binary tree insertion using paramorphism

gistfile1.txt

{-#LANGUAGE TypeFamilies, DeriveFunctor #-}

import Data.Functor.Foldable

data Tree a = Leaf | Node a (Tree a) (Tree a) deriving (Show)
data TreeF a b = LeafF | NodeF a b b deriving (Functor, Show)

type instance Base (Tree a) = TreeF a
instance Recursive (Tree a) where
    project Leaf = LeafF
    project (Node a b c) = NodeF a b c
    
search = fst
stay = snd 

insertAlgebra a LeafF = Node a Leaf Leaf
insertAlgebra a (NodeF b left right)
    | a < b = Node b (search left) (stay right) -- only search in the left, the right side of the tree is untouched
    | a > b = Node b (stay left) (search right) -- only search in the right, the left side of the tree is untouched
    | otherwise = Node b (stay left) (stay right) -- stop the recursion
    
insert x = para (insertAlgebra x)