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."