1 \documentclass[a4paper,leqno]{strayman}
3 \let\numberwithin=\notdef
11 \renewcommand{\ge}{\geqslant}
12 \renewcommand{\le}{\leqslant}
13 \newcommand{\nge}{\ngeqslant}
14 \newcommand{\nle}{\nleqslant}
16 \newcommand{\has}{\sqsupseteq}
17 \newcommand{\isin}{\sqsubseteq}
19 \newcommand{\nothaspatch}{\mathrel{\,\not\!\not\relax\haspatch}}
20 \newcommand{\notpatchisin}{\mathrel{\,\not\!\not\relax\patchisin}}
21 \newcommand{\haspatch}{\sqSupset}
22 \newcommand{\patchisin}{\sqSubset}
24 \newif\ifhidehack\hidehackfalse
25 \DeclareRobustCommand\hidefromedef[2]{%
26 \hidehacktrue\ifhidehack#1\else#2\fi\hidehackfalse}
27 \newcommand{\pa}[1]{\hidefromedef{\varmathbb{#1}}{#1}}
29 \newcommand{\set}[1]{\mathbb{#1}}
30 \newcommand{\pay}[1]{\pa{#1}^+}
31 \newcommand{\pan}[1]{\pa{#1}^-}
33 \newcommand{\p}{\pa{P}}
34 \newcommand{\py}{\pay{P}}
35 \newcommand{\pn}{\pan{P}}
37 \newcommand{\pr}{\pa{R}}
38 \newcommand{\pry}{\pay{R}}
39 \newcommand{\prn}{\pan{R}}
41 %\newcommand{\hasparents}{\underaccent{1}{>}}
42 %\newcommand{\hasparents}{{%
43 % \declareslashed{}{_{_1}}{0}{-0.8}{>}\slashed{>}}}
44 \newcommand{\hasparents}{>_{\mkern-7.0mu _1}}
45 \newcommand{\areparents}{<_{\mkern-14.0mu _1\mkern+5.0mu}}
47 \renewcommand{\implies}{\Rightarrow}
48 \renewcommand{\equiv}{\Leftrightarrow}
49 \renewcommand{\land}{\wedge}
50 \renewcommand{\lor}{\vee}
52 \newcommand{\pancs}{{\mathcal A}}
53 \newcommand{\pends}{{\mathcal E}}
55 \newcommand{\pancsof}[2]{\pancs ( #1 , #2 ) }
56 \newcommand{\pendsof}[2]{\pends ( #1 , #2 ) }
58 \newcommand{\merge}{{\mathcal M}}
59 \newcommand{\mergeof}[4]{\merge(#1,#2,#3,#4)}
60 %\newcommand{\merge}[4]{{#2 {{\frac{ #1 }{ #3 } #4}}}}
62 \newcommand{\patch}{{\mathcal P}}
63 \newcommand{\base}{{\mathcal B}}
65 \newcommand{\patchof}[1]{\patch ( #1 ) }
66 \newcommand{\baseof}[1]{\base ( #1 ) }
68 \newcommand{\eqn}[2]{ #2 \tag*{\mbox{\bf #1}} }
69 \newcommand{\corrolary}[1]{ #1 \tag*{\mbox{\it Corrolary.}} }
71 %\newcommand{\bigforall}{\mathop{\hbox{\huge$\forall$}}}
72 \newcommand{\bigforall}{%
74 {\hbox{\huge$\forall$}}%
75 {\hbox{\Large$\forall$}}%
76 {\hbox{\normalsize$\forall$}}%
77 {\hbox{\scriptsize$\forall$}}}%
80 \newcommand{\Largeexists}{\mathop{\hbox{\Large$\exists$}}}
81 \newcommand{\Largenexists}{\mathop{\hbox{\Large$\nexists$}}}
83 \newcommand{\qed}{\square}
84 \newcommand{\proofstarts}{{\it Proof:}}
85 \newcommand{\proof}[1]{\proofstarts #1 $\qed$}
87 \newcommand{\gathbegin}{\begin{gather} \tag*{}}
88 \newcommand{\gathnext}{\\ \tag*{}}
91 \newcommand{\false}{f}
99 \desclabelstyle{\nextlinelabel}
101 \item[ $ C \hasparents \set X $ ]
102 The parents of commit $C$ are exactly the set
106 $C$ is a descendant of $D$ in the git commit
107 graph. This is a partial order, namely the transitive closure of
108 $ D \in \set X $ where $ C \hasparents \set X $.
110 \item[ $ C \has D $ ]
111 Informally, the tree at commit $C$ contains the change
112 made in commit $D$. Does not take account of deliberate reversions by
113 the user or reversion, rebasing or rewinding in
114 non-Topbloke-controlled branches. For merges and Topbloke-generated
115 anticommits or re-commits, the ``change made'' is only to be thought
116 of as any conflict resolution. This is not a partial order because it
119 \item[ $ \p, \py, \pn $ ]
120 A patch $\p$ consists of two sets of commits $\pn$ and $\py$, which
121 are respectively the base and tip git branches. $\p$ may be used
122 where the context requires a set, in which case the statement
123 is to be taken as applying to both $\py$ and $\pn$.
124 All these sets are distinct. Hence:
126 \item[ $ \patchof{ C } $ ]
127 Either $\p$ s.t. $ C \in \p $, or $\bot$.
128 A function from commits to patches' sets $\p$.
130 \item[ $ \pancsof{C}{\set P} $ ]
131 $ \{ A \; | \; A \le C \land A \in \set P \} $
132 i.e. all the ancestors of $C$
133 which are in $\set P$.
135 \item[ $ \pendsof{C}{\set P} $ ]
136 $ \{ E \; | \; E \in \pancsof{C}{\set P}
137 \land \mathop{\not\exists}_{A \in \pancsof{C}{\set P}}
138 E \neq A \land E \le A \} $
139 i.e. all $\le$-maximal commits in $\pancsof{C}{\set P}$.
141 \item[ $ \baseof{C} $ ]
142 $ \pendsof{C}{\pn} = \{ \baseof{C} \} $ where $ C \in \py $.
143 A partial function from commits to commits.
144 See Unique Base, below.
146 \item[ $ C \haspatch \p $ ]
147 $\displaystyle \bigforall_{D \in \py} D \isin C \equiv D \le C $.
148 ~ Informally, $C$ has the contents of $\p$.
150 \item[ $ C \nothaspatch \p $ ]
151 $\displaystyle \bigforall_{D \in \py} D \not\isin C $.
152 ~ Informally, $C$ has none of the contents of $\p$.
154 Non-Topbloke commits are $\nothaspatch \p$ for all $\p$; if a Topbloke
155 patch is applied to a non-Topbloke branch and then bubbles back to
156 the Topbloke patch itself, we hope that git's merge algorithm will
157 DTRT or that the user will no longer care about the Topbloke patch.
159 \item[ $\displaystyle \mergeof{C}{L}{M}{R} $ ]
160 The contents of a git merge result:
162 $\displaystyle D \isin C \equiv
164 (D \isin L \land D \isin R) \lor D = C : & \true \\
165 (D \not\isin L \land D \not\isin R) \land D \neq C : & \false \\
166 \text{otherwise} : & D \not\isin M
174 We maintain these each time we construct a new commit. \\
176 C \has D \implies C \ge D
178 \[\eqn{Unique Base:}{
179 \bigforall_{C \in \py} \pendsof{C}{\pn} = \{ B \}
181 \[\eqn{Tip Contents:}{
182 \bigforall_{C \in \py} D \isin C \equiv
183 { D \isin \baseof{C} \lor \atop
184 (D \in \py \land D \le C) }
186 \[\eqn{Base Acyclic:}{
187 \bigforall_{B \in \pn} D \isin B \implies D \notin \py
190 \bigforall_{C,\p} C \haspatch \p \lor C \nothaspatch \p
192 \[\eqn{Foreign Inclusion:}{
193 \bigforall_{D \text{ s.t. } \patchof{D} = \bot} D \isin C \equiv D \leq C
196 \section{Some lemmas}
198 \[ \eqn{Exclusive Tip Contents:}{
199 \bigforall_{C \in \py}
200 \neg \Bigl[ D \isin \baseof{C} \land ( D \in \py \land D \le C )
203 Ie, the two limbs of the RHS of Tip Contents are mutually exclusive.
206 Let $B = \baseof{C}$ in $D \isin \baseof{C}$. Now $B \in \pn$.
207 So by Base Acyclic $D \isin B \implies D \notin \py$.
210 \bigforall_{C \in \py} D \isin C \equiv
212 D \in \py : & D \le C \\
213 D \not\in \py : & D \isin \baseof{C}
217 \[ \eqn{Tip Self Inpatch:}{
218 \bigforall_{C \in \py} C \haspatch \p
220 Ie, tip commits contain their own patch.
223 Apply Exclusive Tip Contents to some $D \in \py$:
224 $ \bigforall_{C \in \py}\bigforall_{D \in \py}
225 D \isin C \equiv D \le C $
228 \[ \eqn{Exact Ancestors:}{
229 \bigforall_{ C \hasparents \set{R} }
231 ( \mathop{\hbox{\huge{$\vee$}}}_{R \in \set R} D \le R )
235 \[ \eqn{Transitive Ancestors:}{
236 \left[ \bigforall_{ E \in \pendsof{C}{\set P} } E \le M \right] \equiv
237 \left[ \bigforall_{ A \in \pancsof{C}{\set P} } A \le M \right]
241 The implication from right to left is trivial because
242 $ \pends() \subset \pancs() $.
243 For the implication from left to right:
244 by the definition of $\mathcal E$,
245 for every such $A$, either $A \in \pends()$ which implies
246 $A \le M$ by the LHS directly,
247 or $\exists_{A' \in \pancs()} \; A' \neq A \land A \le A' $
248 in which case we repeat for $A'$. Since there are finitely many
249 commits, this terminates with $A'' \in \pends()$, ie $A'' \le M$
250 by the LHS. And $A \le A''$.
252 \[ \eqn{Calculation Of Ends:}{
253 \bigforall_{C \hasparents \set A}
254 \pendsof{C}{\set P} =
256 \Bigl[ \Largeexists_{A \in \set A}
257 E \in \pendsof{A}{\set P} \Bigr] \land
258 \Bigl[ \Largenexists_{B \in \set A}
259 E \neq B \land E \le B \Bigr]
264 \subsection{No Replay for Merge Results}
266 If we are constructing $C$, with,
274 No Replay is preserved. \proofstarts
276 \subsubsection{For $D=C$:} $D \isin C, D \le C$. OK.
278 \subsubsection{For $D \isin L \land D \isin R$:}
279 $D \isin C$. And $D \isin L \implies D \le L \implies D \le C$. OK.
281 \subsubsection{For $D \neq C \land D \not\isin L \land D \not\isin R$:}
284 \subsubsection{For $D \neq C \land (D \isin L \equiv D \not\isin R)
285 \land D \not\isin M$:}
286 $D \isin C$. Also $D \isin L \lor D \isin R$ so $D \le L \lor D \le
289 \subsubsection{For $D \neq C \land (D \isin L \equiv D \not\isin R)
295 \section{Commit annotation}
297 We annotate each Topbloke commit $C$ with:
301 \baseof{C}, \text{ if } C \in \py
304 \text{ either } C \haspatch \pa{Q} \text{ or } C \nothaspatch \pa{Q}
306 \bigforall_{\pay{Q} \not\ni C} \pendsof{C}{\pay{Q}}
309 We do not annotate $\pendsof{C}{\py}$ for $C \in \py$. Doing so would
310 make it wrong to make plain commits with git because the recorded $\pends$
311 would have to be updated. The annotation is not needed because
312 $\forall_{\py \ni C} \; \pendsof{C}{\py} = \{C\}$.
314 \section{Simple commit}
316 A simple single-parent forward commit $C$ as made by git-commit.
318 \tag*{} C \hasparents \{ A \} \\
319 \tag*{} \patchof{C} = \patchof{A} \\
320 \tag*{} D \isin C \equiv D \isin A \lor D = C
323 \subsection{No Replay}
326 \subsection{Unique Base}
327 If $A, C \in \py$ then $\baseof{C} = \baseof{A}$. $\qed$
329 \subsection{Tip Contents}
330 We need to consider only $A, C \in \py$. From Tip Contents for $A$:
331 \[ D \isin A \equiv D \isin \baseof{A} \lor ( D \in \py \land D \le A ) \]
332 Substitute into the contents of $C$:
333 \[ D \isin C \equiv D \isin \baseof{A} \lor ( D \in \py \land D \le A )
335 Since $D = C \implies D \in \py$,
336 and substituting in $\baseof{C}$, this gives:
337 \[ D \isin C \equiv D \isin \baseof{C} \lor
338 (D \in \py \land D \le A) \lor
339 (D = C \land D \in \py) \]
340 \[ \equiv D \isin \baseof{C} \lor
341 [ D \in \py \land ( D \le A \lor D = C ) ] \]
342 So by Exact Ancestors:
343 \[ D \isin C \equiv D \isin \baseof{C} \lor ( D \in \py \land D \le C
347 \subsection{Base Acyclic}
349 Need to consider only $A, C \in \pn$.
351 For $D = C$: $D \in \pn$ so $D \not\in \py$. OK.
353 For $D \neq C$: $D \isin C \equiv D \isin A$, so by Base Acyclic for
354 $A$, $D \isin C \implies D \not\in \py$. $\qed$
356 \subsection{Coherence and patch inclusion}
358 Need to consider $D \in \py$
360 \subsubsection{For $A \haspatch P, D = C$:}
366 $ D \isin C \equiv \ldots \lor \true \text{ so } D \haspatch C $.
368 \subsubsection{For $A \haspatch P, D \neq C$:}
369 Ancestors: $ D \le C \equiv D \le A $.
371 Contents: $ D \isin C \equiv D \isin A \lor f $
372 so $ D \isin C \equiv D \isin A $.
375 \[ A \haspatch P \implies C \haspatch P \]
377 \subsubsection{For $A \nothaspatch P$:}
379 Firstly, $C \not\in \py$ since if it were, $A \in \py$.
382 Now by contents of $A$, $D \notin A$, so $D \notin C$.
385 \[ A \nothaspatch P \implies C \nothaspatch P \]
388 \subsection{Foreign inclusion:}
390 If $D = C$, trivial. For $D \neq C$:
391 $D \isin C \equiv D \isin A \equiv D \le A \equiv D \le C$. $\qed$
395 Given $L, R^+, R^-$ where
396 $R^+ \in \pry, R^- = \baseof{R^+}$.
397 Construct $C$ which has $\pr$ removed.
398 Used for removing a branch dependency.
400 C \hasparents \{ L \}
402 \patchof{C} = \patchof{L}
404 \mergeof{C}{L}{R^+}{R^-}
407 \subsection{Conditions}
409 \[ \eqn{ Unique Tip }{
410 \pendsof{L}{\pry} = \{ R^+ \}
412 \[ \eqn{ Currently Included }{
419 \subsection{No Replay}
421 By Unique Tip, $R^+ \le L$. By definition of $\base$, $R^- \le R^+$
422 so $R^- \le L$. So $R^+ \le C$ and $R^- \le C$ and No Replay for
423 Merge Results applies. $\qed$
425 \subsection{Desired Contents}
427 \[ D \isin C \equiv [ D \notin \pry \land D \isin L ] \lor D = C \]
430 \subsubsection{For $D = C$:}
432 Trivially $D \isin C$. OK.
434 \subsubsection{For $D \neq C, D \not\le L$:}
436 By No Replay $D \not\isin L$. Also $D \not\le R^-$ hence
437 $D \not\isin R^-$. Thus $D \not\isin C$. OK.
439 \subsubsection{For $D \neq C, D \le L, D \in \pry$:}
441 By Currently Included, $D \isin L$.
443 By Tip Self Inpatch, $D \isin R^+ \equiv D \le R^+$, but by
444 by Unique Tip, $D \le R^+ \equiv D \le L$.
447 By Base Acyclic, $D \not\isin R^-$.
449 Apply $\merge$: $D \not\isin C$. OK.
451 \subsubsection{For $D \neq C, D \le L, D \notin \pry$:}
453 By Tip Contents for $R^+$, $D \isin R^+ \equiv D \isin R^-$.
455 Apply $\merge$: $D \isin C \equiv D \isin L$. OK.
459 \subsection{Unique Base}
461 Need to consider only $C \in \py$, ie $L \in \py$.
465 xxx need to finish anticommit
469 Merge commits $L$ and $R$ using merge base $M$ ($M < L, M < R$):
471 C \hasparents \{ L, R \}
473 \patchof{C} = \patchof{L}
477 We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$.
479 \subsection{Conditions}
481 \[ \eqn{ Tip Merge }{
484 R \in \py : & \baseof{R} \ge \baseof{L}
485 \land [\baseof{L} = M \lor \baseof{L} = \baseof{M}] \\
486 R \in \pn : & R \ge \baseof{L}
487 \land M = \baseof{L} \\
488 \text{otherwise} : & \false
491 \[ \eqn{ Merge Ends }{
492 X \not\haspatch \p \land
496 M \haspatch \p : & \displaystyle
497 \bigforall_{E \in \pendsof{Y}{\py}} E \le M \\
498 M \nothaspatch \p : & \displaystyle
499 \bigforall_{E \in \pendsof{X}{\py}} E \le Y
503 \subsection{No Replay}
505 See No Replay for Merge Results.
507 \subsection{Unique Base}
509 Need to consider only $C \in \py$, ie $L \in \py$,
510 and calculate $\pendsof{C}{\pn}$. So we will consider some
511 putative ancestor $A \in \pn$ and see whether $A \le C$.
513 By Exact Ancestors for C, $A \le C \equiv A \le L \lor A \le R \lor A = C$.
514 But $C \in py$ and $A \in \pn$ so $A \neq C$.
515 Thus $A \le C \equiv A \le L \lor A \le R$.
517 By Unique Base of L and Transitive Ancestors,
518 $A \le L \equiv A \le \baseof{L}$.
520 \subsubsection{For $R \in \py$:}
522 By Unique Base of $R$ and Transitive Ancestors,
523 $A \le R \equiv A \le \baseof{R}$.
525 But by Tip Merge condition on $\baseof{R}$,
526 $A \le \baseof{L} \implies A \le \baseof{R}$, so
527 $A \le \baseof{R} \lor A \le \baseof{L} \equiv A \le \baseof{R}$.
528 Thus $A \le C \equiv A \le \baseof{R}$.
529 That is, $\baseof{C} = \baseof{R}$.
531 \subsubsection{For $R \in \pn$:}
533 By Tip Merge condition on $R$,
534 $A \le \baseof{L} \implies A \le R$, so
535 $A \le R \lor A \le \baseof{L} \equiv A \le R$.
536 Thus $A \le C \equiv A \le R$.
537 That is, $\baseof{C} = R$.
541 \subsection{Coherence and patch inclusion}
543 Need to determine $C \haspatch \p$ based on $L,M,R \haspatch \p$.
544 This involves considering $D \in \py$.
546 \subsubsection{For $L \nothaspatch \p, R \nothaspatch \p$:}
547 $D \not\isin L \land D \not\isin R$. $C \not\in \py$ (otherwise $L
548 \in \py$ ie $L \haspatch \p$ by Tip Self Inpatch). So $D \neq C$.
549 Applying $\merge$ gives $D \not\isin C$ i.e. $C \nothaspatch \p$.
551 \subsubsection{For $L \haspatch \p, R \haspatch \p$:}
552 $D \isin L \equiv D \le L$ and $D \isin R \equiv D \le R$.
553 (Likewise $D \isin X \equiv D \le X$ and $D \isin Y \equiv D \le Y$.)
555 Consider $D = C$: $D \isin C$, $D \le C$, OK for $C \haspatch \p$.
557 For $D \neq C$: $D \le C \equiv D \le L \lor D \le R
558 \equiv D \isin L \lor D \isin R$.
559 (Likewise $D \le C \equiv D \le X \lor D \le Y$.)
561 Consider $D \neq C, D \isin X \land D \isin Y$:
562 By $\merge$, $D \isin C$. Also $D \le X$
563 so $D \le C$. OK for $C \haspatch \p$.
565 Consider $D \neq C, D \not\isin X \land D \not\isin Y$:
566 By $\merge$, $D \not\isin C$.
567 And $D \not\le X \land D \not\le Y$ so $D \not\le C$.
568 OK for $C \haspatch \p$.
570 Remaining case, wlog, is $D \not\isin X \land D \isin Y$.
571 $D \not\le X$ so $D \not\le M$ so $D \not\isin M$.
572 Thus by $\merge$, $D \isin C$. And $D \le Y$ so $D \le C$.
573 OK for $C \haspatch \p$.
575 So indeed $L \haspatch \p \land R \haspatch \p \implies C \haspatch \p$.
577 \subsubsection{For (wlog) $X \not\haspatch \p, Y \haspatch \p$:}
579 $C \haspatch \p \equiv M \nothaspatch \p$.
583 Merge Ends applies. Recall that we are considering $D \in \py$.
584 $D \isin Y \equiv D \le Y$. $D \not\isin X$.
585 We will show for each of
586 various cases that $D \isin C \equiv M \nothaspatch \p \land D \le C$
587 (which suffices by definition of $\haspatch$ and $\nothaspatch$).
589 Consider $D = C$: Thus $C \in \py, L \in \py$, and by Tip
590 Self Inpatch $L \haspatch \p$, so $L=Y, R=X$. By Tip Merge,
591 $M=\baseof{L}$. So by Base Acyclic $D \not\isin M$, i.e.
592 $M \nothaspatch \p$. And indeed $D \isin C$ and $D \le C$. OK.
594 Consider $D \neq C, M \nothaspatch P, D \isin Y$:
595 $D \le Y$ so $D \le C$.
596 $D \not\isin M$ so by $\merge$, $D \isin C$. OK.
598 Consider $D \neq C, M \nothaspatch P, D \not\isin Y$:
599 $D \not\le Y$. If $D \le X$ then
600 $D \in \pancsof{X}{\py}$, so by Merge Ends and
601 Transitive Ancestors $D \le Y$ --- a contradiction, so $D \not\le X$.
602 Thus $D \not\le C$. By $\merge$, $D \not\isin C$. OK.
604 Consider $D \neq C, M \haspatch P, D \isin Y$:
605 $D \le Y$ so $D \in \pancsof{Y}{\py}$ so by Merge Ends
606 and Transitive Ancestors $D \le M$.
607 Thus $D \isin M$. By $\merge$, $D \not\isin C$. OK.
609 Consider $D \neq C, M \haspatch P, D \not\isin Y$:
610 By $\merge$, $D \not\isin C$. OK.