I have the following sequence flow, graphically:

enter image description here

According to my understanding, the diagram describes the following sequence

$$ y[n] = (cx[n] + x[n-1])[n-3](-1) $$

Now I'm very tempted to simplify this further:

$$ y[n] = (cx[n-3] + x[n-4])(-1) $$

I saw something like this done before in some examples, but I wasn't able to find a justification why such a simplification would be valid. So my question:

  • Is that simplification indeed valid?
  • Is there a theorem or so that proves this "distributive property" (or whatever the correct name is)?

Your simplification is valid. You don't need any special theorem or property. The delay operator introduces a time-shift, so you're just following the definition of the delay operator.

What I would suggest editing is the first equation line in your question. Although the reader will likely understand what you mean there, it's not standard notation. One way to present it with standard notation is to define a variable, let's call it $\tilde x[n]$, for the output of the summation block:

$$ \tilde x[n] := cx[n]+x[n-1] \;\;\;\;\;\; (*) $$

Now observe that $$ y[n] = -\tilde x[n-3] $$ and just plug in the definition of $\tilde x[n]$ from $(*)$ to get your answer.


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