X-Git-Url: http://www.chiark.greenend.org.uk/ucgi/~ian/git?a=blobdiff_plain;f=article.tex;h=9286d604f594c43e9da561d475808d59651d941b;hb=7a176b9f9e3ac8838539c91cc2466d4e0f16bc70;hp=f69420fc071f170952b9e5a60ca9a60916658ec5;hpb=ed0c2b492c0fc7fd6b4917a39649421edb8f9605;p=topbloke-formulae.git

diff --git a/article.tex b/article.tex
index f69420f..9286d60 100644
--- a/article.tex
+++ b/article.tex
@@ -301,12 +301,16 @@ xxx proof tbd
    \right]
   \implies
    \left[
-    D \isin C
+    D \le C
    \implies
     \patchof{D} = \bot
    \right]
 }\]
-xxx proof tbd
+\proof{
+Consider some $D \le C$.  If $D = C$, $\patchof{D} = \bot$ trivially.
+If $D \neq C$ then $D \le A$ where $A \in \set A$.  By Foreign
+Contents of $A$, $\patchof{D} = \bot$.
+}
 
 \subsection{No Replay for Merge Results}
 
@@ -461,10 +465,8 @@ $D \isin C \equiv D \isin A \equiv D \le A \equiv D \le C$.  $\qed$
 
 \subsection{Foreign Contents:}
 
-Only relevant if $\patchof{C} = \bot$.  Trivial by Foreign Contents of
-$A$.  $\qed$
-
-xxx fixme not trivial use Totally Foreign Contents
+Only relevant if $\patchof{C} = \bot$, and in that case Totally
+Foreign Contents applies. $\qed$
 
 \section{Create Base}
 
@@ -472,7 +474,7 @@ Given $L$, create a Topbloke base branch initial commit $B$.
 \gathbegin
  B \hasparents \{ L \}
 \gathnext
- \patchof{B} = \pa{B}
+ \patchof{B} = \pan{B}
 \gathnext
  D \isin B \equiv D \isin L \lor D = B
 \end{gather}
@@ -490,9 +492,9 @@ Given $L$, create a Topbloke base branch initial commit $B$.
 
 If $\patchof{L} = \pa{L}$, trivial by Base Acyclic for $L$.
 
-If $\patchof{L} = \bot$, xxx
-
-Trivial from Base Acyclic for $L$.  $\qed$
+If $\patchof{L} = \bot$, consider some $D \isin B$.  $D \neq B$.
+Thus $D \isin L$.  So by No Replay of $D$ in $L$, $D \le L$.
+Thus $D \le B$.
 
 \subsection{Unique Base}
 
@@ -504,9 +506,29 @@ Not applicable. $\qed$
 
 \subsection{Base Acyclic}
 
-xxx
+Consider some $D \isin B$.  If $D = B$, $D \in \pn$, OK.
 
-xxx unfinished
+If $D \neq B$, $D \isin L$.  By No Replay of $D$ in $L$, $D \le L$.
+Thus by Foreign Contents of $L$, $\patchof{D} = \bot$.  OK.
+
+$\qed$
+
+\subsection{Coherence and Patch Inclusion}
+
+Consider some $D \in \p$.
+$B \not\in \py$ so $D \neq B$.  So $D \isin B \equiv D \isin L$.
+
+Thus $L \haspatch \p \implies B \haspatch P$
+and $L \nothaspatch \p \implies B \nothaspatch P$.
+
+$\qed$.
+
+\subsection{Foreign Inclusion}
+
+Consider some $D$ s.t. $\patchof{D} = \bot$.  $D \neq B$
+so $D \isin B \equiv D \isin L$.
+
+By Foreign Inclusion of $D$ xxx up to here
 
 \section{Create Tip}
 
@@ -908,9 +930,8 @@ $\qed$
 
 \subsection{Foreign Contents}
 
-xxx use Totally Foreign Contents
-
-If $\patchof{C} = \bot$, by Foreign Merges
-$\patchof{L} = \patchof{R} = \bot$.
+Only relevant if $\patchof{L} = \bot$, in which case
+$\patchof{C} = \bot$ and by Foreign Merges $\patchof{R} = \bot$,
+so Totally Foreign Contents applies.  $\qed$
 
 \end{document}