Record Fbound : Set := Bound {vNum : positive; dExp : N}.Record Fbound : Set := Bound {vNum : positive; dExp : N}.Record Fbound : Set := Bound {vNum : positive; dExp : N}.Record Fbound : Set := Bound {vNum : positive; dExp : N}.Definition Fbounded (b : Fbound) (d : float) :=
  (Zabs (Fnum d) < Zpos (vNum b))%Z /\ (- dExp b <= Fexp d)%Z.

Theorem LessExpBound : forall (b : Fbound) (p q : float), Fbounded b p -> Fbounded b q -> (Fexp q <= Fexp p)%Z -> (0 <= p)%R -> (p <= q)%R -> exists m : Z, Float m (Fexp q) = p :>R /\ Fbounded b (Float m (Fexp q)).