Split Decisions: Explicit Contexts for Substructural Languages

A central challenge in mechanizing the meta-theory of substructural languages is modeling contexts. Although various ad hoc approaches to this problem exist, we lack a set of good practices and a simple infrastructure that can be leveraged for mechanizing a wide range of substructural systems. In this work, we describe Contexts as Resource Vectors (CARVe), a general syntactic infrastructure for managing substructural contexts, where elements are annotated with tags from a resource algebra denoting their availability. Assumptions persist as contexts are manipulated since we model resource consumption by changing their tags. We may thus define relations between substructural contexts via simultaneous substitutions without the need to split them. Moreover, we establish a series of algebraic properties about context operations that are typically required to carry out proofs in practice. CARVe is implemented in the proof assistant Beluga. To illustrate best practices for using our infrastructure, we give a detailed reformulation of the linear sequent calculus and bidirectional linear λ-calculus in terms of CARVe’s context operations and prove their equivalence using the aforementioned algebraic properties. In addition, we apply CARVe to mechanize a diverse set of systems, from the affine λ-calculus to the session-typed process calculus CP, giving us confidence that CARVe is sufficiently general to mechanize a broad range of substructural systems.