X-Git-Url: http://www.chiark.greenend.org.uk/ucgi/~ian/git?p=topbloke-formulae.git;a=blobdiff_plain;f=article.tex;h=01297a8fb0681c79d7ffc1933fe941f02f77adb5;hp=674e5936f60fc6ae27161506b57b458b7517ccab;hb=e02ac317b82e521bccdbfadda3ff9c11d6a1533d;hpb=fe4881a01172df0a21e3951fc1f01e73f17c50cc diff --git a/article.tex b/article.tex index 674e593..01297a8 100644 --- a/article.tex +++ b/article.tex @@ -10,6 +10,8 @@ \renewcommand{\ge}{\geqslant} \renewcommand{\le}{\leqslant} +\newcommand{\nge}{\ngeqslant} +\newcommand{\nle}{\nleqslant} \newcommand{\has}{\sqsupseteq} \newcommand{\isin}{\sqsubseteq} @@ -53,14 +55,18 @@ \newcommand{\pancsof}[2]{\pancs ( #1 , #2 ) } \newcommand{\pendsof}[2]{\pends ( #1 , #2 ) } -\newcommand{\merge}[4]{{\mathcal M}(#1,#2,#3,#4)} +\newcommand{\merge}{{\mathcal M}} +\newcommand{\mergeof}[4]{\merge(#1,#2,#3,#4)} %\newcommand{\merge}[4]{{#2 {{\frac{ #1 }{ #3 } #4}}}} -\newcommand{\patchof}[1]{{\mathcal P} ( #1 ) } -\newcommand{\baseof}[1]{{\mathcal B} ( #1 ) } +\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}{% @@ -75,7 +81,8 @@ \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*{}} @@ -149,7 +156,7 @@ 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. -\item[ $\displaystyle \merge{C}{L}{M}{R} $ ] +\item[ $\displaystyle \mergeof{C}{L}{M}{R} $ ] The contents of a git merge result: $\displaystyle D \isin C \equiv @@ -199,7 +206,7 @@ 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 \\ @@ -245,19 +252,26 @@ by the LHS. And $A \le A''$. \[ \eqn{Calculation Of Ends:}{ \bigforall_{C \hasparents \set A} \pendsof{C}{\set P} = - \Bigl\{ E \Big| + \left\{ E \Big| \Bigl[ \Largeexists_{A \in \set A} E \in \pendsof{A}{\set P} \Bigr] \land \Bigl[ \Largenexists_{B \in \set A} E \neq B \land E \le B \Bigr] - \Bigr\} + \right\} }\] XXX proof TBD. \subsection{No Replay for Merge Results} -If we are constructing $C$ such that $\merge{C}{L}{M}{R}$, No Replay -is preserved. {\it Proof:} +If we are constructing $C$, with, +\gathbegin + \mergeof{C}{L}{M}{R} +\gathnext + L \le C +\gathnext + R \le C +\end{gather} +No Replay is preserved. \proofstarts \subsubsection{For $D=C$:} $D \isin C, D \le C$. OK. @@ -267,9 +281,6 @@ $D \isin C$. And $D \isin L \implies D \le L \implies D \le C$. OK. \subsubsection{For $D \neq C \land D \not\isin L \land D \not\isin R$:} $D \not\isin C$. OK. -\subsubsection{For $D \neq C \land D \not\isin L \land D \not\isin R$:} -$D \not\isin C$. OK. - \subsubsection{For $D \neq C \land (D \isin L \equiv D \not\isin R) \land D \not\isin M$:} $D \isin C$. Also $D \isin L \lor D \isin R$ so $D \le L \lor D \le @@ -277,8 +288,7 @@ R$ so $D \le C$. OK. \subsubsection{For $D \neq C \land (D \isin L \equiv D \not\isin R) \land D \isin M$:} -$D \not\isin C$. Also $D \isin L \lor D \isin R$ so $D \le L \lor D \le -R$ so $D \le C$. OK. +$D \not\isin C$. OK. $\qed$ @@ -383,7 +393,7 @@ $D \isin C \equiv D \isin A \equiv D \le A \equiv D \le C$. $\qed$ \section{Anticommit} Given $L, R^+, R^-$ where -$\patchof{R^+} = \pry, \patchof{R^-} = \prn$. +$R^+ \in \pry, R^- = \baseof{R^+}$. Construct $C$ which has $\pr$ removed. Used for removing a branch dependency. \gathbegin @@ -391,7 +401,7 @@ Used for removing a branch dependency. \gathnext \patchof{C} = \patchof{L} \gathnext - \merge{C}{L}{R^+}{R^-} + \mergeof{C}{L}{R^+}{R^-} \end{gather} \subsection{Conditions} @@ -399,16 +409,60 @@ Used for removing a branch dependency. \[ \eqn{ Unique Tip }{ \pendsof{L}{\pry} = \{ R^+ \} }\] -\[ \eqn{ Correct Base }{ - \baseof{R^+} = R^- -}\] \[ \eqn{ Currently Included }{ L \haspatch \pry }\] +\[ \eqn{ Not Self }{ + L \not\in \{ R^+ \} +}\] + +\subsection{No Replay} + +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. $\qed$ + +\subsection{Desired Contents} + +\[ D \isin C \equiv [ D \notin \pry \land D \isin L ] \lor D = C \] +\proofstarts + +\subsubsection{For $D = C$:} + +Trivially $D \isin C$. OK. + +\subsubsection{For $D \neq C, D \not\le L$:} + +By No Replay $D \not\isin L$. Also $D \not\le R^-$ hence +$D \not\isin R^-$. Thus $D \not\isin C$. OK. + +\subsubsection{For $D \neq C, D \le L, D \in \pry$:} + +By Currently Included, $D \isin L$. + +By Tip Self Inpatch, $D \isin R^+ \equiv D \le R^+$, but by +by Unique Tip, $D \le R^+ \equiv D \le L$. +So $D \isin R^+$. + +By Base Acyclic, $D \not\isin R^-$. + +Apply $\merge$: $D \not\isin C$. OK. + +\subsubsection{For $D \neq C, D \le L, D \notin \pry$:} + +By Tip Contents for $R^+$, $D \isin R^+ \equiv D \isin R^-$. + +Apply $\merge$: $D \isin C \equiv D \isin L$. OK. +$\qed$ + +\subsection{Unique Base} + +Need to consider only $C \in \py$, ie $L \in \py$. +xxx tbd -xxx want to prove $D \isin C \equiv D \not\in \pry \land D \isin L$. +xxx need to finish anticommit \section{Merge} @@ -418,8 +472,9 @@ Merge commits $L$ and $R$ using merge base $M$ ($M < L, M < R$): \gathnext \patchof{C} = \patchof{L} \gathnext - \merge{C}{L}{M}{R} + \mergeof{C}{L}{M}{R} \end{gather} +We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$. \subsection{Conditions} @@ -433,10 +488,25 @@ Merge commits $L$ and $R$ using merge base $M$ ($M < L, M < R$): \text{otherwise} : & \false \end{cases} }\] +\[ \eqn{ Removal Merge Ends }{ + X \not\haspatch \p \land + Y \haspatch \p \land + M \haspatch \p + \implies + \pendsof{Y}{\py} = \pendsof{M}{\py} +}\] +\[ \eqn{ Addition Merge Ends }{ + X \not\haspatch \p \land + Y \haspatch \p \land + M \nothaspatch \p + \implies \left[ + \bigforall_{E \in \pendsof{X}{\py}} E \le Y + \right] +}\] -\subsection{Merge Results} +\subsection{No Replay} -As above. +See No Replay for Merge Results. \subsection{Unique Base} @@ -472,4 +542,78 @@ That is, $\baseof{C} = R$. $\qed$ +\subsection{Coherence and patch inclusion} + +Need to determine $C \haspatch \p$ based on $L,M,R \haspatch \p$. +This involves considering $D \in \py$. + +\subsubsection{For $L \nothaspatch \p, R \nothaspatch \p$:} +$D \not\isin L \land D \not\isin R$. $C \not\in \py$ (otherwise $L +\in \py$ ie $L \haspatch \p$ by Tip Self Inpatch). So $D \neq C$. +Applying $\merge$ gives $D \not\isin C$ i.e. $C \nothaspatch \p$. + +\subsubsection{For $L \haspatch \p, R \haspatch \p$:} +$D \isin L \equiv D \le L$ and $D \isin R \equiv D \le R$. +(Likewise $D \isin X \equiv D \le X$ and $D \isin Y \equiv D \le Y$.) + +Consider $D = C$: $D \isin C$, $D \le C$, OK for $C \haspatch \p$. + +For $D \neq C$: $D \le C \equiv D \le L \lor D \le R + \equiv D \isin L \lor D \isin R$. +(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 $D \le X$ +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 $D \not\le C$. +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 $D \not\isin M$. +Thus by $\merge$, $D \isin C$. And $D \le Y$ so $D \le C$. +OK for $C \haspatch \p$. + +So indeed $L \haspatch \p \land R \haspatch \p \implies C \haspatch \p$. + +\subsubsection{For (wlog) $X \not\haspatch \p, Y \haspatch \p$:} + +$C \haspatch \p \equiv M \nothaspatch \p$. + +\proofstarts + +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. + +$\qed$ + \end{document}