(** Specialization of inbetween for two consecutive integers. *)
Definition inbetween_int m x l :=
inbetween (Z2R m) (Z2R (m + 1)) x l.

Theorem inbetween_float_new_location :
forall x m e l k,
(0 < k)%Z ->
inbetween_float m e x l ->
inbetween_float (Zdiv m (Zpower beta k)) (e + k) x (new_location (Zpower beta k) (Zmod m (Zpower beta k)) l).