intros p; unfold mBFloat in |- *.
replace (Float 0%nat (- dExp b)) with
((fun p : Z * Z => Float (fst p) (snd p))
(Fnum (Float 0%nat (- dExp b)), Fexp (Float 0%nat (- dExp b)))).
apply in_map with (f := fun p : Z * Z => Float (fst p) (snd p));
auto.
apply mProd_correct; auto.
apply mZlist_correct; auto.
simpl in |- *; auto with zarith.
replace 0%Z with (- (0))%Z; [ idtac | simpl in |- *; auto ].
apply Zle_Zopp; unfold pPred in |- *; apply Zle_Zpred.
red in |- *; simpl in |- *; auto with zarith.
simpl in |- *; auto with zarith.
unfold pPred in |- *; apply Zle_Zpred.
red in |- *; simpl in |- *; auto with zarith.
apply mZlist_correct; auto.
simpl in |- *; auto with zarith.
unfold boundR, boundNat in |- *; simpl in |- *; auto with zarith.
apply Zle_trans with 0%Z; auto with zarith.
case (dExp b); auto with zarith.
simpl in |- *; auto with zarith.
Qed.

Theorem mBPadic_Fbounded :
forall (p : float) (r : R), In p (mBFloat r) -> Fbounded b p.