Abstract
The Church-Rosser theorem in the type-free lambda-calculus is well investigated both for beta-equality and beta-reduction. We provide a new proof of the theorem for beta-equality with no use of parallel reductions, but simply with Takahashi's translation (Gross-Knuth strategy). Based on this, upper bounds for reduction sequences on the theorem are obtained as the fourth level of the Grzegorczyk hierarchy.
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