Proof using.
unfold PB_net. intros.
pb_unfold. repeat (first [break_let | break_match]); repeat find_inversion; auto; simpl.
- left. intuition. left. eauto.
- right. left. intuition.
- right. left. intuition. right. eexists. eexists. intuition eauto.
find_rewrite. auto.
- right. left. intuition. right. eexists. eexists. intuition eauto.
find_rewrite. auto.
- right. right. intuition. eauto.
- left. intuition.
Qed.
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.
Fixpoint outputs_m (tr : list (name * (@input PB_base_params + list (@output PB_base_params)))) :
list (@output base_params) :=
match tr with
| [] => []
| (Primary, inr l) :: tr' => filterMap (fun x => match x with
| RequestResponse i os => Some os
| _ => None
end) l ++ outputs_m tr'
| _ :: tr' => outputs_m tr'
end.
Fixpoint processInputs (d : @data base_params) (l : list (@input base_params)) :
(@data base_params * list (@output base_params)) :=
match l with
| [] => (d, [])
| i :: l' => let (os, d') := @handler _ one_node_params i d in
let (d'', os') := processInputs d' l' in
(d'', os :: os')
end.
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_1_nil_outputs_1_nil :
forall tr,
inputs_1 tr = [] ->
outputs_1 tr = [].