Positive Focusing is Directly Useful

Recently, Miller and Wu introduced the positive $λ$-calculus, a call-by-value $λ$-calculus with sharing obtained by assigning proof terms to the positively polarized focused proofs for minimal intuitionistic logic. The positive $λ$-calculus stands out among $λ$-calculi with sharing for a compactness property related to the sharing of variables. We show that -- thanks to compactness -- the positive calculus neatly captures the core of useful sharing, a technique for the study of reasonable time cost models.