Proof using.
intros. unfold doLeader, advanceCommitIndex in *.
repeat break_match; find_inversion; intuition;
find_apply_lem_hyp key_in_output_list_empty; auto.
Qed.

Lemma handleInput_key_in_output_list :
forall st h i out st' m client id,
handleInput h i st = (out, st', m) ->
~ key_in_output_list client id out.