Proof with auto with typeclass_instances.
intros x y f g H1 H2 Hxy.
apply Rle_trans with (round beta fexp Zrnd_odd x).
right; apply Rnd_odd_pt_unicity with x; try assumption.
apply round_odd_pt.
apply Rle_trans with (round beta fexp Zrnd_odd y).
apply round_le...
right; apply Rnd_odd_pt_unicity with y; try assumption.
apply round_odd_pt.
Qed.
End Fcore_rnd_odd.
Section Odd_prop_aux.
Variable beta : radix.
Hypothesis Even_beta: Zeven (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.

Lemma exists_even_fexp_lt: forall (c:Z->Z), forall (x:R),
(exists f:float beta, F2R f = x /\ (c (ln_beta beta x) < Fexp f)%Z) ->
exists f:float beta, F2R f =x /\ canonic beta c f /\ Zeven (Fnum f) = true.