Abstract
We prove strong normalization of β-reduction+β-expansion for the Calculus of Constructions, thus providing the first strong normalization result for β-reduction+β-expansion in calculi of dependent types and answering in the affirmative a conjecture by Di Cosmo and Ghani. In addition, we prove strong normalization of β-reduction+β-expansion+algebraic reduction for the Algebraic Calculus of Constructions, which extends the Calculus of Constructions with first-order term-rewriting systems. The latter result, which requires the term-rewriting system to be non-duplicating, partially answers in the affirmative another conjecture by Di Cosmo and Ghani.
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References
T. Altenkirch. Constructions, inductive types and strong normalisation. PhD thesis, Laboratory for the Foundations of Computer Science, University of Edinburgh, 1994.
F. Barbanera and M. Fernández. Modularity of termination and conuence in combinations of rewrite systems with λω. In A. Lingas, R. Karlsson, and S. Karlsson, editors, Proceedings of ICALP’93, volume 700 of Lecture Notes in Computer Science, pages 657–668. Springer-Verlag, 1993.
F. Barbanera, M. Fernández, and H. Geuvers. Modularity of strong normalisation and conuence in the algebraic λ-cube. Journal of Functional Programming, 7(6):613–660, November 1997.
H. Barendregt. Introduction to Generalised Type Systems. Journal of Functional Programming, 1(2):125–154, April 1991.
H. Barendregt. Lambda calculi with types. In S. Abramsky, D. Gabbay, and T. Maibaum, editors, Handbook of Logic in Computer Science, pages 117–309. Oxford Science Publications, 1992. Volume 2.
G. Barthe. Existence and uniqueness of normal forms in pure type systems with βη-conversion. Manuscript, 1998.
G. Barthe. The relevance of proof-irrelevance. In K. G. Larsen, S. Skyum, and G. Winskel, editors, Proceedings of ICALP’98, volume 1443 of Lecture Notes in Computer Science, pages 755–768. Springer-Verlag, 1998.
G. Barthe and F. van Raamsdonk. Termination of algebraic type systems: the syntactic approach. In M. Hanus and J. Heering, editors, Proceedings of ALP’ 97-HOA’ 97, volume 1298 of Lecture Notes in Computer Science, pages 174–193. Springer-Verlag, 1997.
R. Di Cosmo. A brief history of rewriting with extensionality. Presented at the International Summer School on Type Theory and Term Rewriting, Glasgow, September 1996.
R. Di Cosmo and N. Ghani. On modular properties of higher order extensional lambda calculi. In P. Degano, R. Gorrieri, and A. Marchetti-Spaccamela, editors, Proceedings of ICALP’97, volume 1256 of Lecture Notes in Computer Science, pages 237–247. Springer-Verlag, 1997.
R. Di Cosmo and D. Kesner. Simulating expansions without expansions. Mathematical Structures in Computer Science, 4(3):315–362, September 1994.
D. van Daalen. The language theory of Automath. PhD thesis, Technical University of Eindhoven, 1980.
G. Dowek, G. Huet, and B. Werner. On the existence of long βη-normal forms in the cube. In H. Geuvers, editor, Informal Proceedings of TYPES’93, pages 115–130, 1993.
H. Geuvers. Logics and type systems. PhD thesis, University of Nijmegen, 1993.
H. Geuvers. A short and exible proof of strong normalisation for the Calculus of Constructions. In P. Dybjer, B. Nordström, and J. Smith, editors, Proceedings of TYPES’94, volume 996 of Lecture Notes in Computer Science, pages 14–38. Springer-Verlag, 1995.
H. Geuvers and B. Werner. On the Church-Rosser property for expressive type systems and its consequence for their metatheoretic study. In Proceedings of LICS’94, pages 320–329. IEEE Computer Society Press, 1994.
N. Ghani. Eta-expansions in dependent type theory—the calculus of constructions. In P. de Groote and J. Hindley, editors, Proceedings of TLCA’97, volume 1210 of Lecture Notes in Computer Science, pages 164–180. Springer-Verlag, 1997.
F. Joachimski. η-expansion in Gödel’s T, Pure Type Systems and calculi with permutative conversions. Manuscript, 1997.
J. W. Klop. Term-rewriting systems. In S. Abramsky, D. Gabbay, and T. Maibaum, editors, Handbook of Logic in Computer Science, pages 1–116. Oxford Science Publications, 1992. Volume 2.
P.-A. Melliès and B. Werner. A generic proof of strong normalisation for pure type systems. In E. Giménez and C. Paulin-Mohring, editors, Proceedings of TYPES’96, volume 1512 of Lecture Notes in Computer Science. Springer-Verlag, 1998.
R. Nederpelt. Strong normalisation in a typed lambda calculus with lambda structured types. PhD thesis, Technical University of Eindhoven, 1973.
G. Pottinger. Strong normalisation for terms of the theory of constructions. Technical Report TR 11-7, Odissey Research Associates, 1987.
D. Prawitz. Ideas and results of proof theory. In J. E. Fenstad, editor, The 2nd Scandinavian Logical Symposium, pages 235–307. North-Holland, 1970.
A. Salvesen. The Church-Rosser property for βη-reduction. Manuscript, 1991.
J. Terlouw. Strong normalization in type systems: a model theoretic approach. Annals of Pure and Applied Logic, 73(1):53–78, May 1995.
H. Xi. Simulating η-expansions with β-reductions in the second-order polymorphic lambda-calculus. In S. Adian and A. Nerode, editors, Proceedings of LFCS’97, volume 1234 of Lecture Notes in Computer Science, pages 399–409. Springer-Verlag, 1997.
H. Zantema. Termination of term rewriting: Interpretation and type elimination. Journal of Symbolic Computation, 17(1):23–50, January 1994.
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Barthe, G. (1999). Expanding the Cube. In: Thomas, W. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 1999. Lecture Notes in Computer Science, vol 1578. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-49019-1_7
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DOI: https://doi.org/10.1007/3-540-49019-1_7
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