A sequent calculus for limit computable mathematics

We prove a kind of Curry-Howard isomorphism, with some set of recursive winning strategies taking the place of typed λ-terms, and some set of classical proofs taking the place of intuitionistic proofs. We introduce an implication-free fragment PA1 of ω-arithmetic, having Exchange rule for sequents dropped. Exchange rule for formulas is, instead, an admissible rule in PA1. Then we show that cut-free proofs of PA1 are tree-isomorphic with recursive winning strategies for a set of games we call “1-backtracking games”. “1-backtracking games” are introduced in [1] as a complete semantics for the implication-free fragment of Limit Computable Mathematics, or LCM ([3], [4]). LCM, in turn, is a subset of Classical Mathematics introduced for software verification whose proofs can be “animated” to detect a formalization error. Our result implies that PA1 is the first formal system sound and complete for the implication-free fragment of LCM. No sound and complete formal system is known for LCM itself.