「モナド則だけみつめていたい」をつくった時の没ネタです。

Stateモナドを簡約すると何がおきてるかわかりやすいかなと思ったけど説明できるほどわかりやすくならなかった。

とりあえず、Stateモナドの定義

newtype State s a = State { runState :: (s -> (a,s)) } 
 
instance Monad (State s) where 
    return a        = State $ \s -> (a,s)
    (State x) >>= f = State $ \s -> let (v,s') = x s in runState (f v) s'

とりあえず、モナド則

join . fmap join = join . join
join . fmap return = join . return = id
return . f = fmap f . return
join . fmap (fmap f) = fmap f . join

return 1

State (\s1 -> (1, s1))

return (return 1)

State (\s2 -> (return 1, s2))
State (\s2 -> (State (\s1 -> (1,s1)) ,s2))

join (return (return 1))

(return (return 1)) >>= id
State (\s2 -> (State (\s1 -> (1,s1)) ,s2)) >>= id
State $ (\s3 -> let (v,s3') = (\s2 -> (State (\s1 -> (1,s1)) ,s2)) s3 in runState (id v) s3')
State $ (\s3 -> let (v,s3') = (State (\s1 -> (1,s1)), s3) in runState v s3')
State (\s3 -> runState ((State (\s1 -> (1,s1))) s3
State (\s2 -> (\s1 -> (1,s1)) s2)
State (\s2 -> (1, s2))

参考文献

http://ja.wikibooks.org/wiki/Haskell/%E5%9C%8F%E8%AB%96#.E3.83.A2.E3.83.8A.E3.83.89.E5.89.87.E3.81.A8.E3.81.9D.E3.81.AE.E9.87.8D.E8.A6.81.E6.80.A7
http://www.sampou.org/haskell/a-a-monads/html/statemonad.html