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{\nequiv}{\nLeftrightarrow}
50 \renewcommand{\land}{\wedge}
51 \renewcommand{\lor}{\vee}
53 \newcommand{\pancs}{{\mathcal A}}
54 \newcommand{\pends}{{\mathcal E}}
56 \newcommand{\pancsof}[2]{\pancs ( #1 , #2 ) }
57 \newcommand{\pendsof}[2]{\pends ( #1 , #2 ) }
59 \newcommand{\merge}{{\mathcal M}}
60 \newcommand{\mergeof}[4]{\merge(#1,#2,#3,#4)}
61 %\newcommand{\merge}[4]{{#2 {{\frac{ #1 }{ #3 } #4}}}}
63 \newcommand{\patch}{{\mathcal P}}
64 \newcommand{\base}{{\mathcal B}}
66 \newcommand{\patchof}[1]{\patch ( #1 ) }
67 \newcommand{\baseof}[1]{\base ( #1 ) }
69 \newcommand{\eqntag}[2]{ #2 \tag*{\mbox{#1}} }
70 \newcommand{\eqn}[2]{ #2 \tag*{\mbox{\bf #1}} }
72 %\newcommand{\bigforall}{\mathop{\hbox{\huge$\forall$}}}
73 \newcommand{\bigforall}{%
75 {\hbox{\huge$\forall$}}%
76 {\hbox{\Large$\forall$}}%
77 {\hbox{\normalsize$\forall$}}%
78 {\hbox{\scriptsize$\forall$}}}%
81 \newcommand{\Largeexists}{\mathop{\hbox{\Large$\exists$}}}
82 \newcommand{\Largenexists}{\mathop{\hbox{\Large$\nexists$}}}
84 \newcommand{\qed}{\square}
85 \newcommand{\proofstarts}{{\it Proof:}}
86 \newcommand{\proof}[1]{\proofstarts #1 $\qed$}
88 \newcommand{\gathbegin}{\begin{gather} \tag*{}}
89 \newcommand{\gathnext}{\\ \tag*{}}
92 \newcommand{\false}{f}
100 \desclabelstyle{\nextlinelabel}
102 \item[ $ C \hasparents \set X $ ]
103 The parents of commit $C$ are exactly the set
107 $C$ is a descendant of $D$ in the git commit
108 graph. This is a partial order, namely the transitive closure of
109 $ D \in \set X $ where $ C \hasparents \set X $.
111 \item[ $ C \has D $ ]
112 Informally, the tree at commit $C$ contains the change
113 made in commit $D$. Does not take account of deliberate reversions by
114 the user or reversion, rebasing or rewinding in
115 non-Topbloke-controlled branches. For merges and Topbloke-generated
116 anticommits or re-commits, the ``change made'' is only to be thought
117 of as any conflict resolution. This is not a partial order because it
120 \item[ $ \p, \py, \pn $ ]
121 A patch $\p$ consists of two sets of commits $\pn$ and $\py$, which
122 are respectively the base and tip git branches. $\p$ may be used
123 where the context requires a set, in which case the statement
124 is to be taken as applying to both $\py$ and $\pn$.
125 None of these sets overlap. Hence:
127 \item[ $ \patchof{ C } $ ]
128 Either $\p$ s.t. $ C \in \p $, or $\bot$.
129 A function from commits to patches' sets $\p$.
131 \item[ $ \pancsof{C}{\set P} $ ]
132 $ \{ A \; | \; A \le C \land A \in \set P \} $
133 i.e. all the ancestors of $C$
134 which are in $\set P$.
136 \item[ $ \pendsof{C}{\set P} $ ]
137 $ \{ E \; | \; E \in \pancsof{C}{\set P}
138 \land \mathop{\not\exists}_{A \in \pancsof{C}{\set P}}
139 E \neq A \land E \le A \} $
140 i.e. all $\le$-maximal commits in $\pancsof{C}{\set P}$.
142 \item[ $ \baseof{C} $ ]
143 $ \pendsof{C}{\pn} = \{ \baseof{C} \} $ where $ C \in \py $.
144 A partial function from commits to commits.
145 See Unique Base, below.
147 \item[ $ C \haspatch \p $ ]
148 $\displaystyle \bigforall_{D \in \py} D \isin C \equiv D \le C $.
149 ~ Informally, $C$ has the contents of $\p$.
151 \item[ $ C \nothaspatch \p $ ]
152 $\displaystyle \bigforall_{D \in \py} D \not\isin C $.
153 ~ Informally, $C$ has none of the contents of $\p$.
155 Non-Topbloke commits are $\nothaspatch \p$ for all $\p$. This
156 includes commits on plain git branches made by applying a Topbloke
158 patch is applied to a non-Topbloke branch and then bubbles back to
159 the relevant Topbloke branches, we hope that
160 if the user still cares about the Topbloke patch,
161 git's merge algorithm will DTRT when trying to re-apply the changes.
163 \item[ $\displaystyle \mergeof{C}{L}{M}{R} $ ]
164 The contents of a git merge result:
166 $\displaystyle D \isin C \equiv
168 (D \isin L \land D \isin R) \lor D = C : & \true \\
169 (D \not\isin L \land D \not\isin R) \land D \neq C : & \false \\
170 \text{otherwise} : & D \not\isin M
178 We maintain these each time we construct a new commit. \\
180 C \has D \implies C \ge D
182 \[\eqn{Unique Base:}{
183 \bigforall_{C \in \py} \pendsof{C}{\pn} = \{ B \}
185 \[\eqn{Tip Contents:}{
186 \bigforall_{C \in \py} D \isin C \equiv
187 { D \isin \baseof{C} \lor \atop
188 (D \in \py \land D \le C) }
190 \[\eqn{Base Acyclic:}{
191 \bigforall_{B \in \pn} D \isin B \implies D \notin \py
194 \bigforall_{C,\p} C \haspatch \p \lor C \nothaspatch \p
196 \[\eqn{Foreign Inclusion:}{
197 \bigforall_{D \text{ s.t. } \patchof{D} = \bot} D \isin C \equiv D \leq C
199 \[\eqn{Foreign Contents:}{
200 \bigforall_{C \text{ s.t. } \patchof{C} = \bot}
201 D \le C \implies \patchof{D} = \bot
204 \section{Some lemmas}
206 \[ \eqn{Alternative (overlapping) formulations defining
207 $\mergeof{C}{L}{M}{R}$:}{
210 D \isin L \equiv D \isin R : & D = C \lor D \isin L \\
211 D \isin L \nequiv D \isin R : & D = C \lor D \not\isin M \\
212 D \isin L \equiv D \isin M : & D = C \lor D \isin R \\
213 D \isin L \nequiv D \isin M : & D = C \lor D \isin L \\
214 \text{as above with L and R exchanged}
220 Original definition is symmetrical in $L$ and $R$.
223 \[ \eqn{Exclusive Tip Contents:}{
224 \bigforall_{C \in \py}
225 \neg \Bigl[ D \isin \baseof{C} \land ( D \in \py \land D \le C )
228 Ie, the two limbs of the RHS of Tip Contents are mutually exclusive.
231 Let $B = \baseof{C}$ in $D \isin \baseof{C}$. Now $B \in \pn$.
232 So by Base Acyclic $D \isin B \implies D \notin \py$.
234 \[ \eqntag{{\it Corollary - equivalent to Tip Contents}}{
235 \bigforall_{C \in \py} D \isin C \equiv
237 D \in \py : & D \le C \\
238 D \not\in \py : & D \isin \baseof{C}
242 \[ \eqn{Tip Self Inpatch:}{
243 \bigforall_{C \in \py} C \haspatch \p
245 Ie, tip commits contain their own patch.
248 Apply Exclusive Tip Contents to some $D \in \py$:
249 $ \bigforall_{C \in \py}\bigforall_{D \in \py}
250 D \isin C \equiv D \le C $
253 \[ \eqn{Exact Ancestors:}{
254 \bigforall_{ C \hasparents \set{R} }
256 ( \mathop{\hbox{\huge{$\vee$}}}_{R \in \set R} D \le R )
261 \[ \eqn{Transitive Ancestors:}{
262 \left[ \bigforall_{ E \in \pendsof{C}{\set P} } E \le M \right] \equiv
263 \left[ \bigforall_{ A \in \pancsof{C}{\set P} } A \le M \right]
267 The implication from right to left is trivial because
268 $ \pends() \subset \pancs() $.
269 For the implication from left to right:
270 by the definition of $\mathcal E$,
271 for every such $A$, either $A \in \pends()$ which implies
272 $A \le M$ by the LHS directly,
273 or $\exists_{A' \in \pancs()} \; A' \neq A \land A \le A' $
274 in which case we repeat for $A'$. Since there are finitely many
275 commits, this terminates with $A'' \in \pends()$, ie $A'' \le M$
276 by the LHS. And $A \le A''$.
279 \[ \eqn{Calculation Of Ends:}{
280 \bigforall_{C \hasparents \set A}
281 \pendsof{C}{\set P} =
285 C \not\in \p : & \displaystyle
287 \Bigl[ \Largeexists_{A \in \set A}
288 E \in \pendsof{A}{\set P} \Bigr] \land
289 \Bigl[ \Largenexists_{B \in \set A}
290 E \neq B \land E \le B \Bigr]
296 \[ \eqn{Totally Foreign Contents:}{
297 \bigforall_{C \hasparents \set A}
299 \patchof{C} = \bot \land
300 \bigforall_{A \in \set A} \patchof{A} = \bot
311 \subsection{No Replay for Merge Results}
313 If we are constructing $C$, with,
321 No Replay is preserved. \proofstarts
323 \subsubsection{For $D=C$:} $D \isin C, D \le C$. OK.
325 \subsubsection{For $D \isin L \land D \isin R$:}
326 $D \isin C$. And $D \isin L \implies D \le L \implies D \le C$. OK.
328 \subsubsection{For $D \neq C \land D \not\isin L \land D \not\isin R$:}
331 \subsubsection{For $D \neq C \land (D \isin L \equiv D \not\isin R)
332 \land D \not\isin M$:}
333 $D \isin C$. Also $D \isin L \lor D \isin R$ so $D \le L \lor D \le
336 \subsubsection{For $D \neq C \land (D \isin L \equiv D \not\isin R)
342 \section{Commit annotation}
344 We annotate each Topbloke commit $C$ with:
348 \baseof{C}, \text{ if } C \in \py
351 \text{ either } C \haspatch \pa{Q} \text{ or } C \nothaspatch \pa{Q}
353 \bigforall_{\pay{Q} \not\ni C} \pendsof{C}{\pay{Q}}
356 $\patchof{C}$, for each kind of Topbloke-generated commit, is stated
357 in the summary in the section for that kind of commit.
359 Whether $\baseof{C}$ is required, and if so what the value is, is
360 stated in the proof of Unique Base for each kind of commit.
362 $C \haspatch \pa{Q}$ or $\nothaspatch \pa{Q}$ is represented as the
363 set $\{ \pa{Q} | C \haspatch \pa{Q} \}$. Whether $C \haspatch \pa{Q}$
365 (in terms of $I \haspatch \pa{Q}$ or $I \nothaspatch \pa{Q}$
366 for the ingredients $I$),
367 in the proof of Coherence for each kind of commit.
369 $\pendsof{C}{\pa{Q}^+}$ is computed, for all Topbloke-generated commits,
370 using the lemma Calculation of Ends, above.
371 We do not annotate $\pendsof{C}{\py}$ for $C \in \py$. Doing so would
372 make it wrong to make plain commits with git because the recorded $\pends$
373 would have to be updated. The annotation is not needed in that case
374 because $\forall_{\py \ni C} \; \pendsof{C}{\py} = \{C\}$.
376 \section{Simple commit}
378 A simple single-parent forward commit $C$ as made by git-commit.
380 \tag*{} C \hasparents \{ A \} \\
381 \tag*{} \patchof{C} = \patchof{A} \\
382 \tag*{} D \isin C \equiv D \isin A \lor D = C
384 This also covers Topbloke-generated commits on plain git branches:
385 Topbloke strips the metadata when exporting.
387 \subsection{No Replay}
390 \subsection{Unique Base}
391 If $A, C \in \py$ then by Calculation of Ends for
392 $C, \py, C \not\in \py$:
393 $\pendsof{C}{\pn} = \pendsof{A}{\pn}$ so
394 $\baseof{C} = \baseof{A}$. $\qed$
396 \subsection{Tip Contents}
397 We need to consider only $A, C \in \py$. From Tip Contents for $A$:
398 \[ D \isin A \equiv D \isin \baseof{A} \lor ( D \in \py \land D \le A ) \]
399 Substitute into the contents of $C$:
400 \[ D \isin C \equiv D \isin \baseof{A} \lor ( D \in \py \land D \le A )
402 Since $D = C \implies D \in \py$,
403 and substituting in $\baseof{C}$, this gives:
404 \[ D \isin C \equiv D \isin \baseof{C} \lor
405 (D \in \py \land D \le A) \lor
406 (D = C \land D \in \py) \]
407 \[ \equiv D \isin \baseof{C} \lor
408 [ D \in \py \land ( D \le A \lor D = C ) ] \]
409 So by Exact Ancestors:
410 \[ D \isin C \equiv D \isin \baseof{C} \lor ( D \in \py \land D \le C
414 \subsection{Base Acyclic}
416 Need to consider only $A, C \in \pn$.
418 For $D = C$: $D \in \pn$ so $D \not\in \py$. OK.
420 For $D \neq C$: $D \isin C \equiv D \isin A$, so by Base Acyclic for
421 $A$, $D \isin C \implies D \not\in \py$.
425 \subsection{Coherence and patch inclusion}
427 Need to consider $D \in \py$
429 \subsubsection{For $A \haspatch P, D = C$:}
435 $ D \isin C \equiv \ldots \lor \true \text{ so } D \haspatch C $.
437 \subsubsection{For $A \haspatch P, D \neq C$:}
438 Ancestors: $ D \le C \equiv D \le A $.
440 Contents: $ D \isin C \equiv D \isin A \lor f $
441 so $ D \isin C \equiv D \isin A $.
444 \[ A \haspatch P \implies C \haspatch P \]
446 \subsubsection{For $A \nothaspatch P$:}
448 Firstly, $C \not\in \py$ since if it were, $A \in \py$.
451 Now by contents of $A$, $D \notin A$, so $D \notin C$.
454 \[ A \nothaspatch P \implies C \nothaspatch P \]
457 \subsection{Foreign inclusion:}
459 If $D = C$, trivial. For $D \neq C$:
460 $D \isin C \equiv D \isin A \equiv D \le A \equiv D \le C$. $\qed$
462 \subsection{Foreign Contents:}
464 Only relevant if $\patchof{C} = \bot$. Trivial by Foreign Contents of
467 xxx fixme not trivial use Totally Foreign Contents
469 \section{Create Base}
471 Given $L$, create a Topbloke base branch initial commit $B$.
473 B \hasparents \{ L \}
477 D \isin B \equiv D \isin L \lor D = B
480 \subsection{Conditions}
482 \[ \eqn{ Ingredients }{
483 \patchof{L} = \pa{L} \lor \patchof{L} = \bot
485 \[ \eqn{ Non-recursion }{
489 \subsection{No Replay}
491 If $\patchof{L} = \pa{L}$, trivial by Base Acyclic for $L$.
493 If $\patchof{L} = \bot$, xxx
495 Trivial from Base Acyclic for $L$. $\qed$
497 \subsection{Unique Base}
499 Not applicable. $\qed$
501 \subsection{Tip Contents}
503 Not applicable. $\qed$
505 \subsection{Base Acyclic}
517 Given $L$ and $\pr$ as represented by $R^+, R^-$.
518 Construct $C$ which has $\pr$ removed.
519 Used for removing a branch dependency.
521 C \hasparents \{ L \}
523 \patchof{C} = \patchof{L}
525 \mergeof{C}{L}{R^+}{R^-}
528 \subsection{Conditions}
530 \[ \eqn{ Ingredients }{
531 R^+ \in \pry \land R^- = \baseof{R^+}
533 \[ \eqn{ Into Base }{
536 \[ \eqn{ Unique Tip }{
537 \pendsof{L}{\pry} = \{ R^+ \}
539 \[ \eqn{ Currently Included }{
543 \subsection{Ordering of ${L, R^+, R^-}$:}
545 By Unique Tip, $R^+ \le L$. By definition of $\base$, $R^- \le R^+$
546 so $R^- \le L$. So $R^+ \le C$ and $R^- \le C$.
549 (Note that $R^+ \not\le R^-$, i.e. the merge base
550 is a descendant, not an ancestor, of the 2nd parent.)
552 \subsection{No Replay}
554 No Replay for Merge Results applies. $\qed$
556 \subsection{Desired Contents}
558 \[ D \isin C \equiv [ D \notin \pry \land D \isin L ] \lor D = C \]
561 \subsubsection{For $D = C$:}
563 Trivially $D \isin C$. OK.
565 \subsubsection{For $D \neq C, D \not\le L$:}
567 By No Replay $D \not\isin L$. Also $D \not\le R^-$ hence
568 $D \not\isin R^-$. Thus $D \not\isin C$. OK.
570 \subsubsection{For $D \neq C, D \le L, D \in \pry$:}
572 By Currently Included, $D \isin L$.
574 By Tip Self Inpatch, $D \isin R^+ \equiv D \le R^+$, but by
575 by Unique Tip, $D \le R^+ \equiv D \le L$.
578 By Base Acyclic, $D \not\isin R^-$.
580 Apply $\merge$: $D \not\isin C$. OK.
582 \subsubsection{For $D \neq C, D \le L, D \notin \pry$:}
584 By Tip Contents for $R^+$, $D \isin R^+ \equiv D \isin R^-$.
586 Apply $\merge$: $D \isin C \equiv D \isin L$. OK.
590 \subsection{Unique Base}
592 Into Base means that $C \in \pn$, so Unique Base is not
595 \subsection{Tip Contents}
597 Again, not applicable. $\qed$
599 \subsection{Base Acyclic}
601 By Base Acyclic for $L$, $D \isin L \implies D \not\in \py$.
602 And by Into Base $C \not\in \py$.
603 Now from Desired Contents, above, $D \isin C
604 \implies D \isin L \lor D = C$, which thus
605 $\implies D \not\in \py$. $\qed$.
607 \subsection{Coherence and Patch Inclusion}
609 Need to consider some $D \in \py$. By Into Base, $D \neq C$.
611 \subsubsection{For $\p = \pr$:}
612 By Desired Contents, above, $D \not\isin C$.
613 So $C \nothaspatch \pr$.
615 \subsubsection{For $\p \neq \pr$:}
616 By Desired Contents, $D \isin C \equiv D \isin L$
617 (since $D \in \py$ so $D \not\in \pry$).
619 If $L \nothaspatch \p$, $D \not\isin L$ so $D \not\isin C$.
620 So $L \nothaspatch \p \implies C \nothaspatch \p$.
622 Whereas if $L \haspatch \p$, $D \isin L \equiv D \le L$.
623 so $L \haspatch \p \implies C \haspatch \p$.
627 \subsection{Foreign Inclusion}
629 Consider some $D$ s.t. $\patchof{D} = \bot$. $D \neq C$.
630 So by Desired Contents $D \isin C \equiv D \isin L$.
631 By Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$.
633 And $D \le C \equiv D \le L$.
634 Thus $D \isin C \equiv D \le C$.
638 \subsection{Foreign Contents:}
640 Not applicable. $\qed$
644 Merge commits $L$ and $R$ using merge base $M$:
646 C \hasparents \{ L, R \}
648 \patchof{C} = \patchof{L}
652 We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$.
654 \subsection{Conditions}
655 \[ \eqn{ Ingredients }{
658 \[ \eqn{ Tip Merge }{
661 R \in \py : & \baseof{R} \ge \baseof{L}
662 \land [\baseof{L} = M \lor \baseof{L} = \baseof{M}] \\
663 R \in \pn : & M = \baseof{L} \\
664 \text{otherwise} : & \false
667 \[ \eqn{ Merge Acyclic }{
672 \[ \eqn{ Removal Merge Ends }{
673 X \not\haspatch \p \land
677 \pendsof{Y}{\py} = \pendsof{M}{\py}
679 \[ \eqn{ Addition Merge Ends }{
680 X \not\haspatch \p \land
684 \bigforall_{E \in \pendsof{X}{\py}} E \le Y
688 \subsection{Non-Topbloke merges}
690 We require both $\patchof{L} = \bot$ and $\patchof{R} = \bot$.
691 I.e. not only is it forbidden to merge into a Topbloke-controlled
692 branch without Topbloke's assistance, it is also forbidden to
693 merge any Topbloke-controlled branch into any plain git branch.
695 Given those conditions, Tip Merge and Merge Acyclic do not apply.
696 And $Y \not\in \py$ so $\neg [ Y \haspatch \p ]$ so neither
697 Merge Ends condition applies. Good.
699 \subsection{No Replay}
701 No Replay for Merge Results applies. $\qed$
703 \subsection{Unique Base}
705 Need to consider only $C \in \py$, ie $L \in \py$,
706 and calculate $\pendsof{C}{\pn}$. So we will consider some
707 putative ancestor $A \in \pn$ and see whether $A \le C$.
709 By Exact Ancestors for C, $A \le C \equiv A \le L \lor A \le R \lor A = C$.
710 But $C \in py$ and $A \in \pn$ so $A \neq C$.
711 Thus $A \le C \equiv A \le L \lor A \le R$.
713 By Unique Base of L and Transitive Ancestors,
714 $A \le L \equiv A \le \baseof{L}$.
716 \subsubsection{For $R \in \py$:}
718 By Unique Base of $R$ and Transitive Ancestors,
719 $A \le R \equiv A \le \baseof{R}$.
721 But by Tip Merge condition on $\baseof{R}$,
722 $A \le \baseof{L} \implies A \le \baseof{R}$, so
723 $A \le \baseof{R} \lor A \le \baseof{L} \equiv A \le \baseof{R}$.
724 Thus $A \le C \equiv A \le \baseof{R}$.
725 That is, $\baseof{C} = \baseof{R}$.
727 \subsubsection{For $R \in \pn$:}
729 By Tip Merge condition on $R$ and since $M \le R$,
730 $A \le \baseof{L} \implies A \le R$, so
731 $A \le R \lor A \le \baseof{L} \equiv A \le R$.
732 Thus $A \le C \equiv A \le R$.
733 That is, $\baseof{C} = R$.
737 \subsection{Coherence and Patch Inclusion}
739 Need to determine $C \haspatch \p$ based on $L,M,R \haspatch \p$.
740 This involves considering $D \in \py$.
742 \subsubsection{For $L \nothaspatch \p, R \nothaspatch \p$:}
743 $D \not\isin L \land D \not\isin R$. $C \not\in \py$ (otherwise $L
744 \in \py$ ie $L \haspatch \p$ by Tip Self Inpatch). So $D \neq C$.
745 Applying $\merge$ gives $D \not\isin C$ i.e. $C \nothaspatch \p$.
747 \subsubsection{For $L \haspatch \p, R \haspatch \p$:}
748 $D \isin L \equiv D \le L$ and $D \isin R \equiv D \le R$.
749 (Likewise $D \isin X \equiv D \le X$ and $D \isin Y \equiv D \le Y$.)
751 Consider $D = C$: $D \isin C$, $D \le C$, OK for $C \haspatch \p$.
753 For $D \neq C$: $D \le C \equiv D \le L \lor D \le R
754 \equiv D \isin L \lor D \isin R$.
755 (Likewise $D \le C \equiv D \le X \lor D \le Y$.)
757 Consider $D \neq C, D \isin X \land D \isin Y$:
758 By $\merge$, $D \isin C$. Also $D \le X$
759 so $D \le C$. OK for $C \haspatch \p$.
761 Consider $D \neq C, D \not\isin X \land D \not\isin Y$:
762 By $\merge$, $D \not\isin C$.
763 And $D \not\le X \land D \not\le Y$ so $D \not\le C$.
764 OK for $C \haspatch \p$.
766 Remaining case, wlog, is $D \not\isin X \land D \isin Y$.
767 $D \not\le X$ so $D \not\le M$ so $D \not\isin M$.
768 Thus by $\merge$, $D \isin C$. And $D \le Y$ so $D \le C$.
769 OK for $C \haspatch \p$.
771 So indeed $L \haspatch \p \land R \haspatch \p \implies C \haspatch \p$.
773 \subsubsection{For (wlog) $X \not\haspatch \p, Y \haspatch \p$:}
775 $M \haspatch \p \implies C \nothaspatch \p$.
776 $M \nothaspatch \p \implies C \haspatch \p$.
780 One of the Merge Ends conditions applies.
781 Recall that we are considering $D \in \py$.
782 $D \isin Y \equiv D \le Y$. $D \not\isin X$.
783 We will show for each of
784 various cases that $D \isin C \equiv M \nothaspatch \p \land D \le C$
785 (which suffices by definition of $\haspatch$ and $\nothaspatch$).
787 Consider $D = C$: Thus $C \in \py, L \in \py$, and by Tip
788 Self Inpatch $L \haspatch \p$, so $L=Y, R=X$. By Tip Merge,
789 $M=\baseof{L}$. So by Base Acyclic $D \not\isin M$, i.e.
790 $M \nothaspatch \p$. And indeed $D \isin C$ and $D \le C$. OK.
792 Consider $D \neq C, M \nothaspatch P, D \isin Y$:
793 $D \le Y$ so $D \le C$.
794 $D \not\isin M$ so by $\merge$, $D \isin C$. OK.
796 Consider $D \neq C, M \nothaspatch P, D \not\isin Y$:
797 $D \not\le Y$. If $D \le X$ then
798 $D \in \pancsof{X}{\py}$, so by Addition Merge Ends and
799 Transitive Ancestors $D \le Y$ --- a contradiction, so $D \not\le X$.
800 Thus $D \not\le C$. By $\merge$, $D \not\isin C$. OK.
802 Consider $D \neq C, M \haspatch P, D \isin Y$:
803 $D \le Y$ so $D \in \pancsof{Y}{\py}$ so by Removal Merge Ends
804 and Transitive Ancestors $D \in \pancsof{M}{\py}$ so $D \le M$.
805 Thus $D \isin M$. By $\merge$, $D \not\isin C$. OK.
807 Consider $D \neq C, M \haspatch P, D \not\isin Y$:
808 By $\merge$, $D \not\isin C$. OK.
812 \subsection{Base Acyclic}
814 This applies when $C \in \pn$.
815 $C \in \pn$ when $L \in \pn$ so by Merge Acyclic, $R \nothaspatch \p$.
817 Consider some $D \in \py$.
819 By Base Acyclic of $L$, $D \not\isin L$. By the above, $D \not\isin
820 R$. And $D \neq C$. So $D \not\isin C$.
824 \subsection{Tip Contents}
826 We need worry only about $C \in \py$.
827 And $\patchof{C} = \patchof{L}$
828 so $L \in \py$ so $L \haspatch \p$. We will use the Unique Base
829 of $C$, and its Coherence and Patch Inclusion, as just proved.
831 Firstly we show $C \haspatch \p$: If $R \in \py$, then $R \haspatch
832 \p$ and by Coherence/Inclusion $C \haspatch \p$ . If $R \not\in \py$
833 then by Tip Merge $M = \baseof{L}$ so by Base Acyclic and definition
834 of $\nothaspatch$, $M \nothaspatch \p$. So by Coherence/Inclusion $C
835 \haspatch \p$ (whether $R \haspatch \p$ or $\nothaspatch$).
837 We will consider an arbitrary commit $D$
838 and prove the Exclusive Tip Contents form.
840 \subsubsection{For $D \in \py$:}
841 $C \haspatch \p$ so by definition of $\haspatch$, $D \isin C \equiv D
844 \subsubsection{For $D \not\in \py, R \not\in \py$:}
846 $D \neq C$. By Tip Contents of $L$,
847 $D \isin L \equiv D \isin \baseof{L}$, and by Tip Merge condition,
848 $D \isin L \equiv D \isin M$. So by definition of $\merge$, $D \isin
849 C \equiv D \isin R$. And $R = \baseof{C}$ by Unique Base of $C$.
850 Thus $D \isin C \equiv D \isin \baseof{C}$. OK.
852 \subsubsection{For $D \not\in \py, R \in \py$:}
857 $D \isin L \equiv D \isin \baseof{L}$ and
858 $D \isin R \equiv D \isin \baseof{R}$.
860 If $\baseof{L} = M$, trivially $D \isin M \equiv D \isin \baseof{L}.$
861 Whereas if $\baseof{L} = \baseof{M}$, by definition of $\base$,
862 $\patchof{M} = \patchof{L} = \py$, so by Tip Contents of $M$,
863 $D \isin M \equiv D \isin \baseof{M} \equiv D \isin \baseof{L}$.
865 So $D \isin M \equiv D \isin L$ and by $\merge$,
866 $D \isin C \equiv D \isin R$. But from Unique Base,
867 $\baseof{C} = R$ so $D \isin C \equiv D \isin \baseof{C}$. OK.
871 \subsection{Foreign Inclusion}
873 Consider some $D$ s.t. $\patchof{D} = \bot$.
874 By Foreign Inclusion of $L, M, R$:
875 $D \isin L \equiv D \le L$;
876 $D \isin M \equiv D \le M$;
877 $D \isin R \equiv D \le R$.
879 \subsubsection{For $D = C$:}
881 $D \isin C$ and $D \le C$. OK.
883 \subsubsection{For $D \neq C, D \isin M$:}
885 Thus $D \le M$ so $D \le L$ and $D \le R$ so $D \isin L$ and $D \isin
886 R$. So by $\merge$, $D \isin C$. And $D \le C$. OK.
888 \subsubsection{For $D \neq C, D \not\isin M, D \isin X$:}
890 By $\merge$, $D \isin C$.
891 And $D \isin X$ means $D \le X$ so $D \le C$.
894 \subsubsection{For $D \neq C, D \not\isin M, D \not\isin L, D \not\isin R$:}
896 By $\merge$, $D \not\isin C$.
897 And $D \not\le L, D \not\le R$ so $D \not\le C$.
902 \subsection{Foreign Contents:}
904 xxx use Totally Foreign Contents
906 If $\patchof{C} = \bot$, by Foreign Merges
907 $\patchof{L} = \patchof{R} = \bot$.