src/HOL/Set.ML
author paulson
Tue Jul 01 10:37:03 1997 +0200 (1997-07-01)
changeset 3469 61d927bd57ec
parent 3420 02dc9c5b035f
child 3582 b87c86b6c291
permissions -rw-r--r--
Now Collect_mem_eq is a default simprule (how could it have ever been omitted?
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@2499
    16
goal Set.thy "!!a. P(a) ==> a : {x.P(x)}";
paulson@2499
    17
by (Asm_simp_tac 1);
clasohm@923
    18
qed "CollectI";
clasohm@923
    19
paulson@2499
    20
val prems = goal Set.thy "!!a. a : {x.P(x)} ==> P(a)";
paulson@2499
    21
by (Asm_full_simp_tac 1);
clasohm@923
    22
qed "CollectD";
clasohm@923
    23
clasohm@923
    24
val [prem] = goal Set.thy "[| !!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
clasohm@923
    30
val [prem] = goal Set.thy "[| !!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
clasohm@923
    42
val prems = goalw Set.thy [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
clasohm@923
    47
val [major,minor] = goalw Set.thy [Ball_def]
clasohm@923
    48
    "[| ! x:A. P(x);  x:A |] ==> P(x)";
clasohm@923
    49
by (rtac (minor RS (major RS spec RS mp)) 1);
clasohm@923
    50
qed "bspec";
clasohm@923
    51
clasohm@923
    52
val major::prems = goalw Set.thy [Ball_def]
clasohm@923
    53
    "[| ! x:A. P(x);  P(x) ==> Q;  x~:A ==> Q |] ==> Q";
clasohm@923
    54
by (rtac (major RS spec RS impCE) 1);
clasohm@923
    55
by (REPEAT (eresolve_tac prems 1));
clasohm@923
    56
qed "ballE";
clasohm@923
    57
clasohm@923
    58
(*Takes assumptions ! x:A.P(x) and a:A; creates assumption P(a)*)
clasohm@923
    59
fun ball_tac i = etac ballE i THEN contr_tac (i+1);
clasohm@923
    60
paulson@2499
    61
AddSIs [ballI];
paulson@2499
    62
AddEs  [ballE];
paulson@2499
    63
clasohm@923
    64
val prems = goalw Set.thy [Bex_def]
clasohm@923
    65
    "[| P(x);  x:A |] ==> ? x:A. P(x)";
clasohm@923
    66
by (REPEAT (ares_tac (prems @ [exI,conjI]) 1));
clasohm@923
    67
qed "bexI";
clasohm@923
    68
clasohm@923
    69
qed_goal "bexCI" Set.thy 
clasohm@923
    70
   "[| ! x:A. ~P(x) ==> P(a);  a:A |] ==> ? x:A.P(x)"
clasohm@923
    71
 (fn prems=>
clasohm@923
    72
  [ (rtac classical 1),
clasohm@923
    73
    (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1))  ]);
clasohm@923
    74
clasohm@923
    75
val major::prems = goalw Set.thy [Bex_def]
clasohm@923
    76
    "[| ? x:A. P(x);  !!x. [| x:A; P(x) |] ==> Q  |] ==> Q";
clasohm@923
    77
by (rtac (major RS exE) 1);
clasohm@923
    78
by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1));
clasohm@923
    79
qed "bexE";
clasohm@923
    80
paulson@2499
    81
AddIs  [bexI];
paulson@2499
    82
AddSEs [bexE];
paulson@2499
    83
paulson@3420
    84
(*Trival rewrite rule*)
paulson@3420
    85
goal Set.thy "(! x:A.P) = ((? x. x:A) --> P)";
paulson@3420
    86
by (simp_tac (!simpset addsimps [Ball_def]) 1);
paulson@3420
    87
qed "ball_triv";
paulson@1816
    88
paulson@1882
    89
(*Dual form for existentials*)
paulson@3420
    90
goal Set.thy "(? x:A.P) = ((? x. x:A) & P)";
paulson@3420
    91
by (simp_tac (!simpset addsimps [Bex_def]) 1);
paulson@3420
    92
qed "bex_triv";
paulson@1882
    93
paulson@3420
    94
Addsimps [ball_triv, bex_triv];
clasohm@923
    95
clasohm@923
    96
(** Congruence rules **)
clasohm@923
    97
clasohm@923
    98
val prems = goal Set.thy
clasohm@923
    99
    "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
clasohm@923
   100
\    (! x:A. P(x)) = (! x:B. Q(x))";
clasohm@923
   101
by (resolve_tac (prems RL [ssubst]) 1);
clasohm@923
   102
by (REPEAT (ares_tac [ballI,iffI] 1
clasohm@923
   103
     ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1));
clasohm@923
   104
qed "ball_cong";
clasohm@923
   105
clasohm@923
   106
val prems = goal Set.thy
clasohm@923
   107
    "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
clasohm@923
   108
\    (? x:A. P(x)) = (? x:B. Q(x))";
clasohm@923
   109
by (resolve_tac (prems RL [ssubst]) 1);
clasohm@923
   110
by (REPEAT (etac bexE 1
clasohm@923
   111
     ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1));
clasohm@923
   112
qed "bex_cong";
clasohm@923
   113
nipkow@1548
   114
section "Subsets";
clasohm@923
   115
clasohm@923
   116
val prems = goalw Set.thy [subset_def] "(!!x.x:A ==> x:B) ==> A <= B";
clasohm@923
   117
by (REPEAT (ares_tac (prems @ [ballI]) 1));
clasohm@923
   118
qed "subsetI";
clasohm@923
   119
paulson@2881
   120
Blast.declConsts (["op <="], [subsetI]);	(*overloading of <=*)
paulson@2881
   121
clasohm@923
   122
(*Rule in Modus Ponens style*)
clasohm@923
   123
val major::prems = goalw Set.thy [subset_def] "[| A <= B;  c:A |] ==> c:B";
clasohm@923
   124
by (rtac (major RS bspec) 1);
clasohm@923
   125
by (resolve_tac prems 1);
clasohm@923
   126
qed "subsetD";
clasohm@923
   127
clasohm@923
   128
(*The same, with reversed premises for use with etac -- cf rev_mp*)
clasohm@923
   129
qed_goal "rev_subsetD" Set.thy "[| c:A;  A <= B |] ==> c:B"
clasohm@923
   130
 (fn prems=>  [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]);
clasohm@923
   131
paulson@1920
   132
(*Converts A<=B to x:A ==> x:B*)
paulson@1920
   133
fun impOfSubs th = th RSN (2, rev_subsetD);
paulson@1920
   134
paulson@1841
   135
qed_goal "contra_subsetD" Set.thy "!!c. [| A <= B; c ~: B |] ==> c ~: A"
paulson@1841
   136
 (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
paulson@1841
   137
paulson@1841
   138
qed_goal "rev_contra_subsetD" Set.thy "!!c. [| c ~: B;  A <= B |] ==> c ~: A"
paulson@1841
   139
 (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
paulson@1841
   140
clasohm@923
   141
(*Classical elimination rule*)
clasohm@923
   142
val major::prems = goalw Set.thy [subset_def] 
clasohm@923
   143
    "[| A <= B;  c~:A ==> P;  c:B ==> P |] ==> P";
clasohm@923
   144
by (rtac (major RS ballE) 1);
clasohm@923
   145
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   146
qed "subsetCE";
clasohm@923
   147
clasohm@923
   148
(*Takes assumptions A<=B; c:A and creates the assumption c:B *)
clasohm@923
   149
fun set_mp_tac i = etac subsetCE i  THEN  mp_tac i;
clasohm@923
   150
paulson@2499
   151
AddSIs [subsetI];
paulson@2499
   152
AddEs  [subsetD, subsetCE];
clasohm@923
   153
paulson@2499
   154
qed_goal "subset_refl" Set.thy "A <= (A::'a set)"
paulson@2891
   155
 (fn _=> [Blast_tac 1]);
paulson@2499
   156
paulson@2499
   157
val prems = goal Set.thy "!!B. [| A<=B;  B<=C |] ==> A<=(C::'a set)";
paulson@2891
   158
by (Blast_tac 1);
clasohm@923
   159
qed "subset_trans";
clasohm@923
   160
clasohm@923
   161
nipkow@1548
   162
section "Equality";
clasohm@923
   163
clasohm@923
   164
(*Anti-symmetry of the subset relation*)
clasohm@923
   165
val prems = goal Set.thy "[| A <= B;  B <= A |] ==> A = (B::'a set)";
clasohm@923
   166
by (rtac (iffI RS set_ext) 1);
clasohm@923
   167
by (REPEAT (ares_tac (prems RL [subsetD]) 1));
clasohm@923
   168
qed "subset_antisym";
clasohm@923
   169
val equalityI = subset_antisym;
clasohm@923
   170
paulson@2881
   171
Blast.declConsts (["op ="], [equalityI]);	(*overloading of equality*)
berghofe@1762
   172
AddSIs [equalityI];
berghofe@1762
   173
clasohm@923
   174
(* Equality rules from ZF set theory -- are they appropriate here? *)
clasohm@923
   175
val prems = goal Set.thy "A = B ==> A<=(B::'a set)";
clasohm@923
   176
by (resolve_tac (prems RL [subst]) 1);
clasohm@923
   177
by (rtac subset_refl 1);
clasohm@923
   178
qed "equalityD1";
clasohm@923
   179
clasohm@923
   180
val prems = goal Set.thy "A = B ==> B<=(A::'a set)";
clasohm@923
   181
by (resolve_tac (prems RL [subst]) 1);
clasohm@923
   182
by (rtac subset_refl 1);
clasohm@923
   183
qed "equalityD2";
clasohm@923
   184
clasohm@923
   185
val prems = goal Set.thy
clasohm@923
   186
    "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";
clasohm@923
   187
by (resolve_tac prems 1);
clasohm@923
   188
by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
clasohm@923
   189
qed "equalityE";
clasohm@923
   190
clasohm@923
   191
val major::prems = goal Set.thy
clasohm@923
   192
    "[| A = B;  [| c:A; c:B |] ==> P;  [| c~:A; c~:B |] ==> P |]  ==>  P";
clasohm@923
   193
by (rtac (major RS equalityE) 1);
clasohm@923
   194
by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
clasohm@923
   195
qed "equalityCE";
clasohm@923
   196
clasohm@923
   197
(*Lemma for creating induction formulae -- for "pattern matching" on p
clasohm@923
   198
  To make the induction hypotheses usable, apply "spec" or "bspec" to
clasohm@923
   199
  put universal quantifiers over the free variables in p. *)
clasohm@923
   200
val prems = goal Set.thy 
clasohm@923
   201
    "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
clasohm@923
   202
by (rtac mp 1);
clasohm@923
   203
by (REPEAT (resolve_tac (refl::prems) 1));
clasohm@923
   204
qed "setup_induction";
clasohm@923
   205
clasohm@923
   206
paulson@2858
   207
section "The empty set -- {}";
paulson@2858
   208
paulson@2858
   209
qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False"
paulson@2891
   210
 (fn _ => [ (Blast_tac 1) ]);
paulson@2858
   211
paulson@2858
   212
Addsimps [empty_iff];
paulson@2858
   213
paulson@2858
   214
qed_goal "emptyE" Set.thy "!!a. a:{} ==> P"
paulson@2858
   215
 (fn _ => [Full_simp_tac 1]);
paulson@2858
   216
paulson@2858
   217
AddSEs [emptyE];
paulson@2858
   218
paulson@2858
   219
qed_goal "empty_subsetI" Set.thy "{} <= A"
paulson@2891
   220
 (fn _ => [ (Blast_tac 1) ]);
paulson@2858
   221
paulson@2858
   222
qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}"
paulson@2858
   223
 (fn [prem]=>
paulson@2935
   224
  [ (blast_tac (!claset addIs [prem RS FalseE]) 1) ]);
paulson@2858
   225
paulson@2858
   226
qed_goal "equals0D" Set.thy "!!a. [| A={};  a:A |] ==> P"
paulson@2891
   227
 (fn _ => [ (Blast_tac 1) ]);
paulson@2858
   228
paulson@2858
   229
goal Set.thy "Ball {} P = True";
paulson@2858
   230
by (simp_tac (HOL_ss addsimps [mem_Collect_eq, Ball_def, empty_def]) 1);
paulson@2858
   231
qed "ball_empty";
paulson@2858
   232
paulson@2858
   233
goal Set.thy "Bex {} P = False";
paulson@2858
   234
by (simp_tac (HOL_ss addsimps [mem_Collect_eq, Bex_def, empty_def]) 1);
paulson@2858
   235
qed "bex_empty";
paulson@2858
   236
Addsimps [ball_empty, bex_empty];
paulson@2858
   237
paulson@2858
   238
paulson@2858
   239
section "The Powerset operator -- Pow";
paulson@2858
   240
paulson@2858
   241
qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)"
paulson@2858
   242
 (fn _ => [ (Asm_simp_tac 1) ]);
paulson@2858
   243
paulson@2858
   244
AddIffs [Pow_iff]; 
paulson@2858
   245
paulson@2858
   246
qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
paulson@2858
   247
 (fn _ => [ (etac CollectI 1) ]);
paulson@2858
   248
paulson@2858
   249
qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B)  ==>  A<=B"
paulson@2858
   250
 (fn _=> [ (etac CollectD 1) ]);
paulson@2858
   251
paulson@2858
   252
val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)
paulson@2858
   253
val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)
paulson@2858
   254
paulson@2858
   255
nipkow@1548
   256
section "Set complement -- Compl";
clasohm@923
   257
paulson@2499
   258
qed_goalw "Compl_iff" Set.thy [Compl_def] "(c : Compl(A)) = (c~:A)"
paulson@2891
   259
 (fn _ => [ (Blast_tac 1) ]);
paulson@2499
   260
paulson@2499
   261
Addsimps [Compl_iff];
paulson@2499
   262
clasohm@923
   263
val prems = goalw Set.thy [Compl_def]
clasohm@923
   264
    "[| c:A ==> False |] ==> c : Compl(A)";
clasohm@923
   265
by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
clasohm@923
   266
qed "ComplI";
clasohm@923
   267
clasohm@923
   268
(*This form, with negated conclusion, works well with the Classical prover.
clasohm@923
   269
  Negated assumptions behave like formulae on the right side of the notional
clasohm@923
   270
  turnstile...*)
clasohm@923
   271
val major::prems = goalw Set.thy [Compl_def]
paulson@2499
   272
    "c : Compl(A) ==> c~:A";
clasohm@923
   273
by (rtac (major RS CollectD) 1);
clasohm@923
   274
qed "ComplD";
clasohm@923
   275
clasohm@923
   276
val ComplE = make_elim ComplD;
clasohm@923
   277
paulson@2499
   278
AddSIs [ComplI];
paulson@2499
   279
AddSEs [ComplE];
paulson@1640
   280
clasohm@923
   281
nipkow@1548
   282
section "Binary union -- Un";
clasohm@923
   283
paulson@2499
   284
qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)"
paulson@2891
   285
 (fn _ => [ Blast_tac 1 ]);
paulson@2499
   286
paulson@2499
   287
Addsimps [Un_iff];
paulson@2499
   288
paulson@2499
   289
goal Set.thy "!!c. c:A ==> c : A Un B";
paulson@2499
   290
by (Asm_simp_tac 1);
clasohm@923
   291
qed "UnI1";
clasohm@923
   292
paulson@2499
   293
goal Set.thy "!!c. c:B ==> c : A Un B";
paulson@2499
   294
by (Asm_simp_tac 1);
clasohm@923
   295
qed "UnI2";
clasohm@923
   296
clasohm@923
   297
(*Classical introduction rule: no commitment to A vs B*)
clasohm@923
   298
qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B"
clasohm@923
   299
 (fn prems=>
paulson@2499
   300
  [ (Simp_tac 1),
paulson@2499
   301
    (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
clasohm@923
   302
clasohm@923
   303
val major::prems = goalw Set.thy [Un_def]
clasohm@923
   304
    "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
clasohm@923
   305
by (rtac (major RS CollectD RS disjE) 1);
clasohm@923
   306
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   307
qed "UnE";
clasohm@923
   308
paulson@2499
   309
AddSIs [UnCI];
paulson@2499
   310
AddSEs [UnE];
paulson@1640
   311
clasohm@923
   312
nipkow@1548
   313
section "Binary intersection -- Int";
clasohm@923
   314
paulson@2499
   315
qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)"
paulson@2891
   316
 (fn _ => [ (Blast_tac 1) ]);
paulson@2499
   317
paulson@2499
   318
Addsimps [Int_iff];
paulson@2499
   319
paulson@2499
   320
goal Set.thy "!!c. [| c:A;  c:B |] ==> c : A Int B";
paulson@2499
   321
by (Asm_simp_tac 1);
clasohm@923
   322
qed "IntI";
clasohm@923
   323
paulson@2499
   324
goal Set.thy "!!c. c : A Int B ==> c:A";
paulson@2499
   325
by (Asm_full_simp_tac 1);
clasohm@923
   326
qed "IntD1";
clasohm@923
   327
paulson@2499
   328
goal Set.thy "!!c. c : A Int B ==> c:B";
paulson@2499
   329
by (Asm_full_simp_tac 1);
clasohm@923
   330
qed "IntD2";
clasohm@923
   331
clasohm@923
   332
val [major,minor] = goal Set.thy
clasohm@923
   333
    "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
clasohm@923
   334
by (rtac minor 1);
clasohm@923
   335
by (rtac (major RS IntD1) 1);
clasohm@923
   336
by (rtac (major RS IntD2) 1);
clasohm@923
   337
qed "IntE";
clasohm@923
   338
paulson@2499
   339
AddSIs [IntI];
paulson@2499
   340
AddSEs [IntE];
clasohm@923
   341
nipkow@1548
   342
section "Set difference";
clasohm@923
   343
paulson@2499
   344
qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)"
paulson@2891
   345
 (fn _ => [ (Blast_tac 1) ]);
clasohm@923
   346
paulson@2499
   347
Addsimps [Diff_iff];
paulson@2499
   348
paulson@2499
   349
qed_goal "DiffI" Set.thy "!!c. [| c : A;  c ~: B |] ==> c : A - B"
paulson@2499
   350
 (fn _=> [ Asm_simp_tac 1 ]);
clasohm@923
   351
paulson@2499
   352
qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A"
paulson@2499
   353
 (fn _=> [ (Asm_full_simp_tac 1) ]);
clasohm@923
   354
paulson@2499
   355
qed_goal "DiffD2" Set.thy "!!c. [| c : A - B;  c : B |] ==> P"
paulson@2499
   356
 (fn _=> [ (Asm_full_simp_tac 1) ]);
paulson@2499
   357
paulson@2499
   358
qed_goal "DiffE" Set.thy "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P"
clasohm@923
   359
 (fn prems=>
clasohm@923
   360
  [ (resolve_tac prems 1),
clasohm@923
   361
    (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]);
clasohm@923
   362
paulson@2499
   363
AddSIs [DiffI];
paulson@2499
   364
AddSEs [DiffE];
clasohm@923
   365
clasohm@923
   366
nipkow@1548
   367
section "Augmenting a set -- insert";
clasohm@923
   368
paulson@2499
   369
qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)"
paulson@2891
   370
 (fn _ => [Blast_tac 1]);
paulson@2499
   371
paulson@2499
   372
Addsimps [insert_iff];
clasohm@923
   373
paulson@2499
   374
qed_goal "insertI1" Set.thy "a : insert a B"
paulson@2499
   375
 (fn _ => [Simp_tac 1]);
paulson@2499
   376
paulson@2499
   377
qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B"
paulson@2499
   378
 (fn _=> [Asm_simp_tac 1]);
clasohm@923
   379
clasohm@923
   380
qed_goalw "insertE" Set.thy [insert_def]
clasohm@923
   381
    "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P"
clasohm@923
   382
 (fn major::prems=>
clasohm@923
   383
  [ (rtac (major RS UnE) 1),
clasohm@923
   384
    (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
clasohm@923
   385
clasohm@923
   386
(*Classical introduction rule*)
clasohm@923
   387
qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
paulson@2499
   388
 (fn prems=>
paulson@2499
   389
  [ (Simp_tac 1),
paulson@2499
   390
    (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
paulson@2499
   391
paulson@2499
   392
AddSIs [insertCI]; 
paulson@2499
   393
AddSEs [insertE];
clasohm@923
   394
nipkow@1548
   395
section "Singletons, using insert";
clasohm@923
   396
clasohm@923
   397
qed_goal "singletonI" Set.thy "a : {a}"
clasohm@923
   398
 (fn _=> [ (rtac insertI1 1) ]);
clasohm@923
   399
paulson@2499
   400
goal Set.thy "!!a. b : {a} ==> b=a";
paulson@2891
   401
by (Blast_tac 1);
clasohm@923
   402
qed "singletonD";
clasohm@923
   403
oheimb@1776
   404
bind_thm ("singletonE", make_elim singletonD);
oheimb@1776
   405
paulson@2499
   406
qed_goal "singleton_iff" thy "(b : {a}) = (b=a)" 
paulson@2891
   407
(fn _ => [Blast_tac 1]);
clasohm@923
   408
paulson@2499
   409
goal Set.thy "!!a b. {a}={b} ==> a=b";
paulson@2935
   410
by (blast_tac (!claset addEs [equalityE]) 1);
clasohm@923
   411
qed "singleton_inject";
clasohm@923
   412
paulson@2858
   413
(*Redundant? But unlike insertCI, it proves the subgoal immediately!*)
paulson@2858
   414
AddSIs [singletonI];   
paulson@2858
   415
    
paulson@2499
   416
AddSDs [singleton_inject];
paulson@2499
   417
nipkow@1531
   418
nipkow@1548
   419
section "The universal set -- UNIV";
nipkow@1531
   420
paulson@1882
   421
qed_goal "UNIV_I" Set.thy "x : UNIV"
paulson@1882
   422
  (fn _ => [rtac ComplI 1, etac emptyE 1]);
paulson@1882
   423
nipkow@1531
   424
qed_goal "subset_UNIV" Set.thy "A <= UNIV"
paulson@1882
   425
  (fn _ => [rtac subsetI 1, rtac UNIV_I 1]);
nipkow@1531
   426
nipkow@1531
   427
nipkow@1548
   428
section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
clasohm@923
   429
paulson@2499
   430
goalw Set.thy [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
paulson@2891
   431
by (Blast_tac 1);
paulson@2499
   432
qed "UN_iff";
paulson@2499
   433
paulson@2499
   434
Addsimps [UN_iff];
paulson@2499
   435
clasohm@923
   436
(*The order of the premises presupposes that A is rigid; b may be flexible*)
paulson@2499
   437
goal Set.thy "!!b. [| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
paulson@2499
   438
by (Auto_tac());
clasohm@923
   439
qed "UN_I";
clasohm@923
   440
clasohm@923
   441
val major::prems = goalw Set.thy [UNION_def]
clasohm@923
   442
    "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
clasohm@923
   443
by (rtac (major RS CollectD RS bexE) 1);
clasohm@923
   444
by (REPEAT (ares_tac prems 1));
clasohm@923
   445
qed "UN_E";
clasohm@923
   446
paulson@2499
   447
AddIs  [UN_I];
paulson@2499
   448
AddSEs [UN_E];
paulson@2499
   449
clasohm@923
   450
val prems = goal Set.thy
clasohm@923
   451
    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
clasohm@923
   452
\    (UN x:A. C(x)) = (UN x:B. D(x))";
clasohm@923
   453
by (REPEAT (etac UN_E 1
clasohm@923
   454
     ORELSE ares_tac ([UN_I,equalityI,subsetI] @ 
clasohm@1465
   455
                      (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
clasohm@923
   456
qed "UN_cong";
clasohm@923
   457
clasohm@923
   458
nipkow@1548
   459
section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
clasohm@923
   460
paulson@2499
   461
goalw Set.thy [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
paulson@2499
   462
by (Auto_tac());
paulson@2499
   463
qed "INT_iff";
paulson@2499
   464
paulson@2499
   465
Addsimps [INT_iff];
paulson@2499
   466
clasohm@923
   467
val prems = goalw Set.thy [INTER_def]
clasohm@923
   468
    "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
clasohm@923
   469
by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
clasohm@923
   470
qed "INT_I";
clasohm@923
   471
paulson@2499
   472
goal Set.thy "!!b. [| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
paulson@2499
   473
by (Auto_tac());
clasohm@923
   474
qed "INT_D";
clasohm@923
   475
clasohm@923
   476
(*"Classical" elimination -- by the Excluded Middle on a:A *)
clasohm@923
   477
val major::prems = goalw Set.thy [INTER_def]
clasohm@923
   478
    "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
clasohm@923
   479
by (rtac (major RS CollectD RS ballE) 1);
clasohm@923
   480
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   481
qed "INT_E";
clasohm@923
   482
paulson@2499
   483
AddSIs [INT_I];
paulson@2499
   484
AddEs  [INT_D, INT_E];
paulson@2499
   485
clasohm@923
   486
val prems = goal Set.thy
clasohm@923
   487
    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
clasohm@923
   488
\    (INT x:A. C(x)) = (INT x:B. D(x))";
clasohm@923
   489
by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
clasohm@923
   490
by (REPEAT (dtac INT_D 1
clasohm@923
   491
     ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
clasohm@923
   492
qed "INT_cong";
clasohm@923
   493
clasohm@923
   494
nipkow@1548
   495
section "Unions over a type; UNION1(B) = Union(range(B))";
clasohm@923
   496
paulson@2499
   497
goalw Set.thy [UNION1_def] "(b: (UN x. B(x))) = (EX x. b: B(x))";
paulson@2499
   498
by (Simp_tac 1);
paulson@2891
   499
by (Blast_tac 1);
paulson@2499
   500
qed "UN1_iff";
paulson@2499
   501
paulson@2499
   502
Addsimps [UN1_iff];
paulson@2499
   503
clasohm@923
   504
(*The order of the premises presupposes that A is rigid; b may be flexible*)
paulson@2499
   505
goal Set.thy "!!b. b: B(x) ==> b: (UN x. B(x))";
paulson@2499
   506
by (Auto_tac());
clasohm@923
   507
qed "UN1_I";
clasohm@923
   508
clasohm@923
   509
val major::prems = goalw Set.thy [UNION1_def]
clasohm@923
   510
    "[| b : (UN x. B(x));  !!x. b: B(x) ==> R |] ==> R";
clasohm@923
   511
by (rtac (major RS UN_E) 1);
clasohm@923
   512
by (REPEAT (ares_tac prems 1));
clasohm@923
   513
qed "UN1_E";
clasohm@923
   514
paulson@2499
   515
AddIs  [UN1_I];
paulson@2499
   516
AddSEs [UN1_E];
paulson@2499
   517
clasohm@923
   518
nipkow@1548
   519
section "Intersections over a type; INTER1(B) = Inter(range(B))";
clasohm@923
   520
paulson@2499
   521
goalw Set.thy [INTER1_def] "(b: (INT x. B(x))) = (ALL x. b: B(x))";
paulson@2499
   522
by (Simp_tac 1);
paulson@2891
   523
by (Blast_tac 1);
paulson@2499
   524
qed "INT1_iff";
paulson@2499
   525
paulson@2499
   526
Addsimps [INT1_iff];
paulson@2499
   527
clasohm@923
   528
val prems = goalw Set.thy [INTER1_def]
clasohm@923
   529
    "(!!x. b: B(x)) ==> b : (INT x. B(x))";
clasohm@923
   530
by (REPEAT (ares_tac (INT_I::prems) 1));
clasohm@923
   531
qed "INT1_I";
clasohm@923
   532
paulson@2499
   533
goal Set.thy "!!b. b : (INT x. B(x)) ==> b: B(a)";
paulson@2499
   534
by (Asm_full_simp_tac 1);
clasohm@923
   535
qed "INT1_D";
clasohm@923
   536
paulson@2499
   537
AddSIs [INT1_I]; 
paulson@2499
   538
AddDs  [INT1_D];
paulson@2499
   539
paulson@2499
   540
nipkow@1548
   541
section "Union";
clasohm@923
   542
paulson@2499
   543
goalw Set.thy [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
paulson@2891
   544
by (Blast_tac 1);
paulson@2499
   545
qed "Union_iff";
paulson@2499
   546
paulson@2499
   547
Addsimps [Union_iff];
paulson@2499
   548
clasohm@923
   549
(*The order of the premises presupposes that C is rigid; A may be flexible*)
paulson@2499
   550
goal Set.thy "!!X. [| X:C;  A:X |] ==> A : Union(C)";
paulson@2499
   551
by (Auto_tac());
clasohm@923
   552
qed "UnionI";
clasohm@923
   553
clasohm@923
   554
val major::prems = goalw Set.thy [Union_def]
clasohm@923
   555
    "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
clasohm@923
   556
by (rtac (major RS UN_E) 1);
clasohm@923
   557
by (REPEAT (ares_tac prems 1));
clasohm@923
   558
qed "UnionE";
clasohm@923
   559
paulson@2499
   560
AddIs  [UnionI];
paulson@2499
   561
AddSEs [UnionE];
paulson@2499
   562
paulson@2499
   563
nipkow@1548
   564
section "Inter";
clasohm@923
   565
paulson@2499
   566
goalw Set.thy [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
paulson@2891
   567
by (Blast_tac 1);
paulson@2499
   568
qed "Inter_iff";
paulson@2499
   569
paulson@2499
   570
Addsimps [Inter_iff];
paulson@2499
   571
clasohm@923
   572
val prems = goalw Set.thy [Inter_def]
clasohm@923
   573
    "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
clasohm@923
   574
by (REPEAT (ares_tac ([INT_I] @ prems) 1));
clasohm@923
   575
qed "InterI";
clasohm@923
   576
clasohm@923
   577
(*A "destruct" rule -- every X in C contains A as an element, but
clasohm@923
   578
  A:X can hold when X:C does not!  This rule is analogous to "spec". *)
paulson@2499
   579
goal Set.thy "!!X. [| A : Inter(C);  X:C |] ==> A:X";
paulson@2499
   580
by (Auto_tac());
clasohm@923
   581
qed "InterD";
clasohm@923
   582
clasohm@923
   583
(*"Classical" elimination rule -- does not require proving X:C *)
clasohm@923
   584
val major::prems = goalw Set.thy [Inter_def]
paulson@2721
   585
    "[| A : Inter(C);  X~:C ==> R;  A:X ==> R |] ==> R";
clasohm@923
   586
by (rtac (major RS INT_E) 1);
clasohm@923
   587
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   588
qed "InterE";
clasohm@923
   589
paulson@2499
   590
AddSIs [InterI];
paulson@2499
   591
AddEs  [InterD, InterE];
paulson@2499
   592
paulson@2499
   593
nipkow@2912
   594
(*** Image of a set under a function ***)
nipkow@2912
   595
nipkow@2912
   596
(*Frequently b does not have the syntactic form of f(x).*)
nipkow@2912
   597
val prems = goalw thy [image_def] "[| b=f(x);  x:A |] ==> b : f``A";
nipkow@2912
   598
by (REPEAT (resolve_tac (prems @ [CollectI,bexI,prem]) 1));
nipkow@2912
   599
qed "image_eqI";
nipkow@2912
   600
nipkow@2912
   601
bind_thm ("imageI", refl RS image_eqI);
nipkow@2912
   602
nipkow@2912
   603
(*The eta-expansion gives variable-name preservation.*)
nipkow@2912
   604
val major::prems = goalw thy [image_def]
nipkow@2912
   605
    "[| b : (%x.f(x))``A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P"; 
nipkow@2912
   606
by (rtac (major RS CollectD RS bexE) 1);
nipkow@2912
   607
by (REPEAT (ares_tac prems 1));
nipkow@2912
   608
qed "imageE";
nipkow@2912
   609
nipkow@2912
   610
AddIs  [image_eqI];
nipkow@2912
   611
AddSEs [imageE]; 
nipkow@2912
   612
nipkow@2912
   613
goalw thy [o_def] "(f o g)``r = f``(g``r)";
paulson@2935
   614
by (Blast_tac 1);
nipkow@2912
   615
qed "image_compose";
nipkow@2912
   616
nipkow@2912
   617
goal thy "f``(A Un B) = f``A Un f``B";
paulson@2935
   618
by (Blast_tac 1);
nipkow@2912
   619
qed "image_Un";
nipkow@2912
   620
nipkow@2912
   621
nipkow@2912
   622
(*** Range of a function -- just a translation for image! ***)
nipkow@2912
   623
nipkow@2912
   624
goal thy "!!b. 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
nipkow@2912
   630
val [major,minor] = goal thy 
nipkow@2912
   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
nipkow@2912
   636
AddIs  [rangeI]; 
nipkow@2912
   637
AddSEs [rangeE]; 
nipkow@2912
   638
oheimb@1776
   639
oheimb@1776
   640
(*** Set reasoning tools ***)
oheimb@1776
   641
oheimb@1776
   642
paulson@2499
   643
(*Each of these has ALREADY been added to !simpset above.*)
paulson@2024
   644
val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff, 
paulson@2499
   645
                 mem_Collect_eq, 
paulson@2499
   646
		 UN_iff, UN1_iff, Union_iff, 
paulson@2499
   647
		 INT_iff, INT1_iff, Inter_iff];
oheimb@1776
   648
paulson@1937
   649
(*Not for Addsimps -- it can cause goals to blow up!*)
paulson@1937
   650
goal Set.thy "(a : (if Q then x else y)) = ((Q --> a:x) & (~Q --> a : y))";
paulson@1937
   651
by (simp_tac (!simpset setloop split_tac [expand_if]) 1);
paulson@1937
   652
qed "mem_if";
paulson@1937
   653
oheimb@1776
   654
val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs;
oheimb@1776
   655
paulson@2499
   656
simpset := !simpset addcongs [ball_cong,bex_cong]
oheimb@1776
   657
                    setmksimps (mksimps mksimps_pairs);
nipkow@3222
   658
nipkow@3222
   659
Addsimps[subset_UNIV, empty_subsetI, subset_refl];
nipkow@3222
   660
nipkow@3222
   661
nipkow@3222
   662
(*** < ***)
nipkow@3222
   663
nipkow@3222
   664
goalw Set.thy [psubset_def] "!!A::'a set. [| A <= B; A ~= B |] ==> A<B";
nipkow@3222
   665
by (Blast_tac 1);
nipkow@3222
   666
qed "psubsetI";
nipkow@3222
   667
nipkow@3222
   668
goalw Set.thy [psubset_def]
nipkow@3222
   669
    "!!x. A < insert x B ==> (x ~: A) & A<=B | x:A & A-{x}<B";
nipkow@3222
   670
by (Auto_tac());
nipkow@3222
   671
qed "psubset_insertD";