`α, β, γ, ...`

as type variables. We'll
@@ -44,46 +44,55 @@ to specify which one of them the box is capturing. But let's keep it simple.) So
(Î±, R) tree (assuming R contains no free type variables)
(Î±, Î±) tree
-The idea is that whatever type the free type variable Î± might be,
-the boxed type will be a box that "contains" an object of type `Î±`.
-For instance, if `Î± list` is our box type, and `Î±` is the type
-`int`, then in this context, `int list` is the type of a boxed integer.
+The idea is that whatever type the free type variable `Î±` might be instantiated to,
+we will be a "type box" of a certain sort that "contains" values of type `Î±`. For instance,
+if `Î± list` is our box type, and `Î±` is the type `int`, then in this context, `int list`
+is the type of a boxed integer.
-Warning: although our initial motivating examples are naturally thought of as "containers" (lists, trees, and so on, with `Î±`s as their "elments"), with later examples we discuss it will less intuitive to describe the box types that way. For example, where `R` is some fixed type, `R -> Î±` is a box type.
+Warning: although our initial motivating examples are readily thought of as "containers" (lists, trees, and so on, with `Î±`s as their "elements"), with later examples we discuss it will be less natural to describe the boxed types that way. For example, where `R` is some fixed type, `R -> Î±` is a box type.
-The *box type* is the type `Î± list` (or as we might just say, `list`); the *boxed type* is some specific instantiantion of the free type variable `Î±`. We'll often write boxed types as a box containing the instance of the free
-type variable. So if our box type is `Î± list`, and `Î±` is instantiated with the specific type `int`, we would write:
+Also, for clarity: the *box type* is the type `Î± list` (or as we might just say, the `list` type operator); the *boxed type* is some specific instantiation of the free type variable `Î±`. We'll often write boxed types as a box containing what the free
+type variable instantiates to. So if our box type is `Î± list`, and `Î±` instantiates to the specific type `int`, we would write:
__Q__

is __bool__

).
Note that the left-hand schema `P` is permitted to itself be a boxed type. That is, where
if `Î± list` is our box type, we can write the second arrow as
`<=< or mcomp : (Q -> `__R__) -> (P -> __Q__) -> (P -> __R__)

-`>=> or mpmoc (m-flipcomp): (P -> `__Q__) -> (Q -> __R__) -> (P -> __R__)

+`>=> or mpmoc (flip mcomp): (P -> `__Q__) -> (Q -> __R__) -> (P -> __R__)

`>>= or mbind : (`__Q__) -> (Q -> __R__) -> (__R__)

-`=<<mdnib (or m-flipbind) (`__Q__) -> (Q -> __R__) -> (__R__)

+`=<< or mdnib (flip mbind) (`__Q__) -> (Q -> __R__) -> (__R__)

+
+`join: P -> `__P__

+
+
+Test1: `join: `__2____P__ -> __P__

-The managerie isn't quite as bewildering as you might suppose. Many of these will
+The menagerie isn't quite as bewildering as you might suppose. Many of these will
be interdefinable. For example, here is how `mcomp` and `mbind` are related: ```
k <=< j â¡
\a. (j a >>= k)
```

.
@@ -139,7 +158,7 @@ are a definition of `mid`, on the one hand, and one of `mcomp`, `mbind`, or `joi
Here are some interdefinitions: TODO. Names in Haskell TODO.
-## Examples
+## Examples ##
To take a trivial (but, as we will see, still useful) example,
consider the identity box type Id: `Î±`. So if `Î±` is type `bool`,
@@ -208,8 +227,7 @@ Contrast that to `m$` (`mapply`, which operates not on two *box-producing functi
As we illustrated in class, there are clear patterns shared between lists and option types and trees, so perhaps you can see why people want to identify the general structures. But it probably isn't obvious yet why it would be useful to do so. To a large extent, this will only emerge over the next few classes. But we'll begin to demonstrate the usefulness of these patterns by talking through a simple example, that uses the Monadic functions of the Option/Maybe box type.
-Safe division
--------------
+## Safe division ##
Integer division presupposes that its second argument
(the divisor) is not zero, upon pain of presupposition failure.
@@ -369,7 +387,7 @@ Compare the new definitions of `safe_add3` and `safe_div3` closely: the definiti
for `safe_add3` shows what it looks like to equip an ordinary operation to
survive in dangerous presupposition-filled world. Note that the new
definition of `safe_add3` does not need to test whether its arguments are
-None objects or real numbers---those details are hidden inside of the
+None values or real numbers---those details are hidden inside of the
`bind` function.
The definition of `safe_div3` shows exactly what extra needs to be said in