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Somewhat similar related question: Can a corollary follow a conjecture?

One writes of a corollary to a theorem, a corollary to a lemma, a corollary to a "proposition", a corollary to a corollary, a corollary to a conjecture (but might that one be confusing sometimes?), perhaps a corollary to an axiom (I don't know that I've ever seen that one, but I don't have any qualms about it), maybe a corollary to a definition, and in some odd contexts a corollary to a statement that is known to be false.

But today I found myself tempted to label a statement a "corollary," when the thing it would be a corollary to is none of the above, but rather a proof. I.e. the proof proves not only the theorem that precedes it, but also the "corollary" that follows it. It's a very terse proof because it just says: put together the several lemmas that precede the theorem, so there!

Can it be, at least in some cases, proper usage to call something a corollary to a proof? Or is there some other word for this situation that is known to everyone except me?

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    $\begingroup$ "Corollary to a proof" has a bad "smell" and probably should be avoided. Instead, add another sentence to the theorem (which is actually being proven). The new sentence will start with the word "Furthermore". $\endgroup$
    – Sam Nead
    Commented Jul 28 at 20:17
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    $\begingroup$ I think Johnstone in the Elephant uses the term "Scholium" for observations that follow from inspecting proofs. $\endgroup$
    – Jonas Frey
    Commented Jul 28 at 20:35
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    $\begingroup$ There is a never-ending war between "getting on with it" and "doing things right". The latter demands that we refactor our work: pulling definitions out of theorem statements, setting notation before it is used, hunting down all instances of the word "clearly", ... The former (namely, getting papers out the door) demands that we stop cleaning up and just get on with it. $\endgroup$
    – Sam Nead
    Commented Jul 28 at 20:38
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    $\begingroup$ @MichaelHardy: "The notation is clearly defined in the proof of Theorem 1." No, defining notation in a proof, although it is used outside the proof, is not clear. As a reader (and as a referee) I go mad if the authors of a paper expect me to read a proof just in order to be able to parse the statement of a result. $\endgroup$
    – Jochen Glueck
    Commented Jul 28 at 21:47
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    $\begingroup$ As a referee if I read "corollary of the proof", or "by the proof of Lemma X" or similar, I usually deduce that the author hasn't made the effort to write down the "right" statement. $\endgroup$
    – YCor
    Commented Jul 28 at 22:39

3 Answers 3

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It's hard to know exactly what the best way to write your paper is without more details. But I can see the following structure being commonly and effectively used:

Theorem If [condition P] then [conclusion Q].

Proof: Blah blah blah. $\square$

Remark: We see from the above proof of the theorem that it in fact suffices to assume only [weaker but more technical condition P'] to reach the same [conclusion Q].

In other words, rather than writing what you want to say as a "corollary to a proof" or a "porism" or a "scholium" or any other 10-dollar word, just put it there as a remark. I view a remark as any standalone comment which provides interesting context for the rest of the paper but which is not seriously needed in anything that follows.

Incidentally, I disagree with those who say it is important to state the stronger but potentially more subtle or cumbersome-to-phrase conclusion before the proof. Stating a theorem in a clean way, giving the proof, and then explaining that it is visible from the argument that the conditions can be weakened somewhat is often a more natural flow of ideas.

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  • $\begingroup$ Ha, I commented "please, before the proof!" while reading the beginning, only to see at the end that my objection had been preëmpted. I do wonder to what extent you think that this kind of postscript is appropriate in, say, a textbook vs. a paper. (Personally, I think either one argues against it: in a textbook, one must be particularly gentle on readers who are new to the subject; whereas, in a paper, I am much more likely to read asynchronously, and so benefit from a note that will tell me how to dip into the proof before I dip into it, than to read it as one whole piece of exposition.) $\endgroup$
    – LSpice
    Commented Jul 28 at 22:58
  • $\begingroup$ @LSpice I've made remarks for things like this also, and I guess my philosophy is that it depends how important/useful it seems. $\endgroup$
    – Kimball
    Commented Jul 29 at 5:11
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    $\begingroup$ I have an example of this in a pre-print. The goal of the paper was to determine which abelian groups had a certain property. The proof ended up using a much weaker but more technical condition than "abelian group", and in particular worked for all quasigroups. My coauthor persuaded me that rewriting the whole paper in terms of quasigroups would obscure the original motivation, and we settled on a remark to the effect that the proof carried over to quasigroups, and perhaps the literature on related properties could be reexamined in that context. $\endgroup$
    – Peter Taylor
    Commented Jul 29 at 6:44
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    $\begingroup$ In the situation you have presented, I would not leave it as just a remark. Instead, I would do something like this: "Examination of the proof shows that we have actually proved a stronger statement, with a weaker hypothesis: Theorem..." In order to do this it might also be necessary to export a definition or two from the proof to this post-proof position. I suppose a reader might wonder why the theorem was not stated like that in the first case, but as you say, there can be expository reasons for doing things that way, e.g. because the "weaker hypothesis" might be annoyingly technical. $\endgroup$
    – Lee Mosher
    Commented Jul 29 at 17:38
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One of the reasons why having a “corollary to a proof” is frowned upon, I think, is that it breaks the following implicit social contract between the writer and the reader: once the proof is closed (as symbolized by the end-of-proof symbol), it is so to speak “sealed”, the reader can forget all about it except for the fact that the asserted statement has been validated. Returning to a past proof to extract more from it than the statement that was enunciated means breaking the seal, which might distress a reader who was very happy to “garbage collect” all the notations and constructions inside the proof. (Maybe this “social contract” can be compared to the principle of “proof irrelevance” in type theories / proof assistants.)

But it is true that it sometimes isn't feasible to “expose” all the necessary machinery from the proof in the form of a technical lemma (that could then be used to proof both the theorem and the would-be-corollary-to-the-proof).

If you don't want to add a “furthermore” statement to the theorem because it would somehow mar its statement or because you want to be able to refer to both statements separately, one possibility is to state, separately but before the proof, both the main theorem and the side statement as an “addendum”, then proceed to prove them both simultaneously. I would suggest something like this:

Theorem 42. Every blueish foobar is cromulent.

Addendum 43. The blueification functor on the category of foobars is fully faithful.

Proof (of theorem 42 and addendum 43). We define the following setup: (…). From the above remarks, theorem 42 follows. Additionally, (…) from which addendum 43 follows. ∎

If the “addendum” statement needs to be relocated to a different part of the text (e.g., because it requires some notation that will be introduced later), I think it can still be called an “addendum”, but maybe there is no alternative but to return to the proof. In this case, I think it's very important to warn the reader in advance that this particular proof should not be “sealed” in their mind because later results depend on it. Maybe something like this could work:

Theorem 42. Every blueish foobar is cromulent.

We now give the proof of theorem 42, but we will return to it later on, as it will also serve to prove addendum 76 below.

Proof. We define the following setup: (…). From the above remarks, theorem 42 follows. ∎

(…)

Definition 75. The blueification functor is (…).

We can now state the following complement to theorem 42 above:

Addendum 76. The blueification functor on the category of foobars is fully faithful.

As mentioned earlier, the proof is a continuation of that of theorem 42.

Proof. Recall the following objects constructed in the proof of theorem 42: (…). We now remark that (…) from which addendum 76 follows. ∎

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    $\begingroup$ This 'contract' only holds among those who assume proof-irrelevance holds always. This is 'officially' what people do, but in reality they do things like what the OP is wanting: state a theorem that a thing exists, the proof constructs such a thing, and then they later use the construction. Or they extract the method of the proof and turn that into a definition. And so on. $\endgroup$
    – David Roberts ♦
    Commented Jul 28 at 23:01
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    $\begingroup$ @DavidRoberts, re, to be sure, writers do this, but do readers benefit from it? I find it hard enough to read the best-written paper, and each abrogation of the sort of contract defined here makes, or risks making, it still harder for me. But I have had it driven home to me by working with different co-authors that one reader's meat is another reader's poison, so I ask genuinely, not rhetorically. $\endgroup$
    – LSpice
    Commented Jul 28 at 23:05
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    $\begingroup$ @LSpice oh, I'm not saying readers benefit, but the number of proofs that claim to show mere existence, but the proof is to give a construction of the thing, is surely nontrivial. If people could be trained into giving the definition/construction first, and then proving it satisfies the required conditions, we would all be better off :-) Referring to notation introduced in a proof is I agree not a great practice. $\endgroup$
    – David Roberts ♦
    Commented Jul 28 at 23:20
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    $\begingroup$ People are not robots. If you state the "corollary" right after the "proof", which is how it is usually done, the reader has not yet forgotten anything, even if they believe in this "social contract". It's really like "before we forget about this proof, here's one more thing to note". $\endgroup$
    – Emil Jeřábek
    Commented Jul 29 at 6:33
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    $\begingroup$ @EmilJeřábek Of course the reader will not really forget everything about the proof as soon as they reach the “end of proof” symbol. What will really happen, however, is that many will read the statement, think “oh this is what I care about”, and then read the proof somewhat carelessly because they're only paying attention to what they decided they cared about (e.g., how a particular object is constructed); telling them ex post facto “oh you should also have cared about this other part” can seem rude. $\endgroup$
    – Gro-Tsen
    Commented Jul 29 at 8:21
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Stating a corollary to a proof is certainly something I have done myself and I have seen others do. It applies in a situation where a proof is showing something stronger than the theorem (etc) it is for, which is then extracted via the corollary.

Before stating a corollary to a proof, one should consider whether it wouldn't be preferable to either strengthen the theorem statement to reflect what it is actually proven, or whether a lemma can be distilled from the proof such that the corollary becomes a corollary of the lemma. These seem like cleaner approaches in general, but in rare cases, may not be the most conductive to the overall clarity of the exposition.

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