How do I use lemmas that are probably already proven, but including the proof myself is quicker than looking up a source?

Question

Suppose I have proven several lemmas during my work that are neither original nor significant (these results were needed for applied engineering/CS research, not mathematics). I believe that if I spend enough time going through articles in minor journals/theses in theoretical CS/applied mathematics, eventually, I will find the statements/proofs of these results. However, it seems that finding these results may take more time than it took to prove the results.

I am curious whether there is a chance that not providing citations could be perceived as plagiarism in one form or another. Would it suffice to state that I do not claim that the results are original without providing a citation?

Answer 1

You do need to make some effort to find in the literature a result you think is known, but after that what you propose is common and I feel is fine. I do it sometimes. In such a situation, I tend to say ``the following is probably known, but we include a proof for completeness.''

Even if you find the all your background lemmas in the literature, it might still make the reader's life easier if you include some or all of their proofs. It is really annoying to have to request seven obscure papers by inter-library loan, and then figure out all the different authors' notation, just to fill in a few pages of proofs.

Answer 2

For completeness, no it is not plagiarism to state a theorem and prove it without knowing whether that theorem is available already in the literature.

Plagiarism is the purposeful appropriation of someone else's words, claiming that they are your own.

Answer 3

Perhaps, they may be best classified as 'mathematical folklore'.

Let me suggest that citing it as 'mathematical folklore' would be incorrect. Based on what you have said in the question, you have proven the result (you know it to be true) and you also are not aware of a proof in the literature, although you believe one exists. However attributing a result to folklore typically implies much more than that: that the result is well-known by many people in the field, but that it doesn't have a canonical published proof in the literature. As you aren't an expert in the literature in this area, these statements go beyond what you can claim; you would need to be aware, and not just suspect, that the result is common knowledge.

I believe that if I spend enough time going through articles in minor journals/theses in theoretical CS/applied mathematics, eventually, I will find the statements/proofs of these results.

A common scenario in applied research! My approach in such situations is to somewhat "downplay" the result; for example, don't state it as a theorem, but as a proposition. And don't claim in your introduction or your list of contributions that you have proven a new result; focus on the new application instead, and the theorems are just there for completeness of the formal development or out of necessity. Finally, depending on how much effort you (or a coauthor) has put into searching the literature, either say that it is not known to your knowledge, or that it may be known, but you include a proof here anyway.

If you do all this and word it carefully, I don't think you are crossing any ethical lines by not citing the result. And you are certainly not committing plagiarism just by not being aware of something.

Answer 4

A neat way around this would be to say it is easily/ readily shown ...

That way you make clear that this is really only a step along the way and make no claim to novelty and it is a common hand waving technique. You could even relocate the bulk of the Lemma to the appendix if you feel that the workings don't add anything.

Though I'd ask if it is such an obvious thing to work out that you'd rather just do it from scratch, it is really necessary to be included. What does the reader gain? Ask yourself how this really contributes to the point your trying to make.

Answer 5

When writing a mathematical paper, it is often a good for readability idea to dissect the proof of a big theorem into a series of lemmata.

In all likelihood, some of them will end up being sufficiently localised and technical that no one has ever stated or proved them in the same form.

Some, on the other hand, will be very general observations which have almost certainly appeared in the work of others. Still, if it is merely a stepping stone towards the main result, you have independently formulated and proved it, and the proof is reasonably simple, then there is no harm in stating it as a proposition and moving on. Some caveats:

1. With a proposition of this kind, you should definitely avoid making it sound like it is some significant contribution of yours. If a reader or referee sees you claiming to have proven some significant fact, only to see something that is obvious and/or well-known, it will make you look real bad. (This does not mean that you should not stress its importance in the paper itself --- it is not unusual for simple observations to be very important.)
2. By extension, if the main result is described by the last paragraph (or trivially follows from a proposition like this), then most likely the paper is not sufficiently original for a research paper.
3. If you know the result is folklore (you have heard other people mention it, for instance), you should say it (and maybe state it as a fact, not a proposition). If you do not know, but are almost sure it is folklore, then you can say that the result is likely folklore. In both cases, if you have not found a proof in the literature, you should at least include a sketch.
4. If you know a very similar, but not completely trivially folklore result, then even if you have a citation, you should at least briefly indicate how the proof should be adapted or how your statement can be derived from the original (just as when citing known non-folklore results with modifications).
5. If the lemma has a reasonably simple statement, but the proof is either complicated or uses many preceding propositions, then you should make sure to check standard textbooks and, failing at that, perhaps ask on mathoverflow or math.se for a citation. If you spend half of the paper on proving a lemma which turns out to be a trivial variation of a well-known classical theorem, this will make you look bad.

Answer 6

I am basing my answer on some assumptions:

• Those results are really well-known;
• Those results can be found in a standard literature on the topic;
• You need those results, as you (as a decent mathematical thesis would) begin from the definitions and need to lay some ground work, but your actual contribution is later and definitely not here.

I would bring the lemmas from some standard book, adjusted to your notation, of course. I would cite the book in the lemma, e.g.,

Lemma 17.34 (Einstein and Feynman, 1892, Lemma 1.2). Each smooth n-dimensional hedgehog can be brushed over if n is even.

Then I would not state the full proof, but the proof idea:

Idea of the proof: If n is odd, the tail cannot be brushed. However, 2k-dimensional hedgehogs can be shown not to have a tail.

You want to advance the knowledge, not gurgle-up decades-old proofs. Of course, you should be able to reproduce the proof, if asked, e.g., during a thesis defence.