src/HOL/Set.ML
 author wenzelm Tue Dec 16 17:58:03 1997 +0100 (1997-12-16) changeset 4423 a129b817b58a parent 4240 8ba60a4cd380 child 4434 75f38104ff80 permissions -rw-r--r--
expandshort;
```     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.overloaded ("op <=", domain_type); (*The <= relation is overloaded*)
```
```   121
```
```   122 (*While (:) is not, its type must be kept
```
```   123   for overloading of = to work.*)
```
```   124 Blast.overloaded ("op :", domain_type);
```
```   125 seq (fn a => Blast.overloaded (a, HOLogic.dest_setT o domain_type))
```
```   126     ["Ball", "Bex"];
```
```   127 (*need UNION, INTER also?*)
```
```   128
```
```   129
```
```   130 (*Rule in Modus Ponens style*)
```
```   131 val major::prems = goalw Set.thy [subset_def] "[| A <= B;  c:A |] ==> c:B";
```
```   132 by (rtac (major RS bspec) 1);
```
```   133 by (resolve_tac prems 1);
```
```   134 qed "subsetD";
```
```   135
```
```   136 (*The same, with reversed premises for use with etac -- cf rev_mp*)
```
```   137 qed_goal "rev_subsetD" Set.thy "[| c:A;  A <= B |] ==> c:B"
```
```   138  (fn prems=>  [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]);
```
```   139
```
```   140 (*Converts A<=B to x:A ==> x:B*)
```
```   141 fun impOfSubs th = th RSN (2, rev_subsetD);
```
```   142
```
```   143 qed_goal "contra_subsetD" Set.thy "!!c. [| A <= B; c ~: B |] ==> c ~: A"
```
```   144  (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
```
```   145
```
```   146 qed_goal "rev_contra_subsetD" Set.thy "!!c. [| c ~: B;  A <= B |] ==> c ~: A"
```
```   147  (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
```
```   148
```
```   149 (*Classical elimination rule*)
```
```   150 val major::prems = goalw Set.thy [subset_def]
```
```   151     "[| A <= B;  c~:A ==> P;  c:B ==> P |] ==> P";
```
```   152 by (rtac (major RS ballE) 1);
```
```   153 by (REPEAT (eresolve_tac prems 1));
```
```   154 qed "subsetCE";
```
```   155
```
```   156 (*Takes assumptions A<=B; c:A and creates the assumption c:B *)
```
```   157 fun set_mp_tac i = etac subsetCE i  THEN  mp_tac i;
```
```   158
```
```   159 AddSIs [subsetI];
```
```   160 AddEs  [subsetD, subsetCE];
```
```   161
```
```   162 qed_goal "subset_refl" Set.thy "A <= (A::'a set)"
```
```   163  (fn _=> [Fast_tac 1]);		(*Blast_tac would try order_refl and fail*)
```
```   164
```
```   165 val prems = goal Set.thy "!!B. [| A<=B;  B<=C |] ==> A<=(C::'a set)";
```
```   166 by (Blast_tac 1);
```
```   167 qed "subset_trans";
```
```   168
```
```   169
```
```   170 section "Equality";
```
```   171
```
```   172 (*Anti-symmetry of the subset relation*)
```
```   173 val prems = goal Set.thy "[| A <= B;  B <= A |] ==> A = (B::'a set)";
```
```   174 by (rtac (iffI RS set_ext) 1);
```
```   175 by (REPEAT (ares_tac (prems RL [subsetD]) 1));
```
```   176 qed "subset_antisym";
```
```   177 val equalityI = subset_antisym;
```
```   178
```
```   179 AddSIs [equalityI];
```
```   180
```
```   181 (* Equality rules from ZF set theory -- are they appropriate here? *)
```
```   182 val prems = goal Set.thy "A = B ==> A<=(B::'a set)";
```
```   183 by (resolve_tac (prems RL [subst]) 1);
```
```   184 by (rtac subset_refl 1);
```
```   185 qed "equalityD1";
```
```   186
```
```   187 val prems = goal Set.thy "A = B ==> B<=(A::'a set)";
```
```   188 by (resolve_tac (prems RL [subst]) 1);
```
```   189 by (rtac subset_refl 1);
```
```   190 qed "equalityD2";
```
```   191
```
```   192 val prems = goal Set.thy
```
```   193     "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";
```
```   194 by (resolve_tac prems 1);
```
```   195 by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
```
```   196 qed "equalityE";
```
```   197
```
```   198 val major::prems = goal Set.thy
```
```   199     "[| A = B;  [| c:A; c:B |] ==> P;  [| c~:A; c~:B |] ==> P |]  ==>  P";
```
```   200 by (rtac (major RS equalityE) 1);
```
```   201 by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
```
```   202 qed "equalityCE";
```
```   203
```
```   204 (*Lemma for creating induction formulae -- for "pattern matching" on p
```
```   205   To make the induction hypotheses usable, apply "spec" or "bspec" to
```
```   206   put universal quantifiers over the free variables in p. *)
```
```   207 val prems = goal Set.thy
```
```   208     "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
```
```   209 by (rtac mp 1);
```
```   210 by (REPEAT (resolve_tac (refl::prems) 1));
```
```   211 qed "setup_induction";
```
```   212
```
```   213
```
```   214 section "The universal set -- UNIV";
```
```   215
```
```   216 qed_goalw "UNIV_I" Set.thy [UNIV_def] "x : UNIV"
```
```   217   (fn _ => [rtac CollectI 1, rtac TrueI 1]);
```
```   218
```
```   219 AddIffs [UNIV_I];
```
```   220
```
```   221 qed_goal "subset_UNIV" Set.thy "A <= UNIV"
```
```   222   (fn _ => [rtac subsetI 1, rtac UNIV_I 1]);
```
```   223
```
```   224 (** Eta-contracting these two rules (to remove P) causes them to be ignored
```
```   225     because of their interaction with congruence rules. **)
```
```   226
```
```   227 goalw Set.thy [Ball_def] "Ball UNIV P = All P";
```
```   228 by (Simp_tac 1);
```
```   229 qed "ball_UNIV";
```
```   230
```
```   231 goalw Set.thy [Bex_def] "Bex UNIV P = Ex P";
```
```   232 by (Simp_tac 1);
```
```   233 qed "bex_UNIV";
```
```   234 Addsimps [ball_UNIV, bex_UNIV];
```
```   235
```
```   236
```
```   237 section "The empty set -- {}";
```
```   238
```
```   239 qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False"
```
```   240  (fn _ => [ (Blast_tac 1) ]);
```
```   241
```
```   242 Addsimps [empty_iff];
```
```   243
```
```   244 qed_goal "emptyE" Set.thy "!!a. a:{} ==> P"
```
```   245  (fn _ => [Full_simp_tac 1]);
```
```   246
```
```   247 AddSEs [emptyE];
```
```   248
```
```   249 qed_goal "empty_subsetI" Set.thy "{} <= A"
```
```   250  (fn _ => [ (Blast_tac 1) ]);
```
```   251
```
```   252 qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}"
```
```   253  (fn [prem]=>
```
```   254   [ (blast_tac (claset() addIs [prem RS FalseE]) 1) ]);
```
```   255
```
```   256 qed_goal "equals0D" Set.thy "!!a. [| A={};  a:A |] ==> P"
```
```   257  (fn _ => [ (Blast_tac 1) ]);
```
```   258
```
```   259 goalw Set.thy [Ball_def] "Ball {} P = True";
```
```   260 by (Simp_tac 1);
```
```   261 qed "ball_empty";
```
```   262
```
```   263 goalw Set.thy [Bex_def] "Bex {} P = False";
```
```   264 by (Simp_tac 1);
```
```   265 qed "bex_empty";
```
```   266 Addsimps [ball_empty, bex_empty];
```
```   267
```
```   268 goal thy "UNIV ~= {}";
```
```   269 by (blast_tac (claset() addEs [equalityE]) 1);
```
```   270 qed "UNIV_not_empty";
```
```   271 AddIffs [UNIV_not_empty];
```
```   272
```
```   273
```
```   274
```
```   275 section "The Powerset operator -- Pow";
```
```   276
```
```   277 qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)"
```
```   278  (fn _ => [ (Asm_simp_tac 1) ]);
```
```   279
```
```   280 AddIffs [Pow_iff];
```
```   281
```
```   282 qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
```
```   283  (fn _ => [ (etac CollectI 1) ]);
```
```   284
```
```   285 qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B)  ==>  A<=B"
```
```   286  (fn _=> [ (etac CollectD 1) ]);
```
```   287
```
```   288 val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)
```
```   289 val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)
```
```   290
```
```   291
```
```   292 section "Set complement -- Compl";
```
```   293
```
```   294 qed_goalw "Compl_iff" Set.thy [Compl_def] "(c : Compl(A)) = (c~:A)"
```
```   295  (fn _ => [ (Blast_tac 1) ]);
```
```   296
```
```   297 Addsimps [Compl_iff];
```
```   298
```
```   299 val prems = goalw Set.thy [Compl_def]
```
```   300     "[| c:A ==> False |] ==> c : Compl(A)";
```
```   301 by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
```
```   302 qed "ComplI";
```
```   303
```
```   304 (*This form, with negated conclusion, works well with the Classical prover.
```
```   305   Negated assumptions behave like formulae on the right side of the notional
```
```   306   turnstile...*)
```
```   307 val major::prems = goalw Set.thy [Compl_def]
```
```   308     "c : Compl(A) ==> c~:A";
```
```   309 by (rtac (major RS CollectD) 1);
```
```   310 qed "ComplD";
```
```   311
```
```   312 val ComplE = make_elim ComplD;
```
```   313
```
```   314 AddSIs [ComplI];
```
```   315 AddSEs [ComplE];
```
```   316
```
```   317
```
```   318 section "Binary union -- Un";
```
```   319
```
```   320 qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)"
```
```   321  (fn _ => [ Blast_tac 1 ]);
```
```   322
```
```   323 Addsimps [Un_iff];
```
```   324
```
```   325 goal Set.thy "!!c. c:A ==> c : A Un B";
```
```   326 by (Asm_simp_tac 1);
```
```   327 qed "UnI1";
```
```   328
```
```   329 goal Set.thy "!!c. c:B ==> c : A Un B";
```
```   330 by (Asm_simp_tac 1);
```
```   331 qed "UnI2";
```
```   332
```
```   333 (*Classical introduction rule: no commitment to A vs B*)
```
```   334 qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B"
```
```   335  (fn prems=>
```
```   336   [ (Simp_tac 1),
```
```   337     (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
```
```   338
```
```   339 val major::prems = goalw Set.thy [Un_def]
```
```   340     "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
```
```   341 by (rtac (major RS CollectD RS disjE) 1);
```
```   342 by (REPEAT (eresolve_tac prems 1));
```
```   343 qed "UnE";
```
```   344
```
```   345 AddSIs [UnCI];
```
```   346 AddSEs [UnE];
```
```   347
```
```   348
```
```   349 section "Binary intersection -- Int";
```
```   350
```
```   351 qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)"
```
```   352  (fn _ => [ (Blast_tac 1) ]);
```
```   353
```
```   354 Addsimps [Int_iff];
```
```   355
```
```   356 goal Set.thy "!!c. [| c:A;  c:B |] ==> c : A Int B";
```
```   357 by (Asm_simp_tac 1);
```
```   358 qed "IntI";
```
```   359
```
```   360 goal Set.thy "!!c. c : A Int B ==> c:A";
```
```   361 by (Asm_full_simp_tac 1);
```
```   362 qed "IntD1";
```
```   363
```
```   364 goal Set.thy "!!c. c : A Int B ==> c:B";
```
```   365 by (Asm_full_simp_tac 1);
```
```   366 qed "IntD2";
```
```   367
```
```   368 val [major,minor] = goal Set.thy
```
```   369     "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
```
```   370 by (rtac minor 1);
```
```   371 by (rtac (major RS IntD1) 1);
```
```   372 by (rtac (major RS IntD2) 1);
```
```   373 qed "IntE";
```
```   374
```
```   375 AddSIs [IntI];
```
```   376 AddSEs [IntE];
```
```   377
```
```   378 section "Set difference";
```
```   379
```
```   380 qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)"
```
```   381  (fn _ => [ (Blast_tac 1) ]);
```
```   382
```
```   383 Addsimps [Diff_iff];
```
```   384
```
```   385 qed_goal "DiffI" Set.thy "!!c. [| c : A;  c ~: B |] ==> c : A - B"
```
```   386  (fn _=> [ Asm_simp_tac 1 ]);
```
```   387
```
```   388 qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A"
```
```   389  (fn _=> [ (Asm_full_simp_tac 1) ]);
```
```   390
```
```   391 qed_goal "DiffD2" Set.thy "!!c. [| c : A - B;  c : B |] ==> P"
```
```   392  (fn _=> [ (Asm_full_simp_tac 1) ]);
```
```   393
```
```   394 qed_goal "DiffE" Set.thy "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P"
```
```   395  (fn prems=>
```
```   396   [ (resolve_tac prems 1),
```
```   397     (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]);
```
```   398
```
```   399 AddSIs [DiffI];
```
```   400 AddSEs [DiffE];
```
```   401
```
```   402
```
```   403 section "Augmenting a set -- insert";
```
```   404
```
```   405 qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)"
```
```   406  (fn _ => [Blast_tac 1]);
```
```   407
```
```   408 Addsimps [insert_iff];
```
```   409
```
```   410 qed_goal "insertI1" Set.thy "a : insert a B"
```
```   411  (fn _ => [Simp_tac 1]);
```
```   412
```
```   413 qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B"
```
```   414  (fn _=> [Asm_simp_tac 1]);
```
```   415
```
```   416 qed_goalw "insertE" Set.thy [insert_def]
```
```   417     "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P"
```
```   418  (fn major::prems=>
```
```   419   [ (rtac (major RS UnE) 1),
```
```   420     (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
```
```   421
```
```   422 (*Classical introduction rule*)
```
```   423 qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
```
```   424  (fn prems=>
```
```   425   [ (Simp_tac 1),
```
```   426     (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
```
```   427
```
```   428 AddSIs [insertCI];
```
```   429 AddSEs [insertE];
```
```   430
```
```   431 section "Singletons, using insert";
```
```   432
```
```   433 qed_goal "singletonI" Set.thy "a : {a}"
```
```   434  (fn _=> [ (rtac insertI1 1) ]);
```
```   435
```
```   436 goal Set.thy "!!a. b : {a} ==> b=a";
```
```   437 by (Blast_tac 1);
```
```   438 qed "singletonD";
```
```   439
```
```   440 bind_thm ("singletonE", make_elim singletonD);
```
```   441
```
```   442 qed_goal "singleton_iff" thy "(b : {a}) = (b=a)"
```
```   443 (fn _ => [Blast_tac 1]);
```
```   444
```
```   445 goal Set.thy "!!a b. {a}={b} ==> a=b";
```
```   446 by (blast_tac (claset() addEs [equalityE]) 1);
```
```   447 qed "singleton_inject";
```
```   448
```
```   449 (*Redundant? But unlike insertCI, it proves the subgoal immediately!*)
```
```   450 AddSIs [singletonI];
```
```   451 AddSDs [singleton_inject];
```
```   452 AddSEs [singletonE];
```
```   453
```
```   454 goal Set.thy "{x. x=a} = {a}";
```
```   455 by (Blast_tac 1);
```
```   456 qed "singleton_conv";
```
```   457 Addsimps [singleton_conv];
```
```   458
```
```   459
```
```   460 section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
```
```   461
```
```   462 goalw Set.thy [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
```
```   463 by (Blast_tac 1);
```
```   464 qed "UN_iff";
```
```   465
```
```   466 Addsimps [UN_iff];
```
```   467
```
```   468 (*The order of the premises presupposes that A is rigid; b may be flexible*)
```
```   469 goal Set.thy "!!b. [| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
```
```   470 by (Auto_tac());
```
```   471 qed "UN_I";
```
```   472
```
```   473 val major::prems = goalw Set.thy [UNION_def]
```
```   474     "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
```
```   475 by (rtac (major RS CollectD RS bexE) 1);
```
```   476 by (REPEAT (ares_tac prems 1));
```
```   477 qed "UN_E";
```
```   478
```
```   479 AddIs  [UN_I];
```
```   480 AddSEs [UN_E];
```
```   481
```
```   482 val prems = goal Set.thy
```
```   483     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
```
```   484 \    (UN x:A. C(x)) = (UN x:B. D(x))";
```
```   485 by (REPEAT (etac UN_E 1
```
```   486      ORELSE ares_tac ([UN_I,equalityI,subsetI] @
```
```   487                       (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
```
```   488 qed "UN_cong";
```
```   489
```
```   490
```
```   491 section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
```
```   492
```
```   493 goalw Set.thy [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
```
```   494 by (Auto_tac());
```
```   495 qed "INT_iff";
```
```   496
```
```   497 Addsimps [INT_iff];
```
```   498
```
```   499 val prems = goalw Set.thy [INTER_def]
```
```   500     "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
```
```   501 by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
```
```   502 qed "INT_I";
```
```   503
```
```   504 goal Set.thy "!!b. [| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
```
```   505 by (Auto_tac());
```
```   506 qed "INT_D";
```
```   507
```
```   508 (*"Classical" elimination -- by the Excluded Middle on a:A *)
```
```   509 val major::prems = goalw Set.thy [INTER_def]
```
```   510     "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
```
```   511 by (rtac (major RS CollectD RS ballE) 1);
```
```   512 by (REPEAT (eresolve_tac prems 1));
```
```   513 qed "INT_E";
```
```   514
```
```   515 AddSIs [INT_I];
```
```   516 AddEs  [INT_D, INT_E];
```
```   517
```
```   518 val prems = goal Set.thy
```
```   519     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
```
```   520 \    (INT x:A. C(x)) = (INT x:B. D(x))";
```
```   521 by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
```
```   522 by (REPEAT (dtac INT_D 1
```
```   523      ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
```
```   524 qed "INT_cong";
```
```   525
```
```   526
```
```   527 section "Union";
```
```   528
```
```   529 goalw Set.thy [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
```
```   530 by (Blast_tac 1);
```
```   531 qed "Union_iff";
```
```   532
```
```   533 Addsimps [Union_iff];
```
```   534
```
```   535 (*The order of the premises presupposes that C is rigid; A may be flexible*)
```
```   536 goal Set.thy "!!X. [| X:C;  A:X |] ==> A : Union(C)";
```
```   537 by (Auto_tac());
```
```   538 qed "UnionI";
```
```   539
```
```   540 val major::prems = goalw Set.thy [Union_def]
```
```   541     "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
```
```   542 by (rtac (major RS UN_E) 1);
```
```   543 by (REPEAT (ares_tac prems 1));
```
```   544 qed "UnionE";
```
```   545
```
```   546 AddIs  [UnionI];
```
```   547 AddSEs [UnionE];
```
```   548
```
```   549
```
```   550 section "Inter";
```
```   551
```
```   552 goalw Set.thy [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
```
```   553 by (Blast_tac 1);
```
```   554 qed "Inter_iff";
```
```   555
```
```   556 Addsimps [Inter_iff];
```
```   557
```
```   558 val prems = goalw Set.thy [Inter_def]
```
```   559     "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
```
```   560 by (REPEAT (ares_tac ([INT_I] @ prems) 1));
```
```   561 qed "InterI";
```
```   562
```
```   563 (*A "destruct" rule -- every X in C contains A as an element, but
```
```   564   A:X can hold when X:C does not!  This rule is analogous to "spec". *)
```
```   565 goal Set.thy "!!X. [| A : Inter(C);  X:C |] ==> A:X";
```
```   566 by (Auto_tac());
```
```   567 qed "InterD";
```
```   568
```
```   569 (*"Classical" elimination rule -- does not require proving X:C *)
```
```   570 val major::prems = goalw Set.thy [Inter_def]
```
```   571     "[| A : Inter(C);  X~:C ==> R;  A:X ==> R |] ==> R";
```
```   572 by (rtac (major RS INT_E) 1);
```
```   573 by (REPEAT (eresolve_tac prems 1));
```
```   574 qed "InterE";
```
```   575
```
```   576 AddSIs [InterI];
```
```   577 AddEs  [InterD, InterE];
```
```   578
```
```   579
```
```   580 (*** Image of a set under a function ***)
```
```   581
```
```   582 (*Frequently b does not have the syntactic form of f(x).*)
```
```   583 val prems = goalw thy [image_def] "[| b=f(x);  x:A |] ==> b : f``A";
```
```   584 by (REPEAT (resolve_tac (prems @ [CollectI,bexI,prem]) 1));
```
```   585 qed "image_eqI";
```
```   586 Addsimps [image_eqI];
```
```   587
```
```   588 bind_thm ("imageI", refl RS image_eqI);
```
```   589
```
```   590 (*The eta-expansion gives variable-name preservation.*)
```
```   591 val major::prems = goalw thy [image_def]
```
```   592     "[| b : (%x. f(x))``A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P";
```
```   593 by (rtac (major RS CollectD RS bexE) 1);
```
```   594 by (REPEAT (ares_tac prems 1));
```
```   595 qed "imageE";
```
```   596
```
```   597 AddIs  [image_eqI];
```
```   598 AddSEs [imageE];
```
```   599
```
```   600 goalw thy [o_def] "(f o g)``r = f``(g``r)";
```
```   601 by (Blast_tac 1);
```
```   602 qed "image_compose";
```
```   603
```
```   604 goal thy "f``(A Un B) = f``A Un f``B";
```
```   605 by (Blast_tac 1);
```
```   606 qed "image_Un";
```
```   607
```
```   608 goal Set.thy "(z : f``A) = (EX x:A. z = f x)";
```
```   609 by (Blast_tac 1);
```
```   610 qed "image_iff";
```
```   611
```
```   612
```
```   613 (*** Range of a function -- just a translation for image! ***)
```
```   614
```
```   615 goal thy "!!b. b=f(x) ==> b : range(f)";
```
```   616 by (EVERY1 [etac image_eqI, rtac UNIV_I]);
```
```   617 bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI));
```
```   618
```
```   619 bind_thm ("rangeI", UNIV_I RS imageI);
```
```   620
```
```   621 val [major,minor] = goal thy
```
```   622     "[| b : range(%x. f(x));  !!x. b=f(x) ==> P |] ==> P";
```
```   623 by (rtac (major RS imageE) 1);
```
```   624 by (etac minor 1);
```
```   625 qed "rangeE";
```
```   626
```
```   627
```
```   628 (*** Set reasoning tools ***)
```
```   629
```
```   630
```
```   631 (** Rewrite rules for boolean case-splitting: faster than
```
```   632 	addsplits[expand_if]
```
```   633 **)
```
```   634
```
```   635 bind_thm ("expand_if_eq1", read_instantiate [("P", "%x. x = ?b")] expand_if);
```
```   636 bind_thm ("expand_if_eq2", read_instantiate [("P", "%x. ?a = x")] expand_if);
```
```   637
```
```   638 bind_thm ("expand_if_mem1",
```
```   639     read_instantiate_sg (sign_of Set.thy) [("P", "%x. x : ?b")] expand_if);
```
```   640 bind_thm ("expand_if_mem2",
```
```   641     read_instantiate_sg (sign_of Set.thy) [("P", "%x. ?a : x")] expand_if);
```
```   642
```
```   643 val expand_ifs = [if_bool_eq, expand_if_eq1, expand_if_eq2,
```
```   644 		  expand_if_mem1, expand_if_mem2];
```
```   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, UN_iff, Union_iff, INT_iff, Inter_iff];
```
```   650
```
```   651 (*Not for Addsimps -- it can cause goals to blow up!*)
```
```   652 goal Set.thy "(a : (if Q then x else y)) = ((Q --> a:x) & (~Q --> a : y))";
```
```   653 by (simp_tac (simpset() addsplits [expand_if]) 1);
```
```   654 qed "mem_if";
```
```   655
```
```   656 val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs;
```
```   657
```
```   658 simpset_ref() := simpset() addcongs [ball_cong,bex_cong]
```
```   659                     setmksimps (mksimps mksimps_pairs);
```
```   660
```
```   661 Addsimps[subset_UNIV, empty_subsetI, subset_refl];
```
```   662
```
```   663
```
```   664 (*** < ***)
```
```   665
```
```   666 goalw Set.thy [psubset_def] "!!A::'a set. [| A <= B; A ~= B |] ==> A<B";
```
```   667 by (Blast_tac 1);
```
```   668 qed "psubsetI";
```
```   669
```
```   670 goalw Set.thy [psubset_def]
```
```   671     "!!x. A < insert x B ==> (x ~: A) & A<=B | x:A & A-{x}<B";
```
```   672 by (Auto_tac());
```
```   673 qed "psubset_insertD";
```
```   674
```
```   675 bind_thm ("psubset_eq", psubset_def RS meta_eq_to_obj_eq);
```