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
687 \[ \eqn{ Foreign Merges }{
688 \patchof{L} = \bot \equiv \patchof{R} = \bot
691 \subsection{Non-Topbloke merges}
693 We require both $\patchof{L} = \bot$ and $\patchof{R} = \bot$
694 (Foreign Merges, above).
695 I.e. not only is it forbidden to merge into a Topbloke-controlled
696 branch without Topbloke's assistance, it is also forbidden to
697 merge any Topbloke-controlled branch into any plain git branch.
699 Given those conditions, Tip Merge and Merge Acyclic do not apply.
700 And $Y \not\in \py$ so $\neg [ Y \haspatch \p ]$ so neither
701 Merge Ends condition applies.
703 So a plain git merge of non-Topbloke branches meets the conditions and
704 is therefore consistent with our scheme.
706 \subsection{No Replay}
708 No Replay for Merge Results applies. $\qed$
710 \subsection{Unique Base}
712 Need to consider only $C \in \py$, ie $L \in \py$,
713 and calculate $\pendsof{C}{\pn}$. So we will consider some
714 putative ancestor $A \in \pn$ and see whether $A \le C$.
716 By Exact Ancestors for C, $A \le C \equiv A \le L \lor A \le R \lor A = C$.
717 But $C \in py$ and $A \in \pn$ so $A \neq C$.
718 Thus $A \le C \equiv A \le L \lor A \le R$.
720 By Unique Base of L and Transitive Ancestors,
721 $A \le L \equiv A \le \baseof{L}$.
723 \subsubsection{For $R \in \py$:}
725 By Unique Base of $R$ and Transitive Ancestors,
726 $A \le R \equiv A \le \baseof{R}$.
728 But by Tip Merge condition on $\baseof{R}$,
729 $A \le \baseof{L} \implies A \le \baseof{R}$, so
730 $A \le \baseof{R} \lor A \le \baseof{L} \equiv A \le \baseof{R}$.
731 Thus $A \le C \equiv A \le \baseof{R}$.
732 That is, $\baseof{C} = \baseof{R}$.
734 \subsubsection{For $R \in \pn$:}
736 By Tip Merge condition on $R$ and since $M \le R$,
737 $A \le \baseof{L} \implies A \le R$, so
738 $A \le R \lor A \le \baseof{L} \equiv A \le R$.
739 Thus $A \le C \equiv A \le R$.
740 That is, $\baseof{C} = R$.
744 \subsection{Coherence and Patch Inclusion}
746 Need to determine $C \haspatch \p$ based on $L,M,R \haspatch \p$.
747 This involves considering $D \in \py$.
749 \subsubsection{For $L \nothaspatch \p, R \nothaspatch \p$:}
750 $D \not\isin L \land D \not\isin R$. $C \not\in \py$ (otherwise $L
751 \in \py$ ie $L \haspatch \p$ by Tip Self Inpatch). So $D \neq C$.
752 Applying $\merge$ gives $D \not\isin C$ i.e. $C \nothaspatch \p$.
754 \subsubsection{For $L \haspatch \p, R \haspatch \p$:}
755 $D \isin L \equiv D \le L$ and $D \isin R \equiv D \le R$.
756 (Likewise $D \isin X \equiv D \le X$ and $D \isin Y \equiv D \le Y$.)
758 Consider $D = C$: $D \isin C$, $D \le C$, OK for $C \haspatch \p$.
760 For $D \neq C$: $D \le C \equiv D \le L \lor D \le R
761 \equiv D \isin L \lor D \isin R$.
762 (Likewise $D \le C \equiv D \le X \lor D \le Y$.)
764 Consider $D \neq C, D \isin X \land D \isin Y$:
765 By $\merge$, $D \isin C$. Also $D \le X$
766 so $D \le C$. OK for $C \haspatch \p$.
768 Consider $D \neq C, D \not\isin X \land D \not\isin Y$:
769 By $\merge$, $D \not\isin C$.
770 And $D \not\le X \land D \not\le Y$ so $D \not\le C$.
771 OK for $C \haspatch \p$.
773 Remaining case, wlog, is $D \not\isin X \land D \isin Y$.
774 $D \not\le X$ so $D \not\le M$ so $D \not\isin M$.
775 Thus by $\merge$, $D \isin C$. And $D \le Y$ so $D \le C$.
776 OK for $C \haspatch \p$.
778 So indeed $L \haspatch \p \land R \haspatch \p \implies C \haspatch \p$.
780 \subsubsection{For (wlog) $X \not\haspatch \p, Y \haspatch \p$:}
782 $M \haspatch \p \implies C \nothaspatch \p$.
783 $M \nothaspatch \p \implies C \haspatch \p$.
787 One of the Merge Ends conditions applies.
788 Recall that we are considering $D \in \py$.
789 $D \isin Y \equiv D \le Y$. $D \not\isin X$.
790 We will show for each of
791 various cases that $D \isin C \equiv M \nothaspatch \p \land D \le C$
792 (which suffices by definition of $\haspatch$ and $\nothaspatch$).
794 Consider $D = C$: Thus $C \in \py, L \in \py$, and by Tip
795 Self Inpatch $L \haspatch \p$, so $L=Y, R=X$. By Tip Merge,
796 $M=\baseof{L}$. So by Base Acyclic $D \not\isin M$, i.e.
797 $M \nothaspatch \p$. And indeed $D \isin C$ and $D \le C$. OK.
799 Consider $D \neq C, M \nothaspatch P, D \isin Y$:
800 $D \le Y$ so $D \le C$.
801 $D \not\isin M$ so by $\merge$, $D \isin C$. OK.
803 Consider $D \neq C, M \nothaspatch P, D \not\isin Y$:
804 $D \not\le Y$. If $D \le X$ then
805 $D \in \pancsof{X}{\py}$, so by Addition Merge Ends and
806 Transitive Ancestors $D \le Y$ --- a contradiction, so $D \not\le X$.
807 Thus $D \not\le C$. By $\merge$, $D \not\isin C$. OK.
809 Consider $D \neq C, M \haspatch P, D \isin Y$:
810 $D \le Y$ so $D \in \pancsof{Y}{\py}$ so by Removal Merge Ends
811 and Transitive Ancestors $D \in \pancsof{M}{\py}$ so $D \le M$.
812 Thus $D \isin M$. By $\merge$, $D \not\isin C$. OK.
814 Consider $D \neq C, M \haspatch P, D \not\isin Y$:
815 By $\merge$, $D \not\isin C$. OK.
819 \subsection{Base Acyclic}
821 This applies when $C \in \pn$.
822 $C \in \pn$ when $L \in \pn$ so by Merge Acyclic, $R \nothaspatch \p$.
824 Consider some $D \in \py$.
826 By Base Acyclic of $L$, $D \not\isin L$. By the above, $D \not\isin
827 R$. And $D \neq C$. So $D \not\isin C$.
831 \subsection{Tip Contents}
833 We need worry only about $C \in \py$.
834 And $\patchof{C} = \patchof{L}$
835 so $L \in \py$ so $L \haspatch \p$. We will use the Unique Base
836 of $C$, and its Coherence and Patch Inclusion, as just proved.
838 Firstly we show $C \haspatch \p$: If $R \in \py$, then $R \haspatch
839 \p$ and by Coherence/Inclusion $C \haspatch \p$ . If $R \not\in \py$
840 then by Tip Merge $M = \baseof{L}$ so by Base Acyclic and definition
841 of $\nothaspatch$, $M \nothaspatch \p$. So by Coherence/Inclusion $C
842 \haspatch \p$ (whether $R \haspatch \p$ or $\nothaspatch$).
844 We will consider an arbitrary commit $D$
845 and prove the Exclusive Tip Contents form.
847 \subsubsection{For $D \in \py$:}
848 $C \haspatch \p$ so by definition of $\haspatch$, $D \isin C \equiv D
851 \subsubsection{For $D \not\in \py, R \not\in \py$:}
853 $D \neq C$. By Tip Contents of $L$,
854 $D \isin L \equiv D \isin \baseof{L}$, and by Tip Merge condition,
855 $D \isin L \equiv D \isin M$. So by definition of $\merge$, $D \isin
856 C \equiv D \isin R$. And $R = \baseof{C}$ by Unique Base of $C$.
857 Thus $D \isin C \equiv D \isin \baseof{C}$. OK.
859 \subsubsection{For $D \not\in \py, R \in \py$:}
864 $D \isin L \equiv D \isin \baseof{L}$ and
865 $D \isin R \equiv D \isin \baseof{R}$.
867 If $\baseof{L} = M$, trivially $D \isin M \equiv D \isin \baseof{L}.$
868 Whereas if $\baseof{L} = \baseof{M}$, by definition of $\base$,
869 $\patchof{M} = \patchof{L} = \py$, so by Tip Contents of $M$,
870 $D \isin M \equiv D \isin \baseof{M} \equiv D \isin \baseof{L}$.
872 So $D \isin M \equiv D \isin L$ and by $\merge$,
873 $D \isin C \equiv D \isin R$. But from Unique Base,
874 $\baseof{C} = R$ so $D \isin C \equiv D \isin \baseof{C}$. OK.
878 \subsection{Foreign Inclusion}
880 Consider some $D$ s.t. $\patchof{D} = \bot$.
881 By Foreign Inclusion of $L, M, R$:
882 $D \isin L \equiv D \le L$;
883 $D \isin M \equiv D \le M$;
884 $D \isin R \equiv D \le R$.
886 \subsubsection{For $D = C$:}
888 $D \isin C$ and $D \le C$. OK.
890 \subsubsection{For $D \neq C, D \isin M$:}
892 Thus $D \le M$ so $D \le L$ and $D \le R$ so $D \isin L$ and $D \isin
893 R$. So by $\merge$, $D \isin C$. And $D \le C$. OK.
895 \subsubsection{For $D \neq C, D \not\isin M, D \isin X$:}
897 By $\merge$, $D \isin C$.
898 And $D \isin X$ means $D \le X$ so $D \le C$.
901 \subsubsection{For $D \neq C, D \not\isin M, D \not\isin L, D \not\isin R$:}
903 By $\merge$, $D \not\isin C$.
904 And $D \not\le L, D \not\le R$ so $D \not\le C$.
909 \subsection{Foreign Contents:}
911 xxx use Totally Foreign Contents
913 If $\patchof{C} = \bot$, by Foreign Merges
914 $\patchof{L} = \patchof{R} = \bot$.