src/HOL/Set.ML
author paulson
Wed Feb 03 13:26:07 1999 +0100 (1999-02-03)
changeset 6171 cd237a10cbf8
parent 6006 d2e271b8d651
child 6291 2c3f72d9f5d1
permissions -rw-r--r--
inj is now a translation of inj_on
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@5316
   102
val prems = Goal
clasohm@923
   103
    "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
clasohm@923
   104
\    (! x:A. P(x)) = (! x:B. Q(x))";
clasohm@923
   105
by (resolve_tac (prems RL [ssubst]) 1);
clasohm@923
   106
by (REPEAT (ares_tac [ballI,iffI] 1
clasohm@923
   107
     ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1));
clasohm@923
   108
qed "ball_cong";
clasohm@923
   109
paulson@5316
   110
val prems = Goal
clasohm@923
   111
    "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
clasohm@923
   112
\    (? x:A. P(x)) = (? x:B. Q(x))";
clasohm@923
   113
by (resolve_tac (prems RL [ssubst]) 1);
clasohm@923
   114
by (REPEAT (etac bexE 1
clasohm@923
   115
     ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1));
clasohm@923
   116
qed "bex_cong";
clasohm@923
   117
nipkow@1548
   118
section "Subsets";
clasohm@923
   119
paulson@5316
   120
val prems = Goalw [subset_def] "(!!x. x:A ==> x:B) ==> A <= B";
clasohm@923
   121
by (REPEAT (ares_tac (prems @ [ballI]) 1));
clasohm@923
   122
qed "subsetI";
clasohm@923
   123
paulson@5649
   124
(*Map the type ('a set => anything) to just 'a.
paulson@5649
   125
  For overloading constants whose first argument has type "'a set" *)
paulson@5649
   126
fun overload_1st_set s = Blast.overloaded (s, HOLogic.dest_setT o domain_type);
paulson@5649
   127
paulson@4059
   128
(*While (:) is not, its type must be kept
paulson@4059
   129
  for overloading of = to work.*)
paulson@4240
   130
Blast.overloaded ("op :", domain_type);
paulson@5649
   131
paulson@5649
   132
overload_1st_set "Ball";		(*need UNION, INTER also?*)
paulson@5649
   133
overload_1st_set "Bex";
paulson@4059
   134
paulson@4469
   135
(*Image: retain the type of the set being expressed*)
paulson@5336
   136
Blast.overloaded ("op ``", domain_type);
paulson@2881
   137
clasohm@923
   138
(*Rule in Modus Ponens style*)
paulson@5316
   139
Goalw [subset_def] "[| A <= B;  c:A |] ==> c:B";
paulson@5316
   140
by (Blast_tac 1);
clasohm@923
   141
qed "subsetD";
clasohm@923
   142
clasohm@923
   143
(*The same, with reversed premises for use with etac -- cf rev_mp*)
clasohm@923
   144
qed_goal "rev_subsetD" Set.thy "[| c:A;  A <= B |] ==> c:B"
clasohm@923
   145
 (fn prems=>  [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]);
clasohm@923
   146
paulson@1920
   147
(*Converts A<=B to x:A ==> x:B*)
paulson@1920
   148
fun impOfSubs th = th RSN (2, rev_subsetD);
paulson@1920
   149
paulson@1841
   150
qed_goal "contra_subsetD" Set.thy "!!c. [| A <= B; c ~: B |] ==> c ~: A"
paulson@1841
   151
 (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
paulson@1841
   152
paulson@1841
   153
qed_goal "rev_contra_subsetD" Set.thy "!!c. [| c ~: B;  A <= B |] ==> c ~: A"
paulson@1841
   154
 (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
paulson@1841
   155
clasohm@923
   156
(*Classical elimination rule*)
paulson@5316
   157
val major::prems = Goalw [subset_def] 
clasohm@923
   158
    "[| A <= B;  c~:A ==> P;  c:B ==> P |] ==> P";
clasohm@923
   159
by (rtac (major RS ballE) 1);
clasohm@923
   160
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   161
qed "subsetCE";
clasohm@923
   162
clasohm@923
   163
(*Takes assumptions A<=B; c:A and creates the assumption c:B *)
clasohm@923
   164
fun set_mp_tac i = etac subsetCE i  THEN  mp_tac i;
clasohm@923
   165
paulson@2499
   166
AddSIs [subsetI];
paulson@2499
   167
AddEs  [subsetD, subsetCE];
clasohm@923
   168
paulson@2499
   169
qed_goal "subset_refl" Set.thy "A <= (A::'a set)"
paulson@4059
   170
 (fn _=> [Fast_tac 1]);		(*Blast_tac would try order_refl and fail*)
paulson@2499
   171
paulson@5316
   172
Goal "[| A<=B;  B<=C |] ==> A<=(C::'a set)";
paulson@2891
   173
by (Blast_tac 1);
clasohm@923
   174
qed "subset_trans";
clasohm@923
   175
clasohm@923
   176
nipkow@1548
   177
section "Equality";
clasohm@923
   178
clasohm@923
   179
(*Anti-symmetry of the subset relation*)
paulson@5316
   180
Goal "[| A <= B;  B <= A |] ==> A = (B::'a set)";
paulson@5318
   181
by (rtac set_ext 1);
paulson@5316
   182
by (blast_tac (claset() addIs [subsetD]) 1);
clasohm@923
   183
qed "subset_antisym";
clasohm@923
   184
val equalityI = subset_antisym;
clasohm@923
   185
berghofe@1762
   186
AddSIs [equalityI];
berghofe@1762
   187
clasohm@923
   188
(* Equality rules from ZF set theory -- are they appropriate here? *)
paulson@5316
   189
Goal "A = B ==> A<=(B::'a set)";
paulson@5316
   190
by (etac ssubst 1);
clasohm@923
   191
by (rtac subset_refl 1);
clasohm@923
   192
qed "equalityD1";
clasohm@923
   193
paulson@5316
   194
Goal "A = B ==> B<=(A::'a set)";
paulson@5316
   195
by (etac ssubst 1);
clasohm@923
   196
by (rtac subset_refl 1);
clasohm@923
   197
qed "equalityD2";
clasohm@923
   198
paulson@5316
   199
val prems = Goal
clasohm@923
   200
    "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";
clasohm@923
   201
by (resolve_tac prems 1);
clasohm@923
   202
by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
clasohm@923
   203
qed "equalityE";
clasohm@923
   204
paulson@5316
   205
val major::prems = Goal
clasohm@923
   206
    "[| A = B;  [| c:A; c:B |] ==> P;  [| c~:A; c~:B |] ==> P |]  ==>  P";
clasohm@923
   207
by (rtac (major RS equalityE) 1);
clasohm@923
   208
by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
clasohm@923
   209
qed "equalityCE";
clasohm@923
   210
clasohm@923
   211
(*Lemma for creating induction formulae -- for "pattern matching" on p
clasohm@923
   212
  To make the induction hypotheses usable, apply "spec" or "bspec" to
clasohm@923
   213
  put universal quantifiers over the free variables in p. *)
paulson@5316
   214
val prems = Goal 
clasohm@923
   215
    "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
clasohm@923
   216
by (rtac mp 1);
clasohm@923
   217
by (REPEAT (resolve_tac (refl::prems) 1));
clasohm@923
   218
qed "setup_induction";
clasohm@923
   219
clasohm@923
   220
paulson@4159
   221
section "The universal set -- UNIV";
paulson@4159
   222
paulson@4159
   223
qed_goalw "UNIV_I" Set.thy [UNIV_def] "x : UNIV"
paulson@4159
   224
  (fn _ => [rtac CollectI 1, rtac TrueI 1]);
paulson@4159
   225
paulson@4434
   226
Addsimps [UNIV_I];
paulson@4434
   227
AddIs    [UNIV_I];  (*unsafe makes it less likely to cause problems*)
paulson@4159
   228
paulson@4159
   229
qed_goal "subset_UNIV" Set.thy "A <= UNIV"
paulson@4159
   230
  (fn _ => [rtac subsetI 1, rtac UNIV_I 1]);
paulson@4159
   231
paulson@4159
   232
(** Eta-contracting these two rules (to remove P) causes them to be ignored
paulson@4159
   233
    because of their interaction with congruence rules. **)
paulson@4159
   234
wenzelm@5069
   235
Goalw [Ball_def] "Ball UNIV P = All P";
paulson@4159
   236
by (Simp_tac 1);
paulson@4159
   237
qed "ball_UNIV";
paulson@4159
   238
wenzelm@5069
   239
Goalw [Bex_def] "Bex UNIV P = Ex P";
paulson@4159
   240
by (Simp_tac 1);
paulson@4159
   241
qed "bex_UNIV";
paulson@4159
   242
Addsimps [ball_UNIV, bex_UNIV];
paulson@4159
   243
paulson@4159
   244
paulson@2858
   245
section "The empty set -- {}";
paulson@2858
   246
paulson@2858
   247
qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False"
paulson@2891
   248
 (fn _ => [ (Blast_tac 1) ]);
paulson@2858
   249
paulson@2858
   250
Addsimps [empty_iff];
paulson@2858
   251
paulson@2858
   252
qed_goal "emptyE" Set.thy "!!a. a:{} ==> P"
paulson@2858
   253
 (fn _ => [Full_simp_tac 1]);
paulson@2858
   254
paulson@2858
   255
AddSEs [emptyE];
paulson@2858
   256
paulson@2858
   257
qed_goal "empty_subsetI" Set.thy "{} <= A"
paulson@2891
   258
 (fn _ => [ (Blast_tac 1) ]);
paulson@2858
   259
paulson@5256
   260
(*One effect is to delete the ASSUMPTION {} <= A*)
paulson@5256
   261
AddIffs [empty_subsetI];
paulson@5256
   262
paulson@2858
   263
qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}"
paulson@2858
   264
 (fn [prem]=>
wenzelm@4089
   265
  [ (blast_tac (claset() addIs [prem RS FalseE]) 1) ]);
paulson@2858
   266
paulson@5256
   267
(*Use for reasoning about disjointness: A Int B = {} *)
paulson@5450
   268
qed_goal "equals0D" Set.thy "!!a. A={} ==> a ~: A"
paulson@2891
   269
 (fn _ => [ (Blast_tac 1) ]);
paulson@2858
   270
paulson@5450
   271
AddDs [equals0D, sym RS equals0D];
paulson@5256
   272
wenzelm@5069
   273
Goalw [Ball_def] "Ball {} P = True";
paulson@4159
   274
by (Simp_tac 1);
paulson@4159
   275
qed "ball_empty";
paulson@4159
   276
wenzelm@5069
   277
Goalw [Bex_def] "Bex {} P = False";
paulson@4159
   278
by (Simp_tac 1);
paulson@4159
   279
qed "bex_empty";
paulson@4159
   280
Addsimps [ball_empty, bex_empty];
paulson@4159
   281
wenzelm@5069
   282
Goal "UNIV ~= {}";
paulson@4159
   283
by (blast_tac (claset() addEs [equalityE]) 1);
paulson@4159
   284
qed "UNIV_not_empty";
paulson@4159
   285
AddIffs [UNIV_not_empty];
paulson@4159
   286
paulson@4159
   287
paulson@2858
   288
paulson@2858
   289
section "The Powerset operator -- Pow";
paulson@2858
   290
paulson@2858
   291
qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)"
paulson@2858
   292
 (fn _ => [ (Asm_simp_tac 1) ]);
paulson@2858
   293
paulson@2858
   294
AddIffs [Pow_iff]; 
paulson@2858
   295
paulson@2858
   296
qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
paulson@2858
   297
 (fn _ => [ (etac CollectI 1) ]);
paulson@2858
   298
paulson@2858
   299
qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B)  ==>  A<=B"
paulson@2858
   300
 (fn _=> [ (etac CollectD 1) ]);
paulson@2858
   301
paulson@2858
   302
val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)
paulson@2858
   303
val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)
paulson@2858
   304
paulson@2858
   305
paulson@5931
   306
section "Set complement";
clasohm@923
   307
paulson@5490
   308
qed_goalw "Compl_iff" Set.thy [Compl_def] "(c : -A) = (c~:A)"
paulson@2891
   309
 (fn _ => [ (Blast_tac 1) ]);
paulson@2499
   310
paulson@2499
   311
Addsimps [Compl_iff];
paulson@2499
   312
paulson@5490
   313
val prems = Goalw [Compl_def] "[| c:A ==> False |] ==> c : -A";
clasohm@923
   314
by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
clasohm@923
   315
qed "ComplI";
clasohm@923
   316
clasohm@923
   317
(*This form, with negated conclusion, works well with the Classical prover.
clasohm@923
   318
  Negated assumptions behave like formulae on the right side of the notional
clasohm@923
   319
  turnstile...*)
paulson@5490
   320
Goalw [Compl_def] "c : -A ==> c~:A";
paulson@5316
   321
by (etac CollectD 1);
clasohm@923
   322
qed "ComplD";
clasohm@923
   323
clasohm@923
   324
val ComplE = make_elim ComplD;
clasohm@923
   325
paulson@2499
   326
AddSIs [ComplI];
paulson@2499
   327
AddSEs [ComplE];
paulson@1640
   328
clasohm@923
   329
nipkow@1548
   330
section "Binary union -- Un";
clasohm@923
   331
paulson@2499
   332
qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)"
paulson@2891
   333
 (fn _ => [ Blast_tac 1 ]);
paulson@2499
   334
paulson@2499
   335
Addsimps [Un_iff];
paulson@2499
   336
paulson@5143
   337
Goal "c:A ==> c : A Un B";
paulson@2499
   338
by (Asm_simp_tac 1);
clasohm@923
   339
qed "UnI1";
clasohm@923
   340
paulson@5143
   341
Goal "c:B ==> c : A Un B";
paulson@2499
   342
by (Asm_simp_tac 1);
clasohm@923
   343
qed "UnI2";
clasohm@923
   344
clasohm@923
   345
(*Classical introduction rule: no commitment to A vs B*)
clasohm@923
   346
qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B"
clasohm@923
   347
 (fn prems=>
paulson@2499
   348
  [ (Simp_tac 1),
paulson@2499
   349
    (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
clasohm@923
   350
paulson@5316
   351
val major::prems = Goalw [Un_def]
clasohm@923
   352
    "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
clasohm@923
   353
by (rtac (major RS CollectD RS disjE) 1);
clasohm@923
   354
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   355
qed "UnE";
clasohm@923
   356
paulson@2499
   357
AddSIs [UnCI];
paulson@2499
   358
AddSEs [UnE];
paulson@1640
   359
clasohm@923
   360
nipkow@1548
   361
section "Binary intersection -- Int";
clasohm@923
   362
paulson@2499
   363
qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)"
paulson@2891
   364
 (fn _ => [ (Blast_tac 1) ]);
paulson@2499
   365
paulson@2499
   366
Addsimps [Int_iff];
paulson@2499
   367
paulson@5143
   368
Goal "[| c:A;  c:B |] ==> c : A Int B";
paulson@2499
   369
by (Asm_simp_tac 1);
clasohm@923
   370
qed "IntI";
clasohm@923
   371
paulson@5143
   372
Goal "c : A Int B ==> c:A";
paulson@2499
   373
by (Asm_full_simp_tac 1);
clasohm@923
   374
qed "IntD1";
clasohm@923
   375
paulson@5143
   376
Goal "c : A Int B ==> c:B";
paulson@2499
   377
by (Asm_full_simp_tac 1);
clasohm@923
   378
qed "IntD2";
clasohm@923
   379
paulson@5316
   380
val [major,minor] = Goal
clasohm@923
   381
    "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
clasohm@923
   382
by (rtac minor 1);
clasohm@923
   383
by (rtac (major RS IntD1) 1);
clasohm@923
   384
by (rtac (major RS IntD2) 1);
clasohm@923
   385
qed "IntE";
clasohm@923
   386
paulson@2499
   387
AddSIs [IntI];
paulson@2499
   388
AddSEs [IntE];
clasohm@923
   389
nipkow@1548
   390
section "Set difference";
clasohm@923
   391
paulson@2499
   392
qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)"
paulson@2891
   393
 (fn _ => [ (Blast_tac 1) ]);
clasohm@923
   394
paulson@2499
   395
Addsimps [Diff_iff];
paulson@2499
   396
paulson@2499
   397
qed_goal "DiffI" Set.thy "!!c. [| c : A;  c ~: B |] ==> c : A - B"
paulson@2499
   398
 (fn _=> [ Asm_simp_tac 1 ]);
clasohm@923
   399
paulson@2499
   400
qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A"
paulson@2499
   401
 (fn _=> [ (Asm_full_simp_tac 1) ]);
clasohm@923
   402
paulson@2499
   403
qed_goal "DiffD2" Set.thy "!!c. [| c : A - B;  c : B |] ==> P"
paulson@2499
   404
 (fn _=> [ (Asm_full_simp_tac 1) ]);
paulson@2499
   405
paulson@2499
   406
qed_goal "DiffE" Set.thy "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P"
clasohm@923
   407
 (fn prems=>
clasohm@923
   408
  [ (resolve_tac prems 1),
clasohm@923
   409
    (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]);
clasohm@923
   410
paulson@2499
   411
AddSIs [DiffI];
paulson@2499
   412
AddSEs [DiffE];
clasohm@923
   413
clasohm@923
   414
nipkow@1548
   415
section "Augmenting a set -- insert";
clasohm@923
   416
paulson@2499
   417
qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)"
paulson@2891
   418
 (fn _ => [Blast_tac 1]);
paulson@2499
   419
paulson@2499
   420
Addsimps [insert_iff];
clasohm@923
   421
paulson@2499
   422
qed_goal "insertI1" Set.thy "a : insert a B"
paulson@2499
   423
 (fn _ => [Simp_tac 1]);
paulson@2499
   424
paulson@2499
   425
qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B"
paulson@2499
   426
 (fn _=> [Asm_simp_tac 1]);
clasohm@923
   427
clasohm@923
   428
qed_goalw "insertE" Set.thy [insert_def]
clasohm@923
   429
    "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P"
clasohm@923
   430
 (fn major::prems=>
clasohm@923
   431
  [ (rtac (major RS UnE) 1),
clasohm@923
   432
    (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
clasohm@923
   433
clasohm@923
   434
(*Classical introduction rule*)
clasohm@923
   435
qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
paulson@2499
   436
 (fn prems=>
paulson@2499
   437
  [ (Simp_tac 1),
paulson@2499
   438
    (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
paulson@2499
   439
paulson@2499
   440
AddSIs [insertCI]; 
paulson@2499
   441
AddSEs [insertE];
clasohm@923
   442
nipkow@1548
   443
section "Singletons, using insert";
clasohm@923
   444
clasohm@923
   445
qed_goal "singletonI" Set.thy "a : {a}"
clasohm@923
   446
 (fn _=> [ (rtac insertI1 1) ]);
clasohm@923
   447
paulson@5143
   448
Goal "b : {a} ==> b=a";
paulson@2891
   449
by (Blast_tac 1);
clasohm@923
   450
qed "singletonD";
clasohm@923
   451
oheimb@1776
   452
bind_thm ("singletonE", make_elim singletonD);
oheimb@1776
   453
paulson@2499
   454
qed_goal "singleton_iff" thy "(b : {a}) = (b=a)" 
paulson@2891
   455
(fn _ => [Blast_tac 1]);
clasohm@923
   456
paulson@5143
   457
Goal "{a}={b} ==> a=b";
wenzelm@4089
   458
by (blast_tac (claset() addEs [equalityE]) 1);
clasohm@923
   459
qed "singleton_inject";
clasohm@923
   460
paulson@2858
   461
(*Redundant? But unlike insertCI, it proves the subgoal immediately!*)
paulson@2858
   462
AddSIs [singletonI];   
paulson@2499
   463
AddSDs [singleton_inject];
paulson@3718
   464
AddSEs [singletonE];
paulson@2499
   465
wenzelm@5069
   466
Goal "{x. x=a} = {a}";
wenzelm@4423
   467
by (Blast_tac 1);
nipkow@3582
   468
qed "singleton_conv";
nipkow@3582
   469
Addsimps [singleton_conv];
nipkow@1531
   470
nipkow@5600
   471
Goal "{x. a=x} = {a}";
nipkow@5600
   472
by(Blast_tac 1);
nipkow@5600
   473
qed "singleton_conv2";
nipkow@5600
   474
Addsimps [singleton_conv2];
nipkow@5600
   475
nipkow@1531
   476
nipkow@1548
   477
section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
clasohm@923
   478
wenzelm@5069
   479
Goalw [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
paulson@2891
   480
by (Blast_tac 1);
paulson@2499
   481
qed "UN_iff";
paulson@2499
   482
paulson@2499
   483
Addsimps [UN_iff];
paulson@2499
   484
clasohm@923
   485
(*The order of the premises presupposes that A is rigid; b may be flexible*)
paulson@5143
   486
Goal "[| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
paulson@4477
   487
by Auto_tac;
clasohm@923
   488
qed "UN_I";
clasohm@923
   489
paulson@5316
   490
val major::prems = Goalw [UNION_def]
clasohm@923
   491
    "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
clasohm@923
   492
by (rtac (major RS CollectD RS bexE) 1);
clasohm@923
   493
by (REPEAT (ares_tac prems 1));
clasohm@923
   494
qed "UN_E";
clasohm@923
   495
paulson@2499
   496
AddIs  [UN_I];
paulson@2499
   497
AddSEs [UN_E];
paulson@2499
   498
paulson@5316
   499
val prems = Goal
clasohm@923
   500
    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
clasohm@923
   501
\    (UN x:A. C(x)) = (UN x:B. D(x))";
clasohm@923
   502
by (REPEAT (etac UN_E 1
clasohm@923
   503
     ORELSE ares_tac ([UN_I,equalityI,subsetI] @ 
clasohm@1465
   504
                      (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
clasohm@923
   505
qed "UN_cong";
clasohm@923
   506
clasohm@923
   507
nipkow@1548
   508
section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
clasohm@923
   509
wenzelm@5069
   510
Goalw [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
paulson@4477
   511
by Auto_tac;
paulson@2499
   512
qed "INT_iff";
paulson@2499
   513
paulson@2499
   514
Addsimps [INT_iff];
paulson@2499
   515
paulson@5316
   516
val prems = Goalw [INTER_def]
clasohm@923
   517
    "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
clasohm@923
   518
by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
clasohm@923
   519
qed "INT_I";
clasohm@923
   520
paulson@5143
   521
Goal "[| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
paulson@4477
   522
by Auto_tac;
clasohm@923
   523
qed "INT_D";
clasohm@923
   524
clasohm@923
   525
(*"Classical" elimination -- by the Excluded Middle on a:A *)
paulson@5316
   526
val major::prems = Goalw [INTER_def]
clasohm@923
   527
    "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
clasohm@923
   528
by (rtac (major RS CollectD RS ballE) 1);
clasohm@923
   529
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   530
qed "INT_E";
clasohm@923
   531
paulson@2499
   532
AddSIs [INT_I];
paulson@2499
   533
AddEs  [INT_D, INT_E];
paulson@2499
   534
paulson@5316
   535
val prems = Goal
clasohm@923
   536
    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
clasohm@923
   537
\    (INT x:A. C(x)) = (INT x:B. D(x))";
clasohm@923
   538
by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
clasohm@923
   539
by (REPEAT (dtac INT_D 1
clasohm@923
   540
     ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
clasohm@923
   541
qed "INT_cong";
clasohm@923
   542
clasohm@923
   543
nipkow@1548
   544
section "Union";
clasohm@923
   545
wenzelm@5069
   546
Goalw [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
paulson@2891
   547
by (Blast_tac 1);
paulson@2499
   548
qed "Union_iff";
paulson@2499
   549
paulson@2499
   550
Addsimps [Union_iff];
paulson@2499
   551
clasohm@923
   552
(*The order of the premises presupposes that C is rigid; A may be flexible*)
paulson@5143
   553
Goal "[| X:C;  A:X |] ==> A : Union(C)";
paulson@4477
   554
by Auto_tac;
clasohm@923
   555
qed "UnionI";
clasohm@923
   556
paulson@5316
   557
val major::prems = Goalw [Union_def]
clasohm@923
   558
    "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
clasohm@923
   559
by (rtac (major RS UN_E) 1);
clasohm@923
   560
by (REPEAT (ares_tac prems 1));
clasohm@923
   561
qed "UnionE";
clasohm@923
   562
paulson@2499
   563
AddIs  [UnionI];
paulson@2499
   564
AddSEs [UnionE];
paulson@2499
   565
paulson@2499
   566
nipkow@1548
   567
section "Inter";
clasohm@923
   568
wenzelm@5069
   569
Goalw [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
paulson@2891
   570
by (Blast_tac 1);
paulson@2499
   571
qed "Inter_iff";
paulson@2499
   572
paulson@2499
   573
Addsimps [Inter_iff];
paulson@2499
   574
paulson@5316
   575
val prems = Goalw [Inter_def]
clasohm@923
   576
    "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
clasohm@923
   577
by (REPEAT (ares_tac ([INT_I] @ prems) 1));
clasohm@923
   578
qed "InterI";
clasohm@923
   579
clasohm@923
   580
(*A "destruct" rule -- every X in C contains A as an element, but
clasohm@923
   581
  A:X can hold when X:C does not!  This rule is analogous to "spec". *)
paulson@5143
   582
Goal "[| A : Inter(C);  X:C |] ==> A:X";
paulson@4477
   583
by Auto_tac;
clasohm@923
   584
qed "InterD";
clasohm@923
   585
clasohm@923
   586
(*"Classical" elimination rule -- does not require proving X:C *)
paulson@5316
   587
val major::prems = Goalw [Inter_def]
paulson@2721
   588
    "[| A : Inter(C);  X~:C ==> R;  A:X ==> R |] ==> R";
clasohm@923
   589
by (rtac (major RS INT_E) 1);
clasohm@923
   590
by (REPEAT (eresolve_tac prems 1));
clasohm@923
   591
qed "InterE";
clasohm@923
   592
paulson@2499
   593
AddSIs [InterI];
paulson@2499
   594
AddEs  [InterD, InterE];
paulson@2499
   595
paulson@2499
   596
nipkow@2912
   597
(*** Image of a set under a function ***)
nipkow@2912
   598
nipkow@2912
   599
(*Frequently b does not have the syntactic form of f(x).*)
paulson@5316
   600
Goalw [image_def] "[| b=f(x);  x:A |] ==> b : f``A";
paulson@5316
   601
by (Blast_tac 1);
nipkow@2912
   602
qed "image_eqI";
nipkow@3909
   603
Addsimps [image_eqI];
nipkow@2912
   604
nipkow@2912
   605
bind_thm ("imageI", refl RS image_eqI);
nipkow@2912
   606
nipkow@2912
   607
(*The eta-expansion gives variable-name preservation.*)
paulson@5316
   608
val major::prems = Goalw [image_def]
wenzelm@3842
   609
    "[| b : (%x. f(x))``A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P"; 
nipkow@2912
   610
by (rtac (major RS CollectD RS bexE) 1);
nipkow@2912
   611
by (REPEAT (ares_tac prems 1));
nipkow@2912
   612
qed "imageE";
nipkow@2912
   613
nipkow@2912
   614
AddIs  [image_eqI];
nipkow@2912
   615
AddSEs [imageE]; 
nipkow@2912
   616
wenzelm@5069
   617
Goal "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
wenzelm@5069
   621
Goal "(z : f``A) = (EX x:A. z = f x)";
paulson@3960
   622
by (Blast_tac 1);
paulson@3960
   623
qed "image_iff";
paulson@3960
   624
paulson@4523
   625
(*This rewrite rule would confuse users if made default.*)
wenzelm@5069
   626
Goal "(f``A <= B) = (ALL x:A. f(x): B)";
paulson@4523
   627
by (Blast_tac 1);
paulson@4523
   628
qed "image_subset_iff";
paulson@4523
   629
paulson@4523
   630
(*Replaces the three steps subsetI, imageE, hyp_subst_tac, but breaks too
paulson@4523
   631
  many existing proofs.*)
paulson@5316
   632
val prems = Goal "(!!x. x:A ==> f(x) : B) ==> f``A <= B";
paulson@4510
   633
by (blast_tac (claset() addIs prems) 1);
paulson@4510
   634
qed "image_subsetI";
paulson@4510
   635
nipkow@2912
   636
nipkow@2912
   637
(*** Range of a function -- just a translation for image! ***)
nipkow@2912
   638
paulson@5143
   639
Goal "b=f(x) ==> b : range(f)";
nipkow@2912
   640
by (EVERY1 [etac image_eqI, rtac UNIV_I]);
nipkow@2912
   641
bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI));
nipkow@2912
   642
nipkow@2912
   643
bind_thm ("rangeI", UNIV_I RS imageI);
nipkow@2912
   644
paulson@5316
   645
val [major,minor] = Goal 
wenzelm@3842
   646
    "[| b : range(%x. f(x));  !!x. b=f(x) ==> P |] ==> P"; 
nipkow@2912
   647
by (rtac (major RS imageE) 1);
nipkow@2912
   648
by (etac minor 1);
nipkow@2912
   649
qed "rangeE";
nipkow@2912
   650
oheimb@1776
   651
oheimb@1776
   652
(*** Set reasoning tools ***)
oheimb@1776
   653
oheimb@1776
   654
paulson@3912
   655
(** Rewrite rules for boolean case-splitting: faster than 
nipkow@4830
   656
	addsplits[split_if]
paulson@3912
   657
**)
paulson@3912
   658
nipkow@4830
   659
bind_thm ("split_if_eq1", read_instantiate [("P", "%x. x = ?b")] split_if);
nipkow@4830
   660
bind_thm ("split_if_eq2", read_instantiate [("P", "%x. ?a = x")] split_if);
paulson@3912
   661
paulson@5237
   662
(*Split ifs on either side of the membership relation.
paulson@5237
   663
	Not for Addsimps -- can cause goals to blow up!*)
nipkow@4830
   664
bind_thm ("split_if_mem1", 
nipkow@4830
   665
    read_instantiate_sg (sign_of Set.thy) [("P", "%x. x : ?b")] split_if);
nipkow@4830
   666
bind_thm ("split_if_mem2", 
nipkow@4830
   667
    read_instantiate_sg (sign_of Set.thy) [("P", "%x. ?a : x")] split_if);
paulson@3912
   668
nipkow@4830
   669
val split_ifs = [if_bool_eq_conj, split_if_eq1, split_if_eq2,
nipkow@4830
   670
		  split_if_mem1, split_if_mem2];
paulson@3912
   671
paulson@3912
   672
wenzelm@4089
   673
(*Each of these has ALREADY been added to simpset() above.*)
paulson@2024
   674
val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff, 
paulson@4159
   675
                 mem_Collect_eq, UN_iff, Union_iff, INT_iff, Inter_iff];
oheimb@1776
   676
oheimb@1776
   677
val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs;
oheimb@1776
   678
wenzelm@4089
   679
simpset_ref() := simpset() addcongs [ball_cong,bex_cong]
oheimb@1776
   680
                    setmksimps (mksimps mksimps_pairs);
nipkow@3222
   681
paulson@5256
   682
Addsimps[subset_UNIV, subset_refl];
nipkow@3222
   683
nipkow@3222
   684
nipkow@3222
   685
(*** < ***)
nipkow@3222
   686
wenzelm@5069
   687
Goalw [psubset_def] "!!A::'a set. [| A <= B; A ~= B |] ==> A<B";
nipkow@3222
   688
by (Blast_tac 1);
nipkow@3222
   689
qed "psubsetI";
nipkow@3222
   690
paulson@5148
   691
Goalw [psubset_def] "A < insert x B ==> (x ~: A) & A<=B | x:A & A-{x}<B";
paulson@4477
   692
by Auto_tac;
nipkow@3222
   693
qed "psubset_insertD";
paulson@4059
   694
paulson@4059
   695
bind_thm ("psubset_eq", psubset_def RS meta_eq_to_obj_eq);