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[ $\set E_{\pc}$ ]
%$ \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 $\pc^{+/-}$.
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.

\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).

\end{basedescript}

\section{Ranking phase}

We run the following algorithm:
\begin{enumerate}
\item Set $\allpatches = \{ \}$.
\item Repeatedly:
\begin{enumerate}
\item Clear out the graph $\hasdirdep$ so it has no edges.
\item Execute $\alg{Rank-Recurse}(\pc_0)$
\item Until $\allpatches$ remains unchanged.
\end{enumerate}
\end{enumerate}

$\alg{Rank-Recurse}(\pc)$ is:
\begin{enumerate}

\item If we have already done $\alg{Rank-Recurse}(\pc)$ in this
ranking iteration, do nothing.  Otherwise:

\item Add $\pc$ to $\allpatches$ if it is not there already.

\item Set
$$
  \set S \iassign h(\pcn)
     \cup 
        \bigcup_{\p \in \allpatches}
        \bigcup_{H \in h(\pn) \lor H \in h(\py)}
         \{ \baseof{E} \; | \; E \in \pendsof{H}{\pcy} \}
$$

and $W \iassign w(h(\pcn))$

\item While $\exists_{S \in \set S} S \ge W$,
update $W \assign S$ and $\set S \assign \set S \, \backslash \{ S \}$

(This will often remove $W$ from $\set S$.  Afterwards, $\set S$
is a collection of heads to be merged into $W$.)

\item Choose an ordering of $\set S$, $S_i$ for $i=1 \ldots n$.

\item For each $S_i$ in turn, choose a corresponding $M_i$
such that $$
   M_i \le S_i \land \left[
   M_i \le W \lor \bigexists_{j<i} M_i \le S_j
   \right]
$$

\item Set $\Gamma \iassign \depsreqof{W}$.

If there are multiple candidates we prefer $M_i \in \pcn$
if available.

\item For each $i \ldots 1..n$, update our putative direct
dependencies:
$$
\Gamma \assign \alg{set-merge}\left[\Gamma,
 \left( \begin{cases} 
  M_i \in \pcn :     & \depsreqof{M_i} \\
  M_i \not\in \pcn : & \{ \}
 \end{cases} \right),
 \depsreqof{S_i}
 \right]
$$

TODO define $\alg{set-merge}$

\item Finalise our putative direct dependencies
$
\Gamma \assign g(\pc, \Gamma)
$

\item For each direct dependency $\pd \in \Gamma$,

\begin{enumerate}
\item Add an edge $\pc \hasdirdep \pd$ to the digraph (adding nodes
as necessary).
If this results in a cycle, abort entirely (as the function $g$ is
inappropriate; a different $g$ could work).
\end{enumerate}
\item Run $\alg{Rank-Recurse}(\pd)$.

\end{enumerate}

\subsection{Results of the ranking phase}

By the end of the ranking phase, we have recorded the following
information:

\begin{itemize}
\item
$ \allpatches, \hasdirdep $ and hence the completion of $\hasdirdep$
into the partial order $\hasdep$.

\item
For each $\pc \in \allpatches$,
the base branch starting point commit $W^{\pcn} = W$.

\item
For each $\pc$,
the direct dependencies $\Gamma^{\pc} = \Gamma$.

\item
For each $\pc$,
the ordered set of base branch sources $\set S^{\pcn} = \set S,
S^{\pcn}_i = S_i$
and corresponding merge bases $M^{\pcn}_i = M_i$.

\end{itemize}

\section{Traversal phase}

For each patch $C \in \allpatches$ in topological order by $\hasdep$,
lowest first:

\begin{enumerate}

\item Optionally, attempt $\alg{Merge-Base}(\pc)$.

\end{enumerate}


\section{Planning phase}

The results of the planning phase consist of: 
\begin{itemize*}
\item{ The relation $\hasdirdep$ and hence the partial order $\hasdep$. }
\item{ For each commit set $\pc$, a confirmed set of sources $\set S_{\pc}$. }
\item{ For each commit set $\pc$, the order in which to merge the sources
        $E_{\pc,j} \in \set E_{\pc}$. }
\item{ For each $E_{\pc,j}$ an intended merge base $M_{\pc,j}$. }
\end{itemize*}

We use a recursive planning algorith, recursing over Topbloke commit
sets (ie, sets $\py$ or $\pn$).  We'll call the commit set we're
processing at each step $\pc$.
At each recursive step 
we make a plan to merge all $\set E_{\pc} = \{ E_{\pc,j \ldots} \}$
and all the direct contributors of $\pc$ (as determined below)
into $\tipzc$, to make $\tipfc$.

We start with $\pc = \pl$ where $\pl = \patchof{L}$.


\subsection{Direct contributors for $\pc = \pcn$}

The direct contributors of $\pcn$ are the commit sets corresponding to
the tip branches for the direct dependencies of the patch $\pc$.  We
need to calculate what the direct dependencies are going to be.

Choose an (arbitrary, but ideally somehow optimal in
a way not discussed here) ordering of $\set E_{\pc}$, $E_{\pc,j}$
($j = 1 \ldots m$).
For brevity we will write $E_j$ for $E_{\pc,j}$.
Remove from that set (and ordering) any $E_j$ which
are $\le$ and $\neq$ some other $E_k$.

Initially let $\set D_0 = \depsreqof{\tipzc}$.
For each $E_j$ starting with $j=1$ choose a corresponding intended
merge base $M_j$ such that $M_j \le E_j \land M_j \le T_{\pc,j-1}$.
Calculate $\set D_j$ as the 3-way merge of the sets $\set D_{j-1}$ and
$\depsreqof{E_j}$ using as a base $\depsreqof{M_j}$.  This will
generate $D_m$ as the putative direct contributors of $\pcn$.

However, the invocation may give instructions that certain direct
dependencies are definitely to be included, or excluded.  As a result
the set of actual direct contributors is some arbitrary set of patches
(strictly, some arbitrary set of Topbloke tip commit sets).

\subsection{Direct contributors for $\pc = \pcy$}

The sole direct contributor of $\pcy$ is $\pcn$.

\subsection{Recursive step}

For each direct contributor $\p$, we add the edge $\pc \hasdirdep \p$
and augment the ordering $\hasdep$ accordingly.

If this would make a cycle in $\hasdep$, we abort . The operation must
then be retried by the user, if desired, but with different or
additional instructions for modifying the direct contributors of some
$\pqn$ involved in the cycle.

For each such $\p$, after updating $\hasdep$, we recursively make a plan
for $\pc' = \p$.



\section{Execution phase}

We process commit sets from the bottom up according to the relation
$\hasdep$.  For each commit set $\pc$ we construct $\tipfc$ from
$\tipzc$, as planned.  By construction, $\hasdep$ has $\patchof{L}$
as its maximum, so this operation will finish by updating
$\tipca{\patchof{L}}$ with $\tipfa{\patchof{L}}$.

After we are done with each commit set $\pc$, the
new tip $\tipfc$ has the following properties:
\[ \eqn{Tip Sources}{
  \bigforall_{E_i \in \set E_{\pc}} \tipfc \ge E_i
}\]
\[ \eqn{Tip Dependencies}{
  \bigforall_{\pc \hasdep \p} \tipfc \ge \tipfa \p
}\]
\[ \eqn{Perfect Contents}{
  \tipfc \haspatch \p \equiv \pc \hasdep \py
}\]

For brevity we will sometimes write $\tipu$ for $\tipuc$, etc.  We will start
out with $\tipc = \tipz$, and at each step of the way construct some
$\tipu$ from $\tipc$.  The final $\tipu$ becomes $\tipf$.

\subsection{Preparation}

Firstly, we will check each $E_i$ for being $\ge \tipc$.  If
it is, are we fast forward to $E_i$
--- formally, $\tipu = \text{max}(\tipc, E_i)$ ---
and drop $E_i$ from the planned ordering.

Then we will merge the direct contributors and the sources' ends.
This generates more commits $\tipuc \in \pc$, but none in any other
commit set.  We maintain
$$
 \bigforall_{\p \isdep \pc}
 \pancsof{\tipcc}{\p} \subset
   \pancsof{\tipfa \p}{\p}
$$
\proof{
 For $\tipcc = \tipzc$, $T$ ...WRONG WE NEED $\tipfa \p$ TO BE IN $\set E$ SOMEHOW
}

\subsection{Merge Contributors for $\pcy$}

Merge $\pcn$ into $\tipc$.  That is, merge with
$L = \tipc, R = \tipfa{\pcn}, M = \baseof{\tipc}$.
to construct $\tipu$.

Merge conditions:

Ingredients satisfied by construction.
Tip Merge satisfied by construction.  Merge Acyclic follows
from Perfect Contents and $\hasdep$ being acyclic.

Removal Merge Ends: For $\p = \pc$, $M \nothaspatch \p$; OK.
For $\p \neq \pc$, by Tip Contents,
$M \haspatch \p \equiv L \haspatch \p$, so we need only
worry about $X = R, Y = L$; ie $L \haspatch \p$,
$M = \baseof{L} \haspatch \p$.
By Tip Contents for $L$, $D \le L \equiv D \le M$.  OK.~~$\qed$

WIP UP TO HERE

Addition Merge Ends: If $\py \isdep \pcn$, we have already
done the execution phase for $\pcn$ and $\py$.  By
Perfect Contents for $\pcn$, $\tipfa \pcn \haspatch \p$ i.e.
$R \haspatch \p$.  So we only need to worry about $Y = R = \tipfa \pcn$.
By Tip Dependencies $\tipfa \pcn \ge \tipfa \py$.
And by Tip Sources $\tipfa \py \ge $

want to prove $E \le \tipfc$ where $E \in \pendsof{\tipcc}{\py}$

$\pancsof{\tipcc}{\py} = $


computed $\tipfa \py$, and by Perfect Contents for $\py$


with $M=M_j, L=T_{\pc,j-1}, R=E_j$,
and calculate what the resulting desired direct dependencies file
(ie, the set of patches $\set D_j$)
would be.  Eventually we 

So, formally, we select somehow an order of sources $S_i$.  For each 


Make use of the following recursive algorithm, Plan 




 recursively make a plan to merge all $E = \pends$

Specifically, in
