intros P H' p q qmp H'0 H'1 H'2 H'3 H'4 H'5.
case (Rle_or_lt (/ 2%nat * q) p); intros Rle1.
exists p; split; auto.
replace (FtoRradix qmp) with (FtoRradix (Fminus radix q p)).
rewrite (Fminus_correct radix); auto with arith; unfold FtoRradix in |- *;
ring.
apply (RoundedModeProjectorIdemEq b radix precision) with (P := P); auto.
rewrite <- Fopp_Fminus.
apply oppBounded; auto.
apply Sterbenz; auto.
apply Rle_trans with (FtoRradix q); auto with real.
apply Rledouble; auto.
apply Rle_trans with (FtoRradix p); auto with real.
cut (CompatibleP b radix P);
[ intros Cp | apply RoundedModeP_inv2 with (1 := H'); auto ].
apply (Cp (q - p)%R (Fminus radix q p) qmp); auto.
rewrite (Fminus_correct radix); auto with arith.
apply RoundedModeBounded with (radix := radix) (P := P) (r := (q - p)%R);
auto; auto.
exists (Fminus radix q qmp); split.
rewrite <- Fopp_Fminus.
apply oppBounded; auto.
apply Sterbenz; auto.
apply RoundedModeBounded with (radix := radix) (P := P) (r := (q - p)%R);
auto; auto.
case MaxEx with (r := (/ 2%nat * FtoR radix q)%R) (3 := pGivesBound);
auto with arith.
intros max H'6.
apply Rle_trans with (FtoRradix max);
[ apply isMax_inv1 with (1 := H'6); auto | idtac ].
apply (RleBoundRoundl b radix precision) with (P := P) (r := (q - p)%R); auto;
fold FtoRradix in |- *.
case H'6; auto.
case MinEx with (r := (/ 2%nat * FtoR radix q)%R) (3 := pGivesBound);
auto with arith.
intros min H'7.
replace (FtoRradix max) with (q - min)%R.
apply Rplus_le_reg_l with (r := (- q)%R).
cut (forall p q : R, (- p + (p - q))%R = (- q)%R);
[ intros tmp; repeat rewrite tmp; clear tmp | intros; ring ].
apply Ropp_le_contravar.
case H'7.
intros H'8 H'9; elim H'9; intros H'10 H'11; apply H'11; clear H'9; auto.
apply Rlt_le; auto.
unfold FtoRradix in |- *;
rewrite (div2IsBetween b radix precision) with (5 := H'7) (6 := H'6);
auto.
ring.
apply Rle_trans with (FtoRradix q); auto with real.
apply (RleBoundRoundr b radix precision) with (P := P) (r := (q - p)%R); auto;
fold FtoRradix in |- *.
apply Rplus_le_reg_l with (r := (- q)%R).
cut (forall p q : R, (- p + (p - q))%R = (- q)%R);
[ intros tmp; repeat rewrite tmp; clear tmp | intros; ring ].
replace (- q + q)%R with (-0)%R; [ auto with real | ring ].
apply Rle_trans with (FtoRradix q); auto with real.
apply Rledouble; auto.
apply Rle_trans with (FtoRradix p); auto with real.
apply (Fminus_correct radix); auto with arith.
Qed.

Theorem radixRangeBoundExp :
forall p q : float,
Fcanonic radix b p ->
Fcanonic radix b q ->
(0 <= p)%R ->
(p < q)%R -> (q < radix * p)%R -> Fexp p = Fexp q \/ Zsucc (Fexp p) = Fexp q.