# Proof or citation?

I'm writing an article. I suppose that I'll submit it to a more or less decent journal (in English). I have doubts about the following: I have a lemma (with quite a trivial proof). I don't want to include this lemma with complete proof due to its triviality and because it has been already proved before in my Ph.D. thesis.

So I may either prove the lemma in the article (this certainly will not be very good) or simply include a citation of my dissertation instead of proof. But the thesis is in Ukrainian and it's not even available freely via the Internet. Which of the two alternatives would you choose?

Do there exist other ways to do this well?

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Just include the proof. If readers find it trivial then they will skip over it. Citing a dissertation in Ukrainian is worse than useless, it will just annoy people. –  Neil Strickland Jul 14 '14 at 13:24
If you wish, you may include the information that the lemma has been proved in your PhD dissertation, in the introduction, or in a line just before stating it. –  Pietro Majer Jul 14 '14 at 13:57
I generally agree with Neil Strickland. However, if you decide to omit the proof of the lemma and only cite your thesis, in my opinion you should mention that the proof is "quite trivial" (if it really is). Why? I believe most of the readers will not look the result up in your thesis. However, knowing that it should be trivial, they will have more motivation to try finding the proof on their own. –  Michał Kukieła Jul 14 '14 at 14:10
"Citing a dissertation in Ukrainian is worse than useless, it will just annoy people": This is funny to compare this remark to comments in the question mathoverflow.net/q/175988/16178. –  Bruno Jul 14 '14 at 14:52
It happened more than once that the same result would get published more than once by the same author in different languages, as long as later publications mention the earlier ones; rumour has it Riesz did that regularly. Some authors published twice because of the contradictory needs for speedy publication for the priority and well-written thoughtful publication for the posterity. I don't think anybody would have a problem with including the proof of your lemma in the 1st English paper that has it. –  Michael Jul 14 '14 at 21:51

A possible solution is writing before your lemma a sentence like the following: "The following result is easy and probably already known, however we include a proof for the convenience of the reader."

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Using this solution, if the referee really thinks the proof is straightforward she may suggest you don't include the proof: it gives you a second opinion on the matter. –  Benoît Kloeckner Sep 27 '14 at 7:27

One should distinguish between short, trivial and elementary proofs. Trivial proofs can be omitted with no harm. Omission of proof in the other two classes does seldom any good to the reader. In your shoes I would put full proof in the preprint on arXiv. It can be easily edited out in the published version.

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On my opinion, the main criterion for a reference is that it must be AVAILABLE. Either on Internet or in most university libraries. An unpublished thesis in Ukrainian which is not available on Internet does not satisfy this criterion. Such a reference is meaningless.

So the real options are: a) to leave the proof to the reader or b) to include it. I do not understand why you call the second option "not very good". What's wrong with including a trivial lemma? A reader for whom it is trivial will just skip the proof.

PS. Why don't you scan your thesis and put in on Internet? Even in Ukrainian.

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I would include the short proof, but also mention that the lemma appeared before.

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If you use the word "trivial" in its usual meaning, then no proof is needed -- usually, "trivial" means "immediately obvious to any reader". But since you ask, it seems that you use the word in some other meaning -- perhaps in the meaning of "elementary", but this is only a guess. If this is so, then what I would do in this situation depends on the length of the proof, both in terms of pages as well as in terms of percentage of the total length of the paper, as well as on whether the lemma is needed in the proof of some key result of your paper or just in some example, etc.. If the proof does not make the paper significantly longer, or if it is not longer than one or two pages, I would certainly include it.

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Why are people so obsessed with papers as short as possible? Lengthy elementary proofs can be moved to the appendix, which can be omitted in the published version. Of course, boring the reader with obvious and burying the important under tedious is wrong, but please bear in mind that it is usually the case that what seems to be obvious to the author is often impenetrable (at first reading at least) for a reader. –  Vít Tuček Jul 14 '14 at 18:15
@VítTuček: I am talking about published journal articles, and space in printed journals is typically limited. -- Obviously you can provide any additional information you like on your personal homepage or on some preprint server like the arXiv etc. -- I didn't think anyone needs to be told about the latter. –  Stefan Kohl Jul 14 '14 at 20:01
I understand. I just think that the emphasis should be on dissemination of knowledge and not on making publisher happy. ;) –  Vít Tuček Jul 14 '14 at 20:14
@VítTuček Sadly some referees and editors want explanatory material removed... –  Yemon Choi Jul 14 '14 at 21:54