Section FRoundP.
Variable b : Fbound.
Variable radix : Z.
Variable precision : nat.
Let FtoRradix := FtoR radix.
Coercion FtoRradix : float >-> R.
Hypothesis radixMoreThanOne : (1 < radix)%Z.
Let radixMoreThanZERO := Zlt_1_O _ (Zlt_le_weak _ _ radixMoreThanOne).
Hint Resolve radixMoreThanZERO: zarith.
Hypothesis precisionGreaterThanOne : 1 < precision.
Hypothesis pGivesBound : Zpos (vNum b) = Zpower_nat radix precision.
intros P H' p q pq H'0.
case
(RoundedModeRep b radix precision)
with (p := Fplus radix p q) (q := pq) (P := P); auto with float arith.
rewrite Fplus_correct; auto with float arith.
simpl in |- *; intros x H'1.
case
(eqExpLess _ radixMoreThanOne b)
with (p := pq) (q := Float x (Fexp (Fplus radix p q)));
auto.
apply (RoundedModeBounded b radix) with (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.
intros P H' p q pq H'0 H'1 H'2.
replace (Z.succ (Zmax (Fexp p) (Fexp q))) with
(Fexp (Float (pPred (vNum b)) (Z.succ (Zmax (Fexp p) (Fexp q)))));
[ idtac | simpl in |- *; auto ].
unfold FtoRradix in |- *; apply eqExpMax; auto.
apply RoundedModeBounded with (radix := radix) (P := P) (r := (p + q)%R);
auto with float arith.
unfold pPred in |- *; apply maxFbounded; auto.
apply Z.le_trans with (Fexp p); auto with float.
apply Z.le_trans with (Z.succ (Fexp p)); auto with float zarith.
replace
(FtoR radix (Float (pPred (vNum b)) (Z.succ (Zmax (Fexp p) (Fexp q))))) with
(radix * Float (pPred (vNum b)) (Zmax (Fexp p) (Fexp q)))%R.
rewrite Fabs_correct; auto with zarith.
unfold FtoRradix in |- *;
apply
RoundedModeMultAbs
with (b := b) (precision := precision) (P := P) (r := (p + q)%R);
auto.
unfold pPred in |- *; apply maxFbounded; auto.
apply Z.le_trans with (Fexp p); auto with float zarith.
apply Rle_trans with (Rabs p + Rabs q)%R.
apply Rabs_triang; auto.
apply
Rle_trans
with
(2%nat * FtoR radix (Float (pPred (vNum b)) (Zmax (Fexp p) (Fexp q))))%R;
auto.
cut (forall r : R, (2%nat * r)%R = (r + r)%R);
[ intros tmp; rewrite tmp; clear tmp | intros; simpl in |- *; ring ].
apply Rplus_le_compat; auto.
rewrite <- (Fabs_correct radix); auto with arith; apply maxMax1; auto;
apply ZmaxLe1.
rewrite <- (Fabs_correct radix); auto with arith; apply maxMax1; auto;
apply ZmaxLe2.
apply Rmult_le_compat; auto with real arith.
replace 0%R with (INR 0); auto with real arith.
apply LeFnumZERO; simpl in |- *; auto; replace 0%Z with (Z_of_nat 0);
auto with zarith.
unfold pPred in |- *; apply Zle_Zpred; auto with zarith.
rewrite INR_IZR_INZ; apply Rle_IZR; simpl in |- *; auto with zarith.
cut (1 < radix)%Z; auto with zarith;intros.
unfold FtoRradix, FtoR in |- *; simpl in |- *.
rewrite powerRZ_Zs; auto with real zarith; ring.
Qed.

Theorem plusExpBound :
forall P,
RoundedModeP b radix P ->
forall p q pq : float,
P (p + q)%R pq ->
Fbounded b p ->
Fbounded b q ->
exists r : float,
Fbounded b r /\
r = pq :>R /\
(Z.min (Fexp p) (Fexp q) <= Fexp r)%Z /\
(Fexp r <= Z.succ (Zmax (Fexp p) (Fexp q)))%Z.