3 \subsection{Alternative (overlapping) formulations of $\mergeof{C}{L}{M}{R}$}
7 D \isin L \equiv D \isin R : & D = C \lor D \isin L \\
8 D \isin L \nequiv D \isin R : & D = C \lor D \not\isin M \\
9 D \isin L \equiv D \isin M : & D = C \lor D \isin R \\
10 D \isin L \nequiv D \isin M : & D = C \lor D \isin L \\
11 \text{as above with L and R exchanged}
14 \proof{ ~ Truth table (ordered by original definition): \\
15 \begin{tabular}{cccc|c|cc}
19 $\isin R$ & $\isin C$ &
20 $L$ vs. $R$ & $L$ vs. $M$
22 y & ? & ? & ? & y & ? & ? \\
23 n & y & y & y & y & $\equiv$ & $\equiv$ \\
24 n & y & n & y & y & $\equiv$ & $\nequiv$ \\
25 n & n & y & n & n & $\equiv$ & $\nequiv$ \\
26 n & n & n & n & n & $\equiv$ & $\equiv$ \\
27 n & y & y & n & n & $\nequiv$ & $\equiv$ \\
28 n & n & y & y & n & $\nequiv$ & $\nequiv$ \\
29 n & y & n & n & y & $\nequiv$ & $\nequiv$ \\
30 n & n & n & y & y & $\nequiv$ & $\equiv$ \\
32 And original definition is symmetrical in $L$ and $R$.
35 \subsection{Exclusive Tip Contents}
36 Given Base Acyclic for $C$,
38 \bigforall_{C \in \py}
39 \neg \Bigl[ D \isin \baseof{C} \land ( D \in \py \land D \le C )
42 Ie, the two limbs of the RHS of Tip Contents are mutually exclusive.
45 Let $B = \baseof{C}$ in $D \isin \baseof{C}$. Now $B \in \pn$.
46 So by Base Acyclic $D \isin B \implies D \notin \py$.
48 \[ \eqntag{{\it Corollary - equivalent to Tip Contents}}{
49 \bigforall_{C \in \py} D \isin C \equiv
51 D \in \py : & D \le C \\
52 D \not\in \py : & D \isin \baseof{C}
56 \subsection{Tip Self Inpatch}
57 Given Exclusive Tip Contents and Base Acyclic for $C$,
59 \bigforall_{C \in \py} C \haspatch \p
61 Ie, tip commits contain their own patch.
64 Apply Exclusive Tip Contents to some $D \in \py$:
65 $ \bigforall_{C \in \py}\bigforall_{D \in \py}
66 D \isin C \equiv D \le C $
69 \subsection{Exact Ancestors}
71 \bigforall_{ C \hasparents \set{R} }
74 ( \mathop{\hbox{\huge{$\vee$}}}_{R \in \set R} D \le R )
80 \subsection{Transitive Ancestors}
82 \left[ \bigforall_{ E \in \pendsof{C}{\set P} } E \le M \right] \equiv
83 \left[ \bigforall_{ A \in \pancsof{C}{\set P} } A \le M \right]
87 The implication from right to left is trivial because
88 $ \pends() \subset \pancs() $.
89 For the implication from left to right:
90 by the definition of $\mathcal E$,
91 for every such $A$, either $A \in \pends()$ which implies
92 $A \le M$ by the LHS directly,
93 or $\exists_{A' \in \pancs()} \; A' \neq A \land A \le A' $
94 in which case we repeat for $A'$. Since there are finitely many
95 commits, this terminates with $A'' \in \pends()$, ie $A'' \le M$
96 by the LHS. And $A \le A''$.
99 \subsection{Calculation of Ends}
101 \bigforall_{C \hasparents \set A}
102 \pendsof{C}{\set P} =
106 C \not\in \p : & \displaystyle
108 \Bigl[ \Largeexists_{A \in \set A}
109 E \in \pendsof{A}{\set P} \Bigr] \land
110 \Bigl[ \Largenexists_{B \in \set A, F \in \pendsof{B}{\p}}
111 E \neq F \land E \le F \Bigr]
116 Trivial for $C \in \set P$. For $C \not\in \set P$,
117 $\pancsof{C}{\set P} = \bigcup_{A \in \set A} \pancsof{A}{\set P}$.
118 So $\pendsof{C}{\set P} \subset \bigcup_{E in \set E} \pendsof{E}{\set P}$.
119 Consider some $E \in \pendsof{A}{\set P}$. If $\exists_{B,F}$ as
120 specified, then either $F$ is going to be in our result and
121 disqualifies $E$, or there is some other $F'$ (or, eventually,
122 an $F''$) which disqualifies $F$.
123 Otherwise, $E$ meets all the conditions for $\pends$.
126 \subsection{Ingredients Prevent Replay}
129 {C \hasparents \set A} \land
135 \Largeexists_{A \in \set A} D \isin A
137 \right] \implies \left[ \bigforall_{D}
138 D \isin C \implies D \le C
142 Trivial for $D = C$. Consider some $D \neq C$, $D \isin C$.
143 By the preconditions, there is some $A$ s.t. $D \in \set A$
144 and $D \isin A$. By No Replay for $A$, $D \le A$. And
145 $A \le C$ so $D \le C$.
148 \subsection{Simple Foreign Inclusion}
151 C \hasparents \{ L \}
153 \bigforall_{D} D \isin C \equiv D \isin L \lor D = C
157 \bigforall_{D \text{ s.t. } \patchof{D} = \bot}
158 D \isin C \equiv D \le C
162 Consider some $D$ s.t. $\patchof{D} = \bot$.
163 If $D = C$, trivially true. For $D \neq C$,
164 by Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$.
165 And by Exact Ancestors $D \le L \equiv D \le C$.
166 So $D \isin C \equiv D \le C$.
169 \subsection{Totally Foreign Contents}
172 C \hasparents \set A \land
173 \patchof{C} = \bot \land
174 \bigforall_{A \in \set A} \patchof{A} = \bot
185 Consider some $D \le C$. If $D = C$, $\patchof{D} = \bot$ trivially.
186 If $D \neq C$ then $D \le A$ where $A \in \set A$. By Foreign
187 Contents of $A$, $\patchof{D} = \bot$.