intros f Hf1 Hf2; unfold Fulp in |- *.
replace (Fnormalize radix b precision f) with f;
[ idtac
| apply
FcanonicUnique with (radix := radix) (b := b) (precision := precision);
auto with float arith zarith ].
2: apply sym_eq; apply FnormalizeCorrect; auto with arith zarith.
replace (Fnormalize radix b precision (FPred b radix precision f)) with
(FPred b radix precision f);
[ idtac
| apply
FcanonicUnique with (radix := radix) (b := b) (precision := precision);
auto with float arith zarith ].
2: apply sym_eq; apply FnormalizeCorrect; auto with arith zarith.
pattern (IZR radix) at 2 in |- *; replace (IZR radix) with (powerRZ radix 1);
[ idtac | simpl in |- *; auto with arith zarith real ].
rewrite <- powerRZ_add; auto with zarith real.
apply Rle_powerRZ; auto with zarith real.
replace (1 + Fexp (FPred b radix precision f))%Z with
(Zsucc (Fexp (FPred b radix precision f))); auto with zarith.
unfold FPred in |- *.
generalize (Z_eq_bool_correct (Fnum f) (- pPred (vNum b)));
case (Z_eq_bool (Fnum f) (- pPred (vNum b))); intros H1;
[ simpl in |- *; auto with zarith | idtac ].
generalize (Z_eq_bool_correct (Fnum f) (nNormMin radix precision));
case (Z_eq_bool (Fnum f) (nNormMin radix precision));
intros H2; [ idtac | simpl in |- *; auto with zarith ].
generalize (Z_eq_bool_correct (Fexp f) (- dExp b));
case (Z_eq_bool (Fexp f) (- dExp b)); intros H3; simpl in |- *;
auto with zarith.
Qed.

Theorem ClosestErrorBoundNormal2_aux :
forall (x : R) (p : float),
Closest b radix x p ->
Fnormal radix b p ->
Fnormal radix b (Fnormalize radix b precision (FPred b radix precision p)) ->
(0 < x)%R ->
(x < p)%R ->
(Rabs (x - p) <= Rabs x * (/ 2%nat * powerRZ radix (Zsucc (- precision))))%R.