src/HOL/Set.ML
 author oheimb Fri Jul 14 16:27:37 2000 +0200 (2000-07-14) changeset 9338 fcf7f29a3447 parent 9186 7b2f4e6538b4 child 9378 12f251a5a3b5 permissions -rw-r--r--
```     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 section "Relating predicates and sets";
```
```    10
```
```    11 Addsimps [Collect_mem_eq];
```
```    12 AddIffs  [mem_Collect_eq];
```
```    13
```
```    14 Goal "P(a) ==> a : {x. P(x)}";
```
```    15 by (Asm_simp_tac 1);
```
```    16 qed "CollectI";
```
```    17
```
```    18 Goal "a : {x. P(x)} ==> P(a)";
```
```    19 by (Asm_full_simp_tac 1);
```
```    20 qed "CollectD";
```
```    21
```
```    22 val [prem] = Goal "[| !!x. (x:A) = (x:B) |] ==> A = B";
```
```    23 by (rtac (prem RS ext RS arg_cong RS box_equals) 1);
```
```    24 by (rtac Collect_mem_eq 1);
```
```    25 by (rtac Collect_mem_eq 1);
```
```    26 qed "set_ext";
```
```    27
```
```    28 val [prem] = Goal "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}";
```
```    29 by (rtac (prem RS ext RS arg_cong) 1);
```
```    30 qed "Collect_cong";
```
```    31
```
```    32 bind_thm ("CollectE", make_elim CollectD);
```
```    33
```
```    34 AddSIs [CollectI];
```
```    35 AddSEs [CollectE];
```
```    36
```
```    37
```
```    38 section "Bounded quantifiers";
```
```    39
```
```    40 val prems = Goalw [Ball_def]
```
```    41     "[| !!x. x:A ==> P(x) |] ==> ALL x:A. P(x)";
```
```    42 by (REPEAT (ares_tac (prems @ [allI,impI]) 1));
```
```    43 qed "ballI";
```
```    44
```
```    45 bind_thms ("strip", [impI, allI, ballI]);
```
```    46
```
```    47 Goalw [Ball_def] "[| ALL x:A. P(x);  x:A |] ==> P(x)";
```
```    48 by (Blast_tac 1);
```
```    49 qed "bspec";
```
```    50
```
```    51 val major::prems = Goalw [Ball_def]
```
```    52     "[| ALL 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 ALL 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 AddXDs [bspec];
```
```    63 (* gives better instantiation for bound: *)
```
```    64 claset_ref() := claset() addWrapper ("bspec", fn tac2 =>
```
```    65 			 (dtac bspec THEN' atac) APPEND' tac2);
```
```    66
```
```    67 (*Normally the best argument order: P(x) constrains the choice of x:A*)
```
```    68 Goalw [Bex_def] "[| P(x);  x:A |] ==> EX x:A. P(x)";
```
```    69 by (Blast_tac 1);
```
```    70 qed "bexI";
```
```    71
```
```    72 (*The best argument order when there is only one x:A*)
```
```    73 Goalw [Bex_def] "[| x:A;  P(x) |] ==> EX x:A. P(x)";
```
```    74 by (Blast_tac 1);
```
```    75 qed "rev_bexI";
```
```    76
```
```    77 val prems = Goal
```
```    78    "[| ALL x:A. ~P(x) ==> P(a);  a:A |] ==> EX x:A. P(x)";
```
```    79 by (rtac classical 1);
```
```    80 by (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1))  ;
```
```    81 qed "bexCI";
```
```    82
```
```    83 val major::prems = Goalw [Bex_def]
```
```    84     "[| EX x:A. P(x);  !!x. [| x:A; P(x) |] ==> Q  |] ==> Q";
```
```    85 by (rtac (major RS exE) 1);
```
```    86 by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1));
```
```    87 qed "bexE";
```
```    88
```
```    89 AddIs  [bexI];
```
```    90 AddSEs [bexE];
```
```    91
```
```    92 (*Trival rewrite rule*)
```
```    93 Goal "(ALL x:A. P) = ((EX x. x:A) --> P)";
```
```    94 by (simp_tac (simpset() addsimps [Ball_def]) 1);
```
```    95 qed "ball_triv";
```
```    96
```
```    97 (*Dual form for existentials*)
```
```    98 Goal "(EX x:A. P) = ((EX x. x:A) & P)";
```
```    99 by (simp_tac (simpset() addsimps [Bex_def]) 1);
```
```   100 qed "bex_triv";
```
```   101
```
```   102 Addsimps [ball_triv, bex_triv];
```
```   103
```
```   104 (** Congruence rules **)
```
```   105
```
```   106 val prems = Goalw [Ball_def]
```
```   107     "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
```
```   108 \    (ALL x:A. P(x)) = (ALL x:B. Q(x))";
```
```   109 by (asm_simp_tac (simpset() addsimps prems) 1);
```
```   110 qed "ball_cong";
```
```   111
```
```   112 val prems = Goalw [Bex_def]
```
```   113     "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
```
```   114 \    (EX x:A. P(x)) = (EX x:B. Q(x))";
```
```   115 by (asm_simp_tac (simpset() addcongs [conj_cong] addsimps prems) 1);
```
```   116 qed "bex_cong";
```
```   117
```
```   118 Addcongs [ball_cong,bex_cong];
```
```   119
```
```   120 section "Subsets";
```
```   121
```
```   122 val prems = Goalw [subset_def] "(!!x. x:A ==> x:B) ==> A <= B";
```
```   123 by (REPEAT (ares_tac (prems @ [ballI]) 1));
```
```   124 qed "subsetI";
```
```   125
```
```   126 (*Map the type ('a set => anything) to just 'a.
```
```   127   For overloading constants whose first argument has type "'a set" *)
```
```   128 fun overload_1st_set s = Blast.overloaded (s, HOLogic.dest_setT o domain_type);
```
```   129
```
```   130 (*While (:) is not, its type must be kept
```
```   131   for overloading of = to work.*)
```
```   132 Blast.overloaded ("op :", domain_type);
```
```   133
```
```   134 overload_1st_set "Ball";		(*need UNION, INTER also?*)
```
```   135 overload_1st_set "Bex";
```
```   136
```
```   137 (*Image: retain the type of the set being expressed*)
```
```   138 Blast.overloaded ("image", domain_type);
```
```   139
```
```   140 (*Rule in Modus Ponens style*)
```
```   141 Goalw [subset_def] "[| A <= B;  c:A |] ==> c:B";
```
```   142 by (Blast_tac 1);
```
```   143 qed "subsetD";
```
```   144 AddXIs [subsetD];
```
```   145
```
```   146 (*The same, with reversed premises for use with etac -- cf rev_mp*)
```
```   147 Goal "[| c:A;  A <= B |] ==> c:B";
```
```   148 by (REPEAT (ares_tac [subsetD] 1)) ;
```
```   149 qed "rev_subsetD";
```
```   150 AddXIs [rev_subsetD];
```
```   151
```
```   152 (*Converts A<=B to x:A ==> x:B*)
```
```   153 fun impOfSubs th = th RSN (2, rev_subsetD);
```
```   154
```
```   155 Goal "[| A <= B; c ~: B |] ==> c ~: A";
```
```   156 by (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ;
```
```   157 qed "contra_subsetD";
```
```   158
```
```   159 Goal "[| c ~: B;  A <= B |] ==> c ~: A";
```
```   160 by (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ;
```
```   161 qed "rev_contra_subsetD";
```
```   162
```
```   163 (*Classical elimination rule*)
```
```   164 val major::prems = Goalw [subset_def]
```
```   165     "[| A <= B;  c~:A ==> P;  c:B ==> P |] ==> P";
```
```   166 by (rtac (major RS ballE) 1);
```
```   167 by (REPEAT (eresolve_tac prems 1));
```
```   168 qed "subsetCE";
```
```   169
```
```   170 (*Takes assumptions A<=B; c:A and creates the assumption c:B *)
```
```   171 fun set_mp_tac i = etac subsetCE i  THEN  mp_tac i;
```
```   172
```
```   173 AddSIs [subsetI];
```
```   174 AddEs  [subsetD, subsetCE];
```
```   175
```
```   176 Goal "A <= (A::'a set)";
```
```   177 by (Fast_tac 1);
```
```   178 qed "subset_refl";		(*Blast_tac would try order_refl and fail*)
```
```   179
```
```   180 Goal "[| A<=B;  B<=C |] ==> A<=(C::'a set)";
```
```   181 by (Blast_tac 1);
```
```   182 qed "subset_trans";
```
```   183
```
```   184
```
```   185 section "Equality";
```
```   186
```
```   187 (*Anti-symmetry of the subset relation*)
```
```   188 Goal "[| A <= B;  B <= A |] ==> A = (B::'a set)";
```
```   189 by (rtac set_ext 1);
```
```   190 by (blast_tac (claset() addIs [subsetD]) 1);
```
```   191 qed "subset_antisym";
```
```   192 bind_thm ("equalityI", subset_antisym);
```
```   193
```
```   194 AddSIs [equalityI];
```
```   195
```
```   196 (* Equality rules from ZF set theory -- are they appropriate here? *)
```
```   197 Goal "A = B ==> A<=(B::'a set)";
```
```   198 by (etac ssubst 1);
```
```   199 by (rtac subset_refl 1);
```
```   200 qed "equalityD1";
```
```   201
```
```   202 Goal "A = B ==> B<=(A::'a set)";
```
```   203 by (etac ssubst 1);
```
```   204 by (rtac subset_refl 1);
```
```   205 qed "equalityD2";
```
```   206
```
```   207 (*Be careful when adding this to the claset as subset_empty is in the simpset:
```
```   208   A={} goes to {}<=A and A<={} and then back to A={} !*)
```
```   209 val prems = Goal
```
```   210     "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";
```
```   211 by (resolve_tac prems 1);
```
```   212 by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
```
```   213 qed "equalityE";
```
```   214
```
```   215 val major::prems = Goal
```
```   216     "[| A = B;  [| c:A; c:B |] ==> P;  [| c~:A; c~:B |] ==> P |]  ==>  P";
```
```   217 by (rtac (major RS equalityE) 1);
```
```   218 by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
```
```   219 qed "equalityCE";
```
```   220
```
```   221 AddEs [equalityCE];
```
```   222
```
```   223 (*Lemma for creating induction formulae -- for "pattern matching" on p
```
```   224   To make the induction hypotheses usable, apply "spec" or "bspec" to
```
```   225   put universal quantifiers over the free variables in p. *)
```
```   226 val prems = Goal
```
```   227     "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
```
```   228 by (rtac mp 1);
```
```   229 by (REPEAT (resolve_tac (refl::prems) 1));
```
```   230 qed "setup_induction";
```
```   231
```
```   232 Goal "A = B ==> (x : A) = (x : B)";
```
```   233 by (Asm_simp_tac 1);
```
```   234 qed "eqset_imp_iff";
```
```   235
```
```   236
```
```   237 section "The universal set -- UNIV";
```
```   238
```
```   239 Goalw [UNIV_def] "x : UNIV";
```
```   240 by (rtac CollectI 1);
```
```   241 by (rtac TrueI 1);
```
```   242 qed "UNIV_I";
```
```   243
```
```   244 Addsimps [UNIV_I];
```
```   245 AddIs    [UNIV_I];  (*unsafe makes it less likely to cause problems*)
```
```   246
```
```   247 Goal "A <= UNIV";
```
```   248 by (rtac subsetI 1);
```
```   249 by (rtac UNIV_I 1);
```
```   250 qed "subset_UNIV";
```
```   251
```
```   252 (** Eta-contracting these two rules (to remove P) causes them to be ignored
```
```   253     because of their interaction with congruence rules. **)
```
```   254
```
```   255 Goalw [Ball_def] "Ball UNIV P = All P";
```
```   256 by (Simp_tac 1);
```
```   257 qed "ball_UNIV";
```
```   258
```
```   259 Goalw [Bex_def] "Bex UNIV P = Ex P";
```
```   260 by (Simp_tac 1);
```
```   261 qed "bex_UNIV";
```
```   262 Addsimps [ball_UNIV, bex_UNIV];
```
```   263
```
```   264
```
```   265 section "The empty set -- {}";
```
```   266
```
```   267 Goalw [empty_def] "(c : {}) = False";
```
```   268 by (Blast_tac 1) ;
```
```   269 qed "empty_iff";
```
```   270
```
```   271 Addsimps [empty_iff];
```
```   272
```
```   273 Goal "a:{} ==> P";
```
```   274 by (Full_simp_tac 1);
```
```   275 qed "emptyE";
```
```   276
```
```   277 AddSEs [emptyE];
```
```   278
```
```   279 Goal "{} <= A";
```
```   280 by (Blast_tac 1) ;
```
```   281 qed "empty_subsetI";
```
```   282
```
```   283 (*One effect is to delete the ASSUMPTION {} <= A*)
```
```   284 AddIffs [empty_subsetI];
```
```   285
```
```   286 val [prem]= Goal "[| !!y. y:A ==> False |] ==> A={}";
```
```   287 by (blast_tac (claset() addIs [prem RS FalseE]) 1) ;
```
```   288 qed "equals0I";
```
```   289
```
```   290 (*Use for reasoning about disjointness: A Int B = {} *)
```
```   291 Goal "A={} ==> a ~: A";
```
```   292 by (Blast_tac 1) ;
```
```   293 qed "equals0D";
```
```   294
```
```   295 Goalw [Ball_def] "Ball {} P = True";
```
```   296 by (Simp_tac 1);
```
```   297 qed "ball_empty";
```
```   298
```
```   299 Goalw [Bex_def] "Bex {} P = False";
```
```   300 by (Simp_tac 1);
```
```   301 qed "bex_empty";
```
```   302 Addsimps [ball_empty, bex_empty];
```
```   303
```
```   304 Goal "UNIV ~= {}";
```
```   305 by (blast_tac (claset() addEs [equalityE]) 1);
```
```   306 qed "UNIV_not_empty";
```
```   307 AddIffs [UNIV_not_empty];
```
```   308
```
```   309
```
```   310
```
```   311 section "The Powerset operator -- Pow";
```
```   312
```
```   313 Goalw [Pow_def] "(A : Pow(B)) = (A <= B)";
```
```   314 by (Asm_simp_tac 1);
```
```   315 qed "Pow_iff";
```
```   316
```
```   317 AddIffs [Pow_iff];
```
```   318
```
```   319 Goalw [Pow_def] "A <= B ==> A : Pow(B)";
```
```   320 by (etac CollectI 1);
```
```   321 qed "PowI";
```
```   322
```
```   323 Goalw [Pow_def] "A : Pow(B)  ==>  A<=B";
```
```   324 by (etac CollectD 1);
```
```   325 qed "PowD";
```
```   326
```
```   327
```
```   328 bind_thm ("Pow_bottom", empty_subsetI RS PowI);        (* {}: Pow(B) *)
```
```   329 bind_thm ("Pow_top", subset_refl RS PowI);             (* A : Pow(A) *)
```
```   330
```
```   331
```
```   332 section "Set complement";
```
```   333
```
```   334 Goalw [Compl_def] "(c : -A) = (c~:A)";
```
```   335 by (Blast_tac 1);
```
```   336 qed "Compl_iff";
```
```   337
```
```   338 Addsimps [Compl_iff];
```
```   339
```
```   340 val prems = Goalw [Compl_def] "[| c:A ==> False |] ==> c : -A";
```
```   341 by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
```
```   342 qed "ComplI";
```
```   343
```
```   344 (*This form, with negated conclusion, works well with the Classical prover.
```
```   345   Negated assumptions behave like formulae on the right side of the notional
```
```   346   turnstile...*)
```
```   347 Goalw [Compl_def] "c : -A ==> c~:A";
```
```   348 by (etac CollectD 1);
```
```   349 qed "ComplD";
```
```   350
```
```   351 bind_thm ("ComplE", make_elim ComplD);
```
```   352
```
```   353 AddSIs [ComplI];
```
```   354 AddSEs [ComplE];
```
```   355
```
```   356
```
```   357 section "Binary union -- Un";
```
```   358
```
```   359 Goalw [Un_def] "(c : A Un B) = (c:A | c:B)";
```
```   360 by (Blast_tac 1);
```
```   361 qed "Un_iff";
```
```   362 Addsimps [Un_iff];
```
```   363
```
```   364 Goal "c:A ==> c : A Un B";
```
```   365 by (Asm_simp_tac 1);
```
```   366 qed "UnI1";
```
```   367
```
```   368 Goal "c:B ==> c : A Un B";
```
```   369 by (Asm_simp_tac 1);
```
```   370 qed "UnI2";
```
```   371
```
```   372 (*Classical introduction rule: no commitment to A vs B*)
```
```   373
```
```   374 val prems = Goal "(c~:B ==> c:A) ==> c : A Un B";
```
```   375 by (Simp_tac 1);
```
```   376 by (REPEAT (ares_tac (prems@[disjCI]) 1)) ;
```
```   377 qed "UnCI";
```
```   378
```
```   379 val major::prems = Goalw [Un_def]
```
```   380     "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
```
```   381 by (rtac (major RS CollectD RS disjE) 1);
```
```   382 by (REPEAT (eresolve_tac prems 1));
```
```   383 qed "UnE";
```
```   384
```
```   385 AddSIs [UnCI];
```
```   386 AddSEs [UnE];
```
```   387
```
```   388
```
```   389 section "Binary intersection -- Int";
```
```   390
```
```   391 Goalw [Int_def] "(c : A Int B) = (c:A & c:B)";
```
```   392 by (Blast_tac 1);
```
```   393 qed "Int_iff";
```
```   394 Addsimps [Int_iff];
```
```   395
```
```   396 Goal "[| c:A;  c:B |] ==> c : A Int B";
```
```   397 by (Asm_simp_tac 1);
```
```   398 qed "IntI";
```
```   399
```
```   400 Goal "c : A Int B ==> c:A";
```
```   401 by (Asm_full_simp_tac 1);
```
```   402 qed "IntD1";
```
```   403
```
```   404 Goal "c : A Int B ==> c:B";
```
```   405 by (Asm_full_simp_tac 1);
```
```   406 qed "IntD2";
```
```   407
```
```   408 val [major,minor] = Goal
```
```   409     "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
```
```   410 by (rtac minor 1);
```
```   411 by (rtac (major RS IntD1) 1);
```
```   412 by (rtac (major RS IntD2) 1);
```
```   413 qed "IntE";
```
```   414
```
```   415 AddSIs [IntI];
```
```   416 AddSEs [IntE];
```
```   417
```
```   418 section "Set difference";
```
```   419
```
```   420 Goalw [set_diff_def] "(c : A-B) = (c:A & c~:B)";
```
```   421 by (Blast_tac 1);
```
```   422 qed "Diff_iff";
```
```   423 Addsimps [Diff_iff];
```
```   424
```
```   425 Goal "[| c : A;  c ~: B |] ==> c : A - B";
```
```   426 by (Asm_simp_tac 1) ;
```
```   427 qed "DiffI";
```
```   428
```
```   429 Goal "c : A - B ==> c : A";
```
```   430 by (Asm_full_simp_tac 1) ;
```
```   431 qed "DiffD1";
```
```   432
```
```   433 Goal "[| c : A - B;  c : B |] ==> P";
```
```   434 by (Asm_full_simp_tac 1) ;
```
```   435 qed "DiffD2";
```
```   436
```
```   437 val prems = Goal "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P";
```
```   438 by (resolve_tac prems 1);
```
```   439 by (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ;
```
```   440 qed "DiffE";
```
```   441
```
```   442 AddSIs [DiffI];
```
```   443 AddSEs [DiffE];
```
```   444
```
```   445
```
```   446 section "Augmenting a set -- insert";
```
```   447
```
```   448 Goalw [insert_def] "a : insert b A = (a=b | a:A)";
```
```   449 by (Blast_tac 1);
```
```   450 qed "insert_iff";
```
```   451 Addsimps [insert_iff];
```
```   452
```
```   453 Goal "a : insert a B";
```
```   454 by (Simp_tac 1);
```
```   455 qed "insertI1";
```
```   456
```
```   457 Goal "!!a. a : B ==> a : insert b B";
```
```   458 by (Asm_simp_tac 1);
```
```   459 qed "insertI2";
```
```   460
```
```   461 val major::prems = Goalw [insert_def]
```
```   462     "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P";
```
```   463 by (rtac (major RS UnE) 1);
```
```   464 by (REPEAT (eresolve_tac (prems @ [CollectE]) 1));
```
```   465 qed "insertE";
```
```   466
```
```   467 (*Classical introduction rule*)
```
```   468 val prems = Goal "(a~:B ==> a=b) ==> a: insert b B";
```
```   469 by (Simp_tac 1);
```
```   470 by (REPEAT (ares_tac (prems@[disjCI]) 1)) ;
```
```   471 qed "insertCI";
```
```   472
```
```   473 AddSIs [insertCI];
```
```   474 AddSEs [insertE];
```
```   475
```
```   476 Goal "(A <= insert x B) = (if x:A then A-{x} <= B else A<=B)";
```
```   477 by Auto_tac;
```
```   478 qed "subset_insert_iff";
```
```   479
```
```   480 section "Singletons, using insert";
```
```   481
```
```   482 Goal "a : {a}";
```
```   483 by (rtac insertI1 1) ;
```
```   484 qed "singletonI";
```
```   485
```
```   486 Goal "b : {a} ==> b=a";
```
```   487 by (Blast_tac 1);
```
```   488 qed "singletonD";
```
```   489
```
```   490 bind_thm ("singletonE", make_elim singletonD);
```
```   491
```
```   492 Goal "(b : {a}) = (b=a)";
```
```   493 by (Blast_tac 1);
```
```   494 qed "singleton_iff";
```
```   495
```
```   496 Goal "{a}={b} ==> a=b";
```
```   497 by (blast_tac (claset() addEs [equalityE]) 1);
```
```   498 qed "singleton_inject";
```
```   499
```
```   500 (*Redundant? But unlike insertCI, it proves the subgoal immediately!*)
```
```   501 AddSIs [singletonI];
```
```   502 AddSDs [singleton_inject];
```
```   503 AddSEs [singletonE];
```
```   504
```
```   505 Goal "{b} = insert a A = (a = b & A <= {b})";
```
```   506 by (blast_tac (claset() addSEs [equalityE]) 1);
```
```   507 qed "singleton_insert_inj_eq";
```
```   508
```
```   509 Goal "(insert a A = {b}) = (a = b & A <= {b})";
```
```   510 by (blast_tac (claset() addSEs [equalityE]) 1);
```
```   511 qed "singleton_insert_inj_eq'";
```
```   512
```
```   513 AddIffs [singleton_insert_inj_eq, singleton_insert_inj_eq'];
```
```   514
```
```   515 Goal "A <= {x} ==> A={} | A = {x}";
```
```   516 by (Fast_tac 1);
```
```   517 qed "subset_singletonD";
```
```   518
```
```   519 Goal "{x. x=a} = {a}";
```
```   520 by (Blast_tac 1);
```
```   521 qed "singleton_conv";
```
```   522 Addsimps [singleton_conv];
```
```   523
```
```   524 Goal "{x. a=x} = {a}";
```
```   525 by (Blast_tac 1);
```
```   526 qed "singleton_conv2";
```
```   527 Addsimps [singleton_conv2];
```
```   528
```
```   529
```
```   530 section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
```
```   531
```
```   532 Goalw [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
```
```   533 by (Blast_tac 1);
```
```   534 qed "UN_iff";
```
```   535
```
```   536 Addsimps [UN_iff];
```
```   537
```
```   538 (*The order of the premises presupposes that A is rigid; b may be flexible*)
```
```   539 Goal "[| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
```
```   540 by Auto_tac;
```
```   541 qed "UN_I";
```
```   542
```
```   543 val major::prems = Goalw [UNION_def]
```
```   544     "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
```
```   545 by (rtac (major RS CollectD RS bexE) 1);
```
```   546 by (REPEAT (ares_tac prems 1));
```
```   547 qed "UN_E";
```
```   548
```
```   549 AddIs  [UN_I];
```
```   550 AddSEs [UN_E];
```
```   551
```
```   552 val prems = Goalw [UNION_def]
```
```   553     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
```
```   554 \    (UN x:A. C(x)) = (UN x:B. D(x))";
```
```   555 by (asm_simp_tac (simpset() addsimps prems) 1);
```
```   556 qed "UN_cong";
```
```   557
```
```   558
```
```   559 section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
```
```   560
```
```   561 Goalw [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
```
```   562 by Auto_tac;
```
```   563 qed "INT_iff";
```
```   564
```
```   565 Addsimps [INT_iff];
```
```   566
```
```   567 val prems = Goalw [INTER_def]
```
```   568     "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
```
```   569 by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
```
```   570 qed "INT_I";
```
```   571
```
```   572 Goal "[| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
```
```   573 by Auto_tac;
```
```   574 qed "INT_D";
```
```   575
```
```   576 (*"Classical" elimination -- by the Excluded Middle on a:A *)
```
```   577 val major::prems = Goalw [INTER_def]
```
```   578     "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
```
```   579 by (rtac (major RS CollectD RS ballE) 1);
```
```   580 by (REPEAT (eresolve_tac prems 1));
```
```   581 qed "INT_E";
```
```   582
```
```   583 AddSIs [INT_I];
```
```   584 AddEs  [INT_D, INT_E];
```
```   585
```
```   586 val prems = Goalw [INTER_def]
```
```   587     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
```
```   588 \    (INT x:A. C(x)) = (INT x:B. D(x))";
```
```   589 by (asm_simp_tac (simpset() addsimps prems) 1);
```
```   590 qed "INT_cong";
```
```   591
```
```   592
```
```   593 section "Union";
```
```   594
```
```   595 Goalw [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
```
```   596 by (Blast_tac 1);
```
```   597 qed "Union_iff";
```
```   598
```
```   599 Addsimps [Union_iff];
```
```   600
```
```   601 (*The order of the premises presupposes that C is rigid; A may be flexible*)
```
```   602 Goal "[| X:C;  A:X |] ==> A : Union(C)";
```
```   603 by Auto_tac;
```
```   604 qed "UnionI";
```
```   605
```
```   606 val major::prems = Goalw [Union_def]
```
```   607     "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
```
```   608 by (rtac (major RS UN_E) 1);
```
```   609 by (REPEAT (ares_tac prems 1));
```
```   610 qed "UnionE";
```
```   611
```
```   612 AddIs  [UnionI];
```
```   613 AddSEs [UnionE];
```
```   614
```
```   615
```
```   616 section "Inter";
```
```   617
```
```   618 Goalw [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
```
```   619 by (Blast_tac 1);
```
```   620 qed "Inter_iff";
```
```   621
```
```   622 Addsimps [Inter_iff];
```
```   623
```
```   624 val prems = Goalw [Inter_def]
```
```   625     "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
```
```   626 by (REPEAT (ares_tac ([INT_I] @ prems) 1));
```
```   627 qed "InterI";
```
```   628
```
```   629 (*A "destruct" rule -- every X in C contains A as an element, but
```
```   630   A:X can hold when X:C does not!  This rule is analogous to "spec". *)
```
```   631 Goal "[| A : Inter(C);  X:C |] ==> A:X";
```
```   632 by Auto_tac;
```
```   633 qed "InterD";
```
```   634
```
```   635 (*"Classical" elimination rule -- does not require proving X:C *)
```
```   636 val major::prems = Goalw [Inter_def]
```
```   637     "[| A : Inter(C);  X~:C ==> R;  A:X ==> R |] ==> R";
```
```   638 by (rtac (major RS INT_E) 1);
```
```   639 by (REPEAT (eresolve_tac prems 1));
```
```   640 qed "InterE";
```
```   641
```
```   642 AddSIs [InterI];
```
```   643 AddEs  [InterD, InterE];
```
```   644
```
```   645
```
```   646 (*** Image of a set under a function ***)
```
```   647
```
```   648 (*Frequently b does not have the syntactic form of f(x).*)
```
```   649 Goalw [image_def] "[| b=f(x);  x:A |] ==> b : f``A";
```
```   650 by (Blast_tac 1);
```
```   651 qed "image_eqI";
```
```   652 Addsimps [image_eqI];
```
```   653
```
```   654 bind_thm ("imageI", refl RS image_eqI);
```
```   655
```
```   656 (*This version's more effective when we already have the required x*)
```
```   657 Goalw [image_def] "[| x:A;  b=f(x) |] ==> b : f``A";
```
```   658 by (Blast_tac 1);
```
```   659 qed "rev_image_eqI";
```
```   660
```
```   661 (*The eta-expansion gives variable-name preservation.*)
```
```   662 val major::prems = Goalw [image_def]
```
```   663     "[| b : (%x. f(x))``A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P";
```
```   664 by (rtac (major RS CollectD RS bexE) 1);
```
```   665 by (REPEAT (ares_tac prems 1));
```
```   666 qed "imageE";
```
```   667
```
```   668 AddIs  [image_eqI];
```
```   669 AddSEs [imageE];
```
```   670
```
```   671 Goal "f``(A Un B) = f``A Un f``B";
```
```   672 by (Blast_tac 1);
```
```   673 qed "image_Un";
```
```   674
```
```   675 Goal "(z : f``A) = (EX x:A. z = f x)";
```
```   676 by (Blast_tac 1);
```
```   677 qed "image_iff";
```
```   678
```
```   679 (*This rewrite rule would confuse users if made default.*)
```
```   680 Goal "(f``A <= B) = (ALL x:A. f(x): B)";
```
```   681 by (Blast_tac 1);
```
```   682 qed "image_subset_iff";
```
```   683
```
```   684 (*Replaces the three steps subsetI, imageE, hyp_subst_tac, but breaks too
```
```   685   many existing proofs.*)
```
```   686 val prems = Goal "(!!x. x:A ==> f(x) : B) ==> f``A <= B";
```
```   687 by (blast_tac (claset() addIs prems) 1);
```
```   688 qed "image_subsetI";
```
```   689
```
```   690
```
```   691 (*** Range of a function -- just a translation for image! ***)
```
```   692
```
```   693 Goal "b=f(x) ==> b : range(f)";
```
```   694 by (EVERY1 [etac image_eqI, rtac UNIV_I]);
```
```   695 bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI));
```
```   696
```
```   697 bind_thm ("rangeI", UNIV_I RS imageI);
```
```   698
```
```   699 val [major,minor] = Goal
```
```   700     "[| b : range(%x. f(x));  !!x. b=f(x) ==> P |] ==> P";
```
```   701 by (rtac (major RS imageE) 1);
```
```   702 by (etac minor 1);
```
```   703 qed "rangeE";
```
```   704
```
```   705
```
```   706 (*** Set reasoning tools ***)
```
```   707
```
```   708
```
```   709 (** Rewrite rules for boolean case-splitting: faster than
```
```   710 	addsplits[split_if]
```
```   711 **)
```
```   712
```
```   713 bind_thm ("split_if_eq1", read_instantiate [("P", "%x. x = ?b")] split_if);
```
```   714 bind_thm ("split_if_eq2", read_instantiate [("P", "%x. ?a = x")] split_if);
```
```   715
```
```   716 (*Split ifs on either side of the membership relation.
```
```   717 	Not for Addsimps -- can cause goals to blow up!*)
```
```   718 bind_thm ("split_if_mem1",
```
```   719     read_instantiate_sg (Theory.sign_of Set.thy) [("P", "%x. x : ?b")] split_if);
```
```   720 bind_thm ("split_if_mem2",
```
```   721     read_instantiate_sg (Theory.sign_of Set.thy) [("P", "%x. ?a : x")] split_if);
```
```   722
```
```   723 bind_thms ("split_ifs", [if_bool_eq_conj, split_if_eq1, split_if_eq2,
```
```   724 		  split_if_mem1, split_if_mem2]);
```
```   725
```
```   726
```
```   727 (*Each of these has ALREADY been added to simpset() above.*)
```
```   728 bind_thms ("mem_simps", [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff,
```
```   729                  mem_Collect_eq, UN_iff, Union_iff, INT_iff, Inter_iff]);
```
```   730
```
```   731 (*Would like to add these, but the existing code only searches for the
```
```   732   outer-level constant, which in this case is just "op :"; we instead need
```
```   733   to use term-nets to associate patterns with rules.  Also, if a rule fails to
```
```   734   apply, then the formula should be kept.
```
```   735   [("uminus", Compl_iff RS iffD1), ("op -", [Diff_iff RS iffD1]),
```
```   736    ("op Int", [IntD1,IntD2]),
```
```   737    ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
```
```   738  *)
```
```   739 val mksimps_pairs =
```
```   740   [("Ball",[bspec])] @ mksimps_pairs;
```
```   741
```
```   742 simpset_ref() := simpset() setmksimps (mksimps mksimps_pairs);
```
```   743
```
```   744 Addsimps[subset_UNIV, subset_refl];
```
```   745
```
```   746
```
```   747 (*** The 'proper subset' relation (<) ***)
```
```   748
```
```   749 Goalw [psubset_def] "!!A::'a set. [| A <= B; A ~= B |] ==> A<B";
```
```   750 by (Blast_tac 1);
```
```   751 qed "psubsetI";
```
```   752 AddSIs [psubsetI];
```
```   753
```
```   754 Goalw [psubset_def]
```
```   755   "(A < insert x B) = (if x:B then A<B else if x:A then A-{x} < B else A<=B)";
```
```   756 by (asm_simp_tac (simpset() addsimps [subset_insert_iff]) 1);
```
```   757 by (Blast_tac 1);
```
```   758 qed "psubset_insert_iff";
```
```   759
```
```   760 bind_thm ("psubset_eq", psubset_def RS meta_eq_to_obj_eq);
```
```   761
```
```   762 bind_thm ("psubset_imp_subset", psubset_eq RS iffD1 RS conjunct1);
```
```   763
```
```   764 Goal"[| (A::'a set) < B; B <= C |] ==> A < C";
```
```   765 by (auto_tac (claset(), simpset() addsimps [psubset_eq]));
```
```   766 qed "psubset_subset_trans";
```
```   767
```
```   768 Goal"[| (A::'a set) <= B; B < C|] ==> A < C";
```
```   769 by (auto_tac (claset(), simpset() addsimps [psubset_eq]));
```
```   770 qed "subset_psubset_trans";
```
```   771
```
```   772 Goalw [psubset_def] "A < B ==> EX b. b : (B - A)";
```
```   773 by (Blast_tac 1);
```
```   774 qed "psubset_imp_ex_mem";
```
```   775
```
```   776
```
```   777 (* attributes *)
```
```   778
```
```   779 local
```
```   780
```
```   781 fun gen_rulify_prems x =
```
```   782   Attrib.no_args (Drule.rule_attribute (fn _ => (standard o
```
```   783     rule_by_tactic (REPEAT (ALLGOALS (resolve_tac [allI, ballI, impI])))))) x;
```
```   784
```
```   785 in
```
```   786
```
```   787 val rulify_prems_attrib_setup =
```
```   788  [Attrib.add_attributes
```
```   789   [("rulify_prems", (gen_rulify_prems, gen_rulify_prems), "put theorem into standard rule form")]];
```
```   790
```
```   791 end;
```