Factorization and normalization, essentially

λ-calculi come with no fixed evaluation strategy. Different strategies may then be considered, and it is important that they satisfy some abstract rewriting property, such as factorization or normalization theorems. In this paper we provide simple proof techniques for these theorems. Our starting point is a revisitation of Takahashi’s technique to prove factorization for head reduction. Our technique is both simpler and more powerful, as it works in cases where Takahashi’s does not. We then pair factorization with two other abstract properties, defining essential systems, and show that normalization follows. Concretely, we apply the technique to four case studies, two classic ones, head and the leftmost-outermost reductions, and two less classic ones, non-deterministic weak call-by-value and least-level reductions.