intros p q pq Hp Hq H1 H2 H3.
cut (RoundedModeP b radix (Closest b radix));
[ intros H4 | apply ClosestRoundedModeP with precision; auto ].
lapply (errorBoundedMultExp b radix precision);
[ intros H'2; lapply H'2;
[ intros H'3; lapply H'3;
[ intros H'4; lapply (H'4 (Closest b radix));
[ intros H'7; elim (H'7 p q pq);
[ intros r E; elim E; intros s E0; elim E0; intros H'15 H'16;
elim H'16; intros H'17 H'18; elim H'18;
intros H'19 H'20; elim H'20; intros H'21 H'22;
elim H'22; intros H'23 H'24; elim H'24;
intros H'25 H'26;
clear H'24 H'22 H'20 H'18 H'16 E0 E H'3 H'2
| clear  H'3 H'2
| clear  H'3 H'2
| clear  H'3 H'2
| clear  H'3 H'2 ]
| clear H'3 H'2 ]
| clear H'3 H'2 ]
| clear H'2 ]
| idtac]; auto.
exists (Fnormalize radix b precision r); exists s.
cut (Fbounded b (Fnormalize radix b precision r));
[ intros H5 | apply FnormalizeBounded; auto with arith ].
split; [ apply FnormalizeCanonic; auto with arith | idtac ].
repeat (split; auto).
unfold FtoRradix in |- *; rewrite <- H'19; unfold FtoRradix in |- *;
apply FnormalizeCorrect; auto.
unfold FtoRradix in |- *; rewrite FnormalizeCorrect; auto with arith.
apply Zlt_le_weak.
apply
RoundedModeErrorExpStrict with b radix precision (Closest b radix) (p * q)%R;
auto with arith.
generalize ClosestCompatible; unfold CompatibleP in |- *; intros H6.
generalize
(H6 b radix (FtoRradix p * FtoRradix q)%R (FtoRradix p * FtoRradix q)%R pq);
intros H9; apply H9; auto.
rewrite FnormalizeCorrect; auto with real arith.
rewrite FnormalizeCorrect; auto with real arith.
rewrite H'21; rewrite H'19; auto.
apply Zle_trans with (Fexp r); auto.
apply FcanonicLeastExp with radix b precision; auto with arith.
rewrite FnormalizeCorrect; auto with real arith.
apply FnormalizeCanonic; auto with arith.
Qed.

Theorem errorBoundedMultClosest :
forall p q pq : float,
Fbounded b p ->
Fbounded b q ->
Closest b radix (p * q) pq ->
(- dExp b <= Fexp p + Fexp q)%Z ->
(- dExp b <= Fexp (Fnormalize radix b precision pq) - precision)%Z ->
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 r - precision)%Z :>Z)).