Translating dependent type theory into higher order logic

  • Bart Jacobs
  • Tom Melham
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 664)


This paper describes a translation of the complex calculus of dependent type theory into the relatively simpler higher order logic originally introduced by Church. In particular, it shows how type dependency as found in Martin-Löf's Intuitionistic Type Theory can be simulated in the formulation of higher order logic mechanized by the HOL theoremproving system. The outcome is a theorem prover for dependent type theory, built on top of HOL, that allows natural and flexible use of set-theoretic notions. A bit more technically, the language of the resulting theorem-prover is the internal language of a (boolean) topos (as formulated by Phoa).


Inference Rule Type Theory Dependent Type High Order Logic Elimination Rule 
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  1. [1]
    A. Church, ‘A Formulation of the Simple Theory of Types', The Journal of Symbolic Logic, vol. 5 (1940), pp. 56–68.Google Scholar
  2. [2]
    A. Felty, ‘Encoding Dependent Types in an Intuitionistic Logic', in Logical Frameworks, edited by G. Huet and G. Plotkin (Cambridge University Press, 1991), pp. 215–251.Google Scholar
  3. [3]
    M. Gordon, ‘HOL: A Machine Oriented Formulation of Higher Order Logic', Technical Report 68, Computer Laboratory, University of Cambridge, revised version (July 1985).Google Scholar
  4. [4]
    M. J. C. Gordon, ‘HOL: A Proof Generating System for Higher-Order Logic', in VLSI Specification, Verification and Synthesis, edited by G. Birtwistle and P. A. Subrahmanyam, Kluwer International Series in Engineering and Computer Science (Kluwer, 1988), pp. 73–128.Google Scholar
  5. [5]
    M. J. C. Gordon and T. F. Melham (eds.), Introduction to HOL: A theorem proving environment for higher order logic, forthcoming book (Cambridge University Press, 1993).Google Scholar
  6. [6]
    M. J. Gordon, A. J. Milner, and C. P. Wadsworth, Edinburgh LCF: A Mechanised Logic of Computation, Lecture Notes in Computer Science, vol. 78 (Springer-Verlag, 1979).Google Scholar
  7. [7]
    F. K. Hanna, N. Daeche, and M. Longley, ‘Specification and Verification Using Dependent Types', IEEE Transactions on Software Engineering, vol. 16, no. 9 (September 1990), pp. 949–964.Google Scholar
  8. [8]
    B. P. F. Jacobs, Categorical Type Theory (Ph.D. dissertation, University of Nijmgen, 1991).Google Scholar
  9. [9]
    M. Leeser, ‘Using Nuprl for the verification and synthesis of hardware', in Mechanized Reasoning and Hardware Design: a Discussion Meeting held at the Royal Society, October 1991, edited by C. A. R. Hoare and M. J. C. Gordon, Prentice Hall International Series in Computer Science (Prentice Hall, 1992), pp. 49–68.Google Scholar
  10. [10]
    A. C. Leisenring, Mathematical Logic and Hubert's e-Symbol, University Mathematical Series (Macdonald & Co., 1969).Google Scholar
  11. [11]
    P. Martin-Löf, Intuitionistic Type Theory (Bibliopolis, Naples, 1984).Google Scholar
  12. [12]
    T. F. Melham, ‘Automating Recursive Type Definitions in Higher Order Logic', in Current Trends in Hardware Verification and Automated Theorem Proving, edited by G. Birtwistle and P. A. Subrahmanyam (Springer-Verlag, 1989), pp. 341–386.Google Scholar
  13. [13]
    B. Nordström, K. Petersson, and J. M. Smith, Programming in Martin-Lof's Type Theory: An Introduction, International Series of Monographs on Computer Science 7 (Oxford University Press, 1990).Google Scholar
  14. [14]
    W. Phoa, ‘An introduction to fibrations, topos theory, the effective topos and modest sets', Technical Report ECS-LFCS-92-208, LFCS, Department of Computer Science, University of Edinburgh (April 1992).Google Scholar
  15. [15]
    G. Sundholm, ‘Proof Theory and Meaning', in Alternatives in Classical Logic, vol. 3 of Handbook of Philosophical Logic, edited by D. Gabbay and F. Guenthner, 4 vols. (D. Reidel, 1983–9), pp. 471–506.Google Scholar
  16. [16]
    A. S. Troelstra, ‘On the Syntax of Martin-Löf's Theories', Theoretical Computer Science, vol. 51, nos. 1–2 (1987), pp. 1–26.Google Scholar
  17. [17]
    A. S. Troelstra and D. van Dalen, Constructivism in Mathematics: An Introduction, Studies in Logic and the Foundations of Mathematics, 2 vols. (North Holland, 1988).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Bart Jacobs
    • 1
  • Tom Melham
    • 2
  1. 1.Mathematical Institute RUUTA UtrechtNL
  2. 2.Computer LaboratoryUniversity of CambridgeCambridgeUK

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