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diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs_rdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs_rdeq.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/rt_transition/lpx_rdeq.ma".
+include "basic_2/rt_computation/lpxs_lpx.ma".
+
+(* UNBOUND PARALLEL RT-COMPUTATION FOR FULL LOCAL ENVIRONMENTS **************)
+
+(* Properties with degree-based equivalence on referred entries *************)
+
+(* Basic_2A1: uses: lleq_lpxs_trans *)
+lemma rdeq_lpxs_trans (h) (o) (G) (T:term): ∀L2,K2. ⦃G, L2⦄ ⊢ ⬈*[h] K2 →
+                                            ∀L1. L1 ≛[h, o, T] L2 →
+                                            ∃∃K1. ⦃G, L1⦄ ⊢ ⬈*[h] K1 & K1 ≛[h, o, T] K2.
+#h #o #G #T #L2 #K2 #H @(lpxs_ind_sn … H) -L2 /2 width=3 by ex2_intro/
+#L #L2 #HL2 #_ #IH #L1 #HT
+elim (rdeq_lpx_trans … HL2 … HT) -L #L #HL1 #HT
+elim (IH … HT) -L2 #K #HLK #HT
+/3 width=3 by lpxs_step_sn, ex2_intro/
+qed-.
+
+(* Basic_2A1: uses: lpxs_nlleq_inv_step_sn *)
+lemma lpxs_rdneq_inv_step_sn (h) (o) (G) (T:term):
+                             ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[h, o, T] L2 → ⊥) →
+                             ∃∃L,L0. ⦃G, L1⦄ ⊢ ⬈[h] L & L1 ≛[h, o, T] L → ⊥ &
+                                     ⦃G, L⦄ ⊢ ⬈*[h] L0 & L0 ≛[h, o, T] L2.
+#h #o #G #T #L1 #L2 #H @(lpxs_ind_sn … H) -L1
+[ #H elim H -H //
+| #L1 #L #H1 #H2 #IH2 #H12 elim (rdeq_dec h o L1 L T) #H
+  [ -H1 -H2 elim IH2 -IH2 /3 width=3 by rdeq_trans/ -H12
+    #L0 #L3 #H1 #H2 #H3 #H4 lapply (rdeq_rdneq_trans … H … H2) -H2
+    #H2 elim (rdeq_lpx_trans … H1 … H) -L
+    #L #H1 #H lapply (rdneq_rdeq_div … H … H2) -H2
+    #H2 elim (rdeq_lpxs_trans … H3 … H) -L0
+    /3 width=8 by rdeq_trans, ex4_2_intro/
+  | -H12 -IH2 /3 width=6 by ex4_2_intro/
+  ]
+]
+qed-.