simple induction v. simple induction cin; auto with arith.
simple induction cin. rewrite BV_full_adder_nil_true_ok. simpl in |- *. auto with arith.
unfold BV_full_adder in |- *. 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.
(****************************************************************)

Theorem BV_full_adder_ok :
forall (v w : BV) (cin : bool),
BV_to_nat (BV_full_adder v w cin) =
BV_to_nat v + BV_to_nat w + bool_to_nat cin.