When we are trying to do a merge of some kind, in general,
we want to merge some source commits $S_0 \ldots S_n$.
We'll write $S_0 = L$.  We require that $L$ is the current git ref
for $\patchof{L}$.

\stdsection{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 Topbloke commit set $\pc$ has as a direct contributors the
commit set $\p$.  This is an acyclic relation.

\item[ $\p \hasdep \pq$ ]
The commit set $\p$ has as direct or indirect contributor the commit
set $\pq$.
Acyclic; the completion of $\hasdirdep$ into a
partial order.

\item[ $\set E_{\pc}$ ]
$ \bigcup_i \pendsof{S_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}

\section{Planning phase}

The planning phase computes: 
\begin{itemize*}
\item{ The relation $\hasdirdep$ and hence the ordering $\hasdep$. }
\item{ For each commit set $\pc$, the order in which to merge
        $E_{\pc,j} \in \set E_{\pc}$. }
\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
$\tipfa{\patchof{L}}$.

After we are done, the result has the following properties:
\[ \eqn{Tip Inputs}{
  \bigforall_{E_i \in \set E_{\pc}} \tipfc \ge E_i
}\]
\[ \eqn{Tip Dependencies}{
  \bigforall_{\pc \hasdep \p} \tipfc \ge \tipfa \p
}\]
\[ \eqn{Perfect Contents}{
  \tipfc \haspatch \p \equiv \pc \hasdep \py
}\]

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 XXX FIXME IS THIS THE BEST THING?
$$
 \bigforall_{\p \isdep \pc}
 \pancsof{\tipcc}{\p} \subset \left[
   \tipfa \p \cup
   \bigcup_{E \in \set E_{\pc}} \pancsof{E}{\p}
  \right]
$$

\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 $\p = \pc$, $M \nothaspatch \p$.
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 $L$, $D \le L \equiv D \le M$.  $\qed$

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 Inputs $\tipfa \py \ge $


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
