author  clasohm 
Wed, 07 Dec 1994 13:12:04 +0100  
changeset 760  f0200e91b272 
parent 537  3a84f846e649 
child 782  200a16083201 
permissions  rwrr 
435  1 
(* Title: ZF/Univ 
0  2 
ID: $Id$ 
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

435  4 
Copyright 1994 University of Cambridge 
0  5 

6 
The cumulative hierarchy and a small universe for recursive types 

7 
*) 

8 

9 
open Univ; 

10 

11 
(*NOT SUITABLE FOR REWRITING  RECURSIVE!*) 

12 
goal Univ.thy "Vfrom(A,i) = A Un (UN j:i. Pow(Vfrom(A,j)))"; 

13 
by (rtac (Vfrom_def RS def_transrec RS ssubst) 1); 

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by (simp_tac ZF_ss 1); 
760  15 
qed "Vfrom"; 
0  16 

17 
(** Monotonicity **) 

18 

19 
goal Univ.thy "!!A B. A<=B ==> ALL j. i<=j > Vfrom(A,i) <= Vfrom(B,j)"; 

20 
by (eps_ind_tac "i" 1); 

21 
by (rtac (impI RS allI) 1); 

22 
by (rtac (Vfrom RS ssubst) 1); 

23 
by (rtac (Vfrom RS ssubst) 1); 

24 
by (etac Un_mono 1); 

25 
by (rtac UN_mono 1); 

26 
by (assume_tac 1); 

27 
by (rtac Pow_mono 1); 

28 
by (etac (bspec RS spec RS mp) 1); 

29 
by (assume_tac 1); 

30 
by (rtac subset_refl 1); 

760  31 
qed "Vfrom_mono_lemma"; 
0  32 

33 
(* [ A<=B; i<=x ] ==> Vfrom(A,i) <= Vfrom(B,x) *) 

34 
val Vfrom_mono = standard (Vfrom_mono_lemma RS spec RS mp); 

35 

36 

37 
(** A fundamental equality: Vfrom does not require ordinals! **) 

38 

39 
goal Univ.thy "Vfrom(A,x) <= Vfrom(A,rank(x))"; 

40 
by (eps_ind_tac "x" 1); 

41 
by (rtac (Vfrom RS ssubst) 1); 

42 
by (rtac (Vfrom RS ssubst) 1); 

27  43 
by (fast_tac (ZF_cs addSIs [rank_lt RS ltD]) 1); 
760  44 
qed "Vfrom_rank_subset1"; 
0  45 

46 
goal Univ.thy "Vfrom(A,rank(x)) <= Vfrom(A,x)"; 

47 
by (eps_ind_tac "x" 1); 

48 
by (rtac (Vfrom RS ssubst) 1); 

49 
by (rtac (Vfrom RS ssubst) 1); 

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by (rtac (subset_refl RS Un_mono) 1); 
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by (rtac UN_least 1); 
27  52 
(*expand rank(x1) = (UN y:x1. succ(rank(y))) in assumptions*) 
53 
by (etac (rank RS equalityD1 RS subsetD RS UN_E) 1); 

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by (rtac subset_trans 1); 
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55 
by (etac UN_upper 2); 
27  56 
by (rtac (subset_refl RS Vfrom_mono RS subset_trans RS Pow_mono) 1); 
57 
by (etac (ltI RS le_imp_subset) 1); 

58 
by (rtac (Ord_rank RS Ord_succ) 1); 

0  59 
by (etac bspec 1); 
60 
by (assume_tac 1); 

760  61 
qed "Vfrom_rank_subset2"; 
0  62 

63 
goal Univ.thy "Vfrom(A,rank(x)) = Vfrom(A,x)"; 

64 
by (rtac equalityI 1); 

65 
by (rtac Vfrom_rank_subset2 1); 

66 
by (rtac Vfrom_rank_subset1 1); 

760  67 
qed "Vfrom_rank_eq"; 
0  68 

69 

70 
(*** Basic closure properties ***) 

71 

72 
goal Univ.thy "!!x y. y:x ==> 0 : Vfrom(A,x)"; 

73 
by (rtac (Vfrom RS ssubst) 1); 

74 
by (fast_tac ZF_cs 1); 

760  75 
qed "zero_in_Vfrom"; 
0  76 

77 
goal Univ.thy "i <= Vfrom(A,i)"; 

78 
by (eps_ind_tac "i" 1); 

79 
by (rtac (Vfrom RS ssubst) 1); 

80 
by (fast_tac ZF_cs 1); 

760  81 
qed "i_subset_Vfrom"; 
0  82 

83 
goal Univ.thy "A <= Vfrom(A,i)"; 

84 
by (rtac (Vfrom RS ssubst) 1); 

85 
by (rtac Un_upper1 1); 

760  86 
qed "A_subset_Vfrom"; 
0  87 

488  88 
val A_into_Vfrom = A_subset_Vfrom RS subsetD > standard; 
89 

0  90 
goal Univ.thy "!!A a i. a <= Vfrom(A,i) ==> a: Vfrom(A,succ(i))"; 
91 
by (rtac (Vfrom RS ssubst) 1); 

92 
by (fast_tac ZF_cs 1); 

760  93 
qed "subset_mem_Vfrom"; 
0  94 

95 
(** Finite sets and ordered pairs **) 

96 

97 
goal Univ.thy "!!a. a: Vfrom(A,i) ==> {a} : Vfrom(A,succ(i))"; 

98 
by (rtac subset_mem_Vfrom 1); 

99 
by (safe_tac ZF_cs); 

760  100 
qed "singleton_in_Vfrom"; 
0  101 

102 
goal Univ.thy 

103 
"!!A. [ a: Vfrom(A,i); b: Vfrom(A,i) ] ==> {a,b} : Vfrom(A,succ(i))"; 

104 
by (rtac subset_mem_Vfrom 1); 

105 
by (safe_tac ZF_cs); 

760  106 
qed "doubleton_in_Vfrom"; 
0  107 

108 
goalw Univ.thy [Pair_def] 

109 
"!!A. [ a: Vfrom(A,i); b: Vfrom(A,i) ] ==> \ 

110 
\ <a,b> : Vfrom(A,succ(succ(i)))"; 

111 
by (REPEAT (ares_tac [doubleton_in_Vfrom] 1)); 

760  112 
qed "Pair_in_Vfrom"; 
0  113 

114 
val [prem] = goal Univ.thy 

115 
"a<=Vfrom(A,i) ==> succ(a) : Vfrom(A,succ(succ(i)))"; 

116 
by (REPEAT (resolve_tac [subset_mem_Vfrom, succ_subsetI] 1)); 

117 
by (rtac (Vfrom_mono RSN (2,subset_trans)) 2); 

118 
by (REPEAT (resolve_tac [prem, subset_refl, subset_succI] 1)); 

760  119 
qed "succ_in_Vfrom"; 
0  120 

121 
(*** 0, successor and limit equations fof Vfrom ***) 

122 

123 
goal Univ.thy "Vfrom(A,0) = A"; 

124 
by (rtac (Vfrom RS ssubst) 1); 

125 
by (fast_tac eq_cs 1); 

760  126 
qed "Vfrom_0"; 
0  127 

128 
goal Univ.thy "!!i. Ord(i) ==> Vfrom(A,succ(i)) = A Un Pow(Vfrom(A,i))"; 

129 
by (rtac (Vfrom RS trans) 1); 

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by (rtac (succI1 RS RepFunI RS Union_upper RSN 
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(2, equalityI RS subst_context)) 1); 
0  132 
by (rtac UN_least 1); 
133 
by (rtac (subset_refl RS Vfrom_mono RS Pow_mono) 1); 

27  134 
by (etac (ltI RS le_imp_subset) 1); 
135 
by (etac Ord_succ 1); 

760  136 
qed "Vfrom_succ_lemma"; 
0  137 

138 
goal Univ.thy "Vfrom(A,succ(i)) = A Un Pow(Vfrom(A,i))"; 

139 
by (res_inst_tac [("x1", "succ(i)")] (Vfrom_rank_eq RS subst) 1); 

140 
by (res_inst_tac [("x1", "i")] (Vfrom_rank_eq RS subst) 1); 

141 
by (rtac (rank_succ RS ssubst) 1); 

142 
by (rtac (Ord_rank RS Vfrom_succ_lemma) 1); 

760  143 
qed "Vfrom_succ"; 
0  144 

145 
(*The premise distinguishes this from Vfrom(A,0); allowing X=0 forces 

146 
the conclusion to be Vfrom(A,Union(X)) = A Un (UN y:X. Vfrom(A,y)) *) 

147 
val [prem] = goal Univ.thy "y:X ==> Vfrom(A,Union(X)) = (UN y:X. Vfrom(A,y))"; 

148 
by (rtac (Vfrom RS ssubst) 1); 

149 
by (rtac equalityI 1); 

150 
(*first inclusion*) 

151 
by (rtac Un_least 1); 

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by (rtac (A_subset_Vfrom RS subset_trans) 1); 
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153 
by (rtac (prem RS UN_upper) 1); 
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ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
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154 
by (rtac UN_least 1); 
6c6d2f6e3185
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155 
by (etac UnionE 1); 
6c6d2f6e3185
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156 
by (rtac subset_trans 1); 
6c6d2f6e3185
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157 
by (etac UN_upper 2); 
0  158 
by (rtac (Vfrom RS ssubst) 1); 
15
6c6d2f6e3185
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159 
by (etac ([UN_upper, Un_upper2] MRS subset_trans) 1); 
0  160 
(*opposite inclusion*) 
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161 
by (rtac UN_least 1); 
0  162 
by (rtac (Vfrom RS ssubst) 1); 
163 
by (fast_tac ZF_cs 1); 

760  164 
qed "Vfrom_Union"; 
0  165 

166 
goal Univ.thy "!!i. Ord(i) ==> i=0  (EX j. i=succ(j))  Limit(i)"; 

27  167 
by (fast_tac (ZF_cs addSIs [non_succ_LimitI, Ord_0_lt]) 1); 
760  168 
qed "Ord_cases_lemma"; 
0  169 

170 
val major::prems = goal Univ.thy 

171 
"[ Ord(i); \ 

172 
\ i=0 ==> P; \ 

173 
\ !!j. i=succ(j) ==> P; \ 

174 
\ Limit(i) ==> P \ 

175 
\ ] ==> P"; 

176 
by (cut_facts_tac [major RS Ord_cases_lemma] 1); 

177 
by (REPEAT (eresolve_tac (prems@[disjE, exE]) 1)); 

760  178 
qed "Ord_cases"; 
0  179 

180 

181 
(*** Vfrom applied to Limit ordinals ***) 

182 

183 
(*NB. limit ordinals are nonempty; 

184 
Vfrom(A,0) = A = A Un (UN y:0. Vfrom(A,y)) *) 

185 
val [limiti] = goal Univ.thy 

186 
"Limit(i) ==> Vfrom(A,i) = (UN y:i. Vfrom(A,y))"; 

27  187 
by (rtac (limiti RS (Limit_has_0 RS ltD) RS Vfrom_Union RS subst) 1); 
0  188 
by (rtac (limiti RS Limit_Union_eq RS ssubst) 1); 
189 
by (rtac refl 1); 

760  190 
qed "Limit_Vfrom_eq"; 
0  191 

27  192 
goal Univ.thy "!!a. [ a: Vfrom(A,j); Limit(i); j<i ] ==> a : Vfrom(A,i)"; 
193 
by (rtac (Limit_Vfrom_eq RS equalityD2 RS subsetD) 1); 

194 
by (REPEAT (ares_tac [ltD RS UN_I] 1)); 

760  195 
qed "Limit_VfromI"; 
27  196 

197 
val prems = goal Univ.thy 

198 
"[ a: Vfrom(A,i); Limit(i); \ 

199 
\ !!x. [ x<i; a: Vfrom(A,x) ] ==> R \ 

200 
\ ] ==> R"; 

201 
by (rtac (Limit_Vfrom_eq RS equalityD1 RS subsetD RS UN_E) 1); 

202 
by (REPEAT (ares_tac (prems @ [ltI, Limit_is_Ord]) 1)); 

760  203 
qed "Limit_VfromE"; 
0  204 

516  205 
val zero_in_VLimit = Limit_has_0 RS ltD RS zero_in_Vfrom; 
484  206 

0  207 
val [major,limiti] = goal Univ.thy 
208 
"[ a: Vfrom(A,i); Limit(i) ] ==> {a} : Vfrom(A,i)"; 

27  209 
by (rtac ([major,limiti] MRS Limit_VfromE) 1); 
210 
by (etac ([singleton_in_Vfrom, limiti] MRS Limit_VfromI) 1); 

0  211 
by (etac (limiti RS Limit_has_succ) 1); 
760  212 
qed "singleton_in_VLimit"; 
0  213 

214 
val Vfrom_UnI1 = Un_upper1 RS (subset_refl RS Vfrom_mono RS subsetD) 

215 
and Vfrom_UnI2 = Un_upper2 RS (subset_refl RS Vfrom_mono RS subsetD); 

216 

217 
(*Hard work is finding a single j:i such that {a,b}<=Vfrom(A,j)*) 

218 
val [aprem,bprem,limiti] = goal Univ.thy 

219 
"[ a: Vfrom(A,i); b: Vfrom(A,i); Limit(i) ] ==> \ 

220 
\ {a,b} : Vfrom(A,i)"; 

27  221 
by (rtac ([aprem,limiti] MRS Limit_VfromE) 1); 
222 
by (rtac ([bprem,limiti] MRS Limit_VfromE) 1); 

223 
by (rtac ([doubleton_in_Vfrom, limiti] MRS Limit_VfromI) 1); 

224 
by (etac Vfrom_UnI1 1); 

225 
by (etac Vfrom_UnI2 1); 

226 
by (REPEAT (ares_tac [limiti, Limit_has_succ, Un_least_lt] 1)); 

760  227 
qed "doubleton_in_VLimit"; 
0  228 

229 
val [aprem,bprem,limiti] = goal Univ.thy 

230 
"[ a: Vfrom(A,i); b: Vfrom(A,i); Limit(i) ] ==> \ 

231 
\ <a,b> : Vfrom(A,i)"; 

232 
(*Infer that a, b occur at ordinals x,xa < i.*) 

27  233 
by (rtac ([aprem,limiti] MRS Limit_VfromE) 1); 
234 
by (rtac ([bprem,limiti] MRS Limit_VfromE) 1); 

235 
by (rtac ([Pair_in_Vfrom, limiti] MRS Limit_VfromI) 1); 

0  236 
(*Infer that succ(succ(x Un xa)) < i *) 
27  237 
by (etac Vfrom_UnI1 1); 
238 
by (etac Vfrom_UnI2 1); 

239 
by (REPEAT (ares_tac [limiti, Limit_has_succ, Un_least_lt] 1)); 

760  240 
qed "Pair_in_VLimit"; 
484  241 

242 
goal Univ.thy "!!i. Limit(i) ==> Vfrom(A,i)*Vfrom(A,i) <= Vfrom(A,i)"; 

516  243 
by (REPEAT (ares_tac [subsetI,Pair_in_VLimit] 1 
484  244 
ORELSE eresolve_tac [SigmaE, ssubst] 1)); 
760  245 
qed "product_VLimit"; 
484  246 

516  247 
val Sigma_subset_VLimit = 
248 
[Sigma_mono, product_VLimit] MRS subset_trans > standard; 

484  249 

516  250 
val nat_subset_VLimit = 
484  251 
[nat_le_Limit RS le_imp_subset, i_subset_Vfrom] MRS subset_trans 
252 
> standard; 

253 

488  254 
goal Univ.thy "!!i. [ n: nat; Limit(i) ] ==> n : Vfrom(A,i)"; 
516  255 
by (REPEAT (ares_tac [nat_subset_VLimit RS subsetD] 1)); 
760  256 
qed "nat_into_VLimit"; 
484  257 

258 
(** Closure under disjoint union **) 

259 

516  260 
val zero_in_VLimit = Limit_has_0 RS ltD RS zero_in_Vfrom > standard; 
484  261 

262 
goal Univ.thy "!!i. Limit(i) ==> 1 : Vfrom(A,i)"; 

516  263 
by (REPEAT (ares_tac [nat_into_VLimit, nat_0I, nat_succI] 1)); 
760  264 
qed "one_in_VLimit"; 
484  265 

266 
goalw Univ.thy [Inl_def] 

267 
"!!A a. [ a: Vfrom(A,i); Limit(i) ] ==> Inl(a) : Vfrom(A,i)"; 

516  268 
by (REPEAT (ares_tac [zero_in_VLimit, Pair_in_VLimit] 1)); 
760  269 
qed "Inl_in_VLimit"; 
484  270 

271 
goalw Univ.thy [Inr_def] 

272 
"!!A b. [ b: Vfrom(A,i); Limit(i) ] ==> Inr(b) : Vfrom(A,i)"; 

516  273 
by (REPEAT (ares_tac [one_in_VLimit, Pair_in_VLimit] 1)); 
760  274 
qed "Inr_in_VLimit"; 
484  275 

276 
goal Univ.thy "!!i. Limit(i) ==> Vfrom(C,i)+Vfrom(C,i) <= Vfrom(C,i)"; 

516  277 
by (fast_tac (sum_cs addSIs [Inl_in_VLimit, Inr_in_VLimit]) 1); 
760  278 
qed "sum_VLimit"; 
484  279 

516  280 
val sum_subset_VLimit = 
281 
[sum_mono, sum_VLimit] MRS subset_trans > standard; 

484  282 

0  283 

284 

285 
(*** Properties assuming Transset(A) ***) 

286 

287 
goal Univ.thy "!!i A. Transset(A) ==> Transset(Vfrom(A,i))"; 

288 
by (eps_ind_tac "i" 1); 

289 
by (rtac (Vfrom RS ssubst) 1); 

290 
by (fast_tac (ZF_cs addSIs [Transset_Union_family, Transset_Un, 

291 
Transset_Pow]) 1); 

760  292 
qed "Transset_Vfrom"; 
0  293 

294 
goal Univ.thy "!!A i. Transset(A) ==> Vfrom(A, succ(i)) = Pow(Vfrom(A,i))"; 

295 
by (rtac (Vfrom_succ RS trans) 1); 

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ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
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296 
by (rtac (Un_upper2 RSN (2,equalityI)) 1); 
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297 
by (rtac (subset_refl RSN (2,Un_least)) 1); 
6c6d2f6e3185
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
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298 
by (rtac (A_subset_Vfrom RS subset_trans) 1); 
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ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
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299 
by (etac (Transset_Vfrom RS (Transset_iff_Pow RS iffD1)) 1); 
760  300 
qed "Transset_Vfrom_succ"; 
0  301 

435  302 
goalw Ordinal.thy [Pair_def,Transset_def] 
0  303 
"!!C. [ <a,b> <= C; Transset(C) ] ==> a: C & b: C"; 
304 
by (fast_tac ZF_cs 1); 

760  305 
qed "Transset_Pair_subset"; 
0  306 

307 
goal Univ.thy 

308 
"!!a b.[ <a,b> <= Vfrom(A,i); Transset(A); Limit(i) ] ==> \ 

309 
\ <a,b> : Vfrom(A,i)"; 

15
6c6d2f6e3185
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
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310 
by (etac (Transset_Pair_subset RS conjE) 1); 
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ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
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311 
by (etac Transset_Vfrom 1); 
516  312 
by (REPEAT (ares_tac [Pair_in_VLimit] 1)); 
760  313 
qed "Transset_Pair_subset_VLimit"; 
0  314 

315 

316 
(*** Closure under product/sum applied to elements  thus Vfrom(A,i) 

317 
is a model of simple type theory provided A is a transitive set 

318 
and i is a limit ordinal 

319 
***) 

320 

187  321 
(*General theorem for membership in Vfrom(A,i) when i is a limit ordinal*) 
322 
val [aprem,bprem,limiti,step] = goal Univ.thy 

323 
"[ a: Vfrom(A,i); b: Vfrom(A,i); Limit(i); \ 

324 
\ !!x y j. [ j<i; 1:j; x: Vfrom(A,j); y: Vfrom(A,j) \ 

325 
\ ] ==> EX k. h(x,y): Vfrom(A,k) & k<i ] ==> \ 

326 
\ h(a,b) : Vfrom(A,i)"; 

327 
(*Infer that a, b occur at ordinals x,xa < i.*) 

328 
by (rtac ([aprem,limiti] MRS Limit_VfromE) 1); 

329 
by (rtac ([bprem,limiti] MRS Limit_VfromE) 1); 

330 
by (res_inst_tac [("j1", "x Un xa Un succ(1)")] (step RS exE) 1); 

331 
by (DO_GOAL [etac conjE, etac Limit_VfromI, rtac limiti, atac] 5); 

332 
by (etac (Vfrom_UnI2 RS Vfrom_UnI1) 4); 

333 
by (etac (Vfrom_UnI1 RS Vfrom_UnI1) 3); 

334 
by (rtac (succI1 RS UnI2) 2); 

335 
by (REPEAT (ares_tac [limiti, Limit_has_0, Limit_has_succ, Un_least_lt] 1)); 

760  336 
qed "in_VLimit"; 
0  337 

338 
(** products **) 

339 

340 
goal Univ.thy 

187  341 
"!!A. [ a: Vfrom(A,j); b: Vfrom(A,j); Transset(A) ] ==> \ 
342 
\ a*b : Vfrom(A, succ(succ(succ(j))))"; 

0  343 
by (dtac Transset_Vfrom 1); 
344 
by (rtac subset_mem_Vfrom 1); 

345 
by (rewtac Transset_def); 

346 
by (fast_tac (ZF_cs addIs [Pair_in_Vfrom]) 1); 

760  347 
qed "prod_in_Vfrom"; 
0  348 

349 
val [aprem,bprem,limiti,transset] = goal Univ.thy 

350 
"[ a: Vfrom(A,i); b: Vfrom(A,i); Limit(i); Transset(A) ] ==> \ 

351 
\ a*b : Vfrom(A,i)"; 

516  352 
by (rtac ([aprem,bprem,limiti] MRS in_VLimit) 1); 
187  353 
by (REPEAT (ares_tac [exI, conjI, prod_in_Vfrom, transset, 
354 
limiti RS Limit_has_succ] 1)); 

760  355 
qed "prod_in_VLimit"; 
0  356 

357 
(** Disjoint sums, aka Quine ordered pairs **) 

358 

359 
goalw Univ.thy [sum_def] 

187  360 
"!!A. [ a: Vfrom(A,j); b: Vfrom(A,j); Transset(A); 1:j ] ==> \ 
361 
\ a+b : Vfrom(A, succ(succ(succ(j))))"; 

0  362 
by (dtac Transset_Vfrom 1); 
363 
by (rtac subset_mem_Vfrom 1); 

364 
by (rewtac Transset_def); 

365 
by (fast_tac (ZF_cs addIs [zero_in_Vfrom, Pair_in_Vfrom, 

366 
i_subset_Vfrom RS subsetD]) 1); 

760  367 
qed "sum_in_Vfrom"; 
0  368 

369 
val [aprem,bprem,limiti,transset] = goal Univ.thy 

370 
"[ a: Vfrom(A,i); b: Vfrom(A,i); Limit(i); Transset(A) ] ==> \ 

371 
\ a+b : Vfrom(A,i)"; 

516  372 
by (rtac ([aprem,bprem,limiti] MRS in_VLimit) 1); 
187  373 
by (REPEAT (ares_tac [exI, conjI, sum_in_Vfrom, transset, 
374 
limiti RS Limit_has_succ] 1)); 

760  375 
qed "sum_in_VLimit"; 
0  376 

377 
(** function space! **) 

378 

379 
goalw Univ.thy [Pi_def] 

187  380 
"!!A. [ a: Vfrom(A,j); b: Vfrom(A,j); Transset(A) ] ==> \ 
381 
\ a>b : Vfrom(A, succ(succ(succ(succ(j)))))"; 

0  382 
by (dtac Transset_Vfrom 1); 
383 
by (rtac subset_mem_Vfrom 1); 

384 
by (rtac (Collect_subset RS subset_trans) 1); 

385 
by (rtac (Vfrom RS ssubst) 1); 

386 
by (rtac (subset_trans RS subset_trans) 1); 

387 
by (rtac Un_upper2 3); 

388 
by (rtac (succI1 RS UN_upper) 2); 

389 
by (rtac Pow_mono 1); 

390 
by (rewtac Transset_def); 

391 
by (fast_tac (ZF_cs addIs [Pair_in_Vfrom]) 1); 

760  392 
qed "fun_in_Vfrom"; 
0  393 

394 
val [aprem,bprem,limiti,transset] = goal Univ.thy 

395 
"[ a: Vfrom(A,i); b: Vfrom(A,i); Limit(i); Transset(A) ] ==> \ 

396 
\ a>b : Vfrom(A,i)"; 

516  397 
by (rtac ([aprem,bprem,limiti] MRS in_VLimit) 1); 
187  398 
by (REPEAT (ares_tac [exI, conjI, fun_in_Vfrom, transset, 
399 
limiti RS Limit_has_succ] 1)); 

760  400 
qed "fun_in_VLimit"; 
0  401 

402 

403 
(*** The set Vset(i) ***) 

404 

405 
goal Univ.thy "Vset(i) = (UN j:i. Pow(Vset(j)))"; 

406 
by (rtac (Vfrom RS ssubst) 1); 

407 
by (fast_tac eq_cs 1); 

760  408 
qed "Vset"; 
0  409 

410 
val Vset_succ = Transset_0 RS Transset_Vfrom_succ; 

411 

412 
val Transset_Vset = Transset_0 RS Transset_Vfrom; 

413 

414 
(** Characterisation of the elements of Vset(i) **) 

415 

27  416 
val [ordi] = goal Univ.thy "Ord(i) ==> ALL b. b : Vset(i) > rank(b) < i"; 
0  417 
by (rtac (ordi RS trans_induct) 1); 
418 
by (rtac (Vset RS ssubst) 1); 

419 
by (safe_tac ZF_cs); 

420 
by (rtac (rank RS ssubst) 1); 

27  421 
by (rtac UN_succ_least_lt 1); 
422 
by (fast_tac ZF_cs 2); 

423 
by (REPEAT (ares_tac [ltI] 1)); 

760  424 
qed "Vset_rank_imp1"; 
0  425 

27  426 
(* [ Ord(i); x : Vset(i) ] ==> rank(x) < i *) 
427 
val VsetD = standard (Vset_rank_imp1 RS spec RS mp); 

0  428 

429 
val [ordi] = goal Univ.thy "Ord(i) ==> ALL b. rank(b) : i > b : Vset(i)"; 

430 
by (rtac (ordi RS trans_induct) 1); 

431 
by (rtac allI 1); 

432 
by (rtac (Vset RS ssubst) 1); 

27  433 
by (fast_tac (ZF_cs addSIs [rank_lt RS ltD]) 1); 
760  434 
qed "Vset_rank_imp2"; 
0  435 

27  436 
goal Univ.thy "!!x i. rank(x)<i ==> x : Vset(i)"; 
437 
by (etac ltE 1); 

438 
by (etac (Vset_rank_imp2 RS spec RS mp) 1); 

439 
by (assume_tac 1); 

760  440 
qed "VsetI"; 
0  441 

27  442 
goal Univ.thy "!!i. Ord(i) ==> b : Vset(i) <> rank(b) < i"; 
0  443 
by (rtac iffI 1); 
27  444 
by (REPEAT (eresolve_tac [asm_rl, VsetD, VsetI] 1)); 
760  445 
qed "Vset_Ord_rank_iff"; 
0  446 

27  447 
goal Univ.thy "b : Vset(a) <> rank(b) < rank(a)"; 
0  448 
by (rtac (Vfrom_rank_eq RS subst) 1); 
449 
by (rtac (Ord_rank RS Vset_Ord_rank_iff) 1); 

760  450 
qed "Vset_rank_iff"; 
0  451 

452 
goal Univ.thy "!!i. Ord(i) ==> rank(Vset(i)) = i"; 

453 
by (rtac (rank RS ssubst) 1); 

454 
by (rtac equalityI 1); 

455 
by (safe_tac ZF_cs); 

456 
by (EVERY' [wtac UN_I, 

457 
etac (i_subset_Vfrom RS subsetD), 

458 
etac (Ord_in_Ord RS rank_of_Ord RS ssubst), 

459 
assume_tac, 

460 
rtac succI1] 3); 

27  461 
by (REPEAT (eresolve_tac [asm_rl, VsetD RS ltD, Ord_trans] 1)); 
760  462 
qed "rank_Vset"; 
0  463 

464 
(** Lemmas for reasoning about sets in terms of their elements' ranks **) 

465 

466 
goal Univ.thy "a <= Vset(rank(a))"; 

15
6c6d2f6e3185
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

467 
by (rtac subsetI 1); 
27  468 
by (etac (rank_lt RS VsetI) 1); 
760  469 
qed "arg_subset_Vset_rank"; 
0  470 

471 
val [iprem] = goal Univ.thy 

472 
"[ !!i. Ord(i) ==> a Int Vset(i) <= b ] ==> a <= b"; 

27  473 
by (rtac ([subset_refl, arg_subset_Vset_rank] MRS 
474 
Int_greatest RS subset_trans) 1); 

15
6c6d2f6e3185
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

475 
by (rtac (Ord_rank RS iprem) 1); 
760  476 
qed "Int_Vset_subset"; 
0  477 

478 
(** Set up an environment for simplification **) 

479 

480 
val rank_rls = [rank_Inl, rank_Inr, rank_pair1, rank_pair2]; 

27  481 
val rank_trans_rls = rank_rls @ (rank_rls RLN (2, [lt_trans])); 
0  482 

483 
val rank_ss = ZF_ss 

27  484 
addsimps [case_Inl, case_Inr, VsetI] 
6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset

485 
addsimps rank_trans_rls; 
0  486 

487 
(** Recursion over Vset levels! **) 

488 

489 
(*NOT SUITABLE FOR REWRITING: recursive!*) 

490 
goalw Univ.thy [Vrec_def] "Vrec(a,H) = H(a, lam x:Vset(rank(a)). Vrec(x,H))"; 

491 
by (rtac (transrec RS ssubst) 1); 

27  492 
by (simp_tac (ZF_ss addsimps [Ord_rank, Ord_succ, VsetD RS ltD RS beta, 
493 
VsetI RS beta, le_refl]) 1); 

760  494 
qed "Vrec"; 
0  495 

496 
(*This form avoids giant explosions in proofs. NOTE USE OF == *) 

497 
val rew::prems = goal Univ.thy 

498 
"[ !!x. h(x)==Vrec(x,H) ] ==> \ 

499 
\ h(a) = H(a, lam x: Vset(rank(a)). h(x))"; 

500 
by (rewtac rew); 

501 
by (rtac Vrec 1); 

760  502 
qed "def_Vrec"; 
0  503 

504 

505 
(*** univ(A) ***) 

506 

507 
goalw Univ.thy [univ_def] "!!A B. A<=B ==> univ(A) <= univ(B)"; 

508 
by (etac Vfrom_mono 1); 

509 
by (rtac subset_refl 1); 

760  510 
qed "univ_mono"; 
0  511 

512 
goalw Univ.thy [univ_def] "!!A. Transset(A) ==> Transset(univ(A))"; 

513 
by (etac Transset_Vfrom 1); 

760  514 
qed "Transset_univ"; 
0  515 

516 
(** univ(A) as a limit **) 

517 

518 
goalw Univ.thy [univ_def] "univ(A) = (UN i:nat. Vfrom(A,i))"; 

15
6c6d2f6e3185
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

519 
by (rtac (Limit_nat RS Limit_Vfrom_eq) 1); 
760  520 
qed "univ_eq_UN"; 
0  521 

522 
goal Univ.thy "!!c. c <= univ(A) ==> c = (UN i:nat. c Int Vfrom(A,i))"; 

15
6c6d2f6e3185
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

523 
by (rtac (subset_UN_iff_eq RS iffD1) 1); 
6c6d2f6e3185
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

524 
by (etac (univ_eq_UN RS subst) 1); 
760  525 
qed "subset_univ_eq_Int"; 
0  526 

527 
val [aprem, iprem] = goal Univ.thy 

528 
"[ a <= univ(X); \ 

529 
\ !!i. i:nat ==> a Int Vfrom(X,i) <= b \ 

530 
\ ] ==> a <= b"; 

15
6c6d2f6e3185
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

531 
by (rtac (aprem RS subset_univ_eq_Int RS ssubst) 1); 
6c6d2f6e3185
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

532 
by (rtac UN_least 1); 
6c6d2f6e3185
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

533 
by (etac iprem 1); 
760  534 
qed "univ_Int_Vfrom_subset"; 
0  535 

536 
val prems = goal Univ.thy 

537 
"[ a <= univ(X); b <= univ(X); \ 

538 
\ !!i. i:nat ==> a Int Vfrom(X,i) = b Int Vfrom(X,i) \ 

539 
\ ] ==> a = b"; 

15
6c6d2f6e3185
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

540 
by (rtac equalityI 1); 
0  541 
by (ALLGOALS 
542 
(resolve_tac (prems RL [univ_Int_Vfrom_subset]) THEN' 

543 
eresolve_tac (prems RL [equalityD1,equalityD2] RL [subset_trans]) THEN' 

544 
rtac Int_lower1)); 

760  545 
qed "univ_Int_Vfrom_eq"; 
0  546 

547 
(** Closure properties **) 

548 

549 
goalw Univ.thy [univ_def] "0 : univ(A)"; 

550 
by (rtac (nat_0I RS zero_in_Vfrom) 1); 

760  551 
qed "zero_in_univ"; 
0  552 

553 
goalw Univ.thy [univ_def] "A <= univ(A)"; 

554 
by (rtac A_subset_Vfrom 1); 

760  555 
qed "A_subset_univ"; 
0  556 

557 
val A_into_univ = A_subset_univ RS subsetD; 

558 

559 
(** Closure under unordered and ordered pairs **) 

560 

561 
goalw Univ.thy [univ_def] "!!A a. a: univ(A) ==> {a} : univ(A)"; 

516  562 
by (REPEAT (ares_tac [singleton_in_VLimit, Limit_nat] 1)); 
760  563 
qed "singleton_in_univ"; 
0  564 

565 
goalw Univ.thy [univ_def] 

566 
"!!A a. [ a: univ(A); b: univ(A) ] ==> {a,b} : univ(A)"; 

516  567 
by (REPEAT (ares_tac [doubleton_in_VLimit, Limit_nat] 1)); 
760  568 
qed "doubleton_in_univ"; 
0  569 

570 
goalw Univ.thy [univ_def] 

571 
"!!A a. [ a: univ(A); b: univ(A) ] ==> <a,b> : univ(A)"; 

516  572 
by (REPEAT (ares_tac [Pair_in_VLimit, Limit_nat] 1)); 
760  573 
qed "Pair_in_univ"; 
0  574 

484  575 
goalw Univ.thy [univ_def] "univ(A)*univ(A) <= univ(A)"; 
516  576 
by (rtac (Limit_nat RS product_VLimit) 1); 
760  577 
qed "product_univ"; 
0  578 

579 

580 
(** The natural numbers **) 

581 

582 
goalw Univ.thy [univ_def] "nat <= univ(A)"; 

583 
by (rtac i_subset_Vfrom 1); 

760  584 
qed "nat_subset_univ"; 
0  585 

586 
(* n:nat ==> n:univ(A) *) 

587 
val nat_into_univ = standard (nat_subset_univ RS subsetD); 

588 

589 
(** instances for 1 and 2 **) 

590 

484  591 
goalw Univ.thy [univ_def] "1 : univ(A)"; 
516  592 
by (rtac (Limit_nat RS one_in_VLimit) 1); 
760  593 
qed "one_in_univ"; 
0  594 

595 
(*unused!*) 

27  596 
goal Univ.thy "succ(1) : univ(A)"; 
0  597 
by (REPEAT (ares_tac [nat_into_univ, nat_0I, nat_succI] 1)); 
760  598 
qed "two_in_univ"; 
0  599 

600 
goalw Univ.thy [bool_def] "bool <= univ(A)"; 

601 
by (fast_tac (ZF_cs addSIs [zero_in_univ,one_in_univ]) 1); 

760  602 
qed "bool_subset_univ"; 
0  603 

604 
val bool_into_univ = standard (bool_subset_univ RS subsetD); 

605 

606 

607 
(** Closure under disjoint union **) 

608 

484  609 
goalw Univ.thy [univ_def] "!!A a. a: univ(A) ==> Inl(a) : univ(A)"; 
516  610 
by (etac (Limit_nat RSN (2,Inl_in_VLimit)) 1); 
760  611 
qed "Inl_in_univ"; 
0  612 

484  613 
goalw Univ.thy [univ_def] "!!A b. b: univ(A) ==> Inr(b) : univ(A)"; 
516  614 
by (etac (Limit_nat RSN (2,Inr_in_VLimit)) 1); 
760  615 
qed "Inr_in_univ"; 
0  616 

484  617 
goalw Univ.thy [univ_def] "univ(C)+univ(C) <= univ(C)"; 
516  618 
by (rtac (Limit_nat RS sum_VLimit) 1); 
760  619 
qed "sum_univ"; 
0  620 

484  621 
val sum_subset_univ = [sum_mono, sum_univ] MRS subset_trans > standard; 
622 

623 

0  624 
(** Closure under binary union  use Un_least **) 
625 
(** Closure under Collect  use (Collect_subset RS subset_trans) **) 

626 
(** Closure under RepFun  use RepFun_subset **) 