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

(* Specifications: *)
Lemma get_IR_output_keys_complete_O :
forall xs k,
In (IRO k) xs ->
In k (get_IR_output_keys xs).
Lemma get_IR_input_keys_complete :
forall xs k,
In (IRI k) xs ->
In k (get_IR_input_keys xs).