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changeset 44:9d0c57803ca0

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Add figure for description delta (apply function example)
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Thu, 25 Dec 2014 14:52:33 +0900
parents e233b786623e
children bfc3fc56ad90
files figure/apply_function.graffle figure/apply_function.pdf figure/apply_function.xbb sigse.pdf sigse.tex
diffstat 5 files changed, 1359 insertions(+), 3 deletions(-) [+]
line wrap: on
line diff
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/figure/apply_function.graffle	Thu Dec 25 14:52:33 2014 +0900
@@ -0,0 +1,1337 @@
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+			<string>ShapedGraphic</string>
+			<key>FontInfo</key>
+			<dict>
+				<key>Font</key>
+				<string>Helvetica</string>
+				<key>Size</key>
+				<real>12</real>
+			</dict>
+			<key>ID</key>
+			<integer>8</integer>
+			<key>Shape</key>
+			<string>Rectangle</string>
+			<key>Style</key>
+			<dict>
+				<key>shadow</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+			</dict>
+			<key>Text</key>
+			<dict>
+				<key>Text</key>
+				<string>{\rtf1\ansi\ansicpg1252\cocoartf1265\cocoasubrtf210
+\cocoascreenfonts1{\fonttbl\f0\fswiss\fcharset0 Helvetica;}
+{\colortbl;\red255\green255\blue255;}
+\pard\tx560\tx1120\tx1680\tx2240\tx2800\tx3360\tx3920\tx4480\tx5040\tx5600\tx6160\tx6720\pardirnatural\qc
+
+\f0\fs36 \cf0 g}</string>
+			</dict>
+		</dict>
+		<dict>
+			<key>Bounds</key>
+			<string>{{68.624973297119141, 292}, {103, 40}}</string>
+			<key>Class</key>
+			<string>ShapedGraphic</string>
+			<key>FontInfo</key>
+			<dict>
+				<key>Font</key>
+				<string>Helvetica</string>
+				<key>Size</key>
+				<real>14</real>
+			</dict>
+			<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
+\cocoascreenfonts1{\fonttbl\f0\fswiss\fcharset0 Helvetica;}
+{\colortbl;\red255\green255\blue255;}
+\pard\tx560\tx1120\tx1680\tx2240\tx2800\tx3360\tx3920\tx4480\tx5040\tx5600\tx6160\tx6720\pardirnatural\qc
+
+\f0\fs28 \cf0 result\
+(version 3)}</string>
+			</dict>
+		</dict>
+		<dict>
+			<key>Bounds</key>
+			<string>{{236.62497329711914, 292}, {83, 40}}</string>
+			<key>Class</key>
+			<string>ShapedGraphic</string>
+			<key>FontInfo</key>
+			<dict>
+				<key>Font</key>
+				<string>Helvetica</string>
+				<key>Size</key>
+				<real>14</real>
+			</dict>
+			<key>HFlip</key>
+			<string>YES</string>
+			<key>ID</key>
+			<integer>30</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
+\cocoascreenfonts1{\fonttbl\f0\fswiss\fcharset0 Helvetica;}
+{\colortbl;\red255\green255\blue255;}
+\pard\tx560\tx1120\tx1680\tx2240\tx2800\tx3360\tx3920\tx4480\tx5040\tx5600\tx6160\tx6720\pardirnatural\qc
+
+\f0\fs28 \cf0 function\
+(version 2)}</string>
+			</dict>
+		</dict>
+		<dict>
+			<key>Bounds</key>
+			<string>{{68.624973297119141, 252}, {451, 40}}</string>
+			<key>Class</key>
+			<string>ShapedGraphic</string>
+			<key>FontInfo</key>
+			<dict>
+				<key>Font</key>
+				<string>Helvetica</string>
+				<key>Size</key>
+				<real>24</real>
+			</dict>
+			<key>ID</key>
+			<integer>26</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>Align</key>
+				<integer>0</integer>
+				<key>Text</key>
+				<string>{\rtf1\ansi\ansicpg1252\cocoartf1265\cocoasubrtf210
+\cocoascreenfonts1{\fonttbl\f0\fswiss\fcharset0 Helvetica;}
+{\colortbl;\red255\green255\blue255;}
+\pard\tx560\tx1120\tx1680\tx2240\tx2800\tx3360\tx3920\tx4480\tx5040\tx5600\tx6160\tx6720\pardirnatural
+
+\f0\fs48 \cf0        y            =        f      (       g           (  x  ) )}</string>
+			</dict>
+			<key>Wrap</key>
+			<string>NO</string>
+		</dict>
+		<dict>
+			<key>Class</key>
+			<string>LineGraphic</string>
+			<key>Head</key>
+			<dict>
+				<key>ID</key>
+				<integer>14</integer>
+			</dict>
+			<key>ID</key>
+			<integer>24</integer>
+			<key>Points</key>
+			<array>
+				<string>{414.62497329711914, 232.00001436225315}</string>
+				<string>{469.12497329711914, 232.00001436225315}</string>
+			</array>
+			<key>Style</key>
+			<dict>
+				<key>stroke</key>
+				<dict>
+					<key>HeadArrow</key>
+					<string>0</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>8</integer>
+			</dict>
+		</dict>
+		<dict>
+			<key>Class</key>
+			<string>LineGraphic</string>
+			<key>Head</key>
+			<dict>
+				<key>ID</key>
+				<integer>14</integer>
+			</dict>
+			<key>ID</key>
+			<integer>23</integer>
+			<key>Points</key>
+			<array>
+				<string>{414.59249740134356, 201.65593330965854}</string>
+				<string>{469.15744919289472, 222.34406669034144}</string>
+			</array>
+			<key>Style</key>
+			<dict>
+				<key>stroke</key>
+				<dict>
+					<key>HeadArrow</key>
+					<string>0</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>7</integer>
+			</dict>
+		</dict>
+		<dict>
+			<key>Class</key>
+			<string>LineGraphic</string>
+			<key>Head</key>
+			<dict>
+				<key>ID</key>
+				<integer>14</integer>
+			</dict>
+			<key>ID</key>
+			<integer>22</integer>
+			<key>Points</key>
+			<array>
+				<string>{414.52338161271501, 171.25945654263191}</string>
+				<string>{469.22656498152327, 212.74054345736806}</string>
+			</array>
+			<key>Style</key>
+			<dict>
+				<key>stroke</key>
+				<dict>
+					<key>HeadArrow</key>
+					<string>0</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>6</integer>
+			</dict>
+		</dict>
+		<dict>
+			<key>Class</key>
+			<string>LineGraphic</string>
+			<key>Head</key>
+			<dict>
+				<key>ID</key>
+				<integer>8</integer>
+			</dict>
+			<key>ID</key>
+			<integer>21</integer>
+			<key>Points</key>
+			<array>
+				<string>{309.12497329711914, 232.00001623872726}</string>
+				<string>{363.62497329711914, 232.00001623872726}</string>
+			</array>
+			<key>Style</key>
+			<dict>
+				<key>stroke</key>
+				<dict>
+					<key>HeadArrow</key>
+					<string>0</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>7</integer>
+			</dict>
+			<key>ID</key>
+			<integer>20</integer>
+			<key>Points</key>
+			<array>
+				<string>{309.12497310611434, 191.97770727366566}</string>
+				<string>{363.62497517476532, 191.93006203286416}</string>
+			</array>
+			<key>Style</key>
+			<dict>
+				<key>stroke</key>
+				<dict>
+					<key>HeadArrow</key>
+					<string>0</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>10</integer>
+			</dict>
+		</dict>
+		<dict>
+			<key>Class</key>
+			<string>LineGraphic</string>
+			<key>Head</key>
+			<dict>
+				<key>ID</key>
+				<integer>6</integer>
+			</dict>
+			<key>ID</key>
+			<integer>19</integer>
+			<key>Points</key>
+			<array>
+				<string>{309.09249740134356, 182.34406669034144}</string>
+				<string>{363.65744919289472, 161.65593330965854}</string>
+			</array>
+			<key>Style</key>
+			<dict>
+				<key>stroke</key>
+				<dict>
+					<key>HeadArrow</key>
+					<string>0</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>10</integer>
+			</dict>
+		</dict>
+		<dict>
+			<key>Class</key>
+			<string>LineGraphic</string>
+			<key>Head</key>
+			<dict>
+				<key>ID</key>
+				<integer>10</integer>
+			</dict>
+			<key>ID</key>
+			<integer>18</integer>
+			<key>Points</key>
+			<array>
+				<string>{172.1136803707216, 163.24080152942756}</string>
+				<string>{212.62497329711914, 172}</string>
+				<string>{258.14370307128587, 184.82217740117372}</string>
+			</array>
+			<key>Style</key>
+			<dict>
+				<key>stroke</key>
+				<dict>
+					<key>HeadArrow</key>
+					<string>0</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>3</integer>
+			</dict>
+		</dict>
+		<dict>
+			<key>Class</key>
+			<string>LineGraphic</string>
+			<key>Head</key>
+			<dict>
+				<key>ID</key>
+				<integer>10</integer>
+			</dict>
+			<key>ID</key>
+			<integer>17</integer>
+			<key>Points</key>
+			<array>
+				<string>{172.12497329711914, 192.00001953595211}</string>
+				<string>{258.12497329711914, 192.00001953595211}</string>
+			</array>
+			<key>Style</key>
+			<dict>
+				<key>stroke</key>
+				<dict>
+					<key>HeadArrow</key>
+					<string>0</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>4</integer>
+			</dict>
+		</dict>
+		<dict>
+			<key>Class</key>
+			<string>LineGraphic</string>
+			<key>Head</key>
+			<dict>
+				<key>ID</key>
+				<integer>11</integer>
+			</dict>
+			<key>ID</key>
+			<integer>16</integer>
+			<key>Points</key>
+			<array>
+				<string>{172.12497329711914, 232.0000196276261}</string>
+				<string>{258.12497329711914, 232.0000196276261}</string>
+			</array>
+			<key>Style</key>
+			<dict>
+				<key>stroke</key>
+				<dict>
+					<key>HeadArrow</key>
+					<string>0</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>5</integer>
+			</dict>
+		</dict>
+		<dict>
+			<key>Bounds</key>
+			<string>{{258.62497329711914, 172}, {50, 40}}</string>
+			<key>Class</key>
+			<string>ShapedGraphic</string>
+			<key>FontInfo</key>
+			<dict>
+				<key>Font</key>
+				<string>Helvetica</string>
+				<key>Size</key>
+				<real>12</real>
+			</dict>
+			<key>ID</key>
+			<integer>10</integer>
+			<key>Shape</key>
+			<string>Rectangle</string>
+			<key>Style</key>
+			<dict>
+				<key>shadow</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+			</dict>
+			<key>Text</key>
+			<dict>
+				<key>Text</key>
+				<string>{\rtf1\ansi\ansicpg1252\cocoartf1265\cocoasubrtf210
+\cocoascreenfonts1{\fonttbl\f0\fswiss\fcharset0 Helvetica;}
+{\colortbl;\red255\green255\blue255;}
+\pard\tx560\tx1120\tx1680\tx2240\tx2800\tx3360\tx3920\tx4480\tx5040\tx5600\tx6160\tx6720\pardirnatural\qc
+
+\f0\fs36 \cf0 f'}</string>
+			</dict>
+		</dict>
+		<dict>
+			<key>Bounds</key>
+			<string>{{364.12497329711914, 172}, {50, 40}}</string>
+			<key>Class</key>
+			<string>ShapedGraphic</string>
+			<key>FontInfo</key>
+			<dict>
+				<key>Font</key>
+				<string>Helvetica</string>
+				<key>Size</key>
+				<real>12</real>
+			</dict>
+			<key>ID</key>
+			<integer>7</integer>
+			<key>Shape</key>
+			<string>Rectangle</string>
+			<key>Style</key>
+			<dict>
+				<key>shadow</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+			</dict>
+			<key>Text</key>
+			<dict>
+				<key>Text</key>
+				<string>{\rtf1\ansi\ansicpg1252\cocoartf1265\cocoasubrtf210
+\cocoascreenfonts1{\fonttbl\f0\fswiss\fcharset0 Helvetica;}
+{\colortbl;\red255\green255\blue255;}
+\pard\tx560\tx1120\tx1680\tx2240\tx2800\tx3360\tx3920\tx4480\tx5040\tx5600\tx6160\tx6720\pardirnatural\qc
+
+\f0\fs36 \cf0 g'}</string>
+			</dict>
+		</dict>
+		<dict>
+			<key>Bounds</key>
+			<string>{{364.12497329711914, 132}, {50, 40}}</string>
+			<key>Class</key>
+			<string>ShapedGraphic</string>
+			<key>FontInfo</key>
+			<dict>
+				<key>Font</key>
+				<string>Helvetica</string>
+				<key>Size</key>
+				<real>18</real>
+			</dict>
+			<key>ID</key>
+			<integer>6</integer>
+			<key>Shape</key>
+			<string>Rectangle</string>
+			<key>Style</key>
+			<dict>
+				<key>shadow</key>
+				<dict>
+					<key>Draws</key>
+					<string>NO</string>
+				</dict>
+			</dict>
+			<key>Text</key>
+			<dict>
+				<key>Text</key>
+				<string>{\rtf1\ansi\ansicpg1252\cocoartf1265\cocoasubrtf210
+\cocoascreenfonts1{\fonttbl\f0\fswiss\fcharset0 Helvetica;}
+{\colortbl;\red255\green255\blue255;}
+\pard\tx560\tx1120\tx1680\tx2240\tx2800\tx3360\tx3920\tx4480\tx5040\tx5600\tx6160\tx6720\pardirnatural\qc
+
+\f0\fs36 \cf0 g''}</string>
+			</dict>
+		</dict>
+	</array>
+	<key>GridInfo</key>
+	<dict/>
+	<key>GuidesLocked</key>
+	<string>NO</string>
+	<key>GuidesVisible</key>
+	<string>YES</string>
+	<key>HPages</key>
+	<integer>1</integer>
+	<key>ImageCounter</key>
+	<integer>1</integer>
+	<key>KeepToScale</key>
+	<false/>
+	<key>Layers</key>
+	<array>
+		<dict>
+			<key>Lock</key>
+			<string>NO</string>
+			<key>Name</key>
+			<string>Layer 1</string>
+			<key>Print</key>
+			<string>YES</string>
+			<key>View</key>
+			<string>YES</string>
+		</dict>
+	</array>
+	<key>LayoutInfo</key>
+	<dict>
+		<key>Animate</key>
+		<string>NO</string>
+		<key>circoMinDist</key>
+		<real>18</real>
+		<key>circoSeparation</key>
+		<real>0.0</real>
+		<key>layoutEngine</key>
+		<string>dot</string>
+		<key>neatoSeparation</key>
+		<real>0.0</real>
+		<key>twopiSeparation</key>
+		<real>0.0</real>
+	</dict>
+	<key>LinksVisible</key>
+	<string>NO</string>
+	<key>MagnetsVisible</key>
+	<string>NO</string>
+	<key>MasterSheets</key>
+	<array/>
+	<key>ModificationDate</key>
+	<string>2014-12-25 05:46:50 +0000</string>
+	<key>Modifier</key>
+	<string>atton</string>
+	<key>NotesVisible</key>
+	<string>NO</string>
+	<key>Orientation</key>
+	<integer>2</integer>
+	<key>OriginVisible</key>
+	<string>NO</string>
+	<key>PageBreaks</key>
+	<string>YES</string>
+	<key>PrintInfo</key>
+	<dict>
+		<key>NSBottomMargin</key>
+		<array>
+			<string>float</string>
+			<string>41</string>
+		</array>
+		<key>NSHorizonalPagination</key>
+		<array>
+			<string>coded</string>
+			<string>BAtzdHJlYW10eXBlZIHoA4QBQISEhAhOU051bWJlcgCEhAdOU1ZhbHVlAISECE5TT2JqZWN0AIWEASqEhAFxlwCG</string>
+		</array>
+		<key>NSLeftMargin</key>
+		<array>
+			<string>float</string>
+			<string>18</string>
+		</array>
+		<key>NSPaperSize</key>
+		<array>
+			<string>size</string>
+			<string>{594.99997329711914, 842}</string>
+		</array>
+		<key>NSPrintReverseOrientation</key>
+		<array>
+			<string>int</string>
+			<string>0</string>
+		</array>
+		<key>NSRightMargin</key>
+		<array>
+			<string>float</string>
+			<string>18</string>
+		</array>
+		<key>NSTopMargin</key>
+		<array>
+			<string>float</string>
+			<string>18</string>
+		</array>
+	</dict>
+	<key>PrintOnePage</key>
+	<false/>
+	<key>ReadOnly</key>
+	<string>NO</string>
+	<key>RowAlign</key>
+	<integer>1</integer>
+	<key>RowSpacing</key>
+	<real>36</real>
+	<key>SheetTitle</key>
+	<string>Canvas 1</string>
+	<key>SmartAlignmentGuidesActive</key>
+	<string>YES</string>
+	<key>SmartDistanceGuidesActive</key>
+	<string>YES</string>
+	<key>UniqueID</key>
+	<integer>1</integer>
+	<key>UseEntirePage</key>
+	<false/>
+	<key>VPages</key>
+	<integer>1</integer>
+	<key>WindowInfo</key>
+	<dict>
+		<key>CurrentSheet</key>
+		<integer>0</integer>
+		<key>ExpandedCanvases</key>
+		<array>
+			<dict>
+				<key>name</key>
+				<string>Canvas 1</string>
+			</dict>
+		</array>
+		<key>Frame</key>
+		<string>{{373, 4}, {892, 874}}</string>
+		<key>ListView</key>
+		<true/>
+		<key>OutlineWidth</key>
+		<integer>142</integer>
+		<key>RightSidebar</key>
+		<false/>
+		<key>ShowRuler</key>
+		<true/>
+		<key>Sidebar</key>
+		<true/>
+		<key>SidebarWidth</key>
+		<integer>120</integer>
+		<key>VisibleRegion</key>
+		<string>{{-99, 0}, {757, 735}}</string>
+		<key>Zoom</key>
+		<real>1</real>
+		<key>ZoomValues</key>
+		<array>
+			<array>
+				<string>Canvas 1</string>
+				<real>1</real>
+				<real>1</real>
+			</array>
+		</array>
+	</dict>
+</dict>
+</plist>
Binary file figure/apply_function.pdf has changed
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/figure/apply_function.xbb	Thu Dec 25 14:52:33 2014 +0900
@@ -0,0 +1,8 @@
+%%Title: ./apply_function.pdf
+%%Creator: extractbb 20130405
+%%BoundingBox: 0 0 538 280
+%%HiResBoundingBox: 0.000000 0.000000 538.000000 280.000000
+%%PDFVersion: 1.3
+%%Pages: 1
+%%CreationDate: Thu Dec 25 14:47:27 2014
+
Binary file sigse.pdf has changed
--- a/sigse.tex	Thu Dec 25 14:05:51 2014 +0900
+++ b/sigse.tex	Thu Dec 25 14:52:33 2014 +0900
@@ -38,7 +38,7 @@
         Shinji Kono\affiref{ie-ryukyu}}
 
 \begin{abstract}
-大学の講義や卒業研究を通して形式手法を学んだ一年ほどの過程やつまづきと，それらに基づいた形式手法を広めるための意見を述べます。
+大学の講義や卒業研究を通して形式手法を学んだ一年ほどの過程やつまづき，それらに基づいた形式手法を広めるための意見を述べます．
 \end{abstract}
 
 % 表題などの出力
@@ -67,9 +67,20 @@
 Delta を用いたプログラムでは，変数は必ず Delta で記述し，関数は必ず Delta を返すようにします．
 プログラムへの変更は，関数か変数のバージョンアップとして表現し，Delta への新しい値の追加として記述します．
 そうすることにより，任意の変数もしくは関数は必ず1つ以上のバージョンを持ちます．
+
 次にバージョンを持った関数の適用を考えます．
-関数の適用は通常の関数の適用と違い，変数と関数のバージョンの違いを考慮したルールに基づいて適用しなくてはなりません．
-例えば，バージョンが3つある変数に対してバージョンが2つある関数を適用する場合，結果の変数のバージョンは2つにするか3つにするか決める必要があります．
+関数の適用は変数と関数のバージョンの違いを考慮したルールに基づいて適用しなくてはなりません．
+例えば，バージョンが1つある変数に対してバージョンが3つある関数と2つある関数を適用する場合，結果の変数のバージョンがいくつになるか決めなくては適用することができません(図\ref{apply_function})．
+
+\begin{figure}[htbp]
+    \begin{center}
+        \includegraphics[scale=0.4]{./figure/apply_function.pdf}
+        \caption{関数適用時のバージョンの組み合せ例}
+        \label{apply_function}
+    \end{center}
+\end{figure}
+
+
 変数と関数のバージョンの組み合せのルールは Monad を用いて記述しました．
 Haskell における Monad はデータ構造とメタな計算との対応付けです\cite{moggi}．
 メタ計算としてバージョンの組み合せの解決を $ \mu $ と $ \eta $ により定義します．