this paper but which have some intriguing connections to some of our results and techniques, are [32] and [20]. We believe that the concept of prelogical relation would have a beneficial impact on the presentation and understanding of their results

We first present a new proof for the standardization theorem, a fundamental theorem in -calculus. Since our proof is largely built upon structural induction on lambda terms, we can extract some bounds for the number of fi-reduction steps in the standard fi-reduction sequences obtained from transforming any given fi-reduction sequences. This result sharpens the standardization theorem and establishes a link between lazy and eager evaluation orders in the context of computational complexity. As an application, we establish a superexponential bound for the number of fi-reduction steps in fi-reduction sequences from any given simply typed -terms. 1 Introduction The standardization theorem of Curry and Feys [CF58] is a very useful result, stating that if u reduces to v for -terms u and v, then there is a standard fi-reduction from u to v. Using this theorem, we can readily prove the normalization theorem, i.e., a -term has a normal form if and only if the leftmost fi-reduction sequence f...