Proof using.
induction l'; intros; simpl in *; intuition.
- destruct l; simpl in *; auto.
specialize (H1 e); intuition.
- break_match; intuition. subst.
simpl in *. intuition. f_equal.
eapply IHl'; eauto.
intros.
specialize (H1 e0); intuition.
subst. specialize (H0 e0); intuition. omega.
Qed.

(* Specifications: *)
Lemma removeAfterIndex_eq :
forall l i,
(forall e, In e l -> eIndex e <= i) ->
removeAfterIndex l i = l.