--- /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/lfsx_lfpxs.ma".
+
+(* STRONGLY NORMALIZING LOCAL ENV.S FOR UNCOUNTED PARALLEL RT-TRANSITION ****)
+(*
+(* Advanced properties ******************************************************)
+
+lemma lsx_lref_be_lpxs: ∀h,o,I,G,K1,V,i,l. l ≤ yinj i → ⦃G, K1⦄ ⊢ ⬊*[h, o] V →
+ ∀K2. G ⊢ ⬊*[h, o, V, 0] K2 → ⦃G, K1⦄ ⊢ ➡*[h, o] K2 →
+ ∀L2. ⬇[i] L2 ≡ K2.ⓑ{I}V → G ⊢ ⬊*[h, o, #i, l] L2.
+#h #o #I #G #K1 #V #i #l #Hli #H @(csx_ind_alt … H) -V
+#V0 #_ #IHV0 #K2 #H @(lsx_ind … H) -K2
+#K0 #HK0 #IHK0 #HK10 #L0 #HLK0 @lsx_intro
+#L2 #HL02 #HnL02 elim (lpx_drop_conf … HLK0 … HL02) -HL02
+#Y #H #HLK2 elim (lpx_inv_pair1 … H) -H
+#K2 #V2 #HK02 #HV02 #H destruct
+elim (eq_term_dec V0 V2) #HnV02 destruct [ -IHV0 -HV02 -HK0 | -IHK0 -HnL02 -HLK0 ]
+[ /4 width=8 by lpxs_strap1, lleq_lref/
+| @(IHV0 … HnV02 … HLK2) -IHV0 -HnV02 -HLK2
+ /3 width=4 by lsx_cpx_trans_O, lsx_lpx_trans, lpxs_cpx_trans, lpxs_strap1/ (**) (* full auto too slow *)
+]
+qed.
+
+lemma lsx_lref_be: ∀h,o,I,G,K,V,i,l. l ≤ yinj i → ⦃G, K⦄ ⊢ ⬊*[h, o] V →
+ G ⊢ ⬊*[h, o, V, 0] K →
+ ∀L. ⬇[i] L ≡ K.ⓑ{I}V → G ⊢ ⬊*[h, o, #i, l] L.
+/2 width=8 by lsx_lref_be_lpxs/ qed.
+*)
+(* Main properties **********************************************************)
+
+theorem csx_lsx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄.
+#h #o #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
+ elim (csx_inv_lref … H) -H [ |*: * ]
+ [ #HL
+ | #I #K #HLK
+ | #I #K #V #HLK #HV
+ ]
+(*
+ elim (lt_or_ge i (|L|)) /2 width=1 by lsx_lref_free/
+ elim (ylt_split i l) /2 width=1 by lsx_lref_skip/
+ #Hli #Hi elim (drop_O1_lt (Ⓕ) … Hi) -Hi
+ #I #K #V #HLK lapply (csx_inv_lref_bind … HLK … H) -H
+ /4 width=6 by lsx_lref_be, fqup_lref/
+*)
+| #a #I #V #T #HG #HL #HT #H destruct
+ elim (csx_fwd_bind_unit … H Void) -H /3 width=1 by lfsx_bind_void/
+| #I #V #T #HG #HL #HT #H destruct
+ elim (csx_fwd_flat … H) -H /3 width=1 by lfsx_flat/
+]
+qed.