intros r min max H' H'0 H'1; split.
case H'0; auto.
intros f H'2.
case (Rle_or_lt f r); intros H'3.
rewrite (Rabs_right (max - r)).
rewrite (Faux.Rabsolu_left1 (f - r)).
apply Rle_trans with (r - min)%R.
apply Rplus_le_reg_l with (r := FtoRradix min).
repeat rewrite Rplus_minus; auto.
apply Rplus_le_reg_l with (r := r).
replace (r + (min + (max - r)))%R with (min + max)%R.
replace (r + r)%R with (2%nat * r)%R; auto.
simpl in |- *; ring.
simpl in |- *; ring.
replace (r - min)%R with (- (min - r))%R.
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.
simpl in |- *; ring.
apply Rplus_le_reg_l with (r := r).
repeat rewrite Rplus_minus; auto.
rewrite Rplus_0_r; auto.
apply Rle_ge; apply Rplus_le_reg_l with (r := r).
repeat rewrite Rplus_minus; auto.
rewrite Rplus_0_r; auto.
apply isMax_inv1 with (1 := H'0); auto.
repeat rewrite Rabs_right.
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 Rle_ge; apply Rplus_le_reg_l with (r := r).
repeat rewrite Rplus_minus; auto.
rewrite Rplus_0_r; auto.
apply isMax_inv1 with (1 := H'0); auto.
Qed.

Theorem ClosestMinOrMax : MinOrMaxP b radix Closest.
red in |- *.