--- /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_computation/csx.ma".
+include "basic_2/rt_computation/cnuw_tdeq.ma".
+include "basic_2/rt_computation/cpmuwe.ma".
+
+(* T-UNBOUND WHD EVALUATION FOR T-BOUND RT-TRANSITION ON TERMS **************)
+
+(* Properties with strong normalization for unbound rt-transition for terms *)
+
+lemma cpmuwe_total_csx (h) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ∃∃T2,n. ⦃G,L⦄ ⊢ T1 ➡*𝐍𝐖*[h,n] T2.
+#h #G #L #T1 #H
+@(csx_ind … H) -T1 #T1 #_ #IHT1
+elim (cnuw_dec_tdeq h G L T1)
+[ -IHT1 #HT1 /3 width=4 by cpmuwe_intro, ex1_2_intro/
+| * #n1 #T0 #HT10 #HnT10
+ elim (IHT1 … HnT10) -IHT1 -HnT10 [| /2 width=2 by cpm_fwd_cpx/ ]
+ #T2 #n2 * #HT02 #HT2 /4 width=5 by cpms_step_sn, cpmuwe_intro, ex1_2_intro/
+]
+qed-.
+
+lemma R_cpmuwe_total_csx (h) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ∃n. R_cpmuwe h G L T1 n.
+#h #G #L #T1 #H
+elim (cpmuwe_total_csx … H) -H #T2 #n #HT12
+/3 width=3 by ex_intro (* 2x *)/
+qed-.