Proof using.
intros.
rewrite app_comm_cons.
replace (xs ++ O k :: ys) with ((xs ++ [O k]) ++ ys) by now rewrite app_ass.
auto using op_equiv_app_tail, op_equivalent_all_Is_O.
Qed.

(* Specifications: *)
Lemma NoDup_get_op_output_keys_In_O :
forall xs ys k,
NoDup (get_op_output_keys (xs ++ O k :: ys)) ->
In (O k) (xs ++ ys) ->
False.