--- /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/rt_transition/cpm_cpx.ma".
+include "basic_2/rt_transition/cnu_tdeq.ma".
+include "basic_2/rt_computation/csx.ma".
+include "basic_2/rt_computation/cpue.ma".
+
+(* EVALUATION FOR T-UNBOUND RT-TRANSITION ON TERMS **************************)
+
+(* Properties with strong normalization for unbound rt-transition for terms *)
+
+lemma cpue_total_csx (h) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ∃T2. ⦃G,L⦄ ⊢ T1 ⥲*[h] 𝐍⦃T2⦄.
+#h #G #L #T1 #H
+@(csx_ind … H) -T1 #T1 #_ #IHT1
+elim (cnu_dec_tdeq h G L T1) [ /3 width=4 by ex2_intro, ex_intro/ ] *
+#n1 #T0 #HT10 #HnT10
+elim (IHT1 … HnT10) -IHT1 -HnT10 [| /2 width=2 by cpm_fwd_cpx/ ]
+#T2 * #n2 #HT02 #HT2 /4 width=7 by cpms_step_sn, ex2_intro, ex_intro/
+qed-.