src/HOL/equalities.ML
author nipkow
Mon Oct 21 09:50:50 1996 +0200 (1996-10-21)
changeset 2115 9709f9188549
parent 2031 03a843f0f447
child 2512 0231e4f467f2
permissions -rw-r--r--
Added trans_tac (see Provers/nat_transitive.ML)
     1 (*  Title:      HOL/equalities
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1994  University of Cambridge
     5 
     6 Equalities involving union, intersection, inclusion, etc.
     7 *)
     8 
     9 writeln"File HOL/equalities";
    10 
    11 AddSIs [equalityI];
    12 
    13 section "{}";
    14 
    15 goal Set.thy "{x.False} = {}";
    16 by (Fast_tac 1);
    17 qed "Collect_False_empty";
    18 Addsimps [Collect_False_empty];
    19 
    20 goal Set.thy "(A <= {}) = (A = {})";
    21 by (Fast_tac 1);
    22 qed "subset_empty";
    23 Addsimps [subset_empty];
    24 
    25 section "insert";
    26 
    27 (*NOT SUITABLE FOR REWRITING since {a} == insert a {}*)
    28 goal Set.thy "insert a A = {a} Un A";
    29 by (Fast_tac 1);
    30 qed "insert_is_Un";
    31 
    32 goal Set.thy "insert a A ~= {}";
    33 by (fast_tac (!claset addEs [equalityCE]) 1);
    34 qed"insert_not_empty";
    35 Addsimps[insert_not_empty];
    36 
    37 bind_thm("empty_not_insert",insert_not_empty RS not_sym);
    38 Addsimps[empty_not_insert];
    39 
    40 goal Set.thy "!!a. a:A ==> insert a A = A";
    41 by (Fast_tac 1);
    42 qed "insert_absorb";
    43 
    44 goal Set.thy "insert x (insert x A) = insert x A";
    45 by (Fast_tac 1);
    46 qed "insert_absorb2";
    47 Addsimps [insert_absorb2];
    48 
    49 goal Set.thy "insert x (insert y A) = insert y (insert x A)";
    50 by (Fast_tac 1);
    51 qed "insert_commute";
    52 
    53 goal Set.thy "(insert x A <= B) = (x:B & A <= B)";
    54 by (Fast_tac 1);
    55 qed "insert_subset";
    56 Addsimps[insert_subset];
    57 
    58 (* use new B rather than (A-{a}) to avoid infinite unfolding *)
    59 goal Set.thy "!!a. a:A ==> ? B. A = insert a B & a ~: B";
    60 by (res_inst_tac [("x","A-{a}")] exI 1);
    61 by (Fast_tac 1);
    62 qed "mk_disjoint_insert";
    63 
    64 goal Set.thy
    65     "!!A. A~={} ==> (UN x:A. insert a (B x)) = insert a (UN x:A. B x)";
    66 by (Fast_tac 1);
    67 qed "UN_insert_distrib";
    68 
    69 goal Set.thy "(UN x. insert a (B x)) = insert a (UN x. B x)";
    70 by (Fast_tac 1);
    71 qed "UN1_insert_distrib";
    72 
    73 section "``";
    74 
    75 goal Set.thy "f``{} = {}";
    76 by (Fast_tac 1);
    77 qed "image_empty";
    78 Addsimps[image_empty];
    79 
    80 goal Set.thy "f``insert a B = insert (f a) (f``B)";
    81 by (Fast_tac 1);
    82 qed "image_insert";
    83 Addsimps[image_insert];
    84 
    85 qed_goal "ball_image" Set.thy "(!y:F``S. P y) = (!x:S. P (F x))"
    86  (fn _ => [Fast_tac 1]);
    87 
    88 goal Set.thy "!!x. x:A ==> insert (f x) (f``A) = f``A";
    89 by (Fast_tac 1);
    90 qed "insert_image";
    91 Addsimps [insert_image];
    92 
    93 goalw Set.thy [image_def]
    94 "(%x. if P x then f x else g x) `` S                    \
    95 \ = (f `` ({x.x:S & P x})) Un (g `` ({x.x:S & ~(P x)}))";
    96 by (split_tac [expand_if] 1);
    97 by (Fast_tac 1);
    98 qed "if_image_distrib";
    99 Addsimps[if_image_distrib];
   100 
   101 
   102 section "range";
   103 
   104 qed_goal "ball_range" Set.thy "(!y:range f. P y) = (!x. P (f x))"
   105  (fn _ => [Fast_tac 1]);
   106 
   107 qed_goalw "image_range" Set.thy [image_def]
   108  "f``range g = range (%x. f (g x))" 
   109  (fn _ => [rtac Collect_cong 1, Fast_tac 1]);
   110 
   111 section "Int";
   112 
   113 goal Set.thy "A Int A = A";
   114 by (Fast_tac 1);
   115 qed "Int_absorb";
   116 Addsimps[Int_absorb];
   117 
   118 goal Set.thy "A Int B  =  B Int A";
   119 by (Fast_tac 1);
   120 qed "Int_commute";
   121 
   122 goal Set.thy "(A Int B) Int C  =  A Int (B Int C)";
   123 by (Fast_tac 1);
   124 qed "Int_assoc";
   125 
   126 goal Set.thy "{} Int B = {}";
   127 by (Fast_tac 1);
   128 qed "Int_empty_left";
   129 Addsimps[Int_empty_left];
   130 
   131 goal Set.thy "A Int {} = {}";
   132 by (Fast_tac 1);
   133 qed "Int_empty_right";
   134 Addsimps[Int_empty_right];
   135 
   136 goal Set.thy "UNIV Int B = B";
   137 by (Fast_tac 1);
   138 qed "Int_UNIV_left";
   139 Addsimps[Int_UNIV_left];
   140 
   141 goal Set.thy "A Int UNIV = A";
   142 by (Fast_tac 1);
   143 qed "Int_UNIV_right";
   144 Addsimps[Int_UNIV_right];
   145 
   146 goal Set.thy "A Int (B Un C)  =  (A Int B) Un (A Int C)";
   147 by (Fast_tac 1);
   148 qed "Int_Un_distrib";
   149 
   150 goal Set.thy "(B Un C) Int A =  (B Int A) Un (C Int A)";
   151 by (Fast_tac 1);
   152 qed "Int_Un_distrib2";
   153 
   154 goal Set.thy "(A<=B) = (A Int B = A)";
   155 by (fast_tac (!claset addSEs [equalityE]) 1);
   156 qed "subset_Int_eq";
   157 
   158 goal Set.thy "(A Int B = UNIV) = (A = UNIV & B = UNIV)";
   159 by (fast_tac (!claset addEs [equalityCE]) 1);
   160 qed "Int_UNIV";
   161 Addsimps[Int_UNIV];
   162 
   163 section "Un";
   164 
   165 goal Set.thy "A Un A = A";
   166 by (Fast_tac 1);
   167 qed "Un_absorb";
   168 Addsimps[Un_absorb];
   169 
   170 goal Set.thy "A Un B  =  B Un A";
   171 by (Fast_tac 1);
   172 qed "Un_commute";
   173 
   174 goal Set.thy "(A Un B) Un C  =  A Un (B Un C)";
   175 by (Fast_tac 1);
   176 qed "Un_assoc";
   177 
   178 goal Set.thy "{} Un B = B";
   179 by (Fast_tac 1);
   180 qed "Un_empty_left";
   181 Addsimps[Un_empty_left];
   182 
   183 goal Set.thy "A Un {} = A";
   184 by (Fast_tac 1);
   185 qed "Un_empty_right";
   186 Addsimps[Un_empty_right];
   187 
   188 goal Set.thy "UNIV Un B = UNIV";
   189 by (Fast_tac 1);
   190 qed "Un_UNIV_left";
   191 Addsimps[Un_UNIV_left];
   192 
   193 goal Set.thy "A Un UNIV = UNIV";
   194 by (Fast_tac 1);
   195 qed "Un_UNIV_right";
   196 Addsimps[Un_UNIV_right];
   197 
   198 goal Set.thy "(insert a B) Un C = insert a (B Un C)";
   199 by (Fast_tac 1);
   200 qed "Un_insert_left";
   201 
   202 goal Set.thy "A Un (insert a B) = insert a (A Un B)";
   203 by (Fast_tac 1);
   204 qed "Un_insert_right";
   205 
   206 goal Set.thy "(A Int B) Un C  =  (A Un C) Int (B Un C)";
   207 by (Fast_tac 1);
   208 qed "Un_Int_distrib";
   209 
   210 goal Set.thy
   211  "(A Int B) Un (B Int C) Un (C Int A) = (A Un B) Int (B Un C) Int (C Un A)";
   212 by (Fast_tac 1);
   213 qed "Un_Int_crazy";
   214 
   215 goal Set.thy "(A<=B) = (A Un B = B)";
   216 by (fast_tac (!claset addSEs [equalityE]) 1);
   217 qed "subset_Un_eq";
   218 
   219 goal Set.thy "(A <= insert b C) = (A <= C | b:A & A-{b} <= C)";
   220 by (Fast_tac 1);
   221 qed "subset_insert_iff";
   222 
   223 goal Set.thy "(A Un B = {}) = (A = {} & B = {})";
   224 by (fast_tac (!claset addEs [equalityCE]) 1);
   225 qed "Un_empty";
   226 Addsimps[Un_empty];
   227 
   228 section "Compl";
   229 
   230 goal Set.thy "A Int Compl(A) = {}";
   231 by (Fast_tac 1);
   232 qed "Compl_disjoint";
   233 Addsimps[Compl_disjoint];
   234 
   235 goal Set.thy "A Un Compl(A) = UNIV";
   236 by (Fast_tac 1);
   237 qed "Compl_partition";
   238 
   239 goal Set.thy "Compl(Compl(A)) = A";
   240 by (Fast_tac 1);
   241 qed "double_complement";
   242 Addsimps[double_complement];
   243 
   244 goal Set.thy "Compl(A Un B) = Compl(A) Int Compl(B)";
   245 by (Fast_tac 1);
   246 qed "Compl_Un";
   247 
   248 goal Set.thy "Compl(A Int B) = Compl(A) Un Compl(B)";
   249 by (Fast_tac 1);
   250 qed "Compl_Int";
   251 
   252 goal Set.thy "Compl(UN x:A. B(x)) = (INT x:A. Compl(B(x)))";
   253 by (Fast_tac 1);
   254 qed "Compl_UN";
   255 
   256 goal Set.thy "Compl(INT x:A. B(x)) = (UN x:A. Compl(B(x)))";
   257 by (Fast_tac 1);
   258 qed "Compl_INT";
   259 
   260 (*Halmos, Naive Set Theory, page 16.*)
   261 
   262 goal Set.thy "((A Int B) Un C = A Int (B Un C)) = (C<=A)";
   263 by (fast_tac (!claset addSEs [equalityE]) 1);
   264 qed "Un_Int_assoc_eq";
   265 
   266 
   267 section "Union";
   268 
   269 goal Set.thy "Union({}) = {}";
   270 by (Fast_tac 1);
   271 qed "Union_empty";
   272 Addsimps[Union_empty];
   273 
   274 goal Set.thy "Union(UNIV) = UNIV";
   275 by (Fast_tac 1);
   276 qed "Union_UNIV";
   277 Addsimps[Union_UNIV];
   278 
   279 goal Set.thy "Union(insert a B) = a Un Union(B)";
   280 by (Fast_tac 1);
   281 qed "Union_insert";
   282 Addsimps[Union_insert];
   283 
   284 goal Set.thy "Union(A Un B) = Union(A) Un Union(B)";
   285 by (Fast_tac 1);
   286 qed "Union_Un_distrib";
   287 Addsimps[Union_Un_distrib];
   288 
   289 goal Set.thy "Union(A Int B) <= Union(A) Int Union(B)";
   290 by (Fast_tac 1);
   291 qed "Union_Int_subset";
   292 
   293 val prems = goal Set.thy
   294    "(Union(C) Int A = {}) = (! B:C. B Int A = {})";
   295 by (fast_tac (!claset addSEs [equalityE]) 1);
   296 qed "Union_disjoint";
   297 
   298 section "Inter";
   299 
   300 goal Set.thy "Inter({}) = UNIV";
   301 by (Fast_tac 1);
   302 qed "Inter_empty";
   303 Addsimps[Inter_empty];
   304 
   305 goal Set.thy "Inter(UNIV) = {}";
   306 by (Fast_tac 1);
   307 qed "Inter_UNIV";
   308 Addsimps[Inter_UNIV];
   309 
   310 goal Set.thy "Inter(insert a B) = a Int Inter(B)";
   311 by (Fast_tac 1);
   312 qed "Inter_insert";
   313 Addsimps[Inter_insert];
   314 
   315 goal Set.thy "Inter(A) Un Inter(B) <= Inter(A Int B)";
   316 by (Fast_tac 1);
   317 qed "Inter_Un_subset";
   318 
   319 goal Set.thy "Inter(A Un B) = Inter(A) Int Inter(B)";
   320 by (best_tac (!claset) 1);
   321 qed "Inter_Un_distrib";
   322 
   323 section "UN and INT";
   324 
   325 (*Basic identities*)
   326 
   327 goal Set.thy "(UN x:{}. B x) = {}";
   328 by (Fast_tac 1);
   329 qed "UN_empty";
   330 Addsimps[UN_empty];
   331 
   332 goal Set.thy "(UN x:UNIV. B x) = (UN x. B x)";
   333 by (Fast_tac 1);
   334 qed "UN_UNIV";
   335 Addsimps[UN_UNIV];
   336 
   337 goal Set.thy "(INT x:{}. B x) = UNIV";
   338 by (Fast_tac 1);
   339 qed "INT_empty";
   340 Addsimps[INT_empty];
   341 
   342 goal Set.thy "(INT x:UNIV. B x) = (INT x. B x)";
   343 by (Fast_tac 1);
   344 qed "INT_UNIV";
   345 Addsimps[INT_UNIV];
   346 
   347 goal Set.thy "(UN x:insert a A. B x) = B a Un UNION A B";
   348 by (Fast_tac 1);
   349 qed "UN_insert";
   350 Addsimps[UN_insert];
   351 
   352 goal Set.thy "(INT x:insert a A. B x) = B a Int INTER A B";
   353 by (Fast_tac 1);
   354 qed "INT_insert";
   355 Addsimps[INT_insert];
   356 
   357 goal Set.thy
   358     "!!A. A~={} ==> (INT x:A. insert a (B x)) = insert a (INT x:A. B x)";
   359 by (Fast_tac 1);
   360 qed "INT_insert_distrib";
   361 
   362 goal Set.thy "(INT x. insert a (B x)) = insert a (INT x. B x)";
   363 by (Fast_tac 1);
   364 qed "INT1_insert_distrib";
   365 
   366 goal Set.thy "Union(range(f)) = (UN x.f(x))";
   367 by (Fast_tac 1);
   368 qed "Union_range_eq";
   369 
   370 goal Set.thy "Inter(range(f)) = (INT x.f(x))";
   371 by (Fast_tac 1);
   372 qed "Inter_range_eq";
   373 
   374 goal Set.thy "Union(B``A) = (UN x:A. B(x))";
   375 by (Fast_tac 1);
   376 qed "Union_image_eq";
   377 
   378 goal Set.thy "Inter(B``A) = (INT x:A. B(x))";
   379 by (Fast_tac 1);
   380 qed "Inter_image_eq";
   381 
   382 goal Set.thy "!!A. a: A ==> (UN y:A. c) = c";
   383 by (Fast_tac 1);
   384 qed "UN_constant";
   385 
   386 goal Set.thy "!!A. a: A ==> (INT y:A. c) = c";
   387 by (Fast_tac 1);
   388 qed "INT_constant";
   389 
   390 goal Set.thy "(UN x.B) = B";
   391 by (Fast_tac 1);
   392 qed "UN1_constant";
   393 Addsimps[UN1_constant];
   394 
   395 goal Set.thy "(INT x.B) = B";
   396 by (Fast_tac 1);
   397 qed "INT1_constant";
   398 Addsimps[INT1_constant];
   399 
   400 goal Set.thy "(UN x:A. B(x)) = Union({Y. ? x:A. Y=B(x)})";
   401 by (Fast_tac 1);
   402 qed "UN_eq";
   403 
   404 (*Look: it has an EXISTENTIAL quantifier*)
   405 goal Set.thy "(INT x:A. B(x)) = Inter({Y. ? x:A. Y=B(x)})";
   406 by (Fast_tac 1);
   407 qed "INT_eq";
   408 
   409 (*Distributive laws...*)
   410 
   411 goal Set.thy "A Int Union(B) = (UN C:B. A Int C)";
   412 by (Fast_tac 1);
   413 qed "Int_Union";
   414 
   415 (* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: 
   416    Union of a family of unions **)
   417 goal Set.thy "(UN x:C. A(x) Un B(x)) = Union(A``C)  Un  Union(B``C)";
   418 by (Fast_tac 1);
   419 qed "Un_Union_image";
   420 
   421 (*Equivalent version*)
   422 goal Set.thy "(UN i:I. A(i) Un B(i)) = (UN i:I. A(i))  Un  (UN i:I. B(i))";
   423 by (Fast_tac 1);
   424 qed "UN_Un_distrib";
   425 
   426 goal Set.thy "A Un Inter(B) = (INT C:B. A Un C)";
   427 by (Fast_tac 1);
   428 qed "Un_Inter";
   429 
   430 goal Set.thy "(INT x:C. A(x) Int B(x)) = Inter(A``C) Int Inter(B``C)";
   431 by (best_tac (!claset) 1);
   432 qed "Int_Inter_image";
   433 
   434 (*Equivalent version*)
   435 goal Set.thy "(INT i:I. A(i) Int B(i)) = (INT i:I. A(i)) Int (INT i:I. B(i))";
   436 by (Fast_tac 1);
   437 qed "INT_Int_distrib";
   438 
   439 (*Halmos, Naive Set Theory, page 35.*)
   440 goal Set.thy "B Int (UN i:I. A(i)) = (UN i:I. B Int A(i))";
   441 by (Fast_tac 1);
   442 qed "Int_UN_distrib";
   443 
   444 goal Set.thy "B Un (INT i:I. A(i)) = (INT i:I. B Un A(i))";
   445 by (Fast_tac 1);
   446 qed "Un_INT_distrib";
   447 
   448 goal Set.thy
   449     "(UN i:I. A(i)) Int (UN j:J. B(j)) = (UN i:I. UN j:J. A(i) Int B(j))";
   450 by (Fast_tac 1);
   451 qed "Int_UN_distrib2";
   452 
   453 goal Set.thy
   454     "(INT i:I. A(i)) Un (INT j:J. B(j)) = (INT i:I. INT j:J. A(i) Un B(j))";
   455 by (Fast_tac 1);
   456 qed "Un_INT_distrib2";
   457 
   458 section "-";
   459 
   460 goal Set.thy "A-A = {}";
   461 by (Fast_tac 1);
   462 qed "Diff_cancel";
   463 Addsimps[Diff_cancel];
   464 
   465 goal Set.thy "{}-A = {}";
   466 by (Fast_tac 1);
   467 qed "empty_Diff";
   468 Addsimps[empty_Diff];
   469 
   470 goal Set.thy "A-{} = A";
   471 by (Fast_tac 1);
   472 qed "Diff_empty";
   473 Addsimps[Diff_empty];
   474 
   475 goal Set.thy "A-UNIV = {}";
   476 by (Fast_tac 1);
   477 qed "Diff_UNIV";
   478 Addsimps[Diff_UNIV];
   479 
   480 goal Set.thy "!!x. x~:A ==> A - insert x B = A-B";
   481 by (Fast_tac 1);
   482 qed "Diff_insert0";
   483 Addsimps [Diff_insert0];
   484 
   485 (*NOT SUITABLE FOR REWRITING since {a} == insert a 0*)
   486 goal Set.thy "A - insert a B = A - B - {a}";
   487 by (Fast_tac 1);
   488 qed "Diff_insert";
   489 
   490 (*NOT SUITABLE FOR REWRITING since {a} == insert a 0*)
   491 goal Set.thy "A - insert a B = A - {a} - B";
   492 by (Fast_tac 1);
   493 qed "Diff_insert2";
   494 
   495 goal Set.thy "insert x A - B = (if x:B then A-B else insert x (A-B))";
   496 by (simp_tac (!simpset setloop split_tac[expand_if]) 1);
   497 by (Fast_tac 1);
   498 qed "insert_Diff_if";
   499 
   500 goal Set.thy "!!x. x:B ==> insert x A - B = A-B";
   501 by (Fast_tac 1);
   502 qed "insert_Diff1";
   503 Addsimps [insert_Diff1];
   504 
   505 val prems = goal Set.thy "a:A ==> insert a (A-{a}) = A";
   506 by (fast_tac (!claset addSIs prems) 1);
   507 qed "insert_Diff";
   508 
   509 goal Set.thy "A Int (B-A) = {}";
   510 by (Fast_tac 1);
   511 qed "Diff_disjoint";
   512 Addsimps[Diff_disjoint];
   513 
   514 goal Set.thy "!!A. A<=B ==> A Un (B-A) = B";
   515 by (Fast_tac 1);
   516 qed "Diff_partition";
   517 
   518 goal Set.thy "!!A. [| A<=B; B<= C |] ==> (B - (C - A)) = (A :: 'a set)";
   519 by (Fast_tac 1);
   520 qed "double_diff";
   521 
   522 goal Set.thy "A - (B Un C) = (A-B) Int (A-C)";
   523 by (Fast_tac 1);
   524 qed "Diff_Un";
   525 
   526 goal Set.thy "A - (B Int C) = (A-B) Un (A-C)";
   527 by (Fast_tac 1);
   528 qed "Diff_Int";
   529 
   530 Addsimps[subset_UNIV, empty_subsetI, subset_refl];
   531 
   532 
   533 (** Miniscoping: pushing in big Unions and Intersections **)
   534 local
   535   fun prover s = prove_goal Set.thy s (fn _ => [Fast_tac 1])
   536 in
   537 val UN1_simps = map prover 
   538                 ["(UN x. insert a (B x)) = insert a (UN x. B x)",
   539                  "(UN x. A x Int B)  = ((UN x.A x) Int B)",
   540                  "(UN x. A Int B x)  = (A Int (UN x.B x))",
   541                  "(UN x. A x Un B)   = ((UN x.A x) Un B)",
   542                  "(UN x. A Un B x)   = (A Un (UN x.B x))",
   543                  "(UN x. A x - B)    = ((UN x.A x) - B)",
   544                  "(UN x. A - B x)    = (A - (INT x.B x))"];
   545 
   546 val INT1_simps = map prover
   547                 ["(INT x. insert a (B x)) = insert a (INT x. B x)",
   548                  "(INT x. A x Int B) = ((INT x.A x) Int B)",
   549                  "(INT x. A Int B x) = (A Int (INT x.B x))",
   550                  "(INT x. A x Un B)  = ((INT x.A x) Un B)",
   551                  "(INT x. A Un B x)  = (A Un (INT x.B x))",
   552                  "(INT x. A x - B)   = ((INT x.A x) - B)",
   553                  "(INT x. A - B x)   = (A - (UN x.B x))"];
   554 
   555 (*Analogous laws for bounded Unions and Intersections are conditional
   556   on the index set's being non-empty.  Thus they are probably NOT worth 
   557   adding as default rewrites.*)
   558 end;
   559 
   560 Addsimps (UN1_simps @ INT1_simps);