-lemma snv_fwd_csn: ∀h,g,L,T. ⦃h, L⦄ ⊩ T :[g] → L ⊢ ⬊* T.
-#h #g #L #T #H elim (snv_fwd_aaa … H) -H /2 width=2/
+(* Advanced forward lemmas **************************************************)
+
+lemma snv_fwd_da: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, o] → ∃d. ⦃G, L⦄ ⊢ T ▪[h, o] d.
+#h #o #G #L #T #H elim (snv_fwd_aaa … H) -H /2 width=2 by aaa_da/
+qed-.
+
+lemma snv_fwd_lstas: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, o] →
+ ∀d. ∃U. ⦃G, L⦄ ⊢ T •*[h, d] U.
+#h #o #G #L #T #H #d elim (snv_fwd_aaa … H) -H
+#A #HT elim (aaa_lstas h … HT d) -HT /2 width=2 by ex_intro/