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A Simpler Proof of the Four Functions Theorem and Some New Variants

The celebrated Four Functions Theorem of Ahlswede and Daykin is a functional correlation inequality on distributive lattices with myriad applications. Ruozzi proved a variant of the inequality and used it to settle a major conjecture in the area of graphical models. We prove a new functional correlation inequality in the same vein which simplifies the proof of both the Four Functions Theorem and of Ruozzi’s inequality and suggests a unified picture for correlation inequalities on distributive lattices.