Definition new_location :=
  if Zeven nb_steps then new_location_even else new_location_odd.

Theorem new_location_correct : 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 k l).