Size-Based Termination for Non-Positive Types in Simply Typed Lambda-Calculus

Author Yuta Takahashi

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Yuta Takahashi
  • Ochanomizu University, Tokyo, Japan


I want to thank Frédéric Blanqui and Ralph Matthes for valuable discussions on size-based termination and non-strictly positive inductive types, respectively. I also thank the anonymous reviewers for their comments which improved the earlier version of this paper.

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Yuta Takahashi. Size-Based Termination for Non-Positive Types in Simply Typed Lambda-Calculus. In 27th International Conference on Types for Proofs and Programs (TYPES 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 239, pp. 12:1-12:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


So far, several typed lambda-calculus systems were combined with algebraic rewrite rules, and the termination (in other words, strong normalisation) problem of the combined systems was discussed. By the size-based approach, Blanqui formulated a termination criterion for simply typed lambda-calculus with algebraic rewrite rules which guarantees, in some specific cases, the termination of the rewrite relation induced by beta-reduction and algebraic rewrite rules on strictly or non-strictly positive inductive types. Using the inflationary fixed-point construction, we extend this termination criterion so that it is possible to show the termination of the rewrite relation induced by some rewrite rules on types which are called non-positive types. In addition, we note that a condition in Blanqui’s proof can be dropped, and this improves the criterion also for non-strictly positive inductive types.

Subject Classification

ACM Subject Classification
  • Theory of computation → Equational logic and rewriting
  • Theory of computation → Type theory
  • termination
  • higher-order rewriting
  • non-positive types
  • inductive types


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