intros p q r H' H'0 H'1 H'2.
cut
(2%nat * Rabs (FtoR radix (Fplus radix p q) - FtoR radix r) <=
Float 1%nat (Fexp r))%R;
[ rewrite Fplus_correct; auto with zarith; intros Rl1 | idtac ].
case errorBoundedPlus with (p := p) (q := q) (pq := r); auto.
intros x H'3; elim H'3; intros H'4 H'5; elim H'5; intros H'6 H'7;
clear H'5 H'3.
unfold FtoRradix in H'4; rewrite <- H'4 in Rl1.
2: apply Rle_trans with (Fulp b radix precision r); auto.
2: apply (ClosestUlp b radix precision); auto.
2: rewrite Fplus_correct; auto with zarith.
2: unfold FtoRradix in |- *; apply FulpLe; auto.
2: apply
RoundedModeBounded
with (radix := radix) (P := Closest b radix) (r := (p + q)%R);
auto.
2: apply ClosestRoundedModeP with (precision := precision); auto.
cut (x = 0%R :>R); [ unfold FtoRradix in |- *; intros Eq1 | idtac ].
replace (FtoR radix r) with (FtoR radix r + 0)%R; [ idtac | ring ].
rewrite <- Eq1.
rewrite H'4; ring.
apply (is_Fzero_rep1 radix).
case (Z_zerop (Fnum x)); simpl in |- *; auto.
intros H'3; Contradict Rl1.
apply Rgt_not_le.
red in |- *; apply Rle_lt_trans with (Rabs (FtoR radix x)).
unfold FtoRradix, FtoR in |- *; simpl in |- *; auto.
rewrite Rabs_mult.
apply Rmult_le_compat; auto with real arith.
generalize H'3; case (Fnum x); simpl in |- *; auto with real zarith.
intros H'5; case H'5; auto.
intros p0 H'5; rewrite Rabs_right; auto with real.
replace 1%R with (INR 1); unfold IZR; repeat rewrite <- INR_IPR; auto with real arith.
intros p0 H'5; rewrite Faux.Rabsolu_left1; auto.
unfold IZR; rewrite Ropp_involutive.
repeat rewrite <- INR_IPR; simpl; replace 1%R with (INR 1); auto with real arith.
unfold IZR; repeat rewrite <- INR_IPR; replace 0%R with (- 0%nat)%R; auto with real.
rewrite Rabs_right; auto with real arith.
apply Rle_powerRZ; auto with real arith.
auto with zarith.
apply Rle_ge; cut (1 < radix)%Z; auto with float real zarith.
cut (forall r : R, (2%nat * r)%R = (r + r)%R);
[ intros tmp; rewrite tmp; clear tmp | intros f; simpl in |- *; ring ].
pattern (Rabs (FtoR radix x)) at 1 in |- *;
replace (Rabs (FtoR radix x)) with (Rabs (FtoR radix x) + 0)%R;
[ idtac | ring ].
apply Rplus_lt_compat_l; auto.
case (Rabs_pos (FtoR radix x)); auto.
rewrite <- Fabs_correct; auto with arith.
intros H'5; Contradict H'3.
cut (Fnum (Fabs x) = 0%Z).
unfold Fabs in |- *; simpl in |- *; case (Fnum x); simpl in |- *; auto;
intros; discriminate.
change (is_Fzero (Fabs x)) in |- *.
apply (is_Fzero_rep2 radix); auto with arith.
Qed.

Theorem plusExact1bis :
forall p q r : float,
Fbounded b p ->
Fbounded b q ->
Closest b radix (p + q) r ->
r <> (p + q)%R :>R -> (Zmin (Fexp p) (Fexp q) < Fexp r)%Z.