intros P H' r q q' H'0 H'1 H'2.
case (Rle_or_lt 0 r); intros Rl0.
rewrite Rabs_right; auto.
apply RoundedModeMult with (P := P) (r := r); auto.
rewrite <- (Rabs_right r); auto with real.
apply Rle_ge; apply RleRoundedR0 with (P := P) (r := r); auto.
rewrite Faux.Rabsolu_left1; auto.
replace (radix * q')%R with (- (radix * - q'))%R;
[ apply Ropp_le_contravar | ring ].
rewrite <- (Fopp_correct radix).
apply RoundedModeMultLess with (P := P) (r := r); auto.
apply oppBounded; auto.
unfold FtoRradix in |- *; rewrite Fopp_correct.
rewrite <- (Ropp_involutive r).
replace (radix * - FtoR radix q')%R with (- (radix * q'))%R;
[ apply Ropp_le_contravar | fold FtoRradix;ring ]; auto.
rewrite <- (Faux.Rabsolu_left1 r); auto.
apply Rlt_le; auto.
apply RleRoundedLessR0 with (P := P) (r := r); auto.
apply Rlt_le; auto.
Qed.

Theorem isMinComp :
forall (r1 r2 : R) (min max : float),
isMin b radix r1 min ->
isMax b radix r1 max -> (min < r2)%R -> (r2 < max)%R -> isMin b radix r2 min.