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For nonnegative functions on a finite Boolean lattice, the condition $\alpha (A)\beta (B)\le \gamma (A\cup B)\delta (A\cap B)$ for all pairs implies $(\sum \alpha )(\sum \beta )\le (\sum \gamma )(\sum \delta )$. Coordinate elimination and the two-point four-functions inequality prove it by induction. Indicators give inequalities for the union and intersection of set families. The Boolean-lattice statement appears in Theorem 3.1 of Nicholas Ruozzi's paper.

The Ahlswede–Daykin inequality, also called the four functions theorem, concerns nonnegative functions $\alpha ,\beta ,\gamma ,\delta$ on $\mathcal{P}([n])$. IfthenProve it by induction on $n$. The case $n=0$ is the single assumed inequality. For the induction step, sum each function over the last coordinate to obtain ${\alpha }^{\prime },{\beta }^{\prime },{\gamma }^{\prime },{\delta }^{\prime }$ on $\mathcal{P}([n-1])$. For fixed $A,B$ in this smaller cube, set $a_i=\alpha (A\cup i\{n\})$, $b_i=\beta (B\cup i\{n\})$, $c_i=\gamma ((A\cup B)\cup i\{n\})$, and $d_i=\delta ((A\cap B)\cup i\{n\})$, where $i\{n\}$ is empty for $i=0$ and $\{n\}$ for $i=1$.

The hypotheses give $a_0{b}_0\le c_0{d}_0$, $a_1{b}_1\le c_1{d}_1$ and $a_0{b}_1,a_1{b}_0\le c_1{d}_0$. The two-point four-functions inequality shows that these imply $(a_0+a_1)(b_0+b_1)\le (c_0+c_1)(d_0+d_1)$. Here is its key algebra: put $x=a_0{b}_1$, $y=a_1{b}_0$, $M=c_1{d}_0$, $N=c_0{d}_1$. Then $x,y\le M$ and $xy\le MN$. If $M>0$, $(M-x)(M-y)\ge 0$ gives $x+y\le M+xy/M\le M+N$; if $M=0$, both cross terms vanish. Adding the two diagonal bounds proves the claim.

Thus the primed functions satisfy the same hypothesis, and the induction hypothesis applies. Their totals equal the original totals, completing the proof.