Proof using.
unfold locks_correct.
intros.
rewrite_update.
now find_higher_order_rewrite.
Qed.

(* Specifications: *)
Lemma locks_correct_true_at_head_of_queue :
forall sigma x,
locks_correct sigma ->
held (sigma (Client x)) = true ->
at_head_of_queue sigma x.
Lemma locks_correct_true_at_head_of_queue :
forall sigma x,
locks_correct sigma ->
held (sigma (Client x)) = true ->
at_head_of_queue sigma x.