(****************************************************************************  IEEE754  :  FroundPlus  Laurent Thery  ******************************************************************************)
Require Export Finduct.
Require Export FroundProp.
Section FRoundP.
Variable b : Fbound.
Variable radix : Z.
Variable precision : nat.
Let FtoRradix := FtoR radix.
Coercion FtoRradix : float >-> R.
Hypothesis radixMoreThanOne : (1 < radix)%Z.
Let radixMoreThanZERO := Zlt_1_O _ (Zlt_le_weak _ _ radixMoreThanOne).
Hint Resolve radixMoreThanZERO: zarith.
Hypothesis precisionGreaterThanOne : 1 < precision.
Hypothesis pGivesBound : Zpos (vNum b) = Zpower_nat radix precision.

Theorem plusExpMin :
forall P,
RoundedModeP b radix P ->
forall p q pq : float,
P (p + q)%R pq ->
exists s : float,
Fbounded b s /\ s = pq :>R /\ (Zmin (Fexp p) (Fexp q) <= Fexp s)%Z.