Definition pt_map_trace_ev (e : @name _ multi_fst * (@input base_fst + (@output base_fst))) :
 option (@name _ multi_snd * (@input base_snd + (@output base_snd))) :=
match e with
|  (n, inl i) => 
  match pt_map_input i with
  | Some i' => Some (tot_map_name n, inl i')
  | None => None
  end
| (n, inr o) => 
  match pt_map_output o with
  | Some o' => Some (tot_map_name n, inr o')
  | None => None
  end
end.

Lemma filterMap_pt_map_trace_ev_outputs_eq : forall out to, filterMap pt_map_trace_ev (map2fst to (map inr out)) = map2fst (tot_map_name to) (map inr (filterMap pt_map_output out)).