Proof using lb_step_ordered_failure_strong_fairness_enabled_pt_map_onet_eventually.
move => l H_neq s H_exec H_fair.
have H_a: ((lb_step_execution lb_step_ordered_failure) /\_ (strong_fairness lb_step_ordered_failure label_silent)) s by auto.
move: H_a {H_exec H_fair}.
apply: always_map_conv_ext => {s}.
rewrite /and_tl /=.
move => x s0 [H_e H_w].
apply lb_step_execution_invar in H_e.
by apply strong_fairness_invar in H_w.
apply: eventually_map_conv_ext.
- exact: extensional_now.
- exact: extensional_now.
- apply extensional_and_tl.
* exact: lb_step_execution_extensional.
* exact: strong_fairness_extensional.
- rewrite /and_tl /=.
move => x s [H_e H_w].
apply lb_step_execution_invar in H_e.
by apply strong_fairness_invar in H_w.
- rewrite /and_tl.
case => /= x s [H_a H_w] H_en.
exact: lb_step_ordered_failure_strong_fairness_enabled_pt_map_onet_eventually.
Qed.
Hypothesis lb_step_ordered_failure_weak_fairness_always_enabled_pt_map_onet_continuously :
forall l, tot_map_label l <> label_silent ->
forall s, lb_step_execution lb_step_ordered_failure s ->
weak_fairness lb_step_ordered_failure label_silent s ->
always (now (enabled lb_step_ordered_failure (tot_map_label l))) (map pt_map_onet_event s) ->
continuously (now (enabled lb_step_ordered_failure l)) s.

Lemma pt_map_onet_tot_map_labeled_event_state_continuously_enabled :
forall l, tot_map_label l <> label_silent ->
forall s, lb_step_execution lb_step_ordered_failure s ->
weak_fairness lb_step_ordered_failure label_silent s ->
continuously (now (enabled lb_step_ordered_failure (tot_map_label l))) (map pt_map_onet_event s) ->
continuously (now (enabled lb_step_ordered_failure l)) s.