Simplifying the Weft Hierarchy

We give simple, self-contained proofs of the basic hardness results for the classes W[t] of the weft hierarchy. We extend these proofs to higher levels of the hierarchy and illuminate the distinctions among its classes. The anti-monotone collapse at W[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathit{1,s}$\end{document}] and the normalization of weft-t formulas arise as by-products of the proofs.