Proof using.
intros.
find_eapply_lem_hyp messages_trigger_labels; eauto.
find_apply_lem_hyp weak_until_until_or_always.
intuition.
- now eauto using until_eventually.
- find_apply_lem_hyp always_continuously.
eapply_prop_hyp weak_fairness continuously; auto.
destruct s.
now find_apply_lem_hyp always_now.
Qed.
Ltac coinductive_case CIH :=
apply W_tl; simpl in *; auto;
apply CIH; simpl in *; auto.

Lemma Nth_app :
forall A (l : list A) l' a n,
Nth l n a ->
Nth (l ++ l') n a.