Formalisation in Constructive Type Theory of Stoughton's Substitution for the Lambda Calculus

Tasistro, Álvaro; Copello, Ernesto; Szász, Nóra

doi:10.1016/j.entcs.2015.04.013

Cited by 3 publications

(9 citation statements)

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“…In English, a name is fresh in the restriction (σ , M) if and only if it is fresh in every image σ y, for every y * M. Now we shall briefly discuss the mechanism in the framework used to rename the bound names in a given term, and so avoid capturing any free variable during the action of substitution. The complete description can be found in [21]. Let χ' be the function that returns the first name not in a given list:…”

Section: Substitutionmentioning

confidence: 99%

“…Whenever a name x syntactically occurs in a term M and is not bound by any abstraction, we shall say x is free in M, and write it x * M. On the other hand, if every occurrence of x is bound by some abstraction (or even if x does not occur at all), we shall say x is fresh in M, and write it x # M as in nominal techniques, e.g., see [23]. Both relations are inductively defined in a standard manner, and in [21] it was proven that both relations are opposite to each other.…”

Section: The Frameworkmentioning

confidence: 99%

“…In [21] a framework was presented for the formal meta-theory of the pure untyped lambda calculus in first-order abstract syntax (FOAS) and using only one sort of names for both free and bound variables 1 . Based upon Stoughton's work on multiple substitutions [19], the authors were able to give a primitive recursive definition of the operation of substitution which does not identify alpha-convertible terms 2 , avoids variable capture, and has a homogeneous treatment in the case of abstractions.…”

Section: Introductionmentioning

confidence: 99%

“…Initially, the sole objective of this work was to formalize a proof of the Strong Normalization Theorem but only for System T, and by using the framework presented in [21]. Of course, the syntax of the pure lambda terms had to be extended to include the term-formers for the natural numbers and the recursion operator 3 .…”

Section: Introductionmentioning

confidence: 99%

“…In the next section we introduce the new framework: its syntax, substitution, conversion theories and logical relations (reducible terms). Some results presented are completely new, and some others are an extension of [21,24] to consider the additional syntax. From Section 2.5 on, and unless the opposite is explicitly stated, all results represent new developments.…”

Section: Introductionmentioning

confidence: 99%

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A Formal Proof of the Strong Normalization Theorem for System T in Agda

Urciuoli

2023

Electron. Proc. Theor. Comput. Sci.

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We present a framework for the formal meta-theory of lambda calculi in first-order syntax, with two sorts of names, one to represent both free and bound variables, and the other for constants, and by using Stoughton's multiple substitutions. On top of the framework we formalize Girard's proof of the Strong Normalization Theorem for both the simply-typed lambda calculus and System T. As to the latter, we also present a simplification of the original proof. The whole development has been machine-checked using the Agda system. * This work is partially supported by Agencia Nacional de Investigación e Innovación (ANII), Uruguay. 1 Both the previous framework and the one presented here use named variables, it bears repeating. In a contrary sense, there are nameless approaches, e.g., or locally nameless syntax [6], which use numbers to identify the variables.2 Or without using Barendregt's variable convention.

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Section: Substitutionmentioning

confidence: 99%

Section: The Frameworkmentioning

confidence: 99%