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

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]

Cancelative FG

Defined in Control.Applicative.Cancelative

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

gtimes :: Integral n => n -> FG a -> FG a

minus :: FG a -> FG a -> 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. Group g => (a -> Int -> 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]

Cancelative FA

Defined in Control.Applicative.Cancelative

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

gtimes :: Integral n => n -> FA a -> FA a

minus :: FA a -> FA a -> FA a

Church-encoded free abelian group combinators

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

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

interpretFA :: Group 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 :: Ord a => FreeAbelianGroup a -> FA a

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