Proof with auto with typeclass_instances.
cut (forall x : R, (0 < x)%R -> Rnd_odd_pt x (round beta fexp Zrnd_odd x)).
intros H x; case (Rle_or_lt 0 x).
intros H1; destruct H1.
now apply H.
rewrite <- H0.
rewrite round_0...
split.
apply generic_format_0.
now left.
intros Hx; apply Rnd_odd_pt_sym.
rewrite <- round_odd_opp.
apply H.
auto with real.
(* *)
intros x Hxp.
generalize (generic_format_round beta fexp Zrnd_odd x).
set (o:=round beta fexp Zrnd_odd x).
intros Ho.
split.
assumption.
(* *)
case (Req_dec o x); intros Hx.
now left.
right.
assert (o=round beta fexp Zfloor x \/ o=round beta fexp Zceil x).
unfold o, round, F2R;simpl.
case (Zrnd_DN_or_UP Zrnd_odd  (scaled_mantissa beta fexp x))...
intros H; rewrite H; now left.
intros H; rewrite H; now right.
split.
destruct H; rewrite H.
left; apply round_DN_pt...
right; apply round_UP_pt...
(* *)
unfold o, Zrnd_odd, round.
case (Req_EM_T (scaled_mantissa beta fexp x)
(Z2R (Zfloor (scaled_mantissa beta fexp x)))).
intros T.
absurd (o=x); trivial.
apply round_generic...
unfold generic_format, F2R; simpl.
rewrite <- (scaled_mantissa_mult_bpow beta fexp) at 1.
apply f_equal2; trivial; rewrite T at 1.
apply f_equal, sym_eq, Ztrunc_floor.
apply Rmult_le_pos.
now left.
apply bpow_ge_0.
intros L.
case_eq (Zeven (Zfloor (scaled_mantissa beta fexp x))).
(* . *)
generalize (generic_format_round beta fexp Zceil x).
unfold generic_format.
set (f:=round beta fexp Zceil x).
set (ef := canonic_exp beta fexp f).
set (mf := Ztrunc (scaled_mantissa beta fexp f)).
exists (Float beta mf ef).
unfold Fcore_generic_fmt.canonic.
rewrite <- H0.
repeat split; try assumption.
apply trans_eq with (negb (Zeven (Zfloor (scaled_mantissa beta fexp x)))).
2: rewrite H1; reflexivity.
apply trans_eq with (negb (Zeven (Fnum
(Float beta  (Zfloor (scaled_mantissa beta fexp x)) (cexp x))))).
2: reflexivity.
case (Rle_lt_or_eq_dec 0 (round beta fexp Zfloor x)).
rewrite <- round_0 with beta fexp Zfloor...
apply round_le...
now left.
intros Y.
generalize (DN_UP_parity_generic beta fexp)...
unfold DN_UP_parity_prop.
intros T; apply T with x; clear T.
unfold generic_format.
rewrite <- (scaled_mantissa_mult_bpow beta fexp x) at 1.
unfold F2R; simpl.
apply Rmult_neq_compat_r.
apply Rgt_not_eq, bpow_gt_0.
rewrite Ztrunc_floor.
assumption.
apply Rmult_le_pos.
now left.
apply bpow_ge_0.
unfold Fcore_generic_fmt.canonic.
simpl.
apply sym_eq, canonic_exp_DN...
unfold Fcore_generic_fmt.canonic.
rewrite <- H0; reflexivity.
reflexivity.
apply trans_eq with (round beta fexp Ztrunc (round beta fexp Zceil x)).
reflexivity.
apply round_generic...
intros Y.
replace  (Fnum {| Fnum := Zfloor (scaled_mantissa beta fexp x); Fexp := cexp x |})
with (Fnum (Float beta 0 (fexp (ln_beta beta 0)))).
generalize (DN_UP_parity_generic beta fexp)...
unfold DN_UP_parity_prop.
intros T; apply T with x; clear T.
unfold generic_format.
rewrite <- (scaled_mantissa_mult_bpow beta fexp x) at 1.
unfold F2R; simpl.
apply Rmult_neq_compat_r.
apply Rgt_not_eq, bpow_gt_0.
rewrite Ztrunc_floor.
assumption.
apply Rmult_le_pos.
now left.
apply bpow_ge_0.
apply canonic_0.
unfold Fcore_generic_fmt.canonic.
rewrite <- H0; reflexivity.
rewrite <- Y; unfold F2R; simpl; ring.
apply trans_eq with (round beta fexp Ztrunc (round beta fexp Zceil x)).
reflexivity.
apply round_generic...
simpl.
apply eq_Z2R, Rmult_eq_reg_r with (bpow (cexp x)).
unfold round, F2R in Y; simpl in Y; rewrite <- Y.
simpl; ring.
apply Rgt_not_eq, bpow_gt_0.
(* . *)
intros Y.
case (Rle_lt_or_eq_dec 0 (round beta fexp Zfloor x)).
rewrite <- round_0 with beta fexp Zfloor...
apply round_le...
now left.
intros Hrx.
set (ef := canonic_exp beta fexp x).
set (mf := Zfloor (scaled_mantissa beta fexp x)).
exists (Float beta mf ef).
unfold Fcore_generic_fmt.canonic.
repeat split; try assumption.
simpl.
apply trans_eq with (cexp (round beta fexp Zfloor x )).
apply sym_eq, canonic_exp_DN...
reflexivity.
intros Hrx; contradict Y.
replace (Zfloor (scaled_mantissa beta fexp x)) with 0%Z.
simpl; discriminate.
apply eq_Z2R, Rmult_eq_reg_r with (bpow (cexp x)).
unfold round, F2R in Hrx; simpl in Hrx; rewrite <- Hrx.
simpl; ring.
apply Rgt_not_eq, bpow_gt_0.
Qed.

Theorem Rnd_odd_pt_unicity :
forall x f1 f2 : R,
Rnd_odd_pt x f1 -> Rnd_odd_pt x f2 ->
f1 = f2.