chiark / gitweb /
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8 \usepackage{mdwlist}
9 %\usepackage{accents}
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17 \newcommand{\isin}{\sqsubseteq}
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20 \newcommand{\notpatchisin}{\mathrel{\,\not\!\not\relax\patchisin}}
21 \newcommand{\haspatch}{\sqSupset}
22 \newcommand{\patchisin}{\sqSubset}
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30 \newcommand{\pay}[1]{\pa{#1}^+}
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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}}
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42 %\newcommand{\hasparents}{{%
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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 ) }
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84 \newcommand{\qed}{\square}
85 \newcommand{\proofstarts}{{\it Proof:}}
86 \newcommand{\proof}[1]{\proofstarts #1 $\qed$}
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91 \newcommand{\true}{t}
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94 \begin{document}
96 \section{Notation}
98 \begin{basedescript}{
99 \desclabelwidth{5em}
100 \desclabelstyle{\nextlinelabel}
101 }
102 \item[ $C \hasparents \set X$ ]
103 The parents of commit $C$ are exactly the set
104 $\set X$.
106 \item[ $C \ge D$ ]
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
118 is not transitive.
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
157 patch.  If 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 167 \begin{cases} 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 171 \end{cases} 172 174 \end{basedescript} 175 \newpage 176 \section{Invariants} 178 We maintain these each time we construct a new commit. \\ 179 $\eqn{No Replay:}{ 180 C \has D \implies C \ge D 181 }$ 182 $\eqn{Unique Base:}{ 183 \bigforall_{C \in \py} \pendsof{C}{\pn} = \{ B \} 184 }$ 185 $\eqn{Tip Contents:}{ 186 \bigforall_{C \in \py} D \isin C \equiv 187 { D \isin \baseof{C} \lor \atop 188 (D \in \py \land D \le C) } 189 }$ 190 $\eqn{Base Acyclic:}{ 191 \bigforall_{B \in \pn} D \isin B \implies D \notin \py 192 }$ 193 $\eqn{Coherence:}{ 194 \bigforall_{C,\p} C \haspatch \p \lor C \nothaspatch \p 195 }$ 196 $\eqn{Foreign Inclusion:}{ 197 \bigforall_{D \text{ s.t. } \patchof{D} = \bot} D \isin C \equiv D \leq C 198 }$ 199 $\eqn{Foreign Contents:}{ 200 \bigforall_{C \text{ s.t. } \patchof{C} = \bot} 201 D \le C \implies \patchof{D} = \bot 202 }$ 204 \section{Some lemmas} 206 $\eqn{Alternative (overlapping) formulations defining 207 \mergeof{C}{L}{M}{R}:}{ 208 D \isin C \equiv 209 \begin{cases} 210 D \isin L \equiv D \isin R : & D = C \lor D \isin L \\ 211 D \isin L \nequiv D \isin R : & D = C \lor D \not\isin M \\ 212 D \isin L \equiv D \isin M : & D = C \lor D \isin R \\ 213 D \isin L \nequiv D \isin M : & D = C \lor D \isin L \\ 214 \text{as above with L and R exchanged} 215 \end{cases} 216 }$ 217 \proof{ 218 Truth table xxx 220 Original definition is symmetrical in$L$and$R$. 221 } 223 $\eqn{Exclusive Tip Contents:}{ 224 \bigforall_{C \in \py} 225 \neg \Bigl[ D \isin \baseof{C} \land ( D \in \py \land D \le C ) 226 \Bigr] 227 }$ 228 Ie, the two limbs of the RHS of Tip Contents are mutually exclusive. 230 \proof{ 231 Let$B = \baseof{C}$in$D \isin \baseof{C}$. Now$B \in \pn$. 232 So by Base Acyclic$D \isin B \implies D \notin \py$. 233 } 234 $\eqntag{{\it Corollary - equivalent to Tip Contents}}{ 235 \bigforall_{C \in \py} D \isin C \equiv 236 \begin{cases} 237 D \in \py : & D \le C \\ 238 D \not\in \py : & D \isin \baseof{C} 239 \end{cases} 240 }$ 242 $\eqn{Tip Self Inpatch:}{ 243 \bigforall_{C \in \py} C \haspatch \p 244 }$ 245 Ie, tip commits contain their own patch. 247 \proof{ 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 $251 } 253 $\eqn{Exact Ancestors:}{ 254 \bigforall_{ C \hasparents \set{R} } 255 D \le C \equiv 256 ( \mathop{\hbox{\huge{\vee}}}_{R \in \set R} D \le R ) 257 \lor D = C 258 }$ 259 xxx proof tbd 261 $\eqn{Transitive Ancestors:}{ 262 \left[ \bigforall_{ E \in \pendsof{C}{\set P} } E \le M \right] \equiv 263 \left[ \bigforall_{ A \in \pancsof{C}{\set P} } A \le M \right] 264 }$ 266 \proof{ 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''$. 277 } 279 $\eqn{Calculation Of Ends:}{ 280 \bigforall_{C \hasparents \set A} 281 \pendsof{C}{\set P} = 282 \begin{cases} 283 C \in \p : & \{ C \} 284 \\ 285 C \not\in \p : & \displaystyle 286 \left\{ E \Big| 287 \Bigl[ \Largeexists_{A \in \set A} 288 E \in \pendsof{A}{\set P} \Bigr] \land 289 \Bigl[ \Largenexists_{B \in \set A} 290 E \neq B \land E \le B \Bigr] 291 \right\} 292 \end{cases} 293 }$ 294 xxx proof tbd 296 $\eqn{Ingredients Prohibit Replay:}{ 297 \left[ 298 {C \hasparents \set A} \land 299 \\ 300 \left( 301 D \isin C \implies 302 D = C \lor 303 \Largeexists_{A \in \set A} D \isin A 304 \right) 305 \right] \implies \left[ 306 D \isin C \implies D \le C 307 \right] 308 }$ 309 \proof{ 310 Trivial for$D = C$. Consider some$D \neq C$,$D \isin C$. 311 By the preconditions, there is some$A$s.t.$D \in \set A$312 and$D \isin A$. By No Replay for$A$,$D \le A$. And 313$A \le C$so$D \le C$. 314 } 316 $\eqn{Totally Foreign Contents:}{ 317 \bigforall_{C \hasparents \set A} 318 \left[ 319 \patchof{C} = \bot \land 320 \bigforall_{A \in \set A} \patchof{A} = \bot 321 \right] 322 \implies 323 \left[ 324 D \le C 325 \implies 326 \patchof{D} = \bot 327 \right] 328 }$ 329 \proof{ 330 Consider some$D \le C$. If$D = C$,$\patchof{D} = \bot$trivially. 331 If$D \neq C$then$D \le A$where$A \in \set A$. By Foreign 332 Contents of$A$,$\patchof{D} = \bot$. 333 } 335 \section{Commit annotation} 337 We annotate each Topbloke commit$C$with: 338 \gathbegin 339 \patchof{C} 340 \gathnext 341 \baseof{C}, \text{ if } C \in \py 342 \gathnext 343 \bigforall_{\pa{Q}} 344 \text{ either } C \haspatch \pa{Q} \text{ or } C \nothaspatch \pa{Q} 345 \gathnext 346 \bigforall_{\pay{Q} \not\ni C} \pendsof{C}{\pay{Q}} 347 \end{gather} 349$\patchof{C}$, for each kind of Topbloke-generated commit, is stated 350 in the summary in the section for that kind of commit. 352 Whether$\baseof{C}$is required, and if so what the value is, is 353 stated in the proof of Unique Base for each kind of commit. 355$C \haspatch \pa{Q}$or$\nothaspatch \pa{Q}$is represented as the 356 set$\{ \pa{Q} | C \haspatch \pa{Q} \}$. Whether$C \haspatch \pa{Q}$357 is in stated 358 (in terms of$I \haspatch \pa{Q}$or$I \nothaspatch \pa{Q}$359 for the ingredients$I$), 360 in the proof of Coherence for each kind of commit. 362$\pendsof{C}{\pa{Q}^+}$is computed, for all Topbloke-generated commits, 363 using the lemma Calculation of Ends, above. 364 We do not annotate$\pendsof{C}{\py}$for$C \in \py$. Doing so would 365 make it wrong to make plain commits with git because the recorded$\pends$366 would have to be updated. The annotation is not needed in that case 367 because$\forall_{\py \ni C} \; \pendsof{C}{\py} = \{C\}$. 369 \section{Simple commit} 371 A simple single-parent forward commit$C$as made by git-commit. 372 \begin{gather} 373 \tag*{} C \hasparents \{ A \} \\ 374 \tag*{} \patchof{C} = \patchof{A} \\ 375 \tag*{} D \isin C \equiv D \isin A \lor D = C 376 \end{gather} 377 This also covers Topbloke-generated commits on plain git branches: 378 Topbloke strips the metadata when exporting. 380 \subsection{No Replay} 382 Ingredients Prohibit Replay applies.$\qed$384 \subsection{Unique Base} 385 If$A, C \in \py$then by Calculation of Ends for 386$C, \py, C \not\in \py$: 387$\pendsof{C}{\pn} = \pendsof{A}{\pn}$so 388$\baseof{C} = \baseof{A}$.$\qed$390 \subsection{Tip Contents} 391 We need to consider only$A, C \in \py$. From Tip Contents for$A$: 392 $D \isin A \equiv D \isin \baseof{A} \lor ( D \in \py \land D \le A )$ 393 Substitute into the contents of$C$: 394 $D \isin C \equiv D \isin \baseof{A} \lor ( D \in \py \land D \le A ) 395 \lor D = C$ 396 Since$D = C \implies D \in \py$, 397 and substituting in$\baseof{C}$, this gives: 398 $D \isin C \equiv D \isin \baseof{C} \lor 399 (D \in \py \land D \le A) \lor 400 (D = C \land D \in \py)$ 401 $\equiv D \isin \baseof{C} \lor 402 [ D \in \py \land ( D \le A \lor D = C ) ]$ 403 So by Exact Ancestors: 404 $D \isin C \equiv D \isin \baseof{C} \lor ( D \in \py \land D \le C 405 )$ 406$\qed$408 \subsection{Base Acyclic} 410 Need to consider only$A, C \in \pn$. 412 For$D = C$:$D \in \pn$so$D \not\in \py$. OK. 414 For$D \neq C$:$D \isin C \equiv D \isin A$, so by Base Acyclic for 415$A$,$D \isin C \implies D \not\in \py$. 417$\qed$419 \subsection{Coherence and patch inclusion} 421 Need to consider$D \in \py$423 \subsubsection{For$A \haspatch P, D = C$:} 425 Ancestors of$C$: 426$ D \le C $. 428 Contents of$C$: 429$ D \isin C \equiv \ldots \lor \true \text{ so } D \haspatch C $. 431 \subsubsection{For$A \haspatch P, D \neq C$:} 432 Ancestors:$ D \le C \equiv D \le A $. 434 Contents:$ D \isin C \equiv D \isin A \lor f $435 so$ D \isin C \equiv D \isin A $. 437 So: 438 $A \haspatch P \implies C \haspatch P$ 440 \subsubsection{For$A \nothaspatch P$:} 442 Firstly,$C \not\in \py$since if it were,$A \in \py$. 443 Thus$D \neq C$. 445 Now by contents of$A$,$D \notin A$, so$D \notin C$. 447 So: 448 $A \nothaspatch P \implies C \nothaspatch P$ 449$\qed$451 \subsection{Foreign inclusion:} 453 If$D = C$, trivial. For$D \neq C$: 454$D \isin C \equiv D \isin A \equiv D \le A \equiv D \le C$.$\qed$456 \subsection{Foreign Contents:} 458 Only relevant if$\patchof{C} = \bot$, and in that case Totally 459 Foreign Contents applies.$\qed$461 \section{Create Base} 463 Given$L$, create a Topbloke base branch initial commit$B$. 464 \gathbegin 465 B \hasparents \{ L \} 466 \gathnext 467 \patchof{B} = \pan{B} 468 \gathnext 469 D \isin B \equiv D \isin L \lor D = B 470 \end{gather} 472 \subsection{Conditions} 474 $\eqn{ Ingredients }{ 475 \patchof{L} = \pa{L} \lor \patchof{L} = \bot 476 }$ 477 $\eqn{ Non-recursion }{ 478 L \not\in \pa{B} 479 }$ 481 \subsection{No Replay} 483 Ingredients Prohibit Replay applies.$\qed$485 \subsection{Unique Base} 487 Not applicable. 489 \subsection{Tip Contents} 491 Not applicable. 493 \subsection{Base Acyclic} 495 Consider some$D \isin B$. If$D = B$,$D \in \pn$, OK. 497 If$D \neq B$,$D \isin L$. By No Replay of$D$in$L$,$D \le L$. 498 Thus by Foreign Contents of$L$,$\patchof{D} = \bot$. OK. 500$\qed$502 \subsection{Coherence and Patch Inclusion} 504 Consider some$D \in \p$. 505$B \not\in \py$so$D \neq B$. So$D \isin B \equiv D \isin L$. 507 Thus$L \haspatch \p \implies B \haspatch P$508 and$L \nothaspatch \p \implies B \nothaspatch P$. 510$\qed$. 512 \subsection{Foreign Inclusion} 514 Consider some$D$s.t.$\patchof{D} = \bot$.$D \neq B$515 so$D \isin B \equiv D \isin L$. 516 By Foreign Inclusion of$D$in$L$,$D \isin L \equiv D \le L$. 517 And by Exact Ancestors$D \le L \equiv D \le B$. 518 So$D \isin B \equiv D \le B$.$\qed$520 \subsection{Foreign Contents} 522 Not applicable. 524 \section{Create Tip} 526 xxx tbd 528 \section{Anticommit} 530 Given$L$and$\pr$as represented by$R^+, R^-$. 531 Construct$C$which has$\pr$removed. 532 Used for removing a branch dependency. 533 \gathbegin 534 C \hasparents \{ L \} 535 \gathnext 536 \patchof{C} = \patchof{L} 537 \gathnext 538 \mergeof{C}{L}{R^+}{R^-} 539 \end{gather} 541 \subsection{Conditions} 543 $\eqn{ Ingredients }{ 544 R^+ \in \pry \land R^- = \baseof{R^+} 545 }$ 546 $\eqn{ Into Base }{ 547 L \in \pn 548 }$ 549 $\eqn{ Unique Tip }{ 550 \pendsof{L}{\pry} = \{ R^+ \} 551 }$ 552 $\eqn{ Currently Included }{ 553 L \haspatch \pry 554 }$ 556 \subsection{Ordering of Ingredients:} 558 By Unique Tip,$R^+ \le L$. By definition of$\base$,$R^- \le R^+$559 so$R^- \le L$. So$R^+ \le C$and$R^- \le C$. 560$\qed$562 (Note that$R^+ \not\le R^-$, i.e. the merge base 563 is a descendant, not an ancestor, of the 2nd parent.) 565 \subsection{No Replay} 567 By definition of$\merge$, 568$D \isin C \implies D \isin L \lor D \isin R^- \lor D = C$. 569 So, by Ordering of Ingredients, 570 Ingredients Prohibit Replay applies.$\qed$572 \subsection{Desired Contents} 574 $D \isin C \equiv [ D \notin \pry \land D \isin L ] \lor D = C$ 575 \proofstarts 577 \subsubsection{For$D = C$:} 579 Trivially$D \isin C$. OK. 581 \subsubsection{For$D \neq C, D \not\le L$:} 583 By No Replay$D \not\isin L$. Also$D \not\le R^-$hence 584$D \not\isin R^-$. Thus$D \not\isin C$. OK. 586 \subsubsection{For$D \neq C, D \le L, D \in \pry$:} 588 By Currently Included,$D \isin L$. 590 By Tip Self Inpatch,$D \isin R^+ \equiv D \le R^+$, but by 591 by Unique Tip,$D \le R^+ \equiv D \le L$. 592 So$D \isin R^+$. 594 By Base Acyclic,$D \not\isin R^-$. 596 Apply$\merge$:$D \not\isin C$. OK. 598 \subsubsection{For$D \neq C, D \le L, D \notin \pry$:} 600 By Tip Contents for$R^+$,$D \isin R^+ \equiv D \isin R^-$. 602 Apply$\merge$:$D \isin C \equiv D \isin L$. OK. 604$\qed$606 \subsection{Unique Base} 608 Into Base means that$C \in \pn$, so Unique Base is not 609 applicable.$\qed$611 \subsection{Tip Contents} 613 Again, not applicable.$\qed$615 \subsection{Base Acyclic} 617 By Base Acyclic for$L$,$D \isin L \implies D \not\in \py$. 618 And by Into Base$C \not\in \py$. 619 Now from Desired Contents, above,$D \isin C
620 \implies D \isin L \lor D = C$, which thus 621$\implies D \not\in \py$.$\qed$. 623 \subsection{Coherence and Patch Inclusion} 625 Need to consider some$D \in \py$. By Into Base,$D \neq C$. 627 \subsubsection{For$\p = \pr$:} 628 By Desired Contents, above,$D \not\isin C$. 629 So$C \nothaspatch \pr$. 631 \subsubsection{For$\p \neq \pr$:} 632 By Desired Contents,$D \isin C \equiv D \isin L$633 (since$D \in \py$so$D \not\in \pry$). 635 If$L \nothaspatch \p$,$D \not\isin L$so$D \not\isin C$. 636 So$L \nothaspatch \p \implies C \nothaspatch \p$. 638 Whereas if$L \haspatch \p$,$D \isin L \equiv D \le L$. 639 so$L \haspatch \p \implies C \haspatch \p$. 641$\qed$643 \subsection{Foreign Inclusion} 645 Consider some$D$s.t.$\patchof{D} = \bot$.$D \neq C$. 646 So by Desired Contents$D \isin C \equiv D \isin L$. 647 By Foreign Inclusion of$D$in$L$,$D \isin L \equiv D \le L$. 649 And$D \le C \equiv D \le L$. 650 Thus$D \isin C \equiv D \le C$. 652$\qed$654 \subsection{Foreign Contents} 656 Not applicable. 658 \section{Merge} 660 Merge commits$L$and$R$using merge base$M$: 661 \gathbegin 662 C \hasparents \{ L, R \} 663 \gathnext 664 \patchof{C} = \patchof{L} 665 \gathnext 666 \mergeof{C}{L}{M}{R} 667 \end{gather} 668 We will occasionally use$X,Y$s.t.$\{X,Y\} = \{L,R\}$. 670 \subsection{Conditions} 671 $\eqn{ Ingredients }{ 672 M \le L, M \le R 673 }$ 674 $\eqn{ Tip Merge }{ 675 L \in \py \implies 676 \begin{cases} 677 R \in \py : & \baseof{R} \ge \baseof{L} 678 \land [\baseof{L} = M \lor \baseof{L} = \baseof{M}] \\ 679 R \in \pn : & M = \baseof{L} \\ 680 \text{otherwise} : & \false 681 \end{cases} 682 }$ 683 $\eqn{ Merge Acyclic }{ 684 L \in \pn 685 \implies 686 R \nothaspatch \p 687 }$ 688 $\eqn{ Removal Merge Ends }{ 689 X \not\haspatch \p \land 690 Y \haspatch \p \land 691 M \haspatch \p 692 \implies 693 \pendsof{Y}{\py} = \pendsof{M}{\py} 694 }$ 695 $\eqn{ Addition Merge Ends }{ 696 X \not\haspatch \p \land 697 Y \haspatch \p \land 698 M \nothaspatch \p 699 \implies \left[ 700 \bigforall_{E \in \pendsof{X}{\py}} E \le Y 701 \right] 702 }$ 703 $\eqn{ Foreign Merges }{ 704 \patchof{L} = \bot \equiv \patchof{R} = \bot 705 }$ 707 \subsection{Non-Topbloke merges} 709 We require both$\patchof{L} = \bot$and$\patchof{R} = \bot$710 (Foreign Merges, above). 711 I.e. not only is it forbidden to merge into a Topbloke-controlled 712 branch without Topbloke's assistance, it is also forbidden to 713 merge any Topbloke-controlled branch into any plain git branch. 715 Given those conditions, Tip Merge and Merge Acyclic do not apply. 716 And$Y \not\in \py$so$\neg [ Y \haspatch \p ]$so neither 717 Merge Ends condition applies. 719 So a plain git merge of non-Topbloke branches meets the conditions and 720 is therefore consistent with our scheme. 722 \subsection{No Replay} 724 By definition of$\merge$, 725$D \isin C \implies D \isin L \lor D \isin R \lor D = C$. 726 So, by Ingredients, 727 Ingredients Prohibit Replay applies.$\qed$729 \subsection{Unique Base} 731 Need to consider only$C \in \py$, ie$L \in \py$, 732 and calculate$\pendsof{C}{\pn}$. So we will consider some 733 putative ancestor$A \in \pn$and see whether$A \le C$. 735 By Exact Ancestors for C,$A \le C \equiv A \le L \lor A \le R \lor A = C$. 736 But$C \in py$and$A \in \pn$so$A \neq C$. 737 Thus$A \le C \equiv A \le L \lor A \le R$. 739 By Unique Base of L and Transitive Ancestors, 740$A \le L \equiv A \le \baseof{L}$. 742 \subsubsection{For$R \in \py$:} 744 By Unique Base of$R$and Transitive Ancestors, 745$A \le R \equiv A \le \baseof{R}$. 747 But by Tip Merge condition on$\baseof{R}$, 748$A \le \baseof{L} \implies A \le \baseof{R}$, so 749$A \le \baseof{R} \lor A \le \baseof{L} \equiv A \le \baseof{R}$. 750 Thus$A \le C \equiv A \le \baseof{R}$. 751 That is,$\baseof{C} = \baseof{R}$. 753 \subsubsection{For$R \in \pn$:} 755 By Tip Merge condition on$R$and since$M \le R$, 756$A \le \baseof{L} \implies A \le R$, so 757$A \le R \lor A \le \baseof{L} \equiv A \le R$. 758 Thus$A \le C \equiv A \le R$. 759 That is,$\baseof{C} = R$. 761$\qed$763 \subsection{Coherence and Patch Inclusion} 765 Need to determine$C \haspatch \p$based on$L,M,R \haspatch \p$. 766 This involves considering$D \in \py$. 768 \subsubsection{For$L \nothaspatch \p, R \nothaspatch \p$:} 769$D \not\isin L \land D \not\isin R$.$C \not\in \py$(otherwise$L
770 \in \py$ie$L \haspatch \p$by Tip Self Inpatch). So$D \neq C$. 771 Applying$\merge$gives$D \not\isin C$i.e.$C \nothaspatch \p$. 773 \subsubsection{For$L \haspatch \p, R \haspatch \p$:} 774$D \isin L \equiv D \le L$and$D \isin R \equiv D \le R$. 775 (Likewise$D \isin X \equiv D \le X$and$D \isin Y \equiv D \le Y$.) 777 Consider$D = C$:$D \isin C$,$D \le C$, OK for$C \haspatch \p$. 779 For$D \neq C$:$D \le C \equiv D \le L \lor D \le R
780  \equiv D \isin L \lor D \isin R$. 781 (Likewise$D \le C \equiv D \le X \lor D \le Y$.) 783 Consider$D \neq C, D \isin X \land D \isin Y$: 784 By$\merge$,$D \isin C$. Also$D \le X$785 so$D \le C$. OK for$C \haspatch \p$. 787 Consider$D \neq C, D \not\isin X \land D \not\isin Y$: 788 By$\merge$,$D \not\isin C$. 789 And$D \not\le X \land D \not\le Y$so$D \not\le C$. 790 OK for$C \haspatch \p$. 792 Remaining case, wlog, is$D \not\isin X \land D \isin Y$. 793$D \not\le X$so$D \not\le M$so$D \not\isin M$. 794 Thus by$\merge$,$D \isin C$. And$D \le Y$so$D \le C$. 795 OK for$C \haspatch \p$. 797 So indeed$L \haspatch \p \land R \haspatch \p \implies C \haspatch \p$. 799 \subsubsection{For (wlog)$X \not\haspatch \p, Y \haspatch \p$:} 801$M \haspatch \p \implies C \nothaspatch \p$. 802$M \nothaspatch \p \implies C \haspatch \p$. 804 \proofstarts 806 One of the Merge Ends conditions applies. 807 Recall that we are considering$D \in \py$. 808$D \isin Y \equiv D \le Y$.$D \not\isin X$. 809 We will show for each of 810 various cases that$D \isin C \equiv M \nothaspatch \p \land D \le C$811 (which suffices by definition of$\haspatch$and$\nothaspatch$). 813 Consider$D = C$: Thus$C \in \py, L \in \py$, and by Tip 814 Self Inpatch$L \haspatch \p$, so$L=Y, R=X$. By Tip Merge, 815$M=\baseof{L}$. So by Base Acyclic$D \not\isin M$, i.e. 816$M \nothaspatch \p$. And indeed$D \isin C$and$D \le C$. OK. 818 Consider$D \neq C, M \nothaspatch P, D \isin Y$: 819$D \le Y$so$D \le C$. 820$D \not\isin M$so by$\merge$,$D \isin C$. OK. 822 Consider$D \neq C, M \nothaspatch P, D \not\isin Y$: 823$D \not\le Y$. If$D \le X$then 824$D \in \pancsof{X}{\py}$, so by Addition Merge Ends and 825 Transitive Ancestors$D \le Y$--- a contradiction, so$D \not\le X$. 826 Thus$D \not\le C$. By$\merge$,$D \not\isin C$. OK. 828 Consider$D \neq C, M \haspatch P, D \isin Y$: 829$D \le Y$so$D \in \pancsof{Y}{\py}$so by Removal Merge Ends 830 and Transitive Ancestors$D \in \pancsof{M}{\py}$so$D \le M$. 831 Thus$D \isin M$. By$\merge$,$D \not\isin C$. OK. 833 Consider$D \neq C, M \haspatch P, D \not\isin Y$: 834 By$\merge$,$D \not\isin C$. OK. 836$\qed$838 \subsection{Base Acyclic} 840 This applies when$C \in \pn$. 841$C \in \pn$when$L \in \pn$so by Merge Acyclic,$R \nothaspatch \p$. 843 Consider some$D \in \py$. 845 By Base Acyclic of$L$,$D \not\isin L$. By the above,$D \not\isin
846 R$. And$D \neq C$. So$D \not\isin C$. 848$\qed$850 \subsection{Tip Contents} 852 We need worry only about$C \in \py$. 853 And$\patchof{C} = \patchof{L}$854 so$L \in \py$so$L \haspatch \p$. We will use the Unique Base 855 of$C$, and its Coherence and Patch Inclusion, as just proved. 857 Firstly we show$C \haspatch \p$: If$R \in \py$, then$R \haspatch
858 \p$and by Coherence/Inclusion$C \haspatch \p$. If$R \not\in \py$859 then by Tip Merge$M = \baseof{L}$so by Base Acyclic and definition 860 of$\nothaspatch$,$M \nothaspatch \p$. So by Coherence/Inclusion$C
861 \haspatch \p$(whether$R \haspatch \p$or$\nothaspatch$). 863 We will consider an arbitrary commit$D$864 and prove the Exclusive Tip Contents form. 866 \subsubsection{For$D \in \py$:} 867$C \haspatch \p$so by definition of$\haspatch$,$D \isin C \equiv D
868 \le C$. OK. 870 \subsubsection{For$D \not\in \py, R \not\in \py$:} 872$D \neq C$. By Tip Contents of$L$, 873$D \isin L \equiv D \isin \baseof{L}$, and by Tip Merge condition, 874$D \isin L \equiv D \isin M$. So by definition of$\merge$,$D \isin
875 C \equiv D \isin R$. And$R = \baseof{C}$by Unique Base of$C$. 876 Thus$D \isin C \equiv D \isin \baseof{C}$. OK. 878 \subsubsection{For$D \not\in \py, R \in \py$:} 880$D \neq C$. 882 By Tip Contents 883$D \isin L \equiv D \isin \baseof{L}$and 884$D \isin R \equiv D \isin \baseof{R}$. 886 If$\baseof{L} = M$, trivially$D \isin M \equiv D \isin \baseof{L}.$887 Whereas if$\baseof{L} = \baseof{M}$, by definition of$\base$, 888$\patchof{M} = \patchof{L} = \py$, so by Tip Contents of$M$, 889$D \isin M \equiv D \isin \baseof{M} \equiv D \isin \baseof{L}$. 891 So$D \isin M \equiv D \isin L$and by$\merge$, 892$D \isin C \equiv D \isin R$. But from Unique Base, 893$\baseof{C} = R$so$D \isin C \equiv D \isin \baseof{C}$. OK. 895$\qed$897 \subsection{Foreign Inclusion} 899 Consider some$D$s.t.$\patchof{D} = \bot$. 900 By Foreign Inclusion of$L, M, R$: 901$D \isin L \equiv D \le L$; 902$D \isin M \equiv D \le M$; 903$D \isin R \equiv D \le R$. 905 \subsubsection{For$D = C$:} 907$D \isin C$and$D \le C$. OK. 909 \subsubsection{For$D \neq C, D \isin M$:} 911 Thus$D \le M$so$D \le L$and$D \le R$so$D \isin L$and$D \isin
912 R$. So by$\merge$,$D \isin C$. And$D \le C$. OK. 914 \subsubsection{For$D \neq C, D \not\isin M, D \isin X$:} 916 By$\merge$,$D \isin C$. 917 And$D \isin X$means$D \le X$so$D \le C$. 918 OK. 920 \subsubsection{For$D \neq C, D \not\isin M, D \not\isin L, D \not\isin R$:} 922 By$\merge$,$D \not\isin C$. 923 And$D \not\le L, D \not\le R$so$D \not\le C$. 924 OK 926$\qed$928 \subsection{Foreign Contents} 930 Only relevant if$\patchof{L} = \bot$, in which case 931$\patchof{C} = \bot$and by Foreign Merges$\patchof{R} = \bot$, 932 so Totally Foreign Contents applies.$\qed\$
934 \end{document}