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