X-Git-Url: http://www.chiark.greenend.org.uk/ucgi/~ian/git?p=topbloke-formulae.git;a=blobdiff_plain;f=article.tex;h=1e7adc26461f2083df2e4199726814428a396255;hp=92e1f1010b77f5022f606af2de2a4842dcf9efd9;hb=588087cf4ec7d7c90cb21ac1b3790b86283ba992;hpb=1f8eb7a95b12515aa6b4d33ccfdabee2ff042e0f

diff --git a/article.tex b/article.tex
index 92e1f10..1e7adc2 100644
--- a/article.tex
+++ b/article.tex
@@ -10,7 +10,13 @@
 
 \usepackage{fancyhdr}
 \pagestyle{fancy}
-\lhead[\rightmark]{}
+\rhead[\rightmark]{}
+\lhead[]{\rightmark}
+\rfoot[\thepage]{\input{revid.inc}}
+\lfoot[\input{revid.inc}]{\thepage}
+
+\let\stdsection\section
+\renewcommand\section{\newpage\stdsection}
 
 \renewcommand{\ge}{\geqslant}
 \renewcommand{\le}{\leqslant}
@@ -24,6 +30,8 @@
 \newcommand{\notpatchisin}{\mathrel{\,\not\!\not\relax\patchisin}}
 \newcommand{\haspatch}{\sqSupset}
 \newcommand{\patchisin}{\sqSubset}
+%\newcommand{\zhaspatch}{\mathrel{\underset{\fullmoon}\sqSupset}}
+\newcommand{\zhaspatch}{\mathrel{\sqSupset_\varnothing\mkern-4mu}}
 
         \newif\ifhidehack\hidehackfalse
         \DeclareRobustCommand\hidefromedef[2]{%
@@ -38,6 +46,18 @@
 \newcommand{\py}{\pay{P}}
 \newcommand{\pn}{\pan{P}}
 
+\newcommand{\pc}{\pa{C}}
+\newcommand{\pcy}{\pay{C}}
+\newcommand{\pcn}{\pan{C}}
+
+\newcommand{\pd}{\pa{D}}
+\newcommand{\pdy}{\pay{D}}
+\newcommand{\pdn}{\pan{D}}
+
+\newcommand{\pl}{\pa{L}}
+\newcommand{\ply}{\pay{L}}
+\newcommand{\pln}{\pan{L}}
+
 \newcommand{\pq}{\pa{Q}}
 \newcommand{\pqy}{\pay{Q}}
 \newcommand{\pqn}{\pan{Q}}
@@ -49,8 +69,8 @@
 %\newcommand{\hasparents}{\underaccent{1}{>}}
 %\newcommand{\hasparents}{{%
 %  \declareslashed{}{_{_1}}{0}{-0.8}{>}\slashed{>}}}
-\newcommand{\hasparents}{>_{\mkern-7.0mu _1}}
-\newcommand{\areparents}{<_{\mkern-14.0mu _1\mkern+5.0mu}}
+\newcommand{\hasparents}{>_{\mkern-7.0mu _{1:}}}
+\newcommand{\areparents}{<_{\mkern-14.0mu _{1:}\mkern+5.0mu}}
 
 \renewcommand{\implies}{\Rightarrow}
 \renewcommand{\equiv}{\Leftrightarrow}
@@ -70,13 +90,39 @@
 
 \newcommand{\patch}{{\mathcal P}}
 \newcommand{\base}{{\mathcal B}}
+\newcommand{\depsreq}{{\mathcal G}}
 
 \newcommand{\patchof}[1]{\patch ( #1 ) }
 \newcommand{\baseof}[1]{\base ( #1 ) }
+\newcommand{\depsreqof}[1]{\depsreq ( #1 ) }
+
+\newcommand{\allpatches}{\Upsilon}
+\newcommand{\assign}{\leftarrow}
+\newcommand{\iassign}{\leftarrow}
+%\newcommand{\assign}{' =}
+
+\newcommand{\alg}[1]{\text{\bf #1}}
+\newcommand{\setmerge}{\alg{merge}}
+\newcommand{\setmergeof}[3]{\setmerge\left\langle #1 \;\middle\langle #2 \middle\rangle\; #3 \right\rangle}
+%\newcommand{\setmergeof}[3]{\setmerge\left\lgroup #1 \;\middle\lmoustache\; #2 \;\middle\rmoustache\; #3 \right\rgroup}
+%\newcommand{\setmergeof}[3]{\setmerge\left\rmoustache #1 \middle\rmoustache #2 \middle\lmoustache #3 \right\lmoustache}
+%\newcommand{\setmergeof}[3]{\setmerge\left\lfloor #1 \middle\lfloor #2 \middle\rfloor #3 \right\rfloor}
 
 \newcommand{\eqntag}[2]{ #2 \tag*{\mbox{#1}} }
 \newcommand{\eqn}[2]{ #2 \tag*{\mbox{\bf #1}} }
 
+\newcommand{\hasdirdep}{\succ_{\mkern-7.0mu _1}}
+\newcommand{\hasdep}{\succ}
+\newcommand{\isdep}{\prec}
+\newcommand{\isdirdep}{\prec_{\mkern-18.0mu _1}{\mkern+10mu}}
+
+\newcommand{\tip}{ T }
+\newcommand{\tipa}[1]{ \tip^{#1} }
+\newcommand{\tipcn}{ \tipa \pcn }
+\newcommand{\tipcy}{ \tipa \pcy }
+\newcommand{\tipdn}{ \tipa \pdn }
+\newcommand{\tipdy}{ \tipa \pdy }
+
 %\newcommand{\bigforall}{\mathop{\hbox{\huge$\forall$}}}
 \newcommand{\bigforall}{%
   \mathop{\mathchoice%
@@ -85,6 +131,13 @@
     {\hbox{\normalsize$\forall$}}%
     {\hbox{\scriptsize$\forall$}}}%
 }
+\newcommand{\bigexists}{%
+  \mathop{\mathchoice%
+    {\hbox{\huge$\exists$}}%
+    {\hbox{\Large$\exists$}}%
+    {\hbox{\normalsize$\exists$}}%
+    {\hbox{\scriptsize$\exists$}}}%
+}
 
 \newcommand{\Largeexists}{\mathop{\hbox{\Large$\exists$}}}
 \newcommand{\Largenexists}{\mathop{\hbox{\Large$\nexists$}}}
@@ -101,947 +154,21 @@
 
 \begin{document}
 
-\section{Notation}
-
-\begin{basedescript}{
-\desclabelwidth{5em}
-\desclabelstyle{\nextlinelabel}
-}
-\item[ $ C \hasparents \set X $ ]
-The parents of commit $C$ are exactly the set
-$\set X$.
-
-\item[ $ C \ge D $ ]
-$C$ is a descendant of $D$ in the git commit
-graph.  This is a partial order, namely the transitive closure of
-$ D \in \set X $ where $ C \hasparents \set X $.
-
-\item[ $ C \has D $ ]
-Informally, the tree at commit $C$ contains the change
-made in commit $D$.  Does not take account of deliberate reversions by
-the user or reversion, rebasing or rewinding in
-non-Topbloke-controlled branches.  For merges and Topbloke-generated
-anticommits or re-commits, the ``change made'' is only to be thought
-of as any conflict resolution.  This is not a partial order because it
-is not transitive.
-
-\item[ $ \p, \py, \pn $ ]
-A patch $\p$ consists of two sets of commits $\pn$ and $\py$, which
-are respectively the base and tip git branches.  $\p$ may be used
-where the context requires a set, in which case the statement
-is to be taken as applying to both $\py$ and $\pn$.
-None of these sets overlap.  Hence:
-
-\item[ $ \patchof{ C } $ ]
-Either $\p$ s.t. $ C \in \p $, or $\bot$.
-A function from commits to patches' sets $\p$.
-
-\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[ $ \pendsof{C}{\set P} $ ]
-$ \{ E \; | \; E \in \pancsof{C}{\set P}
-  \land \mathop{\not\exists}_{A \in \pancsof{C}{\set P}}
-  E \neq A \land E \le A \} $
-i.e. all $\le$-maximal commits in $\pancsof{C}{\set P}$.
-
-\item[ $ \baseof{C} $ ]
-$ \pendsof{C}{\pn} = \{ \baseof{C} \} $ where $ C \in \py $.
-A partial function from commits to commits.
-See Unique Base, below.
-
-\item[ $ C \haspatch \p $ ]
-$\displaystyle \bigforall_{D \in \py} D \isin C \equiv D \le C $.
-~ Informally, $C$ has the contents of $\p$.
-
-\item[ $ C \nothaspatch \p $ ]
-$\displaystyle \bigforall_{D \in \py} D \not\isin C $.
-~ Informally, $C$ has none of the contents of $\p$.
-
-Non-Topbloke commits are $\nothaspatch \p$ for all $\p$.  This
-includes commits on plain git branches made by applying a Topbloke
-patch.  If a Topbloke
-patch is applied to a non-Topbloke branch and then bubbles back to
-the relevant Topbloke branches, we hope that
-if the user still cares about the Topbloke patch,
-git's merge algorithm will DTRT when trying to re-apply the changes.
-
-\item[ $\displaystyle \mergeof{C}{L}{M}{R} $ ]
-The contents of a git merge result:
-
-$\displaystyle D \isin C \equiv
-  \begin{cases}
-    (D \isin L \land D \isin R) \lor D = C : & \true \\
-    (D \not\isin L \land D \not\isin R) \land D \neq C : & \false \\
-    \text{otherwise} : & D \not\isin M
-  \end{cases}
-$
-
-\end{basedescript}
-\newpage
-\section{Invariants}
-
-We maintain these each time we construct a new commit. \\
-\[ \eqn{No Replay:}{
-  C \has D \implies C \ge D
-}\]
-\[\eqn{Unique Base:}{
- \bigforall_{C \in \py} \pendsof{C}{\pn} = \{ B \}
-}\]
-\[\eqn{Tip Contents:}{
-  \bigforall_{C \in \py} D \isin C \equiv
-    { D \isin \baseof{C} \lor \atop
-      (D \in \py \land D \le C) }
-}\]
-\[\eqn{Base Acyclic:}{
-  \bigforall_{B \in \pn} D \isin B \implies D \notin \py
-}\]
-\[\eqn{Coherence:}{
-  \bigforall_{C,\p} C \haspatch \p \lor C \nothaspatch \p
-}\]
-\[\eqn{Foreign Inclusion:}{
-  \bigforall_{D \text{ s.t. } \patchof{D} = \bot} D \isin C \equiv D \leq C
-}\]
-\[\eqn{Foreign Contents:}{
-  \bigforall_{C \text{ s.t. } \patchof{C} = \bot}
-    D \le C \implies \patchof{D} = \bot
-}\]
-
-\section{Some lemmas}
-
-\[ \eqn{Alternative (overlapping) formulations defining
-  $\mergeof{C}{L}{M}{R}$:}{
- D \isin C \equiv
-  \begin{cases}
-    D \isin L  \equiv D \isin R  : & D = C \lor D \isin L     \\
-    D \isin L \nequiv D \isin R  : & D = C \lor D \not\isin M \\
-    D \isin L  \equiv D \isin M  : & D = C \lor D \isin R     \\
-    D \isin L \nequiv D \isin M  : & D = C \lor D \isin L     \\
-    \text{as above with L and R exchanged}
-  \end{cases}
-}\]
-\proof{ ~ Truth table (ordered by original definition): \\
-  \begin{tabular}{cccc|c|cc}
-     $D = C$ &
-          $\isin L$ &
-               $\isin M$ &
-                    $\isin R$ & $\isin C$ &
-                                      $L$ vs. $R$ & $L$ vs. $M$
-  \\\hline
-     y &  ? &  ? &  ?      &      y   & ?         & ?            \\
-     n &  y &  y &  y      &      y   & $\equiv$  & $\equiv$     \\
-     n &  y &  n &  y      &      y   & $\equiv$  & $\nequiv$    \\
-     n &  n &  y &  n      &      n   & $\equiv$  & $\nequiv$    \\
-     n &  n &  n &  n      &      n   & $\equiv$  & $\equiv$     \\
-     n &  y &  y &  n      &      n   & $\nequiv$ & $\equiv$     \\
-     n &  n &  y &  y      &      n   & $\nequiv$ & $\nequiv$    \\
-     n &  y &  n &  n      &      y   & $\nequiv$ & $\nequiv$    \\
-     n &  n &  n &  y      &      y   & $\nequiv$ & $\equiv$     \\
-  \end{tabular} \\
-  And original definition is symmetrical in $L$ and $R$.
-}
-
-\[ \eqn{Exclusive Tip Contents:}{
-  \bigforall_{C \in \py}
-    \neg \Bigl[ D \isin \baseof{C} \land ( D \in \py \land D \le C )
-      \Bigr]
-}\]
-Ie, the two limbs of the RHS of Tip Contents are mutually exclusive.
-
-\proof{
-Let $B = \baseof{C}$ in $D \isin \baseof{C}$.  Now $B \in \pn$.
-So by Base Acyclic $D \isin B \implies D \notin \py$.
-}
-\[ \eqntag{{\it Corollary - equivalent to Tip Contents}}{
-  \bigforall_{C \in \py} D \isin C \equiv
-  \begin{cases}
-    D \in \py : & D \le C \\
-    D \not\in \py : & D \isin \baseof{C}
-  \end{cases}
-}\]
-
-\[ \eqn{Tip Self Inpatch:}{
-  \bigforall_{C \in \py} C \haspatch \p
-}\]
-Ie, tip commits contain their own patch.
-
-\proof{
-Apply Exclusive Tip Contents to some $D \in \py$:
-$ \bigforall_{C \in \py}\bigforall_{D \in \py}
-  D \isin C \equiv D \le C $
-}
-
-\[ \eqn{Exact Ancestors:}{
-  \bigforall_{ C \hasparents \set{R} }
-  D \le C \equiv
-    ( \mathop{\hbox{\huge{$\vee$}}}_{R \in \set R} D \le R )
-    \lor D = C
-}\]
-\proof{ ~ Trivial.}
-
-\[ \eqn{Transitive Ancestors:}{
-  \left[ \bigforall_{ E \in \pendsof{C}{\set P} } E \le M \right] \equiv
-  \left[ \bigforall_{ A \in \pancsof{C}{\set P} } A \le M \right]
-}\]
-
-\proof{
-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
-$A \le M$ by the LHS directly,
-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 with $A'' \in \pends()$, ie $A'' \le M$
-by the LHS.  And $A \le A''$.
-}
-
-\[ \eqn{Calculation Of Ends:}{
-  \bigforall_{C \hasparents \set A}
-    \pendsof{C}{\set P} =
-      \begin{cases}
-       C \in \p : & \{ C \}
-      \\
-       C \not\in \p : & \displaystyle
-       \left\{ E \Big|
-           \Bigl[ \Largeexists_{A \in \set A}
-                       E \in \pendsof{A}{\set P} \Bigr] \land
-           \Bigl[ \Largenexists_{B \in \set A, F \in \pendsof{B}{\p}}
-                       E \neq F \land E \le F \Bigr]
-       \right\}
-      \end{cases}
-}\]
-\proof{
-Trivial for $C \in \set P$.  For $C \not\in \set P$,
-$\pancsof{C}{\set P} = \bigcup_{A \in \set A} \pancsof{A}{\set P}$.
-So $\pendsof{C}{\set P} \subset \bigcup_{E in \set E} \pendsof{E}{\set P}$.
-Consider some $E \in \pendsof{A}{\set P}$.  If $\exists_{B,F}$ as
-specified, then either $F$ is going to be in our result and
-disqualifies $E$, or there is some other $F'$ (or, eventually,
-an $F''$) which disqualifies $F$.
-Otherwise, $E$ meets all the conditions for $\pends$.
-}
-
-\[ \eqn{Ingredients Prevent Replay:}{
-  \left[
-    {C \hasparents \set A} \land
-   \\
-    \left(
-      D \isin C \implies
-       D = C \lor
-       \Largeexists_{A \in \set A} D \isin A
-    \right)
-  \right] \implies \left[
-    D \isin C \implies D \le C
-  \right]
-}\]
-\proof{
-  Trivial for $D = C$.  Consider some $D \neq C$, $D \isin C$.
-  By the preconditions, there is some $A$ s.t. $D \in \set A$
-  and $D \isin A$.  By No Replay for $A$, $D \le A$.  And
-  $A \le C$ so $D \le C$.
-}
-
-\[ \eqn{Simple Foreign Inclusion:}{
-  \left[
-    C \hasparents \{ L \}
-   \land
-    \bigforall_{D} D \isin C \equiv D \isin L \lor D = C
-  \right]
- \implies
-   \bigforall_{D \text{ s.t. } \patchof{D} = \bot}
-     D \isin C \equiv D \le C
-}\]
-\proof{
-Consider some $D$ s.t. $\patchof{D} = \bot$.
-If $D = C$, trivially true.  For $D \neq C$,
-by Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$.
-And by Exact Ancestors $D \le L \equiv D \le C$.
-So $D \isin C \equiv D \le C$.
-}
-
-\[ \eqn{Totally Foreign Contents:}{
-  \bigforall_{C \hasparents \set A}
-   \left[
-    \patchof{C} = \bot \land
-      \bigforall_{A \in \set A} \patchof{A} = \bot
-   \right]
-  \implies
-   \left[
-    D \le C
-   \implies
-    \patchof{D} = \bot
-   \right]
-}\]
-\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
-Contents of $A$, $\patchof{D} = \bot$.
-}
-
-\section{Commit annotation}
-
-We annotate each Topbloke commit $C$ with:
-\gathbegin
- \patchof{C}
-\gathnext
- \baseof{C}, \text{ if } C \in \py
-\gathnext
- \bigforall_{\pq}
-   \text{ either } C \haspatch \pq \text{ or } C \nothaspatch \pq
-\gathnext
- \bigforall_{\pqy \not\ni C} \pendsof{C}{\pqy}
-\end{gather}
-
-$\patchof{C}$, for each kind of Topbloke-generated commit, is stated
-in the summary in the section for that kind of commit.
-
-Whether $\baseof{C}$ is required, and if so what the value is, is
-stated in the proof of Unique Base for each kind of commit.
-
-$C \haspatch \pq$ or $\nothaspatch \pq$ is represented as the
-set $\{ \pq | C \haspatch \pq \}$.  Whether $C \haspatch \pq$
-is in stated
-(in terms of $I \haspatch \pq$ or $I \nothaspatch \pq$
-for the ingredients $I$),
-in the proof of Coherence for each kind of commit.
-
-$\pendsof{C}{\pq^+}$ is computed, for all Topbloke-generated commits,
-using the lemma Calculation of Ends, above.
-We do not annotate $\pendsof{C}{\py}$ for $C \in \py$.  Doing so would
-make it wrong to make plain commits with git because the recorded $\pends$
-would have to be updated.  The annotation is not needed in that case
-because $\forall_{\py \ni C} \; \pendsof{C}{\py} = \{C\}$.
-
-\section{Simple commit}
-
-A simple single-parent forward commit $C$ as made by git-commit.
-\begin{gather}
-\tag*{} C \hasparents \{ L \} \\
-\tag*{} \patchof{C} = \patchof{L} \\
-\tag*{} D \isin C \equiv D \isin L \lor D = C
-\end{gather}
-This also covers Topbloke-generated commits on plain git branches:
-Topbloke strips the metadata when exporting.
-
-\subsection{No Replay}
-
-Ingredients Prevent Replay applies.  $\qed$
-
-\subsection{Unique Base}
-If $L, C \in \py$ then by Calculation of Ends for
-$C, \py, C \not\in \py$:
-$\pendsof{C}{\pn} = \pendsof{L}{\pn}$ so
-$\baseof{C} = \baseof{L}$. $\qed$
-
-\subsection{Tip Contents}
-We need to consider only $L, C \in \py$.  From Tip Contents for $L$:
-\[ D \isin L \equiv D \isin \baseof{L} \lor ( D \in \py \land D \le L ) \]
-Substitute into the contents of $C$:
-\[ D \isin C \equiv D \isin \baseof{L} \lor ( D \in \py \land D \le L )
-    \lor D = C \]
-Since $D = C \implies D \in \py$,
-and substituting in $\baseof{C}$, this gives:
-\[ D \isin C \equiv D \isin \baseof{C} \lor
-    (D \in \py \land D \le L) \lor
-    (D = C \land D \in \py) \]
-\[ \equiv D \isin \baseof{C} \lor
-   [ D \in \py \land ( D \le L \lor D = C ) ] \]
-So by Exact Ancestors:
-\[ D \isin C \equiv D \isin \baseof{C} \lor ( D \in \py \land D \le C
-) \]
-$\qed$
-
-\subsection{Base Acyclic}
-
-Need to consider only $L, C \in \pn$.
-
-For $D = C$: $D \in \pn$ so $D \not\in \py$. OK.
-
-For $D \neq C$: $D \isin C \equiv D \isin L$, so by Base Acyclic for
-$L$, $D \isin C \implies D \not\in \py$.
-
-$\qed$
-
-\subsection{Coherence and patch inclusion}
-
-Need to consider $D \in \py$
-
-\subsubsection{For $L \haspatch P, D = C$:}
-
-Ancestors of $C$:
-$ D \le C $.
-
-Contents of $C$:
-$ D \isin C \equiv \ldots \lor \true \text{ so } D \haspatch C $.
-
-\subsubsection{For $L \haspatch P, D \neq C$:}
-Ancestors: $ D \le C \equiv D \le L $.
-
-Contents: $ D \isin C \equiv D \isin L \lor f $
-so $ D \isin C \equiv D \isin L $.
-
-So:
-\[ L \haspatch P \implies C \haspatch P \]
-
-\subsubsection{For $L \nothaspatch P$:}
-
-Firstly, $C \not\in \py$ since if it were, $L \in \py$.
-Thus $D \neq C$.
-
-Now by contents of $L$, $D \notin L$, so $D \notin C$.
-
-So:
-\[ L \nothaspatch P \implies C \nothaspatch P \]
-$\qed$
-
-\subsection{Foreign Inclusion:}
-
-Simple Foreign Inclusion applies.  $\qed$
-
-\subsection{Foreign Contents:}
-
-Only relevant if $\patchof{C} = \bot$, and in that case Totally
-Foreign Contents applies. $\qed$
-
-\section{Create Base}
-
-Given $L$, create a Topbloke base branch initial commit $B$.
-\gathbegin
- B \hasparents \{ L \}
-\gathnext
- \patchof{B} = \pqn
-\gathnext
- D \isin B \equiv D \isin L \lor D = B
-\end{gather}
-
-\subsection{Conditions}
-
-\[ \eqn{ Ingredients }{
- \patchof{L} = \pa{L} \lor \patchof{L} = \bot
-}\]
-\[ \eqn{ Create Acyclic }{
- L \not\haspatch \pq
-}\]
-
-\subsection{No Replay}
-
-Ingredients Prevent Replay applies.  $\qed$
-
-\subsection{Unique Base}
-
-Not applicable.
-
-\subsection{Tip Contents}
-
-Not applicable.
-
-\subsection{Base Acyclic}
-
-Consider some $D \isin B$.  If $D = B$, $D \in \pqn$.
-If $D \neq B$, $D \isin L$, and by Create Acyclic
-$D \not\in \pqy$.  $\qed$
-
-\subsection{Coherence and Patch Inclusion}
-
-Consider some $D \in \p$.
-$B \not\in \py$ so $D \neq B$.  So $D \isin B \equiv D \isin L$
-and $D \le B \equiv D \le L$.
-
-Thus $L \haspatch \p \implies B \haspatch P$
-and $L \nothaspatch \p \implies B \nothaspatch P$.
-
-$\qed$.
-
-\subsection{Foreign Inclusion}
-
-Simple Foreign Inclusion applies. $\qed$
-
-\subsection{Foreign Contents}
-
-Not applicable.
-
-\section{Create Tip}
-
-Given a Topbloke base $B$, create a tip branch initial commit B.
-\gathbegin
- C \hasparents \{ B \}
-\gathnext
- \patchof{B} = \pqy
-\gathnext
- D \isin C \equiv D \isin B \lor D = C
-\end{gather}
-
-\subsection{Conditions}
-
-\[ \eqn{ Ingredients }{
- \patchof{B} = \pqn
-}\]
-\[ \eqn{ No Sneak }{
- \pendsof{B}{\pqy} = \{ \}
-}\]
-
-\subsection{No Replay}
-
-Ingredients Prevent Replay applies.  $\qed$
-
-\subsection{Unique Base}
-
-Trivially, $\pendsof{C}{\pqn} = \{B\}$ so $\baseof{C} = B$.  $\qed$
-
-\subsection{Tip Contents}
-
-Consider some arbitrary commit $D$.  If $D = C$, trivially satisfied.
-
-If $D \neq C$, $D \isin C \equiv D \isin B$.
-By Base Acyclic of $B$, $D \isin B \implies D \not\in \pqy$.
-So $D \isin C \equiv D \isin \baseof{B}$.
-
-$\qed$
-
-\subsection{Base Acyclic}
-
-Not applicable.
-
-\subsection{Coherence and Patch Inclusion}
-
-$$
-\begin{cases}
-  \p = \pq    \lor B \haspatch \p : & C \haspatch \p \\
-  \p \neq \pq \land B \nothaspatch \p : & C \nothaspatch \p
-\end{cases}
-$$
-
-\proofstarts
-~ Consider some $D \in \py$.
-
-\subsubsection{For $\p = \pq$:}
-
-By Base Acyclic, $D \not\isin B$.  So $D \isin C \equiv D = C$.
-By No Sneak, $D \le B \equiv D = C$.  Thus $C \haspatch \pq$.
-
-\subsubsection{For $\p \neq \pq$:}
-
-$D \neq C$.  So $D \isin C \equiv D \isin B$,
-and $D \le C \equiv D \le B$.
-
-$\qed$
-
-\subsection{Foreign Inclusion}
-
-Simple Foreign Inclusion applies.  $\qed$
-
-\subsection{Foreign Contents}
-
-Not applicable.
-
-\section{Anticommit}
-
-Given $L$ and $\pr$ as represented by $R^+, R^-$.
-Construct $C$ which has $\pr$ removed.
-Used for removing a branch dependency.
-\gathbegin
- C \hasparents \{ L \}
-\gathnext
- \patchof{C} = \patchof{L}
-\gathnext
- \mergeof{C}{L}{R^+}{R^-}
-\end{gather}
-
-\subsection{Conditions}
-
-\[ \eqn{ Ingredients }{
-R^+ \in \pry \land R^- = \baseof{R^+}
-}\]
-\[ \eqn{ Into Base }{
- L \in \pn
-}\]
-\[ \eqn{ Unique Tip }{
- \pendsof{L}{\pry} = \{ R^+ \}
-}\]
-\[ \eqn{ Currently Included }{
- L \haspatch \pry
-}\]
-
-\subsection{Ordering of Ingredients:}
-
-By Unique Tip, $R^+ \le L$.  By definition of $\base$, $R^- \le R^+$
-so $R^- \le L$.  So $R^+ \le C$ and $R^- \le C$.
-$\qed$
-
-(Note that $R^+ \not\le R^-$, i.e. the merge base
-is a descendant, not an ancestor, of the 2nd parent.)
-
-\subsection{No Replay}
-
-By definition of $\merge$,
-$D \isin C \implies D \isin L \lor D \isin R^- \lor D = C$.
-So, by Ordering of Ingredients,
-Ingredients Prevent Replay applies.  $\qed$
-
-\subsection{Desired Contents}
-
-\[ D \isin C \equiv [ D \notin \pry \land D \isin L ] \lor D = C \]
-\proofstarts
-
-\subsubsection{For $D = C$:}
-
-Trivially $D \isin C$.  OK.
-
-\subsubsection{For $D \neq C, D \not\le L$:}
-
-By No Replay $D \not\isin L$.  Also $D \not\le R^-$ hence
-$D \not\isin R^-$.  Thus $D \not\isin C$.  OK.
-
-\subsubsection{For $D \neq C, D \le L, D \in \pry$:}
-
-By Currently Included, $D \isin L$.
-
-By Tip Self Inpatch, $D \isin R^+ \equiv D \le R^+$, but by
-by Unique Tip, $D \le R^+ \equiv D \le L$.
-So $D \isin R^+$.
-
-By Base Acyclic, $D \not\isin R^-$.
-
-Apply $\merge$: $D \not\isin C$.  OK.
-
-\subsubsection{For $D \neq C, D \le L, D \notin \pry$:}
-
-By Tip Contents for $R^+$, $D \isin R^+ \equiv D \isin R^-$.
-
-Apply $\merge$: $D \isin C \equiv D \isin L$.  OK.
-
-$\qed$
-
-\subsection{Unique Base}
-
-Into Base means that $C \in \pn$, so Unique Base is not
-applicable. $\qed$
-
-\subsection{Tip Contents}
-
-Again, not applicable. $\qed$
-
-\subsection{Base Acyclic}
-
-By Base Acyclic for $L$, $D \isin L \implies D \not\in \py$.
-And by Into Base $C \not\in \py$.
-Now from Desired Contents, above, $D \isin C
-\implies D \isin L \lor D = C$, which thus
-$\implies D \not\in \py$.  $\qed$.
-
-\subsection{Coherence and Patch Inclusion}
-
-Need to consider some $D \in \py$.  By Into Base, $D \neq C$.
-
-\subsubsection{For $\p = \pr$:}
-By Desired Contents, above, $D \not\isin C$.
-So $C \nothaspatch \pr$.
-
-\subsubsection{For $\p \neq \pr$:}
-By Desired Contents, $D \isin C \equiv D \isin L$
-(since $D \in \py$ so $D \not\in \pry$).
-
-If $L \nothaspatch \p$, $D \not\isin L$ so $D \not\isin C$.
-So $L \nothaspatch \p \implies C \nothaspatch \p$.
-
-Whereas if $L \haspatch \p$, $D \isin L \equiv D \le L$.
-so $L \haspatch \p \implies C \haspatch \p$.
-
-$\qed$
-
-\subsection{Foreign Inclusion}
-
-Consider some $D$ s.t. $\patchof{D} = \bot$.  $D \neq C$.
-So by Desired Contents $D \isin C \equiv D \isin L$.
-By Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$.
-
-And $D \le C \equiv D \le L$.
-Thus $D \isin C \equiv D \le C$.
-
-$\qed$
-
-\subsection{Foreign Contents}
-
-Not applicable.
-
-\section{Merge}
-
-Merge commits $L$ and $R$ using merge base $M$:
-\gathbegin
- C \hasparents \{ L, R \}
-\gathnext
- \patchof{C} = \patchof{L}
-\gathnext
- \mergeof{C}{L}{M}{R}
-\end{gather}
-We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$.
-
-\subsection{Conditions}
-\[ \eqn{ Ingredients }{
- M \le L, M \le R
-}\]
-\[ \eqn{ Tip Merge }{
- L \in \py \implies
-   \begin{cases}
-      R \in \py : & \baseof{R} \ge \baseof{L}
-              \land [\baseof{L} = M \lor \baseof{L} = \baseof{M}] \\
-      R \in \pn : & M = \baseof{L} \\
-      \text{otherwise} : & \false
-   \end{cases}
-}\]
-\[ \eqn{ Merge Acyclic }{
-    L \in \pn
-   \implies
-    R \nothaspatch \p
-}\]
-\[ \eqn{ Removal Merge Ends }{
-    X \not\haspatch \p \land
-    Y \haspatch \p \land
-    M \haspatch \p
-  \implies
-    \pendsof{Y}{\py} = \pendsof{M}{\py}
-}\]
-\[ \eqn{ Addition Merge Ends }{
-    X \not\haspatch \p \land
-    Y \haspatch \p \land
-    M \nothaspatch \p
-   \implies \left[
-    \bigforall_{E \in \pendsof{X}{\py}} E \le Y
-   \right]
-}\]
-\[ \eqn{ Foreign Merges }{
-    \patchof{L} = \bot \equiv \patchof{R} = \bot
-}\]
-
-\subsection{Non-Topbloke merges}
-
-We require both $\patchof{L} = \bot$ and $\patchof{R} = \bot$
-(Foreign Merges, above).
-I.e. not only is it forbidden to merge into a Topbloke-controlled
-branch without Topbloke's assistance, it is also forbidden to
-merge any Topbloke-controlled branch into any plain git branch.
-
-Given those conditions, Tip Merge and Merge Acyclic do not apply.
-And $Y \not\in \py$ so $\neg [ Y \haspatch \p ]$ so neither
-Merge Ends condition applies.
-
-So a plain git merge of non-Topbloke branches meets the conditions and
-is therefore consistent with our scheme.
-
-\subsection{No Replay}
-
-By definition of $\merge$,
-$D \isin C \implies D \isin L \lor D \isin R \lor D = C$.
-So, by Ingredients,
-Ingredients Prevent Replay applies.  $\qed$
-
-\subsection{Unique Base}
-
-Need to consider only $C \in \py$, ie $L \in \py$,
-and calculate $\pendsof{C}{\pn}$.  So we will consider some
-putative ancestor $A \in \pn$ and see whether $A \le C$.
-
-By Exact Ancestors for C, $A \le C \equiv A \le L \lor A \le R \lor A = C$.
-But $C \in py$ and $A \in \pn$ so $A \neq C$.
-Thus $A \le C \equiv A \le L \lor A \le R$.
-
-By Unique Base of L and Transitive Ancestors,
-$A \le L \equiv A \le \baseof{L}$.
-
-\subsubsection{For $R \in \py$:}
-
-By Unique Base of $R$ and Transitive Ancestors,
-$A \le R \equiv A \le \baseof{R}$.
-
-But by Tip Merge condition on $\baseof{R}$,
-$A \le \baseof{L} \implies A \le \baseof{R}$, so
-$A \le \baseof{R} \lor A \le \baseof{L} \equiv A \le \baseof{R}$.
-Thus $A \le C \equiv A \le \baseof{R}$.
-That is, $\baseof{C} = \baseof{R}$.
-
-\subsubsection{For $R \in \pn$:}
-
-By Tip Merge condition on $R$ and since $M \le R$,
-$A \le \baseof{L} \implies A \le R$, so
-$A \le R \lor A \le \baseof{L} \equiv A \le R$.
-Thus $A \le C \equiv A \le R$.
-That is, $\baseof{C} = R$.
-
-$\qed$
-
-\subsection{Coherence and Patch Inclusion}
-
-Need to determine $C \haspatch \p$ based on $L,M,R \haspatch \p$.
-This involves considering $D \in \py$.
-
-\subsubsection{For $L \nothaspatch \p, R \nothaspatch \p$:}
-$D \not\isin L \land D \not\isin R$.  $C \not\in \py$ (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. $C \nothaspatch \p$.
-
-\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 $C \haspatch \p$.
-
-For $D \neq C$: $D \le C \equiv D \le L \lor D \le R
- \equiv D \isin L \lor D \isin R$.
-(Likewise $D \le C \equiv D \le X \lor D \le Y$.)
-
-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 $C \haspatch \p$.
-
-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 $C \haspatch \p$.
-
-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 $C \haspatch \p$.
-
-So indeed $L \haspatch \p \land R \haspatch \p \implies C \haspatch \p$.
-
-\subsubsection{For (wlog) $X \not\haspatch \p, Y \haspatch \p$:}
-
-$M \haspatch \p \implies C \nothaspatch \p$.
-$M \nothaspatch \p \implies C \haspatch \p$.
-
-\proofstarts
-
-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 $D \isin C \equiv M \nothaspatch \p \land D \le C$
-(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.
-
-Consider $D \neq C, M \nothaspatch P, D \isin Y$:
-$D \le Y$ so $D \le C$.
-$D \not\isin M$ 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 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 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.
-
-Consider $D \neq C, M \haspatch P, D \not\isin Y$:
-By $\merge$, $D \not\isin C$.  OK.
-
-$\qed$
-
-\subsection{Base Acyclic}
-
-This applies when $C \in \pn$.
-$C \in \pn$ when $L \in \pn$ so by Merge Acyclic, $R \nothaspatch \p$.
-
-Consider some $D \in \py$.
-
-By Base Acyclic of $L$, $D \not\isin L$.  By the above, $D \not\isin
-R$.  And $D \neq C$.  So $D \not\isin C$.
-
-$\qed$
-
-\subsection{Tip Contents}
-
-We need worry only about $C \in \py$.
-And $\patchof{C} = \patchof{L}$
-so $L \in \py$ so $L \haspatch \p$.  We will use the Unique Base
-of $C$, and its Coherence and Patch Inclusion, as just proved.
-
-Firstly we show $C \haspatch \p$: If $R \in \py$, then $R \haspatch
-\p$ and by Coherence/Inclusion $C \haspatch \p$ .  If $R \not\in \py$
-then by Tip Merge $M = \baseof{L}$ so by Base Acyclic and definition
-of $\nothaspatch$, $M \nothaspatch \p$.  So by Coherence/Inclusion $C
-\haspatch \p$ (whether $R \haspatch \p$ or $\nothaspatch$).
-
-We will consider an arbitrary commit $D$
-and prove the Exclusive Tip Contents form.
-
-\subsubsection{For $D \in \py$:}
-$C \haspatch \p$ so by definition of $\haspatch$, $D \isin C \equiv D
-\le C$.  OK.
-
-\subsubsection{For $D \not\in \py, R \not\in \py$:}
-
-$D \neq C$.  By Tip Contents of $L$,
-$D \isin L \equiv D \isin \baseof{L}$, and by Tip Merge condition,
-$D \isin L \equiv D \isin M$.  So by definition of $\merge$, $D \isin
-C \equiv D \isin R$.  And $R = \baseof{C}$ by Unique Base of $C$.
-Thus $D \isin C \equiv D \isin \baseof{C}$.  OK.
-
-\subsubsection{For $D \not\in \py, R \in \py$:}
-
-$D \neq C$.
-
-By Tip Contents
-$D \isin L \equiv D \isin \baseof{L}$ and
-$D \isin R \equiv D \isin \baseof{R}$.
-
-If $\baseof{L} = M$, trivially $D \isin M \equiv D \isin \baseof{L}.$
-Whereas if $\baseof{L} = \baseof{M}$, by definition of $\base$,
-$\patchof{M} = \patchof{L} = \py$, so by Tip Contents of $M$,
-$D \isin M \equiv D \isin \baseof{M} \equiv D \isin \baseof{L}$.
-
-So $D \isin M \equiv D \isin L$ and by $\merge$,
-$D \isin C \equiv D \isin R$.  But from Unique Base,
-$\baseof{C} = R$ so $D \isin C \equiv D \isin \baseof{C}$.  OK.
-
-$\qed$
-
-\subsection{Foreign Inclusion}
-
-Consider some $D$ s.t. $\patchof{D} = \bot$.
-By Foreign Inclusion of $L, M, R$:
-$D \isin L \equiv D \le L$;
-$D \isin M \equiv D \le M$;
-$D \isin R \equiv D \le R$.
-
-\subsubsection{For $D = C$:}
-
-$D \isin C$ and $D \le C$.  OK.
-
-\subsubsection{For $D \neq C, D \isin M$:}
-
-Thus $D \le M$ so $D \le L$ and $D \le R$ so $D \isin L$ and $D \isin
-R$.  So by $\merge$, $D \isin C$.  And $D \le C$.  OK.
-
-\subsubsection{For $D \neq C, D \not\isin M, D \isin X$:}
-
-By $\merge$, $D \isin C$.
-And $D \isin X$ means $D \le X$ so $D \le C$.
-OK.
-
-\subsubsection{For $D \neq C, D \not\isin M, D \not\isin L, D \not\isin R$:}
+\chapter{Data model}
 
-By $\merge$, $D \not\isin C$.
-And $D \not\le L, D \not\le R$ so $D \not\le C$.
-OK
+\input{notation.tex}
+\input{invariants.tex}
+\input{lemmas.tex}
+\input{annotations.tex}
 
-$\qed$
+\input{simple.tex}
+\input{create-base.tex}
+\input{create-tip.tex}
+\input{anticommit.tex}
+\input{merge.tex}
 
-\subsection{Foreign Contents}
+\chapter{Update strategy}
 
-Only relevant if $\patchof{L} = \bot$, in which case
-$\patchof{C} = \bot$ and by Foreign Merges $\patchof{R} = \bot$,
-so Totally Foreign Contents applies.  $\qed$
+\input{strategy.tex}
 
 \end{document}