Safe Haskell	None
Language	Haskell2010

Generics.SOP.NS

Contents

Datatypes
Constructing sums
Destructing sums
Application
Lifting / mapping
Dealing with All c
Collapsing
Sequencing
Catamorphism and anamorphism
Expanding sums to products
Transformation of index lists and coercions

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 (K 1))    :: NS (K Int) '[ Char, Bool ]

Constructors

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

Instances

HExpand [k] k (NS k) Source #
Methods hexpand :: SListIN (NS k) k (Prod (NS k) k h) xs => (forall x. f x) -> h f xs -> Prod (NS k) k h f xs Source # hcexpand :: AllN (NS k) k (Prod (NS k) k h) c xs => proxy c -> (forall x. c x => f x) -> h f xs -> Prod (NS k) k h f xs Source #
HApInjs [k] k (NS k) Source #
Methods hapInjs :: SListIN (NS k) k h xs => Prod (NS k) k h f xs -> [h f xs] Source #
HIndex [k] k (NS k) Source #
Methods hindex :: h f xs -> Int Source #
HSequence [k] k (NS k) Source #
Methods hsequence' :: (SListIN (NS k) k h xs, Applicative f) => h ((k :.: *) f g) xs -> f (h g xs) Source #
HCollapse [k] k (NS k) Source #
Methods hcollapse :: SListIN (NS k) k h xs => h (K k a) xs -> CollapseTo (NS k) k h a Source #
HAp [k] k (NS k) Source #
Methods hap :: Prod (NS k) k h ((k -.-> f) g) xs -> h f xs -> h g xs Source #
HTrans [k2] k2 [k1] k1 (NS k1) (NS k2) Source #
Methods htrans :: AllZipN (NS k1) l1 (NS k2) k1 l1 k1 (Prod l1 k1 h1) c xs ys => proxy c -> (forall x y. c x y => f x -> g y) -> h1 f xs -> h2 g ys Source # hcoerce :: (AllZipN (NS k1) l1 (NS k2) k1 l1 k1 (Prod l1 k1 h1) (LiftedCoercible (NS k2) * k1 f g) xs ys, HTrans (NS k1) (NS k2) l1 k1 h1 h2) => h1 f xs -> h2 g ys Source #
All k (Compose k * Eq f) xs => Eq (NS k f xs) Source #
Methods (==) :: NS k f xs -> NS k f xs -> Bool # (/=) :: NS k f xs -> NS k f xs -> Bool #
(All k (Compose k * Eq f) xs, All k (Compose k * Ord f) xs) => Ord (NS k f xs) Source #
Methods compare :: NS k f xs -> NS k f xs -> Ordering # (<) :: NS k f xs -> NS k f xs -> Bool # (<=) :: NS k f xs -> NS k f xs -> Bool # (>) :: NS k f xs -> NS k f xs -> Bool # (>=) :: NS k f xs -> NS k f xs -> Bool # max :: NS k f xs -> NS k f xs -> NS k f xs # min :: NS k f xs -> NS k f xs -> NS k f xs #
All k (Compose k * Show f) xs => Show (NS k f xs) Source #
Methods showsPrec :: Int -> NS k f xs -> ShowS # show :: NS k f xs -> String # showList :: [NS k f xs] -> ShowS #
All k (Compose k * NFData f) xs => NFData (NS k f xs) Source #	Since: 0.2.5.0
Methods rnf :: NS k f xs -> () #
type SListIN [k] k (NS k) Source #
type SListIN [k] k (NS k) = SListI k
type Prod [k] k (NS k) Source #
type Prod [k] k (NS k) = NP k
type CollapseTo [k] k (NS k) a Source #
type CollapseTo [k] k (NS k) a = a
type Same [k2] k2 [k1] k1 (NS k1) Source #
type Same [k2] k2 [k1] k1 (NS k1) = NS k2

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

HExpand [[k]] k (SOP k) Source #
Methods hexpand :: SListIN (SOP k) k (Prod (SOP k) k h) xs => (forall x. f x) -> h f xs -> Prod (SOP k) k h f xs Source # hcexpand :: AllN (SOP k) k (Prod (SOP k) k h) c xs => proxy c -> (forall x. c x => f x) -> h f xs -> Prod (SOP k) k h f xs Source #
HApInjs [[k]] k (SOP k) Source #
Methods hapInjs :: SListIN (SOP k) k h xs => Prod (SOP k) k h f xs -> [h f xs] Source #
HIndex [[k]] k (SOP k) Source #
Methods hindex :: h f xs -> Int Source #
HSequence [[k]] k (SOP k) Source #
Methods hsequence' :: (SListIN (SOP k) k h xs, Applicative f) => h ((k :.: *) f g) xs -> f (h g xs) Source #
HCollapse [[k]] k (SOP k) Source #
Methods hcollapse :: SListIN (SOP k) k h xs => h (K k a) xs -> CollapseTo (SOP k) k h a Source #
HAp [[k]] k (SOP k) Source #
Methods hap :: Prod (SOP k) k h ((k -.-> f) g) xs -> h f xs -> h g xs Source #
HTrans [[k2]] k2 [[k1]] k1 (SOP k1) (SOP k2) Source #
Methods htrans :: AllZipN (SOP k1) l1 (SOP k2) k1 l1 k1 (Prod l1 k1 h1) c xs ys => proxy c -> (forall x y. c x y => f x -> g y) -> h1 f xs -> h2 g ys Source # hcoerce :: (AllZipN (SOP k1) l1 (SOP k2) k1 l1 k1 (Prod l1 k1 h1) (LiftedCoercible (SOP k2) * k1 f g) xs ys, HTrans (SOP k1) (SOP k2) l1 k1 h1 h2) => h1 f xs -> h2 g ys Source #
Eq (NS [k] (NP k f) xss) => Eq (SOP k f xss) Source #
Methods (==) :: SOP k f xss -> SOP k f xss -> Bool # (/=) :: SOP k f xss -> SOP k f xss -> Bool #
Ord (NS [k] (NP k f) xss) => Ord (SOP k f xss) Source #
Methods compare :: SOP k f xss -> SOP k f xss -> Ordering # (<) :: SOP k f xss -> SOP k f xss -> Bool # (<=) :: SOP k f xss -> SOP k f xss -> Bool # (>) :: SOP k f xss -> SOP k f xss -> Bool # (>=) :: SOP k f xss -> SOP k f xss -> Bool # max :: SOP k f xss -> SOP k f xss -> SOP k f xss # min :: SOP k f xss -> SOP k f xss -> SOP k f xss #
Show (NS [k] (NP k f) xss) => Show (SOP k f xss) Source #
Methods showsPrec :: Int -> SOP k f xss -> ShowS # show :: SOP k f xss -> String # showList :: [SOP k f xss] -> ShowS #
NFData (NS [k] (NP k f) xss) => NFData (SOP k f xss) Source #	Since: 0.2.5.0
Methods rnf :: SOP k f xss -> () #
type SListIN [[k]] k (SOP k) Source #
type SListIN [[k]] k (SOP k) = SListI2 k
type Prod [[k]] k (SOP k) Source #
type Prod [[k]] k (SOP k) = POP k
type CollapseTo [[k]] k (SOP k) a Source #
type CollapseTo [[k]] k (SOP k) a = [a]
type Same [[k2]] k2 [[k1]] k1 (SOP k1) Source #
type Same [[k2]] k2 [[k1]] k1 (SOP k1) = SOP k2

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. SListI 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 #

Deprecated: Use shiftInjection instead.

Shift an injection.

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

shiftInjection :: 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 :: SListI 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'_NP :: SListI xs => NP f xs -> NP (K (NS f xs)) xs Source #

apInjs_NP without hcollapse.

>>> apInjs'_NP (I 'x' :* I True :* I 2 :* Nil)
K (Z (I 'x')) :* K (S (Z (I True))) :* K (S (S (Z (I 2)))) :* Nil

Since: 0.2.5.0

apInjs_POP :: SListI 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))))]

apInjs'_POP :: SListI xss => POP f xss -> NP (K (SOP f xss)) xss Source #

apInjs_POP without hcollapse.

Example:

>>> apInjs'_POP (POP ((I 'x' :* Nil) :* (I True :* I 2 :* Nil) :* Nil))
K (SOP (Z (I 'x' :* Nil))) :* K (SOP (S (Z (I True :* I 2 :* Nil)))) :* Nil

Since: 0.2.5.0

Destructing sums

unZ :: NS f '[x] -> f x Source #

Extract the payload from a unary sum.

For larger sums, this function would be partial, so it is only provided with a rather restrictive type.

Example:

>>> unZ (Z (I 'x'))
I 'x'

Since: 0.2.2.0

index_NS :: forall f xs. NS f xs -> Int Source #

Obtain the index from an n-ary sum.

An n-nary sum represents a choice between n different options. This function returns an integer between 0 and n - 1 indicating the option chosen by the given value.

Examples:

>>> index_NS (S (S (Z (I False))))
2
>>> index_NS (Z (K ()))
0

Since: 0.2.4.0

index_SOP :: SOP f xs -> Int Source #

Obtain the index from an n-ary sum of products.

An n-nary sum represents a choice between n different options. This function returns an integer between 0 and n - 1 indicating the option chosen by the given value.

Specification:

index_SOP = index_NS . unSOP

Example:

>>> index_SOP (SOP (S (Z (I True :* I 'x' :* Nil))))
1

Since: 0.2.4.0

Application

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

Specialization of hap.

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

Specialization of hap.

Lifting / mapping

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

Specialization of hliftA.

liftA_SOP :: All SListI xss => (forall a. f a -> g a) -> SOP f xss -> SOP g xss Source #

Specialization of hliftA.

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

Specialization of hliftA2.

liftA2_SOP :: All SListI 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 => 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 => 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 => 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 => 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.

map_NS :: SListI xs => (forall a. f a -> g a) -> NS f xs -> NS g xs Source #

Specialization of hmap, which is equivalent to hliftA.

map_SOP :: All SListI xss => (forall a. f a -> g a) -> SOP f xss -> SOP g xss Source #

Specialization of hmap, which is equivalent to hliftA.

cmap_NS :: All c xs => proxy c -> (forall a. c a => f a -> g a) -> NS f xs -> NS g xs Source #

Specialization of hcmap, which is equivalent to hcliftA.

cmap_SOP :: All2 c xss => proxy c -> (forall a. c a => f a -> g a) -> SOP f xss -> SOP g xss Source #

Specialization of hcmap, which is equivalent to hcliftA.

Dealing with `All c`

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

Deprecated: Use cliftA2_NS instead.

Specialization of hcliftA2'.

Collapsing

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

Specialization of hcollapse.

collapse_SOP :: SListI xss => SOP (K a) xss -> [a] Source #

Specialization of hcollapse.

Sequencing

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

Specialization of hsequence'.

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

Specialization of hsequence'.

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

Specialization of hsequence.

sequence_SOP :: (All SListI xss, Applicative f) => SOP f xss -> f (SOP I xss) Source #

Specialization of hsequence.

Catamorphism and anamorphism

cata_NS :: forall r f xs. (forall y ys. f y -> r (y ': ys)) -> (forall y ys. r ys -> r (y ': ys)) -> NS f xs -> r xs Source #

Catamorphism for NS.

Takes arguments determining what to do for Z and what to do for S. The result type is still indexed over the type-level lit.

Since: 0.2.3.0

ccata_NS :: forall c proxy r f xs. All c xs => proxy c -> (forall y ys. c y => f y -> r (y ': ys)) -> (forall y ys. c y => r ys -> r (y ': ys)) -> NS f xs -> r xs Source #

Constrained catamorphism for NS.

Since: 0.2.3.0

ana_NS :: forall s f xs. SListI xs => (forall r. s '[] -> r) -> (forall y ys. s (y ': ys) -> Either (f y) (s ys)) -> s xs -> NS f xs Source #

Anamorphism for NS.

Since: 0.2.3.0

cana_NS :: forall c proxy s f xs. All c xs => proxy c -> (forall r. s '[] -> r) -> (forall y ys. c y => s (y ': ys) -> Either (f y) (s ys)) -> s xs -> NS f xs Source #

Constrained anamorphism for NS.

Since: 0.2.3.0

Expanding sums to products

expand_NS :: forall f xs. SListI xs => (forall x. f x) -> NS f xs -> NP f xs Source #

Specialization of hexpand.

Since: 0.2.5.0

cexpand_NS :: forall c proxy f xs. All c xs => proxy c -> (forall x. c x => f x) -> NS f xs -> NP f xs Source #

Specialization of hcexpand.

Since: 0.2.5.0

expand_SOP :: forall f xss. All SListI xss => (forall x. f x) -> SOP f xss -> POP f xss Source #

Specialization of hexpand.

Since: 0.2.5.0

cexpand_SOP :: forall c proxy f xss. All2 c xss => proxy c -> (forall x. c x => f x) -> SOP f xss -> POP f xss Source #

Specialization of hcexpand.

Since: 0.2.5.0

Transformation of index lists and coercions

trans_NS :: AllZip c xs ys => proxy c -> (forall x y. c x y => f x -> g y) -> NS f xs -> NS g ys Source #

Specialization of htrans.

Since: 0.3.1.0

trans_SOP :: AllZip2 c xss yss => proxy c -> (forall x y. c x y => f x -> g y) -> SOP f xss -> SOP g yss Source #

Specialization of htrans.

Since: 0.3.1.0

coerce_NS :: forall f g xs ys. AllZip (LiftedCoercible f g) xs ys => NS f xs -> NS g ys Source #

Specialization of hcoerce.

Since: 0.3.1.0

coerce_SOP :: forall f g xss yss. AllZip2 (LiftedCoercible f g) xss yss => SOP f xss -> SOP g yss Source #

Specialization of hcoerce.

Since: 0.3.1.0

fromI_NS :: forall f xs ys. AllZip (LiftedCoercible I f) xs ys => NS I xs -> NS f ys Source #

Specialization of hfromI.

Since: 0.3.1.0

fromI_SOP :: forall f xss yss. AllZip2 (LiftedCoercible I f) xss yss => SOP I xss -> SOP f yss Source #

Specialization of hfromI.

Since: 0.3.1.0

toI_NS :: forall f xs ys. AllZip (LiftedCoercible f I) xs ys => NS f xs -> NS I ys Source #

Specialization of htoI.

Since: 0.3.1.0

toI_SOP :: forall f xss yss. AllZip2 (LiftedCoercible f I) xss yss => SOP f xss -> SOP I yss Source #

Specialization of htoI.

Since: 0.3.1.0

Datatypes

Constructing sums

Destructing sums

Application

Lifting / mapping

Dealing with All c

Collapsing

Sequencing

Catamorphism and anamorphism

Expanding sums to products

Transformation of index lists and coercions

Dealing with `All c`