-lemma cpds_fwd_cpxs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
-#h #g #G #L #T1 #T2 * /3 width=5 by cpxs_trans, lstas_cpxs, cprs_cpxs/
+lemma cpds_fwd_cpxs: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •*➡*[h, g, l] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
+#h #g #G #L #T1 #T2 #l * /3 width=5 by cpxs_trans, lstas_cpxs, cprs_cpxs/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem cpds_conf_eq: ∀h,g,G,L,T0,T1,l. ⦃G, L⦄ ⊢ T0 •*➡*[h, g, l] T1 →
+ ∀T2. ⦃G, L⦄ ⊢ T0 •*➡*[h, g, l] T2 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ➡* T & ⦃G, L⦄ ⊢ T2 ➡* T.
+#h #g #G #L #T0 #T1 #l0 * #U1 #l1 #_ #_ #H1 #HUT1 #T2 * #U2 #l2 #_ #_ #H2 #HUT2 -l1 -l2
+lapply (lstas_mono … H1 … H2) #H destruct -h -l0 /2 width=3 by cprs_conf/