What to do if one reviewer asks to delete and another asks to add more details to a proof?

Question

I received a referee report for my single authored paper recently in Mathematics. Both the referees have asked for a revision. I am having a problem with a proof of a theorem in the paper.

Reviewer A asks to add more details to the proof of the theorem. He/She says it is not possible for him/her to understand one step in the proof.

Reviewer B says delete the proof of the theorem as it is not necessary. It is easy for him/her to follow it without any details.

The editor has asked me to revise the paper taking into account both the reviewers. I am confused now what to do to the proof of the theorem. Shall I keep it and add more details or shall I delete the theorem? If I delete it, then Reviewer A will get offended and if I add more details, then Reviewer B may reject the paper.

What should I do?

Answer 1

In this particular case (and if the journal allows for it), moving the respective part to an appendix (and expanding it) seems to be a solution that should satisfy both reviewers. Those readers who are like Reviewer B and do not require the proof can ignore it and those who are like Reviewer A can read the proof in its full glory.

Moreover, Reviewer A’s request itself gives you an excellent argument against the complete removal requested by Reviewer B: You can argue that you want to keep that part because you consider it relevant for some readers, as evidenced by Reviewer A’s request.

Answer 2

Of course without knowing the paper and detailed reviews it's impossible to tell, but I want to provide a slightly different view than Wrzlprmft.

In my opinion, this is most likely not even a real case of conflicting reviews. The key information I tend to take away from requests such as "delete X" or "greatly expand X" is that the reviewer did not like the current version of X. Notably, I read statements such as "proof XYZ is too high-level and needs to be expanded considerably" not as an endorsement that the proof is useful in general, but as a statement that the current version is not.

If viewed through that lense, both reviewers have actually identified the same problem, they just propose different approaches to fix it. In that sense I doubt that either reviewer will be greatly upset if you take the other reviewer's suggestion - and in the response letter you can always point to the other reviewer to explain why you have chosen a different approach to fix the problematic proof than the one that was suggested.

Answer 3

When an editor asks you to revise a paper taking reviewers’ comments into account, they mean just that — they do not mean “follow all the reviewers’ suggestions precisely”. In my experience as an author, on any issues bigger than typos, it’s fairly usual that I (and/or co-authors) don’t quite agree with a referee’s suggested edits, but their comment helps me understand what was sub-optimal about my/our original version, and so find an alternative way to improve that shortcoming. And when I’m a referee, reading revisions, I never expect authors to have followed my suggestions to the letter — just to have considered my concerns and addressed them in some reasonable way.

So in cases like yours, the kind of things which might help allay both referees’ concerns are:

• Start the proof by explicitly signposting/acknowledging what is easy about it. “This proof will be routine for experts, as a fairly standard application of pseudolinear estimation techniques.” This allays two possible sources of Referee B’s unhappiness: it lets them know that they can safely skip the proof, and it makes clear that you’re not overstating the novelty of this theorem.

• Within the proof, be conscious which details are more difficult (bearing in mind Referee A’s comments), and explain those clearly; but also, to help highlighting them, don’t belabour the parts that really are routine. A step-by-step argument like

  A = B (by the second unitary condition) ≤ C (by convexity) = D (by Kane’s lemma, part (iii)) < E (by Chen’s inequality).

  can often be clearer as

  It follows directly by unitarity, convexity of f, and Kane’s lemma that A ≤ D; then D < F by Chen’s inequality (with ε = 1/7, and the stochasticity hypothesis ensured by Lemma 3.2).

I don’t mean to suggest that those are necessarily applicable in your case — but rather to illustrate how even when the referees’ specific suggestions are incompatible, the concerns behind them may both be reasonable, and it can be possible to find a revision addressing them both at once.