Proof with auto with typeclass_instances.
intros x m e l Hin [He|(Hl,Hf)].
rewrite He in Hin |- *.
apply inbetween_float_round_sign with (2 := Hin).
exact inbetween_int_valid.
rewrite Hl in Hin.
inversion_clear Hin.
rewrite Hl.
replace (choice (Rlt_bool x 0) m loc_Exact) with m.
(* *)
unfold Rabs in H.
destruct (Rcase_abs x) as [Zx|Zx].
rewrite Rlt_bool_true with (1 := Zx).
simpl.
rewrite F2R_Zopp.
rewrite <- H, Ropp_involutive.
apply round_generic...
rewrite Rlt_bool_false.
simpl.
rewrite <- H.
now apply round_generic.
now apply Rge_le.
(* *)
destruct (Rlt_bool_spec x 0) as [Zx|Zx].
(* . *)
apply Zopp_inj.
change (- m = cond_Zopp true (choice true m loc_Exact))%Z.
rewrite <- (Zrnd_Z2R rnd (-m)) at 1.
assert (Z2R (-m) < 0)%R.
rewrite Z2R_opp.
apply Ropp_lt_gt_0_contravar.
apply (Z2R_lt 0).
apply F2R_gt_0_reg with beta e.
rewrite <- H.
apply Rabs_pos_lt.
now apply Rlt_not_eq.
rewrite <- Rlt_bool_true with (1 := H0).
apply inbetween_int_valid.
constructor.
rewrite Rabs_left with (1 := H0).
rewrite Z2R_opp.
apply Ropp_involutive.
(* . *)
change (m = cond_Zopp false (choice false m loc_Exact))%Z.
rewrite <- (Zrnd_Z2R rnd m) at 1.
assert (0 <= Z2R m)%R.
apply (Z2R_le 0).
apply F2R_ge_0_reg with beta e.
rewrite <- H.
apply Rabs_pos.
rewrite <- Rlt_bool_false with (1 := H0).
apply inbetween_int_valid.
constructor.
now apply Rabs_pos_eq.
Qed.
(** Truncating a triple is sufficient to round a real number. *)

Theorem round_trunc_sign_any_correct :
forall x m e l,
inbetween_float beta m e (Rabs x) l ->
(e <= fexp (Zdigits beta m + e))%Z \/ l = loc_Exact ->
round beta fexp rnd x = let '(m', e', l') := truncate (m, e, l) in F2R (Float beta (cond_Zopp (Rlt_bool x 0) (choice (Rlt_bool x 0) m' l')) e').