--- /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/lfpr_lfpr.ma".
+include "basic_2/computation/lfprs_cprs.ma".
+
+(* FOCALIZED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *********************)
+
+(* Advanced properties ******************************************************)
+
+lemma lfprs_strip: ∀L,L1. ⦃L⦄ ➡* ⦃L1⦄ → ∀L2. ⦃L⦄ ➡ ⦃L2⦄ →
+ ∃∃L0. ⦃L1⦄ ➡ ⦃L0⦄ & ⦃L2⦄ ➡* ⦃L0⦄.
+/3 width=3/ qed.
+
+(* Main properties **********************************************************)
+
+theorem lfprs_conf: ∀L,L1. ⦃L⦄ ➡* ⦃L1⦄ → ∀L2. ⦃L⦄ ➡* ⦃L2⦄ →
+ ∃∃L0. ⦃L1⦄ ➡* ⦃L0⦄ & ⦃L2⦄ ➡* ⦃L0⦄.
+/3 width=3/ qed.
+
+theorem lfprs_trans: ∀L1,L. ⦃L1⦄ ➡* ⦃L⦄ → ∀L2. ⦃L⦄ ➡* ⦃L2⦄ → ⦃L1⦄ ➡* ⦃L2⦄.
+/2 width=3/ qed.
+
+lemma lfprs_pair: ∀L1,L2. ⦃L1⦄ ➡* ⦃L2⦄ → ∀V1,V2. L2 ⊢ V1 ➡* V2 →
+ ∀I. ⦃L1. ⓑ{I} V1⦄ ➡* ⦃L2. ⓑ{I} V2⦄.
+#L1 #L2 #H @(lfprs_ind … H) -L2 /2 width=1/
+#L #L2 #_ #HL2 #IHL1 #V1 #V2 #HV12 #I
+@(lfprs_trans … (L.ⓑ{I}V1)) /2 width=1/
+qed.