+Here we describe the update algorithm. This is responsible for
+refreshing patches against updated versions of their dependencies,
+for merging different versions of the various braches created by
+distributed development, and for implementing decisions to add and
+remove dependencies from patches.
+
+Broadly speaking the update proceeds as follows: during the Ranking
+phase we construct the intended graph of dependencies between patches
+(and incidentally select a merge order for the base branch of each
+patch). Then during the Traversal phase we walk that graph from the
+bottom up, constructing for each patch by a series of merges and other
+operations first a new base branch head commit and then a new tip
+branch head commit. These new head commits are maximums - that is,
+each has as ancestors all of its branches' sources and indeed all
+relevant commits in that branch.
+
+We have two possible strategies for constructing new base branch
+heads: we can either Merge (works incrementally even if there the
+patch has multiple dependencies, but may sometimes not be possible) or
+we can Regenerate (trivial if there is a single dependency, and is
+always possible, but may involve the user re-resolving conflicts if
+there are multiple dependencies).
+
+\section{Notation}
+
+\begin{basedescript}{
+\desclabelwidth{5em}
+\desclabelstyle{\nextlinelabel}
+}
+\item[ $\depsreqof{K}$ ]
+The set of direct dependencies (in the form $\py$)
+requested in the commit $K$ ($K \in \pn$) for the patch $\p$.
+
+\item[ $\pc \hasdirdep \p$ ]
+The patch $\pc$ has as a direct dependency the
+patch $\p$. This is an acyclic relation.
+
+\item[ $\p \hasdep \pq$ ]
+The patch $\p$ has as direct or indirect dependency the
+patch $\pq$.
+Acyclic; the completion of $\hasdirdep$ into a
+partial order.
+
+\item[ $\pendsof{\set J}{\p}$ ]
+Convenience notation for
+the $\le$-maximal elements of $\bigcup_{J \in \set J} \pendsof{J}{\p}$
+(where $\set J$ is some set of commits).
+
+\item[ $\pendsof{\set X}{\p} \le T$ ]
+Convenience notation for
+$\bigforall_{E \in \pendsof{\set X}{\p}} E \le T$
+
+\item[ $\allsrcs$ ]
+$\bigcup_{\p \in \allpatches} \set H^{\pn} \cup \set H^{\py}$.
+All the input commits to the update algorithm. (See below.)
+
+\item[ $\set H^{\pc^{_=/-}}$ ]
+
+The existing head commit(s) $\set H$ of the branch $\pc^{+/-}$.
+These are the heads which will be merged and used in this update.
+This will include the current local and remote git refs, as desired.
+Obtained from the function $h$ (see below).
+
+\item[ $W$ ]
+
+During the Traversal phase, when we generate new commits, the working
+head of the branch we are working on. Starts at $W_0$, updated as the
+algorithm progresses, and ultimately used to set $T$.
+
+%\item[ $\set E_{\pc}$ ]
+%$ \bigcup_i \pendsof{S_{\pc,i}}{\pc} $.
+%All the ends of $\pc$ in the sources.
+
+%\item[ $ \tipzc, \tipcc, \tipuc, \tipfc $ ]
+%The git ref for the Topbloke commit set $\pc$: respectively,
+%the original, current, updated, and final values.
+
+\end{basedescript}
+
+\stdsection{Inputs to the update algorithm}
+
+\begin{basedescript}{
+\desclabelwidth{5em}
+\desclabelstyle{\nextlinelabel}
+}
+\item[ $\pc_0$ ]
+The topmost patch which we are trying to update. This and
+all of its dependencies will be updated.
+
+\item[ $h : \pc^{+/-} \mapsto \set H^{\pc^{+/-}}$ ]
+Function for getting the existing heads $\set H$ of the branch $\pc^{+/-}$.
+
+\item[ $w : \pc^{+/-} \mapsto W_0^{\pc^{+/-}}$ ]
+
+Function for getting the existing local head of the branch
+$\pc^{+/-}$. I.e., the current value of the branch ref for $\pc^{+/-}$.
+$W_0^{\pc^{+/-}} \in \set H$.
+
+\item[ $g : \pc, \Gamma \mapsto \Gamma'$ ]
+Function to allow explicit adjustment of the direct dependencies
+of $\pc$. It is provided with a putative set of direct dependencies
+$\Gamma$ computed as an appropriate merge of the dependencies requested by the
+sources and should return the complete actual set $\Gamma'$ of direct
+dependencies to use. This allows the specification of any desired
+(acyclic) relations $\hasdirdep$ and $\hasdep$.
+
+\end{basedescript}
+
+\stdsection{Important variables and values in the update algorithm}
+
+\begin{basedescript}{
+\desclabelwidth{5em}
+\desclabelstyle{\nextlinelabel}
+}
+\item[ $\Gamma_{\pc}$ ]
+The desired direct dependencies of $\pc$, a set of patches.
+
+\item[ $\allpatches$ ]
+The set of all the patches we are dealing with (constructed
+during the update algorithm).
+
+\item[ $\tipcn, \tipcy$ ]
+The new tips of the git branches $\pcn$ and $\pcy$, containing
+all the appropriate commits (and the appropriate other patches),
+as generated by the Traversal phase of the update algorithm.
+
+\item[ $\allreach$ ]
+The set of all reachable commits.
+
+A reachable commit is one we might refer to explicitly in any of these
+algorithms, and any ancestor of such a commit. We explicitly
+enumerate all of the input commits ($\allsrcs$), so the reachable
+commits are originally the input commits and their ancestors.