+(**************************************************************************)
+(* ___ *)
+(* ||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_computation/cprs_cnr.ma".
+include "basic_2/rt_computation/cprs_cprs.ma".
+include "basic_2/rt_computation/cpre.ma".
+
+(* EVALUATION FOR CONTEXT-SENSITIVE PARALLEL R-TRANSITION ON TERMS *********)
+
+(* Properties with context-sensitive parallel r-computation for terms ******)
+
+lemma cpre_cprs_conf (h) (G) (L) (T):
+ ∀T1. ⦃G,L⦄ ⊢ T ➡*[h] T1 → ∀T2. ⦃G,L⦄ ⊢ T ➡*[h] 𝐍⦃T2⦄ → ⦃G,L⦄ ⊢ T1 ➡*[h] 𝐍⦃T2⦄.
+#h #G #L #T0 #T1 #HT01 #T2 * #HT02 #HT2
+elim (cprs_conf … HT01 … HT02) -T0 #T0 #HT10 #HT20
+lapply (cprs_inv_cnr_sn … HT20 HT2) -HT20 #H destruct
+/2 width=1 by cpme_intro/
+qed-.
+
+(* Main properties *********************************************************)
+
+(* Basic_1: was: nf2_pr3_confluence *)
+theorem cpre_mono (h) (G) (L) (T):
+ ∀T1. ⦃G,L⦄ ⊢ T ➡*[h] 𝐍⦃T1⦄ → ∀T2. ⦃G,L⦄ ⊢ T ➡*[h] 𝐍⦃T2⦄ → T1 = T2.
+#h #G #L #T0 #T1 * #HT01 #HT1 #T2 * #HT02 #HT2
+elim (cprs_conf … HT01 … HT02) -T0 #T0 #HT10 #HT20
+>(cprs_inv_cnr_sn … HT10 HT1) -T1
+>(cprs_inv_cnr_sn … HT20 HT2) -T2 //
+qed-.