wip dependency insertion

[topbloke-formulae.git] / article.tex

diff --git a/article.tex b/article.tex
index 5106c14df573f6a916761b8c7608f796a2307f25..7ec582e666ef94e6b55ba1e682e1866a1048cb03 100644 (file)
--- a/article.tex
+++ b/article.tex
@@ -812,10 +812,17 @@ dependency.
  L \in \pqn
 }\]
 \[ \eqn{ Currently Excluded }{
- L \nothaspatch \pry
+ L \nothaspatch \pr
+}\]
+\[ \eqn{ Inserted's Ends }{
+ E \in \pendsof{L}{\pry} \implies E \le R^+
+}\]
+\[ \eqn{ Others' Ends }{
+ \bigforall_{\p \patchisin \L}
+ E \in \pendsof{R^+}{\py} \implies E \le L
 }\]
 \[ \eqn{ Insertion Acyclic }{
- R^+ \nothaspatch \pqy
+ R^+ \nothaspatch \pq
 }\]
 
 \subsection{No Replay}
@@ -832,7 +839,75 @@ Not applicable.
 
 Not applicable.
 
-xxx up to here
+\subsection{Base Acyclic}
+
+Consider some $D \isin C$.  We will show that $D \not\in \pqy$.
+By $\merge$, $D \isin L \lor D \isin R^+ \lor D = C$.
+
+For $D \isin L$, Base Acyclic for L suffices.  For $D \isin R^+$,
+Insertion Acyclic suffices.  For $D = C$, trivial.  $\qed$.
+
+\subsection{Coherence and Patch Inclusion}
+
+$$
+\begin{cases}
+  \p = \pr    \lor L \haspatch \p : & C \haspatch \p \\
+  \p \neq \pr \land L \nothaspatch \p : & C \nothaspatch \p
+\end{cases}
+$$
+
+\proofstarts
+~ Consider some $D \in \py$.
+$D \neq C$ so $D \le C \equiv D \le L \lor D \le R^+$.
+
+\subsubsection{For $\p = \pr$:}
+
+$D \not\isin L$ by Currently Excluded.
+$D \not\isin R^-$ by Base Acyclic.
+So by $\merge$, $D \isin C \equiv D \isin R^+$,
+which by Tip Self Inpatch of $R^+$, $\equiv D \le R^+$.
+
+And by definition of $\pancs$,
+$D \le L \equiv D \in \pancsof{L}{R^+}$.
+Applying Transitive Ancestors to Inserted's Ends gives
+$A \in \pancsof{L}{R^+} \implies A \le R^+$.
+So $D \le L \implies D \le R^+$.
+Thus $D \le C \equiv D \le R^+$.
+
+So $D \isin C \equiv D \le C$, i.e. $C \haspatch \pr$.
+OK.
+
+\subsubsection{For $\p \neq \pr$:}
+
+By Exclusive Tip Contents for $R^+$ ($D \not\in \pry$ case)
+$D \isin R^+ \equiv D \isin R^-$.
+So by $\merge$, $D \isin C \equiv D \isin L$.
+
+If $L \nothaspatch \p$, $D \not\isin L$ so $C \nothaspatch \p$.  OK.
+
+If $L \haspatch \p$, Others' Ends applies; by Transitive
+Ancestors, $A \in \pancsof{R^+}{\py} \implies A \le L$.
+So $D \le R^+ \implies D \le L$,
+since $D \le R^+ \equiv D \in \pancsof{R^+}{\py}$.
+Thus $D \le C \equiv D \le L$.
+And by $\haspatch$, $D \le L \equiv D \isin L$ so
+$D \isin C \equiv D \le C$.  Thus $C \haspatch \p$.
+OK.
+
+$\qed$
+
+\subsection{Foreign Inclusion}
+
+Consider some $D$ s.t. $\patchof{D} = \bot$.
+
+By Tip Contents for $R^+$, $D \isin R^+ \equiv D \isin R^-$.
+So by $\merge$, $D \isin C \equiv D \isin L$.
+
+xxx up to here, need new condition
+
+$D \neq C$.
+
+
 
 \section{Merge}