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Well-founded recursion with copatterns and sized types

Part of: JFP Research Articles

Published online by Cambridge University Press:  03 March 2016

ANDREAS ABEL  and
BRIGITTE PIENTKA
ANDREAS ABEL
Affiliation:
Department of Computer Science and Engineering, Gothenburg University, Sweden (e-mail: andreas.abel@gu.se)
BRIGITTE PIENTKA
Affiliation:
School of Computer Science, McGill University, Montreal, Canada (e-mail: bpientka@cs.mcgill.ca)
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Abstract

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In 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.

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Articles
Information
Journal of Functional Programming , Volume 26 , 2016 , e2
Copyright
Copyright © Cambridge University Press 2016 

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