How to reduce code duplication when dealing with recursive sum types

Congratulations, you just rediscovered anamorphisms!

Here’s your code, rephrased so that it works with the recursion-schemes package. Alas, it’s not shorter, since we need some boilerplate to make the machinery work. (There might be some automagic way to avoid the boilerplate, e.g. using generics. I simply do not know.)

Below, your recurseAfter is replaced with the standard ana.

We first define your recursive type, as well as the functor it is the fixed point of.

{-# LANGUAGE DeriveFunctor, TypeFamilies, LambdaCase #-}
{-# OPTIONS -Wall #-}
module AnaExpr where

import Data.Functor.Foldable

data Expr
  = Variable String
  | Number Int
  | Add [Expr]
  | Sub Expr Expr
  deriving (Show)

data ExprF a
  = VariableF String
  | NumberF Int
  | AddF [a]
  | SubF a a
  deriving (Functor)

Then we connect the two with a few instances so that we can unfold Expr into the isomorphic ExprF Expr, and fold it back.

type instance Base Expr = ExprF
instance Recursive Expr where
   project (Variable s) = VariableF s
   project (Number i) = NumberF i
   project (Add es) = AddF es
   project (Sub e1 e2) = SubF e1 e2
instance Corecursive Expr where
   embed (VariableF s) = Variable s
   embed (NumberF i) = Number i
   embed (AddF es) = Add es
   embed (SubF e1 e2) = Sub e1 e2

Finally, we adapt your original code, and add a couple of tests.

substituteName :: String -> Int -> Expr -> Expr
substituteName name newValue = ana $ \case
    Variable x | x == name -> NumberF newValue
    other                  -> project other

testSub :: Expr
testSub = substituteName "x" 42 (Add [Add [Variable "x"], Number 0])

replaceSubWithAdd :: Expr -> Expr
replaceSubWithAdd = ana $ \case
    Sub x (Number y) -> AddF [x, Number (-y)]
    other            -> project other

testReplace :: Expr
testReplace = replaceSubWithAdd 
   (Add [Sub (Add [Variable "x", Sub (Variable "y") (Number 34)]) (Number 10), Number 4])

An alternative could be to define ExprF a only, and then derive type Expr = Fix ExprF. This saves some of the boilerplate above (e.g. the two instances), at the cost of having to use Fix (VariableF ...) instead of Variable ..., as well as the analogous for the other constructors.

One could further alleviate that using pattern synonyms (at the cost of a little more boilerplate, though).

Update: I finally found the automagic tool, using template Haskell. This makes the whole code reasonably short. Note that the ExprF functor and the two instances above still exist under the hood, and we still have to use them. We only save the hassle of having to define them manually, but that alone saves a lot of effort.

{-# LANGUAGE DeriveFunctor, DeriveTraversable, TypeFamilies, LambdaCase, TemplateHaskell #-}
{-# OPTIONS -Wall #-}
module AnaExpr where

import Data.Functor.Foldable
import Data.Functor.Foldable.TH

data Expr
  = Variable String
  | Number Int
  | Add [Expr]
  | Sub Expr Expr
  deriving (Show)

makeBaseFunctor ''Expr

substituteName :: String -> Int -> Expr -> Expr
substituteName name newValue = ana $ \case
    Variable x | x == name -> NumberF newValue
    other                  -> project other

testSub :: Expr
testSub = substituteName "x" 42 (Add [Add [Variable "x"], Number 0])

replaceSubWithAdd :: Expr -> Expr
replaceSubWithAdd = ana $ \case
    Sub x (Number y) -> AddF [x, Number (-y)]
    other            -> project other

testReplace :: Expr
testReplace = replaceSubWithAdd 
   (Add [Sub (Add [Variable "x", Sub (Variable "y") (Number 34)]) (Number 10), Number 4])