The Ahlswede–Daykin inequality (Ahlswede & Daykin 1978), also known as the four functions theorem (or inequality), is a correlation-type inequality for four functions on a finite distributive lattice.
It is a fundamental tool in statistical mechanics and probabilistic combinatorics (especially random graphs and the probabilistic method).
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 The Ahlswede–Daykin inequality can be used to provide a short proof of both the Holley inequality and the FKG inequality.
For a proof, see the original article (Ahlswede & Daykin 1978) or (Alon & Spencer 2000).
The "four functions theorem" was independently generalized to 2k functions in (Aharoni & Keich 1996) and (Rinott & Saks 1991).