While my question topic is that of mathematical writing of papers, which is a broad subject, the particular question is specific.
I am writing a paper, in which we have a section called "Outline of Proof". (It's Section 2.) The outline is fairly informal, and we omit some technical details, making approximations. However, among these approximations, my co-author wants to state (and label) important definitions and results (lemmas, equations, etc). He then wants to, later in the paper when we are doing the corresponding part carefully and rigorously, refer back to these (say, "by equation (2.4)", or "by Lemma 2.2"). Moreover, he is very against redundancies, so does not like things being stated twice precisely (including in the outline) -- once precisely in the text and once approximately in the outline is fine.
To me, this seems insane. (Of course, I did not use such a phrase when speaking with him!) When I read a paper, I never carefully read the outline: I just read it, and try to get an overview (or 'outline') of the proof; if there are parts that I don't really understand, I don't get hung up on them, trusting that with the more rigorous explanation later I'll be able to make sense of what the authors are saying.
However, I'm a pretty junior author -- 2nd year of PhD -- while he is a first year postdoc. That doesn't mean that I haven't read a reasonable number of papers (and in fact my lack of experience and knowledge means that I am less able to understand poorly written papers); moreover, he has said that he feels writing papers well isn't his best attribute.
So my question is this:
(a) is it standard to read an outline of a proof carefully?
(b) is it standard (or at least not discouraged) to state precisely important, even key, results/definitions that will be referred back to in the main body of the paper when giving proofs?
Just as some extra comments... I'm not here to try to get people to tell me that I'm right and my co-author is wrong and/or being silly! I know that sometimes some people come to Stack Exchange for such comments (see, particular, Workplace/Interpersonal Skills SEs!). I should have perhaps made the following clear: if everything my co-author does is standard in the field, and I'm in the wrong, I definitely want to know that and will accept it! -- I'm here to learn :-) please have no qualms about hearing criticism! (assume that it's constructive, of course)