Safe Haskell	None
Language	Haskell2010

Control.Algebra.Free

Contents

Type level witnesses
Higher algebra type / constraints
Combinators
Day convolution
Free construction in continuation passing style
Various classes (higher algebra types)

Synopsis

class FreeAlgebra1 (m :: (k -> Type) -> k -> Type) where
- liftFree :: AlgebraType0 m f => f a -> m f a
- foldNatFree :: forall d f a. (AlgebraType m d, AlgebraType0 m f) => (forall x. f x -> d x) -> m f a -> d a
- codom1 :: forall f. AlgebraType0 m f => Proof (AlgebraType m (m f)) (m f)
- forget1 :: forall f. AlgebraType m f => Proof (AlgebraType0 m f) (m f)
data Proof (c :: Constraint) (a :: l) where
- Proof :: c => Proof c a
type family AlgebraType0 (f :: k) (a :: l) :: Constraint
type family AlgebraType (f :: k) (a :: l) :: Constraint
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
foldFree1 :: forall m f a. (FreeAlgebra1 m, AlgebraType m f) => m f a -> f a
unFoldNatFree :: (FreeAlgebra1 m, AlgebraType0 m f) => (forall x. m f x -> d x) -> f a -> d a
hoistFree1 :: forall m f g a. (FreeAlgebra1 m, AlgebraType0 m g, AlgebraType0 m f) => (forall x. f x -> g x) -> m f a -> m g a
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
joinFree1 :: forall m f a. (FreeAlgebra1 m, AlgebraType0 m f) => m (m f) a -> m f a
bindFree1 :: forall m f g a. (FreeAlgebra1 m, AlgebraType0 m g, AlgebraType0 m f) => m f a -> (forall x. f x -> m g x) -> m g a
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
cataFree1 :: forall m f a. (FreeAlgebra1 m, AlgebraType m f, Monad f, Traversable (m f)) => Fix (m f) -> f a
newtype DayF f a = DayF {
- runDayF :: Day f f a
}
dayToAp :: Applicative f => Day f f a -> Ap f a
apToDay :: Applicative f => Ap f a -> Day f f a
newtype Free1 (c :: (Type -> Type) -> Constraint) (f :: Type -> Type) a = Free1 {
- runFree1 :: forall g. c g => (forall x. f x -> g x) -> g a
}
class Monad m => MonadList m where
- mempty1 :: m a
- mappend1 :: m a -> m a -> m a
class MonadMaybe m where
- point :: forall a. m a

Documentation

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

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 Source #

Natural transformation that embeds generators into m.

foldNatFree Source #

Arguments

:: (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) Source #

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.

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

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.

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

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.

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

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.

Instances

(forall (f :: Type -> Type). c (Free1 c f)) => FreeAlgebra1 (Free1 c :: (Type -> Type) -> Type -> Type) Source #
Instance details Defined in Control.Algebra.Free Methods liftFree :: AlgebraType0 (Free1 c) f => f a -> Free1 c f a Source # foldNatFree :: (AlgebraType (Free1 c) d, AlgebraType0 (Free1 c) f) => (forall (x :: k). f x -> d x) -> Free1 c f a -> d a Source # codom1 :: AlgebraType0 (Free1 c) f => Proof (AlgebraType (Free1 c) (Free1 c f)) (Free1 c f) Source # forget1 :: AlgebraType (Free1 c) f => Proof (AlgebraType0 (Free1 c) f) (Free1 c f) Source #
Monad m => FreeAlgebra1 (FreeMAction m :: (Type -> Type) -> Type -> Type) Source #
Instance details Defined in Control.Monad.Action Methods liftFree :: AlgebraType0 (FreeMAction m) f => f a -> FreeMAction m f a Source # foldNatFree :: (AlgebraType (FreeMAction m) d, AlgebraType0 (FreeMAction m) f) => (forall (x :: k). f x -> d x) -> FreeMAction m f a -> d a Source # codom1 :: AlgebraType0 (FreeMAction m) f => Proof (AlgebraType (FreeMAction m) (FreeMAction m f)) (FreeMAction m f) Source # forget1 :: AlgebraType (FreeMAction m) f => Proof (AlgebraType0 (FreeMAction m) f) (FreeMAction m f) Source #
FreeAlgebra1 (ReaderT r :: (Type -> Type) -> Type -> Type) Source #	`ReaderT` is a free monad in the class of all `MonadReader` monads.
Instance details Defined in Control.Algebra.Free Methods liftFree :: AlgebraType0 (ReaderT r) f => f a -> ReaderT r f a Source # foldNatFree :: (AlgebraType (ReaderT r) d, AlgebraType0 (ReaderT r) f) => (forall (x :: k). f x -> d x) -> ReaderT r f a -> d a Source # codom1 :: AlgebraType0 (ReaderT r) f => Proof (AlgebraType (ReaderT r) (ReaderT r f)) (ReaderT r f) Source # forget1 :: AlgebraType (ReaderT r) f => Proof (AlgebraType0 (ReaderT r) f) (ReaderT r f) Source #
FreeAlgebra1 MaybeT Source #
Instance details Defined in Control.Algebra.Free Methods liftFree :: AlgebraType0 MaybeT f => f a -> MaybeT f a Source # foldNatFree :: (AlgebraType MaybeT d, AlgebraType0 MaybeT f) => (forall (x :: k). f x -> d x) -> MaybeT f a -> d a Source # codom1 :: AlgebraType0 MaybeT f => Proof (AlgebraType MaybeT (MaybeT f)) (MaybeT f) Source # forget1 :: AlgebraType MaybeT f => Proof (AlgebraType0 MaybeT f) (MaybeT f) Source #
FreeAlgebra1 F Source #
Instance details Defined in Control.Algebra.Free Methods liftFree :: AlgebraType0 F f => f a -> F f a Source # foldNatFree :: (AlgebraType F d, AlgebraType0 F f) => (forall (x :: k). f x -> d x) -> F f a -> d a Source # codom1 :: AlgebraType0 F f => Proof (AlgebraType F (F f)) (F f) Source # forget1 :: AlgebraType F f => Proof (AlgebraType0 F f) (F f) Source #
FreeAlgebra1 Free Source #	`Free` monad is free in the class of monad over the class of functors.
Instance details Defined in Control.Algebra.Free Methods liftFree :: AlgebraType0 Free f => f a -> Free f a Source # foldNatFree :: (AlgebraType Free d, AlgebraType0 Free f) => (forall (x :: k). f x -> d x) -> Free f a -> d a Source # codom1 :: AlgebraType0 Free f => Proof (AlgebraType Free (Free f)) (Free f) Source # forget1 :: AlgebraType Free f => Proof (AlgebraType0 Free f) (Free f) Source #
FreeAlgebra1 Ap Source #
Instance details Defined in Control.Algebra.Free Methods liftFree :: AlgebraType0 Ap f => f a -> Ap f a Source # foldNatFree :: (AlgebraType Ap d, AlgebraType0 Ap f) => (forall (x :: k). f x -> d x) -> Ap f a -> d a Source # codom1 :: AlgebraType0 Ap f => Proof (AlgebraType Ap (Ap f)) (Ap f) Source # forget1 :: AlgebraType Ap f => Proof (AlgebraType0 Ap f) (Ap f) Source #
FreeAlgebra1 Ap Source #
Instance details Defined in Control.Algebra.Free Methods liftFree :: AlgebraType0 Ap f => f a -> Ap f a Source # foldNatFree :: (AlgebraType Ap d, AlgebraType0 Ap f) => (forall (x :: k). f x -> d x) -> Ap f a -> d a Source # codom1 :: AlgebraType0 Ap f => Proof (AlgebraType Ap (Ap f)) (Ap f) Source # forget1 :: AlgebraType Ap f => Proof (AlgebraType0 Ap f) (Ap f) Source #
FreeAlgebra1 Ap Source #	`Ap` is a free in the class of applicative functors, over any functor (`Ap f` is applicative whenever `f` is a functor)
Instance details Defined in Control.Algebra.Free Methods liftFree :: AlgebraType0 Ap f => f a -> Ap f a Source # foldNatFree :: (AlgebraType Ap d, AlgebraType0 Ap f) => (forall (x :: k). f x -> d x) -> Ap f a -> d a Source # codom1 :: AlgebraType0 Ap f => Proof (AlgebraType Ap (Ap f)) (Ap f) Source # forget1 :: AlgebraType Ap f => Proof (AlgebraType0 Ap f) (Ap f) Source #
FreeAlgebra1 Alt Source #
Instance details Defined in Control.Algebra.Free Methods liftFree :: AlgebraType0 Alt f => f a -> Alt f a Source # foldNatFree :: (AlgebraType Alt d, AlgebraType0 Alt f) => (forall (x :: k). f x -> d x) -> Alt f a -> d a Source # codom1 :: AlgebraType0 Alt f => Proof (AlgebraType Alt (Alt f)) (Alt f) Source # forget1 :: AlgebraType Alt f => Proof (AlgebraType0 Alt f) (Alt f) Source #
FreeAlgebra1 Coyoneda Source #
Instance details Defined in Control.Algebra.Free Methods liftFree :: AlgebraType0 Coyoneda f => f a -> Coyoneda f a Source # foldNatFree :: (AlgebraType Coyoneda d, AlgebraType0 Coyoneda f) => (forall (x :: k). f x -> d x) -> Coyoneda f a -> d a Source # codom1 :: AlgebraType0 Coyoneda f => Proof (AlgebraType Coyoneda (Coyoneda f)) (Coyoneda f) Source # forget1 :: AlgebraType Coyoneda f => Proof (AlgebraType0 Coyoneda f) (Coyoneda f) Source #
FreeAlgebra1 ListT Source #
Instance details Defined in Control.Algebra.Free Methods liftFree :: AlgebraType0 ListT f => f a -> ListT f a Source # foldNatFree :: (AlgebraType ListT d, AlgebraType0 ListT f) => (forall (x :: k). f x -> d x) -> ListT f a -> d a Source # codom1 :: AlgebraType0 ListT f => Proof (AlgebraType ListT (ListT f)) (ListT f) Source # forget1 :: AlgebraType ListT f => Proof (AlgebraType0 ListT f) (ListT f) Source #
FreeAlgebra1 DayF Source #	`DayF`, as `Ap` is a free applicative functor, but over applicative functors (`DayF f` is applicative if `f` is an applicative functor).
Instance details Defined in Control.Algebra.Free Methods liftFree :: AlgebraType0 DayF f => f a -> DayF f a Source # foldNatFree :: (AlgebraType DayF d, AlgebraType0 DayF f) => (forall (x :: k). f x -> d x) -> DayF f a -> d a Source # codom1 :: AlgebraType0 DayF f => Proof (AlgebraType DayF (DayF f)) (DayF f) Source # forget1 :: AlgebraType DayF f => Proof (AlgebraType0 DayF f) (DayF f) Source #
FreeAlgebra1 (ExceptT e :: (Type -> Type) -> Type -> Type) Source #	`ExceptT e` is a free algebra among all `MonadError e` monads.
Instance details Defined in Control.Algebra.Free Methods liftFree :: AlgebraType0 (ExceptT e) f => f a -> ExceptT e f a Source # foldNatFree :: (AlgebraType (ExceptT e) d, AlgebraType0 (ExceptT e) f) => (forall (x :: k). f x -> d x) -> ExceptT e f a -> d a Source # codom1 :: AlgebraType0 (ExceptT e) f => Proof (AlgebraType (ExceptT e) (ExceptT e f)) (ExceptT e f) Source # forget1 :: AlgebraType (ExceptT e) f => Proof (AlgebraType0 (ExceptT e) f) (ExceptT e f) Source #
FreeAlgebra1 (StateT s :: (Type -> Type) -> Type -> Type) Source #	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.
Instance details Defined in Control.Algebra.Free Methods liftFree :: AlgebraType0 (StateT s) f => f a -> StateT s f a Source # foldNatFree :: (AlgebraType (StateT s) d, AlgebraType0 (StateT s) f) => (forall (x :: k). f x -> d x) -> StateT s f a -> d a Source # codom1 :: AlgebraType0 (StateT s) f => Proof (AlgebraType (StateT s) (StateT s f)) (StateT s f) Source # forget1 :: AlgebraType (StateT s) f => Proof (AlgebraType0 (StateT s) f) (StateT s f) Source #
FreeAlgebra1 (StateT s :: (Type -> Type) -> Type -> Type) Source #	Strict `StateT` monad transformer is also a free algebra, thus `hoistFreeH` is an isomorphism between the strict and lazy versions.
Instance details Defined in Control.Algebra.Free Methods liftFree :: AlgebraType0 (StateT s) f => f a -> StateT s f a Source # foldNatFree :: (AlgebraType (StateT s) d, AlgebraType0 (StateT s) f) => (forall (x :: k). f x -> d x) -> StateT s f a -> d a Source # codom1 :: AlgebraType0 (StateT s) f => Proof (AlgebraType (StateT s) (StateT s f)) (StateT s f) Source # forget1 :: AlgebraType (StateT s) f => Proof (AlgebraType0 (StateT s) f) (StateT s f) Source #
FreeAlgebra1 (WriterT w :: (Type -> Type) -> Type -> Type) Source #	Lazy `WriterT` is free for algebras of type `MonadWriter`.
Instance details Defined in Control.Algebra.Free Methods liftFree :: AlgebraType0 (WriterT w) f => f a -> WriterT w f a Source # foldNatFree :: (AlgebraType (WriterT w) d, AlgebraType0 (WriterT w) f) => (forall (x :: k). f x -> d x) -> WriterT w f a -> d a Source # codom1 :: AlgebraType0 (WriterT w) f => Proof (AlgebraType (WriterT w) (WriterT w f)) (WriterT w f) Source # forget1 :: AlgebraType (WriterT w) f => Proof (AlgebraType0 (WriterT w) f) (WriterT w f) Source #
FreeAlgebra1 (WriterT w :: (Type -> Type) -> Type -> Type) Source #	Strict `WriterT` monad transformer is a free algebra among all `MonadWriter`s.
Instance details Defined in Control.Algebra.Free Methods liftFree :: AlgebraType0 (WriterT w) f => f a -> WriterT w f a Source # foldNatFree :: (AlgebraType (WriterT w) d, AlgebraType0 (WriterT w) f) => (forall (x :: k). f x -> d x) -> WriterT w f a -> d a Source # codom1 :: AlgebraType0 (WriterT w) f => Proof (AlgebraType (WriterT w) (WriterT w f)) (WriterT w f) Source # forget1 :: AlgebraType (WriterT w) f => Proof (AlgebraType0 (WriterT w) f) (WriterT w f) Source #
FreeAlgebra1 (RWST r w s :: (Type -> Type) -> Type -> Type) Source #
Instance details Defined in Control.Algebra.Free Methods liftFree :: AlgebraType0 (RWST r w s) f => f a -> RWST r w s f a Source # foldNatFree :: (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 Source # codom1 :: AlgebraType0 (RWST r w s) f => Proof (AlgebraType (RWST r w s) (RWST r w s f)) (RWST r w s f) Source # forget1 :: AlgebraType (RWST r w s) f => Proof (AlgebraType0 (RWST r w s) f) (RWST r w s f) Source #
FreeAlgebra1 (RWST r w s :: (Type -> Type) -> Type -> Type) Source #
Instance details Defined in Control.Algebra.Free Methods liftFree :: AlgebraType0 (RWST r w s) f => f a -> RWST r w s f a Source # foldNatFree :: (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 Source # codom1 :: AlgebraType0 (RWST r w s) f => Proof (AlgebraType (RWST r w s) (RWST r w s f)) (RWST r w s f) Source # forget1 :: AlgebraType (RWST r w s) f => Proof (AlgebraType0 (RWST r w s) f) (RWST r w s f) Source #

Type level witnesses

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

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 Source #

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) Source #	Algebras of the same type as `Coyoneda` are all functors.
Instance details Defined in Control.Algebra.Free type AlgebraType0 Coyoneda (g :: l) = ()
type AlgebraType0 DList (a :: l) Source #
Instance details Defined in Data.Algebra.Free type AlgebraType0 DList (a :: l) = ()
type AlgebraType0 Maybe (a :: l) Source #
Instance details Defined in Data.Algebra.Free type AlgebraType0 Maybe (a :: l) = ()
type AlgebraType0 [] (a :: l) Source #
Instance details Defined in Data.Algebra.Free type AlgebraType0 [] (a :: l) = ()
type AlgebraType0 NonEmpty (a :: l) Source #
Instance details Defined in Data.Algebra.Free type AlgebraType0 NonEmpty (a :: l) = ()
type AlgebraType0 Identity (a :: l) Source #
Instance details Defined in Data.Algebra.Free type AlgebraType0 Identity (a :: l) = ()
type AlgebraType0 (Free1 c :: (Type -> Type) -> Type -> Type) (f :: l) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType0 (Free1 c :: (Type -> Type) -> Type -> Type) (f :: l) = ()
type AlgebraType0 (Free Group) (a :: l) Source #
Instance details Defined in Data.Algebra.Free type AlgebraType0 (Free Group) (a :: l) = ()
type AlgebraType0 (Free Monoid) (a :: l) Source #
Instance details Defined in Data.Algebra.Free type AlgebraType0 (Free Monoid) (a :: l) = ()
type AlgebraType0 (Free Semigroup) (a :: l) Source #
Instance details Defined in Data.Algebra.Free type AlgebraType0 (Free Semigroup) (a :: l) = ()
type AlgebraType0 FreeGroupL (a :: Type) Source #
Instance details Defined in Data.Group.Free type AlgebraType0 FreeGroupL (a :: Type) = Eq a
type AlgebraType0 FreeGroup (a :: Type) Source #
Instance details Defined in Data.Group.Free type AlgebraType0 FreeGroup (a :: Type) = Eq a
type AlgebraType0 FreeAbelianSemigroup (a :: Type) Source #
Instance details Defined in Data.Semigroup.Abelian type AlgebraType0 FreeAbelianSemigroup (a :: Type) = Ord a
type AlgebraType0 FreeAbelianMonoid (a :: Type) Source #
Instance details Defined in Data.Monoid.Abelian type AlgebraType0 FreeAbelianMonoid (a :: Type) = Ord a
type AlgebraType0 FreeSemilattice (a :: Type) Source #
Instance details Defined in Data.Semigroup.Semilattice type AlgebraType0 FreeSemilattice (a :: Type) = Ord a
type AlgebraType0 MaybeT (m :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType0 MaybeT (m :: Type -> Type) = Monad m
type AlgebraType0 F (f :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType0 F (f :: Type -> Type) = Functor f
type AlgebraType0 Free (f :: Type -> Type) Source #	Algebras of the same type as `Free` monad is the class of all monads.
Instance details Defined in Control.Algebra.Free type AlgebraType0 Free (f :: Type -> Type) = Functor f
type AlgebraType0 Ap (g :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType0 Ap (g :: Type -> Type) = Functor g
type AlgebraType0 Ap (g :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType0 Ap (g :: Type -> Type) = Functor g
type AlgebraType0 Ap (g :: Type -> Type) Source #	Algebras of the same type as `Ap` are the applicative functors.
Instance details Defined in Control.Algebra.Free type AlgebraType0 Ap (g :: Type -> Type) = Functor g
type AlgebraType0 Alt (f :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType0 Alt (f :: Type -> Type) = Functor f
type AlgebraType0 ListT (f :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType0 ListT (f :: Type -> Type) = Monad f
type AlgebraType0 DayF (g :: Type -> Type) Source #	Algebras of the same type as `DayF` are all the applicative functors.
Instance details Defined in Control.Algebra.Free type AlgebraType0 DayF (g :: Type -> Type) = Applicative g
type AlgebraType0 (FreeMAction m :: (Type -> Type) -> Type -> Type) (f :: Type -> Type) Source #
Instance details Defined in Control.Monad.Action type AlgebraType0 (FreeMAction m :: (Type -> Type) -> Type -> Type) (f :: Type -> Type) = Functor f
type AlgebraType0 (ExceptT e :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) Source #	Algebras of the same type as `ReaderT` monad is the class of all reader monads.
Instance details Defined in Control.Algebra.Free type AlgebraType0 (ExceptT e :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) = Monad m
type AlgebraType0 (StateT s :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) Source #	Algebras of the same type as `StateT` monad is the class of all state monads.
Instance details Defined in Control.Algebra.Free type AlgebraType0 (StateT s :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) = Monad m
type AlgebraType0 (StateT s :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) Source #	Algebras of the same type as `StateT` monad is the class of all state monads.
Instance details Defined in Control.Algebra.Free type AlgebraType0 (StateT s :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) = Monad m
type AlgebraType0 (WriterT w :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) Source #	Algebras of the same type as `WriterT` monad is the class of all writer monads.
Instance details 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) Source #	Algebras of the same type as `WriterT` monad is the class of all writer monads.
Instance details Defined in Control.Algebra.Free type AlgebraType0 (WriterT w :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) = (Monad m, Monoid w)
type AlgebraType0 (ReaderT r :: (k -> Type) -> k -> Type) (m :: Type -> Type) Source #	Algebras of the same type as `ReaderT` monad is the class of all reader monads. TODO: take advantage of poly-kinded `ReaderT`
Instance details Defined in Control.Algebra.Free type AlgebraType0 (ReaderT r :: (k -> Type) -> k -> Type) (m :: Type -> Type) = Monad m
type AlgebraType0 (RWST r w s :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType0 (RWST r w s :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) = (Monad m, Monoid w)
type AlgebraType0 (RWST r w s :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType0 (RWST r w s :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) = (Monad m, Monoid w)

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

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 Identity (a :: l) Source #
Instance details Defined in Data.Algebra.Free type AlgebraType Identity (a :: l) = ()
type AlgebraType [] (m :: Type) Source #
Instance details Defined in Data.Algebra.Free type AlgebraType [] (m :: Type) = Monoid m
type AlgebraType Maybe (m :: Type) Source #
Instance details Defined in Data.Algebra.Free type AlgebraType Maybe (m :: Type) = Pointed m
type AlgebraType NonEmpty (m :: Type) Source #
Instance details Defined in Data.Algebra.Free type AlgebraType NonEmpty (m :: Type) = Semigroup m
type AlgebraType DList (a :: Type) Source #
Instance details Defined in Data.Algebra.Free type AlgebraType DList (a :: Type) = Monoid a
type AlgebraType FreeGroupL (g :: Type) Source #
Instance details Defined in Data.Group.Free type AlgebraType FreeGroupL (g :: Type) = (Eq g, Group g)
type AlgebraType FreeGroup (g :: Type) Source #
Instance details Defined in Data.Group.Free type AlgebraType FreeGroup (g :: Type) = (Eq g, Group g)
type AlgebraType FreeAbelianSemigroup (a :: Type) Source #
Instance details Defined in Data.Semigroup.Abelian type AlgebraType FreeAbelianSemigroup (a :: Type) = (Ord a, AbelianSemigroup a)
type AlgebraType FreeAbelianMonoid (m :: Type) Source #
Instance details Defined in Data.Monoid.Abelian type AlgebraType FreeAbelianMonoid (m :: Type) = (Ord m, Monoid m, AbelianSemigroup m)
type AlgebraType FreeSemilattice (a :: Type) Source #
Instance details Defined in Data.Semigroup.Semilattice type AlgebraType FreeSemilattice (a :: Type)
type AlgebraType (Free Semigroup) (a :: Type) Source #
Instance details Defined in Data.Algebra.Free type AlgebraType (Free Semigroup) (a :: Type) = Semigroup a
type AlgebraType (Free Monoid) (a :: Type) Source #
Instance details Defined in Data.Algebra.Free type AlgebraType (Free Monoid) (a :: Type) = Monoid a
type AlgebraType (Free Group) (a :: Type) Source #
Instance details Defined in Data.Algebra.Free type AlgebraType (Free Group) (a :: Type) = Group a
type AlgebraType MaybeT (m :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType MaybeT (m :: Type -> Type) = (Monad m, MonadMaybe m)
type AlgebraType F (m :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType F (m :: Type -> Type) = Monad m
type AlgebraType Free (m :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType Free (m :: Type -> Type) = Monad m
type AlgebraType Ap (g :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType Ap (g :: Type -> Type) = Applicative g
type AlgebraType Ap (g :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType Ap (g :: Type -> Type) = Applicative g
type AlgebraType Ap (g :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType Ap (g :: Type -> Type) = Applicative g
type AlgebraType Alt (m :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType Alt (m :: Type -> Type) = Alternative m
type AlgebraType Coyoneda (g :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType Coyoneda (g :: Type -> Type) = Functor g
type AlgebraType ListT (m :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType ListT (m :: Type -> Type) = MonadList m
type AlgebraType DayF (g :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType DayF (g :: Type -> Type) = Applicative g
type AlgebraType (Free1 c :: (Type -> Type) -> Type -> Type) (f :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType (Free1 c :: (Type -> Type) -> Type -> Type) (f :: Type -> Type) = c f
type AlgebraType (FreeMAction m :: (Type -> Type) -> Type -> Type) (f :: Type -> Type) Source #
Instance details Defined in Control.Monad.Action type AlgebraType (FreeMAction m :: (Type -> Type) -> Type -> Type) (f :: Type -> Type) = MAction m f
type AlgebraType (ExceptT e :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType (ExceptT e :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) = MonadError e m
type AlgebraType (StateT s :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType (StateT s :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) = MonadState s m
type AlgebraType (StateT s :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType (StateT s :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) = MonadState s m
type AlgebraType (WriterT w :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) Source #
Instance details 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) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType (WriterT w :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) = MonadWriter w m
type AlgebraType (ReaderT r :: (k -> Type) -> k -> Type) (m :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType (ReaderT r :: (k -> Type) -> k -> Type) (m :: Type -> Type) = MonadReader r m
type AlgebraType (RWST r w s :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType (RWST r w s :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) = MonadRWS r w s m
type AlgebraType (RWST r w s :: (Type -> Type) -> Type -> Type) (m :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType (RWST r w s :: (Type -> Type) -> Type -> Type) (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 Source #

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 Source #

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 Source #

unFoldNatFree is an inverse of foldNatFree

It is uniquelly 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 Source #

Arguments

:: (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 Source #

 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 Source #

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 Source #

Arguments

:: (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) Source #

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

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 Source #

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

Day convolution

newtype DayF f a Source #

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

Functor (DayF f) Source #
Instance details Defined in Control.Algebra.Free Methods fmap :: (a -> b) -> DayF f a -> DayF f b # (<$) :: a -> DayF f b -> DayF f a #
Applicative f => Applicative (DayF f) Source #
Instance details 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 #
FreeAlgebra1 DayF Source #	`DayF`, as `Ap` is a free applicative functor, but over applicative functors (`DayF f` is applicative if `f` is an applicative functor).
Instance details Defined in Control.Algebra.Free Methods liftFree :: AlgebraType0 DayF f => f a -> DayF f a Source # foldNatFree :: (AlgebraType DayF d, AlgebraType0 DayF f) => (forall (x :: k). f x -> d x) -> DayF f a -> d a Source # codom1 :: AlgebraType0 DayF f => Proof (AlgebraType DayF (DayF f)) (DayF f) Source # forget1 :: AlgebraType DayF f => Proof (AlgebraType0 DayF f) (DayF f) Source #
type AlgebraType0 DayF (g :: Type -> Type) Source #	Algebras of the same type as `DayF` are all the applicative functors.
Instance details Defined in Control.Algebra.Free type AlgebraType0 DayF (g :: Type -> Type) = Applicative g
type AlgebraType DayF (g :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType DayF (g :: Type -> Type) = Applicative g

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

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

Free construction in continuation passing style

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

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) Source #
Instance details Defined in Control.Algebra.Free Methods liftFree :: AlgebraType0 (Free1 c) f => f a -> Free1 c f a Source # foldNatFree :: (AlgebraType (Free1 c) d, AlgebraType0 (Free1 c) f) => (forall (x :: k). f x -> d x) -> Free1 c f a -> d a Source # codom1 :: AlgebraType0 (Free1 c) f => Proof (AlgebraType (Free1 c) (Free1 c f)) (Free1 c f) Source # forget1 :: AlgebraType (Free1 c) f => Proof (AlgebraType0 (Free1 c) f) (Free1 c f) Source #
(forall (h :: Type -> Type). c h => Monad h) => Monad (Free1 c f) Source #	`Free1` is a monad whenever `c f` implies `Monad f`.
Instance details 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 # fail :: String -> Free1 c f a #
(forall (h :: Type -> Type). c h => Functor h) => Functor (Free1 c f) Source #	`Free1` is a functor whenever `c f` implies `Functor f` .
Instance details 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 => Applicative h) => Applicative (Free1 c f) Source #	`Free1` is an applicative functor whenever `c f` implies `Applicative f`.
Instance details 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 => MonadZip h) => MonadZip (Free1 c f) Source #
Instance details 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) => Alternative (Free1 c f) Source #
Instance details 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 => MonadPlus h) => MonadPlus (Free1 c f) Source #
Instance details 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) (f :: l) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType0 (Free1 c :: (Type -> Type) -> Type -> Type) (f :: l) = ()
type AlgebraType (Free1 c :: (Type -> Type) -> Type -> Type) (f :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free type AlgebraType (Free1 c :: (Type -> Type) -> Type -> Type) (f :: Type -> Type) = c f

Various classes (higher algebra types)

class Monad m => MonadList m where Source #

Algebra type for ListT monad transformer.

Methods

mempty1 :: m a Source #

mappend1 :: m a -> m a -> m a Source #

Instances

Monad m => MonadList (ListT m) Source #
Instance details Defined in Control.Algebra.Free Methods mempty1 :: ListT m a Source # mappend1 :: ListT m a -> ListT m a -> ListT m a Source #

class MonadMaybe m where Source #

A higher version Pointed class.

With QuantifiedConstraints this class will be redundant.

Methods

point :: forall a. m a Source #

Instances

Monad m => MonadMaybe (MaybeT m :: Type -> Type) Source #
Instance details Defined in Control.Algebra.Free Methods point :: MaybeT m a Source #