Control.Arrow.Free

Free arrow

data Arr f a b where

Constructors

Id :: Arr f a a

Arr :: (b -> c) -> Arr f a b -> Arr f a c

Prod :: Arr f a b -> Arr f a c -> Arr f a (b, c)

Instances

Arrow (Arr f)

Defined in Control.Arrow.Free

Methods

arr :: (b -> c) -> Arr f b c

first :: Arr f b c -> Arr f (b, d) (c, d)

second :: Arr f b c -> Arr f (d, b) (d, c)

(***) :: Arr f b c -> Arr f b' c' -> Arr f (b, b') (c, c')

(&&&) :: Arr f b c -> Arr f b c' -> Arr f b (c, c')

FreeAlgebra2 Arr

Defined in Control.Arrow.Free

Methods

liftFree2 :: AlgebraType0 Arr f => f a b -> Arr f a b

foldNatFree2 :: (AlgebraType Arr d, AlgebraType0 Arr f) => (forall (x :: k) (y :: k). f x y -> d x y) -> Arr f a b -> d a b

codom2 :: AlgebraType0 Arr f => Proof (AlgebraType Arr (Arr f)) (Arr f)

forget2 :: AlgebraType Arr f => Proof (AlgebraType0 Arr f) (Arr f)

Category (Arr f :: Type -> Type -> Type)

Defined in Control.Arrow.Free

Methods

id :: Arr f a a

(.) :: Arr f b c -> Arr f a b -> Arr f a c

Semigroup (Arr f o o)

Defined in Control.Arrow.Free

Methods

(<>) :: Arr f o o -> Arr f o o -> Arr f o o

sconcat :: NonEmpty (Arr f o o) -> Arr f o o

stimes :: Integral b => b -> Arr f o o -> Arr f o o

Monoid (Arr f o o)

Defined in Control.Arrow.Free

Methods

mempty :: Arr f o o

mappend :: Arr f o o -> Arr f o o -> Arr f o o

mconcat :: [Arr f o o] -> Arr f o o

type AlgebraType0 Arr (f :: l)

Defined in Control.Arrow.Free

type AlgebraType0 Arr (f :: l) = ()

type AlgebraType Arr (c :: Type -> Type -> Type)

Defined in Control.Arrow.Free

type AlgebraType Arr (c :: Type -> Type -> Type) = Arrow c

arrArr :: (b -> c) -> Arr f b c

mapArr :: f b c -> Arr f a b -> Arr f a c

foldArr :: forall f arr a b. Arrow arr => (forall x y. f x y -> arr x y) -> Arr f a b -> arr a b

Free arrow (CPS style)

newtype A f a b

Free arrow using CPS sytle.

Constructors

A

Fields

• runA :: forall r. Arrow r => (forall x y. f x y -> r x y) -> r a b

Instances

Arrow (A f)

Defined in Control.Arrow.Free

Methods

arr :: (b -> c) -> A f b c

first :: A f b c -> A f (b, d) (c, d)

second :: A f b c -> A f (d, b) (d, c)

(***) :: A f b c -> A f b' c' -> A f (b, b') (c, c')

(&&&) :: A f b c -> A f b c' -> A f b (c, c')

FreeAlgebra2 A

Defined in Control.Arrow.Free

Methods

liftFree2 :: AlgebraType0 A f => f a b -> A f a b

foldNatFree2 :: (AlgebraType A d, AlgebraType0 A f) => (forall (x :: k) (y :: k). f x y -> d x y) -> A f a b -> d a b

codom2 :: AlgebraType0 A f => Proof (AlgebraType A (A f)) (A f)

forget2 :: AlgebraType A f => Proof (AlgebraType0 A f) (A f)

Category (A f :: Type -> Type -> Type)

Defined in Control.Arrow.Free

Methods

id :: A f a a

(.) :: A f b c -> A f a b -> A f a c

Semigroup (A f o o)

Defined in Control.Arrow.Free

Methods

(<>) :: A f o o -> A f o o -> A f o o

sconcat :: NonEmpty (A f o o) -> A f o o

stimes :: Integral b => b -> A f o o -> A f o o

Monoid (A f o o)

Defined in Control.Arrow.Free

Methods

mempty :: A f o o

mappend :: A f o o -> A f o o -> A f o o

mconcat :: [A f o o] -> A f o o

type AlgebraType0 A (f :: l)

Defined in Control.Arrow.Free

type AlgebraType0 A (f :: l) = ()

type AlgebraType A (c :: Type -> Type -> Type)

Defined in Control.Arrow.Free

type AlgebraType A (c :: Type -> Type -> Type) = Arrow c

fromA :: A f a b -> Arr f a b

Inverse of fromA, which also is a specialisatin of hoistFreeH2.

toA :: Arr f a b -> A f a b

Isomorphism from Arr to A, which is a specialisation of hoistFreeH2.

Free interface re-exports

class FreeAlgebra2 (m :: (k -> k -> Type) -> k -> k -> Type) where

Free algebra class similar to FreeAlgebra1 and FreeAlgebra, but for types of kind k -> k -> Type.

Minimal complete definition

liftFree2, foldNatFree2

Methods

liftFree2 :: AlgebraType0 m f => f a b -> m f a b

Lift a graph f satsifying the constraint AlgebraType0 to a free its object m f.

foldNatFree2 :: (AlgebraType m d, AlgebraType0 m f) => (forall (x :: k) (y :: k). f x y -> d x y) -> m f a b -> d a b

This represents the theorem that m f is indeed free object (as in propositions as types). The types of kind k -> k -> Type form a category, where an arrow from f :: k -> k -> Type to d :: k -> k -> Type is represented by type forall x y. f x y -> d x y. foldNatFree2 states that whenever we have such a morphism and d satisfies the constraint AlgebraType m d then we can construct a morphism from m f to d.

foldNatFree2 nat (liftFree2 tr) = nat tr

foldNatFree2 nat . foldNatFree2 nat' = foldNatFree2 (foldNatFree2 nat . nat')

codom2 :: AlgebraType0 m f => Proof (AlgebraType m (m f)) (m f)

A proof that for each f satisfying AlgebraType0 m f, m f satisfies AlgebraType m (m f) constrant. This means that m is a well defined functor from the full sub-category of types of kind k -> k -> Type which satisfy the AlgebraType0 m constraint to the full subcategory of types of the same kind which satifsfy the constraint AlgebraType m.

forget2 :: AlgebraType m f => Proof (AlgebraType0 m f) (m f)

A proof that each type f :: k -> k -> Type satisfying the Algebra m f constraint also satisfies AlgebraType0 m f. This states that there is a well defined forgetful functor from the category of types of kind k -> k -> Type which satisfy the AlgebraType m to the category of types of the same kind which satisfy the AlgebraType0 m constraint.

Instances

FreeAlgebra2 (ListTr :: (k -> k -> Type) -> k -> k -> Type)

Defined in Control.Category.Free.Internal

Methods

liftFree2 :: AlgebraType0 ListTr f => f a b -> ListTr f a b

foldNatFree2 :: (AlgebraType ListTr d, AlgebraType0 ListTr f) => (forall (x :: k0) (y :: k0). f x y -> d x y) -> ListTr f a b -> d a b

codom2 :: AlgebraType0 ListTr f => Proof (AlgebraType ListTr (ListTr f)) (ListTr f)

forget2 :: AlgebraType ListTr f => Proof (AlgebraType0 ListTr f) (ListTr f)

FreeAlgebra2 (Queue :: (k -> k -> Type) -> k -> k -> Type)

Defined in Control.Category.Free.Internal

Methods

liftFree2 :: AlgebraType0 Queue f => f a b -> Queue f a b

foldNatFree2 :: (AlgebraType Queue d, AlgebraType0 Queue f) => (forall (x :: k0) (y :: k0). f x y -> d x y) -> Queue f a b -> d a b

codom2 :: AlgebraType0 Queue f => Proof (AlgebraType Queue (Queue f)) (Queue f)

forget2 :: AlgebraType Queue f => Proof (AlgebraType0 Queue f) (Queue f)

FreeAlgebra2 (C :: (k -> k -> Type) -> k -> k -> Type)

Defined in Control.Category.Free

Methods

liftFree2 :: AlgebraType0 C f => f a b -> C f a b

foldNatFree2 :: (AlgebraType C d, AlgebraType0 C f) => (forall (x :: k0) (y :: k0). f x y -> d x y) -> C f a b -> d a b

codom2 :: AlgebraType0 C f => Proof (AlgebraType C (C f)) (C f)

forget2 :: AlgebraType C f => Proof (AlgebraType0 C f) (C f)

Monad m => FreeAlgebra2 (EffCat m :: (k -> k -> Type) -> k -> k -> Type)

Defined in Control.Category.FreeEffect

Methods

liftFree2 :: AlgebraType0 (EffCat m) f => f a b -> EffCat m f a b

foldNatFree2 :: (AlgebraType (EffCat m) d, AlgebraType0 (EffCat m) f) => (forall (x :: k0) (y :: k0). f x y -> d x y) -> EffCat m f a b -> d a b

codom2 :: AlgebraType0 (EffCat m) f => Proof (AlgebraType (EffCat m) (EffCat m f)) (EffCat m f)

forget2 :: AlgebraType (EffCat m) f => Proof (AlgebraType0 (EffCat m) f) (EffCat m f)

FreeAlgebra2 A

Defined in Control.Arrow.Free

Methods

liftFree2 :: AlgebraType0 A f => f a b -> A f a b

foldNatFree2 :: (AlgebraType A d, AlgebraType0 A f) => (forall (x :: k) (y :: k). f x y -> d x y) -> A f a b -> d a b

codom2 :: AlgebraType0 A f => Proof (AlgebraType A (A f)) (A f)

forget2 :: AlgebraType A f => Proof (AlgebraType0 A f) (A f)

FreeAlgebra2 Arr

Defined in Control.Arrow.Free

Methods

liftFree2 :: AlgebraType0 Arr f => f a b -> Arr f a b

foldNatFree2 :: (AlgebraType Arr d, AlgebraType0 Arr f) => (forall (x :: k) (y :: k). f x y -> d x y) -> Arr f a b -> d a b

codom2 :: AlgebraType0 Arr f => Proof (AlgebraType Arr (Arr f)) (Arr f)

forget2 :: AlgebraType Arr f => Proof (AlgebraType0 Arr f) (Arr f)

wrapFree2 :: (AlgebraType0 m f, FreeAlgebra2 m, Monad (m f a)) => f a (m f a b) -> m f a b

Version of wrap from free package but for graphs.

foldFree2 :: (FreeAlgebra2 m, AlgebraType m f) => m f a b -> f a b

Like foldFree or foldFree1 but for graphs.

A lawful instance will satisfy:

 foldFree2 . liftFree2 == id :: f a b -> f a b

It is the unit of adjuction defined by FreeAlgebra1 class.

hoistFree2 :: (FreeAlgebra2 m, AlgebraType0 m g, AlgebraType0 m f) => (forall (x :: k) (y :: k). f x y -> g x y) -> m f a b -> m g a b

Hoist the underlying graph in the free structure. This is a higher version of a functor (analogous to fmapFree, which defined functor instance for FreeAlgebra instances) and it satisfies the functor laws:

hoistFree2 id = id

hoistFree2 f . hoistFree2 g = hoistFree2 (f . g)

joinFree2 :: (FreeAlgebra2 m, AlgebraType0 m f) => m (m f) a b -> m f a b

FreeAlgebra2 m is a monad on some subcategory of graphs (types of kind k -> k -> Type@), joinFree it is the join of this monad.

foldNatFree2 nat . joinFree2 = foldNatFree2 (foldNatFree2 nat)

This property is analogous to foldMap f . concat = foldMap (foldMap f),

bindFree2 :: (FreeAlgebra2 m, AlgebraType0 m g, AlgebraType0 m f) => m f a b -> (forall (x :: k) (y :: k). f x y -> m g x y) -> m g a b

bind of the monad defined by m on the subcategory of graphs (types of kind k -> k -> Type).

foldNatFree2 nat (bindFree mf nat') = foldNatFree2 (foldNatFree2 nat . nat') mf