Techniques for simplifying expressions

I am hoping to get techniques to use when faced with expressions I think may be simplified. From two symmetrical looks at a problem I ended up with two inequalities. With excess stuff removed and everything is a non-negative integer:

$ x\le\delta(A_s,X_v)+\delta(z,X_s)$ and $ x\le\delta(X_s,A_v)+\delta(z,A_s)$

$ \delta(z)=\begin{cases} 0 & z=2^{j}\ 1 & otherwise \end{cases}$ and $ \delta(x,y,\ldots)=\delta(x)\delta(y,…)$

It would be nice to just have a single bound for $ x$ if it’s not complicated.

So I define delta as:

d[x_] := If[DigitCount[x, 2, 1] == 1, 0, 1]

I want to simplify:

x <= d[Xs]*d[Av] + d[z]*d[As] && x <= d[As]*d[Xv] + d[z]*d[Xs]

or maybe

Min[d[Xs]*d[Av] + d[z]*d[As], d[As]*d[Xv] + d[z]*d[Xs]]

What can I use to investigate this?