Proof using.
induction tr; intros; simpl in *; repeat break_match; auto.
congruence.
Qed.

(* Specifications: *)
Lemma trace_mutex'_unlock_extend :
forall tr h c,
trace_mutual_exclusion' h tr ->
trace_mutual_exclusion' h (tr ++ [(Client c, inl Unlock)]).
Lemma trace_mutex'_unlock_extend :
forall tr h c,
trace_mutual_exclusion' h tr ->
trace_mutual_exclusion' h (tr ++ [(Client c, inl Unlock)]).