data FieldElement =
  FieldElement
    { number :: Int
    , prime :: Int
    }
  deriving (Eq)

instance Show FieldElement where
  show a = "FieldElement_" ++ show (prime a) ++ " " ++ show (number a)

add :: FieldElement -> FieldElement -> FieldElement
add (FieldElement a b) (FieldElement c d)
  | b /= d = error "Distinct Fields"
  | otherwise = FieldElement (mod (a + c) b) b

sub :: FieldElement -> FieldElement -> FieldElement
sub (FieldElement a b) (FieldElement c d)
  | b /= d = error "Distinct Fields"
  | otherwise = FieldElement (mod (a - c) b) b

mul :: FieldElement -> FieldElement -> FieldElement
mul (FieldElement a b) (FieldElement c d)
  | b /= d = error "Distinct Fields"
  | otherwise = FieldElement (mod (a * c) b) b

assert :: Bool -> Bool
assert False = error "WRONG"
assert x = x

aa =
  let a = FieldElement 2 31
      b = FieldElement 15 31
   in assert
        (and
           [ add a b == FieldElement 17 31
           , a /= b
           , sub a b == FieldElement 18 31
           ])

bb =
  let a = FieldElement 19 31
      b = FieldElement 24 31
   in mul a b