Abstract
The notion of a telescope is basic to Automath’s theory structure; telescopes provide the context for theorems. A dependent record type is an internal version of a telescope and is used in to define theories. This paper shows how A. Kopylov defines these record types in terms of dependent intersections, a new type constructor.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
Bibliography
M. Abadi and L. Cardelli. A Theory of Objects. Springer, 1996.
S. F. Allen. A Non-type-theoretic Definition of Martin-Löf’s Types. In D. Gries, editor, Proceedings of the 2 IEEE Symposium on Logic in Computer Science, pages 215–224. IEEE Computer Society Press, June 1987.
S. F. Allen. From dy/dx to []p: a matter of notation. In Proceedings of the Conference on User Interfaces for Theorem Provers, Eindhoven, The Netherlands, 1998.
H. Barendregt and H. Geuvers. Proof-assistants using dependent type systems. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, pages 1149–1238. Elsevier, 2001.
G. Betarte and A. Tasistro. Extension of Martin Löf’s type theory with record types and subtyping. In Twenty-Five Years of Constructive Type Theory, chapter 2, pages 21–39. Oxford Science Publications, 1999.
R. Bloo, F. Kamareddine, and R. Nederpelt. The Barendregt cube with definitions and generalised reduction. Information and Computation, 126(2):123143, 1996.
R. L. Constable, S. F. Allen, H. M. Bromley, W. R. Cleaveland, J. F. Cremer, R. W. Harper, D. J. Howe, T. B. Knoblock, N. P. Mendier, P. Panangaden, J. T. Sasaki, and S. F. Smith. Implementing Mathematics with the NuPRL Development System. Prentice-Hall, NJ, 1986.
R. L. Constable and J. Hickey. NuPRL’s class theory and its applications. In F. L. Bauer and R. Steinbrueggen, editors, Foundations of Secure Computation, NATO ASI Series, Series F: Computer and System Sciences, pages 91116. IOS Press, 2000.
R. L. Constable, P. Jackson, P. Naumov, and J. Uribe. Constructively formalizing automata theory. Proof, Language and Interaction: Essays in Honour of Robert Milner, 1998.
T. Coquand and G. Huet. The calculus of constructions. Information and Computation, 76: 95–120, 1988.
T. Coquand and C. Paulin-Mohring. Inductively defined types, preliminary version. In COLOG ‘88, International Conference on Computer Logic, volume 417 of Lecture Notes in Computer Science, pages 50–66. Springer, Berlin, 1990.
N. G. de Bruijn. The mathematical language Automath: its usage and some of its extensions. In J. P. Seldin and J. R. Hindley, editors, Symposium on Automatic Demonstration, volume 125 of Lecture Notes in Mathematics, pages 29–61. Springer-Verlag, 1970.
N. G. de Bruijn. A survey of the project Automath. In J. P. Seldin and J. R. Hindley, editors, To H. B. Curry: Essays in Combinatory Logic, Lambda Calculus, and Formalism, pages 589–606. Academic Press, 1980.
G. Dowek, A. Felty, H. Herbelin, G. Huet, C. Murthy, C. Parent, C. Paulin-Mohring, and B. Werner. The Coq Proof Assistant User’s Guide. INRIA, Version 5. 8, 1993.
A. Franke and M. Kohlhase. MATUWEB, an agent-based communication layer for distributed automated theorem proving. In Ganzinger [Ganzinger, 1999 ].
H. Ganzinger, editor. Proceedings of the 16th International Conference on Automated Deduction, volume 1632 of Lecture Notes in Artificial Intelligence, Berlin, July 7–10 1999. Trento, Italy.
H. Geuvers, R. Pollack, F. Wiedijk, and J. Zwanenburg. The algebraic hierarchy of the FTA project. In Calculemus 2001 Proceedings, Siena, Italy, 2001.
J.-Y. Girard. Une extension de l’interpretation de Gödel a l’analyse, et son application a l’elimination des coupures dans l’analyse et la theorie des types. In 2nd Scandinavian Logic Symposium, pages 63–69. Springer-Verlag, NY, 1971.
W. Howard. The formulas-as-types notion of construction. In To H.B. Curry: Essays on Combinatory Logic, Lambda-Calculus and Formalism, pages 479–490. Academic Press, NY, 1980.
D. J. Howe. Equality in lazy computation systems. In Proceedings of the 4th IEEE Symposium on Logic in Computer Science, pages 198–203, Asilomar Conference Center, Pacific Grove, California, June 1989. IEEE, IEEE Computer Society Press.
D. J. Howe. Semantic foundations for embedding HOL in NuPRL. In M. Wirsing and M. Nivat, editors, Algebraic Methodology and Software Technology, volume 1101 of Lecture Notes in Computer Science, pages 85–101. Springer-Verlag, Berlin, 1996.
K. Jensen and N. Wirth. PASCAL user manual and report. Springer-Verlag, New York, 1974.
F. Kamareddine and R. P. Nederpelt. A unified approach to type theory through a refined lambda-calculus. Theoretical Computer Science, 136 (1): 183–216, December 1994.
A. Kopylov. Dependent intersection: A new way of defining records in type theory. In Proceedings of 18th IEEE Symposium on Logic in Computer Science, 2003. To appear.
P. Martin-Löf. An intuitionistic theory of types: Predicative part. In Logic Colloquium ‘73, pages 73–118. North-Holland, Amsterdam, 1973.
P. Martin-Löf. Intuitionistic Type Theory. Number 1 in Studies in Proof Theory, Lecture Notes. Bibliopolis, Napoli, 1984.
R. P. Nederpelt, J. H. Geuvers, and R. C. de Vrijer. Selected Papers on Automath, volume 133 of Studies in Logic and The Foundations of Mathematics. Elsevier, Amsterdam, 1994.
A. Nogin. Quotient types: A modular approach. In V. A. Carreno, C. A. Munoz, and S. Tahar, editors, Proceedings of the 15th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2002), volume 2410 of Lecture Notes in Computer Science, pages 263–280. Springer-Verlag, 2002. Available at http://nogin.org/papers/quotients.html.
F. Pfenning and C. SchĂ¼rmann. Twelf — a meta-logical framework for deductive systems. In Ganzinger [Ganzinger, 1999], pages 202206.
R. Pollack. Dependently typed records for representing mathematical structure. In J. Harrison and M. Aagaard, editors, Theorem Proving in Higher Order Logics: 13th International Conference, TPHOLs 2000, volume 1869 of Lecture Notes in Computer Science, pages 461–478. Springer-Verlag, 2000.
D. Prawitz. Natural Deduction. Almquist and Wiksell, Stockholm, 1965.
D. Scott. Constructive validity. In D. L. M. Laudelt, editor, Symposium on Automatic Demonstration, volume 5(3) of Lecture Notes in Mathematics, pages 237–275. Springer-Verlag, New York, 1970.
W. W. Tait. Intensional interpretation of functionals of finite type. Journal of Symbolic Logic, 32 (2): 189–212, 1967.
F. Wiedijk. A contemporary implementation of Automath. Talk presented at the Workshop on 35 years of Automath, Heriot-Watt University, Edinburgh, Scotland, April 10–13, 2002.
J. Zucker. Formalization of classical mathematics in Automath. In Colloque International de Logique, pages 135–145, Paris, 1977. Colloques Internationaux du Centre National de la Recherche Scientifique, CNRS.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Constable, R.L. (2003). Recent Results in Type Theory and Their Relationship to Automath. In: Kamareddine, F.D. (eds) Thirty Five Years of Automating Mathematics. Applied Logic Series, vol 28. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0253-9_3
Download citation
DOI: https://doi.org/10.1007/978-94-017-0253-9_3
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-6440-0
Online ISBN: 978-94-017-0253-9
eBook Packages: Springer Book Archive