L-types for resource awareness: an implicit name approach

Since the early work of Church on $\lambda$I-calculus and Gentzen on structural rules, the control of variable use has gained an important role in computation and logic emerging different resource aware calculi and substructural logics with applications to programming language and compiler design. This paper presents the first formalization of variable control in calculi with implicit names based on de Bruijn indices. We design and implement three calculi: first, a restricted calculus with implicit names; then, a restricted calculus with implicit names and explicit substitution, and finally, an extended calculus with implicit names and resource control. We propose a novel concept of L-types, which are used (a) to define terms and (b) to characterize certain classes of terms in each of the presented calculi. We have adopted to work simultaneously on the design and implementation in Haskell and Agda. The benefit of this strategy is twofold: dependent types enable to express and check properties of programs in the type system and constructive proofs of preservation enable a constructive evaluator for terms (programs).