Require Import Raux Defs Round_pred Float_prop.

Section Generic.

Variable beta : radix.

Notation bpow e := (bpow beta e).

Section Format.

Variable fexp : Z -> Z.

Class Valid_exp :=
  valid_exp :
  forall k : Z,
  ( (fexp k < k)%Z -> (fexp (k + 1) <= k)%Z ) /\
  ( (k <= fexp k)%Z ->
    (fexp (fexp k + 1) <= fexp k)%Z /\
    forall l : Z, (l <= fexp k)%Z -> fexp l = fexp k ).

Context { valid_exp_ : Valid_exp }.

Theorem valid_exp_large :
  forall k l,
  (fexp k < k)%Z -> (k <= l)%Z ->
  (fexp l < l)%Z.
Proof.
intros k l Hk H.
apply Znot_ge_lt.
intros Hl.
apply Z.ge_le in Hl.
assert (H' := proj2 (proj2 (valid_exp l) Hl) k).
omega.
Qed.

Theorem valid_exp_large' :
  forall k l,
  (fexp k < k)%Z -> (l <= k)%Z ->
  (fexp l < k)%Z.
Proof.
intros k l Hk H.
apply Znot_ge_lt.
intros H'.
apply Z.ge_le in H'.
assert (Hl := Z.le_trans _ _ _ H H').
apply valid_exp in Hl.
assert (H1 := proj2 Hl k H').
omega.
Qed.

Definition cexp x :=
  fexp (mag beta x).

Definition canonical (f : float beta) :=
  Fexp f = cexp (F2R f).

Definition scaled_mantissa x :=
  (x * bpow (- cexp x))%R.

Definition generic_format (x : R) :=
  x = F2R (Float beta (Ztrunc (scaled_mantissa x)) (cexp x)).

Theorem generic_format_0 :
  generic_format 0.
Proof.
unfold generic_format, scaled_mantissa.
rewrite Rmult_0_l.
now rewrite Ztrunc_IZR, F2R_0.
Qed.

Theorem cexp_opp :
  forall x,
  cexp (-x) = cexp x.
Proof.
intros x.
unfold cexp.
now rewrite mag_opp.
Qed.

Theorem cexp_abs :
  forall x,
  cexp (Rabs x) = cexp x.
Proof.
intros x.
unfold cexp.
now rewrite mag_abs.
Qed.

Theorem canonical_generic_format :
  forall x,
  generic_format x ->
  exists f : float beta,
  x = F2R f /\ canonical f.
Proof.
intros x Hx.
rewrite Hx.
eexists.
apply (conj eq_refl).
unfold canonical.
now rewrite <- Hx.
Qed.

Theorem generic_format_bpow :
  forall e, (fexp (e + 1) <= e)%Z ->
  generic_format (bpow e).
Proof.
intros e H.
unfold generic_format, scaled_mantissa, cexp.
rewrite mag_bpow.
rewrite <- bpow_plus.
rewrite <- (IZR_Zpower beta (e + - fexp (e + 1))).
rewrite Ztrunc_IZR.
rewrite <- F2R_bpow.
rewrite F2R_change_exp with (1 := H).
now rewrite Zmult_1_l.
now apply Zle_minus_le_0.
Qed.

Theorem generic_format_bpow' :
  forall e, (fexp e <= e)%Z ->
  generic_format (bpow e).
Proof.
intros e He.
apply generic_format_bpow.
destruct (Zle_lt_or_eq _ _ He).
now apply valid_exp_.
rewrite <- H.
apply valid_exp.
rewrite H.
apply Z.le_refl.
Qed.

Theorem generic_format_F2R :
  forall m e,
  ( m <> 0 -> cexp (F2R (Float beta m e)) <= e )%Z ->
  generic_format (F2R (Float beta m e)).
Proof.
intros m e.
destruct (Z.eq_dec m 0) as [Zm|Zm].
intros _.
rewrite Zm, F2R_0.
apply generic_format_0.
unfold generic_format, scaled_mantissa.
set (e' := cexp (F2R (Float beta m e))).
intros He.
specialize (He Zm).
unfold F2R at 3. simpl.
rewrite F2R_change_exp with (1 := He).
apply F2R_eq.
rewrite Rmult_assoc, <- bpow_plus, <- IZR_Zpower, <- mult_IZR.
now rewrite Ztrunc_IZR.
now apply Zle_left.
Qed.

Lemma generic_format_F2R' :
  forall (x : R) (f : float beta),
  F2R f = x ->
  (x <> 0%R -> (cexp x <= Fexp f)%Z) ->
  generic_format x.
Proof.
intros x f H1 H2.
rewrite <- H1; destruct f as (m,e).
apply generic_format_F2R.
simpl in *; intros H3.
rewrite H1; apply H2.
intros Y; apply H3.
apply eq_0_F2R with beta e.
now rewrite H1.
Qed.

Theorem canonical_opp :
  forall m e,
  canonical (Float beta m e) ->
  canonical (Float beta (-m) e).
Proof.
intros m e H.
unfold canonical.
now rewrite F2R_Zopp, cexp_opp.
Qed.

Theorem canonical_abs :
  forall m e,
  canonical (Float beta m e) ->
  canonical (Float beta (Z.abs m) e).
Proof.
intros m e H.
unfold canonical.
now rewrite F2R_Zabs, cexp_abs.
Qed.

Theorem canonical_0 :
  canonical (Float beta 0 (fexp (mag beta 0%R))).
Proof.
unfold canonical; simpl ; unfold cexp.
now rewrite F2R_0.
Qed.

Theorem canonical_unique :
  forall f1 f2,
  canonical f1 ->
  canonical f2 ->
  F2R f1 = F2R f2 ->
  f1 = f2.
Proof.
intros (m1, e1) (m2, e2).
unfold canonical. simpl.
intros H1 H2 H.
rewrite H in H1.
rewrite <- H2 in H1. clear H2.
rewrite H1 in H |- *.
apply (f_equal (fun m => Float beta m e2)).
apply eq_F2R with (1 := H).
Qed.

Theorem scaled_mantissa_generic :
  forall x,
  generic_format x ->
  scaled_mantissa x = IZR (Ztrunc (scaled_mantissa x)).
Proof.
intros x Hx.
unfold scaled_mantissa.
pattern x at 1 3 ; rewrite Hx.
unfold F2R. simpl.
rewrite Rmult_assoc, <- bpow_plus, Zplus_opp_r, Rmult_1_r.
now rewrite Ztrunc_IZR.
Qed.

Theorem scaled_mantissa_mult_bpow :
  forall x,
  (scaled_mantissa x * bpow (cexp x))%R = x.
Proof.
intros x.
unfold scaled_mantissa.
rewrite Rmult_assoc, <- bpow_plus, Zplus_opp_l.
apply Rmult_1_r.
Qed.

Theorem scaled_mantissa_0 :
  scaled_mantissa 0 = 0%R.
Proof.
apply Rmult_0_l.
Qed.

Theorem scaled_mantissa_opp :
  forall x,
  scaled_mantissa (-x) = (-scaled_mantissa x)%R.
Proof.
intros x.
unfold scaled_mantissa.
rewrite cexp_opp.
now rewrite Ropp_mult_distr_l_reverse.
Qed.

Theorem scaled_mantissa_abs :
  forall x,
  scaled_mantissa (Rabs x) = Rabs (scaled_mantissa x).
Proof.
intros x.
unfold scaled_mantissa.
rewrite cexp_abs, Rabs_mult.
apply f_equal.
apply sym_eq.
apply Rabs_pos_eq.
apply bpow_ge_0.
Qed.

Theorem generic_format_opp :
  forall x, generic_format x -> generic_format (-x).
Proof.
intros x Hx.
unfold generic_format.
rewrite scaled_mantissa_opp, cexp_opp.
rewrite Ztrunc_opp.
rewrite F2R_Zopp.
now apply f_equal.
Qed.

Theorem generic_format_abs :
  forall x, generic_format x -> generic_format (Rabs x).
Proof.
intros x Hx.
unfold generic_format.
rewrite scaled_mantissa_abs, cexp_abs.
rewrite Ztrunc_abs.
rewrite F2R_Zabs.
now apply f_equal.
Qed.

Theorem generic_format_abs_inv :
  forall x, generic_format (Rabs x) -> generic_format x.
Proof.
intros x.
unfold generic_format, Rabs.
case Rcase_abs ; intros _.
rewrite scaled_mantissa_opp, cexp_opp, Ztrunc_opp.
intros H.
rewrite <- (Ropp_involutive x) at 1.
rewrite H, F2R_Zopp.
apply Ropp_involutive.
easy.
Qed.

Theorem cexp_fexp :
  forall x ex,
  (bpow (ex - 1) <= Rabs x < bpow ex)%R ->
  cexp x = fexp ex.
Proof.
intros x ex Hx.
unfold cexp.
now rewrite mag_unique with (1 := Hx).
Qed.

Theorem cexp_fexp_pos :
  forall x ex,
  (bpow (ex - 1) <= x < bpow ex)%R ->
  cexp x = fexp ex.
Proof.
intros x ex Hx.
apply cexp_fexp.
rewrite Rabs_pos_eq.
exact Hx.
apply Rle_trans with (2 := proj1 Hx).
apply bpow_ge_0.
Qed.

Theorem mantissa_small_pos :
  forall x ex,
  (bpow (ex - 1) <= x < bpow ex)%R ->
  (ex <= fexp ex)%Z ->
  (0 < x * bpow (- fexp ex) < 1)%R.
Proof.
intros x ex Hx He.
split.
apply Rmult_lt_0_compat.
apply Rlt_le_trans with (2 := proj1 Hx).
apply bpow_gt_0.
apply bpow_gt_0.
apply Rmult_lt_reg_r with (bpow (fexp ex)).
apply bpow_gt_0.
rewrite Rmult_assoc, <- bpow_plus, Zplus_opp_l.
rewrite Rmult_1_r, Rmult_1_l.
apply Rlt_le_trans with (1 := proj2 Hx).
now apply bpow_le.
Qed.

Theorem scaled_mantissa_lt_1 :
  forall x ex,
  (Rabs x < bpow ex)%R ->
  (ex <= fexp ex)%Z ->
  (Rabs (scaled_mantissa x) < 1)%R.
Proof.
intros x ex Ex He.
destruct (Req_dec x 0) as [Zx|Zx].
rewrite Zx, scaled_mantissa_0, Rabs_R0.
now apply IZR_lt.
rewrite <- scaled_mantissa_abs.
unfold scaled_mantissa.
rewrite cexp_abs.
unfold cexp.
destruct (mag beta x) as (ex', Ex').
simpl.
specialize (Ex' Zx).
apply (mantissa_small_pos _ _ Ex').
assert (ex' <= fexp ex)%Z.
apply Z.le_trans with (2 := He).
apply bpow_lt_bpow with beta.
now apply Rle_lt_trans with (2 := Ex).
now rewrite (proj2 (proj2 (valid_exp _) He)).
Qed.

Theorem scaled_mantissa_lt_bpow :
  forall x,
  (Rabs (scaled_mantissa x) < bpow (mag beta x - cexp x))%R.
Proof.
intros x.
destruct (Req_dec x 0) as [Zx|Zx].
rewrite Zx, scaled_mantissa_0, Rabs_R0.
apply bpow_gt_0.
apply Rlt_le_trans with (1 := bpow_mag_gt beta _).
apply bpow_le.
unfold scaled_mantissa.
rewrite mag_mult_bpow with (1 := Zx).
apply Z.le_refl.
Qed.

Theorem mag_generic_gt :
  forall x, (x <> 0)%R ->
  generic_format x ->
  (cexp x < mag beta x)%Z.
Proof.
intros x Zx Gx.
apply Znot_ge_lt.
unfold cexp.
destruct (mag beta x) as (ex,Ex) ; simpl.
specialize (Ex Zx).
intros H.
apply Z.ge_le in H.
generalize (scaled_mantissa_lt_1 x ex (proj2 Ex) H).
contradict Zx.
rewrite Gx.
replace (Ztrunc (scaled_mantissa x)) with Z0.
apply F2R_0.
cut (Z.abs (Ztrunc (scaled_mantissa x)) < 1)%Z.
clear ; zify ; omega.
apply lt_IZR.
rewrite abs_IZR.
now rewrite <- scaled_mantissa_generic.
Qed.

Lemma mantissa_DN_small_pos :
  forall x ex,
  (bpow (ex - 1) <= x < bpow ex)%R ->
  (ex <= fexp ex)%Z ->
  Zfloor (x * bpow (- fexp ex)) = Z0.
Proof.
intros x ex Hx He.
apply Zfloor_imp. simpl.
assert (H := mantissa_small_pos x ex Hx He).
split ; try apply Rlt_le ; apply H.
Qed.

Lemma mantissa_UP_small_pos :
  forall x ex,
  (bpow (ex - 1) <= x < bpow ex)%R ->
  (ex <= fexp ex)%Z ->
  Zceil (x * bpow (- fexp ex)) = 1%Z.
Proof.
intros x ex Hx He.
apply Zceil_imp. simpl.
assert (H := mantissa_small_pos x ex Hx He).
split ; try apply Rlt_le ; apply H.
Qed.

Theorem generic_format_discrete :
  forall x m,
  let e := cexp x in
  (F2R (Float beta m e) < x < F2R (Float beta (m + 1) e))%R ->
  ~ generic_format x.
Proof.
intros x m e (Hx,Hx2) Hf.
apply Rlt_not_le with (1 := Hx2). clear Hx2.
rewrite Hf.
fold e.
apply F2R_le.
apply Zlt_le_succ.
apply lt_IZR.
rewrite <- scaled_mantissa_generic with (1 := Hf).
apply Rmult_lt_reg_r with (bpow e).
apply bpow_gt_0.
now rewrite scaled_mantissa_mult_bpow.
Qed.

Theorem generic_format_canonical :
  forall f, canonical f ->
  generic_format (F2R f).
Proof.
intros (m, e) Hf.
unfold canonical in Hf. simpl in Hf.
unfold generic_format, scaled_mantissa.
rewrite <- Hf.
apply F2R_eq.
unfold F2R. simpl.
rewrite Rmult_assoc, <- bpow_plus, Zplus_opp_r, Rmult_1_r.
now rewrite Ztrunc_IZR.
Qed.

Theorem generic_format_ge_bpow :
  forall emin,
  ( forall e, (emin <= fexp e)%Z ) ->
  forall x,
  (0 < x)%R ->
  generic_format x ->
  (bpow emin <= x)%R.
Proof.
intros emin Emin x Hx Fx.
rewrite Fx.
apply Rle_trans with (bpow (fexp (mag beta x))).
now apply bpow_le.
apply bpow_le_F2R.
apply gt_0_F2R with beta (cexp x).
now rewrite <- Fx.
Qed.

Theorem abs_lt_bpow_prec:
  forall prec,
  (forall e, (e - prec <= fexp e)%Z) ->
  
  forall x,
  (Rabs x < bpow (prec + cexp x))%R.
intros prec Hp x.
case (Req_dec x 0); intros Hxz.
rewrite Hxz, Rabs_R0.
apply bpow_gt_0.
unfold cexp.
destruct (mag beta x) as (ex,Ex) ; simpl.
specialize (Ex Hxz).
apply Rlt_le_trans with (1 := proj2 Ex).
apply bpow_le.
specialize (Hp ex).
omega.
Qed.

Theorem generic_format_bpow_inv' :
  forall e,
  generic_format (bpow e) ->
  (fexp (e + 1) <= e)%Z.
Proof.
intros e He.
apply Znot_gt_le.
contradict He.
unfold generic_format, scaled_mantissa, cexp, F2R. simpl.
rewrite mag_bpow, <- bpow_plus.
apply Rgt_not_eq.
rewrite Ztrunc_floor.
2: apply bpow_ge_0.
rewrite Zfloor_imp with (n := Z0).
rewrite Rmult_0_l.
apply bpow_gt_0.
split.
apply bpow_ge_0.
apply (bpow_lt _ _ 0).
clear -He ; omega.
Qed.

Theorem generic_format_bpow_inv :
  forall e,
  generic_format (bpow e) ->
  (fexp e <= e)%Z.
Proof.
intros e He.
apply generic_format_bpow_inv' in He.
assert (H := valid_exp_large' (e + 1) e).
omega.
Qed.

Section Fcore_generic_round_pos.

Variable rnd : R -> Z.

Class Valid_rnd := {
  Zrnd_le : forall x y, (x <= y)%R -> (rnd x <= rnd y)%Z ;
  Zrnd_IZR : forall n, rnd (IZR n) = n
}.

Context { valid_rnd : Valid_rnd }.

Theorem Zrnd_DN_or_UP :
  forall x, rnd x = Zfloor x \/ rnd x = Zceil x.
Proof.
intros x.
destruct (Zle_or_lt (rnd x) (Zfloor x)) as [Hx|Hx].
left.
apply Zle_antisym with (1 := Hx).
rewrite <- (Zrnd_IZR (Zfloor x)).
apply Zrnd_le.
apply Zfloor_lb.
right.
apply Zle_antisym.
rewrite <- (Zrnd_IZR (Zceil x)).
apply Zrnd_le.
apply Zceil_ub.
rewrite Zceil_floor_neq.
omega.
intros H.
rewrite <- H in Hx.
rewrite Zfloor_IZR, Zrnd_IZR in Hx.
apply Z.lt_irrefl with (1 := Hx).
Qed.

Theorem Zrnd_ZR_or_AW :
  forall x, rnd x = Ztrunc x \/ rnd x = Zaway x.
Proof.
intros x.
unfold Ztrunc, Zaway.
destruct (Zrnd_DN_or_UP x) as [Hx|Hx] ;
  case Rlt_bool.
now right.
now left.
now left.
now right.
Qed.

Definition round x :=
  F2R (Float beta (rnd (scaled_mantissa x)) (cexp x)).

Theorem round_bounded_large_pos :
  forall x ex,
  (fexp ex < ex)%Z ->
  (bpow (ex - 1) <= x < bpow ex)%R ->
  (bpow (ex - 1) <= round x <= bpow ex)%R.
Proof.
intros x ex He Hx.
unfold round, scaled_mantissa.
rewrite (cexp_fexp_pos _ _ Hx).
unfold F2R. simpl.
destruct (Zrnd_DN_or_UP (x * bpow (- fexp ex))) as [Hr|Hr] ; rewrite Hr.
split.
replace (ex - 1)%Z with (ex - 1 + - fexp ex + fexp ex)%Z by ring.
rewrite bpow_plus.
apply Rmult_le_compat_r.
apply bpow_ge_0.
assert (Hf: IZR (Zpower beta (ex - 1 - fexp ex)) = bpow (ex - 1 + - fexp ex)).
apply IZR_Zpower.
omega.
rewrite <- Hf.
apply IZR_le.
apply Zfloor_lub.
rewrite Hf.
rewrite bpow_plus.
apply Rmult_le_compat_r.
apply bpow_ge_0.
apply Hx.
apply Rle_trans with (2 := Rlt_le _ _ (proj2 Hx)).
apply Rmult_le_reg_r with (bpow (- fexp ex)).
apply bpow_gt_0.
rewrite Rmult_assoc, <- bpow_plus, Zplus_opp_r, Rmult_1_r.
apply Zfloor_lb.
split.
apply Rle_trans with (1 := proj1 Hx).
apply Rmult_le_reg_r with (bpow (- fexp ex)).
apply bpow_gt_0.
rewrite Rmult_assoc, <- bpow_plus, Zplus_opp_r, Rmult_1_r.
apply Zceil_ub.
pattern ex at 3 ; replace ex with (ex - fexp ex + fexp ex)%Z by ring.
rewrite bpow_plus.
apply Rmult_le_compat_r.
apply bpow_ge_0.
assert (Hf: IZR (Zpower beta (ex - fexp ex)) = bpow (ex - fexp ex)).
apply IZR_Zpower.
omega.
rewrite <- Hf.
apply IZR_le.
apply Zceil_glb.
rewrite Hf.
unfold Zminus.
rewrite bpow_plus.
apply Rmult_le_compat_r.
apply bpow_ge_0.
apply Rlt_le.
apply Hx.
Qed.

Theorem round_bounded_small_pos :
  forall x ex,
  (ex <= fexp ex)%Z ->
  (bpow (ex - 1) <= x < bpow ex)%R ->
  round x = 0%R \/ round x = bpow (fexp ex).
Proof.
intros x ex He Hx.
unfold round, scaled_mantissa.
rewrite (cexp_fexp_pos _ _ Hx).
unfold F2R. simpl.
destruct (Zrnd_DN_or_UP (x * bpow (-fexp ex))) as [Hr|Hr] ; rewrite Hr.
left.
apply Rmult_eq_0_compat_r.
apply IZR_eq.
apply Zfloor_imp.
refine (let H := _ in conj (Rlt_le _ _ (proj1 H)) (proj2 H)).
now apply mantissa_small_pos.
right.
pattern (bpow (fexp ex)) at 2 ; rewrite <- Rmult_1_l.
apply (f_equal (fun m => (m * bpow (fexp ex))%R)).
apply IZR_eq.
apply Zceil_imp.
refine (let H := _ in conj (proj1 H) (Rlt_le _ _ (proj2 H))).
now apply mantissa_small_pos.
Qed.

Lemma round_le_pos :
  forall x y, (0 < x)%R -> (x <= y)%R -> (round x <= round y)%R.
Proof.
intros x y Hx Hxy.
destruct (mag beta x) as [ex Hex].
destruct (mag beta y) as [ey Hey].
specialize (Hex (Rgt_not_eq _ _ Hx)).
specialize (Hey (Rgt_not_eq _ _ (Rlt_le_trans _ _ _ Hx Hxy))).
rewrite Rabs_pos_eq in Hex.
2: now apply Rlt_le.
rewrite Rabs_pos_eq in Hey.
2: apply Rle_trans with (2:=Hxy); now apply Rlt_le.
assert (He: (ex <= ey)%Z).
  apply bpow_lt_bpow with beta.
  apply Rle_lt_trans with (1 := proj1 Hex).
  now apply Rle_lt_trans with y.
assert (Heq: fexp ex = fexp ey -> (round x <= round y)%R).
  intros H.
  unfold round, scaled_mantissa, cexp.
  rewrite mag_unique_pos with (1 := Hex).
  rewrite mag_unique_pos with (1 := Hey).
  rewrite H.
  apply F2R_le.
  apply Zrnd_le.
  apply Rmult_le_compat_r with (2 := Hxy).
  apply bpow_ge_0.
destruct (Zle_or_lt ey (fexp ey)) as [Hy1|Hy1].
  apply Heq.
  apply valid_exp with (1 := Hy1).
  now apply Z.le_trans with ey.
destruct (Zle_lt_or_eq _ _ He) as [He'|He'].
2: now apply Heq, f_equal.
apply Rle_trans with (bpow (ey - 1)).
2: now apply round_bounded_large_pos.
destruct (Zle_or_lt ex (fexp ex)) as [Hx1|Hx1].
  destruct (round_bounded_small_pos _ _ Hx1 Hex) as [-> | ->].
  apply bpow_ge_0.
  apply bpow_le.
  apply valid_exp, proj2 in Hx1.
  specialize (Hx1 ey).
  omega.
apply Rle_trans with (bpow ex).
now apply round_bounded_large_pos.
apply bpow_le.
now apply Z.lt_le_pred.
Qed.

Theorem round_generic :
  forall x,
  generic_format x ->
  round x = x.
Proof.
intros x Hx.
unfold round.
rewrite scaled_mantissa_generic with (1 := Hx).
rewrite Zrnd_IZR.
now apply sym_eq.
Qed.

Theorem round_0 :
  round 0 = 0%R.
Proof.
unfold round, scaled_mantissa.
rewrite Rmult_0_l.
rewrite Zrnd_IZR.
apply F2R_0.
Qed.

Theorem exp_small_round_0_pos :
  forall x ex,
  (bpow (ex - 1) <= x < bpow ex)%R ->
  round x = 0%R -> (ex <= fexp ex)%Z .
Proof.
intros x ex H H1.
case (Zle_or_lt ex (fexp ex)); trivial; intros V.
contradict H1.
apply Rgt_not_eq.
apply Rlt_le_trans with (bpow (ex-1)).
apply bpow_gt_0.
apply (round_bounded_large_pos); assumption.
Qed.

Lemma generic_format_round_pos :
  forall x,
  (0 < x)%R ->
  generic_format (round x).
Proof.
intros x Hx0.
destruct (mag beta x) as (ex, Hex).
specialize (Hex (Rgt_not_eq _ _ Hx0)).
rewrite Rabs_pos_eq in Hex. 2: now apply Rlt_le.
destruct (Zle_or_lt ex (fexp ex)) as [He|He].
destruct (round_bounded_small_pos _ _ He Hex) as [Hr|Hr] ; rewrite Hr.
apply generic_format_0.
apply generic_format_bpow.
now apply valid_exp.
generalize (round_bounded_large_pos _ _ He Hex).
intros (Hr1, Hr2).
destruct (Rle_or_lt (bpow ex) (round x)) as [Hr|Hr].
rewrite <- (Rle_antisym _ _ Hr Hr2).
apply generic_format_bpow.
now apply valid_exp.
apply generic_format_F2R.
intros _.
rewrite (cexp_fexp_pos (F2R _) _ (conj Hr1 Hr)).
rewrite (cexp_fexp_pos _ _ Hex).
now apply Zeq_le.
Qed.

End Fcore_generic_round_pos.

Theorem round_ext :
  forall rnd1 rnd2,
  ( forall x, rnd1 x = rnd2 x ) ->
  forall x,
  round rnd1 x = round rnd2 x.
Proof.
intros rnd1 rnd2 Hext x.
unfold round.
now rewrite Hext.
Qed.

Section Zround_opp.

Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.

Definition Zrnd_opp x := Z.opp (rnd (-x)).

Global Instance valid_rnd_opp : Valid_rnd Zrnd_opp.
Proof with auto with typeclass_instances.
split.
intros x y Hxy.
unfold Zrnd_opp.
apply Zopp_le_cancel.
rewrite 2!Z.opp_involutive.
apply Zrnd_le...
now apply Ropp_le_contravar.
intros n.
unfold Zrnd_opp.
rewrite <- opp_IZR, Zrnd_IZR...
apply Z.opp_involutive.
Qed.

Theorem round_opp :
  forall x,
  round rnd (- x) = Ropp (round Zrnd_opp x).
Proof.
intros x.
unfold round.
rewrite <- F2R_Zopp, cexp_opp, scaled_mantissa_opp.
apply F2R_eq.
apply sym_eq.
exact (Z.opp_involutive _).
Qed.

End Zround_opp.

Global Instance valid_rnd_DN : Valid_rnd Zfloor.
Proof.
split.
apply Zfloor_le.
apply Zfloor_IZR.
Qed.

Global Instance valid_rnd_UP : Valid_rnd Zceil.
Proof.
split.
apply Zceil_le.
apply Zceil_IZR.
Qed.

Global Instance valid_rnd_ZR : Valid_rnd Ztrunc.
Proof.
split.
apply Ztrunc_le.
apply Ztrunc_IZR.
Qed.

Global Instance valid_rnd_AW : Valid_rnd Zaway.
Proof.
split.
apply Zaway_le.
apply Zaway_IZR.
Qed.

Section monotone.

Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.

Theorem round_DN_or_UP :
  forall x,
  round rnd x = round Zfloor x \/ round rnd x = round Zceil x.
Proof.
intros x.
unfold round.
destruct (Zrnd_DN_or_UP rnd (scaled_mantissa x)) as [Hx|Hx].
left. now rewrite Hx.
right. now rewrite Hx.
Qed.

Theorem round_ZR_or_AW :
  forall x,
  round rnd x = round Ztrunc x \/ round rnd x = round Zaway x.
Proof.
intros x.
unfold round.
destruct (Zrnd_ZR_or_AW rnd (scaled_mantissa x)) as [Hx|Hx].
left. now rewrite Hx.
right. now rewrite Hx.
Qed.

Theorem round_le :
  forall x y, (x <= y)%R -> (round rnd x <= round rnd y)%R.
Proof with auto with typeclass_instances.
intros x y Hxy.
destruct (total_order_T x 0) as [[Hx|Hx]|Hx].
3: now apply round_le_pos.
unfold round.
destruct (Rlt_or_le y 0) as [Hy|Hy].
rewrite <- (Ropp_involutive x), <- (Ropp_involutive y).
rewrite (scaled_mantissa_opp (-x)), (scaled_mantissa_opp (-y)).
rewrite (cexp_opp (-x)), (cexp_opp (-y)).
apply Ropp_le_cancel.
rewrite <- 2!F2R_Zopp.
apply (round_le_pos (Zrnd_opp rnd) (-y) (-x)).
rewrite <- Ropp_0.
now apply Ropp_lt_contravar.
now apply Ropp_le_contravar.
apply Rle_trans with 0%R.
apply F2R_le_0. simpl.
rewrite <- (Zrnd_IZR rnd 0).
apply Zrnd_le...
simpl.
rewrite <- (Rmult_0_l (bpow (- fexp (mag beta x)))).
apply Rmult_le_compat_r.
apply bpow_ge_0.
now apply Rlt_le.
apply F2R_ge_0. simpl.
rewrite <- (Zrnd_IZR rnd 0).
apply Zrnd_le...
apply Rmult_le_pos.
exact Hy.
apply bpow_ge_0.
rewrite Hx.
rewrite round_0...
apply F2R_ge_0.
simpl.
rewrite <- (Zrnd_IZR rnd 0).
apply Zrnd_le...
apply Rmult_le_pos.
now rewrite <- Hx.
apply bpow_ge_0.
Qed.

Theorem round_ge_generic :
  forall x y, generic_format x -> (x <= y)%R -> (x <= round rnd y)%R.
Proof.
intros x y Hx Hxy.
rewrite <- (round_generic rnd x Hx).
now apply round_le.
Qed.

Theorem round_le_generic :
  forall x y, generic_format y -> (x <= y)%R -> (round rnd x <= y)%R.
Proof.
intros x y Hy Hxy.
rewrite <- (round_generic rnd y Hy).
now apply round_le.
Qed.

End monotone.

Theorem round_abs_abs :
  forall P : R -> R -> Prop,
  ( forall rnd (Hr : Valid_rnd rnd) x, (0 <= x)%R -> P x (round rnd x) ) ->
  forall rnd {Hr : Valid_rnd rnd} x, P (Rabs x) (Rabs (round rnd x)).
Proof with auto with typeclass_instances.
intros P HP rnd Hr x.
destruct (Rle_or_lt 0 x) as [Hx|Hx].
rewrite 2!Rabs_pos_eq.
now apply HP.
rewrite <- (round_0 rnd).
now apply round_le.
exact Hx.
rewrite (Rabs_left _ Hx).
rewrite Rabs_left1.
pattern x at 2 ; rewrite <- Ropp_involutive.
rewrite round_opp.
rewrite Ropp_involutive.
apply HP...
rewrite <- Ropp_0.
apply Ropp_le_contravar.
now apply Rlt_le.
rewrite <- (round_0 rnd).
apply round_le...
now apply Rlt_le.
Qed.

Theorem round_bounded_large :
  forall rnd {Hr : Valid_rnd rnd} x ex,
  (fexp ex < ex)%Z ->
  (bpow (ex - 1) <= Rabs x < bpow ex)%R ->
  (bpow (ex - 1) <= Rabs (round rnd x) <= bpow ex)%R.
Proof with auto with typeclass_instances.
intros rnd Hr x ex He.
apply round_abs_abs...
clear rnd Hr x.
intros rnd' Hr x _.
apply round_bounded_large_pos...
Qed.

Theorem exp_small_round_0 :
  forall rnd {Hr : Valid_rnd rnd} x ex,
  (bpow (ex - 1) <= Rabs x < bpow ex)%R ->
   round rnd x = 0%R -> (ex <= fexp ex)%Z .
Proof.
intros rnd Hr x ex H1 H2.
generalize Rabs_R0.
rewrite <- H2 at 1.
apply (round_abs_abs (fun t rt => forall (ex : Z),
(bpow (ex - 1) <= t < bpow ex)%R ->
rt = 0%R -> (ex <= fexp ex)%Z)); trivial.
clear; intros rnd Hr x Hx.
now apply exp_small_round_0_pos.
Qed.

Section monotone_abs.

Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.

Theorem abs_round_ge_generic :
  forall x y, generic_format x -> (x <= Rabs y)%R -> (x <= Rabs (round rnd y))%R.
Proof with auto with typeclass_instances.
intros x y.
apply round_abs_abs...
clear rnd valid_rnd y.
intros rnd' Hrnd y Hy.
apply round_ge_generic...
Qed.

Theorem abs_round_le_generic :
  forall x y, generic_format y -> (Rabs x <= y)%R -> (Rabs (round rnd x) <= y)%R.
Proof with auto with typeclass_instances.
intros x y.
apply round_abs_abs...
clear rnd valid_rnd x.
intros rnd' Hrnd x Hx.
apply round_le_generic...
Qed.

End monotone_abs.

Theorem round_DN_opp :
  forall x,
  round Zfloor (-x) = (- round Zceil x)%R.
Proof.
intros x.
unfold round.
rewrite scaled_mantissa_opp.
rewrite <- F2R_Zopp.
unfold Zceil.
rewrite Z.opp_involutive.
now rewrite cexp_opp.
Qed.

Theorem round_UP_opp :
  forall x,
  round Zceil (-x) = (- round Zfloor x)%R.
Proof.
intros x.
unfold round.
rewrite scaled_mantissa_opp.
rewrite <- F2R_Zopp.
unfold Zceil.
rewrite Ropp_involutive.
now rewrite cexp_opp.
Qed.

Theorem round_ZR_opp :
  forall x,
  round Ztrunc (- x) = Ropp (round Ztrunc x).
Proof.
intros x.
unfold round.
rewrite scaled_mantissa_opp, cexp_opp, Ztrunc_opp.
apply F2R_Zopp.
Qed.

Theorem round_ZR_abs :
  forall x,
  round Ztrunc (Rabs x) = Rabs (round Ztrunc x).
Proof with auto with typeclass_instances.
intros x.
apply sym_eq.
unfold Rabs at 2.
destruct (Rcase_abs x) as [Hx|Hx].
rewrite round_ZR_opp.
apply Rabs_left1.
rewrite <- (round_0 Ztrunc).
apply round_le...
now apply Rlt_le.
apply Rabs_pos_eq.
rewrite <- (round_0 Ztrunc).
apply round_le...
now apply Rge_le.
Qed.

Theorem round_AW_opp :
  forall x,
  round Zaway (- x) = Ropp (round Zaway x).
Proof.
intros x.
unfold round.
rewrite scaled_mantissa_opp, cexp_opp, Zaway_opp.
apply F2R_Zopp.
Qed.

Theorem round_AW_abs :
  forall x,
  round Zaway (Rabs x) = Rabs (round Zaway x).
Proof with auto with typeclass_instances.
intros x.
apply sym_eq.
unfold Rabs at 2.
destruct (Rcase_abs x) as [Hx|Hx].
rewrite round_AW_opp.
apply Rabs_left1.
rewrite <- (round_0 Zaway).
apply round_le...
now apply Rlt_le.
apply Rabs_pos_eq.
rewrite <- (round_0 Zaway).
apply round_le...
now apply Rge_le.
Qed.

Theorem round_ZR_DN :
  forall x,
  (0 <= x)%R ->
  round Ztrunc x = round Zfloor x.
Proof.
intros x Hx.
unfold round, Ztrunc.
case Rlt_bool_spec.
intros H.
elim Rlt_not_le with (1 := H).
rewrite <- (Rmult_0_l (bpow (- cexp x))).
apply Rmult_le_compat_r with (2 := Hx).
apply bpow_ge_0.
easy.
Qed.

Theorem round_ZR_UP :
  forall x,
  (x <= 0)%R ->
  round Ztrunc x = round Zceil x.
Proof.
intros x Hx.
unfold round, Ztrunc.
case Rlt_bool_spec.
easy.
intros [H|H].
elim Rlt_not_le with (1 := H).
rewrite <- (Rmult_0_l (bpow (- cexp x))).
apply Rmult_le_compat_r with (2 := Hx).
apply bpow_ge_0.
rewrite <- H.
now rewrite Zfloor_IZR, Zceil_IZR.
Qed.

Theorem round_AW_UP :
  forall x,
  (0 <= x)%R ->
  round Zaway x = round Zceil x.
Proof.
intros x Hx.
unfold round, Zaway.
case Rlt_bool_spec.
intros H.
elim Rlt_not_le with (1 := H).
rewrite <- (Rmult_0_l (bpow (- cexp x))).
apply Rmult_le_compat_r with (2 := Hx).
apply bpow_ge_0.
easy.
Qed.

Theorem round_AW_DN :
  forall x,
  (x <= 0)%R ->
  round Zaway x = round Zfloor x.
Proof.
intros x Hx.
unfold round, Zaway.
case Rlt_bool_spec.
easy.
intros [H|H].
elim Rlt_not_le with (1 := H).
rewrite <- (Rmult_0_l (bpow (- cexp x))).
apply Rmult_le_compat_r with (2 := Hx).
apply bpow_ge_0.
rewrite <- H.
now rewrite Zfloor_IZR, Zceil_IZR.
Qed.

Theorem generic_format_round :
  forall rnd { Hr : Valid_rnd rnd } x,
  generic_format (round rnd x).
Proof with auto with typeclass_instances.
intros rnd Zrnd x.
destruct (total_order_T x 0) as [[Hx|Hx]|Hx].
rewrite <- (Ropp_involutive x).
destruct (round_DN_or_UP rnd (- - x)) as [Hr|Hr] ; rewrite Hr.
rewrite round_DN_opp.
apply generic_format_opp.
apply generic_format_round_pos...
now apply Ropp_0_gt_lt_contravar.
rewrite round_UP_opp.
apply generic_format_opp.
apply generic_format_round_pos...
now apply Ropp_0_gt_lt_contravar.
rewrite Hx.
rewrite round_0...
apply generic_format_0.
now apply generic_format_round_pos.
Qed.

Theorem round_DN_pt :
  forall x,
  Rnd_DN_pt generic_format x (round Zfloor x).
Proof with auto with typeclass_instances.
intros x.
split.
apply generic_format_round...
split.
pattern x at 2 ; rewrite <- scaled_mantissa_mult_bpow.
unfold round, F2R. simpl.
apply Rmult_le_compat_r.
apply bpow_ge_0.
apply Zfloor_lb.
intros g Hg Hgx.
apply round_ge_generic...
Qed.

Theorem generic_format_satisfies_any :
  satisfies_any generic_format.
Proof.
split.
exact generic_format_0.
exact generic_format_opp.
intros x.
eexists.
apply round_DN_pt.
Qed.

Theorem round_UP_pt :
  forall x,
  Rnd_UP_pt generic_format x (round Zceil x).
Proof.
intros x.
rewrite <- (Ropp_involutive x).
rewrite round_UP_opp.
apply Rnd_UP_pt_opp.
apply generic_format_opp.
apply round_DN_pt.
Qed.

Theorem round_ZR_pt :
  forall x,
  Rnd_ZR_pt generic_format x (round Ztrunc x).
Proof.
intros x.
split ; intros Hx.
rewrite round_ZR_DN with (1 := Hx).
apply round_DN_pt.
rewrite round_ZR_UP with (1 := Hx).
apply round_UP_pt.
Qed.

Lemma round_DN_small_pos :
  forall x ex,
  (bpow (ex - 1) <= x < bpow ex)%R ->
  (ex <= fexp ex)%Z ->
  round Zfloor x = 0%R.
Proof.
intros x ex Hx He.
rewrite <- (F2R_0 beta (cexp x)).
rewrite <- mantissa_DN_small_pos with (1 := Hx) (2 := He).
now rewrite <- cexp_fexp_pos with (1 := Hx).
Qed.

Theorem round_DN_UP_lt :
  forall x, ~ generic_format x ->
  (round Zfloor x < x < round Zceil x)%R.
Proof with auto with typeclass_instances.
intros x Fx.
assert (Hx:(round Zfloor x <= x <= round Zceil x)%R).
split.
apply round_DN_pt.
apply round_UP_pt.
split.
  destruct Hx as [Hx _].
  apply Rnot_le_lt; intro Hle.
  assert (x = round Zfloor x) by now apply Rle_antisym.
  rewrite H in Fx.
  contradict Fx.
  apply generic_format_round...
destruct Hx as [_ Hx].
apply Rnot_le_lt; intro Hle.
assert (x = round Zceil x) by now apply Rle_antisym.
rewrite H in Fx.
contradict Fx.
apply generic_format_round...
Qed.

Lemma round_UP_small_pos :
  forall x ex,
  (bpow (ex - 1) <= x < bpow ex)%R ->
  (ex <= fexp ex)%Z ->
  round Zceil x = (bpow (fexp ex)).
Proof.
intros x ex Hx He.
rewrite <- F2R_bpow.
rewrite <- mantissa_UP_small_pos with (1 := Hx) (2 := He).
now rewrite <- cexp_fexp_pos with (1 := Hx).
Qed.

Theorem generic_format_EM :
  forall x,
  generic_format x \/ ~generic_format x.
Proof with auto with typeclass_instances.
intros x.
destruct (Req_dec (round Zfloor x) x) as [Hx|Hx].
left.
rewrite <- Hx.
apply generic_format_round...
right.
intros H.
apply Hx.
apply round_generic...
Qed.

Section round_large.

Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.

Lemma round_large_pos_ge_bpow :
  forall x e,
  (0 < round rnd x)%R ->
  (bpow e <= x)%R ->
  (bpow e <= round rnd x)%R.
Proof.
intros x e Hd Hex.
destruct (mag beta x) as (ex, He).
assert (Hx: (0 < x)%R).
apply Rlt_le_trans with (2 := Hex).
apply bpow_gt_0.
specialize (He (Rgt_not_eq _ _ Hx)).
rewrite Rabs_pos_eq in He. 2: now apply Rlt_le.
apply Rle_trans with (bpow (ex - 1)).
apply bpow_le.
cut (e < ex)%Z. omega.
apply (lt_bpow beta).
now apply Rle_lt_trans with (2 := proj2 He).
destruct (Zle_or_lt ex (fexp ex)).
destruct (round_bounded_small_pos rnd x ex H He) as [Hr|Hr].
rewrite Hr in Hd.
elim Rlt_irrefl with (1 := Hd).
rewrite Hr.
apply bpow_le.
omega.
apply (round_bounded_large_pos rnd x ex H He).
Qed.

End round_large.

Theorem mag_round_ZR :
  forall x,
  (round Ztrunc x <> 0)%R ->
  (mag beta (round Ztrunc x) = mag beta x :> Z).
Proof with auto with typeclass_instances.
intros x Zr.
destruct (Req_dec x 0) as [Zx|Zx].
rewrite Zx, round_0...
apply mag_unique.
destruct (mag beta x) as (ex, Ex) ; simpl.
specialize (Ex Zx).
rewrite <- round_ZR_abs.
split.
apply round_large_pos_ge_bpow...
rewrite round_ZR_abs.
now apply Rabs_pos_lt.
apply Ex.
apply Rle_lt_trans with (2 := proj2 Ex).
rewrite round_ZR_DN.
apply round_DN_pt.
apply Rabs_pos.
Qed.

Theorem mag_round :
  forall rnd {Hrnd : Valid_rnd rnd} x,
  (round rnd x <> 0)%R ->
  (mag beta (round rnd x) = mag beta x :> Z) \/
  Rabs (round rnd x) = bpow (Z.max (mag beta x) (fexp (mag beta x))).
Proof with auto with typeclass_instances.
intros rnd Hrnd x.
destruct (round_ZR_or_AW rnd x) as [Hr|Hr] ; rewrite Hr ; clear Hr rnd Hrnd.
left.
now apply mag_round_ZR.
intros Zr.
destruct (Req_dec x 0) as [Zx|Zx].
rewrite Zx, round_0...
destruct (mag beta x) as (ex, Ex) ; simpl.
specialize (Ex Zx).
rewrite <- mag_abs.
rewrite <- round_AW_abs.
destruct (Zle_or_lt ex (fexp ex)) as [He|He].
right.
rewrite Z.max_r with (1 := He).
rewrite round_AW_UP with (1 := Rabs_pos _).
now apply round_UP_small_pos.
destruct (round_bounded_large_pos Zaway _ ex He Ex) as (H1,[H2|H2]).
left.
apply mag_unique.
rewrite <- round_AW_abs, Rabs_Rabsolu.
now split.
right.
now rewrite Z.max_l with (1 := Zlt_le_weak _ _ He).
Qed.

Theorem mag_DN :
  forall x,
  (0 < round Zfloor x)%R ->
  (mag beta (round Zfloor x) = mag beta x :> Z).
Proof.
intros x Hd.
assert (0 < x)%R.
apply Rlt_le_trans with (1 := Hd).
apply round_DN_pt.
revert Hd.
rewrite <- round_ZR_DN.
intros Hd.
apply mag_round_ZR.
now apply Rgt_not_eq.
now apply Rlt_le.
Qed.

Theorem cexp_DN :
  forall x,
  (0 < round Zfloor x)%R ->
  cexp (round Zfloor x) = cexp x.
Proof.
intros x Hd.
apply (f_equal fexp).
now apply mag_DN.
Qed.

Theorem scaled_mantissa_DN :
  forall x,
  (0 < round Zfloor x)%R ->
  scaled_mantissa (round Zfloor x) = IZR (Zfloor (scaled_mantissa x)).
Proof.
intros x Hd.
unfold scaled_mantissa.
rewrite cexp_DN with (1 := Hd).
unfold round, F2R. simpl.
now rewrite Rmult_assoc, <- bpow_plus, Zplus_opp_r, Rmult_1_r.
Qed.

Theorem generic_N_pt_DN_or_UP :
  forall x f,
  Rnd_N_pt generic_format x f ->
  f = round Zfloor x \/ f = round Zceil x.
Proof.
intros x f Hxf.
destruct (Rnd_N_pt_DN_or_UP _ _ _ Hxf).
left.
apply Rnd_DN_pt_unique with (1 := H).
apply round_DN_pt.
right.
apply Rnd_UP_pt_unique with (1 := H).
apply round_UP_pt.
Qed.

Section not_FTZ.

Class Exp_not_FTZ :=
  exp_not_FTZ : forall e, (fexp (fexp e + 1) <= fexp e)%Z.

Context { exp_not_FTZ_ : Exp_not_FTZ }.

Theorem subnormal_exponent :
  forall e x,
  (e <= fexp e)%Z ->
  generic_format x ->
  x = F2R (Float beta (Ztrunc (x * bpow (- fexp e))) (fexp e)).
Proof.
intros e x He Hx.
pattern x at 2 ; rewrite Hx.
unfold F2R at 2. simpl.
rewrite Rmult_assoc, <- bpow_plus.
assert (H: IZR (Zpower beta (cexp x + - fexp e)) = bpow (cexp x + - fexp e)).
apply IZR_Zpower.
unfold cexp.
set (ex := mag beta x).
generalize (exp_not_FTZ ex).
generalize (proj2 (proj2 (valid_exp _) He) (fexp ex + 1)%Z).
omega.
rewrite <- H.
rewrite <- mult_IZR, Ztrunc_IZR.
unfold F2R. simpl.
rewrite mult_IZR.
rewrite H.
rewrite Rmult_assoc, <- bpow_plus.
now ring_simplify (cexp x + - fexp e + fexp e)%Z.
Qed.

End not_FTZ.

Section monotone_exp.

Class Monotone_exp :=
  monotone_exp : forall ex ey, (ex <= ey)%Z -> (fexp ex <= fexp ey)%Z.

Context { monotone_exp_ : Monotone_exp }.

Global Instance monotone_exp_not_FTZ : Exp_not_FTZ.
Proof.
intros e.
destruct (Z_lt_le_dec (fexp e) e) as [He|He].
apply monotone_exp.
now apply Zlt_le_succ.
now apply valid_exp.
Qed.

Lemma cexp_le_bpow :
  forall (x : R) (e : Z),
  x <> 0%R ->
  (Rabs x < bpow e)%R ->
  (cexp x <= fexp e)%Z.
Proof.
intros x e Zx Hx.
apply monotone_exp.
now apply mag_le_bpow.
Qed.

Lemma cexp_ge_bpow :
  forall (x : R) (e : Z),
  (bpow (e - 1) <= Rabs x)%R ->
  (fexp e <= cexp x)%Z.
Proof.
intros x e Hx.
apply monotone_exp.
rewrite (Zsucc_pred e).
apply Zlt_le_succ.
now apply mag_gt_bpow.
Qed.

Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.

Theorem mag_round_ge :
  forall x,
  round rnd x <> 0%R ->
  (mag beta x <= mag beta (round rnd x))%Z.
Proof with auto with typeclass_instances.
intros x.
destruct (round_ZR_or_AW rnd x) as [H|H] ; rewrite H ; clear H ; intros Zr.
rewrite mag_round_ZR with (1 := Zr).
apply Z.le_refl.
apply mag_le_abs.
contradict Zr.
rewrite Zr.
apply round_0...
rewrite <- round_AW_abs.
rewrite round_AW_UP.
apply round_UP_pt.
apply Rabs_pos.
Qed.

Theorem cexp_round_ge :
  forall x,
  round rnd x <> 0%R ->
  (cexp x <= cexp (round rnd x))%Z.
Proof with auto with typeclass_instances.
intros x Zr.
unfold cexp.
apply monotone_exp.
now apply mag_round_ge.
Qed.

End monotone_exp.

Section Znearest.