Skip to main content
SpringerLink
Log in

The metatheory of UTT

  • Conference paper
  • First Online:
Types for Proofs and Programs (TYPES 1994)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 996))

Included in the following conference series:

Abstract

This paper outlines the development of the metatheory for Luo's type theory UTT, the type theory implemented in the proof assistant Lego and containing as subsystems Martin-Löfs type theory and the Calculus of Constructions. The approach used is to define a typed operational semantics for the system and to establish the important metatheoretic properties, such as Church-Rosser, strong normalization and subject reduction, for this operational presentation of the theory. These properties are then transferred to the usual presentation by soundness and completeness results. This technique gives a new and simpler development of the metatheory for systems with dependent types and ν-equality.

This material is based upon work supported by the North Atlantic Treaty Organization under a Grant awarded in 1993.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R. Backhouse. On the meaning and construction of the rules in Martin-Löf's theory of types. In A. Avron et al., editors, Workshop on General Logic. LFCS Report Series, ECS-LFCS-88-52, Dept. of Computer Science, University of Edinburgh, 1988.

    Google Scholar 

  2. H. Barendregt. Lambda calculi with types. In S. Abramsky, D. M. Gabbai, and T. S. E. Maibaum, editors, Handbook of Logic in Computer Science, volume 2. Oxford University Press, 1991.

    Google Scholar 

  3. T. Coquand. An analysis of Girard's paradox. In Proc. of the Symposium on Logic in Computer Science, pages 227–236, Boston, June 1986.

    Google Scholar 

  4. T. Coquand. An algorithm for testing conversion in type theory. In G. Huet and G. Plotkin, editors, Logical Frameworks. Cambridge University Press, 1991.

    Google Scholar 

  5. T. Coquand and J. Gallier. A proof of strong normalization for the theory of constructions using a Kripke-like interpretation. In Workshop on Logical Frameworks-Preliminary Proceedings, 1990.

    Google Scholar 

  6. T. Coquand and C. Paulin-Mohring. Inductively defined types. Lecture Notes In Computer Science, 417, 1990.

    Google Scholar 

  7. P. Dybjer. An inversion principle for Martin-Löf's type theory. In P. Dybjer et al., editors, Workshop on Programming Logic. Programming Methodology Group, Report 54, 1989.

    Google Scholar 

  8. H. Geuvers. Logics and Type Systems. PhD thesis, Katholieke Universiteit Nijmegen, Sept. 1993.

    Google Scholar 

  9. H. Goguen. A Typed Operational Semantics for Type Theory. PhD thesis, University of Edinburgh, Aug. 1994.

    Google Scholar 

  10. H. Goguen. Typed operational semantics. To appear in Proceedings of the International Conference on Typed Lambda Calculi and Applications, 1995.

    Google Scholar 

  11. G. Koletsos. Church-Rosser theorem for typed functional systems. Journal of Symbolic Logic, 50(3):782–790, 1985.

    Google Scholar 

  12. C. Löfwall and G. Sjödin. Strong normalisability in Martin-Löf's type theory. Technical Report R91:09, SICS, 1991.

    Google Scholar 

  13. Z. Luo. An Extended Calculus of Constructions. PhD thesis, University of Edinburgh, Nov. 1990.

    Google Scholar 

  14. Z. Luo. A unifying theory of dependent types: the schematic approach. In Proceedings of the Symposium on Logical Foundations of Computer Science, volume 620 of Lecture Notes in Computer Science. Springer-Verlag, 1992.

    Google Scholar 

  15. Z. Luo. Program specification and data refinement in type theory. Mathematical Structures in Computer Science, 3(3), 1993.

    Google Scholar 

  16. Z. Luo. Computation and Reasoning. Oxford University Press, 1994.

    Google Scholar 

  17. P. Martin-Löf. An intuitionistic theory of types, 1972. Unpublished manuscript.

    Google Scholar 

  18. J. McKinna. Deliverables: a categorical approach to program development in type theory. PhD thesis, University of Edinburgh, 1992.

    Google Scholar 

  19. J. McKinna and R. Burstall. Deliverables: a categorical approach to program development in type theory. Proc of Mathematical Foundations of Computer Science, LNCS 711, 1993.

    Google Scholar 

  20. J. McKinna and R. Pollack. Pure type systems formalized. In M. Bezem and J. F. Groote, editors, Proceedings of the International Conference on Typed Lambda Calculi and Applications, pages 289–305. Springer-Verlag, LNCS 664, Mar. 1993.

    Google Scholar 

  21. B. Nordström, K. Petersson, and J. Smith. Programming in Martin-Löf's Type Theory: An Introduction. Oxford University Press, 1990.

    Google Scholar 

  22. E. Ritter. Categorical Abstract Machine for Higher-Order Typed Lambda Calculus. PhD thesis, University of Cambridge, Sept. 1992.

    Google Scholar 

  23. A. Salvesen. The Church-Rosser property for pure type systems with βλ-reduction, Nov. 1991. Unpublished manuscript.

    Google Scholar 

  24. T. Streicher. Semantics of Type Theory: Correctness, Completeness and Independence Results. Birkhäuser, 1991.

    Google Scholar 

  25. W. W. Tait. Intensional interpretation of functionals of finite type I. Journal of Symbolic Logic, 32, 1967.

    Google Scholar 

  26. L. van Benthem Jutting, J. McKinna, and R. Pollack. Typechecking in pure type systems. In H. Barendregt and T. Nipkow, editors, Types for Proofs and Programming. Springer-Verlag, 1993.

    Google Scholar 

  27. B. Werner. Une Théorie des Constructions Inductives. PhD thesis, Université Paris 7, 1994.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. INRIA Sophia-Antipolis, 2004, route des Lucioles, B. P. 93, 06902, Sophia-Antipolis Cedex, France

    Healfdene Goguen

Authors
  1. Healfdene Goguen
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Peter Dybjer Bengt Nordström Jan Smith

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Goguen, H. (1995). The metatheory of UTT . In: Dybjer, P., Nordström, B., Smith, J. (eds) Types for Proofs and Programs. TYPES 1994. Lecture Notes in Computer Science, vol 996. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60579-7_4

Download citation

  • DOI: https://doi.org/10.1007/3-540-60579-7_4

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-60579-9

  • Online ISBN: 978-3-540-47770-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation