(* Properties with strong normalization for unbound rt-transition for terms *)
(* Basic_1: was just: nf2_sn3 *)
-lemma csx_cpre (h) (G) (L):
- ∀T1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ∃T2. ⦃G, L⦄ ⊢ T1 ➡*[h] 𝐍⦃T2⦄.
+lemma cpre_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 (cnr_dec_tdeq h G L T1) [ /3 width=3 by ex_intro, conj/ ] *
+elim (cnr_dec_tdeq h G L T1) [ /3 width=3 by ex_intro, cpme_intro/ ] *
#T0 #HT10 #HnT10
elim (IHT1 … HnT10) -IHT1 -HnT10 [| /2 width=2 by cpm_fwd_cpx/ ]
-#T2 * /4 width=3 by cprs_step_sn, ex_intro, conj/
+#T2 * /4 width=3 by cprs_step_sn, ex_intro, cpme_intro/
qed-.