intros r L; elim L; simpl in |- *; auto.
left; intros f H'; elim H'.
intros a l H'.
elim H';
[ intros H'0; clear H'
| intros H'0; elim H'0; intros min E; elim E; intros H'1 H'2; elim H'2;
intros H'3 H'4; try exact H'4; clear H'2 E H'0 H' ].
case (Rle_or_lt a r); intros H'1.
right; exists a; repeat split; auto.
intros f H'; elim H';
[ intros H'2; rewrite <- H'2; clear H' | intros H'2; clear H' ];
auto with real.
intros H'; Contradict H'; auto with real.
apply Rlt_not_le; auto with real.
left; intros f H'; elim H';
[ intros H'2; rewrite <- H'2; clear H' | intros H'2; clear H' ];
auto.
case (Rle_or_lt a min); intros H'5.
right; exists min; repeat split; auto.
intros f H'; elim H';
[ intros H'0; rewrite <- H'0; clear H' | intros H'0; clear H' ];
auto.
case (Rle_or_lt a r); intros H'6.
right; exists a; repeat split; auto.
intros f H'; elim H';
[ intros H'0; rewrite <- H'0; clear H' | intros H'0; clear H' ];
auto with real.
intros H'; apply Rle_trans with (FtoRradix min); auto with real.
right; exists min; split; auto; split; auto.
intros f H'; elim H';
[ intros H'0; elim H'0; clear H' | intros H'0; clear H' ];
auto.
intros H'; Contradict H'6; auto with real.
apply Rle_not_lt; auto.
Qed.

Theorem MinEx : forall r : R, exists min : float, isMin r min.
intros r.