X-Git-Url: http://www.chiark.greenend.org.uk/ucgi/~ian/git?a=blobdiff_plain;f=article.tex;h=d5451ea30c0e6b127615466679d03dd789417931;hb=7dc335c17ae313c006e2283a35ca214b213ffcd9;hp=992f0d46204d5007e96878b888bf084b01d57dbc;hpb=456d92924b93b80e43c32b9a5046b5a47600be2d;p=topbloke-formulae.git diff --git a/article.tex b/article.tex index 992f0d4..d5e73b1 100644 --- a/article.tex +++ b/article.tex @@ -1,4 +1,5 @@ \documentclass[a4paper,leqno]{strayman} +\errorcontextlines=50 \let\numberwithin=\notdef \usepackage{amsmath} \usepackage{mathabx} @@ -7,8 +8,17 @@ \usepackage{mdwlist} %\usepackage{accents} +\usepackage{fancyhdr} +\pagestyle{fancy} +\lhead[\rightmark]{} + +\let\stdsection\section +\renewcommand\section{\newpage\stdsection} + \renewcommand{\ge}{\geqslant} \renewcommand{\le}{\leqslant} +\newcommand{\nge}{\ngeqslant} +\newcommand{\nle}{\nleqslant} \newcommand{\has}{\sqsupseteq} \newcommand{\isin}{\sqsubseteq} @@ -18,8 +28,12 @@ \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 +41,14 @@ \newcommand{\py}{\pay{P}} \newcommand{\pn}{\pan{P}} +\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{>}}} @@ -35,6 +57,7 @@ \renewcommand{\implies}{\Rightarrow} \renewcommand{\equiv}{\Leftrightarrow} +\renewcommand{\nequiv}{\nLeftrightarrow} \renewcommand{\land}{\wedge} \renewcommand{\lor}{\vee} @@ -44,11 +67,18 @@ \newcommand{\pancsof}[2]{\pancs ( #1 , #2 ) } \newcommand{\pendsof}[2]{\pends ( #1 , #2 ) } -\newcommand{\patchof}[1]{{\mathcal P} ( #1 ) } -\newcommand{\baseof}[1]{{\mathcal B} ( #1 ) } +\newcommand{\merge}{{\mathcal M}} +\newcommand{\mergeof}[4]{\merge(#1,#2,#3,#4)} +%\newcommand{\merge}[4]{{#2 {{\frac{ #1 }{ #3 } #4}}}} +\newcommand{\patch}{{\mathcal P}} +\newcommand{\base}{{\mathcal B}} + +\newcommand{\patchof}[1]{\patch ( #1 ) } +\newcommand{\baseof}[1]{\base ( #1 ) } + +\newcommand{\eqntag}[2]{ #2 \tag*{\mbox{#1}} } \newcommand{\eqn}[2]{ #2 \tag*{\mbox{\bf #1}} } -\newcommand{\corrolary}[1]{ #1 \tag*{\mbox{\it Corrolary.}} } %\newcommand{\bigforall}{\mathop{\hbox{\huge$\forall$}}} \newcommand{\bigforall}{% @@ -59,177 +89,30 @@ {\hbox{\scriptsize$\forall$}}}% } -\newcommand{\proof}[1]{{\it Proof.} #1 $\square$} - -\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 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. - -\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 $\pn$. -All these sets are distinct. 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}} - A \neq E \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 $\p$; if -a patch is merged into a non-Topbloke branch and we inherit -it, we hope that git's merge algorithm will DTRT. - -\end{basedescript} - -\section{Invariants} - -\[ \eqn{No Replay:}{ - C \has D \implies C \ge D -}\] -\[\eqn{Unique Base:}{ - \bigforall_{C \in \py} \pendsof{C}{\pn} = \{ B \} -}\] -\[\eqn{Tip Contents:}{ - \bigforall_{C \in \py} D \isin C \equiv - { D \isin \baseof{C} \lor \atop - (D \in \py \land D \le C) } -}\] -\[\eqn{Base Acyclic:}{ - \bigforall_{B \in \pn} D \isin B \implies D \notin \py -}\] -\[\eqn{Coherence:}{ - \bigforall_{C,\p} C \haspatch \p \lor C \nothaspatch \p -}\] - -\section{Some lemmas} - -\[ \eqn{Exclusive Tip Contents:}{ - \bigforall_{C \in \py} - \neg \left[ D \isin \baseof{C} \land (D \in \py \land D \le C - \right )] -}\] -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 $D \isin B \implies D \notin \py$. -} -\[ \corrolary{ - \bigforall_{C \in \py} D \isin C \equiv - \begin{cases} - D \in \py : & D \le C \\ - D \not\in \py : & D \isin \baseof{C} - \end{cases} -}\] - -\[ \eqn{Tip Self Inpatch:}{ - \bigforall_{C \in \py} C \haspatch \p -}\] -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 $ -} - -\[ \eqn{Exact Ancestors:}{ - \bigforall_{ C \hasparents \set{R} } - D \le C \equiv - ( \mathop{\hbox{\huge{$\vee$}}}_{R \in \set R} D \le R ) - \lor D = C -}\] - -\[ \eqn{Transitive Ancestors:}{ - \left[ \bigforall_{ E \in \pendsof{C}{\set P} } E \le M \right] \equiv - \left[ \bigforall_{ A \in \pancsof{C}{\set P} } A \le M \right] -}\] - -\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 -$A \le C$, 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 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} +\newcommand{\Largeexists}{\mathop{\hbox{\Large$\exists$}}} +\newcommand{\Largenexists}{\mathop{\hbox{\Large$\nexists$}}} -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 -$\forall_{\py \ni C} \pendsof{C}{\py} = \{C\}$. +\newcommand{\qed}{\square} +\newcommand{\proofstarts}{{\it Proof:}} +\newcommand{\proof}[1]{\proofstarts #1 $\qed$} -\section{Test more symbols} +\newcommand{\gathbegin}{\begin{gather} \tag*{}} +\newcommand{\gathnext}{\\ \tag*{}} -$ C \haspatch \p $ +\newcommand{\true}{t} +\newcommand{\false}{f} -$ C \nothaspatch \p $ - -$ \p \patchisin C $ +\begin{document} -$ \p \notpatchisin C $ +\input{notation.tex} +\input{invariants.tex} +\input{lemmas.tex} +\input{annotations.tex} -$ \{ B \} \areparents C $ +\input{simple.tex} +\input{create-base.tex} +\input{create-tip.tex} +\input{anticommit.tex} +\input{merge.tex} \end{document}