Data.Group.Free.Church

This module provides definitions for Church-encoded FreeGroups, FreeAbelianGroups, along with useful combinators.

Church-encoded free groups

newtype FG a

The Church-encoding of a FreeGroup.

This datatype represents the "true" free group in Haskell on some a-valued generators. For more information on why this encoding is preferred, see Dan Doel's article in the Comonad Reader.

While FreeGroup et al are free in a strict language, and are more intuitive, they are not associative wtih respect to bottoms. FG and FA however, are, and should be preferred when working with possibly undefined data.

Constructors

FG

Fields

• runFG :: forall g. Group g => (a -> g) -> g

Instances

Monad FG

Defined in Data.Group.Free.Church

Methods

(>>=) :: FG a -> (a -> FG b) -> FG b

(>>) :: FG a -> FG b -> FG b

return :: a -> FG a

Functor FG

Defined in Data.Group.Free.Church

Methods

fmap :: (a -> b) -> FG a -> FG b

(<$) :: a -> FG b -> FG a

Applicative FG

Defined in Data.Group.Free.Church

Methods

pure :: a -> FG a

(<*>) :: FG (a -> b) -> FG a -> FG b

liftA2 :: (a -> b -> c) -> FG a -> FG b -> FG c

(*>) :: FG a -> FG b -> FG b

(<*) :: FG a -> FG b -> FG a

Alternative FG

Defined in Data.Group.Free.Church

Methods

empty :: FG a

(<|>) :: FG a -> FG a -> FG a

some :: FG a -> FG [a]

many :: FG a -> FG [a]

Cancellative FG

Defined in Control.Applicative.Cancellative

Methods

cancel :: FG a -> FG a

GroupFoldable FG

Defined in Data.Group.Foldable

Methods

goldMap :: Group g => (a -> g) -> FG a -> g

toFG :: FG a -> FG a

Semigroup (FG a)

Defined in Data.Group.Free.Church

Methods

(<>) :: FG a -> FG a -> FG a

sconcat :: NonEmpty (FG a) -> FG a

stimes :: Integral b => b -> FG a -> FG a

Monoid (FG a)

Defined in Data.Group.Free.Church

Methods

mempty :: FG a

mappend :: FG a -> FG a -> FG a

mconcat :: [FG a] -> FG a

Group (FG a)

Defined in Data.Group.Free.Church

Methods

invert :: FG a -> FG a

(~~) :: FG a -> FG a -> FG a

pow :: Integral x => FG a -> x -> FG a

Church-encoded free group combinators

interpretFG :: Group g => FG g -> g

Interpret a Church-encoded free group as a concrete FreeGroup.

reifyFG :: FG a -> FreeGroup a

Convert a Church-encoded free group to a concrete FreeGroup.

reflectFG :: FreeGroup a -> FG a

Convert a concrete FreeGroup to a Church-encoded free group.

presentFG :: Group g => FG g -> (FG g -> g) -> g

Present a Group as a FG modulo relations.

Church-encoded free abelian groups

newtype FA a

The Church-encoding of a FreeAbelianGroup.

This datatype represents the free group on some a-valued generators, along with their exponents in the group.

Constructors

FA

Fields

• runFA :: forall g. Abelian g => (a -> Integer -> g) -> g

Instances

Monad FA

Defined in Data.Group.Free.Church

Methods

(>>=) :: FA a -> (a -> FA b) -> FA b

(>>) :: FA a -> FA b -> FA b

return :: a -> FA a

Functor FA

Defined in Data.Group.Free.Church

Methods

fmap :: (a -> b) -> FA a -> FA b

(<$) :: a -> FA b -> FA a

Applicative FA

Defined in Data.Group.Free.Church

Methods

pure :: a -> FA a

(<*>) :: FA (a -> b) -> FA a -> FA b

liftA2 :: (a -> b -> c) -> FA a -> FA b -> FA c

(*>) :: FA a -> FA b -> FA b

(<*) :: FA a -> FA b -> FA a

Alternative FA

Defined in Data.Group.Free.Church

Methods

empty :: FA a

(<|>) :: FA a -> FA a -> FA a

some :: FA a -> FA [a]

many :: FA a -> FA [a]

Cancellative FA

Defined in Control.Applicative.Cancellative

Methods

cancel :: FA a -> FA a

Semigroup (FA a)

Defined in Data.Group.Free.Church

Methods

(<>) :: FA a -> FA a -> FA a

sconcat :: NonEmpty (FA a) -> FA a

stimes :: Integral b => b -> FA a -> FA a

Monoid (FA a)

Defined in Data.Group.Free.Church

Methods

mempty :: FA a

mappend :: FA a -> FA a -> FA a

mconcat :: [FA a] -> FA a

Group (FA a)

Defined in Data.Group.Free.Church

Methods

invert :: FA a -> FA a

(~~) :: FA a -> FA a -> FA a

pow :: Integral x => FA a -> x -> FA a

Abelian (FA a)

Defined in Data.Group.Free.Church

Church-encoded free abelian group combinators

forgetFA :: Ord a => FA a -> FG a

Forget the commutative structure of a Church-encoded free abelian group, turning it into a standard free group.

interpretFA :: Abelian g => FA g -> g

Interpret a Church-encoded free abelian group as a concrete FreeAbelianGroup.

reifyFA :: Ord a => FA a -> FreeAbelianGroup a

Convert a Church-encoded free abelian group to a concrete FreeAbelianGroup.

reflectFA :: FreeAbelianGroup a -> FA a

Convert a concrete FreeAbelianGroup to a Church-encoded free abelian group.