Proof using.
induction l; intros.
- auto.
- simpl in *. rewrite get_op_input_keys_defn in *. break_match; simpl in *.
+ subst. intuition congruence.
+ intuition.
Qed.

Lemma get_op_output_keys_sound :
forall k l,
In k (get_op_output_keys l) ->
In (O k) l.