src/HOL/Set.ML
author paulson
Thu Oct 15 11:38:39 1998 +0200 (1998-10-15)
changeset 5649 1bac26652f45
parent 5600 34b3366b83ac
child 5931 325300576da7
permissions -rw-r--r--
Uses overload_1st_set to specify overloading
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];
oheimb@5521
    62
(* gives better instantiation for bound: *)
oheimb@5521
    63
claset_ref() := claset() addWrapper ("bspec", fn tac2 =>
oheimb@5521
    64
			 (dtac bspec THEN' atac) APPEND' tac2);
paulson@2499
    65
paulson@5316
    66
Goalw [Bex_def] "[| P(x);  x:A |] ==> ? x:A. P(x)";
paulson@5316
    67
by (Blast_tac 1);
clasohm@923
    68
qed "bexI";
clasohm@923
    69
clasohm@923
    70
qed_goal "bexCI" Set.thy 
oheimb@5521
    71
   "[| ! x:A. ~P(x) ==> P(a);  a:A |] ==> ? x:A. P(x)" (fn prems =>
clasohm@923
    72
  [ (rtac classical 1),
clasohm@923
    73
    (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1))  ]);
clasohm@923
    74
paulson@5316
    75
val major::prems = Goalw [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*)
wenzelm@5069
    85
Goal "(! x:A. P) = ((? x. x:A) --> P)";
wenzelm@4089
    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*)
wenzelm@5069
    90
Goal "(? x:A. P) = ((? x. x:A) & P)";
wenzelm@4089
    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
paulson@5316
    98
val prems = Goal
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
paulson@5316
   106
val prems = Goal
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
paulson@5316
   116
val prems = Goalw [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@5649
   120
(*Map the type ('a set => anything) to just 'a.
paulson@5649
   121
  For overloading constants whose first argument has type "'a set" *)
paulson@5649
   122
fun overload_1st_set s = Blast.overloaded (s, HOLogic.dest_setT o domain_type);
paulson@5649
   123
paulson@4059
   124
(*While (:) is not, its type must be kept
paulson@4059
   125
  for overloading of = to work.*)
paulson@4240
   126
Blast.overloaded ("op :", domain_type);
paulson@5649
   127
paulson@5649
   128
overload_1st_set "Ball";		(*need UNION, INTER also?*)
paulson@5649
   129
overload_1st_set "Bex";
paulson@4059
   130
paulson@4469
   131
(*Image: retain the type of the set being expressed*)
paulson@5336
   132
Blast.overloaded ("op ``", domain_type);
paulson@2881
   133
clasohm@923
   134
(*Rule in Modus Ponens style*)
paulson@5316
   135
Goalw [subset_def] "[| A <= B;  c:A |] ==> c:B";
paulson@5316
   136
by (Blast_tac 1);
clasohm@923
   137
qed "subsetD";
clasohm@923
   138
clasohm@923
   139
(*The same, with reversed premises for use with etac -- cf rev_mp*)
clasohm@923
   140
qed_goal "rev_subsetD" Set.thy "[| c:A;  A <= B |] ==> c:B"
clasohm@923
   141
 (fn prems=>  [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]);
clasohm@923
   142
paulson@1920
   143
(*Converts A<=B to x:A ==> x:B*)
paulson@1920
   144
fun impOfSubs th = th RSN (2, rev_subsetD);
paulson@1920
   145
paulson@1841
   146
qed_goal "contra_subsetD" Set.thy "!!c. [| A <= B; c ~: B |] ==> c ~: A"
paulson@1841
   147
 (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
paulson@1841
   148
paulson@1841
   149
qed_goal "rev_contra_subsetD" Set.thy "!!c. [| c ~: B;  A <= B |] ==> c ~: A"
paulson@1841
   150
 (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
paulson@1841
   151
clasohm@923
   152
(*Classical elimination rule*)
paulson@5316
   153
val major::prems = Goalw [subset_def] 
clasohm@923
   154
    "[| A <= B;  c~:A ==> P;  c:B ==> P |] ==> P";
clasohm@923
   155
by (rtac (major RS ballE) 1);
clasohm@923
   156
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   157
qed "subsetCE";
clasohm@923
   158
clasohm@923
   159
(*Takes assumptions A<=B; c:A and creates the assumption c:B *)
clasohm@923
   160
fun set_mp_tac i = etac subsetCE i  THEN  mp_tac i;
clasohm@923
   161
paulson@2499
   162
AddSIs [subsetI];
paulson@2499
   163
AddEs  [subsetD, subsetCE];
clasohm@923
   164
paulson@2499
   165
qed_goal "subset_refl" Set.thy "A <= (A::'a set)"
paulson@4059
   166
 (fn _=> [Fast_tac 1]);		(*Blast_tac would try order_refl and fail*)
paulson@2499
   167
paulson@5316
   168
Goal "[| A<=B;  B<=C |] ==> A<=(C::'a set)";
paulson@2891
   169
by (Blast_tac 1);
clasohm@923
   170
qed "subset_trans";
clasohm@923
   171
clasohm@923
   172
nipkow@1548
   173
section "Equality";
clasohm@923
   174
clasohm@923
   175
(*Anti-symmetry of the subset relation*)
paulson@5316
   176
Goal "[| A <= B;  B <= A |] ==> A = (B::'a set)";
paulson@5318
   177
by (rtac set_ext 1);
paulson@5316
   178
by (blast_tac (claset() addIs [subsetD]) 1);
clasohm@923
   179
qed "subset_antisym";
clasohm@923
   180
val equalityI = subset_antisym;
clasohm@923
   181
berghofe@1762
   182
AddSIs [equalityI];
berghofe@1762
   183
clasohm@923
   184
(* Equality rules from ZF set theory -- are they appropriate here? *)
paulson@5316
   185
Goal "A = B ==> A<=(B::'a set)";
paulson@5316
   186
by (etac ssubst 1);
clasohm@923
   187
by (rtac subset_refl 1);
clasohm@923
   188
qed "equalityD1";
clasohm@923
   189
paulson@5316
   190
Goal "A = B ==> B<=(A::'a set)";
paulson@5316
   191
by (etac ssubst 1);
clasohm@923
   192
by (rtac subset_refl 1);
clasohm@923
   193
qed "equalityD2";
clasohm@923
   194
paulson@5316
   195
val prems = Goal
clasohm@923
   196
    "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";
clasohm@923
   197
by (resolve_tac prems 1);
clasohm@923
   198
by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
clasohm@923
   199
qed "equalityE";
clasohm@923
   200
paulson@5316
   201
val major::prems = Goal
clasohm@923
   202
    "[| A = B;  [| c:A; c:B |] ==> P;  [| c~:A; c~:B |] ==> P |]  ==>  P";
clasohm@923
   203
by (rtac (major RS equalityE) 1);
clasohm@923
   204
by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
clasohm@923
   205
qed "equalityCE";
clasohm@923
   206
clasohm@923
   207
(*Lemma for creating induction formulae -- for "pattern matching" on p
clasohm@923
   208
  To make the induction hypotheses usable, apply "spec" or "bspec" to
clasohm@923
   209
  put universal quantifiers over the free variables in p. *)
paulson@5316
   210
val prems = Goal 
clasohm@923
   211
    "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
clasohm@923
   212
by (rtac mp 1);
clasohm@923
   213
by (REPEAT (resolve_tac (refl::prems) 1));
clasohm@923
   214
qed "setup_induction";
clasohm@923
   215
clasohm@923
   216
paulson@4159
   217
section "The universal set -- UNIV";
paulson@4159
   218
paulson@4159
   219
qed_goalw "UNIV_I" Set.thy [UNIV_def] "x : UNIV"
paulson@4159
   220
  (fn _ => [rtac CollectI 1, rtac TrueI 1]);
paulson@4159
   221
paulson@4434
   222
Addsimps [UNIV_I];
paulson@4434
   223
AddIs    [UNIV_I];  (*unsafe makes it less likely to cause problems*)
paulson@4159
   224
paulson@4159
   225
qed_goal "subset_UNIV" Set.thy "A <= UNIV"
paulson@4159
   226
  (fn _ => [rtac subsetI 1, rtac UNIV_I 1]);
paulson@4159
   227
paulson@4159
   228
(** Eta-contracting these two rules (to remove P) causes them to be ignored
paulson@4159
   229
    because of their interaction with congruence rules. **)
paulson@4159
   230
wenzelm@5069
   231
Goalw [Ball_def] "Ball UNIV P = All P";
paulson@4159
   232
by (Simp_tac 1);
paulson@4159
   233
qed "ball_UNIV";
paulson@4159
   234
wenzelm@5069
   235
Goalw [Bex_def] "Bex UNIV P = Ex P";
paulson@4159
   236
by (Simp_tac 1);
paulson@4159
   237
qed "bex_UNIV";
paulson@4159
   238
Addsimps [ball_UNIV, bex_UNIV];
paulson@4159
   239
paulson@4159
   240
paulson@2858
   241
section "The empty set -- {}";
paulson@2858
   242
paulson@2858
   243
qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False"
paulson@2891
   244
 (fn _ => [ (Blast_tac 1) ]);
paulson@2858
   245
paulson@2858
   246
Addsimps [empty_iff];
paulson@2858
   247
paulson@2858
   248
qed_goal "emptyE" Set.thy "!!a. a:{} ==> P"
paulson@2858
   249
 (fn _ => [Full_simp_tac 1]);
paulson@2858
   250
paulson@2858
   251
AddSEs [emptyE];
paulson@2858
   252
paulson@2858
   253
qed_goal "empty_subsetI" Set.thy "{} <= A"
paulson@2891
   254
 (fn _ => [ (Blast_tac 1) ]);
paulson@2858
   255
paulson@5256
   256
(*One effect is to delete the ASSUMPTION {} <= A*)
paulson@5256
   257
AddIffs [empty_subsetI];
paulson@5256
   258
paulson@2858
   259
qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}"
paulson@2858
   260
 (fn [prem]=>
wenzelm@4089
   261
  [ (blast_tac (claset() addIs [prem RS FalseE]) 1) ]);
paulson@2858
   262
paulson@5256
   263
(*Use for reasoning about disjointness: A Int B = {} *)
paulson@5450
   264
qed_goal "equals0D" Set.thy "!!a. A={} ==> a ~: A"
paulson@2891
   265
 (fn _ => [ (Blast_tac 1) ]);
paulson@2858
   266
paulson@5450
   267
AddDs [equals0D, sym RS equals0D];
paulson@5256
   268
wenzelm@5069
   269
Goalw [Ball_def] "Ball {} P = True";
paulson@4159
   270
by (Simp_tac 1);
paulson@4159
   271
qed "ball_empty";
paulson@4159
   272
wenzelm@5069
   273
Goalw [Bex_def] "Bex {} P = False";
paulson@4159
   274
by (Simp_tac 1);
paulson@4159
   275
qed "bex_empty";
paulson@4159
   276
Addsimps [ball_empty, bex_empty];
paulson@4159
   277
wenzelm@5069
   278
Goal "UNIV ~= {}";
paulson@4159
   279
by (blast_tac (claset() addEs [equalityE]) 1);
paulson@4159
   280
qed "UNIV_not_empty";
paulson@4159
   281
AddIffs [UNIV_not_empty];
paulson@4159
   282
paulson@4159
   283
paulson@2858
   284
paulson@2858
   285
section "The Powerset operator -- Pow";
paulson@2858
   286
paulson@2858
   287
qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)"
paulson@2858
   288
 (fn _ => [ (Asm_simp_tac 1) ]);
paulson@2858
   289
paulson@2858
   290
AddIffs [Pow_iff]; 
paulson@2858
   291
paulson@2858
   292
qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
paulson@2858
   293
 (fn _ => [ (etac CollectI 1) ]);
paulson@2858
   294
paulson@2858
   295
qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B)  ==>  A<=B"
paulson@2858
   296
 (fn _=> [ (etac CollectD 1) ]);
paulson@2858
   297
paulson@2858
   298
val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)
paulson@2858
   299
val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)
paulson@2858
   300
paulson@2858
   301
nipkow@1548
   302
section "Set complement -- Compl";
clasohm@923
   303
paulson@5490
   304
qed_goalw "Compl_iff" Set.thy [Compl_def] "(c : -A) = (c~:A)"
paulson@2891
   305
 (fn _ => [ (Blast_tac 1) ]);
paulson@2499
   306
paulson@2499
   307
Addsimps [Compl_iff];
paulson@2499
   308
paulson@5490
   309
val prems = Goalw [Compl_def] "[| c:A ==> False |] ==> c : -A";
clasohm@923
   310
by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
clasohm@923
   311
qed "ComplI";
clasohm@923
   312
clasohm@923
   313
(*This form, with negated conclusion, works well with the Classical prover.
clasohm@923
   314
  Negated assumptions behave like formulae on the right side of the notional
clasohm@923
   315
  turnstile...*)
paulson@5490
   316
Goalw [Compl_def] "c : -A ==> c~:A";
paulson@5316
   317
by (etac CollectD 1);
clasohm@923
   318
qed "ComplD";
clasohm@923
   319
clasohm@923
   320
val ComplE = make_elim ComplD;
clasohm@923
   321
paulson@2499
   322
AddSIs [ComplI];
paulson@2499
   323
AddSEs [ComplE];
paulson@1640
   324
clasohm@923
   325
nipkow@1548
   326
section "Binary union -- Un";
clasohm@923
   327
paulson@2499
   328
qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)"
paulson@2891
   329
 (fn _ => [ Blast_tac 1 ]);
paulson@2499
   330
paulson@2499
   331
Addsimps [Un_iff];
paulson@2499
   332
paulson@5143
   333
Goal "c:A ==> c : A Un B";
paulson@2499
   334
by (Asm_simp_tac 1);
clasohm@923
   335
qed "UnI1";
clasohm@923
   336
paulson@5143
   337
Goal "c:B ==> c : A Un B";
paulson@2499
   338
by (Asm_simp_tac 1);
clasohm@923
   339
qed "UnI2";
clasohm@923
   340
clasohm@923
   341
(*Classical introduction rule: no commitment to A vs B*)
clasohm@923
   342
qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B"
clasohm@923
   343
 (fn prems=>
paulson@2499
   344
  [ (Simp_tac 1),
paulson@2499
   345
    (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
clasohm@923
   346
paulson@5316
   347
val major::prems = Goalw [Un_def]
clasohm@923
   348
    "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
clasohm@923
   349
by (rtac (major RS CollectD RS disjE) 1);
clasohm@923
   350
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   351
qed "UnE";
clasohm@923
   352
paulson@2499
   353
AddSIs [UnCI];
paulson@2499
   354
AddSEs [UnE];
paulson@1640
   355
clasohm@923
   356
nipkow@1548
   357
section "Binary intersection -- Int";
clasohm@923
   358
paulson@2499
   359
qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)"
paulson@2891
   360
 (fn _ => [ (Blast_tac 1) ]);
paulson@2499
   361
paulson@2499
   362
Addsimps [Int_iff];
paulson@2499
   363
paulson@5143
   364
Goal "[| c:A;  c:B |] ==> c : A Int B";
paulson@2499
   365
by (Asm_simp_tac 1);
clasohm@923
   366
qed "IntI";
clasohm@923
   367
paulson@5143
   368
Goal "c : A Int B ==> c:A";
paulson@2499
   369
by (Asm_full_simp_tac 1);
clasohm@923
   370
qed "IntD1";
clasohm@923
   371
paulson@5143
   372
Goal "c : A Int B ==> c:B";
paulson@2499
   373
by (Asm_full_simp_tac 1);
clasohm@923
   374
qed "IntD2";
clasohm@923
   375
paulson@5316
   376
val [major,minor] = Goal
clasohm@923
   377
    "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
clasohm@923
   378
by (rtac minor 1);
clasohm@923
   379
by (rtac (major RS IntD1) 1);
clasohm@923
   380
by (rtac (major RS IntD2) 1);
clasohm@923
   381
qed "IntE";
clasohm@923
   382
paulson@2499
   383
AddSIs [IntI];
paulson@2499
   384
AddSEs [IntE];
clasohm@923
   385
nipkow@1548
   386
section "Set difference";
clasohm@923
   387
paulson@2499
   388
qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)"
paulson@2891
   389
 (fn _ => [ (Blast_tac 1) ]);
clasohm@923
   390
paulson@2499
   391
Addsimps [Diff_iff];
paulson@2499
   392
paulson@2499
   393
qed_goal "DiffI" Set.thy "!!c. [| c : A;  c ~: B |] ==> c : A - B"
paulson@2499
   394
 (fn _=> [ Asm_simp_tac 1 ]);
clasohm@923
   395
paulson@2499
   396
qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A"
paulson@2499
   397
 (fn _=> [ (Asm_full_simp_tac 1) ]);
clasohm@923
   398
paulson@2499
   399
qed_goal "DiffD2" Set.thy "!!c. [| c : A - B;  c : B |] ==> P"
paulson@2499
   400
 (fn _=> [ (Asm_full_simp_tac 1) ]);
paulson@2499
   401
paulson@2499
   402
qed_goal "DiffE" Set.thy "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P"
clasohm@923
   403
 (fn prems=>
clasohm@923
   404
  [ (resolve_tac prems 1),
clasohm@923
   405
    (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]);
clasohm@923
   406
paulson@2499
   407
AddSIs [DiffI];
paulson@2499
   408
AddSEs [DiffE];
clasohm@923
   409
clasohm@923
   410
nipkow@1548
   411
section "Augmenting a set -- insert";
clasohm@923
   412
paulson@2499
   413
qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)"
paulson@2891
   414
 (fn _ => [Blast_tac 1]);
paulson@2499
   415
paulson@2499
   416
Addsimps [insert_iff];
clasohm@923
   417
paulson@2499
   418
qed_goal "insertI1" Set.thy "a : insert a B"
paulson@2499
   419
 (fn _ => [Simp_tac 1]);
paulson@2499
   420
paulson@2499
   421
qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B"
paulson@2499
   422
 (fn _=> [Asm_simp_tac 1]);
clasohm@923
   423
clasohm@923
   424
qed_goalw "insertE" Set.thy [insert_def]
clasohm@923
   425
    "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P"
clasohm@923
   426
 (fn major::prems=>
clasohm@923
   427
  [ (rtac (major RS UnE) 1),
clasohm@923
   428
    (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
clasohm@923
   429
clasohm@923
   430
(*Classical introduction rule*)
clasohm@923
   431
qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
paulson@2499
   432
 (fn prems=>
paulson@2499
   433
  [ (Simp_tac 1),
paulson@2499
   434
    (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
paulson@2499
   435
paulson@2499
   436
AddSIs [insertCI]; 
paulson@2499
   437
AddSEs [insertE];
clasohm@923
   438
nipkow@1548
   439
section "Singletons, using insert";
clasohm@923
   440
clasohm@923
   441
qed_goal "singletonI" Set.thy "a : {a}"
clasohm@923
   442
 (fn _=> [ (rtac insertI1 1) ]);
clasohm@923
   443
paulson@5143
   444
Goal "b : {a} ==> b=a";
paulson@2891
   445
by (Blast_tac 1);
clasohm@923
   446
qed "singletonD";
clasohm@923
   447
oheimb@1776
   448
bind_thm ("singletonE", make_elim singletonD);
oheimb@1776
   449
paulson@2499
   450
qed_goal "singleton_iff" thy "(b : {a}) = (b=a)" 
paulson@2891
   451
(fn _ => [Blast_tac 1]);
clasohm@923
   452
paulson@5143
   453
Goal "{a}={b} ==> a=b";
wenzelm@4089
   454
by (blast_tac (claset() addEs [equalityE]) 1);
clasohm@923
   455
qed "singleton_inject";
clasohm@923
   456
paulson@2858
   457
(*Redundant? But unlike insertCI, it proves the subgoal immediately!*)
paulson@2858
   458
AddSIs [singletonI];   
paulson@2499
   459
AddSDs [singleton_inject];
paulson@3718
   460
AddSEs [singletonE];
paulson@2499
   461
wenzelm@5069
   462
Goal "{x. x=a} = {a}";
wenzelm@4423
   463
by (Blast_tac 1);
nipkow@3582
   464
qed "singleton_conv";
nipkow@3582
   465
Addsimps [singleton_conv];
nipkow@1531
   466
nipkow@5600
   467
Goal "{x. a=x} = {a}";
nipkow@5600
   468
by(Blast_tac 1);
nipkow@5600
   469
qed "singleton_conv2";
nipkow@5600
   470
Addsimps [singleton_conv2];
nipkow@5600
   471
nipkow@1531
   472
nipkow@1548
   473
section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
clasohm@923
   474
wenzelm@5069
   475
Goalw [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
paulson@2891
   476
by (Blast_tac 1);
paulson@2499
   477
qed "UN_iff";
paulson@2499
   478
paulson@2499
   479
Addsimps [UN_iff];
paulson@2499
   480
clasohm@923
   481
(*The order of the premises presupposes that A is rigid; b may be flexible*)
paulson@5143
   482
Goal "[| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
paulson@4477
   483
by Auto_tac;
clasohm@923
   484
qed "UN_I";
clasohm@923
   485
paulson@5316
   486
val major::prems = Goalw [UNION_def]
clasohm@923
   487
    "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
clasohm@923
   488
by (rtac (major RS CollectD RS bexE) 1);
clasohm@923
   489
by (REPEAT (ares_tac prems 1));
clasohm@923
   490
qed "UN_E";
clasohm@923
   491
paulson@2499
   492
AddIs  [UN_I];
paulson@2499
   493
AddSEs [UN_E];
paulson@2499
   494
paulson@5316
   495
val prems = Goal
clasohm@923
   496
    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
clasohm@923
   497
\    (UN x:A. C(x)) = (UN x:B. D(x))";
clasohm@923
   498
by (REPEAT (etac UN_E 1
clasohm@923
   499
     ORELSE ares_tac ([UN_I,equalityI,subsetI] @ 
clasohm@1465
   500
                      (prems RL [equalityD1,equalityD2] RL [subsetD])) 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@5316
   531
val prems = Goal
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))";
clasohm@923
   534
by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
clasohm@923
   535
by (REPEAT (dtac INT_D 1
clasohm@923
   536
     ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
clasohm@923
   537
qed "INT_cong";
clasohm@923
   538
clasohm@923
   539
nipkow@1548
   540
section "Union";
clasohm@923
   541
wenzelm@5069
   542
Goalw [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
paulson@2891
   543
by (Blast_tac 1);
paulson@2499
   544
qed "Union_iff";
paulson@2499
   545
paulson@2499
   546
Addsimps [Union_iff];
paulson@2499
   547
clasohm@923
   548
(*The order of the premises presupposes that C is rigid; A may be flexible*)
paulson@5143
   549
Goal "[| X:C;  A:X |] ==> A : Union(C)";
paulson@4477
   550
by Auto_tac;
clasohm@923
   551
qed "UnionI";
clasohm@923
   552
paulson@5316
   553
val major::prems = Goalw [Union_def]
clasohm@923
   554
    "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
clasohm@923
   555
by (rtac (major RS UN_E) 1);
clasohm@923
   556
by (REPEAT (ares_tac prems 1));
clasohm@923
   557
qed "UnionE";
clasohm@923
   558
paulson@2499
   559
AddIs  [UnionI];
paulson@2499
   560
AddSEs [UnionE];
paulson@2499
   561
paulson@2499
   562
nipkow@1548
   563
section "Inter";
clasohm@923
   564
wenzelm@5069
   565
Goalw [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
paulson@2891
   566
by (Blast_tac 1);
paulson@2499
   567
qed "Inter_iff";
paulson@2499
   568
paulson@2499
   569
Addsimps [Inter_iff];
paulson@2499
   570
paulson@5316
   571
val prems = Goalw [Inter_def]
clasohm@923
   572
    "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
clasohm@923
   573
by (REPEAT (ares_tac ([INT_I] @ prems) 1));
clasohm@923
   574
qed "InterI";
clasohm@923
   575
clasohm@923
   576
(*A "destruct" rule -- every X in C contains A as an element, but
clasohm@923
   577
  A:X can hold when X:C does not!  This rule is analogous to "spec". *)
paulson@5143
   578
Goal "[| A : Inter(C);  X:C |] ==> A:X";
paulson@4477
   579
by Auto_tac;
clasohm@923
   580
qed "InterD";
clasohm@923
   581
clasohm@923
   582
(*"Classical" elimination rule -- does not require proving X:C *)
paulson@5316
   583
val major::prems = Goalw [Inter_def]
paulson@2721
   584
    "[| A : Inter(C);  X~:C ==> R;  A:X ==> R |] ==> R";
clasohm@923
   585
by (rtac (major RS INT_E) 1);
clasohm@923
   586
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   587
qed "InterE";
clasohm@923
   588
paulson@2499
   589
AddSIs [InterI];
paulson@2499
   590
AddEs  [InterD, InterE];
paulson@2499
   591
paulson@2499
   592
nipkow@2912
   593
(*** Image of a set under a function ***)
nipkow@2912
   594
nipkow@2912
   595
(*Frequently b does not have the syntactic form of f(x).*)
paulson@5316
   596
Goalw [image_def] "[| b=f(x);  x:A |] ==> b : f``A";
paulson@5316
   597
by (Blast_tac 1);
nipkow@2912
   598
qed "image_eqI";
nipkow@3909
   599
Addsimps [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.*)
paulson@5316
   604
val major::prems = Goalw [image_def]
wenzelm@3842
   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
wenzelm@5069
   613
Goal "f``(A Un B) = f``A Un f``B";
paulson@2935
   614
by (Blast_tac 1);
nipkow@2912
   615
qed "image_Un";
nipkow@2912
   616
wenzelm@5069
   617
Goal "(z : f``A) = (EX x:A. z = f x)";
paulson@3960
   618
by (Blast_tac 1);
paulson@3960
   619
qed "image_iff";
paulson@3960
   620
paulson@4523
   621
(*This rewrite rule would confuse users if made default.*)
wenzelm@5069
   622
Goal "(f``A <= B) = (ALL x:A. f(x): B)";
paulson@4523
   623
by (Blast_tac 1);
paulson@4523
   624
qed "image_subset_iff";
paulson@4523
   625
paulson@4523
   626
(*Replaces the three steps subsetI, imageE, hyp_subst_tac, but breaks too
paulson@4523
   627
  many existing proofs.*)
paulson@5316
   628
val prems = Goal "(!!x. x:A ==> f(x) : B) ==> f``A <= B";
paulson@4510
   629
by (blast_tac (claset() addIs prems) 1);
paulson@4510
   630
qed "image_subsetI";
paulson@4510
   631
nipkow@2912
   632
nipkow@2912
   633
(*** Range of a function -- just a translation for image! ***)
nipkow@2912
   634
paulson@5143
   635
Goal "b=f(x) ==> b : range(f)";
nipkow@2912
   636
by (EVERY1 [etac image_eqI, rtac UNIV_I]);
nipkow@2912
   637
bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI));
nipkow@2912
   638
nipkow@2912
   639
bind_thm ("rangeI", UNIV_I RS imageI);
nipkow@2912
   640
paulson@5316
   641
val [major,minor] = Goal 
wenzelm@3842
   642
    "[| b : range(%x. f(x));  !!x. b=f(x) ==> P |] ==> P"; 
nipkow@2912
   643
by (rtac (major RS imageE) 1);
nipkow@2912
   644
by (etac minor 1);
nipkow@2912
   645
qed "rangeE";
nipkow@2912
   646
oheimb@1776
   647
oheimb@1776
   648
(*** Set reasoning tools ***)
oheimb@1776
   649
oheimb@1776
   650
paulson@3912
   651
(** Rewrite rules for boolean case-splitting: faster than 
nipkow@4830
   652
	addsplits[split_if]
paulson@3912
   653
**)
paulson@3912
   654
nipkow@4830
   655
bind_thm ("split_if_eq1", read_instantiate [("P", "%x. x = ?b")] split_if);
nipkow@4830
   656
bind_thm ("split_if_eq2", read_instantiate [("P", "%x. ?a = x")] split_if);
paulson@3912
   657
paulson@5237
   658
(*Split ifs on either side of the membership relation.
paulson@5237
   659
	Not for Addsimps -- can cause goals to blow up!*)
nipkow@4830
   660
bind_thm ("split_if_mem1", 
nipkow@4830
   661
    read_instantiate_sg (sign_of Set.thy) [("P", "%x. x : ?b")] split_if);
nipkow@4830
   662
bind_thm ("split_if_mem2", 
nipkow@4830
   663
    read_instantiate_sg (sign_of Set.thy) [("P", "%x. ?a : x")] split_if);
paulson@3912
   664
nipkow@4830
   665
val split_ifs = [if_bool_eq_conj, split_if_eq1, split_if_eq2,
nipkow@4830
   666
		  split_if_mem1, split_if_mem2];
paulson@3912
   667
paulson@3912
   668
wenzelm@4089
   669
(*Each of these has ALREADY been added to simpset() above.*)
paulson@2024
   670
val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff, 
paulson@4159
   671
                 mem_Collect_eq, UN_iff, Union_iff, INT_iff, Inter_iff];
oheimb@1776
   672
oheimb@1776
   673
val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs;
oheimb@1776
   674
wenzelm@4089
   675
simpset_ref() := simpset() addcongs [ball_cong,bex_cong]
oheimb@1776
   676
                    setmksimps (mksimps mksimps_pairs);
nipkow@3222
   677
paulson@5256
   678
Addsimps[subset_UNIV, subset_refl];
nipkow@3222
   679
nipkow@3222
   680
nipkow@3222
   681
(*** < ***)
nipkow@3222
   682
wenzelm@5069
   683
Goalw [psubset_def] "!!A::'a set. [| A <= B; A ~= B |] ==> A<B";
nipkow@3222
   684
by (Blast_tac 1);
nipkow@3222
   685
qed "psubsetI";
nipkow@3222
   686
paulson@5148
   687
Goalw [psubset_def] "A < insert x B ==> (x ~: A) & A<=B | x:A & A-{x}<B";
paulson@4477
   688
by Auto_tac;
nipkow@3222
   689
qed "psubset_insertD";
paulson@4059
   690
paulson@4059
   691
bind_thm ("psubset_eq", psubset_def RS meta_eq_to_obj_eq);