(** Computes a new location when the interval containing a real
    number is split into nb_steps subintervals and the real is
    in the k-th one. (Even radix.) *)

Definition new_location_even k l :=
  if Zeq_bool k 0 then
    match l with loc_Exact => l | _ => loc_Inexact Lt end
  else
    loc_Inexact
    match Zcompare (2 * k) nb_steps with
    | Lt => Lt
    | Eq => match l with loc_Exact => Eq | _ => Gt end
    | Gt => Gt
    end.

Theorem new_location_even_correct : Zeven nb_steps = true -> forall x k l, (0 <= k < nb_steps)%Z -> inbetween (start + Z2R k * step) (start + Z2R (k + 1) * step) x l -> inbetween start (start + Z2R nb_steps * step) x (new_location_even k l).