src/HOL/Set.ML
 author clasohm Wed Mar 13 11:55:25 1996 +0100 (1996-03-13 ago) changeset 1574 5a63ab90ee8a parent 1552 6f71b5d46700 child 1618 372880456b5b permissions -rw-r--r--
modified primrec so it can be used in MiniML/Type.thy
```     1 (*  Title:      HOL/set
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   1991  University of Cambridge
```
```     5
```
```     6 For set.thy.  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 val [prem] = goal Set.thy "P(a) ==> a : {x.P(x)}";
```
```    14 by (rtac (mem_Collect_eq RS ssubst) 1);
```
```    15 by (rtac prem 1);
```
```    16 qed "CollectI";
```
```    17
```
```    18 val prems = goal Set.thy "[| a : {x.P(x)} |] ==> P(a)";
```
```    19 by (resolve_tac (prems RL [mem_Collect_eq  RS subst]) 1);
```
```    20 qed "CollectD";
```
```    21
```
```    22 val [prem] = goal Set.thy "[| !!x. (x:A) = (x:B) |] ==> A = B";
```
```    23 by (rtac (prem RS ext RS arg_cong RS box_equals) 1);
```
```    24 by (rtac Collect_mem_eq 1);
```
```    25 by (rtac Collect_mem_eq 1);
```
```    26 qed "set_ext";
```
```    27
```
```    28 val [prem] = goal Set.thy "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}";
```
```    29 by (rtac (prem RS ext RS arg_cong) 1);
```
```    30 qed "Collect_cong";
```
```    31
```
```    32 val CollectE = make_elim CollectD;
```
```    33
```
```    34 section "Bounded quantifiers";
```
```    35
```
```    36 val prems = goalw Set.thy [Ball_def]
```
```    37     "[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)";
```
```    38 by (REPEAT (ares_tac (prems @ [allI,impI]) 1));
```
```    39 qed "ballI";
```
```    40
```
```    41 val [major,minor] = goalw Set.thy [Ball_def]
```
```    42     "[| ! x:A. P(x);  x:A |] ==> P(x)";
```
```    43 by (rtac (minor RS (major RS spec RS mp)) 1);
```
```    44 qed "bspec";
```
```    45
```
```    46 val major::prems = goalw Set.thy [Ball_def]
```
```    47     "[| ! x:A. P(x);  P(x) ==> Q;  x~:A ==> Q |] ==> Q";
```
```    48 by (rtac (major RS spec RS impCE) 1);
```
```    49 by (REPEAT (eresolve_tac prems 1));
```
```    50 qed "ballE";
```
```    51
```
```    52 (*Takes assumptions ! x:A.P(x) and a:A; creates assumption P(a)*)
```
```    53 fun ball_tac i = etac ballE i THEN contr_tac (i+1);
```
```    54
```
```    55 val prems = goalw Set.thy [Bex_def]
```
```    56     "[| P(x);  x:A |] ==> ? x:A. P(x)";
```
```    57 by (REPEAT (ares_tac (prems @ [exI,conjI]) 1));
```
```    58 qed "bexI";
```
```    59
```
```    60 qed_goal "bexCI" Set.thy
```
```    61    "[| ! x:A. ~P(x) ==> P(a);  a:A |] ==> ? x:A.P(x)"
```
```    62  (fn prems=>
```
```    63   [ (rtac classical 1),
```
```    64     (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1))  ]);
```
```    65
```
```    66 val major::prems = goalw Set.thy [Bex_def]
```
```    67     "[| ? x:A. P(x);  !!x. [| x:A; P(x) |] ==> Q  |] ==> Q";
```
```    68 by (rtac (major RS exE) 1);
```
```    69 by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1));
```
```    70 qed "bexE";
```
```    71
```
```    72 (*Trival rewrite rule;   (! x:A.P)=P holds only if A is nonempty!*)
```
```    73 val prems = goal Set.thy
```
```    74     "(! x:A. True) = True";
```
```    75 by (REPEAT (ares_tac [TrueI,ballI,iffI] 1));
```
```    76 qed "ball_rew";
```
```    77
```
```    78 (** Congruence rules **)
```
```    79
```
```    80 val prems = goal Set.thy
```
```    81     "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
```
```    82 \    (! x:A. P(x)) = (! x:B. Q(x))";
```
```    83 by (resolve_tac (prems RL [ssubst]) 1);
```
```    84 by (REPEAT (ares_tac [ballI,iffI] 1
```
```    85      ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1));
```
```    86 qed "ball_cong";
```
```    87
```
```    88 val prems = goal Set.thy
```
```    89     "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
```
```    90 \    (? x:A. P(x)) = (? x:B. Q(x))";
```
```    91 by (resolve_tac (prems RL [ssubst]) 1);
```
```    92 by (REPEAT (etac bexE 1
```
```    93      ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1));
```
```    94 qed "bex_cong";
```
```    95
```
```    96 section "Subsets";
```
```    97
```
```    98 val prems = goalw Set.thy [subset_def] "(!!x.x:A ==> x:B) ==> A <= B";
```
```    99 by (REPEAT (ares_tac (prems @ [ballI]) 1));
```
```   100 qed "subsetI";
```
```   101
```
```   102 (*Rule in Modus Ponens style*)
```
```   103 val major::prems = goalw Set.thy [subset_def] "[| A <= B;  c:A |] ==> c:B";
```
```   104 by (rtac (major RS bspec) 1);
```
```   105 by (resolve_tac prems 1);
```
```   106 qed "subsetD";
```
```   107
```
```   108 (*The same, with reversed premises for use with etac -- cf rev_mp*)
```
```   109 qed_goal "rev_subsetD" Set.thy "[| c:A;  A <= B |] ==> c:B"
```
```   110  (fn prems=>  [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]);
```
```   111
```
```   112 (*Classical elimination rule*)
```
```   113 val major::prems = goalw Set.thy [subset_def]
```
```   114     "[| A <= B;  c~:A ==> P;  c:B ==> P |] ==> P";
```
```   115 by (rtac (major RS ballE) 1);
```
```   116 by (REPEAT (eresolve_tac prems 1));
```
```   117 qed "subsetCE";
```
```   118
```
```   119 (*Takes assumptions A<=B; c:A and creates the assumption c:B *)
```
```   120 fun set_mp_tac i = etac subsetCE i  THEN  mp_tac i;
```
```   121
```
```   122 qed_goal "subset_refl" Set.thy "A <= (A::'a set)"
```
```   123  (fn _=> [ (REPEAT (ares_tac [subsetI] 1)) ]);
```
```   124
```
```   125 val prems = goal Set.thy "[| A<=B;  B<=C |] ==> A<=(C::'a set)";
```
```   126 by (cut_facts_tac prems 1);
```
```   127 by (REPEAT (ares_tac [subsetI] 1 ORELSE set_mp_tac 1));
```
```   128 qed "subset_trans";
```
```   129
```
```   130
```
```   131 section "Equality";
```
```   132
```
```   133 (*Anti-symmetry of the subset relation*)
```
```   134 val prems = goal Set.thy "[| A <= B;  B <= A |] ==> A = (B::'a set)";
```
```   135 by (rtac (iffI RS set_ext) 1);
```
```   136 by (REPEAT (ares_tac (prems RL [subsetD]) 1));
```
```   137 qed "subset_antisym";
```
```   138 val equalityI = subset_antisym;
```
```   139
```
```   140 (* Equality rules from ZF set theory -- are they appropriate here? *)
```
```   141 val prems = goal Set.thy "A = B ==> A<=(B::'a set)";
```
```   142 by (resolve_tac (prems RL [subst]) 1);
```
```   143 by (rtac subset_refl 1);
```
```   144 qed "equalityD1";
```
```   145
```
```   146 val prems = goal Set.thy "A = B ==> B<=(A::'a set)";
```
```   147 by (resolve_tac (prems RL [subst]) 1);
```
```   148 by (rtac subset_refl 1);
```
```   149 qed "equalityD2";
```
```   150
```
```   151 val prems = goal Set.thy
```
```   152     "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";
```
```   153 by (resolve_tac prems 1);
```
```   154 by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
```
```   155 qed "equalityE";
```
```   156
```
```   157 val major::prems = goal Set.thy
```
```   158     "[| A = B;  [| c:A; c:B |] ==> P;  [| c~:A; c~:B |] ==> P |]  ==>  P";
```
```   159 by (rtac (major RS equalityE) 1);
```
```   160 by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
```
```   161 qed "equalityCE";
```
```   162
```
```   163 (*Lemma for creating induction formulae -- for "pattern matching" on p
```
```   164   To make the induction hypotheses usable, apply "spec" or "bspec" to
```
```   165   put universal quantifiers over the free variables in p. *)
```
```   166 val prems = goal Set.thy
```
```   167     "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
```
```   168 by (rtac mp 1);
```
```   169 by (REPEAT (resolve_tac (refl::prems) 1));
```
```   170 qed "setup_induction";
```
```   171
```
```   172
```
```   173 section "Set complement -- Compl";
```
```   174
```
```   175 val prems = goalw Set.thy [Compl_def]
```
```   176     "[| c:A ==> False |] ==> c : Compl(A)";
```
```   177 by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
```
```   178 qed "ComplI";
```
```   179
```
```   180 (*This form, with negated conclusion, works well with the Classical prover.
```
```   181   Negated assumptions behave like formulae on the right side of the notional
```
```   182   turnstile...*)
```
```   183 val major::prems = goalw Set.thy [Compl_def]
```
```   184     "[| c : Compl(A) |] ==> c~:A";
```
```   185 by (rtac (major RS CollectD) 1);
```
```   186 qed "ComplD";
```
```   187
```
```   188 val ComplE = make_elim ComplD;
```
```   189
```
```   190
```
```   191 section "Binary union -- Un";
```
```   192
```
```   193 val prems = goalw Set.thy [Un_def] "c:A ==> c : A Un B";
```
```   194 by (REPEAT (resolve_tac (prems @ [CollectI,disjI1]) 1));
```
```   195 qed "UnI1";
```
```   196
```
```   197 val prems = goalw Set.thy [Un_def] "c:B ==> c : A Un B";
```
```   198 by (REPEAT (resolve_tac (prems @ [CollectI,disjI2]) 1));
```
```   199 qed "UnI2";
```
```   200
```
```   201 (*Classical introduction rule: no commitment to A vs B*)
```
```   202 qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B"
```
```   203  (fn prems=>
```
```   204   [ (rtac classical 1),
```
```   205     (REPEAT (ares_tac (prems@[UnI1,notI]) 1)),
```
```   206     (REPEAT (ares_tac (prems@[UnI2,notE]) 1)) ]);
```
```   207
```
```   208 val major::prems = goalw Set.thy [Un_def]
```
```   209     "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
```
```   210 by (rtac (major RS CollectD RS disjE) 1);
```
```   211 by (REPEAT (eresolve_tac prems 1));
```
```   212 qed "UnE";
```
```   213
```
```   214
```
```   215 section "Binary intersection -- Int";
```
```   216
```
```   217 val prems = goalw Set.thy [Int_def]
```
```   218     "[| c:A;  c:B |] ==> c : A Int B";
```
```   219 by (REPEAT (resolve_tac (prems @ [CollectI,conjI]) 1));
```
```   220 qed "IntI";
```
```   221
```
```   222 val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:A";
```
```   223 by (rtac (major RS CollectD RS conjunct1) 1);
```
```   224 qed "IntD1";
```
```   225
```
```   226 val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:B";
```
```   227 by (rtac (major RS CollectD RS conjunct2) 1);
```
```   228 qed "IntD2";
```
```   229
```
```   230 val [major,minor] = goal Set.thy
```
```   231     "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
```
```   232 by (rtac minor 1);
```
```   233 by (rtac (major RS IntD1) 1);
```
```   234 by (rtac (major RS IntD2) 1);
```
```   235 qed "IntE";
```
```   236
```
```   237
```
```   238 section "Set difference";
```
```   239
```
```   240 qed_goalw "DiffI" Set.thy [set_diff_def]
```
```   241     "[| c : A;  c ~: B |] ==> c : A - B"
```
```   242  (fn prems=> [ (REPEAT (resolve_tac (prems @ [CollectI,conjI]) 1)) ]);
```
```   243
```
```   244 qed_goalw "DiffD1" Set.thy [set_diff_def]
```
```   245     "c : A - B ==> c : A"
```
```   246  (fn [major]=> [ (rtac (major RS CollectD RS conjunct1) 1) ]);
```
```   247
```
```   248 qed_goalw "DiffD2" Set.thy [set_diff_def]
```
```   249     "[| c : A - B;  c : B |] ==> P"
```
```   250  (fn [major,minor]=>
```
```   251      [rtac (minor RS (major RS CollectD RS conjunct2 RS notE)) 1]);
```
```   252
```
```   253 qed_goal "DiffE" Set.thy
```
```   254     "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P"
```
```   255  (fn prems=>
```
```   256   [ (resolve_tac prems 1),
```
```   257     (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]);
```
```   258
```
```   259 qed_goal "Diff_iff" Set.thy "(c : A-B) = (c:A & c~:B)"
```
```   260  (fn _ => [ (fast_tac (HOL_cs addSIs [DiffI] addSEs [DiffE]) 1) ]);
```
```   261
```
```   262 section "The empty set -- {}";
```
```   263
```
```   264 qed_goalw "emptyE" Set.thy [empty_def] "a:{} ==> P"
```
```   265  (fn [prem] => [rtac (prem RS CollectD RS FalseE) 1]);
```
```   266
```
```   267 qed_goal "empty_subsetI" Set.thy "{} <= A"
```
```   268  (fn _ => [ (REPEAT (ares_tac [equalityI,subsetI,emptyE] 1)) ]);
```
```   269
```
```   270 qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}"
```
```   271  (fn prems=>
```
```   272   [ (REPEAT (ares_tac (prems@[empty_subsetI,subsetI,equalityI]) 1
```
```   273       ORELSE eresolve_tac (prems RL [FalseE]) 1)) ]);
```
```   274
```
```   275 qed_goal "equals0D" Set.thy "[| A={};  a:A |] ==> P"
```
```   276  (fn [major,minor]=>
```
```   277   [ (rtac (minor RS (major RS equalityD1 RS subsetD RS emptyE)) 1) ]);
```
```   278
```
```   279
```
```   280 section "Augmenting a set -- insert";
```
```   281
```
```   282 qed_goalw "insertI1" Set.thy [insert_def] "a : insert a B"
```
```   283  (fn _ => [rtac (CollectI RS UnI1) 1, rtac refl 1]);
```
```   284
```
```   285 qed_goalw "insertI2" Set.thy [insert_def] "a : B ==> a : insert b B"
```
```   286  (fn [prem]=> [ (rtac (prem RS UnI2) 1) ]);
```
```   287
```
```   288 qed_goalw "insertE" Set.thy [insert_def]
```
```   289     "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P"
```
```   290  (fn major::prems=>
```
```   291   [ (rtac (major RS UnE) 1),
```
```   292     (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
```
```   293
```
```   294 qed_goal "insert_iff" Set.thy "a : insert b A = (a=b | a:A)"
```
```   295  (fn _ => [fast_tac (HOL_cs addIs [insertI1,insertI2] addSEs [insertE]) 1]);
```
```   296
```
```   297 (*Classical introduction rule*)
```
```   298 qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
```
```   299  (fn [prem]=>
```
```   300   [ (rtac (disjCI RS (insert_iff RS iffD2)) 1),
```
```   301     (etac prem 1) ]);
```
```   302
```
```   303 section "Singletons, using insert";
```
```   304
```
```   305 qed_goal "singletonI" Set.thy "a : {a}"
```
```   306  (fn _=> [ (rtac insertI1 1) ]);
```
```   307
```
```   308 qed_goal "singletonE" Set.thy "[| a: {b};  a=b ==> P |] ==> P"
```
```   309  (fn major::prems=>
```
```   310   [ (rtac (major RS insertE) 1),
```
```   311     (REPEAT (eresolve_tac (prems @ [emptyE]) 1)) ]);
```
```   312
```
```   313 goalw Set.thy [insert_def] "!!a. b : {a} ==> b=a";
```
```   314 by (fast_tac (HOL_cs addSEs [emptyE,CollectE,UnE]) 1);
```
```   315 qed "singletonD";
```
```   316
```
```   317 val singletonE = make_elim singletonD;
```
```   318
```
```   319 val [major] = goal Set.thy "{a}={b} ==> a=b";
```
```   320 by (rtac (major RS equalityD1 RS subsetD RS singletonD) 1);
```
```   321 by (rtac singletonI 1);
```
```   322 qed "singleton_inject";
```
```   323
```
```   324
```
```   325 section "The universal set -- UNIV";
```
```   326
```
```   327 qed_goal "subset_UNIV" Set.thy "A <= UNIV"
```
```   328   (fn _ => [rtac subsetI 1, rtac ComplI 1, etac emptyE 1]);
```
```   329
```
```   330
```
```   331 section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
```
```   332
```
```   333 (*The order of the premises presupposes that A is rigid; b may be flexible*)
```
```   334 val prems = goalw Set.thy [UNION_def]
```
```   335     "[| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
```
```   336 by (REPEAT (resolve_tac (prems @ [bexI,CollectI]) 1));
```
```   337 qed "UN_I";
```
```   338
```
```   339 val major::prems = goalw Set.thy [UNION_def]
```
```   340     "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
```
```   341 by (rtac (major RS CollectD RS bexE) 1);
```
```   342 by (REPEAT (ares_tac prems 1));
```
```   343 qed "UN_E";
```
```   344
```
```   345 val prems = goal Set.thy
```
```   346     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
```
```   347 \    (UN x:A. C(x)) = (UN x:B. D(x))";
```
```   348 by (REPEAT (etac UN_E 1
```
```   349      ORELSE ares_tac ([UN_I,equalityI,subsetI] @
```
```   350                       (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
```
```   351 qed "UN_cong";
```
```   352
```
```   353
```
```   354 section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
```
```   355
```
```   356 val prems = goalw Set.thy [INTER_def]
```
```   357     "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
```
```   358 by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
```
```   359 qed "INT_I";
```
```   360
```
```   361 val major::prems = goalw Set.thy [INTER_def]
```
```   362     "[| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
```
```   363 by (rtac (major RS CollectD RS bspec) 1);
```
```   364 by (resolve_tac prems 1);
```
```   365 qed "INT_D";
```
```   366
```
```   367 (*"Classical" elimination -- by the Excluded Middle on a:A *)
```
```   368 val major::prems = goalw Set.thy [INTER_def]
```
```   369     "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
```
```   370 by (rtac (major RS CollectD RS ballE) 1);
```
```   371 by (REPEAT (eresolve_tac prems 1));
```
```   372 qed "INT_E";
```
```   373
```
```   374 val prems = goal Set.thy
```
```   375     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
```
```   376 \    (INT x:A. C(x)) = (INT x:B. D(x))";
```
```   377 by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
```
```   378 by (REPEAT (dtac INT_D 1
```
```   379      ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
```
```   380 qed "INT_cong";
```
```   381
```
```   382
```
```   383 section "Unions over a type; UNION1(B) = Union(range(B))";
```
```   384
```
```   385 (*The order of the premises presupposes that A is rigid; b may be flexible*)
```
```   386 val prems = goalw Set.thy [UNION1_def]
```
```   387     "b: B(x) ==> b: (UN x. B(x))";
```
```   388 by (REPEAT (resolve_tac (prems @ [TrueI, CollectI RS UN_I]) 1));
```
```   389 qed "UN1_I";
```
```   390
```
```   391 val major::prems = goalw Set.thy [UNION1_def]
```
```   392     "[| b : (UN x. B(x));  !!x. b: B(x) ==> R |] ==> R";
```
```   393 by (rtac (major RS UN_E) 1);
```
```   394 by (REPEAT (ares_tac prems 1));
```
```   395 qed "UN1_E";
```
```   396
```
```   397
```
```   398 section "Intersections over a type; INTER1(B) = Inter(range(B))";
```
```   399
```
```   400 val prems = goalw Set.thy [INTER1_def]
```
```   401     "(!!x. b: B(x)) ==> b : (INT x. B(x))";
```
```   402 by (REPEAT (ares_tac (INT_I::prems) 1));
```
```   403 qed "INT1_I";
```
```   404
```
```   405 val [major] = goalw Set.thy [INTER1_def]
```
```   406     "b : (INT x. B(x)) ==> b: B(a)";
```
```   407 by (rtac (TrueI RS (CollectI RS (major RS INT_D))) 1);
```
```   408 qed "INT1_D";
```
```   409
```
```   410 section "Union";
```
```   411
```
```   412 (*The order of the premises presupposes that C is rigid; A may be flexible*)
```
```   413 val prems = goalw Set.thy [Union_def]
```
```   414     "[| X:C;  A:X |] ==> A : Union(C)";
```
```   415 by (REPEAT (resolve_tac (prems @ [UN_I]) 1));
```
```   416 qed "UnionI";
```
```   417
```
```   418 val major::prems = goalw Set.thy [Union_def]
```
```   419     "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
```
```   420 by (rtac (major RS UN_E) 1);
```
```   421 by (REPEAT (ares_tac prems 1));
```
```   422 qed "UnionE";
```
```   423
```
```   424 section "Inter";
```
```   425
```
```   426 val prems = goalw Set.thy [Inter_def]
```
```   427     "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
```
```   428 by (REPEAT (ares_tac ([INT_I] @ prems) 1));
```
```   429 qed "InterI";
```
```   430
```
```   431 (*A "destruct" rule -- every X in C contains A as an element, but
```
```   432   A:X can hold when X:C does not!  This rule is analogous to "spec". *)
```
```   433 val major::prems = goalw Set.thy [Inter_def]
```
```   434     "[| A : Inter(C);  X:C |] ==> A:X";
```
```   435 by (rtac (major RS INT_D) 1);
```
```   436 by (resolve_tac prems 1);
```
```   437 qed "InterD";
```
```   438
```
```   439 (*"Classical" elimination rule -- does not require proving X:C *)
```
```   440 val major::prems = goalw Set.thy [Inter_def]
```
```   441     "[| A : Inter(C);  A:X ==> R;  X~:C ==> R |] ==> R";
```
```   442 by (rtac (major RS INT_E) 1);
```
```   443 by (REPEAT (eresolve_tac prems 1));
```
```   444 qed "InterE";
```
```   445
```
```   446 section "The Powerset operator -- Pow";
```
```   447
```
```   448 qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
```
```   449  (fn _ => [ (etac CollectI 1) ]);
```
```   450
```
```   451 qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B)  ==>  A<=B"
```
```   452  (fn _=> [ (etac CollectD 1) ]);
```
```   453
```
```   454 val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)
```
```   455 val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)
```