Section Fcalc_round_fexp.
Variable fexp : Z -> Z.
Context { valid_exp : Valid_exp fexp }.
Notation format := (generic_format beta fexp).
(** Relates location and rounding. *)
Definition cond_incr (b : bool) m := if b then (m + 1)%Z else m.
(** Relates location and rounding down. *)
Definition round_sign_DN s l :=
match l with
| loc_Exact => false
| _ => s
end.
(** Relates location and rounding up. *)
Definition round_UP l :=
match l with
| loc_Exact => false
| _ => true
end.
Definition round_sign_UP s l :=
match l with
| loc_Exact => false
| _ => negb s
end.
(** Relates location and rounding toward zero. *)
Definition round_ZR (s : bool) l :=
match l with
| loc_Exact => false
| _ => s
end.
Proof with auto with typeclass_instances.
intros x m l Hl.
inversion_clear Hl as [Hx|l' Hx Hl'].
(* Exact *)
rewrite Hx.
rewrite Zrnd_IZR...
(* 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 IZR_le.
elim Rle_not_lt with (1 := Hx').
apply Rlt_le_trans with (1 := proj2 Hx).
apply IZR_le.
now apply Zlt_le_succ.
rewrite Hm.
now apply Rlt_not_eq.
Qed.
(** Relates location and rounding to nearest. *)
Definition round_N (p : bool) l :=
match l with
| loc_Exact => false
| loc_Inexact Lt => false
| loc_Inexact Eq => p
| loc_Inexact Gt => true
end.
Proof with auto with typeclass_instances.
intros choice x m l Hl.
inversion_clear Hl as [Hx|l' Hx Hl'].
(* Exact *)
rewrite Hx.
rewrite Zrnd_IZR...
(* not Exact *)
unfold Znearest.
assert (Hm: Zfloor x = m).
apply Zfloor_imp.
exact (conj (Rlt_le _ _ (proj1 Hx)) (proj2 Hx)).
rewrite Zceil_floor_neq.
rewrite Hm.
replace (Rcompare (x - IZR m) (/2)) with l'.
now case l'.
rewrite <- Hl'.
rewrite plus_IZR.
rewrite <- (Rcompare_plus_r (- IZR m) x).
apply f_equal.
field.
rewrite Hm.
now apply Rlt_not_eq.
Qed.
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_IZR...
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 - IZR m) (/2)) with l'.
now case l'.
rewrite <- Hl'.
rewrite plus_IZR.
rewrite <- (Rcompare_plus_r (- IZR m) (-x)).
apply f_equal.
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_IZR...
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 - IZR m) (/2)) with l'.
now case l'.
rewrite <- Hl'.
rewrite plus_IZR.
rewrite <- (Rcompare_plus_r (- IZR m) x).
apply f_equal.
field.
rewrite Hm.
now apply Rlt_not_eq.
Qed.
(** Relates location and rounding to nearest even. *)
(** Relates location and rounding to nearest away. *)

Theorem inbetween_float_NA :
forall x m l,
let e := cexp beta fexp x in
inbetween_float beta m e x l ->
round beta fexp ZnearestA x = F2R (Float beta (cond_incr (round_N (Zle_bool 0 m) l) m) e).