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.*"

clearly markedas 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. $\endgroup$ – Terry Tao Jul 13 '15 at 18:16