(* Title: ZF/Constructible/WF_absolute.thy ID: $Id: WF_absolute.thy,v 1.28 2005/06/17 14:15:10 haftmann Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory *) header {*Absoluteness of Well-Founded Recursion*} theory WF_absolute imports WFrec begin subsection{*Transitive closure without fixedpoints*} constdefs rtrancl_alt :: "[i,i]=>i" "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="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) == ∃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)))" rtran_closure :: "[i=>o,i,i] => o" "rtran_closure(M,r,s) == ∀A[M]. is_field(M,r,A) --> (∀p[M]. p ∈ s <-> rtran_closure_mem(M,A,r,p))" tran_closure :: "[i=>o,i,i] => o" "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))" 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, λx. ∃w[M]. w ∈ Z & <w,x> ∈ 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∈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 "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 @{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); 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 @{text 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 (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, 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+) 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^+" "λx 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
lemma alt_rtrancl_lemma1:
[| n ∈ nat; f ∈ succ(n) -> field(r); !!i. i ∈ n ==> 〈f ` i, f ` succ(i)〉 ∈ r |] ==> 〈f ` 0, f ` n〉 ∈ r^*
lemma rtrancl_alt_subset_rtrancl:
rtrancl_alt(field(r), r) ⊆ r^*
lemma rtrancl_subset_rtrancl_alt:
r^* ⊆ rtrancl_alt(field(r), r)
lemma rtrancl_alt_eq_rtrancl:
rtrancl_alt(field(r), r) = r^*
lemma rtran_closure_mem_iff:
[| PROP M_basic(M); 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)))
lemma rtran_closure_rtrancl:
[| PROP M_trancl(M); M(r) |] ==> rtran_closure(M, r, r^*)
lemma rtrancl_closed:
[| PROP M_trancl(M); M(r) |] ==> M(r^*)
lemma rtrancl_abs:
[| PROP M_trancl(M); M(r); M(z) |] ==> rtran_closure(M, r, z) <-> z = r^*
lemma trancl_closed:
[| PROP M_trancl(M); M(r) |] ==> M(r^+)
lemma trancl_abs:
[| PROP M_trancl(M); M(r); M(z) |] ==> tran_closure(M, r, z) <-> z = r^+
lemma wellfounded_trancl_separation':
[| PROP M_trancl(M); M(r); M(Z) |] ==> separation(M, %x. ∃w[M]. w ∈ Z ∧ 〈w, x〉 ∈ r^+)
lemma
[| wf[A](r); r -`` A ⊆ A |] ==> wf[A](r^+)
lemma wellfounded_on_trancl:
[| PROP M_trancl(M); wellfounded_on(M, A, r); r -`` A ⊆ A; M(r); M(A) |] ==> wellfounded_on(M, A, r^+)
lemma wellfounded_trancl:
[| PROP M_trancl(M); wellfounded(M, r); M(r) |] ==> wellfounded(M, r^+)
lemma wfrec_relativize:
[| PROP M_trancl(M); wf(r); M(a); M(r); strong_replacement (M, %x z. ∃y[M]. ∃g[M]. pair(M, x, y, z) ∧ is_recfun (r^+, x, %x 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, %x f. H(x, restrict(f, r -`` {x})), f) ∧ z = H(a, restrict(f, r -`` {a})))
theorem trans_wfrec_relativize:
[| PROP M_trancl(M); 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))
theorem trans_wfrec_abs:
[| PROP M_trancl(M); 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)
lemma trans_eq_pair_wfrec_iff:
[| PROP M_trancl(M); 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)〉)
lemma wfrec_closed_lemma:
[| PROP M_trancl(M); wf(r); M(r); strong_replacement(M, %x y. y = 〈x, wfrec(r, x, H)〉); ∀x[M]. ∀g[M]. function(g) --> M(H(x, g)) |] ==> M(a) --> M(wfrec(r, a, H))
lemma wfrec_replacement_iff:
PROP M_trancl(M) ==> 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, %x y. ∃f[M]. is_recfun(r, x, H, f) ∧ y = 〈x, H(x, f)〉)
theorem trans_wfrec_closed:
[| PROP M_trancl(M); 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))
lemma eq_pair_wfrec_iff:
[| PROP M_trancl(M); wf(r); M(r); M(y); strong_replacement (M, %x z. ∃y[M]. ∃g[M]. pair(M, x, y, z) ∧ is_recfun (r^+, x, %x 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, %x f. H(x, restrict(f, r -`` {x})), f) ∧ y = 〈x, H(x, restrict(f, r -`` {x}))〉)
theorem wfrec_closed:
[| PROP M_trancl(M); wf(r); M(r); M(a); wfrec_replacement(M, MH, r^+); relation2(M, MH, %x f. H(x, restrict(f, r -`` {x}))); ∀x[M]. ∀g[M]. function(g) --> M(H(x, g)) |] ==> M(wfrec(r, a, H))