Definition minimal_contradicting_leader_log net t e :=
    argmin (fun l => fst (fst l)) (contradicting_leader_logs net nodes t e).

Lemma contradicting_leader_logs_on_leader_complete : forall t e t' l ll, In (t', l) ll -> t < t' -> ~ In e l -> In (t', l) (contradicting_leader_logs_on_leader ll t e).