FboundI

# Library FboundI

`Require Export AllFloat.
Section FboundedI_Def.
Variable radix : Z.
Hypothesis radixMoreThanOne : (1 < radix)%Z.
Let FtoRradix := FtoR radix.
Coercion FtoRradix : float >-> R.
Record FboundI : Set := BoundI {vNumInf : nat; vNumSup : nat; dExpI : nat}.
Definition FboundedI (b : FboundI) (d : float) :=
((- vNumInf b <= Fnum d)%Z /\ (Fnum d <= vNumSup b)%Z) /\
(- dExpI b <= Fexp d)%Z.
Theorem isBoundedI :
forall (b : FboundI) (p : float), {FboundedI b p} + {~ FboundedI b p}.
Theorem FboundedIFzero : forall b : FboundI, FboundedI b (Fzero (- dExpI b)).
Definition FnormalI (b : FboundI) (p : float) :=
FboundedI b p /\
((vNumSup b < radix * Fnum p)%R \/ (radix * Fnum p < (- vNumInf b)%Z)%R).
Theorem FnormalINotZero :
forall (b : FboundI) (p : float), FnormalI b p -> ~ is_Fzero p.
Theorem FnormalIUnique_aux :
forall (b : FboundI) (p q : float),
FnormalI b p ->
FnormalI b q -> p = q :>R -> (Fexp p < Fexp q)%Z -> p = q :>float.
Theorem FnormalIUnique :
forall (b : FboundI) (p q : float),
FnormalI b p -> FnormalI b q -> p = q :>R -> p = q :>float.
Definition FsubnormalI (b : FboundI) (p : float) :=
FboundedI b p /\
Fexp p = (- dExpI b)%Z /\
((- vNumInf b)%Z <= radix * Fnum p)%R /\ (radix * Fnum p <= vNumSup b)%R.
Theorem FsubnormalIUnique :
forall (b : FboundI) (p q : float),
FsubnormalI b p -> FsubnormalI b q -> p = q :>R -> p = q :>float.
Theorem NormalIandSubnormalINotEq :
forall (b : FboundI) (p q : float),
FnormalI b p -> FsubnormalI b q -> p <> q :>R.
Definition FcanonicI (b : FboundI) (a : float) :=
FnormalI b a \/ FsubnormalI b a.
Theorem FcanonicIUnique :
forall (b : FboundI) (p q : float),
FcanonicI b p -> FcanonicI b q -> p = q :>R -> p = q :>float.
Theorem Zpower_nat_S :
forall n : nat, Zpower_nat radix (S n) = (radix * Zpower_nat radix n)%Z.
Fixpoint FNIAux (v N q : nat) {struct q} : nat :=
match q with
| O => 0
| S q' =>
match (Zpower_nat radix q' * v ?= radix * N)%Z with
| Datatypes.Lt => q'
| Datatypes.Eq => q'
| _ => FNIAux v N q'
end
end.
Definition FNI (q N : nat) := pred (FNIAux q N (S (S N))).
Theorem FNIAuxLess :
forall v N q : nat,
0 < v -> v <= N -> (Zpower_nat radix (FNIAux v N q) * v <= radix * N)%Z.
Theorem FNILess :
forall q N : nat, 0 < q -> q <= N -> (Zpower_nat radix (FNI q N) * q <= N)%Z.
Theorem FNIAuxMore :
forall v N q : nat,
FNIAux v N q < pred q -> (N < Zpower_nat radix (FNIAux v N q) * v)%Z.
Theorem Zlt_Zpower_nat : forall n : nat, (n < Zpower_nat radix n)%Z.
Theorem FNIMore :
forall N q : nat,
0 < q -> q <= N -> (N < radix * (Zpower_nat radix (FNI q N) * q))%Z.
Definition FnormalizeI (b : FboundI) (p : float) :=
match (0 ?= Fnum p)%Z with
| Datatypes.Eq => Float 0 (- dExpI b)
| Datatypes.Lt =>
Fshift radix
(min (FNI (Zabs_nat (Fnum p)) (vNumSup b))
(Zabs_nat (Fexp p + dExpI b))) p
| Datatypes.Gt =>
Fshift radix
(min (FNI (Zabs_nat (Fnum p)) (vNumInf b))
(Zabs_nat (Fexp p + dExpI b))) p
end.
Theorem FnormalizeICorrect :
forall (b : FboundI) (p : float), FnormalizeI b p = p :>R.
Theorem FnormalizeIBounded :
forall (b : FboundI) (p : float),
FboundedI b p -> FboundedI b (FnormalizeI b p).
Theorem FcanonicIBoundedI :
forall (b : FboundI) (p : float), FcanonicI b p -> FboundedI b p.
Theorem FnormalizeIFcanonicI :
forall (b : FboundI) (p : float),
FboundedI b p -> FcanonicI b (FnormalizeI b p).
End FboundedI_Def.
`