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_sub_bound:
forall x y,
(x < y)%positive ->
(Z.to_pos (Z.pos_sub y (y - x)) < Pos.succ x)%positive.