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

(* Specifications: *)
Lemma removeAfterIndex_covariant :
forall xs ys i x,
sorted xs ->
sorted ys ->
In x (removeAfterIndex xs i) ->
(forall x, In x xs -> In x ys) ->
In x (removeAfterIndex ys i).