author  wenzelm 
Wed, 27 Mar 2013 16:38:25 +0100  
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parent 46823  57bf0cecb366 
child 58871  c399ae4b836f 
permissions  rwrr 
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(* Title: ZF/Constructible/WF_absolute.thy 
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

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*) 

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header {*Absoluteness of WellFounded Recursion*} 
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theory WF_absolute imports WFrec begin 
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subsection{*Transitive closure without fixedpoints*} 

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definition 
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rtrancl_alt :: "[i,i]=>i" where 
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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) \<longrightarrow> \<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) 
13223  30 
done 
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46823  32 
lemma rtrancl_alt_subset_rtrancl: "rtrancl_alt(field(r),r) \<subseteq> 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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46823  37 
lemma rtrancl_subset_rtrancl_alt: "r^* \<subseteq> 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="\<lambda>x\<in>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="\<lambda>i\<in>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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56 
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) 
13223  59 

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definition 
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rtran_closure_mem :: "[i=>o,i,i,i] => o" where 
13324  63 
{*The property of belonging to @{text "rtran_closure(r)"}*} 
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"rtran_closure_mem(M,A,r,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]. 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 \<longrightarrow> 
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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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definition 
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rtran_closure :: "[i=>o,i,i] => o" where 
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"rtran_closure(M,r,s) == 
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\<forall>A[M]. is_field(M,r,A) \<longrightarrow> 
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(\<forall>p[M]. p \<in> s \<longleftrightarrow> rtran_closure_mem(M,A,r,p))" 

13242  80 

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definition 
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tran_closure :: "[i=>o,i,i] => o" where 
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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)" 
13242  85 

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lemma (in M_basic) rtran_closure_mem_iff: 
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"[M(A); M(r); M(p)] 
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==> rtran_closure_mem(M,A,r,p) \<longleftrightarrow> 
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(\<exists>n[M]. n\<in>nat & 
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(\<exists>f[M]. f \<in> succ(n) > A & 

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(\<exists>x[M]. \<exists>y[M]. 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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by (simp add: rtran_closure_mem_def Ord_succ_mem_iff nat_0_le [THEN ltD]) 
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13242  95 

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locale M_trancl = M_basic + 
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assumes rtrancl_separation: 
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"[ M(r); M(A) ] ==> separation (M, rtran_closure_mem(M,A,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 rtran_closure_mem_iff 
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rtrancl_alt_eq_rtrancl [symmetric] rtrancl_alt_def) 

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apply (auto simp add: nat_0_le [THEN ltD] apply_funtype) 
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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 rtran_closure_mem_iff) 
13242  118 
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) \<longleftrightarrow> z = rtrancl(r)" 
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apply (rule iffI) 
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txt{*Proving the righttoleft 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 rtran_closure_mem_iff) 
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apply (auto simp add: nat_0_le [THEN ltD] apply_funtype) 
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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) 
13242  134 

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lemma (in M_trancl) trancl_abs [simp]: 
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"[ M(r); M(z) ] ==> tran_closure(M,r,z) \<longleftrightarrow> z = trancl(r)" 
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by (simp add: tran_closure_def trancl_def) 
13242  138 

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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) 
13242  142 

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text{*Alternative proof of @{text wf_on_trancl}; inspiration for the 
13242  144 
relativized version. Original version is on theory WF.*} 
46823  145 
lemma "[ wf[A](r); r``A \<subseteq> 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) 
13242  150 
done 
151 

152 
lemma (in M_trancl) wellfounded_on_trancl: 

46823  153 
"[ wellfounded_on(M,A,r); r``A \<subseteq> A; M(r); M(A) ] 
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==> wellfounded_on(M,A,r^+)" 
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apply (simp add: wellfounded_on_def) 
13242  156 
apply (safe intro!: equalityI) 
157 
apply (rename_tac Z x) 

13268  158 
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') 
13299  161 
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) 
13242  164 
apply (drule_tac x=w in bspec, assumption, clarify) 
165 
apply (erule tranclE) 

166 
apply (blast dest: transM) (*transM is needed to prove M(xa)*) 

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apply blast 
13242  168 
done 
169 

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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 (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 
13242  177 

13223  178 

13647  179 
text{*Absoluteness for wfrecdefined functions.*} 
13254  180 

181 
(*first use is_recfun, then M_is_recfun*) 

182 

183 
lemma (in M_trancl) wfrec_relativize: 

184 
"[wf(r); M(a); M(r); 

13268  185 
strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M]. 
13254  186 
pair(M,x,y,z) & 
187 
is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r `` {x})), g) & 

188 
y = H(x, restrict(g, r `` {x}))); 

46823  189 
\<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))] 
190 
==> wfrec(r,a,H) = z \<longleftrightarrow> 

13268  191 
(\<exists>f[M]. is_recfun(r^+, a, \<lambda>x f. H(x, restrict(f, r `` {x})), f) & 
13254  192 
z = H(a,restrict(f,r``{a})))" 
193 
apply (frule wf_trancl) 

194 
apply (simp add: wftrec_def wfrec_def, safe) 

195 
apply (frule wf_exists_is_recfun 

196 
[of concl: "r^+" a "\<lambda>x f. H(x, restrict(f, r `` {x}))"]) 

197 
apply (simp_all add: trans_trancl function_restrictI trancl_subset_times) 

13268  198 
apply (clarify, rule_tac x=x in rexI) 
13254  199 
apply (simp_all add: the_recfun_eq trans_trancl trancl_subset_times) 
200 
done 

201 

202 

203 
text{*Assuming @{term r} is transitive simplifies the occurrences of @{text H}. 

204 
The premise @{term "relation(r)"} is necessary 

205 
before we can replace @{term "r^+"} by @{term r}. *} 

206 
theorem (in M_trancl) trans_wfrec_relativize: 

207 
"[wf(r); trans(r); relation(r); M(r); M(a); 

13634  208 
wfrec_replacement(M,MH,r); relation2(M,MH,H); 
46823  209 
\<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))] 
210 
==> wfrec(r,a,H) = z \<longleftrightarrow> (\<exists>f[M]. is_recfun(r,a,H,f) & z = H(a,f))" 

13353  211 
apply (frule wfrec_replacement', assumption+) 
212 
apply (simp cong: is_recfun_cong 

213 
add: wfrec_relativize trancl_eq_r 

214 
is_recfun_restrict_idem domain_restrict_idem) 

215 
done 

13254  216 

13353  217 
theorem (in M_trancl) trans_wfrec_abs: 
218 
"[wf(r); trans(r); relation(r); M(r); M(a); M(z); 

13634  219 
wfrec_replacement(M,MH,r); relation2(M,MH,H); 
46823  220 
\<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))] 
221 
==> is_wfrec(M,MH,r,a,z) \<longleftrightarrow> z=wfrec(r,a,H)" 

13634  222 
by (simp add: trans_wfrec_relativize [THEN iff_sym] is_wfrec_abs, blast) 
223 

13254  224 

225 
lemma (in M_trancl) trans_eq_pair_wfrec_iff: 

226 
"[wf(r); trans(r); relation(r); M(r); M(y); 

13634  227 
wfrec_replacement(M,MH,r); relation2(M,MH,H); 
46823  228 
\<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))] 
229 
==> y = <x, wfrec(r, x, H)> \<longleftrightarrow> 

13268  230 
(\<exists>f[M]. is_recfun(r,x,H,f) & y = <x, H(x,f)>)" 
13293  231 
apply safe 
232 
apply (simp add: trans_wfrec_relativize [THEN iff_sym, of concl: _ x]) 

13254  233 
txt{*converse direction*} 
234 
apply (rule sym) 

235 
apply (simp add: trans_wfrec_relativize, blast) 

236 
done 

237 

238 

239 
subsection{*M is closed under wellfounded recursion*} 

240 

241 
text{*Lemma with the awkward premise mentioning @{text wfrec}.*} 

13634  242 
lemma (in M_trancl) wfrec_closed_lemma [rule_format]: 
13254  243 
"[wf(r); M(r); 
244 
strong_replacement(M, \<lambda>x y. y = \<langle>x, wfrec(r, x, H)\<rangle>); 

46823  245 
\<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g)) ] 
246 
==> M(a) \<longrightarrow> M(wfrec(r,a,H))" 

13254  247 
apply (rule_tac a=a in wf_induct, assumption+) 
248 
apply (subst wfrec, assumption, clarify) 

249 
apply (drule_tac x1=x and x="\<lambda>x\<in>r `` {x}. wfrec(r, x, H)" 

250 
in rspec [THEN rspec]) 

251 
apply (simp_all add: function_lam) 

13505  252 
apply (blast intro: lam_closed dest: pair_components_in_M) 
13254  253 
done 
254 

255 
text{*Eliminates one instance of replacement.*} 

13634  256 
lemma (in M_trancl) wfrec_replacement_iff: 
13353  257 
"strong_replacement(M, \<lambda>x z. 
46823  258 
\<exists>y[M]. pair(M,x,y,z) & (\<exists>g[M]. is_recfun(r,x,H,g) & y = H(x,g))) \<longleftrightarrow> 
13254  259 
strong_replacement(M, 
13268  260 
\<lambda>x y. \<exists>f[M]. is_recfun(r,x,H,f) & y = <x, H(x,f)>)" 
13254  261 
apply simp 
262 
apply (rule strong_replacement_cong, blast) 

263 
done 

264 

265 
text{*Useful version for transitive relations*} 

13634  266 
theorem (in M_trancl) trans_wfrec_closed: 
13254  267 
"[wf(r); trans(r); relation(r); M(r); M(a); 
13634  268 
wfrec_replacement(M,MH,r); relation2(M,MH,H); 
46823  269 
\<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g)) ] 
13254  270 
==> M(wfrec(r,a,H))" 
13353  271 
apply (frule wfrec_replacement', assumption+) 
13254  272 
apply (frule wfrec_replacement_iff [THEN iffD1]) 
273 
apply (rule wfrec_closed_lemma, assumption+) 

274 
apply (simp_all add: wfrec_replacement_iff trans_eq_pair_wfrec_iff) 

275 
done 

276 

13506  277 
subsection{*Absoluteness without assuming transitivity*} 
13254  278 
lemma (in M_trancl) eq_pair_wfrec_iff: 
279 
"[wf(r); M(r); M(y); 

13268  280 
strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M]. 
13254  281 
pair(M,x,y,z) & 
282 
is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r `` {x})), g) & 

283 
y = H(x, restrict(g, r `` {x}))); 

46823  284 
\<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))] 
285 
==> y = <x, wfrec(r, x, H)> \<longleftrightarrow> 

13268  286 
(\<exists>f[M]. is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r `` {x})), f) & 
13254  287 
y = <x, H(x,restrict(f,r``{x}))>)" 
288 
apply safe 

13293  289 
apply (simp add: wfrec_relativize [THEN iff_sym, of concl: _ x]) 
13254  290 
txt{*converse direction*} 
291 
apply (rule sym) 

292 
apply (simp add: wfrec_relativize, blast) 

293 
done 

294 

295 
text{*Full version not assuming transitivity, but maybe not very useful.*} 

13634  296 
theorem (in M_trancl) wfrec_closed: 
13254  297 
"[wf(r); M(r); M(a); 
13353  298 
wfrec_replacement(M,MH,r^+); 
13634  299 
relation2(M,MH, \<lambda>x f. H(x, restrict(f, r `` {x}))); 
46823  300 
\<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g)) ] 
13254  301 
==> M(wfrec(r,a,H))" 
13353  302 
apply (frule wfrec_replacement' 
303 
[of MH "r^+" "\<lambda>x f. H(x, restrict(f, r `` {x}))"]) 

304 
prefer 4 

305 
apply (frule wfrec_replacement_iff [THEN iffD1]) 

306 
apply (rule wfrec_closed_lemma, assumption+) 

307 
apply (simp_all add: eq_pair_wfrec_iff func.function_restrictI) 

13254  308 
done 
309 

13223  310 
end 