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I'm writing a mathematical paper. In it, I use a lemma. The lemma is not hard to prove and I have verified it myself. The proof is too tedious to include in the paper, so I want to just include a citation. I found a paper that includes the result. However, that paper does not actually include a proof. I cannot find any other place where this lemma appears.

I see three options:

  1. State the lemma without proof or citation.

  2. State the lemma without proof, but cite the paper (that states the lemma without proof or citation).

  3. Provide a proof of the lemma.

Which is most appropriate? Option 1 is easiest, but might annoy some readers who don't believe me. Option 2 seems like a cop out. Option 3 is safest, but I don't think it's necessary, as the proof is really just a long and boring calculation.

ADDED: To be clear, the lemma is basically an integral. The proof consists of splitting up the domain of integration to remove absolute values, evaluating each of the parts (easy enough for symbolic integration packages like mathematica), and then joining them back up. This is "obvious", but messy because the expressions are quite long. My writeup is two pages.

Maybe a better way to phrase my question: The result is trivial -- I think so, the authors of the other paper think so, and the journal they published in thinks so. Should I still provide a citation? Is it misleading to cite the other paper without clarifying that it doesn't provide a proof?

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  • It sounds like you used the lemma first and then searched for other appearances of it in publications. Is this the case? If so, did you create it yourself or did you obtain it from somewhere?
    – Mick
    Sep 20, 2017 at 4:00
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    "If only I had the theorems! Then I should find the proofs easily enough." -Riemann . If someone states a lemma/theorem/conjecture you use (even if you reinvented it), you need to cite the source regardless if there is a proof or not. If you easily found a proof, then you should probably be convinced (as long as the original paper acted like it was a fact) that the original writer also knew a proof. Saying that if you use something and the proof isn't widely available, you are probably doing a disservice to the reader to not include it(unless you can show the reader how to reproduce a proof).
    – PVAL
    Sep 20, 2017 at 4:05
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    If not for providing evidence, you should at least provide a proof for making the life of your readers easier. However, you have multiple options if you don't want to include it in the main part of your paper.
    – koalo
    Sep 20, 2017 at 6:36
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    Chemical papers often make use of Supporting Information for these very things: parts the main paper builds on but too irrelevant to be included in it.
    – Jan
    Sep 20, 2017 at 13:18

6 Answers 6

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Citing the other paper seems necessary in any case, as they have stated the lemma before you. This is for attribution. Citing the other paper for evidence seems not appropriate, as there is no proof given there.

If the lemma is not rather obvious (say the obvious proof strategy works in < 5min), then stating it without proof would be very bad form. Put in an appendix if you don't want a boring lengthy proof to spoil the otherwise elegant paper, but put it somewhere people can find it.

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    The obvious retort is "If they can state it without proof, then why can't I?"
    – user80085
    Sep 19, 2017 at 22:37
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    Well, if they had written down the proof as they should have, you wouldnt have this problem now, right?
    – Arno
    Sep 19, 2017 at 22:54
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    Peer review means very little - and if there is no proof for that lemma in the paper, the reviewers cant have checked the proof (of course they could have derived a proof themselves, but how many more peole are supposed to do that?).
    – Arno
    Sep 19, 2017 at 23:00
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    Agree with Arno. I don't see any disadvantage in including the proof. This is the only professional solution here: prove it and cite the lemma. I would also add a note "X stated the lemma without a proof, and thus we shall prove it here", which will sound slightly critical of the original authors, as it should be. It seems user80085 believes too much in the system, no offense, as if the fact that a paper passed peer-review means every aspect of the paper is justified. It's obviously not.
    – Dilworth
    Sep 19, 2017 at 23:30
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    Given that two people have now used this lemma, I think a proof is worth presenting even if it is "trivial". You can send in your writeup (which you say you already have) as an appendix with this paper, and if the reviewers also feel it doesn't belong in this paper, post it as a short paper in arXiv/etc. so the third user of this lemma won't face your problem.
    – Jeffrey Bosboom
    Sep 20, 2017 at 0:55
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If the result is basically trivial (as you say it is), I think how you proceed should consider how standard this type of result would be in the field.

You could put something like:

The following result can be established by standard (but tedious) computation.

if it's the sort of thing you could expect an early graduate student in the field to do as a homework question, or

The following result, which is stated by (Author) in [(paper)], can be established by standard (but tedious) computation.

otherwise.

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    The amount of discussion and contemplation on whether to prove "trivial lemmas" already shows why any professional mathematician should include the proof of these "trivialities", and redeem us from such discussions. Indeed, if it's so trivial, why not just prove it and finish with it?
    – Dilworth
    Sep 21, 2017 at 22:51
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    @Dilworth I don't really agree. I don't think citing something trivial is actually a good thing. If it's trivial, the person who wrote it down doesn't really own it in the same way as a non-trivial result. Also, it's not obvious where you would look for a trivial result, as it could appear in papers on a range of topics. Moreover, often if the result is trivial there is not just one version of it. I think whether you include a proof of the exact variant needed for your paper will depend on how crucial the tiny details are to you.
    – Jessica B
    Sep 22, 2017 at 5:39
  • @Dilworth On the other hand, a useful thing a mathematician could do in such circumstances is write a textbook that includes a proof. But of course the incentive to do so it lower. Alternatively, I like the idea of papers in electronic form where most of the detail is initially hidden and you can expand out more layers as you wish. Then the proof could be included for those that want to read it, without getting in the way for those who don't.
    – Jessica B
    Sep 22, 2017 at 5:42
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    Don't say "by standard techniques" say which standard techniques you're using!
    – Noah Snyder
    Sep 22, 2017 at 13:19
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    @JessicaB, I was not talking about citation indeed. I was talking about proving what you actually state is true. Papers are written not just to inform us of new results, but to validate and establish true statements. Working out tedious details to yield important results is a contribution by itself.
    – Dilworth
    Sep 23, 2017 at 22:40
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Cite the paper when you state the lemma. Then write:

\begin{proof} Split up the domain of integration to remove absolute values, then evaluate each of the parts. \end{proof}

It's a waste of everyone's time to have two pages of a calculus exercise. But it's also a waste of everyone's time to have to guess how the proof goes. The above is the best compromise that makes it clear how the proof goes in the least amount of time.

If the proof were one paragraph instead of two pages then I'd say include it all.

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  • This was my thought also, although perhaps with slightly more explanation if some tricky manipulations are involved (but the OP's description suggests this is not the case), and perhaps also state the result of the evaluation for each of the parts. In addition, I think it would be useful cite the other reference at the beginning and say something to the effect that you (the OP) have confirmed the result stated in [XX] by using the following method, and then give the brief explanation.
    – Dave L Renfro
    Sep 22, 2017 at 17:31
  • Yeah, I don't know the actual calculation. Perhaps another sentence or two would be in order like: "for the third part we use integration by parts with u=blah and dv=blah." But at any rate it seems clear that two pages is too much, but zero sentences is too little.
    – Noah Snyder
    Sep 22, 2017 at 20:19
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Just say you have discovered a truly marvelous proof for it that won't fit in the confines of your paper's length restrictions. ;)

No one will mind, right? They can always work out the proof that you had in mind....

I recommend including a proof in the appendix, if none has previously been published.

A proof without proof is just a statement. If you feel you shouldn't just state something without any proof at all, then don't state "there is a proof" without any proof of that statement.

The historical example I've alluded to is a good illustration of the problems that can arise from the unproven assertion, "I have a proof for this."

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    This is just a joke, followed by a repetition of a suggestion that had already been posted when this answer was written.
    – David Richerby
    Sep 21, 2017 at 10:10
  • @DavidRicherby the joke has a point, though. If you think about the parallel I've drawn, you will gain your own insight into the perils of asserting the existence of a proof without...well, without proof. (Edit: I've made this more explicit in my answer.)
    – Wildcard
    Sep 21, 2017 at 10:34
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I suggest not including the proof in your paper.

If you need to, cite the other paper which states the lemma Then, since you have determined that the proof is "obvious", simply state that.

For example;

  • "Lemma 2 is stated without proof by Bloggs (2007). The proof is trivial and not included here" [The wording "and not included here" is optional, since you won't provide a proof];
  • (If you want to provide some pointer on how to start the proof) "Lemma 2 is stated without proof by Bloggs (2007). If one starts by splitting up the domain of integration to remove absolute values, the proof is trivial."

Any competent mathematician will understand your point, since it is fairly common practice in mathematical journals.

If they so desire, the reader will be able to derive the lemma on their own. In fact, some mathematicians will enjoy doing exactly that as an exercise - why deprive them of that enjoyment?

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    A mathematical paper is normally written for a target audience of mathematicians - or, at least, people with an interest and appreciation of mathematics. It is not necessary to assume a completely uninformed reader, nor is it necessary to spoon-feed the reader.
    – Peter
    Sep 22, 2017 at 9:07
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    I was talking about mathematicians as well. Many mathematicians are "incompetent" in many aspects of mathematics, as they are only human. Writing papers assuming a priori that everyone obviously knows the basics of "Semi-periodic-C*-elimination-theory" is simply bad writing.
    – Dilworth
    Sep 23, 2017 at 22:32
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    to be fair, the OP added this explanation on the actual proof he refers to, only after the discussion here took place.
    – Dilworth
    Sep 23, 2017 at 22:42
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    I disagree. There is value/benefit in brevity just like there is a value/benefits in full proofs. The question is which is more valuable, and what is more appropriate to a journal publication. A journal publication must be correct, leaving out long calculations means there is good chance the authors made a mistake, the reviewer didn't bother to check, and future readers will be unable to follow the proofs. The possible value in brevity is thus smaller than the cons. I also claim that "substantive knowledge" is a relative term, and cannot solely be the point of a mathematical paper.
    – Dilworth
    Sep 24, 2017 at 13:02
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    I disagree, factually. My substantial knowledge of errors that made their way into peer-review publication have always been the result of not including the proofs of "obvious/trivial" lemmas. Not the other way around, as you suggest. Making the effort to prove everything reduces substantially the chance of errors.
    – Dilworth
    Sep 24, 2017 at 13:16
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Could you just give the reader an idea of the tediousness by showing just a part of the proof? Maybe showing the verbosity of just one part of it is the best way to convince any reader that your omission is fully understandable and justified.

Alternatively, if feasible, you could link to an URL containing a (sketch of?) proof, either in LaTeX or, e.g., as a Mathematica nb file.

I would cite the other paper in any case, for attribution, as suggested by others.

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