X-Git-Url: http://www.chiark.greenend.org.uk/ucgi/~ian/git?p=topbloke-formulae.git;a=blobdiff_plain;f=article.tex;h=090ae13120114180e1a0d49e141f518d5cf15f1a;hp=b97f7bb606db5b814b129ec5561dc39ef6c62042;hb=a95c3485b6141faa1833701180536b9ed138c89a;hpb=c60a10efc12a3c6b09c536c424fc9ca24a167350 diff --git a/article.tex b/article.tex index b97f7bb..090ae13 100644 --- a/article.tex +++ b/article.tex @@ -46,6 +46,7 @@ \renewcommand{\implies}{\Rightarrow} \renewcommand{\equiv}{\Leftrightarrow} +\renewcommand{\nequiv}{\nLeftrightarrow} \renewcommand{\land}{\wedge} \renewcommand{\lor}{\vee} @@ -65,8 +66,8 @@ \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}{% @@ -121,7 +122,7 @@ 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: +None of these sets overlap. Hence: \item[ $ \patchof{ C } $ ] Either $\p$ s.t. $ C \in \p $, or $\bot$. @@ -151,10 +152,13 @@ $\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 Topbloke +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 Topbloke patch itself, we hope that git's merge algorithm will -DTRT or that the user will no longer care about the Topbloke patch. +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: @@ -195,6 +199,23 @@ We maintain these each time we construct a new commit. \\ \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 ) @@ -206,7 +227,7 @@ Ie, the two limbs of the RHS of Tip Contents are mutually exclusive. 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{ +\[ \eqntag{{\it Corollary - equivalent to Tip Contents}}{ \bigforall_{C \in \py} D \isin C \equiv \begin{cases} D \in \py : & D \le C \\ @@ -231,6 +252,7 @@ $ \bigforall_{C \in \py}\bigforall_{D \in \py} ( \mathop{\hbox{\huge{$\vee$}}}_{R \in \set R} D \le R ) \lor D = C }\] +xxx proof tbd \[ \eqn{Transitive Ancestors:}{ \left[ \bigforall_{ E \in \pendsof{C}{\set P} } E \le M \right] \equiv @@ -249,17 +271,23 @@ 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. +xxx proof tbd \subsection{No Replay for Merge Results} @@ -306,10 +334,25 @@ We annotate each Topbloke commit $C$ with: \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 because -$\forall_{\py \ni C} \; \pendsof{C}{\py} = \{C\}$. +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} @@ -319,12 +362,17 @@ A simple single-parent forward commit $C$ as made by git-commit. \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 $\baseof{C} = \baseof{A}$. $\qed$ +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 $A$: @@ -351,7 +399,9 @@ 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 -$A$, $D \isin C \implies D \not\in \py$. $\qed$ +$A$, $D \isin C \implies D \not\in \py$. + +$\qed$ \subsection{Coherence and patch inclusion} @@ -390,10 +440,17 @@ $\qed$ 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{Create Base} + +xxx tbd + +\section{Create Tip} + +xxx tbd\ + \section{Anticommit} -Given $L, R^+, R^-$ where -$R^+ \in \pry, R^- = \baseof{R^+}$. +Given $L$ and $\pr$ as represented by $R^+, R^-$. Construct $C$ which has $\pr$ removed. Used for removing a branch dependency. \gathbegin @@ -406,21 +463,31 @@ Used for removing a branch dependency. \subsection{Conditions} +\[ \eqn{ Ingredients }{ +R^+ \in \pry \land R^- = \baseof{R^+} +}\] +\[ \eqn{ Into Base }{ + L \in \pn +}\] \[ \eqn{ Unique Tip }{ \pendsof{L}{\pry} = \{ R^+ \} }\] \[ \eqn{ Currently Included }{ L \haspatch \pry }\] -\[ \eqn{ Not Self }{ - L \not\in \{ R^+ \} -}\] -\subsection{No Replay} +\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 $R^- \le C$ and No Replay for -Merge Results applies. $\qed$ +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. $\qed$ \subsection{Desired Contents} @@ -458,11 +525,51 @@ $\qed$ \subsection{Unique Base} -Need to consider only $C \in \py$, ie $L \in \py$. +Into Base means that $C \in \pn$, so Unique Base is not +applicable. $\qed$ -xxx tbd +\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, $D \neq C$. -xxx need to finish anticommit +\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 $L \haspatch \p \implies C \haspatch \p$. + +$\qed$ + +\section{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 $D \isin C \equiv D \le C$. + +$\qed$ \section{Merge} @@ -483,22 +590,45 @@ We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$. \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} \\ + R \in \pn : & M = \baseof{L} \\ \text{otherwise} : & \false \end{cases} }\] -\[ \eqn{ Merge Ends }{ +\[ \eqn{ Merge Acyclic }{ + L \in \pn + \implies + R \nothaspatch \p +}\] +\[ \eqn{ Removal Merge Ends }{ X \not\haspatch \p \land Y \haspatch \p \land - E \in \pendsof{X}{\py} + M \haspatch \p \implies - E \le Y + \pendsof{Y}{\py} = \pendsof{M}{\py} }\] +\[ \eqn{ Addition Merge Ends }{ + X \not\haspatch \p \land + Y \haspatch \p \land + M \nothaspatch \p + \implies \left[ + \bigforall_{E \in \pendsof{X}{\py}} E \le Y + \right] +}\] + +\subsection{Non-Topbloke merges} + +We require both $\patchof{L} = \bot$ and $\patchof{R} = \bot$. +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. Good. \subsection{No Replay} -See No Replay for Merge Results. +No Replay for Merge Results applies. $\qed$ \subsection{Unique Base} @@ -526,7 +656,7 @@ That is, $\baseof{C} = \baseof{R}$. \subsubsection{For $R \in \pn$:} -By Tip Merge condition on $R$, +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 $A \le C \equiv A \le R$. @@ -534,7 +664,7 @@ That is, $\baseof{C} = R$. $\qed$ -\subsection{Coherence and patch inclusion} +\subsection{Coherence and Patch Inclusion} Need to determine $C \haspatch \p$ based on $L,M,R \haspatch \p$. This involves considering $D \in \py$. @@ -572,23 +702,131 @@ So indeed $L \haspatch \p \land R \haspatch \p \implies C \haspatch \p$. \subsubsection{For (wlog) $X \not\haspatch \p, Y \haspatch \p$:} -$C \haspatch \p \equiv M \nothaspatch \p$. +$M \haspatch \p \implies C \nothaspatch \p$. +$M \nothaspatch \p \implies C \haspatch \p$. \proofstarts -Merge Ends 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 +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. -$D \isin Y \equiv D \le Y$. $D \not\isin X$. Recall that we -are considering $D \in \py$. +\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. + +$\qed$ -Consider $D = C$. Thus $C \in \py, L \in \py$. -But $X \not\haspatch \p$ means xxx wip -But $X \not\haspatch \p$ means $D \not\in X$, +\subsection{Foreign Inclusion} -so we have $L = Y, R = -X$. Thus $R \not\haspatch \p$ and by Tip Self Inpatch $R \not\in -\py$. Thus by Tip Merge $R \in \pn$ and $M = \baseof{L}$. -So by Base Acyclic, $M \nothaspatch \py$. Thus we are expecting -$C \haspatch \py$. And indeed $D \isin C$ and $D \le C$. OK. +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 $D \neq C, D \isin M$:} + +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. + +\subsubsection{For $D \neq C, D \not\isin M, D \isin X$:} + +By $\merge$, $D \isin C$. +And $D \isin X$ means $D \le X$ so $D \le C$. +OK. + +\subsubsection{For $D \neq C, D \not\isin M, D \not\isin L, D \not\isin R$:} + +By $\merge$, $D \not\isin C$. +And $D \not\le L, D \not\le R$ so $D \not\le C$. +OK + +$\qed$ \end{document}