src/HOL/Set.ML
 author nipkow Fri Apr 04 16:33:28 1997 +0200 (1997-04-04) changeset 2912 3fac3e8d5d3e parent 2891 d8f254ad1ab9 child 2935 998cb95fdd43 permissions -rw-r--r--
moved inj and surj from Set to Fun and Inv -> inv.
```     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 Blast.declConsts (["op <="], [subsetI]);	(*overloading of <=*)
```
```   120
```
```   121 (*Rule in Modus Ponens style*)
```
```   122 val major::prems = goalw Set.thy [subset_def] "[| A <= B;  c:A |] ==> c:B";
```
```   123 by (rtac (major RS bspec) 1);
```
```   124 by (resolve_tac prems 1);
```
```   125 qed "subsetD";
```
```   126
```
```   127 (*The same, with reversed premises for use with etac -- cf rev_mp*)
```
```   128 qed_goal "rev_subsetD" Set.thy "[| c:A;  A <= B |] ==> c:B"
```
```   129  (fn prems=>  [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]);
```
```   130
```
```   131 (*Converts A<=B to x:A ==> x:B*)
```
```   132 fun impOfSubs th = th RSN (2, rev_subsetD);
```
```   133
```
```   134 qed_goal "contra_subsetD" Set.thy "!!c. [| A <= B; c ~: B |] ==> c ~: A"
```
```   135  (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
```
```   136
```
```   137 qed_goal "rev_contra_subsetD" Set.thy "!!c. [| c ~: B;  A <= B |] ==> c ~: A"
```
```   138  (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
```
```   139
```
```   140 (*Classical elimination rule*)
```
```   141 val major::prems = goalw Set.thy [subset_def]
```
```   142     "[| A <= B;  c~:A ==> P;  c:B ==> P |] ==> P";
```
```   143 by (rtac (major RS ballE) 1);
```
```   144 by (REPEAT (eresolve_tac prems 1));
```
```   145 qed "subsetCE";
```
```   146
```
```   147 (*Takes assumptions A<=B; c:A and creates the assumption c:B *)
```
```   148 fun set_mp_tac i = etac subsetCE i  THEN  mp_tac i;
```
```   149
```
```   150 AddSIs [subsetI];
```
```   151 AddEs  [subsetD, subsetCE];
```
```   152
```
```   153 qed_goal "subset_refl" Set.thy "A <= (A::'a set)"
```
```   154  (fn _=> [Blast_tac 1]);
```
```   155
```
```   156 val prems = goal Set.thy "!!B. [| A<=B;  B<=C |] ==> A<=(C::'a set)";
```
```   157 by (Blast_tac 1);
```
```   158 qed "subset_trans";
```
```   159
```
```   160
```
```   161 section "Equality";
```
```   162
```
```   163 (*Anti-symmetry of the subset relation*)
```
```   164 val prems = goal Set.thy "[| A <= B;  B <= A |] ==> A = (B::'a set)";
```
```   165 by (rtac (iffI RS set_ext) 1);
```
```   166 by (REPEAT (ares_tac (prems RL [subsetD]) 1));
```
```   167 qed "subset_antisym";
```
```   168 val equalityI = subset_antisym;
```
```   169
```
```   170 Blast.declConsts (["op ="], [equalityI]);	(*overloading of equality*)
```
```   171 AddSIs [equalityI];
```
```   172
```
```   173 (* Equality rules from ZF set theory -- are they appropriate here? *)
```
```   174 val prems = goal Set.thy "A = B ==> A<=(B::'a set)";
```
```   175 by (resolve_tac (prems RL [subst]) 1);
```
```   176 by (rtac subset_refl 1);
```
```   177 qed "equalityD1";
```
```   178
```
```   179 val prems = goal Set.thy "A = B ==> B<=(A::'a set)";
```
```   180 by (resolve_tac (prems RL [subst]) 1);
```
```   181 by (rtac subset_refl 1);
```
```   182 qed "equalityD2";
```
```   183
```
```   184 val prems = goal Set.thy
```
```   185     "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";
```
```   186 by (resolve_tac prems 1);
```
```   187 by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
```
```   188 qed "equalityE";
```
```   189
```
```   190 val major::prems = goal Set.thy
```
```   191     "[| A = B;  [| c:A; c:B |] ==> P;  [| c~:A; c~:B |] ==> P |]  ==>  P";
```
```   192 by (rtac (major RS equalityE) 1);
```
```   193 by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
```
```   194 qed "equalityCE";
```
```   195
```
```   196 (*Lemma for creating induction formulae -- for "pattern matching" on p
```
```   197   To make the induction hypotheses usable, apply "spec" or "bspec" to
```
```   198   put universal quantifiers over the free variables in p. *)
```
```   199 val prems = goal Set.thy
```
```   200     "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
```
```   201 by (rtac mp 1);
```
```   202 by (REPEAT (resolve_tac (refl::prems) 1));
```
```   203 qed "setup_induction";
```
```   204
```
```   205
```
```   206 section "The empty set -- {}";
```
```   207
```
```   208 qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False"
```
```   209  (fn _ => [ (Blast_tac 1) ]);
```
```   210
```
```   211 Addsimps [empty_iff];
```
```   212
```
```   213 qed_goal "emptyE" Set.thy "!!a. a:{} ==> P"
```
```   214  (fn _ => [Full_simp_tac 1]);
```
```   215
```
```   216 AddSEs [emptyE];
```
```   217
```
```   218 qed_goal "empty_subsetI" Set.thy "{} <= A"
```
```   219  (fn _ => [ (Blast_tac 1) ]);
```
```   220
```
```   221 qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}"
```
```   222  (fn [prem]=>
```
```   223   [ (fast_tac (!claset addIs [prem RS FalseE]) 1) ]);
```
```   224
```
```   225 qed_goal "equals0D" Set.thy "!!a. [| A={};  a:A |] ==> P"
```
```   226  (fn _ => [ (Blast_tac 1) ]);
```
```   227
```
```   228 goal Set.thy "Ball {} P = True";
```
```   229 by (simp_tac (HOL_ss addsimps [mem_Collect_eq, Ball_def, empty_def]) 1);
```
```   230 qed "ball_empty";
```
```   231
```
```   232 goal Set.thy "Bex {} P = False";
```
```   233 by (simp_tac (HOL_ss addsimps [mem_Collect_eq, Bex_def, empty_def]) 1);
```
```   234 qed "bex_empty";
```
```   235 Addsimps [ball_empty, bex_empty];
```
```   236
```
```   237 goalw Set.thy [Ball_def] "(!x:A.False) = (A = {})";
```
```   238 by(Blast_tac 1);
```
```   239 qed "ball_False";
```
```   240 Addsimps [ball_False];
```
```   241
```
```   242 (* The dual is probably not helpful:
```
```   243 goal Set.thy "(? x:A.True) = (A ~= {})";
```
```   244 by(Blast_tac 1);
```
```   245 qed "bex_True";
```
```   246 Addsimps [bex_True];
```
```   247 *)
```
```   248
```
```   249
```
```   250 section "The Powerset operator -- Pow";
```
```   251
```
```   252 qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)"
```
```   253  (fn _ => [ (Asm_simp_tac 1) ]);
```
```   254
```
```   255 AddIffs [Pow_iff];
```
```   256
```
```   257 qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
```
```   258  (fn _ => [ (etac CollectI 1) ]);
```
```   259
```
```   260 qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B)  ==>  A<=B"
```
```   261  (fn _=> [ (etac CollectD 1) ]);
```
```   262
```
```   263 val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)
```
```   264 val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)
```
```   265
```
```   266
```
```   267 section "Set complement -- Compl";
```
```   268
```
```   269 qed_goalw "Compl_iff" Set.thy [Compl_def] "(c : Compl(A)) = (c~:A)"
```
```   270  (fn _ => [ (Blast_tac 1) ]);
```
```   271
```
```   272 Addsimps [Compl_iff];
```
```   273
```
```   274 val prems = goalw Set.thy [Compl_def]
```
```   275     "[| c:A ==> False |] ==> c : Compl(A)";
```
```   276 by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
```
```   277 qed "ComplI";
```
```   278
```
```   279 (*This form, with negated conclusion, works well with the Classical prover.
```
```   280   Negated assumptions behave like formulae on the right side of the notional
```
```   281   turnstile...*)
```
```   282 val major::prems = goalw Set.thy [Compl_def]
```
```   283     "c : Compl(A) ==> c~:A";
```
```   284 by (rtac (major RS CollectD) 1);
```
```   285 qed "ComplD";
```
```   286
```
```   287 val ComplE = make_elim ComplD;
```
```   288
```
```   289 AddSIs [ComplI];
```
```   290 AddSEs [ComplE];
```
```   291
```
```   292
```
```   293 section "Binary union -- Un";
```
```   294
```
```   295 qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)"
```
```   296  (fn _ => [ Blast_tac 1 ]);
```
```   297
```
```   298 Addsimps [Un_iff];
```
```   299
```
```   300 goal Set.thy "!!c. c:A ==> c : A Un B";
```
```   301 by (Asm_simp_tac 1);
```
```   302 qed "UnI1";
```
```   303
```
```   304 goal Set.thy "!!c. c:B ==> c : A Un B";
```
```   305 by (Asm_simp_tac 1);
```
```   306 qed "UnI2";
```
```   307
```
```   308 (*Classical introduction rule: no commitment to A vs B*)
```
```   309 qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B"
```
```   310  (fn prems=>
```
```   311   [ (Simp_tac 1),
```
```   312     (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
```
```   313
```
```   314 val major::prems = goalw Set.thy [Un_def]
```
```   315     "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
```
```   316 by (rtac (major RS CollectD RS disjE) 1);
```
```   317 by (REPEAT (eresolve_tac prems 1));
```
```   318 qed "UnE";
```
```   319
```
```   320 AddSIs [UnCI];
```
```   321 AddSEs [UnE];
```
```   322
```
```   323
```
```   324 section "Binary intersection -- Int";
```
```   325
```
```   326 qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)"
```
```   327  (fn _ => [ (Blast_tac 1) ]);
```
```   328
```
```   329 Addsimps [Int_iff];
```
```   330
```
```   331 goal Set.thy "!!c. [| c:A;  c:B |] ==> c : A Int B";
```
```   332 by (Asm_simp_tac 1);
```
```   333 qed "IntI";
```
```   334
```
```   335 goal Set.thy "!!c. c : A Int B ==> c:A";
```
```   336 by (Asm_full_simp_tac 1);
```
```   337 qed "IntD1";
```
```   338
```
```   339 goal Set.thy "!!c. c : A Int B ==> c:B";
```
```   340 by (Asm_full_simp_tac 1);
```
```   341 qed "IntD2";
```
```   342
```
```   343 val [major,minor] = goal Set.thy
```
```   344     "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
```
```   345 by (rtac minor 1);
```
```   346 by (rtac (major RS IntD1) 1);
```
```   347 by (rtac (major RS IntD2) 1);
```
```   348 qed "IntE";
```
```   349
```
```   350 AddSIs [IntI];
```
```   351 AddSEs [IntE];
```
```   352
```
```   353 section "Set difference";
```
```   354
```
```   355 qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)"
```
```   356  (fn _ => [ (Blast_tac 1) ]);
```
```   357
```
```   358 Addsimps [Diff_iff];
```
```   359
```
```   360 qed_goal "DiffI" Set.thy "!!c. [| c : A;  c ~: B |] ==> c : A - B"
```
```   361  (fn _=> [ Asm_simp_tac 1 ]);
```
```   362
```
```   363 qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A"
```
```   364  (fn _=> [ (Asm_full_simp_tac 1) ]);
```
```   365
```
```   366 qed_goal "DiffD2" Set.thy "!!c. [| c : A - B;  c : B |] ==> P"
```
```   367  (fn _=> [ (Asm_full_simp_tac 1) ]);
```
```   368
```
```   369 qed_goal "DiffE" Set.thy "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P"
```
```   370  (fn prems=>
```
```   371   [ (resolve_tac prems 1),
```
```   372     (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]);
```
```   373
```
```   374 AddSIs [DiffI];
```
```   375 AddSEs [DiffE];
```
```   376
```
```   377
```
```   378 section "Augmenting a set -- insert";
```
```   379
```
```   380 qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)"
```
```   381  (fn _ => [Blast_tac 1]);
```
```   382
```
```   383 Addsimps [insert_iff];
```
```   384
```
```   385 qed_goal "insertI1" Set.thy "a : insert a B"
```
```   386  (fn _ => [Simp_tac 1]);
```
```   387
```
```   388 qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B"
```
```   389  (fn _=> [Asm_simp_tac 1]);
```
```   390
```
```   391 qed_goalw "insertE" Set.thy [insert_def]
```
```   392     "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P"
```
```   393  (fn major::prems=>
```
```   394   [ (rtac (major RS UnE) 1),
```
```   395     (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
```
```   396
```
```   397 (*Classical introduction rule*)
```
```   398 qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
```
```   399  (fn prems=>
```
```   400   [ (Simp_tac 1),
```
```   401     (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
```
```   402
```
```   403 AddSIs [insertCI];
```
```   404 AddSEs [insertE];
```
```   405
```
```   406 section "Singletons, using insert";
```
```   407
```
```   408 qed_goal "singletonI" Set.thy "a : {a}"
```
```   409  (fn _=> [ (rtac insertI1 1) ]);
```
```   410
```
```   411 goal Set.thy "!!a. b : {a} ==> b=a";
```
```   412 by (Blast_tac 1);
```
```   413 qed "singletonD";
```
```   414
```
```   415 bind_thm ("singletonE", make_elim singletonD);
```
```   416
```
```   417 qed_goal "singleton_iff" thy "(b : {a}) = (b=a)"
```
```   418 (fn _ => [Blast_tac 1]);
```
```   419
```
```   420 goal Set.thy "!!a b. {a}={b} ==> a=b";
```
```   421 by (fast_tac (!claset addEs [equalityE]) 1);
```
```   422 qed "singleton_inject";
```
```   423
```
```   424 (*Redundant? But unlike insertCI, it proves the subgoal immediately!*)
```
```   425 AddSIs [singletonI];
```
```   426
```
```   427 AddSDs [singleton_inject];
```
```   428
```
```   429
```
```   430 section "The universal set -- UNIV";
```
```   431
```
```   432 qed_goal "UNIV_I" Set.thy "x : UNIV"
```
```   433   (fn _ => [rtac ComplI 1, etac emptyE 1]);
```
```   434
```
```   435 qed_goal "subset_UNIV" Set.thy "A <= UNIV"
```
```   436   (fn _ => [rtac subsetI 1, rtac UNIV_I 1]);
```
```   437
```
```   438
```
```   439 section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
```
```   440
```
```   441 goalw Set.thy [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
```
```   442 by (Blast_tac 1);
```
```   443 qed "UN_iff";
```
```   444
```
```   445 Addsimps [UN_iff];
```
```   446
```
```   447 (*The order of the premises presupposes that A is rigid; b may be flexible*)
```
```   448 goal Set.thy "!!b. [| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
```
```   449 by (Auto_tac());
```
```   450 qed "UN_I";
```
```   451
```
```   452 val major::prems = goalw Set.thy [UNION_def]
```
```   453     "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
```
```   454 by (rtac (major RS CollectD RS bexE) 1);
```
```   455 by (REPEAT (ares_tac prems 1));
```
```   456 qed "UN_E";
```
```   457
```
```   458 AddIs  [UN_I];
```
```   459 AddSEs [UN_E];
```
```   460
```
```   461 val prems = goal Set.thy
```
```   462     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
```
```   463 \    (UN x:A. C(x)) = (UN x:B. D(x))";
```
```   464 by (REPEAT (etac UN_E 1
```
```   465      ORELSE ares_tac ([UN_I,equalityI,subsetI] @
```
```   466                       (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
```
```   467 qed "UN_cong";
```
```   468
```
```   469
```
```   470 section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
```
```   471
```
```   472 goalw Set.thy [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
```
```   473 by (Auto_tac());
```
```   474 qed "INT_iff";
```
```   475
```
```   476 Addsimps [INT_iff];
```
```   477
```
```   478 val prems = goalw Set.thy [INTER_def]
```
```   479     "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
```
```   480 by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
```
```   481 qed "INT_I";
```
```   482
```
```   483 goal Set.thy "!!b. [| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
```
```   484 by (Auto_tac());
```
```   485 qed "INT_D";
```
```   486
```
```   487 (*"Classical" elimination -- by the Excluded Middle on a:A *)
```
```   488 val major::prems = goalw Set.thy [INTER_def]
```
```   489     "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
```
```   490 by (rtac (major RS CollectD RS ballE) 1);
```
```   491 by (REPEAT (eresolve_tac prems 1));
```
```   492 qed "INT_E";
```
```   493
```
```   494 AddSIs [INT_I];
```
```   495 AddEs  [INT_D, INT_E];
```
```   496
```
```   497 val prems = goal Set.thy
```
```   498     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
```
```   499 \    (INT x:A. C(x)) = (INT x:B. D(x))";
```
```   500 by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
```
```   501 by (REPEAT (dtac INT_D 1
```
```   502      ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
```
```   503 qed "INT_cong";
```
```   504
```
```   505
```
```   506 section "Unions over a type; UNION1(B) = Union(range(B))";
```
```   507
```
```   508 goalw Set.thy [UNION1_def] "(b: (UN x. B(x))) = (EX x. b: B(x))";
```
```   509 by (Simp_tac 1);
```
```   510 by (Blast_tac 1);
```
```   511 qed "UN1_iff";
```
```   512
```
```   513 Addsimps [UN1_iff];
```
```   514
```
```   515 (*The order of the premises presupposes that A is rigid; b may be flexible*)
```
```   516 goal Set.thy "!!b. b: B(x) ==> b: (UN x. B(x))";
```
```   517 by (Auto_tac());
```
```   518 qed "UN1_I";
```
```   519
```
```   520 val major::prems = goalw Set.thy [UNION1_def]
```
```   521     "[| b : (UN x. B(x));  !!x. b: B(x) ==> R |] ==> R";
```
```   522 by (rtac (major RS UN_E) 1);
```
```   523 by (REPEAT (ares_tac prems 1));
```
```   524 qed "UN1_E";
```
```   525
```
```   526 AddIs  [UN1_I];
```
```   527 AddSEs [UN1_E];
```
```   528
```
```   529
```
```   530 section "Intersections over a type; INTER1(B) = Inter(range(B))";
```
```   531
```
```   532 goalw Set.thy [INTER1_def] "(b: (INT x. B(x))) = (ALL x. b: B(x))";
```
```   533 by (Simp_tac 1);
```
```   534 by (Blast_tac 1);
```
```   535 qed "INT1_iff";
```
```   536
```
```   537 Addsimps [INT1_iff];
```
```   538
```
```   539 val prems = goalw Set.thy [INTER1_def]
```
```   540     "(!!x. b: B(x)) ==> b : (INT x. B(x))";
```
```   541 by (REPEAT (ares_tac (INT_I::prems) 1));
```
```   542 qed "INT1_I";
```
```   543
```
```   544 goal Set.thy "!!b. b : (INT x. B(x)) ==> b: B(a)";
```
```   545 by (Asm_full_simp_tac 1);
```
```   546 qed "INT1_D";
```
```   547
```
```   548 AddSIs [INT1_I];
```
```   549 AddDs  [INT1_D];
```
```   550
```
```   551
```
```   552 section "Union";
```
```   553
```
```   554 goalw Set.thy [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
```
```   555 by (Blast_tac 1);
```
```   556 qed "Union_iff";
```
```   557
```
```   558 Addsimps [Union_iff];
```
```   559
```
```   560 (*The order of the premises presupposes that C is rigid; A may be flexible*)
```
```   561 goal Set.thy "!!X. [| X:C;  A:X |] ==> A : Union(C)";
```
```   562 by (Auto_tac());
```
```   563 qed "UnionI";
```
```   564
```
```   565 val major::prems = goalw Set.thy [Union_def]
```
```   566     "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
```
```   567 by (rtac (major RS UN_E) 1);
```
```   568 by (REPEAT (ares_tac prems 1));
```
```   569 qed "UnionE";
```
```   570
```
```   571 AddIs  [UnionI];
```
```   572 AddSEs [UnionE];
```
```   573
```
```   574
```
```   575 section "Inter";
```
```   576
```
```   577 goalw Set.thy [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
```
```   578 by (Blast_tac 1);
```
```   579 qed "Inter_iff";
```
```   580
```
```   581 Addsimps [Inter_iff];
```
```   582
```
```   583 val prems = goalw Set.thy [Inter_def]
```
```   584     "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
```
```   585 by (REPEAT (ares_tac ([INT_I] @ prems) 1));
```
```   586 qed "InterI";
```
```   587
```
```   588 (*A "destruct" rule -- every X in C contains A as an element, but
```
```   589   A:X can hold when X:C does not!  This rule is analogous to "spec". *)
```
```   590 goal Set.thy "!!X. [| A : Inter(C);  X:C |] ==> A:X";
```
```   591 by (Auto_tac());
```
```   592 qed "InterD";
```
```   593
```
```   594 (*"Classical" elimination rule -- does not require proving X:C *)
```
```   595 val major::prems = goalw Set.thy [Inter_def]
```
```   596     "[| A : Inter(C);  X~:C ==> R;  A:X ==> R |] ==> R";
```
```   597 by (rtac (major RS INT_E) 1);
```
```   598 by (REPEAT (eresolve_tac prems 1));
```
```   599 qed "InterE";
```
```   600
```
```   601 AddSIs [InterI];
```
```   602 AddEs  [InterD, InterE];
```
```   603
```
```   604
```
```   605 (*** Image of a set under a function ***)
```
```   606
```
```   607 (*Frequently b does not have the syntactic form of f(x).*)
```
```   608 val prems = goalw thy [image_def] "[| b=f(x);  x:A |] ==> b : f``A";
```
```   609 by (REPEAT (resolve_tac (prems @ [CollectI,bexI,prem]) 1));
```
```   610 qed "image_eqI";
```
```   611
```
```   612 bind_thm ("imageI", refl RS image_eqI);
```
```   613
```
```   614 (*The eta-expansion gives variable-name preservation.*)
```
```   615 val major::prems = goalw thy [image_def]
```
```   616     "[| b : (%x.f(x))``A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P";
```
```   617 by (rtac (major RS CollectD RS bexE) 1);
```
```   618 by (REPEAT (ares_tac prems 1));
```
```   619 qed "imageE";
```
```   620
```
```   621 AddIs  [image_eqI];
```
```   622 AddSEs [imageE];
```
```   623
```
```   624 goalw thy [o_def] "(f o g)``r = f``(g``r)";
```
```   625 by (Fast_tac 1);
```
```   626 qed "image_compose";
```
```   627
```
```   628 goal thy "f``(A Un B) = f``A Un f``B";
```
```   629 by (Fast_tac 1);
```
```   630 qed "image_Un";
```
```   631
```
```   632
```
```   633 (*** Range of a function -- just a translation for image! ***)
```
```   634
```
```   635 goal thy "!!b. b=f(x) ==> b : range(f)";
```
```   636 by (EVERY1 [etac image_eqI, rtac UNIV_I]);
```
```   637 bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI));
```
```   638
```
```   639 bind_thm ("rangeI", UNIV_I RS imageI);
```
```   640
```
```   641 val [major,minor] = goal thy
```
```   642     "[| b : range(%x.f(x));  !!x. b=f(x) ==> P |] ==> P";
```
```   643 by (rtac (major RS imageE) 1);
```
```   644 by (etac minor 1);
```
```   645 qed "rangeE";
```
```   646
```
```   647 AddIs  [rangeI];
```
```   648 AddSEs [rangeE];
```
```   649
```
```   650
```
```   651 (*** Set reasoning tools ***)
```
```   652
```
```   653
```
```   654 (*Each of these has ALREADY been added to !simpset above.*)
```
```   655 val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff,
```
```   656                  mem_Collect_eq,
```
```   657 		 UN_iff, UN1_iff, Union_iff,
```
```   658 		 INT_iff, INT1_iff, Inter_iff];
```
```   659
```
```   660 (*Not for Addsimps -- it can cause goals to blow up!*)
```
```   661 goal Set.thy "(a : (if Q then x else y)) = ((Q --> a:x) & (~Q --> a : y))";
```
```   662 by (simp_tac (!simpset setloop split_tac [expand_if]) 1);
```
```   663 qed "mem_if";
```
```   664
```
```   665 val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs;
```
```   666
```
```   667 simpset := !simpset addcongs [ball_cong,bex_cong]
```
```   668                     setmksimps (mksimps mksimps_pairs);
```