intros r min H' H'0.
rewrite <- (FzeroisZero radix b).
case H'; intros H'1.
apply (MonotoneMin b radix) with (p := FtoRradix (Fzero (- dExp b))) (q := r);
auto.
unfold FtoRradix in |- *; rewrite (FzeroisZero radix b); auto.
apply (RoundedModeProjectorIdem (isMin b radix)); auto.
apply MinRoundedModeP with (precision := precision); auto with float.
apply FboundedFzero; auto.
replace (FtoR radix (Fzero (- dExp b))) with (FtoRradix min); auto with real.
apply sym_eq; apply (ProjectMin b radix).
apply FboundedFzero; auto.
rewrite <- H'1 in H'0; rewrite <- (FzeroisZero radix b) in H'0; auto.
Qed.

Theorem RleRoundedR0 :
forall (P : R -> float -> Prop) (p : float) (r : R),
RoundedModeP b radix P -> P r p -> (0 <= r)%R -> (0 <= p)%R.