(****************************************************************************  IEEE754  :  ClosestMult  Laurent Thery, Sylvie Boldo  ******************************************************************************)
Require Export FroundMult.
Require Export ClosestProp.
Section FRoundP.
Variable b : Fbound.
Variable radix : Z.
Variable precision : nat.
Let FtoRradix := FtoR radix.
Local Coercion FtoRradix : float >-> R.
Hypothesis radixMoreThanOne : (1 < radix)%Z.
Hypothesis precisionGreaterThanOne : 1 < precision.
Hypothesis pGivesBound : Zpos (vNum b) = Zpower_nat radix precision.

(* Specifications: *)
Theorem closestLessMultPos :
forall (p : float) (r : R),
Closest b radix r p -> (0 <= r)%R -> (p <= 2%nat * r)%R.