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 lt_succ_neq:
forall x y z,
(x <= y < x + Z.succ z)%Z ->
y <> (x + z)%Z ->
(x <= y < x + z)%Z.