intros P H' p q pq H'0.
case
(RoundedModeRep b radix precision) with (p := Fmult p q) (q := pq) (P := P);
auto with zarith.
rewrite Fmult_correct; auto with zarith.
simpl in |- *; intros x H'1.
case
(eqExpLess _ radixMoreThanOne b)
with (p := pq) (q := Float x (Fexp (Fmult p q)));
auto.
apply RoundedModeBounded with (radix := radix) (P := P) (r := (p * q)%R);
auto.
simpl in |- *; intros x0 H'2; elim H'2; intros H'3 H'4; elim H'4;
intros H'5 H'6; clear H'4 H'2.
exists x0; split; [ idtac | split ]; auto.
unfold FtoRradix in |- *; rewrite H'5; auto.
apply le_IZR; auto.
Qed.

Theorem multExpUpperBound :
forall P,
RoundedModeP b radix P ->
forall p q pq : float,
P (p * q)%R pq ->
Fbounded b p ->
Fbounded b q ->
(- dExp b <= Fexp p + Fexp q)%Z ->
exists r : float,
Fbounded b r /\ r = pq :>R /\ (Fexp r <= precision + (Fexp p + Fexp q))%Z.