(* 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 (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=
"\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=
"\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)
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)
"
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)
"
and M_nat [iff] :
"M(nat)"
lemma (
in M_trancl) 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)))
"
apply (simp add: rtran_closure_mem_def Ord_succ_mem_iff nat_0_le [
THEN ltD])
done
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)
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, \x. \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 (simp add: wf_on_def wf_def)
apply (safe)
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 (simp add: wellfounded_on_def)
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 (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‹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
"\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
🍋‹r
› is transitive simplifies the occurrences of
‹H
›.
The premise
🍋‹relation(r)
› is necessary
before we can replace
🍋‹r^+
› by 🍋‹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
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); 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)
apply (simp add: trans_wfrec_relativize, blast)
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=
"\x\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_trancl) wfrec_replacement_iff:
"strong_replacement(M, \x 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+)
apply (simp_all add: wfrec_replacement_iff trans_eq_pair_wfrec_iff)
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)
apply (simp add: wfrec_relativize, blast)
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^+" "\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
