Reductions in Higher-Order Rewriting and Their Equivalence

Authors Pablo Barenbaum, Eduardo Bonelli

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Pablo Barenbaum
  • National University of Quilmes (CONICET), Bernal, Argentina
  • University of Buenos Aires, Argentina
Eduardo Bonelli
  • Stevens Institute of Technology, Hoboken, NJ, USA

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Pablo Barenbaum and Eduardo Bonelli. Reductions in Higher-Order Rewriting and Their Equivalence. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Proof terms are syntactic expressions that represent computations in term rewriting. They were introduced by Meseguer and exploited by van Oostrom and de Vrijer to study equivalence of reductions in (left-linear) first-order term rewriting systems. We study the problem of extending the notion of proof term to higher-order rewriting, which generalizes the first-order setting by allowing terms with binders and higher-order substitution. In previous works that devise proof terms for higher-order rewriting, such as Bruggink’s, it has been noted that the challenge lies in reconciling composition of proof terms and higher-order substitution (β-equivalence). This led Bruggink to reject "nested" composition, other than at the outermost level. In this paper, we propose a notion of higher-order proof term we dub rewrites that supports nested composition. We then define two notions of equivalence on rewrites, namely permutation equivalence and projection equivalence, and show that they coincide.

Subject Classification

ACM Subject Classification
  • Theory of computation → Equational logic and rewriting
  • Theory of computation → Type theory
  • Term Rewriting
  • Higher-Order Rewriting
  • Proof terms
  • Equivalence of Computations


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