Rank-N types

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Normal Haskell '98 types are considered Rank-1 types. A Haskell '98 type signature such as

a -> b -> a

implies that the type variables are universally quantified like so:

forall a b. a -> b -> a

forall can be floated out of the right-hand side of (->) if it appears there, so:

forall a. a -> (forall b. b -> a)

is also a Rank-1 type because it is equivalent to the previous signature.

However, a forall appearing within the left-hand side of (->) cannot be moved up, and therefore forms another level or rank. The type is labeled "Rank-N" where N is the number of foralls which are nested and cannot be merged with a previous one. For example:

(forall a. a -> a) -> (forall b. b -> b)

is a Rank-2 type because the latter forall can be moved to the start but the former one cannot. Therefore, there are two levels of universal quantification.

Rank-N type reconstruction is undecidable in general, and some explicit type annotations are required in their presence.

Rank-2 or Rank-N types may be specifically enabled by the language extensions {-# LANGUAGE Rank2Types #-} or {-# LANGUAGE RankNTypes #-}.

Also see

Rank-N types on the Haskell' website.