Proof using rmri lci.
intros.
pose proof msg_lift_prop _ leader_completeness_invariant _ ltac:(eauto).
unfold lifted_leader_completeness, leader_completeness in *.
intuition.
- unfold lifted_leader_completeness_directly_committed, leader_completeness_directly_committed in *.
intros.
find_apply_lem_hyp lifted_directly_committed_directly_committed.
eapply_prop_hyp directly_committed directly_committed; eauto.
rewrite msg_deghost_spec'. eauto.
- unfold lifted_leader_completeness_committed, leader_completeness_committed in *.
intros.
find_apply_lem_hyp lifted_committed_committed.
eapply_prop_hyp committed committed; eauto.
rewrite msg_deghost_spec'. eauto.
Qed.
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 msg_lifted_leader_sublog_host_invariant :
forall net,
msg_refined_raft_intermediate_reachable net ->
msg_lifted_leader_sublog_host net.