Proof using rmri lalcii rri.
intros.
pose proof (lift_prop _ (lastApplied_le_commitIndex_invariant)).
find_apply_lem_hyp msg_simulation_1.
find_apply_hyp_hyp.
unfold lastApplied_le_commitIndex in *.
match goal with
| [ H : _ |- _ ] => specialize (H h)
end.
find_rewrite_lem deghost_spec.
find_rewrite_lem msg_deghost_spec.
auto.
Qed.

Lemma lifted_directly_committed_directly_committed :
forall net e,
lifted_directly_committed net e ->
directly_committed (mgv_deghost net) e.