--- /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/csx_lsubr.ma".
+include "basic_2/rt_computation/csx_cpxs.ma".
+include "basic_2/rt_computation/sdsx_rdsx.ma".
+
+(* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
+
+(* Advanced properties ******************************************************)
+
+lemma rdsx_pair_lpxs (h) (I) (G):
+ ∀K1,V1. G ⊢ ⬈*[h,V1] 𝐒⦃K1⦄ →
+ ∀V2. ⦃G,K1⦄ ⊢ ⬈*[h] 𝐒⦃V2⦄ →
+ (∀K. ⦃G,K1⦄ ⊢ ⬈*[h] K → ⦃G,K⦄ ⊢ V1 ⬈*[h] V2) →
+ G ⊢ ⬈*[h,#0] 𝐒⦃K1.ⓑ{I}V2⦄.
+#h #I #G #K1 #V1 #H
+@(rdsx_ind_lpxs … H) -K1 #K1 #_ #IHK #V2 #H
+@(csx_ind_cpxs … H) -V2 #V2 #HV2 #IHV #HK
+@rdsx_intro_lpxs #Y #HY #HnY
+elim (lpxs_inv_pair_sn … HY) -HY #K3 #V3 #HK13 #HV23 #H destruct
+elim (tdeq_dec V2 V3)
+[ -IHV -HV23 #HV23
+ @(rdsx_rdeq_trans … (K3.ⓑ{I}V2)) [| /2 width=1 by rdeq_pair_refl/ ]
+ @(IHK … HK13) -IHK
+ [
+ |
+ | /3 width=3 by lpxs_trans/
+ ]
+| -IHK -HnY #HnV23
+ @(rdsx_lpxs_trans … (K1.ⓑ{I}V3)) [| /2 width=1 by lpxs_bind_refl_dx/ ]
+ @(IHV … HV23 HnV23) -IHV -HnV23
+ #K #HK