Definition lifted_no_entries_past_current_term_host net :=
    forall (h : name) e,
      In e (log (snd (nwState net h))) ->
      eTerm e <= currentTerm (snd (nwState net h)).

Lemma commit_recorded_lift_intro : forall (net : network (params := raft_refined_multi_params)) h e, In e (log (snd (nwState net h))) -> (eIndex e <= lastApplied (snd (nwState net h)) \/ eIndex e <= commitIndex (snd (nwState net h))) -> commit_recorded (deghost net) h e.