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+module HW3 where
+
+import Prelude hiding (Enum(..), sum)
+import Data.List (group)
+import Data.Bifunctor (Bifunctor(second))
+
+--
+-- * Part 1: Run-length lists
+--
+
+-- | Convert a regular list into a run-length list.
+--
+--   >>> compress [1,1,1,2,3,3,3,1,2,2,2,2]
+--   [(3,1),(1,2),(3,3),(1,1),(4,2)]
+-- 
+--   >>> compress "Mississippi"
+--   [(1,'M'),(1,'i'),(2,'s'),(1,'i'),(2,'s'),(1,'i'),(2,'p'),(1,'i')]
+--
+compress :: Eq a => [a] -> [(Int,a)]
+compress = map ((>>=id) $ fmap ((second head) .) (,) . length) . group
+
+-- Yeah this isn't so good pointfree.
+-- I don't normally write code like this, believe me.
+--
+-- So we have:
+--     >>= id = join :: m (m a) -> m a = (a -> a -> b) -> a -> b
+--     temp1 = (,) . length :: [a] -> [a] -> (Int, [a])
+--         (make a tuple and find the length)
+--     temp2 = fmap ((second head) .) temp1 :: [a] -> [a] -> (Int, a)
+--         (transform the tuple's second element into its head)
+--     temp3 = join temp2 :: [a] -> (Int, a)
+--         (ensure both parts of the tuple are the same list)
+
+-- | Convert a run-length list back into a regular list.
+--
+--   >>> decompress [(5,'a'),(3,'b'),(4,'c'),(1,'a'),(2,'b')]
+--   "aaaaabbbccccabb"
+--  
+decompress :: [(Int,a)] -> [a]
+decompress = concatMap (uncurry replicate)
+
+
+--
+-- * Part 2: Natural numbers
+--
+
+-- | The natural numbers.
+data Nat
+   = Zero
+   | Succ Nat
+  deriving (Eq,Show)
+
+fold :: a -> (a -> a) -> Nat -> a
+fold b f Zero = b
+fold b f (Succ n) = f (fold b f n)
+
+-- | The number 1.
+one :: Nat
+one = Succ Zero
+
+-- | The number 2.
+two :: Nat
+two = Succ one
+
+-- | The number 3.
+three :: Nat
+three = Succ two
+
+-- | The number 4.
+four :: Nat
+four = Succ three
+
+
+-- | The predecessor of a natural number.
+--   
+--   >>> pred Zero
+--   Zero
+--   
+--   >>> pred three
+--   Succ (Succ Zero)
+--   
+pred = snd . fold (id, Zero) ((,) Succ . uncurry ($))
+
+
+-- | True if the given value is zero.
+--
+--   >>> isZero Zero
+--   True
+--
+--   >>> isZero two
+--   False
+--
+isZero = fold True (const False)
+
+
+-- | Convert a natural number to an integer. NOTE: We use this function in
+--   tests, but you should not use it in your other definitions!
+--
+--   >>> toInt Zero
+--   0
+--
+--   >>> toInt three
+--   3
+--
+toInt = fold 0 (+1)
+
+
+-- | Add two natural numbers.
+--
+--   >>> add one two
+--   Succ (Succ (Succ Zero))
+--
+--   >>> add Zero one == one
+--   True
+--
+--   >>> add two two == four
+--   True
+--
+--   >>> add two three == add three two
+--   True
+--   
+add = flip fold Succ
+
+
+-- | Subtract the second natural number from the first. Return zero
+--   if the second number is bigger.
+--
+--   >>> sub two one
+--   Succ Zero
+--   
+--   >>> sub three one
+--   Succ (Succ Zero)
+--
+--   >>> sub one one
+--   Zero
+--
+--   >>> sub one three
+--   Zero
+--
+sub = flip $ fold id ((pred.))
+
+-- | Is the left value greater than the right?
+--
+--   >>> gt one two
+--   False
+--
+--   >>> gt two one
+--   True
+--
+--   >>> gt two two
+--   False
+--
+gt = fmap (fmap (not . (==Zero))) sub
+
+-- | Multiply two natural numbers.
+--
+--   >>> mult two Zero
+--   Zero
+--
+--   >>> mult Zero three
+--   Zero
+--
+--   >>> toInt (mult two three)
+--   6
+--
+--   >>> toInt (mult three three)
+--   9
+--
+mult = fold Zero . add
+
+
+-- | Compute the sum of a list of natural numbers.
+--
+--   >>> sum []
+--   Zero
+--   
+--   >>> sum [one,Zero,two]
+--   Succ (Succ (Succ Zero))
+--
+--   >>> toInt (sum [one,two,three])
+--   6
+--
+sum :: [Nat] -> Nat -- Monomorphism restriction hits here
+sum = foldl add Zero
+
+
+-- | An infinite list of all of the *odd* natural numbers, in order.
+--
+--   >>> map toInt (take 5 odds)
+--   [1,3,5,7,9]
+--
+--   >>> toInt (sum (take 100 odds))
+--   10000
+--
+odds = one : map (Succ . Succ) odds