EFast2Sum

# Library EFast2Sum

`Require Export Fast2Sum.
Section EFast.
Variable b : Fbound.
Variable precision : nat.
Let radix := 2%Z.
Let radixMoreThanOne : (1 < radix)%Z.
Let radixMoreThanZERO := Zlt_1_O _ (Zlt_le_weak _ _ radixMoreThanOne).
Hint Resolve radixMoreThanZERO radixMoreThanOne: zarith.
Coercion Local FtoRradix := FtoR radix.
Hypothesis precisionGreaterThanOne : 1 < precision.
Hypothesis pGivesBound : Zpos (vNum b) = Zpower_nat radix precision.
Variable Iplus : float -> float -> float.
Hypothesis
IplusCorrect :
forall p q : float,
Fbounded b p -> Fbounded b q -> Closest b radix (p + q) (Iplus p q).
Hypothesis
IplusComp :
forall p q r s : float,
Fbounded b p ->
Fbounded b q ->
Fbounded b r ->
Fbounded b s -> p = r :>R -> q = s :>R -> Iplus p q = Iplus r s :>R.
Hypothesis IplusSym : forall p q : float, Iplus p q = Iplus q p.
Hypothesis
IplusOp : forall p q : float, Fopp (Iplus p q) = Iplus (Fopp p) (Fopp q).
Variable Iminus : float -> float -> float.
Hypothesis IminusPlus : forall p q : float, Iminus p q = Iplus p (Fopp q).
Theorem IminusComp :
forall p q r s : float,
Fbounded b p ->
Fbounded b q ->
Fbounded b r ->
Fbounded b s -> p = r :>R -> q = s :>R -> Iminus p q = Iminus r s :>R.
Theorem EvenBound :
forall (p : float) (m : Z),
Even m ->
(Zpred (Zpower_nat radix precision) <= m)%Z ->
(m <= Zpower_nat radix (S precision) - radix)%Z ->
Fbounded b p -> exists q : float, Fbounded b q /\ q = Float m (Fexp p) :>R.
Theorem ExtMDekker1 :
forall p q : float,
Fbounded b p ->
Fbounded b q ->
(Fexp q <= Fexp p)%Z ->
(0 <= p)%R -> Iminus (Iplus p q) p = (Iplus p q - p)%R :>R.
Theorem ExtMDekker :
forall p q : float,
Fbounded b p ->
Fbounded b q ->
(Fexp q <= Fexp p)%Z -> Iminus (Iplus p q) p = (Iplus p q - p)%R :>R.
Theorem ExtDekker :
forall p q : float,
Fbounded b p ->
Fbounded b q ->
(Fexp q <= Fexp p)%Z ->
Iminus q (Iminus (Iplus p q) p) = (p + q - Iplus p q)%R :>R.
End EFast.
`