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A general formulation of simultaneous inductive-recursive definitions in type theory

  • Peter Dybjer (a1)

The first example of a simultaneous inductive-recursive definition in intuitionistic type theory is Martin-Löfs universe à la Tarski. A set U0 of codes for small sets is generated inductively at the same time as a function T0, which maps a code to the corresponding small set, is defined by recursion on the way the elements of U0 are generated.

In this paper we argue that there is an underlying general notion of simultaneous inductive-recursive definition which is implicit in Martin-Löf's intuitionistic type theory. We extend previously given schematic formulations of inductive definitions in type theory to encompass a general notion of simultaneous induction-recursion. This enables us to give a unified treatment of several interesting constructions including various universe constructions by Palmgren, Griffor, Rathjen, and Setzer and a constructive version of Aczel's Frege structures. Consistency of a restricted version of the extension is shown by constructing a realisability model in the style of Allen.

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[1]Aczel, Peter, An introduction to inductive definitions, Handbook of mathematical logic (Barwise, Jon, editor), North-Holland, 1977, pp. 739782.
[2]Aczel, Peter, The strength of Martin-Löf's type theory with one universe, Proceedings of the symposium on mathematical logic (Oulu 1974) (Miettinen, S. and Väänanen, J., editors), 1977, Report No 2 of Dept. Philosophy, University of Helsinki, pp. 132.
[3]Aczel, Peter, The type theoretic interpretation of constructive set theory, Logic colloquium '77 (MacIntyre, A., Pacholski, L., and Paris, J., editors), North-Holland, 1978, pp. 5566.
[4]Aczel, Peter, Frege Structures and the Notions of Proposition, Truth, and Set, pp. 3159, North-Holland, 1980, pp. 31–59.
[5]Allen, Stuart, A non-type-theoretic semantics for type-theoretic language, Ph.D. thesis Department of Computer Science, Cornell University, 1987.
[6]Altenkirch, Thorsten, Constructions, inductive types and strong normalization, Ph.D. thesis, The University of Edinburgh, Department of Computer Science, 11 1993.
[7]Backhouse, Roland, On the meaning and construction of the rules in Martin-Löf's theory of types, Proceedings of the workshop on general logic, edinburg, february 1987 (Avron, A., Harper, B., Honsell, F., Mason, I., and Plotkin, G., editors), Laboratory for Foundations of Computer Science, Department of Computer Science, University of Edinburgh, 1988, ECS-LFCS-88-52.
[8]Barendregt, Henk P., The lambda calculus, North-Holland, 1984, Revised edition.
[9]Bird, Richard and Wadler, Philip, Introduction to functional programming, Prentice Hall, 1988.
[10]Bishop, E., Foundations of constructive analysis, McGraw-Hill, 1967.
[11]Coquand, Catarina, From semantics to rules: a machine assisted analysis, Proceedings of CSL '93, LNCS 832 (Börger, Egon, Gurevich, Yuri, and Meinke, Karl, editors), 1993.
[12]Coquand, Catarina, A realizability interpretation of Martin-Löf's type theory, Twenty-five years of constructive type theory (Sambin, G. and Smith, J., editors), Oxford University Press, 1998, pp. 7382.
[13]Coquand, Thierry, An algorithm for testing conversion in type theory, Logical frameworks, Cambridge University Press, 1991, pp. 255279.
[14]Coquand, Thierry, Pattern matching with dependent types, Proceedings of the 1992 workshop on types for proofs and programs (Nordström, Bengt and Petersson, Kent and Plotkin, Gordon, editor), 06 1992.
[15]Coquand, Thierry and Dybjer, Peter, Intuitionistic model constructions and normalization proofs, Mathematical Structures in Computer Science, vol. 7 (1997), pp. 7594.
[16]Coquand, Thierry and Paulin, Christine, Inductively defined types, preliminary version, LNCS 417, COLOG '88, International Conference on Computer Logic, Springer-Verlag, 1990.
[17]Čubrić, Djordje, Dybjer, Peter, and Scott, Philip, Normalization and the Yoneda embedding, Mathematical Structures in Computer Science, vol. 8 (1998), pp. 153192.
[18]Dowek, Gilles, Felty, Amy, Herbelin, Hugo, Huet, Gerard, Paulin, Christine, and Werner, Benjamin, The Coq proof assistant version 5.6, user's guide, Technical report, INRIA Rocquencourt - CNRS ENS Lyon, 1991.
[19]Dybjer, Peter, Inductive sets and families in Martin-Löf's type theory and their set-theoretic semantics, Logical frameworks (Huet, Gerard and Plotkin, Gordon, editors), Cambridge University Press, 1991, pp. 280306.
[20]Dybjer, Peter, Inductive families, Formal Aspects of Computing, vol. 6 (1994), pp. 440465.
[21]Dybjer, Peter, Internal type theory, Types '95, types for proofs and programs, Lecture Notes in Computer Science, no. 1158, Springer, 1996, pp. 120134.
[22]Dybjer, Peter and Setzer, Anton, A finite axiomatization of inductive-recursive definitions, Typed Lambda Calculi and Applications (Girard, J-Y., editor), Lecture Notes in Computer Science, no. 1581, Springer, 1999, pp. 129146.
[23]Giménez, Eduardo, A command for inductive sets in ILF, Master Thesis, Universidad de la República, Montevideo, 1992.
[24]Griffor, Edward and Rathjen, Michael, The strength of some Martin-Löf type theories, Archive for Mathematical Logic, vol. 33 (1994), pp. 337385.
[25]Hedberg, Michael, Type theory and the external logic of programs, Ph.D. thesis, Chalmers University of Technology and University of Göteborg, 1994.
[26]Kameyama, Yukiyoshi, A type-free theory of half-monotone inductive definitions, International Journal of Foundations of Computer Science, vol. 6 (1995), no. 3, pp. 203234.
[27]Martin-Löf, Per, Hauptsatz for the intuitionistic theory of iterated inductive definitions, Proceedings of the Second Scandinavian Logic Symposium (Fenstad, J. E., editor), North-Holland, 1971, pp. 179216.
[28]Martin-Löf, Per, An intuitionistic theory of types: Predicative part, Logic colloquium '73 (Rose, H. E. and Shepherdson, J. C., editors), North-Holland, 1975, pp. 73118.
[29]Martin-Löf, Per, Constructive mathematics and computer programming, Logic, methodology and philosophy of science, vi, 1979, North-Holland, 1982, pp. 153175.
[30]Martin-Löf, Per, Intuitionistic type theory, Bibliopolis, 1984.
[31]Martin-Löf, Per, Amendment to intuitionistic type theory, Notes from a lecture given in Göteborg, 03 1986.
[32]Martin-Löf, Per, An intuitionistic theoy of types, Twenty-five years of constructive type theory (Sambin, G. and Smith, J., editors), Oxford University Press, 1998, Reprinted version of an unpublished report from 1972, pp. 127172.
[33]Mendler, Paul Francis, Predicative type universes and primitive recursion, Proceedings sixth annual synposium on logic in computer science, IEEE Computer Society Press, 1991.
[34]Milner, Robin, Tofte, Mads, and Harper, Robert, The Definition of Standard ML, MIT Press, 1990.
[35]Nordström, Bengt, Petersson, Kent, and Smith, Jan, Programming in Martin-Löfs type theory: an introduction, Oxford University Press, 1990.
[36]Palmgren, Erik, On universes in type theory, Twenty-Five Years of Constructive Type Theory (Sambin, G. and Smith, J., editors), pp. 191204.
[37]Palmgren, Erik, On fixed point operators, inductive definitions and universes in Martin-Löfs type theory, Ph.D. thesis, Uppsala University, 1991.
[38]Parigot, Michel, Programming with proofs: a second order type theory, ESOP'88, 2nd European Symposium on Programming, Nancy, LNCS300 (Ganzinger, H., editor), 03 1988, pp. 145159.
[39]Paulin-Mohring, Christine, Inductive definitions in the system Cog - rules and properties, Proceedings typed λ-calculus and applications, Springer-Verlag, LNCS, 03 1993, pp. 328–245.
[40]Rathjen, Michael, Edward Griffor, R., and Palmgren, Erik, Inaccessibility in constructive set theory and type theory, Annals of Pure and Applied Logic, vol. 94 (1998), pp. 181200.
[41]Sato, Masahiko, Adding proof objects and inductive definition mechanism to Frege structures, Proc. international conference on theoretical aspects of computer science (Ito, T. and Meyer, A., editors), LNCS, no. 526, Springer Verlag, 1991, pp. 5387.
[42]Setzer, Anton, Extending Martin-Löf type theory by one Mahlo-universe, to appear in Archive for Mathematical Logic.
[43]Setzer, Anton, Proof theoretical strength of Martin-Löf type theory with W-type and one universe, Ph.D. thesis, Fakultät für Mathematik der Ludwig-Maximilians-Universität München, 1993.
[44]Smith, Jan, The independence of Peano's fourth axiom from Martin-Löf's type theory without universes, this Journal, vol. 49 (1988). no. 3.
[45]Smith, Jan, Prepositional functions and families of types, Notre Dame Journal of Formal Logic, vol. 30 (1989), no. 3, pp. 442458.
[46]Werner, Benjamin, A normalization proof for an impredicative type system with large elimination over integers, Proceedings of the 1992 workshop on proofs and programs (Nordström, Bengt, Petersson, Kent, and Plotkin, Gordon, editors), Department of Computer Sciences, Chalmers University of Technology, 06 1992, pp. 361377.
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