intros p q pq H H0 H1.
case (plusExpBound b radix precision) with (P := Closest b radix) (5 := H1);
auto with zarith.
apply (ClosestRoundedModeP b radix precision); auto with zarith.
intros r (H2, (H3, (H4, H5))); fold FtoRradix in H3.
case (Req_dec (p + q - pq) 0); intros Hr.
cut (Fbounded b (Fzero (Zmin (Fexp p) (Fexp q)))); [ intros Fbs | idtac ].
exists (Fzero (Zmin (Fexp p) (Fexp q))); exists r; repeat (split; auto).
rewrite (FzeroisReallyZero radix); rewrite <- Hr; rewrite <- H3; auto.
case (Zmin_or (Fexp p) (Fexp q)); intros Hz; rewrite Hz;
apply FboundedZeroSameExp; auto.
case (errorBoundedPlus p q pq); auto.
intros error (H6, (H7, H8)).
exists error; exists r; repeat (split; auto).
rewrite H3; auto.
rewrite H8; auto.
Qed.

Theorem plusExactExpCanonic :
forall c d p q : float,
Fbounded b c ->
Fbounded b d ->
Fbounded b p ->
Fbounded b q ->
Closest b radix (c + d) p ->
q = (c + d - p)%R :>R ->
q <> 0%R :>R ->
ex
(fun r : float =>
ex
(fun s : float =>
Fcanonic radix b s /\
Fbounded b r /\
s = p :>R /\
r = (c + d - s)%R :>R /\
Fexp r = Zmin (Fexp c) (Fexp d) :>Z /\
(Fexp r < Fexp s)%Z /\ (Fexp s <= Zsucc (Zmax (Fexp c) (Fexp d)))%Z)).