unfold nilbv in |- *.
simple induction v. trivial. intros.
rewrite BV_full_adder_carry_eq3. rewrite half_adder_carry_false.
trivial.
Qed. Hint Resolve BV_full_adder_carry_v_nil_false.

Lemma BV_full_adder_sum_sym :
forall (v w : BV) (cin : bool),
BV_full_adder_sum v w cin = BV_full_adder_sum w v cin.