Safe Haskell	None
Language	Haskell2010

Generics.SOP.NS

Contents

Datatypes
Constructing sums
Application
Lifting / mapping
Dealing with All c
Collapsing
Sequencing

Description

n-ary sums (and sums of products)

Synopsis

Datatypes

data NS :: (k -> *) -> [k] -> * where Source

An n-ary sum.

The sum is parameterized by a type constructor f and indexed by a type-level list xs. The length of the list determines the number of choices in the sum and if the i-th element of the list is of type x, then the i-th choice of the sum is of type f x.

The constructor names are chosen to resemble Peano-style natural numbers, i.e., Z is for "zero", and S is for "successor". Chaining S and Z chooses the corresponding component of the sum.

Examples:

Z         :: f x -> NS f (x ': xs)
S . Z     :: f y -> NS f (x ': y ': xs)
S . S . Z :: f z -> NS f (x ': y ': z ': xs)
...

Note that empty sums (indexed by an empty list) have no non-bottom elements.

Two common instantiations of f are the identity functor I and the constant functor K. For I, the sum becomes a direct generalization of the Either type to arbitrarily many choices. For K a, the result is a homogeneous choice type, where the contents of the type-level list are ignored, but its length specifies the number of options.

In the context of the SOP approach to generic programming, an n-ary sum describes the top-level structure of a datatype, which is a choice between all of its constructors.

Examples:

Z (I 'x')      :: NS I       '[ Char, Bool ]
S (Z (I True)) :: NS I       '[ Char, Bool ]
S (Z (I 1))    :: NS (K Int) '[ Char, Bool ]

Constructors

Z :: f x -> NS f (x : xs)
S :: NS f xs -> NS f (x : xs)

Instances

HSequence k [k] (NS k)
HCollapse k [k] (NS k)
HAp k [k] (NS k)
All * Eq (Map * k f xs) => Eq (NS k f xs)
(All * Eq (Map * k f xs), All * Ord (Map * k f xs)) => Ord (NS k f xs)
All * Show (Map * k f xs) => Show (NS k f xs)
type CollapseTo k [k] (NS k) = I
type Prod k [k] (NS k) = NP k

newtype SOP f xss Source

A sum of products.

This is a 'newtype' for an NS of an NP. The elements of the (inner) products are applications of the parameter f. The type SOP is indexed by the list of lists that determines the sizes of both the (outer) sum and all the (inner) products, as well as the types of all the elements of the inner products.

An SOP I reflects the structure of a normal Haskell datatype. The sum structure represents the choice between the different constructors, the product structure represents the arguments of each constructor.

Constructors

SOP (NS (NP f) xss)

Instances

HSequence k [[k]] (SOP k)
HCollapse k [[k]] (SOP k)
HAp k [[k]] (SOP k)
All * Eq (Map * [k] (NP k f) xss) => Eq (SOP k f xss)
(All * Eq (Map * [k] (NP k f) xss), All * Ord (Map * [k] (NP k f) xss)) => Ord (SOP k f xss)
All * Show (Map * [k] (NP k f) xss) => Show (SOP k f xss)
type CollapseTo k [[k]] (SOP k) = []
type Prod k [[k]] (SOP k) = POP k

unSOP :: SOP f xss -> NS (NP f) xss Source

Unwrap a sum of products.

Constructing sums

type Injection f xs = f -.-> K (NS f xs) Source

The type of injections into an n-ary sum.

If you expand the type synonyms and newtypes involved, you get

Injection f xs a = (f -.-> K (NS f xs)) a ~= f a -> K (NS f xs) a ~= f a -> NS f xs

If we pick a to be an element of xs, this indeed corresponds to an injection into the sum.

injections :: forall xs f. SingI xs => NP (Injection f xs) xs Source

Compute all injections into an n-ary sum.

Each element of the resulting product contains one of the injections.

shift :: Injection f xs a -> Injection f (x : xs) a Source

Shift an injection.

Given an injection, return an injection into a sum that is one component larger.

apInjs_NP :: SingI xs => NP f xs -> [NS f xs] Source

Apply injections to a product.

Given a product containing all possible choices, produce a list of sums by applying each injection to the appropriate element.

Example:

>>> apInjs_NP (I 'x' :* I True :* I 2 :* Nil)
[Z (I 'x'), S (Z (I True)), S (S (Z (I 2)))]

apInjs_POP :: SingI xss => POP f xss -> [SOP f xss] Source

Apply injections to a product of product.

This operates on the outer product only. Given a product containing all possible choices (that are products), produce a list of sums (of products) by applying each injection to the appropriate element.

Example:

>>> apInjs_POP (POP ((I 'x' :* Nil) :* (I True :* I 2 :* Nil) :* Nil))
[SOP (Z (I 'x' :* Nil)),SOP (S (Z (I True :* (I 2 :* Nil))))]

Application

ap_NS :: NP (f -.-> g) xs -> NS f xs -> NS g xs Source

Specialization of hap.

ap_SOP :: POP (f -.-> g) xs -> SOP f xs -> SOP g xs Source

Specialization of hap.

Lifting / mapping

liftA_NS :: SingI xs => (forall a. f a -> g a) -> NS f xs -> NS g xs Source

Specialization of hliftA.

liftA_SOP :: SingI xss => (forall a. f a -> g a) -> SOP f xss -> SOP g xss Source

Specialization of hliftA.

liftA2_NS :: SingI xs => (forall a. f a -> g a -> h a) -> NP f xs -> NS g xs -> NS h xs Source

Specialization of hliftA2.

liftA2_SOP :: SingI xss => (forall a. f a -> g a -> h a) -> POP f xss -> SOP g xss -> SOP h xss Source

Specialization of hliftA2.

cliftA_NS :: (All c xs, SingI xs) => Proxy c -> (forall a. c a => f a -> g a) -> NS f xs -> NS g xs Source

Specialization of hcliftA.

cliftA_SOP :: (All2 c xss, SingI xss) => Proxy c -> (forall a. c a => f a -> g a) -> SOP f xss -> SOP g xss Source

Specialization of hcliftA.

cliftA2_NS :: (All c xs, SingI xs) => Proxy c -> (forall a. c a => f a -> g a -> h a) -> NP f xs -> NS g xs -> NS h xs Source

Specialization of hcliftA2.

cliftA2_SOP :: (All2 c xss, SingI xss) => Proxy c -> (forall a. c a => f a -> g a -> h a) -> POP f xss -> SOP g xss -> SOP h xss Source

Specialization of hcliftA2.

Dealing with `All c`

cliftA2'_NS :: (All2 c xss, SingI xss) => Proxy c -> (forall xs. (SingI xs, All c xs) => f xs -> g xs -> h xs) -> NP f xss -> NS g xss -> NS h xss Source

Specialization of hcliftA2'.