Formal metatheory of the Lambda calculus using Stoughton's substitution

Formalization of metatheory of the Lambda Calculus in constructive type theory using the Barendregt variable convention

“…We have also developed formal metatheory of the Lambda Calculus using multiple substitutions on concrete terms in Copello et al (2017). The difference with the present work has to do fundamentally with whether α-conversion is defined using substitution or a more elementary operation -for which, as has been argued, it seems more adequate to choose swapping of names.…”

Formalization in Constructive Type Theory of the Standardization Theorem for the Lambda Calculus using Multiple Substitution

“…This lemma is key to the proof and was originally stated by Kashima for single substitutions as: M st N and P st Q =⇒ M[z := P] st N[z := Q]. By taking the substitution to be an arbitrary (multiple) σ instead of the particular case where we replace just one variable z, we obtain a definition of substitution by structural recursion, and hence we can prove this result using just structural induction (see [3] for a detailed explanation). The substitution lemma is then stated as follows: • Case st-app:…”

“…From now on we present definitions and results not included in the library [2]. Firstly, we have proven that this definition of alpha equivalence is decidable:…”

“…In [3] a formalization of the Lambda Calculus in Martin-Löf's Constructive Type Theory is presented, which uses first-order syntax with one sort of names for both free and bound variables that does not identify α-convertible terms, and a multiple substitution operation introduced by Stoughton in [11]. The approach enables the authors to prove in a completely formal and quite elegant way significant results about the metatheory of the Lambda Calculus, namely the Church-Rosser Theorem and Subject Reduction for the simply typed Lambda Calculus à la Curry.…”

Formal metatheory of second-order abstract syntax

“…The question of formalising and reasoning about abstract syntax was motivated by the development of proof assistants and the realisation that the Barendregt [1984] variable convention -"rename variables as needed to avoid clashes" -is difficult to translate into a formal setting. This has lead to a host of approaches that address the encoding of variable binding in proof assistants and functional languages, such as: higher-order abstract syntax [Pfenning and Elliot 1988;Chlipala 2008]; locally nameless representation [Bird and Paterson 1999;McBride and McKinna 2004;Weirich et al 2011;Charguéraud 2012]; intrinsically-typed encoding [Benton et al 2012;Allais et al 2021;Érdi 2018];and others [Shinwell et al 2003;Urban and Kaliszyk 2011;Copello et al 2017]. Similarly active is the mathematical study of abstract syntax and variable binding: developments include presheaf models [Fiore et al 1999;Hofmann 1999]; nominal sets [Gabbay and Pitts 1999]; monadic approaches [Bellegarde and Hook 1994;Altenkirch and Reus 1999]; and others [Pigozzi and Salibra 1995;Sun 1999;Blanchette et al 2019;Chen and Roşu 2020].…”