See: https://www.reddit.com/r/haskell/comments/12hq6z6/the_free_boolean_cube_an_exploration_of_things/

Cube.hs

module Cube where

import Data.Functor.Const (Const(..))
import Data.Functor.Product (Product(..))
import Data.Functor.Sum (Sum(..))


type (&&) = Product
type (||) = Sum

-- Functor Origin
data Empty s

-- Functor Axes
type Ann     = Const
type Rec f   = f
type Con f a = Ann a && Rec f

-- Functor Cube
type C0   f a = Empty
type CX   f a = Ann a                     -- Const a
type CY   f a =          Rec f            --            f
type CZ   f a =                   Con f a --                 Const a && f
type CXY  f a = Ann a || Rec f            -- Const a || f
type CXZ  f a = Ann a          || Con f a -- Const a      || Const a && f
type CYZ  f a =          Rec f || Con f a --            f || Const a && f
type CXYZ f a = Ann a || Rec f || Con f a -- Const a || f || Const a && f


-- Fixpoint Appendix

-- Let t := Fix (CXY f a)
-- Then:
-- t ~= Fix (Const a || f)
--   ~= (Const a || f) t
--   ~= Const a t + f t
--   ~= a + f t
--   ~= Free f a

-- Let t := Fix (CZ f a)
-- Then:
-- t ~= Fix (Const a && f)
--   ~= (Const a && f) t
--   ~= Const a t * f t
--   ~= a * f t
--   ~= Cofree f a