Section LabeledStepExecution.
Variable A : Type.
Variable L : Type.
Variable trace : Type.
Definition lb_step_relation := A -> L -> A -> list trace -> Prop.
Definition lb_step_ex (step : lb_step_relation) (l : L) (a : A) : Prop :=
exists a' tr, step a l a' tr.
Record event := { evt_a : A ; evt_l : L ; evt_trace : list trace }.
Definition enabled (step : lb_step_relation) (l : L) (e : event) : Prop :=
lb_step_ex step l (evt_a e).
Definition occurred (l : L) (e : event) : Prop := l = evt_l e.
Definition inf_enabled (step : lb_step_relation) (l : L) (s : infseq event) : Prop :=
inf_often (now (enabled step l)) s.
Definition cont_enabled (step : lb_step_relation) (l : L) (s : infseq event) : Prop :=
continuously (now (enabled step l)) s.
Definition inf_occurred (l : L) (s : infseq event) : Prop :=
inf_often (now (occurred l)) s.
Definition strong_fairness (step : lb_step_relation) (silent : L) (s : infseq event) : Prop :=
forall l : L, l <> silent -> inf_enabled step l s -> inf_occurred l s.
Definition weak_fairness (step : lb_step_relation) (silent : L) (s : infseq event) : Prop :=
forall l : L, l <> silent -> cont_enabled step l s -> inf_occurred l s.
Proof using.
unfold strong_fairness. unfold inf_enabled, inf_occurred, inf_often.
intros step e silent s fair l neq alev.
assert (alevt_es: always (eventually (now (enabled step l))) (Cons e s)).
constructor.
constructor 2. destruct alev; assumption.
simpl. assumption.
clear alev. generalize (fair l neq alevt_es); clear fair alevt_es.
intro fair; case (always_Cons fair); trivial.
Qed.
Proof using.
move => step silent.
rewrite /extensional /strong_fairness /inf_enabled /inf_occurred /=.
move => s1 s2 H_eq H_s1 l' H_neq' H_en.
have H_s1l := H_s1 l'.
move: H_s1l.
set s1i := inf_often (now (occurred _)) s1.
move => H_s1l.
suff H_suff: s1i.
move: H_suff.
apply extensional_inf_often => //.
exact: extensional_now.
apply: H_s1l => {s1i}; first by [].
move: H_en.
apply: extensional_inf_often; last exact: exteq_sym.
exact: extensional_now.
Qed.
Proof using.
unfold weak_fairness. unfold cont_enabled, inf_occurred, continuously, inf_often.
intros step e silent s fair a neq eval.
assert (eval_es: eventually (always (now (enabled step a))) (Cons e s)).
apply E_next. assumption.
apply fair in eval_es.
apply always_invar in eval_es.
assumption.
assumption.
Qed.
Proof using.
move => step silent.
rewrite /extensional /weak_fairness /cont_enabled /inf_occurred /=.
move => s1 s2 H_eq H_s1 l' H_neq' H_en.
have H_s1l := H_s1 l'.
move: H_s1l.
set s1i := inf_often (now (occurred _)) s1.
move => H_s1l.
suff H_suff: s1i.
move: H_suff.
apply extensional_inf_often => //.
exact: extensional_now.
apply: H_s1l => {s1i}; first by [].
move: H_en.
apply: extensional_continuously; last exact: exteq_sym.
exact: extensional_now.
Qed.
Proof using.
move => step silent.
case => e s.
rewrite /strong_fairness /weak_fairness /inf_enabled /cont_enabled.
move => H_str l neq H_cont.
apply: H_str; first by [].
exact: continuously_inf_often.
Qed.
CoInductive lb_step_execution (step : lb_step_relation) : infseq event -> Prop :=
Cons_lb_step_exec : forall (e e' : event) (tr : list trace) (s : infseq event),
step (evt_a e) (evt_l e) (evt_a e') tr ->
evt_trace e' = evt_trace e ++ tr ->
lb_step_execution step (Cons e' s) ->
lb_step_execution step (Cons e (Cons e' s)).
Proof using.
intros step x s e. change (lb_step_execution step (tl (Cons x s))).
destruct e; simpl. assumption.
Qed.
Proof using.
move => step.
rewrite /extensional /=.
cofix c.
case => e1; case => e1' s1.
case => e2; case => e2' s2 H_eq.
find_apply_lem_hyp exteq_inversion.
break_and.
find_copy_apply_lem_hyp exteq_inversion.
break_and.
repeat find_rewrite.
move => H_exec.
inversion H_exec; subst.
apply (Cons_lb_step_exec _ tr) => //.
by apply: c; eauto.
Qed.
Definition event_step_star (step : step_relation A trace) (init : A) (e : event) :=
refl_trans_1n_trace step init (evt_a e) (evt_trace e).
Definition step_star_lb_step_reachable (lb_step : lb_step_relation) (step : step_relation A trace) (init : A) :=
forall a l a' tr tr',
refl_trans_1n_trace step init a tr' ->
lb_step a l a' tr ->
refl_trans_1n_trace step init a' (tr' ++ tr).

Lemma step_star_lb_step_execution :
forall lb_step step init,
step_star_lb_step_reachable lb_step step init ->
forall s, event_step_star step init (hd s) ->
lb_step_execution lb_step s ->
always (now (event_step_star step init)) s.