## A simplified proof of the Church-Rosser theorem.(English)Zbl 1338.03017

Summary: Takahashi translation $$\ast$$ is a translation which means reducing all of the redexes in a $$\lambda$$-term simultaneously. In [J. Symb. Comput. 7, No. 2, 113–123 (1989; Zbl 0661.03008); Inf. Comput. 118, No. 1, 120–127 (1995; Zbl 0827.68060)], M. Takahashi gave a simple proof of the Church-Rosser confluence theorem by using the notion of parallel reduction and Takahashi translation. Our aim of this paper is to give a simpler proof of Church-Rosser theorem using only the notion of Takahashi translation.

### MSC:

 03B40 Combinatory logic and lambda calculus

### Citations:

Zbl 0661.03008; Zbl 0827.68060
Full Text:

### References:

 [1] Barendregt, H. P., The Lambda Calculus: Its Syntax and Semantics, 2nd edition, North Holland, Amsterdam, 1984. · Zbl 0551.03007 [2] Hindley, J. R. and J. P. Seldin, Lambda-calculus and Combinators, An Introduction, Cambridge University Press, Cambridge, 2008. · Zbl 1149.03016 [3] Komori, Y. and F. Yamakawa, ‘The system of CLλ and a method to calculate $$β$$ forms without $$β$$-reductions’, to appear. · Zbl 0827.68060 [4] Takahashi, M., Parallel reductions in λ-calculus, Journal of Symbolic Computation, 7, 113-123, (1989) · Zbl 0661.03008 [5] Takahashi, M., Parallel reductions in λ-calculus, Information and Computation, 118, 120-127, (1995) · Zbl 0827.68060 [6] Takahashi, M., Theory of Computation, Computability and Lambda Calculus, Kindai Kagaku Sha, Tokyo, 1991 (in Japanese). · Zbl 0827.68060
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