--- /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/ssta_lpss.ma".
+include "basic_2/unfold/sstas.ma".
+
+(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS *********************)
+
+(* Properties about sn parallel substitution for local environments *********)
+
+lemma sstas_cpss_lpss_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
+ ∀T2. L1 ⊢ T1 ▶* T2 → ∀L2. L1 ⊢ ▶* L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 & L1 ⊢ U1 ▶* U2.
+#h #g #L1 #T1 #U1 #H @(sstas_ind_dx … H) -T1 /2 width=3/
+#T0 #U0 #l0 #HTU0 #_ #IHU01 #T #HT0 #L2 #HL12
+elim (ssta_cpss_lpss_conf … HTU0 … HT0 … HL12) -HTU0 -HT0 #U #HTU #HU0
+elim (IHU01 … HU0 … HL12) -IHU01 -U0 -HL12 /3 width=4/
+qed-.
+
+lemma sstas_cpss_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*[g] U1 →
+ ∀T2. L ⊢ T1 ▶* T2 →
+ ∃∃U2. ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U1 ▶* U2.
+/2 width=3 by sstas_cpss_lpss_conf/ qed-.
+
+lemma sstas_lpss_conf: ∀h,g,L1,T,U1. ⦃h, L1⦄ ⊢ T •*[g] U1 →
+ ∀L2. L1 ⊢ ▶* L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T •*[g] U2 & L1 ⊢ U1 ▶* U2.
+/2 width=3 by sstas_cpss_lpss_conf/ qed-.