To end a proof, I often write "as was to be shown" or "q.e.d". Both of these terms make sense to me as a reader. On the other hand, I feel a little strange to put down $\square$ although I saw it many times here and there. In fact, I learned $\square$ notation here. I wonder if anyone could give me a brief explanation of $\square$ notation in mathematics. Where does it come from? More importantly, how does it logically mean "end" of a proof? Thank you.
Proof Writing – Why Does Drawing $\square$ Mean the End of a Proof?
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The highest degree of rigor would be achieved this way: Write down a list of axioms and a list of rules of inference. Start from an axiom and modify its logical formula using only one rule at a step and at each step clearly stating what rule you have used.
I'm not a logician and my description is probably not very good, but my point is that, whether or not this or some similar approach could be called "absolutely rigorous", it is highly impractical. At some stage in your development you know the law of associativity and need not be reminded of it every time it is used. At a later stage in your development the same is true about, say, the binomial theorem.
So, in practice, rigor is a relative concept. Proofs are written for the reader to check them. The goal of the author is to make this checking as quick and effortless as possible (this is often not true for textbook authors). To this end he must find the right amount of detail. The reader should not lose time by having to check four steps for a statement he could have easily understood in one step. On the other hand, the reader should not be forced to brood a long time over a statement that could also be written up for example with three intermediary steps each taking only a tenth of this time to check.
But this checking process depends on the reader. In my opinion, you cannot talk about rigor without talking about the "mathematical maturity" your average reader has. In a research paper proofs are considered rigorous that would be called handwaving or incomprehensible in an undergraduate textbook.
Of course, giving the right amount of steps in proofs is only one aspect of rigor. Another is not speaking about concepts you haven't properly defined. But this is relative as well. In a research paper about mathematical physics you don't have to clearly state the axioms for the real numbers. In a calculus textbook you do.
In your particular example: If you can expect your readers to know for example that the coefficients of polynomial functions are unique, your proof (or at least that part of it) is rigorous. If not, it is not, and you have to give some explanation.
You ask about when should you be satisfied that a proof is a proof. My opinion: if and only if you are sure you could, if required, fill in all thinkable intermediary steps and trace each fact you use back to the very axioms. If you are already asking yourself "is this really a proof?", it is most certainly not for you (though it is for Spivak, and, if he succeeded, for his intended readership). You have to break it down to steps that are completely obvious to you. In your mathematical development more and more arguments come to fulfill this criterion. For example, when you first learn about induction, you need to clearly state the base case, the induction hypothesis and so on. Once you have seen your share of proofs by induction, you are satisfied and often grateful if the author just writes "by induction we get...".
You see, it depends mostly on culture and translation, and also the habits. When you get used to using a notation you cannot give up on it easily. For instance, the notations you gave for derivates are both acceptable. If I am right, one of them is the notation used by Leibniz, and one of them by Newton. They can both be used and whichever comes easy, you use it. However, the upperline notation is a different case. For instance, in my country we use that as you do too, and I guess it is used in every country but it is not so popular you see. The notation itself is usually used (also in my country) in mathematical olympiads. Also if you want to learn more, I suggest you reading this article https://www.stephenwolfram.com/publications/mathematical-notation-past-future/.
Best Answer
It just means the same thing as q.e.d. Its introduction is usually attributed to Paul Halmos:
(This is quoted in Wikipedia)