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Recursive Families of Inductive Types

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Theorem Proving in Higher Order Logics (TPHOLs 2000)

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Abstract

Families of inductive types defined by recursion arise in the formalization of mathematical theories. An example is the family of term algebras on the type of signatures. Type theory does not allow the direct definition of such families. We state the problem abstractly by defining a notion, strong positivity, that characterizes these families. Then we investigate its solutions. First, we construct a model using wellorderings. Second, we use an extension of type theory, implemented in the proof tool Coq, to construct another model that does not have extensionality problems. Finally, we apply the two level approach: We internalize inductive definitions, so that we can manipulate them and reason about them inside type theory.

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References

  1. H. P. Barendregt. Lambda calculi with types. In S. Abramsky, Dov M. Gabbay, and T. S. E. Maibaum, editors, Handbook of Logic in Computer Science, Volume 2, pages 117–309. Oxford University Press, 1992.

    Google Scholar 

  2. Bruno Barras, Samuel Boutin, Cristina Cornes, Judicaël Courant, Yann Coscoy, David Delahaye, Daniel de Rauglaudre, Jean-Christophe Filliâtre, Eduardo Giménez, Hugo Herbelin, Gérard Huet, Henri Laulhère, César Muñoz, Chetan Murthy, Catherine Parent-Vigouroux, Patrick Loiseleur, Christine Paulin-Mohring, Amokrane Saïbi, and Benjanin Werner. The Coq Proof Assistant Reference Manual. Version 6.3. INRIA, 1999.

    Google Scholar 

  3. Bruno Barras and Benjamin Werner. Coq in Coq. Draft paper, 2000.

    Google Scholar 

  4. G. Barthe, M. Ruys, and H. P. Barendregt. A two-level approach towards lean proof-checking. In S. Berardi and M. Coppo, editors, Types for Proofs and Programs (TYPES’95), volume 1158 of LNCS, pages 16–35. Springer, 1995.

    Google Scholar 

  5. Samuel Boutin. Using reflection to build efficient and certified decision procedures. In Martín Abadi and Takayasu Ito, editors, Theoretical Aspects of Computer Software. Third International Symposium, TACS’97, volume 1281 of LNCS, pages 515–529. Springer, 1997.

    Google Scholar 

  6. Venanzio Capretta. Universal algebra in type theory. In Yves Bertot, Gilles Dowek, André Hirschowits, Christine Paulin, and Laurent Théry, editors, Theorem Proving in Higher Order Logics, 12th International Conference, TPHOLs’ 99, volume 1690 of LNCS, pages 131–148. Springer-Verlag, 1999.

    Google Scholar 

  7. Venanzio Capretta. Equational reasoning in type theory. http://www.cs.kun.nl/~venanzio, 2000.

  8. Thierry Coquand. An analysis of Girard’s paradox. In Proceedings, Symposium on Logic in Computer Science, pages 227–236, Cambridge, Massachusetts, 16–18 June 1986. IEEE Computer Society.

    Google Scholar 

  9. Thierry Coquand and Christine Paulin. Inductively defined types. In P. Martin-Löf, editor, Proceedings of Colog’ 88, volume 417 of LNCS. Springer-Verlag, 1990.

    Google Scholar 

  10. Peter Dybjer. Representing Inductively Defined Sets by Wellorderings in Martin-Löf Type Theory. Theoretical Computer Science, 176:329–335, 1997.

    Article  MATH  Google Scholar 

  11. Herman Geuvers. Inductive and coinductive types with iteration and recursion. In Bengt Nordström, Kent Pettersson, and Gordon Plotkin, editors, Proceedings of the 1992 Workshop on Types for Proofs and Programs, Båstad, Sweden, June 1992, pages 193–217, 1992. ftp://ftp.cs.chalmers.se/pub/cs-reports/baastad.92/proc.dvi.Z.

  12. Eduardo Giménez. A Tutorial on Recursive Types in Coq. Technical Report 0221, Unité de recherche INRIA Rocquencourt, May 1998.

    Google Scholar 

  13. Douglas J. Howe. Computational metatheory in Nuprl. In E. Lusk and R. Overbeek, editors, 9th International Conference on Automated Deduction, volume 310 of LNCS, pages 238–257. Springer-Verlag, 1988.

    Chapter  Google Scholar 

  14. Per Martin-Löf. Constructive mathematics and computer programming. In Logic, Methodology and Philosophy of Science, VI, 1979, pages 153–175. North-Holland, 1982.

    Google Scholar 

  15. Per Martin-Löf. Intuitionistic Type Theory. Bibliopolis, 1984. Notes by Giovanni Sambin of a series of lectures given in Padua, June 1980.

    Google Scholar 

  16. Ralph Matthes. Monotone (co)inductive types and positive fixed-point types. Theoretical Informatics and Applications, 33:309–328, 1999.

    Article  MATH  Google Scholar 

  17. Paul Francis Mendler. Inductive Definition in Type Theory. PhD thesis, Department of Computer Science, Cornell University, Ithaca, New York, 1987.

    Google Scholar 

  18. Bengt Nordström, Kent Petersson, and Jan M. Smith. Programming in Martin-Löf’s Type Theory. Clarendon Press, 1990.

    Google Scholar 

  19. C. Paulin-Mohring. Inductive Definitions in the System Coq-Rules and Properties. In M. Bezem and J.-F. Groote, editors, Proceedings of the conference Typed Lambda Calculi and Applications, volume 664 of LNCS, 1993. LIP research report 92-49.

    Chapter  Google Scholar 

  20. Holger Pfeifer and Harals Rueβ. Polytypic proof construction. In Yves Bertot, Gilleds Dowek, André Hirschowits, Christine Paulin, and Laurent Théry, editors, Theorem Proving in Higher Order Logics, 12th International Conference, TPHOLs’ 99, volume 1690 of LNCS, pages 54–72. Springer-Verlag, 1999.

    Google Scholar 

  21. F. Pfenning and C. Paulin-Mohring. Inductively defined types in the Calculus of Constructions. In Proceedings of Mathematical Foundations of Programming Semantics, volume 442 of LNCS. Springer-Verlag, 1990. technical report CMU-CS-89-209.

    Chapter  Google Scholar 

  22. Mark Ruys. Studies in Mechanical Verification of Mathematical Proofs. PhD thesis, Computer Science Institute, University of Nijmegen, 1999.

    Google Scholar 

  23. Milena Stefanova. Properties of Typing Systems. PhD thesis, Computing Science Institute, University of Nijmegen, 1999.

    Google Scholar 

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Capretta, V. (2000). Recursive Families of Inductive Types. In: Aagaard, M., Harrison, J. (eds) Theorem Proving in Higher Order Logics. TPHOLs 2000. Lecture Notes in Computer Science, vol 1869. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44659-1_5

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  • DOI: https://doi.org/10.1007/3-540-44659-1_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67863-2

  • Online ISBN: 978-3-540-44659-0

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