Abstract
This paper is concerned with the foundations of the Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions by inductive data types. CAC generalizes inductive types equipped with higher-order primitive recursion, by providing definitions of functions by pattern-matching which capture recursor definitions for arbitrary non-dependent and non-polymorphic inductive types satisfying a strictly positivity condition. CAC also generalizes the first-order framework of abstract data types by providing dependent types and higher-order rewrite rules. Full proofs are available at http://www.lri.fr/~blanqui/publis/rta99full.ps.gz.
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References
F. Barbanera, M. Fernández, and H. Geuvers. Modularity of strong normalization in the algebraic-λ-cube. Journal of Functional Programming, 7(6), 1997.
H. Barendregt. Introduction to generalized type systems. Journal of Functional Programming, 1992.
F. Blanqui, J.-P. Jouannaud, and M. Okada. Inductive Data Type Systems, 1998.
V. Breazu-Tannen. Combining algebra and higher-order types. In Third IEEE Annual Symposium on Logic in Computer Science, pages 82–90. 1988.
V. Breazu-Tannen and J. Gallier. Polymorphic rewriting conserves algebraic strong normalization. Theoretical Computer Science, 83 (1):3–28, June 1991.
T. Coquand. Pattern matching with dependent types. In B. Nordström, K. Pettersson, G. Plotkin, editors, Workshop on Types for Proofs and Programs, 1992.
T. Coquand and J. Gallier. A proof of strong normalization for the Theory of Constructions using a Kripke-like interpretation. 1st Intl. Workshop on Logical Frameworks. 1990.
T. Coquand and G. Huet. The Calculus of Constructions. Information and Computation, 76:96–120, 1988.
T. Coquand and C. Paulin-Mohring. Inductively defined types. In P. Martin-Löf and G. Mints, editors, Proceedings of Colog’88, LNCS 417. Springer-Verlag, 1990.
C. Cornes. Conception d’un langage de haut niveau de representation de preuves: Récurrence par filtrage de motifs; Unification en présence de types inductifs primitifs; Synthése de lemmes d’inversion. PhD thesis, Université de Paris 7, 1997.
R. Di Cosmo and D. Kesner. Combining algebraic rewriting, extensional lambda calculi, and fixpoints. Theoretical Computer Science, 169(2):201–220, 1996.
J. Courant. A module calculus for Pure Type Systems. TLCA’97.
G. Dowek, T. Hardin, and C. Kirchner. Theorem proving modulo. Technical Report 3400, INRIA, 1998.
J. Gallier. On Girard’s “Candidats de Réductibilité”. In P.-G. Odifreddi, editor, Logic and Computer Science. North Holland, 1990.
H. Geuvers. A short and flexible proof of strong normalization for the Calculus of Constructions. In P. Dybjer, B. Nordström, and J. Smith, editors, Selected Papers 2nd Intl. Workshop on Types for Proofs and Programs, TYPES’94, Bástad, Sweden, 6-10 June 1994, volume 996 of LNCS, pages 14–38. 1995.
J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1988.
J.-P. Jouannaud and M. Okada. Abstract Data Type Systems. Theoretical Computer Science, 173(2):349–391, February 1997.
J. W. Klop, V. van Oostrom, and F. van Raamsdonk. Combinatory reduction systems: introduction and survey. Theoretical Computer Science, 121(1-2):279–308, December 1993.
T. Nipkow. Higher-order critical pairs. In Proc. 6th IEEE Symp. Logic in Computer Science, Amsterdam, pages 342–349, 1991.
M. Okada. Strong normalizability for the combined system of the typed lambda calculus and an arbitrary convergent term rewrite system. In G. H. Gonnet, editor, Proceedings of the ACM-SIGSAM 1989 International Symposium on Symbolic and Algebraic Computation, pages 357–363. ACM Press, July 1989.
B. Werner. Une Théorie des Constructions Inductives. Thése, Université Paris 7, 1994.
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Blanqui, F., Jouannaud, JP., Okada, M. (1999). The Calculus of Algebraic Constructions. In: Narendran, P., Rusinowitch, M. (eds) Rewriting Techniques and Applications. RTA 1999. Lecture Notes in Computer Science, vol 1631. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48685-2_25
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DOI: https://doi.org/10.1007/3-540-48685-2_25
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