Fexp

# Library Fexp

`Require Export AllFloat.
Require Export List.
Section mf.
Variable b : Fbound.
Variable radix : Z.
Variable precision : nat.
Coercion Local FtoRradix := FtoR radix.
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.
Inductive IsExpansion : list float -> Prop :=
| IsExpansionNil : IsExpansion nil
| IsExpansionSingle :
forall x : float, Fbounded b x -> IsExpansion (x :: nil)
| IsExpansionTop1 :
forall (x : float) (L : list float),
is_Fzero x -> Fbounded b x -> IsExpansion L -> IsExpansion (x :: L)
| IsExpansionTop2 :
forall (x y : float) (L : list float),
Fbounded b x ->
is_Fzero y ->
Fbounded b y -> IsExpansion (x :: L) -> IsExpansion (x :: y :: L)
| IsExpansionTop :
forall (x y : float) (L : list float),
~ is_Fzero x ->
~ is_Fzero y ->
Fbounded b x ->
Fbounded b y ->
(MSB radix x < LSB radix y)%Z ->
IsExpansion (y :: L) -> IsExpansion (x :: y :: L).
Theorem Zlt_MSB_LSB :
forall x y : float,
~ is_Fzero y -> (MSB radix x < LSB radix y)%Z -> (Fabs x < Fabs y)%R.
Theorem IsExpansion_comp1 :
forall (L : list float) (x y : float),
Fbounded b y -> x = y :>R -> IsExpansion (x :: L) -> IsExpansion (y :: L).
Theorem IsExpansion_comp2 :
forall (L : list float) (x y z : float),
Fbounded b z ->
y = z :>R -> IsExpansion (x :: y :: L) -> IsExpansion (x :: z :: L).
Fixpoint expValue (L : list float) : float :=
match L with
| nil => Fzero 0
| x :: L1 => Fplus radix x (expValue L1)
end.
Fixpoint expNormalize (L : list float) : list float :=
match L with
| nil => nil
| x :: L1 =>
match Fnum x with
| Z0 => expNormalize L1
| Zpos p => x :: expNormalize L1
| Zneg p => x :: expNormalize L1
end
end.
Theorem expInv :
forall (a : float) (L : list float), IsExpansion (a :: L) -> IsExpansion L.
Theorem expBoundedInv :
forall (a : float) (L : list float), IsExpansion (a :: L) -> Fbounded b a.
Theorem expNormalizeCorrect :
forall L : list float, expValue L = expValue (expNormalize L) :>R.
Theorem expNormalizeExp :
forall L : list float, IsExpansion L -> IsExpansion (expNormalize L).
Inductive IsNzExpansion : list float -> Prop :=
| IsNzExpansionNil : IsNzExpansion nil
| IsNzExpansionSingle :
forall x : float,
~ is_Fzero x -> Fbounded b x -> IsNzExpansion (x :: nil)
| IsNzExpansionTop :
forall (x y : float) (L : list float),
~ is_Fzero x ->
~ is_Fzero y ->
Fbounded b x ->
Fbounded b y ->
(MSB radix x < LSB radix y)%Z ->
IsNzExpansion (y :: L) -> IsNzExpansion (x :: y :: L).
Theorem isNzExpansionIsExpansion :
forall l : list float, IsNzExpansion l -> IsExpansion l.
Theorem IsNzExpansionInv :
forall (a : float) (l : list float),
IsNzExpansion (a :: l) -> IsNzExpansion l.
Inductive IsNzExpansionR : list float -> Prop :=
| IsNzExpansionNilR : IsNzExpansionR nil
| IsNzExpansionSingleR :
forall x : float,
~ is_Fzero x -> Fbounded b x -> IsNzExpansionR (x :: nil)
| IsNzExpansionTopR :
forall (x y : float) (L : list float),
~ is_Fzero x ->
~ is_Fzero y ->
Fbounded b x ->
Fbounded b y ->
(MSB radix y < LSB radix x)%Z ->
IsNzExpansionR (y :: L) -> IsNzExpansionR (x :: y :: L).
Theorem isNzExpansionRIsNzExpansion :
forall l : list float, IsNzExpansionR l -> IsNzExpansion (rev l).
Theorem IsNzExpansionRInv :
forall (a : float) (l : list float),
IsNzExpansionR (a :: l) -> IsNzExpansionR l.
Theorem mkIsNzAux :
forall (n : nat) (p : float),
(- dExp b <= LSB radix p)%Z ->
(MSB radix p < - dExp b + n)%Z ->
exists l : list float,
IsNzExpansionR l /\
p = expValue l :>R /\
match l with
| a :: _ => ToZeroP b radix p a
| nil => True
end.
Theorem expValueApp :
forall l1 l2 : list float,
expValue (l1 ++ l2) = (expValue l1 + expValue l2)%R :>R.
Theorem expValueRev :
forall l : list float, expValue (rev l) = expValue l :>R.
Theorem existExp :
forall p : float,
(- dExp b <= LSB radix p)%Z ->
exists l : list float, IsExpansion l /\ p = expValue l :>R.
End mf.
`