Proof using lb_step_ordered_failure_weak_fairness_always_enabled_pt_map_onet_continuously.
move => l H_neq s H_exec H_fair.
have H_a: ((lb_step_execution lb_step_ordered_failure) /\_ (weak_fairness lb_step_ordered_failure label_silent)) s by auto.
move: H_a {H_exec H_fair}.
apply: eventually_map_conv_ext => {s}.
- apply extensional_always.
exact: extensional_now.
- apply extensional_always.
exact: extensional_now.
- apply extensional_and_tl.
* exact: lb_step_execution_extensional.
* exact: weak_fairness_extensional.
- rewrite /and_tl /=.
move => x s [H_e H_w].
apply lb_step_execution_invar in H_e.
by apply weak_fairness_invar in H_w.
- case => x s [H_a H_w] H_al.
simpl in *.
exact: lb_step_ordered_failure_weak_fairness_always_enabled_pt_map_onet_continuously.
Qed.

(* Specifications: *)
Lemma pt_map_onet_tot_map_label_event_strong_fairness :
forall s, lb_step_execution lb_step_ordered_failure s ->
strong_fairness lb_step_ordered_failure label_silent s ->
strong_fairness lb_step_ordered_failure label_silent (map pt_map_onet_event s).