# Theory WF_absolute

theory WF_absolute
imports WFrec
```(*  Title:      ZF/Constructible/WF_absolute.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

section ‹Absoluteness of Well-Founded Recursion›

theory WF_absolute imports WFrec begin

subsection‹Transitive closure without fixedpoints›

definition
rtrancl_alt :: "[i,i]=>i" where
"rtrancl_alt(A,r) ==
{p ∈ A*A. ∃n∈nat. ∃f ∈ succ(n) -> A.
(∃x y. p = <x,y> &  f`0 = x & f`n = y) &
(∀i∈n. <f`i, f`succ(i)> ∈ r)}"

lemma alt_rtrancl_lemma1 [rule_format]:
"n ∈ nat
==> ∀f ∈ succ(n) -> field(r).
(∀i∈n. ⟨f`i, f ` succ(i)⟩ ∈ r) ⟶ ⟨f`0, f`n⟩ ∈ 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 (blast intro: alt_rtrancl_lemma1)
done

lemma rtrancl_subset_rtrancl_alt: "r^* ⊆ rtrancl_alt(field(r),r)"
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="λx∈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="λi∈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)

definition
rtran_closure_mem :: "[i=>o,i,i,i] => o" where
―‹The property of belonging to ‹rtran_closure(r)››
"rtran_closure_mem(M,A,r,p) ==
∃nnat[M]. ∃n[M]. ∃n'[M].
omega(M,nnat) & n∈nnat & successor(M,n,n') &
(∃f[M]. typed_function(M,n',A,f) &
(∃x[M]. ∃y[M]. ∃zero[M]. pair(M,x,y,p) & empty(M,zero) &
fun_apply(M,f,zero,x) & fun_apply(M,f,n,y)) &
(∀j[M]. j∈n ⟶
(∃fj[M]. ∃sj[M]. ∃fsj[M]. ∃ffp[M].
fun_apply(M,f,j,fj) & successor(M,j,sj) &
fun_apply(M,f,sj,fsj) & pair(M,fj,fsj,ffp) & ffp ∈ r)))"

definition
rtran_closure :: "[i=>o,i,i] => o" where
"rtran_closure(M,r,s) ==
∀A[M]. is_field(M,r,A) ⟶
(∀p[M]. p ∈ s ⟷ rtran_closure_mem(M,A,r,p))"

definition
tran_closure :: "[i=>o,i,i] => o" where
"tran_closure(M,r,t) ==
∃s[M]. rtran_closure(M,r,s) & composition(M,r,s,t)"

lemma (in M_basic) rtran_closure_mem_iff:
"[|M(A); M(r); M(p)|]
==> rtran_closure_mem(M,A,r,p) ⟷
(∃n[M]. n∈nat &
(∃f[M]. f ∈ succ(n) -> A &
(∃x[M]. ∃y[M]. p = <x,y> & f`0 = x & f`n = y) &
(∀i∈n. <f`i, f`succ(i)> ∈ r)))"
by (simp add: rtran_closure_mem_def Ord_succ_mem_iff nat_0_le [THEN ltD])

locale M_trancl = M_basic +
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, λx.
∃w[M]. ∃wx[M]. ∃rp[M].
w ∈ Z & pair(M,w,x,wx) & tran_closure(M,r,rp) & wx ∈ rp)"

lemma (in M_trancl) rtran_closure_rtrancl:
"M(r) ==> rtran_closure(M,r,rtrancl(r))"
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)"])
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)"

lemma (in M_trancl) wellfounded_trancl_separation':
"[| M(r); M(Z) |] ==> separation (M, λx. ∃w[M]. w ∈ Z & <w,x> ∈ r^+)"
by (insert wellfounded_trancl_separation [of r Z], simp)

text‹Alternative proof of ‹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 (safe intro!: equalityI)
apply (drule_tac x = "{x∈A. ∃w. ⟨w,x⟩ ∈ r^+ & w ∈ 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 (safe intro!: equalityI)
apply (rename_tac Z x)
apply (subgoal_tac "M({x∈A. ∃w[M]. w ∈ Z & ⟨w,x⟩ ∈ r^+})")
prefer 2
apply (blast intro: wellfounded_trancl_separation')
apply (drule_tac x = "{x∈A. ∃w[M]. w ∈ Z & ⟨w,x⟩ ∈ 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 (rule wellfounded_on_subset_A, erule wellfounded_on_trancl)
apply blast
apply (simp_all add: trancl_type [THEN field_rel_subset])
done

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, λx z. ∃y[M]. ∃g[M].
pair(M,x,y,z) &
is_recfun(r^+, x, λx f. H(x, restrict(f, r -`` {x})), g) &
y = H(x, restrict(g, r -`` {x})));
∀x[M]. ∀g[M]. function(g) ⟶ M(H(x,g))|]
==> wfrec(r,a,H) = z ⟷
(∃f[M]. is_recfun(r^+, a, λx 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 "λx 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 ‹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);  relation2(M,MH,H);
∀x[M]. ∀g[M]. function(g) ⟶ M(H(x,g))|]
==> wfrec(r,a,H) = z ⟷ (∃f[M]. is_recfun(r,a,H,f) & z = H(a,f))"
apply (frule wfrec_replacement', assumption+)
apply (simp cong: is_recfun_cong
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);  relation2(M,MH,H);
∀x[M]. ∀g[M]. function(g) ⟶ M(H(x,g))|]
==> is_wfrec(M,MH,r,a,z) ⟷ z=wfrec(r,a,H)"
by (simp add: trans_wfrec_relativize [THEN iff_sym] is_wfrec_abs, blast)

lemma (in M_trancl) trans_eq_pair_wfrec_iff:
"[|wf(r);  trans(r); relation(r); M(r);  M(y);
wfrec_replacement(M,MH,r);  relation2(M,MH,H);
∀x[M]. ∀g[M]. function(g) ⟶ M(H(x,g))|]
==> y = <x, wfrec(r, x, H)> ⟷
(∃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)
done

subsection‹M is closed under well-founded recursion›

text‹Lemma with the awkward premise mentioning ‹wfrec›.›
lemma (in M_trancl) wfrec_closed_lemma [rule_format]:
"[|wf(r); M(r);
strong_replacement(M, λx y. y = ⟨x, wfrec(r, x, H)⟩);
∀x[M]. ∀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="λx∈r -`` {x}. wfrec(r, x, H)"
in rspec [THEN rspec])
apply (blast intro: lam_closed dest: pair_components_in_M)
done

text‹Eliminates one instance of replacement.›
lemma (in M_trancl) wfrec_replacement_iff:
"strong_replacement(M, λx z.
∃y[M]. pair(M,x,y,z) & (∃g[M]. is_recfun(r,x,H,g) & y = H(x,g))) ⟷
strong_replacement(M,
λx y. ∃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_trancl) trans_wfrec_closed:
"[|wf(r); trans(r); relation(r); M(r); M(a);
wfrec_replacement(M,MH,r);  relation2(M,MH,H);
∀x[M]. ∀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+)
done

subsection‹Absoluteness without assuming transitivity›
lemma (in M_trancl) eq_pair_wfrec_iff:
"[|wf(r);  M(r);  M(y);
strong_replacement(M, λx z. ∃y[M]. ∃g[M].
pair(M,x,y,z) &
is_recfun(r^+, x, λx f. H(x, restrict(f, r -`` {x})), g) &
y = H(x, restrict(g, r -`` {x})));
∀x[M]. ∀g[M]. function(g) ⟶ M(H(x,g))|]
==> y = <x, wfrec(r, x, H)> ⟷
(∃f[M]. is_recfun(r^+, x, λx 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)
done

text‹Full version not assuming transitivity, but maybe not very useful.›
theorem (in M_trancl) wfrec_closed:
"[|wf(r); M(r); M(a);
wfrec_replacement(M,MH,r^+);
relation2(M,MH, λx f. H(x, restrict(f, r -`` {x})));
∀x[M]. ∀g[M]. function(g) ⟶ M(H(x,g)) |]
==> M(wfrec(r,a,H))"
apply (frule wfrec_replacement'
[of MH "r^+" "λx f. H(x, restrict(f, r -`` {x}))"])
prefer 4
apply (frule wfrec_replacement_iff [THEN iffD1])
apply (rule wfrec_closed_lemma, assumption+)