intros P H' p q H'0 H'1 H'2 H'3 H'4 r H'5 H'6 H'7.
Casec H'0; intros H'0.
case (Rle_or_lt q (2%nat * p)); intros Rl1.
exists (Fminus radix r p); split; auto.
rewrite <- Fopp_Fminus.
apply oppBounded.
apply Sterbenz; auto.
apply (FcanonicBound radix b); auto with arith.
apply (FcanonicBound radix b); auto with arith.
apply Rmult_le_reg_l with (r := INR 2); auto with real.
rewrite <- Rmult_assoc; rewrite Rinv_r; auto with real arith;
rewrite Rmult_1_l; auto.
apply Rle_trans with (1 := H'7); auto.
apply Rle_trans with (1 := H'6); auto.
apply Rledouble; auto.
apply Rle_trans with (2 := H'6); apply Rlt_le; auto.
rewrite (Fminus_correct radix); auto with arith.
case (Rle_or_lt r (2%nat * p)); intros Rl2.
exists (Fminus radix r p); split; auto.
rewrite <- Fopp_Fminus.
apply oppBounded.
apply Sterbenz; auto.
apply (FcanonicBound radix b); auto with arith.
apply (FcanonicBound radix b); auto with arith.
apply Rmult_le_reg_l with (r := INR 2); auto with real.
rewrite <- Rmult_assoc; rewrite Rinv_r; auto with real arith;
rewrite Rmult_1_l; auto.
apply Rle_trans with (1 := H'6); auto.
apply Rledouble; auto.
apply Rle_trans with (2 := H'6); apply Rlt_le; auto.
rewrite (Fminus_correct radix); auto with arith.
apply ExactMinusIntervalAux with (P := P) (q := q); auto.
exists r; split; auto.
apply (FcanonicBound radix b); auto with arith.
rewrite <- H'0; ring.
Qed.

Theorem ExactMinusInterval :
forall P,
RoundedModeP b radix P ->
forall p q : float,
(0 <= p)%R ->
(p <= q)%R ->
Fbounded b p ->
Fbounded b q ->
(exists r : float, Fbounded b r /\ r = (q - p)%R :>R) ->
forall r : float,
Fbounded b r ->
(p <= r)%R ->
(r <= q)%R -> exists r' : float, Fbounded b r' /\ r' = (r - p)%R :>R.