--- /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/unfold/lcpss_cpss.ma".
+
+(* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
+
+(* Main properties **********************************************************)
+
+theorem lcpss_conf: confluent … lcpss.
+#L0 @(f_ind … length … L0) -L0 #n #IH *
+[ #_ #X1 #H1 #X2 #H2 -n
+ >(lcpss_inv_atom1 … H1) -X1
+ >(lcpss_inv_atom1 … H2) -X2 /2 width=3/
+| #L0 #I #V0 #Hn #X1 #H1 #X2 #H2 destruct
+ elim (lcpss_inv_pair1 … H1) -H1 #L1 #V1 #HL01 #HV01 #H destruct
+ elim (lcpss_inv_pair1 … H2) -H2 #L2 #V2 #HL02 #HV02 #H destruct
+ elim (IH … HL01 … HL02) -IH normalize // #L #HL1 #HL2
+ elim (cpss_conf_lcpss … HV01 … HV02 … HL01 … HL02) -L0 -V0 /3 width=5/
+]
+qed-.
+
+theorem lcpss_trans: Transitive … lcpss.
+#L1 #L #H elim H -L1 -L //
+#I #L1 #L #V1 #V #HL1 #HV1 #IHL1 #X #H
+elim (lcpss_inv_pair1 … H) -H #L2 #V2 #HL2 #HV2 #H destruct
+lapply (cpss_trans_lcpss … HV1 … HL1 … HV2) -V -HL1 /3 width=1/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma cpss_fwd_shift1: ∀L1,L,T1,T. L ⊢ L1 @@ T1 ▶* T →
+ ∃∃L2,T2. L @@ L1 ⊢ ▶* L @@ L2 & L @@ L1 ⊢ T1 ▶* T2 &
+ T = L2 @@ T2.
+#L1 @(lenv_ind_dx … L1) -L1
+[ #L #T1 #T #HT1
+ @ex3_2_intro [3: // |4,5: // |1,2: skip ] (**) (* /2 width=4/ does not work *)
+| #I #L1 #V1 #IH #L #T1 #T >shift_append_assoc #H <append_assoc
+ elim (cpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ elim (IH … HT12) -IH -HT12 #L2 #T #HL12 #HT1 #H destruct
+ lapply (lcpss_trans … HL12 (L.ⓑ{I}V2@@L2) ?) -HL12 /3 width=1/ #HL12
+ @(ex3_2_intro … (⋆.ⓑ{I}V2@@L2)) [4: /2 width=3/ | skip ] <append_assoc // (**) (* explicit constructor *)
+]
+qed-.