Definition msg_lifted_leader_sublog_host (net : ghost_log_network) : Prop :=
    forall leader e h,
      type (snd (nwState net leader)) = Leader ->
      In e (log (snd (nwState net h))) ->
      eTerm e = currentTerm (snd (nwState net leader)) ->
      In e (log (snd (nwState net leader))).

Lemma maxIndex_sanity_lower : forall net, lifted_maxIndex_sanity net -> maxIndex_sanity (deghost (mgv_deghost net)).