Section Odd_prop_aux.
Variable beta : radix.
Hypothesis Even_beta: Z.even (radix_val beta)=true.
Notation bpow e := (bpow beta e).
Variable fexp : Z -> Z.
Variable fexpe : Z -> Z.
Context { valid_exp : Valid_exp fexp }.
Context { exists_NE_ : Exists_NE beta fexp }. (* for underflow reason *)
Context { valid_expe : Valid_exp fexpe }.
Context { exists_NE_e : Exists_NE beta fexpe }. (* for defining rounding to odd *)
Hypothesis fexpe_fexp: forall e, (fexpe e <= fexp e -2)%Z.
Proof with auto with typeclass_instances.
intros c x (g,(Hg1,Hg2)).
exists (Float beta
(Fnum g*Z.pow (radix_val beta) (Fexp g - c (mag beta x)))
(c (mag beta x))).
assert (F2R (Float beta
(Fnum g*Z.pow (radix_val beta) (Fexp g - c (mag beta x)))
(c (mag beta x))) = x).
unfold F2R; simpl.
rewrite mult_IZR, IZR_Zpower.
rewrite Rmult_assoc, <- bpow_plus.
rewrite <- Hg1; unfold F2R.
apply f_equal, f_equal.
ring.
omega.
split; trivial.
split.
unfold canonical, cexp.
now rewrite H.
simpl.
rewrite Z.even_mul.
rewrite Z.even_pow.
rewrite Even_beta.
apply Bool.orb_true_intro.
now right.
omega.
Qed.
Variable choice:Z->bool.
Variable x:R.
Variable d u: float beta.
Hypothesis Hd: Rnd_DN_pt (generic_format beta fexp) x (F2R d).
Hypothesis Cd: canonical beta fexp d.
Hypothesis Hu: Rnd_UP_pt (generic_format beta fexp) x (F2R u).
Hypothesis Cu: canonical beta fexp u.
Hypothesis xPos: (0 < x)%R.
Let m:= ((F2R d+F2R u)/2)%R.
Proof with auto with typeclass_instances.
apply Rnd_DN_pt_unique with (generic_format beta fexp) x...
apply round_DN_pt...
Qed.
Proof with auto with typeclass_instances.
apply Rnd_UP_pt_unique with (generic_format beta fexp) x...
apply round_UP_pt...
Qed.
Proof with auto with typeclass_instances.
rewrite d_eq; apply round_ge_generic...
apply generic_format_0.
now left.
Qed.
Proof with auto with typeclass_instances.
intros Y.
rewrite d_eq; apply mag_DN...
now rewrite <- d_eq.
Qed.
Proof with auto with typeclass_instances.
intros Y.
now rewrite Cd, <- mag_d.
Qed.
Proof with auto with typeclass_instances.
intros Y.
apply generic_format_bpow.
apply valid_exp.
rewrite <- Fexp_d; trivial.
apply Z.lt_le_trans with  (mag beta (F2R d))%Z.
rewrite Cd; apply mag_generic_gt...
now apply Rgt_not_eq.
apply Hd.
apply mag_le; trivial.
apply Hd.
Qed.
Proof with auto with typeclass_instances.
intros Y; apply generic_format_bpow.
apply valid_exp.
apply mag_generic_gt...
now apply Rgt_not_eq.
now apply generic_format_canonical.
Qed.
Proof with auto with typeclass_instances.
intros dPos; apply mag_unique_pos.
split.
apply Rle_trans with (F2R d).
destruct (mag beta (F2R d)) as (e,He).
simpl.
rewrite Rabs_right in He.
apply He.
now apply Rgt_not_eq.
apply Rle_ge; now left.
apply d_le_m.
case m_le_u; intros H.
apply Rlt_le_trans with (1:=H).
rewrite u_eq.
apply round_le_generic...
apply generic_format_bpow.
apply valid_exp.
apply mag_generic_gt...
now apply Rgt_not_eq.
now apply generic_format_canonical.
case (Rle_or_lt x (bpow (mag beta (F2R d)))); trivial; intros Z.
absurd ((bpow (mag beta (F2R d)) <= (F2R d)))%R.
apply Rlt_not_le.
destruct  (mag beta (F2R d)) as (e,He).
simpl in *; rewrite Rabs_right in He.
apply He.
now apply Rgt_not_eq.
apply Rle_ge; now left.
apply Rle_trans with (round beta fexp Zfloor x).
2: right; apply sym_eq, d_eq.
apply round_ge_generic...
apply generic_format_bpow.
apply valid_exp.
apply mag_generic_gt...
now apply Rgt_not_eq.
now apply generic_format_canonical.
now left.
replace m with (F2R d).
destruct (mag beta (F2R d)) as (e,He).
simpl in *; rewrite Rabs_right in He.
apply He.
now apply Rgt_not_eq.
apply Rle_ge; now left.
unfold m in H |- *.
lra.
Qed.
Proof with auto with typeclass_instances.
intros Y.
apply mag_unique_pos.
unfold m; rewrite <- Y, Rplus_0_l.
rewrite u_eq.
destruct (mag beta x) as (e,He).
rewrite Rabs_pos_eq in He by now apply Rlt_le.
rewrite round_UP_small_pos with (ex:=e).
rewrite mag_bpow.
ring_simplify (fexp e + 1 - 1)%Z.
split.
unfold Zminus; rewrite bpow_plus.
unfold Rdiv; apply Rmult_le_compat_l.
apply bpow_ge_0.
simpl; unfold Z.pow_pos; simpl.
rewrite Zmult_1_r; apply Rinv_le.
exact Rlt_0_2.
apply IZR_le.
specialize (radix_gt_1 beta).
omega.
apply Rlt_le_trans with (bpow (fexp e)*1)%R.
2: right; ring.
unfold Rdiv; apply Rmult_lt_compat_l.
apply bpow_gt_0.
lra.
now apply He, Rgt_not_eq.
apply exp_small_round_0_pos with beta (Zfloor) x...
now apply He, Rgt_not_eq.
now rewrite <- d_eq, Y.
Qed.
Proof with auto with typeclass_instances.
intros Y.
rewrite u_eq; unfold round.
eexists; repeat split.
simpl; now rewrite Fexp_d.
Qed.
Proof with auto with typeclass_instances.
intros Y.
specialize (Zeven_ex (radix_val beta)); rewrite Even_beta.
intros (b, Hb); rewrite Zplus_0_r in Hb.
destruct u'_eq as (u', (Hu'1,Hu'2)); trivial.
exists (Fmult (Float beta b (-1)) (Fplus d u'))%R.
split.
rewrite F2R_mult, F2R_plus, Hu'1.
unfold m; rewrite Rmult_comm.
unfold Rdiv; apply f_equal.
unfold F2R; simpl; unfold Z.pow_pos; simpl.
rewrite Zmult_1_r, Hb, mult_IZR.
simpl; field.
apply Rgt_not_eq, Rmult_lt_reg_l with (1 := Rlt_0_2).
rewrite Rmult_0_r, <- (mult_IZR 2), <-Hb.
apply radix_pos.
apply trans_eq with (-1+Fexp (Fplus d u'))%Z.
unfold Fmult.
destruct  (Fplus d u'); reflexivity.
rewrite Zplus_comm; unfold Zminus; apply f_equal2.
2: reflexivity.
rewrite Fexp_Fplus.
rewrite Z.min_l.
now rewrite Fexp_d.
rewrite Hu'2; omega.
Qed.
Proof with auto with typeclass_instances.
intros Y.
specialize (Zeven_ex (radix_val beta)); rewrite Even_beta.
intros (b, Hb); rewrite Zplus_0_r in Hb.
exists (Fmult (Float beta b (-1)) u)%R.
split.
rewrite F2R_mult; unfold m; rewrite <- Y, Rplus_0_l.
rewrite Rmult_comm.
unfold Rdiv; apply f_equal.
unfold F2R; simpl; unfold Z.pow_pos; simpl.
rewrite Zmult_1_r, Hb, mult_IZR.
simpl; field.
apply Rgt_not_eq, Rmult_lt_reg_l with (1 := Rlt_0_2).
rewrite Rmult_0_r, <- (mult_IZR 2), <-Hb.
apply radix_pos.
apply trans_eq with (-1+Fexp u)%Z.
unfold Fmult.
destruct u; reflexivity.
rewrite Zplus_comm, Cu; unfold Zminus; now apply f_equal2.
Qed.
Proof with auto with typeclass_instances.
intros Y.
assert ((fexp (mag beta (F2R u) - 1) <= fexp (mag beta (F2R u))))%Z.
2: omega.
destruct (mag beta x) as (e,He).
rewrite Rabs_right in He.
2: now left.
assert (e <= fexp e)%Z.
apply exp_small_round_0_pos with beta (Zfloor) x...
now apply He, Rgt_not_eq.
now rewrite <- d_eq, Y.
rewrite u_eq, round_UP_small_pos with (ex:=e); trivial.
2: now apply He, Rgt_not_eq.
rewrite mag_bpow.
ring_simplify (fexp e + 1 - 1)%Z.
replace (fexp (fexp e)) with (fexp e).
case exists_NE_; intros V.
contradict V; rewrite Even_beta; discriminate.
rewrite (proj2 (V e)); omega.
apply sym_eq, valid_exp; omega.
Qed.

Lemma Zm:
exists g : float beta, F2R g = m /\ canonical beta fexpe g /\ Z.even (Fnum g) = true.