intros r min max H' H'0 H'1; split.
case H'; auto.
intros f H'2.
case (Rle_or_lt f r); intros H'3.
repeat rewrite Faux.Rabsolu_left1.
apply Ropp_le_contravar.
apply Rplus_le_reg_l with (r := r).
repeat rewrite Rplus_minus; auto.
case H'; auto.
intros H'4 H'5; elim H'5; intros H'6 H'7; apply H'7; clear H'5; auto.
apply Rplus_le_reg_l with (r := r).
repeat rewrite Rplus_minus; auto.
rewrite Rplus_0_r; auto.
apply Rplus_le_reg_l with (r := r).
repeat rewrite Rplus_minus; auto.
rewrite Rplus_0_r; auto.
apply isMin_inv1 with (1 := H'); auto.
rewrite (Faux.Rabsolu_left1 (min - r)).
rewrite (Rabs_right (f - r)).
apply Rle_trans with (max - r)%R.
cut (forall x y : R, (- y + x)%R = (- (y - x))%R);
[ intros Eq0; repeat rewrite Eq0; clear Eq0 | intros; ring ].
cut (forall x y : R, (- (x - y))%R = (y - x)%R);
[ intros Eq0; repeat rewrite Eq0; clear Eq0
| intros; unfold Rminus in |- *; ring ].
apply Rplus_le_reg_l with (r := FtoR radix min).
repeat rewrite Rplus_minus; auto.
apply Rplus_le_reg_l with (r := r).
replace (r + (FtoR radix min + (max - r)))%R with (min + max)%R.
replace (r + r)%R with (2%nat * r)%R; auto.
simpl in |- *; ring.
simpl in |- *; fold FtoRradix; ring.
apply Rplus_le_reg_l with (r := r).
repeat rewrite Rplus_minus; auto.
case H'0; auto.
intros H'4 H'5; elim H'5; intros H'6 H'7; apply H'7; clear H'5; auto.
apply Rlt_le; auto.
apply Rle_ge; apply Rplus_le_reg_l with (r := r).
repeat rewrite Rplus_minus; auto.
rewrite Rplus_0_r; auto.
apply Rlt_le; auto.
apply Rplus_le_reg_l with (r := r).
repeat rewrite Rplus_minus; auto.
rewrite Rplus_0_r; auto.
apply isMin_inv1 with (1 := H'); auto.
Qed.

Theorem ClosestMax :
forall (r : R) (min max : float),
isMin b radix r min ->
isMax b radix r max -> (min + max <= 2%nat * r)%R -> Closest r max.