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
-$W \assign R$.)
+$W \assign R$.
+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:
\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
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}
\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 $L = W,\; R = \tipdy$.
-(Base Dependency Merge.)
\item For each $S \in S^{\pcn}_i$, merge it into $W$.
That is, use $\alg{Merge}$ with $L = W,\; R = S,\; M = M^{\pcn}_i$.
\item
Use $\alg{Create Base}$ with $L$ = $\tipdy,\; \pq = \pc$ to generate $C$
-and set $W \iassign C$. (Recreate Base Beginning.)
+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
Declare that we contain all of the relevant information from the
sources. That is, use $\alg{Pseudo-Merge}$ with $L = W, \;
\set R = \{ W \} \cup \set S^{\pcn}$.
-(Recreate Base Final Declaration.)
+
+ \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}
\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 $M$. (Tip Base Merge.)
+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 $M$.
-(Tip Source Merge.)
+
+ \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