Type-oriented programming/Type variance

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Consider the following type hierarchy:

type A {}

type B : A {}

type F[T] {}

Without type arguments, F is a type operator while F[A] and F[B] are proper types. In general, neither is a subtype of the other. However it may sometimes make sense for them to be in a subtype relation depending on the hierarchy of their argument(s). If we declare F as follows

type F[cov T] {}

then F[B] will be a subtype of F[A] because B is a subtype of A. Conversely, if we declare F as

type F[con T] {}

then F[A] will be a subtype of F[B]. In the former case, we say that F is covariant in its type argument whereas in the latter case its contravariant (since the hierarchy is reversed).

A real-world example of type variance are function types. When we expect a function whose return type is A (that is, of type Func[A]), we can always use a function of type Func[B] in its stead. On the other hand, when we expect a function whose argument is of type B (that is, of type Func[B,X], we can always use a function of type Func[A,X] in its stead. In sum, function types are covariant in their return type and contravariant in their arguments’ types.

NB: The pseudocode can be tried out using the Funcy app, which can be downloaded for free from Apple’s App Store (iOS/macOS), Google Play (Android) or Amazon Appstore. The code to be executed must be placed in a main {} block.