(******************)
unfold BV, lenbv, appbv in |- *. simple induction l. intros. inversion H. simpl in |- *. auto.
intros. simpl in |- *. inversion H0. simpl in |- *.
rewrite (H n l1). rewrite mult_plus_distr_r. repeat rewrite plus_assoc.
rewrite
(plus_permute2 (bool_to_nat a + BV_to_nat l0) (power2 l1 * BV_to_nat n)
(BV_to_nat l0)).
auto.
exact H4.
Qed. Hint Resolve BV_to_nat_app.

Lemma BV_to_nat_app2 :
forall l n : BV,
BV_to_nat (appbv l n) = BV_to_nat l + power2 (lengthbv l) * BV_to_nat n.