--- /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/substitution/lleq_ldrop.ma".
+include "basic_2/computation/llsx.ma".
+
+(* LAZY SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS *****************)
+
+(* Advanced properties ******************************************************)
+
+lemma llsx_lref_free: ∀h,g,G,L,d,i. |L| ≤ i → G ⊢ ⋕⬊*[h, g, #i, d] L.
+#h #g #G #L1 #d #i #HL1 @llsx_intro
+#L2 #HL12 #H elim H -H
+/4 width=8 by llpx_fwd_length, lleq_free, le_repl_sn_conf_aux/
+qed.
+
+lemma llsx_lref_skip: ∀h,g,G,L,d,i. yinj i < d → G ⊢ ⋕⬊*[h, g, #i, d] L.
+#h #g #G #L1 #d #i #HL1 @llsx_intro
+#L2 #HL12 #H elim H -H
+/3 width=6 by llpx_fwd_length, lleq_skip/
+qed.