post(typoclassopedia): Monads#Laws

This commit is contained in:
Mahdi Dibaiee 2017-10-11 21:55:52 +03:30
parent 52fcc103e1
commit 2a22aae326

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@ -651,3 +651,46 @@ instance Monad Maybe where
fmap f a = a >>= (return . f)
join m = m >>= id
```
## Laws
Standard:
```haskell
return a >>= k = k a
m >>= return = m
m >>= (\x -> k x >>= h) = (m >>= k) >>= h
```
In terms of `>=>`:
```haskell
return >=> g = g
g >=> return = g
(g >=> h) >=> k = g >=> (h >=> k)
```
### Exercises
1. Given the definition `g >=> h = \x -> g x >>= h`, prove the equivalence of the above laws and the standard monad laws.
**Solution**:
```haskell
return >=> g
= \x -> return x >>= g
= \x -> g x
= g
g >=> return
= \x -> g x >>= return
= \x -> g x
= g
g >=> (h >=> k)
= \y -> g y >>= (\x -> h x >>= k)
= \y -> (g y >>= h) >>= k
= \y -> (\x -> g x >>= h) y >>= k
= (\x -> g x >>= h) >=> k
= (g >=> h) >=> k
```