Generics.SOP.NS

Anamorphism for NS.

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

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

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)))]

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)))]

Specialization of hap.

Specialization of hap.

Constrained anamorphism for NS.

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.

Constrained catamorphism for NS.

Constrained version of compare_NS.

Constrained version of compare_SOP.

Specialization of hcexpand.

Specialization of hcexpand.

Specialization of hcfoldMap.

Note: We don't need Monoid instance.

Specialization of hcfoldMap.

Specialization of hcliftA2'.

Specialization of hcliftA2.

Specialization of hcliftA2.

Specialization of hcliftA.

Specialization of hcliftA.

Specialization of hcmap, which is equivalent to hcliftA.

Specialization of hcmap, which is equivalent to hcliftA.

Specialization of hcoerce.

Specialization of hcoerce.

Specialization of hcollapse.

Specialization of hcollapse.

Compare two sums with respect to the choice they are making.

A value that chooses the first option is considered smaller than one that chooses the second option.

If the choices are different, then either the first (if the first is smaller than the second) or the third (if the first is larger than the second) argument are called. If both choices are equal, then the second argument is called, which has access to the elements contained in the sums.

Compare two sums of products with respect to the choice in the sum they are making.

Only the sum structure is used for comparison. This is a small wrapper around ccompare_NS for a common special case.

Specialization of hctraverse'.

Note: as NS has exactly one element, the Functor constraint is enough.

Specialization of hctraverse'.

Specialization of hctraverse.

Specialization of hctraverse.

Specialization of hctraverse_.

Note: we don't need Applicative constraint.

Specialization of hctraverse_.

Compute all ejections from an n-ary sum.

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

Specialization of hexpand.

Specialization of hexpand.

Specialization of hfromI.

Specialization of hfromI.

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

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

Compute all injections into an n-ary sum.

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

Specialization of hliftA2.

Specialization of hliftA2.

Specialization of hliftA.

Specialization of hliftA.

Specialization of hmap, which is equivalent to hliftA.

Specialization of hmap, which is equivalent to hliftA.

Specialization of hsequence'.

Specialization of hsequence'.

Specialization of hsequence.

Specialization of hsequence.

Shift an injection.

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

Shift an injection.

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

Specialization of htoI.

Specialization of htoI.

Specialization of htrans.

Specialization of htrans.

Specialization of htraverse'.

Note: as NS has exactly one element, the Functor constraint is enough.

Specialization of htraverse'.

Specialization of htraverse_.

Note: we don't need Applicative constraint.

Specialization of htraverse_.

Unwrap a sum of products.

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'

The type of ejections from an n-ary sum.

An ejection is the pattern matching function for one part of the n-ary sum.

It is the opposite of an Injection.

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.

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 ]

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.

A 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.