In [1], it states that checking $f(..., x_i, ..., \bar{x}_j, ...) = f(..., \bar{x}_j, ..., x_i, ...)$ (variables $x_i$ and $x_j$ are E-symmetric) is equivalent to checking $f_{x_ix_j} = f_{\bar{x}_i\bar{x}_j}$. I am having trouble proving this statement.
Using Boole's expansion theorem, we can write Boolean function $f$ in $n$ variables as:
$$ f(..., x_i, ..., x_j, ...) = x_ix_jf_{x_ix_j} + x_i\bar{x}_jf_{x_i\bar{x}_j} + \bar{x}_ix_jf_{\bar{x}_ix_j} + \bar{x}_i\bar{x}_jf_{\bar{x}_i\bar{x}_j}$$
Negating variable $x_j$ results in:
\begin{align} f(..., x_i, ..., \bar{x}_j, ...) &= x_i\bar{x}_jf_{x_ix_j} + x_ix_jf_{x_i\bar{x}_j} + \bar{x}_i\bar{x}_jf_{\bar{x}_ix_j} + \bar{x}_ix_jf_{\bar{x}_i\bar{x}_j} &(1) \end{align}
Defining $f(..., \bar{x}_j, ..., x_i, ...)$ as the permutation of variable indices $i$ and $j$ in the above expression (1) results in: \begin{align} f(..., \bar{x}_j, ..., x_i, ...) &= x_j\bar{x}_if_{x_ix_j} + x_jx_if_{x_i\bar{x}_j} + \bar{x}_j\bar{x}_if_{\bar{x}_ix_j} + \bar{x}_jx_if_{\bar{x}_i\bar{x}_j} &(2) \end{align}
which shows that checking $f(..., x_i, ..., \bar{x}_j, ...) = f(..., \bar{x}_j, ..., x_i, ...)$ is equivalent to checking $f_{x_ix_j} = f_{\bar{x}_i\bar{x}_j}$.
However, following Definition 1 in [1], they define $f(..., \bar{x}_j, ..., x_i, ...)$ as $\bar{x}_j$ being substituted for $x_i$, and $x_j$ being substituted for $x_i$. This would give:
\begin{align} f(..., \bar{x}_j, ..., x_i, ...) &= \bar{x}_jx_if_{x_ix_j} + \bar{x}_j\bar{x}_if_{x_i\bar{x}_j} + x_jx_if_{\bar{x}_ix_j} + x_j\bar{x}_if_{\bar{x}_i\bar{x}_j} &(3) \end{align}
This causes a problem. As checking $f(..., x_i, ..., \bar{x}_j, ...) = f(..., \bar{x}_j, ..., x_i, ...)$ is NOT equivalent to checking $f_{x_ix_j} = f_{\bar{x}_i\bar{x}_j}$, as the terms in (1) and (3) containing those cofactors are equal. Instead it is equivalent to checking $f_{x_i\bar{x}_j} = f_{\bar{x}_ix_j}$, which is the check for NE-equivalence.
Any guidance would be much appretiated.
Additional clarification on difference. Following Definition 1 in [1], they state:
For example, applying transformation $\tau=(\bar{x}_2, x_1, x_3, \bar{z})$ to function $f(x_1, x_2, x_3) = x_1x_2 + x_2x_3$ results in a new function $f\circ\tau=\overline{\bar{x}_2x_1+x_1x_3}$, which replaces $x_1$ with $\bar{x}_2$, and $x_2$ with $x_1$ respectively, while also negating the output.
However, if I follow the rules that were used to get (2) above, the function would be $f\circ\tau=\overline{x_2\bar{x}_1+\bar{x}_1x_3}$.
[1] X. Zhou, L. Wang, P. Zhao and A. Mishchenko, "Fast Adjustable NPN Classification using Generalized Symmetries," 2018 28th International Conference on Field Programmable Logic and Applications (FPL), Dublin, Ireland, 2018, pp. 1-16, doi: 10.1109/FPL.2018.00008.
