author  paulson 
Fri, 16 Feb 1996 18:00:47 +0100  
changeset 1512  ce37c64244c0 
parent 1461  6bcb44e4d6e5 
child 2033  639de962ded4 
permissions  rwrr 
1461  1 
(* Title: ZF/Univ 
0  2 
ID: $Id$ 
1461  3 
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) *) 

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bind_thm ("Vfrom_mono", (Vfrom_mono_lemma RS spec RS mp)); 
0  35 

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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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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); 
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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 

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bind_thm ("A_into_Vfrom", A_subset_Vfrom RS subsetD); 
488  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 
1461  131 
(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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by (rtac (prem RS UN_upper) 1); 
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by (rtac UN_least 1); 
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155 
by (etac UnionE 1); 
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156 
by (rtac subset_trans 1); 
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157 
by (etac UN_upper 2); 
0  158 
by (rtac (Vfrom RS ssubst) 1); 
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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 
(*** Vfrom applied to Limit ordinals ***) 

167 

168 
(*NB. limit ordinals are nonempty; 

169 
Vfrom(A,0) = A = A Un (UN y:0. Vfrom(A,y)) *) 

170 
val [limiti] = goal Univ.thy 

171 
"Limit(i) ==> Vfrom(A,i) = (UN y:i. Vfrom(A,y))"; 

27  172 
by (rtac (limiti RS (Limit_has_0 RS ltD) RS Vfrom_Union RS subst) 1); 
0  173 
by (rtac (limiti RS Limit_Union_eq RS ssubst) 1); 
174 
by (rtac refl 1); 

760  175 
qed "Limit_Vfrom_eq"; 
0  176 

27  177 
goal Univ.thy "!!a. [ a: Vfrom(A,j); Limit(i); j<i ] ==> a : Vfrom(A,i)"; 
178 
by (rtac (Limit_Vfrom_eq RS equalityD2 RS subsetD) 1); 

179 
by (REPEAT (ares_tac [ltD RS UN_I] 1)); 

760  180 
qed "Limit_VfromI"; 
27  181 

182 
val prems = goal Univ.thy 

1461  183 
"[ a: Vfrom(A,i); Limit(i); \ 
184 
\ !!x. [ x<i; a: Vfrom(A,x) ] ==> R \ 

27  185 
\ ] ==> R"; 
186 
by (rtac (Limit_Vfrom_eq RS equalityD1 RS subsetD RS UN_E) 1); 

187 
by (REPEAT (ares_tac (prems @ [ltI, Limit_is_Ord]) 1)); 

760  188 
qed "Limit_VfromE"; 
0  189 

516  190 
val zero_in_VLimit = Limit_has_0 RS ltD RS zero_in_Vfrom; 
484  191 

0  192 
val [major,limiti] = goal Univ.thy 
193 
"[ a: Vfrom(A,i); Limit(i) ] ==> {a} : Vfrom(A,i)"; 

27  194 
by (rtac ([major,limiti] MRS Limit_VfromE) 1); 
195 
by (etac ([singleton_in_Vfrom, limiti] MRS Limit_VfromI) 1); 

0  196 
by (etac (limiti RS Limit_has_succ) 1); 
760  197 
qed "singleton_in_VLimit"; 
0  198 

199 
val Vfrom_UnI1 = Un_upper1 RS (subset_refl RS Vfrom_mono RS subsetD) 

200 
and Vfrom_UnI2 = Un_upper2 RS (subset_refl RS Vfrom_mono RS subsetD); 

201 

202 
(*Hard work is finding a single j:i such that {a,b}<=Vfrom(A,j)*) 

203 
val [aprem,bprem,limiti] = goal Univ.thy 

204 
"[ a: Vfrom(A,i); b: Vfrom(A,i); Limit(i) ] ==> \ 

205 
\ {a,b} : Vfrom(A,i)"; 

27  206 
by (rtac ([aprem,limiti] MRS Limit_VfromE) 1); 
207 
by (rtac ([bprem,limiti] MRS Limit_VfromE) 1); 

208 
by (rtac ([doubleton_in_Vfrom, limiti] MRS Limit_VfromI) 1); 

209 
by (etac Vfrom_UnI1 1); 

210 
by (etac Vfrom_UnI2 1); 

211 
by (REPEAT (ares_tac [limiti, Limit_has_succ, Un_least_lt] 1)); 

760  212 
qed "doubleton_in_VLimit"; 
0  213 

214 
val [aprem,bprem,limiti] = goal Univ.thy 

215 
"[ a: Vfrom(A,i); b: Vfrom(A,i); Limit(i) ] ==> \ 

216 
\ <a,b> : Vfrom(A,i)"; 

217 
(*Infer that a, b occur at ordinals x,xa < i.*) 

27  218 
by (rtac ([aprem,limiti] MRS Limit_VfromE) 1); 
219 
by (rtac ([bprem,limiti] MRS Limit_VfromE) 1); 

220 
by (rtac ([Pair_in_Vfrom, limiti] MRS Limit_VfromI) 1); 

0  221 
(*Infer that succ(succ(x Un xa)) < i *) 
27  222 
by (etac Vfrom_UnI1 1); 
223 
by (etac Vfrom_UnI2 1); 

224 
by (REPEAT (ares_tac [limiti, Limit_has_succ, Un_least_lt] 1)); 

760  225 
qed "Pair_in_VLimit"; 
484  226 

227 
goal Univ.thy "!!i. Limit(i) ==> Vfrom(A,i)*Vfrom(A,i) <= Vfrom(A,i)"; 

516  228 
by (REPEAT (ares_tac [subsetI,Pair_in_VLimit] 1 
484  229 
ORELSE eresolve_tac [SigmaE, ssubst] 1)); 
760  230 
qed "product_VLimit"; 
484  231 

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232 
bind_thm ("Sigma_subset_VLimit", 
1461  233 
[Sigma_mono, product_VLimit] MRS subset_trans); 
484  234 

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235 
bind_thm ("nat_subset_VLimit", 
1461  236 
[nat_le_Limit RS le_imp_subset, i_subset_Vfrom] MRS subset_trans); 
484  237 

488  238 
goal Univ.thy "!!i. [ n: nat; Limit(i) ] ==> n : Vfrom(A,i)"; 
516  239 
by (REPEAT (ares_tac [nat_subset_VLimit RS subsetD] 1)); 
760  240 
qed "nat_into_VLimit"; 
484  241 

242 
(** Closure under disjoint union **) 

243 

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244 
bind_thm ("zero_in_VLimit", Limit_has_0 RS ltD RS zero_in_Vfrom); 
484  245 

246 
goal Univ.thy "!!i. Limit(i) ==> 1 : Vfrom(A,i)"; 

516  247 
by (REPEAT (ares_tac [nat_into_VLimit, nat_0I, nat_succI] 1)); 
760  248 
qed "one_in_VLimit"; 
484  249 

250 
goalw Univ.thy [Inl_def] 

251 
"!!A a. [ a: Vfrom(A,i); Limit(i) ] ==> Inl(a) : Vfrom(A,i)"; 

516  252 
by (REPEAT (ares_tac [zero_in_VLimit, Pair_in_VLimit] 1)); 
760  253 
qed "Inl_in_VLimit"; 
484  254 

255 
goalw Univ.thy [Inr_def] 

256 
"!!A b. [ b: Vfrom(A,i); Limit(i) ] ==> Inr(b) : Vfrom(A,i)"; 

516  257 
by (REPEAT (ares_tac [one_in_VLimit, Pair_in_VLimit] 1)); 
760  258 
qed "Inr_in_VLimit"; 
484  259 

260 
goal Univ.thy "!!i. Limit(i) ==> Vfrom(C,i)+Vfrom(C,i) <= Vfrom(C,i)"; 

516  261 
by (fast_tac (sum_cs addSIs [Inl_in_VLimit, Inr_in_VLimit]) 1); 
760  262 
qed "sum_VLimit"; 
484  263 

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264 
bind_thm ("sum_subset_VLimit", [sum_mono, sum_VLimit] MRS subset_trans); 
484  265 

0  266 

267 

268 
(*** Properties assuming Transset(A) ***) 

269 

270 
goal Univ.thy "!!i A. Transset(A) ==> Transset(Vfrom(A,i))"; 

271 
by (eps_ind_tac "i" 1); 

272 
by (rtac (Vfrom RS ssubst) 1); 

273 
by (fast_tac (ZF_cs addSIs [Transset_Union_family, Transset_Un, 

1461  274 
Transset_Pow]) 1); 
760  275 
qed "Transset_Vfrom"; 
0  276 

277 
goal Univ.thy "!!A i. Transset(A) ==> Vfrom(A, succ(i)) = Pow(Vfrom(A,i))"; 

278 
by (rtac (Vfrom_succ RS trans) 1); 

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279 
by (rtac (Un_upper2 RSN (2,equalityI)) 1); 
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280 
by (rtac (subset_refl RSN (2,Un_least)) 1); 
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281 
by (rtac (A_subset_Vfrom RS subset_trans) 1); 
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282 
by (etac (Transset_Vfrom RS (Transset_iff_Pow RS iffD1)) 1); 
760  283 
qed "Transset_Vfrom_succ"; 
0  284 

435  285 
goalw Ordinal.thy [Pair_def,Transset_def] 
0  286 
"!!C. [ <a,b> <= C; Transset(C) ] ==> a: C & b: C"; 
287 
by (fast_tac ZF_cs 1); 

760  288 
qed "Transset_Pair_subset"; 
0  289 

290 
goal Univ.thy 

291 
"!!a b.[ <a,b> <= Vfrom(A,i); Transset(A); Limit(i) ] ==> \ 

292 
\ <a,b> : Vfrom(A,i)"; 

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293 
by (etac (Transset_Pair_subset RS conjE) 1); 
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294 
by (etac Transset_Vfrom 1); 
516  295 
by (REPEAT (ares_tac [Pair_in_VLimit] 1)); 
760  296 
qed "Transset_Pair_subset_VLimit"; 
0  297 

298 

299 
(*** Closure under product/sum applied to elements  thus Vfrom(A,i) 

300 
is a model of simple type theory provided A is a transitive set 

301 
and i is a limit ordinal 

302 
***) 

303 

187  304 
(*General theorem for membership in Vfrom(A,i) when i is a limit ordinal*) 
305 
val [aprem,bprem,limiti,step] = goal Univ.thy 

1461  306 
"[ a: Vfrom(A,i); b: Vfrom(A,i); Limit(i); \ 
187  307 
\ !!x y j. [ j<i; 1:j; x: Vfrom(A,j); y: Vfrom(A,j) \ 
1461  308 
\ ] ==> EX k. h(x,y): Vfrom(A,k) & k<i ] ==> \ 
187  309 
\ h(a,b) : Vfrom(A,i)"; 
310 
(*Infer that a, b occur at ordinals x,xa < i.*) 

311 
by (rtac ([aprem,limiti] MRS Limit_VfromE) 1); 

312 
by (rtac ([bprem,limiti] MRS Limit_VfromE) 1); 

828  313 
by (res_inst_tac [("j1", "x Un xa Un 2")] (step RS exE) 1); 
187  314 
by (DO_GOAL [etac conjE, etac Limit_VfromI, rtac limiti, atac] 5); 
315 
by (etac (Vfrom_UnI2 RS Vfrom_UnI1) 4); 

316 
by (etac (Vfrom_UnI1 RS Vfrom_UnI1) 3); 

317 
by (rtac (succI1 RS UnI2) 2); 

318 
by (REPEAT (ares_tac [limiti, Limit_has_0, Limit_has_succ, Un_least_lt] 1)); 

760  319 
qed "in_VLimit"; 
0  320 

321 
(** products **) 

322 

323 
goal Univ.thy 

187  324 
"!!A. [ a: Vfrom(A,j); b: Vfrom(A,j); Transset(A) ] ==> \ 
325 
\ a*b : Vfrom(A, succ(succ(succ(j))))"; 

0  326 
by (dtac Transset_Vfrom 1); 
327 
by (rtac subset_mem_Vfrom 1); 

328 
by (rewtac Transset_def); 

329 
by (fast_tac (ZF_cs addIs [Pair_in_Vfrom]) 1); 

760  330 
qed "prod_in_Vfrom"; 
0  331 

332 
val [aprem,bprem,limiti,transset] = goal Univ.thy 

333 
"[ a: Vfrom(A,i); b: Vfrom(A,i); Limit(i); Transset(A) ] ==> \ 

334 
\ a*b : Vfrom(A,i)"; 

516  335 
by (rtac ([aprem,bprem,limiti] MRS in_VLimit) 1); 
187  336 
by (REPEAT (ares_tac [exI, conjI, prod_in_Vfrom, transset, 
1461  337 
limiti RS Limit_has_succ] 1)); 
760  338 
qed "prod_in_VLimit"; 
0  339 

340 
(** Disjoint sums, aka Quine ordered pairs **) 

341 

342 
goalw Univ.thy [sum_def] 

187  343 
"!!A. [ a: Vfrom(A,j); b: Vfrom(A,j); Transset(A); 1:j ] ==> \ 
344 
\ a+b : Vfrom(A, succ(succ(succ(j))))"; 

0  345 
by (dtac Transset_Vfrom 1); 
346 
by (rtac subset_mem_Vfrom 1); 

347 
by (rewtac Transset_def); 

348 
by (fast_tac (ZF_cs addIs [zero_in_Vfrom, Pair_in_Vfrom, 

1461  349 
i_subset_Vfrom RS subsetD]) 1); 
760  350 
qed "sum_in_Vfrom"; 
0  351 

352 
val [aprem,bprem,limiti,transset] = goal Univ.thy 

353 
"[ a: Vfrom(A,i); b: Vfrom(A,i); Limit(i); Transset(A) ] ==> \ 

354 
\ a+b : Vfrom(A,i)"; 

516  355 
by (rtac ([aprem,bprem,limiti] MRS in_VLimit) 1); 
187  356 
by (REPEAT (ares_tac [exI, conjI, sum_in_Vfrom, transset, 
1461  357 
limiti RS Limit_has_succ] 1)); 
760  358 
qed "sum_in_VLimit"; 
0  359 

360 
(** function space! **) 

361 

362 
goalw Univ.thy [Pi_def] 

187  363 
"!!A. [ a: Vfrom(A,j); b: Vfrom(A,j); Transset(A) ] ==> \ 
364 
\ a>b : Vfrom(A, succ(succ(succ(succ(j)))))"; 

0  365 
by (dtac Transset_Vfrom 1); 
366 
by (rtac subset_mem_Vfrom 1); 

367 
by (rtac (Collect_subset RS subset_trans) 1); 

368 
by (rtac (Vfrom RS ssubst) 1); 

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

370 
by (rtac Un_upper2 3); 

371 
by (rtac (succI1 RS UN_upper) 2); 

372 
by (rtac Pow_mono 1); 

373 
by (rewtac Transset_def); 

374 
by (fast_tac (ZF_cs addIs [Pair_in_Vfrom]) 1); 

760  375 
qed "fun_in_Vfrom"; 
0  376 

377 
val [aprem,bprem,limiti,transset] = goal Univ.thy 

378 
"[ a: Vfrom(A,i); b: Vfrom(A,i); Limit(i); Transset(A) ] ==> \ 

379 
\ a>b : Vfrom(A,i)"; 

516  380 
by (rtac ([aprem,bprem,limiti] MRS in_VLimit) 1); 
187  381 
by (REPEAT (ares_tac [exI, conjI, fun_in_Vfrom, transset, 
1461  382 
limiti RS Limit_has_succ] 1)); 
760  383 
qed "fun_in_VLimit"; 
0  384 

385 

386 
(*** The set Vset(i) ***) 

387 

388 
goal Univ.thy "Vset(i) = (UN j:i. Pow(Vset(j)))"; 

389 
by (rtac (Vfrom RS ssubst) 1); 

390 
by (fast_tac eq_cs 1); 

760  391 
qed "Vset"; 
0  392 

393 
val Vset_succ = Transset_0 RS Transset_Vfrom_succ; 

394 

395 
val Transset_Vset = Transset_0 RS Transset_Vfrom; 

396 

397 
(** Characterisation of the elements of Vset(i) **) 

398 

27  399 
val [ordi] = goal Univ.thy "Ord(i) ==> ALL b. b : Vset(i) > rank(b) < i"; 
0  400 
by (rtac (ordi RS trans_induct) 1); 
401 
by (rtac (Vset RS ssubst) 1); 

402 
by (safe_tac ZF_cs); 

403 
by (rtac (rank RS ssubst) 1); 

27  404 
by (rtac UN_succ_least_lt 1); 
405 
by (fast_tac ZF_cs 2); 

406 
by (REPEAT (ares_tac [ltI] 1)); 

760  407 
qed "Vset_rank_imp1"; 
0  408 

27  409 
(* [ Ord(i); x : Vset(i) ] ==> rank(x) < i *) 
782
200a16083201
added bind_thm for theorems defined by "standard ..."
clasohm
parents:
760
diff
changeset

410 
bind_thm ("VsetD", (Vset_rank_imp1 RS spec RS mp)); 
0  411 

412 
val [ordi] = goal Univ.thy "Ord(i) ==> ALL b. rank(b) : i > b : Vset(i)"; 

413 
by (rtac (ordi RS trans_induct) 1); 

414 
by (rtac allI 1); 

415 
by (rtac (Vset RS ssubst) 1); 

27  416 
by (fast_tac (ZF_cs addSIs [rank_lt RS ltD]) 1); 
760  417 
qed "Vset_rank_imp2"; 
0  418 

27  419 
goal Univ.thy "!!x i. rank(x)<i ==> x : Vset(i)"; 
420 
by (etac ltE 1); 

421 
by (etac (Vset_rank_imp2 RS spec RS mp) 1); 

422 
by (assume_tac 1); 

760  423 
qed "VsetI"; 
0  424 

27  425 
goal Univ.thy "!!i. Ord(i) ==> b : Vset(i) <> rank(b) < i"; 
0  426 
by (rtac iffI 1); 
27  427 
by (REPEAT (eresolve_tac [asm_rl, VsetD, VsetI] 1)); 
760  428 
qed "Vset_Ord_rank_iff"; 
0  429 

27  430 
goal Univ.thy "b : Vset(a) <> rank(b) < rank(a)"; 
0  431 
by (rtac (Vfrom_rank_eq RS subst) 1); 
432 
by (rtac (Ord_rank RS Vset_Ord_rank_iff) 1); 

760  433 
qed "Vset_rank_iff"; 
0  434 

435 
goal Univ.thy "!!i. Ord(i) ==> rank(Vset(i)) = i"; 

436 
by (rtac (rank RS ssubst) 1); 

437 
by (rtac equalityI 1); 

438 
by (safe_tac ZF_cs); 

828  439 
by (EVERY' [rtac UN_I, 
1461  440 
etac (i_subset_Vfrom RS subsetD), 
441 
etac (Ord_in_Ord RS rank_of_Ord RS ssubst), 

442 
assume_tac, 

443 
rtac succI1] 3); 

27  444 
by (REPEAT (eresolve_tac [asm_rl, VsetD RS ltD, Ord_trans] 1)); 
760  445 
qed "rank_Vset"; 
0  446 

447 
(** Lemmas for reasoning about sets in terms of their elements' ranks **) 

448 

449 
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

450 
by (rtac subsetI 1); 
27  451 
by (etac (rank_lt RS VsetI) 1); 
760  452 
qed "arg_subset_Vset_rank"; 
0  453 

454 
val [iprem] = goal Univ.thy 

455 
"[ !!i. Ord(i) ==> a Int Vset(i) <= b ] ==> a <= b"; 

27  456 
by (rtac ([subset_refl, arg_subset_Vset_rank] MRS 
1461  457 
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

458 
by (rtac (Ord_rank RS iprem) 1); 
760  459 
qed "Int_Vset_subset"; 
0  460 

461 
(** Set up an environment for simplification **) 

462 

463 
val rank_rls = [rank_Inl, rank_Inr, rank_pair1, rank_pair2]; 

27  464 
val rank_trans_rls = rank_rls @ (rank_rls RLN (2, [lt_trans])); 
0  465 

466 
val rank_ss = ZF_ss 

27  467 
addsimps [case_Inl, case_Inr, VsetI] 
6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset

468 
addsimps rank_trans_rls; 
0  469 

470 
(** Recursion over Vset levels! **) 

471 

472 
(*NOT SUITABLE FOR REWRITING: recursive!*) 

473 
goalw Univ.thy [Vrec_def] "Vrec(a,H) = H(a, lam x:Vset(rank(a)). Vrec(x,H))"; 

474 
by (rtac (transrec RS ssubst) 1); 

27  475 
by (simp_tac (ZF_ss addsimps [Ord_rank, Ord_succ, VsetD RS ltD RS beta, 
1461  476 
VsetI RS beta, le_refl]) 1); 
760  477 
qed "Vrec"; 
0  478 

479 
(*This form avoids giant explosions in proofs. NOTE USE OF == *) 

480 
val rew::prems = goal Univ.thy 

481 
"[ !!x. h(x)==Vrec(x,H) ] ==> \ 

482 
\ h(a) = H(a, lam x: Vset(rank(a)). h(x))"; 

483 
by (rewtac rew); 

484 
by (rtac Vrec 1); 

760  485 
qed "def_Vrec"; 
0  486 

487 

488 
(*** univ(A) ***) 

489 

490 
goalw Univ.thy [univ_def] "!!A B. A<=B ==> univ(A) <= univ(B)"; 

491 
by (etac Vfrom_mono 1); 

492 
by (rtac subset_refl 1); 

760  493 
qed "univ_mono"; 
0  494 

495 
goalw Univ.thy [univ_def] "!!A. Transset(A) ==> Transset(univ(A))"; 

496 
by (etac Transset_Vfrom 1); 

760  497 
qed "Transset_univ"; 
0  498 

499 
(** univ(A) as a limit **) 

500 

501 
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

502 
by (rtac (Limit_nat RS Limit_Vfrom_eq) 1); 
760  503 
qed "univ_eq_UN"; 
0  504 

505 
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

506 
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

507 
by (etac (univ_eq_UN RS subst) 1); 
760  508 
qed "subset_univ_eq_Int"; 
0  509 

510 
val [aprem, iprem] = goal Univ.thy 

1461  511 
"[ a <= univ(X); \ 
512 
\ !!i. i:nat ==> a Int Vfrom(X,i) <= b \ 

0  513 
\ ] ==> a <= b"; 
15
6c6d2f6e3185
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

514 
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

515 
by (rtac UN_least 1); 
6c6d2f6e3185
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

516 
by (etac iprem 1); 
760  517 
qed "univ_Int_Vfrom_subset"; 
0  518 

519 
val prems = goal Univ.thy 

520 
"[ a <= univ(X); b <= univ(X); \ 

521 
\ !!i. i:nat ==> a Int Vfrom(X,i) = b Int Vfrom(X,i) \ 

522 
\ ] ==> a = b"; 

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

523 
by (rtac equalityI 1); 
0  524 
by (ALLGOALS 
525 
(resolve_tac (prems RL [univ_Int_Vfrom_subset]) THEN' 

526 
eresolve_tac (prems RL [equalityD1,equalityD2] RL [subset_trans]) THEN' 

527 
rtac Int_lower1)); 

760  528 
qed "univ_Int_Vfrom_eq"; 
0  529 

530 
(** Closure properties **) 

531 

532 
goalw Univ.thy [univ_def] "0 : univ(A)"; 

533 
by (rtac (nat_0I RS zero_in_Vfrom) 1); 

760  534 
qed "zero_in_univ"; 
0  535 

536 
goalw Univ.thy [univ_def] "A <= univ(A)"; 

537 
by (rtac A_subset_Vfrom 1); 

760  538 
qed "A_subset_univ"; 
0  539 

540 
val A_into_univ = A_subset_univ RS subsetD; 

541 

542 
(** Closure under unordered and ordered pairs **) 

543 

544 
goalw Univ.thy [univ_def] "!!A a. a: univ(A) ==> {a} : univ(A)"; 

516  545 
by (REPEAT (ares_tac [singleton_in_VLimit, Limit_nat] 1)); 
760  546 
qed "singleton_in_univ"; 
0  547 

548 
goalw Univ.thy [univ_def] 

549 
"!!A a. [ a: univ(A); b: univ(A) ] ==> {a,b} : univ(A)"; 

516  550 
by (REPEAT (ares_tac [doubleton_in_VLimit, Limit_nat] 1)); 
760  551 
qed "doubleton_in_univ"; 
0  552 

553 
goalw Univ.thy [univ_def] 

554 
"!!A a. [ a: univ(A); b: univ(A) ] ==> <a,b> : univ(A)"; 

516  555 
by (REPEAT (ares_tac [Pair_in_VLimit, Limit_nat] 1)); 
760  556 
qed "Pair_in_univ"; 
0  557 

484  558 
goalw Univ.thy [univ_def] "univ(A)*univ(A) <= univ(A)"; 
516  559 
by (rtac (Limit_nat RS product_VLimit) 1); 
760  560 
qed "product_univ"; 
0  561 

562 

563 
(** The natural numbers **) 

564 

565 
goalw Univ.thy [univ_def] "nat <= univ(A)"; 

566 
by (rtac i_subset_Vfrom 1); 

760  567 
qed "nat_subset_univ"; 
0  568 

569 
(* n:nat ==> n:univ(A) *) 

782
200a16083201
added bind_thm for theorems defined by "standard ..."
clasohm
parents:
760
diff
changeset

570 
bind_thm ("nat_into_univ", (nat_subset_univ RS subsetD)); 
0  571 

572 
(** instances for 1 and 2 **) 

573 

484  574 
goalw Univ.thy [univ_def] "1 : univ(A)"; 
516  575 
by (rtac (Limit_nat RS one_in_VLimit) 1); 
760  576 
qed "one_in_univ"; 
0  577 

578 
(*unused!*) 

828  579 
goal Univ.thy "2 : univ(A)"; 
0  580 
by (REPEAT (ares_tac [nat_into_univ, nat_0I, nat_succI] 1)); 
760  581 
qed "two_in_univ"; 
0  582 

583 
goalw Univ.thy [bool_def] "bool <= univ(A)"; 

584 
by (fast_tac (ZF_cs addSIs [zero_in_univ,one_in_univ]) 1); 

760  585 
qed "bool_subset_univ"; 
0  586 

782
200a16083201
added bind_thm for theorems defined by "standard ..."
clasohm
parents:
760
diff
changeset

587 
bind_thm ("bool_into_univ", (bool_subset_univ RS subsetD)); 
0  588 

589 

590 
(** Closure under disjoint union **) 

591 

484  592 
goalw Univ.thy [univ_def] "!!A a. a: univ(A) ==> Inl(a) : univ(A)"; 
516  593 
by (etac (Limit_nat RSN (2,Inl_in_VLimit)) 1); 
760  594 
qed "Inl_in_univ"; 
0  595 

484  596 
goalw Univ.thy [univ_def] "!!A b. b: univ(A) ==> Inr(b) : univ(A)"; 
516  597 
by (etac (Limit_nat RSN (2,Inr_in_VLimit)) 1); 
760  598 
qed "Inr_in_univ"; 
0  599 

484  600 
goalw Univ.thy [univ_def] "univ(C)+univ(C) <= univ(C)"; 
516  601 
by (rtac (Limit_nat RS sum_VLimit) 1); 
760  602 
qed "sum_univ"; 
0  603 

803
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

604 
bind_thm ("sum_subset_univ", [sum_mono, sum_univ] MRS subset_trans); 
484  605 

606 

0  607 
(** Closure under binary union  use Un_least **) 
608 
(** Closure under Collect  use (Collect_subset RS subset_trans) **) 

609 
(** Closure under RepFun  use RepFun_subset **) 

803
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

610 

4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

611 

4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

612 
(*** Finite Branching Closure Properties ***) 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

613 

4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

614 
(** Closure under finite powerset **) 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

615 

4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

616 
goal Univ.thy 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

617 
"!!i. [ b: Fin(Vfrom(A,i)); Limit(i) ] ==> EX j. b <= Vfrom(A,j) & j<i"; 
1461  618 
by (etac Fin_induct 1); 
803
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

619 
by (fast_tac (ZF_cs addSDs [Limit_has_0]) 1); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

620 
by (safe_tac ZF_cs); 
1461  621 
by (etac Limit_VfromE 1); 
803
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

622 
by (assume_tac 1); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

623 
by (res_inst_tac [("x", "xa Un j")] exI 1); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

624 
by (best_tac (ZF_cs addIs [subset_refl RS Vfrom_mono RS subsetD, 
1461  625 
Un_least_lt]) 1); 
803
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

626 
val Fin_Vfrom_lemma = result(); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

627 

4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

628 
goal Univ.thy "!!i. Limit(i) ==> Fin(Vfrom(A,i)) <= Vfrom(A,i)"; 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

629 
by (rtac subsetI 1); 
1461  630 
by (dtac Fin_Vfrom_lemma 1); 
803
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

631 
by (safe_tac ZF_cs); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

632 
by (resolve_tac [Vfrom RS ssubst] 1); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

633 
by (fast_tac (ZF_cs addSDs [ltD]) 1); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

634 
val Fin_VLimit = result(); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

635 

4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

636 
bind_thm ("Fin_subset_VLimit", [Fin_mono, Fin_VLimit] MRS subset_trans); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

637 

4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

638 
goalw Univ.thy [univ_def] "Fin(univ(A)) <= univ(A)"; 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

639 
by (rtac (Limit_nat RS Fin_VLimit) 1); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

640 
val Fin_univ = result(); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

641 

4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

642 
(** Closure under finite powers (functions from a fixed natural number) **) 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

643 

4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

644 
goal Univ.thy 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

645 
"!!i. [ n: nat; Limit(i) ] ==> n > Vfrom(A,i) <= Vfrom(A,i)"; 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

646 
by (eresolve_tac [nat_fun_subset_Fin RS subset_trans] 1); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

647 
by (REPEAT (ares_tac [Fin_subset_VLimit, Sigma_subset_VLimit, 
1461  648 
nat_subset_VLimit, subset_refl] 1)); 
803
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

649 
val nat_fun_VLimit = result(); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

650 

4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

651 
bind_thm ("nat_fun_subset_VLimit", [Pi_mono, nat_fun_VLimit] MRS subset_trans); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

652 

4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

653 
goalw Univ.thy [univ_def] "!!i. n: nat ==> n > univ(A) <= univ(A)"; 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

654 
by (etac (Limit_nat RSN (2,nat_fun_VLimit)) 1); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

655 
val nat_fun_univ = result(); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

656 

4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

657 

4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

658 
(** Closure under finite function space **) 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

659 

4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

660 
(*General but seldomused version; normally the domain is fixed*) 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

661 
goal Univ.thy 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

662 
"!!i. Limit(i) ==> Vfrom(A,i) > Vfrom(A,i) <= Vfrom(A,i)"; 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

663 
by (resolve_tac [FiniteFun.dom_subset RS subset_trans] 1); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

664 
by (REPEAT (ares_tac [Fin_subset_VLimit, Sigma_subset_VLimit, subset_refl] 1)); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

665 
val FiniteFun_VLimit1 = result(); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

666 

4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

667 
goalw Univ.thy [univ_def] "univ(A) > univ(A) <= univ(A)"; 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

668 
by (rtac (Limit_nat RS FiniteFun_VLimit1) 1); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

669 
val FiniteFun_univ1 = result(); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

670 

4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

671 
(*Version for a fixed domain*) 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

672 
goal Univ.thy 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

673 
"!!i. [ W <= Vfrom(A,i); Limit(i) ] ==> W > Vfrom(A,i) <= Vfrom(A,i)"; 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

674 
by (eresolve_tac [subset_refl RSN (2, FiniteFun_mono) RS subset_trans] 1); 
1461  675 
by (etac FiniteFun_VLimit1 1); 
803
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

676 
val FiniteFun_VLimit = result(); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

677 

4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

678 
goalw Univ.thy [univ_def] 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

679 
"!!W. W <= univ(A) ==> W > univ(A) <= univ(A)"; 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

680 
by (etac (Limit_nat RSN (2, FiniteFun_VLimit)) 1); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

681 
val FiniteFun_univ = result(); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

682 

4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

683 
goal Univ.thy 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

684 
"!!W. [ f: W > univ(A); W <= univ(A) ] ==> f : univ(A)"; 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

685 
by (eresolve_tac [FiniteFun_univ RS subsetD] 1); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

686 
by (assume_tac 1); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

687 
val FiniteFun_in_univ = result(); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

688 

4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

689 
(*Remove <= from the rule above*) 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

690 
val FiniteFun_in_univ' = subsetI RSN (2, FiniteFun_in_univ); 
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

691 

4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
782
diff
changeset

692 