Ltac get_invariant_pre inv :=
    match goal with
      | H : raft_intermediate_reachable _ |- _=>
        match (type of H) with
          | context [mkNetwork] =>
            fail 2
        end || copy_apply inv H
    end.Ltac get_invariant_post inv :=
    match goal with
      | H : raft_intermediate_reachable _ |- _=>
        match (type of H) with
          | context [mkNetwork] =>
            copy_apply inv H            
        end
    end.Definition clientCache_to_ks (c : list (clientId * (nat * output))) :=
    map (fun e => (fst e, fst (snd e))) c.

Lemma log_to_ks'_assoc_default_eq : forall l ks ks' c, assoc_default clientId_eq_dec (log_to_ks' l ks) c 0 <= assoc_default clientId_eq_dec ks' c 0 -> assoc_default clientId_eq_dec (log_to_ks' l ks') c 0 = assoc_default clientId_eq_dec ks' c 0.