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 IR_equiv_AAOF_II_neq : forall xs ys target k k', k <> k' -> op_equivalent xs ys -> IR_equivalent (acknowledge_all_ops_func xs target) (acknowledge_all_ops_func ys target) -> IR_equivalent (acknowledge_all_ops_func (I k :: I k' :: xs) target) (acknowledge_all_ops_func (I k' :: I k :: ys) target).