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Dependently Typed Records for Representing Mathematical Structure

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Theorem Proving in Higher Order Logics (TPHOLs 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1869))

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Abstract

Consider a statement about groups “For G a group, ...”. A naive approach to formalize this is to unfold the meaning of group, so that every statement about groups begins with

$$ For\,\,G\,\,a\,\,set,\,\, + \,an\,\,operation\,\,on\,\,G,\,\, + \,\,associative,\,\,e\,\, \in \,\,G,\,... $$
(1)

This “unpackaged” approach can be improved by collecting all the parts of the meaning of group into a context, which need not be explicitly mentioned in every statement. A means of discharging some of the context is provided, so that statements made under that context can be instantiated with particular groups. However once that group context is discharged, all the parts of a group must be mentioned when using any general lemma about groups. Variations on this are supported by many proof tools, e.g. Coq’s Section mechanism [Coq99], Lego’s Discharge [LEG99], Automath contexts and Isabelle locales.

Supported by UK EPSRC grant GR/M75518

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References

  1. P. Aczel. Simple overloading for type theories. Privately circulated notes, 1994.

    Google Scholar 

  2. A. Bailey. The Machine-checked Literate Formalisation of Algebra in Type Theory. PhD thesis, Univ. of Manchester, 1998.

    Google Scholar 

  3. G. Betarte. Dependent Record Types and Formal Abstract Reasoning. PhD thesis, Chalmers Univ. of Technology, 1998.

    Google Scholar 

  4. G. Betarte and A. Tasistro. Extension of Martin-Löf’s type theory with record types and subtyping. In G. Sambin and J. Smith, editors, Twenty Five Years of Constructive Type Theory. Oxford Univ. Press, 1998.

    Google Scholar 

  5. The Coq Project, 1999. http://pauillac.inria.fr/coq/.

  6. J. Courant. MC: A module calculus for Pure Type Systems. Technical Report 1217, CNRS Université Paris Sud 8623: LRI, June 1999.

    Google Scholar 

  7. N. G. de Bruijn. Telescopic mappings in typed lambda calculus. Information and Computation, 91(2):189–204, April 1991.

    Article  MATH  Google Scholar 

  8. Peter Dybjer. A general notion of simultaneous inductive-recursive definition in type theory. Journal of Symbolic Logic, 1997. To appear.

    Google Scholar 

  9. R. Harper and M. Lillibridge. A type-theoretic approach to higher-order modules with sharing. In POPL’94. ACM Press, 1994.

    Google Scholar 

  10. F. Kammüller. Modular structures as dependent types in isabelle. In T. Altenkirch, W. Naraschewski, and B. Reus, editors, TYPES’98, Selected Papers, volume 1657 of LNCS. Springer-Verlag, 1999.

    Google Scholar 

  11. B. Lampson and R. Burstall. Pebble, a kernel language for modules and abstract data types. Information and Computation, 76(2/3), 1988.

    Google Scholar 

  12. The LEGO Proof Assistant, 1999. http://www.dcs.ed.ac.uk/home/lego/.

  13. X. Leroy. Manifest types, modules, and separate compilation. In POPL’94, New York, NY, USA, 1994. ACM Press.

    Google Scholar 

  14. Z. Luo and S. Soloviev. Dependent coercions. In Category Theory in Computer Science, CTCS’99, Electronic Notes in Theoretical Computer Science. Elsevier, 1999.

    Google Scholar 

  15. Z. Luo. Coercive subtyping. Journal of Logic and Computation, 9(1), 1999.

    Google Scholar 

  16. D. MacQueen. Using dependent types to express modular structure. In POPL’86, 1986.

    Google Scholar 

  17. Loic Pottier. Algebra with Coq. See User contributions in Coq release [Coq99], 1999.

    Google Scholar 

  18. A. Saibi. Typing algorithm in type theory with inheritance. POPL’97, 1997.

    Google Scholar 

  19. A. Tasistro. Substitution, Record Types and Subtyping in Type Theory, with Applications to the Theory of Programming. PhD thesis, Chalmers Univ. of Technology, May 1997.

    Google Scholar 

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Pollack, R. (2000). Dependently Typed Records for Representing Mathematical Structure. 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_29

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  • DOI: https://doi.org/10.1007/3-540-44659-1_29

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67863-2

  • Online ISBN: 978-3-540-44659-0

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