Notation bpow e := (bpow beta e).(** Specialization of inbetween for two consecutive floating-point numbers. *)

Definition inbetween_float m e x l :=
  inbetween (F2R (Float beta m e)) (F2R (Float beta (m + 1) e)) x l.

Theorem inbetween_float_bounds : forall x m e l, inbetween_float m e x l -> (F2R (Float beta m e) <= x < F2R (Float beta (m + 1) e))%R.