From Van Laarhoven Isomorphisms in one shot.
The type ∀ f, g : Functor. (g a → f b) → g s → f t is isomorphic to the type (s → a)×(b → t). The Van Laarhoven representation of isomorphisms uses this representation of a pair of function to capture the notion of an isomorphism.
Given a value of type ∀ f, g : Functor. (g a → f b) → g s → f t, the normal way of extracting the two components is to instantiate the value twice using Identity and Constant functors.
extractPair :: (forall f g. (Functor f, Functor g) => (g a -> f b) -> g s -> f t)
-> (s -> a, b -> t)
extractPair l = (getConstant . (l (Constant . runIdentity)) . Identity
,runIdentity . (l (Identity . getConstant)) . Constant)
However, it is possible to extract the two components in one shot.
data PStore i j x = PStore {peek :: j -> x, pos :: i}
instance Functor (PStore i j) where
fmap f (Pstore h i) = PStore (f . h) i
extractPair :: (((s -> a) -> PStore (s -> a) b b) -> (s -> s) -> PStore (s -> a) b t)
-> (s -> a, b -> t)
extractPair l = (f, g)
where
PStore g f = l (PStore id) id
