3 Given $L$ which contains $\pr$ as represented by $R^+, R^-$.
4 Construct $C$ which has $\pr$ removed.
5 Used for removing a branch dependency.
9 \patchof{C} = \patchof{L}
11 \mergeof{C}{L}{R^+}{R^-}
14 \subsection{Conditions}
16 \[ \eqn{ Ingredients }{
17 R^+ \in \pry \land R^- = \baseof{R^+}
22 \[ \eqn{ Unique Tip }{
23 \pendsof{L}{\pry} = \{ R^+ \}
25 \[ \eqn{ Currently Included }{
29 \subsection{Ordering of Ingredients:}
31 By Unique Tip, $R^+ \le L$. By definition of $\base$, $R^- \le R^+$
32 so $R^- \le L$. So $R^+ \le C$ and $R^- \le C$.
35 (Note that $R^+ \not\le R^-$, i.e. the merge base
36 is a descendant, not an ancestor, of the 2nd parent.)
38 \subsection{No Replay}
41 $D \isin C \implies D \isin L \lor D \isin R^- \lor D = C$.
42 So, by Ordering of Ingredients,
43 Ingredients Prevent Replay applies. $\qed$
45 \subsection{Desired Contents}
47 \[ D \isin C \equiv [ D \notin \pry \land D \isin L ] \lor D = C \]
50 \subsubsection{For $D = C$:}
52 Trivially $D \isin C$. OK.
54 \subsubsection{For $D \neq C, D \not\le L$:}
56 By No Replay for $L$, $D \not\isin L$.
57 Also, by Ordering of Ingredients, $D \not\le R^-$ hence
58 $D \not\isin R^-$. Thus $D \not\isin C$. OK.
60 \subsubsection{For $D \neq C, D \le L, D \in \pry$:}
62 By Currently Included, $D \isin L$.
64 By Tip Own Contents for $R^+$, $D \isin R^+ \equiv D \le R^+$, but
65 by Unique Tip, $D \le R^+ \equiv D \le L$.
68 By Base Acyclic for $R^-$, $D \not\isin R^-$.
70 Apply $\merge$: $D \not\isin C$. OK.
72 \subsubsection{For $D \neq C, D \le L, D \notin \pry$:}
74 By Tip Contents for $R^+$, $D \isin R^+ \equiv D \isin R^-$.
76 Apply $\merge$: $D \isin C \equiv D \isin L$. OK.
80 \subsection{Unique Base}
82 Into Base means that $C \in \pln$, so Unique Base is not
85 \subsection{Tip Contents}
87 Again, not applicable.
89 \subsection{Base Acyclic}
91 By Into Base and Base Acyclic for $L$, $D \isin L \implies D \not\in \ply$.
92 And by Into Base $C \not\in \ply$.
93 Now from Desired Contents, above, $D \isin C
94 \implies D \isin L \lor D = C$, which thus
95 $\implies D \not\in \ply$. $\qed$.
97 \subsection{Coherence and Patch Inclusion}
101 \p = \pr : & C \nothaspatch \p \\
102 \p \neq \pr \land L \nothaspatch \p : & C \nothaspatch \p \\
103 \p \neq \pr \land L \haspatch \p : & C \haspatch \p
107 ~ Need to consider some $D \in \py$. By Into Base, $D \neq C$.
109 \subsubsection{For $\p = \pr$:}
110 By Desired Contents, above, $D \not\isin C$.
113 \subsubsection{For $\p \neq \pr$:}
114 By Desired Contents, $D \isin C \equiv D \isin L$
115 (since $D \in \py$ so $D \not\in \pry$).
117 If $L \nothaspatch \p$, $D \not\isin L$ so $D \not\isin C$.
120 Whereas, if $L \haspatch \p$, $D \isin L \equiv D \le L$,
121 so $C \zhaspatch \p$;
122 and $\exists_{F \in \py} F \le L$ and this $F \le C$.
123 Thus $C \haspatch \p$.
128 \subsection{Unique Tips:}
130 Single Parent Unique Tips applies. $\qed$
132 \subsection{Foreign Inclusion}
134 Consider some $D$ s.t. $\patchof{D} = \bot$. $D \neq C$.
135 So by Desired Contents $D \isin C \equiv D \isin L$.
136 By Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$.
138 And $D \le C \equiv D \le L$.
139 Thus $D \isin C \equiv D \le C$.
143 \subsection{Foreign Contents}