On a Local-Step Cut-Elimination Procedure for the Intuitionistic Sequent Calculus
In this paper we investigate, for intuitionistic implicational logic, the relationship between normalization in natural deduction and cut-elimination in a standard sequent calculus. First we identify a subset of proofs in the sequent calculus that correspond to proofs in natural deduction. Then we define a reduction relation on those proofs that exactly corresponds to normalization in natural deduction. The reduction relation is simulated soundly and completely by a cut-elimination procedure which consists of local proof transformations. It follows that the sequent calculus with our cut-elimination procedure is a proper extension that is conservative over natural deduction with normalization.
Unable to display preview. Download preview PDF.
- 6.Gentzen, G.: Untersuchungen über das logische Schliessen. Mathematische Zeitschrift 39, 176–210, 405–431 (1935); English translation in , pp. 68–131Google Scholar
- 8.Howard, W.A.: The formulae-as-types notion of construction. In: Seldin, J.P., Hindley, J.R. (eds.) To H. B. Curry: Essays on Combinatory Logic, Lambda-Calculus and Formalism, pp. 479–490. Academic Press, London (1980)Google Scholar
- 10.Prawitz, D.: Natural Deduction, A Proof-Theoretical Study. Almquist and Wiksell (1965)Google Scholar
- 11.Szabo, M.E. (ed.): The Collected Papers of Gerhard Gentzen. North-Holland (1969)Google Scholar
- 12.Urban, C.: Classical Logic and Computation. PhD thesis, University of Cambridge (2000)Google Scholar