Copyright	(c) 2020 Reed Mullanix Emily Pillmore
License	BSD-style
Maintainer	Reed Mullanix <reedmullanix@gmail.com>, Emily Pillmore <emilypi@cohomolo.gy>
Stability	stable
Portability	non-portable
Safe Haskell	Safe
Language	Haskell2010

Data.Group.Free.Church

Contents

Church-encoded free groups
- Church-encoded free group combinators
Church-encoded free abelian groups
- Church-encoded free abelian group combinators

Description

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

Synopsis

newtype FG a = FG {
- runFG :: forall g. Group g => (a -> g) -> g
}
interpretFG :: Group g => FG g -> g
reifyFG :: FG a -> FreeGroup a
reflectFG :: FreeGroup a -> FG a
presentFG :: Group g => FG g -> (FG g -> g) -> g
newtype FA a = FA {
- runFA :: forall g. Group g => (a -> Int -> g) -> g
}
forgetFA :: Group a => FA a -> FG a
interpretFA :: Group g => FA g -> g
reifyFA :: Ord a => FA a -> FreeAbelianGroup a
reflectFA :: Ord a => FreeAbelianGroup a -> FA a

Church-encoded free groups

newtype FG a Source #

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

Instances details

Monad FG Source #
Instance details 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 Source #
Instance details Defined in Data.Group.Free.Church Methods fmap :: (a -> b) -> FG a -> FG b # (<$) :: a -> FG b -> FG a #
Applicative FG Source #
Instance details 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 Source #
Instance details 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 Source #
Instance details Defined in Control.Applicative.Cancelative Methods cancel :: FG a -> FG a Source #
GroupFoldable FG Source #
Instance details Defined in Data.Group.Foldable Methods goldMap :: Group g => (a -> g) -> FG a -> g Source # toFG :: FG a -> FG a Source #
Semigroup (FG a) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details Defined in Data.Group.Free.Church Methods invert :: FG a -> FG a Source # gtimes :: Integral n => n -> FG a -> FG a Source # minus :: FG a -> FG a -> FG a Source #

Church-encoded free group combinators

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

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

reifyFG :: FG a -> FreeGroup a Source #

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

reflectFG :: FreeGroup a -> FG a Source #

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

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

Present a Group as a FG modulo relations.

Church-encoded free abelian groups

newtype FA a Source #

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

Instances details

Monad FA Source #
Instance details 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 Source #
Instance details Defined in Data.Group.Free.Church Methods fmap :: (a -> b) -> FA a -> FA b # (<$) :: a -> FA b -> FA a #
Applicative FA Source #
Instance details 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 Source #
Instance details 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 Source #
Instance details Defined in Control.Applicative.Cancelative Methods cancel :: FA a -> FA a Source #
Semigroup (FA a) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details Defined in Data.Group.Free.Church Methods invert :: FA a -> FA a Source # gtimes :: Integral n => n -> FA a -> FA a Source # minus :: FA a -> FA a -> FA a Source #

Church-encoded free abelian group combinators

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

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

interpretFA :: Group g => FA g -> g Source #

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

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

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

reflectFA :: Ord a => FreeAbelianGroup a -> FA a Source #

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