A sequent calculus for type theory

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Volumn 4207 LNCS, Issue , 2006, Pages 441-455

  Lengrand, Stéphane ^a,b   Dyckhoff, Roy ^a   McKinna, James ^a  

^a UNIVERSITY OF ST ANDREWS   (United Kingdom)

^b UNIVERSITÉ PARIS DESCARTES   (France)

Author keywords

Proof search; PTS; Sequent calculus; Strong normalisation; Type theory

Indexed keywords

ARTIFICIAL INTELLIGENCE; KNOWLEDGE BASED SYSTEMS;

PROOF-SEARCH; SEQUENT CALCULUS; STRONG NORMALIZATION; TYPE THEORY;

COMPUTER SCIENCE;

EID: 33750309939     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/11874683_29     Document Type: Conference Paper

References (27)

1. [Bar92] H. P. Bareudrogt. Lambda calculi with types. In S. Abramsky, D. M. Gabby, and T. S. E. Maibaum, editors, Hand. Lag. Comput. Sci., volume 2, chapter 2. pages 117-309. Oxford University Press, 1992.
2. [BG99] R. Bloo and H. Genvers. Explicit substitution: on the edge of strong normalisation. Theoret. Comput. Sci., 211(1-2):375-395, 1999.
3. [Blo01] R. Bloo. Pure type systems with explicit substitution. Math. Structures in Comput Sci., 11(1):3-19, 2001.
4. [BR95] R. Bloo and K. H. Hose. Preservation of strong normalisation in named lambda calculi with explicit substitution and garbage collœtion. In Computing Science, in tlu: Netherlands (CSN '95), pages 62-72, Koninklijke Jaarbeurs. Utrecht, 1995.
5. [CH88] T. Coqnand and G. Huet. The calculus of constructions. Inf. Comput., 76(2 3):95-120, 1988.
6. [Coq] The Coq Proof Assistant, http://coq.inria.fr/.
7. [Dow93] G. Dowek. A complete proof synthesis method for type systems of the cube. J. Logic. Comput., 1993.
8. [DP99a] R. Dyckhoff and L. Pinto. Proof search in constructive logics. In Sels and proofs (Leeds, 1997), pages 53-65. Cambridge Univ. Press, Cambridge, 1999.
9. [DP99b] R. Dyckhoff and L. Pinto. Permutability of proofs in intuitionistic sequent calculi. Theoret. Comput. Sci., 212(1-2):141-155, 1999.
10. [DU03] R. Dyckhoff and C. Urban. Strong normalization of Herbelin's explicit substitution calculus with substitution propagation. J. Logic Comput., 13(5):689-706, 2003.
11. [Geu35] G. Gentzen. Investigations into logical deduction. In Gentzen. collected works, pages 68-131. Ed M. E. Szabo, North Holland, (1969), 1935.
12. [GR03] F. Gutiérrez and B. Ruiz. Cut elimination in a class of sequent calculi for pure type systems. In R. de Quciroz, K. Pimentel, and L. Figueiredo, editors, ENTCS, volume 84. Elsevier, 2003.
13. [Her94] H. Herbelin. A lambda-calculus structure isomorphic to Gentzen-style sequent calculus structure. In L. Pacholski and J. Tiuryn, editors, Computer Science Logic, 8th Int. Work., CSL '94, volume 933 of LNCS, pages 61-75. Springer, 1994.
14. [Her95] H. Herbelin. Séquenti qu'on calcule. PhD thesis. Université Paris 7, 1995.
15. [Hue89] G. Huet. The constructive engine. World Scientific Publishing, Commemorative Volume for Gift Siromoney, 1989.
16. [Kik04] K. Kikuchi. A direct proof of strong normalization for an extended Herbelin's calculus. In Y. Karneyama and P. J. Stuckcy, editors, Proc. of the 7th Int. Symp. on Functional and Logic Programming (FLOPS'04), volume 2998 of LNCS, pages 244-259. Springer-Verlag, 2004.
17. [KL80] S. Kamin and J.-J. Lévy. Attempts for generalizing the recursive path orderings. Handwritten paper, University of Illinois, 1980.
18. [Kri] J.-L. Krivine. Un interpréteur du λ-calcul. available at http://www.pps.jussieu.fr/~krivine/.
19. ILDMJ S. Lengrand, R. Dyckhoff, and J. McKiuna. A sequent calculus for type theory - longer version. available at http://www.pps.jussieu.fr/~lengrand/ Work/Reports/Proofs.ps.
20. [LP92] Z. Luo and R. Pollack. LEGO Proof Development System: User's Manual. Technical Report ECS-LFCS-92-211, School of Informatics, University of Edinburgh, 1992.
21. [MNPS91] D. Miller, G. Nadathur, F. Pfenning, and A. Scedrov. Uniform proofs as a foundation for logic programming. Ann. Pure Appl Logic, 51:125-157, 1991.
22. [Mun01] C. Munoz. Proof-term synthesis on dependent-type systems via explicit substitutions. Theor. Comput. Sci., 266(1-2):407-440, 2001.
23. [PD00] L. Pinto and R. Dyckhoff. Sequent calculi for the normal terms of the Λ∏ and Λ∏Σ calculi. In D. Gahniche, editor, ENTCS, volume 17. Elsevier, 2000.
24. [Pol98] E. Poll. Expansion Postponement for Normalising Pure Type Systems. J. Funct. Programming, 8(1):89-96, 1998.
25. [Pra65] D. Prawitz. Natural deduction, a proof-theoretical study. In Acta Universitatis Stockholmiensis, volume 3. Almqvist & Wiksell, 1965.
26. [IMP94] B. van Benthem Jutting, J. McKinna, and R. Pollack. Checking Algorithms for Pure Type Systems. In H. Barendregt and T. Nipkow, editors, Types for Proofs and Programs, volume 806 of LNCS. Springer-Verlag, 1994.
27. [Zuc74] J. Zucker. The correspondence between cut-elimination and normalization. Annals of Mathematical Logic, 7:1-156, 1974.