add some haskell programs to demonstrate numbers

haskell/.gitignore

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+main
+*.hi
+*.o

haskell/Games.hs

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+module Games where
+
+import qualified Data.Set as S
+import Data.Foldable.WithIndex
+import Data.Group
+
+
+data Game = Game (S.Set Game) (S.Set Game)
+  deriving (Eq, Show, Ord)
+
+zero, one, two :: Game
+zero = game 0
+one = game 1
+two = game 2
+
+-- | Construct the game representing the natural number n
+game :: (Num t, Ord t) => t -> Game
+game n
+  | n == 0 = Game S.empty S.empty
+  | n > 0 = Game (S.singleton $ game (n-1)) S.empty
+
+-- | Construct the n-th nimber
+nimber n
+  | n == 0 = zero
+  | n > 0 = Game (S.fromList $ map nimber [0..(n-1)]) (S.fromList $ map nimber [0..(n-1)])
+
+-- | Add games
+add :: Game -> Game -> Game
+add g1@(Game l1 r1) g2@(Game l2 r2) =
+  Game (S.union (S.map (add g2) l1 ) (S.map (add g1) l2))
+       (S.union (S.map (add g2) r1) (S.map (add g1) r2))
+
+-- | Negative of a game
+neg :: Game -> Game
+neg (Game l r) = Game (S.map neg r) (S.map neg l)
+
+-- | Compare games with <=
+leq :: Game -> Game -> Bool
+leq g1@(Game l1 r1) g2@(Game l2 r2) = none (\r -> leq r g1) r2 && none (\l -> leq g2 l) l1
+
+-- | Equality of games
+eq :: Game -> Game -> Bool
+eq g h = leq g h && leq h g
+
+
+-- | Make games a semigroup
+instance Semigroup Game where
+  (<>) = add
+
+instance Monoid Game where
+  mempty = zero
+
+instance Group Game where
+  invert = neg

haskell/Makefile

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+main:
+	ghc -dynamic --make main.hs

haskell/main.hs

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+import Games
+
+
+main :: IO ()
+main = do
+  print $ game 1 <> game 1
+  print $ game 2
+  print $ leq zero one
+  print $ leq zero zero
+  print $ leq one two
+  print $ leq zero two
+  print $ leq (nimber 1) zero
+  print $ leq zero (nimber 1)
+  print $ eq (nimber 1 <> nimber 1) zero
+  print $ eq (nimber 5 <> nimber 3) $ nimber 6
+  print $ eq (nimber 5 <> nimber 1) $ nimber 4
+  print $ eq (game 1 <> game 4) $ game 5
+  print $ eq zero (add (neg (game 10)) (game 10))
+  print $ eq (nimber 5) (neg $ nimber 5)