Repetition-free and infinitary analytic calculi for first-order rational Pavelka logic

We present an analytic hypersequent calculus $\textnormal{G}^3\textnormal{\L}\forall$ for first-order infinite-valued Łukasiewicz logic $\textnormal{\L}\forall$ and for an extension of it, first-order rational Pavelka logic $\textnormal{RPL}\forall$; the calculus is intended for bottom-up proof search. In $\textnormal{G}^3\textnormal{\L}\forall$, 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 $\textnormal{G}^3\textnormal{\L}\forall$ proves any sentence that is provable in at least one of the previously known analytic calculi for $\textnormal{\L}\forall$ or $\textnormal{RPL}\forall$, including Baaz and Metcalfe's hypersequent calculus $\textnormal{G}\textnormal{\L}\forall$ for $\textnormal{\L}\forall$. We study proof-theoretic properties of $\textnormal{G}^3\textnormal{\L}\forall$ and thereby provide foundations for proof search algorithms. We also give the first correct proof of the completeness of the $\textnormal{G}\textnormal{\L}\forall$-based infinitary calculus for prenex $\textnormal{\L}\forall$-sentences, and establish the completeness of a $\textnormal{G}^3\textnormal{\L}\forall$-based infinitary calculus for prenex $\textnormal{RPL}\forall$-sentences.