X-Git-Url: http://www.chiark.greenend.org.uk/ucgi/~ian/git?p=topbloke-formulae.git;a=blobdiff_plain;f=article.tex;h=a57bbec868dc202f99c959e68d802c566573a906;hp=c3c658ce30a417fe861c3a52abd8762ba72d1142;hb=3da86b6d60fbd0edfa34366c2c57a5481790883c;hpb=277226a13b267113aab79a0841ee3aa532d7c3b6 diff --git a/article.tex b/article.tex index c3c658c..a57bbec 100644 --- a/article.tex +++ b/article.tex @@ -34,6 +34,10 @@ \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}} @@ -105,7 +109,7 @@ $\set X$. \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 $ ] @@ -125,18 +129,18 @@ is to be taken as applying to both $\py$ and $\pn$. None of these sets overlap. 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} $ ] @@ -150,13 +154,13 @@ $\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. @@ -169,7 +173,7 @@ $\displaystyle D \isin C \equiv (D \not\isin L \land D \not\isin R) \land D \neq C : & \false \\ \text{otherwise} : & D \not\isin M \end{cases} -$ +$ \end{basedescript} \newpage @@ -214,14 +218,29 @@ We maintain these each time we construct a new commit. \\ \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 $R$. } \[ \eqn{Exclusive Tip Contents:}{ - \bigforall_{C \in \py} + \bigforall_{C \in \py} \neg \Bigl[ D \isin \baseof{C} \land ( D \in \py \land D \le C ) \Bigr] }\] @@ -256,7 +275,7 @@ $ \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 +\proof{ ~ Trivial.} \[ \eqn{Transitive Ancestors:}{ \left[ \bigforall_{ E \in \pendsof{C}{\set P} } E \le M \right] \equiv @@ -266,7 +285,7 @@ xxx proof tbd \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, @@ -284,9 +303,9 @@ 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} + \Bigl[ \Largenexists_{B \in \set A} E \neq B \land E \le B \Bigr] \right\} \end{cases} @@ -313,6 +332,24 @@ xxx proof tbd $A \le C$ so $D \le C$. } +\[ \eqn{Simple Foreign Inclusion:}{ + \left[ + C \hasparents \{ L \} + \land + \bigforall_{D} D \isin C \equiv D \isin L \lor D = C + \right] + \implies + \bigforall_{D \text{ s.t. } \patchof{D} = \bot} + D \isin C \equiv D \le C +}\] +\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 $D \isin C \equiv D \le C$. +} + \[ \eqn{Totally Foreign Contents:}{ \bigforall_{C \hasparents \set A} \left[ @@ -340,10 +377,10 @@ We annotate each Topbloke commit $C$ with: \gathnext \baseof{C}, \text{ if } C \in \py \gathnext - \bigforall_{\pa{Q}} - \text{ either } C \haspatch \pa{Q} \text{ or } C \nothaspatch \pa{Q} + \bigforall_{\pq} + \text{ either } C \haspatch \pq \text{ or } C \nothaspatch \pq \gathnext - \bigforall_{\pay{Q} \not\ni C} \pendsof{C}{\pay{Q}} + \bigforall_{\pqy \not\ni C} \pendsof{C}{\pqy} \end{gather} $\patchof{C}$, for each kind of Topbloke-generated commit, is stated @@ -352,14 +389,14 @@ 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 $C \haspatch \pa{Q}$ +$C \haspatch \pq$ or $\nothaspatch \pq$ is represented as the +set $\{ \pq | C \haspatch \pq \}$. Whether $C \haspatch \pq$ is in stated -(in terms of $I \haspatch \pa{Q}$ or $I \nothaspatch \pa{Q}$ +(in terms of $I \haspatch \pq$ or $I \nothaspatch \pq$ 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, +$\pendsof{C}{\pq^+}$ 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 $\pends$ @@ -393,7 +430,7 @@ We need to consider only $A, C \in \py$. From Tip Contents for $A$: 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 $D = C \implies D \in \py$, +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 @@ -407,7 +444,7 @@ $\qed$ \subsection{Base Acyclic} -Need to consider only $A, C \in \pn$. +Need to consider only $A, C \in \pn$. For $D = C$: $D \in \pn$ so $D \not\in \py$. OK. @@ -439,7 +476,7 @@ So: \subsubsection{For $A \nothaspatch P$:} -Firstly, $C \not\in \py$ since if it were, $A \in \py$. +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 $D \notin C$. @@ -448,10 +485,9 @@ So: \[ A \nothaspatch P \implies C \nothaspatch P \] $\qed$ -\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:} @@ -464,7 +500,7 @@ Given $L$, create a Topbloke base branch initial commit $B$. \gathbegin B \hasparents \{ L \} \gathnext - \patchof{B} = \pan{Q} + \patchof{B} = \pqn \gathnext D \isin B \equiv D \isin L \lor D = B \end{gather} @@ -474,8 +510,8 @@ Given $L$, create a Topbloke base branch initial commit $B$. \[ \eqn{ Ingredients }{ \patchof{L} = \pa{L} \lor \patchof{L} = \bot }\] -\[ \eqn{ Non-recursion }{ - L \not\in \pa{Q} +\[ \eqn{ Create Acyclic }{ + L \not\haspatch \pq }\] \subsection{No Replay} @@ -492,17 +528,15 @@ Not applicable. \subsection{Base Acyclic} -Consider some $D \isin B$. If $D = B$, $D \in \pn$, OK. - -If $D \neq B$, $D \isin L$. By No Replay of $D$ in $L$, $D \le L$. -Thus by Foreign Contents of $L$, $\patchof{D} = \bot$. OK. - -$\qed$ +Consider some $D \isin B$. If $D = B$, $D \in \pqn$. +If $D \neq B$, $D \isin L$, 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 $L \nothaspatch \p \implies B \nothaspatch P$. @@ -511,11 +545,7 @@ $\qed$. \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} @@ -523,7 +553,77 @@ Not applicable. \section{Create Tip} -xxx tbd +Given a Topbloke base $B$, create a tip branch initial commit B. +\gathbegin + C \hasparents \{ B \} +\gathnext + \patchof{B} = \pqy +\gathnext + D \isin C \equiv D \isin B \lor D = C +\end{gather} + +\subsection{Conditions} + +\[ \eqn{ Ingredients }{ + \patchof{B} = \pqn +}\] +\[ \eqn{ No Sneak }{ + \pendsof{B}{\pqy} = \{ \} +}\] + +\subsection{No Replay} + +Ingredients Prevent Replay applies. $\qed$ + +\subsection{Unique Base} + +Trivially, $\pendsof{C}{\pqn} = \{B\}$ so $\baseof{C} = B$. $\qed$ + +\subsection{Tip Contents} + +Consider some arbitrary commit $D$. If $D = C$, trivially satisfied. + +If $D \neq C$, $D \isin C \equiv D \isin B$. +By Base Acyclic of $B$, $D \isin B \implies D \not\in \pqy$. +So $D \isin C \equiv D \isin \baseof{B}$. + +$\qed$ + +\subsection{Base Acyclic} + +Not applicable. + +\subsection{Coherence and Patch Inclusion} + +$$ +\begin{cases} + \p = \pq \lor B \haspatch \p : & C \haspatch \p \\ + \p \neq \pq \land B \nothaspatch \p : & C \nothaspatch \p +\end{cases} +$$ + +\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 $C \haspatch \pq$. + +\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} @@ -588,7 +688,7 @@ $D \not\isin R^-$. 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, $D \not\isin R^-$. @@ -733,7 +833,7 @@ 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, @@ -747,15 +847,15 @@ $A \le R \equiv A \le \baseof{R}$. 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, $\baseof{C} = R$. $\qed$ @@ -763,7 +863,7 @@ $\qed$ \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 @@ -777,20 +877,20 @@ $D \isin L \equiv D \le L$ 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 $C \haspatch \p$. @@ -803,7 +903,7 @@ $M \nothaspatch \p \implies C \haspatch \p$. \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 @@ -816,12 +916,12 @@ $M=\baseof{L}$. 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. @@ -849,7 +949,7 @@ $\qed$ \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.