X-Git-Url: http://www.chiark.greenend.org.uk/ucgi/~ian/git?p=topbloke-formulae.git;a=blobdiff_plain;f=article.tex;h=fac5c82efd6483ac6cc865cbd2db836f2d29d81e;hp=2f512f4c96e2215b002ae886742a107753823bb2;hb=13285d550a2eceb866519b89d1ed3c4894f3db65;hpb=81ac90d1051946e902cd316e19c733ddf26ad1c7

diff --git a/article.tex b/article.tex
index 2f512f4..fac5c82 100644
--- a/article.tex
+++ b/article.tex
@@ -490,10 +490,11 @@ We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$.
 }\]
 \[ \eqn{ Merge Ends }{
     X \not\haspatch \p \land
-    Y \haspatch \p \land
-    E \in \pendsof{X}{\py}
-  \implies
+    Y \haspatch \p
+  \implies \left[
+  \bigforall_{E \in \pendsof{X}{\py}}
     E \le Y
+  \right]
 }\]
 
 \subsection{No Replay}
@@ -536,19 +537,19 @@ $\qed$
 
 \subsection{Coherence and patch inclusion}
 
-Need to determine $C \haspatch P$ based on $L,M,R \haspatch P$.
+Need to determine $C \haspatch \p$ based on $L,M,R \haspatch \p$.
 This involves considering $D \in \py$.  
 
-\subsubsection{For $L \nothaspatch P, R \nothaspatch P$:}
+\subsubsection{For $L \nothaspatch \p, R \nothaspatch \p$:}
 $D \not\isin L \land D \not\isin R$.  $C \not\in \py$ (otherwise $L
-\in \py$ ie $L \haspatch P$ by Tip Self Inpatch).  So $D \neq C$.
-Applying $\merge$ gives $D \not\isin C$ i.e. $C \nothaspatch P$.
+\in \py$ ie $L \haspatch \p$ by Tip Self Inpatch).  So $D \neq C$.
+Applying $\merge$ gives $D \not\isin C$ i.e. $C \nothaspatch \p$.
 
-\subsubsection{For $L \haspatch P, R \haspatch P$:}
+\subsubsection{For $L \haspatch \p, R \haspatch \p$:}
 $D \isin L \equiv D \le L$ and $D \isin R \equiv D \le R$.
 (Likewise $D \isin X \equiv D \le X$ and $D \isin Y \equiv D \le Y$.)
 
-Consider $D = C$: $D \isin C$, $D \le C$, OK for $C \haspatch P$.
+Consider $D = C$: $D \isin C$, $D \le C$, OK for $C \haspatch \p$.
 
 For $D \neq C$: $D \le C \equiv D \le L \lor D \le R
  \equiv D \isin L \lor D \isin R$.  
@@ -556,24 +557,35 @@ For $D \neq C$: $D \le C \equiv D \le L \lor D \le R
 
 Consider $D \neq C, D \isin X \land D \isin Y$:
 By $\merge$, $D \isin C$.  Also $D \le X$ 
-so $D \le C$.  OK for $C \haspatch P$.
+so $D \le C$.  OK for $C \haspatch \p$.
 
 Consider $D \neq C, D \not\isin X \land D \not\isin Y$:
 By $\merge$, $D \not\isin C$.  
 And $D \not\le X \land D \not\le Y$ so $D \not\le C$.  
-OK for $C \haspatch P$.
+OK for $C \haspatch \p$.
 
 Remaining case, wlog, is $D \not\isin X \land D \isin Y$.
 $D \not\le X$ so $D \not\le M$ so $D \not\isin M$.  
 Thus by $\merge$, $D \isin C$.  And $D \le Y$ so $D \le C$.
-OK for $C \haspatch P$.
+OK for $C \haspatch \p$.
 
-So indeed $L \haspatch P \land R \haspatch P \implies C \haspatch P$.
+So indeed $L \haspatch \p \land R \haspatch \p \implies C \haspatch \p$.
 
-\subsubsection{For (wlog) $X \not\haspatch P, Y \haspatch P$:}
+\subsubsection{For (wlog) $X \not\haspatch \p, Y \haspatch \p$:}
 
-$C \haspatch P \equiv C \nothaspatch M$.
+$C \haspatch \p \equiv M \nothaspatch \p$.
 
 \proofstarts
 
+Merge Ends applies.  Recall that we are considering $D \in \py$.
+$D \isin Y \equiv D \le Y$.  $D \not\isin X$.
+We will show for each of
+various cases that $D \isin C \equiv M \nothaspatch \p \land D \le C$
+(which suffices by definition of $\haspatch$ and $\nothaspatch$).
+
+Consider $D = C$.  Thus $C \in \py, L \in \py$, and by Tip
+Self Inpatch $L \haspatch \p$, so $L=Y, R=X$.  By Tip Merge,
+$M=\baseof{L}$.  So by Base Acyclic $D \not\isin M$, i.e.
+$M \nothaspatch \p$.  And indeed $D \isin C$ and $D \le C$.  OK.
+
 \end{document}