src/HOL/Set.ML
 author oheimb Wed Jan 31 10:15:55 2001 +0100 (2001-01-31) changeset 11008 f7333f055ef6 parent 11007 438f31613093 child 11166 eca861fd1eb5 permissions -rw-r--r--
improved theory reference in comment
```     1 (*  Title:      HOL/Set.ML
```
```     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 (*Classical elimination rule*)
```
```   156 val major::prems = Goalw [subset_def]
```
```   157     "[| A <= B;  c~:A ==> P;  c:B ==> P |] ==> P";
```
```   158 by (rtac (major RS ballE) 1);
```
```   159 by (REPEAT (eresolve_tac prems 1));
```
```   160 qed "subsetCE";
```
```   161
```
```   162 (*Takes assumptions A<=B; c:A and creates the assumption c:B *)
```
```   163 fun set_mp_tac i = etac subsetCE i  THEN  mp_tac i;
```
```   164
```
```   165 AddSIs [subsetI];
```
```   166 AddEs  [subsetD, subsetCE];
```
```   167
```
```   168 Goal "[| A <= B; c ~: B |] ==> c ~: A";
```
```   169 by (Blast_tac 1);
```
```   170 qed "contra_subsetD";
```
```   171
```
```   172 Goal "A <= (A::'a set)";
```
```   173 by (Fast_tac 1);
```
```   174 qed "subset_refl";
```
```   175
```
```   176 Goal "[| A<=B;  B<=C |] ==> A<=(C::'a set)";
```
```   177 by (Blast_tac 1);
```
```   178 qed "subset_trans";
```
```   179
```
```   180
```
```   181 section "Equality";
```
```   182
```
```   183 (*Anti-symmetry of the subset relation*)
```
```   184 Goal "[| A <= B;  B <= A |] ==> A = (B::'a set)";
```
```   185 by (rtac set_ext 1);
```
```   186 by (blast_tac (claset() addIs [subsetD]) 1);
```
```   187 qed "subset_antisym";
```
```   188 bind_thm ("equalityI", subset_antisym);
```
```   189
```
```   190 AddSIs [equalityI];
```
```   191
```
```   192 (* Equality rules from ZF set theory -- are they appropriate here? *)
```
```   193 Goal "A = B ==> A<=(B::'a set)";
```
```   194 by (etac ssubst 1);
```
```   195 by (rtac subset_refl 1);
```
```   196 qed "equalityD1";
```
```   197
```
```   198 Goal "A = B ==> B<=(A::'a set)";
```
```   199 by (etac ssubst 1);
```
```   200 by (rtac subset_refl 1);
```
```   201 qed "equalityD2";
```
```   202
```
```   203 (*Be careful when adding this to the claset as subset_empty is in the simpset:
```
```   204   A={} goes to {}<=A and A<={} and then back to A={} !*)
```
```   205 val prems = Goal
```
```   206     "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";
```
```   207 by (resolve_tac prems 1);
```
```   208 by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
```
```   209 qed "equalityE";
```
```   210
```
```   211 val major::prems = Goal
```
```   212     "[| A = B;  [| c:A; c:B |] ==> P;  [| c~:A; c~:B |] ==> P |]  ==>  P";
```
```   213 by (rtac (major RS equalityE) 1);
```
```   214 by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
```
```   215 qed "equalityCE";
```
```   216
```
```   217 AddEs [equalityCE];
```
```   218
```
```   219 (*Lemma for creating induction formulae -- for "pattern matching" on p
```
```   220   To make the induction hypotheses usable, apply "spec" or "bspec" to
```
```   221   put universal quantifiers over the free variables in p. *)
```
```   222 val prems = Goal
```
```   223     "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
```
```   224 by (rtac mp 1);
```
```   225 by (REPEAT (resolve_tac (refl::prems) 1));
```
```   226 qed "setup_induction";
```
```   227
```
```   228 Goal "A = B ==> (x : A) = (x : B)";
```
```   229 by (Asm_simp_tac 1);
```
```   230 qed "eqset_imp_iff";
```
```   231
```
```   232
```
```   233 section "The universal set -- UNIV";
```
```   234
```
```   235 Goalw [UNIV_def] "x : UNIV";
```
```   236 by (rtac CollectI 1);
```
```   237 by (rtac TrueI 1);
```
```   238 qed "UNIV_I";
```
```   239
```
```   240 Addsimps [UNIV_I];
```
```   241 AddIs    [UNIV_I];  (*unsafe makes it less likely to cause problems*)
```
```   242
```
```   243 Goal "EX x. x : UNIV";
```
```   244 by (Simp_tac 1);
```
```   245 qed "UNIV_witness";
```
```   246 AddXIs [UNIV_witness];
```
```   247
```
```   248 Goal "A <= UNIV";
```
```   249 by (rtac subsetI 1);
```
```   250 by (rtac UNIV_I 1);
```
```   251 qed "subset_UNIV";
```
```   252
```
```   253 (** Eta-contracting these two rules (to remove P) causes them to be ignored
```
```   254     because of their interaction with congruence rules. **)
```
```   255
```
```   256 Goalw [Ball_def] "Ball UNIV P = All P";
```
```   257 by (Simp_tac 1);
```
```   258 qed "ball_UNIV";
```
```   259
```
```   260 Goalw [Bex_def] "Bex UNIV P = Ex P";
```
```   261 by (Simp_tac 1);
```
```   262 qed "bex_UNIV";
```
```   263 Addsimps [ball_UNIV, bex_UNIV];
```
```   264
```
```   265
```
```   266 section "The empty set -- {}";
```
```   267
```
```   268 Goalw [empty_def] "(c : {}) = False";
```
```   269 by (Blast_tac 1) ;
```
```   270 qed "empty_iff";
```
```   271
```
```   272 Addsimps [empty_iff];
```
```   273
```
```   274 Goal "a:{} ==> P";
```
```   275 by (Full_simp_tac 1);
```
```   276 qed "emptyE";
```
```   277
```
```   278 AddSEs [emptyE];
```
```   279
```
```   280 Goal "{} <= A";
```
```   281 by (Blast_tac 1) ;
```
```   282 qed "empty_subsetI";
```
```   283
```
```   284 (*One effect is to delete the ASSUMPTION {} <= A*)
```
```   285 AddIffs [empty_subsetI];
```
```   286
```
```   287 val [prem]= Goal "[| !!y. y:A ==> False |] ==> A={}";
```
```   288 by (blast_tac (claset() addIs [prem RS FalseE]) 1) ;
```
```   289 qed "equals0I";
```
```   290
```
```   291 (*Use for reasoning about disjointness: A Int B = {} *)
```
```   292 Goal "A={} ==> a ~: A";
```
```   293 by (Blast_tac 1) ;
```
```   294 qed "equals0D";
```
```   295
```
```   296 Goalw [Ball_def] "Ball {} P = True";
```
```   297 by (Simp_tac 1);
```
```   298 qed "ball_empty";
```
```   299
```
```   300 Goalw [Bex_def] "Bex {} P = False";
```
```   301 by (Simp_tac 1);
```
```   302 qed "bex_empty";
```
```   303 Addsimps [ball_empty, bex_empty];
```
```   304
```
```   305 Goal "UNIV ~= {}";
```
```   306 by (blast_tac (claset() addEs [equalityE]) 1);
```
```   307 qed "UNIV_not_empty";
```
```   308 AddIffs [UNIV_not_empty];
```
```   309
```
```   310
```
```   311
```
```   312 section "The Powerset operator -- Pow";
```
```   313
```
```   314 Goalw [Pow_def] "(A : Pow(B)) = (A <= B)";
```
```   315 by (Asm_simp_tac 1);
```
```   316 qed "Pow_iff";
```
```   317
```
```   318 AddIffs [Pow_iff];
```
```   319
```
```   320 Goalw [Pow_def] "A <= B ==> A : Pow(B)";
```
```   321 by (etac CollectI 1);
```
```   322 qed "PowI";
```
```   323
```
```   324 Goalw [Pow_def] "A : Pow(B)  ==>  A<=B";
```
```   325 by (etac CollectD 1);
```
```   326 qed "PowD";
```
```   327
```
```   328
```
```   329 bind_thm ("Pow_bottom", empty_subsetI RS PowI);        (* {}: Pow(B) *)
```
```   330 bind_thm ("Pow_top", subset_refl RS PowI);             (* A : Pow(A) *)
```
```   331
```
```   332
```
```   333 section "Set complement";
```
```   334
```
```   335 Goalw [Compl_def] "(c : -A) = (c~:A)";
```
```   336 by (Blast_tac 1);
```
```   337 qed "Compl_iff";
```
```   338
```
```   339 Addsimps [Compl_iff];
```
```   340
```
```   341 val prems = Goalw [Compl_def] "[| c:A ==> False |] ==> c : -A";
```
```   342 by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
```
```   343 qed "ComplI";
```
```   344
```
```   345 (*This form, with negated conclusion, works well with the Classical prover.
```
```   346   Negated assumptions behave like formulae on the right side of the notional
```
```   347   turnstile...*)
```
```   348 Goalw [Compl_def] "c : -A ==> c~:A";
```
```   349 by (etac CollectD 1);
```
```   350 qed "ComplD";
```
```   351
```
```   352 bind_thm ("ComplE", make_elim ComplD);
```
```   353
```
```   354 AddSIs [ComplI];
```
```   355 AddSEs [ComplE];
```
```   356
```
```   357
```
```   358 section "Binary union -- Un";
```
```   359
```
```   360 Goalw [Un_def] "(c : A Un B) = (c:A | c:B)";
```
```   361 by (Blast_tac 1);
```
```   362 qed "Un_iff";
```
```   363 Addsimps [Un_iff];
```
```   364
```
```   365 Goal "c:A ==> c : A Un B";
```
```   366 by (Asm_simp_tac 1);
```
```   367 qed "UnI1";
```
```   368
```
```   369 Goal "c:B ==> c : A Un B";
```
```   370 by (Asm_simp_tac 1);
```
```   371 qed "UnI2";
```
```   372
```
```   373 AddXIs [UnI1, UnI2];
```
```   374
```
```   375
```
```   376 (*Classical introduction rule: no commitment to A vs B*)
```
```   377
```
```   378 val prems = Goal "(c~:B ==> c:A) ==> c : A Un B";
```
```   379 by (Simp_tac 1);
```
```   380 by (REPEAT (ares_tac (prems@[disjCI]) 1)) ;
```
```   381 qed "UnCI";
```
```   382
```
```   383 val major::prems = Goalw [Un_def]
```
```   384     "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
```
```   385 by (rtac (major RS CollectD RS disjE) 1);
```
```   386 by (REPEAT (eresolve_tac prems 1));
```
```   387 qed "UnE";
```
```   388
```
```   389 AddSIs [UnCI];
```
```   390 AddSEs [UnE];
```
```   391
```
```   392
```
```   393 section "Binary intersection -- Int";
```
```   394
```
```   395 Goalw [Int_def] "(c : A Int B) = (c:A & c:B)";
```
```   396 by (Blast_tac 1);
```
```   397 qed "Int_iff";
```
```   398 Addsimps [Int_iff];
```
```   399
```
```   400 Goal "[| c:A;  c:B |] ==> c : A Int B";
```
```   401 by (Asm_simp_tac 1);
```
```   402 qed "IntI";
```
```   403
```
```   404 Goal "c : A Int B ==> c:A";
```
```   405 by (Asm_full_simp_tac 1);
```
```   406 qed "IntD1";
```
```   407
```
```   408 Goal "c : A Int B ==> c:B";
```
```   409 by (Asm_full_simp_tac 1);
```
```   410 qed "IntD2";
```
```   411
```
```   412 val [major,minor] = Goal
```
```   413     "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
```
```   414 by (rtac minor 1);
```
```   415 by (rtac (major RS IntD1) 1);
```
```   416 by (rtac (major RS IntD2) 1);
```
```   417 qed "IntE";
```
```   418
```
```   419 AddSIs [IntI];
```
```   420 AddSEs [IntE];
```
```   421
```
```   422 section "Set difference";
```
```   423
```
```   424 Goalw [set_diff_def] "(c : A-B) = (c:A & c~:B)";
```
```   425 by (Blast_tac 1);
```
```   426 qed "Diff_iff";
```
```   427 Addsimps [Diff_iff];
```
```   428
```
```   429 Goal "[| c : A;  c ~: B |] ==> c : A - B";
```
```   430 by (Asm_simp_tac 1) ;
```
```   431 qed "DiffI";
```
```   432
```
```   433 Goal "c : A - B ==> c : A";
```
```   434 by (Asm_full_simp_tac 1) ;
```
```   435 qed "DiffD1";
```
```   436
```
```   437 Goal "[| c : A - B;  c : B |] ==> P";
```
```   438 by (Asm_full_simp_tac 1) ;
```
```   439 qed "DiffD2";
```
```   440
```
```   441 val prems = Goal "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P";
```
```   442 by (resolve_tac prems 1);
```
```   443 by (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ;
```
```   444 qed "DiffE";
```
```   445
```
```   446 AddSIs [DiffI];
```
```   447 AddSEs [DiffE];
```
```   448
```
```   449
```
```   450 section "Augmenting a set -- insert";
```
```   451
```
```   452 Goalw [insert_def] "a : insert b A = (a=b | a:A)";
```
```   453 by (Blast_tac 1);
```
```   454 qed "insert_iff";
```
```   455 Addsimps [insert_iff];
```
```   456
```
```   457 Goal "a : insert a B";
```
```   458 by (Simp_tac 1);
```
```   459 qed "insertI1";
```
```   460
```
```   461 Goal "!!a. a : B ==> a : insert b B";
```
```   462 by (Asm_simp_tac 1);
```
```   463 qed "insertI2";
```
```   464
```
```   465 val major::prems = Goalw [insert_def]
```
```   466     "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P";
```
```   467 by (rtac (major RS UnE) 1);
```
```   468 by (REPEAT (eresolve_tac (prems @ [CollectE]) 1));
```
```   469 qed "insertE";
```
```   470
```
```   471 (*Classical introduction rule*)
```
```   472 val prems = Goal "(a~:B ==> a=b) ==> a: insert b B";
```
```   473 by (Simp_tac 1);
```
```   474 by (REPEAT (ares_tac (prems@[disjCI]) 1)) ;
```
```   475 qed "insertCI";
```
```   476
```
```   477 AddSIs [insertCI];
```
```   478 AddSEs [insertE];
```
```   479
```
```   480 Goal "(A <= insert x B) = (if x:A then A-{x} <= B else A<=B)";
```
```   481 by Auto_tac;
```
```   482 qed "subset_insert_iff";
```
```   483
```
```   484 section "Singletons, using insert";
```
```   485
```
```   486 Goal "a : {a}";
```
```   487 by (rtac insertI1 1) ;
```
```   488 qed "singletonI";
```
```   489
```
```   490 Goal "b : {a} ==> b=a";
```
```   491 by (Blast_tac 1);
```
```   492 qed "singletonD";
```
```   493
```
```   494 bind_thm ("singletonE", make_elim singletonD);
```
```   495
```
```   496 Goal "(b : {a}) = (b=a)";
```
```   497 by (Blast_tac 1);
```
```   498 qed "singleton_iff";
```
```   499
```
```   500 Goal "{a}={b} ==> a=b";
```
```   501 by (blast_tac (claset() addEs [equalityE]) 1);
```
```   502 qed "singleton_inject";
```
```   503
```
```   504 (*Redundant? But unlike insertCI, it proves the subgoal immediately!*)
```
```   505 AddSIs [singletonI];
```
```   506 AddSDs [singleton_inject];
```
```   507 AddSEs [singletonE];
```
```   508
```
```   509 Goal "{b} = insert a A = (a = b & A <= {b})";
```
```   510 by (blast_tac (claset() addSEs [equalityE]) 1);
```
```   511 qed "singleton_insert_inj_eq";
```
```   512
```
```   513 Goal "(insert a A = {b}) = (a = b & A <= {b})";
```
```   514 by (blast_tac (claset() addSEs [equalityE]) 1);
```
```   515 qed "singleton_insert_inj_eq'";
```
```   516
```
```   517 AddIffs [singleton_insert_inj_eq, singleton_insert_inj_eq'];
```
```   518
```
```   519 Goal "A <= {x} ==> A={} | A = {x}";
```
```   520 by (Fast_tac 1);
```
```   521 qed "subset_singletonD";
```
```   522
```
```   523 Goal "{x. x=a} = {a}";
```
```   524 by (Blast_tac 1);
```
```   525 qed "singleton_conv";
```
```   526 Addsimps [singleton_conv];
```
```   527
```
```   528 Goal "{x. a=x} = {a}";
```
```   529 by (Blast_tac 1);
```
```   530 qed "singleton_conv2";
```
```   531 Addsimps [singleton_conv2];
```
```   532
```
```   533 Goal "[| A - {x} <= B; x : A |] ==> A <= insert x B";
```
```   534 by(Blast_tac 1);
```
```   535 qed "diff_single_insert";
```
```   536
```
```   537
```
```   538 section "Unions of families -- UNION x:A. B(x) is Union(B`A)";
```
```   539
```
```   540 Goalw [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
```
```   541 by (Blast_tac 1);
```
```   542 qed "UN_iff";
```
```   543
```
```   544 Addsimps [UN_iff];
```
```   545
```
```   546 (*The order of the premises presupposes that A is rigid; b may be flexible*)
```
```   547 Goal "[| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
```
```   548 by Auto_tac;
```
```   549 qed "UN_I";
```
```   550
```
```   551 val major::prems = Goalw [UNION_def]
```
```   552     "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
```
```   553 by (rtac (major RS CollectD RS bexE) 1);
```
```   554 by (REPEAT (ares_tac prems 1));
```
```   555 qed "UN_E";
```
```   556
```
```   557 AddIs  [UN_I];
```
```   558 AddSEs [UN_E];
```
```   559
```
```   560 val prems = Goalw [UNION_def]
```
```   561     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
```
```   562 \    (UN x:A. C(x)) = (UN x:B. D(x))";
```
```   563 by (asm_simp_tac (simpset() addsimps prems) 1);
```
```   564 qed "UN_cong";
```
```   565 Addcongs [UN_cong];
```
```   566
```
```   567
```
```   568 section "Intersections of families -- INTER x:A. B(x) is Inter(B`A)";
```
```   569
```
```   570 Goalw [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
```
```   571 by Auto_tac;
```
```   572 qed "INT_iff";
```
```   573
```
```   574 Addsimps [INT_iff];
```
```   575
```
```   576 val prems = Goalw [INTER_def]
```
```   577     "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
```
```   578 by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
```
```   579 qed "INT_I";
```
```   580
```
```   581 Goal "[| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
```
```   582 by Auto_tac;
```
```   583 qed "INT_D";
```
```   584
```
```   585 (*"Classical" elimination -- by the Excluded Middle on a:A *)
```
```   586 val major::prems = Goalw [INTER_def]
```
```   587     "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
```
```   588 by (rtac (major RS CollectD RS ballE) 1);
```
```   589 by (REPEAT (eresolve_tac prems 1));
```
```   590 qed "INT_E";
```
```   591
```
```   592 AddSIs [INT_I];
```
```   593 AddEs  [INT_D, INT_E];
```
```   594
```
```   595 val prems = Goalw [INTER_def]
```
```   596     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
```
```   597 \    (INT x:A. C(x)) = (INT x:B. D(x))";
```
```   598 by (asm_simp_tac (simpset() addsimps prems) 1);
```
```   599 qed "INT_cong";
```
```   600 Addcongs [INT_cong];
```
```   601
```
```   602
```
```   603 section "Union";
```
```   604
```
```   605 Goalw [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
```
```   606 by (Blast_tac 1);
```
```   607 qed "Union_iff";
```
```   608
```
```   609 Addsimps [Union_iff];
```
```   610
```
```   611 (*The order of the premises presupposes that C is rigid; A may be flexible*)
```
```   612 Goal "[| X:C;  A:X |] ==> A : Union(C)";
```
```   613 by Auto_tac;
```
```   614 qed "UnionI";
```
```   615
```
```   616 val major::prems = Goalw [Union_def]
```
```   617     "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
```
```   618 by (rtac (major RS UN_E) 1);
```
```   619 by (REPEAT (ares_tac prems 1));
```
```   620 qed "UnionE";
```
```   621
```
```   622 AddIs  [UnionI];
```
```   623 AddSEs [UnionE];
```
```   624
```
```   625
```
```   626 section "Inter";
```
```   627
```
```   628 Goalw [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
```
```   629 by (Blast_tac 1);
```
```   630 qed "Inter_iff";
```
```   631
```
```   632 Addsimps [Inter_iff];
```
```   633
```
```   634 val prems = Goalw [Inter_def]
```
```   635     "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
```
```   636 by (REPEAT (ares_tac ([INT_I] @ prems) 1));
```
```   637 qed "InterI";
```
```   638
```
```   639 (*A "destruct" rule -- every X in C contains A as an element, but
```
```   640   A:X can hold when X:C does not!  This rule is analogous to "spec". *)
```
```   641 Goal "[| A : Inter(C);  X:C |] ==> A:X";
```
```   642 by Auto_tac;
```
```   643 qed "InterD";
```
```   644
```
```   645 (*"Classical" elimination rule -- does not require proving X:C *)
```
```   646 val major::prems = Goalw [Inter_def]
```
```   647     "[| A : Inter(C);  X~:C ==> R;  A:X ==> R |] ==> R";
```
```   648 by (rtac (major RS INT_E) 1);
```
```   649 by (REPEAT (eresolve_tac prems 1));
```
```   650 qed "InterE";
```
```   651
```
```   652 AddSIs [InterI];
```
```   653 AddEs  [InterD, InterE];
```
```   654
```
```   655
```
```   656 (*** Image of a set under a function ***)
```
```   657
```
```   658 (*Frequently b does not have the syntactic form of f(x).*)
```
```   659 Goalw [image_def] "[| b=f(x);  x:A |] ==> b : f`A";
```
```   660 by (Blast_tac 1);
```
```   661 qed "image_eqI";
```
```   662 Addsimps [image_eqI];
```
```   663
```
```   664 bind_thm ("imageI", refl RS image_eqI);
```
```   665
```
```   666 (*This version's more effective when we already have the required x*)
```
```   667 Goalw [image_def] "[| x:A;  b=f(x) |] ==> b : f`A";
```
```   668 by (Blast_tac 1);
```
```   669 qed "rev_image_eqI";
```
```   670
```
```   671 (*The eta-expansion gives variable-name preservation.*)
```
```   672 val major::prems = Goalw [image_def]
```
```   673     "[| b : (%x. f(x))`A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P";
```
```   674 by (rtac (major RS CollectD RS bexE) 1);
```
```   675 by (REPEAT (ares_tac prems 1));
```
```   676 qed "imageE";
```
```   677
```
```   678 AddIs  [image_eqI];
```
```   679 AddSEs [imageE];
```
```   680
```
```   681 Goal "f`(A Un B) = f`A Un f`B";
```
```   682 by (Blast_tac 1);
```
```   683 qed "image_Un";
```
```   684
```
```   685 Goal "(z : f`A) = (EX x:A. z = f x)";
```
```   686 by (Blast_tac 1);
```
```   687 qed "image_iff";
```
```   688
```
```   689 (*This rewrite rule would confuse users if made default.*)
```
```   690 Goal "(f`A <= B) = (ALL x:A. f(x): B)";
```
```   691 by (Blast_tac 1);
```
```   692 qed "image_subset_iff";
```
```   693
```
```   694 Goal "B <= f ` A = (? AA. AA <= A & B = f ` AA)";
```
```   695 by Safe_tac;
```
```   696 by  (Fast_tac 2);
```
```   697 by (res_inst_tac [("x","{a. a : A & f a : B}")] exI 1);
```
```   698 by (Fast_tac 1);
```
```   699 qed "subset_image_iff";
```
```   700
```
```   701 (*Replaces the three steps subsetI, imageE, hyp_subst_tac, but breaks too
```
```   702   many existing proofs.*)
```
```   703 val prems = Goal "(!!x. x:A ==> f(x) : B) ==> f`A <= B";
```
```   704 by (blast_tac (claset() addIs prems) 1);
```
```   705 qed "image_subsetI";
```
```   706
```
```   707 (*** Range of a function -- just a translation for image! ***)
```
```   708
```
```   709 Goal "b=f(x) ==> b : range(f)";
```
```   710 by (EVERY1 [etac image_eqI, rtac UNIV_I]);
```
```   711 bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI));
```
```   712
```
```   713 bind_thm ("rangeI", UNIV_I RS imageI);
```
```   714
```
```   715 val [major,minor] = Goal
```
```   716     "[| b : range(%x. f(x));  !!x. b=f(x) ==> P |] ==> P";
```
```   717 by (rtac (major RS imageE) 1);
```
```   718 by (etac minor 1);
```
```   719 qed "rangeE";
```
```   720 AddXEs [rangeE];
```
```   721
```
```   722
```
```   723 (*** Set reasoning tools ***)
```
```   724
```
```   725
```
```   726 (** Rewrite rules for boolean case-splitting: faster than
```
```   727 	addsplits[split_if]
```
```   728 **)
```
```   729
```
```   730 bind_thm ("split_if_eq1", read_instantiate [("P", "%x. x = ?b")] split_if);
```
```   731 bind_thm ("split_if_eq2", read_instantiate [("P", "%x. ?a = x")] split_if);
```
```   732
```
```   733 (*Split ifs on either side of the membership relation.
```
```   734 	Not for Addsimps -- can cause goals to blow up!*)
```
```   735 bind_thm ("split_if_mem1", inst "P" "%x. x : ?b" split_if);
```
```   736 bind_thm ("split_if_mem2", inst "P" "%x. ?a : x" split_if);
```
```   737
```
```   738 bind_thms ("split_ifs", [if_bool_eq_conj, split_if_eq1, split_if_eq2,
```
```   739 			 split_if_mem1, split_if_mem2]);
```
```   740
```
```   741
```
```   742 (*Each of these has ALREADY been added to simpset() above.*)
```
```   743 bind_thms ("mem_simps", [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff,
```
```   744                  mem_Collect_eq, UN_iff, Union_iff, INT_iff, Inter_iff]);
```
```   745
```
```   746 (*Would like to add these, but the existing code only searches for the
```
```   747   outer-level constant, which in this case is just "op :"; we instead need
```
```   748   to use term-nets to associate patterns with rules.  Also, if a rule fails to
```
```   749   apply, then the formula should be kept.
```
```   750   [("uminus", Compl_iff RS iffD1), ("op -", [Diff_iff RS iffD1]),
```
```   751    ("op Int", [IntD1,IntD2]),
```
```   752    ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
```
```   753  *)
```
```   754 val mksimps_pairs =
```
```   755   [("Ball",[bspec])] @ mksimps_pairs;
```
```   756
```
```   757 simpset_ref() := simpset() setmksimps (mksimps mksimps_pairs);
```
```   758
```
```   759 Addsimps[subset_UNIV, subset_refl];
```
```   760
```
```   761
```
```   762 (*** The 'proper subset' relation (<) ***)
```
```   763
```
```   764 Goalw [psubset_def] "!!A::'a set. [| A <= B; A ~= B |] ==> A<B";
```
```   765 by (Blast_tac 1);
```
```   766 qed "psubsetI";
```
```   767 AddSIs [psubsetI];
```
```   768
```
```   769 Goalw [psubset_def]
```
```   770   "(A < insert x B) = (if x:B then A<B else if x:A then A-{x} < B else A<=B)";
```
```   771 by (asm_simp_tac (simpset() addsimps [subset_insert_iff]) 1);
```
```   772 by (Blast_tac 1);
```
```   773 qed "psubset_insert_iff";
```
```   774
```
```   775 bind_thm ("psubset_eq", psubset_def RS meta_eq_to_obj_eq);
```
```   776
```
```   777 bind_thm ("psubset_imp_subset", psubset_eq RS iffD1 RS conjunct1);
```
```   778
```
```   779 Goal"[| (A::'a set) < B; B <= C |] ==> A < C";
```
```   780 by (auto_tac (claset(), simpset() addsimps [psubset_eq]));
```
```   781 qed "psubset_subset_trans";
```
```   782
```
```   783 Goal"[| (A::'a set) <= B; B < C|] ==> A < C";
```
```   784 by (auto_tac (claset(), simpset() addsimps [psubset_eq]));
```
```   785 qed "subset_psubset_trans";
```
```   786
```
```   787 Goalw [psubset_def] "A < B ==> EX b. b : (B - A)";
```
```   788 by (Blast_tac 1);
```
```   789 qed "psubset_imp_ex_mem";
```
```   790
```
```   791
```
```   792 (* rulify setup *)
```
```   793
```
```   794 Goal "(!!x. x:A ==> P x) == Trueprop (ALL x:A. P x)";
```
```   795 by (simp_tac (simpset () addsimps (Ball_def :: thms "atomize")) 1);
```
```   796 qed "ball_eq";
```
```   797
```
```   798 local
```
```   799   val ss = HOL_basic_ss addsimps
```
```   800     (Drule.norm_hhf_eq :: map Thm.symmetric (ball_eq :: thms "atomize"));
```
```   801 in
```
```   802
```
```   803 structure Rulify = RulifyFun
```
```   804  (val make_meta = Simplifier.simplify ss
```
```   805   val full_make_meta = Simplifier.full_simplify ss);
```
```   806
```
```   807 structure BasicRulify: BASIC_RULIFY = Rulify;
```
```   808 open BasicRulify;
```
```   809
```
```   810 end;
```