FexpAdd

# Library FexpAdd

`Require Export Fexp2.
Section FexpAdd.
Variable b : Fbound.
Variable precision : nat.
Let radix := 2%Z.
Let TMTO : (1 < radix)%Z := TwoMoreThanOne.
Hint Resolve TMTO: zarith.
Let FtoRradix := FtoR radix.
Coercion FtoRradix : float >-> R.
Hypothesis precisionGreaterThanOne : 1 < precision.
Hypothesis pGivesBound : Zpos (vNum b) = Zpower_nat radix precision.
Hypothesis Ngd : (1 <= pPred (vNum b) * (1 - / radix))%R.
Hypothesis Ngd2 : (6%nat <= pPred (vNum b) * (1 - / radix * / radix))%R.
Inductive NearEqual : list float -> list float -> Prop :=
| IsEqual : forall x : list float, NearEqual x x
| OneMoreR :
forall (x : list float) (e : float),
Fbounded b e -> NearEqual x (e :: x).
Hint Resolve IsEqual OneMoreR.
Definition Step (x y i x' y' : float) (input output output' : list float) :=
Fbounded b x' /\
Fbounded b y' /\
NearEqual output output' /\
(x + (y + (sum (i :: input) + sum output)))%R =
(x' + (y' + (sum input + sum output')))%R :>R /\
(Fexp i <= Fexp y')%Z /\
(Fexp y' <= Fexp x')%Z /\
IsExp b (i :: input) /\
(x' = 0%R :>R \/
y' = 0%R :>R \/
(Rabs y' <=
pPred (vNum b) * (Float 1%nat (Fexp x') - Float 1%nat (hdexp b input)))%R /\
(Rabs y' <=
pPred (vNum b) * (Float 1%nat (Fexp x') - Float 1%nat (Fexp i)))%R) /\
(output' = output \/
(Rabs (x' + y') <= 3%nat * radix * / pPred (vNum b) * Rabs (hd b output'))%R) /\
(output' = nil \/
(Rabs (hd b input) <=
3%nat * (radix * (Rabs (hd b output') * / (pPred (vNum b) - 1%nat))))%R /\
(input = nil \/
(Float (pPred (vNum b)) (Fexp (hd b input)) <=
3%nat * (radix * (Rabs (hd b output') * / (pPred (vNum b) - 1%nat))))%R /\
(Rabs (sum input) <=
length input *
(3%nat * (radix * (Rabs (hd b output') * / (pPred (vNum b) - 1%nat)))))%R)).
Theorem Rle_mult_pos :
forall r1 r2 : R, (0 <= r1)%R -> (0 <= r2)%R -> (0 <= r1 * r2)%R.
Theorem AddStep :
forall (x y i : float) (input output : list float),
Fbounded b x ->
Fbounded b y ->
IsExp b (i :: input) ->
(Fexp i <= Fexp y)%Z ->
(Fexp y <= Fexp x)%Z ->
(FtoR radix x = 0%R :>R \/ FtoR radix y = 0%R :>R \/ FtoR radix i = 0%R :>R) \/
(Rabs (FtoR radix y) <=
pPred (vNum b) *
(FtoR radix (Float 1%nat (Fexp x)) - FtoR radix (Float 1%nat (Fexp i))))%R ->
output = nil \/
(Rabs i <=
3%nat * (radix * (Rabs (hd b output) * / (pPred (vNum b) - 1%nat))))%R /\
(Float (pPred (vNum b)) (Fexp i) <=
3%nat * (radix * (Rabs (hd b output) * / (pPred (vNum b) - 1%nat))))%R /\
(Rabs (sum (i :: input)) <=
length (i :: input) *
(3%nat * (radix * (Rabs (hd b output) * / (pPred (vNum b) - 1%nat)))))%R ->
exists x' : float,
(exists y' : float,
(exists output' : list float, Step x y i x' y' input output output')).
Variable input : list float.
Inductive IsRleEpsExp : list float -> Prop :=
| IsRleEpsExpNil : IsRleEpsExp nil
| IsRleEpsExpSingle :
forall x : float, Fbounded b x -> IsRleEpsExp (x :: nil)
| IsRleEpsExpTop :
forall (x y : float) (L : list float),
Fbounded b x ->
Fbounded b y ->
(Rabs x <=
(6%nat * length input + 6%nat) *
/ (pPred (vNum b) - 1%nat - 6%nat * length input) *
Rabs y)%R -> IsRleEpsExp (y :: L) -> IsRleEpsExp (x :: y :: L).
Theorem endof_Rle_length :
forall (P Q : list float) (k : float),
endof input (P ++ k :: Q) -> (length P <= length input - 1)%R.
Theorem FexpAdd_aux :
(6%nat * length input * / (pPred (vNum b) - 1%nat) < 1)%R ->
forall (L output : list float) (x y : float),
endof input L ->
IsExp b L ->
IsRleEpsExp output ->
output = nil \/
(Rabs (x + y) <=
(pPred (vNum b) - 1%nat) * / pPred (vNum b) *
((6%nat * length input + 6%nat) *
/ (pPred (vNum b) - 1%nat - 6%nat * length input)) *
Rabs (hd b output))%R ->
output = nil \/
(exists L1 : list float,
(exists x1 : float,
(exists y1 : float,
IsExp b (L1 ++ L) /\
endof input (L1 ++ L) /\
(x1 + (y1 + sum L1))%R = (x + y)%R :>R /\
(Rabs (sum L1) <=
length L1 *
(3%nat *
(radix * (Rabs (hd b output) * / (pPred (vNum b) - 1%nat)))))%R /\
(Rabs (x1 + y1) <=
3%nat * radix * / pPred (vNum b) * Rabs (hd b output))%R))) ->
Fbounded b x ->
Fbounded b y ->
(hdexp b L <= Fexp y)%Z ->
(Fexp y <= Fexp x)%Z ->
(FtoR radix x = 0%R :>R \/ FtoR radix y = 0%R :>R) \/
(Rabs (FtoR radix y) <=
pPred (vNum b) *
(FtoR radix (Float 1%nat (Fexp x)) - FtoR radix (Float 1%nat (hdexp b L))))%R ->
output = nil \/
L = nil \/
(Rabs (hd b L) <=
3%nat * (radix * (Rabs (hd b output) * / (pPred (vNum b) - 1%nat))))%R /\
(Float (pPred (vNum b)) (Fexp (hd b L)) <=
3%nat * (radix * (Rabs (hd b output) * / (pPred (vNum b) - 1%nat))))%R /\
(Rabs (sum L) <=
length L *
(3%nat * (radix * (Rabs (hd b output) * / (pPred (vNum b) - 1%nat)))))%R ->
exists x' : float,
(exists y' : float,
(exists output' : list float,
Fbounded b x' /\
Fbounded b y' /\
(x + (y + (sum output + sum L)))%R = (x' + (y' + sum output'))%R :>R /\
length output' <= length L + length output /\
(Fexp y' <= Fexp x')%Z /\
IsRleEpsExp output' /\
(output = nil \/
(exists L1 : list float,
(exists x1 : float,
(exists y1 : float,
IsExp b (L1 ++ L) /\
endof input (L1 ++ L) /\
(x1 + (y1 + sum L1))%R = (x + y)%R :>R /\
(Rabs (sum L1) <=
length L1 *
(3%nat *
(radix *
(Rabs (hd b output) * / (pPred (vNum b) - 1%nat)))))%R /\
(Rabs (x1 + y1) <=
3%nat * radix * / pPred (vNum b) * Rabs (hd b output))%R)))) /\
endof output' output /\
(output' = nil \/
(Rabs (x' + y') <=
(pPred (vNum b) - 1%nat) * / pPred (vNum b) *
((6%nat * length input + 6%nat) *
/ (pPred (vNum b) - 1%nat - 6%nat * length input)) *
Rabs (hd b output'))%R))).
Theorem FexpAdd_aux2 :
forall L : list float,
L = input ->
IsExp b input ->
(6%nat * length input * / (pPred (vNum b) - 1%nat) < 1)%R ->
exists output : list float,
(input <> nil -> length output <= S (length input)) /\
sum input = sum output :>R /\ IsRleEpsExp output.
Theorem FexpAdd_main :
IsExp b input ->
(6%nat * length input * / (pPred (vNum b) - 1%nat) < 1)%R ->
exists output : list float,
(input <> nil -> length output <= S (length input)) /\
sum input = sum output :>R /\ IsRleEpsExp output.
End FexpAdd.
`