simple induction v. simple induction w. auto. intros.
rewrite BV_full_adder_sum_eq2. rewrite BV_full_adder_sum_eq3.
rewrite H. auto. simple induction w. intro.
rewrite BV_full_adder_sum_eq2. rewrite BV_full_adder_sum_eq3. rewrite H.
auto. intros. repeat rewrite BV_full_adder_sum_eq4. rewrite H.
do 2 rewrite full_adder_carry_sym1. do 2 rewrite full_adder_sum_sym1. auto.
Qed.

Lemma length_BV_full_adder_sum :
forall (v w : BV) (cin : bool),
lengthbv v = lengthbv w -> lengthbv (BV_full_adder_sum v w cin) = lengthbv v.