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_lt_sub:
forall a b c,
(a < b)%positive ->
(b <= c)%positive ->
(b - a < c)%positive.