X-Git-Url: http://www.chiark.greenend.org.uk/ucgi/~ian/git?p=topbloke-formulae.git;a=blobdiff_plain;f=trav-alg.tex;fp=trav-alg.tex;h=c9647ad7e6ea93d98c08f0177cb709df1c035328;hp=e56a8a912cdf0274b83e4bae94b4a1a679429ac6;hb=6e7093d49c00150e48ebc264fb1a29de2004aeed;hpb=67e27b1cb5487e79f47e828773fdd574b9da9712

diff --git a/trav-alg.tex b/trav-alg.tex
index e56a8a9..c9647ad 100644
--- a/trav-alg.tex
+++ b/trav-alg.tex
@@ -4,8 +4,10 @@
 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:
@@ -44,6 +46,46 @@ such that:
   \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
@@ -63,6 +105,10 @@ 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}
@@ -74,7 +120,6 @@ In some order, perhaps interleaving the two kinds of merge:
 \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 $L = W,\; R = S,\; M = M^{\pcn}_i$.
@@ -103,7 +148,17 @@ Choose a $\hasdep$-maximal direct dependency $\pd$ of $\pc$.
 \item
 
 Use $\alg{Create Base}$ with $L$ = $\tipdy,\; \pq = \pc$ to generate $C$
-and set $W \iassign C$.  (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
 
@@ -114,7 +169,16 @@ Execute the subalgorithm $\alg{Recreate-Recurse}(\pc)$.
 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}
 
@@ -146,10 +210,31 @@ $L = W, \; R = \tipdy, \; M = \baseof{R} = \tipdn$.
 \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}