Proof using multi_map_lb_congr lb_step_ordered_failure_weak_fairness_always_enabled_pt_map_onet_continuously label_tot_mapped.
move => s.
rewrite /weak_fairness => H_exec H_fair l H_neq H_en.
have [l' H_l] := label_tot_mapped l.
rewrite H_l.
apply pt_map_onet_tot_map_label_event_inf_often_occurred.
apply H_fair; first by move => H_eq; rewrite H_eq pt_lb_label_silent_fst_snd in H_l.
rewrite H_l in H_en.
unfold cont_enabled in *.
apply: pt_map_onet_tot_map_labeled_event_state_continuously_enabled => //.
move => H_eq.
by rewrite -H_l in H_eq.
Qed.
Context {overlay_fst : NameOverlayParams (@unlabeled_multi_params _ labeled_multi_fst)}.
Context {overlay_snd : NameOverlayParams (@unlabeled_multi_params _ labeled_multi_snd)}.
Context {overlay_map_congr : NameOverlayParamsTotalMapCongruency overlay_fst overlay_snd name_map}.
Context {fail_msg_fst : FailMsgParams (@unlabeled_multi_params _ labeled_multi_fst)}.
Context {fail_msg_snd : FailMsgParams (@unlabeled_multi_params _ labeled_multi_snd)}.
Context {fail_msg_map_congr : FailMsgParamsPartialMapCongruency fail_msg_fst fail_msg_snd msg_map}.

(* Specifications: *)
Lemma pt_map_onet_hd_step_ordered_failure_star :
forall e, event_step_star step_ordered_failure step_ordered_failure_init e ->
event_step_star step_ordered_failure step_ordered_failure_init (pt_map_onet_event e).