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

Lemma minimal_contradicting_leader_log_elim : forall net t e t' h l, minimal_contradicting_leader_log net t e = Some (t', h, l) -> (t < t' /\ In (t', l) (leaderLogs (fst (nwState net h))) /\ ~ In e l /\ (forall h' t'' l', In (t'', l') (leaderLogs (fst (nwState net h'))) -> (t'' <= t \/ t'' >= t' \/ In e l'))).