Section Phi.
(** Variant of [phi] via [recrbis] *)
Let Phi := fun b (_:int31) =>
match b with D0 => Z.double | D1 => Z.succ_double end.
Definition phibis_aux n x := recrbis_aux n _ Z0 Phi x.
(** Recursive equations satisfied by [phi] *)

Lemma phi_twice_firstl : forall x, firstl x = D0 ->
phi (twice x) = Z.double (phi x).