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Towards a Formally Verified Proof Assistant

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Interactive Theorem Proving (ITP 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8558))

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Abstract

This paper presents a formalization of Nuprl’s metatheory in Coq. It includes a nominal-style definition of the Nuprl language, its reduction rules, a coinductive computational equivalence, and a Curry-style type system where a type is defined as a Partial Equivalence Relation (PER) à la Allen. This type system includes Martin-Löf dependent types, a hierarchy of universes, inductive types and partial types. We then prove that the typehood rules of Nuprl are valid w.r.t. this PER semantics and hence reduce Nuprl’s consistency to Coq’s consistency.

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References

  1. Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.): ITP 2013. LNCS, vol. 7998. Springer, Heidelberg (2013)

    MATH  Google Scholar 

  2. Abbott, M., Altenkirch, T., Ghani, N.: Representing nested inductive types using W-types. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 59–71. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  3. Allen, S.F.: A Non-Type-Theoretic Semantics for Type-Theoretic Language. PhD thesis, Cornell University (1987)

    Google Scholar 

  4. Allen, S.F., Bickford, M., Constable, R.L., Eaton, R., Kreitz, C., Lorigo, L., Moran, E.: Innovations in computational type theory using Nuprl. J. Applied Logic 4(4), 428–469 (2006), http://www.nuprl.org/

    Article  MATH  MathSciNet  Google Scholar 

  5. Anand, A., Rahli, V.: Towards a formally verified proof assistant. Technical report, Cornell University (2014), http://www.nuprl.org/html/Nuprl2Coq/

  6. Aydemir, B.E., Charguéraud, A., Pierce, B.C., Pollack, R., Weirich, S.: Engineering formal metatheory. In: POPL 2008, pp. 3–15. ACM (2008)

    Google Scholar 

  7. Barras, B.: Sets in Coq, Coq in sets. Journal of Formalized Reasoning 3(1), 29–48 (2010)

    MATH  MathSciNet  Google Scholar 

  8. Barras, B., Werner, B.: Coq in Coq. Technical report, INRIA Rocquencourt (1997)

    Google Scholar 

  9. Bertot, Y., Casteran, P.: Interactive Theorem Proving and Program Development. Springer (2004), http://coq.inria.fr/

  10. Bove, A., Dybjer, P., Norell, U.: A brief overview of Agda – A functional language with dependent types. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 73–78. Springer, Heidelberg (2009), http://wiki.portal.chalmers.se/agda/pmwiki.php

    Chapter  Google Scholar 

  11. Brady, E.: Idris —: systems programming meets full dependent types. In: 5th ACM Workshop Programming Languages meets Program Verification, PLPV 2011, pp. 43–54. ACM (2011)

    Google Scholar 

  12. Buisse, A., Dybjer, P.: Towards formalizing categorical models of type theory in type theory. Electr. Notes Theor. Comput. Sci. 196, 137–151 (2008)

    Article  Google Scholar 

  13. Capretta, V.: A polymorphic representation of induction-recursion (2004), http://www.cs.ru.nl/~venanzio/publications/induction_recursion.ps

  14. Constable, R.L., Allen, S.F., Bromley, H.M., Cleaveland, W.R., Cremer, J.F., Harper, R.W., Howe, D.J., Knoblock, T.B., Mendler, N.P., Panangaden, P., Sasaki, J.T., Smith, S.F.: Implementing mathematics with the Nuprl proof development system. Prentice-Hall, Inc., Upper Saddle River (1986)

    Google Scholar 

  15. Crary, K.: Type-Theoretic Methodology for Practical Programming Languages. PhD thesis, Cornell University, Ithaca, NY (August 1998)

    Google Scholar 

  16. Danielsson, N.A.: A formalisation of a dependently typed language as an inductive-recursive family. In: Altenkirch, T., McBride, C. (eds.) TYPES 2006. LNCS, vol. 4502, pp. 93–109. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  17. Dybjer, P.: A general formulation of simultaneous inductive-recursive definitions in type theory. J. Symb. Log. 65(2), 525–549 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  18. Dybjer, P., Setzer, A.: Induction-recursion and initial algebras. Ann. Pure Appl. Logic 124(1-3), 1–47 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. Gonthier, G., Asperti, A., Avigad, J., Bertot, Y., Cohen, C., Garillot, F., Le Roux, S., Mahboubi, A., O’Connor, R., Biha, S.O., Pasca, I., Rideau, L., Solovyev, A., Tassi, E., Théry, L.: A machine-checked proof of the odd order theorem. In: ITP 2013, [1], pp. 163–179

    Google Scholar 

  20. Harrison, J.: Towards Self-verification of HOL Light. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 177–191. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  21. Howe, D.J.: Equality in lazy computation systems. In: Proceedings of Fourth IEEE Symposium on Logic in Computer Science, pp. 198–203. IEEE Computer Society (1989)

    Google Scholar 

  22. Howe, D.J.: Semantic foundations for embedding HOL in Nuprl. In: Wirsing, M., Nivat, M. (eds.) AMAST 1996. LNCS, vol. 1101, pp. 85–101. Springer, Heidelberg (1996)

    Chapter  Google Scholar 

  23. Hur, C.-K., Neis, G., Dreyer, D., Vafeiadis, V.: The power of parameterization in coinductive proof. In: POPL 2013, pp. 193–206. ACM (2013)

    Google Scholar 

  24. Kopylov, A.: Type Theoretical Foundations for Data Structures, Classes, and Objects. PhD thesis, Cornell University, Ithaca, NY (2004)

    Google Scholar 

  25. Kumar, R., Myreen, M.O., Norrish, M., Owens, S.: CakeML: a verified implementation of ML. In: POPL 2014, pp. 179–192. ACM (2014)

    Google Scholar 

  26. Leroy, X.: Formal certification of a compiler back-end or: programming a compiler with a proof assistant. In: POPL 2006, pp. 42–54. ACM (2006)

    Google Scholar 

  27. McBride, C.: Hier soir, an OTT hierarchy (2011), http://sneezy.cs.nott.ac.uk/epilogue/?p=1098

  28. Mendler, P.F.: Inductive Definition in Type Theory. PhD thesis, Cornell University, Ithaca, NY (1988)

    Google Scholar 

  29. Myreen, M.O., Davis, J.: A verified runtime for a verified theorem prover. In: van Eekelen, M., Geuvers, H., Schmaltz, J., Wiedijk, F. (eds.) ITP 2011. LNCS, vol. 6898, pp. 265–280. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  30. Myreen, M.O., Davis, J.: The reflective milawa theorem prover is sound (Down to the machine code that runs it). In: Klein, G., Gamboa, R. (eds.) ITP 2014. LNCS (LNAI), vol. 8558, pp. 413–428. Springer, Heidelberg (2014)

    Google Scholar 

  31. Myreen, M.O., Owens, S., Kumar, R.: Steps towards verified implementations of hol light. In: ITP 2013 [1], pp. 490–495

    Google Scholar 

  32. Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)

    Book  MATH  Google Scholar 

  33. Paulin-Mohring, C.: Inductive definitions in the system Coq - rules and properties. In: Bezem, M., Groote, J.F. (eds.) TLCA 1993. LNCS, vol. 664, pp. 328–345. Springer, Heidelberg (1993)

    Chapter  Google Scholar 

  34. Rahli, V., Bickford, M., Anand, A.: Formal program optimization in Nuprl using computational equivalence and partial types. In: ITP 2013, [1], pp. 261–278

    Google Scholar 

  35. Setzer, A.: Proof theoretical strength of Martin-Löf Type Theory with W-type and one universe. PhD thesis, Ludwig Maximilian University of Munich (1993)

    Google Scholar 

  36. I.A.S. The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Univalent Foundations (2013)

    Google Scholar 

  37. Werner, B.: Sets in types, types in sets. In: Ito, T., Abadi, M. (eds.) TACS 1997. LNCS, vol. 1281, pp. 530–546. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. Cornell University, Ithaca, NY, USA

    Abhishek Anand & Vincent Rahli

Authors
  1. Abhishek Anand
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  2. Vincent Rahli
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Editor information

Editors and Affiliations

  1. NICTA, Sydney, NSW, Australia

    Gerwin Klein

  2. University of Wyoming, Laramie, WY, USA

    Ruben Gamboa

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Anand, A., Rahli, V. (2014). Towards a Formally Verified Proof Assistant. In: Klein, G., Gamboa, R. (eds) Interactive Theorem Proving. ITP 2014. Lecture Notes in Computer Science, vol 8558. Springer, Cham. https://doi.org/10.1007/978-3-319-08970-6_3

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  • DOI: https://doi.org/10.1007/978-3-319-08970-6_3

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-08969-0

  • Online ISBN: 978-3-319-08970-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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