- specifying how to unwrap the monadic object `m`. In the bind
- opertor for the option monad, we unwraped the option monad by
- matching the monadic object `m` with `Some n`--whenever `m`
- happend to be a box containing an integer `n`, this allowed us to
- get our hands on that `n` and feed it to `f`.
-
-So the "option monad" consists of the polymorphic option type, the
-unit function, and the bind function. With the option monad, we can
-think of the "safe division" operation
-
-<pre>
-# let divide num den = if den = 0 then None else Some (num/den);;
-val divide : int -> int -> int option = <fun>
-</pre>
-
-as basically a function from two integers to an integer, except with
-this little bit of option frill, or option plumbing, on the side.
-
-A note on notation: Haskell uses the infix operator `>>=` to stand
-for `bind`. I really hate that symbol. Following Wadler, I prefer to
-infix five-pointed star, or on a keyboard, `*`.
-
-
-The Monad laws
+ specifying how to unbox the monadic value `u`. In the `bind`
+ operator for the Option monad, we unboxed the monadic value by
+ matching it with the pattern `Some x`---whenever `u`
+ happened to be a box containing an integer `x`, this allowed us to
+ get our hands on that `x` and feed it to `f`.
+
+ If the monadic box didn't contain any ordinary value,
+ we instead pass through the empty box unaltered.
+
+ In a more complicated case, like our whimsical "singing box" example
+ from before, if the monadic value happened to be a singing box
+ containing an integer `x`, then the `bind` operation would probably
+ be defined so as to make sure that the result of `f x` was also
+ a singing box. If `f` also wanted to insert a song, you'd have to decide
+ whether both songs would be carried through, or only one of them.
+ (Are you beginning to realize how wierd and wonderful monads
+ can be?)
+
+ There is no single `bind` function that dictates how this must go.
+ For each new monadic type, this has to be worked out in an
+ useful way.
+
+So the "Option/Maybe monad" consists of the polymorphic `option` type, the
+`unit`/return function, and the `bind` function.
+
+
+A note on notation: Haskell uses the infix operator `>>=` to stand for
+`bind`: wherever you see `u >>= f`, that means `bind u f`.
+Wadler uses ⋆, but that hasn't been widely adopted (unfortunately).
+
+Also, if you ever see this notation:
+
+ do
+ x <- u
+ f x
+
+That's a Haskell shorthand for `u >>= (\x -> f x)`, that is, `bind u f`.
+Similarly:
+
+ do
+ x <- u
+ y <- v
+ f x y
+
+is shorthand for `u >>= (\x -> v >>= (\y -> f x y))`, that is, `bind u
+(fun x -> bind v (fun y -> f x y))`. Those who did last week's
+homework may recognize this last expression. You can think of the
+notation like this: take the singing box `u` and evaluate it (which
+includes listening to the song). Take the int contained in the
+singing box (the end result of evaluting `u`) and bind the variable
+`x` to that int. So `x <- u` means "Sing me up an int, which I'll call
+`x`".
+
+(Note that the above "do" notation comes from Haskell. We're mentioning it here
+because you're likely to see it when reading about monads. (See our page on [[Translating between OCaml Scheme and Haskell]].) It won't work in
+OCaml. In fact, the `<-` symbol already means something different in OCaml,
+having to do with mutable record fields. We'll be discussing mutation someday
+soon.)
+
+As we proceed, we'll be seeing a variety of other monad systems. For example, another monad is the List monad. Here the monadic type is:
+
+ # type 'a list
+
+The `unit`/return operation is:
+
+ # let unit x = [x];;
+ val unit : 'a -> 'a list = <fun>
+
+That is, the simplest way to lift an `'a` into an `'a list` is just to make a
+singleton list of that `'a`. Finally, the `bind` operation is:
+
+ # let bind u f = List.concat (List.map f u);;
+ val bind : 'a list -> ('a -> 'b list) -> 'b list = <fun>
+
+What's going on here? Well, consider `List.map f u` first. This goes through all
+the members of the list `u`. There may be just a single member, if `u = unit x`
+for some `x`. Or on the other hand, there may be no members, or many members. In
+any case, we go through them in turn and feed them to `f`. Anything that gets fed
+to `f` will be an `'a`. `f` takes those values, and for each one, returns a `'b list`.
+For example, it might return a list of all that value's divisors. Then we'll
+have a bunch of `'b list`s. The surrounding `List.concat ( )` converts that bunch
+of `'b list`s into a single `'b list`:
+
+ # List.concat [[1]; [1;2]; [1;3]; [1;2;4]]
+ - : int list = [1; 1; 2; 1; 3; 1; 2; 4]
+
+So now we've seen two monads: the Option/Maybe monad, and the List monad. For any
+monadic system, there has to be a specification of the complex monad type,
+which will be parameterized on some simpler type `'a`, and the `unit`/return
+operation, and the `bind` operation. These will be different for different
+monadic systems.
+
+Many monadic systems will also define special-purpose operations that only make
+sense for that system.
+
+Although the `unit` and `bind` operation are defined differently for different
+monadic systems, there are some general rules they always have to follow.
+
+
+The Monad Laws