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

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