Abstract
The proofs of the Church-Rosser theorems for β, η and β ∪ η reduction in untyped λ-calculus are formalized in Isabelle/HOL, an implementation of Higher Order Logic in the generic theorem prover Isabelle. For β-reduction, both the standard proof and the variation by Takahashi are given and compared. All proofs are based on a general theory of commutating relations which supports an almost geometric style of confluence proofs.
Research supported by ESPRIT BRA 6453, TYPES.
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References
H. P. Barendregt. The Lambda Calculus, its Syntax and Semantics. North Holland, 2nd edition, 1984.
D. Briaud. An explicit Eta rewrite rule. In M. Dezani-Ciancaglini and G. Plotkin, editors, Typed Lambda Calculi and Applications, volume 902 of Lect. Notes in Comp. Sci., pages 94–108. Springer-Verlag, 1995.
N. G. de Bruijn. Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser theorem. Indagationes Mathematicae, 34:381–392, 1972.
M. Gordon and T. Melham. Introduction to HOL: a theorem-proving environment for higher-order logic. Cambridge University Press, 1993.
T. Hardin. Eta-conversion for the language of explicit substitutions. In H. Kirchner and G. Levi, editors, Algebraic and Logic Programming, volume 632 of Lect. Notes in Comp. Sci., pages 306–321. Springer-Verlag, 1992.
G. Huet. Residual theory in λ-calculus: A formal development. J. Functional Programming, 4:371–394, 1994.
J. McKinna and R. Pollack. Pure type systems formalized. In M. Bezem and J. Groote, editors, Typed Lambda Calculi and Applications, volume 664 of Lect. Notes in Comp. Sci., pages 289–305. Springer-Verlag, 1993.
L. C. Paulson. Isabelle: A Generic Theorem Prover, volume 828 of Lect. Notes in Comp. Sci. Springer-Verlag, 1994.
F. Pfenning. A proof of the Church-Rosser theorem and its representation in a logical framework. J. Automated Reasoning, 199? To appear.
R. Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.
R. Pollack. Polishing up the Tait-Martin-Löf proof of the Church-Rosser theorem. Unpublished manuscript, Jan. 1995.
O. Rasmussen. The Church-Rosser theorem in Isabelle: A proof porting experiment. Technical Report 364, University of Cambridge, Computer Laboratory, May 1995.
N. Shankar. Metamathematics, Machines, and Gödel's Proof. Cambridge University Press, 1994.
M. Takahashi. Parallel reductions in λ-calculus. Information and Computation, 118:120–127, 1995.
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Nipkow, T. (1996). More Church-Rosser proofs (in Isabelle/HOL). In: McRobbie, M.A., Slaney, J.K. (eds) Automated Deduction — Cade-13. CADE 1996. Lecture Notes in Computer Science, vol 1104. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61511-3_125
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DOI: https://doi.org/10.1007/3-540-61511-3_125
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