Applicative package:LambdaHack

A functor with application, providing operations to

• embed pure expressions (pure), and
• sequence computations and combine their results (<*> and liftA2).

A minimal complete definition must include implementations of pure and of either <*> or liftA2. If it defines both, then they must behave the same as their default definitions:

(<*>) = liftA2 id

liftA2 f x y = f <$> x <*> y

Further, any definition must satisfy the following:

• Identity

  pure id <*> v =
  v

• Composition

  pure (.) <*> u
  <*> v <*> w = u <*> (v
  <*> w)

• Homomorphism

  pure f <*>
  pure x = pure (f x)

• Interchange

  u <*> pure y =
  pure ($ y) <*> u

The other methods have the following default definitions, which may be overridden with equivalent specialized implementations:

• u *> v = (id <$ u)
  <*> v

• u <* v = liftA2 const u v

As a consequence of these laws, the Functor instance for f will satisfy

• fmap f x = pure f <*> x

It may be useful to note that supposing

forall x y. p (q x y) = f x . g y

it follows from the above that

liftA2 p (liftA2 q u v) = liftA2 f u . liftA2 g v

If f is also a Monad, it should satisfy

• pure = return

• m1 <*> m2 = m1 >>= (\x1 -> m2
  >>= (\x2 -> return (x1 x2)))

• (*>) = (>>)

(which implies that pure and <*> satisfy the applicative functor laws).