An Arithmetical Proof of the Strong Normalization for the λ-Calculus with Recursive Equations on Types

  • René David
  • Karim Nour
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4583)


We give an arithmetical proof of the strong normalization of the λ-calculus (and also of the λμ-calculus) where the type system is the one of simple types with recursive equations on types.

The proof using candidates of reducibility is an easy extension of the one without equations but this proof cannot be formalized in Peano arithmetic. The strength of the system needed for such a proof was not known. Our proof shows that it is not more than Peano arithmetic.


Classical Logic Intuitionistic Logic Order Type Recursive Equation Typing Rule 
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  1. Barendregt, H.P.: The Lambda Calculus, Its Syntax and Semantics. North-Holland, Amsterdam (1985)zbMATHGoogle Scholar
  2. Barendregt, H.P.: Lambda Calculi with types. In: Abramsky & al. pp. 117–309 (1992)Google Scholar
  3. Barendregt, H.P., Dekkers, W., Statman, R.: Typed lambda calculus (to appear)Google Scholar
  4. Berger, U., Schwichtenberg, H.: An Inverse of the Evaluation Functional for Typed lambda-calculus. LICS, pp. 203–211 (1991)Google Scholar
  5. Church, A.: A Formulation of the Simple Theory of Types. JSL 5 (1940)Google Scholar
  6. Coppo, M., Dezani, M.: A new type assignment for lambda terms. Archiv. Math. Logik (19), 139–156 (1978)zbMATHCrossRefGoogle Scholar
  7. Coquand, T., Huet, G.: A calculus of constructions. Information and Computation (76), 95–120 (1988)CrossRefGoogle Scholar
  8. David, R.: Normalization without reducibility. Annals of Pure. and Applied Logic (107), 121–130 (2001)zbMATHCrossRefGoogle Scholar
  9. David, R.: A short proof of the strong normalization of the simply typed lambda calculus̃avid
  10. David, R., Nour, K.: A short proof of the strong normalization of the simply typed λμ-calculus. Schedae Informaticae 12, 27–34 (2003)Google Scholar
  11. David, R., Nour, K.: Arithmetical proofs of strong normalization results for the symmetric lambda-mu-calculus. In: Urzyczyn, P. (ed.) TLCA 2005. LNCS, vol. 3461, pp. 162–178. Springer, Heidelberg (2005)Google Scholar
  12. David, R., Nour, K.: Arithmetical proofs of strong normalization results for symmetric λ-calculi. Fundamenta Informaticae (to appear)Google Scholar
  13. de Groote, P.: On the Relation between the Lambda-Mu-Calculus and the Syntactic Theory of Sequential Control. LPAR, pp. 31–43 (1994)Google Scholar
  14. de Groote, P.: A CPS-Translation of the λμ-Calculus. In: Tison, S. (ed.) CAAP 1994. LNCS, vol. 787, pp. 85–99. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  15. Doyen, J.: Quelques propriétés du typage des fonctions des entiers dans les entiers. C.R. Acad. Sci. Paris, t.321, Série 1, 663–665 (1995)Google Scholar
  16. Fortune, S., Leivant, D., O’Donnell, M.: Simple and Second order Types Structures. JACM 30-1, 151–185 (1983)CrossRefGoogle Scholar
  17. Friedman, H.: Equality between functionals. Logic Coll’73, LNM 453, pp. 22–37 (1975)Google Scholar
  18. Girard, J.-Y., Lafont, Y., Taylor, P.: Proofs and Types. Cambridge University Press, Cambridge (1989)zbMATHGoogle Scholar
  19. Gödel, K.: Über eine bisher noch nicht benütztz Erweiterung des finiten Standpunkts. Dialectica 12, 280–287 (1958)zbMATHCrossRefGoogle Scholar
  20. Goldfarb, W.D.: The undecidability of the 2nd order unification problem. TCS (13), 225–230 (1981)zbMATHCrossRefGoogle Scholar
  21. Huet, G.P.: The Undecidability of Unification in Third Order Logic. Information and Control 22(3), 257–267 (1973)CrossRefGoogle Scholar
  22. Joachimski, F., Matthes, R.: Short proofs of normalization for the simply-typed lambda-calculus, permutative conversions and Gödel’s T. Archive for Mathematical Logic 42(1), 59–87 (2003)zbMATHCrossRefGoogle Scholar
  23. Jung, A., Tiuryn, J.A.: A New Characterization of Lambda Definability. In: Bezem, M., Groote, J.F. (eds.) TLCA 1993. LNCS, vol. 664, pp. 245–257. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  24. Krivine, J.-L.: Un algorithme non typable dans le système F. C. R. Acad. Sci. Paris 304(5), (1987)Google Scholar
  25. Krivine, J.-L.: Lambda Calcul: types et modèles. Masson, Paris (1990)Google Scholar
  26. Krivine, J.-L.: Classical Logic, Storage Operators and Second-Order lambda-Calculus. Ann. Pure Appl. Logic 68(1), 53–78 (1994)zbMATHCrossRefGoogle Scholar
  27. Lambek, J.: Cartesian Closed Categories and Typed Lambda-calculi. Combinators and Functional Programming Languages, pp. 136-175 (1985)Google Scholar
  28. Loader, R.: The Undecidability of ł-definability. In: Essays in memory of A. Church, pp. 331–342 (2001)Google Scholar
  29. Mendler, N.P.: Recursive Types and Type Constraints in Second-Order Lambda Calculus. LICS, pp. 30–36 (1987)Google Scholar
  30. Mendler, N.P.: Inductive Types and Type Constraints in the Second-Order Lambda Calculus. Ann. Pure Appl. Logic 51(1-2), 159–172 (1991)CrossRefGoogle Scholar
  31. Parigot, M.: Programming with proofs: a second order type theory. In: Ganzinger, H. (ed.) ESOP 1988. LNCS, vol. 300, pp. 145–159. Springer, Heidelberg (1988)Google Scholar
  32. Parigot, M.: On representation of data in lambda calculus. CSL, pp. 309–321 (1989)Google Scholar
  33. Parigot, M.: Recursive programming with proofs. Theoritical Computer Science 94, 335–356 (1992)zbMATHCrossRefGoogle Scholar
  34. Parigot, M.: Strong Normalization for Second Order Classical Natural Deduction. LICS, pp. 39–46 (1993)Google Scholar
  35. Parigot, M.: λμ-calculus: An algorithmic interpretation of classical natural deduction. Journal of symbolic logic 62(4), 1461–1479 (1997)zbMATHCrossRefGoogle Scholar
  36. Parigot, M.: Proofs of Strong Normalisation for Second Order Classical Natural Deduction. J. Symb. Log. 62(4), 1461–1479 (1997)zbMATHCrossRefGoogle Scholar
  37. Plotkin, G.D.: Lambda-definability and logical relations. Technical report (1973)Google Scholar
  38. Schwichtenberg, H.: Functions definable in the simply-typed lambda calculus. Arch. Math Logik 17, 113–114 (1976)zbMATHCrossRefGoogle Scholar
  39. Statman, R.: The Typed lambda-Calculus is not Elementary Recursive. FOCS, pp. 90–94 (1977)Google Scholar
  40. Statman, R.: λ-definable functionals and βη-conversion. Arch. Math. Logik 23, 21–26 (1983)zbMATHCrossRefGoogle Scholar
  41. Statman, R.: Recursive types and the subject reduction theorem. Technical report, Carnegie Mellon University, pp. 94–164 (March 1994)Google Scholar
  42. Tait, W.W.: Intensional Interpretations of Functionals of Finite Type I. JSL, vol. 32(2) (1967)Google Scholar
  43. Weiermann, A.: A proof of strongly uniform termination for Gödel’s T by methods from local predicativity. Archive fot Mathematical Logic 36, 445–460 (1997)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • René David
    • 1
  • Karim Nour
    • 1
  1. 1.Université de Savoie, Campus Scientifique, 73376 Le Bourget du LacFrance

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