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A few weeks ago, I submitted a mathematical paper in which the main proof contains a sentence that says: “by X and by Y, then Z is true”. It turns out that, luckily, the sentence is true. However, it is not nearly as direct as I am claiming in my paper. I have now been able to develop the argument by which for X and Y, Z is true; and it turns out that it is a half-page long argument.

Beyond never revising already submitted papers, is there anything I can do about it? Should I perhaps notify the editor-in-charge?

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    This is a rather typical thing to happen in my experience. As reviewer I come across this kind of thing all the time, i.e., it is claimed that something holds which indeed holds or at least looks credible to me, but arguments why it holds are not given or not conclusive. Also occasionally it has happened to me that reviewers pointed out that what I claim at some point isn't quite as obvious as my writing makes it seem. No big deal, and in the revision it can be amended. Answer of Buffy is good, just to let you know that you're not alone with this.
    – Christian Hennig
    Commented Apr 6 at 10:43

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You can ask the editor if you can submit an updated version. Say why you have made the change. But you can also wait to see what the reviewers say about the paper. If they raise the issue and suggest revisions, then you can do it then. And, you can add it to a revision even if reviewers don't notice, though you should then point it out.

If you let the editor know then they may take some action, or not, depending on where the review is. Asking the reviewers to start over might not be their best option. But they should have an idea of the status and the proper action.

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    Thank you very much for your answer. I suspect the review process is almost complete, for the journal aims to reply within 10 weeks for papers like mine, and we are in week 11 now… The change is really not substantial—it’s just the development of a not-obvious step in a proof.
    – EoDmnFOr3q
    Commented Apr 5 at 19:13
  • @Buffy -- Needless to say, I like the dog in your profile picture, Buffy.
    – JosephDoggie
    Commented Apr 5 at 19:39
  • I have accepted your answer, although I didn’t even have time to apply it because the paper just got rejected. And I finally made the choice to change jobs ASAP.
    – EoDmnFOr3q
    Commented Apr 6 at 19:22

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