class (Lattice a) <= BooleanAlgebra a where
  complement :: a -> a

instance booleanBoolean :: BooleanAlgebra Boolean

booleanNegate :: Boolean -> Boolean
not :: forall b. (BooleanAlgebra b) => b -> b

class (JoinSemilattice l) <= LowerSemilattice l where
  bottom :: l
class (MeetSemilattice l) <= UpperSemilattice l where
  top :: l

instance lowerSemilatticeBoolean :: LowerSemilattice Boolean
instance upperSemilatticeBoolean :: UpperSemilattice Boolean

class (AbelianAddMonoid g, AddGroup g) <= AbelianAddGroup g where
class (AbelianMonoid g, Group g) <= AbelianGroup g where
class (AbelianMultMonoid g, MultGroup g) <= AbelianMultGroup g where
class (AddMonoid g) <= AddGroup g where
  negate :: g -> g
  (-) :: g -> g -> g
class (Monoid g) <= Group g where
  inverse :: g -> g
class (MultMonoid g) <= MultGroup g where
  recip :: g -> g
  (/) :: g -> g -> g

instance abelianAddGroup :: AbelianAddGroup Number
instance abelianMultGroup :: AbelianMultGroup Number
instance additiveNumber :: AddGroup Number
instance multiplicativeNumber :: MultGroup Number

numDivide :: Number -> Number -> Number
numMinus :: Number -> Number -> Number

class (LowerSemilattice l, UpperSemilattice l) <= Lattice l where

instance latticeBoolean :: Lattice Boolean

class (LeftModule m r, RightModule m r) <= Module m r where

class (AbelianAddSemigroup m, AddMonoid m) <= AbelianAddMonoid m where
class (AbelianSemigroup m, Monoid m) <= AbelianMonoid m where
class (AbelianMultSemigroup m, MultMonoid m) <= AbelianMultMonoid m where
class (AddSemigroup m) <= AddMonoid m where
  zero :: m
class (Semigroup m) <= Monoid m where
  mempty :: m
class (MultSemigroup m) <= MultMonoid m where
  one :: m

instance abelianAddMonoid :: AbelianAddMonoid Number
instance abelianMultMonoid :: AbelianMultMonoid Number
instance additiveNumber :: AddMonoid Number
instance monoidString :: Monoid String
instance multiplicativeNumber :: MultMonoid Number

class (AbelianAddMonoid r, MultMonoid r) <= Rig r where
class (AbelianAddGroup r, MultMonoid r) <= Ring r where
class (AbelianAddGroup r, MultSemigroup r) <= Rng r where

instance ringNumber :: Ring Number

class (AddSemigroup s) <= AbelianAddSemigroup s where
class (MultSemigroup s) <= AbelianMultSemigroup s where
class (Semigroup s) <= AbelianSemigroup s where
class AddSemigroup s where
  (+) :: s -> s -> s
class MultSemigroup s where
  (*) :: s -> s -> s
class Semigroup s where
  (<>) :: s -> s -> s

instance abelianAddSemigroup :: AbelianAddSemigroup Number
instance abelianMultSemigroup :: AbelianMultSemigroup Number
instance additiveNumber :: AddSemigroup Number
instance multiplicativeNumber :: MultSemigroup Number
instance semigroupString :: Semigroup String

numPlus :: Number -> Number -> Number
numTimes :: Number -> Number -> Number
strConcat :: String -> String -> String

class JoinSemilattice j where
  (\/) :: j -> j -> j
class MeetSemilattice m where
  (/\) :: m -> m -> m

instance joinSemilatticeBoolean :: JoinSemilattice Boolean
instance meetSemilatticeBoolean :: MeetSemilattice Boolean

booleanAnd :: Boolean -> Boolean -> Boolean
booleanOr :: Boolean -> Boolean -> Boolean

class (AbelianAddGroup m, Ring r) <= LeftModule m r where
  (*<) :: r -> m -> m

class (AbelianAddGroup m, Ring r) <= RightModule m r where
  (>*) :: m -> r -> m