I've been doing some basic work on a support library for type level programming ( see http://hackage.haskell.org/trac/summer-of-code/ticket/1541 ). I know there have been similar attempts using fundeps ( Edward Kmett showed me some of his work, but I've lost the address... ) but this approach uses type families. Anyway, I would like to hear your critique! Currently I have higher order type functions and ad-hoc parametrized functions. Here's what foldl looks like: type family Foldl ( func :: * -> * -> * ) val list type instance Foldl func val ( Cons first rest ) = Foldl func ( Eval ( func val first ) ) rest type instance Foldl func val Nill = val Notice the use of Eval - this is a trick to enable us to pass around data with kind * -> *, or whatever, and then trip this into becoming a value. Here's an example, using this trick to define factorial: -- multiplication type family Times x y type instance Times x Zero = Zero type instance Times x ( Succ y ) = Sum x ( Times x y ) -- The "first order" function version of Times data TimesL x y -- Where what Eval forced TimesL to become. type instance Eval ( TimesL x y ) = Times x y -- multiplies all the elements of list of Nat together type Product l = Foldl TimesL ( Succ Zero ) l -- here list to creates a list from ( Succ Zero ) to the given number type Factorial x = Product ( ListTo x ) We can now use the function like this: *TPrelude> result ( ) :: Factorial ( Succ ( Succ ( Succ ( Zero ) ) ) ) Succ (Succ (Succ (Succ (Succ (Succ Zero))))) Using the parametrized types kinda reminds me of programming in Erlang: -- What would conventionally be the monad type class, parametized over m type family Bind m ma ( f :: * -> * ) type family Return m a type family Sequence m ma mb Here's the maybe monad: -- Monad instance type instance Bind ( Maybe t ) Nothing f = Nothing type instance Bind ( Maybe t ) ( Just a ) f = Eval ( f a ) type instance Sequence ( Maybe t ) Nothing a = Nothing type instance Sequence ( Maybe t ) ( Just a ) b = b type instance Return ( Maybe t ) a = Just a type instance Eval ( Just x ) = Just x Here's an example: *TPrelude> result ( ) :: Bind ( Maybe Nat ) ( Just Zero ) Just Just Zero For more information and to download the loose collection of module implementing this please see: http://www.killersmurf.com/projects/typelib John Morrice