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=ad68c2ca763c9f9efd2ce59f75301dde9eae624e;hb=588087cf4ec7d7c90cb21ac1b3790b86283ba992;hpb=1bd4718fef57ea9276c2b38a27848b036d43963d diff --git a/article.tex b/article.tex index ad68c2c..1e7adc2 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 +8,20 @@ \usepackage{mdwlist} %\usepackage{accents} +\usepackage{fancyhdr} +\pagestyle{fancy} +\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{\nle}{\nleqslant} \newcommand{\has}{\sqsupseteq} \newcommand{\isin}{\sqsubseteq} @@ -17,9 +30,15 @@ \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{\set}[1]{\mathbb #1} -\newcommand{\pa}[1]{\varmathbb #1} +\newcommand{\set}[1]{\mathbb{#1}} \newcommand{\pay}[1]{\pa{#1}^+} \newcommand{\pan}[1]{\pa{#1}^-} @@ -27,14 +46,35 @@ \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{\hasparents}{>_{\mkern-7.0mu _{1:}}} +\newcommand{\areparents}{<_{\mkern-14.0mu _{1:}\mkern+5.0mu}} \renewcommand{\implies}{\Rightarrow} \renewcommand{\equiv}{\Leftrightarrow} +\renewcommand{\nequiv}{\nLeftrightarrow} \renewcommand{\land}{\wedge} \renewcommand{\lor}{\vee} @@ -44,11 +84,44 @@ \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{\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{\corrolary}[1]{ #1 \tag*{\mbox{\it Corrolary.}} } + +\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}{% @@ -58,241 +131,44 @@ {\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{\qed}{\square} -\newcommand{\proof}[1]{{\it Proof.} #1 $\qed$} +\newcommand{\proofstarts}{{\it Proof:}} +\newcommand{\proof}[1]{\proofstarts #1 $\qed$} \newcommand{\gathbegin}{\begin{gather} \tag*{}} \newcommand{\gathnext}{\\ \tag*{}} -\begin{document} - -\section{Notation} +\newcommand{\true}{t} +\newcommand{\false}{f} -\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 $\pn$. -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 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} -\newpage -\section{Invariants} - -We maintain these each time we construct a new commit. \\ -\[ \eqn{No Replay:}{ - C \has D \implies C \ge D -}\] -\[\eqn{Unique Base:}{ - \bigforall_{C \in \py} \pendsof{C}{\pn} = \{ B \} -}\] -\[\eqn{Tip Contents:}{ - \bigforall_{C \in \py} D \isin C \equiv - { D \isin \baseof{C} \lor \atop - (D \in \py \land D \le C) } -}\] -\[\eqn{Base Acyclic:}{ - \bigforall_{B \in \pn} D \isin B \implies D \notin \py -}\] -\[\eqn{Coherence:}{ - \bigforall_{C,\p} C \haspatch \p \lor C \nothaspatch \p -}\] -\[\eqn{Foreign Inclusion:}{ - \bigforall_{D \text{ s.t. } \patchof{D} = \bot} D \isin C \equiv D \leq C -}\] - -\section{Some lemmas} - -\[ \eqn{Exclusive Tip Contents:}{ - \bigforall_{C \in \py} - \neg \Bigl[ D \isin \baseof{C} \land ( D \in \py \land D \le C ) - \Bigr] -}\] -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 $D \isin B \implies D \notin \py$. -} -\[ \corrolary{ - \bigforall_{C \in \py} D \isin C \equiv - \begin{cases} - D \in \py : & D \le C \\ - D \not\in \py : & D \isin \baseof{C} - \end{cases} -}\] - -\[ \eqn{Tip Self Inpatch:}{ - \bigforall_{C \in \py} C \haspatch \p -}\] -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 -$A \le M$ 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 $A \le A''$. -} - -\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} - -\subsubsection{For $A \haspatch P, D = C$:} -\[ D \isin C \equiv \ldots \lor t \text{ so } D \haspatch C \] -\[ D \le C \] -OK - -\section{Test more symbols} +\begin{document} -$ C \haspatch \p $ +\chapter{Data model} -$ C \nothaspatch \p $ +\input{notation.tex} +\input{invariants.tex} +\input{lemmas.tex} +\input{annotations.tex} -$ \p \patchisin C $ +\input{simple.tex} +\input{create-base.tex} +\input{create-tip.tex} +\input{anticommit.tex} +\input{merge.tex} -$ \p \notpatchisin C $ +\chapter{Update strategy} -$ \{ B \} \areparents C $ +\input{strategy.tex} \end{document}