Portability MPTCs, fundeps provisional Edward Kmett None

Description

Unlike `Free`, this is a true monad transformer.

Synopsis

newtype IterT m a Source

The monad supporting iteration based over a base monad `m`.

``` `IterT` ~ `FreeT` `Identity`
```

Constructors

 IterT FieldsrunIterT :: m (Either a (IterT m a))

Instances

iter :: Either a (Iter a) -> Iter aSource

# Operations

delay :: (Monad f, MonadFree f m) => m a -> m aSource

hoistIterT :: Monad n => (forall a. m a -> n a) -> IterT m b -> IterT n bSource

Lift a monad homomorphism from `m` to `n` into a Monad homomorphism from `IterT m` to `IterT n`.

retract :: Monad m => IterT m a -> m aSource

`retract` is the left inverse of `lift`

``` `retract` . `lift` = `id`
```

fold :: Monad m => (m a -> a) -> IterT m a -> aSource

Tear down a `Free` `Monad` using iteration.

foldM :: (Monad m, Monad n) => (m (n a) -> n a) -> IterT m a -> n aSource

Like `fold` with monadic result.

# IterT ~ FreeT Identity

class Monad m => MonadFree f m | m -> f whereSource

Monads provide substitution (`fmap`) and renormalization (`join`):

`m `>>=` f = `join` (`fmap` f m)`

A free `Monad` is one that does no work during the normalization step beyond simply grafting the two monadic values together.

`[]` is not a free `Monad` (in this sense) because `join [[a]]` smashes the lists flat.

On the other hand, consider:

``` data Tree a = Bin (Tree a) (Tree a) | Tip a
```
``` instance `Monad` Tree where
`return` = Tip
Tip a `>>=` f = f a
Bin l r `>>=` f = Bin (l `>>=` f) (r `>>=` f)
```

This `Monad` is the free `Monad` of Pair:

``` data Pair a = Pair a a
```

And we could make an instance of `MonadFree` for it directly:

``` instance `MonadFree` Pair Tree where
`wrap` (Pair l r) = Bin l r
```

Or we could choose to program with `Free Pair` instead of `Tree` and thereby avoid having to define our own `Monad` instance.

Moreover, Control.Monad.Free.Church provides a `MonadFree` instance that can improve the asymptotic complexity of code that constructs free monads by effectively reassociating the use of (`>>=`). You may also want to take a look at the `kan-extensions` package (http://hackage.haskell.org/package/kan-extensions).

See `Free` for a more formal definition of the free `Monad` for a `Functor`.

Methods

wrap :: f (m a) -> m aSource

``` wrap (fmap f x) ≡ wrap (fmap return x) >>= f