--- /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_csx.ma".
+include "basic_2/rt_computation/jsx_drops.ma".
+include "basic_2/rt_computation/jsx_lsubr.ma".
+
+(* COMPATIBILITY OF STRONG NORMALIZATION FOR UNBOUND RT-TRANSITION **********)
+
+(* Properties with strongly rt-normalizing terms ****************************)
+
+lemma jsx_csx_conf (h) (G):
+ ∀L1,L2. G ⊢ L1 ⊒[h] L2 →
+ ∀T. ⦃G,L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G,L2⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+/3 width=5 by jsx_fwd_lsubr, csx_lsubr_conf/ qed-.
+
+(* Properties with strongly rt-normalizing referred local environments ******)
+
+(* Note: Try by induction on the 2nd premise by generalizing V with f *)
+lemma rsx_jsx_trans (h) (G):
+ ∀L1,V. G ⊢ ⬈*[h,V] 𝐒⦃L1⦄ →
+ ∀L2. G ⊢ L1 ⊒[h] L2 → G ⊢ ⬈*[h,V] 𝐒⦃L2⦄.
+#h #G #L1 #V @(fqup_wf_ind_eq (Ⓕ) … G L1 V) -G -L1 -V
+#G0 #L0 #V0 #IH #G #L1 * *
+[ //
+| #i #HG #HL #HV #H #L2 #HL12 destruct
+ elim (rsx_inv_lref_drops … H) -H [|*: * ]
+ [ #HL1 -IH
+ lapply (jsx_fwd_drops_atom_sn … HL12 … HL1) -L1
+ /2 width=1 by rsx_lref_atom_drops/
+ | #I #K1 #HLK1 -IH
+ elim (jsx_fwd_drops_unit_sn … HL12 … HLK1) -L1 [| // ] #K2 #HK12 #HLK2
+ /2 width=3 by rsx_lref_unit_drops/
+ | #I #K1 #V1 #HLK1 #HV1 #HK1
+ elim (jsx_fwd_drops_pair_sn … HL12 … HLK1) -HL12 [3: // |*: * ]
+ [ #K2 #HK12 #HLK2
+ /4 width=6 by rsx_lref_pair_drops, jsx_csx_conf, fqup_lref/
+ | #K2 #_ #HLK2 #_
+ /2 width=3 by rsx_lref_unit_drops/
+ ]
+ ]
+| //
+| #p #I #V #T #HG #HL #HV #H #L2 #HL12 destruct
+ elim (rsx_inv_bind_void … H) -H #HV #HT
+ /4 width=4 by jsx_bind, rsx_bind_void/
+| #I #V #T #HG #HL #HV #H #L2 #HL12 destruct
+ elim (rsx_inv_flat … H) -H #HV #HT
+ /3 width=4 by rsx_flat/
+]
+qed-.