{-# OPTIONS_FRONTEND -Wno-missing-signatures #-}

import Prelude hiding (elem)

-- Returns the last element of a list:
last :: Data a => [a] -> a
last xs | _ ++ [e] === xs = e
  where e free

data Nat = Z | S Nat
 deriving Show

instance Eq Nat where
  x == y = True -- unintended instance!
{-
  Z   == Z   = True
  S x == S y = x == y
  Z   == S y = False
  S x == Z   = False

  Z   =:= Z   = True
  S x =:= S y = x =:= y
-}

-- Occurs an element in a list?
elem :: Eq a => a -> [a] -> Bool
elem x []     = False
elem x (y:ys) = x == y || elem x ys

-- Complex data with a unique index/hash value
data IVal a = IVal Int a
 deriving Show

instance Eq a => Eq (IVal a) where
  IVal i1 _ == IVal i2 _ = i1 == i2


-- Addition on natural numbers
add :: Nat -> Nat -> Nat
add Z     y = y
add (S x) y = S (add x y)