case MinEx with (r := r) (3 := pGivesBound); auto with arith.
intros min H'.
case MaxEx with (r := r) (3 := pGivesBound); auto with arith.
intros max H'0.
cut (min <= r)%R; [ intros Rl1 | apply isMin_inv1 with (1 := H'); auto ].
cut (r <= max)%R; [ intros Rl2 | apply isMax_inv1 with (1 := H'0) ].
case (Rle_or_lt (r - min) (max - r)); intros H'1.
case H'1; intros H'2; auto.
exists min; split.
apply ClosestMin with (max := max); auto.
replace (2%nat * r)%R with (r + r)%R; [ idtac | simpl in |- *; ring ].
apply Rminus_le; auto.
replace (r + r - (min + max))%R with (r - min - (max - r))%R;
[ idtac | simpl in |- *; ring ].
apply Rle_minus; auto.
right; intros q H'3.
apply ClosestMinEq with (r := r) (max := max); auto.
replace (2%nat * r)%R with (r + r)%R; [ idtac | simpl in |- *; ring ].
apply Rminus_lt; auto.
replace (r + r - (min + max))%R with (r - min - (max - r))%R;
[ idtac | simpl in |- *; ring ].
apply Rlt_minus; auto.
case (FNevenOrFNodd b radix precision min); intros Ev0.
exists min; split; auto.
apply ClosestMin with (max := max); auto.
replace (2%nat * r)%R with (r + r)%R; [ idtac | simpl in |- *; ring ].
apply Rminus_le; auto.
replace (r + r - (min + max))%R with (r - min - (max - r))%R;
[ idtac | simpl in |- *; ring ].
apply Rle_minus; auto.
exists max; split; auto.
apply ClosestMax with (min := min); auto.
replace (2%nat * r)%R with (r + r)%R; [ idtac | simpl in |- *; ring ].
apply Rminus_le; auto.
replace (min + max - (r + r))%R with (max - r - (r - min))%R;
[ idtac | simpl in |- *; ring ].
apply Rle_minus; auto.
rewrite H'2; auto with real.
case (Req_dec min max); intros H'5.
right; intros q H'3.
case (ClosestMinOrMax _ _ H'3); intros isM0.
rewrite <- H'5.
apply MinEq with (1 := isM0); auto.
apply MaxEq with (1 := isM0); auto.
left.
apply FNevenEq with (f1 := FNSucc b radix precision min); auto.
apply FcanonicBound with (radix := radix).
apply FNSuccCanonic; auto with arith.
case H'; auto.
case H'0; auto.
apply MaxEq with (b := b) (r := r); auto.
apply MinMax; auto with arith.
Contradict H'5; auto.
fold FtoRradix in H'5; rewrite H'5 in H'2.
replace (FtoRradix max) with (min + (max - min))%R;
[ rewrite <- H'2 | idtac ]; ring.
apply FNoddSuc; auto.
case H'; auto.
exists max; split; auto.
apply ClosestMax with (min := min); auto.
replace (2%nat * r)%R with (r + r)%R; [ idtac | simpl in |- *; ring ].
apply Rminus_le; auto.
replace (min + max - (r + r))%R with (max - r - (r - min))%R;
[ idtac | simpl in |- *; ring ].
apply Rle_minus; auto with real.
right; intros q H'2.
apply ClosestMaxEq with (r := r) (min := min); auto.
replace (2%nat * r)%R with (r + r)%R; [ idtac | simpl in |- *; ring ].
apply Rminus_lt; auto.
replace (min + max - (r + r))%R with (max - r - (r - min))%R;
[ idtac | simpl in |- *; ring ].
apply Rlt_minus; auto.
Qed.

Theorem EvenClosestCompatible : CompatibleP b radix EvenClosest.
red in |- *; simpl in |- *.