Section FRoundP.
Variable b : Fbound.
Variable radix : Z.
Variable precision : nat.
Let FtoRradix := FtoR radix.
Coercion FtoRradix : float >-> R.
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.
Definition Fulp (p : float) :=
powerRZ radix (Fexp (Fnormalize radix b precision p)).
intros p q H' H'0 H'1; unfold Fulp in |- *.
rewrite
FcanonicUnique
with
(p := Fnormalize radix b precision p)
(q := Fnormalize radix b precision q)
(3 := pGivesBound); auto with float arith.
apply trans_eq with (FtoR radix p).
apply FnormalizeCorrect; auto.
apply trans_eq with (FtoR radix q); auto.
apply sym_eq; apply FnormalizeCorrect; auto.
Qed.
intros p H'; unfold Fulp, FtoRradix, FtoR, Fnormalize in |- *; simpl in |- *;
rewrite Rmult_1_l.
case (Z_zerop (Fnum p)); simpl in |- *; auto.
intros H'0; apply (Rle_powerRZ radix (- dExp b) (Fexp p));
auto with float real zarith.
intros H'0; apply Rle_powerRZ; auto with real zarith arith.
Qed.
intros x; unfold is_Fzero, Fulp, Fnormalize in |- *; case (Z_zerop (Fnum x));
simpl in |- *; auto.
intros H' H'0; Contradict H'; auto.
Qed.

Theorem FulpLe2 :
forall p : float,
Fbounded b p ->
Fnormal radix b (Fnormalize radix b precision p) ->
(Fulp p <= Rabs p * powerRZ radix (Z.succ (- precision)))%R.