Proof with auto with typeclass_instances.
intros m x.
destruct (Req_dec x 0) as [Hx|Hx].
(* . *)
rewrite Hx, round_0...
unfold Rminus.
rewrite Rplus_0_l, Rabs_Ropp, Rabs_R0.
rewrite Rmult_0_r.
apply Rle_refl.
(* . *)
apply relative_error_N.
unfold x.
rewrite <- F2R_Zabs.
apply bpow_le_F2R.
apply F2R_lt_reg with beta emin.
rewrite F2R_0, F2R_Zabs.
now apply Rabs_pos_lt.
Qed.

(* Specifications: *)
Theorem relative_error_N_F2R_emin_ex :
forall m, let x := F2R (Float beta m emin) in
exists eps,
(Rabs eps <= /2 * bpow (-p + 1))%R /\ round beta fexp (Znearest choice) x = (x * (1 + eps))%R.
Theorem relative_error_N_F2R_emin_ex :
forall m, let x := F2R (Float beta m emin) in
exists eps,
(Rabs eps <= /2 * bpow (-p + 1))%R /\ round beta fexp (Znearest choice) x = (x * (1 + eps))%R.