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formatting: a couple of extra [ ]
author
Ian Jackson
<ijackson@chiark.greenend.org.uk>
Mon, 12 Mar 2012 15:53:40 +0000
(15:53 +0000)
committer
Ian Jackson
<ijackson@chiark.greenend.org.uk>
Mon, 12 Mar 2012 15:59:38 +0000
(15:59 +0000)
article.tex
patch
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diff --git
a/article.tex
b/article.tex
index 92e1f1010b77f5022f606af2de2a4842dcf9efd9..a7b27094e0fb09243c554881a1d588a13156b5d0 100644
(file)
--- a/
article.tex
+++ b/
article.tex
@@
-12,6
+12,9
@@
\pagestyle{fancy}
\lhead[\rightmark]{}
\pagestyle{fancy}
\lhead[\rightmark]{}
+\let\stdsection\section
+\renewcommand\section{\newpage\stdsection}
+
\renewcommand{\ge}{\geqslant}
\renewcommand{\le}{\leqslant}
\newcommand{\nge}{\ngeqslant}
\renewcommand{\ge}{\geqslant}
\renewcommand{\le}{\leqslant}
\newcommand{\nge}{\ngeqslant}
@@
-211,8
+214,8
@@
We maintain these each time we construct a new commit. \\
\section{Some lemmas}
\section{Some lemmas}
-\
[ \eqn{Alternative (overlapping) formulations defining
- $\mergeof{C}{L}{M}{R}$:}{
+\
subsection{Alternative (overlapping) formulations of $\mergeof{C}{L}{M}{R}$}
+$$
D \isin C \equiv
\begin{cases}
D \isin L \equiv D \isin R : & D = C \lor D \isin L \\
D \isin C \equiv
\begin{cases}
D \isin L \equiv D \isin R : & D = C \lor D \isin L \\
@@
-221,7
+224,7
@@
We maintain these each time we construct a new commit. \\
D \isin L \nequiv D \isin M : & D = C \lor D \isin L \\
\text{as above with L and R exchanged}
\end{cases}
D \isin L \nequiv D \isin M : & D = C \lor D \isin L \\
\text{as above with L and R exchanged}
\end{cases}
-}\]
+$$
\proof{ ~ Truth table (ordered by original definition): \\
\begin{tabular}{cccc|c|cc}
$D = C$ &
\proof{ ~ Truth table (ordered by original definition): \\
\begin{tabular}{cccc|c|cc}
$D = C$ &
@@
-243,11
+246,12
@@
We maintain these each time we construct a new commit. \\
And original definition is symmetrical in $L$ and $R$.
}
And original definition is symmetrical in $L$ and $R$.
}
-\[ \eqn{Exclusive Tip Contents:}{
+\subsection{Exclusive Tip Contents}
+$$
\bigforall_{C \in \py}
\neg \Bigl[ D \isin \baseof{C} \land ( D \in \py \land D \le C )
\Bigr]
\bigforall_{C \in \py}
\neg \Bigl[ D \isin \baseof{C} \land ( D \in \py \land D \le C )
\Bigr]
-}\]
+$$
Ie, the two limbs of the RHS of Tip Contents are mutually exclusive.
\proof{
Ie, the two limbs of the RHS of Tip Contents are mutually exclusive.
\proof{
@@
-262,9
+266,10
@@
So by Base Acyclic $D \isin B \implies D \notin \py$.
\end{cases}
}\]
\end{cases}
}\]
-\[ \eqn{Tip Self Inpatch:}{
+\subsection{Tip Self Inpatch}
+$$
\bigforall_{C \in \py} C \haspatch \p
\bigforall_{C \in \py} C \haspatch \p
-}\]
+$$
Ie, tip commits contain their own patch.
\proof{
Ie, tip commits contain their own patch.
\proof{
@@
-273,18
+278,20
@@
$ \bigforall_{C \in \py}\bigforall_{D \in \py}
D \isin C \equiv D \le C $
}
D \isin C \equiv D \le C $
}
-\[ \eqn{Exact Ancestors:}{
+\subsection{Exact Ancestors}
+$$
\bigforall_{ C \hasparents \set{R} }
D \le C \equiv
( \mathop{\hbox{\huge{$\vee$}}}_{R \in \set R} D \le R )
\lor D = C
\bigforall_{ C \hasparents \set{R} }
D \le C \equiv
( \mathop{\hbox{\huge{$\vee$}}}_{R \in \set R} D \le R )
\lor D = C
-}\]
+$$
\proof{ ~ Trivial.}
\proof{ ~ Trivial.}
-\[ \eqn{Transitive Ancestors:}{
+\subsection{Transitive Ancestors}
+$$
\left[ \bigforall_{ E \in \pendsof{C}{\set P} } E \le M \right] \equiv
\left[ \bigforall_{ A \in \pancsof{C}{\set P} } A \le M \right]
\left[ \bigforall_{ E \in \pendsof{C}{\set P} } E \le M \right] \equiv
\left[ \bigforall_{ A \in \pancsof{C}{\set P} } A \le M \right]
-}\]
+$$
\proof{
The implication from right to left is trivial because
\proof{
The implication from right to left is trivial because
@@
-299,7
+306,8
@@
commits, this terminates with $A'' \in \pends()$, ie $A'' \le M$
by the LHS. And $A \le A''$.
}
by the LHS. And $A \le A''$.
}
-\[ \eqn{Calculation Of Ends:}{
+\subsection{Calculation Of Ends}
+$$
\bigforall_{C \hasparents \set A}
\pendsof{C}{\set P} =
\begin{cases}
\bigforall_{C \hasparents \set A}
\pendsof{C}{\set P} =
\begin{cases}
@@
-313,7
+321,7
@@
by the LHS. And $A \le A''$.
E \neq F \land E \le F \Bigr]
\right\}
\end{cases}
E \neq F \land E \le F \Bigr]
\right\}
\end{cases}
-}\]
+$$
\proof{
Trivial for $C \in \set P$. For $C \not\in \set P$,
$\pancsof{C}{\set P} = \bigcup_{A \in \set A} \pancsof{A}{\set P}$.
\proof{
Trivial for $C \in \set P$. For $C \not\in \set P$,
$\pancsof{C}{\set P} = \bigcup_{A \in \set A} \pancsof{A}{\set P}$.
@@
-325,7
+333,8
@@
an $F''$) which disqualifies $F$.
Otherwise, $E$ meets all the conditions for $\pends$.
}
Otherwise, $E$ meets all the conditions for $\pends$.
}
-\[ \eqn{Ingredients Prevent Replay:}{
+\subsection{Ingredients Prevent Replay}
+$$
\left[
{C \hasparents \set A} \land
\\
\left[
{C \hasparents \set A} \land
\\
@@
-337,7
+346,7
@@
Otherwise, $E$ meets all the conditions for $\pends$.
\right] \implies \left[
D \isin C \implies D \le C
\right]
\right] \implies \left[
D \isin C \implies D \le C
\right]
-}\]
+$$
\proof{
Trivial for $D = C$. Consider some $D \neq C$, $D \isin C$.
By the preconditions, there is some $A$ s.t. $D \in \set A$
\proof{
Trivial for $D = C$. Consider some $D \neq C$, $D \isin C$.
By the preconditions, there is some $A$ s.t. $D \in \set A$
@@
-345,16
+354,19
@@
Otherwise, $E$ meets all the conditions for $\pends$.
$A \le C$ so $D \le C$.
}
$A \le C$ so $D \le C$.
}
-\[ \eqn{Simple Foreign Inclusion:}{
+\subsection{Simple Foreign Inclusion}
+$$
\left[
C \hasparents \{ L \}
\land
\bigforall_{D} D \isin C \equiv D \isin L \lor D = C
\right]
\implies
\left[
C \hasparents \{ L \}
\land
\bigforall_{D} D \isin C \equiv D \isin L \lor D = C
\right]
\implies
+ \left[
\bigforall_{D \text{ s.t. } \patchof{D} = \bot}
D \isin C \equiv D \le C
\bigforall_{D \text{ s.t. } \patchof{D} = \bot}
D \isin C \equiv D \le C
-}\]
+ \right]
+$$
\proof{
Consider some $D$ s.t. $\patchof{D} = \bot$.
If $D = C$, trivially true. For $D \neq C$,
\proof{
Consider some $D$ s.t. $\patchof{D} = \bot$.
If $D = C$, trivially true. For $D \neq C$,
@@
-363,7
+375,8
@@
And by Exact Ancestors $D \le L \equiv D \le C$.
So $D \isin C \equiv D \le C$.
}
So $D \isin C \equiv D \le C$.
}
-\[ \eqn{Totally Foreign Contents:}{
+\subsection{Totally Foreign Contents}
+$$
\bigforall_{C \hasparents \set A}
\left[
\patchof{C} = \bot \land
\bigforall_{C \hasparents \set A}
\left[
\patchof{C} = \bot \land
@@
-375,7
+388,7
@@
So $D \isin C \equiv D \le C$.
\implies
\patchof{D} = \bot
\right]
\implies
\patchof{D} = \bot
\right]
-}\]
+$$
\proof{
Consider some $D \le C$. If $D = C$, $\patchof{D} = \bot$ trivially.
If $D \neq C$ then $D \le A$ where $A \in \set A$. By Foreign
\proof{
Consider some $D \le C$. If $D = C$, $\patchof{D} = \bot$ trivially.
If $D \neq C$ then $D \le A$ where $A \in \set A$. By Foreign