How to define “type disjunction” (union types)?

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Miles Sabin describes a very nice way to get union type in his recent blog post Unboxed union types in Scala via the Curry-Howard isomorphism:

He first defines negation of types as

type ¬[A] = A => Nothing

using De Morgan’s law this allows him to define union types

type ∨[T, U] = ¬[¬[T] with ¬[U]]

With the following auxiliary constructs 

type ¬¬[A] = ¬[¬[A]]
type |∨|[T, U] = { type λ[X] = ¬¬[X] <:< (T ∨ U) }

you can write union types as follows:

def size[T : (Int |∨| String)#λ](t : T) = t match {
    case i : Int => i
    case s : String => s.length
}

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