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_U.
Qed.

Lemma NoDup_get_op_input_keys_In_I :
forall xs ys k,
NoDup (get_op_input_keys (xs ++ I k :: ys)) ->
In (I k) (xs ++ ys) -> False.