src/HOL/Set.ML
 author paulson Thu Aug 06 10:47:13 1998 +0200 (1998-08-06) changeset 5266 1d11c7e4b999 parent 5256 e6983301ce5e child 5305 513925de8962 permissions -rw-r--r--
Now recognizes both {}= and ={}
```     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 "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 "(! 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 "(? 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 (*Image: retain the type of the set being expressed*)
```
```   130 Blast.overloaded ("op ``", domain_type o domain_type);
```
```   131
```
```   132 (*Rule in Modus Ponens style*)
```
```   133 val major::prems = goalw Set.thy [subset_def] "[| A <= B;  c:A |] ==> c:B";
```
```   134 by (rtac (major RS bspec) 1);
```
```   135 by (resolve_tac prems 1);
```
```   136 qed "subsetD";
```
```   137
```
```   138 (*The same, with reversed premises for use with etac -- cf rev_mp*)
```
```   139 qed_goal "rev_subsetD" Set.thy "[| c:A;  A <= B |] ==> c:B"
```
```   140  (fn prems=>  [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]);
```
```   141
```
```   142 (*Converts A<=B to x:A ==> x:B*)
```
```   143 fun impOfSubs th = th RSN (2, rev_subsetD);
```
```   144
```
```   145 qed_goal "contra_subsetD" Set.thy "!!c. [| A <= B; c ~: B |] ==> c ~: A"
```
```   146  (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
```
```   147
```
```   148 qed_goal "rev_contra_subsetD" Set.thy "!!c. [| c ~: B;  A <= B |] ==> c ~: A"
```
```   149  (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
```
```   150
```
```   151 (*Classical elimination rule*)
```
```   152 val major::prems = goalw Set.thy [subset_def]
```
```   153     "[| A <= B;  c~:A ==> P;  c:B ==> P |] ==> P";
```
```   154 by (rtac (major RS ballE) 1);
```
```   155 by (REPEAT (eresolve_tac prems 1));
```
```   156 qed "subsetCE";
```
```   157
```
```   158 (*Takes assumptions A<=B; c:A and creates the assumption c:B *)
```
```   159 fun set_mp_tac i = etac subsetCE i  THEN  mp_tac i;
```
```   160
```
```   161 AddSIs [subsetI];
```
```   162 AddEs  [subsetD, subsetCE];
```
```   163
```
```   164 qed_goal "subset_refl" Set.thy "A <= (A::'a set)"
```
```   165  (fn _=> [Fast_tac 1]);		(*Blast_tac would try order_refl and fail*)
```
```   166
```
```   167 val prems = goal Set.thy "!!B. [| A<=B;  B<=C |] ==> A<=(C::'a set)";
```
```   168 by (Blast_tac 1);
```
```   169 qed "subset_trans";
```
```   170
```
```   171
```
```   172 section "Equality";
```
```   173
```
```   174 (*Anti-symmetry of the subset relation*)
```
```   175 val prems = goal Set.thy "[| A <= B;  B <= A |] ==> A = (B::'a set)";
```
```   176 by (rtac (iffI RS set_ext) 1);
```
```   177 by (REPEAT (ares_tac (prems RL [subsetD]) 1));
```
```   178 qed "subset_antisym";
```
```   179 val equalityI = subset_antisym;
```
```   180
```
```   181 AddSIs [equalityI];
```
```   182
```
```   183 (* Equality rules from ZF set theory -- are they appropriate here? *)
```
```   184 val prems = goal Set.thy "A = B ==> A<=(B::'a set)";
```
```   185 by (resolve_tac (prems RL [subst]) 1);
```
```   186 by (rtac subset_refl 1);
```
```   187 qed "equalityD1";
```
```   188
```
```   189 val prems = goal Set.thy "A = B ==> B<=(A::'a set)";
```
```   190 by (resolve_tac (prems RL [subst]) 1);
```
```   191 by (rtac subset_refl 1);
```
```   192 qed "equalityD2";
```
```   193
```
```   194 val prems = goal Set.thy
```
```   195     "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";
```
```   196 by (resolve_tac prems 1);
```
```   197 by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
```
```   198 qed "equalityE";
```
```   199
```
```   200 val major::prems = goal Set.thy
```
```   201     "[| A = B;  [| c:A; c:B |] ==> P;  [| c~:A; c~:B |] ==> P |]  ==>  P";
```
```   202 by (rtac (major RS equalityE) 1);
```
```   203 by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
```
```   204 qed "equalityCE";
```
```   205
```
```   206 (*Lemma for creating induction formulae -- for "pattern matching" on p
```
```   207   To make the induction hypotheses usable, apply "spec" or "bspec" to
```
```   208   put universal quantifiers over the free variables in p. *)
```
```   209 val prems = goal Set.thy
```
```   210     "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
```
```   211 by (rtac mp 1);
```
```   212 by (REPEAT (resolve_tac (refl::prems) 1));
```
```   213 qed "setup_induction";
```
```   214
```
```   215
```
```   216 section "The universal set -- UNIV";
```
```   217
```
```   218 qed_goalw "UNIV_I" Set.thy [UNIV_def] "x : UNIV"
```
```   219   (fn _ => [rtac CollectI 1, rtac TrueI 1]);
```
```   220
```
```   221 Addsimps [UNIV_I];
```
```   222 AddIs    [UNIV_I];  (*unsafe makes it less likely to cause problems*)
```
```   223
```
```   224 qed_goal "subset_UNIV" Set.thy "A <= UNIV"
```
```   225   (fn _ => [rtac subsetI 1, rtac UNIV_I 1]);
```
```   226
```
```   227 (** Eta-contracting these two rules (to remove P) causes them to be ignored
```
```   228     because of their interaction with congruence rules. **)
```
```   229
```
```   230 Goalw [Ball_def] "Ball UNIV P = All P";
```
```   231 by (Simp_tac 1);
```
```   232 qed "ball_UNIV";
```
```   233
```
```   234 Goalw [Bex_def] "Bex UNIV P = Ex P";
```
```   235 by (Simp_tac 1);
```
```   236 qed "bex_UNIV";
```
```   237 Addsimps [ball_UNIV, bex_UNIV];
```
```   238
```
```   239
```
```   240 section "The empty set -- {}";
```
```   241
```
```   242 qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False"
```
```   243  (fn _ => [ (Blast_tac 1) ]);
```
```   244
```
```   245 Addsimps [empty_iff];
```
```   246
```
```   247 qed_goal "emptyE" Set.thy "!!a. a:{} ==> P"
```
```   248  (fn _ => [Full_simp_tac 1]);
```
```   249
```
```   250 AddSEs [emptyE];
```
```   251
```
```   252 qed_goal "empty_subsetI" Set.thy "{} <= A"
```
```   253  (fn _ => [ (Blast_tac 1) ]);
```
```   254
```
```   255 (*One effect is to delete the ASSUMPTION {} <= A*)
```
```   256 AddIffs [empty_subsetI];
```
```   257
```
```   258 qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}"
```
```   259  (fn [prem]=>
```
```   260   [ (blast_tac (claset() addIs [prem RS FalseE]) 1) ]);
```
```   261
```
```   262 (*Use for reasoning about disjointness: A Int B = {} *)
```
```   263 qed_goal "equals0E" Set.thy "!!a. [| A={};  a:A |] ==> P"
```
```   264  (fn _ => [ (Blast_tac 1) ]);
```
```   265
```
```   266 AddEs [equals0E, sym RS equals0E];
```
```   267
```
```   268 Goalw [Ball_def] "Ball {} P = True";
```
```   269 by (Simp_tac 1);
```
```   270 qed "ball_empty";
```
```   271
```
```   272 Goalw [Bex_def] "Bex {} P = False";
```
```   273 by (Simp_tac 1);
```
```   274 qed "bex_empty";
```
```   275 Addsimps [ball_empty, bex_empty];
```
```   276
```
```   277 Goal "UNIV ~= {}";
```
```   278 by (blast_tac (claset() addEs [equalityE]) 1);
```
```   279 qed "UNIV_not_empty";
```
```   280 AddIffs [UNIV_not_empty];
```
```   281
```
```   282
```
```   283
```
```   284 section "The Powerset operator -- Pow";
```
```   285
```
```   286 qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)"
```
```   287  (fn _ => [ (Asm_simp_tac 1) ]);
```
```   288
```
```   289 AddIffs [Pow_iff];
```
```   290
```
```   291 qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
```
```   292  (fn _ => [ (etac CollectI 1) ]);
```
```   293
```
```   294 qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B)  ==>  A<=B"
```
```   295  (fn _=> [ (etac CollectD 1) ]);
```
```   296
```
```   297 val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)
```
```   298 val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)
```
```   299
```
```   300
```
```   301 section "Set complement -- Compl";
```
```   302
```
```   303 qed_goalw "Compl_iff" Set.thy [Compl_def] "(c : Compl(A)) = (c~:A)"
```
```   304  (fn _ => [ (Blast_tac 1) ]);
```
```   305
```
```   306 Addsimps [Compl_iff];
```
```   307
```
```   308 val prems = goalw Set.thy [Compl_def]
```
```   309     "[| c:A ==> False |] ==> c : Compl(A)";
```
```   310 by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
```
```   311 qed "ComplI";
```
```   312
```
```   313 (*This form, with negated conclusion, works well with the Classical prover.
```
```   314   Negated assumptions behave like formulae on the right side of the notional
```
```   315   turnstile...*)
```
```   316 val major::prems = goalw Set.thy [Compl_def]
```
```   317     "c : Compl(A) ==> c~:A";
```
```   318 by (rtac (major RS CollectD) 1);
```
```   319 qed "ComplD";
```
```   320
```
```   321 val ComplE = make_elim ComplD;
```
```   322
```
```   323 AddSIs [ComplI];
```
```   324 AddSEs [ComplE];
```
```   325
```
```   326
```
```   327 section "Binary union -- Un";
```
```   328
```
```   329 qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)"
```
```   330  (fn _ => [ Blast_tac 1 ]);
```
```   331
```
```   332 Addsimps [Un_iff];
```
```   333
```
```   334 Goal "c:A ==> c : A Un B";
```
```   335 by (Asm_simp_tac 1);
```
```   336 qed "UnI1";
```
```   337
```
```   338 Goal "c:B ==> c : A Un B";
```
```   339 by (Asm_simp_tac 1);
```
```   340 qed "UnI2";
```
```   341
```
```   342 (*Classical introduction rule: no commitment to A vs B*)
```
```   343 qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B"
```
```   344  (fn prems=>
```
```   345   [ (Simp_tac 1),
```
```   346     (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
```
```   347
```
```   348 val major::prems = goalw Set.thy [Un_def]
```
```   349     "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
```
```   350 by (rtac (major RS CollectD RS disjE) 1);
```
```   351 by (REPEAT (eresolve_tac prems 1));
```
```   352 qed "UnE";
```
```   353
```
```   354 AddSIs [UnCI];
```
```   355 AddSEs [UnE];
```
```   356
```
```   357
```
```   358 section "Binary intersection -- Int";
```
```   359
```
```   360 qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)"
```
```   361  (fn _ => [ (Blast_tac 1) ]);
```
```   362
```
```   363 Addsimps [Int_iff];
```
```   364
```
```   365 Goal "[| c:A;  c:B |] ==> c : A Int B";
```
```   366 by (Asm_simp_tac 1);
```
```   367 qed "IntI";
```
```   368
```
```   369 Goal "c : A Int B ==> c:A";
```
```   370 by (Asm_full_simp_tac 1);
```
```   371 qed "IntD1";
```
```   372
```
```   373 Goal "c : A Int B ==> c:B";
```
```   374 by (Asm_full_simp_tac 1);
```
```   375 qed "IntD2";
```
```   376
```
```   377 val [major,minor] = goal Set.thy
```
```   378     "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
```
```   379 by (rtac minor 1);
```
```   380 by (rtac (major RS IntD1) 1);
```
```   381 by (rtac (major RS IntD2) 1);
```
```   382 qed "IntE";
```
```   383
```
```   384 AddSIs [IntI];
```
```   385 AddSEs [IntE];
```
```   386
```
```   387 section "Set difference";
```
```   388
```
```   389 qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)"
```
```   390  (fn _ => [ (Blast_tac 1) ]);
```
```   391
```
```   392 Addsimps [Diff_iff];
```
```   393
```
```   394 qed_goal "DiffI" Set.thy "!!c. [| c : A;  c ~: B |] ==> c : A - B"
```
```   395  (fn _=> [ Asm_simp_tac 1 ]);
```
```   396
```
```   397 qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A"
```
```   398  (fn _=> [ (Asm_full_simp_tac 1) ]);
```
```   399
```
```   400 qed_goal "DiffD2" Set.thy "!!c. [| c : A - B;  c : B |] ==> P"
```
```   401  (fn _=> [ (Asm_full_simp_tac 1) ]);
```
```   402
```
```   403 qed_goal "DiffE" Set.thy "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P"
```
```   404  (fn prems=>
```
```   405   [ (resolve_tac prems 1),
```
```   406     (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]);
```
```   407
```
```   408 AddSIs [DiffI];
```
```   409 AddSEs [DiffE];
```
```   410
```
```   411
```
```   412 section "Augmenting a set -- insert";
```
```   413
```
```   414 qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)"
```
```   415  (fn _ => [Blast_tac 1]);
```
```   416
```
```   417 Addsimps [insert_iff];
```
```   418
```
```   419 qed_goal "insertI1" Set.thy "a : insert a B"
```
```   420  (fn _ => [Simp_tac 1]);
```
```   421
```
```   422 qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B"
```
```   423  (fn _=> [Asm_simp_tac 1]);
```
```   424
```
```   425 qed_goalw "insertE" Set.thy [insert_def]
```
```   426     "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P"
```
```   427  (fn major::prems=>
```
```   428   [ (rtac (major RS UnE) 1),
```
```   429     (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
```
```   430
```
```   431 (*Classical introduction rule*)
```
```   432 qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
```
```   433  (fn prems=>
```
```   434   [ (Simp_tac 1),
```
```   435     (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
```
```   436
```
```   437 AddSIs [insertCI];
```
```   438 AddSEs [insertE];
```
```   439
```
```   440 section "Singletons, using insert";
```
```   441
```
```   442 qed_goal "singletonI" Set.thy "a : {a}"
```
```   443  (fn _=> [ (rtac insertI1 1) ]);
```
```   444
```
```   445 Goal "b : {a} ==> b=a";
```
```   446 by (Blast_tac 1);
```
```   447 qed "singletonD";
```
```   448
```
```   449 bind_thm ("singletonE", make_elim singletonD);
```
```   450
```
```   451 qed_goal "singleton_iff" thy "(b : {a}) = (b=a)"
```
```   452 (fn _ => [Blast_tac 1]);
```
```   453
```
```   454 Goal "{a}={b} ==> a=b";
```
```   455 by (blast_tac (claset() addEs [equalityE]) 1);
```
```   456 qed "singleton_inject";
```
```   457
```
```   458 (*Redundant? But unlike insertCI, it proves the subgoal immediately!*)
```
```   459 AddSIs [singletonI];
```
```   460 AddSDs [singleton_inject];
```
```   461 AddSEs [singletonE];
```
```   462
```
```   463 Goal "{x. x=a} = {a}";
```
```   464 by (Blast_tac 1);
```
```   465 qed "singleton_conv";
```
```   466 Addsimps [singleton_conv];
```
```   467
```
```   468
```
```   469 section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
```
```   470
```
```   471 Goalw [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
```
```   472 by (Blast_tac 1);
```
```   473 qed "UN_iff";
```
```   474
```
```   475 Addsimps [UN_iff];
```
```   476
```
```   477 (*The order of the premises presupposes that A is rigid; b may be flexible*)
```
```   478 Goal "[| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
```
```   479 by Auto_tac;
```
```   480 qed "UN_I";
```
```   481
```
```   482 val major::prems = goalw Set.thy [UNION_def]
```
```   483     "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
```
```   484 by (rtac (major RS CollectD RS bexE) 1);
```
```   485 by (REPEAT (ares_tac prems 1));
```
```   486 qed "UN_E";
```
```   487
```
```   488 AddIs  [UN_I];
```
```   489 AddSEs [UN_E];
```
```   490
```
```   491 val prems = goal Set.thy
```
```   492     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
```
```   493 \    (UN x:A. C(x)) = (UN x:B. D(x))";
```
```   494 by (REPEAT (etac UN_E 1
```
```   495      ORELSE ares_tac ([UN_I,equalityI,subsetI] @
```
```   496                       (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
```
```   497 qed "UN_cong";
```
```   498
```
```   499
```
```   500 section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
```
```   501
```
```   502 Goalw [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
```
```   503 by Auto_tac;
```
```   504 qed "INT_iff";
```
```   505
```
```   506 Addsimps [INT_iff];
```
```   507
```
```   508 val prems = goalw Set.thy [INTER_def]
```
```   509     "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
```
```   510 by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
```
```   511 qed "INT_I";
```
```   512
```
```   513 Goal "[| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
```
```   514 by Auto_tac;
```
```   515 qed "INT_D";
```
```   516
```
```   517 (*"Classical" elimination -- by the Excluded Middle on a:A *)
```
```   518 val major::prems = goalw Set.thy [INTER_def]
```
```   519     "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
```
```   520 by (rtac (major RS CollectD RS ballE) 1);
```
```   521 by (REPEAT (eresolve_tac prems 1));
```
```   522 qed "INT_E";
```
```   523
```
```   524 AddSIs [INT_I];
```
```   525 AddEs  [INT_D, INT_E];
```
```   526
```
```   527 val prems = goal Set.thy
```
```   528     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
```
```   529 \    (INT x:A. C(x)) = (INT x:B. D(x))";
```
```   530 by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
```
```   531 by (REPEAT (dtac INT_D 1
```
```   532      ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
```
```   533 qed "INT_cong";
```
```   534
```
```   535
```
```   536 section "Union";
```
```   537
```
```   538 Goalw [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
```
```   539 by (Blast_tac 1);
```
```   540 qed "Union_iff";
```
```   541
```
```   542 Addsimps [Union_iff];
```
```   543
```
```   544 (*The order of the premises presupposes that C is rigid; A may be flexible*)
```
```   545 Goal "[| X:C;  A:X |] ==> A : Union(C)";
```
```   546 by Auto_tac;
```
```   547 qed "UnionI";
```
```   548
```
```   549 val major::prems = goalw Set.thy [Union_def]
```
```   550     "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
```
```   551 by (rtac (major RS UN_E) 1);
```
```   552 by (REPEAT (ares_tac prems 1));
```
```   553 qed "UnionE";
```
```   554
```
```   555 AddIs  [UnionI];
```
```   556 AddSEs [UnionE];
```
```   557
```
```   558
```
```   559 section "Inter";
```
```   560
```
```   561 Goalw [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
```
```   562 by (Blast_tac 1);
```
```   563 qed "Inter_iff";
```
```   564
```
```   565 Addsimps [Inter_iff];
```
```   566
```
```   567 val prems = goalw Set.thy [Inter_def]
```
```   568     "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
```
```   569 by (REPEAT (ares_tac ([INT_I] @ prems) 1));
```
```   570 qed "InterI";
```
```   571
```
```   572 (*A "destruct" rule -- every X in C contains A as an element, but
```
```   573   A:X can hold when X:C does not!  This rule is analogous to "spec". *)
```
```   574 Goal "[| A : Inter(C);  X:C |] ==> A:X";
```
```   575 by Auto_tac;
```
```   576 qed "InterD";
```
```   577
```
```   578 (*"Classical" elimination rule -- does not require proving X:C *)
```
```   579 val major::prems = goalw Set.thy [Inter_def]
```
```   580     "[| A : Inter(C);  X~:C ==> R;  A:X ==> R |] ==> R";
```
```   581 by (rtac (major RS INT_E) 1);
```
```   582 by (REPEAT (eresolve_tac prems 1));
```
```   583 qed "InterE";
```
```   584
```
```   585 AddSIs [InterI];
```
```   586 AddEs  [InterD, InterE];
```
```   587
```
```   588
```
```   589 (*** Image of a set under a function ***)
```
```   590
```
```   591 (*Frequently b does not have the syntactic form of f(x).*)
```
```   592 val prems = goalw thy [image_def] "[| b=f(x);  x:A |] ==> b : f``A";
```
```   593 by (REPEAT (resolve_tac (prems @ [CollectI,bexI,prem]) 1));
```
```   594 qed "image_eqI";
```
```   595 Addsimps [image_eqI];
```
```   596
```
```   597 bind_thm ("imageI", refl RS image_eqI);
```
```   598
```
```   599 (*The eta-expansion gives variable-name preservation.*)
```
```   600 val major::prems = goalw thy [image_def]
```
```   601     "[| b : (%x. f(x))``A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P";
```
```   602 by (rtac (major RS CollectD RS bexE) 1);
```
```   603 by (REPEAT (ares_tac prems 1));
```
```   604 qed "imageE";
```
```   605
```
```   606 AddIs  [image_eqI];
```
```   607 AddSEs [imageE];
```
```   608
```
```   609 Goalw [o_def] "(f o g)``r = f``(g``r)";
```
```   610 by (Blast_tac 1);
```
```   611 qed "image_compose";
```
```   612
```
```   613 Goal "f``(A Un B) = f``A Un f``B";
```
```   614 by (Blast_tac 1);
```
```   615 qed "image_Un";
```
```   616
```
```   617 Goal "(z : f``A) = (EX x:A. z = f x)";
```
```   618 by (Blast_tac 1);
```
```   619 qed "image_iff";
```
```   620
```
```   621 (*This rewrite rule would confuse users if made default.*)
```
```   622 Goal "(f``A <= B) = (ALL x:A. f(x): B)";
```
```   623 by (Blast_tac 1);
```
```   624 qed "image_subset_iff";
```
```   625
```
```   626 (*Replaces the three steps subsetI, imageE, hyp_subst_tac, but breaks too
```
```   627   many existing proofs.*)
```
```   628 val prems = goal thy "(!!x. x:A ==> f(x) : B) ==> f``A <= B";
```
```   629 by (blast_tac (claset() addIs prems) 1);
```
```   630 qed "image_subsetI";
```
```   631
```
```   632
```
```   633 (*** Range of a function -- just a translation for image! ***)
```
```   634
```
```   635 Goal "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
```
```   648 (*** Set reasoning tools ***)
```
```   649
```
```   650
```
```   651 (** Rewrite rules for boolean case-splitting: faster than
```
```   652 	addsplits[split_if]
```
```   653 **)
```
```   654
```
```   655 bind_thm ("split_if_eq1", read_instantiate [("P", "%x. x = ?b")] split_if);
```
```   656 bind_thm ("split_if_eq2", read_instantiate [("P", "%x. ?a = x")] split_if);
```
```   657
```
```   658 (*Split ifs on either side of the membership relation.
```
```   659 	Not for Addsimps -- can cause goals to blow up!*)
```
```   660 bind_thm ("split_if_mem1",
```
```   661     read_instantiate_sg (sign_of Set.thy) [("P", "%x. x : ?b")] split_if);
```
```   662 bind_thm ("split_if_mem2",
```
```   663     read_instantiate_sg (sign_of Set.thy) [("P", "%x. ?a : x")] split_if);
```
```   664
```
```   665 val split_ifs = [if_bool_eq_conj, split_if_eq1, split_if_eq2,
```
```   666 		  split_if_mem1, split_if_mem2];
```
```   667
```
```   668
```
```   669 (*Each of these has ALREADY been added to simpset() above.*)
```
```   670 val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff,
```
```   671                  mem_Collect_eq, UN_iff, Union_iff, INT_iff, Inter_iff];
```
```   672
```
```   673 val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs;
```
```   674
```
```   675 simpset_ref() := simpset() addcongs [ball_cong,bex_cong]
```
```   676                     setmksimps (mksimps mksimps_pairs);
```
```   677
```
```   678 Addsimps[subset_UNIV, subset_refl];
```
```   679
```
```   680
```
```   681 (*** < ***)
```
```   682
```
```   683 Goalw [psubset_def] "!!A::'a set. [| A <= B; A ~= B |] ==> A<B";
```
```   684 by (Blast_tac 1);
```
```   685 qed "psubsetI";
```
```   686
```
```   687 Goalw [psubset_def] "A < insert x B ==> (x ~: A) & A<=B | x:A & A-{x}<B";
```
```   688 by Auto_tac;
```
```   689 qed "psubset_insertD";
```
```   690
```
```   691 bind_thm ("psubset_eq", psubset_def RS meta_eq_to_obj_eq);
```