Abstract
We introduce new proof systems G[β] and G ext[β], which are equivalent to the standard equational calculi of λβ- and λβη- conversion, and which may be qualified as ‘analytic’ because it is possible to establish, by purely proof-theoretical methods, that in both of them the transitivity rule admits effective elimination. This key feature, besides its intrinsic conceptual significance, turns out to provide a common logical background to new and comparatively simple demonstrations—rooted in nice proof-theoretical properties of transitivity-free derivations—of a number of well-known and central results concerning β- and βη-reduction. The latter include the Church–Rosser theorem for both reductions, the Standardization theorem for β- reduction, as well as the Normalization (Leftmost reduction) theorem and the Postponement of η-reduction theorem for βη-reduction
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References
Aehlig K. and Joachimski F. (2004). Operational aspects of untyped normalization by evaluation. Math. Struct. Comput. Sci. 14: 587–611
Aspinall D. and Compagnoni A. (2001). Subtyping dependent types. Theor. Comput. Sci. 266: 273–309
Barendregt, H.P.: The Lambda Calculus, its Syntax and Semantics. Revised edition. North-Holland, Amsterdam (1984)
Barendregt H.P. (1992). Lambda Calculi with Types. In: Abramsky, S., Gabbay, D.M. and Maibaum, T.S. (eds) Handbook of Logic in Computer Science, vol II, pp 117–309. Clarendon Press, Oxford
Barendregt, H.P., Bergstra, J., Klop, J.W., Volken, A.: Some notes on lambda reduction. In: Degrees, reductions and representability in the lambda calculus, pp. 13–53. Preprint n. 22, University of Utrecht, Department of Mathematics (1976)
Chen, G.: Dependent type system with subtyping. Type level transitivity elimination. Technical Report LIENS-96-27, Laboratoire d’Informatique, Ecole Normale Supérieure, Paris (1996)
Chen, G.: Subtyping, type conversions and elimination of transitivity. PhD thesis, Université Paris VII (1998)
Chen, G.: Coercive subtyping for the Calculus of Constructions. In: Conference record of POPL 2003. The 30th SIGPLAN-SIGACT Symposium on Principles in Programming Languages, New Orleans, January 2003, pp. 150–159. SIGPLAN Notices 38/1 (2003)
Chen, G., Longo, G.: Subtyping parametric and dependent types. An introduction. http://www.di.ens.fr/~longo/download.html (1999)
Compagnoni A. (2004). Higher-order subtyping and its decidability. Inf. Comput. 191: 41–113
Coquand T. (1991). An algorithm for testing conversion in type theory. In: Huet, G. and Plotkin, G. (eds) Logical Frameworks, pp 255–279. Cambridge University Press, Cambridge
Curien, P.-L., Ghelli, G.: Coherence and subsumption, minimum typing and type-checking in \(F_{\leq}\). Math. Struct. Comput. Sci. 2, 55–91 (1992)
Curry, H.B., Feys, R.: Combinatory Logic. vol. I, North-Holland, Amsterdam (1958)
Curry, H.B., Hindley, J.R., Seldin, J.P.: Combinatory Logic. vol. II, North-Holland, Amsterdam (1972)
David R. (1995). Une preuve simple de résultats classiques en λ-calcul. C. R. Acad. Sci. Paris Sér. I Math. 320: 1401–1406
Filinski A. and Rohde H.K. (2005). Denotational aspects of untyped normalization by evaluation. Theor. Inform. Appl. 39: 423–453
Gonthier, G., Lévy, J.-J., Melliès, P.-A.: An abstract standardisation theorem. In: Proceedings of the 7th Annual IEEE Symposium on Logic in Computer Science (LICS’92), Santa Cruz, California, August 1992, pp. 72–81. IEEE Computer Society Press (1992)
Hindley J.R. and Seldin J.P. (1986). Introduction to Combinators and λ–Calculus. Cambridge University Press, London
Kashima R. (2001). A proof of the standardization theorem in λ-calculus. RIMS Kokyuoroku 1217: 37–44
Kashima, R.: On the standardization theorem for λβη-calculus. Contributed paper, International workshop on Rewriting in Proof and Computation (RPC’01), Sendai, Japan, October 2001.
Klop, J.W.: Combinatory reduction systems. Mathematical Center Tracts, 127. Mathematisch Centrum, Amsterdam (1980)
Lévy, J.-J.: Réductions correctes et optimales dans le λ-calcul. PhD thesis, Université Paris VII (1978)
Longo G., Milsted K. and Soloviev S. (2000). Coherence and transitivity of subtyping as entailment. J. Log. Comput. 10: 493–526
McKinna J. and Pollack R. (1999). Some Lambda-Calculus and Type Theory formalized. J. Autom. Reasoning 23: 373–409
Melliès, P.-A.: Axiomatic rewriting theory I. A diagrammatic standardization theorem. In: Middeldorp, A., van Oostrom, V., van Raamsdonk, F., de Vrijer, R. (eds.) Processes, Terms and Cycles: steps on the road to infinity, pp. 554–638. Lectures Notes in Computer Science 3838, Springer, Berlin (2005)
Minari P. (2004). Analytic combinatory calculi and the elimination of transitivity. Arch. Math. Logic 43: 159–191
Minari, P.: Proof-theoretical methods in combinatory logic and λ-calculus. In: Cooper, S.B., Löwe, B., Torenvliet, L. (eds.) CiE 2005: New Computational Paradigms, pp. 148–157. ILLC X-2005-01, Amsterdam (2005)
Mitschke G. (1979). The standardization theorem for the λ-calculus. Z. Math. Logik Grundlagen Math. 25: 29–31
Takahashi M. (1995). Parallel Reductions in λ-Calculus. Inf. Comput. 118: 120–127
Xi H. (1999). Upper bounds for standardizations and an application. J. Symb. Log. 64: 291–303
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Minari, P. Analytic proof systems for λ-calculus: the elimination of transitivity, and why it matters. Arch. Math. Logic 46, 385–424 (2007). https://doi.org/10.1007/s00153-007-0039-1
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DOI: https://doi.org/10.1007/s00153-007-0039-1