227 lines
7.4 KiB
Idris
227 lines
7.4 KiB
Idris
||| This module defines profunctor costrength with respect to a particular
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||| monoidal structure.
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||| Since the homset profunctor (`Morphism`) is not costrong, very few
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||| profunctors implement this interface.
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||| Unlike Haskell's profunctors library, `Costrong` and `Cochoice` are here
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||| special cases of the interface `GenCostrong`, which defines costrength with
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||| respect to an arbitrary tensor product. When writing implementations for
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||| a profunctor, `GenCostrong Pair` and `GenCostrong Either` should be used instead
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||| of `Costrong` and `Cochoice` respectively.
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module Data.Profunctor.Costrong
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import Data.Morphisms
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import Data.Tensor
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import Data.Profunctor.Functor
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import Data.Profunctor.Types
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%default total
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------------------------------------------------------------------------------
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-- Costrength interface
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------------------------------------------------------------------------------
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||| Profunctor costrength with respect to a tensor product.
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||| These constraints are not required by the interface, but the tensor product
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||| `ten` is generally expected to implement `(Tensor ten i, Symmetric ten)`.
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||| Laws:
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||| * `costrongl = costrongr . dimap swap' swap'`
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||| * `costrongl . dimap unitr.rightToLeft unitr.leftToRight = id`
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||| * `costrongl . lmap (mapSnd f) = costrongl . rmap (mapSnd f)`
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||| * `costrongr . costrongr = costrongr . dimap assocl assocr`
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||| @ ten The tensor product of the monoidal structure
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public export
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interface Profunctor p => GenCostrong (0 ten : Type -> Type -> Type) p where
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||| The left action of a costrong profunctor.
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costrongl : p (a `ten` c) (b `ten` c) -> p a b
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||| The right action of a costrong profunctor.
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costrongr : p (c `ten` a) (c `ten` b) -> p a b
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||| Profunctor costrength with respect to the product (`Pair`).
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public export
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Costrong : (p : Type -> Type -> Type) -> Type
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Costrong = GenCostrong Pair
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||| A special case of `costrongl` with constraint `Costrong`.
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||| This is useful if the typechecker has trouble inferring the tensor product.
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public export
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unfirst : Costrong p => p (a, c) (b, c) -> p a b
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unfirst = costrongl {ten=Pair}
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||| A special case of `costrongr` with constraint `Costrong`.
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||| This is useful if the typechecker has trouble inferring the tensor product.
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public export
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unsecond : Costrong p => p (c, a) (c, b) -> p a b
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unsecond = costrongr {ten=Pair}
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||| Profunctor costrength with respect to the coproduct (`Either`).
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public export
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Cochoice : (p : Type -> Type -> Type) -> Type
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Cochoice = GenCostrong Either
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||| A special case of `costrongl` with constraint `Cochoice`.
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||| This is useful if the typechecker has trouble inferring the tensor product.
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public export
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unleft : Cochoice p => p (Either a c) (Either b c) -> p a b
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unleft = costrongl {ten=Either}
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||| A special case of `costrongr` with constraint `Cochoice`.
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||| This is useful if the typechecker has trouble inferring the tensor product.
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public export
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unright : Cochoice p => p (Either c a) (Either c b) -> p a b
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unright = costrongr {ten=Either}
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------------------------------------------------------------------------------
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-- Implementations
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------------------------------------------------------------------------------
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export
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GenCostrong Pair Tagged where
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costrongl (Tag (x,_)) = Tag x
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costrongr (Tag (_,x)) = Tag x
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export
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GenCostrong Either (Forget r) where
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costrongl (MkForget k) = MkForget (k . Left)
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costrongr (MkForget k) = MkForget (k . Right)
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export
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GenCostrong Pair (Coforget r) where
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costrongl (MkCoforget k) = MkCoforget (fst . k)
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costrongr (MkCoforget k) = MkCoforget (snd . k)
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------------------------------------------------------------------------------
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-- Cotambara
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------------------------------------------------------------------------------
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||| The comonad generated by the reflective subcategory of profunctors that
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||| implement `GenCostrong ten`.
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public export
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data GenCotambara : (ten, p : Type -> Type -> Type) -> Type -> Type -> Type where
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MkCotambara : GenCostrong ten q => q :-> p -> q a b -> GenCotambara ten p a b
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export
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Profunctor (GenCotambara ten p) where
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lmap f (MkCotambara n p) = MkCotambara n (lmap f p)
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rmap f (MkCotambara n p) = MkCotambara n (rmap f p)
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dimap f g (MkCotambara n p) = MkCotambara n (dimap f g p)
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export
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ProfunctorFunctor (GenCotambara ten) where
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promap f (MkCotambara n p) = MkCotambara (f . n) p
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export
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GenCostrong ten (GenCotambara ten p) where
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costrongl (MkCotambara @{costr} n p) = MkCotambara n (costrongl @{costr} p)
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costrongr (MkCotambara @{costr} n p) = MkCotambara n (costrongr @{costr} p)
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export
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ProfunctorComonad (GenCotambara ten) where
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proextract (MkCotambara n p) = n p
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produplicate (MkCotambara n p) = MkCotambara id (MkCotambara n p)
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export
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Functor (GenCotambara ten p a) where
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map = rmap
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||| The comonad generated by the reflective subcategory of profunctors that
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||| implement `Costrong`.
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||| This is a special case of `GenCotambara`.
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public export
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Cotambara : (p : Type -> Type -> Type) -> Type -> Type -> Type
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Cotambara = GenCotambara Pair
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||| The comonad generated by the reflective subcategory of profunctors that
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||| implement `Cochoice`.
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||| This is a special case of `GenCotambara`.
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public export
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CotambaraSum : (p : Type -> Type -> Type) -> Type -> Type -> Type
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CotambaraSum = GenCotambara Either
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export
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cotambara : GenCostrong ten p => p :-> q -> p :-> GenCotambara ten q
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cotambara f = MkCotambara f
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export
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uncotambara : Tensor ten i => Profunctor q => p :-> GenCotambara ten q -> p :-> q
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uncotambara f p = proextract (f p)
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------------------------------------------------------------------------------
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-- Copastro
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------------------------------------------------------------------------------
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||| The monad generated by the reflective subcategory of profunctors that
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public export
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record GenCopastro (ten, p : Type -> Type -> Type) a b where
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constructor MkCopastro
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runCopastro : forall q. GenCostrong ten q => p :-> q -> q a b
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export
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Profunctor (GenCopastro ten p) where
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dimap f g (MkCopastro h) = MkCopastro $ \n => dimap f g (h n)
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lmap f (MkCopastro h) = MkCopastro $ \n => lmap f (h n)
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rmap f (MkCopastro h) = MkCopastro $ \n => rmap f (h n)
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export
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ProfunctorFunctor (GenCopastro ten) where
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promap f (MkCopastro h) = MkCopastro $ \n => h (n . f)
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export
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ProfunctorMonad (GenCopastro ten) where
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propure p = MkCopastro ($ p)
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projoin p = MkCopastro $ \x => runCopastro p (\y => runCopastro y x)
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export
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GenCostrong ten (GenCopastro ten p) where
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costrongl (MkCopastro h) = MkCopastro $ \n => costrongl {ten} (h n)
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costrongr (MkCopastro h) = MkCopastro $ \n => costrongr {ten} (h n)
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export
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ProfunctorAdjunction (GenCopastro ten) (GenCotambara ten) where
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prounit p = MkCotambara id (propure {t=GenCopastro ten} p)
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procounit (MkCopastro h) = proextract (h id)
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||| The monad generated by the reflective subcategory of profunctors that
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|||
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||| This is a special case of `GenCopastro`.
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public export
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Copastro : (p : Type -> Type -> Type) -> Type -> Type -> Type
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Copastro = GenCopastro Pair
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||| The monad generated by the reflective subcategory of profunctors that
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||| This is a special case of `GenCopastro`.
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public export
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CopastroSum : (p : Type -> Type -> Type) -> Type -> Type -> Type
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CopastroSum = GenCopastro Either
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export
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copastro : GenCostrong ten q => p :-> q -> GenCopastro ten p :-> q
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copastro f (MkCopastro h) = h f
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export
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uncopastro : Tensor ten i => GenCopastro ten p :-> q -> p :-> q
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uncopastro f x = f (MkCopastro ($ x))
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