Proof using lmi rri.
intros.
pose proof (lift_prop _ log_matching_invariant).
forward_invariant.
unfold log_matching, log_matching_nw, entries_match_nw_host in *. break_and.
intros.
repeat find_apply_lem_hyp ghost_packet.
forward_nw_invariant. break_and.
match goal with
| [  |- context [ snd (nwState ?net ?h) ] ] =>
replace (snd (nwState net h)) with (nwState (deghost net) h) in * by auto using deghost_spec
end.
match goal with
| [ H : forall _ _ _, In _ ?es -> _,
He : In ?e ?es,
He' : In ?e' ?log,
Hle : eIndex ?e'' <= eIndex ?e
|- In ?e'' ?log ] =>
specialize (H _ _ _ He He')
end.
repeat concludes; break_and. eauto.
Qed.

(* Specifications: *)
Lemma allEntries_gt_0_invariant :
forall net h e,
refined_raft_intermediate_reachable net ->
In e (map snd (allEntries (fst (nwState net h)))) ->
eIndex e > 0.