While floating across the internets, I came across a blog with title
fmap fix return. I immediately run ghci and queried it’s type:
ghci> :t fmap fix return fmap fix return :: a -> a
We know that only value with type
forall a. a -> a (other than bottom) is identity function(
id in Haskell). I found it very interesting,
fmap fix return gives us identity function!
I got a pen and paper and started evaluating expressions to understand how does that give us the identity function.
First, let’s note our actors:
fmap :: Functor f => (a -> b) -> f a -> f b -- instance specific implementation fix :: (a -> a) -> a fix f = let x = f x in x return :: Monad m => a -> m a -- instance specific implementation
A very important thing to realize at this point is which monad and functor instances are used for
fmap. To realize this, observe that we get a function as return value of
fmap. Which means
Functor f => f b is
a -> a, or written in a different style to see it’s functor property:
((->) a) a, so our functor here is
After that, we need to look up functor and monad instances for
(->) a. Since it’s 2:30 AM here, I wanted to derive that too.
instance Functor ((->) a) where fmap = (.)
It’s very easy to derive just by looking specialized version of
fmaps type for
fmap :: (a1 -> b) -> (a -> a1) -> (a -> b).
We can easily prove that it satisfies functor laws:
fmap id f = id . f = f fmap (p . q) <-> (fmap p) . (fmap q) (fmap p) . (fmap q) f = fmap p . (q . f) = p . q . f = fmap (p . q) f
Monad instance can also be derived from types of
instance Monad ((->) a) where return = const >>= fn = \r -> fn (f r) r f
(I found Monad instance of
((->) a) very interesting, especially the
>>= part. I couldn’t come up with a problem that makes use of this instance, I’ll investigate that after some sleep)
It satisfies monad laws:
-- left identity return a >>= f = const a >>= f = \r -> f ((const a) r) r = \r -> f a r = f a -- right identity m >>= return = \r -> return (m r) r = \r -> (const (m r) r) = \r -> m r = m -- associativity (m >>= f) >>= g <-> m >>= (\x -> f x >>= g) let's first write `p` for `m >>= f` = p >>= g = \r1 -> g (p r1) r1 let's also evaluaute p p = m >>= f = \r2 -> f (m r2) r2 substitute new p = \r1 -> g ((\r2 -> f (m r2) r2) r1) r1 = \r1 -> g (f (m r1) r1) r1 = \r -> g (f (m r) r) r now let's also evaluate right hand side of equation m >>= (\x -> f x >>= g) = m >>= (\x -> ( f x >>= g )) -- just added a paren for clarity = m >>= (\x -> (\r -> g (f x r) r)) let write `p` for `\x -> (\r -> g (f x r) r)` = m >>= p = \r1 -> p (m r1) r1 put p back = \r1 -> (\x -> (\r -> g (f x r) r)) (m r1) r1 = \r1 -> (\r -> g (f (m r1) r)) r1 = \r1 -> g (f (m r1) r1)
For those who have no idea what’s above, I’m basically proving some equalities by substituting expressions with their equivalents. This is a benefit for working on a purely functional language, which means referential transparency.
OK, now with these
fmap implementations in mind, let’s evaluate
fmap fix return:
fmap fix return = fix . return = \r -> fix (return r) = \r -> fix (const r) = \r -> (\f = let x = f x in x) (const r) = \r -> let x = (const r) x in x since `const a _ = a`, we have x = r here = \r -> r
.. which is the identity function.
fmap fix return demystified.