Proof using.
induction xs; intros; simpl; auto using f_equal.
Qed.

(* Specifications: *)
Lemma deduplicate_log'_snoc_drop_keys :
forall es ks e n,
assoc clientId_eq_dec ks (eClient e) = Some n ->
eId e <= n ->
deduplicate_log' (es ++ [e]) ks = deduplicate_log' es ks.