case (MinExList r (mBFloat r)).
intros H'0; absurd (Fopp (boundR r) <= r)%R; auto.
apply Rlt_not_le.
apply H'0.
apply mBFadic_correct3; auto.
(* A minimum always exists *)
apply Rlt_le.
apply boundRCorrect2; auto.
intros H'0; elim H'0; intros min E; elim E; intros H'1 H'2; elim H'2;
intros H'3 H'4; clear H'2 E H'0.
exists min; split; auto.
apply mBPadic_Fbounded with (r := r); auto.
split; auto.
intros f H'0 H'2.
case (Req_dec f 0); intros H'6.
replace (FtoRradix f) with (FtoRradix (Float 0%nat (- dExp b))).
apply H'4; auto.
apply mBFadic_correct4; auto.
replace (FtoRradix (Float 0%nat (- dExp b))) with (FtoRradix f); auto.
rewrite H'6.
unfold FtoRradix, FtoR in |- *; simpl in |- *; auto with real.
rewrite H'6.
unfold FtoRradix, FtoR in |- *; simpl in |- *; auto with real.
case (Rle_or_lt f (Fopp (boundR r))); intros H'5.
apply Rle_trans with (FtoRradix (Fopp (boundR r))); auto.
apply H'4; auto.
apply mBFadic_correct3; auto.
apply Rlt_le.
apply boundRCorrect2; auto.
case (Rle_or_lt (boundR r) f); intros H'7.
Contradict H'2; apply Rlt_not_le.
apply Rlt_le_trans with (FtoRradix (boundR r)); auto.
apply boundRCorrect1; auto.
apply H'4; auto.
apply mBFadic_correct1; auto.
Contradict H'6; unfold FtoRradix in |- *; apply is_Fzero_rep1; auto.
Qed.

Theorem MaxEx : forall r : R, exists max : float, isMax r max.
intros r; case (MinEx r).