include "basic_2/rt_computation/csx_cpxs.ma".
include "basic_2/rt_computation/jsx_rsx.ma".
-(* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
+(* STRONGLY NORMALIZING REFERRED LOCAL ENVS FOR EXTENDED RT-TRANSITION ******)
(* Forward lemmas with strongly rt-normalizing terms ************************)
-fact rsx_fwd_lref_pair_csx_aux (h) (G):
- ∀L. G ⊢ ⬈*[h,#0] 𝐒❪L❫ →
- â\88\80I,K,V. L = K.â\93\91[I]V â\86\92 â\9dªG,Kâ\9d« â\8a¢ â¬\88*[h] ð\9d\90\92â\9dªVâ\9d«.
-#h #G #L #H
+fact rsx_fwd_lref_pair_csx_aux (G):
+ ∀L. G ⊢ ⬈*𝐒[#0] L →
+ â\88\80I,K,V. L = K.â\93\91[I]V â\86\92 â\9d¨G,Kâ\9d© â\8a¢ â¬\88*ð\9d\90\92 V.
+#G #L #H
@(rsx_ind … H) -L #L #_ #IH #I #K #V1 #H destruct
@csx_intro #V2 #HV12 #HnV12
@(IH … I) -IH [1,4: // | -HnV12 | -G #H ]
[ /2 width=1 by lpx_pair/
-| elim (reqx_inv_zero_pair_sn … H) -H #Y #X #_ #H1 #H2 destruct -I
+| elim (reqg_inv_zero_pair_sn … H) -H #Y #X #_ #H1 #H2 destruct -I
/2 width=1 by/
]
qed-.
-lemma rsx_fwd_lref_pair_csx (h) (G):
- ∀I,K,V. G ⊢ ⬈*[h,#0] 𝐒❪K.ⓑ[I]V❫ → ❪G,K❫ ⊢ ⬈*[h] 𝐒❪V❫.
+lemma rsx_fwd_lref_pair_csx (G):
+ ∀I,K,V. G ⊢ ⬈*𝐒[#0] K.ⓑ[I]V → ❨G,K❩ ⊢ ⬈*𝐒 V.
/2 width=4 by rsx_fwd_lref_pair_csx_aux/ qed-.
-lemma rsx_fwd_lref_pair_csx_drops (h) (G):
- ∀I,K,V,i,L. ⇩*[i] L ≘ K.ⓑ[I]V → G ⊢ ⬈*[h,#i] 𝐒❪L❫ → ❪G,K❫ ⊢ ⬈*[h] 𝐒❪V❫.
-#h #G #I #K #V #i elim i -i
+lemma rsx_fwd_lref_pair_csx_drops (G):
+ ∀I,K,V,i,L. ⇩[i] L ≘ K.ⓑ[I]V → G ⊢ ⬈*𝐒[#i] L → ❨G,K❩ ⊢ ⬈*𝐒 V.
+#G #I #K #V #i elim i -i
[ #L #H >(drops_fwd_isid … H) -H
/2 width=2 by rsx_fwd_lref_pair_csx/
| #i #IH #L #H1 #H2
(* Inversion lemmas with strongly rt-normalizing terms **********************)
-lemma rsx_inv_lref_pair (h) (G):
- ∀I,K,V. G ⊢ ⬈*[h,#0] 𝐒❪K.ⓑ[I]V❫ →
- â\88§â\88§ â\9dªG,Kâ\9d« â\8a¢ â¬\88*[h] ð\9d\90\92â\9dªVâ\9d« & G â\8a¢ â¬\88*[h,V] ð\9d\90\92â\9dªKâ\9d«.
+lemma rsx_inv_lref_pair (G):
+ ∀I,K,V. G ⊢ ⬈*𝐒[#0] K.ⓑ[I]V →
+ â\88§â\88§ â\9d¨G,Kâ\9d© â\8a¢ â¬\88*ð\9d\90\92 V & G â\8a¢ â¬\88*ð\9d\90\92[V] K.
/3 width=2 by rsx_fwd_lref_pair_csx, rsx_fwd_pair, conj/ qed-.
-lemma rsx_inv_lref_pair_drops (h) (G):
- ∀I,K,V,i,L. ⇩*[i] L ≘ K.ⓑ[I]V → G ⊢ ⬈*[h,#i] 𝐒❪L❫ →
- â\88§â\88§ â\9dªG,Kâ\9d« â\8a¢ â¬\88*[h] ð\9d\90\92â\9dªVâ\9d« & G â\8a¢ â¬\88*[h,V] ð\9d\90\92â\9dªKâ\9d«.
+lemma rsx_inv_lref_pair_drops (G):
+ ∀I,K,V,i,L. ⇩[i] L ≘ K.ⓑ[I]V → G ⊢ ⬈*𝐒[#i] L →
+ â\88§â\88§ â\9d¨G,Kâ\9d© â\8a¢ â¬\88*ð\9d\90\92 V & G â\8a¢ â¬\88*ð\9d\90\92[V] K.
/3 width=5 by rsx_fwd_lref_pair_csx_drops, rsx_fwd_lref_pair_drops, conj/ qed-.
-lemma rsx_inv_lref_drops (h) (G):
- ∀L,i. G ⊢ ⬈*[h,#i] 𝐒❪L❫ →
+lemma rsx_inv_lref_drops (G):
+ ∀L,i. G ⊢ ⬈*𝐒[#i] L →
∨∨ ⇩*[Ⓕ,𝐔❨i❩] L ≘ ⋆
- | ∃∃I,K. ⇩*[i] L ≘ K.ⓤ[I]
- | ∃∃I,K,V. ⇩*[i] L ≘ K.ⓑ[I]V & ❪G,K❫ ⊢ ⬈*[h] 𝐒❪V❫ & G ⊢ ⬈*[h,V] 𝐒❪K❫.
-#h #G #L #i #H elim (drops_F_uni L i)
+ | ∃∃I,K. ⇩[i] L ≘ K.ⓤ[I]
+ | ∃∃I,K,V. ⇩[i] L ≘ K.ⓑ[I]V & ❨G,K❩ ⊢ ⬈*𝐒 V & G ⊢ ⬈*𝐒[V] K.
+#G #L #i #H elim (drops_F_uni L i)
[ /2 width=1 by or3_intro0/
| * * /4 width=10 by rsx_fwd_lref_pair_csx_drops, rsx_fwd_lref_pair_drops, ex3_3_intro, ex1_2_intro, or3_intro2, or3_intro1/
]
(* Note: swapping the eliminations to avoid rsx_cpx_trans: no solution found *)
(* Basic_2A1: uses: lsx_lref_be_lpxs *)
-lemma rsx_lref_pair_lpxs (h) (G):
- â\88\80K1,V. â\9dªG,K1â\9d« â\8a¢ â¬\88*[h] ð\9d\90\92â\9dªVâ\9d« →
- ∀K2. G ⊢ ⬈*[h,V] 𝐒❪K2❫ → ❪G,K1❫ ⊢ ⬈*[h] K2 →
- ∀I. G ⊢ ⬈*[h,#0] 𝐒❪K2.ⓑ[I]V❫.
-#h #G #K1 #V #H
+lemma rsx_lref_pair_lpxs (G):
+ â\88\80K1,V. â\9d¨G,K1â\9d© â\8a¢ â¬\88*ð\9d\90\92 V →
+ ∀K2. G ⊢ ⬈*𝐒[V] K2 → ❨G,K1❩ ⊢ ⬈* K2 →
+ ∀I. G ⊢ ⬈*𝐒[#0] K2.ⓑ[I]V.
+#G #K1 #V #H
@(csx_ind_cpxs … H) -V #V0 #_ #IHV0 #K2 #H
@(rsx_ind … H) -K2 #K0 #HK0 #IHK0 #HK10 #I
@rsx_intro #Y #HY #HnY
elim (lpx_inv_pair_sn … HY) -HY #K2 #V2 #HK02 #HV02 #H destruct
elim (teqx_dec V0 V2) #HnV02 destruct [ -IHV0 -HV02 -HK0 | -IHK0 -HnY ]
-[ /5 width=5 by rsx_reqx_trans, lpxs_step_dx, reqx_pair/
+[ /5 width=5 by rsx_reqx_trans, lpxs_step_dx, reqg_pair, reqg_refl/
| @(IHV0 … HnV02) -IHV0 -HnV02
[ /2 width=3 by lpxs_cpx_trans/
| /3 width=3 by rsx_lpx_trans, rsx_cpx_trans/
]
qed.
-lemma rsx_lref_pair (h) (G):
- â\88\80K,V. â\9dªG,Kâ\9d« â\8a¢ â¬\88*[h] ð\9d\90\92â\9dªVâ\9d« â\86\92 G â\8a¢ â¬\88*[h,V] ð\9d\90\92â\9dªKâ\9d« â\86\92 â\88\80I. G â\8a¢ â¬\88*[h,#0] ð\9d\90\92â\9dªK.â\93\91[I]Vâ\9d«.
+lemma rsx_lref_pair (G):
+ â\88\80K,V. â\9d¨G,Kâ\9d© â\8a¢ â¬\88*ð\9d\90\92 V â\86\92 G â\8a¢ â¬\88*ð\9d\90\92[V] K â\86\92 â\88\80I. G â\8a¢ â¬\88*ð\9d\90\92[#0] K.â\93\91[I]V.
/2 width=3 by rsx_lref_pair_lpxs/ qed.
(* Basic_2A1: uses: lsx_lref_be *)
-lemma rsx_lref_pair_drops (h) (G):
- â\88\80K,V. â\9dªG,Kâ\9d« â\8a¢ â¬\88*[h] ð\9d\90\92â\9dªVâ\9d« â\86\92 G â\8a¢ â¬\88*[h,V] ð\9d\90\92â\9dªKâ\9d« →
- ∀I,i,L. ⇩*[i] L ≘ K.ⓑ[I]V → G ⊢ ⬈*[h,#i] 𝐒❪L❫.
-#h #G #K #V #HV #HK #I #i elim i -i
+lemma rsx_lref_pair_drops (G):
+ â\88\80K,V. â\9d¨G,Kâ\9d© â\8a¢ â¬\88*ð\9d\90\92 V â\86\92 G â\8a¢ â¬\88*ð\9d\90\92[V] K →
+ ∀I,i,L. ⇩[i] L ≘ K.ⓑ[I]V → G ⊢ ⬈*𝐒[#i] L.
+#G #K #V #HV #HK #I #i elim i -i
[ #L #H >(drops_fwd_isid … H) -H /2 width=1 by rsx_lref_pair/
| #i #IH #L #H
elim (drops_inv_bind2_isuni_next … H) -H // #J #Y #HY #H destruct
(* Main properties with strongly rt-normalizing terms ***********************)
(* Basic_2A1: uses: csx_lsx *)
-theorem csx_rsx (h) (G): ∀L,T. ❪G,L❫ ⊢ ⬈*[h] 𝐒❪T❫ → G ⊢ ⬈*[h,T] 𝐒❪L❫.
-#h #G #L #T @(fqup_wf_ind_eq (Ⓣ) … G L T) -G -L -T
+theorem csx_rsx (G):
+ ∀L,T. ❨G,L❩ ⊢ ⬈*𝐒 T → G ⊢ ⬈*𝐒[T] L.
+#G #L #T @(fqup_wf_ind_eq (Ⓣ) … G L T) -G -L -T
#Z #Y #X #IH #G #L * *
[ //
| #i #HG #HL #HT #H destruct