\usepackage{mathabx}
\usepackage{txfonts}
\usepackage{amsfonts}
+\usepackage{eufrak}
\usepackage{mdwlist}
+\usepackage{hyperref}
+%\usepackage{lastpage}
%\usepackage{accents}
+\usepackage{fancyhdr}
+\pagestyle{fancyplain}
+\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}
\newcommand{\nge}{\ngeqslant}
\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]{%
\hidehacktrue\ifhidehack#1\else#2\fi\hidehackfalse}
\newcommand{\pa}[1]{\hidefromedef{\varmathbb{#1}}{#1}}
+\newcommand{\stmt}[1]{\mathfrak{#1}}
+
\newcommand{\set}[1]{\mathbb{#1}}
\newcommand{\pay}[1]{\pa{#1}^+}
\newcommand{\pan}[1]{\pa{#1}^-}
\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}}
+
\newcommand{\pr}{\pa{R}}
\newcommand{\pry}{\pay{R}}
\newcommand{\prn}{\pan{R}}
%\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{\hasparent}{>_{\mkern-7.0mu _{1}}}
+\newcommand{\isparent}{<_{\mkern-18.0mu _{1}\mkern+9.0mu}}
+
+\newcommand{\hasparents}{>_{\mkern-7.0mu _{1:}}}
+\newcommand{\areparents}{<_{\mkern-14.0mu _{1:}\mkern+5.0mu}}
\renewcommand{\implies}{\Rightarrow}
\renewcommand{\equiv}{\Leftrightarrow}
\newcommand{\pancsof}[2]{\pancs ( #1 , #2 ) }
\newcommand{\pendsof}[2]{\pends ( #1 , #2 ) }
-\newcommand{\merge}{{\mathcal M}}
-\newcommand{\mergeof}[4]{\merge(#1,#2,#3,#4)}
+%\newcommand{\commitmerge}{\text{\commitmergename}}
+\newcommand{\commitmergeof}[4]{#1 \has \stmtmergeof{#2}{#3}{#4}}
%\newcommand{\merge}[4]{{#2 {{\frac{ #1 }{ #3 } #4}}}}
+\newcommand{\commitmergename}{Git Merge}
\newcommand{\patch}{{\mathcal P}}
\newcommand{\base}{{\mathcal B}}
+\newcommand{\depsreq}{{\mathcal G}}
+
+\newcommand{\allsrcs}{\set U}
+\newcommand{\allreach}{\set O}
+\newcommand{\allreachof}[1]{\set O^{#1}}
\newcommand{\patchof}[1]{\patch ( #1 ) }
\newcommand{\baseof}[1]{\base ( #1 ) }
+\newcommand{\depsreqof}[1]{\depsreq ( #1 ) }
+
+\newcommand{\foreign}{\pa F}
+\newcommand{\isforeign}[1]{#1 \in \foreign}
+
+\newcommand{\allpatches}{\Upsilon}
+\newcommand{\assign}{\leftarrow}
+\newcommand{\iassign}{\leftarrow}
+%\newcommand{\assign}{' =}
+
+\newcommand{\mergeof}[3]{\left\langle #1 \;\middle\langle #2 \middle\rangle\; #3 \right\rangle}
+
+\newcommand{\alg}[1]{\text{\bf #1}}
+\newcommand{\setmerge}{\mergeof{}{}{}}
+\newcommand{\setmergeof}[3]{\mergeof{#1}{#2}{#3}}
+\newcommand{\stmtmergeof}[3]{\mergeof{#1}{#2}{#3}}
+
+%\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{\tippn}{ \tipa \pn }
+\newcommand{\tippy}{ \tipa \py }
+
%\newcommand{\bigforall}{\mathop{\hbox{\huge$\forall$}}}
\newcommand{\bigforall}{%
\mathop{\mathchoice%
{\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$\nexists$}}}
\newcommand{\proofstarts}{{\it Proof:}}
\newcommand{\proof}[1]{\proofstarts #1 $\qed$}
-\newcommand{\gathbegin}{\begin{gather} \tag*{}}
-\newcommand{\gathnext}{\\ \tag*{}}
+\newcommand{\commitproof}[1]{{\bf Proof of commit generation conditions:}
+ #1 $\qed$}
+\newcommand{\condproof}[2]{
-\newcommand{\true}{t}
-\newcommand{\false}{f}
-
-\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 xxx
-
- 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 $
-}
+{\it #1:} #2
-\[ \eqn{Exact Ancestors:}{
- \bigforall_{ C \hasparents \set{R} }
- D \le C \equiv
- ( \mathop{\hbox{\huge{$\vee$}}}_{R \in \set R} D \le R )
- \lor D = C
-}\]
-xxx proof tbd
-
-\[ \eqn{Transitive Ancestors:}{
- \left[ \bigforall_{ E \in \pendsof{C}{\set P} } E \le M \right] \equiv
- \left[ \bigforall_{ A \in \pancsof{C}{\set P} } A \le M \right]
-}\]
-
-\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 $A \le A''$.
}
-\[ \eqn{Calculation Of Ends:}{
- \bigforall_{C \hasparents \set A}
- \pendsof{C}{\set P} =
- \begin{cases}
- C \in \p : & \{ C \}
- \\
- C \not\in \p : & \displaystyle
- \left\{ E \Big|
- \Bigl[ \Largeexists_{A \in \set A}
- E \in \pendsof{A}{\set P} \Bigr] \land
- \Bigl[ \Largenexists_{B \in \set A}
- E \neq B \land E \le B \Bigr]
- \right\}
- \end{cases}
-}\]
-xxx proof tbd
-
-\[ \eqn{Totally Foreign Contents:}{
- \bigforall_{C \hasparents \set A}
- \left[
- \patchof{C} = \bot \land
- \bigforall_{A \in \set A} \patchof{A} = \bot
- \right]
- \implies
- \left[
- D \isin C
- \implies
- \patchof{D} = \bot
- \right]
-}\]
-xxx proof tbd
-
-\subsection{No Replay for Merge Results}
-
-If we are constructing $C$, with,
-\gathbegin
- \mergeof{C}{L}{M}{R}
-\gathnext
- L \le C
-\gathnext
- R \le C
-\end{gather}
-No Replay is preserved. \proofstarts
-
-\subsubsection{For $D=C$:} $D \isin C, D \le C$. OK.
-
-\subsubsection{For $D \isin L \land D \isin R$:}
-$D \isin C$. And $D \isin L \implies D \le L \implies D \le C$. OK.
-
-\subsubsection{For $D \neq C \land D \not\isin L \land D \not\isin R$:}
-$D \not\isin C$. OK.
-
-\subsubsection{For $D \neq C \land (D \isin L \equiv D \not\isin R)
- \land D \not\isin M$:}
-$D \isin C$. Also $D \isin L \lor D \isin R$ so $D \le L \lor D \le
-R$ so $D \le C$. OK.
-
-\subsubsection{For $D \neq C \land (D \isin L \equiv D \not\isin R)
- \land D \isin M$:}
-$D \not\isin C$. OK.
-
-$\qed$
-
-\section{Commit annotation}
-
-We annotate each Topbloke commit $C$ with:
-\gathbegin
- \patchof{C}
-\gathnext
- \baseof{C}, \text{ if } C \in \py
-\gathnext
- \bigforall_{\pa{Q}}
- \text{ either } C \haspatch \pa{Q} \text{ or } C \nothaspatch \pa{Q}
-\gathnext
- \bigforall_{\pay{Q} \not\ni C} \pendsof{C}{\pay{Q}}
-\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.
-
-$C \haspatch \pa{Q}$ or $\nothaspatch \pa{Q}$ is represented as the
-set $\{ \pa{Q} | C \haspatch \pa{Q} \}$. Whether $C \haspatch \pa{Q}$
-is in stated
-(in terms of $I \haspatch \pa{Q}$ or $I \nothaspatch \pa{Q}$
-for the ingredients $I$),
-in the proof of Coherence for each kind of commit.
-
-$\pendsof{C}{\pa{Q}^+}$ 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 \{ A \} \\
-\tag*{} \patchof{C} = \patchof{A} \\
-\tag*{} D \isin C \equiv D \isin A \lor D = C
-\end{gather}
-This also covers Topbloke-generated commits on plain git branches:
-Topbloke strips the metadata when exporting.
-
-\subsection{No Replay}
-Trivial.
-
-\subsection{Unique Base}
-If $A, C \in \py$ then by Calculation of Ends for
-$C, \py, C \not\in \py$:
-$\pendsof{C}{\pn} = \pendsof{A}{\pn}$ so
-$\baseof{C} = \baseof{A}$. $\qed$
-
-\subsection{Tip Contents}
-We need to consider only $A, C \in \py$. From Tip Contents for $A$:
-\[ D \isin A \equiv D \isin \baseof{A} \lor ( D \in \py \land D \le A ) \]
-Substitute into the contents of $C$:
-\[ D \isin C \equiv D \isin \baseof{A} \lor ( D \in \py \land D \le A )
- \lor D = C \]
-Since $D = C \implies D \in \py$,
-and substituting in $\baseof{C}$, this gives:
-\[ D \isin C \equiv D \isin \baseof{C} \lor
- (D \in \py \land D \le A) \lor
- (D = C \land D \in \py) \]
-\[ \equiv D \isin \baseof{C} \lor
- [ D \in \py \land ( D \le A \lor D = C ) ] \]
-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 $A, C \in \pn$.
-
-For $D = C$: $D \in \pn$ so $D \not\in \py$. OK.
-
-For $D \neq C$: $D \isin C \equiv D \isin A$, so by Base Acyclic for
-$A$, $D \isin C \implies D \not\in \py$.
-
-$\qed$
-
-\subsection{Coherence and patch inclusion}
-
-Need to consider $D \in \py$
-
-\subsubsection{For $A \haspatch P, D = C$:}
-
-Ancestors of $C$:
-$ D \le C $.
-
-Contents of $C$:
-$ D \isin C \equiv \ldots \lor \true \text{ so } D \haspatch C $.
-
-\subsubsection{For $A \haspatch P, D \neq C$:}
-Ancestors: $ D \le C \equiv D \le A $.
-
-Contents: $ D \isin C \equiv D \isin A \lor f $
-so $ D \isin C \equiv D \isin A $.
-
-So:
-\[ A \haspatch P \implies C \haspatch P \]
-
-\subsubsection{For $A \nothaspatch P$:}
-
-Firstly, $C \not\in \py$ since if it were, $A \in \py$.
-Thus $D \neq C$.
-
-Now by contents of $A$, $D \notin A$, so $D \notin C$.
-
-So:
-\[ A \nothaspatch P \implies C \nothaspatch P \]
-$\qed$
-
-\subsection{Foreign inclusion:}
-
-If $D = C$, trivial. For $D \neq C$:
-$D \isin C \equiv D \isin A \equiv D \le A \equiv D \le C$. $\qed$
-
-\subsection{Foreign Contents:}
-
-Only relevant if $\patchof{C} = \bot$. Trivial by Foreign Contents of
-$A$. $\qed$
-
-xxx fixme not trivial use Totally Foreign Contents
-
-\section{Create Base}
-
-Given $L$, create a Topbloke base branch initial commit $B$.
-\gathbegin
- B \hasparents \{ L \}
-\gathnext
- \patchof{B} = \pa{B}
-\gathnext
- D \isin B \equiv D \isin L \lor D = B
-\end{gather}
-
-\subsection{Conditions}
-
-\[ \eqn{ Ingredients }{
- \patchof{L} = \pa{L} \lor \patchof{L} = \bot
-}\]
-\[ \eqn{ Non-recursion }{
- L \not\in \pa{B}
-}\]
-
-\subsection{No Replay}
-
-If $\patchof{L} = \pa{L}$, trivial by Base Acyclic for $L$.
-
-If $\patchof{L} = \bot$, xxx
-
-Trivial from Base Acyclic for $L$. $\qed$
-
-\subsection{Unique Base}
-
-Not applicable. $\qed$
-
-\subsection{Tip Contents}
-
-Not applicable. $\qed$
-
-\subsection{Base Acyclic}
-
-xxx
-
-xxx unfinished
-
-\section{Create Tip}
-
-xxx tbd
-
-\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 ${L, R^+, R^-}$:}
-
-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}
-
-No Replay for Merge Results applies. $\qed$
+\newcommand{\statement}[2]{\[\eqn{ #1 }{ #2 }\]}
-\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. $\qed$
-
-\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}
-
-No Replay for Merge Results 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 $D \neq C, D \isin M$:}
+\newcommand{\gathbegin}{\begin{gather} \tag*{}}
+\newcommand{\gathnext}{\\ \tag*{}}
-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.
+\newcommand{\true}{t}
+\newcommand{\false}{f}
-\subsubsection{For $D \neq C, D \not\isin M, D \isin X$:}
+\begin{document}
-By $\merge$, $D \isin C$.
-And $D \isin X$ means $D \le X$ so $D \le C$.
-OK.
+\chapter{Data model}
-\subsubsection{For $D \neq C, D \not\isin M, D \not\isin L, D \not\isin R$:}
+\input{notation.tex}
+\input{invariants.tex}
+\input{lemmas.tex}
+\input{annotations.tex}
-By $\merge$, $D \not\isin C$.
-And $D \not\le L, D \not\le R$ so $D \not\le C$.
-OK
+\input{simple.tex}
+\input{create-base.tex}
+\input{create-tip.tex}
+\input{anticommit.tex}
+\input{merge.tex}
+\input{pseudomerge.tex}
-$\qed$
+\chapter{Update strategy}
-\subsection{Foreign Contents}
+\input{strategy.tex}
+\input{ranking.tex}
+\input{traversal.tex}
-xxx use Totally Foreign Contents
+%\AtEndDocument{\par\bigskip\centerline{\Large\bfseries Document ends here}}
-If $\patchof{C} = \bot$, by Foreign Merges
-$\patchof{L} = \patchof{R} = \bot$.
+\par$\bigboxdot$
\end{document}