Basic proof theory
Permutability of proofs in intuitionistic sequent calculi
Theoretical Computer Science - Special issue: Gentzen
Terminiation of permutative conversions in intuitionistic Gentzen calculi
Theoretical Computer Science - Special issue: Gentzen
A Lambda-Calculus Structure Isomorphic to Gentzen-Style Sequent Calculus Structure
CSL '94 Selected Papers from the 8th International Workshop on Computer Science Logic
Towards a canonical classical natural deduction system
CSL'10/EACSL'10 Proceedings of the 24th international conference/19th annual conference on Computer science logic
A calculus of multiary sequent terms
ACM Transactions on Computational Logic (TOCL)
On a local-step cut-elimination procedure for the intuitionistic sequent calculus
LPAR'06 Proceedings of the 13th international conference on Logic for Programming, Artificial Intelligence, and Reasoning
Structural proof theory as rewriting
RTA'06 Proceedings of the 17th international conference on Term Rewriting and Applications
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Variants of Herbelin's λ-calculus, here collectively named Herbelin calculi, have proved useful both in foundational studies and as internal languages for the efficient representation of λ-terms. An obvious requirement of both these two kinds of applications is a clear understanding of the relationship between cut-elimination in Herbelin calculi and normalisation in the λ-calculus. However,this understanding is not complete so far. Our previous work showed that λ is isomorphic to a Herbelin calculus,here named λP, only admitting cuts that are both left- and right-permuted. In this paper we consider a generalisation λPh admitting any kind of right-permuted cut. We show that there is a natural deduction system λNh which conservatively extends λ and is isomorphic to λPh. The idea is to build in the natural deduction system a distinction between applicative term and application, together with a distinction between head and tail application. This is suggested by examining how natural deduction proofs are mapped to sequent calculus derivations according to a translation due to Prawitz. In addition to β, λNh includes a reduction rule that mirrors left permutation of cuts, but without performing any append of lists/spines.