Uniplate-style traversals.

Toy example: Consider our favourite data type

 data Expr e 
   = Kst Int 
   | Var String 
   | Add e e 
   deriving (Eq,Show,Functor,Foldable,Traversable)

 instance ShowF Expr where showsPrecF = showsPrec

and write a function simplifying additions with zero:

 simplifyAdd :: Mu Expr -> Mu Expr
 simplifyAdd = transform worker where
   worker expr = case expr of
     Fix (Add x (Fix (Kst 0))) -> x    -- 0+x = x
     Fix (Add (Fix (Kst 0)) y) -> y    -- x+0 = 0
     _ -> expr

Unfortunately, all these Fix wrappers are rather ugly; but they are straightforward to put in, and in principle one could use Template Haskell quasi-quotation to generate patterns.

The list of all substructures. Together with list-comprehension syntax this is a powerful query tool. For example the following is how you get the list of all variable names in an expression:

 variables expr = [ s | Fix (Var s) <- universe expr ]