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

(* Specifications: *)
Lemma last_holder_unlock_none :
forall tr c,
last_holder tr = Some c ->
last_holder (tr ++ [(Client c, inl Unlock)]) = None.
Lemma last_holder_unlock_none :
forall tr c,
last_holder tr = Some c ->
last_holder (tr ++ [(Client c, inl Unlock)]) = None.