laws.md

Various Laws

Functor Laws

Identity

fmap id = id

Composition

fmap (g . f) = fmap g . fmap f

Applicative Functor Laws

Identity

pure id <*> v = v

Homomorphism (function application order doesn't matter)

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

Interchange (wrapping order doesn't matter for application)

u :: Applicative u <*> pure v = pure ($ v) <*> u

Composition

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

Monad Laws

Identity

pure a >>= f = f

m >>= pure = m

Associativity

(m >>= f) >>= g = (\x -> f x >>= g) Essentially stating that you can move around paren order and it won't matter Are all monads monoids? or semigroups?

Verifying instance implementations

• Implement the Arbitrary instance for your new type ( can just use the arbitrary instances of inner prelude types )

• Write a function which represents one of the laws

• Then call quickCheck which will run many different tests to verify that your implementation is correct

Sources:

• https://mmhaskell.com/monads/laws