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16 months ago, I finished a paper A and submitted it to a journal X. It is currently still under review. (For the journal's defence, it must be said that the paper is quite long - 89 pages).

This paper contains a proposition (let's call it A.1) whose proof relies on a lemma (let's call it A.2). However neither this proof, nor even the statement of Lemma A.2, have been given explicitly in paper A. Instead, I just wrote: "the proof of Proposition A.1 is analogous to the proof of Proposition a.1 from my earlier paper a".

(That earlier paper contains a detailed proof of Proposition a.1, which is done by using a lemma a.2. The results a.1 and a.2 are basically special cases of A.1 and A.2; the statements, and proofs, of the results A.x are very closely parallel to the statements and proofs of the results a.x - you can obtain the former from the latter by a simple search-and-replace.)

In the meanwhile, I wrote another paper B, that heavily relies on the results of paper A. In particular, at some point, I actually needed lemma A.2. So I made explicit the statement of A.2, and refered to paper A for the proof.

I submitted paper B to a journal Y, where it was rejected. The editor said that the result was not good enough for the journal (fine; I'll aim lower next time), but also said that the reviewers complained about the reference to paper A for the proof of A.2 (which did not explicitly appear there). They were understandably confused by this; reading my paper failed to convince them that my argument has no gaps.

Is it acceptable to email the editor of journal X and send them an updated version of paper A, where I make explicit the proof of Proposition A.1 (by stating and proving Lemma A.2)?

Now to rebut some of the obvious answers:

  • Redoing the proof of lemma A.2 in paper B is of course technically possible, but would require reintroducing a lot of formalism from paper A (Proposition A.1 appears on page 83 of paper A... and the whole paper B is only 26 pages long!) Besides, I feel that Lemma A.2 really belongs in paper A (it is after all an essential step to prove Proposition A.1, so it is already implicitly present there). I would really like to avoid this option at all costs.

  • Waiting for the answer from the journal X would mean holding up the publication of paper B, which is very costly for me: in the current climate, I feel that publishing as much as possible as early as possible is critical to secure a permanent job.

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    Is there some reason you don't just update paper A on the ArXiv for now and incorporate the changes into the revision you'll send Journal X after you hear back from the referee? – Alexander Woo May 1 '18 at 2:03
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I think you are focusing too much on the details, which may be confusing the issue. It sounds to me like your goal is to make sure reviewers of paper B are convinced by your arguments, rather than to write up a proper proof of A.2.

It is a perfectly acceptable and common thing to refer to paper A for a proof of say Lemma 1 even if Lemma 1 is not explicitly stated or proved in paper A. (E.g., "by a mild extension of the proof of X in [A], one gets...") What is important here is two things:

  • your explanation of exactly how it follows from [A] is clear to someone who has not read [A]

  • that there really is nothing new to check and you are not just being lazy

From what you wrote, it sounds like at least the first part was an issue for you. And if you can't make the proof of your lemma transparent in such a way, then you should write down the proof, either in paper A or paper B (possibly in an appendix).

Since you've made it clear you don't want to do the latter, and this sounds like a minor change to paper A (just adding in a small amount of details), it's fine to revise your paper A and update it on the arXiv (so make sure your paper B references the new arXiv version) but there's no need to revise your submission of paper A for a minor change. You can just point out you made this change upon resubmission after getting the referee reports.

  • OK, this is helpful! For some reason I never thought I could update the paper on arXiv without notifying the editor. (In fact, I did not think of adding a proof as being a "minor change".) – Ilia Smilga May 1 '18 at 14:06

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