intros r p H'.
case (Rle_or_lt p r); intros H'1.
left; split; auto.
case H'; auto.
split; auto.
intros f H'0 H'2.
apply Rplus_le_reg_l with (r := (- r)%R).
cut (forall x y : R, (- y + x)%R = (- (y - x))%R);
[ intros Eq0; repeat rewrite Eq0; clear Eq0 | intros; ring ].
apply Ropp_le_contravar.
rewrite <- (Rabs_right (r - FtoR radix p)).
rewrite <- (Rabs_right (r - FtoR radix f)).
cut (forall x y : R, Rabs (x - y) = Rabs (y - x));
[ intros Eq0; repeat rewrite (Eq0 r); clear Eq0
| intros x y; rewrite <- (Rabs_Ropp (x - y)); rewrite Ropp_minus_distr ];
auto.
elim H'; auto.
apply Rle_ge; apply Rplus_le_reg_l with (r := FtoR radix f).
repeat rewrite Rplus_minus; auto.
rewrite Rplus_0_r; auto.
apply Rle_ge; apply Rplus_le_reg_l with (r := FtoR radix p).
repeat rewrite Rplus_minus; auto.
rewrite Rplus_0_r; auto.
right; split; auto.
case H'; auto.
split; auto.
apply Rlt_le; auto.
intros f H'0 H'2.
apply Rplus_le_reg_l with (r := (- r)%R).
cut (forall x y : R, (- y + x)%R = (- (y - x))%R);
[ intros Eq0; repeat rewrite Eq0; clear Eq0 | intros; ring ].
rewrite <- (Faux.Rabsolu_left1 (r - FtoR radix p)).
rewrite <- (Faux.Rabsolu_left1 (r - FtoR radix f)).
cut (forall x y : R, Rabs (x - y) = Rabs (y - x));
[ intros Eq0; repeat rewrite (Eq0 r); clear Eq0
| intros x y; rewrite <- (Rabs_Ropp (x - y)); rewrite Ropp_minus_distr ];
auto.
elim H'; auto.
apply Rplus_le_reg_l with (r := FtoR radix f).
repeat rewrite Rplus_minus; auto.
rewrite Rplus_0_r; auto.
apply Rplus_le_reg_l with (r := FtoR radix p).
repeat rewrite Rplus_minus; auto.
rewrite Rplus_0_r; auto.
apply Rlt_le; auto.
Qed.

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