Definition trace (state: Type): Type := stream (state * MetaState state).
Identity Coercion trace_stream: trace >-> stream.
Definition singleton_trace {state: Type} s ms: trace state :=
fin_stream ((s, ms) :: nil).
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.
Inductive begin_end_state {state: Type}: state -> trace state -> MetaState state -> Prop :=
| begin_end_empty: forall s, begin_end_state s empty_stream (Terminating s)
| begin_end_fin: forall s ms s' ms' tr n,
is_n_stream (S n) tr ->
tr 0 = Some (s, ms) ->
tr n = Some (s', ms') ->
begin_end_state s tr ms'
| begin_end_inf: forall s ms tr,
is_inf_stream tr ->
tr 0 = Some (s, ms) ->
begin_end_state s tr NonTerminating.
Definition begin_state {state: Type} (tr: trace state) (s: state): Prop :=
exists ms, begin_end_state s tr ms.
Definition end_state {state: Type} (tr: trace state) (ms: MetaState state): Prop :=
exists s, begin_end_state s tr ms.
Definition ctrace2trace
{cmd: Type}
{state: Type}: trace (cmd * state) -> trace state :=
stream_map (fun p => match p with ((c, s), mcs) => (s, lift_function snd mcs) end).
Definition traces (state: Type): Type := Ensemble (trace state).
Definition traces_app {state: Type} (d1 d2: traces state): traces state :=
fun tr =>
(exists tr1 tr2, d1 tr1 /\ d2 tr2 /\ tr = stream_app tr1 tr2) \/
(exists tr1, d1 tr1 /\ (end_state tr1 NonTerminating \/ end_state tr1 Error) /\ tr = tr1).
Fixpoint traces_power {state: Type} (d: traces state) (n: nat): traces state :=
match n with
| 0 => is_empty_stream
| S n => traces_app (traces_power d n) d
end.
Definition traces_pstar {state: Type} (d: traces state): traces state :=
fun tr => exists n, traces_power d n tr.
Definition traces_pomega {state: Type} (d: traces state): traces state :=
fun tr => exists f, tr = stream_capp f /\ forall n, d (f n).

Lemma traces_app_mono {state: Type}: forall (d1 d2 d3 d4: traces state),
Included _ d1 d3 ->
Included _ d2 d4 ->
Included _ (traces_app d1 d2) (traces_app d3 d4).