--- /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/static/lsubss_ssta.ma".
+include "basic_2/reducibility/xpr.ma".
+
+(* EXTENDED PARALLEL REDUCTION ON TERMS *************************************)
+
+(* Properties on lenv ref for stratified type assignment ********************)
+
+lemma lsubss_xpr_trans: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 →
+ ∀T1,T2. ⦃h, L2⦄ ⊢ T1 •➡[g] T2 → ⦃h, L1⦄ ⊢ T1 •➡[g] T2.
+#h #g #L1 #L2 #HL12 #T1 #T2 * [ | #l ] #HT12
+[ lapply (lsubss_fwd_lsubs2 … HL12) -HL12 /3 width=3/
+| /3 width=4/
+]
+qed-.