I'm trying to use NCAlgebra to simplify some expressions and I'm having trouble with simplifying conjugates. Let's say I have two variables $a, b$ and a third variable $c$ which I want to conjugate with. We know $$c(ab)c^{-1} = (cac^{-1})(cbc^{-1})$$ so that if say $cac^{-1} = a$ and $cbc^{-1} = a^{-1}$ then we expect $c(ab)c^{-1} = 1$.
I've looked into the documentation and I can't seem to find a way to tell NCAlgebra to simplify (if possible) using this identity. I've tried to add an explicit rule and a more general replacement rule. The expressions I'm interested in are more complicated but I can't get a working example even in this simple case. Even when I explicit give it a rule it doesn't work:
SetNonCommutative[a, b, c];
NCReplaceRepeated[c ** (a ** b) ** inv[c],
{c ** a ** inv[c] -> a,
c ** b ** inv[c] -> inv[a],
c ** x_ ** y_ ** inv[c] -> (c ** x ** inv[c]) ** (c ** y ** inv[c]),
c ** (a ** b) **
inv[c] -> (c ** a ** inv[c]) ** (c ** b ** inv[c])}]
In short I would like NCAlgebra to try and use the fact that $c(ab)c^{-1} = (cac^{-1})(cbc^{-1})$ in order to look for simplifications. In some expressions this might not lead to a useful simplification (in which case $c(xy)c^{-1}$ is the simplest form period) but I would like for NCAlgebra to try and look as far as possible.