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Say I find a Math Proposition in another paper, which is just stated, and no proof (nor even sketch of proof) is given.

If I am writing a paper referencing that paper, is it good or bad (from reviewer's point of view) for me to fill in the proof of that proposition?

If I do so, what is a good way for me to indicate it in the paper? (that the proof written is by me, though the proposition is by the other paper)

The proof is not trivial, though it is not particularly difficult either.

Or should I just privately verify if the proposition is correct? And just cite the proposition (without proof) in my paper?

Thanks.

I am concerned about this issue since I followed the news story in the case of Grigori Perelman's proof of Poincare conjecture; the authors who filled in "gaps" in his proof did not go well with the reviewers and the general public. (https://www.newyorker.com/magazine/2006/08/28/manifold-destiny)

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    I really dislike how the authors of that article keep writing "the Poincaré" for "the Poincaré conjecture". – user9646 Feb 25 '18 at 16:14
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    @NajibIdrissi That is kind of annoying, though I note the authors quote Perelman as stating “I never set out to be the sole solver of the Poincaré.” Assuming this is a complete and accurate quote, and the article's prior description of Perelman's English as being fluent is reasonable (I've never heard him talk, personally), the authors may have simply accepted that "the Poincaré" must be an accurate and acceptable way of referencing the problem. – zibadawa timmy Feb 25 '18 at 16:51
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    Not all authors who 'fill in "gaps" in a proof' are jealous... as Najib said, one would have to be rather obnoxious to face that kind of disapproval from the mathematical community. For instance, don't go around telling people that the cited paper is full of holes, or that you are the one who contributed 100% of the proof of that proposition, or that your paper finally makes the cited paper complete, or something like that... – user21820 Feb 26 '18 at 11:54
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    I think you mis-read that New Yorker article. Your assertion "the authors who filled in "gaps" ... did not go well with the reviewers" is not correct. To the contrary, what that article is about is that the stakes for understanding and filling in the gaps of Perelman's proof were so high that there were vicious academic fights for the credit of that understanding and gap-filling. – Lee Mosher Feb 26 '18 at 15:28
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Let's call the other paper [A].

First of all, you might like to check whether a proof already appears in some other paper or book, [B]. You can search MathSciNet or other databases for papers which cite [A]. If so then you can cite both ("The following proposition was stated in [A]; see [B] for a proof.")

Otherwise, whether to give a proof is at your discretion. If you think the reader would find it helpful to see the proof, and it doesn't distract from the main purpose of your paper, and it isn't excessively long, then sure, you can include it. People often use phrasing like:

We make use of the following proposition, which is stated in [A]. Since [A] does not include a proof, we give one here.

Another option is to put the proof in an appendix, or to write it up as a separate note which you post on arXiv or something.

The Perelman case is about something different - people disagreeing over whether the original paper actually solved the problem, and how much of the credit for the result is due to those who filled in whatever gaps there may have been. In this case, you describe the proof of the proposition as "not particularly difficult", so it's likely that the authors of [A] did in fact know how to prove it, and so would most of their readers who took the time to do it. It's reasonable to give all the credit for this proposition to the authors of [A]. There's nothing wrong with you including a proof, and as long as you don't try to take credit for the result, no controversy will result.

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    Otherwise, whether to give a proof is at your discretion. If you think the reader would find it helpful... I agree, and my rough guideline for when it is helpful is if the idea/details of the proof are used in other parts of your paper. If it the proof is completely tangential, I would typically not include a proof, unless I found a proof I really liked. – Kimball Feb 25 '18 at 17:46
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I don't think the situation is at all comparable with the news article you linked. You're not claiming that you should get or share the credit for this proposition, you're not saying that the cited work lacks critical steps that you're fixing. (I also don't think that the proposition in question is a century-old, world-famous conjecture.)

If the proof is really "not particularly difficult", then I see no problem either way: cite the proposition, and then either write your own proof or leave it at that.

But you shouldn't worry about any kind of backlash if you decide to write your own proof. People rewrite proofs all the time for a variety of reasons (you want to match it to your notations, your hypotheses are slightly different, you want to reuse the steps and ideas...) You would have to write this in a rather obnoxious way to encounter any repercussions.

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    The first sentence is definitely true. The problem with the paper that "filled in the gaps" of Perelman's proof wasn't that it filled in gaps. That there were details missing that needed to be filled in was known. The problem was that it surgically downplayed what Perelman had actually contributed and tried to steal all of the credit. And for a major result, at that; even non-topologists were well aware of this conjecture, especially once Perelman's preprints started appearing. – zibadawa timmy Feb 25 '18 at 17:02
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    +1 for "You would have to write this in a rather obnoxious way to encounter any repercussions." Overall, I thought @Nate Eldredge gave a better answer, but this single sentence really stands out! – Dave L Renfro Feb 25 '18 at 20:20

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