X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week7.mdwn;h=7544425c4bbcca2c3a1b72cf0cf088199a18586f;hp=8fb3a3cbc1cf3d3832b97aa2703c864aa62e38ec;hb=d9e25980d9b3e62e89ab731ed5fbc33126df57e6;hpb=b3fcd749ac69521ca598d3846043bb7aff1ac290;ds=sidebyside

diff --git a/week7.mdwn b/week7.mdwn
index 8fb3a3cb..7544425c 100644
--- a/week7.mdwn
+++ b/week7.mdwn
@@ -402,11 +402,16 @@ them from hurting the people that use them or themselves.
 
 *	**Associativity: bind obeys a kind of associativity**. Like this:
 
-		(u >>= f) >>= g == u >>= (fun x -> f x >>= g)
+		(u >>= f) >>= g  ==  u >>= (fun x -> f x >>= g)
 
 	If you don't understand why the lambda form is necessary (the
 	"fun x -> ..." part), you need to look again at the type of `bind`.
 
+	Wadler and others try to make this look nicer by phrasing it like this,
+	where U, V, and W are schematic for any expressions with the relevant monadic type:
+
+		(U >>= fun x -> V) >>= fun y -> W  ==  U >>= fun x -> (V >>= fun y -> W)
+
 	Some examples of associativity in the Option monad (bear in
 	mind that in the Ocaml implementation of integer division, 2/3
 	evaluates to zero, throwing away the remainder):
@@ -426,15 +431,15 @@ them from hurting the people that use them or themselves.
 		# Some 3 >>= (fun x -> divide 2 x >>= divide 6);;
 		- : int option = None
 
-Of course, associativity must hold for *arbitrary* functions of
-type `'a -> 'b m`, where `m` is the monad type.  It's easy to
-convince yourself that the `bind` operation for the Option monad
-obeys associativity by dividing the inputs into cases: if `u`
-matches `None`, both computations will result in `None`; if
-`u` matches `Some x`, and `f x` evalutes to `None`, then both
-computations will again result in `None`; and if the value of
-`f x` matches `Some y`, then both computations will evaluate
-to `g y`.
+	Of course, associativity must hold for *arbitrary* functions of
+	type `'a -> 'b m`, where `m` is the monad type.  It's easy to
+	convince yourself that the `bind` operation for the Option monad
+	obeys associativity by dividing the inputs into cases: if `u`
+	matches `None`, both computations will result in `None`; if
+	`u` matches `Some x`, and `f x` evalutes to `None`, then both
+	computations will again result in `None`; and if the value of
+	`f x` matches `Some y`, then both computations will evaluate
+	to `g y`.
 
 *	**Right identity: unit is a right identity for bind.**  That is, 
 	`u >>= unit == u` for all monad objects `u`.  For instance,
@@ -461,10 +466,12 @@ Theory](/advanced_topics/monads_in_category_theory).
 
 See also:
 
-*	[Haskell Wikibook on Monad Laws](http://www.haskell.org/haskellwiki/Monad_Laws).
-*	[Haskell Wikibook on Understanding Monads](http://en.wikibooks.org/wiki/Haskell/Understanding_monads)
-*	[Haskell Wikibook on Advanced Monads](http://en.wikibooks.org/wiki/Haskell/Advanced_monads)
-*	[Haskell Wikibook on do-notation](http://en.wikibooks.org/wiki/Haskell/do_Notation)
+*	[Haskell wikibook on Monad Laws](http://www.haskell.org/haskellwiki/Monad_Laws).
+*	[Yet Another Haskell Tutorial on Monad Laws](http://en.wikibooks.org/wiki/Haskell/YAHT/Monads#Definition)
+*	[Haskell wikibook on Understanding Monads](http://en.wikibooks.org/wiki/Haskell/Understanding_monads)
+*	[Haskell wikibook on Advanced Monads](http://en.wikibooks.org/wiki/Haskell/Advanced_monads)
+*	[Haskell wikibook on do-notation](http://en.wikibooks.org/wiki/Haskell/do_Notation)
+*	[Yet Another Haskell Tutorial on do-notation](http://en.wikibooks.org/wiki/Haskell/YAHT/Monads#Do_Notation)
 
 
 Here are some papers that introduced monads into functional programming: