Ltac coinductive_case CIH :=
    apply W_tl; simpl in *; auto;
    apply CIH; simpl in *; auto.

Lemma clients_move_way_up_in_queue : forall n n' c s, n' <= n -> event_step_star step_async step_async_init (hd s) -> lb_step_execution lb_step_async s -> weak_fairness lb_step_async label_silent s -> (Nth (queue (nwState (evt_a (hd s)) Server)) (S n) c /\ (S n = 0 -> In (mkPacket Server (Client c) Locked) (nwPackets (evt_a (hd s))))) -> eventually (fun s => Nth (queue (nwState (evt_a (hd s)) Server)) n' c /\ (n' = 0 -> In (mkPacket Server (Client c) Locked) (nwPackets (evt_a (hd s))))) s.