Proof using.
intros.
eapply In_cons_2_3_neq; eauto using in_cons; try congruence.
intro. find_inversion. eauto using get_IR_output_keys_complete_O.
Qed.

Lemma IRU_I_preserved_IU :
forall xs ys k' ir,
(~ In k' (get_IR_output_keys ir)) ->
(forall k, In (IRU k) (IRI k' :: IRU k' :: ir) ->
In (I k) (xs ++ I k' :: ys)) ->
forall k, In (IRU k) ir ->
In (I k) (xs ++ ys).