A fundamental tool in statistical mechanics and probabilistic combinatorics (especially random graphs and the probabilistic method), the Ahlswede–Daykin inequality , also known as the four functions theorem (or inequality), is a correlation-type inequality for four functions on a finite distributive lattice. It states that if are nonnegative functions on a finite distributive lattice such that for all x, y in the lattice, then for all subsets X, Y of the lattice, where and For a proof, see the original article or .