idris2-profunctors/Data/Profunctor/Costrong.idr

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2023-03-06 22:47:03 -05:00
module Data.Profunctor.Costrong
import Data.Morphisms
import Data.Tensor
import Data.Profunctor.Functor
import Data.Profunctor.Types
%default total
public export
interface Profunctor p => GenCostrong (0 ten : Type -> Type -> Type) p where
costrongl : p (a `ten` c) (b `ten` c) -> p a b
costrongr : p (c `ten` a) (c `ten` b) -> p a b
public export
Costrong : (p : Type -> Type -> Type) -> Type
Costrong = GenCostrong Pair
public export
unfirst : Costrong p => p (a, c) (b, c) -> p a b
unfirst = costrongl {ten=Pair}
public export
unsecond : Costrong p => p (c, a) (c, b) -> p a b
unsecond = costrongr {ten=Pair}
public export
Cochoice : (p : Type -> Type -> Type) -> Type
Cochoice = GenCostrong Either
public export
unleft : Cochoice p => p (Either a c) (Either b c) -> p a b
unleft = costrongl {ten=Either}
public export
unright : Cochoice p => p (Either c a) (Either c b) -> p a b
unright = costrongr {ten=Either}
-- Implementations
export
GenCostrong Pair Tagged where
costrongl (Tag (x,_)) = Tag x
costrongr (Tag (_,x)) = Tag x
-- Tambara
public export
data GenCotambara : (ten, p : Type -> Type -> Type) -> Type -> Type -> Type where
MkCotambara : GenCostrong ten q => q :-> p -> q a b -> GenCotambara ten p a b
export
Profunctor (GenCotambara ten p) where
lmap f (MkCotambara n p) = MkCotambara n (lmap f p)
rmap f (MkCotambara n p) = MkCotambara n (rmap f p)
dimap f g (MkCotambara n p) = MkCotambara n (dimap f g p)
export
ProfunctorFunctor (GenCotambara ten) where
promap f (MkCotambara n p) = MkCotambara (f . n) p
export
GenCostrong ten (GenCotambara ten p) where
costrongl (MkCotambara @{costr} n p) = MkCotambara n (costrongl @{costr} p)
costrongr (MkCotambara @{costr} n p) = MkCotambara n (costrongr @{costr} p)
export
ProfunctorComonad (GenCotambara ten) where
proextract (MkCotambara n p) = n p
produplicate (MkCotambara n p) = MkCotambara id (MkCotambara n p)
export
Functor (GenCotambara ten p a) where
map = rmap
export
gencotambara : GenCostrong ten p => p :-> q -> p :-> GenCotambara ten q
gencotambara f = MkCotambara f
export
ungencotambara : Tensor ten i => Profunctor q => p :-> GenCotambara ten q -> p :-> q
ungencotambara f p = proextract (f p)
public export
Cotambara : (p : Type -> Type -> Type) -> Type -> Type -> Type
Cotambara = GenCotambara Pair
export
cotambara : Costrong p => p :-> q -> p :-> Cotambara q
cotambara = gencotambara
export
uncotambara : Profunctor q => p :-> Cotambara q -> p :-> q
uncotambara = ungencotambara
public export
CotambaraSum : (p : Type -> Type -> Type) -> Type -> Type -> Type
CotambaraSum = GenCotambara Either
export
cotambaraSum : Cochoice p => p :-> q -> p :-> CotambaraSum q
cotambaraSum = gencotambara
export
uncotambaraSum : Profunctor q => p :-> CotambaraSum q -> p :-> q
uncotambaraSum = ungencotambara
-- Copastro
public export
record GenCopastro (ten, p : Type -> Type -> Type) a b where
constructor MkCopastro
runCopastro : forall q. GenCostrong ten q => p :-> q -> q a b
export
Profunctor (GenCopastro ten p) where
dimap f g (MkCopastro h) = MkCopastro $ \n => dimap f g (h n)
lmap f (MkCopastro h) = MkCopastro $ \n => lmap f (h n)
rmap f (MkCopastro h) = MkCopastro $ \n => rmap f (h n)
export
ProfunctorFunctor (GenCopastro ten) where
promap f (MkCopastro h) = MkCopastro $ \n => h (n . f)
export
ProfunctorMonad (GenCopastro ten) where
propure p = MkCopastro ($ p)
projoin p = MkCopastro $ \x => runCopastro p (\y => runCopastro y x)
export
GenCostrong ten (GenCopastro ten p) where
costrongl (MkCopastro h) = MkCopastro $ \n => costrongl {ten} (h n)
costrongr (MkCopastro h) = MkCopastro $ \n => costrongr {ten} (h n)
export
ProfunctorAdjunction (GenCopastro ten) (GenCotambara ten) where
prounit p = MkCotambara id (propure {t=GenCopastro ten} p)
procounit (MkCopastro h) = proextract (h id)
export
gencopastro : GenCostrong ten q => p :-> q -> GenCopastro ten p :-> q
gencopastro f (MkCopastro h) = h f
export
ungencopastro : Tensor ten i => GenCopastro ten p :-> q -> p :-> q
ungencopastro f x = f (MkCopastro ($ x))
public export
Copastro : (p : Type -> Type -> Type) -> Type -> Type -> Type
Copastro = GenCopastro Pair
export
copastro : Costrong q => p :-> q -> Copastro p :-> q
copastro = gencopastro
export
uncopastro : Copastro p :-> q -> p :-> q
uncopastro = ungencopastro
public export
CopastroSum : (p : Type -> Type -> Type) -> Type -> Type -> Type
CopastroSum = GenCopastro Either
export
copastroSum : Cochoice q => p :-> q -> CopastroSum p :-> q
copastroSum = gencopastro
export
uncopastroSum : CopastroSum p :-> q -> p :-> q
uncopastroSum = ungencopastro