header {*Absoluteness for Well-Founded Relations and Well-Founded Recursion*}
theory WF_absolute = WFrec:
subsection{*Every well-founded relation is a subset of some inverse image of
an ordinal*}
lemma wf_rvimage_Ord: "Ord(i) \<Longrightarrow> wf(rvimage(A, f, Memrel(i)))"
by (blast intro: wf_rvimage wf_Memrel)
constdefs
wfrank :: "[i,i]=>i"
"wfrank(r,a) == wfrec(r, a, %x f. \<Union>y \<in> r-``{x}. succ(f`y))"
constdefs
wftype :: "i=>i"
"wftype(r) == \<Union>y \<in> range(r). succ(wfrank(r,y))"
lemma wfrank: "wf(r) ==> wfrank(r,a) = (\<Union>y \<in> r-``{a}. succ(wfrank(r,y)))"
by (subst wfrank_def [THEN def_wfrec], simp_all)
lemma Ord_wfrank: "wf(r) ==> Ord(wfrank(r,a))"
apply (rule_tac a=a in wf_induct, assumption)
apply (subst wfrank, assumption)
apply (rule Ord_succ [THEN Ord_UN], blast)
done
lemma wfrank_lt: "[|wf(r); <a,b> \<in> r|] ==> wfrank(r,a) < wfrank(r,b)"
apply (rule_tac a1 = b in wfrank [THEN ssubst], assumption)
apply (rule UN_I [THEN ltI])
apply (simp add: Ord_wfrank vimage_iff)+
done
lemma Ord_wftype: "wf(r) ==> Ord(wftype(r))"
by (simp add: wftype_def Ord_wfrank)
lemma wftypeI: "\<lbrakk>wf(r); x \<in> field(r)\<rbrakk> \<Longrightarrow> wfrank(r,x) \<in> wftype(r)"
apply (simp add: wftype_def)
apply (blast intro: wfrank_lt [THEN ltD])
done
lemma wf_imp_subset_rvimage:
"[|wf(r); r \<subseteq> A*A|] ==> \<exists>i f. Ord(i) & r <= rvimage(A, f, Memrel(i))"
apply (rule_tac x="wftype(r)" in exI)
apply (rule_tac x="\<lambda>x\<in>A. wfrank(r,x)" in exI)
apply (simp add: Ord_wftype, clarify)
apply (frule subsetD, assumption, clarify)
apply (simp add: rvimage_iff wfrank_lt [THEN ltD])
apply (blast intro: wftypeI)
done
theorem wf_iff_subset_rvimage:
"relation(r) ==> wf(r) <-> (\<exists>i f A. Ord(i) & r <= rvimage(A, f, Memrel(i)))"
by (blast dest!: relation_field_times_field wf_imp_subset_rvimage
intro: wf_rvimage_Ord [THEN wf_subset])
subsection{*Transitive closure without fixedpoints*}
constdefs
rtrancl_alt :: "[i,i]=>i"
"rtrancl_alt(A,r) ==
{p \<in> A*A. \<exists>n\<in>nat. \<exists>f \<in> succ(n) -> A.
(\<exists>x y. p = <x,y> & f`0 = x & f`n = y) &
(\<forall>i\<in>n. <f`i, f`succ(i)> \<in> r)}"
lemma alt_rtrancl_lemma1 [rule_format]:
"n \<in> nat
==> \<forall>f \<in> succ(n) -> field(r).
(\<forall>i\<in>n. \<langle>f`i, f ` succ(i)\<rangle> \<in> r) --> \<langle>f`0, f`n\<rangle> \<in> r^*"
apply (induct_tac n)
apply (simp_all add: apply_funtype rtrancl_refl, clarify)
apply (rename_tac n f)
apply (rule rtrancl_into_rtrancl)
prefer 2 apply assumption
apply (drule_tac x="restrict(f,succ(n))" in bspec)
apply (blast intro: restrict_type2)
apply (simp add: Ord_succ_mem_iff nat_0_le [THEN ltD] leI [THEN ltD] ltI)
done
lemma rtrancl_alt_subset_rtrancl: "rtrancl_alt(field(r),r) <= r^*"
apply (simp add: rtrancl_alt_def)
apply (blast intro: alt_rtrancl_lemma1)
done
lemma rtrancl_subset_rtrancl_alt: "r^* <= rtrancl_alt(field(r),r)"
apply (simp add: rtrancl_alt_def, clarify)
apply (frule rtrancl_type [THEN subsetD], clarify, simp)
apply (erule rtrancl_induct)
txt{*Base case, trivial*}
apply (rule_tac x=0 in bexI)
apply (rule_tac x="lam x:1. xa" in bexI)
apply simp_all
txt{*Inductive step*}
apply clarify
apply (rename_tac n f)
apply (rule_tac x="succ(n)" in bexI)
apply (rule_tac x="lam i:succ(succ(n)). if i=succ(n) then z else f`i" in bexI)
apply (simp add: Ord_succ_mem_iff nat_0_le [THEN ltD] leI [THEN ltD] ltI)
apply (blast intro: mem_asym)
apply typecheck
apply auto
done
lemma rtrancl_alt_eq_rtrancl: "rtrancl_alt(field(r),r) = r^*"
by (blast del: subsetI
intro: rtrancl_alt_subset_rtrancl rtrancl_subset_rtrancl_alt)
constdefs
rtran_closure_mem :: "[i=>o,i,i,i] => o"
--{*The property of belonging to @{text "rtran_closure(r)"}*}
"rtran_closure_mem(M,A,r,p) ==
\<exists>nnat[M]. \<exists>n[M]. \<exists>n'[M].
omega(M,nnat) & n\<in>nnat & successor(M,n,n') &
(\<exists>f[M]. typed_function(M,n',A,f) &
(\<exists>x[M]. \<exists>y[M]. \<exists>zero[M]. pair(M,x,y,p) & empty(M,zero) &
fun_apply(M,f,zero,x) & fun_apply(M,f,n,y)) &
(\<forall>j[M]. j\<in>n -->
(\<exists>fj[M]. \<exists>sj[M]. \<exists>fsj[M]. \<exists>ffp[M].
fun_apply(M,f,j,fj) & successor(M,j,sj) &
fun_apply(M,f,sj,fsj) & pair(M,fj,fsj,ffp) & ffp \<in> r)))"
rtran_closure :: "[i=>o,i,i] => o"
"rtran_closure(M,r,s) ==
\<forall>A[M]. is_field(M,r,A) -->
(\<forall>p[M]. p \<in> s <-> rtran_closure_mem(M,A,r,p))"
tran_closure :: "[i=>o,i,i] => o"
"tran_closure(M,r,t) ==
\<exists>s[M]. rtran_closure(M,r,s) & composition(M,r,s,t)"
lemma (in M_axioms) rtran_closure_mem_iff:
"[|M(A); M(r); M(p)|]
==> rtran_closure_mem(M,A,r,p) <->
(\<exists>n[M]. n\<in>nat &
(\<exists>f[M]. f \<in> succ(n) -> A &
(\<exists>x[M]. \<exists>y[M]. p = <x,y> & f`0 = x & f`n = y) &
(\<forall>i\<in>n. <f`i, f`succ(i)> \<in> r)))"
by (simp add: rtran_closure_mem_def Ord_succ_mem_iff nat_0_le [THEN ltD])
locale M_trancl = M_axioms +
assumes rtrancl_separation:
"[| M(r); M(A) |] ==> separation (M, rtran_closure_mem(M,A,r))"
and wellfounded_trancl_separation:
"[| M(r); M(Z) |] ==>
separation (M, \<lambda>x.
\<exists>w[M]. \<exists>wx[M]. \<exists>rp[M].
w \<in> Z & pair(M,w,x,wx) & tran_closure(M,r,rp) & wx \<in> rp)"
lemma (in M_trancl) rtran_closure_rtrancl:
"M(r) ==> rtran_closure(M,r,rtrancl(r))"
apply (simp add: rtran_closure_def rtran_closure_mem_iff
rtrancl_alt_eq_rtrancl [symmetric] rtrancl_alt_def)
apply (auto simp add: nat_0_le [THEN ltD] apply_funtype)
done
lemma (in M_trancl) rtrancl_closed [intro,simp]:
"M(r) ==> M(rtrancl(r))"
apply (insert rtrancl_separation [of r "field(r)"])
apply (simp add: rtrancl_alt_eq_rtrancl [symmetric]
rtrancl_alt_def rtran_closure_mem_iff)
done
lemma (in M_trancl) rtrancl_abs [simp]:
"[| M(r); M(z) |] ==> rtran_closure(M,r,z) <-> z = rtrancl(r)"
apply (rule iffI)
txt{*Proving the right-to-left implication*}
prefer 2 apply (blast intro: rtran_closure_rtrancl)
apply (rule M_equalityI)
apply (simp add: rtran_closure_def rtrancl_alt_eq_rtrancl [symmetric]
rtrancl_alt_def rtran_closure_mem_iff)
apply (auto simp add: nat_0_le [THEN ltD] apply_funtype)
done
lemma (in M_trancl) trancl_closed [intro,simp]:
"M(r) ==> M(trancl(r))"
by (simp add: trancl_def comp_closed rtrancl_closed)
lemma (in M_trancl) trancl_abs [simp]:
"[| M(r); M(z) |] ==> tran_closure(M,r,z) <-> z = trancl(r)"
by (simp add: tran_closure_def trancl_def)
lemma (in M_trancl) wellfounded_trancl_separation':
"[| M(r); M(Z) |] ==> separation (M, \<lambda>x. \<exists>w[M]. w \<in> Z & <w,x> \<in> r^+)"
by (insert wellfounded_trancl_separation [of r Z], simp)
text{*Alternative proof of @{text wf_on_trancl}; inspiration for the
relativized version. Original version is on theory WF.*}
lemma "[| wf[A](r); r-``A <= A |] ==> wf[A](r^+)"
apply (simp add: wf_on_def wf_def)
apply (safe intro!: equalityI)
apply (drule_tac x = "{x\<in>A. \<exists>w. \<langle>w,x\<rangle> \<in> r^+ & w \<in> Z}" in spec)
apply (blast elim: tranclE)
done
lemma (in M_trancl) wellfounded_on_trancl:
"[| wellfounded_on(M,A,r); r-``A <= A; M(r); M(A) |]
==> wellfounded_on(M,A,r^+)"
apply (simp add: wellfounded_on_def)
apply (safe intro!: equalityI)
apply (rename_tac Z x)
apply (subgoal_tac "M({x\<in>A. \<exists>w[M]. w \<in> Z & \<langle>w,x\<rangle> \<in> r^+})")
prefer 2
apply (blast intro: wellfounded_trancl_separation')
apply (drule_tac x = "{x\<in>A. \<exists>w[M]. w \<in> Z & \<langle>w,x\<rangle> \<in> r^+}" in rspec, safe)
apply (blast dest: transM, simp)
apply (rename_tac y w)
apply (drule_tac x=w in bspec, assumption, clarify)
apply (erule tranclE)
apply (blast dest: transM) (*transM is needed to prove M(xa)*)
apply blast
done
lemma (in M_trancl) wellfounded_trancl:
"[|wellfounded(M,r); M(r)|] ==> wellfounded(M,r^+)"
apply (rotate_tac -1)
apply (simp add: wellfounded_iff_wellfounded_on_field)
apply (rule wellfounded_on_subset_A, erule wellfounded_on_trancl)
apply blast
apply (simp_all add: trancl_type [THEN field_rel_subset])
done
text{*Relativized to M: Every well-founded relation is a subset of some
inverse image of an ordinal. Key step is the construction (in M) of a
rank function.*}
locale M_wfrank = M_trancl +
assumes wfrank_separation:
"M(r) ==>
separation (M, \<lambda>x.
\<forall>rplus[M]. tran_closure(M,r,rplus) -->
~ (\<exists>f[M]. M_is_recfun(M, %x f y. is_range(M,f,y), rplus, x, f)))"
and wfrank_strong_replacement:
"M(r) ==>
strong_replacement(M, \<lambda>x z.
\<forall>rplus[M]. tran_closure(M,r,rplus) -->
(\<exists>y[M]. \<exists>f[M]. pair(M,x,y,z) &
M_is_recfun(M, %x f y. is_range(M,f,y), rplus, x, f) &
is_range(M,f,y)))"
and Ord_wfrank_separation:
"M(r) ==>
separation (M, \<lambda>x.
\<forall>rplus[M]. tran_closure(M,r,rplus) -->
~ (\<forall>f[M]. \<forall>rangef[M].
is_range(M,f,rangef) -->
M_is_recfun(M, \<lambda>x f y. is_range(M,f,y), rplus, x, f) -->
ordinal(M,rangef)))"
text{*Proving that the relativized instances of Separation or Replacement
agree with the "real" ones.*}
lemma (in M_wfrank) wfrank_separation':
"M(r) ==>
separation
(M, \<lambda>x. ~ (\<exists>f[M]. is_recfun(r^+, x, %x f. range(f), f)))"
apply (insert wfrank_separation [of r])
apply (simp add: relativize2_def is_recfun_abs [of "%x. range"])
done
lemma (in M_wfrank) wfrank_strong_replacement':
"M(r) ==>
strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>f[M].
pair(M,x,y,z) & is_recfun(r^+, x, %x f. range(f), f) &
y = range(f))"
apply (insert wfrank_strong_replacement [of r])
apply (simp add: relativize2_def is_recfun_abs [of "%x. range"])
done
lemma (in M_wfrank) Ord_wfrank_separation':
"M(r) ==>
separation (M, \<lambda>x.
~ (\<forall>f[M]. is_recfun(r^+, x, \<lambda>x. range, f) --> Ord(range(f))))"
apply (insert Ord_wfrank_separation [of r])
apply (simp add: relativize2_def is_recfun_abs [of "%x. range"])
done
text{*This function, defined using replacement, is a rank function for
well-founded relations within the class M.*}
constdefs
wellfoundedrank :: "[i=>o,i,i] => i"
"wellfoundedrank(M,r,A) ==
{p. x\<in>A, \<exists>y[M]. \<exists>f[M].
p = <x,y> & is_recfun(r^+, x, %x f. range(f), f) &
y = range(f)}"
lemma (in M_wfrank) exists_wfrank:
"[| wellfounded(M,r); M(a); M(r) |]
==> \<exists>f[M]. is_recfun(r^+, a, %x f. range(f), f)"
apply (rule wellfounded_exists_is_recfun)
apply (blast intro: wellfounded_trancl)
apply (rule trans_trancl)
apply (erule wfrank_separation')
apply (erule wfrank_strong_replacement')
apply (simp_all add: trancl_subset_times)
done
lemma (in M_wfrank) M_wellfoundedrank:
"[| wellfounded(M,r); M(r); M(A) |] ==> M(wellfoundedrank(M,r,A))"
apply (insert wfrank_strong_replacement' [of r])
apply (simp add: wellfoundedrank_def)
apply (rule strong_replacement_closed)
apply assumption+
apply (rule univalent_is_recfun)
apply (blast intro: wellfounded_trancl)
apply (rule trans_trancl)
apply (simp add: trancl_subset_times, blast)
done
lemma (in M_wfrank) Ord_wfrank_range [rule_format]:
"[| wellfounded(M,r); a\<in>A; M(r); M(A) |]
==> \<forall>f[M]. is_recfun(r^+, a, %x f. range(f), f) --> Ord(range(f))"
apply (drule wellfounded_trancl, assumption)
apply (rule wellfounded_induct, assumption, erule (1) transM)
apply simp
apply (blast intro: Ord_wfrank_separation', clarify)
txt{*The reasoning in both cases is that we get @{term y} such that
@{term "\<langle>y, x\<rangle> \<in> r^+"}. We find that
@{term "f`y = restrict(f, r^+ -`` {y})"}. *}
apply (rule OrdI [OF _ Ord_is_Transset])
txt{*An ordinal is a transitive set...*}
apply (simp add: Transset_def)
apply clarify
apply (frule apply_recfun2, assumption)
apply (force simp add: restrict_iff)
txt{*...of ordinals. This second case requires the induction hyp.*}
apply clarify
apply (rename_tac i y)
apply (frule apply_recfun2, assumption)
apply (frule is_recfun_imp_in_r, assumption)
apply (frule is_recfun_restrict)
(*simp_all won't work*)
apply (simp add: trans_trancl trancl_subset_times)+
apply (drule spec [THEN mp], assumption)
apply (subgoal_tac "M(restrict(f, r^+ -`` {y}))")
apply (drule_tac x="restrict(f, r^+ -`` {y})" in rspec)
apply assumption
apply (simp add: function_apply_equality [OF _ is_recfun_imp_function])
apply (blast dest: pair_components_in_M)
done
lemma (in M_wfrank) Ord_range_wellfoundedrank:
"[| wellfounded(M,r); r \<subseteq> A*A; M(r); M(A) |]
==> Ord (range(wellfoundedrank(M,r,A)))"
apply (frule wellfounded_trancl, assumption)
apply (frule trancl_subset_times)
apply (simp add: wellfoundedrank_def)
apply (rule OrdI [OF _ Ord_is_Transset])
prefer 2
txt{*by our previous result the range consists of ordinals.*}
apply (blast intro: Ord_wfrank_range)
txt{*We still must show that the range is a transitive set.*}
apply (simp add: Transset_def, clarify, simp)
apply (rename_tac x i f u)
apply (frule is_recfun_imp_in_r, assumption)
apply (subgoal_tac "M(u) & M(i) & M(x)")
prefer 2 apply (blast dest: transM, clarify)
apply (rule_tac a=u in rangeI)
apply (rule_tac x=u in ReplaceI)
apply simp
apply (rule_tac x="restrict(f, r^+ -`` {u})" in rexI)
apply (blast intro: is_recfun_restrict trans_trancl dest: apply_recfun2)
apply simp
apply blast
txt{*Unicity requirement of Replacement*}
apply clarify
apply (frule apply_recfun2, assumption)
apply (simp add: trans_trancl is_recfun_cut)
done
lemma (in M_wfrank) function_wellfoundedrank:
"[| wellfounded(M,r); M(r); M(A)|]
==> function(wellfoundedrank(M,r,A))"
apply (simp add: wellfoundedrank_def function_def, clarify)
txt{*Uniqueness: repeated below!*}
apply (drule is_recfun_functional, assumption)
apply (blast intro: wellfounded_trancl)
apply (simp_all add: trancl_subset_times trans_trancl)
done
lemma (in M_wfrank) domain_wellfoundedrank:
"[| wellfounded(M,r); M(r); M(A)|]
==> domain(wellfoundedrank(M,r,A)) = A"
apply (simp add: wellfoundedrank_def function_def)
apply (rule equalityI, auto)
apply (frule transM, assumption)
apply (frule_tac a=x in exists_wfrank, assumption+, clarify)
apply (rule_tac b="range(f)" in domainI)
apply (rule_tac x=x in ReplaceI)
apply simp
apply (rule_tac x=f in rexI, blast, simp_all)
txt{*Uniqueness (for Replacement): repeated above!*}
apply clarify
apply (drule is_recfun_functional, assumption)
apply (blast intro: wellfounded_trancl)
apply (simp_all add: trancl_subset_times trans_trancl)
done
lemma (in M_wfrank) wellfoundedrank_type:
"[| wellfounded(M,r); M(r); M(A)|]
==> wellfoundedrank(M,r,A) \<in> A -> range(wellfoundedrank(M,r,A))"
apply (frule function_wellfoundedrank [of r A], assumption+)
apply (frule function_imp_Pi)
apply (simp add: wellfoundedrank_def relation_def)
apply blast
apply (simp add: domain_wellfoundedrank)
done
lemma (in M_wfrank) Ord_wellfoundedrank:
"[| wellfounded(M,r); a \<in> A; r \<subseteq> A*A; M(r); M(A) |]
==> Ord(wellfoundedrank(M,r,A) ` a)"
by (blast intro: apply_funtype [OF wellfoundedrank_type]
Ord_in_Ord [OF Ord_range_wellfoundedrank])
lemma (in M_wfrank) wellfoundedrank_eq:
"[| is_recfun(r^+, a, %x. range, f);
wellfounded(M,r); a \<in> A; M(f); M(r); M(A)|]
==> wellfoundedrank(M,r,A) ` a = range(f)"
apply (rule apply_equality)
prefer 2 apply (blast intro: wellfoundedrank_type)
apply (simp add: wellfoundedrank_def)
apply (rule ReplaceI)
apply (rule_tac x="range(f)" in rexI)
apply blast
apply simp_all
txt{*Unicity requirement of Replacement*}
apply clarify
apply (drule is_recfun_functional, assumption)
apply (blast intro: wellfounded_trancl)
apply (simp_all add: trancl_subset_times trans_trancl)
done
lemma (in M_wfrank) wellfoundedrank_lt:
"[| <a,b> \<in> r;
wellfounded(M,r); r \<subseteq> A*A; M(r); M(A)|]
==> wellfoundedrank(M,r,A) ` a < wellfoundedrank(M,r,A) ` b"
apply (frule wellfounded_trancl, assumption)
apply (subgoal_tac "a\<in>A & b\<in>A")
prefer 2 apply blast
apply (simp add: lt_def Ord_wellfoundedrank, clarify)
apply (frule exists_wfrank [of concl: _ b], erule (1) transM, assumption)
apply clarify
apply (rename_tac fb)
apply (frule is_recfun_restrict [of concl: "r^+" a])
apply (rule trans_trancl, assumption)
apply (simp_all add: r_into_trancl trancl_subset_times)
txt{*Still the same goal, but with new @{text is_recfun} assumptions.*}
apply (simp add: wellfoundedrank_eq)
apply (frule_tac a=a in wellfoundedrank_eq, assumption+)
apply (simp_all add: transM [of a])
txt{*We have used equations for wellfoundedrank and now must use some
for @{text is_recfun}. *}
apply (rule_tac a=a in rangeI)
apply (simp add: is_recfun_type [THEN apply_iff] vimage_singleton_iff
r_into_trancl apply_recfun r_into_trancl)
done
lemma (in M_wfrank) wellfounded_imp_subset_rvimage:
"[|wellfounded(M,r); r \<subseteq> A*A; M(r); M(A)|]
==> \<exists>i f. Ord(i) & r <= rvimage(A, f, Memrel(i))"
apply (rule_tac x="range(wellfoundedrank(M,r,A))" in exI)
apply (rule_tac x="wellfoundedrank(M,r,A)" in exI)
apply (simp add: Ord_range_wellfoundedrank, clarify)
apply (frule subsetD, assumption, clarify)
apply (simp add: rvimage_iff wellfoundedrank_lt [THEN ltD])
apply (blast intro: apply_rangeI wellfoundedrank_type)
done
lemma (in M_wfrank) wellfounded_imp_wf:
"[|wellfounded(M,r); relation(r); M(r)|] ==> wf(r)"
by (blast dest!: relation_field_times_field wellfounded_imp_subset_rvimage
intro: wf_rvimage_Ord [THEN wf_subset])
lemma (in M_wfrank) wellfounded_on_imp_wf_on:
"[|wellfounded_on(M,A,r); relation(r); M(r); M(A)|] ==> wf[A](r)"
apply (simp add: wellfounded_on_iff_wellfounded wf_on_def)
apply (rule wellfounded_imp_wf)
apply (simp_all add: relation_def)
done
theorem (in M_wfrank) wf_abs [simp]:
"[|relation(r); M(r)|] ==> wellfounded(M,r) <-> wf(r)"
by (blast intro: wellfounded_imp_wf wf_imp_relativized)
theorem (in M_wfrank) wf_on_abs [simp]:
"[|relation(r); M(r); M(A)|] ==> wellfounded_on(M,A,r) <-> wf[A](r)"
by (blast intro: wellfounded_on_imp_wf_on wf_on_imp_relativized)
text{*absoluteness for wfrec-defined functions.*}
(*first use is_recfun, then M_is_recfun*)
lemma (in M_trancl) wfrec_relativize:
"[|wf(r); M(a); M(r);
strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M].
pair(M,x,y,z) &
is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), g) &
y = H(x, restrict(g, r -`` {x})));
\<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|]
==> wfrec(r,a,H) = z <->
(\<exists>f[M]. is_recfun(r^+, a, \<lambda>x f. H(x, restrict(f, r -`` {x})), f) &
z = H(a,restrict(f,r-``{a})))"
apply (frule wf_trancl)
apply (simp add: wftrec_def wfrec_def, safe)
apply (frule wf_exists_is_recfun
[of concl: "r^+" a "\<lambda>x f. H(x, restrict(f, r -`` {x}))"])
apply (simp_all add: trans_trancl function_restrictI trancl_subset_times)
apply (clarify, rule_tac x=x in rexI)
apply (simp_all add: the_recfun_eq trans_trancl trancl_subset_times)
done
text{*Assuming @{term r} is transitive simplifies the occurrences of @{text H}.
The premise @{term "relation(r)"} is necessary
before we can replace @{term "r^+"} by @{term r}. *}
theorem (in M_trancl) trans_wfrec_relativize:
"[|wf(r); trans(r); relation(r); M(r); M(a);
wfrec_replacement(M,MH,r); relativize2(M,MH,H);
\<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|]
==> wfrec(r,a,H) = z <-> (\<exists>f[M]. is_recfun(r,a,H,f) & z = H(a,f))"
apply (frule wfrec_replacement', assumption+)
apply (simp cong: is_recfun_cong
add: wfrec_relativize trancl_eq_r
is_recfun_restrict_idem domain_restrict_idem)
done
theorem (in M_trancl) trans_wfrec_abs:
"[|wf(r); trans(r); relation(r); M(r); M(a); M(z);
wfrec_replacement(M,MH,r); relativize2(M,MH,H);
\<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|]
==> is_wfrec(M,MH,r,a,z) <-> z=wfrec(r,a,H)"
apply (simp add: trans_wfrec_relativize [THEN iff_sym] is_wfrec_abs, blast)
done
lemma (in M_trancl) trans_eq_pair_wfrec_iff:
"[|wf(r); trans(r); relation(r); M(r); M(y);
wfrec_replacement(M,MH,r); relativize2(M,MH,H);
\<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|]
==> y = <x, wfrec(r, x, H)> <->
(\<exists>f[M]. is_recfun(r,x,H,f) & y = <x, H(x,f)>)"
apply safe
apply (simp add: trans_wfrec_relativize [THEN iff_sym, of concl: _ x])
txt{*converse direction*}
apply (rule sym)
apply (simp add: trans_wfrec_relativize, blast)
done
subsection{*M is closed under well-founded recursion*}
text{*Lemma with the awkward premise mentioning @{text wfrec}.*}
lemma (in M_wfrank) wfrec_closed_lemma [rule_format]:
"[|wf(r); M(r);
strong_replacement(M, \<lambda>x y. y = \<langle>x, wfrec(r, x, H)\<rangle>);
\<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)) |]
==> M(a) --> M(wfrec(r,a,H))"
apply (rule_tac a=a in wf_induct, assumption+)
apply (subst wfrec, assumption, clarify)
apply (drule_tac x1=x and x="\<lambda>x\<in>r -`` {x}. wfrec(r, x, H)"
in rspec [THEN rspec])
apply (simp_all add: function_lam)
apply (blast intro: lam_closed dest: pair_components_in_M )
done
text{*Eliminates one instance of replacement.*}
lemma (in M_wfrank) wfrec_replacement_iff:
"strong_replacement(M, \<lambda>x z.
\<exists>y[M]. pair(M,x,y,z) & (\<exists>g[M]. is_recfun(r,x,H,g) & y = H(x,g))) <->
strong_replacement(M,
\<lambda>x y. \<exists>f[M]. is_recfun(r,x,H,f) & y = <x, H(x,f)>)"
apply simp
apply (rule strong_replacement_cong, blast)
done
text{*Useful version for transitive relations*}
theorem (in M_wfrank) trans_wfrec_closed:
"[|wf(r); trans(r); relation(r); M(r); M(a);
wfrec_replacement(M,MH,r); relativize2(M,MH,H);
\<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)) |]
==> M(wfrec(r,a,H))"
apply (frule wfrec_replacement', assumption+)
apply (frule wfrec_replacement_iff [THEN iffD1])
apply (rule wfrec_closed_lemma, assumption+)
apply (simp_all add: wfrec_replacement_iff trans_eq_pair_wfrec_iff)
done
section{*Absoluteness without assuming transitivity*}
lemma (in M_trancl) eq_pair_wfrec_iff:
"[|wf(r); M(r); M(y);
strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M].
pair(M,x,y,z) &
is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), g) &
y = H(x, restrict(g, r -`` {x})));
\<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|]
==> y = <x, wfrec(r, x, H)> <->
(\<exists>f[M]. is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), f) &
y = <x, H(x,restrict(f,r-``{x}))>)"
apply safe
apply (simp add: wfrec_relativize [THEN iff_sym, of concl: _ x])
txt{*converse direction*}
apply (rule sym)
apply (simp add: wfrec_relativize, blast)
done
text{*Full version not assuming transitivity, but maybe not very useful.*}
theorem (in M_wfrank) wfrec_closed:
"[|wf(r); M(r); M(a);
wfrec_replacement(M,MH,r^+);
relativize2(M,MH, \<lambda>x f. H(x, restrict(f, r -`` {x})));
\<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)) |]
==> M(wfrec(r,a,H))"
apply (frule wfrec_replacement'
[of MH "r^+" "\<lambda>x f. H(x, restrict(f, r -`` {x}))"])
prefer 4
apply (frule wfrec_replacement_iff [THEN iffD1])
apply (rule wfrec_closed_lemma, assumption+)
apply (simp_all add: eq_pair_wfrec_iff func.function_restrictI)
done
end