Research proof index

If $Z_1, Z_2$ are independent, then $d[Z_1; Z_2]$ is equal to $d[\pi(Z_1);\pi(Z_2)] + d[Z_1|\pi(Z_1); Z_2 |\pi(Z_2)]$ plus $I( Z_1 - Z_2 : (\pi(Z_1), \pi(Z_2)) | \pi(Z_1 - Z_2) ).$

Let $Y_1,Y_2,Y_3$ and $Y_4$ be independent $G$-valued random variables. Then $d[Y_1-Y_3; Y_2-Y_4] + d[Y_1|Y_1-Y_3; Y_2|Y_2-Y_4]$ $+ I[Y_1-Y_2 : Y_2 - Y_4 | Y_1-Y_2-Y_3+Y_4] = d[Y_1; Y_2] + d[Y_3; Y_4].$

For four measurable random variables $Y_0,Y_1,Y_2,Y_3$ in a finite abelian group, the conditional mutual-information term used in the fibring identity can be reduced to $I\!\left[(Y_0-Y_1,Y_2-Y_3):(Y_0-Y_2,Y_1-Y_3)\,\middle|\,Y_0-Y_1-Y_2+Y_3\right]=I[Y_0-Y_1:Y_1-Y_3\mid Y_0-Y_1-Y_2+Y_3].$