Proof using.
unfold msg_refined_raft_net_invariant_init, commit_invariant.
split.
- unfold commit_invariant_host, commit_recorded_committed, commit_recorded, committed. simpl.
intuition.
- unfold commit_invariant_nw; simpl; intuition.
Qed.

Lemma msg_lifted_lastApplied_le_commitIndex :
forall net h,
msg_refined_raft_intermediate_reachable net ->
lastApplied (snd (nwState net h)) <= commitIndex (snd (nwState net h)).