[Tex/LaTex] Which packages/practices are relevant for writing Structured Derivations? (similar to Dijkstra’s calculational style of proofs)

Question

Does anyone know of any packages that support this style of writing proofs?

The only things that I've found are: tex-ewd, a bunch of plain TeX macros on CTAN written specifically with Dijkstra in mind; and a Windows-only version of LyX that's been extended with support with structured derivations. (It's too bad that the LyX-SD is limited to Windows, because I'd guess that writing proofs in LyX would be wonderful)

In the case that there's nothing out there and that I have to roll my own, are there any packages that are similar enough that I could learn the "right way" to go about making a Structured Derivations package for LaTeX?

And I offer a pre-emptive "thank you" 🙂 I'll be really grateful for any help.

If you need more information to be helpful (or even if I've just piqued your interest) here are some good resources:

• the Wikipedia page
• A sort of home page for an associated project.
• A Powerpoint presentation with lots of examples. And colors! http://www.risc.jku.at/conferences/cadgme2009/Back/TeachingMathLinzShort.pdf
• The paper containing the details and formal grammar. The example that I attached to this question is a screenshot from this paper. http://www.tara.tcd.ie/bitstream/2262/41529/1/PEER_stage2_10.1007%252Fs00165-009-0136-5.pdf

Answer 1

I'd be tempted to use a sequence of itemize and enumerate environment, using the enumitem package to help with formatting

In the code below I've used the resume feature provided by the enumitem package to help keep the numbering of the enumerate environment without having to hardcode any of the numbers.

I've also used the optional argument of the \item command, which can be used, for example

\item[$\Vdash$]

If you plan to use this a lot though, you could always make this into a \newcommand

\documentclass{article}
\usepackage{geometry}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{enumitem}

% global settings for enumerate environment
\setlist[enumerate]{label*=(\arabic*),leftmargin=0mm}

% itemize environment, when at nested depths 2, 3, 4
\setlist[itemize,2]{label=\textbullet,leftmargin=1cm}
\setlist[itemize,3]{label=\textbullet,leftmargin=1cm}
\setlist[itemize,4]{label=\textbullet,leftmargin=1cm}

\begin{document}
\begin{itemize}
    \item Show that $f$ is continuous at $x_0$, when
        \begin{enumerate}
            \item\label{item:range} $f(x)=\sqrt{x}$ when $x\geq 0$, and
            \item\label{item:pos} $x_0>0$
        \end{enumerate}
    \item[$\Vdash$] $f$ is continuous at $x_0$
    \item[$\equiv$] $\{\text{definition of continuity}\}$

            $(\forall\epsilon>0\cdot \exists\delta>0\cdot\forall x\geq 0\cdot |x-x_0|<\delta\Rightarrow|f(x)-f(x_0)|<\epsilon)$
    \item[$\equiv$] $\{\text{choose arbitrary }\epsilon \text{generilization rule}\}$
    \begin{itemize}
        \item Show that $(\exists\delta>0\cdot\forall x\geq 0\cdot |x-x_0|<\delta\Rightarrow |f(x)-f(x_0)|<\epsilon)$, when
        \begin{enumerate}[resume]
            \item $\epsilon>0$
        \end{enumerate}
        \item[$\Vdash$] $\{\text{find a suitable value for } \delta\text{ witness rule}\}$
        \begin{itemize}
            \item Show that $\delta>0\wedge(\forall x\geq 0\cdot|x-x_0|<\delta\Rightarrow|f(x)-f(x_0)|<\epsilon)$, when
            \begin{enumerate}[resume]
                \item $\delta=?$ $\sharp$ $\delta=\epsilon\cdot\sqrt{x_0}$
            \end{enumerate}
            \item[$\Vdash$] $\{\text{show the two conjuncts separately, use generalization rule for second conjunct}\}$
            \begin{itemize}
                \item Show that $|f(x)-f(x_0)|<\epsilon$, when
                \begin{enumerate}[resume]
                    \item\label{item:xnonneg} $x\geq 0$, and
                    \item $|x-x_0|<\delta$
                \end{enumerate}
                \item[$\Vdash$] $|f(x)-f(x_0)|$
                \item[$=$] $\{$assumption \ref{item:range}, $f(x)$ and $f(x_0)$ are defined by assumptions \ref{item:xnonneg} and \ref{item:range}$\}$
                \item[] $|\sqrt{x}-\sqrt{x_0}|$
                \item[$=$] $\{$extend with conjugate value for $\sqrt{x}-\sqrt{x_0}$, i.e., with $\sqrt{x}+\sqrt{x_0}>0$$\}$
                \item[] $\dfrac{|x-x_0|}{\sqrt{x}+\sqrt{x_0}}$
                \item[$\leq$] $\{$make divisor smaller, $\sqrt{x}\geq 0$ $\}$
            \end{itemize}
        \end{itemize}
    \end{itemize}
\end{itemize}
\end{document}