strategy: ranking: proof of termination

[topbloke-formulae.git] / strategy.tex

diff --git a/strategy.tex b/strategy.tex
index 48d93d77954256085eb3d44c0dcf0eb52272d065..e2bcf46fc50dfa768b2108b930b9728a0d39cae7 100644 (file)
--- a/strategy.tex
+++ b/strategy.tex
@@ -155,16 +155,19 @@ if available.
 \item For each $i \ldots 1..n$, update our putative direct
 dependencies:
 $$
-\Gamma \assign \alg{set-merge}\left[\Gamma,
- \left( \begin{cases} 
-  M_i \in \pcn :     & \depsreqof{M_i} \\
-  M_i \not\in \pcn : & \{ \}
- \end{cases} \right),
- \depsreqof{S_i}
- \right]
+\Gamma \assign \setmergeof{
+    \Gamma
+  }{
+    \begin{cases}
+     M_i \in \pcn :     & \depsreqof{M_i} \\
+     M_i \not\in \pcn : & \{ \}
+    \end{cases}
+  }{
+    \depsreqof{S_i}
+  }
 $$
 
-TODO define $\alg{set-merge}$
+TODO define $\setmerge$
 
 \item Finalise our putative direct dependencies
 $
@@ -178,8 +181,8 @@ $
 as necessary).
 If this results in a cycle, abort entirely (as the function $g$ is
 inappropriate; a different $g$ could work).
-\end{enumerate}
 \item Run $\alg{Rank-Recurse}(\pd)$.
+\end{enumerate}
 
 \end{enumerate}
 
@@ -209,6 +212,18 @@ and corresponding merge bases $M^{\pcn}_i = M_i$.
 
 \end{itemize}
 
+\subsection{Proof of termination}
+
+$\alg{Rank-Recurse}(\pc)$ recurses but only downwards through the
+finite graph $\hasdirdep$, so it must terminate.  
+
+The whole ranking algorithm iterates but each iteration involves
+adding one or more patches to $\allpatches$.  Since there are
+finitely many patches and we never remove anything from $\allpatches$
+this must complete eventually.
+
+$\qed$
+
 \section{Traversal phase}
 
 For each patch $C \in \allpatches$ in topological order by $\hasdep$,