Safe positive induction in the programming logic TK

  • Martin C. Henson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 592)


We describe an alternative schema of induction for the programming logic TK based on safe positive induction. This replaces the original schema based on the well founded part of a relation. We show how the new schema can be included into the realizability definition and how the soundness of realizability can be extended to allow for the derivation of recursive programs from proofs of specifications which use the new schema. We further show how systems of mutual induction can be handled naturally with the new schema. In particular we show how useful systems of mutually recursive combinators can be derived which realize the principles of mutual induction.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [CoP 89]
    Coquand, T., Paulin, C., Inductively defined types, Proc. of the workshop on programming logic, 1989.Google Scholar
  2. [Dyb 87]
    Dybjer, P., Inductively defined sets in Martin-Löf's type theory, Proc. of the workshop on General Logic, Edinburgh, 1987.Google Scholar
  3. [Dyb 90]
    Dybjer, P., An inversion principle for Martin-Lors type theory, Manuscript, 1990.Google Scholar
  4. [Fef 79]
    Feferman, S., Constructive theories of functions and classes, Logic Coll. '78, pp 159–224, North Holland, 1979.Google Scholar
  5. [HaN 87]
    Hayashi S. & Nakano, H., The PX system—A computational logic, Publications of the Research Institute for Mathematical Sciences, Kyoto University, Tokyo, 1987.Google Scholar
  6. [Hen 89a]
    Henson, M. C., Program development in the constructive set theory TK, Formal Aspects of Computing, 1, pp 173–192, 1989.CrossRefGoogle Scholar
  7. [Hen 89b]
    Henson, M. C., Realizability models for program construction, Proc. Conf. on Mathematics of Program Construction, Gröningen, LNCS 375, pp 256–272, Springer, 1989.Google Scholar
  8. [Hen 90]
    Henson, M. C., Information Loss in the Programming Logic TK, Proc. IFIP TC2 Working Conf. on Programming Concepts and Methods, pp 509–545, Elsevier, 1990Google Scholar
  9. [HeT 88]
    Henson, M. C. & Turner, R., A. constructive set theory for program development, Proc. 8th Conf. on FST & TCS, Bangalore, LNCS 338, pp 329–347, Springer, 1988.Google Scholar
  10. [Mar 71]
    Martin-Löf, P., Haupsatz for the intuitionistic theory of iterated inductive definitions, Proc. 2nd Scandinavian Logic Symp., pp 179–216, 1971.Google Scholar
  11. [Mar 84]
    Martin-Löf, P., Intuitionistic type theory, Bibliopolis, 1984.Google Scholar
  12. [Mos 74]
    Moschovakis, Y. N., Elementary induction on abstract structures. North Holland, 1974.Google Scholar
  13. [Pet 84]
    Petersson, K., The subset type former and the type of small types in Martin-Löf's theory of types, Tech. rep. 3, University of Göteborg, Programming methodology group, 1984.Google Scholar
  14. [SaG90]
    Sanderson, M. T. & Ghosh-Roy, R., TKE: the TK proof development environment, private communication, University of Essex, 1990.Google Scholar
  15. [Sal 89]
    Salvesen, A., On information discharging and retrieval in Martin-Lofs type theory, Ph.D. thesis, University of Oslo, 1989.Google Scholar
  16. [Tur 85]
    Turner, D. A., Miranda — A non-strict functional language with polymorphic types, in: Proc. IFIP Int. Conf. on functional programming languages and computer architecture, Nancy, LNCS 201, Springer Verlag, pp 445–472, 1985.Google Scholar
  17. [WaH 90]
    Wadler, P., Hudak, P. (eds.) et al, Report on the programming language Haskell, University of Glasgow, Technical Report, 1990.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Martin C. Henson
    • 1
    • 2
  1. 1.Department of Computer ScienceUniversity of EssexColchesterEngland
  2. 2.Department of Computer ScienceUniversity of OtagoDunedinNew Zealand

Personalised recommendations