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A λ-calculus structure isomorphic to Gentzen-style sequent calculus structure

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Book cover Computer Science Logic (CSL 1994)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 933))

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Abstract

We consider a λ-calculus for which applicative terms have no longer the form (...((u u 1) u 2)... u n) but the form (u [u 1;...;u n]), for which [u 1;...;u n] is a list of terms. While the structure of the usual λ-calculus is isomorphic to the structure of natural deduction, this new structure is isomorphic to the structure of Gentzen-style sequent calculus. To express the basis of the isomorphism, we consider intuitionistic logic with the implication as sole connective. However we do not consider Gentzen's calculus LJ, but a calculus LJT which leads to restrict the notion of cut-free proofs in LJ. We need also to explicitly consider, in a simply typed version of this λ-calculus, a substitution operator and a list concatenation operator. By this way, each elementary step of cutelimination exactly matches with a β-reduction, a substitution propagation step or a concatenation computation step.

Though it is possible to extend the isomorphism to classical logic and to other connectives, we do not treat of it in this paper.

This research was partly supported by ESPRIT Basic Research Action “Types for Proofs and Programs” and by Programme de Recherche Coordonnées “Mécanisation du raisonnement”.

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Authors and Affiliations

  1. LITP, University Paris 7, 2 place Jussieu, 78252, Paris Cedex 05, France

    Hugo Herbelin

  2. INRIA-Rocquencourt, B.P. 105, 78153, Le Chesnay Cedex, France

    Hugo Herbelin

Authors
  1. Hugo Herbelin
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Leszek Pacholski Jerzy Tiuryn

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© 1995 Springer-Verlag Berlin Heidelberg

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Herbelin, H. (1995). A λ-calculus structure isomorphic to Gentzen-style sequent calculus structure. In: Pacholski, L., Tiuryn, J. (eds) Computer Science Logic. CSL 1994. Lecture Notes in Computer Science, vol 933. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0022247

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  • DOI: https://doi.org/10.1007/BFb0022247

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-60017-6

  • Online ISBN: 978-3-540-49404-1

  • eBook Packages: Springer Book Archive

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