cut (FtoR radix (Fzero (- dExp b)) = 0%R);
[ intros Eq0 | unfold FtoR in |- *; simpl in |- * ];
auto with real.
simpl in |- *; intros p q p' q' H' H'0; case H'0; clear H'0.
intros H'0; elim H'0; intros H'1 H'2; clear H'0; intros H'0.
case H'0; intros H'3; elim H'3; clear H'3; auto.
intros H'3 H'4.
apply (MonotoneMin b radix) with (p := p) (q := q); auto.
intros H'3 H'4.
apply Rle_trans with p; [ apply isMin_inv1 with (1 := H'2); auto | idtac ].
apply Rle_trans with q; [ idtac | apply isMax_inv1 with (1 := H'4) ]; auto.
apply Rlt_le; auto.
intros H'0; elim H'0; intros H'1 H'2; clear H'0.
intros H'0; case H'0; clear H'0; intros H'0; case H'0; intros H'3 H'4;
clear H'0.
apply Rle_trans with (FtoRradix (Fzero (- dExp b))); auto.
elim H'2.
intros H'0 H'5; elim H'5; intros H'6 H'7; apply H'7; clear H'5; auto.
repeat split; simpl in |- *; auto with zarith.
rewrite Eq0; auto.
elim H'4.
intros H'0 H'5; elim H'5; intros H'6 H'7; apply H'7; clear H'5; auto.
repeat split; simpl in |- *; auto with zarith.
rewrite Eq0; auto.
apply (MonotoneMax b radix) with (p := p) (q := q); auto.
Qed.

Theorem ToZeroRoundedModeP : RoundedModeP ToZeroP.
repeat split.