Research proof index

Double contraction. Contracting the last k of k+2 index pairs leaves two free pairs, with value a 2×2 generalized Kronecker delta times the factorial factor.

Symbol-level double contraction, two free pairs.

Full contraction. Iterating the snoc contraction over all four indices: ∑_f δ^{f}_{f} = 4!. Here f ranges over all maps Fin 4 → Fin 4.

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 $I\!\left[(Y_0-Y_1,Y_2-Y_3):(Y_0-Y_2,Y_1-Y_3)\,\middle|\,Y_0-Y_1-Y_2+Y_3\right]=I[Y_0-Y_1:Y_1-Y_3\mid Y_0-Y_1-Y_2+Y_3].$

Selector fibers inside an optional output are disjoint, so their finite sum is bounded by the probability of returning any some value.