License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FSCD.2023.24
URN: urn:nbn:de:0030-drops-180086
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18008/
Altenkirch, Thorsten ;
Kaposi, Ambrus ;
Šinkarovs, Artjoms ;
Végh, Tamás
Combinatory Logic and Lambda Calculus Are Equal, Algebraically
Abstract
It is well-known that extensional lambda calculus is equivalent to extensional combinatory logic. In this paper we describe a formalisation of this fact in Cubical Agda. The distinguishing features of our formalisation are the following: (i) Both languages are defined as generalised algebraic theories, the syntaxes are intrinsically typed and quotiented by conversion; we never mention preterms or break the quotients in our construction. (ii) Typing is a parameter, thus the un(i)typed and simply typed variants are special cases of the same proof. (iii) We define syntaxes as quotient inductive-inductive types (QIITs) in Cubical Agda; we prove the equivalence and (via univalence) the equality of these QIITs; we do not rely on any axioms, the conversion functions all compute and can be experimented with.
BibTeX - Entry
@InProceedings{altenkirch_et_al:LIPIcs.FSCD.2023.24,
author = {Altenkirch, Thorsten and Kaposi, Ambrus and \v{S}inkarovs, Artjoms and V\'{e}gh, Tam\'{a}s},
title = {{Combinatory Logic and Lambda Calculus Are Equal, Algebraically}},
booktitle = {8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)},
pages = {24:1--24:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-277-8},
ISSN = {1868-8969},
year = {2023},
volume = {260},
editor = {Gaboardi, Marco and van Raamsdonk, Femke},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/18008},
URN = {urn:nbn:de:0030-drops-180086},
doi = {10.4230/LIPIcs.FSCD.2023.24},
annote = {Keywords: Combinatory logic, lambda calculus, quotient inductive types, Cubical Agda}
}
