Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T03:32:02.040Z Has data issue: false hasContentIssue false
  • English
  • Français

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/
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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