src/HOL/Set.ML
author paulson
Tue Sep 15 15:04:07 1998 +0200 (1998-09-15)
changeset 5490 85855f65d0c6
parent 5450 fe9d103464a4
child 5521 7970832271cc
permissions -rw-r--r--
From Compl(A) to -A
clasohm@1465
     1
(*  Title:      HOL/set
clasohm@923
     2
    ID:         $Id$
clasohm@1465
     3
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
clasohm@923
     4
    Copyright   1991  University of Cambridge
clasohm@923
     5
paulson@1985
     6
Set theory for higher-order logic.  A set is simply a predicate.
clasohm@923
     7
*)
clasohm@923
     8
clasohm@923
     9
open Set;
clasohm@923
    10
nipkow@1548
    11
section "Relating predicates and sets";
nipkow@1548
    12
paulson@3469
    13
Addsimps [Collect_mem_eq];
paulson@3469
    14
AddIffs  [mem_Collect_eq];
paulson@2499
    15
paulson@5143
    16
Goal "P(a) ==> a : {x. P(x)}";
paulson@2499
    17
by (Asm_simp_tac 1);
clasohm@923
    18
qed "CollectI";
clasohm@923
    19
paulson@5316
    20
Goal "a : {x. P(x)} ==> P(a)";
paulson@2499
    21
by (Asm_full_simp_tac 1);
clasohm@923
    22
qed "CollectD";
clasohm@923
    23
paulson@5316
    24
val [prem] = Goal "[| !!x. (x:A) = (x:B) |] ==> A = B";
clasohm@923
    25
by (rtac (prem RS ext RS arg_cong RS box_equals) 1);
clasohm@923
    26
by (rtac Collect_mem_eq 1);
clasohm@923
    27
by (rtac Collect_mem_eq 1);
clasohm@923
    28
qed "set_ext";
clasohm@923
    29
paulson@5316
    30
val [prem] = Goal "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}";
clasohm@923
    31
by (rtac (prem RS ext RS arg_cong) 1);
clasohm@923
    32
qed "Collect_cong";
clasohm@923
    33
clasohm@923
    34
val CollectE = make_elim CollectD;
clasohm@923
    35
paulson@2499
    36
AddSIs [CollectI];
paulson@2499
    37
AddSEs [CollectE];
paulson@2499
    38
paulson@2499
    39
nipkow@1548
    40
section "Bounded quantifiers";
clasohm@923
    41
paulson@5316
    42
val prems = Goalw [Ball_def]
clasohm@923
    43
    "[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)";
clasohm@923
    44
by (REPEAT (ares_tac (prems @ [allI,impI]) 1));
clasohm@923
    45
qed "ballI";
clasohm@923
    46
paulson@5316
    47
Goalw [Ball_def] "[| ! x:A. P(x);  x:A |] ==> P(x)";
paulson@5316
    48
by (Blast_tac 1);
clasohm@923
    49
qed "bspec";
clasohm@923
    50
paulson@5316
    51
val major::prems = Goalw [Ball_def]
clasohm@923
    52
    "[| ! x:A. P(x);  P(x) ==> Q;  x~:A ==> Q |] ==> Q";
clasohm@923
    53
by (rtac (major RS spec RS impCE) 1);
clasohm@923
    54
by (REPEAT (eresolve_tac prems 1));
clasohm@923
    55
qed "ballE";
clasohm@923
    56
clasohm@923
    57
(*Takes assumptions ! x:A.P(x) and a:A; creates assumption P(a)*)
clasohm@923
    58
fun ball_tac i = etac ballE i THEN contr_tac (i+1);
clasohm@923
    59
paulson@2499
    60
AddSIs [ballI];
paulson@2499
    61
AddEs  [ballE];
paulson@2499
    62
paulson@5316
    63
Goalw [Bex_def] "[| P(x);  x:A |] ==> ? x:A. P(x)";
paulson@5316
    64
by (Blast_tac 1);
clasohm@923
    65
qed "bexI";
clasohm@923
    66
clasohm@923
    67
qed_goal "bexCI" Set.thy 
wenzelm@3842
    68
   "[| ! x:A. ~P(x) ==> P(a);  a:A |] ==> ? x:A. P(x)"
clasohm@923
    69
 (fn prems=>
clasohm@923
    70
  [ (rtac classical 1),
clasohm@923
    71
    (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1))  ]);
clasohm@923
    72
paulson@5316
    73
val major::prems = Goalw [Bex_def]
clasohm@923
    74
    "[| ? x:A. P(x);  !!x. [| x:A; P(x) |] ==> Q  |] ==> Q";
clasohm@923
    75
by (rtac (major RS exE) 1);
clasohm@923
    76
by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1));
clasohm@923
    77
qed "bexE";
clasohm@923
    78
paulson@2499
    79
AddIs  [bexI];
paulson@2499
    80
AddSEs [bexE];
paulson@2499
    81
paulson@3420
    82
(*Trival rewrite rule*)
wenzelm@5069
    83
Goal "(! x:A. P) = ((? x. x:A) --> P)";
wenzelm@4089
    84
by (simp_tac (simpset() addsimps [Ball_def]) 1);
paulson@3420
    85
qed "ball_triv";
paulson@1816
    86
paulson@1882
    87
(*Dual form for existentials*)
wenzelm@5069
    88
Goal "(? x:A. P) = ((? x. x:A) & P)";
wenzelm@4089
    89
by (simp_tac (simpset() addsimps [Bex_def]) 1);
paulson@3420
    90
qed "bex_triv";
paulson@1882
    91
paulson@3420
    92
Addsimps [ball_triv, bex_triv];
clasohm@923
    93
clasohm@923
    94
(** Congruence rules **)
clasohm@923
    95
paulson@5316
    96
val prems = Goal
clasohm@923
    97
    "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
clasohm@923
    98
\    (! x:A. P(x)) = (! x:B. Q(x))";
clasohm@923
    99
by (resolve_tac (prems RL [ssubst]) 1);
clasohm@923
   100
by (REPEAT (ares_tac [ballI,iffI] 1
clasohm@923
   101
     ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1));
clasohm@923
   102
qed "ball_cong";
clasohm@923
   103
paulson@5316
   104
val prems = Goal
clasohm@923
   105
    "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
clasohm@923
   106
\    (? x:A. P(x)) = (? x:B. Q(x))";
clasohm@923
   107
by (resolve_tac (prems RL [ssubst]) 1);
clasohm@923
   108
by (REPEAT (etac bexE 1
clasohm@923
   109
     ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1));
clasohm@923
   110
qed "bex_cong";
clasohm@923
   111
nipkow@1548
   112
section "Subsets";
clasohm@923
   113
paulson@5316
   114
val prems = Goalw [subset_def] "(!!x. x:A ==> x:B) ==> A <= B";
clasohm@923
   115
by (REPEAT (ares_tac (prems @ [ballI]) 1));
clasohm@923
   116
qed "subsetI";
clasohm@923
   117
paulson@4059
   118
(*While (:) is not, its type must be kept
paulson@4059
   119
  for overloading of = to work.*)
paulson@4240
   120
Blast.overloaded ("op :", domain_type);
paulson@4240
   121
seq (fn a => Blast.overloaded (a, HOLogic.dest_setT o domain_type))
paulson@4059
   122
    ["Ball", "Bex"];
paulson@4059
   123
(*need UNION, INTER also?*)
paulson@4059
   124
paulson@4469
   125
(*Image: retain the type of the set being expressed*)
paulson@5336
   126
Blast.overloaded ("op ``", domain_type);
paulson@2881
   127
clasohm@923
   128
(*Rule in Modus Ponens style*)
paulson@5316
   129
Goalw [subset_def] "[| A <= B;  c:A |] ==> c:B";
paulson@5316
   130
by (Blast_tac 1);
clasohm@923
   131
qed "subsetD";
clasohm@923
   132
clasohm@923
   133
(*The same, with reversed premises for use with etac -- cf rev_mp*)
clasohm@923
   134
qed_goal "rev_subsetD" Set.thy "[| c:A;  A <= B |] ==> c:B"
clasohm@923
   135
 (fn prems=>  [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]);
clasohm@923
   136
paulson@1920
   137
(*Converts A<=B to x:A ==> x:B*)
paulson@1920
   138
fun impOfSubs th = th RSN (2, rev_subsetD);
paulson@1920
   139
paulson@1841
   140
qed_goal "contra_subsetD" Set.thy "!!c. [| A <= B; c ~: B |] ==> c ~: A"
paulson@1841
   141
 (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
paulson@1841
   142
paulson@1841
   143
qed_goal "rev_contra_subsetD" Set.thy "!!c. [| c ~: B;  A <= B |] ==> c ~: A"
paulson@1841
   144
 (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
paulson@1841
   145
clasohm@923
   146
(*Classical elimination rule*)
paulson@5316
   147
val major::prems = Goalw [subset_def] 
clasohm@923
   148
    "[| A <= B;  c~:A ==> P;  c:B ==> P |] ==> P";
clasohm@923
   149
by (rtac (major RS ballE) 1);
clasohm@923
   150
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   151
qed "subsetCE";
clasohm@923
   152
clasohm@923
   153
(*Takes assumptions A<=B; c:A and creates the assumption c:B *)
clasohm@923
   154
fun set_mp_tac i = etac subsetCE i  THEN  mp_tac i;
clasohm@923
   155
paulson@2499
   156
AddSIs [subsetI];
paulson@2499
   157
AddEs  [subsetD, subsetCE];
clasohm@923
   158
paulson@2499
   159
qed_goal "subset_refl" Set.thy "A <= (A::'a set)"
paulson@4059
   160
 (fn _=> [Fast_tac 1]);		(*Blast_tac would try order_refl and fail*)
paulson@2499
   161
paulson@5316
   162
Goal "[| A<=B;  B<=C |] ==> A<=(C::'a set)";
paulson@2891
   163
by (Blast_tac 1);
clasohm@923
   164
qed "subset_trans";
clasohm@923
   165
clasohm@923
   166
nipkow@1548
   167
section "Equality";
clasohm@923
   168
clasohm@923
   169
(*Anti-symmetry of the subset relation*)
paulson@5316
   170
Goal "[| A <= B;  B <= A |] ==> A = (B::'a set)";
paulson@5318
   171
by (rtac set_ext 1);
paulson@5316
   172
by (blast_tac (claset() addIs [subsetD]) 1);
clasohm@923
   173
qed "subset_antisym";
clasohm@923
   174
val equalityI = subset_antisym;
clasohm@923
   175
berghofe@1762
   176
AddSIs [equalityI];
berghofe@1762
   177
clasohm@923
   178
(* Equality rules from ZF set theory -- are they appropriate here? *)
paulson@5316
   179
Goal "A = B ==> A<=(B::'a set)";
paulson@5316
   180
by (etac ssubst 1);
clasohm@923
   181
by (rtac subset_refl 1);
clasohm@923
   182
qed "equalityD1";
clasohm@923
   183
paulson@5316
   184
Goal "A = B ==> B<=(A::'a set)";
paulson@5316
   185
by (etac ssubst 1);
clasohm@923
   186
by (rtac subset_refl 1);
clasohm@923
   187
qed "equalityD2";
clasohm@923
   188
paulson@5316
   189
val prems = Goal
clasohm@923
   190
    "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";
clasohm@923
   191
by (resolve_tac prems 1);
clasohm@923
   192
by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
clasohm@923
   193
qed "equalityE";
clasohm@923
   194
paulson@5316
   195
val major::prems = Goal
clasohm@923
   196
    "[| A = B;  [| c:A; c:B |] ==> P;  [| c~:A; c~:B |] ==> P |]  ==>  P";
clasohm@923
   197
by (rtac (major RS equalityE) 1);
clasohm@923
   198
by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
clasohm@923
   199
qed "equalityCE";
clasohm@923
   200
clasohm@923
   201
(*Lemma for creating induction formulae -- for "pattern matching" on p
clasohm@923
   202
  To make the induction hypotheses usable, apply "spec" or "bspec" to
clasohm@923
   203
  put universal quantifiers over the free variables in p. *)
paulson@5316
   204
val prems = Goal 
clasohm@923
   205
    "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
clasohm@923
   206
by (rtac mp 1);
clasohm@923
   207
by (REPEAT (resolve_tac (refl::prems) 1));
clasohm@923
   208
qed "setup_induction";
clasohm@923
   209
clasohm@923
   210
paulson@4159
   211
section "The universal set -- UNIV";
paulson@4159
   212
paulson@4159
   213
qed_goalw "UNIV_I" Set.thy [UNIV_def] "x : UNIV"
paulson@4159
   214
  (fn _ => [rtac CollectI 1, rtac TrueI 1]);
paulson@4159
   215
paulson@4434
   216
Addsimps [UNIV_I];
paulson@4434
   217
AddIs    [UNIV_I];  (*unsafe makes it less likely to cause problems*)
paulson@4159
   218
paulson@4159
   219
qed_goal "subset_UNIV" Set.thy "A <= UNIV"
paulson@4159
   220
  (fn _ => [rtac subsetI 1, rtac UNIV_I 1]);
paulson@4159
   221
paulson@4159
   222
(** Eta-contracting these two rules (to remove P) causes them to be ignored
paulson@4159
   223
    because of their interaction with congruence rules. **)
paulson@4159
   224
wenzelm@5069
   225
Goalw [Ball_def] "Ball UNIV P = All P";
paulson@4159
   226
by (Simp_tac 1);
paulson@4159
   227
qed "ball_UNIV";
paulson@4159
   228
wenzelm@5069
   229
Goalw [Bex_def] "Bex UNIV P = Ex P";
paulson@4159
   230
by (Simp_tac 1);
paulson@4159
   231
qed "bex_UNIV";
paulson@4159
   232
Addsimps [ball_UNIV, bex_UNIV];
paulson@4159
   233
paulson@4159
   234
paulson@2858
   235
section "The empty set -- {}";
paulson@2858
   236
paulson@2858
   237
qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False"
paulson@2891
   238
 (fn _ => [ (Blast_tac 1) ]);
paulson@2858
   239
paulson@2858
   240
Addsimps [empty_iff];
paulson@2858
   241
paulson@2858
   242
qed_goal "emptyE" Set.thy "!!a. a:{} ==> P"
paulson@2858
   243
 (fn _ => [Full_simp_tac 1]);
paulson@2858
   244
paulson@2858
   245
AddSEs [emptyE];
paulson@2858
   246
paulson@2858
   247
qed_goal "empty_subsetI" Set.thy "{} <= A"
paulson@2891
   248
 (fn _ => [ (Blast_tac 1) ]);
paulson@2858
   249
paulson@5256
   250
(*One effect is to delete the ASSUMPTION {} <= A*)
paulson@5256
   251
AddIffs [empty_subsetI];
paulson@5256
   252
paulson@2858
   253
qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}"
paulson@2858
   254
 (fn [prem]=>
wenzelm@4089
   255
  [ (blast_tac (claset() addIs [prem RS FalseE]) 1) ]);
paulson@2858
   256
paulson@5256
   257
(*Use for reasoning about disjointness: A Int B = {} *)
paulson@5450
   258
qed_goal "equals0D" Set.thy "!!a. A={} ==> a ~: A"
paulson@2891
   259
 (fn _ => [ (Blast_tac 1) ]);
paulson@2858
   260
paulson@5450
   261
AddDs [equals0D, sym RS equals0D];
paulson@5256
   262
wenzelm@5069
   263
Goalw [Ball_def] "Ball {} P = True";
paulson@4159
   264
by (Simp_tac 1);
paulson@4159
   265
qed "ball_empty";
paulson@4159
   266
wenzelm@5069
   267
Goalw [Bex_def] "Bex {} P = False";
paulson@4159
   268
by (Simp_tac 1);
paulson@4159
   269
qed "bex_empty";
paulson@4159
   270
Addsimps [ball_empty, bex_empty];
paulson@4159
   271
wenzelm@5069
   272
Goal "UNIV ~= {}";
paulson@4159
   273
by (blast_tac (claset() addEs [equalityE]) 1);
paulson@4159
   274
qed "UNIV_not_empty";
paulson@4159
   275
AddIffs [UNIV_not_empty];
paulson@4159
   276
paulson@4159
   277
paulson@2858
   278
paulson@2858
   279
section "The Powerset operator -- Pow";
paulson@2858
   280
paulson@2858
   281
qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)"
paulson@2858
   282
 (fn _ => [ (Asm_simp_tac 1) ]);
paulson@2858
   283
paulson@2858
   284
AddIffs [Pow_iff]; 
paulson@2858
   285
paulson@2858
   286
qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
paulson@2858
   287
 (fn _ => [ (etac CollectI 1) ]);
paulson@2858
   288
paulson@2858
   289
qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B)  ==>  A<=B"
paulson@2858
   290
 (fn _=> [ (etac CollectD 1) ]);
paulson@2858
   291
paulson@2858
   292
val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)
paulson@2858
   293
val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)
paulson@2858
   294
paulson@2858
   295
nipkow@1548
   296
section "Set complement -- Compl";
clasohm@923
   297
paulson@5490
   298
qed_goalw "Compl_iff" Set.thy [Compl_def] "(c : -A) = (c~:A)"
paulson@2891
   299
 (fn _ => [ (Blast_tac 1) ]);
paulson@2499
   300
paulson@2499
   301
Addsimps [Compl_iff];
paulson@2499
   302
paulson@5490
   303
val prems = Goalw [Compl_def] "[| c:A ==> False |] ==> c : -A";
clasohm@923
   304
by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
clasohm@923
   305
qed "ComplI";
clasohm@923
   306
clasohm@923
   307
(*This form, with negated conclusion, works well with the Classical prover.
clasohm@923
   308
  Negated assumptions behave like formulae on the right side of the notional
clasohm@923
   309
  turnstile...*)
paulson@5490
   310
Goalw [Compl_def] "c : -A ==> c~:A";
paulson@5316
   311
by (etac CollectD 1);
clasohm@923
   312
qed "ComplD";
clasohm@923
   313
clasohm@923
   314
val ComplE = make_elim ComplD;
clasohm@923
   315
paulson@2499
   316
AddSIs [ComplI];
paulson@2499
   317
AddSEs [ComplE];
paulson@1640
   318
clasohm@923
   319
nipkow@1548
   320
section "Binary union -- Un";
clasohm@923
   321
paulson@2499
   322
qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)"
paulson@2891
   323
 (fn _ => [ Blast_tac 1 ]);
paulson@2499
   324
paulson@2499
   325
Addsimps [Un_iff];
paulson@2499
   326
paulson@5143
   327
Goal "c:A ==> c : A Un B";
paulson@2499
   328
by (Asm_simp_tac 1);
clasohm@923
   329
qed "UnI1";
clasohm@923
   330
paulson@5143
   331
Goal "c:B ==> c : A Un B";
paulson@2499
   332
by (Asm_simp_tac 1);
clasohm@923
   333
qed "UnI2";
clasohm@923
   334
clasohm@923
   335
(*Classical introduction rule: no commitment to A vs B*)
clasohm@923
   336
qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B"
clasohm@923
   337
 (fn prems=>
paulson@2499
   338
  [ (Simp_tac 1),
paulson@2499
   339
    (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
clasohm@923
   340
paulson@5316
   341
val major::prems = Goalw [Un_def]
clasohm@923
   342
    "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
clasohm@923
   343
by (rtac (major RS CollectD RS disjE) 1);
clasohm@923
   344
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   345
qed "UnE";
clasohm@923
   346
paulson@2499
   347
AddSIs [UnCI];
paulson@2499
   348
AddSEs [UnE];
paulson@1640
   349
clasohm@923
   350
nipkow@1548
   351
section "Binary intersection -- Int";
clasohm@923
   352
paulson@2499
   353
qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)"
paulson@2891
   354
 (fn _ => [ (Blast_tac 1) ]);
paulson@2499
   355
paulson@2499
   356
Addsimps [Int_iff];
paulson@2499
   357
paulson@5143
   358
Goal "[| c:A;  c:B |] ==> c : A Int B";
paulson@2499
   359
by (Asm_simp_tac 1);
clasohm@923
   360
qed "IntI";
clasohm@923
   361
paulson@5143
   362
Goal "c : A Int B ==> c:A";
paulson@2499
   363
by (Asm_full_simp_tac 1);
clasohm@923
   364
qed "IntD1";
clasohm@923
   365
paulson@5143
   366
Goal "c : A Int B ==> c:B";
paulson@2499
   367
by (Asm_full_simp_tac 1);
clasohm@923
   368
qed "IntD2";
clasohm@923
   369
paulson@5316
   370
val [major,minor] = Goal
clasohm@923
   371
    "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
clasohm@923
   372
by (rtac minor 1);
clasohm@923
   373
by (rtac (major RS IntD1) 1);
clasohm@923
   374
by (rtac (major RS IntD2) 1);
clasohm@923
   375
qed "IntE";
clasohm@923
   376
paulson@2499
   377
AddSIs [IntI];
paulson@2499
   378
AddSEs [IntE];
clasohm@923
   379
nipkow@1548
   380
section "Set difference";
clasohm@923
   381
paulson@2499
   382
qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)"
paulson@2891
   383
 (fn _ => [ (Blast_tac 1) ]);
clasohm@923
   384
paulson@2499
   385
Addsimps [Diff_iff];
paulson@2499
   386
paulson@2499
   387
qed_goal "DiffI" Set.thy "!!c. [| c : A;  c ~: B |] ==> c : A - B"
paulson@2499
   388
 (fn _=> [ Asm_simp_tac 1 ]);
clasohm@923
   389
paulson@2499
   390
qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A"
paulson@2499
   391
 (fn _=> [ (Asm_full_simp_tac 1) ]);
clasohm@923
   392
paulson@2499
   393
qed_goal "DiffD2" Set.thy "!!c. [| c : A - B;  c : B |] ==> P"
paulson@2499
   394
 (fn _=> [ (Asm_full_simp_tac 1) ]);
paulson@2499
   395
paulson@2499
   396
qed_goal "DiffE" Set.thy "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P"
clasohm@923
   397
 (fn prems=>
clasohm@923
   398
  [ (resolve_tac prems 1),
clasohm@923
   399
    (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]);
clasohm@923
   400
paulson@2499
   401
AddSIs [DiffI];
paulson@2499
   402
AddSEs [DiffE];
clasohm@923
   403
clasohm@923
   404
nipkow@1548
   405
section "Augmenting a set -- insert";
clasohm@923
   406
paulson@2499
   407
qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)"
paulson@2891
   408
 (fn _ => [Blast_tac 1]);
paulson@2499
   409
paulson@2499
   410
Addsimps [insert_iff];
clasohm@923
   411
paulson@2499
   412
qed_goal "insertI1" Set.thy "a : insert a B"
paulson@2499
   413
 (fn _ => [Simp_tac 1]);
paulson@2499
   414
paulson@2499
   415
qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B"
paulson@2499
   416
 (fn _=> [Asm_simp_tac 1]);
clasohm@923
   417
clasohm@923
   418
qed_goalw "insertE" Set.thy [insert_def]
clasohm@923
   419
    "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P"
clasohm@923
   420
 (fn major::prems=>
clasohm@923
   421
  [ (rtac (major RS UnE) 1),
clasohm@923
   422
    (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
clasohm@923
   423
clasohm@923
   424
(*Classical introduction rule*)
clasohm@923
   425
qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
paulson@2499
   426
 (fn prems=>
paulson@2499
   427
  [ (Simp_tac 1),
paulson@2499
   428
    (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
paulson@2499
   429
paulson@2499
   430
AddSIs [insertCI]; 
paulson@2499
   431
AddSEs [insertE];
clasohm@923
   432
nipkow@1548
   433
section "Singletons, using insert";
clasohm@923
   434
clasohm@923
   435
qed_goal "singletonI" Set.thy "a : {a}"
clasohm@923
   436
 (fn _=> [ (rtac insertI1 1) ]);
clasohm@923
   437
paulson@5143
   438
Goal "b : {a} ==> b=a";
paulson@2891
   439
by (Blast_tac 1);
clasohm@923
   440
qed "singletonD";
clasohm@923
   441
oheimb@1776
   442
bind_thm ("singletonE", make_elim singletonD);
oheimb@1776
   443
paulson@2499
   444
qed_goal "singleton_iff" thy "(b : {a}) = (b=a)" 
paulson@2891
   445
(fn _ => [Blast_tac 1]);
clasohm@923
   446
paulson@5143
   447
Goal "{a}={b} ==> a=b";
wenzelm@4089
   448
by (blast_tac (claset() addEs [equalityE]) 1);
clasohm@923
   449
qed "singleton_inject";
clasohm@923
   450
paulson@2858
   451
(*Redundant? But unlike insertCI, it proves the subgoal immediately!*)
paulson@2858
   452
AddSIs [singletonI];   
paulson@2499
   453
AddSDs [singleton_inject];
paulson@3718
   454
AddSEs [singletonE];
paulson@2499
   455
wenzelm@5069
   456
Goal "{x. x=a} = {a}";
wenzelm@4423
   457
by (Blast_tac 1);
nipkow@3582
   458
qed "singleton_conv";
nipkow@3582
   459
Addsimps [singleton_conv];
nipkow@1531
   460
nipkow@1531
   461
nipkow@1548
   462
section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
clasohm@923
   463
wenzelm@5069
   464
Goalw [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
paulson@2891
   465
by (Blast_tac 1);
paulson@2499
   466
qed "UN_iff";
paulson@2499
   467
paulson@2499
   468
Addsimps [UN_iff];
paulson@2499
   469
clasohm@923
   470
(*The order of the premises presupposes that A is rigid; b may be flexible*)
paulson@5143
   471
Goal "[| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
paulson@4477
   472
by Auto_tac;
clasohm@923
   473
qed "UN_I";
clasohm@923
   474
paulson@5316
   475
val major::prems = Goalw [UNION_def]
clasohm@923
   476
    "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
clasohm@923
   477
by (rtac (major RS CollectD RS bexE) 1);
clasohm@923
   478
by (REPEAT (ares_tac prems 1));
clasohm@923
   479
qed "UN_E";
clasohm@923
   480
paulson@2499
   481
AddIs  [UN_I];
paulson@2499
   482
AddSEs [UN_E];
paulson@2499
   483
paulson@5316
   484
val prems = Goal
clasohm@923
   485
    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
clasohm@923
   486
\    (UN x:A. C(x)) = (UN x:B. D(x))";
clasohm@923
   487
by (REPEAT (etac UN_E 1
clasohm@923
   488
     ORELSE ares_tac ([UN_I,equalityI,subsetI] @ 
clasohm@1465
   489
                      (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
clasohm@923
   490
qed "UN_cong";
clasohm@923
   491
clasohm@923
   492
nipkow@1548
   493
section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
clasohm@923
   494
wenzelm@5069
   495
Goalw [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
paulson@4477
   496
by Auto_tac;
paulson@2499
   497
qed "INT_iff";
paulson@2499
   498
paulson@2499
   499
Addsimps [INT_iff];
paulson@2499
   500
paulson@5316
   501
val prems = Goalw [INTER_def]
clasohm@923
   502
    "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
clasohm@923
   503
by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
clasohm@923
   504
qed "INT_I";
clasohm@923
   505
paulson@5143
   506
Goal "[| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
paulson@4477
   507
by Auto_tac;
clasohm@923
   508
qed "INT_D";
clasohm@923
   509
clasohm@923
   510
(*"Classical" elimination -- by the Excluded Middle on a:A *)
paulson@5316
   511
val major::prems = Goalw [INTER_def]
clasohm@923
   512
    "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
clasohm@923
   513
by (rtac (major RS CollectD RS ballE) 1);
clasohm@923
   514
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   515
qed "INT_E";
clasohm@923
   516
paulson@2499
   517
AddSIs [INT_I];
paulson@2499
   518
AddEs  [INT_D, INT_E];
paulson@2499
   519
paulson@5316
   520
val prems = Goal
clasohm@923
   521
    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
clasohm@923
   522
\    (INT x:A. C(x)) = (INT x:B. D(x))";
clasohm@923
   523
by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
clasohm@923
   524
by (REPEAT (dtac INT_D 1
clasohm@923
   525
     ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
clasohm@923
   526
qed "INT_cong";
clasohm@923
   527
clasohm@923
   528
nipkow@1548
   529
section "Union";
clasohm@923
   530
wenzelm@5069
   531
Goalw [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
paulson@2891
   532
by (Blast_tac 1);
paulson@2499
   533
qed "Union_iff";
paulson@2499
   534
paulson@2499
   535
Addsimps [Union_iff];
paulson@2499
   536
clasohm@923
   537
(*The order of the premises presupposes that C is rigid; A may be flexible*)
paulson@5143
   538
Goal "[| X:C;  A:X |] ==> A : Union(C)";
paulson@4477
   539
by Auto_tac;
clasohm@923
   540
qed "UnionI";
clasohm@923
   541
paulson@5316
   542
val major::prems = Goalw [Union_def]
clasohm@923
   543
    "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
clasohm@923
   544
by (rtac (major RS UN_E) 1);
clasohm@923
   545
by (REPEAT (ares_tac prems 1));
clasohm@923
   546
qed "UnionE";
clasohm@923
   547
paulson@2499
   548
AddIs  [UnionI];
paulson@2499
   549
AddSEs [UnionE];
paulson@2499
   550
paulson@2499
   551
nipkow@1548
   552
section "Inter";
clasohm@923
   553
wenzelm@5069
   554
Goalw [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
paulson@2891
   555
by (Blast_tac 1);
paulson@2499
   556
qed "Inter_iff";
paulson@2499
   557
paulson@2499
   558
Addsimps [Inter_iff];
paulson@2499
   559
paulson@5316
   560
val prems = Goalw [Inter_def]
clasohm@923
   561
    "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
clasohm@923
   562
by (REPEAT (ares_tac ([INT_I] @ prems) 1));
clasohm@923
   563
qed "InterI";
clasohm@923
   564
clasohm@923
   565
(*A "destruct" rule -- every X in C contains A as an element, but
clasohm@923
   566
  A:X can hold when X:C does not!  This rule is analogous to "spec". *)
paulson@5143
   567
Goal "[| A : Inter(C);  X:C |] ==> A:X";
paulson@4477
   568
by Auto_tac;
clasohm@923
   569
qed "InterD";
clasohm@923
   570
clasohm@923
   571
(*"Classical" elimination rule -- does not require proving X:C *)
paulson@5316
   572
val major::prems = Goalw [Inter_def]
paulson@2721
   573
    "[| A : Inter(C);  X~:C ==> R;  A:X ==> R |] ==> R";
clasohm@923
   574
by (rtac (major RS INT_E) 1);
clasohm@923
   575
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   576
qed "InterE";
clasohm@923
   577
paulson@2499
   578
AddSIs [InterI];
paulson@2499
   579
AddEs  [InterD, InterE];
paulson@2499
   580
paulson@2499
   581
nipkow@2912
   582
(*** Image of a set under a function ***)
nipkow@2912
   583
nipkow@2912
   584
(*Frequently b does not have the syntactic form of f(x).*)
paulson@5316
   585
Goalw [image_def] "[| b=f(x);  x:A |] ==> b : f``A";
paulson@5316
   586
by (Blast_tac 1);
nipkow@2912
   587
qed "image_eqI";
nipkow@3909
   588
Addsimps [image_eqI];
nipkow@2912
   589
nipkow@2912
   590
bind_thm ("imageI", refl RS image_eqI);
nipkow@2912
   591
nipkow@2912
   592
(*The eta-expansion gives variable-name preservation.*)
paulson@5316
   593
val major::prems = Goalw [image_def]
wenzelm@3842
   594
    "[| b : (%x. f(x))``A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P"; 
nipkow@2912
   595
by (rtac (major RS CollectD RS bexE) 1);
nipkow@2912
   596
by (REPEAT (ares_tac prems 1));
nipkow@2912
   597
qed "imageE";
nipkow@2912
   598
nipkow@2912
   599
AddIs  [image_eqI];
nipkow@2912
   600
AddSEs [imageE]; 
nipkow@2912
   601
wenzelm@5069
   602
Goal "f``(A Un B) = f``A Un f``B";
paulson@2935
   603
by (Blast_tac 1);
nipkow@2912
   604
qed "image_Un";
nipkow@2912
   605
wenzelm@5069
   606
Goal "(z : f``A) = (EX x:A. z = f x)";
paulson@3960
   607
by (Blast_tac 1);
paulson@3960
   608
qed "image_iff";
paulson@3960
   609
paulson@4523
   610
(*This rewrite rule would confuse users if made default.*)
wenzelm@5069
   611
Goal "(f``A <= B) = (ALL x:A. f(x): B)";
paulson@4523
   612
by (Blast_tac 1);
paulson@4523
   613
qed "image_subset_iff";
paulson@4523
   614
paulson@4523
   615
(*Replaces the three steps subsetI, imageE, hyp_subst_tac, but breaks too
paulson@4523
   616
  many existing proofs.*)
paulson@5316
   617
val prems = Goal "(!!x. x:A ==> f(x) : B) ==> f``A <= B";
paulson@4510
   618
by (blast_tac (claset() addIs prems) 1);
paulson@4510
   619
qed "image_subsetI";
paulson@4510
   620
nipkow@2912
   621
nipkow@2912
   622
(*** Range of a function -- just a translation for image! ***)
nipkow@2912
   623
paulson@5143
   624
Goal "b=f(x) ==> b : range(f)";
nipkow@2912
   625
by (EVERY1 [etac image_eqI, rtac UNIV_I]);
nipkow@2912
   626
bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI));
nipkow@2912
   627
nipkow@2912
   628
bind_thm ("rangeI", UNIV_I RS imageI);
nipkow@2912
   629
paulson@5316
   630
val [major,minor] = Goal 
wenzelm@3842
   631
    "[| b : range(%x. f(x));  !!x. b=f(x) ==> P |] ==> P"; 
nipkow@2912
   632
by (rtac (major RS imageE) 1);
nipkow@2912
   633
by (etac minor 1);
nipkow@2912
   634
qed "rangeE";
nipkow@2912
   635
oheimb@1776
   636
oheimb@1776
   637
(*** Set reasoning tools ***)
oheimb@1776
   638
oheimb@1776
   639
paulson@3912
   640
(** Rewrite rules for boolean case-splitting: faster than 
nipkow@4830
   641
	addsplits[split_if]
paulson@3912
   642
**)
paulson@3912
   643
nipkow@4830
   644
bind_thm ("split_if_eq1", read_instantiate [("P", "%x. x = ?b")] split_if);
nipkow@4830
   645
bind_thm ("split_if_eq2", read_instantiate [("P", "%x. ?a = x")] split_if);
paulson@3912
   646
paulson@5237
   647
(*Split ifs on either side of the membership relation.
paulson@5237
   648
	Not for Addsimps -- can cause goals to blow up!*)
nipkow@4830
   649
bind_thm ("split_if_mem1", 
nipkow@4830
   650
    read_instantiate_sg (sign_of Set.thy) [("P", "%x. x : ?b")] split_if);
nipkow@4830
   651
bind_thm ("split_if_mem2", 
nipkow@4830
   652
    read_instantiate_sg (sign_of Set.thy) [("P", "%x. ?a : x")] split_if);
paulson@3912
   653
nipkow@4830
   654
val split_ifs = [if_bool_eq_conj, split_if_eq1, split_if_eq2,
nipkow@4830
   655
		  split_if_mem1, split_if_mem2];
paulson@3912
   656
paulson@3912
   657
wenzelm@4089
   658
(*Each of these has ALREADY been added to simpset() above.*)
paulson@2024
   659
val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff, 
paulson@4159
   660
                 mem_Collect_eq, UN_iff, Union_iff, INT_iff, Inter_iff];
oheimb@1776
   661
oheimb@1776
   662
val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs;
oheimb@1776
   663
wenzelm@4089
   664
simpset_ref() := simpset() addcongs [ball_cong,bex_cong]
oheimb@1776
   665
                    setmksimps (mksimps mksimps_pairs);
nipkow@3222
   666
paulson@5256
   667
Addsimps[subset_UNIV, subset_refl];
nipkow@3222
   668
nipkow@3222
   669
nipkow@3222
   670
(*** < ***)
nipkow@3222
   671
wenzelm@5069
   672
Goalw [psubset_def] "!!A::'a set. [| A <= B; A ~= B |] ==> A<B";
nipkow@3222
   673
by (Blast_tac 1);
nipkow@3222
   674
qed "psubsetI";
nipkow@3222
   675
paulson@5148
   676
Goalw [psubset_def] "A < insert x B ==> (x ~: A) & A<=B | x:A & A-{x}<B";
paulson@4477
   677
by Auto_tac;
nipkow@3222
   678
qed "psubset_insertD";
paulson@4059
   679
paulson@4059
   680
bind_thm ("psubset_eq", psubset_def RS meta_eq_to_obj_eq);