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 O_IRO_preserved : forall xs ys zs k' ir, NoDup (get_op_output_keys (xs ++ I k' :: ys ++ O k' :: zs)) -> (forall k, In (O k) (xs ++ I k' :: ys ++ O k' :: zs) -> In (IRO k) (IRI k' :: IRO k' :: ir)) -> forall k, In (O k) (xs ++ ys ++ zs) -> In (IRO k) ir.