[Haskell-cafe] Re: Monads from Functors

[Ben Franksen]

Sebastian Fischer wrote:
>  > {-# LANGUAGE Rank2Types #-}
> 
> Dear Haskellers,
> 
> I just realized that we get instances of `Monad` from pointed functors
> and instances of `MonadPlus` from alternative functors.
> 
> Is this folklore?
> 
>  > import Control.Monad
>  > import Control.Applicative
> 
> In fact, every unary type constructor gives rise to a monad by the
> continuation monad transformer.
> 
>  > newtype ContT t a = ContT { unContT :: forall r . (a -> t r) -> t r }
>  >
>  > instance Monad (ContT t)
>  >  where
>  >   return x = ContT ($x)
>  >   m >>= f  = ContT (\k -> unContT m (\x -> unContT (f x) k))
> 
> Both the `mtl` package and the `transformers` package use the same
> `Monad` instance for their `ContT` type but require `t` to be an
> instance of `Monad`. Why? [^1]

Maybe because this is needed to prove monad laws?

Hm, let's try it.

  m >>= return
= ContT (\k -> unContT m (\x -> unContT (return x) k))
= ContT (\k -> unContT m (\x -> unContT (ContT ($x)) k))
= ContT (\k -> unContT m (\x -> ($x) k))
= ContT (\k -> unContT m (\x -> k x))
= ContT (\k -> unContT m k)
= ContT (unContT m)
= m

  return x >>= f
= ContT (\k -> unContT (return x) (\x' -> unContT (f x') k))
= ContT (\k -> unContT (ContT ($x)) (\x' -> unContT (f x') k))
= ContT (\k -> ($x) (\x' -> unContT (f x') k))
= ContT (\k -> (\x' -> unContT (f x') k) x)
= ContT (\k -> unContT (f x) k)
= ContT (unContT (f x))
= f x

(m >>= f) >>= g
= ContT (\k -> unContT m (\x -> unContT (f x) k)) >>= g
= ContT (\q -> unContT (ContT (\k -> unContT m (\x -> unContT (f x) k)))
(\y -> unContT (g y) q))
= ContT (\q -> (\k -> unContT m (\x -> unContT (f x) k)) (\y -> unContT (g
y) q))
= ContT (\q -> unContT m (\x -> unContT (f x) (\y -> unContT (g y) q)))
= ContT (\q -> unContT m (\x -> (\k -> unContT (f x) (\y -> unContT (g y)
k)) q))
= ContT (\q -> unContT m (\x -> unContT (ContT (\k -> unContT (f x) (\y ->
unContT (g y) k))) q))
= ContT (\q -> unContT m (\x -> unContT (f x >>= g) q))
= ContT (\q -> unContT m (\x -> unContT ((\y -> f y >>= g) x) q))
= m >>= (\y -> f y >>= g)

Uff.

So, that wasn't the reason. It really is a monad.

Cheers
Ben