patch-theory.tex

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\documentclass{article}
%\usepackage{color}

\usepackage{verbatim}
\usepackage{hyperref}
\usepackage{fancyvrb}
\newenvironment{code}{\comment}{\endcomment}
% \newenvironment{code}{\color{blue}\verbatim}{\endverbatim}

\newcommand{\commutes}{\longleftrightarrow}

\begin{document}


\section{Patch properties}

Darcs is built on a hierarchy of patch types.  At the lowest level are
``primitive'' patches, and from these building blocks, a whole hierarchy of
patch types are built.  Each of these patch types must support a number of
functions, which must obey a number of laws.

\subsection{Properties of identity}

\newtheorem{prp}{Property}

\begin{prp}[Identity commutes trivially]
  The identity patch must commute with any patch without modifying said patch.
\end{prp}


\begin{prp}[Inverse doesn't commute]
  A patch and its inverse will always commute, unless that patch is an
  identity patch (or an identity-like patch that has no effect).
\end{prp}


\subsection{Commute properties}

\begin{prp}[Recommute]
  $AB \commutes B'A'$ if and only if $B'A' \commutes AB$
\end{prp}


\begin{prp}[Commute inverses]
  $AB \commutes B'A'$ if and only if $B^{-1}A^{-1} \commutes A'^{-1}B'^{-1}$
\end{prp}


\begin{prp}[Patch and inverse]
  If $AB \commutes B'A'$ then $A^{-1}B' \commutes BA'^{-1}$
\end{prp}

This property is only true of primitive patches.


\begin{prp}[Permutivity]
  (to be added...)
\end{prp}


\end{document}