introduce and use Totally Foreign Contents

[topbloke-formulae.git] / article.tex

diff --git a/article.tex b/article.tex
index bb050ae6273ad9cba01131a4131c4b4ba75e1622..3f066dc59b5275790551472e2735cf50ceeffc73 100644 (file)
--- a/article.tex
+++ b/article.tex
@@ -317,6 +317,24 @@ xxx proof tbd
   $A \le C$ so $D \le C$.
 }
 
+\[ \eqn{Simple Foreign Inclusion:}{
+  \left[
+    C \hasparents \{ L \}
+   \land
+    \bigforall_{D} D \isin C \equiv D \isin L \lor D = C
+  \right]
+ \implies
+   \bigforall_{D \text{ s.t. } \patchof{D} = \bot}
+     D \isin C \equiv D \le C
+}\]
+\proof{
+Consider some $D$ s.t. $\patchof{D} = \bot$.
+If $D = C$, trivially true.  For $D \neq C$,
+by Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$.
+And by Exact Ancestors $D \le L \equiv D \le C$.
+So $D \isin C \equiv D \le C$.
+}
+
 \[ \eqn{Totally Foreign Contents:}{
   \bigforall_{C \hasparents \set A}
    \left[
@@ -513,11 +531,7 @@ $\qed$.
 
 \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 $D$ in $L$, $D \isin L \equiv D \le L$.
-And by Exact Ancestors $D \le L \equiv D \le B$.
-So $D \isin B \equiv D \le B$.  $\qed$
+Simple Foreign Inclusion applies. $\qed$
 
 \subsection{Foreign Contents}
 
@@ -567,13 +581,28 @@ Not applicable.
 
 \subsection{Coherence and Patch Inclusion}
 
-Consider some $D \in \py$.
+$$
+\begin{cases}
+  \p = \pq    \lor B \haspatch \p : & C \haspatch \p \\
+  \p \neq \pq \land B \nothaspatch \p : & C \nothaspatch \p
+\end{cases}
+$$
+
+\proofstarts
+~ Consider some $D \in \py$.
 
 \subsubsection{For $\p = \pq$:}
 
 By Base Acyclic, $D \not\isin B$.  So $D \isin C \equiv D = C$.
 By No Sneak, $D \le B \equiv D = C$.  Thus $C \haspatch \pq$.
 
+\subsubsection{For $\p \neq \pq$:}
+
+$D \neq C$.  So $D \isin C \equiv D \isin B$,
+and $D \le C \equiv D \le B$.
+
+$\qed$
+
 xxx up to here
 
 \section{Anticommit}