Proof using name_map_bijective multi_map_lb_congr multi_map_congr label_eq_dec.
cofix c.
move => s H_exec.
rewrite -pt_map_onet_event_Map_unfold {1}/pt_map_onet_event /=.
inversion H_exec; subst => /=.
rewrite -pt_map_onet_event_Map_unfold /= /pt_map_onet_event /=.
case (label_eq_dec (tot_map_label (evt_l e)) label_silent) => H_eq.
apply: Cons_lb_step_exec.
- rewrite H_eq.
destruct e, e'.
destruct evt_a, evt_a0.
simpl in *.
by eapply lb_step_ordered_failure_pt_mapped_simulation_1_silent; eauto.
- destruct e, e'.
destruct evt_a, evt_a0.
apply (lb_step_ordered_failure_pt_mapped_simulation_1_silent H_eq) in H.
break_and.
simpl in *.
rewrite H0 filterMap_app.
by aggressive_rewrite_goal.
- pose s' := Cons e' s0.
rewrite (pt_map_onet_event_Map_unfold s').
exact: c.
apply: Cons_lb_step_exec => /=.
- destruct e, e'.
destruct evt_a, evt_a0.
simpl in *.
by eapply lb_step_ordered_failure_pt_mapped_simulation_1_non_silent; eauto.
- by rewrite H0 filterMap_app.
- pose s' := Cons e' s0.
rewrite (pt_map_onet_event_Map_unfold s').
exact: c.
Qed.

(* Specifications: *)
Lemma pt_map_onet_tot_map_label_event_inf_often_occurred :
forall l s,
inf_often (now (occurred l)) s ->
inf_often (now (occurred (tot_map_label l))) (map pt_map_onet_event s).