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Proof normalization modulo

Published online by Cambridge University Press:  12 March 2014

Gilles Dowek  and
Benjamin Werner
Gilles Dowek
Affiliation:
École Polytechnique and Inria, Lix, École Polytechnique, 91128 Palaiseau Cedex, France, E-mail: [email protected], URL: http://www.lix.polytechnique.fr/~dowek/
Benjamin Werner
Affiliation:
École Polytechnique and Inria, Lix, École Polytechnique, 91128 Palaiseau Cedex, France, E-mail: [email protected], URL: http://www.lix.polytechnique.fr/~werner/
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Abstract

We define a generic notion of cut that applies to many first-order theories. We prove a generic cut elimination theorem showing that the cut elimination property holds for all theories having a so-called pre-model. As a corollary, we retrieve cut elimination for several axiomatic theories, including Church's simple type theory.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 4 , December 2003 , pp. 1289 - 1316
Copyright
Copyright © Association for Symbolic Logic 2003

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References

REFERENCES

[1] Bailin, S.C., A normalization theorem for set theory, this Journal, vol. 53 (1988), no. 3, pp. 673–695.Google Scholar
[2] Barendregt, H., Lambda calculi with types, Handbook of logic in computer science (Abramsky, S., Gabbay, D.M., and Maibaum, T.S.E., editors), vol. 2, Oxford University Press, 1992.Google Scholar
[3] Boyer, R.S. and Moore, J.S., A computational logic, Academic Press, 1979.Google Scholar
[4] Church, A., A formulation of the simple theory of types, this Journal, vol. 5 (1940), no. 1, pp. 56–68.Google Scholar
[5] Coquand, Th. and Huet, G., The calculus of constructions, Information and Computation, vol. 76 (1988), pp. 95–120.Google Scholar
[6] Crabbé, M., Non-normalisation de ZF, Manuscript, 1974.Google Scholar
[7] Crabbé, M., Stratification and cut-elimination, this Journal, vol. 56 (1991), no. 1, pp. 213–226.Google Scholar
[8] Curry, H.B., An analysis of the logical substitution, American Journal of Mathematics, vol. 51 (1929), pp. 263–384.Google Scholar
[9] Dowek, G., Proof normalization for a first-order formulation of higher-order logic, Theorem proving in higher-order logics (Gunter, E.L. and Felty, A., editors), Lecture Notes in Computer Science, vol. 1275, Springer-Verlag, 1997, pp. 105–119. Rapport de Recherche 3383, Institut National de Recherche en Informatique et en Automatique, 1998.CrossRefGoogle Scholar
[10] Dowek, G., About folding-unfolding cuts and cuts modulo, Journal of Logic and Computation, vol. 11 (2001), no. 3, pp. 419–429.Google Scholar
[11] Dowek, G., Hardin, Th., and Kirchner, C., Theorem proving modulo, Journal of Automated Reasoning , (to appear).Google Scholar
[12] Ekman, J., Normal proofs in set theory, Doctoral thesis, Chalmers university of technology and University of Göteborg, 1994.Google Scholar
[13] Enderton, H.B., A mathematical introduction to logic, Academic Press, 1972.Google Scholar
[14] Fay, M.J., First-order unification in an equational theory, Fourth workshop on automated deduction, 1979, pp. 161–167.Google Scholar
[15] Gallier, J., Logic in computer science, Harper and Row, 1986.Google Scholar
[16] Girard, J.Y., Intérpretation fonctionnelle et élimination des coupures dans l'arithmétique d'ordre supérieur, Doctoral thesis, Université de Paris 7, 1972.Google Scholar
[17] Girard, J.Y., Lafont, Y., and Taylor, P., Proofs and types, Cambridge University Press, 1989.Google Scholar
[18] Hallnäs, L., On normalization of proofs in set theory, Doctoral thesis, University of Stockholm, 1983,Google Scholar
[19] Henkin, L., Banishing the rule of substitution for functional variables, this Journal, vol. 18 (1953), no. 3, pp. 201–208.Google Scholar
[20] Hullot, J.-M., Canonical forms and unification, Conference on automated deduction (Bibel, W. and Kowalski, R., editors), Lecture Notes in Computer Science, vol. 87, Springer-Verlag, 1980, pp. 318–334.Google Scholar
[21] Krivine, J.-L. and Parigot, M., Programming with proofs. Journal of Information Processing and Cybernetics, vol. 26 (1990), no. 3, pp. 149–167.Google Scholar
[22] Martin-Löf, P., Intuitionistic type theory, Bibliopolis, 1984.Google Scholar
[23] Nederpelt, R.P., Geuvers, J.H., and de Vrijer, R.C. (editors), Selected papers on automath. Studies in Logic and the Foundations of Mathematics, vol. 133, North-Holland, 1994.Google Scholar
[24] Owre, S. and N.Shankar, The formal semantics of PVS, Technical Report CSL-97-2R, SRI, 1999.Google Scholar
[25] Paulin-Mohring, Ch., Inductive definitions in the system Coq - rules and properties. Typed lambda calculi and applications (Bezem, M. and Groote, J.-F., editors), Lecture Notes in Computer Science, vol. 664, Springer-Verlag, 1993, pp. 328–345.Google Scholar
[26] Plotkin, G., Building-in equational theories, Machine Intelligence, vol. 7 (1972), pp. 73–90.Google Scholar
[27] Prawitz, D., Natural deduction, a proof-theoretical study, Almqvist & Wiksell, 1965.Google Scholar
[28] Stickel, M., Automated deduction by theory resolution. Journal of Automated Reasoning, vol. 4 (1985), no. 1, pp. 285–289.Google Scholar
[29] Tait, W.W., Intensional interpretation of functionals of finite type I, this Journal, vol. 32 (1967), no. 2, pp. 198–212.Google Scholar
[30] Tait, W.W., A realizability interpretation of the theory of species, Logic colloquium, Lecture Notes in Mathematics, vol. 435, 1975, pp. 240–251.Google Scholar
[31] Werner, B., Une théorie des constructions inductives. Doctoral thesis, Université Paris 7, 1994.Google Scholar