1.1 --- a/src/HOL/Set.ML Thu Jan 09 10:22:42 1997 +0100
1.2 +++ b/src/HOL/Set.ML Thu Jan 09 10:23:39 1997 +0100
1.3 @@ -10,13 +10,14 @@
1.4
1.5 section "Relating predicates and sets";
1.6
1.7 -val [prem] = goal Set.thy "P(a) ==> a : {x.P(x)}";
1.8 -by (stac mem_Collect_eq 1);
1.9 -by (rtac prem 1);
1.10 +AddIffs [mem_Collect_eq];
1.11 +
1.12 +goal Set.thy "!!a. P(a) ==> a : {x.P(x)}";
1.13 +by (Asm_simp_tac 1);
1.14 qed "CollectI";
1.15
1.16 -val prems = goal Set.thy "[| a : {x.P(x)} |] ==> P(a)";
1.17 -by (resolve_tac (prems RL [mem_Collect_eq RS subst]) 1);
1.18 +val prems = goal Set.thy "!!a. a : {x.P(x)} ==> P(a)";
1.19 +by (Asm_full_simp_tac 1);
1.20 qed "CollectD";
1.21
1.22 val [prem] = goal Set.thy "[| !!x. (x:A) = (x:B) |] ==> A = B";
1.23 @@ -31,6 +32,10 @@
1.24
1.25 val CollectE = make_elim CollectD;
1.26
1.27 +AddSIs [CollectI];
1.28 +AddSEs [CollectE];
1.29 +
1.30 +
1.31 section "Bounded quantifiers";
1.32
1.33 val prems = goalw Set.thy [Ball_def]
1.34 @@ -52,6 +57,9 @@
1.35 (*Takes assumptions ! x:A.P(x) and a:A; creates assumption P(a)*)
1.36 fun ball_tac i = etac ballE i THEN contr_tac (i+1);
1.37
1.38 +AddSIs [ballI];
1.39 +AddEs [ballE];
1.40 +
1.41 val prems = goalw Set.thy [Bex_def]
1.42 "[| P(x); x:A |] ==> ? x:A. P(x)";
1.43 by (REPEAT (ares_tac (prems @ [exI,conjI]) 1));
1.44 @@ -69,6 +77,9 @@
1.45 by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1));
1.46 qed "bexE";
1.47
1.48 +AddIs [bexI];
1.49 +AddSEs [bexE];
1.50 +
1.51 (*Trival rewrite rule; (! x:A.P)=P holds only if A is nonempty!*)
1.52 goalw Set.thy [Ball_def] "(! x:A. True) = True";
1.53 by (Simp_tac 1);
1.54 @@ -134,12 +145,14 @@
1.55 (*Takes assumptions A<=B; c:A and creates the assumption c:B *)
1.56 fun set_mp_tac i = etac subsetCE i THEN mp_tac i;
1.57
1.58 -qed_goal "subset_refl" Set.thy "A <= (A::'a set)"
1.59 - (fn _=> [ (REPEAT (ares_tac [subsetI] 1)) ]);
1.60 +AddSIs [subsetI];
1.61 +AddEs [subsetD, subsetCE];
1.62
1.63 -val prems = goal Set.thy "[| A<=B; B<=C |] ==> A<=(C::'a set)";
1.64 -by (cut_facts_tac prems 1);
1.65 -by (REPEAT (ares_tac [subsetI] 1 ORELSE set_mp_tac 1));
1.66 +qed_goal "subset_refl" Set.thy "A <= (A::'a set)"
1.67 + (fn _=> [Fast_tac 1]);
1.68 +
1.69 +val prems = goal Set.thy "!!B. [| A<=B; B<=C |] ==> A<=(C::'a set)";
1.70 +by (Fast_tac 1);
1.71 qed "subset_trans";
1.72
1.73
1.74 @@ -189,6 +202,11 @@
1.75
1.76 section "Set complement -- Compl";
1.77
1.78 +qed_goalw "Compl_iff" Set.thy [Compl_def] "(c : Compl(A)) = (c~:A)"
1.79 + (fn _ => [ (Fast_tac 1) ]);
1.80 +
1.81 +Addsimps [Compl_iff];
1.82 +
1.83 val prems = goalw Set.thy [Compl_def]
1.84 "[| c:A ==> False |] ==> c : Compl(A)";
1.85 by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
1.86 @@ -198,32 +216,36 @@
1.87 Negated assumptions behave like formulae on the right side of the notional
1.88 turnstile...*)
1.89 val major::prems = goalw Set.thy [Compl_def]
1.90 - "[| c : Compl(A) |] ==> c~:A";
1.91 + "c : Compl(A) ==> c~:A";
1.92 by (rtac (major RS CollectD) 1);
1.93 qed "ComplD";
1.94
1.95 val ComplE = make_elim ComplD;
1.96
1.97 -qed_goal "Compl_iff" Set.thy "(c : Compl(A)) = (c~:A)"
1.98 - (fn _ => [ (fast_tac (!claset addSIs [ComplI] addSEs [ComplE]) 1) ]);
1.99 +AddSIs [ComplI];
1.100 +AddSEs [ComplE];
1.101
1.102
1.103 section "Binary union -- Un";
1.104
1.105 -val prems = goalw Set.thy [Un_def] "c:A ==> c : A Un B";
1.106 -by (REPEAT (resolve_tac (prems @ [CollectI,disjI1]) 1));
1.107 +qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)"
1.108 + (fn _ => [ Fast_tac 1 ]);
1.109 +
1.110 +Addsimps [Un_iff];
1.111 +
1.112 +goal Set.thy "!!c. c:A ==> c : A Un B";
1.113 +by (Asm_simp_tac 1);
1.114 qed "UnI1";
1.115
1.116 -val prems = goalw Set.thy [Un_def] "c:B ==> c : A Un B";
1.117 -by (REPEAT (resolve_tac (prems @ [CollectI,disjI2]) 1));
1.118 +goal Set.thy "!!c. c:B ==> c : A Un B";
1.119 +by (Asm_simp_tac 1);
1.120 qed "UnI2";
1.121
1.122 (*Classical introduction rule: no commitment to A vs B*)
1.123 qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B"
1.124 (fn prems=>
1.125 - [ (rtac classical 1),
1.126 - (REPEAT (ares_tac (prems@[UnI1,notI]) 1)),
1.127 - (REPEAT (ares_tac (prems@[UnI2,notE]) 1)) ]);
1.128 + [ (Simp_tac 1),
1.129 + (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
1.130
1.131 val major::prems = goalw Set.thy [Un_def]
1.132 "[| c : A Un B; c:A ==> P; c:B ==> P |] ==> P";
1.133 @@ -231,23 +253,27 @@
1.134 by (REPEAT (eresolve_tac prems 1));
1.135 qed "UnE";
1.136
1.137 -qed_goal "Un_iff" Set.thy "(c : A Un B) = (c:A | c:B)"
1.138 - (fn _ => [ (fast_tac (!claset addSIs [UnCI] addSEs [UnE]) 1) ]);
1.139 +AddSIs [UnCI];
1.140 +AddSEs [UnE];
1.141
1.142
1.143 section "Binary intersection -- Int";
1.144
1.145 -val prems = goalw Set.thy [Int_def]
1.146 - "[| c:A; c:B |] ==> c : A Int B";
1.147 -by (REPEAT (resolve_tac (prems @ [CollectI,conjI]) 1));
1.148 +qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)"
1.149 + (fn _ => [ (Fast_tac 1) ]);
1.150 +
1.151 +Addsimps [Int_iff];
1.152 +
1.153 +goal Set.thy "!!c. [| c:A; c:B |] ==> c : A Int B";
1.154 +by (Asm_simp_tac 1);
1.155 qed "IntI";
1.156
1.157 -val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:A";
1.158 -by (rtac (major RS CollectD RS conjunct1) 1);
1.159 +goal Set.thy "!!c. c : A Int B ==> c:A";
1.160 +by (Asm_full_simp_tac 1);
1.161 qed "IntD1";
1.162
1.163 -val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:B";
1.164 -by (rtac (major RS CollectD RS conjunct2) 1);
1.165 +goal Set.thy "!!c. c : A Int B ==> c:B";
1.166 +by (Asm_full_simp_tac 1);
1.167 qed "IntD2";
1.168
1.169 val [major,minor] = goal Set.thy
1.170 @@ -257,53 +283,54 @@
1.171 by (rtac (major RS IntD2) 1);
1.172 qed "IntE";
1.173
1.174 -qed_goal "Int_iff" Set.thy "(c : A Int B) = (c:A & c:B)"
1.175 - (fn _ => [ (fast_tac (!claset addSIs [IntI] addSEs [IntE]) 1) ]);
1.176 -
1.177 +AddSIs [IntI];
1.178 +AddSEs [IntE];
1.179
1.180 section "Set difference";
1.181
1.182 -qed_goalw "DiffI" Set.thy [set_diff_def]
1.183 - "[| c : A; c ~: B |] ==> c : A - B"
1.184 - (fn prems=> [ (REPEAT (resolve_tac (prems @ [CollectI,conjI]) 1)) ]);
1.185 +qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)"
1.186 + (fn _ => [ (Fast_tac 1) ]);
1.187
1.188 -qed_goalw "DiffD1" Set.thy [set_diff_def]
1.189 - "c : A - B ==> c : A"
1.190 - (fn [major]=> [ (rtac (major RS CollectD RS conjunct1) 1) ]);
1.191 +Addsimps [Diff_iff];
1.192 +
1.193 +qed_goal "DiffI" Set.thy "!!c. [| c : A; c ~: B |] ==> c : A - B"
1.194 + (fn _=> [ Asm_simp_tac 1 ]);
1.195
1.196 -qed_goalw "DiffD2" Set.thy [set_diff_def]
1.197 - "[| c : A - B; c : B |] ==> P"
1.198 - (fn [major,minor]=>
1.199 - [rtac (minor RS (major RS CollectD RS conjunct2 RS notE)) 1]);
1.200 +qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A"
1.201 + (fn _=> [ (Asm_full_simp_tac 1) ]);
1.202
1.203 -qed_goal "DiffE" Set.thy
1.204 - "[| c : A - B; [| c:A; c~:B |] ==> P |] ==> P"
1.205 +qed_goal "DiffD2" Set.thy "!!c. [| c : A - B; c : B |] ==> P"
1.206 + (fn _=> [ (Asm_full_simp_tac 1) ]);
1.207 +
1.208 +qed_goal "DiffE" Set.thy "[| c : A - B; [| c:A; c~:B |] ==> P |] ==> P"
1.209 (fn prems=>
1.210 [ (resolve_tac prems 1),
1.211 (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]);
1.212
1.213 -qed_goal "Diff_iff" Set.thy "(c : A-B) = (c:A & c~:B)"
1.214 - (fn _ => [ (fast_tac (!claset addSIs [DiffI] addSEs [DiffE]) 1) ]);
1.215 +AddSIs [DiffI];
1.216 +AddSEs [DiffE];
1.217
1.218 section "The empty set -- {}";
1.219
1.220 -qed_goalw "emptyE" Set.thy [empty_def] "a:{} ==> P"
1.221 - (fn [prem] => [rtac (prem RS CollectD RS FalseE) 1]);
1.222 +qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False"
1.223 + (fn _ => [ (Fast_tac 1) ]);
1.224 +
1.225 +Addsimps [empty_iff];
1.226 +
1.227 +qed_goal "emptyE" Set.thy "!!a. a:{} ==> P"
1.228 + (fn _ => [Full_simp_tac 1]);
1.229 +
1.230 +AddSEs [emptyE];
1.231
1.232 qed_goal "empty_subsetI" Set.thy "{} <= A"
1.233 - (fn _ => [ (REPEAT (ares_tac [equalityI,subsetI,emptyE] 1)) ]);
1.234 + (fn _ => [ (Fast_tac 1) ]);
1.235
1.236 qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}"
1.237 - (fn prems=>
1.238 - [ (REPEAT (ares_tac (prems@[empty_subsetI,subsetI,equalityI]) 1
1.239 - ORELSE eresolve_tac (prems RL [FalseE]) 1)) ]);
1.240 + (fn [prem]=>
1.241 + [ (fast_tac (!claset addIs [prem RS FalseE]) 1) ]);
1.242
1.243 -qed_goal "equals0D" Set.thy "[| A={}; a:A |] ==> P"
1.244 - (fn [major,minor]=>
1.245 - [ (rtac (minor RS (major RS equalityD1 RS subsetD RS emptyE)) 1) ]);
1.246 -
1.247 -qed_goal "empty_iff" Set.thy "(c : {}) = False"
1.248 - (fn _ => [ (fast_tac (!claset addSEs [emptyE]) 1) ]);
1.249 +qed_goal "equals0D" Set.thy "!!a. [| A={}; a:A |] ==> P"
1.250 + (fn _ => [ (Fast_tac 1) ]);
1.251
1.252 goal Set.thy "Ball {} P = True";
1.253 by (simp_tac (HOL_ss addsimps [mem_Collect_eq, Ball_def, empty_def]) 1);
1.254 @@ -317,11 +344,16 @@
1.255
1.256 section "Augmenting a set -- insert";
1.257
1.258 -qed_goalw "insertI1" Set.thy [insert_def] "a : insert a B"
1.259 - (fn _ => [rtac (CollectI RS UnI1) 1, rtac refl 1]);
1.260 +qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)"
1.261 + (fn _ => [Fast_tac 1]);
1.262 +
1.263 +Addsimps [insert_iff];
1.264
1.265 -qed_goalw "insertI2" Set.thy [insert_def] "a : B ==> a : insert b B"
1.266 - (fn [prem]=> [ (rtac (prem RS UnI2) 1) ]);
1.267 +qed_goal "insertI1" Set.thy "a : insert a B"
1.268 + (fn _ => [Simp_tac 1]);
1.269 +
1.270 +qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B"
1.271 + (fn _=> [Asm_simp_tac 1]);
1.272
1.273 qed_goalw "insertE" Set.thy [insert_def]
1.274 "[| a : insert b A; a=b ==> P; a:A ==> P |] ==> P"
1.275 @@ -329,37 +361,35 @@
1.276 [ (rtac (major RS UnE) 1),
1.277 (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
1.278
1.279 -qed_goal "insert_iff" Set.thy "a : insert b A = (a=b | a:A)"
1.280 - (fn _ => [fast_tac (!claset addIs [insertI1,insertI2] addSEs [insertE]) 1]);
1.281 -
1.282 (*Classical introduction rule*)
1.283 qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
1.284 - (fn [prem]=>
1.285 - [ (rtac (disjCI RS (insert_iff RS iffD2)) 1),
1.286 - (etac prem 1) ]);
1.287 + (fn prems=>
1.288 + [ (Simp_tac 1),
1.289 + (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
1.290 +
1.291 +AddSIs [insertCI];
1.292 +AddSEs [insertE];
1.293
1.294 section "Singletons, using insert";
1.295
1.296 qed_goal "singletonI" Set.thy "a : {a}"
1.297 (fn _=> [ (rtac insertI1 1) ]);
1.298
1.299 -goalw Set.thy [insert_def] "!!a. b : {a} ==> b=a";
1.300 -by (fast_tac (!claset addSEs [emptyE,CollectE,UnE]) 1);
1.301 +goal Set.thy "!!a. b : {a} ==> b=a";
1.302 +by (Fast_tac 1);
1.303 qed "singletonD";
1.304
1.305 bind_thm ("singletonE", make_elim singletonD);
1.306
1.307 -qed_goal "singleton_iff" thy "(b : {a}) = (b=a)" (fn _ => [
1.308 - rtac iffI 1,
1.309 - etac singletonD 1,
1.310 - hyp_subst_tac 1,
1.311 - rtac singletonI 1]);
1.312 +qed_goal "singleton_iff" thy "(b : {a}) = (b=a)"
1.313 +(fn _ => [Fast_tac 1]);
1.314
1.315 -val [major] = goal Set.thy "{a}={b} ==> a=b";
1.316 -by (rtac (major RS equalityD1 RS subsetD RS singletonD) 1);
1.317 -by (rtac singletonI 1);
1.318 +goal Set.thy "!!a b. {a}={b} ==> a=b";
1.319 +by (fast_tac (!claset addEs [equalityE]) 1);
1.320 qed "singleton_inject";
1.321
1.322 +AddSDs [singleton_inject];
1.323 +
1.324
1.325 section "The universal set -- UNIV";
1.326
1.327 @@ -372,10 +402,15 @@
1.328
1.329 section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
1.330
1.331 +goalw Set.thy [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
1.332 +by (Fast_tac 1);
1.333 +qed "UN_iff";
1.334 +
1.335 +Addsimps [UN_iff];
1.336 +
1.337 (*The order of the premises presupposes that A is rigid; b may be flexible*)
1.338 -val prems = goalw Set.thy [UNION_def]
1.339 - "[| a:A; b: B(a) |] ==> b: (UN x:A. B(x))";
1.340 -by (REPEAT (resolve_tac (prems @ [bexI,CollectI]) 1));
1.341 +goal Set.thy "!!b. [| a:A; b: B(a) |] ==> b: (UN x:A. B(x))";
1.342 +by (Auto_tac());
1.343 qed "UN_I";
1.344
1.345 val major::prems = goalw Set.thy [UNION_def]
1.346 @@ -384,6 +419,9 @@
1.347 by (REPEAT (ares_tac prems 1));
1.348 qed "UN_E";
1.349
1.350 +AddIs [UN_I];
1.351 +AddSEs [UN_E];
1.352 +
1.353 val prems = goal Set.thy
1.354 "[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \
1.355 \ (UN x:A. C(x)) = (UN x:B. D(x))";
1.356 @@ -395,15 +433,19 @@
1.357
1.358 section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
1.359
1.360 +goalw Set.thy [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
1.361 +by (Auto_tac());
1.362 +qed "INT_iff";
1.363 +
1.364 +Addsimps [INT_iff];
1.365 +
1.366 val prems = goalw Set.thy [INTER_def]
1.367 "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
1.368 by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
1.369 qed "INT_I";
1.370
1.371 -val major::prems = goalw Set.thy [INTER_def]
1.372 - "[| b : (INT x:A. B(x)); a:A |] ==> b: B(a)";
1.373 -by (rtac (major RS CollectD RS bspec) 1);
1.374 -by (resolve_tac prems 1);
1.375 +goal Set.thy "!!b. [| b : (INT x:A. B(x)); a:A |] ==> b: B(a)";
1.376 +by (Auto_tac());
1.377 qed "INT_D";
1.378
1.379 (*"Classical" elimination -- by the Excluded Middle on a:A *)
1.380 @@ -413,6 +455,9 @@
1.381 by (REPEAT (eresolve_tac prems 1));
1.382 qed "INT_E";
1.383
1.384 +AddSIs [INT_I];
1.385 +AddEs [INT_D, INT_E];
1.386 +
1.387 val prems = goal Set.thy
1.388 "[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \
1.389 \ (INT x:A. C(x)) = (INT x:B. D(x))";
1.390 @@ -424,10 +469,16 @@
1.391
1.392 section "Unions over a type; UNION1(B) = Union(range(B))";
1.393
1.394 +goalw Set.thy [UNION1_def] "(b: (UN x. B(x))) = (EX x. b: B(x))";
1.395 +by (Simp_tac 1);
1.396 +by (Fast_tac 1);
1.397 +qed "UN1_iff";
1.398 +
1.399 +Addsimps [UN1_iff];
1.400 +
1.401 (*The order of the premises presupposes that A is rigid; b may be flexible*)
1.402 -val prems = goalw Set.thy [UNION1_def]
1.403 - "b: B(x) ==> b: (UN x. B(x))";
1.404 -by (REPEAT (resolve_tac (prems @ [TrueI, CollectI RS UN_I]) 1));
1.405 +goal Set.thy "!!b. b: B(x) ==> b: (UN x. B(x))";
1.406 +by (Auto_tac());
1.407 qed "UN1_I";
1.408
1.409 val major::prems = goalw Set.thy [UNION1_def]
1.410 @@ -436,25 +487,43 @@
1.411 by (REPEAT (ares_tac prems 1));
1.412 qed "UN1_E";
1.413
1.414 +AddIs [UN1_I];
1.415 +AddSEs [UN1_E];
1.416 +
1.417
1.418 section "Intersections over a type; INTER1(B) = Inter(range(B))";
1.419
1.420 +goalw Set.thy [INTER1_def] "(b: (INT x. B(x))) = (ALL x. b: B(x))";
1.421 +by (Simp_tac 1);
1.422 +by (Fast_tac 1);
1.423 +qed "INT1_iff";
1.424 +
1.425 +Addsimps [INT1_iff];
1.426 +
1.427 val prems = goalw Set.thy [INTER1_def]
1.428 "(!!x. b: B(x)) ==> b : (INT x. B(x))";
1.429 by (REPEAT (ares_tac (INT_I::prems) 1));
1.430 qed "INT1_I";
1.431
1.432 -val [major] = goalw Set.thy [INTER1_def]
1.433 - "b : (INT x. B(x)) ==> b: B(a)";
1.434 -by (rtac (TrueI RS (CollectI RS (major RS INT_D))) 1);
1.435 +goal Set.thy "!!b. b : (INT x. B(x)) ==> b: B(a)";
1.436 +by (Asm_full_simp_tac 1);
1.437 qed "INT1_D";
1.438
1.439 +AddSIs [INT1_I];
1.440 +AddDs [INT1_D];
1.441 +
1.442 +
1.443 section "Union";
1.444
1.445 +goalw Set.thy [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
1.446 +by (Fast_tac 1);
1.447 +qed "Union_iff";
1.448 +
1.449 +Addsimps [Union_iff];
1.450 +
1.451 (*The order of the premises presupposes that C is rigid; A may be flexible*)
1.452 -val prems = goalw Set.thy [Union_def]
1.453 - "[| X:C; A:X |] ==> A : Union(C)";
1.454 -by (REPEAT (resolve_tac (prems @ [UN_I]) 1));
1.455 +goal Set.thy "!!X. [| X:C; A:X |] ==> A : Union(C)";
1.456 +by (Auto_tac());
1.457 qed "UnionI";
1.458
1.459 val major::prems = goalw Set.thy [Union_def]
1.460 @@ -463,8 +532,18 @@
1.461 by (REPEAT (ares_tac prems 1));
1.462 qed "UnionE";
1.463
1.464 +AddIs [UnionI];
1.465 +AddSEs [UnionE];
1.466 +
1.467 +
1.468 section "Inter";
1.469
1.470 +goalw Set.thy [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
1.471 +by (Fast_tac 1);
1.472 +qed "Inter_iff";
1.473 +
1.474 +Addsimps [Inter_iff];
1.475 +
1.476 val prems = goalw Set.thy [Inter_def]
1.477 "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
1.478 by (REPEAT (ares_tac ([INT_I] @ prems) 1));
1.479 @@ -472,10 +551,8 @@
1.480
1.481 (*A "destruct" rule -- every X in C contains A as an element, but
1.482 A:X can hold when X:C does not! This rule is analogous to "spec". *)
1.483 -val major::prems = goalw Set.thy [Inter_def]
1.484 - "[| A : Inter(C); X:C |] ==> A:X";
1.485 -by (rtac (major RS INT_D) 1);
1.486 -by (resolve_tac prems 1);
1.487 +goal Set.thy "!!X. [| A : Inter(C); X:C |] ==> A:X";
1.488 +by (Auto_tac());
1.489 qed "InterD";
1.490
1.491 (*"Classical" elimination rule -- does not require proving X:C *)
1.492 @@ -485,8 +562,17 @@
1.493 by (REPEAT (eresolve_tac prems 1));
1.494 qed "InterE";
1.495
1.496 +AddSIs [InterI];
1.497 +AddEs [InterD, InterE];
1.498 +
1.499 +
1.500 section "The Powerset operator -- Pow";
1.501
1.502 +qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)"
1.503 + (fn _ => [ (Asm_simp_tac 1) ]);
1.504 +
1.505 +AddIffs [Pow_iff];
1.506 +
1.507 qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
1.508 (fn _ => [ (etac CollectI 1) ]);
1.509
1.510 @@ -501,8 +587,11 @@
1.511 (*** Set reasoning tools ***)
1.512
1.513
1.514 +(*Each of these has ALREADY been added to !simpset above.*)
1.515 val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff,
1.516 - mem_Collect_eq];
1.517 + mem_Collect_eq,
1.518 + UN_iff, UN1_iff, Union_iff,
1.519 + INT_iff, INT1_iff, Inter_iff];
1.520
1.521 (*Not for Addsimps -- it can cause goals to blow up!*)
1.522 goal Set.thy "(a : (if Q then x else y)) = ((Q --> a:x) & (~Q --> a : y))";
1.523 @@ -511,6 +600,5 @@
1.524
1.525 val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs;
1.526
1.527 -simpset := !simpset addsimps mem_simps
1.528 - addcongs [ball_cong,bex_cong]
1.529 +simpset := !simpset addcongs [ball_cong,bex_cong]
1.530 setmksimps (mksimps mksimps_pairs);