simple induction v. simple induction w. auto. intros.
rewrite BV_full_adder_carry_eq2. rewrite BV_full_adder_carry_eq3.
rewrite H; auto. simple induction w. intros. rewrite BV_full_adder_carry_eq2.
rewrite BV_full_adder_carry_eq3.
rewrite H. auto. intros. do 2 rewrite BV_full_adder_carry_eq4.
rewrite H. rewrite full_adder_carry_sym1. auto.
Qed.

Lemma BV_full_adder_sym :
forall (v w : BV) (cin : bool),
BV_full_adder v w cin = BV_full_adder w v cin.