\let\numberwithin=\notdef
\usepackage{amsmath}
\usepackage{mathabx}
-\usepackage{stmaryrd}
-\usepackage{slashed}
\usepackage{txfonts}
\usepackage{amsfonts}
\usepackage{mdwlist}
\newcommand{\has}{\sqsupseteq}
\newcommand{\isin}{\sqsubseteq}
-\newcommand{\nothaspatch}{{%
- \declareslashed{}{\sslash}{-0.04}{0}{\sqSupset}\slashed{\sqSupset}}}
-\newcommand{\notpatchisin}{{%
- \declareslashed{}{\sslash}{-0.04}{0}{\sqSubset}\slashed{\sqSubset}}}
+\newcommand{\nothaspatch}{\mathrel{\,\not\!\not\relax\haspatch}}
+\newcommand{\notpatchisin}{\mathrel{\,\not\!\not\relax\patchisin}}
\newcommand{\haspatch}{\sqSupset}
\newcommand{\patchisin}{\sqSubset}
\newcommand{\areparents}{<_{\mkern-14.0mu _1\mkern+5.0mu}}
\renewcommand{\implies}{\Rightarrow}
+\renewcommand{\equiv}{\Leftrightarrow}
+\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}}
-\renewcommand{\land}{\wedge}
+\newcommand{\pancsof}[2]{\pancs ( #1 , #2 ) }
+\newcommand{\pendsof}[2]{\pends ( #1 , #2 ) }
+
+\newcommand{\patchof}[1]{{\mathcal P} ( #1 ) }
+\newcommand{\baseof}[1]{{\mathcal B} ( #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}{%
+ \mathop{\mathchoice%
+ {\hbox{\huge$\forall$}}%
+ {\hbox{\Large$\forall$}}%
+ {\hbox{\normalsize$\forall$}}%
+ {\hbox{\scriptsize$\forall$}}}%
+}
+
+\newcommand{\proof}[1]{{\it Proof.} #1 $\square$}
\begin{document}
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.
+All these sets are distinct. Hence:
+
+\item[ $ \patchof{ C } $ ]
+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 \} $
+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}}
+ A \neq E \land E \le A \} $
+i.e. all $\le$-maximal commits in $\pancsof{C}{\set P}$.
+
+\item[ $ \baseof{C} $ ]
+$ \pendsof{C}{\pn} = \{ \baseof{C} \} $ where $ C \in \py $.
+A partial function from commits to commits.
+See ``unique base'', below.
+
+\item[ $ C \haspatch \p $ ]
+$\displaystyle \bigforall_{D \in \py} D \isin C \equiv D \le C $.
+~ Informally, $C$ has the contents of $\p$.
+
+\item[ $ C \nothaspatch \p $ ]
+$\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 patch is merged into a non-Topbloke branch and we inherit
+it, we hope that git's merge algorithm will DTRT.
\end{basedescript}
\section{Invariants}
-\[ C \has D \implies C \ge D \tag*{\mbox{No replay:}} \]
+\[ \eqn{No Replay:}{
+ C \has D \implies C \ge D
+}\]
+\[\eqn{Unique Base:}{
+ \bigforall_{C \in \py} \pendsof{C}{\pn} = \{ B \}
+}\]
+\[\eqn{Tip Contents:}{
+ \bigforall_{C \in \py} D \isin C \equiv
+ { D \isin \baseof{C} \lor \atop
+ (D \in \py \land D \le C) }
+}\]
+\[\eqn{Base Acyclic:}{
+ \bigforall_{B \in \pn} D \isin B \implies D \notin \py
+}\]
+\[\eqn{Coherence:}{
+ \bigforall_{C,\p} C \haspatch \p \lor C \nothaspatch \p
+}\]
+
+\section{Some lemmas}
+
+\[ \eqn{Exclusive Tip Contents:}{
+ \bigforall_{C \in \py}
+ \neg \left[ D \isin \baseof{C} \land (D \in \py \land D \le C
+ \right )]
+}\]
+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 $D \isin B \implies D \notin \py$.
+}
+\[ \corrolary{
+ \bigforall_{C \in \py} D \isin C \equiv
+ \begin{cases}
+ D \in \py : & D \le C \\
+ D \not\in \py : & D \isin \baseof{C}
+ \end{cases}
+}\]
+
+\[ \eqn{Tip Self Inpatch:}{
+ \bigforall_{C \in \py} C \haspatch \p
+}\]
+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 $
+}
-Unique base: \[ \mathop{\forall}_{C \in \py} \pends{C}{\pn} = \{ B \} \]
+\[ \eqn{Exact Ancestors:}{
+ \bigforall_{ C \hasparents \set{R} }
+ D \le C \equiv
+ ( \mathop{\hbox{\huge{$\vee$}}}_{R \in \set R} D \le R )
+ \lor D = C
+}\]
+
+\[ \eqn{Transitive Ancestors:}{
+ \left[ \bigforall_{ E \in \pendsof{C}{\set P} } E \le M \right] \equiv
+ \left[ \bigforall_{ A \in \pancsof{C}{\set P} } A \le M \right]
+}\]
+
+\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
+$A \le C$, 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 $A \le A''$.
+}
+
+\section{Commit annotation}
+
+We annotate each Topbloke commit $C$ with:
+\begin{gather}
+\tag*{} \patchof{C} \\
+\tag*{} \baseof{C}, \text{ if } C \in \py \\
+\tag*{} \bigforall_{\pa{Q}}
+ \text{ either } C \haspatch \pa{Q} \text{ or } C \nothaspatch \pa{Q} \\
+\tag*{} \bigforall_{\pay{Q} \not\ni C} \pendsof{C}{\pay{Q}}
+\end{gather}
+
+We do not annotate $\pendsof{C}{\py}$ for $C \in \py$ doing so would
+break making plain commits with git because the recorded $\pends$
+would have to be updated. The annotation is not needed because
+$\forall_{\py \ni C} \pendsof{C}{\py} = \{C\}$.
\section{Test more symbols}
$ C \nothaspatch \p $
+$ \p \patchisin C $
+
+$ \p \notpatchisin C $
+
$ \{ B \} \areparents C $
\end{document}
~ian / topbloke-formulae.git / blobdiff