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fix typo
[topbloke-formulae.git]
/
simple.tex
diff --git
a/simple.tex
b/simple.tex
index 28dc6e53bd5f15a9b14d27dc323d67340958df02..5f6e184aad199a55782ac4c8e2884d632fa05663 100644
(file)
--- a/
simple.tex
+++ b/
simple.tex
@@
-47,36
+47,47
@@
$L$, $D \isin C \implies D \not\in \py$.
$\qed$
$\qed$
-\subsection{Coherence and
patch i
nclusion}
+\subsection{Coherence and
Patch I
nclusion}
-Need to consider $D \in \py$
+$$
+\begin{cases}
+ L \haspatch \p : & C \haspatch \p \\
+ L \nothaspatch \p : & C \nothaspatch \p
+\end{cases}
+$$
+\proofstarts
-\subsubsection{For $L \haspatch P, D = C$:}
+Firstly, if $L \haspatch \p$, $\exists_{F \in \py} F \le L$
+and this $F$ is also $\le C$
+so $C \zhaspatch \p \implies C \haspatch \p$.
+We will prove $\zhaspatch$
+
+We need to consider $D \in \py$.
+
+\subsubsection{For $L \haspatch \p, D = C$:}
Ancestors of $C$:
$ D \le C $.
Contents of $C$:
Ancestors of $C$:
$ D \le C $.
Contents of $C$:
-$ D \isin C \equiv \ldots \lor \true \text{ so } D \haspatch C $.
+$ D \isin C \equiv \ldots \lor \true$. So $ D \zhaspatch C $.
+OK.
-\subsubsection{For $L \haspatch
P
, D \neq C$:}
+\subsubsection{For $L \haspatch
\p
, D \neq C$:}
Ancestors: $ D \le C \equiv D \le L $.
Contents: $ D \isin C \equiv D \isin L \lor f $
Ancestors: $ D \le C \equiv D \le L $.
Contents: $ D \isin C \equiv D \isin L \lor f $
-so $ D \isin C \equiv D \isin L $.
-
-So:
-\[ L \haspatch P \implies C \haspatch P \]
+so $ D \isin C \equiv D \isin L $, i.e. $ C \zhaspatch P $.
+OK.
-\subsubsection{For $L \nothaspatch
P
$:}
+\subsubsection{For $L \nothaspatch
\p
$:}
Firstly, $C \not\in \py$ since if it were, $L \in \py$.
Thus $D \neq C$.
Firstly, $C \not\in \py$ since if it were, $L \in \py$.
Thus $D \neq C$.
-Now by contents of $L$, $D \notin L$, so $D \notin C$.
+Now by $\nothaspatch$, $D \not\isin L$ so $D \not\isin C$.
+OK.
-So:
-\[ L \nothaspatch P \implies C \nothaspatch P \]
$\qed$
\subsection{Foreign Inclusion:}
$\qed$
\subsection{Foreign Inclusion:}