Formal Representation of Mathematics in a Dependently Typed Set Theory

  • Feryal Fulya Horozal
  • Chad E. Brown
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4573)


We have formalized material from an introductory real analysis textbook in the proof assistant Scunak. Scunak is a system based on set theory encoded in a dependent type theory. We use the formalized material to illustrate some interesting aspects of the relationship between informal presentations of mathematics and their formal representation. We focus especially on a representative example proved using the system.


Type Theory Dependent Type Proof Assistant Type Check Type Conversion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Autexier, S., Fiedler, A.: Textbook proofs meet formal logic - the problem of underspecification and granularity. In: Kohlhase, M. (ed.) MKM 2005. LNCS (LNAI), vol. 3863, pp. 96–110. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  2. 2.
    Bartle, R.G., Sherbert, D.R.: Introduction to Real Analysis. John Wiley and Sons, New York (1982)zbMATHGoogle Scholar
  3. 3.
    Baur, J.: Syntax und semantik mathematischer texte. Diploma thesis, Saarland University, Saarbrücken, Germany (1999)Google Scholar
  4. 4.
    Bertot, Y., Castéran, P.: Interactive Theorem Proving and Program Development. Coq’Art: The Calculus of Inductive Constructions. In: Texts in Theoretical Computer Science. An EATCS Series, Springer, Heidelberg (2004)Google Scholar
  5. 5.
    Brown, E.C.: Combining Type Theory and Untyped Set Theory. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 205–219. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. 6.
    Brown, C.E.: Encoding functional relations in Scunak. In: LFMTP 2006 (September 2006)Google Scholar
  7. 7.
    Brown, C.E.: Scunak users manual (2006),
  8. 8.
    Brown, C.E.: Verifying and invalidating textbook proofs using scunak. In: Borwein, J.M., Farmer, W.M. (eds.) MKM 2006. LNCS (LNAI), vol. 4108, pp. 110–123. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    Brown, C.E.: Dependently Typed Set Theory. SEKI-Working-Paper SWP–2006–03, SEKI Publications, Saarland Univ., (2006) ISSN 1860–5931Google Scholar
  10. 10.
    Kotowicz, J.: Real sequences and basic operations on them. Journal of Formalized Mathematics 1 (1989)Google Scholar
  11. 11.
    Lane, S.M.: Mathematics, Form, and Function. Springer, Heidelberg (1986)Google Scholar
  12. 12.
    Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  13. 13.
    Reed, J.: Proof irrelevance and strict definitions in a logical framework. Technical Report 02-153, School of Computer Science, Carnegie Mellon University (2002)Google Scholar
  14. 14.
    Rudnicki, P.: An overview of the mizar project. In: Workshop on Types for Proofs and Programs, pp. 311–332 (1992)Google Scholar
  15. 15.
    Wiedijk, F.: Mizar: An impression,

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Feryal Fulya Horozal
    • 1
  • Chad E. Brown
    • 1
  1. 1.Universitüat des Saarlandes, SaarbrüuckenGermany

Personalised recommendations