Safe Haskell | None |
---|---|

Language | Haskell2010 |

n-ary sums (and sums of products)

- data NS :: (k -> *) -> [k] -> * where
- newtype SOP f xss = SOP (NS (NP f) xss)
- unSOP :: SOP f xss -> NS (NP f) xss
- type Injection f xs = f -.-> K (NS f xs)
- injections :: forall xs f. SingI xs => NP (Injection f xs) xs
- shift :: Injection f xs a -> Injection f (x : xs) a
- apInjs_NP :: SingI xs => NP f xs -> [NS f xs]
- apInjs_POP :: SingI xss => POP f xss -> [SOP f xss]
- ap_NS :: NP (f -.-> g) xs -> NS f xs -> NS g xs
- ap_SOP :: POP (f -.-> g) xs -> SOP f xs -> SOP g xs
- liftA_NS :: SingI xs => (forall a. f a -> g a) -> NS f xs -> NS g xs
- liftA_SOP :: SingI xss => (forall a. f a -> g a) -> SOP f xss -> SOP g xss
- liftA2_NS :: SingI xs => (forall a. f a -> g a -> h a) -> NP f xs -> NS g xs -> NS h xs
- liftA2_SOP :: SingI xss => (forall a. f a -> g a -> h a) -> POP f xss -> SOP g xss -> SOP h xss
- cliftA_NS :: (All c xs, SingI xs) => Proxy c -> (forall a. c a => f a -> g a) -> NS f xs -> NS g xs
- cliftA_SOP :: (All2 c xss, SingI xss) => Proxy c -> (forall a. c a => f a -> g a) -> SOP f xss -> SOP g xss
- 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
- 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
- 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
- collapse_NS :: NS (K a) xs -> a
- collapse_SOP :: SingI xss => SOP (K a) xss -> [a]
- sequence'_NS :: Applicative f => NS (f :.: g) xs -> f (NS g xs)
- sequence'_SOP :: (SingI xss, Applicative f) => SOP (f :.: g) xss -> f (SOP g xss)
- sequence_NS :: (SingI xs, Applicative f) => NS f xs -> f (NS I xs)
- sequence_SOP :: (SingI xss, Applicative f) => SOP f xss -> f (SOP I xss)

# 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

, 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.`K`

a

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 ]

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

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.`SOP`

`I`

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 |

# 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:*

`>>>`

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

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:*

`>>>`

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

# Application

# 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

`All`

ccliftA2'_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'`

.

# Collapsing

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

Specialization of `hcollapse`

.

# Sequencing

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

Specialization of `hsequence'`

.

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

Specialization of `hsequence'`

.

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

Specialization of `hsequence`

.

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

Specialization of `hsequence`

.