Class TraceRelation `{State : Type} `{Event : Type} (step : step_relation State Event) :=
{
init : State;
T : (list Event) -> Prop;
T_dec : forall l, {T l} + {~ T l};
R : State -> Prop;
R_monotonic : forall s s' tr o, refl_trans_1n_trace step init s tr ->
step s s' o ->
R s ->
R s';
T_false_init : ~ T [];
T_implies_R : forall tr s s' o,
refl_trans_1n_trace step init s tr ->
~ T tr ->
step s s' o ->
T (tr ++ o) ->
R s'
}.
Section TraceRelations.
Context `{TR : TraceRelation}.

Theorem trace_relations_work :
forall s tr,
refl_trans_1n_trace step init s tr ->
T tr -> R s.