Proof using.
unfold get_IR_input_keys.
intros.
eapply filterMap_In; eauto. auto.
Qed.

Lemma op_equivalent_all_Is_O_middle :
forall xs ys k,
(forall k', In (I k') xs -> k <> k') ->
op_equivalent (xs ++ O k :: ys) (O k :: xs ++ ys).