Proof with auto with typeclass_instances.
intros choice x m l Hl.
simpl.
unfold Rabs in Hl.
destruct (Rcase_abs x) as [Zx|Zx].
(* *)
rewrite Rlt_bool_true with (1 := Zx).
simpl.
rewrite <- (Ropp_involutive x).
rewrite Znearest_opp.
apply f_equal.
inversion_clear Hl as [Hx|l' Hx Hl'].
rewrite Hx.
apply Zrnd_Z2R...
assert (Hm: Zfloor (-x) = m).
apply Zfloor_imp.
exact (conj (Rlt_le _ _ (proj1 Hx)) (proj2 Hx)).
unfold Znearest.
rewrite Zceil_floor_neq.
rewrite Hm.
replace (Rcompare (- x - Z2R m) (/2)) with l'.
now case l'.
rewrite <- Hl'.
rewrite Z2R_plus.
rewrite <- (Rcompare_plus_r (- Z2R m) (-x)).
apply f_equal.
simpl (Z2R 1).
field.
rewrite Hm.
now apply Rlt_not_eq.
(* *)
generalize (Rge_le _ _ Zx).
clear Zx. intros Zx.
rewrite Rlt_bool_false with (1 := Zx).
simpl.
inversion_clear Hl as [Hx|l' Hx Hl'].
rewrite Hx.
apply Zrnd_Z2R...
assert (Hm: Zfloor x = m).
apply Zfloor_imp.
exact (conj (Rlt_le _ _ (proj1 Hx)) (proj2 Hx)).
unfold Znearest.
rewrite Zceil_floor_neq.
rewrite Hm.
replace (Rcompare (x - Z2R m) (/2)) with l'.
now case l'.
rewrite <- Hl'.
rewrite Z2R_plus.
rewrite <- (Rcompare_plus_r (- Z2R m) x).
apply f_equal.
simpl (Z2R 1).
field.
rewrite Hm.
now apply Rlt_not_eq.
Qed.
(** Relates location and rounding to nearest even. *)

Theorem inbetween_int_NE_sign :
forall x m l,
inbetween_int m (Rabs x) l ->
ZnearestE x = cond_Zopp (Rlt_bool x 0) (cond_incr (round_N (negb (Zeven m)) l) m).