Proof with auto with typeclass_instances.
intros choice u v Fu H.
(* . *)
assert (V: ((succ u = 0 /\ u = 0) \/ u < succ u)%R).
specialize (succ_ge_id u); intros P; destruct P as [P|P].
now right.
case (Req_dec u 0); intros Zu.
left; split; trivial.
now rewrite <- P.
right; now apply succ_gt_id.
(* *)
destruct V as [(V1,V2)|V].
rewrite V2; apply round_le_generic...
apply generic_format_0.
left; apply Rlt_le_trans with (1:=H).
rewrite V1,V2; right; field.
(* *)
assert (T: (u < (u + succ u) / 2 < succ u)%R) by lra.
destruct T as (T1,T2).
apply Rnd_N_pt_monotone with F v ((u + succ u) / 2)%R...
apply round_N_pt...
apply Rnd_DN_pt_N with (succ u)%R.
pattern u at 3; replace u with (round beta fexp Zfloor ((u + succ u) / 2)).
apply round_DN_pt...
apply round_DN_eq_betw; trivial.
split; try left; assumption.
pattern (succ u) at 2; replace (succ u) with (round beta fexp Zceil ((u + succ u) / 2)).
apply round_UP_pt...
apply round_UP_eq_betw; trivial.
apply generic_format_succ...
rewrite pred_succ; trivial.
split; try left; assumption.
right; field.
Qed.

(* Specifications: *)
Theorem round_N_ge_midp: forall choice u v,
F u ->  ((u + pred u)/2 < v)%R
-> (u <= round beta fexp (Znearest choice)  v)%R.
Theorem round_N_ge_midp: forall choice u v,
F u ->  ((u + pred u)/2 < v)%R
-> (u <= round beta fexp (Znearest choice)  v)%R.