src/HOL/equalities.ML
author paulson
Tue Sep 24 13:53:18 1996 +0200 (1996-09-24)
changeset 2021 dd5866263153
parent 1917 27b71d839d50
child 2024 909153d8318f
permissions -rw-r--r--
Added miniscoping for UN and INT
     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 ":";
    26 
    27 goal Set.thy "x ~: {}";
    28 by (Fast_tac 1);
    29 qed "in_empty";
    30 Addsimps[in_empty];
    31 
    32 goal Set.thy "x : insert y A = (x=y | x:A)";
    33 by (Fast_tac 1);
    34 qed "in_insert";
    35 Addsimps[in_insert];
    36 
    37 section "insert";
    38 
    39 (*NOT SUITABLE FOR REWRITING since {a} == insert a {}*)
    40 goal Set.thy "insert a A = {a} Un A";
    41 by (Fast_tac 1);
    42 qed "insert_is_Un";
    43 
    44 goal Set.thy "insert a A ~= {}";
    45 by (fast_tac (!claset addEs [equalityCE]) 1);
    46 qed"insert_not_empty";
    47 Addsimps[insert_not_empty];
    48 
    49 bind_thm("empty_not_insert",insert_not_empty RS not_sym);
    50 Addsimps[empty_not_insert];
    51 
    52 goal Set.thy "!!a. a:A ==> insert a A = A";
    53 by (Fast_tac 1);
    54 qed "insert_absorb";
    55 
    56 goal Set.thy "insert x (insert x A) = insert x A";
    57 by (Fast_tac 1);
    58 qed "insert_absorb2";
    59 Addsimps [insert_absorb2];
    60 
    61 goal Set.thy "insert x (insert y A) = insert y (insert x A)";
    62 by (Fast_tac 1);
    63 qed "insert_commute";
    64 
    65 goal Set.thy "(insert x A <= B) = (x:B & A <= B)";
    66 by (Fast_tac 1);
    67 qed "insert_subset";
    68 Addsimps[insert_subset];
    69 
    70 (* use new B rather than (A-{a}) to avoid infinite unfolding *)
    71 goal Set.thy "!!a. a:A ==> ? B. A = insert a B & a ~: B";
    72 by (res_inst_tac [("x","A-{a}")] exI 1);
    73 by (Fast_tac 1);
    74 qed "mk_disjoint_insert";
    75 
    76 goal Set.thy
    77     "!!A. A~={} ==> (UN x:A. insert a (B x)) = insert a (UN x:A. B x)";
    78 by (Fast_tac 1);
    79 qed "UN_insert_distrib";
    80 
    81 goal Set.thy "(UN x. insert a (B x)) = insert a (UN x. B x)";
    82 by (Fast_tac 1);
    83 qed "UN1_insert_distrib";
    84 
    85 section "``";
    86 
    87 goal Set.thy "f``{} = {}";
    88 by (Fast_tac 1);
    89 qed "image_empty";
    90 Addsimps[image_empty];
    91 
    92 goal Set.thy "f``insert a B = insert (f a) (f``B)";
    93 by (Fast_tac 1);
    94 qed "image_insert";
    95 Addsimps[image_insert];
    96 
    97 qed_goal "ball_image" Set.thy "(!y:F``S. P y) = (!x:S. P (F x))"
    98  (fn _ => [Fast_tac 1]);
    99 
   100 goal Set.thy "!!x. x:A ==> insert (f x) (f``A) = f``A";
   101 by (Fast_tac 1);
   102 qed "insert_image";
   103 Addsimps [insert_image];
   104 
   105 goalw Set.thy [image_def]
   106 "(%x. if P x then f x else g x) `` S                    \
   107 \ = (f `` ({x.x:S & P x})) Un (g `` ({x.x:S & ~(P x)}))";
   108 by(split_tac [expand_if] 1);
   109 by(Fast_tac 1);
   110 qed "if_image_distrib";
   111 Addsimps[if_image_distrib];
   112 
   113 
   114 section "range";
   115 
   116 qed_goal "ball_range" Set.thy "(!y:range f. P y) = (!x. P (f x))"
   117  (fn _ => [Fast_tac 1]);
   118 
   119 qed_goalw "image_range" Set.thy [image_def]
   120  "f``range g = range (%x. f (g x))" 
   121  (fn _ => [rtac Collect_cong 1, Fast_tac 1]);
   122 
   123 section "Int";
   124 
   125 goal Set.thy "A Int A = A";
   126 by (Fast_tac 1);
   127 qed "Int_absorb";
   128 Addsimps[Int_absorb];
   129 
   130 goal Set.thy "A Int B  =  B Int A";
   131 by (Fast_tac 1);
   132 qed "Int_commute";
   133 
   134 goal Set.thy "(A Int B) Int C  =  A Int (B Int C)";
   135 by (Fast_tac 1);
   136 qed "Int_assoc";
   137 
   138 goal Set.thy "{} Int B = {}";
   139 by (Fast_tac 1);
   140 qed "Int_empty_left";
   141 Addsimps[Int_empty_left];
   142 
   143 goal Set.thy "A Int {} = {}";
   144 by (Fast_tac 1);
   145 qed "Int_empty_right";
   146 Addsimps[Int_empty_right];
   147 
   148 goal Set.thy "UNIV Int B = B";
   149 by (Fast_tac 1);
   150 qed "Int_UNIV_left";
   151 Addsimps[Int_UNIV_left];
   152 
   153 goal Set.thy "A Int UNIV = A";
   154 by (Fast_tac 1);
   155 qed "Int_UNIV_right";
   156 Addsimps[Int_UNIV_right];
   157 
   158 goal Set.thy "A Int (B Un C)  =  (A Int B) Un (A Int C)";
   159 by (Fast_tac 1);
   160 qed "Int_Un_distrib";
   161 
   162 goal Set.thy "(B Un C) Int A =  (B Int A) Un (C Int A)";
   163 by (Fast_tac 1);
   164 qed "Int_Un_distrib2";
   165 
   166 goal Set.thy "(A<=B) = (A Int B = A)";
   167 by (fast_tac (!claset addSEs [equalityE]) 1);
   168 qed "subset_Int_eq";
   169 
   170 goal Set.thy "(A Int B = UNIV) = (A = UNIV & B = UNIV)";
   171 by (fast_tac (!claset addEs [equalityCE]) 1);
   172 qed "Int_UNIV";
   173 Addsimps[Int_UNIV];
   174 
   175 section "Un";
   176 
   177 goal Set.thy "A Un A = A";
   178 by (Fast_tac 1);
   179 qed "Un_absorb";
   180 Addsimps[Un_absorb];
   181 
   182 goal Set.thy "A Un B  =  B Un A";
   183 by (Fast_tac 1);
   184 qed "Un_commute";
   185 
   186 goal Set.thy "(A Un B) Un C  =  A Un (B Un C)";
   187 by (Fast_tac 1);
   188 qed "Un_assoc";
   189 
   190 goal Set.thy "{} Un B = B";
   191 by (Fast_tac 1);
   192 qed "Un_empty_left";
   193 Addsimps[Un_empty_left];
   194 
   195 goal Set.thy "A Un {} = A";
   196 by (Fast_tac 1);
   197 qed "Un_empty_right";
   198 Addsimps[Un_empty_right];
   199 
   200 goal Set.thy "UNIV Un B = UNIV";
   201 by (Fast_tac 1);
   202 qed "Un_UNIV_left";
   203 Addsimps[Un_UNIV_left];
   204 
   205 goal Set.thy "A Un UNIV = UNIV";
   206 by (Fast_tac 1);
   207 qed "Un_UNIV_right";
   208 Addsimps[Un_UNIV_right];
   209 
   210 goal Set.thy "(insert a B) Un C = insert a (B Un C)";
   211 by (Fast_tac 1);
   212 qed "Un_insert_left";
   213 
   214 goal Set.thy "A Un (insert a B) = insert a (A Un B)";
   215 by (Fast_tac 1);
   216 qed "Un_insert_right";
   217 
   218 goal Set.thy "(A Int B) Un C  =  (A Un C) Int (B Un C)";
   219 by (Fast_tac 1);
   220 qed "Un_Int_distrib";
   221 
   222 goal Set.thy
   223  "(A Int B) Un (B Int C) Un (C Int A) = (A Un B) Int (B Un C) Int (C Un A)";
   224 by (Fast_tac 1);
   225 qed "Un_Int_crazy";
   226 
   227 goal Set.thy "(A<=B) = (A Un B = B)";
   228 by (fast_tac (!claset addSEs [equalityE]) 1);
   229 qed "subset_Un_eq";
   230 
   231 goal Set.thy "(A <= insert b C) = (A <= C | b:A & A-{b} <= C)";
   232 by (Fast_tac 1);
   233 qed "subset_insert_iff";
   234 
   235 goal Set.thy "(A Un B = {}) = (A = {} & B = {})";
   236 by (fast_tac (!claset addEs [equalityCE]) 1);
   237 qed "Un_empty";
   238 Addsimps[Un_empty];
   239 
   240 section "Compl";
   241 
   242 goal Set.thy "A Int Compl(A) = {}";
   243 by (Fast_tac 1);
   244 qed "Compl_disjoint";
   245 Addsimps[Compl_disjoint];
   246 
   247 goal Set.thy "A Un Compl(A) = UNIV";
   248 by (Fast_tac 1);
   249 qed "Compl_partition";
   250 
   251 goal Set.thy "Compl(Compl(A)) = A";
   252 by (Fast_tac 1);
   253 qed "double_complement";
   254 Addsimps[double_complement];
   255 
   256 goal Set.thy "Compl(A Un B) = Compl(A) Int Compl(B)";
   257 by (Fast_tac 1);
   258 qed "Compl_Un";
   259 
   260 goal Set.thy "Compl(A Int B) = Compl(A) Un Compl(B)";
   261 by (Fast_tac 1);
   262 qed "Compl_Int";
   263 
   264 goal Set.thy "Compl(UN x:A. B(x)) = (INT x:A. Compl(B(x)))";
   265 by (Fast_tac 1);
   266 qed "Compl_UN";
   267 
   268 goal Set.thy "Compl(INT x:A. B(x)) = (UN x:A. Compl(B(x)))";
   269 by (Fast_tac 1);
   270 qed "Compl_INT";
   271 
   272 (*Halmos, Naive Set Theory, page 16.*)
   273 
   274 goal Set.thy "((A Int B) Un C = A Int (B Un C)) = (C<=A)";
   275 by (fast_tac (!claset addSEs [equalityE]) 1);
   276 qed "Un_Int_assoc_eq";
   277 
   278 
   279 section "Union";
   280 
   281 goal Set.thy "Union({}) = {}";
   282 by (Fast_tac 1);
   283 qed "Union_empty";
   284 Addsimps[Union_empty];
   285 
   286 goal Set.thy "Union(UNIV) = UNIV";
   287 by (Fast_tac 1);
   288 qed "Union_UNIV";
   289 Addsimps[Union_UNIV];
   290 
   291 goal Set.thy "Union(insert a B) = a Un Union(B)";
   292 by (Fast_tac 1);
   293 qed "Union_insert";
   294 Addsimps[Union_insert];
   295 
   296 goal Set.thy "Union(A Un B) = Union(A) Un Union(B)";
   297 by (Fast_tac 1);
   298 qed "Union_Un_distrib";
   299 Addsimps[Union_Un_distrib];
   300 
   301 goal Set.thy "Union(A Int B) <= Union(A) Int Union(B)";
   302 by (Fast_tac 1);
   303 qed "Union_Int_subset";
   304 
   305 val prems = goal Set.thy
   306    "(Union(C) Int A = {}) = (! B:C. B Int A = {})";
   307 by (fast_tac (!claset addSEs [equalityE]) 1);
   308 qed "Union_disjoint";
   309 
   310 section "Inter";
   311 
   312 goal Set.thy "Inter({}) = UNIV";
   313 by (Fast_tac 1);
   314 qed "Inter_empty";
   315 Addsimps[Inter_empty];
   316 
   317 goal Set.thy "Inter(UNIV) = {}";
   318 by (Fast_tac 1);
   319 qed "Inter_UNIV";
   320 Addsimps[Inter_UNIV];
   321 
   322 goal Set.thy "Inter(insert a B) = a Int Inter(B)";
   323 by (Fast_tac 1);
   324 qed "Inter_insert";
   325 Addsimps[Inter_insert];
   326 
   327 goal Set.thy "Inter(A) Un Inter(B) <= Inter(A Int B)";
   328 by (Fast_tac 1);
   329 qed "Inter_Un_subset";
   330 
   331 goal Set.thy "Inter(A Un B) = Inter(A) Int Inter(B)";
   332 by (best_tac (!claset) 1);
   333 qed "Inter_Un_distrib";
   334 
   335 section "UN and INT";
   336 
   337 (*Basic identities*)
   338 
   339 goal Set.thy "(UN x:{}. B x) = {}";
   340 by (Fast_tac 1);
   341 qed "UN_empty";
   342 Addsimps[UN_empty];
   343 
   344 goal Set.thy "(UN x:UNIV. B x) = (UN x. B x)";
   345 by (Fast_tac 1);
   346 qed "UN_UNIV";
   347 Addsimps[UN_UNIV];
   348 
   349 goal Set.thy "(INT x:{}. B x) = UNIV";
   350 by (Fast_tac 1);
   351 qed "INT_empty";
   352 Addsimps[INT_empty];
   353 
   354 goal Set.thy "(INT x:UNIV. B x) = (INT x. B x)";
   355 by (Fast_tac 1);
   356 qed "INT_UNIV";
   357 Addsimps[INT_UNIV];
   358 
   359 goal Set.thy "(UN x:insert a A. B x) = B a Un UNION A B";
   360 by (Fast_tac 1);
   361 qed "UN_insert";
   362 Addsimps[UN_insert];
   363 
   364 goal Set.thy "(INT x:insert a A. B x) = B a Int INTER A B";
   365 by (Fast_tac 1);
   366 qed "INT_insert";
   367 Addsimps[INT_insert];
   368 
   369 goal Set.thy
   370     "!!A. A~={} ==> (INT x:A. insert a (B x)) = insert a (INT x:A. B x)";
   371 by (Fast_tac 1);
   372 qed "INT_insert_distrib";
   373 
   374 goal Set.thy "(INT x. insert a (B x)) = insert a (INT x. B x)";
   375 by (Fast_tac 1);
   376 qed "INT1_insert_distrib";
   377 
   378 goal Set.thy "Union(range(f)) = (UN x.f(x))";
   379 by (Fast_tac 1);
   380 qed "Union_range_eq";
   381 
   382 goal Set.thy "Inter(range(f)) = (INT x.f(x))";
   383 by (Fast_tac 1);
   384 qed "Inter_range_eq";
   385 
   386 goal Set.thy "Union(B``A) = (UN x:A. B(x))";
   387 by (Fast_tac 1);
   388 qed "Union_image_eq";
   389 
   390 goal Set.thy "Inter(B``A) = (INT x:A. B(x))";
   391 by (Fast_tac 1);
   392 qed "Inter_image_eq";
   393 
   394 goal Set.thy "!!A. a: A ==> (UN y:A. c) = c";
   395 by (Fast_tac 1);
   396 qed "UN_constant";
   397 
   398 goal Set.thy "!!A. a: A ==> (INT y:A. c) = c";
   399 by (Fast_tac 1);
   400 qed "INT_constant";
   401 
   402 goal Set.thy "(UN x.B) = B";
   403 by (Fast_tac 1);
   404 qed "UN1_constant";
   405 Addsimps[UN1_constant];
   406 
   407 goal Set.thy "(INT x.B) = B";
   408 by (Fast_tac 1);
   409 qed "INT1_constant";
   410 Addsimps[INT1_constant];
   411 
   412 goal Set.thy "(UN x:A. B(x)) = Union({Y. ? x:A. Y=B(x)})";
   413 by (Fast_tac 1);
   414 qed "UN_eq";
   415 
   416 (*Look: it has an EXISTENTIAL quantifier*)
   417 goal Set.thy "(INT x:A. B(x)) = Inter({Y. ? x:A. Y=B(x)})";
   418 by (Fast_tac 1);
   419 qed "INT_eq";
   420 
   421 (*Distributive laws...*)
   422 
   423 goal Set.thy "A Int Union(B) = (UN C:B. A Int C)";
   424 by (Fast_tac 1);
   425 qed "Int_Union";
   426 
   427 (* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: 
   428    Union of a family of unions **)
   429 goal Set.thy "(UN x:C. A(x) Un B(x)) = Union(A``C)  Un  Union(B``C)";
   430 by (Fast_tac 1);
   431 qed "Un_Union_image";
   432 
   433 (*Equivalent version*)
   434 goal Set.thy "(UN i:I. A(i) Un B(i)) = (UN i:I. A(i))  Un  (UN i:I. B(i))";
   435 by (Fast_tac 1);
   436 qed "UN_Un_distrib";
   437 
   438 goal Set.thy "A Un Inter(B) = (INT C:B. A Un C)";
   439 by (Fast_tac 1);
   440 qed "Un_Inter";
   441 
   442 goal Set.thy "(INT x:C. A(x) Int B(x)) = Inter(A``C) Int Inter(B``C)";
   443 by (best_tac (!claset) 1);
   444 qed "Int_Inter_image";
   445 
   446 (*Equivalent version*)
   447 goal Set.thy "(INT i:I. A(i) Int B(i)) = (INT i:I. A(i)) Int (INT i:I. B(i))";
   448 by (Fast_tac 1);
   449 qed "INT_Int_distrib";
   450 
   451 (*Halmos, Naive Set Theory, page 35.*)
   452 goal Set.thy "B Int (UN i:I. A(i)) = (UN i:I. B Int A(i))";
   453 by (Fast_tac 1);
   454 qed "Int_UN_distrib";
   455 
   456 goal Set.thy "B Un (INT i:I. A(i)) = (INT i:I. B Un A(i))";
   457 by (Fast_tac 1);
   458 qed "Un_INT_distrib";
   459 
   460 goal Set.thy
   461     "(UN i:I. A(i)) Int (UN j:J. B(j)) = (UN i:I. UN j:J. A(i) Int B(j))";
   462 by (Fast_tac 1);
   463 qed "Int_UN_distrib2";
   464 
   465 goal Set.thy
   466     "(INT i:I. A(i)) Un (INT j:J. B(j)) = (INT i:I. INT j:J. A(i) Un B(j))";
   467 by (Fast_tac 1);
   468 qed "Un_INT_distrib2";
   469 
   470 section "-";
   471 
   472 goal Set.thy "A-A = {}";
   473 by (Fast_tac 1);
   474 qed "Diff_cancel";
   475 Addsimps[Diff_cancel];
   476 
   477 goal Set.thy "{}-A = {}";
   478 by (Fast_tac 1);
   479 qed "empty_Diff";
   480 Addsimps[empty_Diff];
   481 
   482 goal Set.thy "A-{} = A";
   483 by (Fast_tac 1);
   484 qed "Diff_empty";
   485 Addsimps[Diff_empty];
   486 
   487 goal Set.thy "A-UNIV = {}";
   488 by (Fast_tac 1);
   489 qed "Diff_UNIV";
   490 Addsimps[Diff_UNIV];
   491 
   492 goal Set.thy "!!x. x~:A ==> A - insert x B = A-B";
   493 by (Fast_tac 1);
   494 qed "Diff_insert0";
   495 Addsimps [Diff_insert0];
   496 
   497 (*NOT SUITABLE FOR REWRITING since {a} == insert a 0*)
   498 goal Set.thy "A - insert a B = A - B - {a}";
   499 by (Fast_tac 1);
   500 qed "Diff_insert";
   501 
   502 (*NOT SUITABLE FOR REWRITING since {a} == insert a 0*)
   503 goal Set.thy "A - insert a B = A - {a} - B";
   504 by (Fast_tac 1);
   505 qed "Diff_insert2";
   506 
   507 goal Set.thy "insert x A - B = (if x:B then A-B else insert x (A-B))";
   508 by (simp_tac (!simpset setloop split_tac[expand_if]) 1);
   509 by (Fast_tac 1);
   510 qed "insert_Diff_if";
   511 
   512 goal Set.thy "!!x. x:B ==> insert x A - B = A-B";
   513 by (Fast_tac 1);
   514 qed "insert_Diff1";
   515 Addsimps [insert_Diff1];
   516 
   517 val prems = goal Set.thy "a:A ==> insert a (A-{a}) = A";
   518 by (fast_tac (!claset addSIs prems) 1);
   519 qed "insert_Diff";
   520 
   521 goal Set.thy "A Int (B-A) = {}";
   522 by (Fast_tac 1);
   523 qed "Diff_disjoint";
   524 Addsimps[Diff_disjoint];
   525 
   526 goal Set.thy "!!A. A<=B ==> A Un (B-A) = B";
   527 by (Fast_tac 1);
   528 qed "Diff_partition";
   529 
   530 goal Set.thy "!!A. [| A<=B; B<= C |] ==> (B - (C - A)) = (A :: 'a set)";
   531 by (Fast_tac 1);
   532 qed "double_diff";
   533 
   534 goal Set.thy "A - (B Un C) = (A-B) Int (A-C)";
   535 by (Fast_tac 1);
   536 qed "Diff_Un";
   537 
   538 goal Set.thy "A - (B Int C) = (A-B) Un (A-C)";
   539 by (Fast_tac 1);
   540 qed "Diff_Int";
   541 
   542 Addsimps[subset_UNIV, empty_subsetI, subset_refl];
   543 
   544 
   545 (** Miniscoping: pushing in big Unions and Intersections **)
   546 local
   547   fun prover s = prove_goal Set.thy s (fn _ => [Fast_tac 1])
   548 in
   549 val UN1_simps = map prover 
   550 		["(UN x. insert a (B x)) = insert a (UN x. B x)",
   551 		 "(UN x. A x Int B)  = ((UN x.A x) Int B)",
   552 		 "(UN x. A Int B x)  = (A Int (UN x.B x))",
   553 		 "(UN x. A x Un B)   = ((UN x.A x) Un B)",
   554 		 "(UN x. A Un B x)   = (A Un (UN x.B x))",
   555 		 "(UN x. A x - B)    = ((UN x.A x) - B)",
   556 		 "(UN x. A - B x)    = (A - (INT x.B x))"];
   557 
   558 val INT1_simps = map prover
   559 		["(INT x. insert a (B x)) = insert a (INT x. B x)",
   560 		 "(INT x. A x Int B) = ((INT x.A x) Int B)",
   561 		 "(INT x. A Int B x) = (A Int (INT x.B x))",
   562 		 "(INT x. A x Un B)  = ((INT x.A x) Un B)",
   563 		 "(INT x. A Un B x)  = (A Un (INT x.B x))",
   564 		 "(INT x. A x - B)   = ((INT x.A x) - B)",
   565 		 "(INT x. A - B x)   = (A - (UN x.B x))"];
   566 
   567 (*Analogous laws for bounded Unions and Intersections are conditional
   568   on the index set's being non-empty.  Thus they are probably NOT worth 
   569   adding as default rewrites.*)
   570 end;
   571 
   572 Addsimps (UN1_simps @ INT1_simps);