Definition weak_sanity pli ll ll' :=
    pli = 0 ->
    (exists e, eIndex e = 0 /\ In e ll) \/
    ll = ll'.Definition logs_leaderLogs_nw_weak net :=
    forall p t n pli plt es ci e,
      In p (nwPackets net) ->
      pBody p = AppendEntries t n pli plt es ci ->
      In e es ->
      exists leader ll es' ll',
        In (eTerm e, ll) (leaderLogs (fst (nwState net leader))) /\
        Prefix ll' ll /\
        removeAfterIndex es (eIndex e) = es' ++ ll' /\
        (forall e', In e' es' -> eTerm e' = eTerm e) /\
        weak_sanity pli ll ll'.

Lemma logs_leaderLogs_nw_weaken : forall net, logs_leaderLogs_nw net -> logs_leaderLogs_nw_weak net.