Two Decreasing Measures for Simply Typed λ-Terms

Authors Pablo Barenbaum, Cristian Sottile

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Pablo Barenbaum
  • ICC, Universidad de Buenos Aires, Argentina
  • Universidad Nacional de Quilmes (CONICET), Buenos Aires, Argentina
Cristian Sottile
  • ICC, Universidad de Buenos Aires (CONICET), Argentina
  • Universidad Nacional de Quilmes, Buenos Aires, Argentina


To Giulio Manzonetto for fruitful discussions that led to the development of this work. To Eduardo Bonelli and the anonymous reviewers for feedback on earlier versions of this paper. The second author would like to thank his advisors Alejandro Díaz-Caro and Pablo E. Martínez López.

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Pablo Barenbaum and Cristian Sottile. Two Decreasing Measures for Simply Typed λ-Terms. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 11:1-11:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


This paper defines two decreasing measures for terms of the simply typed λ-calculus, called the 𝒲-measure and the 𝒯^{𝐦}-measure. A decreasing measure is a function that maps each typable λ-term to an element of a well-founded ordering, in such a way that contracting any β-redex decreases the value of the function, entailing strong normalization. Both measures are defined constructively, relying on an auxiliary calculus, a non-erasing variant of the λ-calculus. In this system, dubbed the λ^{𝐦}-calculus, each β-step creates a "wrapper" containing a copy of the argument that cannot be erased and cannot interact with the context in any other way. Both measures rely crucially on the observation, known to Turing and Prawitz, that contracting a redex cannot create redexes of higher degree, where the degree of a redex is defined as the height of the type of its λ-abstraction. The 𝒲-measure maps each λ-term to a natural number, and it is obtained by evaluating the term in the λ^{𝐦}-calculus and counting the number of remaining wrappers. The 𝒯^{𝐦}-measure maps each λ-term to a structure of nested multisets, where the nesting depth is proportional to the maximum redex degree.

Subject Classification

ACM Subject Classification
  • Theory of computation → Equational logic and rewriting
  • Theory of computation → Lambda calculus
  • Lambda Calculus
  • Rewriting
  • Termination
  • Strong Normalization
  • Simple Types


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