Abstract
We have presented efficient syntax directed presentations of two subclasses of PTS:
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the semi-full systems, via the ⊢ sdsf relation
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the functional systems, via the ⊢f relation
The only remaining defect in these presentations lies in the possible failure of tests for conversion in the application rule. Thus for normalizing functional and semi-full systems, everything has been said.
For non-functional systems the situation is less clear. We know of no a priori bound on the amount of reduction necessary to correctly type λ-abstractions, so we must be content with the collective completeness of the family of syntax directed systems ⊢sd−n.
We have made little impact on the Expansion Postponement problem, which we leave as future work. We can however bask in the relative peace of mind gained from the machine-checked presentation of most (i.e. those not concerning schematic judgments) of the above results.
This work was supported by the ESPRIT Basic Research Actions on Logical Frame-works and Types for Proofs and Programs, and by grants from the British Science and Engineering Research Council.
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References
Henk Barendregt. Introduction to generalised type systems. J. Functional Programming, 1(2):125–154, April 1991.
Henk Barendregt. Lambda calculi with types. In Gabbai Abramsky and Maibaum, editors, Handbook of Logic in Computer Science, volume II. Oxford University Press, 1992.
Stefano Berardi. Type Dependence and Constructive Mathematics. PhD thesis, Dipartimento di Informatica, Torino, Italy, 1990.
Thierry Coquand and Gérard Huet. The calculus of constructions. Information and Computation, 76(2/3):95–120, February/March 1988.
Tat-Hung Chan. An algorithm for checking PL/CV arithmetical inferences. Technical Report 77-236, Computer Science Department, Cornell University, Ithaca, New York, 1977.
Herman Geuvers. The calculus of constructions and higher order logic. In preparation.
Herman Geuvers. Logics and Type Systems. PhD thesis, Department of Mathematics and Computer Science, University of Nijmegen, 1993. To appear.
Herman Geuvers and Mark-Jan Nederhof. A modular proof of strong normalization for the calculus of constructions. Journal of Functional Programming, 1(2):155–189, April 1991.
Leen Helmink. Goal directed proof construction in type theory. In Logical Frameworks. Cambridge University Press, 1991.
Robert Harper, Furio Honsell, and Gordon Plotkin. A framework for defining logics. In Proceedings of the Symposium on Logic in Computer Science, pages 194–204, Ithaca, New York, June 1987.
Robert Harper, Furio Honsell, and Gordon Plotkin. A framework for defining logics. Journal of the ACM, 40(1):143–184, 1992. Preliminary version in LICS'87.
Robert Harper and Robert Pollack. Type checking with universes. Theoretical Computer Science, 89:107–136, 1991.
Gérard Huet. Extending the calculus of constructions with Type:Type. unpublished manuscript, April 1987.
Gérard Huet. The constructive engine. In R. Narasimhan, editor, A Perspective in Theoretical Computer Science. World Scientific Publishing, 1989. Commemorative Volume for Gift Siromoney.
Zhaohui Luo and Robert Pollack. LEGO proof development system: User's manual. Technical Report ECS-LFCS-92-211, LFCS, Computer Science Dept., University of Edinburgh, The King's Buildings, Edinburgh EH9 3JZ, May 1992. Updated version.
Zhaohui Luo. An Extended Calculus of Constructions. PhD thesis, Department of Computer Science, University of Edinburgh, June 1990.
Per Martin-Löf. An intuitionistic theory of types. Technical report, University of Stockholm, 1972.
James McKinna and Robert Pollack. Pure type systems formalized. In M.Bezem and J.F.Groote, editors, Proceedings of the International Conference on Typed Lambda Calculi and Applications, TLCA'93, pages 289–305. Springer-Verlag, LNCS 664, March 1993.
James McKinna and Robert Pollack. Pure type systems formalized. Available by anonymous ftp from ftp.dcs.ed.ac.uk, directory export/lego, file PTSproofs.tar.Z, 1994.
Frank Pfenning. Elf: A language for logic definition and verified metaprogramming. In Proceedings of the Fourth Annual Symposium on Logic in Computer Science, Asilomar, California, June 1989.
R. Pollack. Typechecking in pure type systems. In Informal Proceedings of the 1992 Workshop on Types for Proofs and Programs, Båstad, Sweden, pages 271–288, June 1992. available by ftp.
Robert Pollack. The Theory of LEGO; A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994. In preparation.
L.S. van Benthem Jutting. Typing in pure type systems. Information and Computation, 105(1):30–41, July 1993.
D. T. van Daalen. The Language Theory of Automath. PhD thesis, Technische Hogeschool Eindhoven, 1980.
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van Benthem Jutting, L.S., McKinna, J., Pollack, R. (1994). Checking algorithms for Pure Type Systems. In: Barendregt, H., Nipkow, T. (eds) Types for Proofs and Programs. TYPES 1993. Lecture Notes in Computer Science, vol 806. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58085-9_71
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