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On the Strength of Proof-Irrelevant Type Theories

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Automated Reasoning (IJCAR 2006)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 4130))

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Abstract

We present a type theory with some proof-irrelevance built into the conversion rule. We argue that this feature is particularly useful when type theory is used as the logical formalism underlying a theorem prover. We also show a close relation with the subset types of the theory of PVS. Finally we show that in these theories, because of the additional extentionality, the axiom of choice implies the decidability of equality, that is, almost classical logic.

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References

  1. Altenkirch, T.: Proving strong normalization for CC by modifying realizability semantics. In: Barendregt, H., Nipkow, T. (eds.) TYPES 1993. LNCS, vol. 806, Springer, Heidelberg (1994)

    Google Scholar 

  2. Altenkirch, T.: Extensional Equality in Intensional Type Theory. LICS (1999)

    Google Scholar 

  3. Barendregt, H.: Lambda Calculi with Types.Technical Report 91-19, Catholic University Nijmegen, 1991.In Handbook of Logic in Computer Science, Vol II, Elsevier (1992)

    Google Scholar 

  4. Barthe, G.: The relevance of proof-irrelevance. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 755–768. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  5. Blanqui, F.: Definitions by rewriting in the Calculus of Constructions. MSCS, vol. 15(1) (2003)

    Google Scholar 

  6. Caldwell, J.: Moving Proofs-as-Programs into Practice. In: Proceedings of the 12th IEEE International Conference on Automated Software Engineering, IEEE, Los Alamitos (1997)

    Google Scholar 

  7. Chen, C., Xi, H.: Combining Programming with Theorem Proving. In: ICFP 2005 (2005)

    Google Scholar 

  8. The Coq Development Team. The Coq Proof-Assistant User’s Manual, INRIA, http://coq.inria.fr/

  9. Courtieu, P.: Normalized Types. In: Fribourg, L. (ed.) CSL 2001 and EACSL 2001. LNCS, vol. 2142, Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  10. Gimenez, E.: A Tutorial on Recursive Types in Coq. INRIA Technical Report (1999)

    Google Scholar 

  11. Gonthier, G.: A computer-checked proof of the Four Colour Theorem. Manuscript (2005)

    Google Scholar 

  12. Grégoire, B., Leroy, X.: A compiled implementation of strong reduction. In: proceedings of ICFP (2002)

    Google Scholar 

  13. Compilation des termes de preuves: un (nouveau) mariage entre Coq et Ocaml. Thése de doctorat, Université Paris 7 (2003)

    Google Scholar 

  14. Théry, L., Werner, B., Grégoire, B.: A computational approach to Pocklington certificates in type theory. In: Hagiya, M., Wadler, P. (eds.) FLOPS 2006. LNCS, vol. 3945, Springer, Heidelberg (2006)

    Google Scholar 

  15. Hofmann, M., Streicher, T.: A groupoid model refutes uniqueness of identity proofs. LICS 1994, Paris (1994)

    Google Scholar 

  16. Luo, Z.: ECC: An Extended Calculus of Constructions. In: Proc. of IEEE LICS 1989 (1989)

    Google Scholar 

  17. Martin-Löf, P.: Intuitionistic Type Theory. Studies in Proof Theory, Bibliopolis (1984)

    Google Scholar 

  18. McBride, C.: Elimination with a Motive. In: Callaghan, P., Luo, Z., McKinna, J., Pollack, R. (eds.) TYPES 2000. LNCS, vol. 2277, Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  19. McKinna, J., Pollack, R.: Pure Type Systems formalized. In: Bezem, M., Groote, J.F. (eds.) TLCA 1993. LNCS, vol. 664, Springer, Heidelberg (1993)

    Chapter  Google Scholar 

  20. Melliès, P.-A., Werner, B.: A Generic Normalization Proof for Pure Type System. In: Giménez, E. (ed.) TYPES 1996. LNCS, vol. 1512, Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  21. Miquel, A., Werner, B.: The not so simple proof-irrelevant model of CC. In: Geuvers, H., Wiedijk, F. (eds.) TYPES 2002. LNCS, vol. 2646, Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  22. Nogin, A., Kopilov, A.: Formalizing Type Operations Using the Image Type Constructor. In: WoLLIC. ENTCS (to appear, 2006)

    Google Scholar 

  23. Owre, S., Shankar, N.: The Formal Semantics of PVS. SRI Technical Report CSL-97-2R. Revised (March 1999)

    Google Scholar 

  24. Paulin-Mohring, C.: Extraction de Programmes dans le Calcul des Constructions. Thèse de doctorat, Université Paris 7 (1989)

    Google Scholar 

  25. Pfenning, F.: Intensionality, Extensionality, and Proof Irrelevance in Modal Type Theory. In: Proceedings of LICS, IEEE, Los Alamitos (2001)

    Google Scholar 

  26. Werner, B.: Sets in Types, Types in Sets. In: Ito, T., Abadi, M. (eds.) TACS 1997. LNCS, vol. 1281, Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  27. Hongwei Xi, Dependent Types in Practical Programming, Ph.D, CMU (1998)

    Google Scholar 

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Author information

Authors and Affiliations

  1. INRIA-Futurs and LIX, Ecole Polytechnique, France

    Benjamin Werner

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  1. Benjamin Werner
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Editors and Affiliations

  1. Department of Computer Science, Artificial Intelligence Research Group, University of Koblenz-Landau, Universitätsstr. 1, 56070, Koblenz

    Ulrich Furbach

  2. SRI International, MS EL256,, 333 Ravenswood Avenue, 94025-3493, Menlo Park, CA, USA

    Natarajan Shankar

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© 2006 Springer-Verlag Berlin Heidelberg

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Werner, B. (2006). On the Strength of Proof-Irrelevant Type Theories. In: Furbach, U., Shankar, N. (eds) Automated Reasoning. IJCAR 2006. Lecture Notes in Computer Science(), vol 4130. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11814771_49

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  • DOI: https://doi.org/10.1007/11814771_49

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-37187-8

  • Online ISBN: 978-3-540-37188-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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