intros p r H'; case (Rle_or_lt 0 r); intros H'1.
repeat rewrite Rabs_right; auto with real.
apply closestLessMultPos; auto.
apply Rle_ge;
apply (RleRoundedR0 b radix precision) with (P := Closest b radix) (r := r);
auto.
apply ClosestRoundedModeP with (precision := precision); auto.
repeat rewrite Faux.Rabsolu_left1; auto.
replace (2%nat * - r)%R with (- (2%nat * r))%R;
[ apply Ropp_le_contravar | ring ].
apply closestLessMultNeg; auto.
apply Rlt_le; auto.
apply Rlt_le; auto.
apply
(RleRoundedLessR0 b radix precision) with (P := Closest b radix) (r := r);
auto.
apply ClosestRoundedModeP with (precision := precision); auto.
apply Rlt_le; auto.
Qed.

Theorem errorBoundedMultClosest_aux :
forall p q pq : float,
Fbounded b p ->
Fbounded b q ->
Closest b radix (p * q) pq ->
(- dExp b <= Fexp p + Fexp q)%Z ->
(p * q - pq)%R <> 0%R :>R ->
ex
(fun r : float =>
ex
(fun s : float =>
Fcanonic radix b r /\
Fbounded b r /\
Fbounded b s /\
r = pq :>R /\
s = (p * q - r)%R :>R /\
Fexp s = (Fexp p + Fexp q)%Z :>Z /\
(Fexp s <= Fexp r)%Z /\ (Fexp r <= precision + (Fexp p + Fexp q))%Z)).