Lincx: A Linear Logical Framework with First-class Contexts

Linear logic provides an elegant framework for modelling stateful, imperative and concurrent systems by viewing a context of assumptions as a set of resources. However, mechanizing the meta-theory of such systems remains a challenge, as we need to manage and reason about mixed contexts of linear and intuitionistic assumptions.

We present Lincx, a contextual linear logical framework with first-class mixed contexts. Lincx allows us to model (linear) abstract syntax trees as syntactic structures that may depend on intuitionistic and linear assumptions. It can also serve as a foundation for reasoning about such structures. Lincx extends the linear logical framework LLF with first-class (linear) contexts and an equational theory of context joins that can otherwise be very tedious and intricate to develop. This work may be also viewed as a generalization of contextual LF that supports both intuitionistic and linear variables, functions, and assumptions.

We describe a decidable type-theoretic foundation for Lincx that only characterizes canonical forms and show that our equational theory of context joins is associative and commutative. Finally, we outline how Lincx may serve as a practical foundation for mechanizing the meta-theory of stateful systems.