X-Git-Url: http://www.chiark.greenend.org.uk/ucgi/~ian/git?p=topbloke-formulae.git;a=blobdiff_plain;f=article.tex;h=1a1e9d14363d2b3b629a21e6320029754bdcc614;hp=36233ae4b7f147a2c67c7f2b96798c5feca1b4e0;hb=11f4dd11cebc837935410fb336c610a906fd1f27;hpb=259a2d309ccfbde19838df143ff1fc88843c7d4d diff --git a/article.tex b/article.tex index 36233ae..1a1e9d1 100644 --- a/article.tex +++ b/article.tex @@ -8,6 +8,13 @@ \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} @@ -109,7 +116,7 @@ $\set X$. \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 +graph. This is a partial order, namely the transitive closure of $ D \in \set X $ where $ C \hasparents \set X $. \item[ $ C \has D $ ] @@ -126,21 +133,21 @@ 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: +All of these sets are disjoint. Hence: \item[ $ \patchof{ C } $ ] -Either $\p$ s.t. $ C \in \p $, or $\bot$. +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 \} $ +$ \{ 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 \} $ + E \neq A \land E \le A \} $ i.e. all $\le$-maximal commits in $\pancsof{C}{\set P}$. \item[ $ \baseof{C} $ ] @@ -154,13 +161,13 @@ $\displaystyle \bigforall_{D \in \py} D \isin C \equiv D \le C $. \item[ $ C \nothaspatch \p $ ] $\displaystyle \bigforall_{D \in \py} D \not\isin C $. -~ Informally, $C$ has none of the contents of $\p$. +~ 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 +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. @@ -173,7 +180,7 @@ $\displaystyle D \isin C \equiv (D \not\isin L \land D \not\isin R) \land D \neq C : & \false \\ \text{otherwise} : & D \not\isin M \end{cases} -$ +$ \end{basedescript} \newpage @@ -207,8 +214,8 @@ We maintain these each time we construct a new commit. \\ \section{Some lemmas} -\[ \eqn{Alternative (overlapping) formulations defining - $\mergeof{C}{L}{M}{R}$:}{ +\subsection{Alternative (overlapping) formulations of $\mergeof{C}{L}{M}{R}$} +$$ D \isin C \equiv \begin{cases} D \isin L \equiv D \isin R : & D = C \lor D \isin L \\ @@ -217,18 +224,34 @@ We maintain these each time we construct a new commit. \\ 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 $R$. +$$ +\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} +\subsection{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{ @@ -243,9 +266,10 @@ So by Base Acyclic $D \isin B \implies D \notin \py$. \end{cases} }\] -\[ \eqn{Tip Self Inpatch:}{ +\subsection{Tip Self Inpatch} +$$ \bigforall_{C \in \py} C \haspatch \p -}\] +$$ Ie, tip commits contain their own patch. \proof{ @@ -254,23 +278,25 @@ $ \bigforall_{C \in \py}\bigforall_{D \in \py} D \isin C \equiv D \le C $ } -\[ \eqn{Exact Ancestors:}{ +\subsection{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 +$$ +\proof{ ~ Trivial.} -\[ \eqn{Transitive Ancestors:}{ +\subsection{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: +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, @@ -280,7 +306,8 @@ commits, this terminates with $A'' \in \pends()$, ie $A'' \le M$ by the LHS. And $A \le A''$. } -\[ \eqn{Calculation Of Ends:}{ +\subsection{Calculation Of Ends} +$$ \bigforall_{C \hasparents \set A} \pendsof{C}{\set P} = \begin{cases} @@ -288,16 +315,26 @@ by the LHS. And $A \le A''$. \\ C \not\in \p : & \displaystyle \left\{ E \Big| - \Bigl[ \Largeexists_{A \in \set A} + \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] + \Bigl[ \Largenexists_{B \in \set A, F \in \pendsof{B}{\p}} + E \neq F \land E \le F \Bigr] \right\} \end{cases} -}\] -xxx proof tbd +$$ +\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:}{ +\subsection{Ingredients Prevent Replay} +$$ \left[ {C \hasparents \set A} \land \\ @@ -309,7 +346,7 @@ xxx proof tbd \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. $D \in \set A$ @@ -317,7 +354,29 @@ xxx proof tbd $A \le C$ so $D \le C$. } -\[ \eqn{Totally Foreign Contents:}{ +\subsection{Simple Foreign Inclusion} +$$ + \left[ + C \hasparents \{ L \} + \land + \bigforall_{D} D \isin C \equiv D \isin L \lor D = C + \right] + \implies + \left[ + \bigforall_{D \text{ s.t. } \patchof{D} = \bot} + D \isin C \equiv D \le C + \right] +$$ +\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 $D \isin C \equiv D \le C$. +} + +\subsection{Totally Foreign Contents} +$$ \bigforall_{C \hasparents \set A} \left[ \patchof{C} = \bot \land @@ -329,7 +388,7 @@ xxx proof tbd \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 @@ -374,9 +433,9 @@ because $\forall_{\py \ni C} \; \pendsof{C}{\py} = \{C\}$. 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 +\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. @@ -386,24 +445,24 @@ Topbloke strips the metadata when exporting. Ingredients Prevent Replay applies. $\qed$ \subsection{Unique Base} -If $A, C \in \py$ then by Calculation of Ends for +If $L, C \in \py$ then by Calculation of Ends for $C, \py, C \not\in \py$: -$\pendsof{C}{\pn} = \pendsof{A}{\pn}$ so -$\baseof{C} = \baseof{A}$. $\qed$ +$\pendsof{C}{\pn} = \pendsof{L}{\pn}$ so +$\baseof{C} = \baseof{L}$. $\qed$ \subsection{Tip Contents} -We need to consider only $A, C \in \py$. From Tip Contents for $A$: -\[ D \isin A \equiv D \isin \baseof{A} \lor ( D \in \py \land D \le A ) \] +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{A} \lor ( D \in \py \land D \le A ) +\[ D \isin C \equiv D \isin \baseof{L} \lor ( D \in \py \land D \le L ) \lor D = C \] -Since $D = C \implies D \in \py$, +Since $D = C \implies D \in \py$, and substituting in $\baseof{C}$, this gives: \[ D \isin C \equiv D \isin \baseof{C} \lor - (D \in \py \land D \le A) \lor + (D \in \py \land D \le L) \lor (D = C \land D \in \py) \] \[ \equiv D \isin \baseof{C} \lor - [ D \in \py \land ( D \le A \lor D = C ) ] \] + [ D \in \py \land ( D \le L \lor D = C ) ] \] So by Exact Ancestors: \[ D \isin C \equiv D \isin \baseof{C} \lor ( D \in \py \land D \le C ) \] @@ -411,12 +470,12 @@ $\qed$ \subsection{Base Acyclic} -Need to consider only $A, C \in \pn$. +Need to consider only $L, C \in \pn$. For $D = C$: $D \in \pn$ so $D \not\in \py$. OK. -For $D \neq C$: $D \isin C \equiv D \isin A$, so by Base Acyclic for -$A$, $D \isin C \implies D \not\in \py$. +For $D \neq C$: $D \isin C \equiv D \isin L$, so by Base Acyclic for +$L$, $D \isin C \implies D \not\in \py$. $\qed$ @@ -424,7 +483,7 @@ $\qed$ Need to consider $D \in \py$ -\subsubsection{For $A \haspatch P, D = C$:} +\subsubsection{For $L \haspatch P, D = C$:} Ancestors of $C$: $ D \le C $. @@ -432,30 +491,29 @@ $ 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 $. +\subsubsection{For $L \haspatch P, D \neq C$:} +Ancestors: $ D \le C \equiv D \le L $. -Contents: $ D \isin C \equiv D \isin A \lor f $ -so $ D \isin C \equiv D \isin A $. +Contents: $ D \isin C \equiv D \isin L \lor f $ +so $ D \isin C \equiv D \isin L $. So: -\[ A \haspatch P \implies C \haspatch P \] +\[ L \haspatch P \implies C \haspatch P \] -\subsubsection{For $A \nothaspatch P$:} +\subsubsection{For $L \nothaspatch P$:} -Firstly, $C \not\in \py$ since if it were, $A \in \py$. +Firstly, $C \not\in \py$ since if it were, $L \in \py$. Thus $D \neq C$. -Now by contents of $A$, $D \notin A$, so $D \notin C$. +Now by contents of $L$, $D \notin L$, so $D \notin C$. So: -\[ A \nothaspatch P \implies C \nothaspatch P \] +\[ L \nothaspatch P \implies C \nothaspatch P \] $\qed$ -\subsection{Foreign inclusion:} +\subsection{Foreign Inclusion:} -If $D = C$, trivial. For $D \neq C$: -$D \isin C \equiv D \isin A \equiv D \le A \equiv D \le C$. $\qed$ +Simple Foreign Inclusion applies. $\qed$ \subsection{Foreign Contents:} @@ -475,11 +533,8 @@ Given $L$, create a Topbloke base branch initial commit $B$. \subsection{Conditions} -\[ \eqn{ Ingredients }{ - \patchof{L} = \pa{L} \lor \patchof{L} = \bot -}\] \[ \eqn{ Create Acyclic }{ - L \not\haspatch \pq + \pendsof{L}{\pqy} = \{ \} }\] \subsection{No Replay} @@ -497,13 +552,15 @@ 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 +If $D \neq B$, $D \isin L$, so by No Replay $D \le L$ +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 $D \isin B \equiv D \isin L$. +$B \not\in \py$ 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 $L \nothaspatch \p \implies B \nothaspatch P$. @@ -512,11 +569,7 @@ $\qed$. \subsection{Foreign Inclusion} -Consider some $D$ s.t. $\patchof{D} = \bot$. $D \neq B$ -so $D \isin B \equiv D \isin L$. -By Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$. -And by Exact Ancestors $D \le L \equiv D \le B$. -So $D \isin B \equiv D \le B$. $\qed$ +Simple Foreign Inclusion applies. $\qed$ \subsection{Foreign Contents} @@ -548,7 +601,7 @@ Ingredients Prevent Replay applies. $\qed$ \subsection{Unique Base} -Trivially, $\pendsof{C}{\pqn} = \{B\}$ so $\baseof{C} = B$. +Trivially, $\pendsof{C}{\pqn} = \{B\}$ so $\baseof{C} = B$. $\qed$ \subsection{Tip Contents} @@ -566,14 +619,35 @@ Not applicable. \subsection{Coherence and Patch Inclusion} -Consider some $D \in \py$. +$$ +\begin{cases} + \p = \pq \lor B \haspatch \p : & C \haspatch \p \\ + \p \neq \pq \land B \nothaspatch \p : & C \nothaspatch \p +\end{cases} +$$ + +\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 $C \haspatch \pq$. -xxx up to here +\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} @@ -638,7 +712,7 @@ $D \not\isin R^-$. Thus $D \not\isin C$. OK. By Currently Included, $D \isin L$. By Tip Self Inpatch, $D \isin R^+ \equiv D \le R^+$, but by -by Unique Tip, $D \le R^+ \equiv D \le L$. +by Unique Tip, $D \le R^+ \equiv D \le L$. So $D \isin R^+$. By Base Acyclic, $D \not\isin R^-$. @@ -767,7 +841,7 @@ 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. +is therefore consistent with our model. \subsection{No Replay} @@ -783,7 +857,7 @@ 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 $A \neq C$. +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, @@ -797,15 +871,15 @@ $A \le R \equiv A \le \baseof{R}$. 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 $A \le C \equiv A \le \baseof{R}$. +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 $A \le C \equiv A \le R$. +$A \le R \lor A \le \baseof{L} \equiv A \le R$. +Thus $A \le C \equiv A \le R$. That is, $\baseof{C} = R$. $\qed$ @@ -813,7 +887,7 @@ $\qed$ \subsection{Coherence and Patch Inclusion} Need to determine $C \haspatch \p$ based on $L,M,R \haspatch \p$. -This involves considering $D \in \py$. +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 @@ -827,20 +901,20 @@ $D \isin L \equiv D \le L$ and $D \isin R \equiv D \le R$. Consider $D = C$: $D \isin C$, $D \le C$, OK for $C \haspatch \p$. For $D \neq C$: $D \le C \equiv D \le L \lor D \le R - \equiv D \isin L \lor D \isin R$. + \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 $D \le X$ +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 $D \not\le C$. +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 $D \not\isin M$. +$D \not\le X$ 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 $C \haspatch \p$. @@ -853,7 +927,7 @@ $M \nothaspatch \p \implies C \haspatch \p$. \proofstarts -One of the Merge Ends conditions applies. +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 @@ -866,12 +940,12 @@ $M=\baseof{L}$. 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 $D \le C$. +$D \le Y$ 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 +$D \in \pancsof{X}{\py}$, 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. @@ -899,7 +973,7 @@ $\qed$ \subsection{Tip Contents} -We need worry only about $C \in \py$. +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.