Section Odd_prop.
Variable beta : radix.
Hypothesis Even_beta: Z.even (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 round_N_odd :
forall x,
round beta fexp (Znearest choice) (round beta fexpe Zrnd_odd x) =
round beta fexp (Znearest choice) x.