(****************************************************************)
(* Les registres: *)

Fixpoint R1 (st : nat) : BV :=
  match st return BV with
  | O => V1
  | S t =>
      appbv (highs (R1 t))
        (Mux (lowbit (R1 t))
           (consbv (lowbit (BV_full_adder_sum (R2 t) V2 false)) nilbv)
           (consbv (lowbit (R2 t)) nilbv))
  end
 
 with R2 (st : nat) : BV :=
  match st return BV with
  | O => BV_null size
  | 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))
  end.

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)).