Abstract
We give a simple and direct proof that super-consistency implies cut elimination in deduction modulo. This proof can be seen as a simplification of the proof that super-consistency implies proof normalization. It also takes ideas from the semantic proofs of cut elimination that proceed by proving the completeness of the cut free calculus. In particular, it gives a generalization, to all super-consistent theories, of the notion of V-complex, introduced in the semantic cut elimination proofs for simple type theory.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Andrews, P.B.: Resolution in type theory. The Journal of Symbolic Logic 36(3), 414–432 (1971)
Church, A.: A formulation of the simple theory of types. Journal of Symbolic Logic 5, 56–68 (1940)
De Marco, M., Lipton, J.: Completeness and cut elimination in Church’s intuitionistic theory of types. Journal of Logic and Computation 15, 821–854 (2005)
Dowek, G.: Truth value algebras and proof normalization. In: TYPES 2006, Springer, Heidelberg (to appear, 2006)
Dowek, G., Hardin, T., Kirchner, C.: Hol-lambda-sigma: an intentional first-order expression of higher-order logic. Mathematical Structures in Computer Science 11, 1–25 (2001)
Dowek, G., Hardin, T., Kirchner, C.: Theorem proving modulo. Journal of Automated Reasoning 31, 33–72 (2003)
Dowek, G., Werner, B.: Proof normalization modulo. The. Journal of Symbolic Logic 68(4), 1289–1316 (2003)
Girard, J.-Y.: Une extension de l’interprétation de Gödel à l’analyse et son application à l’élimination des coupures dans l’analyse et la théorie des types. In: Proceedings of the Second Scandinavian Logic Symposium (Univ. Oslo, Oslo, 1970) vol. 63 of Studies in Logic and the Foundations of Mathematics, pp. 63–92. North-Holland, Amsterdam (1971)
Hermant, O.: Semantic cut elimination in the intuitionistic sequent calculus. In: Urzyczyn, P. (ed.) TLCA 2005. LNCS, vol. 3461, pp. 221–233. Springer, Heidelberg (2005)
Okada, M.: Phase semantic cut-elimination and normalization proofs of first- and higher-order linear logic. Theoretical Computer Science 227, 333–396 (1999)
Okada, M.: A uniform semantic proof for cut-elimination and completeness of various first and higher order logics. Theoretical Computer Science 281, 471–498 (2002)
Prawitz, D.: Hauptsatz for higher order logic. The Journal of Symbolic Logic 33, 452–457 (1968)
Takahashi, M.-o.: A proof of cut-elimination theorem in simple type-theory. Journal of the Mathematical Society of Japan 19(4), 399–410 (1967)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Dowek, G., Hermant, O. (2007). A Simple Proof That Super-Consistency Implies Cut Elimination. In: Baader, F. (eds) Term Rewriting and Applications. RTA 2007. Lecture Notes in Computer Science, vol 4533. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73449-9_9
Download citation
DOI: https://doi.org/10.1007/978-3-540-73449-9_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73447-5
Online ISBN: 978-3-540-73449-9
eBook Packages: Computer ScienceComputer Science (R0)