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A Lambda-calculus Structure Isomorphic to Gentzen-style Sequent Calculus Structure

Abstract : We consider a lambda-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 lambda-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 lambda-calculus, a substitution operator and a list concatenation operator. By this way, each elementary step of cut-elimination exactly matches with a beta-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.
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Submitted on : Tuesday, May 5, 2009 - 4:39:36 PM
Last modification on : Saturday, November 20, 2021 - 3:50:10 AM
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  • HAL Id : inria-00381525, version 1


Hugo Herbelin. A Lambda-calculus Structure Isomorphic to Gentzen-style Sequent Calculus Structure. Computer Science Logic, Sep 1994, Kazimierz, Poland. pp.61--75. ⟨inria-00381525⟩



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