+(**************************************************************************)
+(* ___ *)
+(* ||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/static/lfdeq_lfeq.ma".
+include "basic_2/rt_transition/lfpx_lpx.ma".
+include "basic_2/rt_computation/lfsx_lfsx.ma".
+
+(* STRONGLY NORMALIZING LOCAL ENV.S FOR UNCOUNTED PARALLEL RT-TRANSITION ****)
+
+(* Properties with uncounted rt-transition **********************************)
+
+lemma lfsx_intro_lpx: ∀h,o,G,L1,T.
+ (∀L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[h, o, T] L2 → ⊥) → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄) →
+ G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄.
+#h #o #G #L1 #T #HT
+@lfsx_intro #L2 #H
+elim (lfpx_inv_lpx_lfeq … H) -H
+/6 width=3 by lfsx_lfdeq_trans, lfdeq_trans, lfeq_lfdeq/
+qed-.
+
+lemma lfsx_lpx_trans: ∀h,o,G,L1,T. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
+ ∀L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄.
+/3 width=3 by lfsx_lfpx_trans, lfpx_lpx/ qed-.
+
+(* Eliminators with uncounted rt-transition *********************************)
+
+lemma lfsx_ind_lpx: ∀h,o,G,T. ∀R:predicate lenv.
+ (∀L1. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
+ (∀L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[h, o, T] L2 → ⊥) → R L2) →
+ R L1
+ ) →
+ ∀L. G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄ → R L.
+/5 width=6 by lfsx_ind, lfpx_lpx/ qed-.