Documentation updates (#76)

* More clearly explain why we do not have an AST in the implementation

* Improve the description of type lattice.
Cross reference to future chapters.

* Minor update

* Add lattice visual
Revise the text describing meet.

* minor update

* Add a lattice visual to the text

* Update the lattice visual.
Add description of the new edges in the SoN graph

4 files changed, 290 insertions, 13 deletions

diff --git a/chapter06/README.md b/chapter06/README.md
index d3fe6ed9..6f7f3baa 100644
--- a/chapter06/README.md
+++ b/chapter06/README.md
@@ -4,11 +4,11 @@ In this chapter we do not add new language features. Our goal is to add peephole
 
 ## Type System Revision
 
-We now need to extend the type system to introduce another lattice structure, in addition to the Integer lattice.
-As before we denote  "top" by T and "bottom" by ⊥.
+We now need to extend the type system to introduce another lattice structure, in addition to the Integer sub-lattice.
+As before we denote  "Top" by T and "Bottom" by ⊥.
 
-* "ctrl" represents a live control, whereas "~ctrl" represents a dead control.
-* "INT" refers to the integer type.
+* "Ctrl" represents a live control, whereas "~Ctrl" represents a dead control.
+* "INT" refers to an integer value, note that integer values have their own sub-lattice.
 
 |       | ⊥ | Ctrl | ~Ctrl | T     | INT |
 |-------|---|------|-------|-------|-----|
@@ -17,11 +17,16 @@ As before we denote  "top" by T and "bottom" by ⊥.
 | ~Ctrl | ⊥ | Ctrl | ~Ctrl | ~Ctrl | ⊥   |
 | ⊥     | ⊥ | ⊥    | ⊥     | ⊥     | ⊥   |
 
-* "top" meets any type is that type,
-* "bottom" meets any type is "bottom",
-* "top", "bottom", "ctrl", "~ctrl" are classed as simple types,
-* "ctrl" meets "~ctrl" is "ctrl"
-* Unless covered above, a simple type meets non-simple results in "bottom"
+* "Top" meets any type is that type,
+* "Bottom" meets any type is "Bottom",
+* "Top", "Bottom", "Ctrl", "~Ctrl" are classed as simple types,
+* "Ctrl" meets "~Ctrl" is "Ctrl"
+* Unless covered above, a simple type meets non-simple results in "Bottom"
+
+The entire revised Lattice is shown below:
+
+![Lattice](./docs/lattice.svg)
+
 
 ## Peephole of `if` 
 
@@ -33,7 +38,7 @@ As before we denote  "top" by T and "bottom" by ⊥.
 * When we peephole an `If` node we test if the predicate is a constant.  If so,
   then one branch or the other dead, depending on if the constant is `0` or not.
 * When we add the `Proj` nodes, in this scenario, we already know one of the projections is dead.  The peephole of the
-  relevant `Proj` node replaces the `Proj` with a `Constant` of type "~ctrl" indicating dead control.
+  relevant `Proj` node replaces the `Proj` with a `Constant` of type "~Ctrl" indicating dead control.
 * At this point our dead code elimination would kill the `If` node since it has no uses
    - but we cannot do that because we need to continue the parse.  The
   important observation is that *we __can__ kill the `If` node after __both__
@@ -41,11 +46,11 @@ As before we denote  "top" by T and "bottom" by ⊥.
   only see the `Proj` nodes.
 * To ensure this, we add a dummy user to `If` before creating the first `Proj` and remove it immediately after, allowing the 
   second `Proj` to trigger a dead code elimination of the `If`.
-* The live `Proj` gets replaced by the parent of `If`, whereas the dead `Proj` gets replaced by "~ctrl".
+* The live `Proj` gets replaced by the parent of `If`, whereas the dead `Proj` gets replaced by "~Ctrl".
 * The parsing then continues as normal.
 * The other changes are when we reach the merge point with a dead control.
 * Again keeping with our strategy we merge as normal, including creating `Phi` nodes.
-* The `Region` may have have one of its inputs as "~ctrl".  This will be seen
+* The `Region` may have have one of its inputs as "~Ctrl".  This will be seen
   by the `Phi` which then optimizes itself.  If the `Phi` has just one live
   input, the `Phi` peephole replaces itself with the remaining input.  
 * Finally, at the end, when the `Region` node is peepholed, we see that it has only one live input and no Phi uses
@@ -60,7 +65,7 @@ deleted.  Conversely, we cannot collapse a Region until it has no dependent
 `Phi`s.
 
 When processing `If` we do not remove control inputs to a `Region`, instead the
-dead control input is simply set to `Constant(~ctrl)`.  The peephole logic in
+dead control input is simply set to `Constant(~Ctrl)`.  The peephole logic in
 a `Phi` notices this and replaces itself with the live input.
 
 ## Discussion
diff --git a/chapter06/docs/build-graphs.sh b/chapter06/docs/build-graphs.sh
index 5238dbfc..44d2de3a 100644
--- a/chapter06/docs/build-graphs.sh
+++ b/chapter06/docs/build-graphs.sh
@@ -9,3 +9,5 @@ dot -Tsvg  06-graph3-false.gv > 06-graph3-false.svg
 dot -Tsvg  06-graph4-nonconst.gv > 06-graph4-nonconst.svg
 dot -Tsvg  06-graph4-true.gv > 06-graph4-true.svg
 dot -Tsvg  06-graph4-arg2.gv > 06-graph4-arg2.svg
+dot -Tsvg  lattice.gv > lattice.svg
+
diff --git a/chapter06/docs/lattice.gv b/chapter06/docs/lattice.gv
new file mode 100644
index 00000000..2cf27452
--- /dev/null
+++ b/chapter06/docs/lattice.gv
@@ -0,0 +1,64 @@
+digraph Lattice {
+    // Define nodes
+    splines=false;
+    node [shape=circle, style=filled, fillcolor=lightblue];
+    nodesep=0.6;
+
+    TOP [label="⊤"];
+    BOT [label="⊥"];
+
+    XCTRL [label="~Ctrl"];
+    CTRL [label="Ctrl"];
+
+    INT_TOP [label="IntTop"];
+    INT_BOT [label="IntBot"];
+
+    INT_MIN [label="-n"];
+    INT_MINDOTS [label="..." shape=none style=none];
+    INT_MINUS [label="-1"];
+    INT_ZERO [label="0"];
+    INT_PLUS [label="1"];
+    INT_MAXDOTS [label="..." shape=none style=none];
+    INT_MAX [label="+n"];
+
+    TUPLE [label="TUPLE"];
+
+    // Define edges
+    edge [arrowhead=none];
+
+    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];
+
+    TOP          -> TUPLE;
+    TUPLE        -> BOT;
+
+    TOP -> XCTRL;
+    XCTRL -> CTRL;
+    CTRL -> BOT;
+
+    // Additional customization
+    {rank=same; TOP}
+    {rank=same; INT_TOP; XCTRL}
+    {rank=same; INT_MIN; INT_MINDOTS;
+                INT_MINUS; INT_ZERO;
+                INT_PLUS; INT_MAXDOTS;
+                INT_MAX; TUPLE}
+    {rank=same; INT_BOT; CTRL}
+    {rank=same; BOT;}
+}
\ No newline at end of file
diff --git a/chapter06/docs/lattice.svg b/chapter06/docs/lattice.svg
new file mode 100644
index 00000000..713b54c1
--- /dev/null
+++ b/chapter06/docs/lattice.svg
@@ -0,0 +1,206 @@
+<?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: Lattice Pages: 1 -->
+<svg width="702pt" height="478pt"
+ viewBox="0.00 0.00 702.04 478.27" 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(4 474.27)">
+<title>Lattice</title>
+<polygon fill="white" stroke="transparent" points="-4,4 -4,-474.27 698.04,-474.27 698.04,4 -4,4"/>
+<!-- TOP -->
+<g id="node1" class="node">
+<title>TOP</title>
+<ellipse fill="lightblue" stroke="black" cx="277.5" cy="-450.77" rx="19.5" ry="19.5"/>
+<text text-anchor="middle" x="277.5" y="-447.07" font-family="Times,serif" font-size="14.00">⊤</text>
+</g>
+<!-- XCTRL -->
+<g id="node3" class="node">
+<title>XCTRL</title>
+<ellipse fill="lightblue" stroke="black" cx="46.5" cy="-354.98" rx="36.29" ry="36.29"/>
+<text text-anchor="middle" x="46.5" y="-351.28" font-family="Times,serif" font-size="14.00">~Ctrl</text>
+</g>
+<!-- TOP&#45;&gt;XCTRL -->
+<g id="edge25" class="edge">
+<title>TOP&#45;&gt;XCTRL</title>
+<path fill="none" stroke="black" d="M259.76,-442.57C221.3,-426.95 129.74,-389.78 80.01,-369.59"/>
+</g>
+<!-- INT_TOP -->
+<g id="node5" class="node">
+<title>INT_TOP</title>
+<ellipse fill="lightblue" stroke="black" cx="277.5" cy="-354.98" rx="40.09" ry="40.09"/>
+<text text-anchor="middle" x="277.5" y="-351.28" font-family="Times,serif" font-size="14.00">IntTop</text>
+</g>
+<!-- TOP&#45;&gt;INT_TOP -->
+<g id="edge1" class="edge">
+<title>TOP&#45;&gt;INT_TOP</title>
+<path fill="none" stroke="black" d="M277.5,-431.24C277.5,-420.99 277.5,-407.84 277.5,-395.42"/>
+</g>
+<!-- TUPLE -->
+<g id="node14" class="node">
+<title>TUPLE</title>
+<ellipse fill="lightblue" stroke="black" cx="650.5" cy="-235.14" rx="43.59" ry="43.59"/>
+<text text-anchor="middle" x="650.5" y="-231.44" font-family="Times,serif" font-size="14.00">TUPLE</text>
+</g>
+<!-- TOP&#45;&gt;TUPLE -->
+<g id="edge23" class="edge">
+<title>TOP&#45;&gt;TUPLE</title>
+<path fill="none" stroke="black" d="M294.49,-440.95C350.93,-408.32 533.03,-303.05 612.65,-257.02"/>
+</g>
+<!-- BOT -->
+<g id="node2" class="node">
+<title>BOT</title>
+<ellipse fill="lightblue" stroke="black" cx="277.5" cy="-19.5" rx="19.5" ry="19.5"/>
+<text text-anchor="middle" x="277.5" y="-15.8" font-family="Times,serif" font-size="14.00">⊥</text>
+</g>
+<!-- CTRL -->
+<g id="node4" class="node">
+<title>CTRL</title>
+<ellipse fill="lightblue" stroke="black" cx="50.5" cy="-115.29" rx="28.7" ry="28.7"/>
+<text text-anchor="middle" x="50.5" y="-111.59" font-family="Times,serif" font-size="14.00">Ctrl</text>
+</g>
+<!-- XCTRL&#45;&gt;CTRL -->
+<g id="edge26" class="edge">
+<title>XCTRL&#45;&gt;CTRL</title>
+<path fill="none" stroke="black" d="M47.11,-318.58C47.9,-270.77 49.29,-187.82 50.02,-144.12"/>
+</g>
+<!-- CTRL&#45;&gt;BOT -->
+<g id="edge27" class="edge">
+<title>CTRL&#45;&gt;BOT</title>
+<path fill="none" stroke="black" d="M76.98,-103.35C123.76,-84.02 219.83,-44.32 259.63,-27.88"/>
+</g>
+<!-- INT_MIN -->
+<g id="node7" class="node">
+<title>INT_MIN</title>
+<ellipse fill="lightblue" stroke="black" cx="19.5" cy="-235.14" rx="19.5" ry="19.5"/>
+<text text-anchor="middle" x="19.5" y="-231.44" font-family="Times,serif" font-size="14.00">&#45;n</text>
+</g>
+<!-- INT_TOP&#45;&gt;INT_MIN -->
+<g id="edge2" class="edge">
+<title>INT_TOP&#45;&gt;INT_MIN</title>
+<path fill="none" stroke="black" d="M241.08,-337.34C184.83,-311.65 78.92,-263.28 37.12,-244.18"/>
+</g>
+<!-- INT_MINDOTS -->
+<g id="node8" class="node">
+<title>INT_MINDOTS</title>
+<text text-anchor="middle" x="108.5" y="-231.44" font-family="Times,serif" font-size="14.00">...</text>
+</g>
+<!-- INT_TOP&#45;&gt;INT_MINDOTS -->
+<g id="edge3" class="edge">
+<title>INT_TOP&#45;&gt;INT_MINDOTS</title>
+<path fill="none" stroke="black" d="M244.91,-331.26C212.21,-308.45 162.56,-273.83 133.04,-253.25"/>
+</g>
+<!-- INT_MINUS -->
+<g id="node9" class="node">
+<title>INT_MINUS</title>
+<ellipse fill="lightblue" stroke="black" cx="197.5" cy="-235.14" rx="19.5" ry="19.5"/>
+<text text-anchor="middle" x="197.5" y="-231.44" font-family="Times,serif" font-size="14.00">&#45;1</text>
+</g>
+<!-- INT_TOP&#45;&gt;INT_MINUS -->
+<g id="edge4" class="edge">
+<title>INT_TOP&#45;&gt;INT_MINUS</title>
+<path fill="none" stroke="black" d="M255.21,-321.15C239.86,-298.54 220.15,-269.51 208.19,-251.89"/>
+</g>
+<!-- INT_ZERO -->
+<g id="node10" class="node">
+<title>INT_ZERO</title>
+<ellipse fill="lightblue" stroke="black" cx="277.5" cy="-235.14" rx="18" ry="18"/>
+<text text-anchor="middle" x="277.5" y="-231.44" font-family="Times,serif" font-size="14.00">0</text>
+</g>
+<!-- INT_TOP&#45;&gt;INT_ZERO -->
+<g id="edge5" class="edge">
+<title>INT_TOP&#45;&gt;INT_ZERO</title>
+<path fill="none" stroke="black" d="M277.5,-314.41C277.5,-293.68 277.5,-269.36 277.5,-253.39"/>
+</g>
+<!-- INT_PLUS -->
+<g id="node11" class="node">
+<title>INT_PLUS</title>
+<ellipse fill="lightblue" stroke="black" cx="356.5" cy="-235.14" rx="18" ry="18"/>
+<text text-anchor="middle" x="356.5" y="-231.44" font-family="Times,serif" font-size="14.00">1</text>
+</g>
+<!-- INT_TOP&#45;&gt;INT_PLUS -->
+<g id="edge6" class="edge">
+<title>INT_TOP&#45;&gt;INT_PLUS</title>
+<path fill="none" stroke="black" d="M299.5,-321.15C315.03,-298 335.07,-268.1 346.78,-250.63"/>
+</g>
+<!-- INT_MAXDOTS -->
+<g id="node12" class="node">
+<title>INT_MAXDOTS</title>
+<text text-anchor="middle" x="444.5" y="-231.44" font-family="Times,serif" font-size="14.00">...</text>
+</g>
+<!-- INT_TOP&#45;&gt;INT_MAXDOTS -->
+<g id="edge7" class="edge">
+<title>INT_TOP&#45;&gt;INT_MAXDOTS</title>
+<path fill="none" stroke="black" d="M310.09,-330.98C342.46,-308.14 391.33,-273.65 420.37,-253.16"/>
+</g>
+<!-- INT_MAX -->
+<g id="node13" class="node">
+<title>INT_MAX</title>
+<ellipse fill="lightblue" stroke="black" cx="539.5" cy="-235.14" rx="24.9" ry="24.9"/>
+<text text-anchor="middle" x="539.5" y="-231.44" font-family="Times,serif" font-size="14.00">+n</text>
+</g>
+<!-- INT_TOP&#45;&gt;INT_MAX -->
+<g id="edge8" class="edge">
+<title>INT_TOP&#45;&gt;INT_MAX</title>
+<path fill="none" stroke="black" d="M313.93,-337.59C369.05,-312.8 471.99,-266.5 517.22,-246.16"/>
+</g>
+<!-- INT_BOT -->
+<g id="node6" class="node">
+<title>INT_BOT</title>
+<ellipse fill="lightblue" stroke="black" cx="277.5" cy="-115.29" rx="40.09" ry="40.09"/>
+<text text-anchor="middle" x="277.5" y="-111.59" font-family="Times,serif" font-size="14.00">IntBot</text>
+</g>
+<!-- INT_BOT&#45;&gt;BOT -->
+<g id="edge16" class="edge">
+<title>INT_BOT&#45;&gt;BOT</title>
+<path fill="none" stroke="black" d="M277.5,-74.79C277.5,-62.37 277.5,-49.25 277.5,-39.01"/>
+</g>
+<!-- INT_MIN&#45;&gt;INT_BOT -->
+<g id="edge9" class="edge">
+<title>INT_MIN&#45;&gt;INT_BOT</title>
+<path fill="none" stroke="black" d="M37.23,-226.04C79.13,-206.9 184.92,-158.58 241.11,-132.91"/>
+</g>
+<!-- INT_MIN&#45;&gt;INT_MINDOTS -->
+<!-- INT_MINDOTS&#45;&gt;INT_BOT -->
+<g id="edge10" class="edge">
+<title>INT_MINDOTS&#45;&gt;INT_BOT</title>
+<path fill="none" stroke="black" d="M133.07,-217C162.55,-196.45 212.07,-161.91 244.77,-139.11"/>
+</g>
+<!-- INT_MINDOTS&#45;&gt;INT_MINUS -->
+<!-- INT_MINUS&#45;&gt;INT_BOT -->
+<g id="edge11" class="edge">
+<title>INT_MINUS&#45;&gt;INT_BOT</title>
+<path fill="none" stroke="black" d="M208.29,-218.24C220.23,-200.65 239.81,-171.81 255.1,-149.29"/>
+</g>
+<!-- INT_MINUS&#45;&gt;INT_ZERO -->
+<!-- INT_ZERO&#45;&gt;INT_BOT -->
+<g id="edge12" class="edge">
+<title>INT_ZERO&#45;&gt;INT_BOT</title>
+<path fill="none" stroke="black" d="M277.5,-217C277.5,-201.04 277.5,-176.66 277.5,-155.88"/>
+</g>
+<!-- INT_ZERO&#45;&gt;INT_PLUS -->
+<!-- INT_PLUS&#45;&gt;INT_BOT -->
+<g id="edge13" class="edge">
+<title>INT_PLUS&#45;&gt;INT_BOT</title>
+<path fill="none" stroke="black" d="M346.81,-219.69C335.11,-202.23 315.06,-172.33 299.53,-149.16"/>
+</g>
+<!-- INT_PLUS&#45;&gt;INT_MAXDOTS -->
+<!-- INT_MAXDOTS&#45;&gt;INT_BOT -->
+<g id="edge14" class="edge">
+<title>INT_MAXDOTS&#45;&gt;INT_BOT</title>
+<path fill="none" stroke="black" d="M420.22,-217C391.18,-196.51 342.45,-162.13 310.14,-139.33"/>
+</g>
+<!-- INT_MAXDOTS&#45;&gt;INT_MAX -->
+<!-- INT_MAX&#45;&gt;INT_BOT -->
+<g id="edge15" class="edge">
+<title>INT_MAX&#45;&gt;INT_BOT</title>
+<path fill="none" stroke="black" d="M517.16,-224.09C471.99,-203.77 369.42,-157.64 314.25,-132.82"/>
+</g>
+<!-- TUPLE&#45;&gt;BOT -->
+<g id="edge24" class="edge">
+<title>TUPLE&#45;&gt;BOT</title>
+<path fill="none" stroke="black" d="M612.68,-213.27C533.04,-167.23 350.75,-61.85 294.41,-29.27"/>
+</g>
+</g>
+</svg>