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.

Lemma int_round_odd_shifts:
forall x p, 0 <= p ->
int_round_odd x p =
Z.shiftl (if Z.eqb (x mod 2^p) 0 then Z.shiftr x p else Z.lor (Z.shiftr x p) 1) p.