In a mathematics article, how much of the argument can be given outside of "proof" environments?

Question

I'm writing an article containing one main theorem, with the other results essentially corollaries of that theorem. The largest part of the article is thus spend proving this main theorem.

It feels natural to me to write the prove as one long "story", most of which is told outside of "proof environments". So something like this

bla bla, consider this and that construction. Notice that bla bla. But we saw before that such and such, and conclude:

Proposition 5: (concisely write down the conclusion from the arguments laid out above)

(there is no proof environment here, the "proof" was essentially given in the block of text above the proposition) Keeping this mind, we now observe that bla bla...............

So this style results in a section without any actual proof environments, and propositions/lemma's in the section are essential "milestones" in the argument.

Is this style frowned upon, or considered annoying/difficult to read? To me it feels more natural than the more common style of mostly just a string of lemma's leading to a theorem.

Answer 1

It is hard to judge in the abstract, of course. But there are a number of ways to organize such a paper. It is probably best to be clear when you are giving formal arguments (i.e. proofs) and when you are just giving informal discussion.

One way is to list the main theorem early and state that you intend to prove it. You can precede this with some of your supporting lemmas (with proof, perhaps) and follow it with others. You can then show how the lemmas fit together to prove the theorem if needed.

But, I think, that a sequence of lemmas with no reveal about why you are stating and proving them followed by the theorem statement isn't very compelling reading.

Organize it so that your reader wants to read it and so that the purpose of each supporting piece is clear.

Sometimes the lemmas have independent interest. But sometimes the technique of the proof is much more interesting than the statement of any theorem. Take that into consideration.

You don't want your readers constantly asking themselves "What is the point of this?".

Answer 2

Explicit assumptions

Be clear what the assumptions of the theorem are. This is easy, unless you want to introduce side remarks with different assumptions along the way. Avoid adding assumptions midway through the process, or if you do, be very explicit about them. Certainly include them in the statement of the main theorem, or if they are cumbersome, write a specific assumptions environment. Hidden assumptions make your result very difficult to use for those who want to use its statement, not the proof.

Will anyone want to refer to parts of your argument or find them?

If you think that someone might want to refer to specific parts of your argument, then the usual lemma-proof-lemma style might be preferable. It makes it very easy to find a statement and its proof. In the more narrative and freeflowing style, one has to hope that the result has been quantified as a (say) lemma, and then has to look carefully up and down in the hope of finding a proof.

So, at the very least, be explicit about where to find the proof of any given statement. Write it immediately above or below the statement. Do not assume a reader has read or will read through your paper in detail. It will make life easier for even those will read everything.

Given this, I would avoid starting a proof, unless it immediately follows a lemma or such, is encased in a proof environment, or starts almost immediately from the beginning of a (sub)section. Otherwise the proof as an entity will be hard to find.

Otherwise

I have referred to, paraphrasing from memory so details are not correct, "the following lemma which has been on page XX of paper YY, but has not been stated explicitly. It follows from equation ZZ and the next equation."

Explicit structure makes it easier for others to use your work.