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 => in_applied_entries client id (snd s)
    }.

Theorem output_implies_applied : forall failed net tr, step_failure_star step_failure_init (failed, net) tr -> key_in_output_trace client id tr -> in_applied_entries client id net.