+
+lemma aaa_cpms_total (h) (G) (L) (n) (A):
+ ∀T. ⦃G, L⦄ ⊢ T ⁝ A → ∃U. ⦃G,L⦄ ⊢ T ➡*[n,h] U.
+#h #G #L #n elim n -n
+[ /2 width=3 by ex_intro/
+| #n #IH #A #T1 #HT1 <plus_SO
+ elim (IH … HT1) -IH #T0 #HT10
+ lapply (cpms_aaa_conf … HT1 … HT10) -HT1 #HT0
+ elim (aaa_cpm_SO h … HT0) -HT0 #T2 #HT02
+ /3 width=4 by cpms_step_dx, ex_intro/
+]
+qed-.