Techniques for simplifying expressions – Mathematica Stack Exchange

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:

$$xledelta(A_s,X_v)+delta(z,X_s)$$ and $$xledelta(X_s,A_v)+delta(z,A_s)$$

$$delta(z)=begin{cases} 0 & z=2^{j}\ 1 & otherwise end{cases}$$ 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?