--- /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_2A/notation/relations/predeval_4.ma".
+include "basic_2A/computation/cprs.ma".
+include "basic_2A/computation/csx.ma".
+
+(* EVALUATION FOR CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS *************)
+
+definition cpre: relation4 genv lenv term term ≝
+ λG,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 ∧ ⦃G, L⦄ ⊢ ➡ 𝐍⦃T2⦄.
+
+interpretation "evaluation for context-sensitive parallel reduction (term)"
+ 'PRedEval G L T1 T2 = (cpre G L T1 T2).
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was just: nf2_sn3 *)
+lemma csx_cpre: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 → ∃T2. ⦃G, L⦄ ⊢ T1 ➡* 𝐍⦃T2⦄.
+#h #g #G #L #T1 #H @(csx_ind … H) -T1
+#T1 #_ #IHT1 elim (cnr_dec G L T1) /3 width=3 by ex_intro, conj/
+* #T #H1T1 #H2T1 elim (IHT1 … H2T1) -IHT1 -H2T1 /2 width=2 by cpr_cpx/
+#T2 * /4 width=3 by cprs_strap2, ex_intro, conj/
+qed.