Program Instance TR : TraceRelation step_failure :=
      {
        init := step_failure_init;
        T := key_in_output_trace client id ;
        T_dec := key_in_output_trace_dec client id ;
        R := fun s => before_func (has_key client id) (has_key client id') (applied_entries (nwState (snd s)))
      }.

Theorem output_greatest_id : forall failed net tr, step_failure_star step_failure_init (failed, net) tr -> key_in_output_trace client id tr -> before_func (has_key client id) (has_key client id') (applied_entries (nwState net)).