I'm learning Haskell, and one exercise was to implement squashing of identity-multiplications, e.g.: 

\$5 + (2*1) => 5 + 2\$
   
Here is my implementation:
    
    squashMulId :: (Ring a, Eq a) => RingExpr a -> RingExpr a
    squashMulId (Mul x y)
        | x == (Lit (mulId))    = squashMulId y
        | y == (Lit (mulId))    = squashMulId x
        | otherwise             = if sqX == x && sqY == y then (Mul sqX sqY) else squashMulId (Mul sqX sqY)
                                  where sqX = squashMulId x
                                        sqY = squashMulId y
    squashMulId (AddInv x)      = (AddInv (squashMulId x))
    squashMulId (Add x y)       = (Add (squashMulId x) (squashMulId y))
    squashMulId x               = x
    
    squashMulIdInt :: RingExpr Integer -> RingExpr Integer
    squashMulIdInt = squashMulId
    
    tests_squash = test [
                          "right" ~: (Add (Lit 5) (Lit 2)) ~=? squashMulIdInt (Add (Lit 5) (Mul (Lit 2) (Lit 1)) )
                         ,"left" ~: (Add (Lit 5) (Lit 2)) ~=? squashMulIdInt (Add (Lit 5) (Mul (Lit 1) (Lit 2)) )
                         ,"recursive" ~: (Add (Lit 5) (Lit 2)) ~=? squashMulIdInt (Add (Lit 5) (Mul (Lit 1) (Mul (Lit 2) (Mul (Lit 1) (Lit 1)))))
                        ]


How can I improve on this implementation? Is it idiomatic? Can it be written in a more succinct way?


The following code/definitions were given:

    {-# OPTIONS_GHC -Wall #-}
    module SQMULID where
    
    import Data.Char
    import Data.List
    import Data.Maybe
    import Test.HUnit
    import Test.QuickCheck
    import Test.QuickCheck.Gen
    
    
    class Ring a where
      addId  :: a            -- additive identity
      addInv :: a -> a       -- additive inverse
      mulId  :: a            -- multiplicative identity
    
      add :: a -> a -> a     -- addition
      mul :: a -> a -> a     -- multiplication
    
    -- | A datatype for storing and manipulating ring expressions.
    data RingExpr a = Lit a
                    | AddId
                    | AddInv (RingExpr a)
                    | MulId
                    | Add (RingExpr a) (RingExpr a)
                    | Mul (RingExpr a) (RingExpr a)
      deriving (Show, Eq)
    
    instance Ring (RingExpr a) where
      addId  = AddId
      addInv = AddInv
      mulId  = MulId
    
      add = Add
      mul = Mul
    
    -- The canonical instance for integers:
    instance Ring Integer where
      addId  = 0
      addInv = negate
      mulId  = 1
    
      add = (+)
      mul = (*)