Generics.SOP.NS

n-ary sums (and sums of products)

Datatypes

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

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

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

Unwrap a sum of products.

Constructing sums

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

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 -> K (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

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

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]

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]

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

Specialization of hap.

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

Specialization of hap.

Lifting / mapping

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

Specialization of hliftA.

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

Specialization of hliftA.

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

Specialization of hliftA2.

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

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

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

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

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

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

Specialization of hcliftA2'.

Collapsing

collapse_NS :: NS (K a) xs -> a

Specialization of hcollapse.

collapse_SOP :: SingI xss => SOP (K a) xss -> [a]

Specialization of hcollapse.

Sequencing

sequence'_NS :: Applicative f => NS (f :.: g) xs -> f (NS g xs)

Specialization of hsequence'.

sequence'_SOP :: (SingI xss, Applicative f) => SOP (f :.: g) xss -> f (SOP g xss)

Specialization of hsequence'.

sequence_NS :: (SingI xs, Applicative f) => NS f xs -> f (NS I xs)

Specialization of hsequence.

sequence_SOP :: (SingI xss, Applicative f) => SOP f xss -> f (SOP I xss)

Specialization of hsequence.