src/ZF/Constructible/WF_absolute.thy
author paulson
Tue Jul 09 15:39:44 2002 +0200 (2002-07-09)
changeset 13323 2c287f50c9f3
parent 13306 6eebcddee32b
child 13324 39d1b3a4c6f4
permissions -rw-r--r--
More relativization, reflection and proofs of separation
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header {*Absoluteness for Well-Founded Relations and Well-Founded Recursion*}
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theory WF_absolute = WFrec:
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subsection{*Every well-founded relation is a subset of some inverse image of
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      an ordinal*}
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lemma wf_rvimage_Ord: "Ord(i) \<Longrightarrow> wf(rvimage(A, f, Memrel(i)))"
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by (blast intro: wf_rvimage wf_Memrel)
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constdefs
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  wfrank :: "[i,i]=>i"
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    "wfrank(r,a) == wfrec(r, a, %x f. \<Union>y \<in> r-``{x}. succ(f`y))"
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constdefs
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  wftype :: "i=>i"
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    "wftype(r) == \<Union>y \<in> range(r). succ(wfrank(r,y))"
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lemma wfrank: "wf(r) ==> wfrank(r,a) = (\<Union>y \<in> r-``{a}. succ(wfrank(r,y)))"
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by (subst wfrank_def [THEN def_wfrec], simp_all)
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lemma Ord_wfrank: "wf(r) ==> Ord(wfrank(r,a))"
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apply (rule_tac a="a" in wf_induct, assumption)
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apply (subst wfrank, assumption)
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apply (rule Ord_succ [THEN Ord_UN], blast)
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done
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lemma wfrank_lt: "[|wf(r); <a,b> \<in> r|] ==> wfrank(r,a) < wfrank(r,b)"
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apply (rule_tac a1 = "b" in wfrank [THEN ssubst], assumption)
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apply (rule UN_I [THEN ltI])
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apply (simp add: Ord_wfrank vimage_iff)+
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done
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lemma Ord_wftype: "wf(r) ==> Ord(wftype(r))"
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by (simp add: wftype_def Ord_wfrank)
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lemma wftypeI: "\<lbrakk>wf(r);  x \<in> field(r)\<rbrakk> \<Longrightarrow> wfrank(r,x) \<in> wftype(r)"
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apply (simp add: wftype_def)
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apply (blast intro: wfrank_lt [THEN ltD])
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done
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lemma wf_imp_subset_rvimage:
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     "[|wf(r); r \<subseteq> A*A|] ==> \<exists>i f. Ord(i) & r <= rvimage(A, f, Memrel(i))"
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apply (rule_tac x="wftype(r)" in exI)
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apply (rule_tac x="\<lambda>x\<in>A. wfrank(r,x)" in exI)
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apply (simp add: Ord_wftype, clarify)
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apply (frule subsetD, assumption, clarify)
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apply (simp add: rvimage_iff wfrank_lt [THEN ltD])
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apply (blast intro: wftypeI)
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done
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theorem wf_iff_subset_rvimage:
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  "relation(r) ==> wf(r) <-> (\<exists>i f A. Ord(i) & r <= rvimage(A, f, Memrel(i)))"
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by (blast dest!: relation_field_times_field wf_imp_subset_rvimage
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          intro: wf_rvimage_Ord [THEN wf_subset])
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subsection{*Transitive closure without fixedpoints*}
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constdefs
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  rtrancl_alt :: "[i,i]=>i"
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    "rtrancl_alt(A,r) ==
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       {p \<in> A*A. \<exists>n\<in>nat. \<exists>f \<in> succ(n) -> A.
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                 (\<exists>x y. p = <x,y> &  f`0 = x & f`n = y) &
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                       (\<forall>i\<in>n. <f`i, f`succ(i)> \<in> r)}"
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lemma alt_rtrancl_lemma1 [rule_format]:
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    "n \<in> nat
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     ==> \<forall>f \<in> succ(n) -> field(r).
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         (\<forall>i\<in>n. \<langle>f`i, f ` succ(i)\<rangle> \<in> r) --> \<langle>f`0, f`n\<rangle> \<in> r^*"
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apply (induct_tac n)
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apply (simp_all add: apply_funtype rtrancl_refl, clarify)
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apply (rename_tac n f)
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apply (rule rtrancl_into_rtrancl)
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 prefer 2 apply assumption
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apply (drule_tac x="restrict(f,succ(n))" in bspec)
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 apply (blast intro: restrict_type2)
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apply (simp add: Ord_succ_mem_iff nat_0_le [THEN ltD] leI [THEN ltD] ltI)
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done
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lemma rtrancl_alt_subset_rtrancl: "rtrancl_alt(field(r),r) <= r^*"
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apply (simp add: rtrancl_alt_def)
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apply (blast intro: alt_rtrancl_lemma1)
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done
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lemma rtrancl_subset_rtrancl_alt: "r^* <= rtrancl_alt(field(r),r)"
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apply (simp add: rtrancl_alt_def, clarify)
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apply (frule rtrancl_type [THEN subsetD], clarify, simp)
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apply (erule rtrancl_induct)
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 txt{*Base case, trivial*}
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 apply (rule_tac x=0 in bexI)
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  apply (rule_tac x="lam x:1. xa" in bexI)
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   apply simp_all
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txt{*Inductive step*}
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apply clarify
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apply (rename_tac n f)
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apply (rule_tac x="succ(n)" in bexI)
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 apply (rule_tac x="lam i:succ(succ(n)). if i=succ(n) then z else f`i" in bexI)
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  apply (simp add: Ord_succ_mem_iff nat_0_le [THEN ltD] leI [THEN ltD] ltI)
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  apply (blast intro: mem_asym)
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 apply typecheck
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 apply auto
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done
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lemma rtrancl_alt_eq_rtrancl: "rtrancl_alt(field(r),r) = r^*"
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by (blast del: subsetI
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	  intro: rtrancl_alt_subset_rtrancl rtrancl_subset_rtrancl_alt)
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constdefs
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  rtran_closure :: "[i=>o,i,i] => o"
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    "rtran_closure(M,r,s) ==
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        \<forall>A. M(A) --> is_field(M,r,A) -->
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 	 (\<forall>p. M(p) -->
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          (p \<in> s <->
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           (\<exists>n\<in>nat. M(n) &
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            (\<exists>n'. M(n') & successor(M,n,n') &
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             (\<exists>f. M(f) & typed_function(M,n',A,f) &
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              (\<exists>x\<in>A. M(x) & (\<exists>y\<in>A. M(y) & pair(M,x,y,p) &
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                   fun_apply(M,f,0,x) & fun_apply(M,f,n,y))) &
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              (\<forall>i\<in>n. M(i) -->
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                (\<forall>i'. M(i') --> successor(M,i,i') -->
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                 (\<forall>fi. M(fi) --> fun_apply(M,f,i,fi) -->
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                  (\<forall>fi'. M(fi') --> fun_apply(M,f,i',fi') -->
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                   (\<forall>q. M(q) --> pair(M,fi,fi',q) --> q \<in> r))))))))))"
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  tran_closure :: "[i=>o,i,i] => o"
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    "tran_closure(M,r,t) ==
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         \<exists>s[M]. rtran_closure(M,r,s) & composition(M,r,s,t)"
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locale M_trancl = M_axioms +
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  assumes rtrancl_separation:
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	 "[| M(r); M(A) |] ==>
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	  separation (M, \<lambda>p. 
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	      \<exists>nnat[M]. \<exists>n[M]. \<exists>n'[M]. 
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               omega(M,nnat) & n\<in>nnat & successor(M,n,n') &
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	       (\<exists>f[M]. 
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		typed_function(M,n',A,f) &
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		(\<exists>x[M]. \<exists>y[M]. \<exists>zero[M]. pair(M,x,y,p) & empty(M,zero) &
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		  fun_apply(M,f,zero,x) & fun_apply(M,f,n,y)) &
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		  (\<forall>j[M]. j\<in>n --> 
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		    (\<exists>fj[M]. \<exists>sj[M]. \<exists>fsj[M]. \<exists>ffp[M]. 
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		      fun_apply(M,f,j,fj) & successor(M,j,sj) &
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		      fun_apply(M,f,sj,fsj) & pair(M,fj,fsj,ffp) & ffp \<in> r))))"
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      and wellfounded_trancl_separation:
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	 "[| M(r); M(Z) |] ==> 
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	  separation (M, \<lambda>x. 
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	      \<exists>w[M]. \<exists>wx[M]. \<exists>rp[M]. 
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	       w \<in> Z & pair(M,w,x,wx) & tran_closure(M,r,rp) & wx \<in> rp)"
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lemma (in M_trancl) rtran_closure_rtrancl:
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     "M(r) ==> rtran_closure(M,r,rtrancl(r))"
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apply (simp add: rtran_closure_def rtrancl_alt_eq_rtrancl [symmetric]
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                 rtrancl_alt_def field_closed typed_apply_abs apply_closed
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                 Ord_succ_mem_iff M_nat  nat_0_le [THEN ltD], clarify)
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apply (rule iffI)
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 apply clarify
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 apply simp
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 apply (rename_tac n f)
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 apply (rule_tac x=n in bexI)
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  apply (rule_tac x=f in exI)
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  apply simp
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  apply (blast dest: finite_fun_closed dest: transM)
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 apply assumption
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apply clarify
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apply (simp add: nat_0_le [THEN ltD] apply_funtype, blast)
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done
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lemma (in M_trancl) rtrancl_closed [intro,simp]:
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     "M(r) ==> M(rtrancl(r))"
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apply (insert rtrancl_separation [of r "field(r)"])
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apply (simp add: rtrancl_alt_eq_rtrancl [symmetric]
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                 rtrancl_alt_def field_closed typed_apply_abs apply_closed
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                 Ord_succ_mem_iff M_nat nat_into_M
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                 nat_0_le [THEN ltD] leI [THEN ltD] ltI apply_funtype)
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done
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lemma (in M_trancl) rtrancl_abs [simp]:
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     "[| M(r); M(z) |] ==> rtran_closure(M,r,z) <-> z = rtrancl(r)"
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apply (rule iffI)
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 txt{*Proving the right-to-left implication*}
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 prefer 2 apply (blast intro: rtran_closure_rtrancl)
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apply (rule M_equalityI)
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apply (simp add: rtran_closure_def rtrancl_alt_eq_rtrancl [symmetric]
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                 rtrancl_alt_def field_closed typed_apply_abs apply_closed
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                 Ord_succ_mem_iff M_nat
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                 nat_0_le [THEN ltD] leI [THEN ltD] ltI apply_funtype)
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 prefer 2 apply assumption
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 prefer 2 apply blast
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apply (rule iffI, clarify)
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apply (simp add: nat_0_le [THEN ltD]  apply_funtype, blast, clarify, simp)
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 apply (rename_tac n f)
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 apply (rule_tac x=n in bexI)
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  apply (rule_tac x=f in exI)
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  apply (blast dest!: finite_fun_closed, assumption)
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done
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lemma (in M_trancl) trancl_closed [intro,simp]:
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     "M(r) ==> M(trancl(r))"
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by (simp add: trancl_def comp_closed rtrancl_closed)
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lemma (in M_trancl) trancl_abs [simp]:
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     "[| M(r); M(z) |] ==> tran_closure(M,r,z) <-> z = trancl(r)"
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by (simp add: tran_closure_def trancl_def)
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lemma (in M_trancl) wellfounded_trancl_separation':
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     "[| M(r); M(Z) |] ==> separation (M, \<lambda>x. \<exists>w[M]. w \<in> Z & <w,x> \<in> r^+)"
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by (insert wellfounded_trancl_separation [of r Z], simp) 
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text{*Alternative proof of @{text wf_on_trancl}; inspiration for the
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      relativized version.  Original version is on theory WF.*}
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lemma "[| wf[A](r);  r-``A <= A |] ==> wf[A](r^+)"
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apply (simp add: wf_on_def wf_def)
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apply (safe intro!: equalityI)
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apply (drule_tac x = "{x\<in>A. \<exists>w. \<langle>w,x\<rangle> \<in> r^+ & w \<in> Z}" in spec)
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apply (blast elim: tranclE)
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done
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lemma (in M_trancl) wellfounded_on_trancl:
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     "[| wellfounded_on(M,A,r);  r-``A <= A; M(r); M(A) |]
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      ==> wellfounded_on(M,A,r^+)"
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apply (simp add: wellfounded_on_def)
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apply (safe intro!: equalityI)
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apply (rename_tac Z x)
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apply (subgoal_tac "M({x\<in>A. \<exists>w[M]. w \<in> Z & \<langle>w,x\<rangle> \<in> r^+})")
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 prefer 2
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 apply (blast intro: wellfounded_trancl_separation') 
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apply (drule_tac x = "{x\<in>A. \<exists>w[M]. w \<in> Z & \<langle>w,x\<rangle> \<in> r^+}" in rspec, safe)
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apply (blast dest: transM, simp)
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apply (rename_tac y w)
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apply (drule_tac x=w in bspec, assumption, clarify)
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apply (erule tranclE)
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  apply (blast dest: transM)   (*transM is needed to prove M(xa)*)
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 apply blast
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done
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lemma (in M_trancl) wellfounded_trancl:
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     "[|wellfounded(M,r); M(r)|] ==> wellfounded(M,r^+)"
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apply (rotate_tac -1)
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apply (simp add: wellfounded_iff_wellfounded_on_field)
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apply (rule wellfounded_on_subset_A, erule wellfounded_on_trancl)
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   apply blast
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  apply (simp_all add: trancl_type [THEN field_rel_subset])
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done
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text{*Relativized to M: Every well-founded relation is a subset of some
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inverse image of an ordinal.  Key step is the construction (in M) of a
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rank function.*}
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(*NEEDS RELATIVIZATION*)
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locale M_wfrank = M_trancl +
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  assumes wfrank_separation':
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     "M(r) ==>
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	separation
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	   (M, \<lambda>x. ~ (\<exists>f[M]. is_recfun(r^+, x, %x f. range(f), f)))"
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 and wfrank_strong_replacement':
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     "M(r) ==>
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      strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>f[M]. 
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		  pair(M,x,y,z) & is_recfun(r^+, x, %x f. range(f), f) &
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		  y = range(f))"
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 and Ord_wfrank_separation:
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     "M(r) ==>
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      separation (M, \<lambda>x. ~ (\<forall>f. M(f) \<longrightarrow>
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                       is_recfun(r^+, x, \<lambda>x. range, f) \<longrightarrow> Ord(range(f))))"
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text{*This function, defined using replacement, is a rank function for
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well-founded relations within the class M.*}
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constdefs
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 wellfoundedrank :: "[i=>o,i,i] => i"
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    "wellfoundedrank(M,r,A) ==
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        {p. x\<in>A, \<exists>y[M]. \<exists>f[M]. 
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                       p = <x,y> & is_recfun(r^+, x, %x f. range(f), f) &
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                       y = range(f)}"
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lemma (in M_wfrank) exists_wfrank:
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    "[| wellfounded(M,r); M(a); M(r) |]
paulson@13268
   284
     ==> \<exists>f[M]. is_recfun(r^+, a, %x f. range(f), f)"
paulson@13251
   285
apply (rule wellfounded_exists_is_recfun)
paulson@13251
   286
      apply (blast intro: wellfounded_trancl)
paulson@13251
   287
     apply (rule trans_trancl)
paulson@13251
   288
    apply (erule wfrank_separation')
paulson@13251
   289
   apply (erule wfrank_strong_replacement')
paulson@13251
   290
apply (simp_all add: trancl_subset_times)
paulson@13223
   291
done
paulson@13223
   292
paulson@13268
   293
lemma (in M_wfrank) M_wellfoundedrank:
paulson@13251
   294
    "[| wellfounded(M,r); M(r); M(A) |] ==> M(wellfoundedrank(M,r,A))"
paulson@13251
   295
apply (insert wfrank_strong_replacement' [of r])
paulson@13251
   296
apply (simp add: wellfoundedrank_def)
paulson@13251
   297
apply (rule strong_replacement_closed)
paulson@13242
   298
   apply assumption+
paulson@13251
   299
 apply (rule univalent_is_recfun)
paulson@13251
   300
   apply (blast intro: wellfounded_trancl)
paulson@13251
   301
  apply (rule trans_trancl)
paulson@13254
   302
 apply (simp add: trancl_subset_times, blast)
paulson@13223
   303
done
paulson@13223
   304
paulson@13268
   305
lemma (in M_wfrank) Ord_wfrank_range [rule_format]:
paulson@13251
   306
    "[| wellfounded(M,r); a\<in>A; M(r); M(A) |]
paulson@13242
   307
     ==> \<forall>f. M(f) --> is_recfun(r^+, a, %x f. range(f), f) --> Ord(range(f))"
paulson@13251
   308
apply (drule wellfounded_trancl, assumption)
paulson@13251
   309
apply (rule wellfounded_induct, assumption+)
paulson@13254
   310
  apply simp
paulson@13254
   311
 apply (blast intro: Ord_wfrank_separation, clarify)
paulson@13242
   312
txt{*The reasoning in both cases is that we get @{term y} such that
paulson@13251
   313
   @{term "\<langle>y, x\<rangle> \<in> r^+"}.  We find that
paulson@13242
   314
   @{term "f`y = restrict(f, r^+ -`` {y})"}. *}
paulson@13242
   315
apply (rule OrdI [OF _ Ord_is_Transset])
paulson@13242
   316
 txt{*An ordinal is a transitive set...*}
paulson@13251
   317
 apply (simp add: Transset_def)
paulson@13242
   318
 apply clarify
paulson@13251
   319
 apply (frule apply_recfun2, assumption)
paulson@13242
   320
 apply (force simp add: restrict_iff)
paulson@13251
   321
txt{*...of ordinals.  This second case requires the induction hyp.*}
paulson@13251
   322
apply clarify
paulson@13242
   323
apply (rename_tac i y)
paulson@13251
   324
apply (frule apply_recfun2, assumption)
paulson@13251
   325
apply (frule is_recfun_imp_in_r, assumption)
paulson@13251
   326
apply (frule is_recfun_restrict)
paulson@13242
   327
    (*simp_all won't work*)
paulson@13251
   328
    apply (simp add: trans_trancl trancl_subset_times)+
paulson@13242
   329
apply (drule spec [THEN mp], assumption)
paulson@13242
   330
apply (subgoal_tac "M(restrict(f, r^+ -`` {y}))")
paulson@13251
   331
 apply (drule_tac x="restrict(f, r^+ -`` {y})" in spec)
paulson@13242
   332
 apply (simp add: function_apply_equality [OF _ is_recfun_imp_function])
paulson@13242
   333
apply (blast dest: pair_components_in_M)
paulson@13223
   334
done
paulson@13223
   335
paulson@13268
   336
lemma (in M_wfrank) Ord_range_wellfoundedrank:
paulson@13251
   337
    "[| wellfounded(M,r); r \<subseteq> A*A;  M(r); M(A) |]
paulson@13242
   338
     ==> Ord (range(wellfoundedrank(M,r,A)))"
paulson@13251
   339
apply (frule wellfounded_trancl, assumption)
paulson@13251
   340
apply (frule trancl_subset_times)
paulson@13242
   341
apply (simp add: wellfoundedrank_def)
paulson@13242
   342
apply (rule OrdI [OF _ Ord_is_Transset])
paulson@13242
   343
 prefer 2
paulson@13251
   344
 txt{*by our previous result the range consists of ordinals.*}
paulson@13251
   345
 apply (blast intro: Ord_wfrank_range)
paulson@13242
   346
txt{*We still must show that the range is a transitive set.*}
paulson@13247
   347
apply (simp add: Transset_def, clarify, simp)
paulson@13293
   348
apply (rename_tac x i f u)
paulson@13251
   349
apply (frule is_recfun_imp_in_r, assumption)
paulson@13251
   350
apply (subgoal_tac "M(u) & M(i) & M(x)")
paulson@13251
   351
 prefer 2 apply (blast dest: transM, clarify)
paulson@13251
   352
apply (rule_tac a=u in rangeI)
paulson@13293
   353
apply (rule_tac x=u in ReplaceI)
paulson@13293
   354
  apply simp 
paulson@13293
   355
  apply (rule_tac x="restrict(f, r^+ -`` {u})" in rexI)
paulson@13293
   356
   apply (blast intro: is_recfun_restrict trans_trancl dest: apply_recfun2)
paulson@13293
   357
  apply simp 
paulson@13293
   358
apply blast 
paulson@13251
   359
txt{*Unicity requirement of Replacement*}
paulson@13242
   360
apply clarify
paulson@13251
   361
apply (frule apply_recfun2, assumption)
paulson@13293
   362
apply (simp add: trans_trancl is_recfun_cut)
paulson@13223
   363
done
paulson@13223
   364
paulson@13268
   365
lemma (in M_wfrank) function_wellfoundedrank:
paulson@13251
   366
    "[| wellfounded(M,r); M(r); M(A)|]
paulson@13242
   367
     ==> function(wellfoundedrank(M,r,A))"
paulson@13251
   368
apply (simp add: wellfoundedrank_def function_def, clarify)
paulson@13242
   369
txt{*Uniqueness: repeated below!*}
paulson@13242
   370
apply (drule is_recfun_functional, assumption)
paulson@13251
   371
     apply (blast intro: wellfounded_trancl)
paulson@13251
   372
    apply (simp_all add: trancl_subset_times trans_trancl)
paulson@13223
   373
done
paulson@13223
   374
paulson@13268
   375
lemma (in M_wfrank) domain_wellfoundedrank:
paulson@13251
   376
    "[| wellfounded(M,r); M(r); M(A)|]
paulson@13242
   377
     ==> domain(wellfoundedrank(M,r,A)) = A"
paulson@13251
   378
apply (simp add: wellfoundedrank_def function_def)
paulson@13242
   379
apply (rule equalityI, auto)
paulson@13251
   380
apply (frule transM, assumption)
paulson@13251
   381
apply (frule_tac a=x in exists_wfrank, assumption+, clarify)
paulson@13293
   382
apply (rule_tac b="range(f)" in domainI)
paulson@13293
   383
apply (rule_tac x=x in ReplaceI)
paulson@13293
   384
  apply simp 
paulson@13268
   385
  apply (rule_tac x=f in rexI, blast, simp_all)
paulson@13242
   386
txt{*Uniqueness (for Replacement): repeated above!*}
paulson@13242
   387
apply clarify
paulson@13242
   388
apply (drule is_recfun_functional, assumption)
paulson@13251
   389
    apply (blast intro: wellfounded_trancl)
paulson@13251
   390
    apply (simp_all add: trancl_subset_times trans_trancl)
paulson@13223
   391
done
paulson@13223
   392
paulson@13268
   393
lemma (in M_wfrank) wellfoundedrank_type:
paulson@13251
   394
    "[| wellfounded(M,r);  M(r); M(A)|]
paulson@13242
   395
     ==> wellfoundedrank(M,r,A) \<in> A -> range(wellfoundedrank(M,r,A))"
paulson@13251
   396
apply (frule function_wellfoundedrank [of r A], assumption+)
paulson@13251
   397
apply (frule function_imp_Pi)
paulson@13251
   398
 apply (simp add: wellfoundedrank_def relation_def)
paulson@13251
   399
 apply blast
paulson@13242
   400
apply (simp add: domain_wellfoundedrank)
paulson@13223
   401
done
paulson@13223
   402
paulson@13268
   403
lemma (in M_wfrank) Ord_wellfoundedrank:
paulson@13251
   404
    "[| wellfounded(M,r); a \<in> A; r \<subseteq> A*A;  M(r); M(A) |]
paulson@13242
   405
     ==> Ord(wellfoundedrank(M,r,A) ` a)"
paulson@13242
   406
by (blast intro: apply_funtype [OF wellfoundedrank_type]
paulson@13242
   407
                 Ord_in_Ord [OF Ord_range_wellfoundedrank])
paulson@13223
   408
paulson@13268
   409
lemma (in M_wfrank) wellfoundedrank_eq:
paulson@13242
   410
     "[| is_recfun(r^+, a, %x. range, f);
paulson@13251
   411
         wellfounded(M,r);  a \<in> A; M(f); M(r); M(A)|]
paulson@13242
   412
      ==> wellfoundedrank(M,r,A) ` a = range(f)"
paulson@13251
   413
apply (rule apply_equality)
paulson@13251
   414
 prefer 2 apply (blast intro: wellfoundedrank_type)
paulson@13242
   415
apply (simp add: wellfoundedrank_def)
paulson@13242
   416
apply (rule ReplaceI)
paulson@13268
   417
  apply (rule_tac x="range(f)" in rexI) 
paulson@13251
   418
  apply blast
paulson@13268
   419
 apply simp_all
paulson@13251
   420
txt{*Unicity requirement of Replacement*}
paulson@13242
   421
apply clarify
paulson@13242
   422
apply (drule is_recfun_functional, assumption)
paulson@13251
   423
    apply (blast intro: wellfounded_trancl)
paulson@13251
   424
    apply (simp_all add: trancl_subset_times trans_trancl)
paulson@13223
   425
done
paulson@13223
   426
paulson@13247
   427
paulson@13268
   428
lemma (in M_wfrank) wellfoundedrank_lt:
paulson@13247
   429
     "[| <a,b> \<in> r;
paulson@13251
   430
         wellfounded(M,r); r \<subseteq> A*A;  M(r); M(A)|]
paulson@13247
   431
      ==> wellfoundedrank(M,r,A) ` a < wellfoundedrank(M,r,A) ` b"
paulson@13251
   432
apply (frule wellfounded_trancl, assumption)
paulson@13247
   433
apply (subgoal_tac "a\<in>A & b\<in>A")
paulson@13247
   434
 prefer 2 apply blast
paulson@13251
   435
apply (simp add: lt_def Ord_wellfoundedrank, clarify)
paulson@13251
   436
apply (frule exists_wfrank [of concl: _ b], assumption+, clarify)
paulson@13247
   437
apply (rename_tac fb)
paulson@13251
   438
apply (frule is_recfun_restrict [of concl: "r^+" a])
paulson@13251
   439
    apply (rule trans_trancl, assumption)
paulson@13251
   440
   apply (simp_all add: r_into_trancl trancl_subset_times)
paulson@13247
   441
txt{*Still the same goal, but with new @{text is_recfun} assumptions.*}
paulson@13251
   442
apply (simp add: wellfoundedrank_eq)
paulson@13247
   443
apply (frule_tac a=a in wellfoundedrank_eq, assumption+)
paulson@13247
   444
   apply (simp_all add: transM [of a])
paulson@13247
   445
txt{*We have used equations for wellfoundedrank and now must use some
paulson@13247
   446
    for  @{text is_recfun}. *}
paulson@13251
   447
apply (rule_tac a=a in rangeI)
paulson@13251
   448
apply (simp add: is_recfun_type [THEN apply_iff] vimage_singleton_iff
paulson@13251
   449
                 r_into_trancl apply_recfun r_into_trancl)
paulson@13247
   450
done
paulson@13247
   451
paulson@13247
   452
paulson@13268
   453
lemma (in M_wfrank) wellfounded_imp_subset_rvimage:
paulson@13251
   454
     "[|wellfounded(M,r); r \<subseteq> A*A; M(r); M(A)|]
paulson@13247
   455
      ==> \<exists>i f. Ord(i) & r <= rvimage(A, f, Memrel(i))"
paulson@13247
   456
apply (rule_tac x="range(wellfoundedrank(M,r,A))" in exI)
paulson@13247
   457
apply (rule_tac x="wellfoundedrank(M,r,A)" in exI)
paulson@13251
   458
apply (simp add: Ord_range_wellfoundedrank, clarify)
paulson@13251
   459
apply (frule subsetD, assumption, clarify)
paulson@13247
   460
apply (simp add: rvimage_iff wellfoundedrank_lt [THEN ltD])
paulson@13251
   461
apply (blast intro: apply_rangeI wellfoundedrank_type)
paulson@13247
   462
done
paulson@13247
   463
paulson@13268
   464
lemma (in M_wfrank) wellfounded_imp_wf:
paulson@13251
   465
     "[|wellfounded(M,r); relation(r); M(r)|] ==> wf(r)"
paulson@13247
   466
by (blast dest!: relation_field_times_field wellfounded_imp_subset_rvimage
paulson@13247
   467
          intro: wf_rvimage_Ord [THEN wf_subset])
paulson@13247
   468
paulson@13268
   469
lemma (in M_wfrank) wellfounded_on_imp_wf_on:
paulson@13251
   470
     "[|wellfounded_on(M,A,r); relation(r); M(r); M(A)|] ==> wf[A](r)"
paulson@13251
   471
apply (simp add: wellfounded_on_iff_wellfounded wf_on_def)
paulson@13247
   472
apply (rule wellfounded_imp_wf)
paulson@13251
   473
apply (simp_all add: relation_def)
paulson@13247
   474
done
paulson@13247
   475
paulson@13247
   476
paulson@13268
   477
theorem (in M_wfrank) wf_abs [simp]:
paulson@13247
   478
     "[|relation(r); M(r)|] ==> wellfounded(M,r) <-> wf(r)"
paulson@13251
   479
by (blast intro: wellfounded_imp_wf wf_imp_relativized)
paulson@13247
   480
paulson@13268
   481
theorem (in M_wfrank) wf_on_abs [simp]:
paulson@13247
   482
     "[|relation(r); M(r); M(A)|] ==> wellfounded_on(M,A,r) <-> wf[A](r)"
paulson@13251
   483
by (blast intro: wellfounded_on_imp_wf_on wf_on_imp_relativized)
paulson@13247
   484
paulson@13254
   485
paulson@13254
   486
text{*absoluteness for wfrec-defined functions.*}
paulson@13254
   487
paulson@13254
   488
(*first use is_recfun, then M_is_recfun*)
paulson@13254
   489
paulson@13254
   490
lemma (in M_trancl) wfrec_relativize:
paulson@13254
   491
  "[|wf(r); M(a); M(r);  
paulson@13268
   492
     strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M].
paulson@13254
   493
          pair(M,x,y,z) & 
paulson@13254
   494
          is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), g) & 
paulson@13254
   495
          y = H(x, restrict(g, r -`` {x}))); 
paulson@13254
   496
     \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|] 
paulson@13254
   497
   ==> wfrec(r,a,H) = z <-> 
paulson@13268
   498
       (\<exists>f[M]. is_recfun(r^+, a, \<lambda>x f. H(x, restrict(f, r -`` {x})), f) & 
paulson@13254
   499
            z = H(a,restrict(f,r-``{a})))"
paulson@13254
   500
apply (frule wf_trancl) 
paulson@13254
   501
apply (simp add: wftrec_def wfrec_def, safe)
paulson@13254
   502
 apply (frule wf_exists_is_recfun 
paulson@13254
   503
              [of concl: "r^+" a "\<lambda>x f. H(x, restrict(f, r -`` {x}))"]) 
paulson@13254
   504
      apply (simp_all add: trans_trancl function_restrictI trancl_subset_times)
paulson@13268
   505
 apply (clarify, rule_tac x=x in rexI) 
paulson@13254
   506
 apply (simp_all add: the_recfun_eq trans_trancl trancl_subset_times)
paulson@13254
   507
done
paulson@13254
   508
paulson@13254
   509
paulson@13254
   510
text{*Assuming @{term r} is transitive simplifies the occurrences of @{text H}.
paulson@13254
   511
      The premise @{term "relation(r)"} is necessary 
paulson@13254
   512
      before we can replace @{term "r^+"} by @{term r}. *}
paulson@13254
   513
theorem (in M_trancl) trans_wfrec_relativize:
paulson@13254
   514
  "[|wf(r);  trans(r);  relation(r);  M(r);  M(a);
paulson@13293
   515
     strong_replacement(M, \<lambda>x z. \<exists>y[M]. 
paulson@13293
   516
                pair(M,x,y,z) & (\<exists>g[M]. is_recfun(r,x,H,g) & y = H(x,g))); 
paulson@13254
   517
     \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|] 
paulson@13268
   518
   ==> wfrec(r,a,H) = z <-> (\<exists>f[M]. is_recfun(r,a,H,f) & z = H(a,f))" 
paulson@13254
   519
by (simp cong: is_recfun_cong
paulson@13254
   520
         add: wfrec_relativize trancl_eq_r
paulson@13254
   521
               is_recfun_restrict_idem domain_restrict_idem)
paulson@13254
   522
paulson@13254
   523
paulson@13254
   524
lemma (in M_trancl) trans_eq_pair_wfrec_iff:
paulson@13254
   525
  "[|wf(r);  trans(r); relation(r); M(r);  M(y); 
paulson@13293
   526
     strong_replacement(M, \<lambda>x z. \<exists>y[M]. 
paulson@13293
   527
                pair(M,x,y,z) & (\<exists>g[M]. is_recfun(r,x,H,g) & y = H(x,g))); 
paulson@13254
   528
     \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|] 
paulson@13254
   529
   ==> y = <x, wfrec(r, x, H)> <-> 
paulson@13268
   530
       (\<exists>f[M]. is_recfun(r,x,H,f) & y = <x, H(x,f)>)"
paulson@13293
   531
apply safe 
paulson@13293
   532
 apply (simp add: trans_wfrec_relativize [THEN iff_sym, of concl: _ x]) 
paulson@13254
   533
txt{*converse direction*}
paulson@13254
   534
apply (rule sym)
paulson@13254
   535
apply (simp add: trans_wfrec_relativize, blast) 
paulson@13254
   536
done
paulson@13254
   537
paulson@13254
   538
paulson@13254
   539
subsection{*M is closed under well-founded recursion*}
paulson@13254
   540
paulson@13254
   541
text{*Lemma with the awkward premise mentioning @{text wfrec}.*}
paulson@13268
   542
lemma (in M_wfrank) wfrec_closed_lemma [rule_format]:
paulson@13254
   543
     "[|wf(r); M(r); 
paulson@13254
   544
        strong_replacement(M, \<lambda>x y. y = \<langle>x, wfrec(r, x, H)\<rangle>);
paulson@13254
   545
        \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)) |] 
paulson@13254
   546
      ==> M(a) --> M(wfrec(r,a,H))"
paulson@13254
   547
apply (rule_tac a=a in wf_induct, assumption+)
paulson@13254
   548
apply (subst wfrec, assumption, clarify)
paulson@13254
   549
apply (drule_tac x1=x and x="\<lambda>x\<in>r -`` {x}. wfrec(r, x, H)" 
paulson@13254
   550
       in rspec [THEN rspec]) 
paulson@13254
   551
apply (simp_all add: function_lam) 
paulson@13254
   552
apply (blast intro: dest: pair_components_in_M ) 
paulson@13254
   553
done
paulson@13254
   554
paulson@13254
   555
text{*Eliminates one instance of replacement.*}
paulson@13268
   556
lemma (in M_wfrank) wfrec_replacement_iff:
paulson@13268
   557
     "strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M]. 
paulson@13254
   558
                pair(M,x,y,z) & is_recfun(r,x,H,g) & y = H(x,g)) <->
paulson@13254
   559
      strong_replacement(M, 
paulson@13268
   560
           \<lambda>x y. \<exists>f[M]. is_recfun(r,x,H,f) & y = <x, H(x,f)>)"
paulson@13254
   561
apply simp 
paulson@13254
   562
apply (rule strong_replacement_cong, blast) 
paulson@13254
   563
done
paulson@13254
   564
paulson@13254
   565
text{*Useful version for transitive relations*}
paulson@13268
   566
theorem (in M_wfrank) trans_wfrec_closed:
paulson@13254
   567
     "[|wf(r); trans(r); relation(r); M(r); M(a);
paulson@13254
   568
        strong_replacement(M, 
paulson@13268
   569
             \<lambda>x z. \<exists>y[M]. \<exists>g[M].
paulson@13254
   570
                    pair(M,x,y,z) & is_recfun(r,x,H,g) & y = H(x,g)); 
paulson@13254
   571
        \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)) |] 
paulson@13254
   572
      ==> M(wfrec(r,a,H))"
paulson@13254
   573
apply (frule wfrec_replacement_iff [THEN iffD1]) 
paulson@13254
   574
apply (rule wfrec_closed_lemma, assumption+) 
paulson@13254
   575
apply (simp_all add: wfrec_replacement_iff trans_eq_pair_wfrec_iff) 
paulson@13254
   576
done
paulson@13254
   577
paulson@13254
   578
section{*Absoluteness without assuming transitivity*}
paulson@13254
   579
lemma (in M_trancl) eq_pair_wfrec_iff:
paulson@13254
   580
  "[|wf(r);  M(r);  M(y); 
paulson@13268
   581
     strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M].
paulson@13254
   582
          pair(M,x,y,z) & 
paulson@13254
   583
          is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), g) & 
paulson@13254
   584
          y = H(x, restrict(g, r -`` {x}))); 
paulson@13254
   585
     \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|] 
paulson@13254
   586
   ==> y = <x, wfrec(r, x, H)> <-> 
paulson@13268
   587
       (\<exists>f[M]. is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), f) & 
paulson@13254
   588
            y = <x, H(x,restrict(f,r-``{x}))>)"
paulson@13254
   589
apply safe  
paulson@13293
   590
 apply (simp add: wfrec_relativize [THEN iff_sym, of concl: _ x]) 
paulson@13254
   591
txt{*converse direction*}
paulson@13254
   592
apply (rule sym)
paulson@13254
   593
apply (simp add: wfrec_relativize, blast) 
paulson@13254
   594
done
paulson@13254
   595
paulson@13268
   596
lemma (in M_wfrank) wfrec_closed_lemma [rule_format]:
paulson@13254
   597
     "[|wf(r); M(r); 
paulson@13254
   598
        strong_replacement(M, \<lambda>x y. y = \<langle>x, wfrec(r, x, H)\<rangle>);
paulson@13254
   599
        \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)) |] 
paulson@13254
   600
      ==> M(a) --> M(wfrec(r,a,H))"
paulson@13254
   601
apply (rule_tac a=a in wf_induct, assumption+)
paulson@13254
   602
apply (subst wfrec, assumption, clarify)
paulson@13254
   603
apply (drule_tac x1=x and x="\<lambda>x\<in>r -`` {x}. wfrec(r, x, H)" 
paulson@13254
   604
       in rspec [THEN rspec]) 
paulson@13254
   605
apply (simp_all add: function_lam) 
paulson@13254
   606
apply (blast intro: dest: pair_components_in_M ) 
paulson@13254
   607
done
paulson@13254
   608
paulson@13254
   609
text{*Full version not assuming transitivity, but maybe not very useful.*}
paulson@13268
   610
theorem (in M_wfrank) wfrec_closed:
paulson@13254
   611
     "[|wf(r); M(r); M(a);
paulson@13268
   612
     strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M].
paulson@13254
   613
          pair(M,x,y,z) & 
paulson@13254
   614
          is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), g) & 
paulson@13254
   615
          y = H(x, restrict(g, r -`` {x}))); 
paulson@13254
   616
        \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)) |] 
paulson@13254
   617
      ==> M(wfrec(r,a,H))"
paulson@13254
   618
apply (frule wfrec_replacement_iff [THEN iffD1]) 
paulson@13254
   619
apply (rule wfrec_closed_lemma, assumption+) 
paulson@13254
   620
apply (simp_all add: eq_pair_wfrec_iff) 
paulson@13254
   621
done
paulson@13254
   622
paulson@13223
   623
end