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

Lemma get_op_input_keys_app :
forall xs ys,
get_op_input_keys (xs ++ ys) = get_op_input_keys xs ++ get_op_input_keys ys.