Definition term_of (m : msg) :=
    match m with
      | RequestVote t _ _ _ => Some t
      | RequestVoteReply t _ => Some t
      | AppendEntries t _ _ _ _ _ => Some t
      | AppendEntriesReply t _ _ => Some t
    end.

Lemma applied_entries_safe_update : forall sigma h st, lastApplied st = lastApplied (sigma h) -> removeAfterIndex (log st) (lastApplied (sigma h)) = removeAfterIndex (log (sigma h)) (lastApplied (sigma h)) -> applied_entries (update name_eq_dec sigma h st) = applied_entries sigma.