  We present a hypersequent calculus $\text{G}^3\text{\L}\forall$ for
first-order infinite-valued {\L}ukasiewicz logic and for an extension of it,
first-order rational Pavelka logic; the calculus is intended for bottom-up
proof search. In $\text{G}^3\text{\L}\forall$, there are no structural rules,
all the rules are invertible, and designations of multisets of formulas are not
repeated in any premise of the rules. The calculus $\text{G}^3\text{\L}\forall$
proves any sentence that is provable in at least one of the previously known
hypersequent calculi for the given logics. We study proof-theoretic properties
of $\text{G}^3\text{\L}\forall$ and thereby provide foundations for proof
search algorithms.
