Studia Logica

, Volume 54, Issue 2, pp 199–230

A note on the proof theory the λII-calculus

  • David J. Pym

DOI: 10.1007/BF01063152

Cite this article as:
Pym, D.J. Stud Logica (1995) 54: 199. doi:10.1007/BF01063152


The λII-calculus, a theory of first-order dependent function types in Curry-Howard-de Bruijn correspondence with a fragment of minimal first-order logic, is defined as a system of (linearized) natural deduction. In this paper, we present a Gentzen-style sequent calculus for the λII-calculus and prove the cut-elimination theorem.

The cut-elimination result builds upon the existence of normal forms for the natural deduction system and can be considered to be analogous to a proof provided by Prawitz for first-order logic. The type-theoretic setting considered here elegantly illustrates the distinction between the processes of normalization in a natural deduction system and cut-elimination in a Gentzen-style sequent calculus.

We consider an application of the cut-free calculus, via the subformula property, to proof-search in the λII-calculus. For this application, the normalization result for the natural deduction calculus alone is inadequate, a (cut-free) calculus with the subformula property being required.

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • David J. Pym
    • 1
  1. 1.Queen Mary and Westfield CollegeUniversity of London EnglandUK