Boolean equation simplifier in Haskell

simplifier.hs

module Main where

data Cuit
  = Cuit :+: Cuit 
  | Cuit :*: Cuit 
  | Neg Cuit
  | Var String

(-|) = Neg

instance Show Cuit where 
  show (a :+: b) = "(" ++ show a ++ "+" ++ show b ++ ")"
  show (a :*: b) = "(" ++ show a ++ "." ++ show b ++ ")"
  show (Neg a)   = show a ++ "'"
  show (Var a)   = a

instance Eq Cuit where 
  (a :+: b) == (a1 :+: b1) = (a == a1) && (b == b1)
  (a :*: b) == (a1 :*: b1) = (a == a1) && (b == b1)
  (Neg a) == (Neg b) = a == b
  (Var a) == (Var b) = a == b

search :: Eq a => [(a,b)] -> a -> Maybe b 
search list name = 
  let res = filter ((== name) . fst) list in
  if null res then
    Nothing
  else
    Just $ snd $ head res

solve :: Cuit -> [(String, Cuit)] -> Cuit 

solve (a :+: b) list = 
  case (solve a list, solve b list) of 
    (la :*: ra, lb :*: rb) | la == lb -> la :*: (ra :+: rb)
    (a,b) -> a :+: b

solve (a :*: b) list = solve a list :*: solve b list
solve (Neg a)   list = Neg (solve a list)
solve (Var a)   list =
  case search list a of 
    Just a -> solve a list 
    Nothing -> Var a

-- Definitions

main :: IO ()
main = print $ 
  solve (Var "X" :+: Var "Y") 
        [("X", Var "W" :*: Var "a")
       ,("Y", Var "W" :*: (-|) (Var "b"))
       ,("W", Var "U" :+: Var "V")
       ,("U", Var "T" :+: (-|) (Var "a"))
       ,("V", Var "T" :*: (-|) (Var "b"))
       ,("T", Var "b" :+: (-|) (Var "a"))]