X-Git-Url: http://www.chiark.greenend.org.uk/ucgi/~ian/git?p=topbloke-formulae.git;a=blobdiff_plain;f=article.tex;h=e6447d92380b854b40c0c1bf9d61825564f79a4d;hp=3512d9c5418954713d1ea10b36c82c829ce29308;hb=1dba870613dc3668dbbcb5cf5afd61f9cd310a1d;hpb=4e90b170345fbb32ce672b73e1711af787831439 diff --git a/article.tex b/article.tex index 3512d9c..e6447d9 100644 --- a/article.tex +++ b/article.tex @@ -1,4 +1,5 @@ \documentclass[a4paper,leqno]{strayman} +\errorcontextlines=50 \let\numberwithin=\notdef \usepackage{amsmath} \usepackage{mathabx} @@ -9,6 +10,8 @@ \renewcommand{\ge}{\geqslant} \renewcommand{\le}{\leqslant} +\newcommand{\nge}{\ngeqslant} +\newcommand{\nle}{\nleqslant} \newcommand{\has}{\sqsupseteq} \newcommand{\isin}{\sqsubseteq} @@ -18,8 +21,12 @@ \newcommand{\haspatch}{\sqSupset} \newcommand{\patchisin}{\sqSubset} -\newcommand{\set}[1]{\mathbb #1} -\newcommand{\pa}[1]{\varmathbb #1} + \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{\pay}[1]{\pa{#1}^+} \newcommand{\pan}[1]{\pa{#1}^-} @@ -27,6 +34,10 @@ \newcommand{\py}{\pay{P}} \newcommand{\pn}{\pan{P}} +\newcommand{\pr}{\pa{R}} +\newcommand{\pry}{\pay{R}} +\newcommand{\prn}{\pan{R}} + %\newcommand{\hasparents}{\underaccent{1}{>}} %\newcommand{\hasparents}{{% % \declareslashed{}{_{_1}}{0}{-0.8}{>}\slashed{>}}} @@ -35,16 +46,28 @@ \renewcommand{\implies}{\Rightarrow} \renewcommand{\equiv}{\Leftrightarrow} +\renewcommand{\nequiv}{\nLeftrightarrow} \renewcommand{\land}{\wedge} \renewcommand{\lor}{\vee} -\newcommand{\pancs}[2]{{\mathcal A} ( #1 , #2 ) } -\newcommand{\pends}[2]{{\mathcal E} ( #1 , #2 ) } +\newcommand{\pancs}{{\mathcal A}} +\newcommand{\pends}{{\mathcal E}} + +\newcommand{\pancsof}[2]{\pancs ( #1 , #2 ) } +\newcommand{\pendsof}[2]{\pends ( #1 , #2 ) } + +\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{\patchof}[1]{{\mathcal P} ( #1 ) } -\newcommand{\baseof}[1]{{\mathcal B} ( #1 ) } +\newcommand{\patchof}[1]{\patch ( #1 ) } +\newcommand{\baseof}[1]{\base ( #1 ) } -\newcommand{\eqn}[2]{ #2 \tag*{\mbox{#1}} } +\newcommand{\eqntag}[2]{ #2 \tag*{\mbox{#1}} } +\newcommand{\eqn}[2]{ #2 \tag*{\mbox{\bf #1}} } %\newcommand{\bigforall}{\mathop{\hbox{\huge$\forall$}}} \newcommand{\bigforall}{% @@ -55,6 +78,19 @@ {\hbox{\scriptsize$\forall$}}}% } +\newcommand{\Largeexists}{\mathop{\hbox{\Large$\exists$}}} +\newcommand{\Largenexists}{\mathop{\hbox{\Large$\nexists$}}} + +\newcommand{\qed}{\square} +\newcommand{\proofstarts}{{\it Proof:}} +\newcommand{\proof}[1]{\proofstarts #1 $\qed$} + +\newcommand{\gathbegin}{\begin{gather} \tag*{}} +\newcommand{\gathnext}{\\ \tag*{}} + +\newcommand{\true}{t} +\newcommand{\false}{f} + \begin{document} \section{Notation} @@ -75,76 +111,703 @@ $ 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 in non-Topbloke-controlled branches; these are considered -normal, forward, commits. For merges and Topbloke-generated -anticommits, the ``change made'' is only to be thought of as any -conflict resolution. This is not a partial order because it is not -transitive. +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: +None of these sets overlap. Hence: \item[ $ \patchof{ C } $ ] Either $\p$ s.t. $ C \in \p $, or $\bot$. -A function from commits to sets $\p$. +A function from commits to patches' sets $\p$. -\item[ $ \pancs{C}{\set 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[ $ \pends{C}{\set P} $ ] -$ \{ E \; | \; E \in \pancs{C}{\set P} - \land \mathop{\not\exists}_{A \in \pancs{C}{\set P}} - A \neq E \land E \le A \} $ -i.e. all $\le$-maximal commits in $\pancs{C}{\set P}$. +\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 \} $ +i.e. all $\le$-maximal commits in $\pancsof{C}{\set P}$. \item[ $ \baseof{C} $ ] -$ \pends{C}{\pn} = \{ \baseof{C} \} $ where $ C \in \py $. +$ \pendsof{C}{\pn} = \{ \baseof{C} \} $ where $ C \in \py $. A partial function from commits to commits. -See ``unique base'', below. +See Unique Base, below. \item[ $ C \haspatch \p $ ] -$ \bigforall_{D \in \py} D \isin C \equiv D \le C $. -Informally, $C$ has the contents of $\p$. +$\displaystyle \bigforall_{D \in \py} D \isin C \equiv D \le C $. +~ Informally, $C$ has the contents of $\p$. -\item[ $\displaystyle C \nothaspatch \p $ ] +\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 +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: + +$\displaystyle D \isin C \equiv + \begin{cases} + (D \isin L \land D \isin R) \lor D = C : & \true \\ + (D \not\isin L \land D \not\isin R) \land D \neq C : & \false \\ + \text{otherwise} : & D \not\isin M + \end{cases} +$ \end{basedescript} - +\newpage \section{Invariants} -\[ \eqn{No replay:}{ +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} \pends{C}{\pn} = \{ B \} +\[\eqn{Unique Base:}{ + \bigforall_{C \in \py} \pendsof{C}{\pn} = \{ B \} }\] -\[\eqn{Tip contents:}{ +\[\eqn{Tip Contents:}{ \bigforall_{C \in \py} D \isin C \equiv { D \isin \baseof{C} \lor \atop (D \in \py \land D \le C) } }\] -\[\eqn{Base non-circ:}{ +\[\eqn{Base Acyclic:}{ \bigforall_{B \in \pn} D \isin B \implies D \notin \py }\] +\[\eqn{Coherence:}{ + \bigforall_{C,\p} C \haspatch \p \lor C \nothaspatch \p +}\] +\[\eqn{Foreign Inclusion:}{ + \bigforall_{D \text{ s.t. } \patchof{D} = \bot} D \isin C \equiv D \leq C +}\] + +\section{Some lemmas} + +\[ \eqn{Alternative (overlapping) formulations defining + $\mergeof{C}{L}{M}{R}$:}{ + D \isin C \equiv + \begin{cases} + D \isin L \equiv D \isin R : & D = C \lor D \isin L \\ + D \isin L \nequiv D \isin R : & D = C \lor D \not\isin M \\ + D \isin L \equiv D \isin M : & D = C \lor D \isin R \\ + D \isin L \nequiv D \isin M : & D = C \lor D \isin L \\ + \text{as above with L and R exchanged} + \end{cases} +}\] +\proof{ + Truth table xxx + + Original definition is symmetrical in $L$ and $R$. +} + +\[ \eqn{Exclusive Tip Contents:}{ + \bigforall_{C \in \py} + \neg \Bigl[ D \isin \baseof{C} \land ( D \in \py \land D \le C ) + \Bigr] +}\] +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 $D \isin B \implies D \notin \py$. +} +\[ \eqntag{{\it Corollary - equivalent to Tip Contents}}{ + \bigforall_{C \in \py} D \isin C \equiv + \begin{cases} + D \in \py : & D \le C \\ + D \not\in \py : & D \isin \baseof{C} + \end{cases} +}\] + +\[ \eqn{Tip Self Inpatch:}{ + \bigforall_{C \in \py} C \haspatch \p +}\] +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 $ +} + +\[ \eqn{Exact Ancestors:}{ + \bigforall_{ C \hasparents \set{R} } + D \le C \equiv + ( \mathop{\hbox{\huge{$\vee$}}}_{R \in \set R} D \le R ) + \lor D = C +}\] +xxx proof tbd + +\[ \eqn{Transitive Ancestors:}{ + \left[ \bigforall_{ E \in \pendsof{C}{\set P} } E \le M \right] \equiv + \left[ \bigforall_{ A \in \pancsof{C}{\set P} } A \le M \right] +}\] + +\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 $A \le A''$. +} + +\[ \eqn{Calculation Of Ends:}{ + \bigforall_{C \hasparents \set A} + \pendsof{C}{\set P} = + \begin{cases} + C \in \p : & \{ C \} + \\ + C \not\in \p : & \displaystyle + \left\{ E \Big| + \Bigl[ \Largeexists_{A \in \set A} + E \in \pendsof{A}{\set P} \Bigr] \land + \Bigl[ \Largenexists_{B \in \set A} + E \neq B \land E \le B \Bigr] + \right\} + \end{cases} +}\] +xxx proof tbd + +\subsection{No Replay for Merge Results} + +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. + +\subsubsection{For $D \isin L \land D \isin R$:} +$D \isin C$. 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 $D \neq C \land (D \isin L \equiv D \not\isin R) + \land D \not\isin M$:} +$D \isin C$. Also $D \isin L \lor D \isin R$ so $D \le L \lor D \le +R$ so $D \le C$. OK. + +\subsubsection{For $D \neq C \land (D \isin L \equiv D \not\isin R) + \land D \isin M$:} +$D \not\isin C$. OK. + +$\qed$ + +\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} + +$\patchof{C}$, 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 in that case +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} +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 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 $A$: +\[ D \isin A \equiv D \isin \baseof{A} \lor ( D \in \py \land D \le A ) \] +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 and patch inclusion} + +Need to consider $D \in \py$ + +\subsubsection{For $A \haspatch P, D = C$:} + +Ancestors of $C$: +$ 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 $. + +Contents: $ D \isin C \equiv D \isin A \lor f $ +so $ D \isin C \equiv D \isin A $. + +So: +\[ A \haspatch P \implies C \haspatch P \] + +\subsubsection{For $A \nothaspatch P$:} + +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 $D \notin C$. + +So: +\[ A \nothaspatch P \implies C \nothaspatch P \] +$\qed$ + +\subsection{Foreign inclusion:} + +If $D = C$, trivial. For $D \neq C$: +$D \isin C \equiv D \isin A \equiv D \le A \equiv D \le C$. $\qed$ + +\section{Anticommit} + +Given $L, R^+, R^-$ where +$R^+ \in \pry, R^- = \baseof{R^+}$. +Construct $C$ which has $\pr$ removed. +Used for removing a branch dependency. +\gathbegin + C \hasparents \{ L \} +\gathnext + \patchof{C} = \patchof{L} +\gathnext + \mergeof{C}{L}{R^+}{R^-} +\end{gather} + +\subsection{Conditions} + +\[ \eqn{ Into Base }{ + L \in \pn +}\] +\[ \eqn{ Unique Tip }{ + \pendsof{L}{\pry} = \{ R^+ \} +}\] +\[ \eqn{ Currently Included }{ + L \haspatch \pry +}\] + +\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 $R^- \le C$. + +(Note that the merge base $R^+ \not\le R^-$, i.e. the merge base is +later than one of the branches to be merged.) + +\subsection{No Replay} + +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} + +Into Base means that $C \in \pn$, so Unique Base is not +applicable. $\qed$ + +\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, $D \neq C$. + +\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 $L \haspatch \p \implies C \haspatch \p$. + +\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} + +Merge commits $L$ and $R$ using merge base $M$ ($M < L, M < R$): +\gathbegin + C \hasparents \{ L, R \} +\gathnext + \patchof{C} = \patchof{L} +\gathnext + \mergeof{C}{L}{M}{R} +\end{gather} +We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$. + +\subsection{Conditions} + +\[ \eqn{ Tip Merge }{ + L \in \py \implies + \begin{cases} + R \in \py : & \baseof{R} \ge \baseof{L} + \land [\baseof{L} = M \lor \baseof{L} = \baseof{M}] \\ + R \in \pn : & M = \baseof{L} \\ + \text{otherwise} : & \false + \end{cases} +}\] +\[ \eqn{ Merge Acyclic }{ + L \in \pn + \implies + R \nothaspatch \p +}\] +\[ \eqn{ Removal Merge Ends }{ + X \not\haspatch \p \land + Y \haspatch \p \land + M \haspatch \p + \implies + \pendsof{Y}{\py} = \pendsof{M}{\py} +}\] +\[ \eqn{ Addition Merge Ends }{ + X \not\haspatch \p \land + Y \haspatch \p \land + M \nothaspatch \p + \implies \left[ + \bigforall_{E \in \pendsof{X}{\py}} E \le Y + \right] +}\] + +\subsection{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} + +No Replay for Merge Results applies. $\qed$ + +\subsection{Unique Base} + +Need to consider only $C \in \py$, ie $L \in \py$, +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 $A \neq C$. +Thus $A \le C \equiv A \le L \lor A \le R$. + +By Unique Base of L and Transitive Ancestors, +$A \le L \equiv A \le \baseof{L}$. + +\subsubsection{For $R \in \py$:} + +By Unique Base of $R$ and Transitive Ancestors, +$A \le R \equiv A \le \baseof{R}$. + +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 $A \le C \equiv A \le \baseof{R}$. +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 $A \le C \equiv A \le R$. +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) $X \not\haspatch \p, Y \haspatch \p$:} + +$M \haspatch \p \implies C \nothaspatch \p$. +$M \nothaspatch \p \implies C \haspatch \p$. + +\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. + +$\qed$ + +\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. + +\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. + +$\qed$ + +\subsection{Foreign Inclusion} + +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 $D \neq C, D \isin M$:} -\section{Test more symbols} +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. -$ C \haspatch \p $ +\subsubsection{For $D \neq C, D \not\isin M, D \isin X$:} -$ C \nothaspatch \p $ +By $\merge$, $D \isin C$. +And $D \isin X$ means $D \le X$ so $D \le C$. +OK. -$ \p \patchisin C $ +\subsubsection{For $D \neq C, D \not\isin M, D \not\isin L, D \not\isin R$:} -$ \p \notpatchisin C $ +By $\merge$, $D \not\isin C$. +And $D \not\le L, D \not\le R$ so $D \not\le C$. +OK -$ \{ B \} \areparents C $ +$\qed$ \end{document}