[Haskell-beginners] Type classes and synonyms

[pbrowne]

On Sun, Nov 22, 2009 at 03:32:07PM +0000, Brent Yorgey wrote:
> A "free theorem" is a *property* which
> is satisfied by any function with a particular type.  For example, any
> function with the type
> 
>   foo :: [a] -> Maybe a
> 
> necessarily satisfies
> 
>   foo (map f xs) = fmap f (foo xs)


A theorem is a statement which has been proved on the basis of
previously established theorems or axiom. So, should the definition of
the function not satisfy the signature? I may be confusing terminology
here. I am coming of an OBJ2/Maude/CafeOBJ background to Haskell. In
CafeOBJ a *property* of an operation could, for example, be
associativity. I am having difficulty in adjusting to Haskells level of
formality.

>> In Haskell the *theorems for free* means roughly that the
>> definition of a class, instance or a function follows from the supplied
>> types because they are the only types that don’t have undefined argumens
>>  or undefined return types.

Question: Is my generalization of applying the "free theorem" concept to
classes and instances correct?

Pat