a-conjecture-of-mine

diff --git a/Haskell/app/Main.hs b/Haskell/app/Main.hs
@@ -5,13 +5,13 @@
 
 module Main where
 
-import Numeric
+import Numeric (readDec)
 import Numeric.Natural
 import System.Environment
 import System.Exit
 import Control.Monad (foldM)
-import Data.Map (Map, insert, lookup, empty)
-import qualified Data.Map as Map
+import Data.Map (Map, insert, empty)
+import qualified Data.Map
 
 main :: IO Int
 main = do
@@ -19,12 +19,13 @@ main = do
 
     if length args > 0
         then case readDec (args !! 0) :: [(Natural, String)] of
-            [(max, "")] -> if counter' max empty then exitFailure else exitSuccess
-                    
+            [(max, "")] ->
+                if counter' max then exitFailure else exitSuccess
             _ -> exitInvalidInput
             
         else exitInvalidInput
 
+-- Calculates the sum of the digits of `n`.
 sum' :: Natural -> Int
 sum' n
     | n < 10 = fromEnum n
@@ -46,19 +47,28 @@ test' a b sums =
         Nothing -> retry a
 
     where retry x = test' a b $ insert x (sum' x) sums
-          lookup x = Map.lookup x sums
+          lookup x = Data.Map.lookup x sums
 
-counter' :: Natural -> Map Natural Int -> Bool
-counter' max sums =
-    case foldM f (max, sums) [0..max] of
-        Nothing  -> True
-        Just _   -> False
-    where f (a, sums) b = iter' b max sums
+-- Checks if there is any counterexample in
+-- [(a, b) | a <- [0..max], b <- [0..max]].
+--
+-- Returns `True` if a counter example was found.
+-- Otherwise returns `False`.
+counter' :: Natural -> Bool
+counter' max =
+    case foldM f empty [0..max] of
+        Nothing -> True
+        Just _  -> False
+    where f accSums a = iter' a max accSums
 
-iter' :: Natural -> Natural -> Map Natural Int -> Maybe (Natural, Map Natural Int)
-iter' a max sums = foldM f sums [a..max] >>= continue
-    where continue updated = Just (a - 1, updated)
-          f sums b = test' a b sums
+-- Checks if there is any counter example in [(a, b)| b <- [a..max]].
+-- Returns `Nothing` if a counter example was found.
+--
+-- Otherwise returns `Just accSums`, where `accSums` maps every
+-- computed value to the sum of it's digits.
+iter' :: Natural -> Natural -> Map Natural Int -> Maybe (Map Natural Int)
+iter' a max sums = foldM f sums [a..max]
+    where f accSums b = test' a b accSums
 
 exitInvalidInput :: IO Int
-exitInvalidInput = exitWith (ExitFailure 2)-
\ No newline at end of file
+exitInvalidInput = exitWith $ ExitFailure 2+
\ No newline at end of file