Proof using.
intros n c.
cofix CIH.
destruct s.
destruct e.
intros Hexec HNth.
invcs Hexec.
invcs H1.
- unfold runGenHandler, NetHandler in *.
break_match.
+ coinductive_case CIH.
find_rewrite. simpl.
now rewrite_update.
+ find_apply_lem_hyp ServerNetHandler_lbcases.
intuition.
* coinductive_case CIH.
repeat find_rewrite. simpl.
rewrite_update.
break_exists.
intuition; repeat find_rewrite; try solve_by_inversion.
now eauto using Nth_app.
* exfalso. clear CIH.
subst.
invcs HNth.
repeat find_reverse_rewrite. simpl in *.
repeat find_rewrite. now solve_by_inversion.
* clear CIH.
apply W0. simpl.
fold LockServ_MultiParams in *. (* typeclass stuff *)
repeat find_rewrite. simpl.
rewrite_update. repeat find_rewrite.
intuition eauto using Nth_tl.
break_exists.
intuition. subst.
find_apply_lem_hyp Nth_tl.
repeat find_rewrite.
invcs HNth. auto.
* coinductive_case CIH.
repeat find_rewrite. simpl.
now rewrite_update.
- unfold runGenHandler in *.
find_apply_lem_hyp InputHandler_cases.
intuition.
+ break_exists. break_and. subst.
coinductive_case CIH.
repeat find_rewrite. simpl.
now rewrite_update.
+ coinductive_case CIH.
repeat find_rewrite. simpl.
update_destruct; subst; now rewrite_update.
- coinductive_case CIH.
Qed.

Lemma MsgUnlock_moves_client :
forall n c s,
lb_step_execution lb_step_async s ->
Nth (queue (nwState (evt_a (hd s)) Server)) (S n) c ->
now (occurred MsgUnlock) s ->
next (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.