--- /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/grammar/lpx_sn_lpx_sn.ma".
+include "basic_2/substitution/fsupp.ma".
+include "basic_2/substitution/lpss_ldrop.ma".
+
+(* SN PARALLEL SUBSTITUTION FOR LOCAL ENVIRONMENTS **************************)
+
+(* Main properties on context-sensitive parallel substitution for terms *****)
+
+theorem cpss_trans_lpss: lpx_sn_transitive cpss cpss.
+#L0 #T0 @(fsupp_wf_ind … L0 T0) -L0 -T0 #L0 #T0 #IH #L1 * [|*]
+[ #I #HL #HT #T #H1 #L2 #HL12 #T2 #HT2 destruct
+ elim (cpss_inv_atom1 … H1) -H1
+ [ #H destruct
+ elim (cpss_inv_atom1 … HT2) -HT2
+ [ #H destruct //
+ | * #K2 #V #V2 #i #HLK2 #HV2 #HVT2 #H destruct
+ elim (lpss_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
+ elim (lpss_inv_pair2 … H) -H #K1 #V1 #HK12 #HV1 #H destruct
+ lapply (fsupp_lref … HLK1) /3 width=9/
+ ]
+ | * #K1 #V1 #V #i #HLK1 #HV1 #HVT #H destruct
+ elim (lpss_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
+ elim (lpss_inv_pair1 … H) -H #K2 #W2 #HK12 #_ #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
+ elim (cpss_inv_lift1 … HT2 … HLK2 … HVT) -L2 -T
+ lapply (fsupp_lref … HLK1) /3 width=9/
+ ]
+| #a #I #V1 #T1 #HL #HT #X1 #H1 #L2 #HL12 #X2 #H2
+ elim (cpss_inv_bind1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
+ elim (cpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct /4 width=5/
+| #I #V1 #T1 #HL #HT #X1 #H1 #L2 #HL12 #X2 #H2
+ elim (cpss_inv_flat1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
+ elim (cpss_inv_flat1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct /3 width=5/
+]
+qed-.
+
+(* Basic_1: was only: subst1_trans *)
+theorem cpss_trans: ∀L. Transitive … (cpss L).
+/2 width=5 by cpss_trans_lpss/ qed-.
+
+(* Properties on context-sensitive parallel substitution for terms **********)
+
+(* Basic_1: was only: subst1_subst1 *)
+lemma lpss_cpss_trans: ∀L1,L2. L1 ⊢ ▶* L2 →
+ ∀T1,T2. L2 ⊢ T1 ▶* T2 → L1 ⊢ T1 ▶* T2.
+/2 width=5 by cpss_trans_lpss/ qed-.