intros.
case (Zle_or_lt (Fexp p) (Fexp q)); auto; intros Z1.
absurd (powerRZ radix (Fexp p) <= powerRZ radix (Fexp q))%R.
2: apply Rle_powerRZ; auto with real arith.
apply Rgt_not_le.
red in |- *; apply Rlt_le_trans with (2%nat * powerRZ radix (Fexp q))%R.
apply Rltdouble; auto with real arith.
apply Rle_trans with (2%nat * Fabs q)%R.
apply Rmult_le_compat_l; auto with real arith.
replace 0%R with (INR 0); auto with real arith.
replace (powerRZ radix (Fexp q)) with (FtoRradix (Float 1%nat (Fexp q)));
auto.
apply (Fop.RleFexpFabs radix); auto with real zarith.
unfold FtoRradix, FtoR in |- *; simpl in |- *; ring.
rewrite (Fabs_correct radix); auto with arith.
replace (FtoR radix q) with (x - p)%R; auto.
apply ClosestExp; auto.
Qed.

Theorem ClosestIdem :
forall p q : float, Fbounded b p -> Closest b radix p q -> p = q :>R.