Definition good_op_move (x y : op) : Prop :=
    (forall k k', ~ (x = O k /\ y = I k')) /\
    (forall k, ~ (x = I k /\ y = O k)).

Lemma in_before_preserved_IU : forall xs ys k', ~ acknowledged_op k' (xs ++ ys) -> (forall k, In (O k) (xs ++ I k' :: ys) -> before (I k) (O k) (xs ++ I k' :: ys)) -> forall k, In (O k) (xs ++ ys) -> before (I k) (O k) (xs ++ ys).