How to (progressively) number everything (paragraphs, definitions, theorems, remarks, examples, etc.) within a section

Question

My aim is to format a text according to the following guidelines:

[as a reference, please cf the pictures below, which are taken from Jean-Michel Bony, Cours d'analyse, Ellipses ed. 2016, actually from its 9th chapter]

1. the main numbered "units" are chapters and sections, and they both work as usual;
2. each section is made up of "parts" which should be numbered progressively in the form: chapter#.section#.#.
3. as a "part" of a section, one should be able to use different environments, such as theorem-like environments (for theorems, propositions, corollaries, lemmas, definitions, remarks, etc.), paragraph-like environments, etc.; each environment should be able to keep its "natural properties": for example the spacing before and after (so that the space before a paragraph is greater than the one before a theorem), or the title format (so that a "theorem" should start with Theorem thm# (thmname). (with "Theorem" before the number), everything bold; a paragraph with par#. parname, everything bold; an "exercise" with Exercise exer#., italic; an "exemple" with exem#. Exemple: exemname, italic; etc.; each part number: thm#, par#, exer#, exem# should follow rule 2);
4. subsections are "numbered" by letters;
5. one should not be forced to number everything (such as in the last picture, where the "parts" Conjugaison complexe et parité, Translation and Dilatation appear to be spaced as each of them were a paragraph, but still they're not numbered).

Is there a simple way to do so? At least is there a simple way to set the same "counter" for everything?

PS: I apologize in advance whether my question is somewhat inappropriate and or ill-formulated; this is my first question ever, please be patient…

Answer 1

I think that for theorems, exercises and other environments of the amsthm package, you will find solutions here: Theorem numbering as chapter.section.subsection.theorem number. More formatting options for theorem titles will be in amsthm's doc.

"Parts" of sections should be "subsections" to me. You can change their numbering format (A, B, C) by adding this to your preamble: \renewcommand{\thesubsection}{\Alph{subsection}}. To add a dot at the end of all ((sub)sub)sections's number, check this out: https://tex.stackexchange.com/a/24440/158599.

If you don't want to number a "part" (chapter, section, subsection, etc.), just use its starred version. For instance: \section*{My section} instead of \section{My section}. For more advanced formatting, I recommend to use the titlesec package.

However, the "Dilatation", "Translation" paragraphs etc. rather seem to be unnumbered amsthm environments. Check out the amsthm's doc to see how to create your unnumbered environment with your customized theorem style (p. 3 and § 4.3, p. 9).