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 reboot_preserves_committed : forall (net net' : ghost_log_network) h e t, (forall h', nwState net' h' = update name_eq_dec (nwState net) h (fst (nwState net h), reboot (snd (nwState net h))) h') -> lifted_committed net e t -> lifted_committed net' e t.