+++ /dev/null
-\section{Traversal phase --- algorithm}
-
-(In general, unless stated otherwise below, when we generate a new
-commit $C$ using one of the commit kind recipies, we update
-$W \assign C$. In any such case where we say we're going to Merge
-with $L = W$, if $R \ge W$ we do not Merge but instead simply set
-$W \assign R$.
-
-For each commit generation operation called for by the traversal
-algorithms, we prove that the commit generation preconditions are met.)
-
-For each patch $\pc \in \allpatches$ in topological order by $\hasdep$,
-lowest first:
-
-\begin{enumerate}
-
-\item Optionally, attempt
- $\alg{Merge-Base}(\pc)$. This may or may not succeed.
-
-\item If this didn't succeed, or was not attempted, execute
- $\alg{Recreate-Base}(\pc)$.
-
-\item Then in any case, execute
- $\alg{Merge-Tip}(\pc)$.
-
-\end{enumerate}
-
-After processing each $\pc$ we will have created $\tipcn$ and $\tipcy$
-such that:
-
-\statement{Correct Base}{
- \baseof{\tipcy} = \tipcn
-}
-\statement{Base Correct Contents}{
- \tipcn \haspatch \pa E
- \equiv
- \pa E \isdep \pc
-}
-\statement{Tip Covers Reachable}{
- \tipcy \ge \pendsof{\allreachof{\pcy}}{\pcy}
-}
-\statement{Base Covers Inputs' Bases}{
- \bigforall_{E \in \pendsof{\allsrcs}{\pcy}} \tipcn \ge \baseof{E}
-}
-\statement{Base Covers Base Inputs}{
- \bigforall_{H \in \set H^{\pn}} \tipcn \ge H
-}
-
-\subsection{Reachability and coverage}
-
-We ensure Tip Covers Reachable as follows:
-
-\begin{itemize}
-\item We do not generate any commits $\in \py$ other than
- during $\alg{Merge-Tip}(\py)$;
-\item So at the start of $\alg{Merge-Tip}(\py)$,
- $ \pendsof{\allreach}{\py} = \pendsof{\allsrcs}{\py} $
-\item $\alg{Merge-tip}$ arranges that when it is done
- $\tippy \ge \pendsof{\allreach}{\py}$ --- see below.
-\end{itemize}
-
-A corrolary is as follows:
-\statement{Tip Covers Superior Reachable} {
- \bigforall_{\pd \isdep \pc}
- \tipdy \ge \pendsof{\allreachof{\pcy}}{\pdy}
-}
-\proof{
- No commits $\in \pdy$ are created other than during
- $\alg{Merge-Tip}(\pd)$, which runs (and has thus completed)
- before $\alg{Merge-Tip}(\pcy)$
- So $\pendsof{\allreachof{\pcy}}{\pdy} =
- \pendsof{\allreachof{\pdy}}{\pdy}$.
-}
-
-\subsection{Traversal 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{$\alg{Merge-Base}(\pc)$}
-
-This algorithm attempts to construct a suitably updated version of the
-base branch $\pcn$ using some existing version of $\pcn$ as a starting
-point.
-
-It should be executed noninteractively. Specifically, if any step
-fails with a merge conflict, the whole thing should be abandoned.
-This avoids asking the user to resolve confusing conflicts. It also
-avoids asking the user to pointlessly resolve conflicts in situations
-where we will later discover that $\alg{Merge-Base}$ wasn't feasible
-after all.
-
-If $\pc$ has only one direct dependency, this algorithm should not be
-used as in that case $\alg{Recreate-Base}$ is trivial and guaranteed
-to generate a perfect answer, whereas this algorithm might involve
-merges and therefore might not produce a perfect answer if the
-situation is complicated.
-
-For \alg{Merge-Base} 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}.
-
-Initially, set $W \iassign W^{\pcn}$.
-
-\subsubsection{Bases and sources}
-
-In some order, perhaps interleaving the two kinds of merge:
-
-\begin{enumerate}
-
-\item For each $\hasdep$-maximal $\pd \isdirdep \pc$, find a merge base
-$M \le W,\; \le \tipdy$ and merge $\tipdy$ into $W$.
-That is, use $\alg{Merge}$ with $L = W,\; R = \tipdy$.
-
-\item For each $S \in S^{\pcn}_i$, merge it into $W$.
-That is, use $\alg{Merge}$ with $L = W,\; R = S,\; M = M^{\pcn}_i$.
-(Base Sibling Merge.)
-
-\end{enumerate}
-
-\subsubsection{Fixup}
-
-Execute $\alg{Fixup-Base}(W,\pc)$.
-
-TODO define Fixup-Base
-
-\subsubsection{Result}
-
-If all of that was successful, let $\tipcn = W$.
-
-\subsection{$\alg{Recreate-Base}(\pc)$}
-
-\begin{enumerate}
-
-\item
-
-Choose a $\hasdep$-maximal direct dependency $\pd$ of $\pc$.
-
-\item
-
-Use $\alg{Create Base}$ with $L$ = $\tipdy,\; \pq = \pc$ to generate $C$
-and set $W \iassign C$.
-
- \commitproof{
- 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.
- }
-
-\item
-
-Execute the subalgorithm $\alg{Recreate-Recurse}(\pc)$.
-
-\item
-
-Declare that we contain all of the relevant information from the
-sources. That is, use $\alg{Pseudo-Merge}$ with $L = W, \;
-\set R = \{ W \} \cup \set S^{\pcn}$.
-
- \commitproof{
- Base Only: $\patchof{W} = \patchof{L} = \pn$. OK.
-
- Unique Tips:
- Want to prove that for any $\p \isin C$, $\tipdy$ is a suitable $T$.
- WIP TODO
-
- WIP TODO INCOMPLETE
- }
-
-\end{enumerate}
-
-\subsubsection{$\alg{Recreate-Recurse}(\pd)$}
-
-\begin{enumerate}
-
-\item Is $W \haspatch \pd$ ? If so, there is nothing to do: return.
-
-\item TODO what about non-Topbloke base branches
-
-\item For all $\hasdep$-maximal $\pd' \isdirdep \pd$,
-execute $\alg{Recreate-Recurse}(\pd')$.
-
-\item Use $\alg{Pseudo-Merge}$ with $L = W,\; \set R = \{ \tipdn \}$.
-(Recreate Base Dependency Base Declaration.)
-
-\item Use $\alg{Merge}$ to apply $\pd$ to $W$. That is,
-$L = W, \; R = \tipdy, \; M = \baseof{R} = \tipdn$.
-(Recreate Reapply.)
-
-\end{enumerate}
-
-
-\subsection{$\alg{Merge-Tip}(\pc)$}
-
-\begin{enumerate}
-
-\item TODO CHOOSE/REFINE W AND S as was done during Ranking for bases
-
-\item $\alg{Merge}$ from $\tipcn$. That is, $L = W, \;
-R = \tipcn$ and choose any suitable $M$.
-
- \commitproof{
- $L = W$, $R = \tipcn$.
- TODO TBD
-
- Afterwards, $\baseof{W} = \tipcn$.
- }
-
-\item For each source $S \in \set S^{\pcy}$,
-$\alg{Merge}$ with $L = W, \; R = S$ and any suitable $M$.
-
- \commitproof{
- 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.
- }
-
-\end{enumerate}
~ian / topbloke-formulae.git / blobdiff