src/HOL/Set.ML
author paulson
Wed Mar 03 11:15:18 1999 +0100 (1999-03-03)
changeset 6301 08245f5a436d
parent 6291 2c3f72d9f5d1
child 6394 3d9fd50fcc43
permissions -rw-r--r--
expandshort
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
nipkow@1548
     9
section "Relating predicates and sets";
nipkow@1548
    10
paulson@3469
    11
Addsimps [Collect_mem_eq];
paulson@3469
    12
AddIffs  [mem_Collect_eq];
paulson@2499
    13
paulson@5143
    14
Goal "P(a) ==> a : {x. P(x)}";
paulson@2499
    15
by (Asm_simp_tac 1);
clasohm@923
    16
qed "CollectI";
clasohm@923
    17
paulson@5316
    18
Goal "a : {x. P(x)} ==> P(a)";
paulson@2499
    19
by (Asm_full_simp_tac 1);
clasohm@923
    20
qed "CollectD";
clasohm@923
    21
paulson@5316
    22
val [prem] = Goal "[| !!x. (x:A) = (x:B) |] ==> A = B";
clasohm@923
    23
by (rtac (prem RS ext RS arg_cong RS box_equals) 1);
clasohm@923
    24
by (rtac Collect_mem_eq 1);
clasohm@923
    25
by (rtac Collect_mem_eq 1);
clasohm@923
    26
qed "set_ext";
clasohm@923
    27
paulson@5316
    28
val [prem] = Goal "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}";
clasohm@923
    29
by (rtac (prem RS ext RS arg_cong) 1);
clasohm@923
    30
qed "Collect_cong";
clasohm@923
    31
clasohm@923
    32
val CollectE = make_elim CollectD;
clasohm@923
    33
paulson@2499
    34
AddSIs [CollectI];
paulson@2499
    35
AddSEs [CollectE];
paulson@2499
    36
paulson@2499
    37
nipkow@1548
    38
section "Bounded quantifiers";
clasohm@923
    39
paulson@5316
    40
val prems = Goalw [Ball_def]
clasohm@923
    41
    "[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)";
clasohm@923
    42
by (REPEAT (ares_tac (prems @ [allI,impI]) 1));
clasohm@923
    43
qed "ballI";
clasohm@923
    44
paulson@5316
    45
Goalw [Ball_def] "[| ! x:A. P(x);  x:A |] ==> P(x)";
paulson@5316
    46
by (Blast_tac 1);
clasohm@923
    47
qed "bspec";
clasohm@923
    48
paulson@5316
    49
val major::prems = Goalw [Ball_def]
clasohm@923
    50
    "[| ! x:A. P(x);  P(x) ==> Q;  x~:A ==> Q |] ==> Q";
clasohm@923
    51
by (rtac (major RS spec RS impCE) 1);
clasohm@923
    52
by (REPEAT (eresolve_tac prems 1));
clasohm@923
    53
qed "ballE";
clasohm@923
    54
clasohm@923
    55
(*Takes assumptions ! x:A.P(x) and a:A; creates assumption P(a)*)
clasohm@923
    56
fun ball_tac i = etac ballE i THEN contr_tac (i+1);
clasohm@923
    57
paulson@2499
    58
AddSIs [ballI];
paulson@2499
    59
AddEs  [ballE];
oheimb@5521
    60
(* gives better instantiation for bound: *)
oheimb@5521
    61
claset_ref() := claset() addWrapper ("bspec", fn tac2 =>
oheimb@5521
    62
			 (dtac bspec THEN' atac) APPEND' tac2);
paulson@2499
    63
paulson@6006
    64
(*Normally the best argument order: P(x) constrains the choice of x:A*)
paulson@5316
    65
Goalw [Bex_def] "[| P(x);  x:A |] ==> ? x:A. P(x)";
paulson@5316
    66
by (Blast_tac 1);
clasohm@923
    67
qed "bexI";
clasohm@923
    68
paulson@6006
    69
(*The best argument order when there is only one x:A*)
paulson@6006
    70
Goalw [Bex_def] "[| x:A;  P(x) |] ==> ? x:A. P(x)";
paulson@6006
    71
by (Blast_tac 1);
paulson@6006
    72
qed "rev_bexI";
paulson@6006
    73
clasohm@923
    74
qed_goal "bexCI" Set.thy 
oheimb@5521
    75
   "[| ! x:A. ~P(x) ==> P(a);  a:A |] ==> ? x:A. P(x)" (fn prems =>
clasohm@923
    76
  [ (rtac classical 1),
clasohm@923
    77
    (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1))  ]);
clasohm@923
    78
paulson@5316
    79
val major::prems = Goalw [Bex_def]
clasohm@923
    80
    "[| ? x:A. P(x);  !!x. [| x:A; P(x) |] ==> Q  |] ==> Q";
clasohm@923
    81
by (rtac (major RS exE) 1);
clasohm@923
    82
by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1));
clasohm@923
    83
qed "bexE";
clasohm@923
    84
paulson@2499
    85
AddIs  [bexI];
paulson@2499
    86
AddSEs [bexE];
paulson@2499
    87
paulson@3420
    88
(*Trival rewrite rule*)
wenzelm@5069
    89
Goal "(! x:A. P) = ((? x. x:A) --> P)";
wenzelm@4089
    90
by (simp_tac (simpset() addsimps [Ball_def]) 1);
paulson@3420
    91
qed "ball_triv";
paulson@1816
    92
paulson@1882
    93
(*Dual form for existentials*)
wenzelm@5069
    94
Goal "(? x:A. P) = ((? x. x:A) & P)";
wenzelm@4089
    95
by (simp_tac (simpset() addsimps [Bex_def]) 1);
paulson@3420
    96
qed "bex_triv";
paulson@1882
    97
paulson@3420
    98
Addsimps [ball_triv, bex_triv];
clasohm@923
    99
clasohm@923
   100
(** Congruence rules **)
clasohm@923
   101
paulson@6291
   102
val prems = Goalw [Ball_def]
clasohm@923
   103
    "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
clasohm@923
   104
\    (! x:A. P(x)) = (! x:B. Q(x))";
paulson@6291
   105
by (asm_simp_tac (simpset() addsimps prems) 1);
clasohm@923
   106
qed "ball_cong";
clasohm@923
   107
paulson@6291
   108
val prems = Goalw [Bex_def]
clasohm@923
   109
    "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
clasohm@923
   110
\    (? x:A. P(x)) = (? x:B. Q(x))";
paulson@6291
   111
by (asm_simp_tac (simpset() addcongs [conj_cong] addsimps prems) 1);
clasohm@923
   112
qed "bex_cong";
clasohm@923
   113
paulson@6291
   114
Addcongs [ball_cong,bex_cong];
paulson@6291
   115
nipkow@1548
   116
section "Subsets";
clasohm@923
   117
paulson@5316
   118
val prems = Goalw [subset_def] "(!!x. x:A ==> x:B) ==> A <= B";
clasohm@923
   119
by (REPEAT (ares_tac (prems @ [ballI]) 1));
clasohm@923
   120
qed "subsetI";
clasohm@923
   121
paulson@5649
   122
(*Map the type ('a set => anything) to just 'a.
paulson@5649
   123
  For overloading constants whose first argument has type "'a set" *)
paulson@5649
   124
fun overload_1st_set s = Blast.overloaded (s, HOLogic.dest_setT o domain_type);
paulson@5649
   125
paulson@4059
   126
(*While (:) is not, its type must be kept
paulson@4059
   127
  for overloading of = to work.*)
paulson@4240
   128
Blast.overloaded ("op :", domain_type);
paulson@5649
   129
paulson@5649
   130
overload_1st_set "Ball";		(*need UNION, INTER also?*)
paulson@5649
   131
overload_1st_set "Bex";
paulson@4059
   132
paulson@4469
   133
(*Image: retain the type of the set being expressed*)
paulson@5336
   134
Blast.overloaded ("op ``", domain_type);
paulson@2881
   135
clasohm@923
   136
(*Rule in Modus Ponens style*)
paulson@5316
   137
Goalw [subset_def] "[| A <= B;  c:A |] ==> c:B";
paulson@5316
   138
by (Blast_tac 1);
clasohm@923
   139
qed "subsetD";
clasohm@923
   140
clasohm@923
   141
(*The same, with reversed premises for use with etac -- cf rev_mp*)
clasohm@923
   142
qed_goal "rev_subsetD" Set.thy "[| c:A;  A <= B |] ==> c:B"
clasohm@923
   143
 (fn prems=>  [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]);
clasohm@923
   144
paulson@1920
   145
(*Converts A<=B to x:A ==> x:B*)
paulson@1920
   146
fun impOfSubs th = th RSN (2, rev_subsetD);
paulson@1920
   147
paulson@1841
   148
qed_goal "contra_subsetD" Set.thy "!!c. [| A <= B; c ~: B |] ==> c ~: A"
paulson@1841
   149
 (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
paulson@1841
   150
paulson@1841
   151
qed_goal "rev_contra_subsetD" Set.thy "!!c. [| c ~: B;  A <= B |] ==> c ~: A"
paulson@1841
   152
 (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
paulson@1841
   153
clasohm@923
   154
(*Classical elimination rule*)
paulson@5316
   155
val major::prems = Goalw [subset_def] 
clasohm@923
   156
    "[| A <= B;  c~:A ==> P;  c:B ==> P |] ==> P";
clasohm@923
   157
by (rtac (major RS ballE) 1);
clasohm@923
   158
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   159
qed "subsetCE";
clasohm@923
   160
clasohm@923
   161
(*Takes assumptions A<=B; c:A and creates the assumption c:B *)
clasohm@923
   162
fun set_mp_tac i = etac subsetCE i  THEN  mp_tac i;
clasohm@923
   163
paulson@2499
   164
AddSIs [subsetI];
paulson@2499
   165
AddEs  [subsetD, subsetCE];
clasohm@923
   166
paulson@2499
   167
qed_goal "subset_refl" Set.thy "A <= (A::'a set)"
paulson@4059
   168
 (fn _=> [Fast_tac 1]);		(*Blast_tac would try order_refl and fail*)
paulson@2499
   169
paulson@5316
   170
Goal "[| A<=B;  B<=C |] ==> A<=(C::'a set)";
paulson@2891
   171
by (Blast_tac 1);
clasohm@923
   172
qed "subset_trans";
clasohm@923
   173
clasohm@923
   174
nipkow@1548
   175
section "Equality";
clasohm@923
   176
clasohm@923
   177
(*Anti-symmetry of the subset relation*)
paulson@5316
   178
Goal "[| A <= B;  B <= A |] ==> A = (B::'a set)";
paulson@5318
   179
by (rtac set_ext 1);
paulson@5316
   180
by (blast_tac (claset() addIs [subsetD]) 1);
clasohm@923
   181
qed "subset_antisym";
clasohm@923
   182
val equalityI = subset_antisym;
clasohm@923
   183
berghofe@1762
   184
AddSIs [equalityI];
berghofe@1762
   185
clasohm@923
   186
(* Equality rules from ZF set theory -- are they appropriate here? *)
paulson@5316
   187
Goal "A = B ==> A<=(B::'a set)";
paulson@5316
   188
by (etac ssubst 1);
clasohm@923
   189
by (rtac subset_refl 1);
clasohm@923
   190
qed "equalityD1";
clasohm@923
   191
paulson@5316
   192
Goal "A = B ==> B<=(A::'a set)";
paulson@5316
   193
by (etac ssubst 1);
clasohm@923
   194
by (rtac subset_refl 1);
clasohm@923
   195
qed "equalityD2";
clasohm@923
   196
paulson@5316
   197
val prems = Goal
clasohm@923
   198
    "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";
clasohm@923
   199
by (resolve_tac prems 1);
clasohm@923
   200
by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
clasohm@923
   201
qed "equalityE";
clasohm@923
   202
paulson@5316
   203
val major::prems = Goal
clasohm@923
   204
    "[| A = B;  [| c:A; c:B |] ==> P;  [| c~:A; c~:B |] ==> P |]  ==>  P";
clasohm@923
   205
by (rtac (major RS equalityE) 1);
clasohm@923
   206
by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
clasohm@923
   207
qed "equalityCE";
clasohm@923
   208
clasohm@923
   209
(*Lemma for creating induction formulae -- for "pattern matching" on p
clasohm@923
   210
  To make the induction hypotheses usable, apply "spec" or "bspec" to
clasohm@923
   211
  put universal quantifiers over the free variables in p. *)
paulson@5316
   212
val prems = Goal 
clasohm@923
   213
    "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
clasohm@923
   214
by (rtac mp 1);
clasohm@923
   215
by (REPEAT (resolve_tac (refl::prems) 1));
clasohm@923
   216
qed "setup_induction";
clasohm@923
   217
clasohm@923
   218
paulson@4159
   219
section "The universal set -- UNIV";
paulson@4159
   220
paulson@4159
   221
qed_goalw "UNIV_I" Set.thy [UNIV_def] "x : UNIV"
paulson@4159
   222
  (fn _ => [rtac CollectI 1, rtac TrueI 1]);
paulson@4159
   223
paulson@4434
   224
Addsimps [UNIV_I];
paulson@4434
   225
AddIs    [UNIV_I];  (*unsafe makes it less likely to cause problems*)
paulson@4159
   226
paulson@4159
   227
qed_goal "subset_UNIV" Set.thy "A <= UNIV"
paulson@4159
   228
  (fn _ => [rtac subsetI 1, rtac UNIV_I 1]);
paulson@4159
   229
paulson@4159
   230
(** Eta-contracting these two rules (to remove P) causes them to be ignored
paulson@4159
   231
    because of their interaction with congruence rules. **)
paulson@4159
   232
wenzelm@5069
   233
Goalw [Ball_def] "Ball UNIV P = All P";
paulson@4159
   234
by (Simp_tac 1);
paulson@4159
   235
qed "ball_UNIV";
paulson@4159
   236
wenzelm@5069
   237
Goalw [Bex_def] "Bex UNIV P = Ex P";
paulson@4159
   238
by (Simp_tac 1);
paulson@4159
   239
qed "bex_UNIV";
paulson@4159
   240
Addsimps [ball_UNIV, bex_UNIV];
paulson@4159
   241
paulson@4159
   242
paulson@2858
   243
section "The empty set -- {}";
paulson@2858
   244
paulson@2858
   245
qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False"
paulson@2891
   246
 (fn _ => [ (Blast_tac 1) ]);
paulson@2858
   247
paulson@2858
   248
Addsimps [empty_iff];
paulson@2858
   249
paulson@2858
   250
qed_goal "emptyE" Set.thy "!!a. a:{} ==> P"
paulson@2858
   251
 (fn _ => [Full_simp_tac 1]);
paulson@2858
   252
paulson@2858
   253
AddSEs [emptyE];
paulson@2858
   254
paulson@2858
   255
qed_goal "empty_subsetI" Set.thy "{} <= A"
paulson@2891
   256
 (fn _ => [ (Blast_tac 1) ]);
paulson@2858
   257
paulson@5256
   258
(*One effect is to delete the ASSUMPTION {} <= A*)
paulson@5256
   259
AddIffs [empty_subsetI];
paulson@5256
   260
paulson@2858
   261
qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}"
paulson@2858
   262
 (fn [prem]=>
wenzelm@4089
   263
  [ (blast_tac (claset() addIs [prem RS FalseE]) 1) ]);
paulson@2858
   264
paulson@5256
   265
(*Use for reasoning about disjointness: A Int B = {} *)
paulson@5450
   266
qed_goal "equals0D" Set.thy "!!a. A={} ==> a ~: A"
paulson@2891
   267
 (fn _ => [ (Blast_tac 1) ]);
paulson@2858
   268
paulson@5450
   269
AddDs [equals0D, sym RS equals0D];
paulson@5256
   270
wenzelm@5069
   271
Goalw [Ball_def] "Ball {} P = True";
paulson@4159
   272
by (Simp_tac 1);
paulson@4159
   273
qed "ball_empty";
paulson@4159
   274
wenzelm@5069
   275
Goalw [Bex_def] "Bex {} P = False";
paulson@4159
   276
by (Simp_tac 1);
paulson@4159
   277
qed "bex_empty";
paulson@4159
   278
Addsimps [ball_empty, bex_empty];
paulson@4159
   279
wenzelm@5069
   280
Goal "UNIV ~= {}";
paulson@4159
   281
by (blast_tac (claset() addEs [equalityE]) 1);
paulson@4159
   282
qed "UNIV_not_empty";
paulson@4159
   283
AddIffs [UNIV_not_empty];
paulson@4159
   284
paulson@4159
   285
paulson@2858
   286
paulson@2858
   287
section "The Powerset operator -- Pow";
paulson@2858
   288
paulson@2858
   289
qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)"
paulson@2858
   290
 (fn _ => [ (Asm_simp_tac 1) ]);
paulson@2858
   291
paulson@2858
   292
AddIffs [Pow_iff]; 
paulson@2858
   293
paulson@2858
   294
qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
paulson@2858
   295
 (fn _ => [ (etac CollectI 1) ]);
paulson@2858
   296
paulson@2858
   297
qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B)  ==>  A<=B"
paulson@2858
   298
 (fn _=> [ (etac CollectD 1) ]);
paulson@2858
   299
paulson@2858
   300
val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)
paulson@2858
   301
val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)
paulson@2858
   302
paulson@2858
   303
paulson@5931
   304
section "Set complement";
clasohm@923
   305
paulson@5490
   306
qed_goalw "Compl_iff" Set.thy [Compl_def] "(c : -A) = (c~:A)"
paulson@2891
   307
 (fn _ => [ (Blast_tac 1) ]);
paulson@2499
   308
paulson@2499
   309
Addsimps [Compl_iff];
paulson@2499
   310
paulson@5490
   311
val prems = Goalw [Compl_def] "[| c:A ==> False |] ==> c : -A";
clasohm@923
   312
by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
clasohm@923
   313
qed "ComplI";
clasohm@923
   314
clasohm@923
   315
(*This form, with negated conclusion, works well with the Classical prover.
clasohm@923
   316
  Negated assumptions behave like formulae on the right side of the notional
clasohm@923
   317
  turnstile...*)
paulson@5490
   318
Goalw [Compl_def] "c : -A ==> c~:A";
paulson@5316
   319
by (etac CollectD 1);
clasohm@923
   320
qed "ComplD";
clasohm@923
   321
clasohm@923
   322
val ComplE = make_elim ComplD;
clasohm@923
   323
paulson@2499
   324
AddSIs [ComplI];
paulson@2499
   325
AddSEs [ComplE];
paulson@1640
   326
clasohm@923
   327
nipkow@1548
   328
section "Binary union -- Un";
clasohm@923
   329
paulson@2499
   330
qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)"
paulson@2891
   331
 (fn _ => [ Blast_tac 1 ]);
paulson@2499
   332
paulson@2499
   333
Addsimps [Un_iff];
paulson@2499
   334
paulson@5143
   335
Goal "c:A ==> c : A Un B";
paulson@2499
   336
by (Asm_simp_tac 1);
clasohm@923
   337
qed "UnI1";
clasohm@923
   338
paulson@5143
   339
Goal "c:B ==> c : A Un B";
paulson@2499
   340
by (Asm_simp_tac 1);
clasohm@923
   341
qed "UnI2";
clasohm@923
   342
clasohm@923
   343
(*Classical introduction rule: no commitment to A vs B*)
clasohm@923
   344
qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B"
clasohm@923
   345
 (fn prems=>
paulson@2499
   346
  [ (Simp_tac 1),
paulson@2499
   347
    (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
clasohm@923
   348
paulson@5316
   349
val major::prems = Goalw [Un_def]
clasohm@923
   350
    "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
clasohm@923
   351
by (rtac (major RS CollectD RS disjE) 1);
clasohm@923
   352
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   353
qed "UnE";
clasohm@923
   354
paulson@2499
   355
AddSIs [UnCI];
paulson@2499
   356
AddSEs [UnE];
paulson@1640
   357
clasohm@923
   358
nipkow@1548
   359
section "Binary intersection -- Int";
clasohm@923
   360
paulson@2499
   361
qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)"
paulson@2891
   362
 (fn _ => [ (Blast_tac 1) ]);
paulson@2499
   363
paulson@2499
   364
Addsimps [Int_iff];
paulson@2499
   365
paulson@5143
   366
Goal "[| c:A;  c:B |] ==> c : A Int B";
paulson@2499
   367
by (Asm_simp_tac 1);
clasohm@923
   368
qed "IntI";
clasohm@923
   369
paulson@5143
   370
Goal "c : A Int B ==> c:A";
paulson@2499
   371
by (Asm_full_simp_tac 1);
clasohm@923
   372
qed "IntD1";
clasohm@923
   373
paulson@5143
   374
Goal "c : A Int B ==> c:B";
paulson@2499
   375
by (Asm_full_simp_tac 1);
clasohm@923
   376
qed "IntD2";
clasohm@923
   377
paulson@5316
   378
val [major,minor] = Goal
clasohm@923
   379
    "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
clasohm@923
   380
by (rtac minor 1);
clasohm@923
   381
by (rtac (major RS IntD1) 1);
clasohm@923
   382
by (rtac (major RS IntD2) 1);
clasohm@923
   383
qed "IntE";
clasohm@923
   384
paulson@2499
   385
AddSIs [IntI];
paulson@2499
   386
AddSEs [IntE];
clasohm@923
   387
nipkow@1548
   388
section "Set difference";
clasohm@923
   389
paulson@2499
   390
qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)"
paulson@2891
   391
 (fn _ => [ (Blast_tac 1) ]);
clasohm@923
   392
paulson@2499
   393
Addsimps [Diff_iff];
paulson@2499
   394
paulson@2499
   395
qed_goal "DiffI" Set.thy "!!c. [| c : A;  c ~: B |] ==> c : A - B"
paulson@2499
   396
 (fn _=> [ Asm_simp_tac 1 ]);
clasohm@923
   397
paulson@2499
   398
qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A"
paulson@2499
   399
 (fn _=> [ (Asm_full_simp_tac 1) ]);
clasohm@923
   400
paulson@2499
   401
qed_goal "DiffD2" Set.thy "!!c. [| c : A - B;  c : B |] ==> P"
paulson@2499
   402
 (fn _=> [ (Asm_full_simp_tac 1) ]);
paulson@2499
   403
paulson@2499
   404
qed_goal "DiffE" Set.thy "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P"
clasohm@923
   405
 (fn prems=>
clasohm@923
   406
  [ (resolve_tac prems 1),
clasohm@923
   407
    (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]);
clasohm@923
   408
paulson@2499
   409
AddSIs [DiffI];
paulson@2499
   410
AddSEs [DiffE];
clasohm@923
   411
clasohm@923
   412
nipkow@1548
   413
section "Augmenting a set -- insert";
clasohm@923
   414
paulson@2499
   415
qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)"
paulson@2891
   416
 (fn _ => [Blast_tac 1]);
paulson@2499
   417
paulson@2499
   418
Addsimps [insert_iff];
clasohm@923
   419
paulson@2499
   420
qed_goal "insertI1" Set.thy "a : insert a B"
paulson@2499
   421
 (fn _ => [Simp_tac 1]);
paulson@2499
   422
paulson@2499
   423
qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B"
paulson@2499
   424
 (fn _=> [Asm_simp_tac 1]);
clasohm@923
   425
clasohm@923
   426
qed_goalw "insertE" Set.thy [insert_def]
clasohm@923
   427
    "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P"
clasohm@923
   428
 (fn major::prems=>
clasohm@923
   429
  [ (rtac (major RS UnE) 1),
clasohm@923
   430
    (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
clasohm@923
   431
clasohm@923
   432
(*Classical introduction rule*)
clasohm@923
   433
qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
paulson@2499
   434
 (fn prems=>
paulson@2499
   435
  [ (Simp_tac 1),
paulson@2499
   436
    (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
paulson@2499
   437
paulson@2499
   438
AddSIs [insertCI]; 
paulson@2499
   439
AddSEs [insertE];
clasohm@923
   440
nipkow@1548
   441
section "Singletons, using insert";
clasohm@923
   442
clasohm@923
   443
qed_goal "singletonI" Set.thy "a : {a}"
clasohm@923
   444
 (fn _=> [ (rtac insertI1 1) ]);
clasohm@923
   445
paulson@5143
   446
Goal "b : {a} ==> b=a";
paulson@2891
   447
by (Blast_tac 1);
clasohm@923
   448
qed "singletonD";
clasohm@923
   449
oheimb@1776
   450
bind_thm ("singletonE", make_elim singletonD);
oheimb@1776
   451
paulson@2499
   452
qed_goal "singleton_iff" thy "(b : {a}) = (b=a)" 
paulson@2891
   453
(fn _ => [Blast_tac 1]);
clasohm@923
   454
paulson@5143
   455
Goal "{a}={b} ==> a=b";
wenzelm@4089
   456
by (blast_tac (claset() addEs [equalityE]) 1);
clasohm@923
   457
qed "singleton_inject";
clasohm@923
   458
paulson@2858
   459
(*Redundant? But unlike insertCI, it proves the subgoal immediately!*)
paulson@2858
   460
AddSIs [singletonI];   
paulson@2499
   461
AddSDs [singleton_inject];
paulson@3718
   462
AddSEs [singletonE];
paulson@2499
   463
wenzelm@5069
   464
Goal "{x. x=a} = {a}";
wenzelm@4423
   465
by (Blast_tac 1);
nipkow@3582
   466
qed "singleton_conv";
nipkow@3582
   467
Addsimps [singleton_conv];
nipkow@1531
   468
nipkow@5600
   469
Goal "{x. a=x} = {a}";
paulson@6301
   470
by (Blast_tac 1);
nipkow@5600
   471
qed "singleton_conv2";
nipkow@5600
   472
Addsimps [singleton_conv2];
nipkow@5600
   473
nipkow@1531
   474
nipkow@1548
   475
section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
clasohm@923
   476
wenzelm@5069
   477
Goalw [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
paulson@2891
   478
by (Blast_tac 1);
paulson@2499
   479
qed "UN_iff";
paulson@2499
   480
paulson@2499
   481
Addsimps [UN_iff];
paulson@2499
   482
clasohm@923
   483
(*The order of the premises presupposes that A is rigid; b may be flexible*)
paulson@5143
   484
Goal "[| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
paulson@4477
   485
by Auto_tac;
clasohm@923
   486
qed "UN_I";
clasohm@923
   487
paulson@5316
   488
val major::prems = Goalw [UNION_def]
clasohm@923
   489
    "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
clasohm@923
   490
by (rtac (major RS CollectD RS bexE) 1);
clasohm@923
   491
by (REPEAT (ares_tac prems 1));
clasohm@923
   492
qed "UN_E";
clasohm@923
   493
paulson@2499
   494
AddIs  [UN_I];
paulson@2499
   495
AddSEs [UN_E];
paulson@2499
   496
paulson@6291
   497
val prems = Goalw [UNION_def]
clasohm@923
   498
    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
clasohm@923
   499
\    (UN x:A. C(x)) = (UN x:B. D(x))";
paulson@6291
   500
by (asm_simp_tac (simpset() addsimps prems) 1);
clasohm@923
   501
qed "UN_cong";
clasohm@923
   502
clasohm@923
   503
nipkow@1548
   504
section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
clasohm@923
   505
wenzelm@5069
   506
Goalw [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
paulson@4477
   507
by Auto_tac;
paulson@2499
   508
qed "INT_iff";
paulson@2499
   509
paulson@2499
   510
Addsimps [INT_iff];
paulson@2499
   511
paulson@5316
   512
val prems = Goalw [INTER_def]
clasohm@923
   513
    "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
clasohm@923
   514
by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
clasohm@923
   515
qed "INT_I";
clasohm@923
   516
paulson@5143
   517
Goal "[| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
paulson@4477
   518
by Auto_tac;
clasohm@923
   519
qed "INT_D";
clasohm@923
   520
clasohm@923
   521
(*"Classical" elimination -- by the Excluded Middle on a:A *)
paulson@5316
   522
val major::prems = Goalw [INTER_def]
clasohm@923
   523
    "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
clasohm@923
   524
by (rtac (major RS CollectD RS ballE) 1);
clasohm@923
   525
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   526
qed "INT_E";
clasohm@923
   527
paulson@2499
   528
AddSIs [INT_I];
paulson@2499
   529
AddEs  [INT_D, INT_E];
paulson@2499
   530
paulson@6291
   531
val prems = Goalw [INTER_def]
clasohm@923
   532
    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
clasohm@923
   533
\    (INT x:A. C(x)) = (INT x:B. D(x))";
paulson@6291
   534
by (asm_simp_tac (simpset() addsimps prems) 1);
clasohm@923
   535
qed "INT_cong";
clasohm@923
   536
clasohm@923
   537
nipkow@1548
   538
section "Union";
clasohm@923
   539
wenzelm@5069
   540
Goalw [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
paulson@2891
   541
by (Blast_tac 1);
paulson@2499
   542
qed "Union_iff";
paulson@2499
   543
paulson@2499
   544
Addsimps [Union_iff];
paulson@2499
   545
clasohm@923
   546
(*The order of the premises presupposes that C is rigid; A may be flexible*)
paulson@5143
   547
Goal "[| X:C;  A:X |] ==> A : Union(C)";
paulson@4477
   548
by Auto_tac;
clasohm@923
   549
qed "UnionI";
clasohm@923
   550
paulson@5316
   551
val major::prems = Goalw [Union_def]
clasohm@923
   552
    "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
clasohm@923
   553
by (rtac (major RS UN_E) 1);
clasohm@923
   554
by (REPEAT (ares_tac prems 1));
clasohm@923
   555
qed "UnionE";
clasohm@923
   556
paulson@2499
   557
AddIs  [UnionI];
paulson@2499
   558
AddSEs [UnionE];
paulson@2499
   559
paulson@2499
   560
nipkow@1548
   561
section "Inter";
clasohm@923
   562
wenzelm@5069
   563
Goalw [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
paulson@2891
   564
by (Blast_tac 1);
paulson@2499
   565
qed "Inter_iff";
paulson@2499
   566
paulson@2499
   567
Addsimps [Inter_iff];
paulson@2499
   568
paulson@5316
   569
val prems = Goalw [Inter_def]
clasohm@923
   570
    "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
clasohm@923
   571
by (REPEAT (ares_tac ([INT_I] @ prems) 1));
clasohm@923
   572
qed "InterI";
clasohm@923
   573
clasohm@923
   574
(*A "destruct" rule -- every X in C contains A as an element, but
clasohm@923
   575
  A:X can hold when X:C does not!  This rule is analogous to "spec". *)
paulson@5143
   576
Goal "[| A : Inter(C);  X:C |] ==> A:X";
paulson@4477
   577
by Auto_tac;
clasohm@923
   578
qed "InterD";
clasohm@923
   579
clasohm@923
   580
(*"Classical" elimination rule -- does not require proving X:C *)
paulson@5316
   581
val major::prems = Goalw [Inter_def]
paulson@2721
   582
    "[| A : Inter(C);  X~:C ==> R;  A:X ==> R |] ==> R";
clasohm@923
   583
by (rtac (major RS INT_E) 1);
clasohm@923
   584
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   585
qed "InterE";
clasohm@923
   586
paulson@2499
   587
AddSIs [InterI];
paulson@2499
   588
AddEs  [InterD, InterE];
paulson@2499
   589
paulson@2499
   590
nipkow@2912
   591
(*** Image of a set under a function ***)
nipkow@2912
   592
nipkow@2912
   593
(*Frequently b does not have the syntactic form of f(x).*)
paulson@5316
   594
Goalw [image_def] "[| b=f(x);  x:A |] ==> b : f``A";
paulson@5316
   595
by (Blast_tac 1);
nipkow@2912
   596
qed "image_eqI";
nipkow@3909
   597
Addsimps [image_eqI];
nipkow@2912
   598
nipkow@2912
   599
bind_thm ("imageI", refl RS image_eqI);
nipkow@2912
   600
nipkow@2912
   601
(*The eta-expansion gives variable-name preservation.*)
paulson@5316
   602
val major::prems = Goalw [image_def]
wenzelm@3842
   603
    "[| b : (%x. f(x))``A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P"; 
nipkow@2912
   604
by (rtac (major RS CollectD RS bexE) 1);
nipkow@2912
   605
by (REPEAT (ares_tac prems 1));
nipkow@2912
   606
qed "imageE";
nipkow@2912
   607
nipkow@2912
   608
AddIs  [image_eqI];
nipkow@2912
   609
AddSEs [imageE]; 
nipkow@2912
   610
wenzelm@5069
   611
Goal "f``(A Un B) = f``A Un f``B";
paulson@2935
   612
by (Blast_tac 1);
nipkow@2912
   613
qed "image_Un";
nipkow@2912
   614
wenzelm@5069
   615
Goal "(z : f``A) = (EX x:A. z = f x)";
paulson@3960
   616
by (Blast_tac 1);
paulson@3960
   617
qed "image_iff";
paulson@3960
   618
paulson@4523
   619
(*This rewrite rule would confuse users if made default.*)
wenzelm@5069
   620
Goal "(f``A <= B) = (ALL x:A. f(x): B)";
paulson@4523
   621
by (Blast_tac 1);
paulson@4523
   622
qed "image_subset_iff";
paulson@4523
   623
paulson@4523
   624
(*Replaces the three steps subsetI, imageE, hyp_subst_tac, but breaks too
paulson@4523
   625
  many existing proofs.*)
paulson@5316
   626
val prems = Goal "(!!x. x:A ==> f(x) : B) ==> f``A <= B";
paulson@4510
   627
by (blast_tac (claset() addIs prems) 1);
paulson@4510
   628
qed "image_subsetI";
paulson@4510
   629
nipkow@2912
   630
nipkow@2912
   631
(*** Range of a function -- just a translation for image! ***)
nipkow@2912
   632
paulson@5143
   633
Goal "b=f(x) ==> b : range(f)";
nipkow@2912
   634
by (EVERY1 [etac image_eqI, rtac UNIV_I]);
nipkow@2912
   635
bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI));
nipkow@2912
   636
nipkow@2912
   637
bind_thm ("rangeI", UNIV_I RS imageI);
nipkow@2912
   638
paulson@5316
   639
val [major,minor] = Goal 
wenzelm@3842
   640
    "[| b : range(%x. f(x));  !!x. b=f(x) ==> P |] ==> P"; 
nipkow@2912
   641
by (rtac (major RS imageE) 1);
nipkow@2912
   642
by (etac minor 1);
nipkow@2912
   643
qed "rangeE";
nipkow@2912
   644
oheimb@1776
   645
oheimb@1776
   646
(*** Set reasoning tools ***)
oheimb@1776
   647
oheimb@1776
   648
paulson@3912
   649
(** Rewrite rules for boolean case-splitting: faster than 
nipkow@4830
   650
	addsplits[split_if]
paulson@3912
   651
**)
paulson@3912
   652
nipkow@4830
   653
bind_thm ("split_if_eq1", read_instantiate [("P", "%x. x = ?b")] split_if);
nipkow@4830
   654
bind_thm ("split_if_eq2", read_instantiate [("P", "%x. ?a = x")] split_if);
paulson@3912
   655
paulson@5237
   656
(*Split ifs on either side of the membership relation.
paulson@5237
   657
	Not for Addsimps -- can cause goals to blow up!*)
nipkow@4830
   658
bind_thm ("split_if_mem1", 
nipkow@4830
   659
    read_instantiate_sg (sign_of Set.thy) [("P", "%x. x : ?b")] split_if);
nipkow@4830
   660
bind_thm ("split_if_mem2", 
nipkow@4830
   661
    read_instantiate_sg (sign_of Set.thy) [("P", "%x. ?a : x")] split_if);
paulson@3912
   662
nipkow@4830
   663
val split_ifs = [if_bool_eq_conj, split_if_eq1, split_if_eq2,
nipkow@4830
   664
		  split_if_mem1, split_if_mem2];
paulson@3912
   665
paulson@3912
   666
wenzelm@4089
   667
(*Each of these has ALREADY been added to simpset() above.*)
paulson@2024
   668
val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff, 
paulson@4159
   669
                 mem_Collect_eq, UN_iff, Union_iff, INT_iff, Inter_iff];
oheimb@1776
   670
oheimb@1776
   671
val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs;
oheimb@1776
   672
paulson@6291
   673
simpset_ref() := simpset() setmksimps (mksimps mksimps_pairs);
nipkow@3222
   674
paulson@5256
   675
Addsimps[subset_UNIV, subset_refl];
nipkow@3222
   676
nipkow@3222
   677
nipkow@3222
   678
(*** < ***)
nipkow@3222
   679
wenzelm@5069
   680
Goalw [psubset_def] "!!A::'a set. [| A <= B; A ~= B |] ==> A<B";
nipkow@3222
   681
by (Blast_tac 1);
nipkow@3222
   682
qed "psubsetI";
nipkow@3222
   683
paulson@5148
   684
Goalw [psubset_def] "A < insert x B ==> (x ~: A) & A<=B | x:A & A-{x}<B";
paulson@4477
   685
by Auto_tac;
nipkow@3222
   686
qed "psubset_insertD";
paulson@4059
   687
paulson@4059
   688
bind_thm ("psubset_eq", psubset_def RS meta_eq_to_obj_eq);