chiark / gitweb /
 author Ian Jackson Thu, 1 Mar 2012 14:32:37 +0000 (14:32 +0000) committer Ian Jackson Thu, 1 Mar 2012 14:32:37 +0000 (14:32 +0000)
 article.tex patch | blob | history

index ad68c2c..f40f6a2 100644 (file)
@@ -276,12 +276,42 @@ For $D = C$: $D \in \pn$ so $D \not\in \py$. OK.
For $D \neq C$: $D \isin C \equiv D \isin A$, so by Base Acyclic for
$A$, $D \isin C \implies D \not\in \py$. $\qed$

-\subsection{Coherence}
+\subsection{Coherence and patch inclusion}
+
+Need to consider $D \in \py$

\subsubsection{For $A \haspatch P, D = C$:}
-$D \isin C \equiv \ldots \lor t \text{ so } D \haspatch C$
-$D \le C$
-OK
+
+Ancestors of $C$:
+$D \le C$.
+
+Contents of $C$:
+$D \isin C \equiv \ldots \lor t \text{ so } D \haspatch C$.
+
+\subsubsection{For $A \haspatch P, D \neq C$:}
+Ancestors: $D \le C \equiv D \le A$.
+
+Contents: $D \isin C \equiv D \isin A \lor f$
+so $D \isin C \equiv D \isin A$.
+
+So:
+$A \haspatch P \implies C \haspatch P$
+
+\subsubsection{For $A \nothaspatch P$:}
+
+Firstly, $C \not\in \py$ since if it were, $A \in \py$.
+Thus $D \neq C$.
+
+Now by contents of $A$, $D \notin A$, so $D \notin C$.
+
+So:
+$A \nothaspatch P \implies C \nothaspatch P$
+$\qed$
+
+\subsection{Foreign inclusion:}
+
+If $D = C$, trivial.  For $D \neq C$:
+$D \isin C \equiv D \isin A \equiv D \le A \equiv D \le C$.  $\qed$

\section{Test more symbols}