src/HOL/Set.ML
author oheimb
Mon Sep 21 23:12:31 1998 +0200 (1998-09-21)
changeset 5521 7970832271cc
parent 5490 85855f65d0c6
child 5600 34b3366b83ac
permissions -rw-r--r--
added wrapper for bspec
     1 (*  Title:      HOL/set
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1991  University of Cambridge
     5 
     6 Set theory for higher-order logic.  A set is simply a predicate.
     7 *)
     8 
     9 open Set;
    10 
    11 section "Relating predicates and sets";
    12 
    13 Addsimps [Collect_mem_eq];
    14 AddIffs  [mem_Collect_eq];
    15 
    16 Goal "P(a) ==> a : {x. P(x)}";
    17 by (Asm_simp_tac 1);
    18 qed "CollectI";
    19 
    20 Goal "a : {x. P(x)} ==> P(a)";
    21 by (Asm_full_simp_tac 1);
    22 qed "CollectD";
    23 
    24 val [prem] = Goal "[| !!x. (x:A) = (x:B) |] ==> A = B";
    25 by (rtac (prem RS ext RS arg_cong RS box_equals) 1);
    26 by (rtac Collect_mem_eq 1);
    27 by (rtac Collect_mem_eq 1);
    28 qed "set_ext";
    29 
    30 val [prem] = Goal "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}";
    31 by (rtac (prem RS ext RS arg_cong) 1);
    32 qed "Collect_cong";
    33 
    34 val CollectE = make_elim CollectD;
    35 
    36 AddSIs [CollectI];
    37 AddSEs [CollectE];
    38 
    39 
    40 section "Bounded quantifiers";
    41 
    42 val prems = Goalw [Ball_def]
    43     "[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)";
    44 by (REPEAT (ares_tac (prems @ [allI,impI]) 1));
    45 qed "ballI";
    46 
    47 Goalw [Ball_def] "[| ! 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     "[| ! 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 ! 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 (* gives better instantiation for bound: *)
    63 claset_ref() := claset() addWrapper ("bspec", fn tac2 =>
    64 			 (dtac bspec THEN' atac) APPEND' tac2);
    65 
    66 Goalw [Bex_def] "[| P(x);  x:A |] ==> ? x:A. P(x)";
    67 by (Blast_tac 1);
    68 qed "bexI";
    69 
    70 qed_goal "bexCI" Set.thy 
    71    "[| ! x:A. ~P(x) ==> P(a);  a:A |] ==> ? x:A. P(x)" (fn prems =>
    72   [ (rtac classical 1),
    73     (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1))  ]);
    74 
    75 val major::prems = Goalw [Bex_def]
    76     "[| ? x:A. P(x);  !!x. [| x:A; P(x) |] ==> Q  |] ==> Q";
    77 by (rtac (major RS exE) 1);
    78 by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1));
    79 qed "bexE";
    80 
    81 AddIs  [bexI];
    82 AddSEs [bexE];
    83 
    84 (*Trival rewrite rule*)
    85 Goal "(! x:A. P) = ((? x. x:A) --> P)";
    86 by (simp_tac (simpset() addsimps [Ball_def]) 1);
    87 qed "ball_triv";
    88 
    89 (*Dual form for existentials*)
    90 Goal "(? x:A. P) = ((? x. x:A) & P)";
    91 by (simp_tac (simpset() addsimps [Bex_def]) 1);
    92 qed "bex_triv";
    93 
    94 Addsimps [ball_triv, bex_triv];
    95 
    96 (** Congruence rules **)
    97 
    98 val prems = Goal
    99     "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
   100 \    (! x:A. P(x)) = (! x:B. Q(x))";
   101 by (resolve_tac (prems RL [ssubst]) 1);
   102 by (REPEAT (ares_tac [ballI,iffI] 1
   103      ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1));
   104 qed "ball_cong";
   105 
   106 val prems = Goal
   107     "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
   108 \    (? x:A. P(x)) = (? x:B. Q(x))";
   109 by (resolve_tac (prems RL [ssubst]) 1);
   110 by (REPEAT (etac bexE 1
   111      ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1));
   112 qed "bex_cong";
   113 
   114 section "Subsets";
   115 
   116 val prems = Goalw [subset_def] "(!!x. x:A ==> x:B) ==> A <= B";
   117 by (REPEAT (ares_tac (prems @ [ballI]) 1));
   118 qed "subsetI";
   119 
   120 (*While (:) is not, its type must be kept
   121   for overloading of = to work.*)
   122 Blast.overloaded ("op :", domain_type);
   123 seq (fn a => Blast.overloaded (a, HOLogic.dest_setT o domain_type))
   124     ["Ball", "Bex"];
   125 (*need UNION, INTER also?*)
   126 
   127 (*Image: retain the type of the set being expressed*)
   128 Blast.overloaded ("op ``", domain_type);
   129 
   130 (*Rule in Modus Ponens style*)
   131 Goalw [subset_def] "[| A <= B;  c:A |] ==> c:B";
   132 by (Blast_tac 1);
   133 qed "subsetD";
   134 
   135 (*The same, with reversed premises for use with etac -- cf rev_mp*)
   136 qed_goal "rev_subsetD" Set.thy "[| c:A;  A <= B |] ==> c:B"
   137  (fn prems=>  [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]);
   138 
   139 (*Converts A<=B to x:A ==> x:B*)
   140 fun impOfSubs th = th RSN (2, rev_subsetD);
   141 
   142 qed_goal "contra_subsetD" Set.thy "!!c. [| A <= B; c ~: B |] ==> c ~: A"
   143  (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
   144 
   145 qed_goal "rev_contra_subsetD" Set.thy "!!c. [| c ~: B;  A <= B |] ==> c ~: A"
   146  (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
   147 
   148 (*Classical elimination rule*)
   149 val major::prems = Goalw [subset_def] 
   150     "[| A <= B;  c~:A ==> P;  c:B ==> P |] ==> P";
   151 by (rtac (major RS ballE) 1);
   152 by (REPEAT (eresolve_tac prems 1));
   153 qed "subsetCE";
   154 
   155 (*Takes assumptions A<=B; c:A and creates the assumption c:B *)
   156 fun set_mp_tac i = etac subsetCE i  THEN  mp_tac i;
   157 
   158 AddSIs [subsetI];
   159 AddEs  [subsetD, subsetCE];
   160 
   161 qed_goal "subset_refl" Set.thy "A <= (A::'a set)"
   162  (fn _=> [Fast_tac 1]);		(*Blast_tac would try order_refl and fail*)
   163 
   164 Goal "[| A<=B;  B<=C |] ==> A<=(C::'a set)";
   165 by (Blast_tac 1);
   166 qed "subset_trans";
   167 
   168 
   169 section "Equality";
   170 
   171 (*Anti-symmetry of the subset relation*)
   172 Goal "[| A <= B;  B <= A |] ==> A = (B::'a set)";
   173 by (rtac set_ext 1);
   174 by (blast_tac (claset() addIs [subsetD]) 1);
   175 qed "subset_antisym";
   176 val equalityI = subset_antisym;
   177 
   178 AddSIs [equalityI];
   179 
   180 (* Equality rules from ZF set theory -- are they appropriate here? *)
   181 Goal "A = B ==> A<=(B::'a set)";
   182 by (etac ssubst 1);
   183 by (rtac subset_refl 1);
   184 qed "equalityD1";
   185 
   186 Goal "A = B ==> B<=(A::'a set)";
   187 by (etac ssubst 1);
   188 by (rtac subset_refl 1);
   189 qed "equalityD2";
   190 
   191 val prems = Goal
   192     "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";
   193 by (resolve_tac prems 1);
   194 by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
   195 qed "equalityE";
   196 
   197 val major::prems = Goal
   198     "[| A = B;  [| c:A; c:B |] ==> P;  [| c~:A; c~:B |] ==> P |]  ==>  P";
   199 by (rtac (major RS equalityE) 1);
   200 by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
   201 qed "equalityCE";
   202 
   203 (*Lemma for creating induction formulae -- for "pattern matching" on p
   204   To make the induction hypotheses usable, apply "spec" or "bspec" to
   205   put universal quantifiers over the free variables in p. *)
   206 val prems = Goal 
   207     "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
   208 by (rtac mp 1);
   209 by (REPEAT (resolve_tac (refl::prems) 1));
   210 qed "setup_induction";
   211 
   212 
   213 section "The universal set -- UNIV";
   214 
   215 qed_goalw "UNIV_I" Set.thy [UNIV_def] "x : UNIV"
   216   (fn _ => [rtac CollectI 1, rtac TrueI 1]);
   217 
   218 Addsimps [UNIV_I];
   219 AddIs    [UNIV_I];  (*unsafe makes it less likely to cause problems*)
   220 
   221 qed_goal "subset_UNIV" Set.thy "A <= UNIV"
   222   (fn _ => [rtac subsetI 1, rtac UNIV_I 1]);
   223 
   224 (** Eta-contracting these two rules (to remove P) causes them to be ignored
   225     because of their interaction with congruence rules. **)
   226 
   227 Goalw [Ball_def] "Ball UNIV P = All P";
   228 by (Simp_tac 1);
   229 qed "ball_UNIV";
   230 
   231 Goalw [Bex_def] "Bex UNIV P = Ex P";
   232 by (Simp_tac 1);
   233 qed "bex_UNIV";
   234 Addsimps [ball_UNIV, bex_UNIV];
   235 
   236 
   237 section "The empty set -- {}";
   238 
   239 qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False"
   240  (fn _ => [ (Blast_tac 1) ]);
   241 
   242 Addsimps [empty_iff];
   243 
   244 qed_goal "emptyE" Set.thy "!!a. a:{} ==> P"
   245  (fn _ => [Full_simp_tac 1]);
   246 
   247 AddSEs [emptyE];
   248 
   249 qed_goal "empty_subsetI" Set.thy "{} <= A"
   250  (fn _ => [ (Blast_tac 1) ]);
   251 
   252 (*One effect is to delete the ASSUMPTION {} <= A*)
   253 AddIffs [empty_subsetI];
   254 
   255 qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}"
   256  (fn [prem]=>
   257   [ (blast_tac (claset() addIs [prem RS FalseE]) 1) ]);
   258 
   259 (*Use for reasoning about disjointness: A Int B = {} *)
   260 qed_goal "equals0D" Set.thy "!!a. A={} ==> a ~: A"
   261  (fn _ => [ (Blast_tac 1) ]);
   262 
   263 AddDs [equals0D, sym RS equals0D];
   264 
   265 Goalw [Ball_def] "Ball {} P = True";
   266 by (Simp_tac 1);
   267 qed "ball_empty";
   268 
   269 Goalw [Bex_def] "Bex {} P = False";
   270 by (Simp_tac 1);
   271 qed "bex_empty";
   272 Addsimps [ball_empty, bex_empty];
   273 
   274 Goal "UNIV ~= {}";
   275 by (blast_tac (claset() addEs [equalityE]) 1);
   276 qed "UNIV_not_empty";
   277 AddIffs [UNIV_not_empty];
   278 
   279 
   280 
   281 section "The Powerset operator -- Pow";
   282 
   283 qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)"
   284  (fn _ => [ (Asm_simp_tac 1) ]);
   285 
   286 AddIffs [Pow_iff]; 
   287 
   288 qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
   289  (fn _ => [ (etac CollectI 1) ]);
   290 
   291 qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B)  ==>  A<=B"
   292  (fn _=> [ (etac CollectD 1) ]);
   293 
   294 val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)
   295 val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)
   296 
   297 
   298 section "Set complement -- Compl";
   299 
   300 qed_goalw "Compl_iff" Set.thy [Compl_def] "(c : -A) = (c~:A)"
   301  (fn _ => [ (Blast_tac 1) ]);
   302 
   303 Addsimps [Compl_iff];
   304 
   305 val prems = Goalw [Compl_def] "[| c:A ==> False |] ==> c : -A";
   306 by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
   307 qed "ComplI";
   308 
   309 (*This form, with negated conclusion, works well with the Classical prover.
   310   Negated assumptions behave like formulae on the right side of the notional
   311   turnstile...*)
   312 Goalw [Compl_def] "c : -A ==> c~:A";
   313 by (etac CollectD 1);
   314 qed "ComplD";
   315 
   316 val ComplE = make_elim ComplD;
   317 
   318 AddSIs [ComplI];
   319 AddSEs [ComplE];
   320 
   321 
   322 section "Binary union -- Un";
   323 
   324 qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)"
   325  (fn _ => [ Blast_tac 1 ]);
   326 
   327 Addsimps [Un_iff];
   328 
   329 Goal "c:A ==> c : A Un B";
   330 by (Asm_simp_tac 1);
   331 qed "UnI1";
   332 
   333 Goal "c:B ==> c : A Un B";
   334 by (Asm_simp_tac 1);
   335 qed "UnI2";
   336 
   337 (*Classical introduction rule: no commitment to A vs B*)
   338 qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B"
   339  (fn prems=>
   340   [ (Simp_tac 1),
   341     (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
   342 
   343 val major::prems = Goalw [Un_def]
   344     "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
   345 by (rtac (major RS CollectD RS disjE) 1);
   346 by (REPEAT (eresolve_tac prems 1));
   347 qed "UnE";
   348 
   349 AddSIs [UnCI];
   350 AddSEs [UnE];
   351 
   352 
   353 section "Binary intersection -- Int";
   354 
   355 qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)"
   356  (fn _ => [ (Blast_tac 1) ]);
   357 
   358 Addsimps [Int_iff];
   359 
   360 Goal "[| c:A;  c:B |] ==> c : A Int B";
   361 by (Asm_simp_tac 1);
   362 qed "IntI";
   363 
   364 Goal "c : A Int B ==> c:A";
   365 by (Asm_full_simp_tac 1);
   366 qed "IntD1";
   367 
   368 Goal "c : A Int B ==> c:B";
   369 by (Asm_full_simp_tac 1);
   370 qed "IntD2";
   371 
   372 val [major,minor] = Goal
   373     "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
   374 by (rtac minor 1);
   375 by (rtac (major RS IntD1) 1);
   376 by (rtac (major RS IntD2) 1);
   377 qed "IntE";
   378 
   379 AddSIs [IntI];
   380 AddSEs [IntE];
   381 
   382 section "Set difference";
   383 
   384 qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)"
   385  (fn _ => [ (Blast_tac 1) ]);
   386 
   387 Addsimps [Diff_iff];
   388 
   389 qed_goal "DiffI" Set.thy "!!c. [| c : A;  c ~: B |] ==> c : A - B"
   390  (fn _=> [ Asm_simp_tac 1 ]);
   391 
   392 qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A"
   393  (fn _=> [ (Asm_full_simp_tac 1) ]);
   394 
   395 qed_goal "DiffD2" Set.thy "!!c. [| c : A - B;  c : B |] ==> P"
   396  (fn _=> [ (Asm_full_simp_tac 1) ]);
   397 
   398 qed_goal "DiffE" Set.thy "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P"
   399  (fn prems=>
   400   [ (resolve_tac prems 1),
   401     (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]);
   402 
   403 AddSIs [DiffI];
   404 AddSEs [DiffE];
   405 
   406 
   407 section "Augmenting a set -- insert";
   408 
   409 qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)"
   410  (fn _ => [Blast_tac 1]);
   411 
   412 Addsimps [insert_iff];
   413 
   414 qed_goal "insertI1" Set.thy "a : insert a B"
   415  (fn _ => [Simp_tac 1]);
   416 
   417 qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B"
   418  (fn _=> [Asm_simp_tac 1]);
   419 
   420 qed_goalw "insertE" Set.thy [insert_def]
   421     "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P"
   422  (fn major::prems=>
   423   [ (rtac (major RS UnE) 1),
   424     (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
   425 
   426 (*Classical introduction rule*)
   427 qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
   428  (fn prems=>
   429   [ (Simp_tac 1),
   430     (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
   431 
   432 AddSIs [insertCI]; 
   433 AddSEs [insertE];
   434 
   435 section "Singletons, using insert";
   436 
   437 qed_goal "singletonI" Set.thy "a : {a}"
   438  (fn _=> [ (rtac insertI1 1) ]);
   439 
   440 Goal "b : {a} ==> b=a";
   441 by (Blast_tac 1);
   442 qed "singletonD";
   443 
   444 bind_thm ("singletonE", make_elim singletonD);
   445 
   446 qed_goal "singleton_iff" thy "(b : {a}) = (b=a)" 
   447 (fn _ => [Blast_tac 1]);
   448 
   449 Goal "{a}={b} ==> a=b";
   450 by (blast_tac (claset() addEs [equalityE]) 1);
   451 qed "singleton_inject";
   452 
   453 (*Redundant? But unlike insertCI, it proves the subgoal immediately!*)
   454 AddSIs [singletonI];   
   455 AddSDs [singleton_inject];
   456 AddSEs [singletonE];
   457 
   458 Goal "{x. x=a} = {a}";
   459 by (Blast_tac 1);
   460 qed "singleton_conv";
   461 Addsimps [singleton_conv];
   462 
   463 
   464 section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
   465 
   466 Goalw [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
   467 by (Blast_tac 1);
   468 qed "UN_iff";
   469 
   470 Addsimps [UN_iff];
   471 
   472 (*The order of the premises presupposes that A is rigid; b may be flexible*)
   473 Goal "[| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
   474 by Auto_tac;
   475 qed "UN_I";
   476 
   477 val major::prems = Goalw [UNION_def]
   478     "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
   479 by (rtac (major RS CollectD RS bexE) 1);
   480 by (REPEAT (ares_tac prems 1));
   481 qed "UN_E";
   482 
   483 AddIs  [UN_I];
   484 AddSEs [UN_E];
   485 
   486 val prems = Goal
   487     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
   488 \    (UN x:A. C(x)) = (UN x:B. D(x))";
   489 by (REPEAT (etac UN_E 1
   490      ORELSE ares_tac ([UN_I,equalityI,subsetI] @ 
   491                       (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
   492 qed "UN_cong";
   493 
   494 
   495 section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
   496 
   497 Goalw [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
   498 by Auto_tac;
   499 qed "INT_iff";
   500 
   501 Addsimps [INT_iff];
   502 
   503 val prems = Goalw [INTER_def]
   504     "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
   505 by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
   506 qed "INT_I";
   507 
   508 Goal "[| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
   509 by Auto_tac;
   510 qed "INT_D";
   511 
   512 (*"Classical" elimination -- by the Excluded Middle on a:A *)
   513 val major::prems = Goalw [INTER_def]
   514     "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
   515 by (rtac (major RS CollectD RS ballE) 1);
   516 by (REPEAT (eresolve_tac prems 1));
   517 qed "INT_E";
   518 
   519 AddSIs [INT_I];
   520 AddEs  [INT_D, INT_E];
   521 
   522 val prems = Goal
   523     "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
   524 \    (INT x:A. C(x)) = (INT x:B. D(x))";
   525 by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
   526 by (REPEAT (dtac INT_D 1
   527      ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
   528 qed "INT_cong";
   529 
   530 
   531 section "Union";
   532 
   533 Goalw [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
   534 by (Blast_tac 1);
   535 qed "Union_iff";
   536 
   537 Addsimps [Union_iff];
   538 
   539 (*The order of the premises presupposes that C is rigid; A may be flexible*)
   540 Goal "[| X:C;  A:X |] ==> A : Union(C)";
   541 by Auto_tac;
   542 qed "UnionI";
   543 
   544 val major::prems = Goalw [Union_def]
   545     "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
   546 by (rtac (major RS UN_E) 1);
   547 by (REPEAT (ares_tac prems 1));
   548 qed "UnionE";
   549 
   550 AddIs  [UnionI];
   551 AddSEs [UnionE];
   552 
   553 
   554 section "Inter";
   555 
   556 Goalw [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
   557 by (Blast_tac 1);
   558 qed "Inter_iff";
   559 
   560 Addsimps [Inter_iff];
   561 
   562 val prems = Goalw [Inter_def]
   563     "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
   564 by (REPEAT (ares_tac ([INT_I] @ prems) 1));
   565 qed "InterI";
   566 
   567 (*A "destruct" rule -- every X in C contains A as an element, but
   568   A:X can hold when X:C does not!  This rule is analogous to "spec". *)
   569 Goal "[| A : Inter(C);  X:C |] ==> A:X";
   570 by Auto_tac;
   571 qed "InterD";
   572 
   573 (*"Classical" elimination rule -- does not require proving X:C *)
   574 val major::prems = Goalw [Inter_def]
   575     "[| A : Inter(C);  X~:C ==> R;  A:X ==> R |] ==> R";
   576 by (rtac (major RS INT_E) 1);
   577 by (REPEAT (eresolve_tac prems 1));
   578 qed "InterE";
   579 
   580 AddSIs [InterI];
   581 AddEs  [InterD, InterE];
   582 
   583 
   584 (*** Image of a set under a function ***)
   585 
   586 (*Frequently b does not have the syntactic form of f(x).*)
   587 Goalw [image_def] "[| b=f(x);  x:A |] ==> b : f``A";
   588 by (Blast_tac 1);
   589 qed "image_eqI";
   590 Addsimps [image_eqI];
   591 
   592 bind_thm ("imageI", refl RS image_eqI);
   593 
   594 (*The eta-expansion gives variable-name preservation.*)
   595 val major::prems = Goalw [image_def]
   596     "[| b : (%x. f(x))``A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P"; 
   597 by (rtac (major RS CollectD RS bexE) 1);
   598 by (REPEAT (ares_tac prems 1));
   599 qed "imageE";
   600 
   601 AddIs  [image_eqI];
   602 AddSEs [imageE]; 
   603 
   604 Goal "f``(A Un B) = f``A Un f``B";
   605 by (Blast_tac 1);
   606 qed "image_Un";
   607 
   608 Goal "(z : f``A) = (EX x:A. z = f x)";
   609 by (Blast_tac 1);
   610 qed "image_iff";
   611 
   612 (*This rewrite rule would confuse users if made default.*)
   613 Goal "(f``A <= B) = (ALL x:A. f(x): B)";
   614 by (Blast_tac 1);
   615 qed "image_subset_iff";
   616 
   617 (*Replaces the three steps subsetI, imageE, hyp_subst_tac, but breaks too
   618   many existing proofs.*)
   619 val prems = Goal "(!!x. x:A ==> f(x) : B) ==> f``A <= B";
   620 by (blast_tac (claset() addIs prems) 1);
   621 qed "image_subsetI";
   622 
   623 
   624 (*** Range of a function -- just a translation for image! ***)
   625 
   626 Goal "b=f(x) ==> b : range(f)";
   627 by (EVERY1 [etac image_eqI, rtac UNIV_I]);
   628 bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI));
   629 
   630 bind_thm ("rangeI", UNIV_I RS imageI);
   631 
   632 val [major,minor] = Goal 
   633     "[| b : range(%x. f(x));  !!x. b=f(x) ==> P |] ==> P"; 
   634 by (rtac (major RS imageE) 1);
   635 by (etac minor 1);
   636 qed "rangeE";
   637 
   638 
   639 (*** Set reasoning tools ***)
   640 
   641 
   642 (** Rewrite rules for boolean case-splitting: faster than 
   643 	addsplits[split_if]
   644 **)
   645 
   646 bind_thm ("split_if_eq1", read_instantiate [("P", "%x. x = ?b")] split_if);
   647 bind_thm ("split_if_eq2", read_instantiate [("P", "%x. ?a = x")] split_if);
   648 
   649 (*Split ifs on either side of the membership relation.
   650 	Not for Addsimps -- can cause goals to blow up!*)
   651 bind_thm ("split_if_mem1", 
   652     read_instantiate_sg (sign_of Set.thy) [("P", "%x. x : ?b")] split_if);
   653 bind_thm ("split_if_mem2", 
   654     read_instantiate_sg (sign_of Set.thy) [("P", "%x. ?a : x")] split_if);
   655 
   656 val split_ifs = [if_bool_eq_conj, split_if_eq1, split_if_eq2,
   657 		  split_if_mem1, split_if_mem2];
   658 
   659 
   660 (*Each of these has ALREADY been added to simpset() above.*)
   661 val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff, 
   662                  mem_Collect_eq, UN_iff, Union_iff, INT_iff, Inter_iff];
   663 
   664 val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs;
   665 
   666 simpset_ref() := simpset() addcongs [ball_cong,bex_cong]
   667                     setmksimps (mksimps mksimps_pairs);
   668 
   669 Addsimps[subset_UNIV, subset_refl];
   670 
   671 
   672 (*** < ***)
   673 
   674 Goalw [psubset_def] "!!A::'a set. [| A <= B; A ~= B |] ==> A<B";
   675 by (Blast_tac 1);
   676 qed "psubsetI";
   677 
   678 Goalw [psubset_def] "A < insert x B ==> (x ~: A) & A<=B | x:A & A-{x}<B";
   679 by Auto_tac;
   680 qed "psubset_insertD";
   681 
   682 bind_thm ("psubset_eq", psubset_def RS meta_eq_to_obj_eq);