X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2Fweek1_advanced_notes.mdwn;h=c5e589f23c678f7f54173ec964d7d641ae4aafdf;hp=2742a7fc330cbda2e135bd2f6b91c934a9aecadc;hb=54c413718dba4d80448dfb652f84ad344f0b860b;hpb=536b3ca117f2fbd3fecc229e62a63a41294428d3

diff --git a/topics/week1_advanced_notes.mdwn b/topics/week1_advanced_notes.mdwn
index 2742a7fc..c5e589f2 100644
--- a/topics/week1_advanced_notes.mdwn
+++ b/topics/week1_advanced_notes.mdwn
@@ -165,12 +165,50 @@ Function composition, which mathematicians write as `f` ○ `g`, is defined as
 
 We've already come across the `id` function, namely λ `x. x`.
 
-Other common functions are `fst`, which takes two arguments and returns the first of them; `snd`, which takes two arguments and returns the second of them; and `swap`, which takes two arguments and returns them both but with their positions swapped. These functions can be defined like this:
+Other common functions are `fst`, which takes two arguments and returns the first of them; `snd`, which takes two arguments and returns the second of them; and `swap`, which takes two arguments and returns them both but with their positions swapped. A fourth function is `dup`, which takes one argument and returns it twice.
+These functions can be defined like this:
 
     let
       fst (x, y) = x;
       snd (x, y) = y;
-      swap (x, y) = (y, x)
-    in (fst, snd, swap)
+      swap (x, y) = (y, x);
+      dup x = (x, x)
+    in (fst, snd, swap, dup)
+
+
+### Sections ###
+
+OCaml and Haskell have a convenient bit of syntax for the common case where you want a function like this:
+
+    lambda x. 10 - x
+
+or like this:
+
+    lambda x. x & ys
+
+or like this:
+
+    lambda (x, y). x + y
+
+They permit you to appreviate the first λ-expression as simply `(10 - )`. We know there's an argument missing, because the infix operator `-` demands two arguments, but we've only supplied one. So `(10 - )` expresses a function that takes an argument `x` and evaluates to `10 - x`. In other words, it expresses λ`x. 10 - x`. Similarly, `( & ys)` expresses a function that takes an argument `x` and evaluates to `x & ys`.
+
+All of this only works with infix operators like `-`, `&` and `+`. You can't write `(1 swap)` or `(swap 1)` to mean λ`x. swap (1, x)`.
+
+Can you guess what our shortcut for the last function will be? It's `( + )`. That
+expresses a function that takes two arguments `(x, y)` and evaluates to `x + y`.
+
+Wait a second, you say. Isn't that just what `+` does *already*? Why am I making a distinction between `+` and `( + )`? The difference is that bare `+` without any parentheses is an *infix* operator that comes between its arguments. Whereas when we wrap it with parentheses, it loses its special infix syntax and then just behaves like a plain variable denoting a function, like `swap`. Thus whereas we write:
+
+    x + y
+
+if we want to use `( + )`, we have to instead write:
+
+    ( + ) (x, y)
+
+It may not be obvious now why this would ever be useful, but sometimes it will be.
+
+All of these shorthands `(10 - )`, `( & ys)` and `( + )` are called "sections". I don't know exactly why.
+
+Confession: actually, what I described here diverges *a bit* from how OCaml and Haskell treat `( + )`. They wouldn't really write `( + ) (x, y)` like I did. Instead they'd write `( + ) x y`. We will look at the difference between these next week.