# When to postpone a proof?

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.

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. $\square$

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."

• As always, look at authors whose writing style you like, and learn from them. Jul 13, 2015 at 12:48
• Example 1 is a good habit that too few people adopt. I'd like to know first what one proves, instead of digging through 50 pages of technicalities then finding out that the main theorems are uninteresting.
– user41593
Jul 13, 2015 at 12:56
• @PerAlexandersson I agree, and even further - if you cannot state the main theorems without complicated definitions, find a special case or corollary and state that instead. Jul 13, 2015 at 14:35
• I can't help but think that the answer will depend on the particular bit of mathematics you are presenting. You want to tell the best mathematical story that you can, and formulaic guidelines will be unsatisfying. Jul 13, 2015 at 15:02
• @darijgrinberg One common compromise is to state a simple but interesting special case of the main theorem (possibly one that is already known in the literature) in the introduction, noting that it will follow from a more general but harder to state result that will be given later. Another compromise is to give an informal version of the theorem in the introduction (which is clearly marked as being informal), promising a more precise statement later, but one has to be careful here to not be so vague and sloppy with the wording that the informal version is more confusing than enlightening. Jul 13, 2015 at 18:16