Chapter 4: External Argument and Comparisons

Co-authored-by: Dibyendu Majumdar <mobile@majumdar.org.uk>
Co-authored-by: Yichen Yan <oraluben@outlook.com>
Co-authored-by: Thalia Archibald <thalia@archibald.dev>
Co-authored-by: Tamas Hadhazy <helsgithub15@gmail.com>
Co-authored-by: Hels15 <helsgithub15@gmail.com>
Co-authored-by: XmiliaH <45106915+XmiliaH@users.noreply.github.com>
Co-authored-by: tannal <tannal2409@gmail.com>
Co-authored-by: hadha <helsgithub15@gmail.com>
Co-authored-by: Tamás Hadházy <85063808+Hels15@users.noreply.github.com>

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+<text text-anchor="middle" x="42.8484" y="-126.4265" font-family="Helvetica,sans-Serif" font-size="8.00" fill="#000000">0</text>
+</g>
+<!-- Return7 -->
+<g id="node3" class="node">
+<title>Return7</title>
+<polygon fill="#ffff00" stroke="#000000" points="80.1025,-52 25.8975,-52 25.8975,-16 80.1025,-16 80.1025,-52"/>
+<text text-anchor="middle" x="53" y="-29.8" font-family="Times,serif" font-size="14.00" fill="#000000">Return</text>
+</g>
+<!-- Return7&#45;&gt;Start2 -->
+<g id="edge2" class="edge">
+<title>Return7-&gt;Start2:p0</title>
+<path fill="none" stroke="#ff0000" d="M64.3196,-52.2322C69.9396,-62.3585 76.1294,-75.4211 79,-88 82.5597,-103.599 83.0135,-108.5115 79,-124 74.2474,-142.3411 60.2215,-145.475 55.5405,-159.004"/>
+<polygon fill="#ff0000" stroke="#ff0000" points="52.0641,-158.5835 54,-169 58.9824,-159.6498 52.0641,-158.5835"/>
+<text text-anchor="middle" x="62.096" y="-54.6322" font-family="Helvetica,sans-Serif" font-size="8.00" fill="#000000">0</text>
+</g>
+<!-- Return7&#45;&gt;Con_6 -->
+<g id="edge3" class="edge">
+<title>Return7-&gt;Con_6</title>
+<path fill="none" stroke="#000000" d="M50.4766,-52.1686C49.4071,-59.869 48.1353,-69.0257 46.9468,-77.5834"/>
+<polygon fill="#000000" stroke="#000000" points="43.4664,-77.2003 45.5574,-87.5867 50.3999,-78.1633 43.4664,-77.2003"/>
+<text text-anchor="middle" x="48.253" y="-54.5686" font-family="Helvetica,sans-Serif" font-size="8.00" fill="#000000">1</text>
+</g>
+</g>
+</svg>
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diff --git a/docs/chapter04/04-dce3.svg b/docs/chapter04/04-dce3.svg
new file mode 100644
index 00000000..86fa6f0d
--- /dev/null
+++ b/docs/chapter04/04-dce3.svg
@@ -0,0 +1,116 @@
+<?xml version="1.0" encoding="UTF-8" standalone="no"?>
+<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
+<!-- Generated by graphviz version 2.40.1 (20161225.0304)
+ -->
+<!-- Title: chapter04 Pages: 1 -->
+<svg width="458pt" height="178pt" viewBox="0.00 0.00 458.00 178.00" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
+<g id="graph0" class="graph" transform="scale(1 1) rotate(0) translate(4 174)">
+<title>chapter04</title>
+<polygon fill="#ffffff" stroke="transparent" points="-4,4 -4,-174 454,-174 454,4 -4,4"/>
+<g id="clust1" class="cluster">
+<title>cluster_Nodes</title>
+<polygon fill="none" stroke="#000000" points="8,-8 8,-162 442,-162 442,-8 8,-8"/>
+</g>
+<!-- Start2 -->
+<g id="node1" class="node">
+<title>Start2</title>
+<polygon fill="#ffff00" stroke="transparent" points="25,-124 25,-150 66,-150 66,-124 25,-124"/>
+<polygon fill="none" stroke="#000000" points="25,-124 25,-150 66,-150 66,-124 25,-124"/>
+<text text-anchor="start" x="32.2805" y="-132.4" font-family="Times,serif" font-size="14.00" fill="#000000">Start</text>
+<polygon fill="none" stroke="#000000" points="25,-92 25,-124 66,-124 66,-92 25,-92"/>
+<polygon fill="#ffff00" stroke="transparent" points="30,-97 30,-119 61,-119 61,-97 30,-97"/>
+<polygon fill="none" stroke="#000000" points="30,-97 30,-119 61,-119 61,-97 30,-97"/>
+<text text-anchor="start" x="32.6718" y="-103.4" font-family="Times,serif" font-size="14.00" fill="#000000">$ctrl</text>
+</g>
+<!-- Con_5 -->
+<g id="node2" class="node">
+<title>Con_5</title>
+<ellipse fill="none" stroke="#000000" cx="335" cy="-121" rx="27" ry="18"/>
+<text text-anchor="middle" x="335" y="-116.8" font-family="Times,serif" font-size="14.00" fill="#000000">2</text>
+</g>
+<!-- Con_6 -->
+<g id="node3" class="node">
+<title>Con_6</title>
+<ellipse fill="none" stroke="#000000" cx="407" cy="-121" rx="27" ry="18"/>
+<text text-anchor="middle" x="407" y="-116.8" font-family="Times,serif" font-size="14.00" fill="#000000">1</text>
+</g>
+<!-- Add7 -->
+<g id="node4" class="node">
+<title>Add7</title>
+<ellipse fill="none" stroke="#000000" cx="371" cy="-34" rx="27" ry="18"/>
+<text text-anchor="middle" x="371" y="-29.8" font-family="Times,serif" font-size="14.00" fill="#000000">+</text>
+</g>
+<!-- Add7&#45;&gt;Con_5 -->
+<g id="edge1" class="edge">
+<title>Add7-&gt;Con_5</title>
+<path fill="none" stroke="#000000" d="M363.7149,-51.6057C358.6889,-63.7518 351.9107,-80.1326 346.2018,-93.9289"/>
+<polygon fill="#000000" stroke="#000000" points="342.8638,-92.842 342.2743,-103.4204 349.3319,-95.5185 342.8638,-92.842"/>
+<text text-anchor="middle" x="361.4913" y="-54.0057" font-family="Helvetica,sans-Serif" font-size="8.00" fill="#000000">1</text>
+</g>
+<!-- Add7&#45;&gt;Con_6 -->
+<g id="edge2" class="edge">
+<title>Add7-&gt;Con_6</title>
+<path fill="none" stroke="#000000" d="M378.2851,-51.6057C383.3111,-63.7518 390.0893,-80.1326 395.7982,-93.9289"/>
+<polygon fill="#000000" stroke="#000000" points="392.6681,-95.5185 399.7257,-103.4204 399.1362,-92.842 392.6681,-95.5185"/>
+<text text-anchor="middle" x="376.0615" y="-54.0057" font-family="Helvetica,sans-Serif" font-size="8.00" fill="#000000">2</text>
+</g>
+<!-- Con_8 -->
+<g id="node5" class="node">
+<title>Con_8</title>
+<ellipse fill="none" stroke="#000000" cx="191" cy="-121" rx="27" ry="18"/>
+<text text-anchor="middle" x="191" y="-116.8" font-family="Times,serif" font-size="14.00" fill="#000000">3</text>
+</g>
+<!-- Con_9 -->
+<g id="node6" class="node">
+<title>Con_9</title>
+<ellipse fill="none" stroke="#000000" cx="263" cy="-121" rx="27" ry="18"/>
+<text text-anchor="middle" x="263" y="-116.8" font-family="Times,serif" font-size="14.00" fill="#000000">3</text>
+</g>
+<!-- Add10 -->
+<g id="node7" class="node">
+<title>Add10</title>
+<ellipse fill="none" stroke="#000000" cx="227" cy="-34" rx="27" ry="18"/>
+<text text-anchor="middle" x="227" y="-29.8" font-family="Times,serif" font-size="14.00" fill="#000000">+</text>
+</g>
+<!-- Add10&#45;&gt;Con_8 -->
+<g id="edge3" class="edge">
+<title>Add10-&gt;Con_8</title>
+<path fill="none" stroke="#000000" d="M219.7149,-51.6057C214.6889,-63.7518 207.9107,-80.1326 202.2018,-93.9289"/>
+<polygon fill="#000000" stroke="#000000" points="198.8638,-92.842 198.2743,-103.4204 205.3319,-95.5185 198.8638,-92.842"/>
+<text text-anchor="middle" x="217.4913" y="-54.0057" font-family="Helvetica,sans-Serif" font-size="8.00" fill="#000000">1</text>
+</g>
+<!-- Add10&#45;&gt;Con_9 -->
+<g id="edge4" class="edge">
+<title>Add10-&gt;Con_9</title>
+<path fill="none" stroke="#000000" d="M234.2851,-51.6057C239.3111,-63.7518 246.0893,-80.1326 251.7982,-93.9289"/>
+<polygon fill="#000000" stroke="#000000" points="248.6681,-95.5185 255.7257,-103.4204 255.1362,-92.842 248.6681,-95.5185"/>
+<text text-anchor="middle" x="232.0615" y="-54.0057" font-family="Helvetica,sans-Serif" font-size="8.00" fill="#000000">2</text>
+</g>
+<!-- Con_11 -->
+<g id="node8" class="node">
+<title>Con_11</title>
+<ellipse fill="none" stroke="#000000" cx="119" cy="-121" rx="27" ry="18"/>
+<text text-anchor="middle" x="119" y="-116.8" font-family="Times,serif" font-size="14.00" fill="#000000">6</text>
+</g>
+<!-- Return12 -->
+<g id="node9" class="node">
+<title>Return12</title>
+<polygon fill="#ffff00" stroke="#000000" points="109.1025,-52 54.8975,-52 54.8975,-16 109.1025,-16 109.1025,-52"/>
+<text text-anchor="middle" x="82" y="-29.8" font-family="Times,serif" font-size="14.00" fill="#000000">Return</text>
+</g>
+<!-- Return12&#45;&gt;Start2 -->
+<g id="edge5" class="edge">
+<title>Return12-&gt;Start2:p0</title>
+<path fill="none" stroke="#ff0000" d="M65.0542,-52.052C57.6135,-61.4899 49.9318,-73.7894 47.1181,-86.8747"/>
+<polygon fill="#ff0000" stroke="#ff0000" points="43.6189,-86.6762 46,-97 50.5766,-87.4446 43.6189,-86.6762"/>
+<text text-anchor="middle" x="62.8306" y="-54.452" font-family="Helvetica,sans-Serif" font-size="8.00" fill="#000000">0</text>
+</g>
+<!-- Return12&#45;&gt;Con_11 -->
+<g id="edge6" class="edge">
+<title>Return12-&gt;Con_11</title>
+<path fill="none" stroke="#000000" d="M89.6664,-52.0265C94.8444,-64.2017 101.7707,-80.4878 107.5926,-94.1772"/>
+<polygon fill="#000000" stroke="#000000" points="104.4613,-95.7577 111.5959,-103.5903 110.903,-93.0181 104.4613,-95.7577"/>
+<text text-anchor="middle" x="87.4428" y="-54.4265" font-family="Helvetica,sans-Serif" font-size="8.00" fill="#000000">1</text>
+</g>
+</g>
+</svg>
\ No newline at end of file
diff --git a/docs/chapter04/04-grammar.md b/docs/chapter04/04-grammar.md
new file mode 100644
index 00000000..48facd4c
--- /dev/null
+++ b/docs/chapter04/04-grammar.md
@@ -0,0 +1,76 @@
+# Grammar for Chapter 1
+
+```antlrv4
+grammar SimpleLanguage;
+
+program
+    : statement+ EOF
+    ;
+
+statement
+    : returnStatement
+    | declStatement
+    | blockStatment
+    | expressionStatement
+    | metaStatement
+    ;
+
+metaStatement
+    : '#showGraph' ';'
+    ;
+
+expressionStatement
+    : IDENTIFIER '=' expression ';'
+    ;
+
+blockStatement
+    : '{' statement+ '}'
+    ;
+
+declStatement
+    : 'int' IDENTIFIER '=' expression ';'
+    ;
+
+returnStatement
+    : 'return' expression ';'
+    ;
+
+expression
+    : comparisonExpression
+    ;
+
+comparisonExpression
+    : additiveExpression (('==' | '!='| '>'| '<'| '>='| '<=') additiveExpression)*
+    ;
+
+additiveExpression
+    : multiplicativeExpression (('+' | '-') multiplicativeExpression)*
+    ;
+
+multiplicativeExpression
+    : unaryExpression (('*' | '/') unaryExpression)*
+    ;
+
+unaryExpression
+    : ('-') unaryExpression
+    | primaryExpression
+    ;
+
+primaryExpression
+    : IDENTIFIER
+    | INTEGER_LITERAL
+    | '(' expression ')'
+    ;
+
+INTEGER_LITERAL
+    : [1-9][0-9]*
+    | [0]
+    ;
+
+IDENTIFIER
+    : NON_DIGIT (NON_DIGIT | DIGIT)*
+    ;
+
+NON_DIGIT: [a-zA-Z_];
+DEC_DIGIT: [0-9];
+```
diff --git a/docs/chapter04/04-post-peephole.gv b/docs/chapter04/04-post-peephole.gv
new file mode 100644
index 00000000..0a9f8e93
--- /dev/null
+++ b/docs/chapter04/04-post-peephole.gv
@@ -0,0 +1,43 @@
+digraph chapter04 {
+/*
+return 1+arg+2; #showGraph;
+*/
+	rankdir=BT;
+	ordering="in";
+	concentrate="true";
+	subgraph cluster_Nodes {
+		Start2 [ shape=plaintext label=<
+			<TABLE BORDER="0" CELLBORDER="1" CELLSPACING="0" CELLPADDING="4">
+			<TR><TD BGCOLOR="yellow">Start</TD></TR>
+			<TR><TD>
+				<TABLE BORDER="0" CELLBORDER="1" CELLSPACING="0">
+				<TR><TD PORT="p0" BGCOLOR="yellow">$ctrl</TD><TD PORT="p1">arg</TD></TR>
+				</TABLE>
+			</TD></TR>
+			</TABLE>>
+		];
+		Con_10 [ label="#3" ];
+		Add11 [ label="+" ];
+		Return12 [ shape=box style=filled fillcolor=yellow label="Return" ];
+	}
+	node [shape=plaintext];
+	subgraph cluster_Scope1_0 {
+		Scope1_0 [label=<
+			<TABLE BORDER="0" CELLBORDER="1" CELLSPACING="0">
+			<TR><TD BGCOLOR="cyan">0</TD><TD PORT="Scope1_0_$ctrl">$ctrl</TD><TD PORT="Scope1_0_arg">arg</TD></TR>
+			</TABLE>>];
+	subgraph cluster_Scope1_1 {
+		Scope1_1 [label=<
+			<TABLE BORDER="0" CELLBORDER="1" CELLSPACING="0">
+			<TR><TD BGCOLOR="cyan">1</TD></TR>
+			</TABLE>>];
+	}
+	}
+	edge [ fontname=Helvetica, fontsize=8 ];
+	Add11 -> Start2:p1[taillabel=1];
+	Add11 -> Con_10[taillabel=2];
+	Return12 -> Start2:p0[taillabel=0; color=red];
+	Return12 -> Add11[taillabel=1];
+	edge [style=dashed color=cornflowerblue];
+	Scope1_0:"Scope1_0_arg" -> Start2:p1;
+}
diff --git a/docs/chapter04/04-post-peephole.svg b/docs/chapter04/04-post-peephole.svg
new file mode 100644
index 00000000..550c16ef
--- /dev/null
+++ b/docs/chapter04/04-post-peephole.svg
@@ -0,0 +1,108 @@
+<?xml version="1.0" encoding="UTF-8" standalone="no"?>
+<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN"
+ "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
+<!-- Generated by graphviz version 2.43.0 (0)
+ -->
+<!-- Title: chapter04 Pages: 1 -->
+<svg width="417pt" height="248pt"
+ viewBox="0.00 0.00 417.00 248.00" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
+<g id="graph0" class="graph" transform="scale(1 1) rotate(0) translate(4 244)">
+<title>chapter04</title>
+<polygon fill="white" stroke="transparent" points="-4,4 -4,-244 413,-244 413,4 -4,4"/>
+<g id="clust1" class="cluster">
+<title>cluster_Nodes</title>
+<polygon fill="none" stroke="black" points="8,-8 8,-232 193,-232 193,-8 8,-8"/>
+</g>
+<g id="clust2" class="cluster">
+<title>cluster_Scope1_0</title>
+<polygon fill="none" stroke="black" points="201,-72 201,-140 401,-140 401,-72 201,-72"/>
+</g>
+<g id="clust3" class="cluster">
+<title>cluster_Scope1_1</title>
+<polygon fill="none" stroke="black" points="323,-80 323,-132 393,-132 393,-80 323,-80"/>
+</g>
+<!-- Start2 -->
+<g id="node1" class="node">
+<title>Start2</title>
+<polygon fill="yellow" stroke="transparent" points="97,-195 97,-220 178,-220 178,-195 97,-195"/>
+<polygon fill="none" stroke="black" points="97,-195 97,-220 178,-220 178,-195 97,-195"/>
+<text text-anchor="start" x="119" y="-203.8" font-family="Times,serif" font-size="14.00">Start</text>
+<polygon fill="none" stroke="black" points="97,-164 97,-195 178,-195 178,-164 97,-164"/>
+<polygon fill="yellow" stroke="transparent" points="102,-169 102,-190 142,-190 142,-169 102,-169"/>
+<polygon fill="none" stroke="black" points="102,-169 102,-190 142,-190 142,-169 102,-169"/>
+<text text-anchor="start" x="105" y="-175.8" font-family="Times,serif" font-size="14.00">$ctrl</text>
+<polygon fill="none" stroke="black" points="142,-169 142,-190 173,-190 173,-169 142,-169"/>
+<text text-anchor="start" x="145" y="-175.8" font-family="Times,serif" font-size="14.00">arg</text>
+</g>
+<!-- Con_10 -->
+<g id="node2" class="node">
+<title>Con_10</title>
+<ellipse fill="none" stroke="black" cx="43" cy="-192" rx="27" ry="18"/>
+<text text-anchor="middle" x="43" y="-188.3" font-family="Times,serif" font-size="14.00">#3</text>
+</g>
+<!-- Add11 -->
+<g id="node3" class="node">
+<title>Add11</title>
+<ellipse fill="none" stroke="black" cx="158" cy="-106" rx="27" ry="18"/>
+<text text-anchor="middle" x="158" y="-102.3" font-family="Times,serif" font-size="14.00">+</text>
+</g>
+<!-- Add11&#45;&gt;Start2 -->
+<g id="edge1" class="edge">
+<title>Add11&#45;&gt;Start2:p1</title>
+<path fill="none" stroke="black" d="M158,-124.09C158,-133.55 158,-145.77 158,-157.69"/>
+<polygon fill="black" stroke="black" points="154.5,-158 158,-168 161.5,-158 154.5,-158"/>
+<text text-anchor="middle" x="155" y="-126.69" font-family="Helvetica,sans-Serif" font-size="8.00">1</text>
+</g>
+<!-- Add11&#45;&gt;Con_10 -->
+<g id="edge2" class="edge">
+<title>Add11&#45;&gt;Con_10</title>
+<path fill="none" stroke="black" d="M139.03,-119.12C123,-129.54 99.59,-145.19 80,-160 75.68,-163.26 71.19,-166.85 66.88,-170.4"/>
+<polygon fill="black" stroke="black" points="64.34,-167.96 58.92,-177.06 68.84,-173.32 64.34,-167.96"/>
+<text text-anchor="middle" x="136.03" y="-121.72" font-family="Helvetica,sans-Serif" font-size="8.00">2</text>
+</g>
+<!-- Return12 -->
+<g id="node4" class="node">
+<title>Return12</title>
+<polygon fill="yellow" stroke="black" points="165,-52 99,-52 99,-16 165,-16 165,-52"/>
+<text text-anchor="middle" x="132" y="-30.3" font-family="Times,serif" font-size="14.00">Return</text>
+</g>
+<!-- Return12&#45;&gt;Start2 -->
+<g id="edge3" class="edge">
+<title>Return12&#45;&gt;Start2:p0</title>
+<path fill="none" stroke="red" d="M126.36,-52.13C124.59,-58.32 122.88,-65.4 122,-72 116.87,-110.66 121.12,-123.21 121.89,-157.66"/>
+<polygon fill="red" stroke="red" points="118.39,-158.04 122,-168 125.39,-157.96 118.39,-158.04"/>
+<text text-anchor="middle" x="123.36" y="-54.73" font-family="Helvetica,sans-Serif" font-size="8.00">0</text>
+</g>
+<!-- Return12&#45;&gt;Add11 -->
+<g id="edge4" class="edge">
+<title>Return12&#45;&gt;Add11</title>
+<path fill="none" stroke="black" d="M138.43,-52.3C141.4,-60.3 144.99,-69.98 148.29,-78.85"/>
+<polygon fill="black" stroke="black" points="145.06,-80.22 151.83,-88.38 151.63,-77.78 145.06,-80.22"/>
+<text text-anchor="middle" x="135.43" y="-54.9" font-family="Helvetica,sans-Serif" font-size="8.00">1</text>
+</g>
+<!-- Scope1_0 -->
+<g id="node5" class="node">
+<title>Scope1_0</title>
+<polygon fill="cyan" stroke="transparent" points="218,-95 218,-116 234,-116 234,-95 218,-95"/>
+<polygon fill="none" stroke="black" points="218,-95 218,-116 234,-116 234,-95 218,-95"/>
+<text text-anchor="start" x="221" y="-101.8" font-family="Times,serif" font-size="14.00">0</text>
+<polygon fill="none" stroke="black" points="234,-95 234,-116 274,-116 274,-95 234,-95"/>
+<text text-anchor="start" x="237" y="-101.8" font-family="Times,serif" font-size="14.00">$ctrl</text>
+<polygon fill="none" stroke="black" points="274,-95 274,-116 305,-116 305,-95 274,-95"/>
+<text text-anchor="start" x="277" y="-101.8" font-family="Times,serif" font-size="14.00">arg</text>
+</g>
+<!-- Scope1_0&#45;&gt;Start2 -->
+<g id="edge5" class="edge">
+<title>Scope1_0:Scope1_0_arg&#45;&gt;Start2:p1</title>
+<path fill="none" stroke="cornflowerblue" stroke-dasharray="5,2" d="M290,-117C290,-172.03 239,-178.4 184.25,-178.95"/>
+<polygon fill="cornflowerblue" stroke="cornflowerblue" points="183.98,-175.45 174,-179 184.02,-182.45 183.98,-175.45"/>
+</g>
+<!-- Scope1_1 -->
+<g id="node6" class="node">
+<title>Scope1_1</title>
+<polygon fill="cyan" stroke="transparent" points="350,-95 350,-116 366,-116 366,-95 350,-95"/>
+<polygon fill="none" stroke="black" points="350,-95 350,-116 366,-116 366,-95 350,-95"/>
+<text text-anchor="start" x="353" y="-101.8" font-family="Times,serif" font-size="14.00">1</text>
+</g>
+</g>
+</svg>
diff --git a/docs/chapter04/04-pre-peephole.gv b/docs/chapter04/04-pre-peephole.gv
new file mode 100644
index 00000000..aefd81c4
--- /dev/null
+++ b/docs/chapter04/04-pre-peephole.gv
@@ -0,0 +1,47 @@
+digraph chapter04 {
+/*
+return 1+arg+2; #showGraph;
+*/
+	rankdir=BT;
+	ordering="in";
+	concentrate="true";
+	subgraph cluster_Nodes {
+		Start2 [ shape=plaintext label=<
+			<TABLE BORDER="0" CELLBORDER="1" CELLSPACING="0" CELLPADDING="4">
+			<TR><TD BGCOLOR="yellow">Start</TD></TR>
+			<TR><TD>
+				<TABLE BORDER="0" CELLBORDER="1" CELLSPACING="0">
+				<TR><TD PORT="p0" BGCOLOR="yellow">$ctrl</TD><TD PORT="p1">arg</TD></TR>
+				</TABLE>
+			</TD></TR>
+			</TABLE>>
+		];
+		Con_5 [ label="#1" ];
+		Con_6 [ label="#2" ];
+		Add7 [ label="+" ];
+		Add8 [ label="+" ];
+		Return9 [ shape=box style=filled fillcolor=yellow label="Return" ];
+	}
+	node [shape=plaintext];
+	subgraph cluster_Scope1_0 {
+		Scope1_0 [label=<
+			<TABLE BORDER="0" CELLBORDER="1" CELLSPACING="0">
+			<TR><TD BGCOLOR="cyan">0</TD><TD PORT="Scope1_0_$ctrl">$ctrl</TD><TD PORT="Scope1_0_arg">arg</TD></TR>
+			</TABLE>>];
+	subgraph cluster_Scope1_1 {
+		Scope1_1 [label=<
+			<TABLE BORDER="0" CELLBORDER="1" CELLSPACING="0">
+			<TR><TD BGCOLOR="cyan">1</TD></TR>
+			</TABLE>>];
+	}
+	}
+	edge [ fontname=Helvetica, fontsize=8 ];
+	Add7 -> Start2:p1[taillabel=1];
+	Add7 -> Con_6[taillabel=2];
+	Add8 -> Con_5[taillabel=1];
+	Add8 -> Add7[taillabel=2];
+	Return9 -> Start2:p0[taillabel=0; color=red];
+	Return9 -> Add8[taillabel=1];
+	edge [style=dashed color=cornflowerblue];
+	Scope1_0:"Scope1_0_arg" -> Start2:p1;
+}
diff --git a/docs/chapter04/04-pre-peephole.svg b/docs/chapter04/04-pre-peephole.svg
new file mode 100644
index 00000000..9ec38d27
--- /dev/null
+++ b/docs/chapter04/04-pre-peephole.svg
@@ -0,0 +1,134 @@
+<?xml version="1.0" encoding="UTF-8" standalone="no"?>
+<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN"
+ "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
+<!-- Generated by graphviz version 2.43.0 (0)
+ -->
+<!-- Title: chapter04 Pages: 1 -->
+<svg width="417pt" height="320pt"
+ viewBox="0.00 0.00 417.00 320.00" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
+<g id="graph0" class="graph" transform="scale(1 1) rotate(0) translate(4 316)">
+<title>chapter04</title>
+<polygon fill="white" stroke="transparent" points="-4,4 -4,-316 413,-316 413,4 -4,4"/>
+<g id="clust1" class="cluster">
+<title>cluster_Nodes</title>
+<polygon fill="none" stroke="black" points="8,-8 8,-304 193,-304 193,-8 8,-8"/>
+</g>
+<g id="clust2" class="cluster">
+<title>cluster_Scope1_0</title>
+<polygon fill="none" stroke="black" points="201,-144 201,-212 401,-212 401,-144 201,-144"/>
+</g>
+<g id="clust3" class="cluster">
+<title>cluster_Scope1_1</title>
+<polygon fill="none" stroke="black" points="323,-152 323,-204 393,-204 393,-152 323,-152"/>
+</g>
+<!-- Start2 -->
+<g id="node1" class="node">
+<title>Start2</title>
+<polygon fill="yellow" stroke="transparent" points="97,-267 97,-292 178,-292 178,-267 97,-267"/>
+<polygon fill="none" stroke="black" points="97,-267 97,-292 178,-292 178,-267 97,-267"/>
+<text text-anchor="start" x="119" y="-275.8" font-family="Times,serif" font-size="14.00">Start</text>
+<polygon fill="none" stroke="black" points="97,-236 97,-267 178,-267 178,-236 97,-236"/>
+<polygon fill="yellow" stroke="transparent" points="102,-241 102,-262 142,-262 142,-241 102,-241"/>
+<polygon fill="none" stroke="black" points="102,-241 102,-262 142,-262 142,-241 102,-241"/>
+<text text-anchor="start" x="105" y="-247.8" font-family="Times,serif" font-size="14.00">$ctrl</text>
+<polygon fill="none" stroke="black" points="142,-241 142,-262 173,-262 173,-241 142,-241"/>
+<text text-anchor="start" x="145" y="-247.8" font-family="Times,serif" font-size="14.00">arg</text>
+</g>
+<!-- Con_5 -->
+<g id="node2" class="node">
+<title>Con_5</title>
+<ellipse fill="none" stroke="black" cx="115" cy="-178" rx="27" ry="18"/>
+<text text-anchor="middle" x="115" y="-174.3" font-family="Times,serif" font-size="14.00">#1</text>
+</g>
+<!-- Con_6 -->
+<g id="node3" class="node">
+<title>Con_6</title>
+<ellipse fill="none" stroke="black" cx="43" cy="-264" rx="27" ry="18"/>
+<text text-anchor="middle" x="43" y="-260.3" font-family="Times,serif" font-size="14.00">#2</text>
+</g>
+<!-- Add7 -->
+<g id="node4" class="node">
+<title>Add7</title>
+<ellipse fill="none" stroke="black" cx="43" cy="-178" rx="27" ry="18"/>
+<text text-anchor="middle" x="43" y="-174.3" font-family="Times,serif" font-size="14.00">+</text>
+</g>
+<!-- Add7&#45;&gt;Start2 -->
+<g id="edge1" class="edge">
+<title>Add7&#45;&gt;Start2:p1</title>
+<path fill="none" stroke="black" d="M56.63,-193.93C62.95,-200.18 70.86,-207.11 79,-212 107.43,-229.09 147.88,-208.4 156.39,-229.99"/>
+<polygon fill="black" stroke="black" points="152.96,-230.68 158,-240 159.87,-229.57 152.96,-230.68"/>
+<text text-anchor="middle" x="53.63" y="-196.53" font-family="Helvetica,sans-Serif" font-size="8.00">1</text>
+</g>
+<!-- Add7&#45;&gt;Con_6 -->
+<g id="edge2" class="edge">
+<title>Add7&#45;&gt;Con_6</title>
+<path fill="none" stroke="black" d="M43,-196.4C43,-207.74 43,-222.77 43,-235.68"/>
+<polygon fill="black" stroke="black" points="39.5,-235.9 43,-245.9 46.5,-235.9 39.5,-235.9"/>
+<text text-anchor="middle" x="40" y="-199" font-family="Helvetica,sans-Serif" font-size="8.00">2</text>
+</g>
+<!-- Add8 -->
+<g id="node5" class="node">
+<title>Add8</title>
+<ellipse fill="none" stroke="black" cx="115" cy="-106" rx="27" ry="18"/>
+<text text-anchor="middle" x="115" y="-102.3" font-family="Times,serif" font-size="14.00">+</text>
+</g>
+<!-- Add8&#45;&gt;Con_5 -->
+<g id="edge3" class="edge">
+<title>Add8&#45;&gt;Con_5</title>
+<path fill="none" stroke="black" d="M115,-124.3C115,-132.02 115,-141.29 115,-149.89"/>
+<polygon fill="black" stroke="black" points="111.5,-149.9 115,-159.9 118.5,-149.9 111.5,-149.9"/>
+<text text-anchor="middle" x="112" y="-126.9" font-family="Helvetica,sans-Serif" font-size="8.00">1</text>
+</g>
+<!-- Add8&#45;&gt;Add7 -->
+<g id="edge4" class="edge">
+<title>Add8&#45;&gt;Add7</title>
+<path fill="none" stroke="black" d="M100.43,-121.17C90.25,-131.06 76.48,-144.45 64.97,-155.64"/>
+<polygon fill="black" stroke="black" points="62.53,-153.13 57.8,-162.62 67.41,-158.15 62.53,-153.13"/>
+<text text-anchor="middle" x="97.43" y="-123.77" font-family="Helvetica,sans-Serif" font-size="8.00">2</text>
+</g>
+<!-- Return9 -->
+<g id="node6" class="node">
+<title>Return9</title>
+<polygon fill="yellow" stroke="black" points="166,-52 100,-52 100,-16 166,-16 166,-52"/>
+<text text-anchor="middle" x="133" y="-30.3" font-family="Times,serif" font-size="14.00">Return</text>
+</g>
+<!-- Return9&#45;&gt;Start2 -->
+<g id="edge5" class="edge">
+<title>Return9&#45;&gt;Start2:p0</title>
+<path fill="none" stroke="red" d="M140.48,-52.3C144.47,-62.48 148.97,-75.73 151,-88 159.98,-142.37 172.54,-161.27 151,-212 145.75,-224.37 132.06,-223.86 125.55,-230.39"/>
+<polygon fill="red" stroke="red" points="122.18,-229.41 122,-240 128.74,-231.83 122.18,-229.41"/>
+<text text-anchor="middle" x="137.48" y="-54.9" font-family="Helvetica,sans-Serif" font-size="8.00">0</text>
+</g>
+<!-- Return9&#45;&gt;Add8 -->
+<g id="edge6" class="edge">
+<title>Return9&#45;&gt;Add8</title>
+<path fill="none" stroke="black" d="M128.55,-52.3C126.55,-60.1 124.13,-69.49 121.9,-78.17"/>
+<polygon fill="black" stroke="black" points="118.5,-77.34 119.4,-87.9 125.28,-79.08 118.5,-77.34"/>
+<text text-anchor="middle" x="125.55" y="-54.9" font-family="Helvetica,sans-Serif" font-size="8.00">1</text>
+</g>
+<!-- Scope1_0 -->
+<g id="node7" class="node">
+<title>Scope1_0</title>
+<polygon fill="cyan" stroke="transparent" points="218,-167 218,-188 234,-188 234,-167 218,-167"/>
+<polygon fill="none" stroke="black" points="218,-167 218,-188 234,-188 234,-167 218,-167"/>
+<text text-anchor="start" x="221" y="-173.8" font-family="Times,serif" font-size="14.00">0</text>
+<polygon fill="none" stroke="black" points="234,-167 234,-188 274,-188 274,-167 234,-167"/>
+<text text-anchor="start" x="237" y="-173.8" font-family="Times,serif" font-size="14.00">$ctrl</text>
+<polygon fill="none" stroke="black" points="274,-167 274,-188 305,-188 305,-167 274,-167"/>
+<text text-anchor="start" x="277" y="-173.8" font-family="Times,serif" font-size="14.00">arg</text>
+</g>
+<!-- Scope1_0&#45;&gt;Start2 -->
+<g id="edge7" class="edge">
+<title>Scope1_0:Scope1_0_arg&#45;&gt;Start2:p1</title>
+<path fill="none" stroke="cornflowerblue" stroke-dasharray="5,2" d="M290,-189C290,-244.03 239,-250.4 184.25,-250.95"/>
+<polygon fill="cornflowerblue" stroke="cornflowerblue" points="183.98,-247.45 174,-251 184.02,-254.45 183.98,-247.45"/>
+</g>
+<!-- Scope1_1 -->
+<g id="node8" class="node">
+<title>Scope1_1</title>
+<polygon fill="cyan" stroke="transparent" points="350,-167 350,-188 366,-188 366,-167 350,-167"/>
+<polygon fill="none" stroke="black" points="350,-167 350,-188 366,-188 366,-167 350,-167"/>
+<text text-anchor="start" x="353" y="-173.8" font-family="Times,serif" font-size="14.00">1</text>
+</g>
+</g>
+</svg>
diff --git a/docs/chapter04/Makefile b/docs/chapter04/Makefile
new file mode 100644
index 00000000..12659b8f
--- /dev/null
+++ b/docs/chapter04/Makefile
@@ -0,0 +1,12 @@
+GV := $(wildcard *.gv)
+SVG := $(patsubst %.gv,%.svg,$(GV))
+
+.PHONY: all
+all: $(SVG)
+
+%.svg: %.gv
+	dot -Tsvg $< > $@
+
+.PHONY: clean
+clean:
+	@rm -f $(SVG)
diff --git a/docs/chapter04/README.md b/docs/chapter04/README.md
new file mode 100644
index 00000000..2e6cc580
--- /dev/null
+++ b/docs/chapter04/README.md
@@ -0,0 +1,430 @@
+# Chapter 4: External Argument and Comparisons
+
+# Table of Contents
+
+1. [Extensions to Intermediate Representation](#extensions-to-intermediate-representation)
+2. [Projection Nodes](#projection-nodes)
+3. [Visualisation](#visualisation)
+5. [Initial values](#initial-values)
+6. [When Is Control Not-Null?](#when-is-control-not-null)
+5. [Note Regarding Visualizations](#note-regarding-visualizations)
+6. [Changes to Type System](#changes-to-type-system)
+7. [Tuple Types](#tuple-types)
+8. [$ctrl name binding](#ctrl-name-binding)
+9. [More Peephole Optimizations](#more-peephole-optimizations)
+10. [Peephole Walkthrough](#peephole-walkthrough)
+11. [Dead Code Elimination(DCE)](#dead-code-eliminationdce)
+
+You can also read [this chapter](https://github.com/SeaOfNodes/Simple/tree/linear-chapter04) in a linear Git revision history on the [linear](https://github.com/SeaOfNodes/Simple/tree/linear) branch and [compare](https://github.com/SeaOfNodes/Simple/compare/linear-chapter03...linear-chapter04) it to the previous chapter.
+
+
+In this chapter we extend the language grammar with following features:
+
+* The program receives a single argument named `arg` of integer type from external environment.
+* Expressions support comparison operators.
+* We introduce a scope sensitive name binding `$ctrl` for the current incoming
+  control node.  Thus, our Return is no longer hard-wired to Start, instead it
+  tracks the in scope `$ctrl` binding.
+
+Here is the [complete language grammar](04-grammar.md) for this chapter.
+
+
+
+## Extensions to Intermediate Representation
+
+In this chapter we add some new nodes and revise some existing nodes.
+Following are revised or new nodes
+
+| Node Name | Type           | Description                                    | Inputs                | Value                                                                      |
+|-----------|----------------|------------------------------------------------|-----------------------|----------------------------------------------------------------------------|
+| MultiNode | Abstract class | A node that has a tuple result                 |                       | A tuple                                                                    |
+| Start     | Control        | Start of function, now a MultiNode             |                       | A tuple with a ctrl token and an `arg` data node                           |
+| Proj      | Data           | Projection nodes extract values from MultiNode | A MultiNode and index | Result is the extracted value from the input MultiNode at offset index     |
+| Bool      | Data           | Represents results of a comparison operator    | Two data nodes        | Result is a comparison, represented as integer value where 1=true, 0=false |
+| Not       | Data           | Logical not                                    | One data node         | Result converts 0 to 1 and vice versa                                      |
+
+Below is our list of Nodes from [Chapter 3](../chapter03/README.md):
+
+| Node Name | Type         | Description                        | Inputs                                                           | Value                        |
+|-----------|--------------|------------------------------------|------------------------------------------------------------------|------------------------------|
+| Return    | Control      | End of function                    | Predecessor control node, Data node value                        | Return value of the function |
+| Constant  | Data         | Constants such as integer literals | None, however Start node is set as input to enable graph walking | Value of the constant        |
+| Add       | Data         | Add two values                     | Two data nodes, values are added, order not important            | Result of the add operation  |
+| Sub       | Data         | Subtract a value from another      | Two data nodes, values are subtracted, order matters             | Result of the subtract       |
+| Mul       | Data         | Multiply two values                | Two data nodes, values are multiplied, order not important       | Result of the multiply       |
+| Div       | Data         | Divide a value by another          | Two data nodes, values are divided, order matters                | Result of the division       |
+| Minus     | Data         | Negate a value                     | One data node, value is negated                                  | Result of the unary minus    |
+| Scope     | Symbol Table | Represents scopes in the graph     | All nodes that define variables                                  | None                         |
+
+## Projection Nodes
+
+We add projection nodes that are used to extract a specific tuple member from a multi-valued node.
+Each projection node contains an index of the field to extract from its input node.
+Projection nodes allow us to maintain labeled use-def edges as simple Node references.
+
+In the visuals, projection nodes are shown as rectangular boxes inside the node to which they are attached.
+
+### Visualisation
+
+![ProjNode](./projnode.png)
+
+In the visual above, the projection nodes have been tagged by the names
+associated with the outputs of the Start node. Both MultiNode that contain
+Control and Control nodes are yellow.  `$ctrl` and `arg` are outputs(users) of
+StartNode(`START`) and have their MultiNode as their only input in slot 0.  The
+`label` has no semantics and is used during debug printing.
+
+```java 
+public ProjNode(MultiNode ctrl, int idx, String label) {
+    super(ctrl);
+```
+
+```java
+_scope.define(ScopeNode.CTRL, new ProjNode(START, 0, ScopeNode.CTRL).peephole());
+_scope.define(ScopeNode.ARG0, new ProjNode(START, 1, ScopeNode.ARG0).peephole());
+```
+
+These projection nodes are going to extract the types for `CTRL` and `ARG0` from the `START` MultiNode.
+The MultiNode holds the types as a TypeTuple:
+
+```java
+public StartNode(Type[] args) {
+    super();
+    _args = new TypeTuple(args);
+```
+
+Extracting the types is based on the `_idx`:
+```java
+return t instanceof TypeTuple tt ? tt._types[_idx] : Type.BOTTOM;
+```
+
+### Initial values
+
+An argument is always provided, and defaults to `TypeInteger.BOT`.  Individual
+tests can (and do) pass in other integer values by calling `new Parser` with
+an argument:
+
+```java
+Parser parser = new Parser("return arg; #showGraph;", TypeInteger.constant(2));
+```
+
+The graph:
+
+![ProjNode](./04-arg1.svg)
+
+- `$ctrl` and `arg` are both defined in the outermost symbol table, at lexical depth 0.
+- There is an empty global-scope symbol table created at depth 1; in the future
+  global variables will be defined here.
+- There is no line for the scope `$ctrl` because control is dead after the
+  return.  It is manually set to null, which calls `setDef` to kill the node
+  (and killed nodes are not part of the program and not visualized).
+
+```
+    ctrl(null);             // Kill control
+```
+
+- The `arg` projection node is not included in the cluster because arg is a
+  constant.  The `arg` was originally defined by a ProjNode; but when peephole
+  is called the computed type is a constant, and the ProjNode gets replaced
+  with a ConstantNode:
+    
+```java
+public final Node peephole( ) {
+    Type type = _type = compute(); 
+    /*   type = {TypeInteger@1527} "2"  */
+
+    if (_disablePeephole)
+        return this;
+    
+    // ProjNode is not a constant unlike its type.
+    if (!(this instanceof ConstantNode) && type.isConstant())
+        /* Create Constant(2) */
+        return deadCodeElim(new ConstantNode(type).peephole());
+    
+    /* won't get called */
+    ...
+```
+
+If you don't pass an argument you get `TypeInteger.BOT`
+
+```java 
+    Parser parser = new Parser("return arg; #showGraph;");
+    ...
+    
+public Parser(String source) {
+    this(source, TypeInteger.BOT);
+}
+    
+```
+
+We clearly see this is not a constant:
+
+```java
+public final static TypeInteger BOT = new TypeInteger(false, 1); 
+   /* _is_con = false */
+```
+
+So the `arg` projection node remains in the cluster:
+
+![ArgNode](./04-arg2.svg)
+
+
+#### When Is Control Not-Null?
+
+In later chapters, `$ctrl`  will point to other control nodes, such as
+`If` and `Region` (and `Loop`), but for now with no other control flow it always gets set to null after a `Return`.
+
+#### Note Regarding Visualizations
+
+From this chapter onwards we omit the edges from Constants to Start node, mainly to reduce
+clutter and help draw reader's attention to the more important aspects of the graph.
+
+
+
+## Changes to Type System
+
+In [Chapter 2](../chapter02/README.md) we introduced the Type System.
+
+We have annotated Nodes with Types and the Type annotation serves two purposes:
+
+* It defines the set of operations allowed on the Node, and
+* it defines the set of values the Node takes on.
+
+We mentioned in [Chapter 2](../chapter02/README.md) that the set of values
+associated with a Type at a specific Node can be conveniently represented as a
+[lattice](https://en.wikipedia.org/wiki/Lattice_(order)).
+
+The type itself is identified by the Java class `Type` and subtypes.  The type
+implementation uses Java classes as convenient to deal with a Types' internal
+structure - but the Java classes have no relation to a Type's place in the
+lattice.
+
+In this chapter we extend the Type hierarchy as follows:
+
+```
+Type              (Enhanced - now has Control, and a global Top and Bottom)
++-- TypeInteger   (Enhanced - now has Top and Bottom types)
++-- TypeTuple     (New - represents multi-valued result)
+```
+
+Our enhanced Lattice looks like this:
+
+![Lattice](./lattice.svg)
+
+
+In this chapter we introduce the possibility of a program input variable named
+`arg`; all we know about this variable is that it is some integer value, but we
+do not know whether it is a constant or not.
+
+To support the requirements for non-constant integer values, we enhance
+`TypeInteger` to allow it to represent `IntTop` and `IntBot` integer types in
+addition to the earlier constant value.
+
+Now that integer values may be constants or non-constants, we need to introduce
+the *meet* operator over our lattice. The `meet` operator describes rules that
+define the resulting type when we combine integer values.  In the lattice
+diagram you can start from the two elements being `meet` and follow the
+two arrows down the graph to the first point they meet.
+
+|        | IntBot | Con1   | Con2   | IntTop |
+|--------|--------|--------|--------|--------|
+| IntBot | IntBot | IntBot | IntBot | IntBot |
+| Con1   | IntBot | Con1   | IntBot | Con1   |
+| IntTop | IntBot | Con1   | Con2   | IntTop |
+
+In the table above `Con1` and `Con2` represent two distinct integer values, and serve to explain the rules below.
+
+* The `meet` of `IntTop` with any integer constant is that constant.
+* The `meet` of `IntBot` with any integer value is `IntBot`.
+* The `meet` of any integer with itself is that integer.
+* The `meet` of two unrelated integer constants is `IntBot`.
+
+Currently, all our integer valued nodes are either a constant or a `IntBot` integer type.
+When we start optimizing loops, we will start seeing `IntTop` values.
+
+
+### Tuple Types
+
+Tuple types need a little more explaination: they are a fixed-size grouping of
+types; literally a `Type[]`.  Tuples represent a collection of otherwise
+unrelated types, and come from `MultiNode`s.  `ProjNode`s will take the
+appropriate `Type` array element out of a Tuple.  The `StartNode` now produces
+a 2-element `TypeTuple` with control and the type of `arg`: `[ctrl, TypeInteger.INTBOT]`
+
+The lattice `meet` operator on Tuples is done element by element; each array
+element recursively calls `meet`.  Tuples of mixed sizes are a internal
+compiler error, and the `meet` uses Bottom for them.
+
+
+## `$ctrl` name binding
+
+In previous chapters, we had a hard coded control input edge from Start to
+Return. In this chapter we no longer have such a hard-wired edge.  Instead, we
+track the current in-scope control node via the name `$ctrl`.  This means that
+when we need to create an edge to the predecessor control node, we simply look
+up this name in the current scope.
+
+This introduces the idea that the control flow subgraph is a Petri net model.[^1]
+The control token moves virtually from node to node as execution proceeds.  The
+initial control token is in Start, it then moves via the Proj node to Return.
+In later chapters we will see how the token moves across branches.
+
+
+## More Peephole Optimizations
+
+Now that we have non-constant integer values, we can do additional
+optimizations, rearranging algebraic expressions to enable constant
+folding. For example:
+
+```
+return 1 + arg + 2;
+```
+
+We would expect the compiler to output `arg+3` here, but as it stands what we get is:
+
+![Graph1](./04-pre-peephole.svg)
+
+We need to perform some algebraic simplifications to enable better outcome. For
+example, we need to rearrange the expression as follows:
+
+```
+arg + (1 + 2)
+```
+
+This then enables constant folding and the final outcome is shown below.  By
+canonicalizing expressions we fold common addressing math constants, remove
+algebraic identities and generally simplify the code.
+
+![Graph1](./04-post-peephole.svg)
+
+Here is a list of peepholes introduced in this Chapter, more will be introduced in later chapters:
+
+| Before                | After                 | Description                                    |
+|-----------------------|-----------------------|------------------------------------------------|
+| (arg + 0 )            |  arg                  | Add of zero identity                           |
+| (arg * 1 )            |  arg                  | Multiple of one identity                       |
+| (con + arg)           | (arg + con)           | Move constants to right, to encourage folding  |
+| (con * arg)           | (arg * con)           | Move constants to right, to encourage folding  |
+| (con1 + (arg + con2)) | (arg + (con1 + con2)) | Move constants to right, to encourage folding  |
+| ((arg1 + con) + arg2) | ((arg1 + arg2) + con) | Move constants to right, to encourage folding  |
+| (arg + arg)           | (arg * 2)             | Sum-of-products form                           |
+| (arg - arg)           | 0                     | Sub of same is 0
+| (con / 1)             | con                   | Division by one
+| (arg == arg)          | 1                     | Compare of same
+| (arg != arg)          | 0                     | Compare of same
+| (arg < arg)           | 0                     | Compare of same
+| (arg > arg)           | 0                     | Compare of same
+| (arg <= arg)          | 1                     | Compare of same
+| (arg >= arg)          | 1                     | Compare of same
+
+
+## Peephole Walkthrough
+
+The peephole optimizations introduced in this chapter are local. They are triggered during parsing
+as new nodes are created, before the newly created node has any uses.
+
+For example, when we parse a unary expression, and create a Node for the parsed expression,
+`peephole()` is invoked on the newly created `MinusNode`:
+
+```java
+private Node parseUnary() {
+    if (match("-")) return new MinusNode(parseUnary()).peephole();
+    return parsePrimary();
+}
+```
+
+As another example, here is the parsing of comparison operators, also introduced in this chapter:
+
+```java
+private Node parseComparison() {
+    var lhs = parseAddition();
+    if (match("==")) return new BoolNode.EQNode(lhs, parseComparison()).peephole();
+    if (match("!=")) return new BoolNode.NENode(lhs, parseComparison()).peephole();
+    if (match("<=")) return new BoolNode.LENode(lhs, parseComparison()).peephole();
+    if (match("<" )) return new BoolNode.LTNode(lhs, parseComparison()).peephole();
+    if (match(">=")) return new BoolNode.GENode(lhs, parseComparison()).peephole();
+    if (match(">" )) return new BoolNode.GTNode(lhs, parseComparison()).peephole();
+    return lhs;
+}
+```
+
+The implementation of `peephole()` is in the main `Node` class, and is thus shared
+by all subclasses of `Node`.
+
+```java
+public final Node peephole( ) {
+    // Compute initial or improved Type
+    Type type = _type = compute();
+
+    // Replace constant computations from non-constants with a constant node
+    if (!(this instanceof ConstantNode) && type.isConstant())
+        return deadCodeElim(new ConstantNode(type).peephole());
+
+    // Ask each node for a better replacement
+    Node n = idealize();
+    if( n != null )         // Something changed
+        // Recursively optimize
+        return deadCodeElim(n.peephole());
+
+    return this;            // No progress
+}
+```
+
+The peephole method does following:
+
+- Compute a Type for the node.
+- If the Type is a Constant and the node is not a ConstantNode, replace with a
+  ConstantNode, recursively invoking peephole on the new constant.
+- Otherwise, ask the Node for a better replacement via a call to `idealize()`.
+  The "better replacement" is things like `(1+2)` becomes `3` and `1+(x+2))`
+  becomes `(x+(1+2))`.
+- - Each Node subtype is responsible for deciding what the `idealize()` step
+  should do. If there is a better replacement then the node returns a non
+  `null` value.
+- - If we see a non `null` value, something changed, so we invoke `peephole()`
+  again and then also run dead code elimination on the (now dead and replaced)
+  `this` node.
+
+`peephole()` calls `deadCodeElim()` which is shown below.
+
+
+## Dead Code Elimination(DCE)
+
+We call `kill` every time a Node makes the 1-user-to-0-users transition during
+graph editing.  This commonly happens when replacing via peeps and exiting
+scopes.  This recursive `kill` amounts to Dead Code Elimination in many other
+compilers; here we call it in a very fine-grained way.  Many Nodes will be
+created by the parser, only to be killed before the very next lexeme is parsed.
+
+For more information please read the [appendix section](appendix/dce.md#appendix) on dead code elimination.
+
+The suggestively called `deadCodeElim` function is a thin wrapper over `kill`
+used by `peepholeOpt` to avoid prematurely killing a Node which only briefly
+makes the 1->0 users transition.
+
+```java
+// m is the new Node, self is the old.
+// Return 'm', which may have zero uses but is alive nonetheless.
+// If self has zero uses (and is not 'm'), {@link #kill} self.
+private Node deadCodeElim(Node m) {
+    // If self is going dead and not being returned here (Nodes returned
+    // from peephole commonly have no uses (yet)), then kill self.
+    if( m != this && isUnused() ) {
+        // Killing self - and since self recursively kills self's inputs we
+        // might end up killing 'm', which we are returning as a live Node.
+        // So we add a bogus extra null output edge to stop kill().
+        m.addUse(null); // Add bogus null use to keep m alive
+        kill();            // Kill self because replacing with 'm'
+        m.delUse(null);    // Remove bogus null.
+    }
+    return m;
+}
+```
+
+Note the temporary add of a bogus user to the Node `m`.  The reason this is
+done is that we know `m` is the new replacement and is alive, but since it is
+not yet hooked into the graph, it has no users yet.  By adding a bogus user we
+prevent it being mistaken for dead and being killed.
+
+[^1]: Click, C. (1995).
+   Combining Analyses, Combining Optimizations, 131.
diff --git a/docs/chapter04/lattice.gv b/docs/chapter04/lattice.gv
new file mode 100644
index 00000000..6efef107
--- /dev/null
+++ b/docs/chapter04/lattice.gv
@@ -0,0 +1,97 @@
+digraph Lattice {
+    compound = true;
+    // Define nodes
+    splines=false;
+    node [shape=circle, style=filled];
+    nodesep=0.6;
+
+    TOP [label=<<B>⊤</B >>, fillcolor=black, fontcolor=white];
+    BOT [label=<<B>⊥</B>>, fillcolor=black, fontcolor=white];
+
+    // ----------------------
+    // Control
+    CTRL  [label="ctrl", fillcolor=yellow, shape=oval];
+
+    TOP   -> CTRL;
+    CTRL  -> BOT;
+
+    // ----------------------
+    // Integers
+    INT_TOP [label=<<B>⊤</B >:int>, fillcolor=lightblue, shape=oval];
+    INT_BOT [label=<<B>⊥</B>:int>, fillcolor=lightblue, shape=oval];
+
+    INT_MIN     [label="-n" , fillcolor=lightblue];
+    INT_MINDOTS [label="...", fillcolor=lightblue shape=oval];
+    INT_MINUS   [label="-1" , fillcolor=lightblue];
+    INT_ZERO    [label="0"  , fillcolor=lightblue];
+    INT_PLUS    [label="1"  , fillcolor=lightblue];
+    INT_MAXDOTS [label="...", fillcolor=lightblue, shape=oval];
+    INT_MAX     [label="+n" , fillcolor=lightblue];
+
+    TOP         -> INT_TOP;
+    INT_TOP     -> INT_MIN;
+    INT_TOP     -> INT_MINDOTS;
+    INT_TOP     -> INT_MINUS;
+    INT_TOP     -> INT_ZERO;
+    INT_TOP     -> INT_PLUS;
+    INT_TOP     -> INT_MAXDOTS;
+    INT_TOP     -> INT_MAX;
+    INT_MIN     -> INT_BOT;
+    INT_MINDOTS -> INT_BOT;
+    INT_MINUS   -> INT_BOT;
+    INT_ZERO    -> INT_BOT;
+    INT_PLUS    -> INT_BOT;
+    INT_MAXDOTS -> INT_BOT;
+    INT_MAX     -> INT_BOT;
+    INT_BOT     -> BOT;
+
+    INT_MIN -> INT_MINDOTS -> INT_MINUS -> INT_ZERO -> INT_PLUS -> INT_MAXDOTS -> INT_MAX [style=invis];
+    
+    // ----------------------
+    // Tuples
+
+    TUPLE_TOP [label=<<B>⊤</B >:tuple>, fillcolor=pink, shape=oval];
+    TUPLE_BOT [label=<<B>⊥</B>:tuple>, fillcolor=pink, shape=oval];
+
+    TUPLE1 [shape=none, fillcolor=none, label=<
+      <TABLE BORDER="0" CELLBORDER="1" CELLSPACING="0">
+        <TH><TD BGCOLOR="pink">tuple</TD></TH>
+        <TR><TD BGCOLOR="black" ><FONT color="white">type</FONT></TD></TR>
+      </TABLE>
+    >];
+    TUPLE2 [shape=none, fillcolor=none, label=<
+      <TABLE BORDER="0" CELLBORDER="1" CELLSPACING="0">
+        <TH><TD BGCOLOR="pink">tuple</TD></TH>
+        <TR><TD BGCOLOR="black" ><FONT color="white">type</FONT></TD></TR>
+        <TR><TD BGCOLOR="black" ><FONT color="white">type</FONT></TD></TR>
+      </TABLE>
+    >];
+    TUPLEN [shape=none, fillcolor=none, label=<
+      <TABLE BORDER="0" CELLBORDER="1" CELLSPACING="0">
+        <TH><TD BGCOLOR="pink">tuple</TD></TH>
+        <TR><TD BGCOLOR="black" ><FONT color="white">type</FONT></TD></TR>
+        <TR><TD BGCOLOR="black" ><FONT color="white">...</FONT></TD></TR>
+        <TR><TD BGCOLOR="black" ><FONT color="white">type</FONT></TD></TR>
+      </TABLE>
+    >];
+
+
+    TOP -> TUPLE_TOP;
+    TUPLE_TOP -> TUPLE1;
+    TUPLE_TOP -> TUPLE2;
+    TUPLE_TOP -> TUPLEN;
+    TUPLE1 -> TUPLE_BOT;
+    TUPLE2 -> TUPLE_BOT;
+    TUPLEN -> TUPLE_BOT;
+    TUPLE_BOT -> BOT;
+
+    // Additional customization
+    {rank=same; TOP}
+    {rank=same; INT_TOP; TUPLE_TOP}
+    {rank=same;
+      CTRL;
+      INT_MIN; INT_MINDOTS; INT_MINUS; INT_ZERO; INT_PLUS; INT_MAXDOTS; INT_MAX;
+    }
+    {rank=same; INT_BOT, TUPLE_BOT}
+    {rank=same; BOT;}
+}
diff --git a/docs/chapter04/lattice.svg b/docs/chapter04/lattice.svg
new file mode 100644
index 00000000..35cb9296
--- /dev/null
+++ b/docs/chapter04/lattice.svg
@@ -0,0 +1,305 @@
+<?xml version="1.0" encoding="UTF-8" standalone="no"?>

+<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN"

+ "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">

+<!-- Generated by graphviz version 12.2.0 (20241103.1931)

+ -->

+<!-- Title: Lattice Pages: 1 -->

+<svg width="952pt" height="406pt"

+ viewBox="0.00 0.00 952.00 406.50" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">

+<g id="graph0" class="graph" transform="scale(1 1) rotate(0) translate(4 402.5)">

+<title>Lattice</title>

+<polygon fill="white" stroke="none" points="-4,4 -4,-402.5 948,-402.5 948,4 -4,4"/>

+<!-- TOP -->

+<g id="node1" class="node">

+<title>TOP</title>

+<ellipse fill="black" stroke="black" cx="374" cy="-377.37" rx="21.12" ry="21.12"/>

+<text text-anchor="start" x="366.12" y="-372.95" font-family="Times New Roman,serif" font-weight="bold" font-size="14.00" fill="white">⊤</text>

+</g>

+<!-- CTRL -->

+<g id="node3" class="node">

+<title>CTRL</title>

+<ellipse fill="yellow" stroke="black" cx="27" cy="-199.25" rx="27" ry="18"/>

+<text text-anchor="middle" x="27" y="-194.2" font-family="Times New Roman,serif" font-size="14.00">ctrl</text>

+</g>

+<!-- TOP&#45;&gt;CTRL -->

+<g id="edge1" class="edge">

+<title>TOP&#45;&gt;CTRL</title>

+<path fill="none" stroke="black" d="M354.85,-367.54C297.95,-338.34 129.51,-251.87 58.81,-215.58"/>

+<polygon fill="black" stroke="black" points="60.56,-212.54 50.07,-211.09 57.36,-218.77 60.56,-212.54"/>

+</g>

+<!-- INT_TOP -->

+<g id="node4" class="node">

+<title>INT_TOP</title>

+<ellipse fill="lightblue" stroke="black" cx="374" cy="-302.25" rx="33.95" ry="18"/>

+<text text-anchor="start" x="357.12" y="-298.2" font-family="Times New Roman,serif" font-weight="bold" font-size="14.00">⊤</text>

+<text text-anchor="start" x="372.88" y="-298.2" font-family="Times New Roman,serif" font-size="14.00">:int</text>

+</g>

+<!-- TOP&#45;&gt;INT_TOP -->

+<g id="edge3" class="edge">

+<title>TOP&#45;&gt;INT_TOP</title>

+<path fill="none" stroke="black" d="M374,-356C374,-348.51 374,-339.87 374,-331.79"/>

+<polygon fill="black" stroke="black" points="377.5,-332.05 374,-322.05 370.5,-332.05 377.5,-332.05"/>

+</g>

+<!-- TUPLE_TOP -->

+<g id="node13" class="node">

+<title>TUPLE_TOP</title>

+<ellipse fill="pink" stroke="black" cx="771" cy="-302.25" rx="42.65" ry="18"/>

+<text text-anchor="start" x="747.75" y="-298.2" font-family="Times New Roman,serif" font-weight="bold" font-size="14.00">⊤</text>

+<text text-anchor="start" x="763.5" y="-298.2" font-family="Times New Roman,serif" font-size="14.00">:tuple</text>

+</g>

+<!-- TOP&#45;&gt;TUPLE_TOP -->

+<g id="edge25" class="edge">

+<title>TOP&#45;&gt;TUPLE_TOP</title>

+<path fill="none" stroke="black" d="M395.06,-372.49C455.97,-361.27 633.5,-328.57 721.03,-312.45"/>

+<polygon fill="black" stroke="black" points="721.57,-315.91 730.77,-310.66 720.3,-309.03 721.57,-315.91"/>

+</g>

+<!-- BOT -->

+<g id="node2" class="node">

+<title>BOT</title>

+<ellipse fill="black" stroke="black" cx="374" cy="-21.12" rx="21.12" ry="21.12"/>

+<text text-anchor="start" x="366.12" y="-16.7" font-family="Times New Roman,serif" font-weight="bold" font-size="14.00" fill="white">⊥</text>

+</g>

+<!-- CTRL&#45;&gt;BOT -->

+<g id="edge2" class="edge">

+<title>CTRL&#45;&gt;BOT</title>

+<path fill="none" stroke="black" d="M48.71,-188.11C108.7,-157.31 276.83,-71 344.87,-36.08"/>

+<polygon fill="black" stroke="black" points="346.26,-39.3 353.55,-31.62 343.06,-33.07 346.26,-39.3"/>

+</g>

+<!-- INT_MIN -->

+<g id="node6" class="node">

+<title>INT_MIN</title>

+<ellipse fill="lightblue" stroke="black" cx="116" cy="-199.25" rx="18.6" ry="18.6"/>

+<text text-anchor="middle" x="116" y="-194.2" font-family="Times New Roman,serif" font-size="14.00">&#45;n</text>

+</g>

+<!-- INT_TOP&#45;&gt;INT_MIN -->

+<g id="edge4" class="edge">

+<title>INT_TOP&#45;&gt;INT_MIN</title>

+<path fill="none" stroke="black" d="M347.36,-290.82C298.27,-271.6 193.98,-230.78 143.94,-211.18"/>

+<polygon fill="black" stroke="black" points="145.28,-207.95 134.69,-207.56 142.73,-214.47 145.28,-207.95"/>

+</g>

+<!-- INT_MINDOTS -->

+<g id="node7" class="node">

+<title>INT_MINDOTS</title>

+<ellipse fill="lightblue" stroke="black" cx="205" cy="-199.25" rx="27" ry="18"/>

+<text text-anchor="middle" x="205" y="-194.2" font-family="Times New Roman,serif" font-size="14.00">...</text>

+</g>

+<!-- INT_TOP&#45;&gt;INT_MINDOTS -->

+<g id="edge5" class="edge">

+<title>INT_TOP&#45;&gt;INT_MINDOTS</title>

+<path fill="none" stroke="black" d="M352.25,-288.25C322.2,-270.29 267.88,-237.83 234.28,-217.75"/>

+<polygon fill="black" stroke="black" points="236.25,-214.85 225.87,-212.72 232.66,-220.85 236.25,-214.85"/>

+</g>

+<!-- INT_MINUS -->

+<g id="node8" class="node">

+<title>INT_MINUS</title>

+<ellipse fill="lightblue" stroke="black" cx="294" cy="-199.25" rx="18.6" ry="18.6"/>

+<text text-anchor="middle" x="294" y="-194.2" font-family="Times New Roman,serif" font-size="14.00">&#45;1</text>

+</g>

+<!-- INT_TOP&#45;&gt;INT_MINUS -->

+<g id="edge6" class="edge">

+<title>INT_TOP&#45;&gt;INT_MINUS</title>

+<path fill="none" stroke="black" d="M361.5,-285.47C348.27,-268.76 327.25,-242.23 312.18,-223.21"/>

+<polygon fill="black" stroke="black" points="315.02,-221.14 306.06,-215.48 309.53,-225.49 315.02,-221.14"/>

+</g>

+<!-- INT_ZERO -->

+<g id="node9" class="node">

+<title>INT_ZERO</title>

+<ellipse fill="lightblue" stroke="black" cx="374" cy="-199.25" rx="18" ry="18"/>

+<text text-anchor="middle" x="374" y="-194.2" font-family="Times New Roman,serif" font-size="14.00">0</text>

+</g>

+<!-- INT_TOP&#45;&gt;INT_ZERO -->

+<g id="edge7" class="edge">

+<title>INT_TOP&#45;&gt;INT_ZERO</title>

+<path fill="none" stroke="black" d="M374,-284.12C374,-269.02 374,-246.74 374,-228.93"/>

+<polygon fill="black" stroke="black" points="377.5,-228.98 374,-218.98 370.5,-228.98 377.5,-228.98"/>

+</g>

+<!-- INT_PLUS -->

+<g id="node10" class="node">

+<title>INT_PLUS</title>

+<ellipse fill="lightblue" stroke="black" cx="453" cy="-199.25" rx="18" ry="18"/>

+<text text-anchor="middle" x="453" y="-194.2" font-family="Times New Roman,serif" font-size="14.00">1</text>

+</g>

+<!-- INT_TOP&#45;&gt;INT_PLUS -->

+<g id="edge8" class="edge">

+<title>INT_TOP&#45;&gt;INT_PLUS</title>

+<path fill="none" stroke="black" d="M386.34,-285.47C399.49,-268.67 420.4,-241.93 435.3,-222.88"/>

+<polygon fill="black" stroke="black" points="437.93,-225.2 441.33,-215.17 432.41,-220.89 437.93,-225.2"/>

+</g>

+<!-- INT_MAXDOTS -->

+<g id="node11" class="node">

+<title>INT_MAXDOTS</title>

+<ellipse fill="lightblue" stroke="black" cx="541" cy="-199.25" rx="27" ry="18"/>

+<text text-anchor="middle" x="541" y="-194.2" font-family="Times New Roman,serif" font-size="14.00">...</text>

+</g>

+<!-- INT_TOP&#45;&gt;INT_MAXDOTS -->

+<g id="edge9" class="edge">

+<title>INT_TOP&#45;&gt;INT_MAXDOTS</title>

+<path fill="none" stroke="black" d="M395.83,-288.04C425.56,-270.07 478.84,-237.84 511.93,-217.83"/>

+<polygon fill="black" stroke="black" points="513.46,-221 520.2,-212.83 509.84,-215.01 513.46,-221"/>

+</g>

+<!-- INT_MAX -->

+<g id="node12" class="node">

+<title>INT_MAX</title>

+<ellipse fill="lightblue" stroke="black" cx="632" cy="-199.25" rx="21.15" ry="21.15"/>

+<text text-anchor="middle" x="632" y="-194.2" font-family="Times New Roman,serif" font-size="14.00">+n</text>

+</g>

+<!-- INT_TOP&#45;&gt;INT_MAX -->

+<g id="edge10" class="edge">

+<title>INT_TOP&#45;&gt;INT_MAX</title>

+<path fill="none" stroke="black" d="M400.64,-290.82C449,-271.89 550.91,-231.99 601.78,-212.08"/>

+<polygon fill="black" stroke="black" points="602.96,-215.38 610.99,-208.47 600.41,-208.86 602.96,-215.38"/>

+</g>

+<!-- INT_BOT -->

+<g id="node5" class="node">

+<title>INT_BOT</title>

+<ellipse fill="lightblue" stroke="black" cx="374" cy="-96.25" rx="33.95" ry="18"/>

+<text text-anchor="start" x="357.12" y="-92.2" font-family="Times New Roman,serif" font-weight="bold" font-size="14.00">⊥</text>

+<text text-anchor="start" x="372.88" y="-92.2" font-family="Times New Roman,serif" font-size="14.00">:int</text>

+</g>

+<!-- INT_BOT&#45;&gt;BOT -->

+<g id="edge18" class="edge">

+<title>INT_BOT&#45;&gt;BOT</title>

+<path fill="none" stroke="black" d="M374,-77.92C374,-70.74 374,-62.18 374,-53.95"/>

+<polygon fill="black" stroke="black" points="377.5,-54.22 374,-44.22 370.5,-54.22 377.5,-54.22"/>

+</g>

+<!-- INT_MIN&#45;&gt;INT_BOT -->

+<g id="edge11" class="edge">

+<title>INT_MIN&#45;&gt;INT_BOT</title>

+<path fill="none" stroke="black" d="M133.33,-191.46C174.74,-175.25 280.22,-133.96 336.8,-111.81"/>

+<polygon fill="black" stroke="black" points="337.98,-115.11 346.02,-108.2 335.43,-108.59 337.98,-115.11"/>

+</g>

+<!-- INT_MIN&#45;&gt;INT_MINDOTS -->

+<!-- INT_MINDOTS&#45;&gt;INT_BOT -->

+<g id="edge12" class="edge">

+<title>INT_MINDOTS&#45;&gt;INT_BOT</title>

+<path fill="none" stroke="black" d="M224.72,-186.46C253.64,-169.18 307.88,-136.76 342.4,-116.13"/>

+<polygon fill="black" stroke="black" points="343.92,-119.3 350.71,-111.17 340.33,-113.29 343.92,-119.3"/>

+</g>

+<!-- INT_MINDOTS&#45;&gt;INT_MINUS -->

+<!-- INT_MINUS&#45;&gt;INT_BOT -->

+<g id="edge13" class="edge">

+<title>INT_MINUS&#45;&gt;INT_BOT</title>

+<path fill="none" stroke="black" d="M305.12,-184.2C317.82,-168.18 338.81,-141.67 354.29,-122.13"/>

+<polygon fill="black" stroke="black" points="356.98,-124.37 360.45,-114.35 351.5,-120.02 356.98,-124.37"/>

+</g>

+<!-- INT_MINUS&#45;&gt;INT_ZERO -->

+<!-- INT_ZERO&#45;&gt;INT_BOT -->

+<g id="edge14" class="edge">

+<title>INT_ZERO&#45;&gt;INT_BOT</title>

+<path fill="none" stroke="black" d="M374,-181.12C374,-166.02 374,-143.74 374,-125.93"/>

+<polygon fill="black" stroke="black" points="377.5,-125.98 374,-115.98 370.5,-125.98 377.5,-125.98"/>

+</g>

+<!-- INT_ZERO&#45;&gt;INT_PLUS -->

+<!-- INT_PLUS&#45;&gt;INT_BOT -->

+<g id="edge15" class="edge">

+<title>INT_PLUS&#45;&gt;INT_BOT</title>

+<path fill="none" stroke="black" d="M442.35,-184.63C429.9,-168.71 409.04,-142.04 393.63,-122.35"/>

+<polygon fill="black" stroke="black" points="396.42,-120.23 387.5,-114.51 390.9,-124.54 396.42,-120.23"/>

+</g>

+<!-- INT_PLUS&#45;&gt;INT_MAXDOTS -->

+<!-- INT_MAXDOTS&#45;&gt;INT_BOT -->

+<g id="edge16" class="edge">

+<title>INT_MAXDOTS&#45;&gt;INT_BOT</title>

+<path fill="none" stroke="black" d="M521.52,-186.46C492.93,-169.18 439.34,-136.76 405.22,-116.13"/>

+<polygon fill="black" stroke="black" points="407.39,-113.35 397.03,-111.17 403.77,-119.34 407.39,-113.35"/>

+</g>

+<!-- INT_MAXDOTS&#45;&gt;INT_MAX -->

+<!-- INT_MAX&#45;&gt;INT_BOT -->

+<g id="edge17" class="edge">

+<title>INT_MAX&#45;&gt;INT_BOT</title>

+<path fill="none" stroke="black" d="M612.61,-190.66C569.87,-173.93 466.96,-133.64 411.28,-111.84"/>

+<polygon fill="black" stroke="black" points="412.8,-108.68 402.21,-108.29 410.25,-115.2 412.8,-108.68"/>

+</g>

+<!-- TUPLE1 -->

+<g id="node15" class="node">

+<title>TUPLE1</title>

+<polygon fill="none" stroke="none" points="750,-225.75 696,-225.75 696,-172.75 750,-172.75 750,-225.75"/>

+<polygon fill="pink" stroke="none" points="706.5,-199.25 706.5,-221.75 739.5,-221.75 739.5,-199.25 706.5,-199.25"/>

+<polygon fill="none" stroke="black" points="706.5,-199.25 706.5,-221.75 739.5,-221.75 739.5,-199.25 706.5,-199.25"/>

+<text text-anchor="start" x="709.5" y="-205.45" font-family="Times New Roman,serif" font-size="14.00">tuple</text>

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+</g>

+<!-- TUPLE_TOP&#45;&gt;TUPLE1 -->

+<g id="edge26" class="edge">

+<title>TUPLE_TOP&#45;&gt;TUPLE1</title>

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+<polygon fill="black" stroke="black" points="743.07,-234.33 735.62,-226.8 736.75,-237.34 743.07,-234.33"/>

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+<!-- TUPLE2 -->

+<g id="node16" class="node">

+<title>TUPLE2</title>

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+</g>

+<!-- TUPLE_TOP&#45;&gt;TUPLE2 -->

+<g id="edge27" class="edge">

+<title>TUPLE_TOP&#45;&gt;TUPLE2</title>

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+<polygon fill="black" stroke="black" points="800.41,-248.65 801.62,-238.13 794.11,-245.6 800.41,-248.65"/>

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+<!-- TUPLEN -->

+<g id="node17" class="node">

+<title>TUPLEN</title>

+<polygon fill="none" stroke="none" points="944,-248.25 890,-248.25 890,-150.25 944,-150.25 944,-248.25"/>

+<polygon fill="pink" stroke="none" points="900.5,-221.75 900.5,-244.25 933.5,-244.25 933.5,-221.75 900.5,-221.75"/>

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+<polygon fill="none" stroke="black" points="900.5,-154.25 900.5,-176.75 933.5,-176.75 933.5,-154.25 900.5,-154.25"/>

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+</g>

+<!-- TUPLE_TOP&#45;&gt;TUPLEN -->

+<g id="edge28" class="edge">

+<title>TUPLE_TOP&#45;&gt;TUPLEN</title>

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+<!-- TUPLE_BOT -->

+<g id="node14" class="node">

+<title>TUPLE_BOT</title>

+<ellipse fill="pink" stroke="black" cx="771" cy="-96.25" rx="42.65" ry="18"/>

+<text text-anchor="start" x="747.75" y="-92.2" font-family="Times New Roman,serif" font-weight="bold" font-size="14.00">⊥</text>

+<text text-anchor="start" x="763.5" y="-92.2" font-family="Times New Roman,serif" font-size="14.00">:tuple</text>

+</g>

+<!-- TUPLE_BOT&#45;&gt;BOT -->

+<g id="edge32" class="edge">

+<title>TUPLE_BOT&#45;&gt;BOT</title>

+<path fill="none" stroke="black" d="M732.57,-88.17C654.42,-73.77 478.23,-41.32 405.9,-28"/>

+<polygon fill="black" stroke="black" points="406.78,-24.6 396.31,-26.23 405.51,-31.49 406.78,-24.6"/>

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+<!-- TUPLE1&#45;&gt;TUPLE_BOT -->

+<g id="edge29" class="edge">

+<title>TUPLE1&#45;&gt;TUPLE_BOT</title>

+<path fill="none" stroke="black" d="M735.11,-172.76C742.13,-158 750.96,-139.41 758.11,-124.38"/>

+<polygon fill="black" stroke="black" points="761.18,-126.06 762.31,-115.53 754.86,-123.06 761.18,-126.06"/>

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+<!-- TUPLE2&#45;&gt;TUPLE_BOT -->

+<g id="edge30" class="edge">

+<title>TUPLE2&#45;&gt;TUPLE_BOT</title>

+<path fill="none" stroke="black" d="M802.17,-161.5C796.24,-149.27 789.75,-135.89 784.23,-124.51"/>

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+<!-- TUPLEN&#45;&gt;TUPLE_BOT -->

+<g id="edge31" class="edge">

+<title>TUPLEN&#45;&gt;TUPLE_BOT</title>

+<path fill="none" stroke="black" d="M890.22,-179.72C865.26,-162.45 828.04,-136.71 801.84,-118.58"/>

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diff --git a/docs/chapter04/projnode.png b/docs/chapter04/projnode.png
new file mode 100644
index 00000000..bf7493f2
--- /dev/null
+++ b/docs/chapter04/projnode.png