Abstract
We present a denotational semantics of a type system with dependent types, where types are interpreted as finitary projections. We prove then the correctness of a type-checking algorithm w.r.t. this semantics. In this way, we can justify some simple optimisation in this algorithm. We then sketch how to extend this semantics to allow a simple record mechanism with manifest fields.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Augustsson, L. Cayenne, A language with dependent types. in Proceedings of ICFP’98, ACM Press, 1998, 239–250.
Barendregt, H. and Rezus, A. Semantics for classical AUTOMATH and related systems. Inform. and Control 59 (1983), no. 1–3, 127–147.
Selected papers on Automath. Edited by R. P. Nederpelt, J. H. Geuvers and R. C. de Vrijer with the assistance of L. S. van Benthem Jutting and D. T. van Daalen. Studies in Logic and the Foundations of Mathematics, 133. North-Holland Publishing Co., Amsterdam, 1994
Cardelli, L. A polymorphic lambda-calculus with Type:Type. SRC Research Report 10, Digital Equipment Corporation Systems Research Center, May 1, 1986.
Coquand, C. A realizability interpretation of Martin-Löf’s type theory. Twenty-five years of constructive type theory (Venice, 1995), 73–82, Oxford Logic Guides, 36, Oxford Univ. Press, New York, 1998.
Coquand, Th. An algorithm for type-checking dependent types. Mathematics of program construction (Kloster Irsee, 1995). Sci. Comput. Programming 26 (1996), no. 1–3, 167–177.
Hickey J. Formal Objects in Type Theory Using Very Dependent Types. in Informal proceedings of Third Workshop on Foundations of Object-Oriented Languages (FOOL 3), 1996.
Jutting, L.S. van Benthem, McKinna J. and Pollack R. Checking Algorithms for Pure Type Systems. In Types for Proofs and Programs, LNCS 806, H. Barendregt and T. Nipkow (Eds.) 1993, 19–61.
Milner, R. A theory of type polymorphism in programming. J. Comput. System Sci. 17 (1978), no. 3, 348–375.
Scott, D. Lectures on a Mathematical Theory of Computation. in Theoretical foundations of programming methodology. Papers presented at the NATO Summer School, Munich, 1981. Edited by Manfred Broy and Gunther Schmidt. NATO Advanced Study Institute Series C: Mathematical and Physical Sciences, 91. D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1982, 145–292.
van Daalen, D. The language theory of Automath, PhD thesis, Eindhoven, 1980.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Coquand, T., Takeyama, M. (2002). An Implementation of Type:Type. In: Callaghan, P., Luo, Z., McKinna, J., Pollack, R., Pollack, R. (eds) Types for Proofs and Programs. TYPES 2000. Lecture Notes in Computer Science, vol 2277. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45842-5_4
Download citation
DOI: https://doi.org/10.1007/3-540-45842-5_4
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43287-6
Online ISBN: 978-3-540-45842-5
eBook Packages: Springer Book Archive