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

Lemma get_op_input_keys_on_Os_nil :
forall l,
(forall o, In o l -> exists k, o = O k) ->
get_op_input_keys l = [].