author  paulson 
Fri, 16 Feb 1996 18:00:47 +0100  
changeset 1512  ce37c64244c0 
parent 187  8729bfdcb638 
permissions  rwrr 
0  1 
(* Title: ZF/univ 
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ID: $Id$ 

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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

4 
Copyright 1992 University of Cambridge 

5 

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The cumulative hierarchy and a small universe for recursive types 

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*) 

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); 
0  15 
val Vfrom = result(); 
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); 

31 
val Vfrom_mono_lemma = result(); 

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 

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(** 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); 
0  44 
val Vfrom_rank_subset1 = result(); 
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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51 
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); 
1c0926788772
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diff
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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); 

61 
val Vfrom_rank_subset2 = result(); 

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); 

67 
val Vfrom_rank_eq = result(); 

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); 

75 
val zero_in_Vfrom = result(); 

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); 

81 
val i_subset_Vfrom = result(); 

82 

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

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

85 
by (rtac Un_upper1 1); 

86 
val A_subset_Vfrom = result(); 

87 

88 
goal Univ.thy "!!A a i. a <= Vfrom(A,i) ==> a: Vfrom(A,succ(i))"; 

89 
by (rtac (Vfrom RS ssubst) 1); 

90 
by (fast_tac ZF_cs 1); 

91 
val subset_mem_Vfrom = result(); 

92 

93 
(** Finite sets and ordered pairs **) 

94 

95 
goal Univ.thy "!!a. a: Vfrom(A,i) ==> {a} : Vfrom(A,succ(i))"; 

96 
by (rtac subset_mem_Vfrom 1); 

97 
by (safe_tac ZF_cs); 

98 
val singleton_in_Vfrom = result(); 

99 

100 
goal Univ.thy 

101 
"!!A. [ a: Vfrom(A,i); b: Vfrom(A,i) ] ==> {a,b} : Vfrom(A,succ(i))"; 

102 
by (rtac subset_mem_Vfrom 1); 

103 
by (safe_tac ZF_cs); 

104 
val doubleton_in_Vfrom = result(); 

105 

106 
goalw Univ.thy [Pair_def] 

107 
"!!A. [ a: Vfrom(A,i); b: Vfrom(A,i) ] ==> \ 

108 
\ <a,b> : Vfrom(A,succ(succ(i)))"; 

109 
by (REPEAT (ares_tac [doubleton_in_Vfrom] 1)); 

110 
val Pair_in_Vfrom = result(); 

111 

112 
val [prem] = goal Univ.thy 

113 
"a<=Vfrom(A,i) ==> succ(a) : Vfrom(A,succ(succ(i)))"; 

114 
by (REPEAT (resolve_tac [subset_mem_Vfrom, succ_subsetI] 1)); 

115 
by (rtac (Vfrom_mono RSN (2,subset_trans)) 2); 

116 
by (REPEAT (resolve_tac [prem, subset_refl, subset_succI] 1)); 

117 
val succ_in_Vfrom = result(); 

118 

119 
(*** 0, successor and limit equations fof Vfrom ***) 

120 

121 
goal Univ.thy "Vfrom(A,0) = A"; 

122 
by (rtac (Vfrom RS ssubst) 1); 

123 
by (fast_tac eq_cs 1); 

124 
val Vfrom_0 = result(); 

125 

126 
goal Univ.thy "!!i. Ord(i) ==> Vfrom(A,succ(i)) = A Un Pow(Vfrom(A,i))"; 

127 
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  130 
by (rtac UN_least 1); 
131 
by (rtac (subset_refl RS Vfrom_mono RS Pow_mono) 1); 

27  132 
by (etac (ltI RS le_imp_subset) 1); 
133 
by (etac Ord_succ 1); 

0  134 
val Vfrom_succ_lemma = result(); 
135 

136 
goal Univ.thy "Vfrom(A,succ(i)) = A Un Pow(Vfrom(A,i))"; 

137 
by (res_inst_tac [("x1", "succ(i)")] (Vfrom_rank_eq RS subst) 1); 

138 
by (res_inst_tac [("x1", "i")] (Vfrom_rank_eq RS subst) 1); 

139 
by (rtac (rank_succ RS ssubst) 1); 

140 
by (rtac (Ord_rank RS Vfrom_succ_lemma) 1); 

141 
val Vfrom_succ = result(); 

142 

143 
(*The premise distinguishes this from Vfrom(A,0); allowing X=0 forces 

144 
the conclusion to be Vfrom(A,Union(X)) = A Un (UN y:X. Vfrom(A,y)) *) 

145 
val [prem] = goal Univ.thy "y:X ==> Vfrom(A,Union(X)) = (UN y:X. Vfrom(A,y))"; 

146 
by (rtac (Vfrom RS ssubst) 1); 

147 
by (rtac equalityI 1); 

148 
(*first inclusion*) 

149 
by (rtac Un_least 1); 

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by (rtac (A_subset_Vfrom RS subset_trans) 1); 
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by (rtac (prem RS UN_upper) 1); 
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by (rtac UN_least 1); 
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153 
by (etac UnionE 1); 
1c0926788772
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by (rtac subset_trans 1); 
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155 
by (etac UN_upper 2); 
0  156 
by (rtac (Vfrom RS ssubst) 1); 
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157 
by (etac ([UN_upper, Un_upper2] MRS subset_trans) 1); 
0  158 
(*opposite inclusion*) 
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159 
by (rtac UN_least 1); 
0  160 
by (rtac (Vfrom RS ssubst) 1); 
161 
by (fast_tac ZF_cs 1); 

162 
val Vfrom_Union = result(); 

163 

164 
(*** Limit ordinals  general properties ***) 

165 

166 
goalw Univ.thy [Limit_def] "!!i. Limit(i) ==> Union(i) = i"; 

27  167 
by (fast_tac (eq_cs addSIs [ltI] addSEs [ltE] addEs [Ord_trans]) 1); 
0  168 
val Limit_Union_eq = result(); 
169 

170 
goalw Univ.thy [Limit_def] "!!i. Limit(i) ==> Ord(i)"; 

171 
by (etac conjunct1 1); 

172 
val Limit_is_Ord = result(); 

173 

27  174 
goalw Univ.thy [Limit_def] "!!i. Limit(i) ==> 0 < i"; 
175 
by (etac (conjunct2 RS conjunct1) 1); 

0  176 
val Limit_has_0 = result(); 
177 

27  178 
goalw Univ.thy [Limit_def] "!!i. [ Limit(i); j<i ] ==> succ(j) < i"; 
0  179 
by (fast_tac ZF_cs 1); 
180 
val Limit_has_succ = result(); 

181 

182 
goalw Univ.thy [Limit_def] "Limit(nat)"; 

27  183 
by (safe_tac (ZF_cs addSIs (ltI::nat_typechecks))); 
184 
by (etac ltD 1); 

0  185 
val Limit_nat = result(); 
186 

187 
goalw Univ.thy [Limit_def] 

37  188 
"!!i. [ 0<i; ALL y. succ(y) ~= i ] ==> Limit(i)"; 
0  189 
by (safe_tac subset_cs); 
27  190 
by (rtac (not_le_iff_lt RS iffD1) 2); 
191 
by (fast_tac (lt_cs addEs [lt_anti_sym]) 4); 

192 
by (REPEAT (eresolve_tac [asm_rl, ltE, Ord_succ] 1)); 

0  193 
val non_succ_LimitI = result(); 
194 

195 
goal Univ.thy "!!i. Ord(i) ==> i=0  (EX j. i=succ(j))  Limit(i)"; 

27  196 
by (fast_tac (ZF_cs addSIs [non_succ_LimitI, Ord_0_lt]) 1); 
0  197 
val Ord_cases_lemma = result(); 
198 

199 
val major::prems = goal Univ.thy 

200 
"[ Ord(i); \ 

201 
\ i=0 ==> P; \ 

202 
\ !!j. i=succ(j) ==> P; \ 

203 
\ Limit(i) ==> P \ 

204 
\ ] ==> P"; 

205 
by (cut_facts_tac [major RS Ord_cases_lemma] 1); 

206 
by (REPEAT (eresolve_tac (prems@[disjE, exE]) 1)); 

207 
val Ord_cases = result(); 

208 

209 

210 
(*** Vfrom applied to Limit ordinals ***) 

211 

212 
(*NB. limit ordinals are nonempty; 

213 
Vfrom(A,0) = A = A Un (UN y:0. Vfrom(A,y)) *) 

214 
val [limiti] = goal Univ.thy 

215 
"Limit(i) ==> Vfrom(A,i) = (UN y:i. Vfrom(A,y))"; 

27  216 
by (rtac (limiti RS (Limit_has_0 RS ltD) RS Vfrom_Union RS subst) 1); 
0  217 
by (rtac (limiti RS Limit_Union_eq RS ssubst) 1); 
218 
by (rtac refl 1); 

219 
val Limit_Vfrom_eq = result(); 

220 

27  221 
goal Univ.thy "!!a. [ a: Vfrom(A,j); Limit(i); j<i ] ==> a : Vfrom(A,i)"; 
222 
by (rtac (Limit_Vfrom_eq RS equalityD2 RS subsetD) 1); 

223 
by (REPEAT (ares_tac [ltD RS UN_I] 1)); 

224 
val Limit_VfromI = result(); 

225 

226 
val prems = goal Univ.thy 

227 
"[ a: Vfrom(A,i); Limit(i); \ 

228 
\ !!x. [ x<i; a: Vfrom(A,x) ] ==> R \ 

229 
\ ] ==> R"; 

230 
by (rtac (Limit_Vfrom_eq RS equalityD1 RS subsetD RS UN_E) 1); 

231 
by (REPEAT (ares_tac (prems @ [ltI, Limit_is_Ord]) 1)); 

232 
val Limit_VfromE = result(); 

0  233 

234 
val [major,limiti] = goal Univ.thy 

235 
"[ a: Vfrom(A,i); Limit(i) ] ==> {a} : Vfrom(A,i)"; 

27  236 
by (rtac ([major,limiti] MRS Limit_VfromE) 1); 
237 
by (etac ([singleton_in_Vfrom, limiti] MRS Limit_VfromI) 1); 

0  238 
by (etac (limiti RS Limit_has_succ) 1); 
239 
val singleton_in_Vfrom_limit = result(); 

240 

241 
val Vfrom_UnI1 = Un_upper1 RS (subset_refl RS Vfrom_mono RS subsetD) 

242 
and Vfrom_UnI2 = Un_upper2 RS (subset_refl RS Vfrom_mono RS subsetD); 

243 

244 
(*Hard work is finding a single j:i such that {a,b}<=Vfrom(A,j)*) 

245 
val [aprem,bprem,limiti] = goal Univ.thy 

246 
"[ a: Vfrom(A,i); b: Vfrom(A,i); Limit(i) ] ==> \ 

247 
\ {a,b} : Vfrom(A,i)"; 

27  248 
by (rtac ([aprem,limiti] MRS Limit_VfromE) 1); 
249 
by (rtac ([bprem,limiti] MRS Limit_VfromE) 1); 

250 
by (rtac ([doubleton_in_Vfrom, limiti] MRS Limit_VfromI) 1); 

251 
by (etac Vfrom_UnI1 1); 

252 
by (etac Vfrom_UnI2 1); 

253 
by (REPEAT (ares_tac [limiti, Limit_has_succ, Un_least_lt] 1)); 

0  254 
val doubleton_in_Vfrom_limit = result(); 
255 

256 
val [aprem,bprem,limiti] = goal Univ.thy 

257 
"[ a: Vfrom(A,i); b: Vfrom(A,i); Limit(i) ] ==> \ 

258 
\ <a,b> : Vfrom(A,i)"; 

259 
(*Infer that a, b occur at ordinals x,xa < i.*) 

27  260 
by (rtac ([aprem,limiti] MRS Limit_VfromE) 1); 
261 
by (rtac ([bprem,limiti] MRS Limit_VfromE) 1); 

262 
by (rtac ([Pair_in_Vfrom, limiti] MRS Limit_VfromI) 1); 

0  263 
(*Infer that succ(succ(x Un xa)) < i *) 
27  264 
by (etac Vfrom_UnI1 1); 
265 
by (etac Vfrom_UnI2 1); 

266 
by (REPEAT (ares_tac [limiti, Limit_has_succ, Un_least_lt] 1)); 

0  267 
val Pair_in_Vfrom_limit = result(); 
268 

269 

270 
(*** Properties assuming Transset(A) ***) 

271 

272 
goal Univ.thy "!!i A. Transset(A) ==> Transset(Vfrom(A,i))"; 

273 
by (eps_ind_tac "i" 1); 

274 
by (rtac (Vfrom RS ssubst) 1); 

275 
by (fast_tac (ZF_cs addSIs [Transset_Union_family, Transset_Un, 

276 
Transset_Pow]) 1); 

277 
val Transset_Vfrom = result(); 

278 

279 
goal Univ.thy "!!A i. Transset(A) ==> Vfrom(A, succ(i)) = Pow(Vfrom(A,i))"; 

280 
by (rtac (Vfrom_succ RS trans) 1); 

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by (rtac (Un_upper2 RSN (2,equalityI)) 1); 
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by (rtac (subset_refl RSN (2,Un_least)) 1); 
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283 
by (rtac (A_subset_Vfrom RS subset_trans) 1); 
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284 
by (etac (Transset_Vfrom RS (Transset_iff_Pow RS iffD1)) 1); 
0  285 
val Transset_Vfrom_succ = result(); 
286 

287 
goalw Ord.thy [Pair_def,Transset_def] 

288 
"!!C. [ <a,b> <= C; Transset(C) ] ==> a: C & b: C"; 

289 
by (fast_tac ZF_cs 1); 

290 
val Transset_Pair_subset = result(); 

291 

292 
goal Univ.thy 

293 
"!!a b.[ <a,b> <= Vfrom(A,i); Transset(A); Limit(i) ] ==> \ 

294 
\ <a,b> : Vfrom(A,i)"; 

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295 
by (etac (Transset_Pair_subset RS conjE) 1); 
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296 
by (etac Transset_Vfrom 1); 
0  297 
by (REPEAT (ares_tac [Pair_in_Vfrom_limit] 1)); 
298 
val Transset_Pair_subset_Vfrom_limit = result(); 

299 

300 

301 
(*** Closure under product/sum applied to elements  thus Vfrom(A,i) 

302 
is a model of simple type theory provided A is a transitive set 

303 
and i is a limit ordinal 

304 
***) 

305 

187  306 
(*General theorem for membership in Vfrom(A,i) when i is a limit ordinal*) 
307 
val [aprem,bprem,limiti,step] = goal Univ.thy 

308 
"[ a: Vfrom(A,i); b: Vfrom(A,i); Limit(i); \ 

309 
\ !!x y j. [ j<i; 1:j; x: Vfrom(A,j); y: Vfrom(A,j) \ 

310 
\ ] ==> EX k. h(x,y): Vfrom(A,k) & k<i ] ==> \ 

311 
\ h(a,b) : Vfrom(A,i)"; 

312 
(*Infer that a, b occur at ordinals x,xa < i.*) 

313 
by (rtac ([aprem,limiti] MRS Limit_VfromE) 1); 

314 
by (rtac ([bprem,limiti] MRS Limit_VfromE) 1); 

315 
by (res_inst_tac [("j1", "x Un xa Un succ(1)")] (step RS exE) 1); 

316 
by (DO_GOAL [etac conjE, etac Limit_VfromI, rtac limiti, atac] 5); 

317 
by (etac (Vfrom_UnI2 RS Vfrom_UnI1) 4); 

318 
by (etac (Vfrom_UnI1 RS Vfrom_UnI1) 3); 

319 
by (rtac (succI1 RS UnI2) 2); 

320 
by (REPEAT (ares_tac [limiti, Limit_has_0, Limit_has_succ, Un_least_lt] 1)); 

321 
val in_Vfrom_limit = result(); 

0  322 

323 
(** products **) 

324 

325 
goal Univ.thy 

187  326 
"!!A. [ a: Vfrom(A,j); b: Vfrom(A,j); Transset(A) ] ==> \ 
327 
\ a*b : Vfrom(A, succ(succ(succ(j))))"; 

0  328 
by (dtac Transset_Vfrom 1); 
329 
by (rtac subset_mem_Vfrom 1); 

330 
by (rewtac Transset_def); 

331 
by (fast_tac (ZF_cs addIs [Pair_in_Vfrom]) 1); 

332 
val prod_in_Vfrom = result(); 

333 

334 
val [aprem,bprem,limiti,transset] = goal Univ.thy 

335 
"[ a: Vfrom(A,i); b: Vfrom(A,i); Limit(i); Transset(A) ] ==> \ 

336 
\ a*b : Vfrom(A,i)"; 

187  337 
by (rtac ([aprem,bprem,limiti] MRS in_Vfrom_limit) 1); 
338 
by (REPEAT (ares_tac [exI, conjI, prod_in_Vfrom, transset, 

339 
limiti RS Limit_has_succ] 1)); 

0  340 
val prod_in_Vfrom_limit = result(); 
341 

342 
(** Disjoint sums, aka Quine ordered pairs **) 

343 

344 
goalw Univ.thy [sum_def] 

187  345 
"!!A. [ a: Vfrom(A,j); b: Vfrom(A,j); Transset(A); 1:j ] ==> \ 
346 
\ a+b : Vfrom(A, succ(succ(succ(j))))"; 

0  347 
by (dtac Transset_Vfrom 1); 
348 
by (rtac subset_mem_Vfrom 1); 

349 
by (rewtac Transset_def); 

350 
by (fast_tac (ZF_cs addIs [zero_in_Vfrom, Pair_in_Vfrom, 

351 
i_subset_Vfrom RS subsetD]) 1); 

352 
val sum_in_Vfrom = result(); 

353 

354 
val [aprem,bprem,limiti,transset] = goal Univ.thy 

355 
"[ a: Vfrom(A,i); b: Vfrom(A,i); Limit(i); Transset(A) ] ==> \ 

356 
\ a+b : Vfrom(A,i)"; 

187  357 
by (rtac ([aprem,bprem,limiti] MRS in_Vfrom_limit) 1); 
358 
by (REPEAT (ares_tac [exI, conjI, sum_in_Vfrom, transset, 

359 
limiti RS Limit_has_succ] 1)); 

0  360 
val sum_in_Vfrom_limit = result(); 
361 

362 
(** function space! **) 

363 

364 
goalw Univ.thy [Pi_def] 

187  365 
"!!A. [ a: Vfrom(A,j); b: Vfrom(A,j); Transset(A) ] ==> \ 
366 
\ a>b : Vfrom(A, succ(succ(succ(succ(j)))))"; 

0  367 
by (dtac Transset_Vfrom 1); 
368 
by (rtac subset_mem_Vfrom 1); 

369 
by (rtac (Collect_subset RS subset_trans) 1); 

370 
by (rtac (Vfrom RS ssubst) 1); 

371 
by (rtac (subset_trans RS subset_trans) 1); 

372 
by (rtac Un_upper2 3); 

373 
by (rtac (succI1 RS UN_upper) 2); 

374 
by (rtac Pow_mono 1); 

375 
by (rewtac Transset_def); 

376 
by (fast_tac (ZF_cs addIs [Pair_in_Vfrom]) 1); 

377 
val fun_in_Vfrom = result(); 

378 

379 
val [aprem,bprem,limiti,transset] = goal Univ.thy 

380 
"[ a: Vfrom(A,i); b: Vfrom(A,i); Limit(i); Transset(A) ] ==> \ 

381 
\ a>b : Vfrom(A,i)"; 

187  382 
by (rtac ([aprem,bprem,limiti] MRS in_Vfrom_limit) 1); 
383 
by (REPEAT (ares_tac [exI, conjI, fun_in_Vfrom, transset, 

384 
limiti RS Limit_has_succ] 1)); 

0  385 
val fun_in_Vfrom_limit = result(); 
386 

387 

388 
(*** The set Vset(i) ***) 

389 

390 
goal Univ.thy "Vset(i) = (UN j:i. Pow(Vset(j)))"; 

391 
by (rtac (Vfrom RS ssubst) 1); 

392 
by (fast_tac eq_cs 1); 

393 
val Vset = result(); 

394 

395 
val Vset_succ = Transset_0 RS Transset_Vfrom_succ; 

396 

397 
val Transset_Vset = Transset_0 RS Transset_Vfrom; 

398 

399 
(** Characterisation of the elements of Vset(i) **) 

400 

27  401 
val [ordi] = goal Univ.thy "Ord(i) ==> ALL b. b : Vset(i) > rank(b) < i"; 
0  402 
by (rtac (ordi RS trans_induct) 1); 
403 
by (rtac (Vset RS ssubst) 1); 

404 
by (safe_tac ZF_cs); 

405 
by (rtac (rank RS ssubst) 1); 

27  406 
by (rtac UN_succ_least_lt 1); 
407 
by (fast_tac ZF_cs 2); 

408 
by (REPEAT (ares_tac [ltI] 1)); 

0  409 
val Vset_rank_imp1 = result(); 
410 

27  411 
(* [ Ord(i); x : Vset(i) ] ==> rank(x) < i *) 
412 
val VsetD = standard (Vset_rank_imp1 RS spec RS mp); 

0  413 

414 
val [ordi] = goal Univ.thy "Ord(i) ==> ALL b. rank(b) : i > b : Vset(i)"; 

415 
by (rtac (ordi RS trans_induct) 1); 

416 
by (rtac allI 1); 

417 
by (rtac (Vset RS ssubst) 1); 

27  418 
by (fast_tac (ZF_cs addSIs [rank_lt RS ltD]) 1); 
0  419 
val Vset_rank_imp2 = result(); 
420 

27  421 
goal Univ.thy "!!x i. rank(x)<i ==> x : Vset(i)"; 
422 
by (etac ltE 1); 

423 
by (etac (Vset_rank_imp2 RS spec RS mp) 1); 

424 
by (assume_tac 1); 

425 
val VsetI = result(); 

0  426 

27  427 
goal Univ.thy "!!i. Ord(i) ==> b : Vset(i) <> rank(b) < i"; 
0  428 
by (rtac iffI 1); 
27  429 
by (REPEAT (eresolve_tac [asm_rl, VsetD, VsetI] 1)); 
0  430 
val Vset_Ord_rank_iff = result(); 
431 

27  432 
goal Univ.thy "b : Vset(a) <> rank(b) < rank(a)"; 
0  433 
by (rtac (Vfrom_rank_eq RS subst) 1); 
434 
by (rtac (Ord_rank RS Vset_Ord_rank_iff) 1); 

435 
val Vset_rank_iff = result(); 

436 

437 
goal Univ.thy "!!i. Ord(i) ==> rank(Vset(i)) = i"; 

438 
by (rtac (rank RS ssubst) 1); 

439 
by (rtac equalityI 1); 

440 
by (safe_tac ZF_cs); 

441 
by (EVERY' [rtac UN_I, 

442 
etac (i_subset_Vfrom RS subsetD), 

443 
etac (Ord_in_Ord RS rank_of_Ord RS ssubst), 

444 
assume_tac, 

445 
rtac succI1] 3); 

27  446 
by (REPEAT (eresolve_tac [asm_rl, VsetD RS ltD, Ord_trans] 1)); 
0  447 
val rank_Vset = result(); 
448 

449 
(** Lemmas for reasoning about sets in terms of their elements' ranks **) 

450 

451 
goal Univ.thy "a <= Vset(rank(a))"; 

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ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
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diff
changeset

452 
by (rtac subsetI 1); 
27  453 
by (etac (rank_lt RS VsetI) 1); 
0  454 
val arg_subset_Vset_rank = result(); 
455 

456 
val [iprem] = goal Univ.thy 

457 
"[ !!i. Ord(i) ==> a Int Vset(i) <= b ] ==> a <= b"; 

27  458 
by (rtac ([subset_refl, arg_subset_Vset_rank] MRS 
459 
Int_greatest RS subset_trans) 1); 

14
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ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
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parents:
6
diff
changeset

460 
by (rtac (Ord_rank RS iprem) 1); 
0  461 
val Int_Vset_subset = result(); 
462 

463 
(** Set up an environment for simplification **) 

464 

465 
val rank_rls = [rank_Inl, rank_Inr, rank_pair1, rank_pair2]; 

27  466 
val rank_trans_rls = rank_rls @ (rank_rls RLN (2, [lt_trans])); 
0  467 

468 
val rank_ss = ZF_ss 

27  469 
addsimps [case_Inl, case_Inr, VsetI] 
6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset

470 
addsimps rank_trans_rls; 
0  471 

472 
(** Recursion over Vset levels! **) 

473 

474 
(*NOT SUITABLE FOR REWRITING: recursive!*) 

475 
goalw Univ.thy [Vrec_def] "Vrec(a,H) = H(a, lam x:Vset(rank(a)). Vrec(x,H))"; 

476 
by (rtac (transrec RS ssubst) 1); 

27  477 
by (simp_tac (ZF_ss addsimps [Ord_rank, Ord_succ, VsetD RS ltD RS beta, 
478 
VsetI RS beta, le_refl]) 1); 

0  479 
val Vrec = result(); 
480 

481 
(*This form avoids giant explosions in proofs. NOTE USE OF == *) 

482 
val rew::prems = goal Univ.thy 

483 
"[ !!x. h(x)==Vrec(x,H) ] ==> \ 

484 
\ h(a) = H(a, lam x: Vset(rank(a)). h(x))"; 

485 
by (rewtac rew); 

486 
by (rtac Vrec 1); 

487 
val def_Vrec = result(); 

488 

489 

490 
(*** univ(A) ***) 

491 

492 
goalw Univ.thy [univ_def] "!!A B. A<=B ==> univ(A) <= univ(B)"; 

493 
by (etac Vfrom_mono 1); 

494 
by (rtac subset_refl 1); 

495 
val univ_mono = result(); 

496 

497 
goalw Univ.thy [univ_def] "!!A. Transset(A) ==> Transset(univ(A))"; 

498 
by (etac Transset_Vfrom 1); 

499 
val Transset_univ = result(); 

500 

501 
(** univ(A) as a limit **) 

502 

503 
goalw Univ.thy [univ_def] "univ(A) = (UN i:nat. Vfrom(A,i))"; 

14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

504 
by (rtac (Limit_nat RS Limit_Vfrom_eq) 1); 
0  505 
val univ_eq_UN = result(); 
506 

507 
goal Univ.thy "!!c. c <= univ(A) ==> c = (UN i:nat. c Int Vfrom(A,i))"; 

14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

508 
by (rtac (subset_UN_iff_eq RS iffD1) 1); 
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

509 
by (etac (univ_eq_UN RS subst) 1); 
0  510 
val subset_univ_eq_Int = result(); 
511 

512 
val [aprem, iprem] = goal Univ.thy 

513 
"[ a <= univ(X); \ 

514 
\ !!i. i:nat ==> a Int Vfrom(X,i) <= b \ 

515 
\ ] ==> a <= b"; 

14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

516 
by (rtac (aprem RS subset_univ_eq_Int RS ssubst) 1); 
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

517 
by (rtac UN_least 1); 
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

518 
by (etac iprem 1); 
0  519 
val univ_Int_Vfrom_subset = result(); 
520 

521 
val prems = goal Univ.thy 

522 
"[ a <= univ(X); b <= univ(X); \ 

523 
\ !!i. i:nat ==> a Int Vfrom(X,i) = b Int Vfrom(X,i) \ 

524 
\ ] ==> a = b"; 

14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

525 
by (rtac equalityI 1); 
0  526 
by (ALLGOALS 
527 
(resolve_tac (prems RL [univ_Int_Vfrom_subset]) THEN' 

528 
eresolve_tac (prems RL [equalityD1,equalityD2] RL [subset_trans]) THEN' 

529 
rtac Int_lower1)); 

530 
val univ_Int_Vfrom_eq = result(); 

531 

532 
(** Closure properties **) 

533 

534 
goalw Univ.thy [univ_def] "0 : univ(A)"; 

535 
by (rtac (nat_0I RS zero_in_Vfrom) 1); 

536 
val zero_in_univ = result(); 

537 

538 
goalw Univ.thy [univ_def] "A <= univ(A)"; 

539 
by (rtac A_subset_Vfrom 1); 

540 
val A_subset_univ = result(); 

541 

542 
val A_into_univ = A_subset_univ RS subsetD; 

543 

544 
(** Closure under unordered and ordered pairs **) 

545 

546 
goalw Univ.thy [univ_def] "!!A a. a: univ(A) ==> {a} : univ(A)"; 

547 
by (rtac singleton_in_Vfrom_limit 1); 

548 
by (REPEAT (ares_tac [Ord_nat,Limit_nat] 1)); 

549 
val singleton_in_univ = result(); 

550 

551 
goalw Univ.thy [univ_def] 

552 
"!!A a. [ a: univ(A); b: univ(A) ] ==> {a,b} : univ(A)"; 

553 
by (rtac doubleton_in_Vfrom_limit 1); 

554 
by (REPEAT (ares_tac [Ord_nat,Limit_nat] 1)); 

555 
val doubleton_in_univ = result(); 

556 

557 
goalw Univ.thy [univ_def] 

558 
"!!A a. [ a: univ(A); b: univ(A) ] ==> <a,b> : univ(A)"; 

559 
by (rtac Pair_in_Vfrom_limit 1); 

560 
by (REPEAT (ares_tac [Ord_nat,Limit_nat] 1)); 

561 
val Pair_in_univ = result(); 

562 

563 
goal Univ.thy "univ(A)*univ(A) <= univ(A)"; 

564 
by (REPEAT (ares_tac [subsetI,Pair_in_univ] 1 

565 
ORELSE eresolve_tac [SigmaE, ssubst] 1)); 

566 
val product_univ = result(); 

567 

568 
val Sigma_subset_univ = standard 

569 
(Sigma_mono RS (product_univ RSN (2,subset_trans))); 

570 

571 
goalw Univ.thy [univ_def] 

572 
"!!a b.[ <a,b> <= univ(A); Transset(A) ] ==> <a,b> : univ(A)"; 

14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

573 
by (etac Transset_Pair_subset_Vfrom_limit 1); 
0  574 
by (REPEAT (ares_tac [Ord_nat,Limit_nat] 1)); 
575 
val Transset_Pair_subset_univ = result(); 

576 

577 

578 
(** The natural numbers **) 

579 

580 
goalw Univ.thy [univ_def] "nat <= univ(A)"; 

581 
by (rtac i_subset_Vfrom 1); 

582 
val nat_subset_univ = result(); 

583 

584 
(* n:nat ==> n:univ(A) *) 

585 
val nat_into_univ = standard (nat_subset_univ RS subsetD); 

586 

587 
(** instances for 1 and 2 **) 

588 

14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

589 
goal Univ.thy "1 : univ(A)"; 
0  590 
by (REPEAT (ares_tac [nat_into_univ, nat_0I, nat_succI] 1)); 
591 
val one_in_univ = result(); 

592 

593 
(*unused!*) 

27  594 
goal Univ.thy "succ(1) : univ(A)"; 
0  595 
by (REPEAT (ares_tac [nat_into_univ, nat_0I, nat_succI] 1)); 
596 
val two_in_univ = result(); 

597 

598 
goalw Univ.thy [bool_def] "bool <= univ(A)"; 

599 
by (fast_tac (ZF_cs addSIs [zero_in_univ,one_in_univ]) 1); 

600 
val bool_subset_univ = result(); 

601 

602 
val bool_into_univ = standard (bool_subset_univ RS subsetD); 

603 

604 

605 
(** Closure under disjoint union **) 

606 

607 
goalw Univ.thy [Inl_def] "!!A a. a: univ(A) ==> Inl(a) : univ(A)"; 

608 
by (REPEAT (ares_tac [zero_in_univ,Pair_in_univ] 1)); 

609 
val Inl_in_univ = result(); 

610 

611 
goalw Univ.thy [Inr_def] "!!A b. b: univ(A) ==> Inr(b) : univ(A)"; 

612 
by (REPEAT (ares_tac [one_in_univ, Pair_in_univ] 1)); 

613 
val Inr_in_univ = result(); 

614 

615 
goal Univ.thy "univ(C)+univ(C) <= univ(C)"; 

616 
by (REPEAT (ares_tac [subsetI,Inl_in_univ,Inr_in_univ] 1 

617 
ORELSE eresolve_tac [sumE, ssubst] 1)); 

618 
val sum_univ = result(); 

619 

620 
val sum_subset_univ = standard 

621 
(sum_mono RS (sum_univ RSN (2,subset_trans))); 

622 

623 

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 **) 

627 

628 