idris2-profunctors/Data/Profunctor/Traversing.idr

234 lines
7.2 KiB
Idris

module Data.Profunctor.Traversing
import Control.Applicative.Const
import Control.Monad.Identity
import Data.Morphisms
import Data.Tensor
import Data.Profunctor
%default total
------------------------------------------------------------------------------
-- Default implementation machinery
------------------------------------------------------------------------------
[FoldablePair] Foldable (Pair c) where
foldr op init (_, x) = x `op` init
foldl op init (_, x) = init `op` x
null _ = False
[TraversablePair] Traversable (Pair c) using FoldablePair where
traverse f (l, r) = (l,) <$> f r
[FoldableIdentity] Foldable Identity where
foldr f i (Id x) = f x i
foldl f i (Id x) = f i x
null _ = False
[TraversableIdentity] Traversable Identity using FoldableIdentity where
traverse f (Id x) = map Id (f x)
record Bazaar a b t where
constructor MkBazaar
runBazaar : forall f. Applicative f => (a -> f b) -> f t
Functor (Bazaar a b) where
map f (MkBazaar g) = MkBazaar (map f . g)
Applicative (Bazaar a b) where
pure a = MkBazaar $ \_ => pure a
mf <*> ma = MkBazaar $ \k => runBazaar mf k <*> runBazaar ma k
sell : a -> Bazaar a b b
sell a = MkBazaar ($ a)
record Baz t b a where
constructor MkBaz
runBaz : forall f. Applicative f => (a -> f b) -> f t
Functor (Baz t b) where
map f (MkBaz g) = MkBaz (g . (. f))
sold : Baz t a a -> t
sold m = runIdentity (runBaz m Id)
Foldable (Baz t b) where
foldr f i bz = runBaz bz @{appEndo} f i
where
-- Equivalent to `Const (Endomorphism acc)`
appEndo : Applicative (\_ => acc -> acc)
appEndo = MkApplicative @{MkFunctor (const id)} (const id) (.)
Traversable (Baz t b) where
traverse f bz = map (\m => MkBaz (runBazaar m)) $ runBaz bz @{Compose} $ \x => sell <$> f x
public export
wanderDef : Profunctor p => (forall f. Traversable f => p a b -> p (f a) (f b))
-> (forall f. Applicative f => (a -> f b) -> s -> f t) -> p a b -> p s t
wanderDef tr f = dimap (\s => MkBaz $ \afb => f afb s) sold . tr
------------------------------------------------------------------------------
-- Traversing interface
------------------------------------------------------------------------------
||| The interface of profunctors that implement `wander`.
||| NOTE: Definitions in terms of `wander` are much more efficient!
|||
||| Laws:
||| * `traverse' = wander traverse`
||| * `traverse' . lmap f = lmap (map f) . traverse'`
||| * `traverse' . rmap f = rmap (map f) . traverse'`
||| * `traverse' . traverse' = traverse' @{Compose}`
||| * `dimap Id runIdentity . traverse' = id`
public export
interface (Strong p, Choice p) => Traversing p where
traverse' : Traversable f => p a b -> p (f a) (f b)
traverse' = wander traverse
wander : (forall f. Applicative f => (a -> f b) -> s -> f t) -> p a b -> p s t
wander = wanderDef traverse'
------------------------------------------------------------------------------
-- Implementations
------------------------------------------------------------------------------
public export
Traversing Morphism where
traverse' (Mor f) = Mor (map f)
wander f (Mor p) = Mor (runIdentity . (f $ Id . p))
||| A named implementation of `Traversing` for function types.
||| Use this to avoid having to use a type wrapper like `Morphism`.
public export
[Function] Traversing (\a,b => a -> b) using Strong.Function where
traverse' = map
wander f g = runIdentity . (f $ Id . g)
public export
Applicative f => Traversing (Kleislimorphism f) where
traverse' (Kleisli p) = Kleisli (traverse p)
wander f (Kleisli p) = Kleisli (f p)
public export
Applicative f => Traversing (Star f) where
traverse' (MkStar p) = MkStar (traverse p)
wander f (MkStar p) = MkStar (f p)
public export
Monoid r => Traversing (Forget r) where
traverse' (MkForget k) = MkForget (foldMap k)
wander f (MkForget k) = MkForget (runConst . f (MkConst . k))
------------------------------------------------------------------------------
-- CofreeTraversing
------------------------------------------------------------------------------
||| The comonad generated by the reflective subcategory of profunctors that
||| implement `Traversing`.
public export
record CofreeTraversing p a b where
constructor MkCFT
runCFT : forall f. Traversable f => p (f a) (f b)
public export
Profunctor p => Profunctor (CofreeTraversing p) where
lmap f (MkCFT p) = MkCFT (lmap (map f) p)
rmap g (MkCFT p) = MkCFT (rmap (map g) p)
dimap f g (MkCFT p) = MkCFT (dimap (map f) (map g) p)
public export
Profunctor p => GenStrong Pair (CofreeTraversing p) where
strongr (MkCFT p) = MkCFT (p @{Compose @{%search} @{TraversablePair}})
strongl = dimap swap' swap' . strongr {p=CofreeTraversing p}
public export
Profunctor p => GenStrong Either (CofreeTraversing p) where
strongr (MkCFT p) = MkCFT (p @{Compose {f=Either c}})
strongl = dimap swap' swap' . strongr {p=CofreeTraversing p}
public export
Profunctor p => Traversing (CofreeTraversing p) where
traverse' (MkCFT p) = MkCFT (p @{Compose})
public export
ProfunctorFunctor CofreeTraversing where
promap f (MkCFT p) = MkCFT (f p)
public export
ProfunctorComonad CofreeTraversing where
proextract (MkCFT p) = dimap Id runIdentity $ p @{TraversableIdentity}
produplicate (MkCFT p) = MkCFT $ MkCFT $ p @{Compose}
public export
Profunctor p => Functor (CofreeTraversing p a) where
map = rmap
public export
cofreeTraversing : Traversing p => p :-> q -> p :-> CofreeTraversing q
cofreeTraversing f p = MkCFT $ f $ traverse' p
public export
uncofreeTraversing : Profunctor q => p :-> CofreeTraversing q -> p :-> q
uncofreeTraversing f p = proextract $ f p
------------------------------------------------------------------------------
-- FreeTraversing
------------------------------------------------------------------------------
||| The monad generated by the reflective subcategory of profunctors that
||| implement `Traversing`.
public export
data FreeTraversing : (p : Type -> Type -> Type) -> Type -> Type -> Type where
MkFT : Traversable f => (f y -> b) -> p x y -> (a -> f x) -> FreeTraversing p a b
public export
Profunctor (FreeTraversing p) where
lmap f (MkFT l m r) = MkFT l m (r . f)
rmap f (MkFT l m r) = MkFT (f . l) m r
dimap f g (MkFT l m r) = MkFT (g . l) m (r . f)
public export
GenStrong Pair (FreeTraversing p) where
strongr (MkFT l m r) = MkFT @{Compose @{TraversablePair}} (map l) m (map r)
strongl = dimap swap' swap' . strongr {p=FreeTraversing p}
public export
GenStrong Either (FreeTraversing p) where
strongr (MkFT l m r) = MkFT @{Compose {t=Either c}} (map l) m (map r)
strongl = dimap swap' swap' . strongr {p=FreeTraversing p}
public export
Traversing (FreeTraversing p) where
traverse' (MkFT l m r) = MkFT @{Compose} (map l) m (map r)
public export
ProfunctorFunctor FreeTraversing where
promap f (MkFT l m r) = MkFT l (f m) r
public export
ProfunctorMonad FreeTraversing where
propure p = MkFT @{TraversableIdentity} runIdentity p Id
projoin (MkFT l' (MkFT l m r) r') = MkFT @{Compose} (l' . map l) m (map r . r')
public export
freeTraversing : Traversing q => p :-> q -> FreeTraversing p :-> q
freeTraversing fn (MkFT {f} l m r) = dimap r l (traverse' {f} (fn m))
public export
unfreeTraversing : FreeTraversing p :-> q -> p :-> q
unfreeTraversing f p = f (MkFT @{TraversableIdentity} runIdentity p Id)