intros P H' r q q' H'0 H'1.
replace (radix * q')%R with (FtoRradix (Float (Fnum q') (Fexp q' + 1%nat))).
intros H'2; case H'2.
intros H'3; case H'; intros H'4 H'5; elim H'5; intros H'6 H'7; elim H'7;
intros H'8 H'9; apply H'9 with (1 := H'3); clear H'7 H'5;
auto.
apply RoundedModeProjectorIdem; auto.
apply FBoundedScale; auto.
intros H'3.
replace (FtoRradix q) with (FtoRradix (Float (Fnum q') (Fexp q' + 1%nat)));
auto with real.
apply (RoundedProjector _ _ P H'); auto.
apply FBoundedScale; auto.
unfold FtoRradix in H'3; rewrite H'3; auto.
case H'.
intros H'4 H'5; elim H'5; intros H'6 H'7; clear H'5.
unfold FtoRradix in |- *; rewrite FvalScale; auto.
rewrite powerRZ_1; auto.
Qed.

Theorem RleMinR0 :
forall (r : R) (min : float),
(0 <= r)%R -> isMin b radix r min -> (0 <= min)%R.