A Finitary Subsystem of the Polymorphic λ-Calculus

  • Thorsten Altenkirch
  • Thierry Coquand
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2044)


We give a finitary normalisation proof for the restriction of system F where we quantify only over first-order type. As an application, the functions representable in this fragment are exactly the ones provably total in Peano Arithmetic. This is inspired by the reduction of π1 1-comprehension to inductive definitions presented in [Buch2] and this complements a result of [Leiv]. The argument uses a finitary model of a fragment of the system AF2 considered in Kriv,Leiv.


Kripke Model Proof Theory Finitary Model Peano Arithmetic Implicative Algebra 
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    W. Buchholz. The ωμ+1-rule. In Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies, volume 897 of Lecture Notes in Mathematics, pages 188–233. 1981.Google Scholar
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    W. Buchholz and K. Schütte. Proof theory of impredicative subsystems of analysis. Studies in Proof Theory. Monographs, 2. Bibliopolis, Naples, 1988.Google Scholar
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    M.D. Gladstone. A reduction of the recursion scheme. J. Symbolic Logic 32 1967 505–508.zbMATHCrossRefMathSciNetGoogle Scholar
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    J.L. Krivine. Lambda-calcul. Types et modèles. Masson, Paris, 1990.Google Scholar
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    D. Leivant. Peano’s Lambda Calculus: The Functional Abstraction Implicit in Arithmetic to be published in the Church Memorial Volume.Google Scholar
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    A. Troelstra. Metamathematical Investigations of Intuitionistic Arithmetic and Analysis. Lecture Notes in Mathematics 344, 1973.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Thorsten Altenkirch
    • 1
  • Thierry Coquand
    • 2
  1. 1.School of Computer Science and Information TechnologyUniversity of NottinghamUK
  2. 2.Department of Computing ScienceChalmers University of TechnologySweden

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