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

1 files changed, 18 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