Definition sequential_trace {state: Type} (tr: trace state) : Prop :=
  forall k s ms s' ms',
    tr k = Some (s, ms) ->
    tr (S k) = Some (s' , ms') ->
    ms = Terminating s'.Definition sound_trace {state: Type} (R: state -> MetaState state -> Prop) (tr: trace state) : Prop :=
  forall k s ms,
    tr k = Some (s, ms) ->
    R s ms.

(* Specifications: *)
Lemma sound_trace_and {state: Type}: forall R1 R2 (tr: trace state), sound_trace (fun s ms => R1 s ms /\ R2 s ms) tr <-> sound_trace R1 tr /\ sound_trace R2 tr.
Lemma trace_app_sequential1 {state: Type}: forall (tr1 tr2: trace state), sequential_trace (stream_app tr1 tr2) -> sequential_trace tr1.
Lemma trace_app_sequential2 {state: Type}: forall (tr1 tr2: trace state), is_fin_stream tr1 -> sequential_trace (stream_app tr1 tr2) -> sequential_trace tr2.
Lemma trace_app_sound1 {state: Type}: forall R (tr1 tr2: trace state), sound_trace R (stream_app tr1 tr2) -> sound_trace R tr1.
Lemma trace_app_sound2 {state: Type}: forall R (tr1 tr2: trace state), is_fin_stream tr1 -> sound_trace R (stream_app tr1 tr2) -> sound_trace R tr2.