Control.Algebra.Free

Documentation

class FreeAlgebra1 (m :: (k -> Type) -> k -> Type) where

Higher kinded version of FreeAlgebra. Instances includes free functors, free applicative functors, free monads, state monads etc.

A lawful instance should guarantee that foldNatFree is an isomorphism with inverses unFoldNatFree.

This guaranties that m is a left adjoint functor from the category of types of kind Type -> Type which satisfy AlgebraType0 m constraint, to the category of types of kind Type -> Type which satisfy the AlgebraType m constraint. This functor is left adjoin to the forgetful functor (which is well defined if the laws on AlgebraType0 family are satisfied. This in turn guarantees that m composed with this forgetful functor is a monad. In result we get monadic operations:

• return = liftFree

• (>>=)  = bindFree1

• join   = joinFree1

For m such that AlgebraType0 subsumes Monad this class implies:

• MFunctor via hoist = hoistFree1
• MMonad via embed = flip bindFree1
• MonadTrans via lift = liftFree

Minimal complete definition

liftFree, foldNatFree

Methods

liftFree :: AlgebraType0 m f => f a -> m f a

Natural transformation that embeds generators into m.

foldNatFree

Arguments

:: forall d f a. (AlgebraType m d, AlgebraType0 m f)

=> (forall x. f x -> d x)

a natural transformation which embeds generators of m into d

-> m f a -> d a

a morphism from m f to d

The freeness property.

foldNatFree nat (liftFree m) = nat m

foldNatFree nat . foldNatFree nat' = foldNatFree (foldNatFree nat . nat')

codom1 :: forall f. AlgebraType0 m f => Proof (AlgebraType m (m f)) (m f)

A proof that AlgebraType m (m f) holds for all AlgebraType0 f => f. Together with hoistFree1 this proves that FreeAlgebra m => m is a functor from the full subcategory of types of kind Type -> Type which satisfy AlgebraType0 m f to ones that satisfy AlgebraType m f.

default codom1 :: forall a. AlgebraType m (m a) => Proof (AlgebraType m (m a)) (m a)

forget1 :: forall f. AlgebraType m f => Proof (AlgebraType0 m f) (m f)

A proof that the forgetful functor from the full subcategory of types of kind Type -> Type satisfying AlgebraType m f constraint to types satisfying AlgebraType0 m f is well defined.

default forget1 :: forall a. AlgebraType0 m a => Proof (AlgebraType0 m a) (m a)

Instances

FreeAlgebra1 Alt

Defined in Control.Algebra.Free

Methods

liftFree :: forall f (a :: k). AlgebraType0 Alt f => f a -> Alt f a

foldNatFree :: forall d f (a :: k). (AlgebraType Alt d, AlgebraType0 Alt f) => (forall (x :: k). f x -> d x) -> Alt f a -> d a

codom1 :: forall (f :: k -> Type). AlgebraType0 Alt f => Proof (AlgebraType Alt (Alt f)) (Alt f)

forget1 :: forall (f :: k -> Type). AlgebraType Alt f => Proof (AlgebraType0 Alt f) (Alt f)

FreeAlgebra1 Ap

Ap is a free in the class of applicative functors, over any functor (Ap f is applicative whenever f is a functor)

Defined in Control.Algebra.Free

Methods

liftFree :: forall f (a :: k). AlgebraType0 Ap f => f a -> Ap f a

foldNatFree :: forall d f (a :: k). (AlgebraType Ap d, AlgebraType0 Ap f) => (forall (x :: k). f x -> d x) -> Ap f a -> d a

codom1 :: forall (f :: k -> Type). AlgebraType0 Ap f => Proof (AlgebraType Ap (Ap f)) (Ap f)

forget1 :: forall (f :: k -> Type). AlgebraType Ap f => Proof (AlgebraType0 Ap f) (Ap f)

FreeAlgebra1 Ap

Defined in Control.Algebra.Free

Methods

liftFree :: forall f (a :: k). AlgebraType0 Ap f => f a -> Ap f a

foldNatFree :: forall d f (a :: k). (AlgebraType Ap d, AlgebraType0 Ap f) => (forall (x :: k). f x -> d x) -> Ap f a -> d a

codom1 :: forall (f :: k -> Type). AlgebraType0 Ap f => Proof (AlgebraType Ap (Ap f)) (Ap f)

forget1 :: forall (f :: k -> Type). AlgebraType Ap f => Proof (AlgebraType0 Ap f) (Ap f)

FreeAlgebra1 Ap

Defined in Control.Algebra.Free

Methods

liftFree :: forall f (a :: k). AlgebraType0 Ap f => f a -> Ap f a

foldNatFree :: forall d f (a :: k). (AlgebraType Ap d, AlgebraType0 Ap f) => (forall (x :: k). f x -> d x) -> Ap f a -> d a

codom1 :: forall (f :: k -> Type). AlgebraType0 Ap f => Proof (AlgebraType Ap (Ap f)) (Ap f)

forget1 :: forall (f :: k -> Type). AlgebraType Ap f => Proof (AlgebraType0 Ap f) (Ap f)

FreeAlgebra1 Free

Free monad is free in the class of monad over the class of functors.

Defined in Control.Algebra.Free

Methods

liftFree :: forall f (a :: k). AlgebraType0 Free f => f a -> Free f a

foldNatFree :: forall d f (a :: k). (AlgebraType Free d, AlgebraType0 Free f) => (forall (x :: k). f x -> d x) -> Free f a -> d a

codom1 :: forall (f :: k -> Type). AlgebraType0 Free f => Proof (AlgebraType Free (Free f)) (Free f)

forget1 :: forall (f :: k -> Type). AlgebraType Free f => Proof (AlgebraType0 Free f) (Free f)

FreeAlgebra1 F

Defined in Control.Algebra.Free

Methods

liftFree :: forall f (a :: k). AlgebraType0 F f => f a -> F f a

foldNatFree :: forall d f (a :: k). (AlgebraType F d, AlgebraType0 F f) => (forall (x :: k). f x -> d x) -> F f a -> d a

codom1 :: forall (f :: k -> Type). AlgebraType0 F f => Proof (AlgebraType F (F f)) (F f)

forget1 :: forall (f :: k -> Type). AlgebraType F f => Proof (AlgebraType0 F f) (F f)

FreeAlgebra1 DayF

DayF, as Ap is a free applicative functor, but over applicative functors (DayF f is applicative if f is an applicative functor).

Defined in Control.Algebra.Free

Methods

liftFree :: forall f (a :: k). AlgebraType0 DayF f => f a -> DayF f a

foldNatFree :: forall d f (a :: k). (AlgebraType DayF d, AlgebraType0 DayF f) => (forall (x :: k). f x -> d x) -> DayF f a -> d a

codom1 :: forall (f :: k -> Type). AlgebraType0 DayF f => Proof (AlgebraType DayF (DayF f)) (DayF f)

forget1 :: forall (f :: k -> Type). AlgebraType DayF f => Proof (AlgebraType0 DayF f) (DayF f)

FreeAlgebra1 Coyoneda

Defined in Control.Algebra.Free

Methods

liftFree :: forall f (a :: k). AlgebraType0 Coyoneda f => f a -> Coyoneda f a

foldNatFree :: forall d f (a :: k). (AlgebraType Coyoneda d, AlgebraType0 Coyoneda f) => (forall (x :: k). f x -> d x) -> Coyoneda f a -> d a

codom1 :: forall (f :: k -> Type). AlgebraType0 Coyoneda f => Proof (AlgebraType Coyoneda (Coyoneda f)) (Coyoneda f)

forget1 :: forall (f :: k -> Type). AlgebraType Coyoneda f => Proof (AlgebraType0 Coyoneda f) (Coyoneda f)

FreeAlgebra1 ListT

Defined in Control.Algebra.Free

Methods

liftFree :: forall f (a :: k). AlgebraType0 ListT f => f a -> ListT f a

foldNatFree :: forall d f (a :: k). (AlgebraType ListT d, AlgebraType0 ListT f) => (forall (x :: k). f x -> d x) -> ListT f a -> d a

codom1 :: forall (f :: k -> Type). AlgebraType0 ListT f => Proof (AlgebraType ListT (ListT f)) (ListT f)

forget1 :: forall (f :: k -> Type). AlgebraType ListT f => Proof (AlgebraType0 ListT f) (ListT f)

FreeAlgebra1 MaybeT

Defined in Control.Algebra.Free

Methods

liftFree :: forall f (a :: k). AlgebraType0 MaybeT f => f a -> MaybeT f a

foldNatFree :: forall d f (a :: k). (AlgebraType MaybeT d, AlgebraType0 MaybeT f) => (forall (x :: k). f x -> d x) -> MaybeT f a -> d a

codom1 :: forall (f :: k -> Type). AlgebraType0 MaybeT f => Proof (AlgebraType MaybeT (MaybeT f)) (MaybeT f)

forget1 :: forall (f :: k -> Type). AlgebraType MaybeT f => Proof (AlgebraType0 MaybeT f) (MaybeT f)

(forall (f :: Type -> Type). c (Free1 c f)) => FreeAlgebra1 (Free1 c :: (Type -> Type) -> Type -> TYPE LiftedRep)

Defined in Control.Algebra.Free

Methods

liftFree :: forall f (a :: k). AlgebraType0 (Free1 c) f => f a -> Free1 c f a

foldNatFree :: forall d f (a :: k). (AlgebraType (Free1 c) d, AlgebraType0 (Free1 c) f) => (forall (x :: k). f x -> d x) -> Free1 c f a -> d a

codom1 :: forall (f :: k -> Type). AlgebraType0 (Free1 c) f => Proof (AlgebraType (Free1 c) (Free1 c f)) (Free1 c f)

forget1 :: forall (f :: k -> Type). AlgebraType (Free1 c) f => Proof (AlgebraType0 (Free1 c) f) (Free1 c f)

Monad m => FreeAlgebra1 (FreeMAction m :: (Type -> Type) -> Type -> TYPE LiftedRep)

Defined in Control.Monad.Action

Methods

liftFree :: forall f (a :: k). AlgebraType0 (FreeMAction m) f => f a -> FreeMAction m f a

foldNatFree :: forall d f (a :: k). (AlgebraType (FreeMAction m) d, AlgebraType0 (FreeMAction m) f) => (forall (x :: k). f x -> d x) -> FreeMAction m f a -> d a

codom1 :: forall (f :: k -> Type). AlgebraType0 (FreeMAction m) f => Proof (AlgebraType (FreeMAction m) (FreeMAction m f)) (FreeMAction m f)

forget1 :: forall (f :: k -> Type). AlgebraType (FreeMAction m) f => Proof (AlgebraType0 (FreeMAction m) f) (FreeMAction m f)

FreeAlgebra1 (ExceptT e :: (Type -> Type) -> Type -> Type)

ExceptT e is a free algebra among all MonadError e monads.

Defined in Control.Algebra.Free

Methods

liftFree :: forall f (a :: k). AlgebraType0 (ExceptT e) f => f a -> ExceptT e f a

foldNatFree :: forall d f (a :: k). (AlgebraType (ExceptT e) d, AlgebraType0 (ExceptT e) f) => (forall (x :: k). f x -> d x) -> ExceptT e f a -> d a

codom1 :: forall (f :: k -> Type). AlgebraType0 (ExceptT e) f => Proof (AlgebraType (ExceptT e) (ExceptT e f)) (ExceptT e f)

forget1 :: forall (f :: k -> Type). AlgebraType (ExceptT e) f => Proof (AlgebraType0 (ExceptT e) f) (ExceptT e f)

FreeAlgebra1 (ReaderT r :: (Type -> Type) -> Type -> TYPE LiftedRep)

ReaderT is a free monad in the class of all MonadReader monads.

Defined in Control.Algebra.Free

Methods

liftFree :: forall f (a :: k). AlgebraType0 (ReaderT r) f => f a -> ReaderT r f a

foldNatFree :: forall d f (a :: k). (AlgebraType (ReaderT r) d, AlgebraType0 (ReaderT r) f) => (forall (x :: k). f x -> d x) -> ReaderT r f a -> d a

codom1 :: forall (f :: k -> Type). AlgebraType0 (ReaderT r) f => Proof (AlgebraType (ReaderT r) (ReaderT r f)) (ReaderT r f)

forget1 :: forall (f :: k -> Type). AlgebraType (ReaderT r) f => Proof (AlgebraType0 (ReaderT r) f) (ReaderT r f)

FreeAlgebra1 (StateT s :: (Type -> Type) -> Type -> TYPE LiftedRep)

Lazy StateT monad transformer is a free algebra in the class of monads which satisfy the MonadState constraint. Note that this instance captures that StateT s is a monad transformer:

 liftFree = lift

This is also true for all the other monad transformers.

Defined in Control.Algebra.Free

Methods

liftFree :: forall f (a :: k). AlgebraType0 (StateT s) f => f a -> StateT s f a

foldNatFree :: forall d f (a :: k). (AlgebraType (StateT s) d, AlgebraType0 (StateT s) f) => (forall (x :: k). f x -> d x) -> StateT s f a -> d a

codom1 :: forall (f :: k -> Type). AlgebraType0 (StateT s) f => Proof (AlgebraType (StateT s) (StateT s f)) (StateT s f)

forget1 :: forall (f :: k -> Type). AlgebraType (StateT s) f => Proof (AlgebraType0 (StateT s) f) (StateT s f)

FreeAlgebra1 (StateT s :: (Type -> Type) -> Type -> TYPE LiftedRep)

Strict StateT monad transformer is also a free algebra, thus hoistFreeH is an isomorphism between the strict and lazy versions.

Defined in Control.Algebra.Free

Methods

liftFree :: forall f (a :: k). AlgebraType0 (StateT s) f => f a -> StateT s f a

foldNatFree :: forall d f (a :: k). (AlgebraType (StateT s) d, AlgebraType0 (StateT s) f) => (forall (x :: k). f x -> d x) -> StateT s f a -> d a

codom1 :: forall (f :: k -> Type). AlgebraType0 (StateT s) f => Proof (AlgebraType (StateT s) (StateT s f)) (StateT s f)

forget1 :: forall (f :: k -> Type). AlgebraType (StateT s) f => Proof (AlgebraType0 (StateT s) f) (StateT s f)

FreeAlgebra1 (WriterT w :: (Type -> Type) -> Type -> Type)

Lazy WriterT is free for algebras of type MonadWriter.

Defined in Control.Algebra.Free

Methods

liftFree :: forall f (a :: k). AlgebraType0 (WriterT w) f => f a -> WriterT w f a

foldNatFree :: forall d f (a :: k). (AlgebraType (WriterT w) d, AlgebraType0 (WriterT w) f) => (forall (x :: k). f x -> d x) -> WriterT w f a -> d a

codom1 :: forall (f :: k -> Type). AlgebraType0 (WriterT w) f => Proof (AlgebraType (WriterT w) (WriterT w f)) (WriterT w f)

forget1 :: forall (f :: k -> Type). AlgebraType (WriterT w) f => Proof (AlgebraType0 (WriterT w) f) (WriterT w f)

FreeAlgebra1 (WriterT w :: (Type -> Type) -> Type -> Type)

Strict WriterT monad transformer is a free algebra among all MonadWriters.

Defined in Control.Algebra.Free

Methods

liftFree :: forall f (a :: k). AlgebraType0 (WriterT w) f => f a -> WriterT w f a

foldNatFree :: forall d f (a :: k). (AlgebraType (WriterT w) d, AlgebraType0 (WriterT w) f) => (forall (x :: k). f x -> d x) -> WriterT w f a -> d a

codom1 :: forall (f :: k -> Type). AlgebraType0 (WriterT w) f => Proof (AlgebraType (WriterT w) (WriterT w f)) (WriterT w f)

forget1 :: forall (f :: k -> Type). AlgebraType (WriterT w) f => Proof (AlgebraType0 (WriterT w) f) (WriterT w f)

FreeAlgebra1 (RWST r w s :: (Type -> Type) -> Type -> TYPE LiftedRep)

Defined in Control.Algebra.Free

Methods

liftFree :: forall f (a :: k). AlgebraType0 (RWST r w s) f => f a -> RWST r w s f a

foldNatFree :: forall d f (a :: k). (AlgebraType (RWST r w s) d, AlgebraType0 (RWST r w s) f) => (forall (x :: k). f x -> d x) -> RWST r w s f a -> d a

codom1 :: forall (f :: k -> Type). AlgebraType0 (RWST r w s) f => Proof (AlgebraType (RWST r w s) (RWST r w s f)) (RWST r w s f)

forget1 :: forall (f :: k -> Type). AlgebraType (RWST r w s) f => Proof (AlgebraType0 (RWST r w s) f) (RWST r w s f)

FreeAlgebra1 (RWST r w s :: (Type -> Type) -> Type -> TYPE LiftedRep)

Defined in Control.Algebra.Free

Methods

liftFree :: forall f (a :: k). AlgebraType0 (RWST r w s) f => f a -> RWST r w s f a

foldNatFree :: forall d f (a :: k). (AlgebraType (RWST r w s) d, AlgebraType0 (RWST r w s) f) => (forall (x :: k). f x -> d x) -> RWST r w s f a -> d a

codom1 :: forall (f :: k -> Type). AlgebraType0 (RWST r w s) f => Proof (AlgebraType (RWST r w s) (RWST r w s f)) (RWST r w s f)

forget1 :: forall (f :: k -> Type). AlgebraType (RWST r w s) f => Proof (AlgebraType0 (RWST r w s) f) (RWST r w s f)

Type level witnesses

data Proof (c :: Constraint) (a :: l) where

A proof that constraint c holds for type a.

Constructors

Proof :: c => Proof c a

Higher algebra type / constraints

type family AlgebraType0 (f :: k) (a :: l) :: Constraint

Type family which limits Hask to its full subcategory which satisfies a given constraints. Some free algebras, like free groups, or free abelian semigroups have additional constraints on on generators, like Eq or Ord.

Instances

type AlgebraType0 Coyoneda (g :: l)

Algebras of the same type as Coyoneda are all functors.

Defined in Control.Algebra.Free

type AlgebraType0 Coyoneda (g :: l) = ()

type AlgebraType0 FreeGroup (a :: Type)

Defined in Data.Group.Free

type AlgebraType0 FreeGroup (a :: Type) = Eq a

type AlgebraType0 FreeGroupL (a :: Type)

Defined in Data.Group.Free

type AlgebraType0 FreeGroupL (a :: Type) = Eq a

type AlgebraType0 FreeAbelianMonoid (a :: Type)

Defined in Data.Monoid.Abelian

type AlgebraType0 FreeAbelianMonoid (a :: Type) = Ord a

type AlgebraType0 FreeAbelianSemigroup (a :: Type)

Defined in Data.Semigroup.Abelian

type AlgebraType0 FreeAbelianSemigroup (a :: Type) = Ord a

type AlgebraType0 FreeSemilattice (a :: Type)

Defined in Data.Semigroup.Semilattice

type AlgebraType0 FreeSemilattice (a :: Type) = Ord a

type AlgebraType0 DList (a :: l)

Defined in Data.Algebra.Free

type AlgebraType0 DList (a :: l) = ()

type AlgebraType0 DNonEmpty (a :: l)

Defined in Data.Algebra.Free

type AlgebraType0 DNonEmpty (a :: l) = ()

type AlgebraType0 Identity (a :: l)

Defined in Data.Algebra.Free

type AlgebraType0 Identity (a :: l) = ()

type AlgebraType0 NonEmpty (a :: l)

Defined in Data.Algebra.Free

type AlgebraType0 NonEmpty (a :: l) = ()

type AlgebraType0 Maybe (a :: l)

Defined in Data.Algebra.Free

type AlgebraType0 Maybe (a :: l) = ()

type AlgebraType0 [] (a :: l)

Defined in Data.Algebra.Free

type AlgebraType0 [] (a :: l) = ()

type AlgebraType0 (Free1 c :: (Type -> Type) -> Type -> TYPE LiftedRep) (f :: l)

Defined in Control.Algebra.Free

type AlgebraType0 (Free1 c :: (Type -> Type) -> Type -> TYPE LiftedRep) (f :: l) = ()

type AlgebraType0 (Free Monoid) (a :: l)

Defined in Data.Algebra.Free

type AlgebraType0 (Free Monoid) (a :: l) = ()

type AlgebraType0 (Free Semigroup) (a :: l)

Defined in Data.Algebra.Free

type AlgebraType0 (Free Semigroup) (a :: l) = ()

type AlgebraType0 (Free Group) (a :: l)

Defined in Data.Algebra.Free

type AlgebraType0 (Free Group) (a :: l) = ()

type AlgebraType0 Alt (f :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType0 Alt (f :: Type -> Type) = Functor f

type AlgebraType0 Ap (g :: Type -> Type)

Algebras of the same type as Ap are the applicative functors.

Defined in Control.Algebra.Free

type AlgebraType0 Ap (g :: Type -> Type) = Functor g

type AlgebraType0 Ap (g :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType0 Ap (g :: Type -> Type) = Functor g

type AlgebraType0 Ap (g :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType0 Ap (g :: Type -> Type) = Functor g

type AlgebraType0 F (f :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType0 F (f :: Type -> Type) = Functor f

type AlgebraType0 DayF (g :: Type -> Type)

Algebras of the same type as DayF are all the applicative functors.

Defined in Control.Algebra.Free

type AlgebraType0 DayF (g :: Type -> Type) = Applicative g

type AlgebraType0 Free (f :: Type -> Type)

Algebras of the same type as Free monad is the class of all monads.

Defined in Control.Algebra.Free

type AlgebraType0 Free (f :: Type -> Type) = Functor f

type AlgebraType0 ListT (f :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType0 ListT (f :: Type -> Type) = Monad f

type AlgebraType0 MaybeT (m :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType0 MaybeT (m :: Type -> Type) = Monad m

type AlgebraType0 (FreeMAction m :: (Type -> Type) -> Type -> TYPE LiftedRep) (f :: Type -> Type)

Defined in Control.Monad.Action

type AlgebraType0 (FreeMAction m :: (Type -> Type) -> Type -> TYPE LiftedRep) (f :: Type -> Type) = Functor f

type AlgebraType0 (ReaderT r :: (Type -> Type) -> Type -> TYPE LiftedRep) (m :: Type -> Type)

Algebras of the same type as ReaderT monad is the class of all reader monads.

TODO: take advantage of poly-kinded ReaderT

Defined in Control.Algebra.Free

type AlgebraType0 (ReaderT r :: (Type -> Type) -> Type -> TYPE LiftedRep) (m :: Type -> Type) = Monad m

type AlgebraType0 (StateT s :: (Type -> Type) -> Type -> TYPE LiftedRep) (m :: Type -> Type)

Algebras of the same type as StateT monad is the class of all state monads.

Defined in Control.Algebra.Free

type AlgebraType0 (StateT s :: (Type -> Type) -> Type -> TYPE LiftedRep) (m :: Type -> Type) = Monad m

type AlgebraType0 (StateT s :: (Type -> Type) -> Type -> TYPE LiftedRep) (m :: Type -> Type)

Algebras of the same type as StateT monad is the class of all state monads.

Defined in Control.Algebra.Free

type AlgebraType0 (StateT s :: (Type -> Type) -> Type -> TYPE LiftedRep) (m :: Type -> Type) = Monad m

type AlgebraType0 (ExceptT e :: (Type -> Type) -> Type -> Type) (m :: Type -> Type)

Algebras of the same type as ReaderT monad is the class of all reader monads.

Defined in Control.Algebra.Free

type AlgebraType0 (ExceptT e :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) = Monad m

type AlgebraType0 (WriterT w :: (Type -> Type) -> Type -> Type) (m :: Type -> Type)

Algebras of the same type as WriterT monad is the class of all writer monads.

Defined in Control.Algebra.Free

type AlgebraType0 (WriterT w :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) = (Monad m, Monoid w)

type AlgebraType0 (WriterT w :: (Type -> Type) -> Type -> Type) (m :: Type -> Type)

Algebras of the same type as WriterT monad is the class of all writer monads.

Defined in Control.Algebra.Free

type AlgebraType0 (WriterT w :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) = (Monad m, Monoid w)

type AlgebraType0 (RWST r w s :: (Type -> Type) -> Type -> TYPE LiftedRep) (m :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType0 (RWST r w s :: (Type -> Type) -> Type -> TYPE LiftedRep) (m :: Type -> Type) = (Monad m, Monoid w)

type AlgebraType0 (RWST r w s :: (Type -> Type) -> Type -> TYPE LiftedRep) (m :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType0 (RWST r w s :: (Type -> Type) -> Type -> TYPE LiftedRep) (m :: Type -> Type) = (Monad m, Monoid w)

type family AlgebraType (f :: k) (a :: l) :: Constraint

Type family which for each free algebra m returns a type level lambda from types to constraints. It is describe the class of algebras for which this free algebra is free.

A lawful instance for this type family must guarantee that the constraint AlgebraType0 m f is implied by the AlgebraType m f constraint. This guarantees that there exists a forgetful functor from the category of types of kind * -> * which satisfy AlgebraType m constrain to the category of types of kind * -> * which satisfy the 'AlgebraType0 m constraint.

Instances

type AlgebraType FreeGroup (g :: Type)

Defined in Data.Group.Free

type AlgebraType FreeGroup (g :: Type) = (Eq g, Group g)

type AlgebraType FreeGroupL (g :: Type)

Defined in Data.Group.Free

type AlgebraType FreeGroupL (g :: Type) = (Eq g, Group g)

type AlgebraType FreeAbelianMonoid (m :: Type)

Defined in Data.Monoid.Abelian

type AlgebraType FreeAbelianMonoid (m :: Type) = (Ord m, Monoid m, AbelianSemigroup m)

type AlgebraType FreeAbelianSemigroup (a :: Type)

Defined in Data.Semigroup.Abelian

type AlgebraType FreeAbelianSemigroup (a :: Type) = (Ord a, AbelianSemigroup a)

type AlgebraType FreeSemilattice (a :: Type)

Defined in Data.Semigroup.Semilattice

type AlgebraType FreeSemilattice (a :: Type)

type AlgebraType Maybe (m :: Type)

Defined in Data.Algebra.Free

type AlgebraType Maybe (m :: Type) = Pointed m

type AlgebraType Identity (a :: l)

Defined in Data.Algebra.Free

type AlgebraType Identity (a :: l) = ()

type AlgebraType DList (a :: TYPE LiftedRep)

Defined in Data.Algebra.Free

type AlgebraType DList (a :: TYPE LiftedRep) = Monoid a

type AlgebraType DNonEmpty (m :: TYPE LiftedRep)

Defined in Data.Algebra.Free

type AlgebraType DNonEmpty (m :: TYPE LiftedRep) = Semigroup m

type AlgebraType NonEmpty (m :: TYPE LiftedRep)

Defined in Data.Algebra.Free

type AlgebraType NonEmpty (m :: TYPE LiftedRep) = Semigroup m

type AlgebraType [] (m :: TYPE LiftedRep)

Defined in Data.Algebra.Free

type AlgebraType [] (m :: TYPE LiftedRep) = Monoid m

type AlgebraType (Free Monoid) (a :: TYPE LiftedRep)

Defined in Data.Algebra.Free

type AlgebraType (Free Monoid) (a :: TYPE LiftedRep) = Monoid a

type AlgebraType (Free Semigroup) (a :: TYPE LiftedRep)

Defined in Data.Algebra.Free

type AlgebraType (Free Semigroup) (a :: TYPE LiftedRep) = Semigroup a

type AlgebraType (Free Group) (a :: TYPE LiftedRep)

Defined in Data.Algebra.Free

type AlgebraType (Free Group) (a :: TYPE LiftedRep) = Group a

type AlgebraType Alt (m :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType Alt (m :: Type -> Type) = Alternative m

type AlgebraType Ap (g :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType Ap (g :: Type -> Type) = Applicative g

type AlgebraType Ap (g :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType Ap (g :: Type -> Type) = Applicative g

type AlgebraType Ap (g :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType Ap (g :: Type -> Type) = Applicative g

type AlgebraType F (m :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType F (m :: Type -> Type) = Monad m

type AlgebraType DayF (g :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType DayF (g :: Type -> Type) = Applicative g

type AlgebraType Free (m :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType Free (m :: Type -> Type) = Monad m

type AlgebraType Coyoneda (g :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType Coyoneda (g :: Type -> Type) = Functor g

type AlgebraType ListT (m :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType ListT (m :: Type -> Type) = MonadList m

type AlgebraType MaybeT (m :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType MaybeT (m :: Type -> Type) = (Monad m, MonadMaybe m)

type AlgebraType (Free1 c :: (Type -> Type) -> Type -> TYPE LiftedRep) (f :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType (Free1 c :: (Type -> Type) -> Type -> TYPE LiftedRep) (f :: Type -> Type) = c f

type AlgebraType (FreeMAction m :: (Type -> Type) -> Type -> TYPE LiftedRep) (f :: Type -> Type)

Defined in Control.Monad.Action

type AlgebraType (FreeMAction m :: (Type -> Type) -> Type -> TYPE LiftedRep) (f :: Type -> Type) = MAction m f

type AlgebraType (ReaderT r :: (Type -> Type) -> Type -> TYPE LiftedRep) (m :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType (ReaderT r :: (Type -> Type) -> Type -> TYPE LiftedRep) (m :: Type -> Type) = MonadReader r m

type AlgebraType (StateT s :: (Type -> Type) -> Type -> TYPE LiftedRep) (m :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType (StateT s :: (Type -> Type) -> Type -> TYPE LiftedRep) (m :: Type -> Type) = MonadState s m

type AlgebraType (StateT s :: (Type -> Type) -> Type -> TYPE LiftedRep) (m :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType (StateT s :: (Type -> Type) -> Type -> TYPE LiftedRep) (m :: Type -> Type) = MonadState s m

type AlgebraType (ExceptT e :: (Type -> Type) -> Type -> Type) (m :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType (ExceptT e :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) = MonadError e m

type AlgebraType (WriterT w :: (Type -> Type) -> Type -> Type) (m :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType (WriterT w :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) = MonadWriter w m

type AlgebraType (WriterT w :: (Type -> Type) -> Type -> Type) (m :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType (WriterT w :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) = MonadWriter w m

type AlgebraType (RWST r w s :: (Type -> Type) -> Type -> TYPE LiftedRep) (m :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType (RWST r w s :: (Type -> Type) -> Type -> TYPE LiftedRep) (m :: Type -> Type) = MonadRWS r w s m

type AlgebraType (RWST r w s :: (Type -> Type) -> Type -> TYPE LiftedRep) (m :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType (RWST r w s :: (Type -> Type) -> Type -> TYPE LiftedRep) (m :: Type -> Type) = MonadRWS r w s m

Combinators

wrapFree :: forall (m :: (Type -> Type) -> Type -> Type) (f :: Type -> Type) a. (FreeAlgebra1 m, AlgebraType0 m f, Monad (m f)) => f (m f a) -> m f a

Anything that carries FreeAlgebra1 constraint is also an instance of MonadFree, but not vice versa. You can use wrap to define a MonadFree instance. ContT is an example of a monad which does have an FreeAlgebra1 instance, but has an MonadFree instance.

The Monad constrain will be satisfied for many monads through the 'AlgebraType m' constraint.

foldFree1 :: forall m f a. (FreeAlgebra1 m, AlgebraType m f) => m f a -> f a

FreeAlgebra1 m implies that m f is a foldable.

 foldFree1 . liftFree == id :: f a -> f a

foldFree1 is the unit of the adjunction imposed by FreeAlgebra1 constraint.

It can be specialized to:

• lowerCoyoneda :: Functor f => Coyoneda f a -> f a

• retractAp :: Applicative f => Ap f a -> f a

• retract :: Monad f => Free f a -> f a

unFoldNatFree :: (FreeAlgebra1 m, AlgebraType0 m f) => (forall x. m f x -> d x) -> f a -> d a

unFoldNatFree is an inverse of foldNatFree

It is uniquely determined by its universal property (by Yonneda lemma):

unFoldNatFree id = ruturnFree1

Note that unFoldNatFree id is the unit of the adjunction imposed by the FreeAlgebra1 constraint.

hoistFree1

Arguments

:: forall m f g a. (FreeAlgebra1 m, AlgebraType0 m g, AlgebraType0 m f)

=> (forall x. f x -> g x)

a natural transformation f ~> g

-> m f a

-> m g a

This is a functor instance for m when considered as an endofuctor of some subcategory of Type -> Type (e.g. endofunctors of Hask) and it satisfies the functor laws:

hoistFree1 id = id

hoistFree1 f . hoistFree1 g = hoistFree1 (f . g)

It can be specialized to:

• hoistAp :: (forall a. f a -> g a) -> Ap f b -> Ap g b

• hoistFree :: Functor g => (forall a. f a -> g a) -> Free f b -> Free g b

• Control.Monad.Morph.hoist for FreeAlgebra1 m => m such that AlgebraType0 m subsumes Monad m, e.g. StateT, WriterT or ReaderT.

hoistFreeH :: forall m n f a. (FreeAlgebra1 m, FreeAlgebra1 n, AlgebraType0 m f, AlgebraType0 n f, AlgebraType m (n f)) => m f a -> n f a

 hoistFreeH . hoistFreeH = hoistFreeH

and when FreeAlgebra1 m ~ FreeAlgebra1 n:

 hoistFreeH = id

joinFree1 :: forall m f a. (FreeAlgebra1 m, AlgebraType0 m f) => m (m f) a -> m f a

joinFree1 makes m a monad in some subcatgory of types of kind Type -> Type (usually the endo-functor category of Hask). It is just a specialization of foldFree1.

bindFree1

Arguments

:: forall m f g a. (FreeAlgebra1 m, AlgebraType0 m g, AlgebraType0 m f)

=> m f a

-> (forall x. f x -> m g x)

natural transformation f ~> m g

-> m g a

Bind operator for the joinFree1 monad, this is just foldNatFree in disguise.

For StateT, WriterT or ReaderT (or any FreeAlgebra1 m => m such that AlgebraType0 m subsumes Monad m), this is the >>= version of Control.Monad.Morph.embed.

assocFree1 :: forall m f a. (FreeAlgebra1 m, AlgebraType m f, Functor (m (m f))) => m f (m f a) -> m (m f) (f a)

iterFree1 :: forall m f a. (FreeAlgebra1 m, AlgebraType0 m f, AlgebraType m Identity) => (forall x. f x -> x) -> m f a -> a

Specialization of foldNatFree @_ @Identity; it will further specialize to:

• \_ -> runIdentity . lowerCoyoneda

• iterAp :: Functor g => (g a -> a) -> Ap g a -> a

• iter :: Functor f => (f a -> a) -> Free f a -> a

cataFree1 :: forall m f a. (FreeAlgebra1 m, AlgebraType m f, Monad f, Traversable (m f)) => Fix (m f) -> f a

Fix (m f) is the initial algebra of type AlgebraType m and AlgebraType0 f.

Day convolution

newtype DayF f a

Day f f newtype wrapper. It is isomorphic with Ap f for applicative functors f via dayToAp (and apToDay).

Constructors

DayF

Fields

• runDayF :: Day f f a

Instances

FreeAlgebra1 DayF

DayF, as Ap is a free applicative functor, but over applicative functors (DayF f is applicative if f is an applicative functor).

Defined in Control.Algebra.Free

Methods

liftFree :: forall f (a :: k). AlgebraType0 DayF f => f a -> DayF f a

foldNatFree :: forall d f (a :: k). (AlgebraType DayF d, AlgebraType0 DayF f) => (forall (x :: k). f x -> d x) -> DayF f a -> d a

codom1 :: forall (f :: k -> Type). AlgebraType0 DayF f => Proof (AlgebraType DayF (DayF f)) (DayF f)

forget1 :: forall (f :: k -> Type). AlgebraType DayF f => Proof (AlgebraType0 DayF f) (DayF f)

Applicative f => Applicative (DayF f)

Defined in Control.Algebra.Free

Methods

pure :: a -> DayF f a

(<*>) :: DayF f (a -> b) -> DayF f a -> DayF f b

liftA2 :: (a -> b -> c) -> DayF f a -> DayF f b -> DayF f c

(*>) :: DayF f a -> DayF f b -> DayF f b

(<*) :: DayF f a -> DayF f b -> DayF f a

Functor (DayF f)

Defined in Control.Algebra.Free

Methods

fmap :: (a -> b) -> DayF f a -> DayF f b

(<$) :: a -> DayF f b -> DayF f a

type AlgebraType DayF (g :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType DayF (g :: Type -> Type) = Applicative g

type AlgebraType0 DayF (g :: Type -> Type)

Algebras of the same type as DayF are all the applicative functors.

Defined in Control.Algebra.Free

type AlgebraType0 DayF (g :: Type -> Type) = Applicative g

dayToAp :: Applicative f => Day f f a -> Ap f a

apToDay :: Applicative f => Ap f a -> Day f f a

Free construction in continuation passing style

newtype Free1 (c :: (Type -> Type) -> Constraint) (f :: Type -> Type) a

Free construction for kinds Type -> Type. Free1 Functor is isomorhpic to Coyoneda via hoistFreeH, and Free1 Applicative is isomorphic to Ap (also via hoistFreeH).

Note: useful instance are only provided for ghc-8.6 using quantified constraints.

Constructors

Free1

Fields

• runFree1 :: forall g. c g => (forall x. f x -> g x) -> g a

Instances

(forall (f :: Type -> Type). c (Free1 c f)) => FreeAlgebra1 (Free1 c :: (Type -> Type) -> Type -> TYPE LiftedRep)

Defined in Control.Algebra.Free

Methods

liftFree :: forall f (a :: k). AlgebraType0 (Free1 c) f => f a -> Free1 c f a

foldNatFree :: forall d f (a :: k). (AlgebraType (Free1 c) d, AlgebraType0 (Free1 c) f) => (forall (x :: k). f x -> d x) -> Free1 c f a -> d a

codom1 :: forall (f :: k -> Type). AlgebraType0 (Free1 c) f => Proof (AlgebraType (Free1 c) (Free1 c f)) (Free1 c f)

forget1 :: forall (f :: k -> Type). AlgebraType (Free1 c) f => Proof (AlgebraType0 (Free1 c) f) (Free1 c f)

(forall (h :: Type -> Type). c h => MonadZip h, c (Free1 c f)) => MonadZip (Free1 c f)

Defined in Control.Algebra.Free

Methods

mzip :: Free1 c f a -> Free1 c f b -> Free1 c f (a, b)

mzipWith :: (a -> b -> c0) -> Free1 c f a -> Free1 c f b -> Free1 c f c0

munzip :: Free1 c f (a, b) -> (Free1 c f a, Free1 c f b)

(forall (h :: Type -> Type). c h => Alternative h, c (Free1 c f)) => Alternative (Free1 c f)

Defined in Control.Algebra.Free

Methods

empty :: Free1 c f a

(<|>) :: Free1 c f a -> Free1 c f a -> Free1 c f a

some :: Free1 c f a -> Free1 c f [a]

many :: Free1 c f a -> Free1 c f [a]

(forall (h :: Type -> Type). c h => Applicative h, c (Free1 c f)) => Applicative (Free1 c f)

Free1 is an applicative functor whenever c f implies Applicative f.

Defined in Control.Algebra.Free

Methods

pure :: a -> Free1 c f a

(<*>) :: Free1 c f (a -> b) -> Free1 c f a -> Free1 c f b

liftA2 :: (a -> b -> c0) -> Free1 c f a -> Free1 c f b -> Free1 c f c0

(*>) :: Free1 c f a -> Free1 c f b -> Free1 c f b

(<*) :: Free1 c f a -> Free1 c f b -> Free1 c f a

(forall (h :: Type -> Type). c h => Functor h) => Functor (Free1 c f)

Free1 is a functor whenever c f implies Functor f .

Defined in Control.Algebra.Free

Methods

fmap :: (a -> b) -> Free1 c f a -> Free1 c f b

(<$) :: a -> Free1 c f b -> Free1 c f a

(forall (h :: Type -> Type). c h => Monad h, c (Free1 c f)) => Monad (Free1 c f)

Free1 is a monad whenever c f implies Monad f.

Defined in Control.Algebra.Free

Methods

(>>=) :: Free1 c f a -> (a -> Free1 c f b) -> Free1 c f b

(>>) :: Free1 c f a -> Free1 c f b -> Free1 c f b

return :: a -> Free1 c f a

(forall (h :: Type -> Type). c h => MonadPlus h, c (Free1 c f)) => MonadPlus (Free1 c f)

Defined in Control.Algebra.Free

Methods

mzero :: Free1 c f a

mplus :: Free1 c f a -> Free1 c f a -> Free1 c f a

type AlgebraType0 (Free1 c :: (Type -> Type) -> Type -> TYPE LiftedRep) (f :: l)

Defined in Control.Algebra.Free

type AlgebraType0 (Free1 c :: (Type -> Type) -> Type -> TYPE LiftedRep) (f :: l) = ()

type AlgebraType (Free1 c :: (Type -> Type) -> Type -> TYPE LiftedRep) (f :: Type -> Type)

Defined in Control.Algebra.Free

type AlgebraType (Free1 c :: (Type -> Type) -> Type -> TYPE LiftedRep) (f :: Type -> Type) = c f

Various classes (higher algebra types)

class Monad m => MonadList m where

Algebra type for ListT monad transformer.

Methods

mempty1 :: m a

mappend1 :: m a -> m a -> m a

Instances

Monad m => MonadList (ListT m)

Defined in Control.Algebra.Free

Methods

mempty1 :: ListT m a

mappend1 :: ListT m a -> ListT m a -> ListT m a

class MonadMaybe m where

A higher version Pointed class.

With QuantifiedConstraints this class will be redundant.

Methods

point :: forall a. m a

Instances

Monad m => MonadMaybe (MaybeT m :: Type -> Type)

Defined in Control.Algebra.Free

Methods

point :: forall (a :: k). MaybeT m a