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_eqn2 : forall x, firstr x = D1 ->
phi x = Z.succ_double (phi (shiftr x)).