creating theorem-like environments within 'notes' (see example)

Question

I would like to create the following style to write mathematics:

In fact the above I created myself but some numbers and spacing I had to set manually (and I would like this to be automatic of course).

So the following are important to me:

• I want the page to consists of 'notes' that are numbered like 1.1, 1.2 etc., where the first number refers to the section/paragraph (so in my example there are four notes in section 1). With every note I want to have the possibility to add a title that appears boldface (for example note 1.1).
• Within a note I want to have the possibility to add a theorem-like environment (in my example you can find 'remark', 'lemma' etc.). Preferably I would like to have one single command for this, and then the possibility to name it the way I want (so lemma or proposition or maybe something else). Also I want them to be numbered within the note they occur (see example), unless I choose them to be non-numbered (so in my example the 'remark' is not numbered).
• Between these theorem-like environments I want to include some small space (I chose \medskip in the example). But I don't want this space to get doubled whenever I have two of these environments one after the other.
• Last but not least: I want the title of a section, the first word of a note, and the heading of the theorem-like environments to be aligned (say one paragraph indent).

Any help or advice is appreciated. Of course, partial solutions are also welcome and I can provide details in case something is not clear.

Thanks a lot in advance! AYK

Answer 1

This might be an approximation to your wish; however, I find the numbering scheme too fussy: I see no reason for statements to be numbered under notes.

If \newnote is immediately followed by \begin, a theorem-like statement is assumed.

\documentclass{article}

\usepackage{amsthm}
\usepackage{thmtools}
\usepackage{chngcntr}

\usepackage{lipsum}

\makeatletter
\newif\ifnotebegin
\newcounter{note}[section]
\renewcommand{\thenote}{\thesection.\arabic{note}}
\newcommand{\newnote}{\@ifnextchar\begin\xnew@note\new@note}
\newcommand{\new@note}[1][]{%
  \par\addvspace{\topsep}%
  \refstepcounter{note}%
  \global\notebeginfalse
  \noindent\formatnote{#1}%
}
\newcommand{\formatnote}[1]{%
  \makebox[\parindent][l]{\normalfont\bfseries\thenote}%
  \if\relax\detokenize{#1}\relax\else\textbf{#1}\ \fi
}
\newcommand{\xnew@note}{%
  \refstepcounter{note}%
  \global\notebegintrue
}
\newcommand\notenumberorindent{%
  \ifnotebegin
    \formatnote{}%
  \else
    \hspace*{\parindent}%
  \fi
  \global\notebeginfalse
}
\makeatother

\declaretheoremstyle[
  headfont=\normalfont\itshape,
  notefont=\normalfont\itshape,
  bodyfont=\normalfont\itshape,
  notebraces={\textnormal{(}}{\textnormal{)}},
  headformat=\notenumberorindent\NAME\ \textup{({\bfseries\NUMBER})}\NOTE,
  headpunct=\ ---,
]{ayk}

\declaretheorem[
  style=ayk,
  within=note,
  name=Lemma,
]{lemma}
\declaretheorem[
  style=ayk,
  sibling=lemma,
  name=Proposition,
]{proposition}

\counterwithin{equation}{lemma}
\AtBeginDocument{\setlength{\parindent}{2em}}

\begin{document}

\section{First notes}

\newnote[Hello]
\lipsum[2]

\begin{lemma}[Some Lemma]
A lemma about something
A lemma about something
A lemma about something
A lemma about something
\end{lemma}

\begin{lemma}
Some other lemma Bla bla
\end{lemma}

\begin{proposition}
Some Proposition
\begin{equation}
  E = mc^2 
\end{equation}
\end{proposition}

\newnote
This note has no title. \lipsum[2]

\begin{lemma}[Some Lemma]
A lemma about something
A lemma about something
A lemma about something
A lemma about something
\end{lemma}

\begin{lemma}
Some other lemma Bla bla
\end{lemma}

\begin{proposition}
Some Proposition
\begin{equation}
  E = mc^2 
\end{equation}
\end{proposition}

\newnote\begin{lemma}[Some Lemma]
A lemma about something
A lemma about something
A lemma about something
A lemma about something
\end{lemma}

\begin{lemma}
Some other lemma Bla bla
\end{lemma}

\begin{proposition}
Some Proposition
\begin{equation}
  E = mc^2 
\end{equation}
\end{proposition}

\end{document}

Answer 2

Only preliminary solution... indentation not done yet.

\newtheorem{foo}{Foo}[section] will define an environment named foo, titled Foo and a counter named (you guess it -- foo ;-)), which is reset each time the section counter is stepped.

\newtheorem{otherfoo}[foo]{Otherfoo} will define an environment named otherfoo, titled Otherfoo and no counter -- it will use the counter foo instead, i.e. foo and otherfoo share the counters.

Since the counter formats should be changed, this has to be done manually, counterwithin from chngcntr is no help, here, in my point of view.

\documentclass{article}

%\usepackage{chngcntr}
\usepackage{amsthm}

%\usepackage{xpatch}

\newtheorem{notes}{Note}[section]
\newtheorem{lemma}{\normalfont\textit{Lemma}}[notes]
\newtheorem{proposition}[lemma]{\normalfont\textit{Proposition}}

\renewcommand{\thelemma}{(\thenotes.\arabic{lemma})}
\renewcommand{\theequation}{\thenotes.\arabic{lemma}.\arabic{equation}}


\usepackage{blindtext}
\begin{document}

\section{First notes}

\begin{notes}{Hello}
\blindtext
\begin{lemma}{Some Lemma}
\blindtext
\end{lemma}

\begin{lemma}{Some other lemma}
Bla bla
\end{lemma}

\begin{proposition}{Some Proposition}
\begin{equation}
  E = mc^2 
\end{equation}
\end{proposition}

\begin{proposition}{Some Proposition}
\begin{equation}
  E^2 = (mc^2)^2 +\left(pc\right)^2
\end{equation}

\end{proposition}

\end{notes}


\end{document}

Answer 3

A solution based on ntheorem, mathtools and etoolbox:

\documentclass[twoside,a4paper,11pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{amsfonts}
\usepackage[hmargin=3cm]{geometry}
\setlength\parindent{1.1cm}
\usepackage{microtype}
\usepackage{mathtools, amssymb}
\newtagform{bold}{\bfseries(}{)}
\usetagform{bold}
\usepackage{etoolbox}
\usepackage{titlesec}
\titleformat{\section}{\Large\bfseries}{\rlap{§\thesection.}}{1.1cm}{\lsstyle\MakeUppercase}
\titleformat{\subsection}[runin]{\normalsize\bfseries}{\rlap{\thesubsection.}}{1.1cm}{\mbox{}}
\usepackage[thref, amsmath, thmmarks]{ntheorem}%
\makeatletter
\newtheoremstyle{myplain}%
{\item[\hskip\labelsep \theorem@headerfont ##1\ \textup{\bfseries(##2)}\ \theorem@separator]}%
{\item[\hskip\labelsep \theorem@headerfont ##1\ \textup{\bfseries(##2)}\ (##3)\theorem@separator]}
\makeatother
\theoremstyle{myplain}
\theoremheaderfont{\itshape}
\theorembodyfont{\itshape}
\theoremseparator{\textemdash}
\newtheorem{lemma}{\hspace*{1.1cm}Lemma}[subsection]%
\newtheorem{prop}[lemma]{Proposition}
\newtheorem{defn}[lemma]{Definition}

\theoremstyle{nonumberplain}
\theorembodyfont{\upshape}
\theoremseparator{ \textemdash}
\newtheorem{remark}{\hspace*{1.1cm}Remark}
\theoremsymbol{\ensuremath{\diamondsuit}}
\newtheorem{proof}{\hspace*{1.1cm}Proof}

\usepackage{lipsum}
\usepackage{cleveref}

\newcommand\note[1][]{\subsection{#1}\ifblank{#1}{\hskip-\labelsep}{\relax}}% 

\usepackage{chngcntr}
\counterwithin{equation}{lemma}

\begin{document}

\section{Some section}
\note[Some optional title]
\lipsum[1]

\note[]
\lipsum[3]

\begin{remark}
\lipsum[5]
\end{remark}

\note
\lipsum[4]
\begin{lemma}\label{lemma1}\lipsum[3]
\end{lemma}

\begin{proof}
A brilliant proof. Blah blah blah. A brilliant proof. Blah blah blah. A brilliant proof. Blah blah blah. A brilliant proof with an unlikely final formula
 \[n! \sim_\infty \sqrt{2\pi n}\Bigl(\frac{n}{\mathrm e}\Bigr)^n.\] %
\end{proof}
\lipsum[5]
\begin{prop}A very interesting proposition, with an amazing formula:
\begin{equation}\label{quadratic}
  x_0 ,x_1 =\frac{-b\pm\sqrt{b^2-4ac}}{2a}
\end{equation}
\end{prop}

\end{document}