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.

Lemma m_le_u: (m <= F2R u)%R.