Proof with auto with typeclass_instances.
set (o:=round beta fexpe Zrnd_odd x).
case (generic_format_EM beta fexp x); intros Hx.
replace o with x; trivial.
unfold o; apply sym_eq, round_generic...
now apply generic_format_fexpe_fexp.
assert (K1:(F2R d <= o)%R).
apply round_ge_generic...
apply generic_format_fexpe_fexp, Hd.
apply Hd.
assert (K2:(o <= F2R u)%R).
apply round_le_generic...
apply generic_format_fexpe_fexp, Hu.
apply Hu.
assert (P:(x <> m -> o=m -> (forall P:Prop, P))).
intros Y1 Y2.
assert (H:(Rnd_odd_pt beta fexpe x o)).
apply round_odd_pt...
destruct H as (_,H); destruct H.
absurd (x=m)%R; try trivial.
now rewrite <- Y2, H.
destruct H as (_,(k,(Hk1,(Hk2,Hk3)))).
destruct Zm as (k',(Hk'1,(Hk'2,Hk'3))).
absurd (true=false).
discriminate.
rewrite <- Hk3, <- Hk'3.
apply f_equal, f_equal.
apply canonic_unicity with fexpe...
now rewrite Hk'1, <- Y2.
assert (generic_format beta fexp o -> (forall P:Prop, P)).
intros Y.
assert (H:(Rnd_odd_pt beta fexpe x o)).
apply round_odd_pt...
destruct H as (_,H); destruct H.
absurd (generic_format beta fexp x); trivial.
now rewrite <- H.
destruct H as (_,(k,(Hk1,(Hk2,Hk3)))).
destruct (exists_even_fexp_lt fexpe o) as (k',(Hk'1,(Hk'2,Hk'3))).
eexists; split.
apply sym_eq, Y.
simpl; unfold canonic_exp.
apply Zle_lt_trans with (1:=fexpe_fexp _).
omega.
absurd (true=false).
discriminate.
rewrite <- Hk3, <- Hk'3.
apply f_equal, f_equal.
apply canonic_unicity with fexpe...
now rewrite Hk'1, <- Hk1.
case K1; clear K1; intros K1.
2: apply H; rewrite <- K1; apply Hd.
case K2; clear K2; intros K2.
2: apply H; rewrite K2; apply Hu.
case (Rle_or_lt  x m); intros Y;[destruct Y|idtac].
(* . *)
apply trans_eq with (F2R d).
apply round_N_eq_DN_pt with (F2R u)...
apply DN_odd_d_aux; split; try left; assumption.
apply UP_odd_d_aux; split; try left; assumption.
assert (o <= (F2R d + F2R u) / 2)%R.
apply round_le_generic...
apply Fm.
now left.
destruct H1; trivial.
apply P.
now apply Rlt_not_eq.
trivial.
apply sym_eq, round_N_eq_DN_pt with (F2R u)...
(* . *)
replace o with x.
reflexivity.
apply sym_eq, round_generic...
rewrite H0; apply Fm.
(* . *)
apply trans_eq with (F2R u).
apply round_N_eq_UP_pt with (F2R d)...
apply DN_odd_d_aux; split; try left; assumption.
apply UP_odd_d_aux; split; try left; assumption.
assert ((F2R d + F2R u) / 2 <= o)%R.
apply round_ge_generic...
apply Fm.
now left.
destruct H0; trivial.
apply P.
now apply Rgt_not_eq.
rewrite <- H0; trivial.
apply sym_eq, round_N_eq_UP_pt with (F2R d)...
Qed.
End Odd_prop_aux.
Section Odd_prop.
Variable beta : radix.
Hypothesis Even_beta: Zeven (radix_val beta)=true.
Variable fexp : Z -> Z.
Variable fexpe : Z -> Z.
Variable choice:Z->bool.
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.

Theorem canonizer: forall f, generic_format beta fexp f
-> exists g : float beta, f = F2R g /\ canonic beta fexp g.