Proof using.
induction xs; intros.
- simpl in *. intuition.
- simpl in *.
break_match; simpl in *; intuition;
subst; do_bool;
match goal with
| e : entry, H : forall _, _ = _ \/ _ -> _ |- _ =>
specialize (H e)
end;
intuition.
+ eauto using removeAfterIndex_le_In.
+ find_apply_hyp_hyp. intuition.
match goal with
| _ : eIndex ?e <= ?li, _ : eIndex ?e > eIndex ?e' |- _ =>
assert (eIndex e' <= li) by omega
end.
eauto using removeAfterIndex_le_In.
Qed.

(* Specifications: *)
Lemma removeAfterIndex_le :
forall xs i j,
i <= j ->
removeAfterIndex xs i = removeAfterIndex (removeAfterIndex xs j) i.