User blog comment:Rgetar/Three properties of standard forms/@comment-35470197-20181124090716

Good! Especially, the ill-definedness derived from the rule \(f(x)[n] = f(x[n])\) in Property 3 is a very common mistake, which I often needed to explain others so many times. Now I can refer to your blog post :P

By the way, I could not understand the last explanation. Even if you choose the "right" computation of \(f(x)\) in your sense, the result heavily depends on the choice of the fundamental sequence \(x[n]\). Therefore this is kind of another problem. Say, defining the fundamental sequence \(y[n]\) of \(y = f(x)\) by using the expression of \(y\) with \(f\) and \(x\), which is not necessarily reconstructible from \(y\) without the assumption of injectivity of \(f\), is ill-defined.