When to postpone a proof?

Question

One possible practice in writing mathematics is to prove every theorem and lemma right after stating it. A long, technical proof — and sometimes even a short one — can interrupt the flow of the presentation, so postponing the proof can improve clarity. But if too many proofs are postponed in a long(ish) paper, it can be difficult to keep track of what depends on unproven facts and what remains to be proven, not to mention the danger of circular proofs.

Are there good guidelines for deciding when to postpone a proof? What should I take into account when making such decisions? Below are some examples of situations where I might postpone a proof, and I would like to expand my understanding by more principles and examples.

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Example 1: Giving main theorems in the introduction as soon as the necessary notions are defined instead of giving them only when you are ready to prove them.

Example 2: Pausing to explain to convince the reader that this is a natural thing to prove.

Lemma 1: Every strong polar bear is a weak penguin.

Before embarking on the proof, let us see why we should not hope for much more. A strong polar bear can fail to be a strong penguin and a weak polar bear can fail to be a penguin in the first place. Moreover, a penguin, even a strong one, need not be a polar bear of any kind. (For counterexamples, see the works of Euler and Gauss.)

Proof of lemma 1: Take a finite igloo containing the polar bear and a penguin…

Example 3: Clear exposition of the main argument. The following could be presented in the introduction or right after it, but the proofs of the lemmas would follow in subsequent sections.

Lemma 1: $A\implies B$.

Lemma 2: $B\implies C$.

Lemma 3: $C\implies D$.

Theorem 4: $A\implies D$.

Proof: Combining lemmas 1—3, we obtain $A\implies B\implies C\implies D$ as claimed. $\square$

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Edit: So far the discussion has mostly been around example 1. The actual question has not been addressed so far. An answer to "When to state the main theorem of a paper?" gives only a partial answer to my question. Let me quote Tom Church's comment below (emphasis mine): "Your Example 1 is dragging this discussion in the wrong direction, since everyone is going to agree with it: who would object to stating the main theorems in the introduction? The real question is about the structure of arguments within the body of the paper, and when it is appropriate to postpone a proof there. This is an important question, which too many authors neglect to think about when outlining their papers; I hope we'll get to hear a discussion on this point."

Answer 1

On my opinion, the modern style is as in Example 1. Long ago, people were expected to read the whole paper. And in the end of it to see "Thus we proved...".

Nowadays they want to see quickly what is there. Because there are too many papers. So is it very common nowadays to write an introduction with all definitions and precise statement of results, and discussion of their significance, and examples (if they are simple), and then to start proving the things. Many people just will not read after the introduction. Sometimes because they can prove the statements themselves. Sometimes because they are not so much interested.

Answer 2

I agree with most comments that more or less always the reader should know at least a good example of what you prove in the paper by reading the introduction. I think usually if you cannot state the main result in the introduction because it's too technical, probably the best thing is to have the section after the introduction contain the necessary definitions and accurate statements. It's not much fun for the reader to look through fifty pages of paper for the statement, and then discover that the technical conditions don't allow them to use it the way they wanted.

I quite often postpone proofs in the style of Example 3. Of course there is the danger of writing a circular argument, but this is something that both the author and reader should be easily able to check for: if not, then the paper isn't explained well enough. The advantage of this is that often in fifty-plus pages you have several proof strategies, maybe even a couple which use superficially similar methods but with an important difference. I find it much easier to understand a paper if I can take the lemmas as black boxes for understanding the main result, and then think about each lemma individually without having to keep my place mentally in the main result. This style of writing (I find) also discourages the author from opening a proof environment on page 3 which continues for the next forty pages, leaving the reader with the impression that they have to keep in memory all constant choices, definitions and equations made outside lemmas, which is usually false - but the important ones aren't indicated. It also discourages the author from stating a 'lemma' which is true in the context of the proof but false in general (in other words, with implicit assumptions) which I find very annoying.

Of course, with Example 3 one has to do more explanation: first an overview of the proof strategy, then a few sentences of why each lemma might be true or what it's used for, then an overview of the lemma proof (if it's long, which can easily be the case), which would probably be more compressed in the linear style (you probably don't explain why the lemma is true or how it's used if you are about to prove it). I don't think this is a good thing; most papers would be better for an extra five percent length in explaining what, why, how and perhaps why the obvious approach doesn't work.