The Dialectica interpretation of first-order classical affine logic

We give a Dialectica-style interpretation of first-order classical affine logic. By moving to a contraction-free logic, the translation (a.k.a. D-translation) of a first-order formula into a higher-type ∃∀- formula can be made symmetric with respect to duality, including exponentials. It turned out that the propositional part of our D-translation uses the same construction as de Paiva's dialectica category struck G signℂ and we show how our D-translation extends double struck G signℂ to the first-order setting in terms of an indexed category. Furthermore the combination of Girard's ?!-translation and our D-translation results in the essentially equivalent ∃∀-formulas as the double-negation translation and Gödel's original D-translation.