  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.
