unfold nilbv in |- *. simple induction v; auto with arith. unfold BV_full_adder in |- *. intros.
rewrite BV_full_adder_sum_eq3. rewrite BV_full_adder_carry_eq3.
rewrite (app_eq2 bool). rewrite half_adder_carry_true.
simpl in |- *. elim a. unfold appbv in H. rewrite H. simpl in |- *. auto with arith.
rewrite BV_full_adder_sum_v_nil_false.
rewrite BV_full_adder_carry_v_nil_false. rewrite BV_to_nat_app2.
simpl in |- *. elim mult_n_O. elim plus_n_O. trivial with arith.
Qed.

Lemma BV_full_adder_nil_ok :
forall (v : BV) (cin : bool),
BV_to_nat (BV_full_adder v nilbv cin) = BV_to_nat v + bool_to_nat cin.