* **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,