+lemma drop_inv_O1_aux: ∀d,e,L2,L1. ↑[d, e] L2 ≡ L1 → d = 0 →
+ ∀K,I,V. L1 = K. 𝕓{I} V →
+ (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
+ (0 < e ∧ ↑[d, e - 1] L2 ≡ K).
+#d #e #L2 #L1 #H elim H -H d e L2 L1
+[ /3/
+| #L1 #L2 #I #V #e #HL12 #_ #_ #K #J #W #H destruct -L1 I V /3/
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #_ #H elim (plus_S_eq_O_false … H)
+]
+qed.
+
+lemma drop_inv_O1: ∀e,L2,K,I,V. ↑[0, e] L2 ≡ K. 𝕓{I} V →
+ (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
+ (0 < e ∧ ↑[0, e - 1] L2 ≡ K).
+/2/ qed.
+
+lemma drop_inv_drop1: ∀e,L2,K,I,V.
+ ↑[0, e] L2 ≡ K. 𝕓{I} V → 0 < e → ↑[0, e - 1] L2 ≡ K.
+#e #L2 #K #I #V #H #He
+elim (drop_inv_O1 … H) -H * // #H destruct -e;
+elim (lt_refl_false … He)
+qed.
+