intros p q p' q' H' H'0 H'1.
change (p' <= q')%R in |- *.
case (Rle_or_lt p p'); intros Rl0.
case (Rle_or_lt p q'); intros Rl1.
apply Rplus_le_reg_l with (r := (- p)%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 (p - p')).
rewrite <- (Faux.Rabsolu_left1 (p - q')).
cut (forall x y : R, Rabs (x - y) = Rabs (y - x));
[ intros Eq0; repeat rewrite (Eq0 p); clear Eq0
| intros x y; rewrite <- (Rabs_Ropp (x - y)); rewrite Ropp_minus_distr ];
auto.
elim H'0; auto.
intros H'2 H'3; apply H'3; auto.
case H'1; auto.
apply Rplus_le_reg_l with (r := FtoR radix q').
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.
case (Rle_or_lt p' q); intros Rl2.
apply Rplus_le_reg_l with (r := (- q)%R).
cut (forall x y : R, (- y + x)%R = (- (y - x))%R);
[ intros Eq0; repeat rewrite Eq0; clear Eq0 | intros; ring ].
rewrite <- (Rabs_right (q - p')).
2: apply Rle_ge; apply Rplus_le_reg_l with (r := FtoR radix p').
2: repeat rewrite Rplus_minus; auto.
2: rewrite Rplus_0_r; auto.
rewrite <- (Rabs_right (q - q')).
2: apply Rle_ge; apply Rplus_le_reg_l with (r := FtoR radix q').
2: repeat rewrite Rplus_minus; auto.
2: rewrite Rplus_0_r; auto.
2: apply Rle_trans with (1 := Rlt_le _ _ Rl1); apply Rlt_le; auto.
cut (forall x y : R, Rabs (x - y) = Rabs (y - x));
[ intros Eq0; repeat rewrite (Eq0 q); clear Eq0
| intros x y; rewrite <- (Rabs_Ropp (x - y)); rewrite Ropp_minus_distr ];
auto.
apply Ropp_le_contravar.
elim H'1; auto.
intros H'2 H'3; apply H'3; auto.
case H'0; auto.
case (Rle_or_lt (p - q') (p' - q)); intros Rl3.
absurd (Rabs (p' - p) <= Rabs (q' - p))%R.
apply Rgt_not_le.
rewrite (Faux.Rabsolu_left1 (q' - p)).
2: apply Rplus_le_reg_l with (r := p).
2: repeat rewrite Rplus_minus; auto.
2: rewrite Rplus_0_r; apply Rlt_le; auto.
rewrite (Rabs_right (p' - p)).
2: apply Rle_ge; apply Rplus_le_reg_l with (r := p).
2: rewrite Rplus_0_r; auto.
cut (forall x y : R, (- (y - x))%R = (x - y)%R);
[ intros Eq0; repeat rewrite Eq0; clear Eq0 | intros; ring ].
red in |- *; apply Rle_lt_trans with (1 := Rl3).
replace (p' - p)%R with (p' - q + (q - p))%R.
pattern (p' - q)%R at 1 in |- *; replace (p' - q)%R with (p' - q + 0)%R.
apply Rplus_lt_compat_l; auto.
apply Rplus_lt_reg_l with (r := p).
repeat rewrite Rplus_minus; auto.
rewrite Rplus_0_r; auto.
rewrite Rplus_0_r; auto.
ring.
replace (p + (p' - p))%R with (FtoRradix p'); auto; ring.
case H'0; intros H'2 H'3; apply H'3; auto.
case H'1; auto.
absurd (Rabs (q' - q) <= Rabs (p' - q))%R.
apply Rgt_not_le.
rewrite (Faux.Rabsolu_left1 (q' - q)).
2: apply Rplus_le_reg_l with (r := q).
2: repeat rewrite Rplus_minus; auto.
2: rewrite Rplus_0_r; apply Rlt_le; auto.
2: apply Rlt_trans with (1 := Rl1); auto.
rewrite (Rabs_right (p' - q)).
2: apply Rle_ge; apply Rplus_le_reg_l with (r := q).
2: rewrite Rplus_0_r; apply Rlt_le; auto.
cut (forall x y : R, (- (y - x))%R = (x - y)%R);
[ intros Eq0; repeat rewrite Eq0; clear Eq0 | intros; ring ].
red in |- *; apply Rlt_trans with (1 := Rl3).
replace (q - q')%R with (p - q' + (q - p))%R.
pattern (p - q')%R at 1 in |- *; replace (p - q')%R with (p - q' + 0)%R.
apply Rplus_lt_compat_l; auto.
apply Rplus_lt_reg_l with (r := p).
repeat rewrite Rplus_minus; auto.
rewrite Rplus_0_r; auto.
rewrite Rplus_0_r; auto.
ring.
replace (q + (p' - q))%R with (FtoRradix p'); auto; ring.
case H'1; intros H'2 H'3; apply H'3; auto.
case H'0; auto.
case (Rle_or_lt p q'); intros Rl1.
apply Rle_trans with p; auto.
apply Rlt_le; auto.
apply Rplus_le_reg_l with (r := (- q)%R).
cut (forall x y : R, (- y + x)%R = (- (y - x))%R);
[ intros Eq0; repeat rewrite Eq0; clear Eq0 | intros; ring ].
rewrite <- (Rabs_right (q - p')).
rewrite <- (Rabs_right (q - q')).
apply Ropp_le_contravar.
cut (forall x y : R, Rabs (x - y) = Rabs (y - x));
[ intros Eq0; repeat rewrite (Eq0 q); clear Eq0
| intros x y; rewrite <- (Rabs_Ropp (x - y)); rewrite Ropp_minus_distr ];
auto.
elim H'1; auto.
intros H'2 H'3; apply H'3; auto.
case H'0; auto.
apply Rle_ge; apply Rplus_le_reg_l with (r := FtoR radix q').
repeat rewrite Rplus_minus; auto.
rewrite Rplus_0_r; auto.
apply Rlt_le; apply Rlt_trans with (1 := Rl1); auto.
apply Rle_ge; apply Rplus_le_reg_l with (r := FtoR radix p').
repeat rewrite Rplus_minus; auto.
rewrite Rplus_0_r; auto.
apply Rlt_le; apply Rlt_trans with (1 := Rl0); auto.
Qed.

Theorem ClosestRoundedModeP : RoundedModeP b radix Closest.
split; try exact ClosestTotal.