Fixpoint Pos_is_pow2 (p : positive) : bool :=
  match p with
  | (p'~1)%positive => false
  | (p'~0)%positive => Pos_is_pow2 p'
  | 1%positive      => true
  end.(** ** Power of two closed upper bound [q <= 2^z] *)

Definition Qbound_le_ZExp2 (q : Q) : Z :=
  match Qnum q with
  (* The -1000 is a compromise between a tight Archimedian and avoiding too large numbers *)
  | Z0 => -1000
  | Zneg p => 0
  | Zpos p => (Z.pos (Pos_log2ceil_plus1 p) - Z.pos (Pos_log2floor_plus1 (Qden q)))%Z
  end.

(* Specifications: *)
Lemma Qbound_le_ZExp2_spec : forall (q : Q), (q <= 2^(Qbound_le_ZExp2 q))%Q.