--- /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/reducibility/tpr_tpss.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
+
+(* Properties concerning parallel unfold on local environments **************)
+
+lemma ltpr_ltps_conf: ∀L1,K1,d,e. L1 [d, e] ▶ K1 → ∀L2. L1 ➡ L2 →
+ ∃∃K2. L2 [d, e] ▶* K2 & K1 ➡ K2.
+#L1 #K1 #d #e #H elim H -L1 -K1 -d -e
+[ /2 width=3/
+| #L1 #I #V1 #X #H
+ elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct /3 width=5/
+| #L1 #K1 #I #V1 #W1 #e #_ #HVW1 #IHLK1 #X #H
+ elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
+ elim (IHLK1 … HL12) -L1 #K2 #HLK2 #HK12
+ elim (tpr_tps_ltpr … HV12 … HVW1 … HK12) -V1 /3 width=5/
+| #L1 #K1 #I #V1 #W1 #d #e #_ #HVW1 #IHLK1 #X #H
+ elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
+ elim (IHLK1 … HL12) -L1 #K2 #HLK2 #HK12
+ elim (tpr_tps_ltpr … HV12 … HVW1 … HK12) -V1 /3 width=5/
+]
+qed.
+
+lemma ltpr_ltpss_conf: ∀L1,K1,d,e. L1 [d, e] ▶* K1 → ∀L2. L1 ➡ L2 →
+ ∃∃K2. L2 [d, e] ▶* K2 & K1 ➡ K2.
+#L1 #K1 #d #e #H @(ltpss_ind … H) -K1 /2 width=3/
+#K #K1 #_ #HK1 #IHK #L2 #HL12
+elim (IHK … HL12) -L1 #K2 #HLK2 #HK2
+elim (ltpr_ltps_conf … HK1 … HK2) -K #K #HK2 #HK1
+lapply (ltpss_trans_eq … HLK2 HK2) -K2 /2 width=3/
+qed.