Proof using.
induction n; intros; simpl in *; auto.
- break_match; try discriminate.
unfold value in *.
find_inversion.
constructor.
- break_match; try discriminate.
subst. constructor.
eauto.
Qed.

Lemma Nth_nth_error :
forall n l (x : A),
Nth l n x ->
nth_error l n = Some x.