\section{Traversal phase --- proofs}
For each operation called for by the traversal algorithms, we prove
that the commit generation preconditions are met.
WIP WHAT ABOUT PROVING ALL THE TRAVERSAL RESULTS
\subsection{Traversal Lemmas}
Firstly, some lemmas.
\statement{Tip Correct Contents}{
\tipcy \haspatch \pa E
\equiv
\pa E = \pc \lor \pa E \isdep \pc
}
\proof{
For $\pc = \pa E$, Tip Own Contents suffices.
For $\pc \neq \pa E$, Exclusive Tip Contents
gives $D \isin \tipcy \equiv D \isin \baseof{\tipcy}$
which by Correct Base $\equiv D \isin \tipcn$.
}
\subsection{Base Dependency Merge, Base Sibling Merge}
We do not prove that the preconditions are met. Instead, we check
them at runtime. If they turn out not to be met, we abandon
\alg{Merge-Base} and resort to \alg{Recreate-Base}.
TODO COMPLETE MERGE-BASE STUFF
WIP WHAT ABOUT PROVING ALL THE TRAVERSAL RESULTS
\subsection{Recreate Base Beginning}
To recap we are executing Create Base with
$L = \tipdy$ and $\pq = \pc$.
\subsubsection{Create Acyclic}
By Tip Correct Contents of $L$,
$L \haspatch \pa E \equiv \pa E = \pd \lor \pa E \isdep \pd$.
Now $\pd \isdirdep \pc$,
so by Coherence, and setting $\pa E = \pc$,
$L \nothaspatch \pc$. I.e. $L \nothaspatch \pq$. OK.
That's everything for Create Base. $\qed$
\subsection{Recreate Base Final Declaration}
\subsubsection{Base Only} $\patchof{W} = \patchof{L} = \pn$. OK.
\subsubsection{Unique Tips}
Want to prove that for any $\p \isin C$, $\tipdy$ is a suitable $T$.
WIP
\subsection{Tip Base Merge}
$L = W$, $R = \tipcn$.
TODO TBD
Afterwards, $\baseof{W} = \tipcn$.
\subsection{Tip Source Merge}
In fact, we do this backwards: $L = S$, $R = W$. Since $S \in \pcy$,
the resulting $C \in \pcy$ and the remaining properties of the Merge
commit construction are symmetrical in $L$ and $R$ so this is fine.
By the results of Tip Base Merge, $\baseof{W} = \tipcn$.
By Base Ends Supreme, $\tipcn \ge \baseof{S}$ i.e.
$\baseof{R} \ge \baseof{L}$.
Either $\baseof{L} = \baseof{M}$, or we must choose a different $M$ in
which case $M = \baseof{S}$ will suffice.