haskellfunctional-programmingrecursion-schemes# Chaining values with catamorphisms

Suppose I have definitions as follows (where `cata`

is the catamorphism):

```
type Algebra f a = f a -> a
newtype Fix f = Fx (f (Fix f))
unFix :: Fix f -> f (Fix f)
unFix (Fx x) = x
cata :: Functor f => (f a -> a) -> Fix f -> a
cata alg = alg . fmap (cata alg) . unFix
```

I was wondering if there would be some way to modify the definition of `cata`

so that I could chain some object such as an `int`

through it such that I could generate unique handles for things within the alg function, i.e. "a0", "a1", "a2", ..., etc.

**Edit:** To make this more clear, I'd like to be able to have some function `cata'`

such that when I have something similar to the following definitions

```
data IntF a
= Const Int
| Add a a
instance Functor IntF where
fmap eval (Const i) = Const i
fmap eval (x `Add` y) = eval x `Add` eval y
alg :: Int -> Algebra IntF String
alg n (Const i) = "a" ++ show n
alg n (s1 `Add` s2) = s1 ++ " && " ++ s2
eval = cata' alg
addExpr = Fx $ (Fx $ Const 5) `Add` (Fx $ Const 4)
run = eval addExpr
```

then `run`

evaluates to "a0 && a1" or something similar, i.e. the two constants don't get labeled the same thing.

Solution

Just sequence them as monads.

```
newtype Ctr a = Ctr { runCtr :: Int -> (a, Int) } -- is State Int
instance Functor Ctr
instance Applicative Ctr
instance Monad Ctr
type MAlgebra m f a = f (m a) -> m a
fresh :: Ctr Int
fresh = Ctr (\i -> (i, i+1))
data IntF a
= Val
| Add a a
malg :: IntF (Ctr String) -> Ctr String
malg Val = (\x -> "a" ++ show x) <$> fresh
malg (Add x y) = (\a b -> a ++ " && " ++ b) <$> x <*> y
go = cata malg
```

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