{- Setup -}

data Expr =
    Number Integer
  | Plus Expr Expr
  | Negate Expr
  deriving Show

eval :: Expr -> Integer
eval (Number x) = x
eval (Plus e1 e2) = eval e1 + eval e2
eval (Negate e) = - eval e

ite True x y = x
ite False x y = y

data Nat =
    Zero
  | Plus1 Nat
  deriving Show

nat0, nat1, nat2, nat3 :: Nat
nat0 = Zero
nat1 = Plus1 nat0
nat2 = Plus1 nat1
nat3 = Plus1 nat2


{- Exercise 1 -}

conj :: Bool -> Bool -> Bool
conj a b = undefined

disj :: Bool -> Bool -> Bool
disj a b = undefined

impl :: Bool -> Bool -> Bool
impl a b = undefined

not2 :: Bool -> Bool
not2 b = undefined

{- Exercise 2 -}

neg :: Expr -> Expr
neg e = undefined

normalize :: Expr -> Expr
normalize e = undefined

showNat :: Nat -> String
showNat n = undefined

nat :: Integer -> Nat
nat n = undefined

exprToNat :: Expr -> Nat
exprToNat e = undefined


{- Tests -}

tests = do
  test "normalize (Plus (Number 2) (Negate (Number 1))) = Plus (Number 2) (Number (-1))" "Plus (Number 2) (Number (-1))" (normalize (Plus (Number 2) (Negate (Number 1))))
  test "showNat Zero = \"0\"" "\"0\"" (showNat Zero)
  test "showNat nat3 = \"1+1+1\"" "\"1+1+1\"" (showNat nat3)
  test "nat 2 = Plus1 (Plus1 Zero)" "Plus1 (Plus1 Zero)" (nat 2)
  test "showNat (nat 5) = \"1+1+1+1+1\"" "\"1+1+1+1+1\"" (showNat (nat 5))
  test "exprToNat (Plus (Number 1) (Number 1)) = nat2" "Plus1 (Plus1 Zero)" (exprToNat (Plus (Number 1) (Number 1)))
  test "exprToNat (Plus (Number 2) (Negate (Number 1))) = nat1" "Plus1 Zero" (exprToNat (Plus (Number 2) (Negate (Number 1))))
  test "conj True False" "False" (conj True False)
  test "conj True True" "True" (conj True True)
  test "disj False True" "True" (disj False True)
  test "disj False False" "False" (disj False False)
  test "impl False True" "True" (impl False True)
  test "impl True False" "False" (impl True False)
  test "not2 True" "False" (not2 True)

test name e c = do
  putStr ("*** " ++ name ++ ": ")
  if show c == e then putStrLn "OK"
  else putStrLn ("ERROR; expected '" ++ e ++ "', but found '" ++ show c ++ "'")