X-Git-Url: http://www.chiark.greenend.org.uk/ucgi/~ian/git?p=topbloke-formulae.git;a=blobdiff_plain;f=article.tex;h=4c5f6751682c83a13e478fdb11d64496abd1bd22;hp=9ca8012f6589cffebe568008215f152fafe1083c;hb=1059b2343f8b2979915db881a067cd3307cdead2;hpb=7a8a90bdecc23aff58e3cb88bc93580d8450dc0e diff --git a/article.tex b/article.tex index 9ca8012..4c5f675 100644 --- a/article.tex +++ b/article.tex @@ -1,4 +1,5 @@ \documentclass[a4paper,leqno]{strayman} +\errorcontextlines=50 \let\numberwithin=\notdef \usepackage{amsmath} \usepackage{mathabx} @@ -18,8 +19,12 @@ \newcommand{\haspatch}{\sqSupset} \newcommand{\patchisin}{\sqSubset} -\newcommand{\set}[1]{\mathbb #1} -\newcommand{\pa}[1]{\varmathbb #1} + \newif\ifhidehack\hidehackfalse + \DeclareRobustCommand\hidefromedef[2]{% + \hidehacktrue\ifhidehack#1\else#2\fi\hidehackfalse} + \newcommand{\pa}[1]{\hidefromedef{\varmathbb{#1}}{#1}} + +\newcommand{\set}[1]{\mathbb{#1}} \newcommand{\pay}[1]{\pa{#1}^+} \newcommand{\pan}[1]{\pa{#1}^-} @@ -27,6 +32,10 @@ \newcommand{\py}{\pay{P}} \newcommand{\pn}{\pan{P}} +\newcommand{\pr}{\pa{R}} +\newcommand{\pry}{\pay{R}} +\newcommand{\prn}{\pan{R}} + %\newcommand{\hasparents}{\underaccent{1}{>}} %\newcommand{\hasparents}{{% % \declareslashed{}{_{_1}}{0}{-0.8}{>}\slashed{>}}} @@ -38,11 +47,20 @@ \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{\merge}[4]{{\mathcal M}(#1,#2,#3,#4)} +%\newcommand{\merge}[4]{{#2 {{\frac{ #1 }{ #3 } #4}}}} + +\newcommand{\patch}{{\mathcal P}} +\newcommand{\base}{{\mathcal B}} -\newcommand{\patchof}[1]{{\mathcal P} ( #1 ) } -\newcommand{\baseof}[1]{{\mathcal B} ( #1 ) } +\newcommand{\patchof}[1]{\patch ( #1 ) } +\newcommand{\baseof}[1]{\base ( #1 ) } \newcommand{\eqn}[2]{ #2 \tag*{\mbox{\bf #1}} } \newcommand{\corrolary}[1]{ #1 \tag*{\mbox{\it Corrolary.}} } @@ -56,7 +74,17 @@ {\hbox{\scriptsize$\forall$}}}% } -\newcommand{\proof}[1]{{\it Proof.} #1 $\square$} +\newcommand{\Largeexists}{\mathop{\hbox{\Large$\exists$}}} +\newcommand{\Largenexists}{\mathop{\hbox{\Large$\nexists$}}} + +\newcommand{\qed}{\square} +\newcommand{\proof}[1]{{\it Proof.} #1 $\qed$} + +\newcommand{\gathbegin}{\begin{gather} \tag*{}} +\newcommand{\gathnext}{\\ \tag*{}} + +\newcommand{\true}{t} +\newcommand{\false}{f} \begin{document} @@ -78,11 +106,11 @@ $ 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 in non-Topbloke-controlled branches; these are considered -normal, forward, commits. For merges and Topbloke-generated -anticommits, the ``change made'' is only to be thought of as any -conflict resolution. This is not a partial order because it is not -transitive. +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 @@ -93,23 +121,23 @@ 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}} - A \neq E \land E \le A \} $ -i.e. all $\le$-maximal commits 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}} + E \neq A \land E \le A \} $ +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. +See Unique Base, below. \item[ $ C \haspatch \p $ ] $\displaystyle \bigforall_{D \in \py} D \isin C \equiv D \le C $. @@ -119,19 +147,32 @@ $\displaystyle \bigforall_{D \in \py} D \isin C \equiv D \le C $. $\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 $\p$; if -a patch is merged into a non-Topbloke branch and we inherit -it, we hope that git's merge algorithm will DTRT. +Non-Topbloke commits are $\nothaspatch \p$ for all $\p$; if a Topbloke +patch is applied to a non-Topbloke branch and then bubbles back to +the Topbloke patch itself, we hope that git's merge algorithm will +DTRT or that the user will no longer care about the Topbloke patch. -\end{basedescript} +\item[ $\displaystyle \merge{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. \\ \[ \eqn{No Replay:}{ C \has D \implies C \ge D }\] \[\eqn{Unique Base:}{ - \bigforall_{C \in \py} \pends{C}{\pn} = \{ B \} + \bigforall_{C \in \py} \pendsof{C}{\pn} = \{ B \} }\] \[\eqn{Tip Contents:}{ \bigforall_{C \in \py} D \isin C \equiv @@ -141,13 +182,19 @@ it, we hope that git's merge algorithm will DTRT. \[\eqn{Base Acyclic:}{ \bigforall_{B \in \pn} D \isin B \implies D \notin \py }\] +\[\eqn{Coherence:}{ + \bigforall_{C,\p} C \haspatch \p \lor C \nothaspatch \p +}\] +\[\eqn{Foreign Inclusion:}{ + \bigforall_{D \text{ s.t. } \patchof{D} = \bot} D \isin C \equiv D \leq C +}\] \section{Some lemmas} \[ \eqn{Exclusive Tip Contents:}{ \bigforall_{C \in \py} - \neg \left[ D \isin \baseof{C} \land (D \in \py \land D \le C - \right )] + \neg \Bigl[ D \isin \baseof{C} \land ( D \in \py \land D \le C ) + \Bigr] }\] Ie, the two limbs of the RHS of Tip Contents are mutually exclusive. @@ -182,28 +229,262 @@ $ \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 +$A \le M$ 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. +commits, this terminates with $A'' \in \pends()$, ie $A'' \le M$ +by the LHS. And $A \le A''$. } +\[ \eqn{Calculation Of Ends:}{ + \bigforall_{C \hasparents \set A} + \pendsof{C}{\set P} = + \left\{ E \Big| + \Bigl[ \Largeexists_{A \in \set A} + E \in \pendsof{A}{\set P} \Bigr] \land + \Bigl[ \Largenexists_{B \in \set A} + E \neq B \land E \le B \Bigr] + \right\} +}\] +XXX proof TBD. + +\subsection{No Replay for Merge Results} + +If we are constructing $C$, given +\gathbegin + \merge{C}{L}{M}{R} +\gathnext + L \le C +\gathnext + R \le C +\end{gather} +No Replay is preserved. {\it Proof:} + +\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} + +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 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} + +\subsection{No Replay} +Trivial. + +\subsection{Unique Base} +If $A, C \in \py$ then $\baseof{C} = \baseof{A}$. $\qed$ + +\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 $D \neq C$: +$D \isin C \equiv D \isin A \equiv D \le A \equiv D \le C$. $\qed$ + +\section{Anticommit} + +Given $L, R^+, R^-$ where +$R^+ \in \pry, R^- = \baseof{R^+}$. +Construct $C$ which has $\pr$ removed. +Used for removing a branch dependency. +\gathbegin + C \hasparents \{ L \} +\gathnext + \patchof{C} = \patchof{L} +\gathnext + \merge{C}{L}{R^+}{R^-} +\end{gather} + +\subsection{Conditions} + +\[ \eqn{ Unique Tip }{ + \pendsof{L}{\pry} = \{ R^+ \} +}\] +\[ \eqn{ Currently Included }{ + L \haspatch \pry +}\] + +\subsection{Desired Contents} + +xxx need to prove $D \isin C \equiv D \not\in \pry \land D \isin L$. + +\subsection{No Replay} + +By Unique Tip, $R^+ \le L$. By definition of $\base$, $R^- \le R^+$ +so $R^- \le L$. So $R^+ \le C$ and $R^- \le C$ and No Replay for +Merge Results applies. $\qed$ + +\subsection{Unique Base} + +Need to consider only $C \in \py$, ie $L \in \py$. + +xxx tbd + +\section{Merge} + +Merge commits $L$ and $R$ using merge base $M$ ($M < L, M < R$): +\gathbegin + C \hasparents \{ L, R \} +\gathnext + \patchof{C} = \patchof{L} +\gathnext + \merge{C}{L}{M}{R} +\end{gather} + +\subsection{Conditions} + +\[ \eqn{ Tip Merge }{ + L \in \py \implies + \begin{cases} + R \in \py : & \baseof{R} \ge \baseof{L} + \land [\baseof{L} = M \lor \baseof{L} = \baseof{M}] \\ + R \in \pn : & R \ge \baseof{L} + \land M = \baseof{L} \\ + \text{otherwise} : & \false + \end{cases} +}\] + +\subsection{No Replay} + +See No Replay for Merge Results. + +\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, +$A \le L \equiv A \le \baseof{L}$. -\section{Test more symbols} +\subsubsection{For $R \in \py$:} -$ C \haspatch \p $ +By Unique Base of $R$ and Transitive Ancestors, +$A \le R \equiv A \le \baseof{R}$. -$ C \nothaspatch \p $ +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, $\baseof{C} = \baseof{R}$. -$ \p \patchisin C $ +\subsubsection{For $R \in \pn$:} -$ \p \notpatchisin C $ +By Tip Merge condition on $R$, +$A \le \baseof{L} \implies A \le R$, so +$A \le R \lor A \le \baseof{L} \equiv A \le R$. +Thus $A \le C \equiv A \le R$. +That is, $\baseof{C} = R$. -$ \{ B \} \areparents C $ +$\qed$ \end{document}