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A direct proof of the finite developments theorem

Published online by Cambridge University Press:  12 March 2014

Roel de Vrijer*
Affiliation:
Interfakultaire Vakgroep Logika en Grondslagen van de Wiskunde, Universiteit van Amsterdam, Amsterdam, The Netherlands

Extract

Let M be a term of the type free λ-calculus and let be a set of occurrences of redexes in M. A reduction sequence from M which first contracts a member of and afterwards only residuals of is called a development (of M with respect to ). The finite developments theorem says that developments are always finite.

There are several proofs of this theorem in the literature. A plausible strategy is to define some kind of measure for pairs (M, ), which—if M′ results from M by contracting a redex occurrence in and ′ is the set of residuals of in M′— decreases in passing from (M, ) to (M′, ′). This procedure is followed as a matter of fact in the proofs in Hyland [4] and in Barendregt [1] (both are covered in Klop [5]). If, as in the latter proof, the natural numbers are used as measures, then the measure of (M, ) will actually denote an upper bound of the number of reduction steps in a development of M with respect to .

In the present proof we straightforwardly define for each pair (M, ) a natural number, which can easily be seen to indicate the exact number of reduction steps in a development of maximal length of M with respect to .

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1985

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References

REFERENCES

[1]Barendregt, H. P., The lambda calculus, North-Holland, Amsterdam, 1981.Google Scholar
[2]van Daalen, D. T., The language theory of Automath, Dissertation, Technological University of Eindhoven, Eindhoven, 1980.Google Scholar
[3]Hindley, R., Reductions of residuals are finite, Transactions of the American Mathematical Society, vol. 240 (1978), pp. 345361.CrossRefGoogle Scholar
[4]Hyland, M., A simple proof of the Church-Rosser theorem, Typescript, Oxford University, Oxford, 1973.Google Scholar
[5]Klop, J. W., Combinatory reduction systems, Dissertation, Mathematical Centre Tracts 127, Mathematisch Centrum, Amsterdam, 1980.Google Scholar