Proof using overlay_map_congr name_map_bijective multi_map_congr fail_msg_map_congr.
move => e.
rewrite /= /pt_map_onet_event /= /event_step_star /=.
move => H_star.
destruct e, evt_a.
simpl in *.
exact: step_ordered_failure_pt_mapped_simulation_star_1.
Qed.

(* Specifications: *)
Lemma pt_map_onet_hd_step_ordered_failure_star_always :
forall s, event_step_star step_ordered_failure step_ordered_failure_init (hd s) ->
lb_step_execution lb_step_ordered_failure s ->
always (now (event_step_star step_ordered_failure step_ordered_failure_init)) (map pt_map_onet_event s).