src/HOL/Set.ML
 author wenzelm Fri Oct 10 19:02:28 1997 +0200 (1997-10-10) changeset 3842 b55686a7b22c parent 3718 d78cf498a88c child 3896 ee8ebb74ec00 permissions -rw-r--r--
fixed dots;
```     1 (*  Title:      HOL/set
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   1991  University of Cambridge
```
```     5
```
```     6 Set theory for higher-order logic.  A set is simply a predicate.
```
```     7 *)
```
```     8
```
```     9 open Set;
```
```    10
```
```    11 section "Relating predicates and sets";
```
```    12
```
```    13 Addsimps [Collect_mem_eq];
```
```    14 AddIffs  [mem_Collect_eq];
```
```    15
```
```    16 goal Set.thy "!!a. P(a) ==> a : {x. P(x)}";
```
```    17 by (Asm_simp_tac 1);
```
```    18 qed "CollectI";
```
```    19
```
```    20 val prems = goal Set.thy "!!a. a : {x. P(x)} ==> P(a)";
```
```    21 by (Asm_full_simp_tac 1);
```
```    22 qed "CollectD";
```
```    23
```
```    24 val [prem] = goal Set.thy "[| !!x. (x:A) = (x:B) |] ==> A = B";
```
```    25 by (rtac (prem RS ext RS arg_cong RS box_equals) 1);
```
```    26 by (rtac Collect_mem_eq 1);
```
```    27 by (rtac Collect_mem_eq 1);
```
```    28 qed "set_ext";
```
```    29
```
```    30 val [prem] = goal Set.thy "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}";
```
```    31 by (rtac (prem RS ext RS arg_cong) 1);
```
```    32 qed "Collect_cong";
```
```    33
```
```    34 val CollectE = make_elim CollectD;
```
```    35
```
```    36 AddSIs [CollectI];
```
```    37 AddSEs [CollectE];
```
```    38
```
```    39
```
```    40 section "Bounded quantifiers";
```
```    41
```
```    42 val prems = goalw Set.thy [Ball_def]
```
```    43     "[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)";
```
```    44 by (REPEAT (ares_tac (prems @ [allI,impI]) 1));
```
```    45 qed "ballI";
```
```    46
```
```    47 val [major,minor] = goalw Set.thy [Ball_def]
```
```    48     "[| ! x:A. P(x);  x:A |] ==> P(x)";
```
```    49 by (rtac (minor RS (major RS spec RS mp)) 1);
```
```    50 qed "bspec";
```
```    51
```
```    52 val major::prems = goalw Set.thy [Ball_def]
```
```    53     "[| ! x:A. P(x);  P(x) ==> Q;  x~:A ==> Q |] ==> Q";
```
```    54 by (rtac (major RS spec RS impCE) 1);
```
```    55 by (REPEAT (eresolve_tac prems 1));
```
```    56 qed "ballE";
```
```    57
```
```    58 (*Takes assumptions ! x:A.P(x) and a:A; creates assumption P(a)*)
```
```    59 fun ball_tac i = etac ballE i THEN contr_tac (i+1);
```
```    60
```
```    61 AddSIs [ballI];
```
```    62 AddEs  [ballE];
```
```    63
```
```    64 val prems = goalw Set.thy [Bex_def]
```
```    65     "[| P(x);  x:A |] ==> ? x:A. P(x)";
```
```    66 by (REPEAT (ares_tac (prems @ [exI,conjI]) 1));
```
```    67 qed "bexI";
```
```    68
```
```    69 qed_goal "bexCI" Set.thy
```
```    70    "[| ! x:A. ~P(x) ==> P(a);  a:A |] ==> ? x:A. P(x)"
```
```    71  (fn prems=>
```
```    72   [ (rtac classical 1),
```
```    73     (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1))  ]);
```
```    74
```
```    75 val major::prems = goalw Set.thy [Bex_def]
```
```    76     "[| ? x:A. P(x);  !!x. [| x:A; P(x) |] ==> Q  |] ==> Q";
```
```    77 by (rtac (major RS exE) 1);
```
```    78 by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1));
```
```    79 qed "bexE";
```
```    80
```
```    81 AddIs  [bexI];
```
```    82 AddSEs [bexE];
```
```    83
```
```    84 (*Trival rewrite rule*)
```
```    85 goal Set.thy "(! x:A. P) = ((? x. x:A) --> P)";
```
```    86 by (simp_tac (!simpset addsimps [Ball_def]) 1);
```
```    87 qed "ball_triv";
```
```    88
```
```    89 (*Dual form for existentials*)
```
```    90 goal Set.thy "(? x:A. P) = ((? x. x:A) & P)";
```
```    91 by (simp_tac (!simpset addsimps [Bex_def]) 1);
```
```    92 qed "bex_triv";
```
```    93
```
```    94 Addsimps [ball_triv, bex_triv];
```
```    95
```
```    96 (** Congruence rules **)
```
```    97
```
```    98 val prems = goal Set.thy
```
```    99     "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
```
```   100 \    (! x:A. P(x)) = (! x:B. Q(x))";
```
```   101 by (resolve_tac (prems RL [ssubst]) 1);
```
```   102 by (REPEAT (ares_tac [ballI,iffI] 1
```
```   103      ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1));
```
```   104 qed "ball_cong";
```
```   105
```
```   106 val prems = goal Set.thy
```
```   107     "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
```
```   108 \    (? x:A. P(x)) = (? x:B. Q(x))";
```
```   109 by (resolve_tac (prems RL [ssubst]) 1);
```
```   110 by (REPEAT (etac bexE 1
```
```   111      ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1));
```
```   112 qed "bex_cong";
```
```   113
```
```   114 section "Subsets";
```
```   115
```
```   116 val prems = goalw Set.thy [subset_def] "(!!x. x:A ==> x:B) ==> A <= B";
```
```   117 by (REPEAT (ares_tac (prems @ [ballI]) 1));
```
```   118 qed "subsetI";
```
```   119
```
```   120 Blast.declConsts (["op <="], [subsetI]);	(*overloading of <=*)
```
```   121
```
```   122 (*Rule in Modus Ponens style*)
```
```   123 val major::prems = goalw Set.thy [subset_def] "[| A <= B;  c:A |] ==> c:B";
```
```   124 by (rtac (major RS bspec) 1);
```
```   125 by (resolve_tac prems 1);
```
```   126 qed "subsetD";
```
```   127
```
```   128 (*The same, with reversed premises for use with etac -- cf rev_mp*)
```
```   129 qed_goal "rev_subsetD" Set.thy "[| c:A;  A <= B |] ==> c:B"
```
```   130  (fn prems=>  [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]);
```
```   131
```
```   132 (*Converts A<=B to x:A ==> x:B*)
```
```   133 fun impOfSubs th = th RSN (2, rev_subsetD);
```
```   134
```
```   135 qed_goal "contra_subsetD" Set.thy "!!c. [| A <= B; c ~: B |] ==> c ~: A"
```
```   136  (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
```
```   137
```
```   138 qed_goal "rev_contra_subsetD" Set.thy "!!c. [| c ~: B;  A <= B |] ==> c ~: A"
```
```   139  (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
```
```   140
```
```   141 (*Classical elimination rule*)
```
```   142 val major::prems = goalw Set.thy [subset_def]
```
```   143     "[| A <= B;  c~:A ==> P;  c:B ==> P |] ==> P";
```
```   144 by (rtac (major RS ballE) 1);
```
```   145 by (REPEAT (eresolve_tac prems 1));
```
```   146 qed "subsetCE";
```
```   147
```
```   148 (*Takes assumptions A<=B; c:A and creates the assumption c:B *)
```
```   149 fun set_mp_tac i = etac subsetCE i  THEN  mp_tac i;
```
```   150
```
```   151 AddSIs [subsetI];
```
```   152 AddEs  [subsetD, subsetCE];
```
```   153
```
```   154 qed_goal "subset_refl" Set.thy "A <= (A::'a set)"
```
```   155  (fn _=> [Blast_tac 1]);
```
```   156
```
```   157 val prems = goal Set.thy "!!B. [| A<=B;  B<=C |] ==> A<=(C::'a set)";
```
```   158 by (Blast_tac 1);
```
```   159 qed "subset_trans";
```
```   160
```
```   161
```
```   162 section "Equality";
```
```   163
```
```   164 (*Anti-symmetry of the subset relation*)
```
```   165 val prems = goal Set.thy "[| A <= B;  B <= A |] ==> A = (B::'a set)";
```
```   166 by (rtac (iffI RS set_ext) 1);
```
```   167 by (REPEAT (ares_tac (prems RL [subsetD]) 1));
```
```   168 qed "subset_antisym";
```
```   169 val equalityI = subset_antisym;
```
```   170
```
```   171 Blast.declConsts (["op ="], [equalityI]);	(*overloading of equality*)
```
```   172 AddSIs [equalityI];
```
```   173
```
```   174 (* Equality rules from ZF set theory -- are they appropriate here? *)
```
```   175 val prems = goal Set.thy "A = B ==> A<=(B::'a set)";
```
```   176 by (resolve_tac (prems RL [subst]) 1);
```
```   177 by (rtac subset_refl 1);
```
```   178 qed "equalityD1";
```
```   179
```
```   180 val prems = goal Set.thy "A = B ==> B<=(A::'a set)";
```
```   181 by (resolve_tac (prems RL [subst]) 1);
```
```   182 by (rtac subset_refl 1);
```
```   183 qed "equalityD2";
```
```   184
```
```   185 val prems = goal Set.thy
```
```   186     "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";
```
```   187 by (resolve_tac prems 1);
```
```   188 by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
```
```   189 qed "equalityE";
```
```   190
```
```   191 val major::prems = goal Set.thy
```
```   192     "[| A = B;  [| c:A; c:B |] ==> P;  [| c~:A; c~:B |] ==> P |]  ==>  P";
```
```   193 by (rtac (major RS equalityE) 1);
```
```   194 by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
```
```   195 qed "equalityCE";
```
```   196
```
```   197 (*Lemma for creating induction formulae -- for "pattern matching" on p
```
```   198   To make the induction hypotheses usable, apply "spec" or "bspec" to
```
```   199   put universal quantifiers over the free variables in p. *)
```
```   200 val prems = goal Set.thy
```
```   201     "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
```
```   202 by (rtac mp 1);
```
```   203 by (REPEAT (resolve_tac (refl::prems) 1));
```
```   204 qed "setup_induction";
```
```   205
```
```   206
```
```   207 section "The empty set -- {}";
```
```   208
```
```   209 qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False"
```
```   210  (fn _ => [ (Blast_tac 1) ]);
```
```   211
```
```   212 Addsimps [empty_iff];
```
```   213
```
```   214 qed_goal "emptyE" Set.thy "!!a. a:{} ==> P"
```
```   215  (fn _ => [Full_simp_tac 1]);
```
```   216
```
```   217 AddSEs [emptyE];
```
```   218
```
```   219 qed_goal "empty_subsetI" Set.thy "{} <= A"
```
```   220  (fn _ => [ (Blast_tac 1) ]);
```
```   221
```
```   222 qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}"
```
```   223  (fn [prem]=>
```
```   224   [ (blast_tac (!claset addIs [prem RS FalseE]) 1) ]);
```
```   225
```
```   226 qed_goal "equals0D" Set.thy "!!a. [| A={};  a:A |] ==> P"
```
```   227  (fn _ => [ (Blast_tac 1) ]);
```
```   228
```
```   229 goal Set.thy "Ball {} P = True";
```
```   230 by (simp_tac (HOL_ss addsimps [mem_Collect_eq, Ball_def, empty_def]) 1);
```
```   231 qed "ball_empty";
```
```   232
```
```   233 goal Set.thy "Bex {} P = False";
```
```   234 by (simp_tac (HOL_ss addsimps [mem_Collect_eq, Bex_def, empty_def]) 1);
```
```   235 qed "bex_empty";
```
```   236 Addsimps [ball_empty, bex_empty];
```
```   237
```
```   238
```
```   239 section "The Powerset operator -- Pow";
```
```   240
```
```   241 qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)"
```
```   242  (fn _ => [ (Asm_simp_tac 1) ]);
```
```   243
```
```   244 AddIffs [Pow_iff];
```
```   245
```
```   246 qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
```
```   247  (fn _ => [ (etac CollectI 1) ]);
```
```   248
```
```   249 qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B)  ==>  A<=B"
```
```   250  (fn _=> [ (etac CollectD 1) ]);
```
```   251
```
```   252 val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)
```
```   253 val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)
```
```   254
```
```   255
```
```   256 section "Set complement -- Compl";
```
```   257
```
```   258 qed_goalw "Compl_iff" Set.thy [Compl_def] "(c : Compl(A)) = (c~:A)"
```
```   259  (fn _ => [ (Blast_tac 1) ]);
```
```   260
```
```   261 Addsimps [Compl_iff];
```
```   262
```
```   263 val prems = goalw Set.thy [Compl_def]
```
```   264     "[| c:A ==> False |] ==> c : Compl(A)";
```
```   265 by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
```
```   266 qed "ComplI";
```
```   267
```
```   268 (*This form, with negated conclusion, works well with the Classical prover.
```
```   269   Negated assumptions behave like formulae on the right side of the notional
```
```   270   turnstile...*)
```
```   271 val major::prems = goalw Set.thy [Compl_def]
```
```   272     "c : Compl(A) ==> c~:A";
```
```   273 by (rtac (major RS CollectD) 1);
```
```   274 qed "ComplD";
```
```   275
```
```   276 val ComplE = make_elim ComplD;
```
```   277
```
```   278 AddSIs [ComplI];
```
```   279 AddSEs [ComplE];
```
```   280
```
```   281
```
```   282 section "Binary union -- Un";
```
```   283
```
```   284 qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)"
```
```   285  (fn _ => [ Blast_tac 1 ]);
```
```   286
```
```   287 Addsimps [Un_iff];
```
```   288
```
```   289 goal Set.thy "!!c. c:A ==> c : A Un B";
```
```   290 by (Asm_simp_tac 1);
```
```   291 qed "UnI1";
```
```   292
```
```   293 goal Set.thy "!!c. c:B ==> c : A Un B";
```
```   294 by (Asm_simp_tac 1);
```
```   295 qed "UnI2";
```
```   296
```
```   297 (*Classical introduction rule: no commitment to A vs B*)
```
```   298 qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B"
```
```   299  (fn prems=>
```
```   300   [ (Simp_tac 1),
```
```   301     (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
```
```   302
```
```   303 val major::prems = goalw Set.thy [Un_def]
```
```   304     "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
```
```   305 by (rtac (major RS CollectD RS disjE) 1);
```
```   306 by (REPEAT (eresolve_tac prems 1));
```
```   307 qed "UnE";
```
```   308
```
```   309 AddSIs [UnCI];
```
```   310 AddSEs [UnE];
```
```   311
```
```   312
```
```   313 section "Binary intersection -- Int";
```
```   314
```
```   315 qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)"
```
```   316  (fn _ => [ (Blast_tac 1) ]);
```
```   317
```
```   318 Addsimps [Int_iff];
```
```   319
```
```   320 goal Set.thy "!!c. [| c:A;  c:B |] ==> c : A Int B";
```
```   321 by (Asm_simp_tac 1);
```
```   322 qed "IntI";
```
```   323
```
```   324 goal Set.thy "!!c. c : A Int B ==> c:A";
```
```   325 by (Asm_full_simp_tac 1);
```
```   326 qed "IntD1";
```
```   327
```
```   328 goal Set.thy "!!c. c : A Int B ==> c:B";
```
```   329 by (Asm_full_simp_tac 1);
```
```   330 qed "IntD2";
```
```   331
```
```   332 val [major,minor] = goal Set.thy
```
```   333     "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
```
```   334 by (rtac minor 1);
```
```   335 by (rtac (major RS IntD1) 1);
```
```   336 by (rtac (major RS IntD2) 1);
```
```   337 qed "IntE";
```
```   338
```
```   339 AddSIs [IntI];
```
```   340 AddSEs [IntE];
```
```   341
```
```   342 section "Set difference";
```
```   343
```
```   344 qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)"
```
```   345  (fn _ => [ (Blast_tac 1) ]);
```
```   346
```
```   347 Addsimps [Diff_iff];
```
```   348
```
```   349 qed_goal "DiffI" Set.thy "!!c. [| c : A;  c ~: B |] ==> c : A - B"
```
```   350  (fn _=> [ Asm_simp_tac 1 ]);
```
```   351
```
```   352 qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A"
```
```   353  (fn _=> [ (Asm_full_simp_tac 1) ]);
```
```   354
```
```   355 qed_goal "DiffD2" Set.thy "!!c. [| c : A - B;  c : B |] ==> P"
```
```   356  (fn _=> [ (Asm_full_simp_tac 1) ]);
```
```   357
```
```   358 qed_goal "DiffE" Set.thy "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P"
```
```   359  (fn prems=>
```
```   360   [ (resolve_tac prems 1),
```
```   361     (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]);
```
```   362
```
```   363 AddSIs [DiffI];
```
```   364 AddSEs [DiffE];
```
```   365
```
```   366
```
```   367 section "Augmenting a set -- insert";
```
```   368
```
```   369 qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)"
```
```   370  (fn _ => [Blast_tac 1]);
```
```   371
```
```   372 Addsimps [insert_iff];
```
```   373
```
```   374 qed_goal "insertI1" Set.thy "a : insert a B"
```
```   375  (fn _ => [Simp_tac 1]);
```
```   376
```
```   377 qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B"
```
```   378  (fn _=> [Asm_simp_tac 1]);
```
```   379
```
```   380 qed_goalw "insertE" Set.thy [insert_def]
```
```   381     "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P"
```
```   382  (fn major::prems=>
```
```   383   [ (rtac (major RS UnE) 1),
```
```   384     (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
```
```   385
```
```   386 (*Classical introduction rule*)
```
```   387 qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
```
```   388  (fn prems=>
```
```   389   [ (Simp_tac 1),
```
```   390     (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
```
```   391
```
```   392 AddSIs [insertCI];
```
```   393 AddSEs [insertE];
```
```   394
```
```   395 section "Singletons, using insert";
```
```   396
```
```   397 qed_goal "singletonI" Set.thy "a : {a}"
```
```   398  (fn _=> [ (rtac insertI1 1) ]);
```
```   399
```
```   400 goal Set.thy "!!a. b : {a} ==> b=a";
```
```   401 by (Blast_tac 1);
```
```   402 qed "singletonD";
```
```   403
```
```   404 bind_thm ("singletonE", make_elim singletonD);
```
```   405
```
```   406 qed_goal "singleton_iff" thy "(b : {a}) = (b=a)"
```
```   407 (fn _ => [Blast_tac 1]);
```
```   408
```
```   409 goal Set.thy "!!a b. {a}={b} ==> a=b";
```
```   410 by (blast_tac (!claset addEs [equalityE]) 1);
```
```   411 qed "singleton_inject";
```
```   412
```
```   413 (*Redundant? But unlike insertCI, it proves the subgoal immediately!*)
```
```   414 AddSIs [singletonI];
```
```   415 AddSDs [singleton_inject];
```
```   416 AddSEs [singletonE];
```
```   417
```
```   418 goal Set.thy "{x. x=a} = {a}";
```
```   419 by(Blast_tac 1);
```
```   420 qed "singleton_conv";
```
```   421 Addsimps [singleton_conv];
```
```   422
```
```   423 section "The universal set -- UNIV";
```
```   424
```
```   425 qed_goal "UNIV_I" Set.thy "x : UNIV"
```
```   426   (fn _ => [rtac ComplI 1, etac emptyE 1]);
```
```   427
```
```   428 qed_goal "subset_UNIV" Set.thy "A <= UNIV"
```
```   429   (fn _ => [rtac subsetI 1, rtac UNIV_I 1]);
```
```   430
```
```   431
```
```   432 section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
```
```   433
```
```   434 goalw Set.thy [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
```
```   435 by (Blast_tac 1);
```
```   436 qed "UN_iff";
```
```   437
```
```   438 Addsimps [UN_iff];
```
```   439
```
```   440 (*The order of the premises presupposes that A is rigid; b may be flexible*)
```
```   441 goal Set.thy "!!b. [| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
```
```   442 by (Auto_tac());
```
```   443 qed "UN_I";
```
```   444
```
```   445 val major::prems = goalw Set.thy [UNION_def]
```
```   446     "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
```
```   447 by (rtac (major RS CollectD RS bexE) 1);
```
```   448 by (REPEAT (ares_tac prems 1));
```
```   449 qed "UN_E";
```
```   450
```
```   451 AddIs  [UN_I];
```
```   452 AddSEs [UN_E];
```
```   453
```
```   454 val prems = goal Set.thy
```
```   455     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
```
```   456 \    (UN x:A. C(x)) = (UN x:B. D(x))";
```
```   457 by (REPEAT (etac UN_E 1
```
```   458      ORELSE ares_tac ([UN_I,equalityI,subsetI] @
```
```   459                       (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
```
```   460 qed "UN_cong";
```
```   461
```
```   462
```
```   463 section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
```
```   464
```
```   465 goalw Set.thy [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
```
```   466 by (Auto_tac());
```
```   467 qed "INT_iff";
```
```   468
```
```   469 Addsimps [INT_iff];
```
```   470
```
```   471 val prems = goalw Set.thy [INTER_def]
```
```   472     "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
```
```   473 by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
```
```   474 qed "INT_I";
```
```   475
```
```   476 goal Set.thy "!!b. [| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
```
```   477 by (Auto_tac());
```
```   478 qed "INT_D";
```
```   479
```
```   480 (*"Classical" elimination -- by the Excluded Middle on a:A *)
```
```   481 val major::prems = goalw Set.thy [INTER_def]
```
```   482     "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
```
```   483 by (rtac (major RS CollectD RS ballE) 1);
```
```   484 by (REPEAT (eresolve_tac prems 1));
```
```   485 qed "INT_E";
```
```   486
```
```   487 AddSIs [INT_I];
```
```   488 AddEs  [INT_D, INT_E];
```
```   489
```
```   490 val prems = goal Set.thy
```
```   491     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
```
```   492 \    (INT x:A. C(x)) = (INT x:B. D(x))";
```
```   493 by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
```
```   494 by (REPEAT (dtac INT_D 1
```
```   495      ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
```
```   496 qed "INT_cong";
```
```   497
```
```   498
```
```   499 section "Unions over a type; UNION1(B) = Union(range(B))";
```
```   500
```
```   501 goalw Set.thy [UNION1_def] "(b: (UN x. B(x))) = (EX x. b: B(x))";
```
```   502 by (Simp_tac 1);
```
```   503 by (Blast_tac 1);
```
```   504 qed "UN1_iff";
```
```   505
```
```   506 Addsimps [UN1_iff];
```
```   507
```
```   508 (*The order of the premises presupposes that A is rigid; b may be flexible*)
```
```   509 goal Set.thy "!!b. b: B(x) ==> b: (UN x. B(x))";
```
```   510 by (Auto_tac());
```
```   511 qed "UN1_I";
```
```   512
```
```   513 val major::prems = goalw Set.thy [UNION1_def]
```
```   514     "[| b : (UN x. B(x));  !!x. b: B(x) ==> R |] ==> R";
```
```   515 by (rtac (major RS UN_E) 1);
```
```   516 by (REPEAT (ares_tac prems 1));
```
```   517 qed "UN1_E";
```
```   518
```
```   519 AddIs  [UN1_I];
```
```   520 AddSEs [UN1_E];
```
```   521
```
```   522
```
```   523 section "Intersections over a type; INTER1(B) = Inter(range(B))";
```
```   524
```
```   525 goalw Set.thy [INTER1_def] "(b: (INT x. B(x))) = (ALL x. b: B(x))";
```
```   526 by (Simp_tac 1);
```
```   527 by (Blast_tac 1);
```
```   528 qed "INT1_iff";
```
```   529
```
```   530 Addsimps [INT1_iff];
```
```   531
```
```   532 val prems = goalw Set.thy [INTER1_def]
```
```   533     "(!!x. b: B(x)) ==> b : (INT x. B(x))";
```
```   534 by (REPEAT (ares_tac (INT_I::prems) 1));
```
```   535 qed "INT1_I";
```
```   536
```
```   537 goal Set.thy "!!b. b : (INT x. B(x)) ==> b: B(a)";
```
```   538 by (Asm_full_simp_tac 1);
```
```   539 qed "INT1_D";
```
```   540
```
```   541 AddSIs [INT1_I];
```
```   542 AddDs  [INT1_D];
```
```   543
```
```   544
```
```   545 section "Union";
```
```   546
```
```   547 goalw Set.thy [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
```
```   548 by (Blast_tac 1);
```
```   549 qed "Union_iff";
```
```   550
```
```   551 Addsimps [Union_iff];
```
```   552
```
```   553 (*The order of the premises presupposes that C is rigid; A may be flexible*)
```
```   554 goal Set.thy "!!X. [| X:C;  A:X |] ==> A : Union(C)";
```
```   555 by (Auto_tac());
```
```   556 qed "UnionI";
```
```   557
```
```   558 val major::prems = goalw Set.thy [Union_def]
```
```   559     "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
```
```   560 by (rtac (major RS UN_E) 1);
```
```   561 by (REPEAT (ares_tac prems 1));
```
```   562 qed "UnionE";
```
```   563
```
```   564 AddIs  [UnionI];
```
```   565 AddSEs [UnionE];
```
```   566
```
```   567
```
```   568 section "Inter";
```
```   569
```
```   570 goalw Set.thy [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
```
```   571 by (Blast_tac 1);
```
```   572 qed "Inter_iff";
```
```   573
```
```   574 Addsimps [Inter_iff];
```
```   575
```
```   576 val prems = goalw Set.thy [Inter_def]
```
```   577     "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
```
```   578 by (REPEAT (ares_tac ([INT_I] @ prems) 1));
```
```   579 qed "InterI";
```
```   580
```
```   581 (*A "destruct" rule -- every X in C contains A as an element, but
```
```   582   A:X can hold when X:C does not!  This rule is analogous to "spec". *)
```
```   583 goal Set.thy "!!X. [| A : Inter(C);  X:C |] ==> A:X";
```
```   584 by (Auto_tac());
```
```   585 qed "InterD";
```
```   586
```
```   587 (*"Classical" elimination rule -- does not require proving X:C *)
```
```   588 val major::prems = goalw Set.thy [Inter_def]
```
```   589     "[| A : Inter(C);  X~:C ==> R;  A:X ==> R |] ==> R";
```
```   590 by (rtac (major RS INT_E) 1);
```
```   591 by (REPEAT (eresolve_tac prems 1));
```
```   592 qed "InterE";
```
```   593
```
```   594 AddSIs [InterI];
```
```   595 AddEs  [InterD, InterE];
```
```   596
```
```   597
```
```   598 (*** Image of a set under a function ***)
```
```   599
```
```   600 (*Frequently b does not have the syntactic form of f(x).*)
```
```   601 val prems = goalw thy [image_def] "[| b=f(x);  x:A |] ==> b : f``A";
```
```   602 by (REPEAT (resolve_tac (prems @ [CollectI,bexI,prem]) 1));
```
```   603 qed "image_eqI";
```
```   604
```
```   605 bind_thm ("imageI", refl RS image_eqI);
```
```   606
```
```   607 (*The eta-expansion gives variable-name preservation.*)
```
```   608 val major::prems = goalw thy [image_def]
```
```   609     "[| b : (%x. f(x))``A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P";
```
```   610 by (rtac (major RS CollectD RS bexE) 1);
```
```   611 by (REPEAT (ares_tac prems 1));
```
```   612 qed "imageE";
```
```   613
```
```   614 AddIs  [image_eqI];
```
```   615 AddSEs [imageE];
```
```   616
```
```   617 goalw thy [o_def] "(f o g)``r = f``(g``r)";
```
```   618 by (Blast_tac 1);
```
```   619 qed "image_compose";
```
```   620
```
```   621 goal thy "f``(A Un B) = f``A Un f``B";
```
```   622 by (Blast_tac 1);
```
```   623 qed "image_Un";
```
```   624
```
```   625
```
```   626 (*** Range of a function -- just a translation for image! ***)
```
```   627
```
```   628 goal thy "!!b. b=f(x) ==> b : range(f)";
```
```   629 by (EVERY1 [etac image_eqI, rtac UNIV_I]);
```
```   630 bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI));
```
```   631
```
```   632 bind_thm ("rangeI", UNIV_I RS imageI);
```
```   633
```
```   634 val [major,minor] = goal thy
```
```   635     "[| b : range(%x. f(x));  !!x. b=f(x) ==> P |] ==> P";
```
```   636 by (rtac (major RS imageE) 1);
```
```   637 by (etac minor 1);
```
```   638 qed "rangeE";
```
```   639
```
```   640 AddIs  [rangeI];
```
```   641 AddSEs [rangeE];
```
```   642
```
```   643
```
```   644 (*** Set reasoning tools ***)
```
```   645
```
```   646
```
```   647 (*Each of these has ALREADY been added to !simpset above.*)
```
```   648 val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff,
```
```   649                  mem_Collect_eq,
```
```   650 		 UN_iff, UN1_iff, Union_iff,
```
```   651 		 INT_iff, INT1_iff, Inter_iff];
```
```   652
```
```   653 (*Not for Addsimps -- it can cause goals to blow up!*)
```
```   654 goal Set.thy "(a : (if Q then x else y)) = ((Q --> a:x) & (~Q --> a : y))";
```
```   655 by (simp_tac (!simpset setloop split_tac [expand_if]) 1);
```
```   656 qed "mem_if";
```
```   657
```
```   658 val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs;
```
```   659
```
```   660 simpset := !simpset addcongs [ball_cong,bex_cong]
```
```   661                     setmksimps (mksimps mksimps_pairs);
```
```   662
```
```   663 Addsimps[subset_UNIV, empty_subsetI, subset_refl];
```
```   664
```
```   665
```
```   666 (*** < ***)
```
```   667
```
```   668 goalw Set.thy [psubset_def] "!!A::'a set. [| A <= B; A ~= B |] ==> A<B";
```
```   669 by (Blast_tac 1);
```
```   670 qed "psubsetI";
```
```   671
```
```   672 goalw Set.thy [psubset_def]
```
```   673     "!!x. A < insert x B ==> (x ~: A) & A<=B | x:A & A-{x}<B";
```
```   674 by (Auto_tac());
```
```   675 qed "psubset_insertD";
```