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For example, alpha equivalent lambda terms receive equal representation; substitution is represented by application in the ambient higher-order logic. ... For example, using higher-order syntax, the terms of the untyped lambda calculus may be represented as a type exp with two con- structors app: exp x exp — exp (binary constructor for application) and lam ...
The solution we propose is based on CINNI, a new calculus of explicit substitutions that makes use of a term representation that contains both the standard named notation and de Bruijn's indexed notation ... The calculus is parametric in the syntax of the object language, which allows us to apply it to different object languages such as λ-calculus, Abadi and Cardelli's object calculus (ς-calculus) and Milner's ... In particular, we are indebted to an anonymous referee for pointing out that the representation used by CINNI for pure terms is equivalent to Berkling's representation, and for providing us with the corresponding ...doi:10.1016/s1571-0661(05)80125-2 fatcat:ggrxnrgmpnhb5ie6rxdwfw2w5q
AbstractWe present the complete development, in Gallina, of the residual theory of β-reduction in pure λ-calculus. ... It is a specific concrete syntax for its abstract framework, the Calculus of Inductive Constructions (Paulin-Mohring, 1993). ... Lambda terms We represent our λ-terms with de Bruijn's indexes  . We need a minimum of arithmetical properties, concerning addition, and the standard orderings <, ≤ and > on natural numbers. ...doi:10.1017/s0956796800001106 fatcat:rvlumdmsmvcgha663kezkoiwtm
It is based on a typed lambda-calculus and it is in a certain sense an attractive mixture of systems like de Bruijn’s Automath, Martin- Léf’s intuitionistic theory of types and Girard’s system F. ... The authors give both the formal syntax and the backgrounds of the calculus of constructions. ...
with de Bruijn's notation (see  for general algorithms for this notation). ... As DCGs obey Prolog term syntax, their transposition must obey AProlog term syntax and typing. ... Thanks must also go to David Trotman and Luke Hornof for their careful reading of the paper. ...doi:10.1016/0743-1066(95)00040-q fatcat:ph7ll7eoinekvll3i6o5ktyqsa
Lecture Notes in Computer Science
We introduce a compressed de Bruijn representation of lambda terms and define its bijections to standard representations. ... Compact combinatorial generation algorithms are given for several families of lambda terms, including open, closed, simply typed and linear terms as well as type inference and normal order reduction algorithms ... We will devise here a simple and safe interpreter for lambda terms supporting normal order β-reduction by using de Bruijn terms, which also ensures that terms are unique up to α-equivalence. ...doi:10.1007/978-3-319-19686-2_9 fatcat:jjoiignyj5g7nirrll45xougnq
For instance, our definition of α-conversion is just syntax directed and we prove it to be an equivalence relation in an easy way, whereas in  the latter was included as part of the definition and ... In  , Alley Stoughton proposed a notion of (simultaneous) substitution for the Lambda calculus as formulated in its original syntax -i.e. with only one sort of symbols (names) for variables-and without ... Another alternative is of course de Bruijn's nameless syntax  or its more up-to-date version locally nameless syntax [2, 4] , which uses names for the free or global variables and the indices counting ...doi:10.1016/j.entcs.2015.04.013 fatcat:o4ksce3hjvdi3drtwjicacuuoi
For doing so, first, the original Huet algorithm for the simply typed λ-calculus with names is adapted to the language of the λcalculus in de Bruijn's notation, since this is the notation used by the λσ-calculus ... We compare two different styles of Higher-Order Unification (HOU): the classical HOU algorithm of Huet for the simply typed λ-calculus and HOU based on the λσ-calculus of explicit substitutions. ... Acknowledgements We would like to thank the careful reading, constructive corrections and detailed suggestions of one of the referees. ...doi:10.1016/j.jal.2006.10.001 fatcat:3ym4njp7wnfkbd4dm4virhbh5m
This paper presents an overview of Cαml's binding specification language and of the code that Cαml produces. ... Cαml is meant to help writers of interpreters, compilers, or other programs-that-manipulate-programs deal with α-conversion in a safe and concise style. ... Acknowledgement The author wishes to thank James Cheney, Jean-Christophe Filliâtre, Sébastien Hinderer, James Leifer, Randy Pollack, Yann Régis-Gianas, Didier Rémy, and Francesco Zappa-Nardelli, as well ...doi:10.1016/j.entcs.2005.11.039 fatcat:oxhhz4qxm5f7pmxzkuod6ollka
Lecture Notes in Computer Science
The original system was developed by Curien  inspired by the equivalence of the theories of typed .~-calculus and Cartesian Closed Categories as shown by Lambek  and Scott . ... In this paper we introduce a new set of categorical combinators which is more efficient than the previous ones, in terms of compilation and execution complexity. ... Acknowledgements This work was developed under the supervision of Dr.Simon Thompson to whom the author is indebted for many discussions, suggestions, and comments. ...doi:10.1007/3-540-18317-5_5 fatcat:odbwrmdiujhvvb5ntyyhs7mnxe
De Bruijn's view on propositions-as-types and proofs-as-terms De Bruijn reinvented the propositions-as-types concept, where he was somewhat influenced by Heyting. ... Informally, it is easy to say that one works "modulo renaming of bound variables", which means that one is not really considering terms, but equivalence classes of terms, where the equivalence relation ...doi:10.1016/j.indag.2013.09.003 fatcat:tgs7v5i6t5gxjl3s5c472obqje
More precisely, we sketch the general structure of a system of typed lambda calculus and show that this system has enough expressive power for the description of various existing systems, ranging from ... These advantages range from generatising reduction and substitution to representing Mathematics and are investigated in detail in various articles cited in the bibliography. ... Kamareddine is grateful to the Department of Mathematics and Computing Science, Eindhoven University of Technology, for their financial support and hospitality from October 1991 to September 1992, and ...doi:10.1016/0304-3975(94)00127-5 fatcat:zl2l6whaqzdlxeyqxfgyzismia
In this paper we focus on the backgrounds and on some uncommon aspects of the syntax of Automath. ... The 'mathematical language' Automath, conceived by N.G. de Bruijn in 1968, was the first theorem prover actually working and was used for checking many specimina of mathematical content. ... Conversion is the equivalence relation generated by β-reduction. Since λs and Πs are identified in Automath, β-reduction and β-conversion also apply to Π-types. ...arXiv:2203.01173v2 fatcat:3p6xtl5enfhwdfag3gefkgyvnu
This paper studies con uence of extensional and non-extensional -calculi with explicit substitutions, where extensionality is interpreted by Á-expansion. ... Our method makes it possible to treat at the same time many well-known calculi such as ; ⇑ ; ; s; v; f ; d and dn . ... for explaining me some key points in the theory of calculi with explicit substitutions. ...doi:10.1016/s0304-3975(98)00166-2 fatcat:5sl7ye4knnbmjkhwihubmrb7um
Lecture Notes in Computer Science
to formalize: variable binding at both the term and type levels, syntactic forms with variable numbers of components (including binders), and proofs demanding complex induction principles. ... Based on the metatheory of System F<:, a typed lambda-calculus with second-order polymorphism, subtyping, and records, these benchmarks embody many aspects of programming languages that are challenging ... Another popular concrete representation is de Bruijn's nameless representation. ...doi:10.1007/11541868_4 fatcat:6eprs3kxvjg67ak2zhuipl5qyu
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