Proof using.
intros. unfold handleInput, handleTimeout, handleClientRequest, tryToBecomeLeader in *.
repeat break_match; find_inversion; intuition eauto using key_in_output_list_empty;
unfold key_in_output_list in *; break_exists; simpl in *; intuition; congruence.
Qed.

Lemma has_key_own_key :
forall e,
has_key (eClient e) (eId e) e = true.