auto with arith. Qed.

Lemma R2_eq2 :
forall t : nat,
R2 (S t) =
appbv
(highs (Mux (lowbit (R1 t)) (BV_full_adder_sum (R2 t) V2 false) (R2 t)))
(Mux (lowbit (R1 t)) (consbv (BV_full_adder_carry (R2 t) V2 false) nilbv)
(consbv false nilbv)).