Substitution for Non-Wellfounded Syntax with Binders Through Monoidal Categories

Authors Ralph Matthes , Kobe Wullaert , Benedikt Ahrens

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Author Details

Ralph Matthes
  • IRIT, Université de Toulouse, CNRS, Toulouse INP, UT3, Toulouse, France
Kobe Wullaert
  • Delft University of Technology, The Netherlands
Benedikt Ahrens
  • Delft University of Technology, The Netherlands
  • University of Birmingham, United Kingdom


We thank Henning Basold for pointing us to the work on completely iterative algebras, leading to a simpler proof of Theorem 4. We also thank Thomas Lamiaux for valuable comments on a draft of this paper. We gratefully acknowledge the work by the Coq development team in providing the Coq proof assistant and surrounding infrastructure, as well as their support in keeping UniMath compatible with Coq. Not least, we thank the anonymous FSCD reviewers for their thoughtful feedback on our submission.

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Ralph Matthes, Kobe Wullaert, and Benedikt Ahrens. Substitution for Non-Wellfounded Syntax with Binders Through Monoidal Categories. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 25:1-25:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We describe a generic construction of non-wellfounded syntax involving variable binding and its monadic substitution operation. Our construction of the syntax and its substitution takes place in category theory, notably by using monoidal categories and strong functors between them. A language is specified by a multi-sorted binding signature, say Σ. First, we provide sufficient criteria for Σ to generate a language of possibly infinite terms, through ω-continuity. Second, we construct a monadic substitution operation for the language generated by Σ. A cornerstone in this construction is a mild generalization of the notion of heterogeneous substitution systems developed by Matthes and Uustalu; such a system encapsulates the necessary corecursion scheme for implementing substitution. The results are formalized in the Coq proof assistant, through the UniMath library of univalent mathematics.

Subject Classification

ACM Subject Classification
  • Theory of computation → Type theory
  • Theory of computation → Logic and verification
  • Non-wellfounded syntax
  • Substitution
  • Monoidal categories
  • Actegories
  • Tensorial strength
  • Proof assistant Coq
  • UniMath library


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