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New Proofs of Important Theorems of Untyped Extensional λ Calculus


This paper contains new proofs of the following two theorems of the untyped extensional λ-calculus: the Curry theorem stating that any λ-term has a βη-normal form if and only if it has a β-normal form and the normalization theorem for βη-reduction. The proposed approach is based on the following well-known results: the postponement theorem of η-reduction and the strong normalization property of η-reduction that allow one to naturally extend some propositions from the usual λ-calculus to the extensional case.

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  1. 1.

    H. P. Barendregt, The Lambda Calculus, Its Syntax and Semantics [Russian translation], Mir, Moscow (1985).

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  2. 2.

    H. B. Curry, R. Feyes, and W. Craig, Combinatory Logic, Vol. 1, North-Holland, Amsterdam (1958).

  3. 3.

    H. B. Curry, J. R. Hindley, and J. P. Seldinn, Combinatory Logic, Vol. 2, North-Holland, Amsterdam (1972).

  4. 4.

    H. P. Barendregt, J. Bergstra, J. W. Klop, and H. Voken, “Some notes on lambda reduction,” in: Degrees, Reductions, and Representability in the Lambda Calculus, Univ. of Utrecht, Dep. of Math., Preprint No. 22 (1976), pp. 13–53.

  5. 5.

    J. W. Klop, “Combinatory reduction systems,” Ph.D. Thesis, Univ. of Utrecht (1980).

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Correspondence to A. A. Lyaletsky.

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Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 53–63, July–August, 2014.

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Lyaletsky, A.A. New Proofs of Important Theorems of Untyped Extensional λ Calculus. Cybern Syst Anal 50, 529–537 (2014).

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  • untyped extensional λ-calculus
  • postponement of η-reduction
  • theorem on βη-normal form
  • normalization theorem for βη-reduction