Safe Haskell	None
Language	GHC2024

Proarrow.Optic.PowerGrate

Contents

n-ary aggregation

Description

A power grate is a Grate whose exponent is a fixed tensor power of the focus: the witness Pow n presents s as a ** ... ** a (n times), i.e. the exponential by a finite arity, which is also the reader applicative for n readers. That fixed, finite shape is a genuine decomposition, so a power grate is also a __fixed-arity Traversal__ (PowerGrateFl <: GrateFl, KaleidoFl, MonTravFl): it zips, aggregates, folds, sets and traverses like those do.

Its distinctive power over a grate or kaleidoscope is the eliminator: powerGrateP distributes an arbitrary MonoidalProfunctor -- the Applicative/zip structure (one and **) alone -- rather than the full StrongDistributiveProfunctor a traversal needs or the traversable carrier a kaleidoscope needs. With a fixed arity, any functor carrier Costar f distributes, by unzipping f (a ** ... ** a) into f a ** ... ** f a.

The aggregation is stated over an abstract MonoidalProfunctor r, not the Hask-specific Costar f = f a -> b: powerGrateOf works at any monoidal profunctor carrier (the hom (~>) gives over; an applicative Star f combines the foci through f).

Synopsis

class (MonTravFl p q, GrateFl p q) => PowerGrateFl (p :: k +-> k) (q :: k +-> k) where
- powerGrateP :: forall r (s :: k) (a :: k) (b :: k) (t :: k). MonoidalProfunctor r => p s a -> q b t -> r a b -> r s t
type PowerGrate (s :: k) (t :: k) (a :: k) (b :: k) = Optic (Prostrong (PowerGrateFl :: (k +-> k) -> (k +-> k) -> Constraint)) s t a b
type PowerGrate' (s :: k) (a :: k) = PowerGrate s s a a
powerGrateOf :: forall {k} (c :: (k -> k -> Type) -> Constraint) (s :: k) (t :: k) (a :: k) (b :: k) r. (Monoidal k, MonoidalProfunctor r, (Ob a, Ob b) => c (ExOptic (PowerGrateFl :: (k +-> k) -> (k +-> k) -> Constraint) a b)) => Optic c s t a b -> r a b -> r s t
zipWithOf :: forall {k} (c :: (k -> k -> Type) -> Constraint) (s :: k) (t :: k) (a :: k) (b :: k). (Monoidal k, Ob a, (Ob a, Ob b) => c (ExOptic (KaleidoFl :: (k +-> k) -> (k +-> k) -> Constraint) a b)) => Optic c s t a b -> ((a ** a) ~> b) -> (s ** s) ~> t
data Nat
- = Z
- | S Nat
type family Tensor (n :: Nat) (a :: k) :: k where ...
class KnownNat (n :: Nat) where
- natCase :: (n ~ 'Z => r) -> (forall (m :: Nat). (n ~ 'S m, KnownNat m) => r) -> r
data Pow (n :: Nat) (s :: k) (a :: k) where
- Pow :: forall (n :: Nat) {k} (s :: k) (a :: k). Ob a => (s ~> Tensor n a) -> Pow n s a
data CoPow (n :: Nat) (b :: k) (t :: k) where
- CoPow :: forall (n :: Nat) {k} (b :: k) (t :: k). Ob b => (Tensor n b ~> t) -> CoPow n b t
powerGrate :: forall {k} (n :: Nat) (s :: k) (t :: k) (a :: k) (b :: k). (CopyDiscard k, HasCoproducts k, KnownNat n, Ob a, Ob b) => (s ~> Tensor n a) -> (Tensor n b ~> t) -> PowerGrate s t a b

Documentation

class (MonTravFl p q, GrateFl p q) => PowerGrateFl (p :: k +-> k) (q :: k +-> k) where Source Github #

The power-grate flavor: distribute any MonoidalProfunctor r through the witness pair. TravFl is a superclass: every power-grate witness is a traversal witness (instantiate r at a StrongDistributiveProfunctor, a special MonoidalProfunctor), so it folds, sets, and traverses. The extra power is distributing the non-SDP monoidal profunctors as well. KaleidoFl is a superclass too: a tensor power is an applicative functor (the reader applicative).

Methods

powerGrateP :: forall r (s :: k) (a :: k) (b :: k) (t :: k). MonoidalProfunctor r => p s a -> q b t -> r a b -> r s t Source Github #

Instances

Instances details

CategoryOf k => PowerGrateFl (Id :: k -> k -> Type) (Id :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods powerGrateP :: forall r (s :: k) (a :: k) (b :: k) (t :: k). MonoidalProfunctor r => Id s a -> Id b t -> r a b -> r s t Source Github #
(CopyDiscard k, HasCoproducts k, KnownNat n) => PowerGrateFl (Pow n :: k -> k -> Type) (CoPow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods powerGrateP :: forall r (s :: k) (a :: k) (b :: k) (t :: k). MonoidalProfunctor r => Pow n s a -> CoPow n b t -> r a b -> r s t Source Github #
(PowerGrateFl f g, PowerGrateFl f' g') => PowerGrateFl (f :.: f' :: i -> i -> Type) (g' :.: g :: i -> i -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods powerGrateP :: forall r (s :: i) (a :: i) (b :: i) (t :: i). MonoidalProfunctor r => (f :.: f') s a -> (g' :.: g) b t -> r a b -> r s t Source Github #

type PowerGrate (s :: k) (t :: k) (a :: k) (b :: k) = Optic (Prostrong (PowerGrateFl :: (k +-> k) -> (k +-> k) -> Constraint)) s t a b Source Github #

type PowerGrate' (s :: k) (a :: k) = PowerGrate s s a a Source Github #

powerGrateOf :: forall {k} (c :: (k -> k -> Type) -> Constraint) (s :: k) (t :: k) (a :: k) (b :: k) r. (Monoidal k, MonoidalProfunctor r, (Ob a, Ob b) => c (ExOptic (PowerGrateFl :: (k +-> k) -> (k +-> k) -> Constraint) a b)) => Optic c s t a b -> r a b -> r s t Source Github #

Distribute any MonoidalProfunctor through a power grate (or any stronger optic). At the hom (~>) this is over; at an applicative Star f the foci are combined through f.

Accepts any encoding (cf. traverseOf), including (%)-composites.

zipWithOf :: forall {k} (c :: (k -> k -> Type) -> Constraint) (s :: k) (t :: k) (a :: k) (b :: k). (Monoidal k, Ob a, (Ob a, Ob b) => c (ExOptic (KaleidoFl :: (k +-> k) -> (k +-> k) -> Constraint) a b)) => Optic c s t a b -> ((a ** a) ~> b) -> (s ** s) ~> t Source Github #

Zip two sources through a Kaleidoscope (or any stronger optic, a Grate in particular, in any encoding): combine the foci pairwise. This is kaleidoscopeOf at the carrier RepCostar (Pow 2), the costar of the binary tensor power -- a binary combination (a ** a) ~> b of foci, which the optic's applicative lifts by liftA2.

`n`-ary aggregation

data Nat Source Github #

A Peano natural, the arity of a Pow witness.

Constructors

Z
S Nat

type family Tensor (n :: Nat) (a :: k) :: k where ... Source Github #

The n-fold tensor power of a: a ** a ** ... ** a (n times, terminated by Unit).

Equations

Tensor 'Z (a :: k) = Unit :: k
Tensor ('S n) (a :: k) = a ** Tensor n a

class KnownNat (n :: Nat) where Source Github #

Case analysis on a type-level Nat: the single method from which every tensor-power operation below is defined by recursion on n.

Methods

natCase :: (n ~ 'Z => r) -> (forall (m :: Nat). (n ~ 'S m, KnownNat m) => r) -> r Source Github #

Instances

Instances details

KnownNat 'Z Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods natCase :: ('Z ~ 'Z => r) -> (forall (m :: Nat). ('Z ~ 'S m, KnownNat m) => r) -> r Source Github #
KnownNat n => KnownNat ('S n) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods natCase :: ('S n ~ 'Z => r) -> (forall (m :: Nat). ('S n ~ 'S m, KnownNat m) => r) -> r Source Github #

data Pow (n :: Nat) (s :: k) (a :: k) where Source Github #

The arity-n aggregation witness: s presents n foci via the tensor power.

Constructors

Pow :: forall (n :: Nat) {k} (s :: k) (a :: k). Ob a => (s ~> Tensor n a) -> Pow n s a

Instances

Instances details

(CopyDiscard k, KnownNat n) => Strong (Tensor :: k -> (k, k) -> Type) (Pow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods act :: forall (a :: k) (x :: k) (y :: k). Ob a => Pow n x y -> Pow n (Act (Tensor :: k -> (k, k) -> Type) a x) (Act (Tensor :: k -> (k, k) -> Type) a y) Source Github #
(SymMonoidal k, KnownNat n) => MonoidalProfunctor (Pow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods one :: Pow n (Unit :: k) (Unit :: k) Source Github # () :: forall (x1 :: k) (x2 :: k) (y1 :: k) (y2 :: k). Pow n x1 x2 -> Pow n y1 y2 -> Pow n (x1 y1) (x2 ** y2) Source Github #
(Monoidal k, KnownNat n) => Profunctor (Pow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods dimap :: forall (c :: k) (a :: k) (b :: k) (d :: k). (c ~> a) -> (b ~> d) -> Pow n a b -> Pow n c d Source Github # lmap :: forall (c :: k) (a :: k) (b :: k). (c ~> a) -> Pow n a b -> Pow n c b Source Github # rmap :: forall (b :: k) (d :: k) (a :: k). (b ~> d) -> Pow n a b -> Pow n a d Source Github # (\\) :: forall (a :: k) (b :: k) r. ((Ob a, Ob b) => r) -> Pow n a b -> r Source Github #
(Monoidal k, KnownNat n) => Representable (Pow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods index :: forall (a :: k) (b :: k). Pow n a b -> a ~> ((Pow n :: k -> k -> Type) % b) Source Github # tabulate :: forall (b :: k) (a :: k). Ob b => (a ~> ((Pow n :: k -> k -> Type) % b)) -> Pow n a b Source Github # repMap :: forall (a :: k) (b :: k). (a ~> b) -> ((Pow n :: k -> k -> Type) % a) ~> ((Pow n :: k -> k -> Type) % b) Source Github # repUniv :: forall (a :: k). Ob a => Pow n ((Pow n :: k -> k -> Type) % a) a Source Github #
(Monoidal k, KnownNat n) => Proadjunction (Pow n :: k -> k -> Type) (CoPow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods unit :: forall (a :: k). Ob a => ((CoPow n :: k -> k -> Type) :.: (Pow n :: k -> k -> Type)) a a Source Github # counit :: ((Pow n :: k -> k -> Type) :.: (CoPow n :: k -> k -> Type)) :~> ((~>) :: CAT k) Source Github #
(Monoidal k, KnownNat n) => FoldFl (Pow n :: k -> k -> Type) (CoPow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods foldMapP :: forall (m :: k) (s :: k) (a :: k). Monoid m => Pow n s a -> (a ~> m) -> s ~> m Source Github #
(Monoidal k, KnownNat n) => Traversable (Pow n :: k -> k -> Type) Source Github #	`Pow n` is the representable profunctor of the tensor power `Tensor n`, which is the reader applicative for `n` readers; the instances below make it a `StrongDistributiveProfunctor`, hence a kaleidoscope witness. \| A tensor power is a fixed-shape traversable: distribute the carrier over the `n` copies.
Instance details Defined in Proarrow.Optic.PowerGrate Methods traverse :: forall (p :: k +-> k). StrongDistributiveProfunctor p => ((Pow n :: k -> k -> Type) :.: p) :~> (p :.: (Pow n :: k -> k -> Type)) Source Github #
(Monoidal k, HasCoproducts k, KnownNat n) => GlassFl (Pow n :: k -> k -> Type) (CoPow n :: k -> k -> Type) Source Github #	A power grate is a glass: ignore the source, and for each of the `n` positions feed the consumer the selector "project this focus". The selectors come from `splitPow` of `sl`, the consumer is copied `n` times with `powCopy`, `powZip` pairs them, and `powDist` applies each. Everything is stated with the `CopyDiscard` structure that `CCC` now provides, so the tensor and the product never have to be identified by hand.
Instance details Defined in Proarrow.Optic.PowerGrate Methods glassP :: forall (s :: k) (a :: k) (b :: k) (t :: k). CCC k => Pow n s a -> CoPow n b t -> (s && ((s ~~> a) ~~> b)) ~> t Source Github #
(CopyDiscard k, HasCoproducts k, KnownNat n) => GrateFl (Pow n :: k -> k -> Type) (CoPow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods zipWithP :: forall (s :: k) (a :: k) (b :: k) (t :: k). (Closed k, SymMonoidal k) => Pow n s a -> CoPow n b t -> forall (x :: k). Ob x => ((x ~~> a) ~> b) -> (x ~~> s) ~> t Source Github #
(CopyDiscard k, HasCoproducts k, KnownNat n) => CotravFl (Pow n :: k -> k -> Type) (CoPow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods cotravP :: forall r (s :: k) (a :: k) (b :: k) (t :: k). Cotraversable r => Pow n s a -> CoPow n b t -> r a b -> r s t Source Github #
(CopyDiscard k, HasCoproducts k, KnownNat n) => KaleidoFl (Pow n :: k -> k -> Type) (CoPow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods kaleidoP :: forall r (s :: k) (a :: k) (b :: k) (t :: k). Kaleidoscopic r => Pow n s a -> CoPow n b t -> r a b -> r s t Source Github #
(CopyDiscard k, HasCoproducts k, KnownNat n) => PowerGrateFl (Pow n :: k -> k -> Type) (CoPow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods powerGrateP :: forall r (s :: k) (a :: k) (b :: k) (t :: k). MonoidalProfunctor r => Pow n s a -> CoPow n b t -> r a b -> r s t Source Github #
(Monoidal k, KnownNat n) => SetterFl (Pow n :: k -> k -> Type) (CoPow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods overP :: forall (s :: k) (a :: k) (b :: k) (t :: k). Pow n s a -> CoPow n b t -> (a ~> b) -> s ~> t Source Github #
(Monoidal k, KnownNat n) => MonTravFl (Pow n :: k -> k -> Type) (CoPow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods monTravP :: forall r (s :: k) (a :: k) (b :: k) (t :: k). StrongDistributiveProfunctor r => Pow n s a -> CoPow n b t -> r a b -> r s t Source Github #
(Monoidal k, KnownNat n) => TravFl (Pow n :: k -> k -> Type) (CoPow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods travP :: forall r (s :: k) (a :: k) (b :: k) (t :: k). (StrongDistributiveProfunctor r, Strong (ProdAction :: k -> (PROD k, k) -> Type) r) => Pow n s a -> CoPow n b t -> r a b -> r s t Source Github #
(CopyDiscard k, HasCoproducts k, KnownNat n) => Strong (CoprodAction :: k -> (COPROD k, k) -> Type) (Pow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods act :: forall (a :: COPROD k) (x :: k) (y :: k). Ob a => Pow n x y -> Pow n (Act (CoprodAction :: k -> (COPROD k, k) -> Type) a x) (Act (CoprodAction :: k -> (COPROD k, k) -> Type) a y) Source Github #
(SymMonoidal k, HasCoproducts k, KnownNat n) => MonoidalProfunctor (Coprod (Pow n :: k -> k -> Type) :: COPROD k -> COPROD k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods one :: Coprod (Pow n :: k -> k -> Type) (Unit :: COPROD k) (Unit :: COPROD k) Source Github # () :: forall (x1 :: COPROD k) (x2 :: COPROD k) (y1 :: COPROD k) (y2 :: COPROD k). Coprod (Pow n :: k -> k -> Type) x1 x2 -> Coprod (Pow n :: k -> k -> Type) y1 y2 -> Coprod (Pow n :: k -> k -> Type) (x1 y1) (x2 ** y2) Source Github #
type (Pow n :: k -> k -> Type) % (a :: k) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate type (Pow n :: k -> k -> Type) % (a :: k) = Tensor n a

data CoPow (n :: Nat) (b :: k) (t :: k) where Source Github #

The dual of Pow: t is rebuilt from n foci.

Constructors

CoPow :: forall (n :: Nat) {k} (b :: k) (t :: k). Ob b => (Tensor n b ~> t) -> CoPow n b t

Instances

Instances details

(Monoidal k, KnownNat n) => Profunctor (CoPow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods dimap :: forall (c :: k) (a :: k) (b :: k) (d :: k). (c ~> a) -> (b ~> d) -> CoPow n a b -> CoPow n c d Source Github # lmap :: forall (c :: k) (a :: k) (b :: k). (c ~> a) -> CoPow n a b -> CoPow n c b Source Github # rmap :: forall (b :: k) (d :: k) (a :: k). (b ~> d) -> CoPow n a b -> CoPow n a d Source Github # (\\) :: forall (a :: k) (b :: k) r. ((Ob a, Ob b) => r) -> CoPow n a b -> r Source Github #
(Monoidal k, KnownNat n) => Proadjunction (Pow n :: k -> k -> Type) (CoPow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods unit :: forall (a :: k). Ob a => ((CoPow n :: k -> k -> Type) :.: (Pow n :: k -> k -> Type)) a a Source Github # counit :: ((Pow n :: k -> k -> Type) :.: (CoPow n :: k -> k -> Type)) :~> ((~>) :: CAT k) Source Github #
(Monoidal k, KnownNat n) => FoldFl (Pow n :: k -> k -> Type) (CoPow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods foldMapP :: forall (m :: k) (s :: k) (a :: k). Monoid m => Pow n s a -> (a ~> m) -> s ~> m Source Github #
(Monoidal k, HasCoproducts k, KnownNat n) => GlassFl (Pow n :: k -> k -> Type) (CoPow n :: k -> k -> Type) Source Github #	A power grate is a glass: ignore the source, and for each of the `n` positions feed the consumer the selector "project this focus". The selectors come from `splitPow` of `sl`, the consumer is copied `n` times with `powCopy`, `powZip` pairs them, and `powDist` applies each. Everything is stated with the `CopyDiscard` structure that `CCC` now provides, so the tensor and the product never have to be identified by hand.
Instance details Defined in Proarrow.Optic.PowerGrate Methods glassP :: forall (s :: k) (a :: k) (b :: k) (t :: k). CCC k => Pow n s a -> CoPow n b t -> (s && ((s ~~> a) ~~> b)) ~> t Source Github #
(CopyDiscard k, HasCoproducts k, KnownNat n) => GrateFl (Pow n :: k -> k -> Type) (CoPow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods zipWithP :: forall (s :: k) (a :: k) (b :: k) (t :: k). (Closed k, SymMonoidal k) => Pow n s a -> CoPow n b t -> forall (x :: k). Ob x => ((x ~~> a) ~> b) -> (x ~~> s) ~> t Source Github #
(CopyDiscard k, HasCoproducts k, KnownNat n) => CotravFl (Pow n :: k -> k -> Type) (CoPow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods cotravP :: forall r (s :: k) (a :: k) (b :: k) (t :: k). Cotraversable r => Pow n s a -> CoPow n b t -> r a b -> r s t Source Github #
(CopyDiscard k, HasCoproducts k, KnownNat n) => KaleidoFl (Pow n :: k -> k -> Type) (CoPow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods kaleidoP :: forall r (s :: k) (a :: k) (b :: k) (t :: k). Kaleidoscopic r => Pow n s a -> CoPow n b t -> r a b -> r s t Source Github #
(CopyDiscard k, HasCoproducts k, KnownNat n) => PowerGrateFl (Pow n :: k -> k -> Type) (CoPow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods powerGrateP :: forall r (s :: k) (a :: k) (b :: k) (t :: k). MonoidalProfunctor r => Pow n s a -> CoPow n b t -> r a b -> r s t Source Github #
(Monoidal k, KnownNat n) => SetterFl (Pow n :: k -> k -> Type) (CoPow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods overP :: forall (s :: k) (a :: k) (b :: k) (t :: k). Pow n s a -> CoPow n b t -> (a ~> b) -> s ~> t Source Github #
(Monoidal k, KnownNat n) => MonTravFl (Pow n :: k -> k -> Type) (CoPow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods monTravP :: forall r (s :: k) (a :: k) (b :: k) (t :: k). StrongDistributiveProfunctor r => Pow n s a -> CoPow n b t -> r a b -> r s t Source Github #
(Monoidal k, KnownNat n) => TravFl (Pow n :: k -> k -> Type) (CoPow n :: k -> k -> Type) Source Github #
Instance details Defined in Proarrow.Optic.PowerGrate Methods travP :: forall r (s :: k) (a :: k) (b :: k) (t :: k). (StrongDistributiveProfunctor r, Strong (ProdAction :: k -> (PROD k, k) -> Type) r) => Pow n s a -> CoPow n b t -> r a b -> r s t Source Github #

powerGrate :: forall {k} (n :: Nat) (s :: k) (t :: k) (a :: k) (b :: k). (CopyDiscard k, HasCoproducts k, KnownNat n, Ob a, Ob b) => (s ~> Tensor n a) -> (Tensor n b ~> t) -> PowerGrate s t a b Source Github #

Build an n-ary power grate from a tensor-power decomposition of s and recomposition of t.

Documentation

Instances

n-ary aggregation

Instances

Instances

Instances

`n`-ary aggregation