Finite Proofs for Infinitary Formulas

Recent work has shown that the infinitary logic of hereand-there is closely related to the input language of the ASP grounder gringo. A formal system axiomatizing that logic exists, but a proof in that system may include infinitely many formulas. In this note, we define a correspondence between the validity of infinitary formulas in the logic of here-and-there and the provability of formulas in some finite deductive systems. On the basis of this correspondence, we can use finite proofs to justify the validity of infinitary formulas.