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Add description Monad
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Sun, 08 Feb 2015 19:57:44 +0900
parents cff9f6f0ff25
children 086fc8bb6ea9
files category.tex fig/delta_example.xbb fig/functor.xbb fig/monad_laws.graffle fig/monad_laws.pdf fig/monad_laws.xbb fig/morphism_composition_law.xbb fig/natural_transformation.xbb fig/non_delta_example.xbb fig/ryukyu.xbb fig/simple_category.xbb
diffstat 11 files changed, 2003 insertions(+), 9 deletions(-) [+]
line wrap: on
line diff
--- a/category.tex	Sun Feb 08 15:43:20 2015 +0900
+++ b/category.tex	Sun Feb 08 19:57:44 2015 +0900
@@ -217,7 +217,56 @@
 
 % }}}
 
+% {{{ Monad
+
 \section{Monad}
-\section{Kleisli Triple}
+\label{section:monad}
+\ref{section:natural_transformation}節では functor 間の可換性 natural transformation について述べた。
+
+category, functor, natural transformation を用いて Monad を定義する。
+
+category C における monad とは以下の性質を持つ $ triple (T, \eta, \mu) $ である。
+
+\begin{itemize}
+    \item $ T : C \rightarrow C $
+
+        category C から C への functor T
+
+    \item $ \eta : id_C \rightarrow T $
+
+        C から T への natural transformation $ \eta $
+
+    \item $ \mu : T^2 \rightarrow T $
+
+        $ T^2 $ から $ T $ への natural transformation $ \mu $
+
+        なお、 $ T^2 $ とは $ TT $ と同義であるとする。つまり functor T による mapping を2回行なうものである。
+        回数を n 回とすることで $ T^n $ と記述する。
+\end{itemize}
+
+この $ triple (T, \eta, \mu) $ が図\ref{fig:monad_laws}の可換図を満たす。
+なお、Aに対する $ \mu $ を $ \mu_A $ と記述する。
+同じように A に対する $ \eta $ を $ \eta_A $ と記述する
+
+\begin{figure}[htbp]
+    \begin{center}
+        \includegraphics[scale=1.0]{fig/monad_laws.pdf}
+        \caption{$ triple (T, \eta, \mu) $ が Monad であるために満たすべき可換図}
+        \label{fig:monad_laws}
+    \end{center}
+\end{figure}
+
+図\ref{fig:monad_laws} で示した可換図は monad が満たすべき Monad則であり、T に対する演算の可換性を強制する。
+$ \eta $ は T を1つ増やすオペレーションであり、$ \mu $ は T を 2 つから1つに減らすオペレーションである。
+
+左側の可換図は T が 3つある場合の可換性である。
+TTT に対して、 (TT)T のよう外側の2つのTに $ \mu $ を行なってから結果の TT に対して $ \mu $ を行なっても、 T(TT) のように内側の2つのTに $ \mu $ を行なってから結果の TT に対して $ \mu $ を行なっても可換であることを示す。
+つまり、TTTを減らす演算は左から行なっても右から行なっても良いことを示す。
+
+そして右側の可換図は T が1つである場合の可換性である。
+T A に対して $ \eta $ を行ない TT A としてから $ \mu $ を行なっても、T A の内部の A に対して $ \eta $ を行ない TT A としてか $ \mu $ を行なっても、 T A の id と同じであることを示している。
+つまり、Tに対する演算における左右の単位元となるような性質を $ triple (T, \eta, \mu) $ に強制することとなる。
+
+% }}}
+
 \section{Monads in Functional Programming}
-
--- a/fig/delta_example.xbb	Sun Feb 08 15:43:20 2015 +0900
+++ b/fig/delta_example.xbb	Sun Feb 08 19:57:44 2015 +0900
@@ -4,5 +4,5 @@
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 %%PDFVersion: 1.3
 %%Pages: 1
-%%CreationDate: Sun Feb  8 15:38:39 2015
+%%CreationDate: Sun Feb  8 16:05:26 2015
 
--- a/fig/functor.xbb	Sun Feb 08 15:43:20 2015 +0900
+++ b/fig/functor.xbb	Sun Feb 08 19:57:44 2015 +0900
@@ -4,5 +4,5 @@
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+				<dict>
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+					<string>NO</string>
+				</dict>
+				<key>stroke</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+			</dict>
+			<key>Text</key>
+			<dict>
+				<key>Pad</key>
+				<integer>0</integer>
+				<key>Text</key>
+				<string>{\rtf1\ansi\ansicpg1252\cocoartf1265\cocoasubrtf210
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+
+\f0\fs12 \cf0 T A}</string>
+				<key>VerticalPad</key>
+				<integer>0</integer>
+			</dict>
+			<key>Wrap</key>
+			<string>NO</string>
+		</dict>
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+			<key>Shape</key>
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+				<dict>
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+				<dict>
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+					<string>NO</string>
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+				<key>stroke</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
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+			<key>Text</key>
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+			<key>Head</key>
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+		<dict>
+			<key>Class</key>
+			<string>LineGraphic</string>
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+					<string>FilledArrow</string>
+					<key>Legacy</key>
+					<true/>
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+		<dict>
+			<key>Class</key>
+			<string>LineGraphic</string>
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+					<true/>
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+			<dict>
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+		<dict>
+			<key>Class</key>
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+					<true/>
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+			<dict>
+				<key>ID</key>
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+		<dict>
+			<key>Class</key>
+			<string>LineGraphic</string>
+			<key>Head</key>
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+			<key>Points</key>
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+				<key>stroke</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
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+			<key>Text</key>
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+
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+			<key>Bounds</key>
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+		<dict>
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+					<string>NO</string>
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+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
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+			<key>Text</key>
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+				<string>{\rtf1\ansi\ansicpg1252\cocoartf1265\cocoasubrtf210
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+					<string>NO</string>
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+					<key>Draws</key>
+					<string>NO</string>
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+			<key>Text</key>
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+				<string>{\rtf1\ansi\ansicpg1252\cocoartf1265\cocoasubrtf210
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+
+\f0\fs24 \cf0 T A}</string>
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+		<dict>
+			<key>Bounds</key>
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+					<key>Draws</key>
+					<string>NO</string>
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+				<key>stroke</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
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+			<key>Text</key>
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+				<string>{\rtf1\ansi\ansicpg1252\cocoartf1265\cocoasubrtf210
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+			<key>Bounds</key>
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+			<string>Resize</string>
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+				<string>Helvetica</string>
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+					<key>Draws</key>
+					<string>NO</string>
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+					<string>NO</string>
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+					<key>Draws</key>
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+					<key>Draws</key>
+					<string>NO</string>
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+			<key>Text</key>
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+\f0\fs24 \cf0 \uc0\u956 }</string>
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+			<string>NO</string>
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+			<string>YES</string>
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+				<key>Font</key>
+				<string>Helvetica</string>
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+					<key>Draws</key>
+					<string>NO</string>
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+				<key>stroke</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
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+			<key>Text</key>
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+
+\f0\fs12 \cf0 T A}</string>
+				<key>VerticalPad</key>
+				<integer>0</integer>
+			</dict>
+			<key>Wrap</key>
+			<string>NO</string>
+		</dict>
+		<dict>
+			<key>Bounds</key>
+			<string>{{39.989234924316406, 77.999998837800916}, {18, 14}}</string>
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+					<string>NO</string>
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+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
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+				<key>stroke</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+			</dict>
+			<key>Text</key>
+			<dict>
+				<key>Pad</key>
+				<integer>0</integer>
+				<key>Text</key>
+				<string>{\rtf1\ansi\ansicpg1252\cocoartf1265\cocoasubrtf210
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+{\colortbl;\red255\green255\blue255;}
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+
+\f0\fs24 \cf0 T \uc0\u956 }</string>
+				<key>VerticalPad</key>
+				<integer>0</integer>
+			</dict>
+			<key>Wrap</key>
+			<string>NO</string>
+		</dict>
+		<dict>
+			<key>Bounds</key>
+			<string>{{135.500000466958, 34.928230363717738}, {9, 7}}</string>
+			<key>Class</key>
+			<string>ShapedGraphic</string>
+			<key>FitText</key>
+			<string>YES</string>
+			<key>Flow</key>
+			<string>Resize</string>
+			<key>FontInfo</key>
+			<dict>
+				<key>Font</key>
+				<string>Helvetica</string>
+				<key>Size</key>
+				<real>6</real>
+			</dict>
+			<key>ID</key>
+			<integer>47</integer>
+			<key>Shape</key>
+			<string>Rectangle</string>
+			<key>Style</key>
+			<dict>
+				<key>fill</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+				<key>shadow</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+				<key>stroke</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+			</dict>
+			<key>Text</key>
+			<dict>
+				<key>Pad</key>
+				<integer>0</integer>
+				<key>Text</key>
+				<string>{\rtf1\ansi\ansicpg1252\cocoartf1265\cocoasubrtf210
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+
+\f0\fs12 \cf0 T A}</string>
+				<key>VerticalPad</key>
+				<integer>0</integer>
+			</dict>
+			<key>Wrap</key>
+			<string>NO</string>
+		</dict>
+		<dict>
+			<key>Bounds</key>
+			<string>{{127.75000278838547, 25.928227437960945}, {8, 14}}</string>
+			<key>Class</key>
+			<string>ShapedGraphic</string>
+			<key>FitText</key>
+			<string>YES</string>
+			<key>Flow</key>
+			<string>Resize</string>
+			<key>ID</key>
+			<integer>46</integer>
+			<key>Shape</key>
+			<string>Rectangle</string>
+			<key>Style</key>
+			<dict>
+				<key>fill</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+				<key>shadow</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+				<key>stroke</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+			</dict>
+			<key>Text</key>
+			<dict>
+				<key>Pad</key>
+				<integer>0</integer>
+				<key>Text</key>
+				<string>{\rtf1\ansi\ansicpg1252\cocoartf1265\cocoasubrtf210
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+
+\f0\fs24 \cf0 \uc0\u956 }</string>
+				<key>VerticalPad</key>
+				<integer>0</integer>
+			</dict>
+			<key>Wrap</key>
+			<string>NO</string>
+		</dict>
+		<dict>
+			<key>Bounds</key>
+			<string>{{135.2500023258817, 136.7511961309412}, {5, 7}}</string>
+			<key>Class</key>
+			<string>ShapedGraphic</string>
+			<key>FitText</key>
+			<string>YES</string>
+			<key>Flow</key>
+			<string>Resize</string>
+			<key>FontInfo</key>
+			<dict>
+				<key>Font</key>
+				<string>Helvetica</string>
+				<key>Size</key>
+				<real>6</real>
+			</dict>
+			<key>ID</key>
+			<integer>45</integer>
+			<key>Shape</key>
+			<string>Rectangle</string>
+			<key>Style</key>
+			<dict>
+				<key>fill</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+				<key>shadow</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+				<key>stroke</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+			</dict>
+			<key>Text</key>
+			<dict>
+				<key>Pad</key>
+				<integer>0</integer>
+				<key>Text</key>
+				<string>{\rtf1\ansi\ansicpg1252\cocoartf1265\cocoasubrtf210
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+{\colortbl;\red255\green255\blue255;}
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+
+\f0\fs12 \cf0 A}</string>
+				<key>VerticalPad</key>
+				<integer>0</integer>
+			</dict>
+			<key>Wrap</key>
+			<string>NO</string>
+		</dict>
+		<dict>
+			<key>Bounds</key>
+			<string>{{73.999996244600581, 113.31100599605436}, {4, 7}}</string>
+			<key>Class</key>
+			<string>ShapedGraphic</string>
+			<key>FitText</key>
+			<string>YES</string>
+			<key>Flow</key>
+			<string>Resize</string>
+			<key>FontInfo</key>
+			<dict>
+				<key>Font</key>
+				<string>Helvetica</string>
+				<key>Size</key>
+				<real>6</real>
+			</dict>
+			<key>ID</key>
+			<integer>44</integer>
+			<key>Shape</key>
+			<string>Rectangle</string>
+			<key>Style</key>
+			<dict>
+				<key>fill</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+				<key>shadow</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+				<key>stroke</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+			</dict>
+			<key>Text</key>
+			<dict>
+				<key>Pad</key>
+				<integer>0</integer>
+				<key>Text</key>
+				<string>{\rtf1\ansi\ansicpg1252\cocoartf1265\cocoasubrtf210
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+{\colortbl;\red255\green255\blue255;}
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+
+\f0\fs12 \cf0 2}</string>
+				<key>VerticalPad</key>
+				<integer>0</integer>
+			</dict>
+			<key>Wrap</key>
+			<string>NO</string>
+		</dict>
+		<dict>
+			<key>Bounds</key>
+			<string>{{202.0000017072305, 34.92822933740662}, {4, 7}}</string>
+			<key>Class</key>
+			<string>ShapedGraphic</string>
+			<key>FitText</key>
+			<string>YES</string>
+			<key>Flow</key>
+			<string>Resize</string>
+			<key>FontInfo</key>
+			<dict>
+				<key>Font</key>
+				<string>Helvetica</string>
+				<key>Size</key>
+				<real>6</real>
+			</dict>
+			<key>ID</key>
+			<integer>43</integer>
+			<key>Shape</key>
+			<string>Rectangle</string>
+			<key>Style</key>
+			<dict>
+				<key>fill</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+				<key>shadow</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+				<key>stroke</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+			</dict>
+			<key>Text</key>
+			<dict>
+				<key>Pad</key>
+				<integer>0</integer>
+				<key>Text</key>
+				<string>{\rtf1\ansi\ansicpg1252\cocoartf1265\cocoasubrtf210
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+
+\f0\fs12 \cf0 2}</string>
+				<key>VerticalPad</key>
+				<integer>0</integer>
+			</dict>
+			<key>Wrap</key>
+			<string>NO</string>
+		</dict>
+		<dict>
+			<key>Bounds</key>
+			<string>{{74.000002750833417, 34.928231099480442}, {4, 7}}</string>
+			<key>Class</key>
+			<string>ShapedGraphic</string>
+			<key>FitText</key>
+			<string>YES</string>
+			<key>Flow</key>
+			<string>Resize</string>
+			<key>FontInfo</key>
+			<dict>
+				<key>Font</key>
+				<string>Helvetica</string>
+				<key>Size</key>
+				<real>6</real>
+			</dict>
+			<key>ID</key>
+			<integer>42</integer>
+			<key>Shape</key>
+			<string>Rectangle</string>
+			<key>Style</key>
+			<dict>
+				<key>fill</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+				<key>shadow</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+				<key>stroke</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+			</dict>
+			<key>Text</key>
+			<dict>
+				<key>Pad</key>
+				<integer>0</integer>
+				<key>Text</key>
+				<string>{\rtf1\ansi\ansicpg1252\cocoartf1265\cocoasubrtf210
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+{\colortbl;\red255\green255\blue255;}
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+
+\f0\fs12 \cf0 3}</string>
+				<key>VerticalPad</key>
+				<integer>0</integer>
+			</dict>
+			<key>Wrap</key>
+			<string>NO</string>
+		</dict>
+		<dict>
+			<key>Class</key>
+			<string>LineGraphic</string>
+			<key>Head</key>
+			<dict>
+				<key>ID</key>
+				<integer>34</integer>
+			</dict>
+			<key>ID</key>
+			<integer>38</integer>
+			<key>Points</key>
+			<array>
+				<string>{204.00001869184746, 60}</string>
+				<string>{204.00001869184746, 110}</string>
+			</array>
+			<key>Style</key>
+			<dict>
+				<key>stroke</key>
+				<dict>
+					<key>HeadArrow</key>
+					<string>FilledArrow</string>
+					<key>Legacy</key>
+					<true/>
+					<key>LineType</key>
+					<integer>1</integer>
+					<key>TailArrow</key>
+					<string>0</string>
+				</dict>
+			</dict>
+			<key>Tail</key>
+			<dict>
+				<key>ID</key>
+				<integer>11</integer>
+			</dict>
+		</dict>
+		<dict>
+			<key>Class</key>
+			<string>LineGraphic</string>
+			<key>Head</key>
+			<dict>
+				<key>ID</key>
+				<integer>33</integer>
+			</dict>
+			<key>ID</key>
+			<integer>37</integer>
+			<key>Points</key>
+			<array>
+				<string>{76.092549489918511, 60}</string>
+				<string>{76.423083382484833, 110}</string>
+			</array>
+			<key>Style</key>
+			<dict>
+				<key>stroke</key>
+				<dict>
+					<key>HeadArrow</key>
+					<string>FilledArrow</string>
+					<key>Legacy</key>
+					<true/>
+					<key>LineType</key>
+					<integer>1</integer>
+					<key>TailArrow</key>
+					<string>0</string>
+				</dict>
+			</dict>
+			<key>Tail</key>
+			<dict>
+				<key>ID</key>
+				<integer>9</integer>
+			</dict>
+		</dict>
+		<dict>
+			<key>Bounds</key>
+			<string>{{127.75000092946176, 127.75119320518438}, {8, 14}}</string>
+			<key>Class</key>
+			<string>ShapedGraphic</string>
+			<key>FitText</key>
+			<string>YES</string>
+			<key>Flow</key>
+			<string>Resize</string>
+			<key>ID</key>
+			<integer>36</integer>
+			<key>Shape</key>
+			<string>Rectangle</string>
+			<key>Style</key>
+			<dict>
+				<key>fill</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+				<key>shadow</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+				<key>stroke</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+			</dict>
+			<key>Text</key>
+			<dict>
+				<key>Pad</key>
+				<integer>0</integer>
+				<key>Text</key>
+				<string>{\rtf1\ansi\ansicpg1252\cocoartf1265\cocoasubrtf210
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+
+\f0\fs24 \cf0 \uc0\u956 }</string>
+				<key>VerticalPad</key>
+				<integer>0</integer>
+			</dict>
+			<key>Wrap</key>
+			<string>NO</string>
+		</dict>
+		<dict>
+			<key>Class</key>
+			<string>LineGraphic</string>
+			<key>Head</key>
+			<dict>
+				<key>ID</key>
+				<integer>34</integer>
+			</dict>
+			<key>ID</key>
+			<integer>35</integer>
+			<key>Points</key>
+			<array>
+				<string>{94.999999999999943, 123.99010942994431}</string>
+				<string>{184.99999999999994, 123.94325936125944}</string>
+			</array>
+			<key>Style</key>
+			<dict>
+				<key>stroke</key>
+				<dict>
+					<key>HeadArrow</key>
+					<string>FilledArrow</string>
+					<key>Legacy</key>
+					<true/>
+					<key>LineType</key>
+					<integer>1</integer>
+					<key>TailArrow</key>
+					<string>0</string>
+				</dict>
+			</dict>
+			<key>Tail</key>
+			<dict>
+				<key>ID</key>
+				<integer>33</integer>
+			</dict>
+		</dict>
+		<dict>
+			<key>Bounds</key>
+			<string>{{184.99999999999994, 110}, {38, 28}}</string>
+			<key>Class</key>
+			<string>ShapedGraphic</string>
+			<key>ID</key>
+			<integer>34</integer>
+			<key>Shape</key>
+			<string>Rectangle</string>
+			<key>Style</key>
+			<dict>
+				<key>shadow</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+				<key>stroke</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+			</dict>
+			<key>Text</key>
+			<dict>
+				<key>Text</key>
+				<string>{\rtf1\ansi\ansicpg1252\cocoartf1265\cocoasubrtf210
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+{\colortbl;\red255\green255\blue255;}
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+
+\f0\fs24 \cf0 T A}</string>
+			</dict>
+		</dict>
+		<dict>
+			<key>Bounds</key>
+			<string>{{56.999999999999943, 110}, {38, 28}}</string>
+			<key>Class</key>
+			<string>ShapedGraphic</string>
+			<key>ID</key>
+			<integer>33</integer>
+			<key>Shape</key>
+			<string>Rectangle</string>
+			<key>Style</key>
+			<dict>
+				<key>shadow</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+				<key>stroke</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+			</dict>
+			<key>Text</key>
+			<dict>
+				<key>Text</key>
+				<string>{\rtf1\ansi\ansicpg1252\cocoartf1265\cocoasubrtf210
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+{\colortbl;\red255\green255\blue255;}
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+
+\f0\fs24 \cf0 T  A}</string>
+			</dict>
+		</dict>
+		<dict>
+			<key>Class</key>
+			<string>LineGraphic</string>
+			<key>Head</key>
+			<dict>
+				<key>ID</key>
+				<integer>11</integer>
+			</dict>
+			<key>ID</key>
+			<integer>12</integer>
+			<key>Points</key>
+			<array>
+				<string>{94.999999999999943, 45.999990631971997}</string>
+				<string>{184.99999999999994, 45.999990631971997}</string>
+			</array>
+			<key>Style</key>
+			<dict>
+				<key>stroke</key>
+				<dict>
+					<key>HeadArrow</key>
+					<string>FilledArrow</string>
+					<key>Legacy</key>
+					<true/>
+					<key>LineType</key>
+					<integer>1</integer>
+					<key>TailArrow</key>
+					<string>0</string>
+				</dict>
+			</dict>
+			<key>Tail</key>
+			<dict>
+				<key>ID</key>
+				<integer>9</integer>
+			</dict>
+		</dict>
+		<dict>
+			<key>Bounds</key>
+			<string>{{184.99999999999994, 32}, {38, 28}}</string>
+			<key>Class</key>
+			<string>ShapedGraphic</string>
+			<key>ID</key>
+			<integer>11</integer>
+			<key>Shape</key>
+			<string>Rectangle</string>
+			<key>Style</key>
+			<dict>
+				<key>shadow</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+				<key>stroke</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+			</dict>
+			<key>Text</key>
+			<dict>
+				<key>Text</key>
+				<string>{\rtf1\ansi\ansicpg1252\cocoartf1265\cocoasubrtf210
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+{\colortbl;\red255\green255\blue255;}
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+
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