Including proofs of known theorems in master's thesis

Question

I am currently writing my master's thesis in computer science. In my topic, I had a lot of papers to read and my main result relies also heavily on some theorems from especially one paper.

In the common literature there are some standard theorems for which the proofs are usually omitted bescause it is common knowledge or radically shortened as in "an easy application of the KKT theorem" and the authors do not want to waste any space for that.

In a master's thesis, however, I would assume that it is good practice to write down those proofs more extensively, since it also shows that you really understand your topic in-depth.

My question is: Would you agree on that and would a citation as in

Proposition 3.14 (see [5]). A nice theorem.
Proof. My extended proof.

be sufficient?

I checked out When should one include the proof of known results in a mathematical PhD thesis? already where the answers suggest that my intuition is right here (for a PhD thesis). My thesis advisor also agrees (which is probably most important), however, I am unsure to what degree this is appropriate.

Answer 1

There are three reasons to include a proof in your master's thesis - two of them good, and one of them bad.

1: As part of your background section

If your work relies on important results in your field, including those theorems' proofs in your introduction and background sections makes sense. This is true even if the proofs are well known. A thesis needs to show that you understand your field thoroughly to your committee, and as a bonus, including your field's well-known results will make your thesis a good introduction to your topic for someone coming in from another field. Theses are actually read this way!

When you cite proofs in this way, there's no need to give an expanded proof. Paraphrase or quote the standard proof (citing it clearly) without much commentary. You're just giving an overview of what others have already accomplished.

2: Because the details of the proofs are important for your own work

In the sections describing your new contributions to the field, your work might depend on the specific details of a previous proof. Either the detail is directly relevant to your own proofs, or the intuition behind the proof you're citing is similar to your own approach. Calling out these specific details is helpful.

When you cite proofs in this way, it makes sense to expand them - but only by focusing on the specific details you want to discuss. Briefly describe the rest of the proof. And again, clearly cite the proof as it's not your proof, you're just commenting on it.

3. Because you want your thesis to be long and detailed

Part of good writing is knowing which details are relevant and concisely sticking to those details. Don't include well-known proofs just for the sake of padding out your thesis or because you're including proofs by default.

Answer 2

It would be a good idea to make it slightly more conspicuous that the theorem (and proof) are not original, e.g.:

The following theorem is due to [5]; for the clarity of our exposition we give a more detailed version of the succinct proof in [5].

This leaves no doubt in the reader's mind that the work is not original, and also explains why you chose to include the proof.

Answer 3

1. If you leave the proof in the main text, make it obvious that your work is not original (or at least that A proof of the thereom was done earlier). Prominent caveat.

2. You could also put it in an appendix.

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I am less negative and more positive than Buffy on the benefit of showing this explication. Theses can serve a lot of purposes. Just make it clear that you are not claiming some discovery, but showing an exercise. But I think there can be benefit in it, both to you and to following lab mates--they will have the same issues dealing with the sparse literature that you did.

Answer 4

I'm not sure why you assume that the proofs are necessary. I would think that a citation to the theorem is enough, especially as you say, the proofs are "common knowledge" or easily derived. It seems like just padding.

However, there are exceptions. If the main ideas in your thesis would be made more understandable or otherwise enhanced by some proof technique of one of the cited theorems then certainly include such a proof. But if there are, then, fewer such proofs you can make a bigger deal of the citation as user Tom van der Zanden suggests. But note that I'm referring to something in the proof itself, not just the theorem.

This would make the thesis a bit tighter and put more of the focus of it on your own work rather than just explicating the work of others.