Proof using.
induction xs; intros.
- intuition.
- simpl in *. break_if; simpl in *; intuition.
subst. do_bool. omega.
Qed.

(* Specifications: *)
Lemma removeAfterIndex_In_le :
forall xs i x,
sorted xs ->
In x (removeAfterIndex xs i) ->
eIndex x <= i.