intros.
generalize dependent i.
generalize dependent v.
induction xs;
intros.
rewrite Coqlib.nth_error_nil in H.
discriminate.
destruct i; simpl in *;
now auto.
Qed.

Lemma forall_and:
forall {A : Type} (f g : A -> Prop),
(forall x : A, f x /\ g x) <->
(forall x : A, f x) /\ (forall x : A, g x).