Encoding Product and Coproduct in Scala

gistfile1.scala

type Product[F[_], G[_], A] = (F[A], G[A])

trait Prod[F[_], G[_]] {
  type and[H[_]] = Prod[F, ({type λ[α] = Prod[G, H, α]})#λ]
  type apply[A] = (F[A], G[A])
}

type product[F[_], G[_]] = Prod[F, G]

type Coproduct[F[_], G[_], A] = Either[F[A], G[A]]

trait Coprod[F[_], G[_]] {
  type or[H[_]] = Coprod[F, ({type λ[α] = Corod[G, H, α]})#λ]
  type apply[A] = Either[F[A], G[A]]
}

type coproduct[F[_], G[_]] = Coprod[F, G]

// the encoding for `Val :+: Add :+: Mult :+: Div` from "Data types a la carte" would be
type Arithmetic[A] = coproduct[Val, Add]#or[Mult]#or[Div]#apply[A]