You Could Have Invented Monads! (And Maybe You Already Have.)
f :: a -> b
f' :: a -> m a
unit :: a -> m a
f' * g' = (bind f') . (bind g')
bind f xs = concat (map f xs)
bind unit xs = concat (map unit xs)
unit x = [x]
bind unit xs
= concat (map unit xs)
= concat (map unit [x1, x2, ...])
= concat [unit x1, unit x2, ...]
= concat [[x1], [x2], ...]
= [x1, x2, ...]
= xs
f' = lift f
lift f = unit . f
unit (or return) can directly act on an ordinary value only, but not on a monadic value. To act on a monadic value, you need to bind it.
How come we do not need to lift return?
f :: a -> b
liftM :: Monad m => (a -> b) -> m a -> m b
return :: a -> m a
(liftM f) :: m a -> m b
(>>=) :: Monad m => m a -> (a -> m b) -> m b
lifeM cannot be applied to return at all.
unit (or return) is neither a pure function nor a monadic function. Instead, it is an half-monadic function, meaning that while its input is an ordinary value, its output is a monadic value.
(bind return xs) -> ys
(bind return) applies to xs.
return applies to x.
liftM is merely fmap implemented with (>>=) and return …
— Wikibooks on Haskell/Understanding monads
— Me@2016-01-26 03:05:50 PM
2016.01.30 Saturday (c) All rights reserved by ACHK
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