From 456d92924b93b80e43c32b9a5046b5a47600be2d Mon Sep 17 00:00:00 2001
From: Ian Jackson <ijackson@chiark.greenend.org.uk>
Date: Thu, 1 Mar 2012 03:22:24 +0000
Subject: [PATCH] ends, work, etc.

---
 article.tex | 58 ++++++++++++++++++++++++++++++++++++++---------------
 1 file changed, 42 insertions(+), 16 deletions(-)

diff --git a/article.tex b/article.tex
index 9ca8012..992f0d4 100644
--- a/article.tex
+++ b/article.tex
@@ -38,8 +38,11 @@
 \renewcommand{\land}{\wedge}
 \renewcommand{\lor}{\vee}
 
-\newcommand{\pancs}[2]{{\mathcal A} ( #1 , #2 ) }
-\newcommand{\pends}[2]{{\mathcal E} ( #1 , #2 ) }
+\newcommand{\pancs}{{\mathcal A}}
+\newcommand{\pends}{{\mathcal E}}
+
+\newcommand{\pancsof}[2]{\pancs ( #1 , #2 ) }
+\newcommand{\pendsof}[2]{\pends ( #1 , #2 ) }
 
 \newcommand{\patchof}[1]{{\mathcal P} ( #1 ) }
 \newcommand{\baseof}[1]{{\mathcal B} ( #1 ) }
@@ -93,21 +96,21 @@ All these sets are distinct.  Hence:
 
 \item[ $ \patchof{ C } $ ]
 Either $\p$ s.t. $ C \in \p $, or $\bot$.  
-A function from commits to sets $\p$.
+A function from commits to patches' sets $\p$.
 
-\item[ $ \pancs{C}{\set P} $ ]
+\item[ $ \pancsof{C}{\set P} $ ]
 $ \{ A \; | \; A \le C \land A \in \set P \} $ 
 i.e. all the ancestors of $C$
 which are in $\set P$.
 
-\item[ $ \pends{C}{\set P} $ ]
-$ \{ E \; | \; E \in \pancs{C}{\set P}
-  \land \mathop{\not\exists}_{A \in \pancs{C}{\set P}}
+\item[ $ \pendsof{C}{\set P} $ ]
+$ \{ E \; | \; E \in \pancsof{C}{\set P}
+  \land \mathop{\not\exists}_{A \in \pancsof{C}{\set P}}
   A \neq E \land E \le A \} $ 
-i.e. all $\le$-maximal commits in $\pancs{C}{\set P}$.
+i.e. all $\le$-maximal commits in $\pancsof{C}{\set P}$.
 
 \item[ $ \baseof{C} $ ]
-$ \pends{C}{\pn} = \{ \baseof{C} \} $ where $ C \in \py $.
+$ \pendsof{C}{\pn} = \{ \baseof{C} \} $ where $ C \in \py $.
 A partial function from commits to commits.
 See ``unique base'', below.
 
@@ -131,7 +134,7 @@ it, we hope that git's merge algorithm will DTRT.
   C \has D \implies C \ge D
 }\]
 \[\eqn{Unique Base:}{
- \bigforall_{C \in \py} \pends{C}{\pn} = \{ B \}
+ \bigforall_{C \in \py} \pendsof{C}{\pn} = \{ B \}
 }\]
 \[\eqn{Tip Contents:}{
   \bigforall_{C \in \py} D \isin C \equiv
@@ -141,6 +144,9 @@ it, we hope that git's merge algorithm will DTRT.
 \[\eqn{Base Acyclic:}{
   \bigforall_{B \in \pn} D \isin B \implies D \notin \py
 }\]
+\[\eqn{Coherence:}{
+  \bigforall_{C,\p} C \haspatch \p \lor C \nothaspatch \p
+}\]
 
 \section{Some lemmas}
 
@@ -182,18 +188,38 @@ $ \bigforall_{C \in \py}\bigforall_{D \in \py}
 }\]
 
 \[ \eqn{Transitive Ancestors:}{
-  \left[ \bigforall_{ E \in \pends{C}{\set P} } E \le C \right] \implies
-  \left[ \bigforall_{ A \in \pancs{C}{\set P} } A \le C \right]
+  \left[ \bigforall_{ E \in \pendsof{C}{\set P} } E \le M \right] \equiv
+  \left[ \bigforall_{ A \in \pancsof{C}{\set P} } A \le M \right]
 }\]
 
 \proof{
-By the definition of $\mathcal E$,
-for every such $A$, either $A \in \pends{C}{\set P}$ which implies
-$A \le C$, or $\exists_{A' \in \pancs{C}{\set P}} \; A' \neq A \land A \le C $
+The implication from right to left is trivial because
+$ \pends() \subset \pancs() $.
+For the implication from left to right: 
+by the definition of $\mathcal E$,
+for every such $A$, either $A \in \pends()$ which implies
+$A \le C$, or $\exists_{A' \in \pancs()} \; A' \neq A \land A \le A' $
 in which case we repeat for $A'$.  Since there are finitely many
-commits, this terminates.
+commits, this terminates with $A'' \in \pends()$, ie $A'' \le M$
+by the LHS.  And $A \le A''$.
 }
 
+\section{Commit annotation}
+
+We annotate each Topbloke commit $C$ with:
+\begin{gather}
+\tag*{} \patchof{C} \\
+\tag*{} \baseof{C}, \text{ if } C \in \py \\
+\tag*{} \bigforall_{\pa{Q}} 
+        \text{ either } C \haspatch \pa{Q} \text{ or } C \nothaspatch \pa{Q} \\
+\tag*{} \bigforall_{\pay{Q} \not\ni C} \pendsof{C}{\pay{Q}}
+\end{gather}
+
+We do not annotate $\pendsof{C}{\py}$ for $C \in \py$ doing so would
+break making plain commits with git because the recorded $\pends$
+would have to be updated.  The annotation is not needed because
+$\forall_{\py \ni C} \pendsof{C}{\py} = \{C\}$.
+
 \section{Test more symbols}
 
 $ C \haspatch \p $
-- 
2.30.2