Thesis II: Explaining Chains of Inequalities

This post is on how I like to explain long chains of inequalities in papers, and the changes I made to kaobook to move these explanations to the margins.

My usual setup

Usually, I like to tag a sequence of inequalities in my papers with explanations of how each one arises. For example, the page below mentions the lemmas or properties that are used to obtain one line of the chain of inequalities from the previous line.

In papers, I like to explain how a chain of inequalities arise as in this figure.

The first chain at the top of the page is obtained using something like the following

\begin{align*}
\x{t}
&= \sum_e \xe{t}
= [...] \\
&\le [...] \tag{by \cref{lem:ApxLoad}} \\
&\le [...] \tag{by \cref{lem:OracleReturn}}\\
&\le [...] \tag{\qedhere}
\end{align*}

The final \qedhere is just to make sure that the box that finishes the proof environment stays on the last line of the align* environment and doesn’t create a new blank line in the proof which, imo, looks ugly.

Directly using this in kaobook

When implemented directly in kaobook, it looks like this.

First attempt at using the above strategy directly in kaobook.

The point of kaobook is to use the margins to explain things, so this was a great opportunity to use the explanations I already had in my papers directly to fill up the margins. For this, I added the following in my headers (before \begin{document}), in particular I separately included it in a formatting.tex file that I include before the \begin{document}. I stole this by frankensmashing various stack exchange answers together.

\makeatletter

\newcommand{\leqnomode}{\tagsleft@true\let\veqno\@@leqno}
\newcommand{\reqnomode}{\tagsleft@false\let\veqno\@@eqno}

\if@twoside%
\def\maketag@@@#1{%
\iftagsleft@%
\hbox{\llap{\kern-\marginparsep\m@th\normalfont#1}\kern1sp}%
\else%
\hbox{\rlap{\kern\marginparsep\m@th\normalfont#1}\kern1sp}%
\fi%
}
\def\taggableenvs{equation,equation*,align,align*,alignat,alignat*}
\@for\env:=\taggableenvs\do{%
  \AtBeginEnvironment{\env}{\Ifthispageodd{\reqnomode}{\leqnomode}}%
}
\else%
\def\maketag@@@#1{\hbox{\rlap{\kern\marginparsep\m@th\normalfont#1}\kern1sp}}%
\fi
\makeatother

It checks if you are in two-sided mode, and moves the tag outside the correct margin. Then, the same page looks the like following in two-sided and one-sided mode respectively.

Equation tags for the same page in verso and recto with the above code in kaobook.

I haven’t had the misfortune of trying to read this for the first time to see if it helps explain things better, but I think (more like hope) it does. Maybe once I forget the things here I’ll come back and see if it helps.