Lambda terms for natural deduction, sequent calculus and cut elimination

@article{Barendregt2000LambdaTF,
  title={Lambda terms for natural deduction, sequent calculus and cut elimination},
  author={HENK P. Barendregt and Silvia Ghilezan},
  journal={J. Funct. Program.},
  year={2000},
  volume={10},
  pages={121-134}
}
It is well known that there is an isomorphism between natural deduction derivations and typed lambda terms. Moreover, normalising these terms corresponds to eliminating cuts in the equivalent sequent calculus derivations. Several papers have been written on this topic. The correspondence between sequent calculus derivations and natural deduction derivations is, however, not a one-one map, which causes some syntactic technicalities. The correspondence is best explained by two extensionally… Expand
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References

SHOWING 1-10 OF 23 REFERENCES
A Lambda-Calculus Structure Isomorphic to Gentzen-Style Sequent Calculus Structure
TLDR
A λ-calculus for which applicative terms have no longer the form (...((u u1) u2)... un) but the form [u [u1;...;un], for which [u 1;... ;un] is a list of terms is considered. Expand
The typed λ-calculus is not elementary recursive
  • R. Statman
  • Computer Science
  • 18th Annual Symposium on Foundations of Computer Science (sfcs 1977)
  • 1977
TLDR
It is shown that in general this question cannot be answered by a Turing machine in elementary time, and the computational complexity of related questions concerning the typed λ-calculus (for example, the question of whether a given type contains a closed term). Expand
Constructive Logics Part I: A Tutorial on Proof Systems and Typed gamma-Calculi
TLDR
This paper has attempted to cover the basic material on natural deduction, sequent calculus, and typed λ-calculus, but also to provide an introduction to Girard's linear logic, one of the most exciting developments in logic these past six years. Expand
A general method for proving the normalization theorem for first and second order typed λ-calculi
In this paper we describe a method for proving the normalization property for a large variety of typed lambda calculi of first and second order, which is based on a proof of equivalence of twoExpand
Intersection and Union Types: Syntax and Semantics
Type assignment systems with intersection and union types are introduced. Although the subject reduction property with respect to s-reduction does not hold for a natural deduction-like system, weExpand
Permutability of Proofs in Intuitionistic Sequent Calculi
TLDR
It is proved, using a folklore theorem, that two derivations in a cut-free sequent calculus for intuitionistic propositional logic are inter-permutable iff they determine the same natural deduction. Expand
Bounds for cut elimination in intuitionistic propositional logic
TLDR
This article shall construct a (different) cut free deduction J(d) for the case of intuitionistic propositional logic and derive considerably sharper upper bounds for l(J(d)), and use the methods developed for this purpose in order to set up an effective decision method. Expand
Strong Normalisation of Cut-Elimination in Classical Logic
TLDR
In this paper a strongly normalizing cut-elimination procedure is presented for classical logic and it is suggested that the symmetric reducibility candidates developed by Barbanera and Berardi should be adapted. Expand
Marginalia on Sequent Calculi
The paper discusses the relationship between normal natural deductions and cutfree proofs in Gentzen (sequent) calculi in the absence of term labeling. For Gentzen calculi this is the usual version;Expand
Termination of Permutative Conversions in Intuitionistic Gentzen Calculi
It is shown that permutative conversions terminate for the cut-free intuitionistic Gentzen (i.e. sequent) calculus; this proves a conjecture by Dyckhoff and Pinto. The main technical tool is a termExpand
...
1
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3
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