Skip to main content
Springer Nature Link
Log in

Dependent Types at Work

  • Chapter
Language Engineering and Rigorous Software Development (LerNet 2008)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 5520))

Abstract

In these lecture notes we give an introduction to functional programming with dependent types. We use the dependently typed programming language Agda which is an extension of Martin-Löf type theory. First we show how to do simply typed functional programming in the style of Haskell and ML. Some differences between Agda’s type system and the Hindley-Milner type system of Haskell and ML are also discussed. Then we show how to use dependent types for programming and we explain the basic ideas behind type-checking dependent types. We go on to explain the Curry-Howard identification of propositions and types. This is what makes Agda a programming logic and not only a programming language. According to Curry-Howard, we identify programs and proofs, something which is possible only by requiring that all program terminate. However, at the end of these notes we present a method for encoding partial and general recursive functions as total functions using dependent types.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Abel, A., Coquand, T., Dybjer, P.: On the algebraic foundation of proof assistants for intuitionistic type theory. In: Garrigue, J., Hermenegildo, M.V. (eds.) FLOPS 2008. LNCS, vol. 4989, pp. 3–13. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  2. Abel, A., Coquand, T., Dybjer, P.: Verifying a semantic beta-eta-conversion test for Martin-Löf type theory. In: Audebaud, P., Paulin-Mohring, C. (eds.) MPC 2008. LNCS, vol. 5133, pp. 29–56. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  3. Agda wiki (2008), appserv.cs.chalmers.se/users/ulfn/wiki/agda.php

  4. Bertot, Y., Castéran, P.: Interactive Theorem Proving and Program Develpment. Coq’Art: The Calculus of Inductive Constructions. Springer, Heidelberg (2004)

    Book  MATH  Google Scholar 

  5. Bove, A., Capretta, V.: Modelling general recursion in type theory. Mathematical Structures in Computer Science 15, 671–708 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Constable, R.L., et al.: Implementing Mathematics with the NuPRL Proof Development System. Prentice-Hall, Englewood Cliffs (1986)

    Google Scholar 

  7. Coquand, T.: An algorithm for testing conversion in type theory. In: Logical Frameworks, pp. 255–279. Cambridge University Press, Cambridge (1991)

    Chapter  Google Scholar 

  8. Coquand, T., Kinoshita, Y., Nordström, B., Takeyama, M.: A simple type-theoretic language: Mini-tt. In: Levy, J.-J., Bertot, Y., Huet, G., Plotkin, G. (eds.) From Semantics to Computer Science: Essays in Honor of Gilles Kahn, pp. 139–164. Cambridge University Press, Cambridge (2008)

    Google Scholar 

  9. Dybjer, P.: Inductive sets and families in Martin-Löf’s type theory and their set-theoretic semantics. In: Logical Frameworks, pp. 280–306. Cambridge University Press, Cambridge (1991)

    Chapter  Google Scholar 

  10. Dybjer, P.: Inductive families. Formal Aspects of Computing 6, 440–465 (1994)

    Article  MATH  Google Scholar 

  11. Dybjer, P.: A general formulation of simultaneous inductive-recursive definitions in type theory. Journal of Symbolic Logic 65(2) (June 2000)

    Google Scholar 

  12. Dybjer, P., Setzer, A.: A finite axiomatization of inductive-recursive definitions. In: Girard, J.-Y. (ed.) TLCA 1999. LNCS, vol. 1581, pp. 129–146. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  13. Dybjer, P., Setzer, A.: Indexed induction-recursion. Journal of Logic and Algebraic Programming 66(1), 1–49 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Girard, J.Y.: Une extension de l’interprétation de Gödel à l’analyse, et son application à l’elimination des coupures dans l’analyse et la théorie des types. In: Fenstad, J.E. (ed.) Proceedings of the Second Scandinavian Logic Symposium, pp. 63–92. North-Holland Publishing Company, Amsterdam (1971)

    Chapter  Google Scholar 

  15. Gödel, K.: Über eine bisher noch nicht benutze erweitrung des finiten standpunktes. Dialectica 12 (1958)

    Google Scholar 

  16. Hinze, R.: A new approach to generic functional programming. In: Proceedings of the 27th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, Boston, Massachusetts (January 2000)

    Google Scholar 

  17. Jansson, P., Jeuring, J.: PolyP — a polytypic programming language extension. In: POPL 1997, pp. 470–482. ACM Press, New York (1997)

    Google Scholar 

  18. Martin-Löf, P.: Hauptsatz for the Intuitionistic Theory of Iterated Inductive Definitions. In: Fenstad, J.E. (ed.) Proceedings of the Second Scandinavian Logic Symposium, pp. 179–216. North-Holland Publishing Company, Amsterdam (1971)

    Chapter  Google Scholar 

  19. Martin-Löf, P.: An Intuitionistic Theory of Types: Predicative Part. In: Rose, H.E., Shepherdson, J.C. (eds.) Logic Colloquium 1973, pp. 73–118. North-Holland Publishing Company, Amsterdam (1975)

    Google Scholar 

  20. Martin-Löf, P.: Constructive mathematics and computer programming. In: Logic, Methodology and Philosophy of Science, VI, 1979, pp. 153–175. North-Holland, Amsterdam (1982)

    Google Scholar 

  21. Martin-Löf, P.: Intuitionistic Type Theory. Bibliopolis (1984)

    Google Scholar 

  22. McBride, C.: Epigram: Practical programming with dependent types. In: Vene, V., Uustalu, T. (eds.) AFP 2004. LNCS, vol. 3622, pp. 130–170. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  23. Milner, R., Tofte, M., Harper, R., MacQueen, D.: The Definition of Standard ML. MIT Press, Cambridge (1997)

    Google Scholar 

  24. Nordström, B., Petersson, K., Smith, J.M.: Programming in Martin-Löf’s Type Theory. An Introduction. Oxford University Press, Oxford (1990)

    MATH  Google Scholar 

  25. Norell, U.: Towards a practical programming language based on dependent type theory. PhD thesis, Department of Computer Science and Engineering, Chalmers University of Technology, SE-412 96 Göteborg, Sweden (September 2007)

    Google Scholar 

  26. Peyton Jones, S. (ed.): Haskell 98 Language and Libraries The Revised Report. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  27. Peyton Jones, S., Vytiniotis, D., Weirich, S., Washburn, G.: Simple unification-based type inference for GADTs. In: ICFP 2006: Proceedings of the Eleventh ACM SIGPLAN International Conference on Functional Programming, pp. 50–61. ACM Press, New York (2006)

    Chapter  Google Scholar 

  28. Pfenning, F., Xi, H.: Dependent Types in practical programming. In: Proc. 26th ACM Symp. on Principles of Prog. Lang., pp. 214–227 (1999)

    Google Scholar 

  29. Plotkin, G.D.: LCF considered as a programming language. Theor. Comput. Sci. 5(3), 225–255 (1977)

    Article  MathSciNet  Google Scholar 

  30. Scott, D.: Constructive validity. In: Symposium on Automatic Demonstration. Lecture Notes in Mathematics, vol. 125, pp. 237–275. Springer, Berlin (1970)

    Chapter  Google Scholar 

  31. Taha, W., Sheard, T.: Metaml and multi-stage programming with explicit annotations. Theor. Comput. Sci. 248(1-2), 211–242 (2000)

    Article  MATH  Google Scholar 

  32. The Coq development team. The Coq proof assistant (2008), coq.inria.fr/

  33. Thompson, S.: Type Theory and Functional Programming. Addison-Wesley, Reading (1991)

    MATH  Google Scholar 

  34. Wahlstedt, D.: Dependent Type Theory with Parameterized First-Order Data Types and Well-Founded Recursion. PhD thesis, Chalmers University of Technology (2007) ISBN 978-91-7291-979-2

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Chalmers University of Technology, Göteborg, Sweden

    Ana Bove & Peter Dybjer

Authors
  1. Ana Bove
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter Dybjer
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science and Engineering, Chalmers University of Technology, Rännvägen 6B, 412 96, Göteborg, Sweden

    Ana Bove

  2. CCTC, Departamento de Informática, Universidade do Minho, Campus de Gualtar, 4700-320, Braga, Portugal

    Luís Soares Barbosa

  3. Facultad de Ingeniería, Instituto de Computación, Universidad de la República, Julio Herrera y Reissig 565 - Piso 5, 11300, Montevideo, Uruguay

    Alberto Pardo

  4. Departamento de Informática, Universidade do Minho, CCTC, Campus de Gualtar, 4700-320, Braga, Portugal

    Jorge Sousa Pinto

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Bove, A., Dybjer, P. (2009). Dependent Types at Work. In: Bove, A., Barbosa, L.S., Pardo, A., Pinto, J.S. (eds) Language Engineering and Rigorous Software Development. LerNet 2008. Lecture Notes in Computer Science, vol 5520. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03153-3_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-03153-3_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03152-6

  • Online ISBN: 978-3-642-03153-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics