Toggling verbosity in LaTeX derivations

March 18, 2024

\[\newcommand{\Fcal}{\mathcal{F}} \newcommand{\RR}{\mathbb{R}} \newcommand{\EE}{\mathbb{E}}\]

I find it helpful to write proofs so that they can optionally be made more verbose (e.g. to fill in certain details in a derivation). There’s a very simple trick you can use to accomplish this and it’s already built in to LaTeX — no additional packages needed. This idea was taken from this reddit comment.

Put the following in your preamble:

% Verbose LaTeX doc.
\newif\ifverbose % Define a new Boolean value "verbose".
% Initially it is false.
\verbosefalse % set verbose to true/false here.

Any content you would like to restrict to the “verbose” version of your paper should then be wrapped in \ifverbose <content> \if.

Demo

For the sake of example, consider the following short proof of the second Borel-Cantelli lemma (the proof details do not matter for the purposes of this post).

Claim: If \((E_k)_{k=1}^\infty\) are independent events and \(\sum_{k=1}^\infty \Pr(E_k) = \infty\), then \[ \Pr \left ( \limsup_{n \to \infty} E_n \right ) = 1. \] Proof. Writing out the probability of \(\limsup_{n} E_n\), we have $$\begin{align} \Pr \left ( \limsup_{n \to \infty} E_n \right )&\equiv \Pr \left ( \bigcap_{m=1}^\infty \bigcup_{k = m}^\infty E_k \right ) \\ &{\color{lightgray}{= \lim_{m \to \infty}\Pr \left ( \bigcup_{k = m}^\infty E_k \right )}} \\ & \color{lightgray}{=1- \lim_{m \to \infty}\Pr \left ( \bigcap_{k = m}^\infty E_k^c \right ) }\\ & =1- \lim_{m \to \infty} \lim_{K \to \infty} \Pr \left ( \bigcap_{k = m}^K E_k^c \right ) \\ &= 1- \lim_{m \to \infty} \lim_{K \to \infty} \prod_{k=m}^K \left [1-\Pr \left ( E_k \right ) \right ]\\ &\geq 1- \lim_{m \to \infty} \lim_{K \to \infty} \exp \left \{ -\sum_{k=m}^K \Pr(E_k) \right \}\\ &= \color{lightgray}{1- \lim_{m \to \infty} \exp \left \{ -\sum_{k=m}^\infty \Pr(E_k) \right \}}\\ &= \color{lightgray}{1- \lim_{m \to \infty} 0}\\ &= 1, \end{align}$$ which completes the proof. \(\square\)

If the above derivations are written with the following TeX, then toggling between \verbosefalse and verbosetrue in your preamble will remove/include the lines in gray above, respectively.

\begin{align}
  \Pr \left ( \limsup_{n \to \infty} E_n \right )
  &\equiv \Pr \left ( \bigcap_{m=1}^\infty \bigcup_{k = m}^\infty E_k \right ) \\
  \ifverbose %BEGIN VERBOSE
  &= \lim_{m \to \infty}\Pr \left ( \bigcup_{k = m}^\infty E_k \right ) \\
  & =1- \lim_{m \to \infty}\Pr \left ( \bigcap_{k = m}^\infty E_k^c \right ) \\
  \fi %END VERBOSE
  & =1- \lim_{m \to \infty} \lim_{K \to \infty}  \Pr \left ( \bigcap_{k = m}^K E_k^c \right ) \\
  &= 1- \lim_{m \to \infty} \lim_{K \to \infty} \prod_{k=m}^K \left [1-\Pr \left ( E_k \right ) \right ]\\
  &\geq 1- \lim_{m \to \infty} \lim_{K \to \infty} \exp \left \{ -\sum_{k=m}^K \Pr(E_k) \right \}\\
  \ifverbose %BEGIN VERBOSE
  &= 1- \lim_{m \to \infty} \exp \left \{ -\sum_{k=m}^\infty \Pr(E_k) \right \}\\
  &= 1- \lim_{m \to \infty} 0\\
  \fi %END VERBOSE
  &= 1,
\end{align}

That’s all there is to it.

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Bonus: if you use yasnippet in emacs or similar tools in neovim etc., it is very handy to have a snippet that creates the following.

\ifverbose %BEGIN VERBOSE
  %<content goes here>
\fi %END VERBOSE