X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2Fweek4_fixed_point_combinators.mdwn;h=ca9ab649f6a131f43585358bca880423dabfa486;hp=601ad813bdd87e87521b80a0965e8803cbc1219a;hb=a95160473037089ea4deb34eceff948391b0cee4;hpb=a29958c3e11fa8ae383aacff54ce7f405def0705

diff --git a/topics/week4_fixed_point_combinators.mdwn b/topics/week4_fixed_point_combinators.mdwn
index 601ad813..ca9ab649 100644
--- a/topics/week4_fixed_point_combinators.mdwn
+++ b/topics/week4_fixed_point_combinators.mdwn
@@ -281,7 +281,7 @@ saying that we are looking for a fixed point for `h`:
 
     h LENGTH <~~> LENGTH
 
-Replacing `h` with its definition, we have
+Replacing `h` with its definition, we have:
 
     (\xs. (empty? xs) 0 (succ (LENGTH (tail xs)))) <~~> LENGTH
 
@@ -289,10 +289,21 @@ If we can find a value for `LENGTH` that satisfies this constraint, we'll
 have a function we can use to compute the length of an arbitrary list.
 All we have to do is find a fixed point for `h`.
 
-The strategy we will present will turn out to be a general way of
+Let's reinforce this. The left-hand side has the form:
+
+    (\body. Φ[...body...]) LENGTH
+
+which beta-reduces to:
+
+    Φ[...LENGTH...]
+
+where that whole formula is convertible with the term LENGTH itself. In other words, the term `Φ[...LENGTH...]` contains (a term that convertible with) itself --- despite being only finitely long. (If it had to contain a term *syntactically identical to* itself, this could not be achieved.)
+
+The key to achieving all this is finding a fixed point for `h`. The strategy we will present will turn out to be a general way of
 finding a fixed point for any lambda term.
 
 
+<a id=deriving-y></a>
 ## Deriving Y, a fixed point combinator ##
 
 How shall we begin?  Well, we need to find an argument to supply to
@@ -533,7 +544,7 @@ to *the tail* of the list we were evaluating its application to at the previous
 
 ## Fixed-point Combinators Are a Bit Intoxicating ##
 
-[[tatto|/images/y-combinator-fixed.jpg]]
+[[tatto|/images/y-combinator-fixed.png]]
 
 There's a tendency for people to say "Y-combinator" to refer to fixed-point combinators generally. We'll probably fall into that usage ourselves. Speaking correctly, though, the Y-combinator is only one of many fixed-point combinators.
 
@@ -553,9 +564,9 @@ then this is a fixed-point combinator:
 For those of you who like to watch ultra slow-mo movies of bullets
 piercing apples, here's a stepwise computation of the application of a
 recursive function.  We'll use a function `sink`, which takes one
-argument.  If the argument is boolean true (i.e., `\x y. x`), it
+argument.  If the argument is boolean true (i.e., `\y n. y`), it
 returns itself (a copy of `sink`); if the argument is boolean false
-(`\x y. y`), it returns `I`.  That is, we want the following behavior:
+(`\y n. n`), it returns `I`.  That is, we want the following behavior:
 
     sink false <~~> I
     sink true false <~~> I