Hostname: page-component-65f69f4695-bd75b Total loading time: 0 Render date: 2025-06-30T13:28:55.686Z Has data issue: false hasContentIssue false
  • English
  • Français

Pure Type System conversion is always typable

Part of: JFP Research Articles

Published online by Cambridge University Press:  16 March 2012

VINCENT SILES  and
HUGO HERBELIN
VINCENT SILES
Affiliation:
Ecole Polytechnique/INRIA/Laboratoire PPS, Equipe πr2 (e-mail: vincent.siles@polytechnique.edu)
HUGO HERBELIN
Affiliation:
INRIA/Laboratoire PPS, Equipe πr2 (e-mail: hugo.herbelin@inria.fr)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Pure Type Systems are usually described in two different ways, one that uses an external notion of computation like beta-reduction, and one that relies on a typed judgment of equality, directly in the typing system. For a long time, the question was open to know whether both presentations described the same theory. A first step towards this equivalence has been made by Adams for a particular class of Pure Type Systems (PTS) called functional. Then, his result has been relaxed to all semi-full PTSs in previous work. In this paper, we finally give a positive answer to the general question, and prove that equivalence holds for any Pure Type System.

Information

Type
Articles
Information
Journal of Functional Programming , Volume 22 , Issue 2 , March 2012 , pp. 153 - 180
Copyright
Copyright © Cambridge University Press 2012

References

Abel, A. (2010) Towards normalization by evaluation for the βη-Calculus of constructions. In Proceedings of the 10th International Symposium on Functional and Logic Programming (FLOPS 2010), Sendai, Japan, April 19-21, 2010, Blume, M., Kobayashi, N. & Vidal, G. (eds), Lecture Notes in Computer Science, vol. 6009. Springer-Verlag, pp. 224–239.Google Scholar
Abel, A., Coquand, T., & Dybjer, P. (2007) Normalization by evaluation for Martin-Löf type theory with typed equality judgements. In Proceedings of the 22nd IEEE Symposium on Logic in Computer Science (LICS 2007), Wroclaw, Poland, 10-12 July 2007, IEEE Computer Society Press, pp. 3–12.Google Scholar
Adams, R. (2006) Pure Type Tystems with judgemental equality. J. Funct. Program., 16 (2), 219–246.CrossRefGoogle Scholar
Barendregt, H. (1991) Introduction to generalized type systems. J. Funct. Program., 1 (2), 125–154.Google Scholar
Barendregt, H. P. (1992) Lambda calculi with types. In Handbook of Logic in Computer Science, Abramsky, S., Gabbay, D. M. & Maibaum, T. S. E. (eds), Oxford University Press, pp. 117–309.Google Scholar
Berardi, S. (1990) Type Dependence and Constructive Mathematics. Ph.D. thesis, Mathematical Institute Torino, Italy.Google Scholar
Coq Development Team. (2010) The Coq Proof Assistant Reference Manual. http://coq.inria.fr/refman/.Google Scholar
de Bruijn, N. G. (1972) Lambda-Calculus notation with nameless dummies: A tool for automatic formula manipulation with application to the Church-Rosser theorem. Indagtiones Mathematica, 34 (5), 381–392.CrossRefGoogle Scholar
Geuvers, H. (1993) Logics and Type Systems. Ph.D. thesis, Katholieke Universiteit Nijmegen, The Netherlands.Google Scholar
Geuvers, H. & Nederhof, M-J. (1991) Modular proof of strong normalization for the calculus of constructions. J. Funct. Program. 1 (2), 155–189.CrossRefGoogle Scholar
Geuvers, H. & Werner, B. (1994) On the Church-Rosser property for expressive type systems and its consequences for their metatheoretic study. In Logic in Computer Science, pp. 320–329.Google Scholar
Goguen, H. (1994) A Typed Operational Semantics for Type Theory. Ph.D. thesis, University of Edinburgh, Scotland.Google Scholar
Luo, Z. (1989) ECC: An extended calculus of constructions. In Proceedings of the Fourth Annual Symposium on Logic in Computer Science. Piscataway, NJ, USA: IEEE Press, pp. 385–395.Google Scholar
Martin-Löf, P. (1984) Intuitionistic Type Theory. Berkeley, CA: Bibliopolis.Google Scholar
Melliès, P.-A. and Werner, B. (1997) A Generic Normalization Proof for Pure Type Systems. TYPES'96, Paulin-Mohring, C. & Gimenez, E. (eds), LNCS, Springer-Verlag.Google Scholar
Nordstrom, B., Petersson, K. & Smith, J. M. (1990) Programming in Martin-Löf's Type Theory: An introduction. New York: Oxford University Press.Google Scholar
Norell, U. (2007 September) Towards a Practical Programming Language Based on Dependent Type Theory. Ph.D. thesis, Department of Computer Science and Engineering, Chalmers University of Technology, SE-412 96 Göteborg, Sweden.Google Scholar
Pollack, R. (1994) The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. Ph.D. thesis, University of Edinburgh, Scotland.Google Scholar
Siles, V. (2010) Formalization of Equivalence Between PTS and PTSe. http://www.lix.polytechnique.fr/vsiles/coq/PTSATR.html.Google Scholar
Siles, V. & Herbelin, H. (2010) Equality is typable in Semi-Full Pure Type Systems. In Proceedings of the 25th Annual IEEE Symposium on Login in Computer Science (LICS '10), Edinburgh, UK, 11-14 July 2010. IEEE Compuer Society Press.Google Scholar
Streicher, T. (1991) Semantics of Type Theory: Correctness, Completeness, and Independence Results. Birkhauser Boston Inc., Cambridge, MA, USA.CrossRefGoogle Scholar
Terlouw, J. (1989) Een nadere Bewijstheoretische Analyse van GSTTs (Manuscript, in Dutch). Tech. Rep. University of Nijmegen, The Netherlands.Google Scholar
van Benthem Jutting, L. S., McKinna, J. & Pollack, R. (1993) Checking algorithms for Pure Type Systems. In TYPES, pp. 19–61.Google Scholar
Werner, B. (1994) Une théorie des Constructions Inductives. Ph.D. thesis, Université Paris, France.Google Scholar
Werner, B., & Lee, G. (2011) A proof-irrelevant model of CC with predicate induction and judgemental equality. Logical Methods Comput. Sci. 7 (4).Google Scholar
Submit a response

Discussions

No Discussions have been published for this article.