+(**************************************************************************)
+(* ___ *)
+(* ||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/cpx_rdeq.ma".
+include "basic_2/rt_computation/cpxs_tdeq.ma".
+
+(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
+
+(* Properties with degree-based equivalence for local environments **********)
+
+(* Basic_2A1: was just: lleq_cpxs_trans *)
+lemma rdeq_cpxs_trans: ∀h,o,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈*[h] T1 →
+ ∀L2. L2 ≛[h, o, T0] L0 →
+ ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈*[h] T & T ≛[h, o] T1.
+#h #o #G #L0 #T0 #T1 #H @(cpxs_ind_dx … H) -T0 /2 width=3 by ex2_intro/
+#T0 #T #HT0 #_ #IH #L2 #HL2
+elim (rdeq_cpx_trans … HL2 … HT0) #U1 #H1 #H2
+elim (IH L2) -IH /2 width=4 by cpx_rdeq_conf_dx/ -L0 #U2 #H3 #H4
+elim (tdeq_cpxs_trans … H2 … H3) -T #U0 #H2 #H3
+/3 width=5 by cpxs_strap2, tdeq_trans, ex2_intro/
+qed-.
+
+(* Basic_2A1: was just: cpxs_lleq_conf *)
+lemma cpxs_rdeq_conf: ∀h,o,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈*[h] T1 →
+ ∀L2. L0 ≛[h, o, T0] L2 →
+ ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈*[h] T & T ≛[h, o] T1.
+/3 width=3 by rdeq_cpxs_trans, rdeq_sym/ qed-.
+
+(* Basic_2A1: was just: cpxs_lleq_conf_dx *)
+lemma cpxs_rdeq_conf_dx: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈*[h] T2 →
+ ∀L1. L1 ≛[h, o, T1] L2 → L1 ≛[h, o, T2] L2.
+#h #o #G #L2 #T1 #T2 #H @(cpxs_ind … H) -T2 /3 width=6 by cpx_rdeq_conf_dx/
+qed-.
+
+(* Basic_2A1: was just: lleq_conf_sn *)
+lemma cpxs_rdeq_conf_sn: ∀h,o,G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ⬈*[h] T2 →
+ ∀L2. L1 ≛[h, o, T1] L2 → L1 ≛[h, o, T2] L2.
+/4 width=6 by cpxs_rdeq_conf_dx, rdeq_sym/ qed-.