complement :: (a -> Bool) -> a -> Bool complement p x = not (p x)
By the signature, the first argument is a function (predicate) which when given a value returns a Bool? And the second argument is just a value? And the function returns a Bool?
Indeed. In the type expression, the lower-case identifiers are type variables, while the upper-case identifiers are types. Thus, "a" could be instantiated to any type, with the constraint that both appearances of "a" are the same type.
map (complement odd) [1,2,3,4,5,6]
Typically, you would use function composition here: map (not.odd) [1..6] If you really want a seperate complement function, you could define it using a section: complement = (not.)
By similar reasoning the always function would seem to have a signature
a -> (b -> a)
where the first argument is just a value and the return value is a function that when given a possibly different value just returns the value originally given to always?
Is that reasoning OK? Are
a -> (b -> a) and a -> b -> a the same signature?
Yes. Function application (->) is right-associative in a type expression. What about a value expression? f a b === (f a) b Looks like an inconsistency? Not if you think about it. :-) Of course, this is what curried functions and partial application are all about.
So the inferred type is usually pretty accurate? These signatures are a bit confusing. Is there a good tutorial?
http://haskell.org/tutorial, particularly chapter 2.