Fbound

# Library Fbound

`Require Export Fop.
Section Fbounded_Def.
Variable radix : Z.
Hypothesis radixMoreThanOne : (1 < radix)%Z.
Let radixMoreThanZERO := Zlt_1_O _ (Zlt_le_weak _ _ radixMoreThanOne).
Hint Resolve radixMoreThanZERO: zarith.
Coercion Local FtoRradix := FtoR radix.
Coercion Z_of_N: N >-> Z.
Record Fbound : Set := Bound {vNum : positive; dExp : N}.
Definition Fbounded (b : Fbound) (d : float) :=
(Zabs (Fnum d) < Zpos (vNum b))%Z /\ (- dExp b <= Fexp d)%Z.
Theorem FboundedNum :
forall (b : Fbound) (p : float),
Fbounded b p -> (Zabs (Fnum p) < Zpos (vNum b))%Z.
Theorem FboundedExp :
forall (b : Fbound) (p : float), Fbounded b p -> (- dExp b <= Fexp p)%Z.
Hint Resolve FboundedNum FboundedExp: float.
Theorem isBounded :
forall (b : Fbound) (p : float), {Fbounded b p} + {~ Fbounded b p}.
Theorem FzeroisZero : forall b : Fbound, Fzero (- dExp b) = 0%R :>R.
Theorem FboundedFzero : forall b : Fbound, Fbounded b (Fzero (- dExp b)).
Hint Unfold Fbounded.
Theorem FboundedZeroSameExp :
forall (b : Fbound) (p : float), Fbounded b p -> Fbounded b (Fzero (Fexp p)).
Theorem FBoundedScale :
forall (b : Fbound) (p : float) (n : nat),
Fbounded b p -> Fbounded b (Float (Fnum p) (Fexp p + n)).
Theorem FvalScale :
forall (b : Fbound) (p : float) (n : nat),
Float (Fnum p) (Fexp p + n) = (powerRZ radix n * p)%R :>R.
Theorem oppBounded :
forall (b : Fbound) (x : float), Fbounded b x -> Fbounded b (Fopp x).
Theorem oppBoundedInv :
forall (b : Fbound) (x : float), Fbounded b (Fopp x) -> Fbounded b x.
Theorem FopRepAux :
forall (b : Fbound) (z : Z) (p : R),
ex (fun r : float => r = (- p)%R :>R /\ Fbounded b r /\ Fexp r = z) ->
ex (fun r : float => r = p :>R /\ Fbounded b r /\ Fexp r = z).
Theorem absFBounded :
forall (b : Fbound) (f : float), Fbounded b f -> Fbounded b (Fabs f).
Theorem FboundedEqExpPos :
forall (b : Fbound) (p q : float),
Fbounded b p ->
p = q :>R -> (Fexp p <= Fexp q)%R -> (0 <= q)%R -> Fbounded b q.
Theorem FboundedEqExp :
forall (b : Fbound) (p q : float),
Fbounded b p -> p = q :>R -> (Fexp p <= Fexp q)%R -> Fbounded b q.
Theorem eqExpLess :
forall (b : Fbound) (p q : float),
Fbounded b p ->
p = q :>R ->
exists r : float, Fbounded b r /\ r = q :>R /\ (Fexp q <= Fexp r)%R.
Theorem FboundedShiftLess :
forall (b : Fbound) (f : float) (n m : nat),
m <= n -> Fbounded b (Fshift radix n f) -> Fbounded b (Fshift radix m f).
Theorem eqExpMax :
forall (b : Fbound) (p q : float),
Fbounded b p ->
Fbounded b q ->
(Fabs p <= q)%R ->
exists r : float, Fbounded b r /\ r = p :>R /\ (Fexp r <= Fexp q)%Z.
Theorem Zle_monotony_contra_abs_pow :
forall x y z n : Z,
(0 < z)%Z ->
(Rabs (x * powerRZ z n) <= Rabs (y * powerRZ z n))%R -> (Zabs x <= Zabs y)%Z.
Theorem LessExpBound :
forall (b : Fbound) (p q : float),
Fbounded b p ->
Fbounded b q ->
(Fexp q <= Fexp p)%Z ->
(0 <= p)%R ->
(p <= q)%R ->
exists m : Z,
Float m (Fexp q) = p :>R /\ Fbounded b (Float m (Fexp q)).
Theorem maxFbounded :
forall (b : Fbound) (z : Z),
(- dExp b <= z)%Z -> Fbounded b (Float (Zpred (Zpos (vNum b))) z).
Theorem maxMax :
forall (b : Fbound) (p : float) (z : Z),
Fbounded b p ->
(Fexp p <= z)%Z -> (Fabs p < Float (Zpos (vNum b)) z)%R.
End Fbounded_Def.
Hint Resolve FboundedFzero oppBounded absFBounded maxFbounded FboundedNum
FboundedExp: float.
`