Definition set_queue {W O} (l : list input) :=
    modify (W := W)(O := O) (fun d => Build_PB_data l (state d)).Definition set_state {W O} (st : data) :=
    modify (W := W)(O := O) (fun d => Build_PB_data (queue d) st).Ltac pb_unfold := unfold set_queue, set_state in *; monad_unfold.Instance PB_base_params : BaseParams :=
    Build_BaseParams
      PB_data
      PB_input
      PB_output.Instance PB_multi_params : MultiParams PB_base_params :=
    Build_MultiParams
      PB_base_params
      msg_eq_dec
      name_eq_dec
      all_nodes_all
      NoDup_all_nodes
      PB_init
      PB_net
      PB_input_handler.Definition inputs_1 (tr : list ((@input base_params) * (@output base_params))) :
      list (@input base_params) :=
    map (@fst _ _) tr.Definition inputs_m (tr : list (name * (@input PB_base_params + list (@output PB_base_params)))) :
    list (@input base_params) :=
    filterMap (fun x => match x with
                          | (Primary, inl (Request i)) => Some i
                          | _ => None
                        end)
              tr.Definition outputs_1 (tr : list ((@input base_params) * (@output base_params))) :
    list (@output base_params) :=
    map (@snd _ _) tr.Definition correspond (st : @data base_params) (sigma : name -> @data PB_base_params) tr_1 tr_m :=
    let (d, os) := processInputs (state (sigma Primary)) (queue (sigma Primary)) in
    outputs_m tr_m ++ os = outputs_1 tr_1 /\
    d = st.

Lemma inputs_m_backup : forall i l, inputs_m ((Backup, inl i) :: l) = inputs_m l.