-lemma ldrop_O1: ∀L,i. i < |L| → ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V.
+lemma ldrop_O1_le: ∀i,L. i ≤ |L| → ∃K. ⇩[0, i] L ≡ K.
+#i @(nat_ind_plus … i) -i /2 width=2/
+#i #IHi *
+[ #H lapply (le_n_O_to_eq … H) -H >commutative_plus normalize #H destruct
+| #L #I #V normalize #H
+ elim (IHi L ?) -IHi /2 width=1/ -H /3 width=2/
+]
+qed.
+
+lemma ldrop_O1_lt: ∀L,i. i < |L| → ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V.