed "UnI2";
```
```   304
```
```   305 (*Classical introduction rule: no commitment to A vs B*)
```
```   306 qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B"
```
```   307  (fn prems=>
```
```   308   [ (Simp_tac 1),
```
```   309     (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
```
```   310
```
```   311 val major::prems = goalw Set.thy [Un_def]
```
```   312     "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
```
```   313 by (rtac (major RS CollectD RS disjE) 1);
```
```   314 by (REPEAT (eresolve_tac prems 1));
```
```   315 qed "UnE";
```
```   316
```
```   317 AddSIs [UnCI];
```
```   318 AddSEs [UnE];
```
```   319
```
```   320
```
```   321 section "Binary intersection -- Int";
```
```   322
```
```   323 qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)"
```
```   324  (fn _ => [ (Fast_tac 1) ]);
```
```   325
```
```   326 Addsimps [Int_iff];
```
```   327
```
```   328 goal Set.thy "!!c. [| c:A;  c:B |] ==> c : A Int B";
```
```   329 by (Asm_simp_tac 1);
```
```   330 qed "IntI";
```
```   331
```
```   332 goal Set.thy "!!c. c : A Int B ==> c:A";
```
```   333 by (Asm_full_simp_tac 1);
```
```   334 qed "IntD1";
```
```   335
```
```   336 goal Set.thy "!!c. c : A Int B ==> c:B";
```
```   337 by (Asm_full_simp_tac 1);
```
```   338 qed "IntD2";
```
```   339
```
```   340 val [major,minor] = goal Set.thy
```
```   341     "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
```
```   342 by (rtac minor 1);
```
```   343 by (rtac (major RS IntD1) 1);
```
```   344 by (rtac (major RS IntD2) 1);
```
```   345 qed "IntE";
```
```   346
```
```   347 AddSIs [IntI];
```
```   348 AddSEs [IntE];
```
```   349
```
```   350 section "Set difference";
```
```   351
```
```   352 qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)"
```
```   353  (fn _ => [ (Fast_tac 1) ]);
```
```   354
```
```   355 Addsimps [Diff_iff];
```
```   356
```
```   357 qed_goal "DiffI" Set.thy "!!c. [| c : A;  c ~: B |] ==> c : A - B"
```
```   358  (fn _=> [ Asm_simp_tac 1 ]);
```
```   359
```
```   360 qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A"
```
```   361  (fn _=> [ (Asm_full_simp_tac 1) ]);
```
```   362
```
```   363 qed_goal "DiffD2" Set.thy "!!c. [| c : A - B;  c : B |] ==> P"
```
```   364  (fn _=> [ (Asm_full_simp_tac 1) ]);
```
```   365
```
```   366 qed_goal "DiffE" Set.thy "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P"
```
```   367  (fn prems=>
```
```   368   [ (resolve_tac prems 1),
```
```   369     (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]);
```
```   370
```
```   371 AddSIs [DiffI];
```
```   372 AddSEs [DiffE];
```
```   373
```
```   374
```
```   375 section "Augmenting a set -- insert";
```
```   376
```
```   377 qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)"
```
```   378  (fn _ => [Fast_tac 1]);
```
```   379
```
```   380 Addsimps [insert_iff];
```
```   381
```
```   382 qed_goal "insertI1" Set.thy "a : insert a B"
```
```   383  (fn _ => [Simp_tac 1]);
```
```   384
```
```   385 qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B"
```
```   386  (fn _=> [Asm_simp_tac 1]);
```
```   387
```
```   388 qed_goalw "insertE" Set.thy [insert_def]
```
```   389     "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P"
```
```   390  (fn major::prems=>
```
```   391   [ (rtac (major RS UnE) 1),
```
```   392     (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
```
```   393
```
```   394 (*Classical introduction rule*)
```
```   395 qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
```
```   396  (fn prems=>
```
```   397   [ (Simp_tac 1),
```
```   398     (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
```
```   399
```
```   400 AddSIs [insertCI];
```
```   401 AddSEs [insertE];
```
```   402
```
```   403 section "Singletons, using insert";
```
```   404
```
```   405 qed_goal "singletonI" Set.thy "a : {a}"
```
```   406  (fn _=> [ (rtac insertI1 1) ]);
```
```   407
```
```   408 goal Set.thy "!!a. b : {a} ==> b=a";
```
```   409 by (Fast_tac 1);
```
```   410 qed "singletonD";
```
```   411
```
```   412 bind_thm ("singletonE", make_elim singletonD);
```
```   413
```
```   414 qed_goal "singleton_iff" thy "(b : {a}) = (b=a)"
```
```   415 (fn _ => [Fast_tac 1]);
```
```   416
```
```   417 goal Set.thy "!!a b. {a}={b} ==> a=b";
```
```   418 by (fast_tac (!claset addEs [equalityE]) 1);
```
```   419 qed "singleton_inject";
```
```   420
```
```   421 (*Redundant? But unlike insertCI, it proves the subgoal immediately!*)
```
```   422 AddSIs [singletonI];
```
```   423
```
```   424 AddSDs [singleton_inject];
```
```   425
```
```   426
```
```   427 section "The universal set -- UNIV";
```
```   428
```
```   429 qed_goal "UNIV_I" Set.thy "x : UNIV"
```
```   430   (fn _ => [rtac ComplI 1, etac emptyE 1]);
```
```   431
```
```   432 qed_goal "subset_UNIV" Set.thy "A <= UNIV"
```
```   433   (fn _ => [rtac subsetI 1, rtac UNIV_I 1]);
```
```   434
```
```   435
```
```   436 section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
```
```   437
```
```   438 goalw Set.thy [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
```
```   439 by (Fast_tac 1);
```
```   440 qed "UN_iff";
```
```   441
```
```   442 Addsimps [UN_iff];
```
```   443
```
```   444 (*The order of the premises presupposes that A is rigid; b may be flexible*)
```
```   445 goal Set.thy "!!b. [| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
```
```   446 by (Auto_tac());
```
```   447 qed "UN_I";
```
```   448
```
```   449 val major::prems = goalw Set.thy [UNION_def]
```
```   450     "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
```
```   451 by (rtac (major RS CollectD RS bexE) 1);
```
```   452 by (REPEAT (ares_tac prems 1));
```
```   453 qed "UN_E";
```
```   454
```
```   455 AddIs  [UN_I];
```
```   456 AddSEs [UN_E];
```
```   457
```
```   458 val prems = goal Set.thy
```
```   459     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
```
```   460 \    (UN x:A. C(x)) = (UN x:B. D(x))";
```
```   461 by (REPEAT (etac UN_E 1
```
```   462      ORELSE ares_tac ([UN_I,equalityI,subsetI] @
```
```   463                       (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
```
```   464 qed "UN_cong";
```
```   465
```
```   466
```
```   467 section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
```
```   468
```
```   469 goalw Set.thy [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
```
```   470 by (Auto_tac());
```
```   471 qed "INT_iff";
```
```   472
```
```   473 Addsimps [INT_iff];
```
```   474
```
```   475 val prems = goalw Set.thy [INTER_def]
```
```   476     "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
```
```   477 by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
```
```   478 qed "INT_I";
```
```   479
```
```   480 goal Set.thy "!!b. [| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
```
```   481 by (Auto_tac());
```
```   482 qed "INT_D";
```
```   483
```
```   484 (*"Classical" elimination -- by the Excluded Middle on a:A *)
```
```   485 val major::prems = goalw Set.thy [INTER_def]
```
```   486     "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
```
```   487 by (rtac (major RS CollectD RS ballE) 1);
```
```   488 by (REPEAT (eresolve_tac prems 1));
```
```   489 qed "INT_E";
```
```   490
```
```   491 AddSIs [INT_I];
```
```   492 AddEs  [INT_D, INT_E];
```
```   493
```
```   494 val prems = goal Set.thy
```
```   495     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
```
```   496 \    (INT x:A. C(x)) = (INT x:B. D(x))";
```
```   497 by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
```
```   498 by (REPEAT (dtac INT_D 1
```
```   499      ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
```
```   500 qed "INT_cong";
```
```   501
```
```   502
```
```   503 section "Unions over a type; UNION1(B) = Union(range(B))";
```
```   504
```
```   505 goalw Set.thy [UNION1_def] "(b: (UN x. B(x))) = (EX x. b: B(x))";
```
```   506 by (Simp_tac 1);
```
```   507 by (Fast_tac 1);
```
```   508 qed "UN1_iff";
```
```   509
```
```   510 Addsimps [UN1_iff];
```
```   511
```
```   512 (*The order of the premises presupposes that A is rigid; b may be flexible*)
```
```   513 goal Set.thy "!!b. b: B(x) ==> b: (UN x. B(x))";
```
```   514 by (Auto_tac());
```
```   515 qed "UN1_I";
```
```   516
```
```   517 val major::prems = goalw Set.thy [UNION1_def]
```
```   518     "[| b : (UN x. B(x));  !!x. b: B(x) ==> R |] ==> R";
```
```   519 by (rtac (major RS UN_E) 1);
```
```   520 by (REPEAT (ares_tac prems 1));
```
```   521 qed "UN1_E";
```
```   522
```
```   523 AddIs  [UN1_I];
```
```   524 AddSEs [UN1_E];
```
```   525
```
```   526
```
```   527 section "Intersections over a type; INTER1(B) = Inter(range(B))";
```
```   528
```
```   529 goalw Set.thy [INTER1_def] "(b: (INT x. B(x))) = (ALL x. b: B(x))";
```
```   530 by (Simp_tac 1);
```
```   531 by (Fast_tac 1);
```
```   532 qed "INT1_iff";
```
```   533
```
```   534 Addsimps [INT1_iff];
```
```   535
```
```   536 val prems = goalw Set.thy [INTER1_def]
```
```   537     "(!!x. b: B(x)) ==> b : (INT x. B(x))";
```
```   538 by (REPEAT (ares_tac (INT_I::prems) 1));
```
```   539 qed "INT1_I";
```
```   540
```
```   541 goal Set.thy "!!b. b : (INT x. B(x)) ==> b: B(a)";
```
```   542 by (Asm_full_simp_tac 1);
```
```   543 qed "INT1_D";
```
```   544
```
```   545 AddSIs [INT1_I];
```
```   546 AddDs  [INT1_D];
```
```   547
```
```   548
```
```   549 section "Union";
```
```   550
```
```   551 goalw Set.thy [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
```
```   552 by (Fast_tac 1);
```
```   553 qed "Union_iff";
```
```   554
```
```   555 Addsimps [Union_iff];
```
```   556
```
```   557 (*The order of the premises presupposes that C is rigid; A may be flexible*)
```
```   558 goal Set.thy "!!X. [| X:C;  A:X |] ==> A : Union(C)";
```
```   559 by (Auto_tac());
```
```   560 qed "UnionI";
```
```   561
```
```   562 val major::prems = goalw Set.thy [Union_def]
```
```   563     "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
```
```   564 by (rtac (major RS UN_E) 1);
```
```   565 by (REPEAT (ares_tac prems 1));
```
```   566 qed "UnionE";
```
```   567
```
```   568 AddIs  [UnionI];
```
```   569 AddSEs [UnionE];
```
```   570
```
```   571
```
```   572 section "Inter";
```
```   573
```
```   574 goalw Set.thy [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
```
```   575 by (Fast_tac 1);
```
```   576 qed "Inter_iff";
```
```   577
```
```   578 Addsimps [Inter_iff];
```
```   579
```
```   580 val prems = goalw Set.thy [Inter_def]
```
```   581     "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
```
```   582 by (REPEAT (ares_tac ([INT_I] @ prems) 1));
```
```   583 qed "InterI";
```
```   584
```
```   585 (*A "destruct" rule -- every X in C contains A as an element, but
```
```   586   A:X can hold when X:C does not!  This rule is analogous to "spec". *)
```
```   587 goal Set.thy "!!X. [| A : Inter(C);  X:C |] ==> A:X";
```
```   588 by (Auto_tac());
```
```   589 qed "InterD";
```
```   590
```
```   591 (*"Classical" elimination rule -- does not require proving X:C *)
```
```   592 val major::prems = goalw Set.thy [Inter_def]
```
```   593     "[| A : Inter(C);  X~:C ==> R;  A:X ==> R |] ==> R";
```
```   594 by (rtac (major RS INT_E) 1);
```
```   595 by (REPEAT (eresolve_tac prems 1));
```
```   596 qed "InterE";
```
```   597
```
```   598 AddSIs [InterI];
```
```   599 AddEs  [InterD, InterE];
```
```   600
```
```   601
```
```   602
```
```   603 (*** Set reasoning tools ***)
```
```   604
```
```   605
```
```   606 (*Each of these has ALREADY been added to !simpset above.*)
```
```   607 val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff,
```
```   608                  mem_Collect_eq,
```
```   609 		 UN_iff, UN1_iff, Union_iff,
```
```   610 		 INT_iff, INT1_iff, Inter_iff];
```
```   611
```
```   612 (*Not for Addsimps -- it can cause goals to blow up!*)
```
```   613 goal Set.thy "(a : (if Q then x else y)) = ((Q --> a:x) & (~Q --> a : y))";
```
```   614 by (simp_tac (!simpset setloop split_tac [expand_if]) 1);
```
```   615 qed "mem_if";
```
```   616
```
```   617 val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs;
```
```   618
```
```   619 simpset := !simpset addcongs [ball_cong,bex_cong]
```
```   620                     setmksimps (mksimps mksimps_pairs);
```