From: Ian Jackson Date: Thu, 8 Mar 2012 17:05:09 +0000 (+0000) Subject: wip merge complex - reformulate merge ends conditions X-Git-Tag: f0.2~125 X-Git-Url: http://www.chiark.greenend.org.uk/ucgi/~ian/git?p=topbloke-formulae.git;a=commitdiff_plain;h=99ce58341cfec01e6c331ac3da875a98fa93baed wip merge complex - reformulate merge ends conditions --- diff --git a/article.tex b/article.tex index 1feeb48..b71b188 100644 --- a/article.tex +++ b/article.tex @@ -488,16 +488,20 @@ We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$. \text{otherwise} : & \false \end{cases} }\] -\[ \eqn{ Merge Ends }{ +\[ \eqn{ Removal Merge Ends }{ X \not\haspatch \p \land - Y \haspatch \p + Y \haspatch \p \land + M \haspatch \p \implies - \begin{cases} - M \haspatch \p : & \pendsof{Y}{\py} = \pendsof{M}{\py} - \\ - M \nothaspatch \p : & \displaystyle - \bigforall_{E \in \pendsof{X}{\py}} E \le Y - \end{cases} + \pendsof{Y}{\py} = \pendsof{M}{\py} +}\] +\[ \eqn{ Addition Merge Ends }{ + X \not\haspatch \p \land + Y \haspatch \p \land + M \nothaspatch \p + \implies \left[ + \bigforall_{E \in \pendsof{X}{\py}} E \le Y + \right] }\] \subsection{No Replay} @@ -580,7 +584,8 @@ $C \haspatch \p \equiv M \nothaspatch \p$. \proofstarts -Merge Ends applies. Recall that we are considering $D \in \py$. +One of the Merge Ends conditions 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 $D \isin C \equiv M \nothaspatch \p \land D \le C$ @@ -597,12 +602,12 @@ $D \not\isin M$ so by $\merge$, $D \isin C$. OK. Consider $D \neq C, M \nothaspatch P, D \not\isin Y$: $D \not\le Y$. If $D \le X$ then -$D \in \pancsof{X}{\py}$, so by Merge Ends and +$D \in \pancsof{X}{\py}$, so by Addition Merge Ends and Transitive Ancestors $D \le Y$ --- a contradiction, so $D \not\le X$. Thus $D \not\le C$. By $\merge$, $D \not\isin C$. OK. Consider $D \neq C, M \haspatch P, D \isin Y$: -$D \le Y$ so $D \in \pancsof{Y}{\py}$ so by Merge Ends +$D \le Y$ so $D \in \pancsof{Y}{\py}$ so by Removal Merge Ends and Transitive Ancestors $D \in \pancsof{M}{\py}$ so $D \le M$. Thus $D \isin M$. By $\merge$, $D \not\isin C$. OK.