A Cut-Free Sequent Calculus for Pure Type Systems Verifying the Structural Rules of Gentzen/Kleene

  • Francisco Gutiérrez
  • Blas Ruiz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2664)


In this paper, a new notion for sequent calculus (à la Gentzen) for Pure Type Systems (PTS) is introduced. This new calculus, \( \mathcal{K} \) , is equivalent to the standard PTS, and it has a cut-free subsystem, \( \mathcal{K}^{{\text{cf}}} \), that will be proved to hold non-trivial properties such as the structural rules of Gentzen/Kleene: thinning, contraction, and interchange.

An interpretation of completeness of the \( \mathcal{K}^{{\text{cf}}} \) system yields the concept of Cut Elimination, (CE), and it is an essential technique in proof theory; thus we think that it will have a deep impact on PTS and in logical frameworks based on PTS.


lambda calculus with types pure type systems sequent calculi cut elimination 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. Geuvers, M. Nederhof, Modular proof of Strong Normalization for the Calculus of Constructions, Journal of Functional Programming 1 (1991) 15–189.MathSciNetGoogle Scholar
  2. 2.
    H. P. Barendregt, Lambda Calculi with Types, in: S. Abramsky, D. Gabbay, T. S. Maibaum (Eds.), Handbook of Logic in Computer Science, Oxford University Press, 1992, Ch. 2.2, pp. 117–309.Google Scholar
  3. 3.
    F. Pfenning, Logical frameworks, in: A. Robinson, A. Voronkov (Eds.), Handbook of Automated Reasoning, Vol. II, Elsevier Science, 2001, Ch. 17, pp. 1063–1147.Google Scholar
  4. 4.
    H. Barendregt, H. Geuvers, Proof-assistants using dependent type systems, in: A. Robinson, A. Voronkov (Eds.), Handbook of Automated Reasoning, Vol. II, Elsevier Science, 2001, Ch. 18, pp. 1149–1238.Google Scholar
  5. 5.
    G. Gentzen, Untersuchungen über das Logische Schliessen, Math. Zeitschrift 39 (1935) 176–210,405–431, translation in [24].CrossRefMathSciNetGoogle Scholar
  6. 6.
    F. Barbanera, M. Dezani-Ciancaglini, U. de’Liguoro, Intersection and union types: Syntax and semantics, Information and Computation 119(2) (1995) 202–230.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    H. P. Barendregt, S. Ghilezan, Lambda terms for natural deduction, secuent calculus and cut elimination, Journal of Functional Programming 10(1) (2000) 121–134.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    M. Baaz, A. Leitsch, Comparing the complexity of cut-elimination methods, Lecture Notes in Computer Science 2183 (2001) 49–67.CrossRefMathSciNetGoogle Scholar
  9. 9.
    D. Galmiche, D. J. Pym, Proof-search in type-theoretic languages: an introduction, Theoretical Computer Science 232(1–2) (2000) 5–53.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    S. C. Kleene, Introduction to Metamathematics, D. van Nostrand, Princeton, New Jersey, 1952.MATHGoogle Scholar
  11. 11.
    D. Pym, A note on the proof theory of the λΠ-calculus, Studia Logica 54 (1995) 199–230.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    L. van Benthem Jutting, Typing in Pure Type Systems, Information and Computation 105(1) (1993) 30–41.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    B. C. Ruiz, Sistemas de Tipos Puros con Universos, Ph.D. thesis, Universidad de Málaga (1999).Google Scholar
  14. 14.
    B. C. Ruiz, Condensing lemmas in Pure Type Systems with Universes, in: A. M. Haeberer (Ed.), 7th International Conference on Algebraic Methodology and Software Technology (AMAST’98) Proceedings, Vol. 1548 of LNCS, Springer-Verlag, 1999, pp. 422–437.Google Scholar
  15. 15.
    J. Gallier, Constructive logics. I. A tutorial on proof systems and typed lambda-calculi, Theoretical Computer Science 110(2) (1993) 249–339.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    M. Baaz, A. Leitsch, Methods of cut elimination, Tec. rep., 11th European Summer School in Logic, Language and Information. Utrecht University (August 9–20 1999). URL http://www.let.uu.nl/esslli/Courses/baaz-leitsch.html
  17. 17.
    H. Yokouchi, Completeness of type assignment systems with intersection, union, and type quantifiers, Theoretical Computer Science 272 (2002) 341–398.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    F. Gutiérrez, B. C. Ruiz, Sequent Calculi for Pure Type Systems, Tech. Report 06/02, Dept. de Lenguajes y Ciencias de la Computación, Universidad de Málaga (Spain), http://polaris.lcc.uma.es/~blas/publicaciones/ (may 2002).Google Scholar
  19. 19.
    H. Geuvers, Logics and type systems, Ph.D. thesis, Computer Science Institute, Katholieke Universiteit Nijmegen (1993).Google Scholar
  20. 20.
    F. Gutiérrez, B. C. Ruiz, Order functional PTS, in: 11th International Workshop on Functional and Logic Programming (WFLP’2002), Vol. 76 of ENTCS, Elsevier, 2002, pp. 1–16, http://www.elsevier.com/gej-ng/31/29/23/126/23/23/76012.pdf.
  21. 21.
    E. Poll, Expansion Postponement for Normalising Pure Type Systems, Journal of Functional Programming 8(1) (1998) 89–96.MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    B. C. Ruiz, The Expansion Postponement Problem for Pure Type Systems with Universes, in: 9th International Workshop on Functional and Logic Programming (WFLP’2000), Dpto. de Sistemas Informáticos y Computación, Technical University of Valencia (Tech. Rep.), 2000, pp. 210–224, september 28–30, Benicassim, Spain.Google Scholar
  23. 23.
    G. Barthe, B. Ruiz, Tipos Principales y Cierre Semi-completo para Sistemas de Tipos Puros Extendidos, in: 2001 Joint Conference on Declarative Programming (APPIA-GULP-PRODE’01), Évora, Portugal, 2001, pp. 149–163.Google Scholar
  24. 24.
    G. Gentzen, Investigations into logical deductions, in: M. Szabo (Ed.), The Collected Papers of Gerhard Gentzen, North-Holland, 1969, pp. 68–131.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Francisco Gutiérrez
    • 1
  • Blas Ruiz
    • 1
  1. 1.Departamento de Lenguajes y Ciencias de la ComputaciónUniversidad de MálagaMálagaSpain

Personalised recommendations