Well-founded recursion with copatterns and sized types

Abstract

AbstractIn this paper, we study strong normalization of a core language based on System ${\mathsf{F}_\omega}$ which supports programming with finite and infinite structures. Finite data such as finite lists and trees is defined via constructors and manipulated via pattern matching, while infinite data such as streams and infinite trees is defined by observations and synthesized via copattern matching. Taking a type-based approach to strong normalization, we track size information about finite and infinite data in the type. We exploit the duality of pattern and copatterns to give a unifying semantic framework which allows us to elegantly and uniformly support both well-founded induction and coinduction by rewriting. The strong normalization proof is structured around Girard's reducibility candidates. As such, our system allows for non-determinism and does not rely on coverage. Since System ${\mathsf{F}_\omega}$ is general enough that it can be the target of compilation for the Calculus of Constructions, this work is a significant step towards representing observation-based infinite data in proof assistants such as Coq and Agda.