--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/rsx_drops.ma".
+include "basic_2/rt_computation/rsx_lpxs.ma".
+include "basic_2/rt_computation/jsx.ma".
+
+(* COMPATIBILITY OF STRONG NORMALIZATION FOR UNBOUND RT-TRANSITION **********)
+
+(* Properties with strongly normalizing referred local environments *********)
+
+(* Basic_2A1: uses: lsx_cpx_trans_lcosx *)
+lemma rsx_cpx_trans_jsx (h) (G):
+ ∀L0,T1,T2. ⦃G,L0⦄ ⊢ T1 ⬈[h] T2 →
+ ∀f,L. G ⊢ L0 ⊒[h,f] L →
+ G ⊢ ⬈*[h,T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h,T2] 𝐒⦃L⦄.
+#h #G #L0 #T1 #T2 #H @(cpx_ind … H) -G -L0 -T1 -T2
+[ //
+| //
+| #I0 #G #K0 #V1 #V2 #W2 #_ #IH #HVW2 #g #L #HK0 #HL
+ elim (jsx_inv_pair_sn_gen … HK0) -HK0 *
+ [ #f #K #HK0 #H1 #H2 destruct
+ /4 width=8 by rsx_lifts, rsx_fwd_pair, drops_refl, drops_drop/
+ | #f #K #HV1 #HK0 #H1 #H2 destruct
+ /4 width=8 by rsx_lifts, drops_refl, drops_drop/
+ ]
+| #I0 #G #K0 #T #U #i #_ #IH #HTU #g #L #HK0 #HL
+ elim (jsx_fwd_bind_sn … HK0) -HK0 #I #K #HK0 #H destruct
+ /6 width=8 by rsx_inv_lifts, rsx_lifts, drops_refl, drops_drop/
+| #p #I0 #G #L0 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #f #L #HL0 #HL
+ elim (rsx_inv_bind_void … HL) -HL
+ /4 width=2 by jsx_pair, rsx_bind_void/
+| #I0 #G #L0 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #f #L #HL0 #HL
+ elim (rsx_inv_flat … HL) -HL /3 width=2 by rsx_flat/
+| #G #L0 #V #U1 #T1 #T2 #HTU1 #_ #IHT12 #f #L #HL0 #HL
+ elim (rsx_inv_bind_void … HL) -HL #HV #HU1
+ /5 width=8 by rsx_inv_lifts, drops_refl, drops_drop/
+| #G #L0 #V #T1 #T2 #_ #IHT12 #f #L #HL0 #HL
+ elim (rsx_inv_flat … HL) -HL /2 width=2 by/
+| #G #L0 #V1 #V2 #T #_ #IHV12 #f #L #HL0 #HL
+ elim (rsx_inv_flat … HL) -HL /2 width=2 by/
+| #p #G #L0 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #f #L #HL0 #HL
+ elim (rsx_inv_flat … HL) -HL #HV1 #HL
+ elim (rsx_inv_bind_void … HL) -HL #HW1 #HT1
+ /4 width=2 by jsx_pair, rsx_bind_void, rsx_flat/
+| #p #G #L0 #V1 #V2 #U2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #HVU2 #f #L #HL0 #HL
+ elim (rsx_inv_flat … HL) -HL #HV1 #HL
+ elim (rsx_inv_bind_void … HL) -HL #HW1 #HT1
+ /6 width=8 by jsx_pair, rsx_lifts, rsx_bind_void, rsx_flat, drops_refl, drops_drop/
+]
+qed-.
+
+(* Advanced properties of strongly normalizing referred local environments **)
+
+(* Basic_2A1: uses: lsx_cpx_trans_O *)
+lemma rsx_cpx_trans (h) (G):
+ ∀L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 →
+ G ⊢ ⬈*[h,T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h,T2] 𝐒⦃L⦄.
+/3 width=6 by rsx_cpx_trans_jsx, jsx_refl/ qed-.
+
+lemma rsx_cpxs_trans (h) (G):
+ ∀L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 →
+ G ⊢ ⬈*[h,T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h,T2] 𝐒⦃L⦄.
+#h #G #L #T1 #T2 #H
+@(cpxs_ind_dx ???????? H) -T1 //
+/3 width=3 by rsx_cpx_trans/
+qed-.