src/ZF/Constructible/Datatype_absolute.thy
 author paulson Thu Jul 18 15:21:42 2002 +0200 (2002-07-18) changeset 13395 4eb948d1eb4e parent 13386 f3e9e8b21aba child 13397 6e5f4d911435 permissions -rw-r--r--
absoluteness for "formula" and "eclose"
1 header {*Absoluteness Properties for Recursive Datatypes*}
3 theory Datatype_absolute = Formula + WF_absolute:
6 subsection{*The lfp of a continuous function can be expressed as a union*}
8 constdefs
9   directed :: "i=>o"
10    "directed(A) == A\<noteq>0 & (\<forall>x\<in>A. \<forall>y\<in>A. x \<union> y \<in> A)"
12   contin :: "(i=>i) => o"
13    "contin(h) == (\<forall>A. directed(A) --> h(\<Union>A) = (\<Union>X\<in>A. h(X)))"
15 lemma bnd_mono_iterates_subset: "[|bnd_mono(D, h); n \<in> nat|] ==> h^n (0) <= D"
16 apply (induct_tac n)
17  apply (simp_all add: bnd_mono_def, blast)
18 done
20 lemma bnd_mono_increasing [rule_format]:
21      "[|i \<in> nat; j \<in> nat; bnd_mono(D,h)|] ==> i \<le> j --> h^i(0) \<subseteq> h^j(0)"
22 apply (rule_tac m=i and n=j in diff_induct, simp_all)
23 apply (blast del: subsetI
24 	     intro: bnd_mono_iterates_subset bnd_monoD2 [of concl: h] )
25 done
27 lemma directed_iterates: "bnd_mono(D,h) ==> directed({h^n (0). n\<in>nat})"
28 apply (simp add: directed_def, clarify)
29 apply (rename_tac i j)
30 apply (rule_tac x="i \<union> j" in bexI)
31 apply (rule_tac i = i and j = j in Ord_linear_le)
32 apply (simp_all add: subset_Un_iff [THEN iffD1] le_imp_subset
33                      subset_Un_iff2 [THEN iffD1])
34 apply (simp_all add: subset_Un_iff [THEN iff_sym] bnd_mono_increasing
35                      subset_Un_iff2 [THEN iff_sym])
36 done
39 lemma contin_iterates_eq:
40     "[|bnd_mono(D, h); contin(h)|]
41      ==> h(\<Union>n\<in>nat. h^n (0)) = (\<Union>n\<in>nat. h^n (0))"
42 apply (simp add: contin_def directed_iterates)
43 apply (rule trans)
44 apply (rule equalityI)
46  apply safe
47  apply (erule_tac [2] natE)
48   apply (rule_tac a="succ(x)" in UN_I)
49    apply simp_all
50 apply blast
51 done
53 lemma lfp_subset_Union:
54      "[|bnd_mono(D, h); contin(h)|] ==> lfp(D,h) <= (\<Union>n\<in>nat. h^n(0))"
55 apply (rule lfp_lowerbound)
57 apply (simp add: contin_def bnd_mono_iterates_subset UN_subset_iff)
58 done
60 lemma Union_subset_lfp:
61      "bnd_mono(D,h) ==> (\<Union>n\<in>nat. h^n(0)) <= lfp(D,h)"
63 apply (rule ballI)
64 apply (induct_tac n, simp_all)
65 apply (rule subset_trans [of _ "h(lfp(D,h))"])
66  apply (blast dest: bnd_monoD2 [OF _ _ lfp_subset] )
67 apply (erule lfp_lemma2)
68 done
70 lemma lfp_eq_Union:
71      "[|bnd_mono(D, h); contin(h)|] ==> lfp(D,h) = (\<Union>n\<in>nat. h^n(0))"
72 by (blast del: subsetI
73           intro: lfp_subset_Union Union_subset_lfp)
76 subsubsection{*Some Standard Datatype Constructions Preserve Continuity*}
78 lemma contin_imp_mono: "[|X\<subseteq>Y; contin(F)|] ==> F(X) \<subseteq> F(Y)"
80 apply (drule_tac x="{X,Y}" in spec)
81 apply (simp add: directed_def subset_Un_iff2 Un_commute)
82 done
84 lemma sum_contin: "[|contin(F); contin(G)|] ==> contin(\<lambda>X. F(X) + G(X))"
85 by (simp add: contin_def, blast)
87 lemma prod_contin: "[|contin(F); contin(G)|] ==> contin(\<lambda>X. F(X) * G(X))"
88 apply (subgoal_tac "\<forall>B C. F(B) \<subseteq> F(B \<union> C)")
89  prefer 2 apply (simp add: Un_upper1 contin_imp_mono)
90 apply (subgoal_tac "\<forall>B C. G(C) \<subseteq> G(B \<union> C)")
91  prefer 2 apply (simp add: Un_upper2 contin_imp_mono)
92 apply (simp add: contin_def, clarify)
93 apply (rule equalityI)
94  prefer 2 apply blast
95 apply clarify
96 apply (rename_tac B C)
97 apply (rule_tac a="B \<union> C" in UN_I)
98  apply (simp add: directed_def, blast)
99 done
101 lemma const_contin: "contin(\<lambda>X. A)"
102 by (simp add: contin_def directed_def)
104 lemma id_contin: "contin(\<lambda>X. X)"
109 subsection {*Absoluteness for "Iterates"*}
111 constdefs
113   iterates_MH :: "[i=>o, [i,i]=>o, i, i, i, i] => o"
114    "iterates_MH(M,isF,v,n,g,z) ==
115         is_nat_case(M, v, \<lambda>m u. \<exists>gm[M]. fun_apply(M,g,m,gm) & isF(gm,u),
116                     n, z)"
118   iterates_replacement :: "[i=>o, [i,i]=>o, i] => o"
119    "iterates_replacement(M,isF,v) ==
120       \<forall>n[M]. n\<in>nat -->
121          wfrec_replacement(M, iterates_MH(M,isF,v), Memrel(succ(n)))"
123 lemma (in M_axioms) iterates_MH_abs:
124   "[| relativize1(M,isF,F); M(n); M(g); M(z) |]
125    ==> iterates_MH(M,isF,v,n,g,z) <-> z = nat_case(v, \<lambda>m. F(g`m), n)"
126 by (simp add: nat_case_abs [of _ "\<lambda>m. F(g ` m)"]
127               relativize1_def iterates_MH_def)
129 lemma (in M_axioms) iterates_imp_wfrec_replacement:
130   "[|relativize1(M,isF,F); n \<in> nat; iterates_replacement(M,isF,v)|]
131    ==> wfrec_replacement(M, \<lambda>n f z. z = nat_case(v, \<lambda>m. F(f`m), n),
132                        Memrel(succ(n)))"
133 by (simp add: iterates_replacement_def iterates_MH_abs)
135 theorem (in M_trancl) iterates_abs:
136   "[| iterates_replacement(M,isF,v); relativize1(M,isF,F);
137       n \<in> nat; M(v); M(z); \<forall>x[M]. M(F(x)) |]
138    ==> is_wfrec(M, iterates_MH(M,isF,v), Memrel(succ(n)), n, z) <->
139        z = iterates(F,n,v)"
140 apply (frule iterates_imp_wfrec_replacement, assumption+)
141 apply (simp add: wf_Memrel trans_Memrel relation_Memrel nat_into_M
142                  relativize2_def iterates_MH_abs
143                  iterates_nat_def recursor_def transrec_def
144                  eclose_sing_Ord_eq nat_into_M
145          trans_wfrec_abs [of _ _ _ _ "\<lambda>n g. nat_case(v, \<lambda>m. F(g`m), n)"])
146 done
149 lemma (in M_wfrank) iterates_closed [intro,simp]:
150   "[| iterates_replacement(M,isF,v); relativize1(M,isF,F);
151       n \<in> nat; M(v); \<forall>x[M]. M(F(x)) |]
152    ==> M(iterates(F,n,v))"
153 apply (frule iterates_imp_wfrec_replacement, assumption+)
154 apply (simp add: wf_Memrel trans_Memrel relation_Memrel nat_into_M
155                  relativize2_def iterates_MH_abs
156                  iterates_nat_def recursor_def transrec_def
157                  eclose_sing_Ord_eq nat_into_M
158          trans_wfrec_closed [of _ _ _ "\<lambda>n g. nat_case(v, \<lambda>m. F(g`m), n)"])
159 done
162 subsection {*lists without univ*}
164 lemmas datatype_univs = Inl_in_univ Inr_in_univ
165                         Pair_in_univ nat_into_univ A_into_univ
167 lemma list_fun_bnd_mono: "bnd_mono(univ(A), \<lambda>X. {0} + A*X)"
168 apply (rule bnd_monoI)
169  apply (intro subset_refl zero_subset_univ A_subset_univ
170 	      sum_subset_univ Sigma_subset_univ)
171 apply (rule subset_refl sum_mono Sigma_mono | assumption)+
172 done
174 lemma list_fun_contin: "contin(\<lambda>X. {0} + A*X)"
175 by (intro sum_contin prod_contin id_contin const_contin)
177 text{*Re-expresses lists using sum and product*}
178 lemma list_eq_lfp2: "list(A) = lfp(univ(A), \<lambda>X. {0} + A*X)"
180 apply (rule equalityI)
181  apply (rule lfp_lowerbound)
182   prefer 2 apply (rule lfp_subset)
183  apply (clarify, subst lfp_unfold [OF list_fun_bnd_mono])
184  apply (simp add: Nil_def Cons_def)
185  apply blast
186 txt{*Opposite inclusion*}
187 apply (rule lfp_lowerbound)
188  prefer 2 apply (rule lfp_subset)
189 apply (clarify, subst lfp_unfold [OF list.bnd_mono])
190 apply (simp add: Nil_def Cons_def)
191 apply (blast intro: datatype_univs
192              dest: lfp_subset [THEN subsetD])
193 done
195 text{*Re-expresses lists using "iterates", no univ.*}
196 lemma list_eq_Union:
197      "list(A) = (\<Union>n\<in>nat. (\<lambda>X. {0} + A*X) ^ n (0))"
198 by (simp add: list_eq_lfp2 lfp_eq_Union list_fun_bnd_mono list_fun_contin)
201 constdefs
202   is_list_functor :: "[i=>o,i,i,i] => o"
203     "is_list_functor(M,A,X,Z) ==
204         \<exists>n1[M]. \<exists>AX[M].
205          number1(M,n1) & cartprod(M,A,X,AX) & is_sum(M,n1,AX,Z)"
207 lemma (in M_axioms) list_functor_abs [simp]:
208      "[| M(A); M(X); M(Z) |] ==> is_list_functor(M,A,X,Z) <-> (Z = {0} + A*X)"
209 by (simp add: is_list_functor_def singleton_0 nat_into_M)
212 subsection {*formulas without univ*}
214 lemma formula_fun_bnd_mono:
215      "bnd_mono(univ(0), \<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X)))"
216 apply (rule bnd_monoI)
217  apply (intro subset_refl zero_subset_univ A_subset_univ
218 	      sum_subset_univ Sigma_subset_univ nat_subset_univ)
219 apply (rule subset_refl sum_mono Sigma_mono | assumption)+
220 done
222 lemma formula_fun_contin:
223      "contin(\<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X)))"
224 by (intro sum_contin prod_contin id_contin const_contin)
227 text{*Re-expresses formulas using sum and product*}
228 lemma formula_eq_lfp2:
229     "formula = lfp(univ(0), \<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X)))"
231 apply (rule equalityI)
232  apply (rule lfp_lowerbound)
233   prefer 2 apply (rule lfp_subset)
234  apply (clarify, subst lfp_unfold [OF formula_fun_bnd_mono])
235  apply (simp add: Member_def Equal_def Neg_def And_def Forall_def)
236  apply blast
237 txt{*Opposite inclusion*}
238 apply (rule lfp_lowerbound)
239  prefer 2 apply (rule lfp_subset, clarify)
240 apply (subst lfp_unfold [OF formula.bnd_mono, simplified])
241 apply (simp add: Member_def Equal_def Neg_def And_def Forall_def)
242 apply (elim sumE SigmaE, simp_all)
243 apply (blast intro: datatype_univs dest: lfp_subset [THEN subsetD])+
244 done
246 text{*Re-expresses formulas using "iterates", no univ.*}
247 lemma formula_eq_Union:
248      "formula =
249       (\<Union>n\<in>nat. (\<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X))) ^ n (0))"
250 by (simp add: formula_eq_lfp2 lfp_eq_Union formula_fun_bnd_mono
251               formula_fun_contin)
254 constdefs
255   is_formula_functor :: "[i=>o,i,i] => o"
256     "is_formula_functor(M,X,Z) ==
257         \<exists>nat'[M]. \<exists>natnat[M]. \<exists>natnatsum[M]. \<exists>XX[M]. \<exists>X3[M]. \<exists>X4[M].
258           omega(M,nat') & cartprod(M,nat',nat',natnat) &
259           is_sum(M,natnat,natnat,natnatsum) &
260           cartprod(M,X,X,XX) & is_sum(M,XX,X,X3) & is_sum(M,X,X3,X4) &
261           is_sum(M,natnatsum,X4,Z)"
263 lemma (in M_axioms) formula_functor_abs [simp]:
264      "[| M(X); M(Z) |]
265       ==> is_formula_functor(M,X,Z) <->
266           Z = ((nat*nat) + (nat*nat)) + (X + (X*X + X))"
270 subsection{*@{term M} Contains the List and Formula Datatypes*}
272 constdefs
273   is_list_n :: "[i=>o,i,i,i] => o"
274     "is_list_n(M,A,n,Z) ==
275       \<exists>zero[M]. \<exists>sn[M]. \<exists>msn[M].
276        empty(M,zero) &
277        successor(M,n,sn) & membership(M,sn,msn) &
278        is_wfrec(M, iterates_MH(M, is_list_functor(M,A),zero), msn, n, Z)"
280   mem_list :: "[i=>o,i,i] => o"
281     "mem_list(M,A,l) ==
282       \<exists>n[M]. \<exists>listn[M].
283        finite_ordinal(M,n) & is_list_n(M,A,n,listn) & l \<in> listn"
285   is_list :: "[i=>o,i,i] => o"
286     "is_list(M,A,Z) == \<forall>l[M]. l \<in> Z <-> mem_list(M,A,l)"
288 constdefs
289   is_formula_n :: "[i=>o,i,i] => o"
290     "is_formula_n(M,n,Z) ==
291       \<exists>zero[M]. \<exists>sn[M]. \<exists>msn[M].
292        empty(M,zero) &
293        successor(M,n,sn) & membership(M,sn,msn) &
294        is_wfrec(M, iterates_MH(M, is_formula_functor(M),zero), msn, n, Z)"
296   mem_formula :: "[i=>o,i] => o"
297     "mem_formula(M,p) ==
298       \<exists>n[M]. \<exists>formn[M].
299        finite_ordinal(M,n) & is_formula_n(M,n,formn) & p \<in> formn"
301   is_formula :: "[i=>o,i] => o"
302     "is_formula(M,Z) == \<forall>p[M]. p \<in> Z <-> mem_formula(M,p)"
304 locale (open) M_datatypes = M_wfrank +
305  assumes list_replacement1:
306    "M(A) ==> iterates_replacement(M, is_list_functor(M,A), 0)"
307   and list_replacement2:
308    "M(A) ==> strong_replacement(M,
309          \<lambda>n y. n\<in>nat &
310                (\<exists>sn[M]. \<exists>msn[M]. successor(M,n,sn) & membership(M,sn,msn) &
311                is_wfrec(M, iterates_MH(M,is_list_functor(M,A), 0),
312                         msn, n, y)))"
313   and formula_replacement1:
314    "iterates_replacement(M, is_formula_functor(M), 0)"
315   and formula_replacement2:
316    "strong_replacement(M,
317          \<lambda>n y. n\<in>nat &
318                (\<exists>sn[M]. \<exists>msn[M]. successor(M,n,sn) & membership(M,sn,msn) &
319                is_wfrec(M, iterates_MH(M,is_formula_functor(M), 0),
320                         msn, n, y)))"
323 subsubsection{*Absoluteness of the List Construction*}
325 lemma (in M_datatypes) list_replacement2':
326   "M(A) ==> strong_replacement(M, \<lambda>n y. n\<in>nat & y = (\<lambda>X. {0} + A * X)^n (0))"
327 apply (insert list_replacement2 [of A])
328 apply (rule strong_replacement_cong [THEN iffD1])
329 apply (rule conj_cong [OF iff_refl iterates_abs [of "is_list_functor(M,A)"]])
330 apply (simp_all add: list_replacement1 relativize1_def)
331 done
333 lemma (in M_datatypes) list_closed [intro,simp]:
334      "M(A) ==> M(list(A))"
335 apply (insert list_replacement1)
336 by  (simp add: RepFun_closed2 list_eq_Union
337                list_replacement2' relativize1_def
338                iterates_closed [of "is_list_functor(M,A)"])
339 lemma (in M_datatypes) is_list_n_abs [simp]:
340      "[|M(A); n\<in>nat; M(Z)|]
341       ==> is_list_n(M,A,n,Z) <-> Z = (\<lambda>X. {0} + A * X)^n (0)"
342 apply (insert list_replacement1)
343 apply (simp add: is_list_n_def relativize1_def nat_into_M
344                  iterates_abs [of "is_list_functor(M,A)" _ "\<lambda>X. {0} + A*X"])
345 done
347 lemma (in M_datatypes) mem_list_abs [simp]:
348      "M(A) ==> mem_list(M,A,l) <-> l \<in> list(A)"
349 apply (insert list_replacement1)
350 apply (simp add: mem_list_def relativize1_def list_eq_Union
351                  iterates_closed [of "is_list_functor(M,A)"])
352 done
354 lemma (in M_datatypes) list_abs [simp]:
355      "[|M(A); M(Z)|] ==> is_list(M,A,Z) <-> Z = list(A)"
356 apply (simp add: is_list_def, safe)
357 apply (rule M_equalityI, simp_all)
358 done
360 subsubsection{*Absoluteness of Formulas*}
362 lemma (in M_datatypes) formula_replacement2':
363   "strong_replacement(M, \<lambda>n y. n\<in>nat & y = (\<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X)))^n (0))"
364 apply (insert formula_replacement2)
365 apply (rule strong_replacement_cong [THEN iffD1])
366 apply (rule conj_cong [OF iff_refl iterates_abs [of "is_formula_functor(M)"]])
367 apply (simp_all add: formula_replacement1 relativize1_def)
368 done
370 lemma (in M_datatypes) formula_closed [intro,simp]:
371      "M(formula)"
372 apply (insert formula_replacement1)
373 apply  (simp add: RepFun_closed2 formula_eq_Union
374                   formula_replacement2' relativize1_def
375                   iterates_closed [of "is_formula_functor(M)"])
376 done
378 lemma (in M_datatypes) is_formula_n_abs [simp]:
379      "[|n\<in>nat; M(Z)|]
380       ==> is_formula_n(M,n,Z) <->
381           Z = (\<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X)))^n (0)"
382 apply (insert formula_replacement1)
383 apply (simp add: is_formula_n_def relativize1_def nat_into_M
384                  iterates_abs [of "is_formula_functor(M)" _
385                         "\<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X))"])
386 done
388 lemma (in M_datatypes) mem_formula_abs [simp]:
389      "mem_formula(M,l) <-> l \<in> formula"
390 apply (insert formula_replacement1)
391 apply (simp add: mem_formula_def relativize1_def formula_eq_Union
392                  iterates_closed [of "is_formula_functor(M)"])
393 done
395 lemma (in M_datatypes) formula_abs [simp]:
396      "[|M(Z)|] ==> is_formula(M,Z) <-> Z = formula"
397 apply (simp add: is_formula_def, safe)
398 apply (rule M_equalityI, simp_all)
399 done
402 subsection{*Absoluteness for @{text \<epsilon>}-Closure: the @{term eclose} Operator*}
404 text{*Re-expresses eclose using "iterates"*}
405 lemma eclose_eq_Union:
406      "eclose(A) = (\<Union>n\<in>nat. Union^n (A))"
408 apply (rule UN_cong)
409 apply (rule refl)
410 apply (induct_tac n)
413 done
415 constdefs
416   is_eclose_n :: "[i=>o,i,i,i] => o"
417     "is_eclose_n(M,A,n,Z) ==
418       \<exists>sn[M]. \<exists>msn[M].
419        successor(M,n,sn) & membership(M,sn,msn) &
420        is_wfrec(M, iterates_MH(M, big_union(M), A), msn, n, Z)"
422   mem_eclose :: "[i=>o,i,i] => o"
423     "mem_eclose(M,A,l) ==
424       \<exists>n[M]. \<exists>eclosen[M].
425        finite_ordinal(M,n) & is_eclose_n(M,A,n,eclosen) & l \<in> eclosen"
427   is_eclose :: "[i=>o,i,i] => o"
428     "is_eclose(M,A,Z) == \<forall>u[M]. u \<in> Z <-> mem_eclose(M,A,u)"
431 locale (open) M_eclose = M_wfrank +
432  assumes eclose_replacement1:
433    "M(A) ==> iterates_replacement(M, big_union(M), A)"
434   and eclose_replacement2:
435    "M(A) ==> strong_replacement(M,
436          \<lambda>n y. n\<in>nat &
437                (\<exists>sn[M]. \<exists>msn[M]. successor(M,n,sn) & membership(M,sn,msn) &
438                is_wfrec(M, iterates_MH(M,big_union(M), A),
439                         msn, n, y)))"
441 lemma (in M_eclose) eclose_replacement2':
442   "M(A) ==> strong_replacement(M, \<lambda>n y. n\<in>nat & y = Union^n (A))"
443 apply (insert eclose_replacement2 [of A])
444 apply (rule strong_replacement_cong [THEN iffD1])
445 apply (rule conj_cong [OF iff_refl iterates_abs [of "big_union(M)"]])
446 apply (simp_all add: eclose_replacement1 relativize1_def)
447 done
449 lemma (in M_eclose) eclose_closed [intro,simp]:
450      "M(A) ==> M(eclose(A))"
451 apply (insert eclose_replacement1)
452 by  (simp add: RepFun_closed2 eclose_eq_Union
453                eclose_replacement2' relativize1_def
454                iterates_closed [of "big_union(M)"])
456 lemma (in M_eclose) is_eclose_n_abs [simp]:
457      "[|M(A); n\<in>nat; M(Z)|] ==> is_eclose_n(M,A,n,Z) <-> Z = Union^n (A)"
458 apply (insert eclose_replacement1)
459 apply (simp add: is_eclose_n_def relativize1_def nat_into_M
460                  iterates_abs [of "big_union(M)" _ "Union"])
461 done
463 lemma (in M_eclose) mem_eclose_abs [simp]:
464      "M(A) ==> mem_eclose(M,A,l) <-> l \<in> eclose(A)"
465 apply (insert eclose_replacement1)
466 apply (simp add: mem_eclose_def relativize1_def eclose_eq_Union
467                  iterates_closed [of "big_union(M)"])
468 done
470 lemma (in M_eclose) eclose_abs [simp]:
471      "[|M(A); M(Z)|] ==> is_eclose(M,A,Z) <-> Z = eclose(A)"
472 apply (simp add: is_eclose_def, safe)
473 apply (rule M_equalityI, simp_all)
474 done
479 subsection {*Absoluteness for @{term transrec}*}
482 text{* @{term "transrec(a,H) \<equiv> wfrec(Memrel(eclose({a})), a, H)"} *}
483 constdefs
485   is_transrec :: "[i=>o, [i,i,i]=>o, i, i] => o"
486    "is_transrec(M,MH,a,z) ==
487       \<exists>sa[M]. \<exists>esa[M]. \<exists>mesa[M].
488        upair(M,a,a,sa) & is_eclose(M,sa,esa) & membership(M,esa,mesa) &
489        is_wfrec(M,MH,mesa,a,z)"
491   transrec_replacement :: "[i=>o, [i,i,i]=>o, i] => o"
492    "transrec_replacement(M,MH,a) ==
493       \<exists>sa[M]. \<exists>esa[M]. \<exists>mesa[M].
494        upair(M,a,a,sa) & is_eclose(M,sa,esa) & membership(M,esa,mesa) &
495        wfrec_replacement(M,MH,mesa)"
497 (*????????????????Ordinal.thy*)
498 lemma Transset_trans_Memrel:
499     "\<forall>j\<in>i. Transset(j) ==> trans(Memrel(i))"
500 by (unfold Transset_def trans_def, blast)
502 text{*The condition @{term "Ord(i)"} lets us use the simpler
503   @{text "trans_wfrec_abs"} rather than @{text "trans_wfrec_abs"},
504   which I haven't even proved yet. *}
505 theorem (in M_eclose) transrec_abs:
506   "[|Ord(i);  M(i);  M(z);
507      transrec_replacement(M,MH,i);  relativize2(M,MH,H);
508      \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|]
509    ==> is_transrec(M,MH,i,z) <-> z = transrec(i,H)"
510 by (simp add: trans_wfrec_abs transrec_replacement_def is_transrec_def
511        transrec_def eclose_sing_Ord_eq wf_Memrel trans_Memrel relation_Memrel)
514 theorem (in M_eclose) transrec_closed:
515      "[|Ord(i);  M(i);  M(z);
516 	transrec_replacement(M,MH,i);  relativize2(M,MH,H);
517 	\<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|]
518       ==> M(transrec(i,H))"
519 by (simp add: trans_wfrec_closed transrec_replacement_def is_transrec_def
520        transrec_def eclose_sing_Ord_eq wf_Memrel trans_Memrel relation_Memrel)
525 end