`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. It won't work in
+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.)
* **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):
# 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,
different types. But it's possible to make the connection between
monads and monoids much closer. This is discussed in [Monads in Category
Theory](/advanced_topics/monads_in_category_theory).
-See also <http://www.haskell.org/haskellwiki/Monad_Laws>.
+
+See also:
+
+* [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: