Basic Understanding of Convolution Operation
Since $y[n-n_0] = y[n] \ast \delta [n - n_0]$ and convolution is both associative and distributive, is it not the case that

$$
y[n-n_0] = \left(x[n] \ast r[n] \right) \ast \delta[n-n_0] = x[n] \ast r[n-n_0] = x[n-n_0] \ast r[n]?
$$