Section Fbounded_Def.
Variable radix : Z.
Hypothesis radixMoreThanOne : (1 < radix)%Z.
Let radixMoreThanZERO := Zlt_1_O _ (Zlt_le_weak _ _ radixMoreThanOne).
Hint Resolve radixMoreThanZERO: zarith.
Let FtoRradix := FtoR radix.
Local Coercion FtoRradix : float >-> R.
Coercion Z_of_N: N >-> Z.
Record Fbound : Set := Bound {vNum : positive; dExp : N}.
Definition Fbounded (b : Fbound) (d : float) :=
(Z.abs (Fnum d) < Zpos (vNum b))%Z /\ (- dExp b <= Fexp d)%Z.
intros b p H; case H; intros H1 H2; case H1; auto.
Qed.
intros b p H; case H; auto.
Qed.
Hint Resolve FboundedNum FboundedExp: float.
intros b p; case (Z_le_gt_dec (Zpos (vNum b)) (Z.abs (Fnum p)));
intros H'.
right; red in |- *; intros H'3; Contradict H'; auto with float zarith.
case (Z_le_gt_dec (- dExp b) (Fexp p)); intros H'1.
left; repeat split; auto with zarith.
right; red in |- *; intros H'3; Contradict H'1; auto with float zarith.
Qed.
intros b; unfold FtoRradix, FtoR in |- *; simpl in |- *; auto with real.
Qed.
intros b; repeat (split; simpl in |- *).
replace 0%Z with (- 0%nat)%Z; [ idtac | simpl in |- *; auto ].
apply Zeq_le; auto with arith.
Qed.
Hint Unfold Fbounded.

Theorem FboundedZeroSameExp :
forall (b : Fbound) (p : float), Fbounded b p -> Fbounded b (Fzero (Fexp p)).