intros P H p q pq H0 H1 H2 H3 H4 H5.
case (multExpUpperBound P H p q pq); auto; intros r (H'1, (H'2, H'3)).
case (Req_dec (p * q - pq) 0); intros H6.
case (Req_dec pq 0); intros H7.
cut (Fbounded b (Fzero (Fexp p + Fexp q))); [ intros Fb1 | idtac ].
exists (Fzero (Fexp p + Fexp q)); exists (Fzero (Fexp p + Fexp q));
repeat (split; simpl in |- *; auto with zarith).
rewrite H7; unfold FtoRradix in |- *; apply FzeroisReallyZero.
unfold FtoRradix in |- *; rewrite FzeroisReallyZero; rewrite <- H7.
pattern (FtoRradix pq) at 1 in |- *; rewrite H7; auto with real.
repeat (split; auto); simpl in |- *; auto with arith zarith.
case (errorBoundedMultExp_aux (Fnum p) (Fnum q)); auto with float real zarith.
exists (Fnum pq); exists (Fexp pq - (Fexp p + Fexp q))%Z; split.
apply Rmult_eq_reg_l with (powerRZ radix (Fexp p + Fexp q));
auto with real zarith.
repeat rewrite (fun x y => Rmult_comm (powerRZ x y)).
apply trans_eq with (p * q)%R; auto.
rewrite <- (Fmult_correct radix); auto with real zarith;
unfold FtoRradix, FtoR, Fmult in |- *; simpl in |- *;
rewrite Rmult_IZR; auto.
apply trans_eq with (FtoRradix pq); auto with real.
rewrite Rmult_assoc; rewrite <- powerRZ_add; auto with real zarith.
replace (Fexp pq - (Fexp p + Fexp q) + (Fexp p + Fexp q))%Z with (Fexp pq);
auto; ring.
cut (Fbounded b pq); [ intros Z1; case Z1 | idtac ]; auto with real zarith.
apply (RoundedModeBounded b radix P (p * q)); auto.
intros nx (ex, (H'4, (H'5, (H'6, H'7)))).
cut (FtoRradix pq = FtoRradix (Float nx (ex + (Fexp p + Fexp q))) :>R);
[ intros Eq1 | idtac ].
exists (Float nx (ex + (Fexp p + Fexp q))); exists (Fzero (Fexp p + Fexp q));
repeat (split; simpl in |- *; auto with real zarith).
rewrite <- Eq1; rewrite H6; apply (FzeroisReallyZero radix).
replace (FtoRradix pq) with (p * q)%R.
unfold FtoRradix in |- *; unfold FtoR in |- *; simpl in |- *.
rewrite powerRZ_add; auto with zarith real.
repeat rewrite <- Rmult_assoc; rewrite <- H'4; rewrite powerRZ_add;
[ ring | auto with zarith real ].
replace (FtoRradix p * FtoRradix q)%R with
(pq + (FtoRradix p * FtoRradix q - FtoRradix pq))%R;
[ rewrite H6 | idtac ]; ring.
case (errorBoundedMultPos P H p q pq); auto.
intros s (H'4, (H'5, H'6)).
exists r; exists s; repeat (split; auto with zarith).
rewrite H'2; auto.
apply Zlt_le_weak;
apply RoundedModeErrorExpStrict with b radix precision P (p * q)%R;
auto.
cut (CompatibleP b radix P);
[ intros H'7 | case H; try intros H'7 (H'8, (H'9, H'10)); auto ].
apply H'7 with (p * q)%R pq; auto with real.
fold FtoRradix in |- *; rewrite H'2; auto with real.
fold FtoRradix in |- *; rewrite H'4; auto with real.
Qed.

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