  In this paper, we present a~generalisation of proof simulation procedures for
Frege systems by Bonet and Buss to some logics for which the deduction theorem
does not hold. In particular, we study the case of finite-valued \L{}ukasiewicz
logics. To this end, we provide proof systems that augment Avron's Frege system
for \L{}ukasiewicz three-valued logic with nested and general versions of the
disjunction elimination rule, respectively. For these systems we provide upper
bounds on speed-ups w.r.t.\ both the number of steps in proofs and the length
of proofs. We also consider Tamminga's natural deduction and Avron's
hypersequent calculus for 3-valued \L{}ukasiewicz logic and generalise our
results considering the disjunction elimination rule to all finite-valued
\L{}ukasiewicz logics.
