Proof with auto with typeclass_instances.
intros x m l Hl.
inversion_clear Hl as [Hx|l' Hx Hl'].
(* Exact *)
rewrite Hx.
rewrite Zrnd_Z2R...
(* not Exact *)
unfold Ztrunc.
assert (Hm: Zfloor x = m).
apply Zfloor_imp.
exact (conj (Rlt_le _ _ (proj1 Hx)) (proj2 Hx)).
rewrite Zceil_floor_neq.
rewrite Hm.
unfold cond_incr.
simpl.
case Rlt_bool_spec ; intros Hx' ;
case Zlt_bool_spec ; intros Hm' ; try apply refl_equal.
elim Rlt_not_le with (1 := Hx').
apply Rlt_le.
apply Rle_lt_trans with (2 := proj1 Hx).
now apply (Z2R_le 0).
elim Rle_not_lt with (1 := Hx').
apply Rlt_le_trans with (1 := proj2 Hx).
apply (Z2R_le _ 0).
now apply Zlt_le_succ.
rewrite Hm.
now apply Rlt_not_eq.
Qed.

(* Specifications: *)
Theorem inbetween_float_ZR :
forall x m l,
let e := canonic_exp beta fexp x in
inbetween_float beta m e x l ->
round beta fexp Ztrunc x = F2R (Float beta (cond_incr (round_ZR (Zlt_bool m 0) l) m) e).
Theorem inbetween_int_ZR_sign :
forall x m l,
inbetween_int m (Rabs x) l ->
Ztrunc x = cond_Zopp (Rlt_bool x 0) m.
Theorem inbetween_float_ZR_sign :
forall x m l,
let e := canonic_exp beta fexp x in
inbetween_float beta m e (Rabs x) l ->
round beta fexp Ztrunc x = F2R (Float beta (cond_Zopp (Rlt_bool x 0) m) e).