Definition int_round_odd (x: Z) (p: Z) :=
(if Z.eqb (x mod 2^p) 0 || Z.odd (x / 2^p) then x / 2^p else x / 2^p + 1) * 2^p.

Theorem BofZ_round_odd:
forall x p,
prec > 1 ->
Z.abs x <= 2^emax - 2^(emax-prec) ->
0 <= p <= emax - prec ->
2^(prec + p + 1) <= Z.abs x ->
BofZ x = BofZ (int_round_odd x p).