# 5. Recursion schemes¶

Traversing an EADT explicitly (see Explicit recursive traversal) can be
tedious. Another approach consists in using dedicated composable combinators
called **recursion schemes**.

The well known `map`

and `fold`

functions are examples of recursion schemes
for lists: these functions handle the recursive traversal of the data structure
and are parameterized by the functions performing the actual work. Recursion
schemes are a generalization of this approach.

The best introduction to recursion schemes I’ve read can be found here: https://blog.sumtypeofway.com/an-introduction-to-recursion-schemes/

To avoid paraphrasing, I recommend that you read it before continuing.

See also: https://bartoszmilewski.com/2013/06/10/understanding-f-algebras/

## 5.1. Catamorphism: Show example¶

Suppose we rewrite our `Show`

class like this:

```
class FunctorShow (f :: * -> *) where
functorShow :: f String -> String
```

We can define instances for `NilF`

and `ConsF`

:

```
instance FunctorShow NilF where
functorShow _ = "Nil"
instance (Show a) => FunctorShow (ConsF a) where
functorShow (ConsF a l) = show a ++ " : " ++ l
```

Note that there is no recursive call in the definition of `ConsF`

’s instance:
it is because we are going to use a recursion scheme that will handle the
recursion.

We also need an instance to handle the generic `VariantF`

type:

```
instance (AlgVariantF FunctorShow String xs) => FunctorShow (VariantF xs) where
functorShow = algVariantF @FunctorShow functorShow
```

Finally we can define a generic `eadtShow`

function that uses the catamorphism
recursion scheme with the `functorShow`

class method.

```
eadtShow ::
( Functor (VariantF xs)
, FunctorShow (VariantF xs)
) => EADT xs -> String
eadtShow = cata functorShow
```

We can test it:

```
intList :: List Int
intList = Cons (10 :: Int) $ Cons (20 :: Int) $ Cons (30 :: Int) Nil
mixedList :: EADT '[ConsF Int, ConsF Float, ConsF String, NilF]
mixedList = Cons @Int 10 $ Cons @Float 5.0 $ Cons "Test" Nil
> putStrLn $ eadtShow intList
10 : 20 : 30 : Nil
> putStrLn $ eadtShow mixedList
10 : 5.0 : "Test" : Nil
```

## 5.2. Catamorphism: List (a -> a) mapping example¶

Similarily to the example above, suppose that we want to implement mapping over an EADT list. We can use the following type-class:

```
class MapEADT a xs (f :: * -> *) where
-- map the outer constructor of an EADT
mapEADT1 :: (a -> a) -> f (EADT xs) -> EADT xs
```

We need some instances to handle our EADT constructors:

```
instance (NilF :<: xs) => MapEADT a xs NilF where
mapEADT1 _ NilF = Nil
instance (ConsF a :<: xs) => MapEADT a xs (ConsF a) where
mapEADT1 f (ConsF a x) = Cons (f a) x
```

And a additional instance to traverse the `VariantF`

combinator datatype:

```
instance (AlgEADT (MapEADT a xs) xs) => MapEADT a xs (VariantF xs) where
mapEADT1 f = algVariantF @(MapEADT a xs) (mapEADT1 f)
```

Now we can define the `mapEADT`

function by using the catamorphism combinator:

```
-- recursively map an EADT
mapEADT :: ( Functor (VariantF xs)
, MapEADT a xs (VariantF xs)
) => (a -> a) -> EADT xs -> EADT xs
mapEADT f = cata (mapEADT1 f)
```

We can test it:

```
intList :: List Int
intList = Cons (10 :: Int) $ Cons (20 :: Int) $ Cons (30 :: Int) Nil
> putStrLn $ eadtShow $ mapEADT ((+5) :: Int -> Int) intList
15 : 25 : 35 : Nil
```

## 5.3. Catamorphism: List (a -> b) mapping example¶

Similarily, we can also support mapping with a function that changes the EADT type as follow:

```
class TransEADT a b xs xs' (f :: * -> *) where
transEADT1 :: (a -> b) -> f (EADT xs) -> EADT xs'
instance (NilF :<: xs') => TransEADT a b xs xs' NilF where
transEADT1 _ NilF = Nil
instance (ConsF b :<: xs', xs ~ xs') => TransEADT a b xs xs' (ConsF a) where
transEADT1 f (ConsF a x) = Cons (f a) x
instance TransEADT a b xs xs' (VariantF []) where
transEADT1 _ _ = undefined
instance
( TransEADT a b xs xs' f
, TransEADT a b xs xs' (VariantF fs)
) => TransEADT a b xs xs' (VariantF (f : fs)) where
transEADT1 f v = case popVariantFHead v of
Right u -> transEADT1 f u
Left w -> transEADT1 f w
transEADT :: ( Functor (VariantF xs)
, TransEADT a b xs' xs' (VariantF xs)
) => (a -> b) -> EADT xs -> EADT xs'
transEADT f = cata (transEADT1 f)
```

Note that we need to specify the resulting type as it can be anything fulfilling the constraints:

```
> putStrLn $ eadtShow $ (transEADT (fromIntegral :: Int -> Float) intList :: List Float)
10.0 : 20.0 : 30.0 : Nil
```