Analytic proof systems for λ-calculus: the elimination of transitivity, and why it matters

Abstract

We introduce new proof systems G[β] and G ext[β], which are equivalent to the standard equational calculi of λβ- and λβη- conversion, and which may be qualified as ‘analytic’ because it is possible to establish, by purely proof-theoretical methods, that in both of them the transitivity rule admits effective elimination. This key feature, besides its intrinsic conceptual significance, turns out to provide a common logical background to new and comparatively simple demonstrations—rooted in nice proof-theoretical properties of transitivity-free derivations—of a number of well-known and central results concerning β- and βη-reduction. The latter include the Church–Rosser theorem for both reductions, the Standardization theorem for β- reduction, as well as the Normalization (Leftmost reduction) theorem and the Postponement of η-reduction theorem for βη-reduction

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Correspondence to Pierluigi Minari.

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Minari, P. Analytic proof systems for λ-calculus: the elimination of transitivity, and why it matters. Arch. Math. Logic 46, 385–424 (2007). https://doi.org/10.1007/s00153-007-0039-1

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Keywords

  • Lambda-calculus
  • Extensionality
  • Elimination of transitivity
  • Equational proof systems
  • Lambda reduction

Mathematics Subject Classification (2000)

  • 03B40
  • 03F03
  • 03F05
  • 03F07