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.

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.