src/ZF/Constructible/WF_absolute.thy
 author paulson Thu Jul 11 13:43:24 2002 +0200 (2002-07-11) changeset 13348 374d05460db4 parent 13339 0f89104dd377 child 13350 626b79677dfa permissions -rw-r--r--
Separation/Replacement up to M_wfrank!
1 header {*Absoluteness for Well-Founded Relations and Well-Founded Recursion*}
3 theory WF_absolute = WFrec:
5 subsection{*Every well-founded relation is a subset of some inverse image of
6       an ordinal*}
8 lemma wf_rvimage_Ord: "Ord(i) \<Longrightarrow> wf(rvimage(A, f, Memrel(i)))"
9 by (blast intro: wf_rvimage wf_Memrel)
12 constdefs
13   wfrank :: "[i,i]=>i"
14     "wfrank(r,a) == wfrec(r, a, %x f. \<Union>y \<in> r-``{x}. succ(f`y))"
16 constdefs
17   wftype :: "i=>i"
18     "wftype(r) == \<Union>y \<in> range(r). succ(wfrank(r,y))"
20 lemma wfrank: "wf(r) ==> wfrank(r,a) = (\<Union>y \<in> r-``{a}. succ(wfrank(r,y)))"
21 by (subst wfrank_def [THEN def_wfrec], simp_all)
23 lemma Ord_wfrank: "wf(r) ==> Ord(wfrank(r,a))"
24 apply (rule_tac a=a in wf_induct, assumption)
25 apply (subst wfrank, assumption)
26 apply (rule Ord_succ [THEN Ord_UN], blast)
27 done
29 lemma wfrank_lt: "[|wf(r); <a,b> \<in> r|] ==> wfrank(r,a) < wfrank(r,b)"
30 apply (rule_tac a1 = b in wfrank [THEN ssubst], assumption)
31 apply (rule UN_I [THEN ltI])
32 apply (simp add: Ord_wfrank vimage_iff)+
33 done
35 lemma Ord_wftype: "wf(r) ==> Ord(wftype(r))"
36 by (simp add: wftype_def Ord_wfrank)
38 lemma wftypeI: "\<lbrakk>wf(r);  x \<in> field(r)\<rbrakk> \<Longrightarrow> wfrank(r,x) \<in> wftype(r)"
40 apply (blast intro: wfrank_lt [THEN ltD])
41 done
44 lemma wf_imp_subset_rvimage:
45      "[|wf(r); r \<subseteq> A*A|] ==> \<exists>i f. Ord(i) & r <= rvimage(A, f, Memrel(i))"
46 apply (rule_tac x="wftype(r)" in exI)
47 apply (rule_tac x="\<lambda>x\<in>A. wfrank(r,x)" in exI)
48 apply (simp add: Ord_wftype, clarify)
49 apply (frule subsetD, assumption, clarify)
50 apply (simp add: rvimage_iff wfrank_lt [THEN ltD])
51 apply (blast intro: wftypeI)
52 done
54 theorem wf_iff_subset_rvimage:
55   "relation(r) ==> wf(r) <-> (\<exists>i f A. Ord(i) & r <= rvimage(A, f, Memrel(i)))"
56 by (blast dest!: relation_field_times_field wf_imp_subset_rvimage
57           intro: wf_rvimage_Ord [THEN wf_subset])
60 subsection{*Transitive closure without fixedpoints*}
62 constdefs
63   rtrancl_alt :: "[i,i]=>i"
64     "rtrancl_alt(A,r) ==
65        {p \<in> A*A. \<exists>n\<in>nat. \<exists>f \<in> succ(n) -> A.
66                  (\<exists>x y. p = <x,y> &  f`0 = x & f`n = y) &
67                        (\<forall>i\<in>n. <f`i, f`succ(i)> \<in> r)}"
69 lemma alt_rtrancl_lemma1 [rule_format]:
70     "n \<in> nat
71      ==> \<forall>f \<in> succ(n) -> field(r).
72          (\<forall>i\<in>n. \<langle>f`i, f ` succ(i)\<rangle> \<in> r) --> \<langle>f`0, f`n\<rangle> \<in> r^*"
73 apply (induct_tac n)
74 apply (simp_all add: apply_funtype rtrancl_refl, clarify)
75 apply (rename_tac n f)
76 apply (rule rtrancl_into_rtrancl)
77  prefer 2 apply assumption
78 apply (drule_tac x="restrict(f,succ(n))" in bspec)
79  apply (blast intro: restrict_type2)
80 apply (simp add: Ord_succ_mem_iff nat_0_le [THEN ltD] leI [THEN ltD] ltI)
81 done
83 lemma rtrancl_alt_subset_rtrancl: "rtrancl_alt(field(r),r) <= r^*"
85 apply (blast intro: alt_rtrancl_lemma1)
86 done
88 lemma rtrancl_subset_rtrancl_alt: "r^* <= rtrancl_alt(field(r),r)"
89 apply (simp add: rtrancl_alt_def, clarify)
90 apply (frule rtrancl_type [THEN subsetD], clarify, simp)
91 apply (erule rtrancl_induct)
92  txt{*Base case, trivial*}
93  apply (rule_tac x=0 in bexI)
94   apply (rule_tac x="lam x:1. xa" in bexI)
95    apply simp_all
96 txt{*Inductive step*}
97 apply clarify
98 apply (rename_tac n f)
99 apply (rule_tac x="succ(n)" in bexI)
100  apply (rule_tac x="lam i:succ(succ(n)). if i=succ(n) then z else f`i" in bexI)
101   apply (simp add: Ord_succ_mem_iff nat_0_le [THEN ltD] leI [THEN ltD] ltI)
102   apply (blast intro: mem_asym)
103  apply typecheck
104  apply auto
105 done
107 lemma rtrancl_alt_eq_rtrancl: "rtrancl_alt(field(r),r) = r^*"
108 by (blast del: subsetI
109 	  intro: rtrancl_alt_subset_rtrancl rtrancl_subset_rtrancl_alt)
112 constdefs
114   rtran_closure_mem :: "[i=>o,i,i,i] => o"
115     --{*The property of belonging to @{text "rtran_closure(r)"}*}
116     "rtran_closure_mem(M,A,r,p) ==
117 	      \<exists>nnat[M]. \<exists>n[M]. \<exists>n'[M].
118                omega(M,nnat) & n\<in>nnat & successor(M,n,n') &
119 	       (\<exists>f[M]. typed_function(M,n',A,f) &
120 		(\<exists>x[M]. \<exists>y[M]. \<exists>zero[M]. pair(M,x,y,p) & empty(M,zero) &
121 		  fun_apply(M,f,zero,x) & fun_apply(M,f,n,y)) &
122 		  (\<forall>j[M]. j\<in>n -->
123 		    (\<exists>fj[M]. \<exists>sj[M]. \<exists>fsj[M]. \<exists>ffp[M].
124 		      fun_apply(M,f,j,fj) & successor(M,j,sj) &
125 		      fun_apply(M,f,sj,fsj) & pair(M,fj,fsj,ffp) & ffp \<in> r)))"
127   rtran_closure :: "[i=>o,i,i] => o"
128     "rtran_closure(M,r,s) ==
129         \<forall>A[M]. is_field(M,r,A) -->
130  	 (\<forall>p[M]. p \<in> s <-> rtran_closure_mem(M,A,r,p))"
132   tran_closure :: "[i=>o,i,i] => o"
133     "tran_closure(M,r,t) ==
134          \<exists>s[M]. rtran_closure(M,r,s) & composition(M,r,s,t)"
136 lemma (in M_axioms) rtran_closure_mem_iff:
137      "[|M(A); M(r); M(p)|]
138       ==> rtran_closure_mem(M,A,r,p) <->
139           (\<exists>n[M]. n\<in>nat &
140            (\<exists>f[M]. f \<in> succ(n) -> A &
141             (\<exists>x[M]. \<exists>y[M]. p = <x,y> & f`0 = x & f`n = y) &
142                            (\<forall>i\<in>n. <f`i, f`succ(i)> \<in> r)))"
143 apply (simp add: rtran_closure_mem_def typed_apply_abs
144                  Ord_succ_mem_iff nat_0_le [THEN ltD], blast)
145 done
147 locale M_trancl = M_axioms +
148   assumes rtrancl_separation:
149 	 "[| M(r); M(A) |] ==> separation (M, rtran_closure_mem(M,A,r))"
150       and wellfounded_trancl_separation:
151 	 "[| M(r); M(Z) |] ==>
152 	  separation (M, \<lambda>x.
153 	      \<exists>w[M]. \<exists>wx[M]. \<exists>rp[M].
154 	       w \<in> Z & pair(M,w,x,wx) & tran_closure(M,r,rp) & wx \<in> rp)"
157 lemma (in M_trancl) rtran_closure_rtrancl:
158      "M(r) ==> rtran_closure(M,r,rtrancl(r))"
159 apply (simp add: rtran_closure_def rtran_closure_mem_iff
160                  rtrancl_alt_eq_rtrancl [symmetric] rtrancl_alt_def)
161 apply (auto simp add: nat_0_le [THEN ltD] apply_funtype)
162 done
164 lemma (in M_trancl) rtrancl_closed [intro,simp]:
165      "M(r) ==> M(rtrancl(r))"
166 apply (insert rtrancl_separation [of r "field(r)"])
167 apply (simp add: rtrancl_alt_eq_rtrancl [symmetric]
168                  rtrancl_alt_def rtran_closure_mem_iff)
169 done
171 lemma (in M_trancl) rtrancl_abs [simp]:
172      "[| M(r); M(z) |] ==> rtran_closure(M,r,z) <-> z = rtrancl(r)"
173 apply (rule iffI)
174  txt{*Proving the right-to-left implication*}
175  prefer 2 apply (blast intro: rtran_closure_rtrancl)
176 apply (rule M_equalityI)
177 apply (simp add: rtran_closure_def rtrancl_alt_eq_rtrancl [symmetric]
178                  rtrancl_alt_def rtran_closure_mem_iff)
179 apply (auto simp add: nat_0_le [THEN ltD] apply_funtype)
180 done
182 lemma (in M_trancl) trancl_closed [intro,simp]:
183      "M(r) ==> M(trancl(r))"
184 by (simp add: trancl_def comp_closed rtrancl_closed)
186 lemma (in M_trancl) trancl_abs [simp]:
187      "[| M(r); M(z) |] ==> tran_closure(M,r,z) <-> z = trancl(r)"
188 by (simp add: tran_closure_def trancl_def)
190 lemma (in M_trancl) wellfounded_trancl_separation':
191      "[| M(r); M(Z) |] ==> separation (M, \<lambda>x. \<exists>w[M]. w \<in> Z & <w,x> \<in> r^+)"
192 by (insert wellfounded_trancl_separation [of r Z], simp)
194 text{*Alternative proof of @{text wf_on_trancl}; inspiration for the
195       relativized version.  Original version is on theory WF.*}
196 lemma "[| wf[A](r);  r-``A <= A |] ==> wf[A](r^+)"
197 apply (simp add: wf_on_def wf_def)
198 apply (safe intro!: equalityI)
199 apply (drule_tac x = "{x\<in>A. \<exists>w. \<langle>w,x\<rangle> \<in> r^+ & w \<in> Z}" in spec)
200 apply (blast elim: tranclE)
201 done
203 lemma (in M_trancl) wellfounded_on_trancl:
204      "[| wellfounded_on(M,A,r);  r-``A <= A; M(r); M(A) |]
205       ==> wellfounded_on(M,A,r^+)"
207 apply (safe intro!: equalityI)
208 apply (rename_tac Z x)
209 apply (subgoal_tac "M({x\<in>A. \<exists>w[M]. w \<in> Z & \<langle>w,x\<rangle> \<in> r^+})")
210  prefer 2
211  apply (blast intro: wellfounded_trancl_separation')
212 apply (drule_tac x = "{x\<in>A. \<exists>w[M]. w \<in> Z & \<langle>w,x\<rangle> \<in> r^+}" in rspec, safe)
213 apply (blast dest: transM, simp)
214 apply (rename_tac y w)
215 apply (drule_tac x=w in bspec, assumption, clarify)
216 apply (erule tranclE)
217   apply (blast dest: transM)   (*transM is needed to prove M(xa)*)
218  apply blast
219 done
221 lemma (in M_trancl) wellfounded_trancl:
222      "[|wellfounded(M,r); M(r)|] ==> wellfounded(M,r^+)"
223 apply (rotate_tac -1)
225 apply (rule wellfounded_on_subset_A, erule wellfounded_on_trancl)
226    apply blast
227   apply (simp_all add: trancl_type [THEN field_rel_subset])
228 done
230 text{*Relativized to M: Every well-founded relation is a subset of some
231 inverse image of an ordinal.  Key step is the construction (in M) of a
232 rank function.*}
235 locale M_wfrank = M_trancl +
236   assumes wfrank_separation:
237      "M(r) ==>
238       separation (M, \<lambda>x.
239          \<forall>rplus[M]. tran_closure(M,r,rplus) -->
240          ~ (\<exists>f[M]. M_is_recfun(M, rplus, x, %x f y. is_range(M,f,y), f)))"
241  and wfrank_strong_replacement:
242      "M(r) ==>
243       strong_replacement(M, \<lambda>x z.
244          \<forall>rplus[M]. tran_closure(M,r,rplus) -->
245          (\<exists>y[M]. \<exists>f[M]. pair(M,x,y,z)  &
246                         M_is_recfun(M, rplus, x, %x f y. is_range(M,f,y), f) &
247                         is_range(M,f,y)))"
248  and Ord_wfrank_separation:
249      "M(r) ==>
250       separation (M, \<lambda>x.
251          \<forall>rplus[M]. tran_closure(M,r,rplus) -->
252           ~ (\<forall>f[M]. \<forall>rangef[M].
253              is_range(M,f,rangef) -->
254              M_is_recfun(M, rplus, x, \<lambda>x f y. is_range(M,f,y), f) -->
255              ordinal(M,rangef)))"
257 text{*Proving that the relativized instances of Separation or Replacement
258 agree with the "real" ones.*}
260 lemma (in M_wfrank) wfrank_separation':
261      "M(r) ==>
262       separation
263 	   (M, \<lambda>x. ~ (\<exists>f[M]. is_recfun(r^+, x, %x f. range(f), f)))"
264 apply (insert wfrank_separation [of r])
265 apply (simp add: is_recfun_iff_M [of concl: _ _ "%x. range", THEN iff_sym])
266 done
268 lemma (in M_wfrank) wfrank_strong_replacement':
269      "M(r) ==>
270       strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>f[M].
271 		  pair(M,x,y,z) & is_recfun(r^+, x, %x f. range(f), f) &
272 		  y = range(f))"
273 apply (insert wfrank_strong_replacement [of r])
274 apply (simp add: is_recfun_iff_M [of concl: _ _ "%x. range", THEN iff_sym])
275 done
277 lemma (in M_wfrank) Ord_wfrank_separation':
278      "M(r) ==>
279       separation (M, \<lambda>x.
280          ~ (\<forall>f[M]. is_recfun(r^+, x, \<lambda>x. range, f) --> Ord(range(f))))"
281 apply (insert Ord_wfrank_separation [of r])
282 apply (simp add: is_recfun_iff_M [of concl: _ _ "%x. range", THEN iff_sym])
283 done
285 text{*This function, defined using replacement, is a rank function for
286 well-founded relations within the class M.*}
287 constdefs
288  wellfoundedrank :: "[i=>o,i,i] => i"
289     "wellfoundedrank(M,r,A) ==
290         {p. x\<in>A, \<exists>y[M]. \<exists>f[M].
291                        p = <x,y> & is_recfun(r^+, x, %x f. range(f), f) &
292                        y = range(f)}"
294 lemma (in M_wfrank) exists_wfrank:
295     "[| wellfounded(M,r); M(a); M(r) |]
296      ==> \<exists>f[M]. is_recfun(r^+, a, %x f. range(f), f)"
297 apply (rule wellfounded_exists_is_recfun)
298       apply (blast intro: wellfounded_trancl)
299      apply (rule trans_trancl)
300     apply (erule wfrank_separation')
301    apply (erule wfrank_strong_replacement')
303 done
305 lemma (in M_wfrank) M_wellfoundedrank:
306     "[| wellfounded(M,r); M(r); M(A) |] ==> M(wellfoundedrank(M,r,A))"
307 apply (insert wfrank_strong_replacement' [of r])
309 apply (rule strong_replacement_closed)
310    apply assumption+
311  apply (rule univalent_is_recfun)
312    apply (blast intro: wellfounded_trancl)
313   apply (rule trans_trancl)
314  apply (simp add: trancl_subset_times, blast)
315 done
317 lemma (in M_wfrank) Ord_wfrank_range [rule_format]:
318     "[| wellfounded(M,r); a\<in>A; M(r); M(A) |]
319      ==> \<forall>f[M]. is_recfun(r^+, a, %x f. range(f), f) --> Ord(range(f))"
320 apply (drule wellfounded_trancl, assumption)
321 apply (rule wellfounded_induct, assumption+)
322   apply simp
323  apply (blast intro: Ord_wfrank_separation', clarify)
324 txt{*The reasoning in both cases is that we get @{term y} such that
325    @{term "\<langle>y, x\<rangle> \<in> r^+"}.  We find that
326    @{term "f`y = restrict(f, r^+ -`` {y})"}. *}
327 apply (rule OrdI [OF _ Ord_is_Transset])
328  txt{*An ordinal is a transitive set...*}
330  apply clarify
331  apply (frule apply_recfun2, assumption)
332  apply (force simp add: restrict_iff)
333 txt{*...of ordinals.  This second case requires the induction hyp.*}
334 apply clarify
335 apply (rename_tac i y)
336 apply (frule apply_recfun2, assumption)
337 apply (frule is_recfun_imp_in_r, assumption)
338 apply (frule is_recfun_restrict)
339     (*simp_all won't work*)
340     apply (simp add: trans_trancl trancl_subset_times)+
341 apply (drule spec [THEN mp], assumption)
342 apply (subgoal_tac "M(restrict(f, r^+ -`` {y}))")
343  apply (drule_tac x="restrict(f, r^+ -`` {y})" in rspec)
344 apply assumption
345  apply (simp add: function_apply_equality [OF _ is_recfun_imp_function])
346 apply (blast dest: pair_components_in_M)
347 done
349 lemma (in M_wfrank) Ord_range_wellfoundedrank:
350     "[| wellfounded(M,r); r \<subseteq> A*A;  M(r); M(A) |]
351      ==> Ord (range(wellfoundedrank(M,r,A)))"
352 apply (frule wellfounded_trancl, assumption)
353 apply (frule trancl_subset_times)
355 apply (rule OrdI [OF _ Ord_is_Transset])
356  prefer 2
357  txt{*by our previous result the range consists of ordinals.*}
358  apply (blast intro: Ord_wfrank_range)
359 txt{*We still must show that the range is a transitive set.*}
360 apply (simp add: Transset_def, clarify, simp)
361 apply (rename_tac x i f u)
362 apply (frule is_recfun_imp_in_r, assumption)
363 apply (subgoal_tac "M(u) & M(i) & M(x)")
364  prefer 2 apply (blast dest: transM, clarify)
365 apply (rule_tac a=u in rangeI)
366 apply (rule_tac x=u in ReplaceI)
367   apply simp
368   apply (rule_tac x="restrict(f, r^+ -`` {u})" in rexI)
369    apply (blast intro: is_recfun_restrict trans_trancl dest: apply_recfun2)
370   apply simp
371 apply blast
372 txt{*Unicity requirement of Replacement*}
373 apply clarify
374 apply (frule apply_recfun2, assumption)
375 apply (simp add: trans_trancl is_recfun_cut)
376 done
378 lemma (in M_wfrank) function_wellfoundedrank:
379     "[| wellfounded(M,r); M(r); M(A)|]
380      ==> function(wellfoundedrank(M,r,A))"
381 apply (simp add: wellfoundedrank_def function_def, clarify)
382 txt{*Uniqueness: repeated below!*}
383 apply (drule is_recfun_functional, assumption)
384      apply (blast intro: wellfounded_trancl)
385     apply (simp_all add: trancl_subset_times trans_trancl)
386 done
388 lemma (in M_wfrank) domain_wellfoundedrank:
389     "[| wellfounded(M,r); M(r); M(A)|]
390      ==> domain(wellfoundedrank(M,r,A)) = A"
391 apply (simp add: wellfoundedrank_def function_def)
392 apply (rule equalityI, auto)
393 apply (frule transM, assumption)
394 apply (frule_tac a=x in exists_wfrank, assumption+, clarify)
395 apply (rule_tac b="range(f)" in domainI)
396 apply (rule_tac x=x in ReplaceI)
397   apply simp
398   apply (rule_tac x=f in rexI, blast, simp_all)
399 txt{*Uniqueness (for Replacement): repeated above!*}
400 apply clarify
401 apply (drule is_recfun_functional, assumption)
402     apply (blast intro: wellfounded_trancl)
403     apply (simp_all add: trancl_subset_times trans_trancl)
404 done
406 lemma (in M_wfrank) wellfoundedrank_type:
407     "[| wellfounded(M,r);  M(r); M(A)|]
408      ==> wellfoundedrank(M,r,A) \<in> A -> range(wellfoundedrank(M,r,A))"
409 apply (frule function_wellfoundedrank [of r A], assumption+)
410 apply (frule function_imp_Pi)
411  apply (simp add: wellfoundedrank_def relation_def)
412  apply blast
414 done
416 lemma (in M_wfrank) Ord_wellfoundedrank:
417     "[| wellfounded(M,r); a \<in> A; r \<subseteq> A*A;  M(r); M(A) |]
418      ==> Ord(wellfoundedrank(M,r,A) ` a)"
419 by (blast intro: apply_funtype [OF wellfoundedrank_type]
420                  Ord_in_Ord [OF Ord_range_wellfoundedrank])
422 lemma (in M_wfrank) wellfoundedrank_eq:
423      "[| is_recfun(r^+, a, %x. range, f);
424          wellfounded(M,r);  a \<in> A; M(f); M(r); M(A)|]
425       ==> wellfoundedrank(M,r,A) ` a = range(f)"
426 apply (rule apply_equality)
427  prefer 2 apply (blast intro: wellfoundedrank_type)
429 apply (rule ReplaceI)
430   apply (rule_tac x="range(f)" in rexI)
431   apply blast
432  apply simp_all
433 txt{*Unicity requirement of Replacement*}
434 apply clarify
435 apply (drule is_recfun_functional, assumption)
436     apply (blast intro: wellfounded_trancl)
437     apply (simp_all add: trancl_subset_times trans_trancl)
438 done
441 lemma (in M_wfrank) wellfoundedrank_lt:
442      "[| <a,b> \<in> r;
443          wellfounded(M,r); r \<subseteq> A*A;  M(r); M(A)|]
444       ==> wellfoundedrank(M,r,A) ` a < wellfoundedrank(M,r,A) ` b"
445 apply (frule wellfounded_trancl, assumption)
446 apply (subgoal_tac "a\<in>A & b\<in>A")
447  prefer 2 apply blast
448 apply (simp add: lt_def Ord_wellfoundedrank, clarify)
449 apply (frule exists_wfrank [of concl: _ b], assumption+, clarify)
450 apply (rename_tac fb)
451 apply (frule is_recfun_restrict [of concl: "r^+" a])
452     apply (rule trans_trancl, assumption)
453    apply (simp_all add: r_into_trancl trancl_subset_times)
454 txt{*Still the same goal, but with new @{text is_recfun} assumptions.*}
456 apply (frule_tac a=a in wellfoundedrank_eq, assumption+)
457    apply (simp_all add: transM [of a])
458 txt{*We have used equations for wellfoundedrank and now must use some
459     for  @{text is_recfun}. *}
460 apply (rule_tac a=a in rangeI)
461 apply (simp add: is_recfun_type [THEN apply_iff] vimage_singleton_iff
462                  r_into_trancl apply_recfun r_into_trancl)
463 done
466 lemma (in M_wfrank) wellfounded_imp_subset_rvimage:
467      "[|wellfounded(M,r); r \<subseteq> A*A; M(r); M(A)|]
468       ==> \<exists>i f. Ord(i) & r <= rvimage(A, f, Memrel(i))"
469 apply (rule_tac x="range(wellfoundedrank(M,r,A))" in exI)
470 apply (rule_tac x="wellfoundedrank(M,r,A)" in exI)
471 apply (simp add: Ord_range_wellfoundedrank, clarify)
472 apply (frule subsetD, assumption, clarify)
473 apply (simp add: rvimage_iff wellfoundedrank_lt [THEN ltD])
474 apply (blast intro: apply_rangeI wellfoundedrank_type)
475 done
477 lemma (in M_wfrank) wellfounded_imp_wf:
478      "[|wellfounded(M,r); relation(r); M(r)|] ==> wf(r)"
479 by (blast dest!: relation_field_times_field wellfounded_imp_subset_rvimage
480           intro: wf_rvimage_Ord [THEN wf_subset])
482 lemma (in M_wfrank) wellfounded_on_imp_wf_on:
483      "[|wellfounded_on(M,A,r); relation(r); M(r); M(A)|] ==> wf[A](r)"
484 apply (simp add: wellfounded_on_iff_wellfounded wf_on_def)
485 apply (rule wellfounded_imp_wf)
487 done
490 theorem (in M_wfrank) wf_abs [simp]:
491      "[|relation(r); M(r)|] ==> wellfounded(M,r) <-> wf(r)"
492 by (blast intro: wellfounded_imp_wf wf_imp_relativized)
494 theorem (in M_wfrank) wf_on_abs [simp]:
495      "[|relation(r); M(r); M(A)|] ==> wellfounded_on(M,A,r) <-> wf[A](r)"
496 by (blast intro: wellfounded_on_imp_wf_on wf_on_imp_relativized)
499 text{*absoluteness for wfrec-defined functions.*}
501 (*first use is_recfun, then M_is_recfun*)
503 lemma (in M_trancl) wfrec_relativize:
504   "[|wf(r); M(a); M(r);
505      strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M].
506           pair(M,x,y,z) &
507           is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), g) &
508           y = H(x, restrict(g, r -`` {x})));
509      \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|]
510    ==> wfrec(r,a,H) = z <->
511        (\<exists>f[M]. is_recfun(r^+, a, \<lambda>x f. H(x, restrict(f, r -`` {x})), f) &
512             z = H(a,restrict(f,r-``{a})))"
513 apply (frule wf_trancl)
514 apply (simp add: wftrec_def wfrec_def, safe)
515  apply (frule wf_exists_is_recfun
516               [of concl: "r^+" a "\<lambda>x f. H(x, restrict(f, r -`` {x}))"])
517       apply (simp_all add: trans_trancl function_restrictI trancl_subset_times)
518  apply (clarify, rule_tac x=x in rexI)
519  apply (simp_all add: the_recfun_eq trans_trancl trancl_subset_times)
520 done
523 text{*Assuming @{term r} is transitive simplifies the occurrences of @{text H}.
524       The premise @{term "relation(r)"} is necessary
525       before we can replace @{term "r^+"} by @{term r}. *}
526 theorem (in M_trancl) trans_wfrec_relativize:
527   "[|wf(r);  trans(r);  relation(r);  M(r);  M(a);
528      strong_replacement(M, \<lambda>x z. \<exists>y[M].
529                 pair(M,x,y,z) & (\<exists>g[M]. is_recfun(r,x,H,g) & y = H(x,g)));
530      \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|]
531    ==> wfrec(r,a,H) = z <-> (\<exists>f[M]. is_recfun(r,a,H,f) & z = H(a,f))"
532 by (simp cong: is_recfun_cong
534                is_recfun_restrict_idem domain_restrict_idem)
537 lemma (in M_trancl) trans_eq_pair_wfrec_iff:
538   "[|wf(r);  trans(r); relation(r); M(r);  M(y);
539      strong_replacement(M, \<lambda>x z. \<exists>y[M].
540                 pair(M,x,y,z) & (\<exists>g[M]. is_recfun(r,x,H,g) & y = H(x,g)));
541      \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|]
542    ==> y = <x, wfrec(r, x, H)> <->
543        (\<exists>f[M]. is_recfun(r,x,H,f) & y = <x, H(x,f)>)"
544 apply safe
545  apply (simp add: trans_wfrec_relativize [THEN iff_sym, of concl: _ x])
546 txt{*converse direction*}
547 apply (rule sym)
548 apply (simp add: trans_wfrec_relativize, blast)
549 done
552 subsection{*M is closed under well-founded recursion*}
554 text{*Lemma with the awkward premise mentioning @{text wfrec}.*}
555 lemma (in M_wfrank) wfrec_closed_lemma [rule_format]:
556      "[|wf(r); M(r);
557         strong_replacement(M, \<lambda>x y. y = \<langle>x, wfrec(r, x, H)\<rangle>);
558         \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)) |]
559       ==> M(a) --> M(wfrec(r,a,H))"
560 apply (rule_tac a=a in wf_induct, assumption+)
561 apply (subst wfrec, assumption, clarify)
562 apply (drule_tac x1=x and x="\<lambda>x\<in>r -`` {x}. wfrec(r, x, H)"
563        in rspec [THEN rspec])
565 apply (blast intro: dest: pair_components_in_M )
566 done
568 text{*Eliminates one instance of replacement.*}
569 lemma (in M_wfrank) wfrec_replacement_iff:
570      "strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M].
571                 pair(M,x,y,z) & is_recfun(r,x,H,g) & y = H(x,g)) <->
572       strong_replacement(M,
573            \<lambda>x y. \<exists>f[M]. is_recfun(r,x,H,f) & y = <x, H(x,f)>)"
574 apply simp
575 apply (rule strong_replacement_cong, blast)
576 done
578 text{*Useful version for transitive relations*}
579 theorem (in M_wfrank) trans_wfrec_closed:
580      "[|wf(r); trans(r); relation(r); M(r); M(a);
581         strong_replacement(M,
582              \<lambda>x z. \<exists>y[M]. \<exists>g[M].
583                     pair(M,x,y,z) & is_recfun(r,x,H,g) & y = H(x,g));
584         \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)) |]
585       ==> M(wfrec(r,a,H))"
586 apply (frule wfrec_replacement_iff [THEN iffD1])
587 apply (rule wfrec_closed_lemma, assumption+)
588 apply (simp_all add: wfrec_replacement_iff trans_eq_pair_wfrec_iff)
589 done
591 section{*Absoluteness without assuming transitivity*}
592 lemma (in M_trancl) eq_pair_wfrec_iff:
593   "[|wf(r);  M(r);  M(y);
594      strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M].
595           pair(M,x,y,z) &
596           is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), g) &
597           y = H(x, restrict(g, r -`` {x})));
598      \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|]
599    ==> y = <x, wfrec(r, x, H)> <->
600        (\<exists>f[M]. is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), f) &
601             y = <x, H(x,restrict(f,r-``{x}))>)"
602 apply safe
603  apply (simp add: wfrec_relativize [THEN iff_sym, of concl: _ x])
604 txt{*converse direction*}
605 apply (rule sym)
606 apply (simp add: wfrec_relativize, blast)
607 done
609 lemma (in M_wfrank) wfrec_closed_lemma [rule_format]:
610      "[|wf(r); M(r);
611         strong_replacement(M, \<lambda>x y. y = \<langle>x, wfrec(r, x, H)\<rangle>);
612         \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)) |]
613       ==> M(a) --> M(wfrec(r,a,H))"
614 apply (rule_tac a=a in wf_induct, assumption+)
615 apply (subst wfrec, assumption, clarify)
616 apply (drule_tac x1=x and x="\<lambda>x\<in>r -`` {x}. wfrec(r, x, H)"
617        in rspec [THEN rspec])
619 apply (blast intro: dest: pair_components_in_M )
620 done
622 text{*Full version not assuming transitivity, but maybe not very useful.*}
623 theorem (in M_wfrank) wfrec_closed:
624      "[|wf(r); M(r); M(a);
625      strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M].
626           pair(M,x,y,z) &
627           is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), g) &
628           y = H(x, restrict(g, r -`` {x})));
629         \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)) |]
630       ==> M(wfrec(r,a,H))"
631 apply (frule wfrec_replacement_iff [THEN iffD1])
632 apply (rule wfrec_closed_lemma, assumption+)