X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2Fweek1_advanced_notes.mdwn;h=8b6289a690abb48e7b2b4f7d8795889f3aba7924;hp=2742a7fc330cbda2e135bd2f6b91c934a9aecadc;hb=8d0fcac289637208d8f22cbfba091e47ecb1a39f;hpb=536b3ca117f2fbd3fecc229e62a63a41294428d3

diff --git a/topics/week1_advanced_notes.mdwn b/topics/week1_advanced_notes.mdwn
index 2742a7fc..8b6289a6 100644
--- a/topics/week1_advanced_notes.mdwn
+++ b/topics/week1_advanced_notes.mdwn
@@ -102,7 +102,7 @@ If we get to the `y & ys` line in the pattern list, and the pattern-match succee
 Sometimes it's useful to bind variables against overlapping parts of a structure. For instance, suppose I'm writing a pattern that is to be matched against multivalues like `([10, 20], 'true)`. And suppose I want to end up with `ys` bound to `[10, 20]`, `x` bound to `10`, and `xs` bound to `[20]`. Using the techniques introduced so far, I have two options. First, I could bind `ys` against `[10, 20]`, and then initiate a second pattern-match to break that up into `10` and `[20]`. Like this:
 
     case ([10, 20], 'true) of
-      [ys, _] then case ys of
+      (ys, _) then case ys of
                      x & xs then ...;
                      ...
                    end;
@@ -112,7 +112,7 @@ Sometimes it's useful to bind variables against overlapping parts of a structure
 Alternatively, I could directly bind `x` against `10` and `xs` against `[20]`. But then I would have to re-cons them together again to get `ys`. Like this:
 
     case ([10, 20], 'true) of
-      [x & xs, _] then let
+      (x & xs, _) then let
                          ys match x & xs
                        in ...;
       ...
@@ -121,7 +121,7 @@ Alternatively, I could directly bind `x` against `10` and `xs` against `[20]`. B
 Both of these strategies work. But they are a bit inefficient. I said you didn't really need to worry about efficiency in this seminar. But these are also a bit cumbersome to write. There's a special syntax that enables us to bind all three of `ys`, `x`, and `xs` in the desired way, despite the fact that they will be matching against overlapping, rather than discrete, parts of the value `[10, 20]`. The special syntax looks like this:
 
     case ([10, 20], 'true) of
-      [(x & xs) as ys, _] then ...
+      ((x & xs) as ys, _) then ...
       ...
     end
 
@@ -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.