Generics.SOP.Classes

Classes for generalized combinators on SOP types.

In the SOP approach to generic programming, we're predominantly concerned with four structured datatypes:

  NP  :: (k -> *) -> ( [k]  -> *)   -- n-ary product
  NS  :: (k -> *) -> ( [k]  -> *)   -- n-ary sum
  POP :: (k -> *) -> ([[k]] -> *)   -- product of products
  SOP :: (k -> *) -> ([[k]] -> *)   -- sum of products

All of these have a kind that fits the following pattern:

  (k -> *) -> (l -> *)

These four types support similar interfaces. In order to allow reusing the same combinator names for all of these types, we define various classes in this module that allow the necessary generalization.

The classes typically lift concepts that exist for kinds * or * -> * to datatypes of kind (k -> *) -> (l -> *). This module also derives a number of derived combinators.

The actual instances are defined in Generics.SOP.NP and Generics.SOP.NS.

Documentation

class HPure h where

A generalization of pure or return to higher kinds.

Methods

hpure :: SingI xs => (forall a. f a) -> h f xs

Corresponds to pure directly.

Instances:

hpure, pure_NP  :: SingI xs  => (forall a. f a) -> NP  f xs
hpure, pure_POP :: SingI xss => (forall a. f a) -> POP f xss

hcpure :: (SingI xs, AllMap h c xs) => Proxy c -> (forall a. c a => f a) -> h f xs

A variant of hpure that allows passing in a constrained argument.

Calling hcpure f s where s :: h f xs causes f to be applied at all the types that are contained in xs. Therefore, the constraint c has to be satisfied for all elements of xs, which is what AllMap h c xs states.

Morally, hpure is a special case of hcpure where the constraint is empty. However, it is in the nature of how AllMap is defined as well as current GHC limitations that it is tricky to prove to GHC in general that AllMap h c NoConstraint xs is always satisfied. Therefore, we typically define hpure separately and directly, and make it a member of the class.

Instances:

hcpure, cpure_NP  :: (SingI xs,  All  c xs ) => Proxy c -> (forall a. c a => f a) -> NP  f xs
hcpure, cpure_POP :: (SingI xss, All2 c xss) => Proxy c -> (forall a. c a => f a) -> POP f xss

Instances

HPure k [[k]] (POP k)

HPure k [k] (NP k)

Typeable (((k -> *) -> k -> *) -> Constraint) (HPure k k)

newtype (f -.-> g) a infixr 1

Lifted functions.

Constructors

Fn

Fields

apFn :: f a -> g a

Instances

Typeable ((k -> *) -> (k -> *) -> k -> *) ((-.->) k)

fn :: (f a -> f' a) -> (f -.-> f') a

Construct a lifted function.

Same as Fn. Only available for uniformity with the higher-arity versions.

fn_2 :: (f a -> f' a -> f'' a) -> (f -.-> (f' -.-> f'')) a

Construct a binary lifted function.

fn_3 :: (f a -> f' a -> f'' a -> f''' a) -> (f -.-> (f' -.-> (f'' -.-> f'''))) a

Construct a ternary lifted function.

fn_4 :: (f a -> f' a -> f'' a -> f''' a -> f'''' a) -> (f -.-> (f' -.-> (f'' -.-> (f''' -.-> f'''')))) a

Construct a quarternary lifted function.

type family Prod h :: (k -> *) -> l -> *

Maps a structure containing sums to the corresponding product structure.

Instances

type Prod k [[k]] (POP k) = POP k

type Prod k [k] (NP k) = NP k

type Prod k [[k]] (SOP k) = POP k

type Prod k [k] (NS k) = NP k

class (Prod (Prod h) ~ Prod h, HPure (Prod h)) => HAp h where

A generalization of <*>.

Methods

hap :: Prod h (f -.-> g) xs -> h f xs -> h g xs

Corresponds to <*>.

For products as well as products or products, the correspondence is rather direct. We combine a structure containing (lifted) functions and a compatible structure containing corresponding arguments into a compatible structure containing results.

The same combinator can also be used to combine a product structure of functions with a sum structure of arguments, which then results in another sum structure of results. The sum structure determines which part of the product structure will be used.

Instances:

hap, ap_NP  :: NP  (f -.-> g) xs  -> NP  f xs  -> NP  g xs
hap, ap_NS  :: NP  (f -.-> g) xs  -> NS  f xs  -> NS  g xs
hap, ap_POP :: POP (f -.-> g) xss -> POP f xss -> POP g xss
hap, ap_SOP :: POP (f -.-> g) xss -> SOP f xss -> SOP g xss

Instances

HAp k [[k]] (POP k)

HAp k [k] (NP k)

HAp k [[k]] (SOP k)

HAp k [k] (NS k)

Typeable (((k -> *) -> k -> *) -> Constraint) (HAp k k)

hliftA :: (SingI xs, HAp h) => (forall a. f a -> f' a) -> h f xs -> h f' xs

A generalized form of liftA, which in turn is a generalized map.

Takes a lifted function and applies it to every element of a structure while preserving its shape.

Specification:

hliftA f xs = hpure (fn f) ` hap ` xs

Instances:

hliftA, liftA_NP  :: SingI xs  => (forall a. f a -> f' a) -> NP  f xs  -> NP  f' xs
hliftA, liftA_NS  :: SingI xs  => (forall a. f a -> f' a) -> NS  f xs  -> NS  f' xs
hliftA, liftA_POP :: SingI xss => (forall a. f a -> f' a) -> POP f xss -> POP f' xss
hliftA, liftA_SOP :: SingI xss => (forall a. f a -> f' a) -> SOP f xss -> SOP f' xss

hliftA2 :: (SingI xs, HAp h, HAp (Prod h)) => (forall a. f a -> f' a -> f'' a) -> Prod h f xs -> h f' xs -> h f'' xs

A generalized form of liftA2, which in turn is a generalized zipWith.

Takes a lifted binary function and uses it to combine two structures of equal shape into a single structure.

It either takes two product structures to a product structure, or one product and one sum structure to a sum structure.

Specification:

hliftA2 f xs ys = hpure (fn_2 f) ` hap ` xs ` hap ` ys

Instances:

hliftA2, liftA2_NP  :: SingI xs  => (forall a. f a -> f' a -> f'' a) -> NP  f xs  -> NP  f' xs  -> NP  f'' xs
hliftA2, liftA2_NS  :: SingI xs  => (forall a. f a -> f' a -> f'' a) -> NP  f xs  -> NS  f' xs  -> NS  f'' xs
hliftA2, liftA2_POP :: SingI xss => (forall a. f a -> f' a -> f'' a) -> POP f xss -> POP f' xss -> POP f'' xss
hliftA2, liftA2_SOP :: SingI xss => (forall a. f a -> f' a -> f'' a) -> POP f xss -> SOP f' xss -> SOP f'' xss

hliftA3 :: (SingI xs, HAp h, HAp (Prod h)) => (forall a. f a -> f' a -> f'' a -> f''' a) -> Prod h f xs -> Prod h f' xs -> h f'' xs -> h f''' xs

A generalized form of liftA3, which in turn is a generalized zipWith3.

Takes a lifted ternary function and uses it to combine three structures of equal shape into a single structure.

It either takes three product structures to a product structure, or two product structures and one sum structure to a sum structure.

Specification:

hliftA3 f xs ys zs = hpure (fn_3 f) ` hap ` xs ` hap ` ys ` hap ` zs

Instances:

hliftA3, liftA3_NP  :: SingI xs  => (forall a. f a -> f' a -> f'' a -> f''' a) -> NP  f xs  -> NP  f' xs  -> NP  f'' xs  -> NP  f''' xs
hliftA3, liftA3_NS  :: SingI xs  => (forall a. f a -> f' a -> f'' a -> f''' a) -> NP  f xs  -> NP  f' xs  -> NS  f'' xs  -> NS  f''' xs
hliftA3, liftA3_POP :: SingI xss => (forall a. f a -> f' a -> f'' a -> f''' a) -> POP f xss -> POP f' xss -> POP f'' xss -> POP f''' xs
hliftA3, liftA3_SOP :: SingI xss => (forall a. f a -> f' a -> f'' a -> f''' a) -> POP f xss -> POP f' xss -> SOP f'' xss -> SOP f''' xs

hcliftA :: (AllMap (Prod h) c xs, SingI xs, HAp h) => Proxy c -> (forall a. c a => f a -> f' a) -> h f xs -> h f' xs

Variant of hliftA that takes a constrained function.

Specification:

hcliftA p f xs = hcpure p (fn f) ` hap ` xs

hcliftA2 :: (AllMap (Prod h) c xs, SingI xs, HAp h, HAp (Prod h)) => Proxy c -> (forall a. c a => f a -> f' a -> f'' a) -> Prod h f xs -> h f' xs -> h f'' xs

Variant of hcliftA2 that takes a constrained function.

Specification:

hcliftA2 p f xs ys = hcpure p (fn_2 f) ` hap ` xs ` hap ` ys

hcliftA3 :: (AllMap (Prod h) c xs, SingI xs, HAp h, HAp (Prod h)) => Proxy c -> (forall a. c a => f a -> f' a -> f'' a -> f''' a) -> Prod h f xs -> Prod h f' xs -> h f'' xs -> h f''' xs

Variant of hcliftA3 that takes a constrained function.

Specification:

hcliftA3 p f xs ys zs = hcpure p (fn_3 f) ` hap ` xs ` hap ` ys ` hap ` zs

type family CollapseTo h :: * -> *

Maps products to lists, and sums to identities.

Instances

type CollapseTo k [[k]] (POP k) = (:.:) * * [] []

type CollapseTo k [k] (NP k) = []

type CollapseTo k [[k]] (SOP k) = []

type CollapseTo k [k] (NS k) = I

class HCollapse h where

A class for collapsing a heterogeneous structure into a homogeneous one.

Methods

hcollapse :: SingI xs => h (K a) xs -> CollapseTo h a

Collapse a heterogeneous structure with homogeneous elements into a homogeneous structure.

If a heterogeneous structure is instantiated to the constant functor K, then it is in fact homogeneous. This function maps such a value to a simpler Haskell datatype reflecting that. An NS (K a) contains a single a, and an NP (K a) contains a list of as.

Instances:

hcollapse, collapse_NP  :: NP  (K a) xs  ->  [a]
hcollapse, collapse_NS  :: NS  (K a) xs  ->   a
hcollapse, collapse_POP :: POP (K a) xss -> [[a]]
hcollapse, collapse_SOP :: SOP (K a) xss ->  [a]

Instances

HCollapse k [[k]] (POP k)

HCollapse k [k] (NP k)

HCollapse k [[k]] (SOP k)

HCollapse k [k] (NS k)

Typeable (((k -> *) -> k -> *) -> Constraint) (HCollapse k k)

class HAp h => HSequence h where

A generalization of sequenceA.

Methods

hsequence' :: (SingI xs, Applicative f) => h (f :.: g) xs -> f (h g xs)

Corresponds to sequenceA.

Lifts an applicative functor out of a structure.

Instances:

hsequence', sequence'_NP  :: (SingI xs , Applicative f) => NP  (f :.: g) xs  -> f (NP  g xs )
hsequence', sequence'_NS  :: (SingI xs , Applicative f) => NS  (f :.: g) xs  -> f (NS  g xs )
hsequence', sequence'_POP :: (SingI xss, Applicative f) => POP (f :.: g) xss -> f (POP g xss)
hsequence', sequence'_SOP :: (SingI xss, Applicative f) => SOP (f :.: g) xss -> f (SOP g xss)

Instances

HSequence k [[k]] (POP k)

HSequence k [k] (NP k)

HSequence k [[k]] (SOP k)

HSequence k [k] (NS k)

Typeable (((k -> *) -> k -> *) -> Constraint) (HSequence k k)

hsequence :: (SingI xs, HSequence h) => Applicative f => h f xs -> f (h I xs)

Special case of hsequence' where g = I.

hsequenceK :: (SingI xs, Applicative f, HSequence h) => h (K (f a)) xs -> f (h (K a) xs)

Special case of hsequence' where g = K a.