{-# LANGUAGE
    DeriveGeneric
  , DeriveDataTypeable
  , DerivingVia
  , FlexibleContexts
  , FlexibleInstances
  , GeneralizedNewtypeDeriving
  , MultiParamTypeClasses
  , ScopedTypeVariables
  , StandaloneDeriving
  , UndecidableInstances
#-}

{-|
Module: Data.Act

An "Act" of a semigroup \( S \) on a type \( X \) gives a way to transform terms of type \( X \) by terms of type \( S \),
in a way that is compatible with the semigroup operation on \( S \).

In the special case that there is a unique way of going from one term of type \( X \) to another
through a transformation by a term of type \( S \), we say that \( X \) is a torsor under \( S \).

For example, the plane has an action by translations. Given any two points, there is a unique translation
that takes the first point to the second. Note that an unmarked plane (like a blank piece of paper)
has no designated origin or reference point, whereas the set of translations is a plane with a given origin
(the zero translation). This is the distinction between an affine space (an unmarked plane) and a vector space.
Enforcing this distinction in the types can help to avoid confusing absolute points with translation vectors.


Simple 'Act' and 'Torsor' instances can be derived through self-actions:

> > newtype Seconds   = Seconds { getSeconds :: Double }
> >   deriving ( Act TimeDelta, Torsor TimeDelta )
> >     via TimeDelta
> > newtype TimeDelta = TimeDelta { timeDeltaInSeconds :: Seconds }
> >   deriving ( Semigroup, Monoid, Group )
> >     via Sum Double

-}

module Data.Act
  ( Act(..)
  , transportAction
  , Trivial(..)
  , Torsor(..)
  , intertwiner
  )
  where

-- base

import Data.Coerce
  ( coerce )
import Data.Data
  ( Data )
import Data.Functor.Const
  ( Const(..) )
import Data.Functor.Contravariant
  ( Op(..) )
import Data.Monoid
  ( Any(..), All(..)
  , Sum(..), Product(..)
  , Ap(..), Endo(..)
  )
import Data.Semigroup
  ( Max(..), Min(..), Dual(..) )
import GHC.Generics
  ( Generic, Generic1 )

-- deepseq

import Control.DeepSeq
  ( NFData )

-- acts

import Data.Group
  ( Group(..) )

-----------------------------------------------------------------


-- | A left __act__ (or left __semigroup action__) of a semigroup @s@ on @x@ consists of an operation

-- 

-- @(•) :: s -> x -> x@

--

-- such that:

-- 

-- @a • ( b • x ) = ( a <> b ) • x@

--

-- In case @s@ is also a 'Monoid', we additionally require:

--

-- @mempty • x = x@

--

-- The synonym @ act = (•) @ is also provided.

class Semigroup s => Act s x where
  {-# MINIMAL (•) | act #-}
  -- | Left action of a semigroup.

  (•), act :: s -> x -> x
  (•) = s -> x -> x
forall s x. Act s x => s -> x -> x
act
  act = s -> x -> x
forall s x. Act s x => s -> x -> x
(•)

infixr 5 •
infixr 5 `act`

-- | Transport an act:

--

-- <<img/transport.svg>>

transportAction :: ( a -> b ) -> ( b -> a ) -> ( g -> b -> b ) -> ( g -> a -> a )
transportAction :: (a -> b) -> (b -> a) -> (g -> b -> b) -> g -> a -> a
transportAction a -> b
to b -> a
from g -> b -> b
actBy g
g = b -> a
from (b -> a) -> (a -> b) -> a -> a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. g -> b -> b
actBy g
g (b -> b) -> (a -> b) -> a -> b
forall b c a. (b -> c) -> (a -> b) -> a -> c
. a -> b
to

-- | Natural left action of a semigroup on itself.

instance Semigroup s => Act s s where
  • :: s -> s -> s
(•) = s -> s -> s
forall s. Semigroup s => s -> s -> s
(<>)

-- | Trivial act of a semigroup on any type (acting by the identity).

newtype Trivial a = Trivial { Trivial a -> a
getTrivial :: a }
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[Trivial a] -> ShowS
Trivial a -> String
(Int -> Trivial a -> ShowS)
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-> Show (Trivial a)
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showList :: [Trivial a] -> ShowS
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show :: Trivial a -> String
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showsPrec :: Int -> Trivial a -> ShowS
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(forall d e. (Data d, Data e) => c (t d e))
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gunfold :: (forall b r. Data b => c (b -> r) -> c r)
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(forall b r. Data b => c (b -> r) -> c r)
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(forall d b. Data d => c (d -> b) -> d -> c b)
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forall x. Rep (Trivial a) x -> Trivial a
forall x. Trivial a -> Rep (Trivial a) x
forall a.
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forall k (f :: k -> *).
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forall a. Eq a => Trivial a -> Trivial a -> Bool
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-> Enum (Trivial a)
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enumFrom :: Trivial a -> [Trivial a]
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fromEnum :: Trivial a -> Int
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toEnum :: Int -> Trivial a
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pred :: Trivial a -> Trivial a
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succ :: Trivial a -> Trivial a
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maxBound :: Trivial a
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minBound :: Trivial a
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NFData )
instance Semigroup s => Act s ( Trivial a ) where
  act :: s -> Trivial a -> Trivial a
act s
_ = Trivial a -> Trivial a
forall a. a -> a
id

deriving via Any instance Act Any Bool
deriving via All instance Act All Bool
deriving via ( Sum     a ) instance Num a => Act ( Sum     a ) a
deriving via ( Product a ) instance Num a => Act ( Product a ) a
deriving via ( Max     a ) instance Ord a => Act ( Max     a ) a
deriving via ( Min     a ) instance Ord a => Act ( Min     a ) a

instance {-# OVERLAPPING #-} Act () x where
  act :: () -> x -> x
act ()
_ = x -> x
forall a. a -> a
id
instance ( Act s1 x1, Act s2 x2 )
      => Act ( s1, s2 ) ( x1,x2 ) where
  act :: (s1, s2) -> (x1, x2) -> (x1, x2)
act ( s1
s1, s2
s2 ) ( x1
x1, x2
x2 ) =
    ( s1 -> x1 -> x1
forall s x. Act s x => s -> x -> x
act s1
s1 x1
x1, s2 -> x2 -> x2
forall s x. Act s x => s -> x -> x
act s2
s2 x2
x2 )
instance ( Act s1 x1, Act s2 x2, Act s3 x3 )
      => Act ( s1, s2, s3 ) ( x1, x2, x3 ) where
  act :: (s1, s2, s3) -> (x1, x2, x3) -> (x1, x2, x3)
act ( s1
s1, s2
s2, s3
s3 ) ( x1
x1, x2
x2, x3
x3 ) =
    ( s1 -> x1 -> x1
forall s x. Act s x => s -> x -> x
act s1
s1 x1
x1, s2 -> x2 -> x2
forall s x. Act s x => s -> x -> x
act s2
s2 x2
x2, s3 -> x3 -> x3
forall s x. Act s x => s -> x -> x
act s3
s3 x3
x3 )
instance ( Act s1 x1, Act s2 x2, Act s3 x3, Act s4 x4 )
      => Act ( s1, s2, s3, s4 ) ( x1, x2, x3, x4 ) where
  act :: (s1, s2, s3, s4) -> (x1, x2, x3, x4) -> (x1, x2, x3, x4)
act ( s1
s1, s2
s2, s3
s3, s4
s4 ) ( x1
x1, x2
x2, x3
x3, x4
x4 ) =
    ( s1 -> x1 -> x1
forall s x. Act s x => s -> x -> x
act s1
s1 x1
x1, s2 -> x2 -> x2
forall s x. Act s x => s -> x -> x
act s2
s2 x2
x2, s3 -> x3 -> x3
forall s x. Act s x => s -> x -> x
act s3
s3 x3
x3, s4 -> x4 -> x4
forall s x. Act s x => s -> x -> x
act s4
s4 x4
x4 )
instance ( Act s1 x1, Act s2 x2, Act s3 x3, Act s4 x4, Act s5 x5 )
      => Act ( s1, s2, s3, s4, s5 ) ( x1, x2, x3, x4, x5 ) where
  act :: (s1, s2, s3, s4, s5)
-> (x1, x2, x3, x4, x5) -> (x1, x2, x3, x4, x5)
act ( s1
s1, s2
s2, s3
s3, s4
s4, s5
s5 ) ( x1
x1, x2
x2, x3
x3, x4
x4, x5
x5 ) =
    ( s1 -> x1 -> x1
forall s x. Act s x => s -> x -> x
act s1
s1 x1
x1, s2 -> x2 -> x2
forall s x. Act s x => s -> x -> x
act s2
s2 x2
x2, s3 -> x3 -> x3
forall s x. Act s x => s -> x -> x
act s3
s3 x3
x3, s4 -> x4 -> x4
forall s x. Act s x => s -> x -> x
act s4
s4 x4
x4, s5 -> x5 -> x5
forall s x. Act s x => s -> x -> x
act s5
s5 x5
x5 )

deriving newtype instance Act s a => Act s ( Const a b )

-- | Acting through a functor using @fmap@.

instance ( Act s x, Functor f ) => Act s ( Ap f x ) where
  act :: s -> Ap f x -> Ap f x
act s
s = (f x -> f x) -> Ap f x -> Ap f x
coerce ( (x -> x) -> f x -> f x
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap ( s -> x -> x
forall s x. Act s x => s -> x -> x
act s
s ) :: f x -> f x )

-- | Acting through the contravariant function arrow functor.

instance ( Semigroup s, Act s a ) => Act ( Dual s ) ( Op b a ) where
  act :: Dual s -> Op b a -> Op b a
act ( Dual s
s ) = ((a -> b) -> a -> b) -> Op b a -> Op b a
coerce ( ( (a -> b) -> (a -> a) -> a -> b
forall b c a. (b -> c) -> (a -> b) -> a -> c
. s -> a -> a
forall s x. Act s x => s -> x -> x
act s
s ) :: ( a -> b ) -> ( a -> b ) )

-- | Acting through a function arrow: both covariant and contravariant actions.

instance ( Semigroup s, Act s a, Act t b ) => Act ( Dual s, t ) ( a -> b ) where
  act :: (Dual s, t) -> (a -> b) -> a -> b
act ( Dual s
s, t
t ) a -> b
p = t -> b -> b
forall s x. Act s x => s -> x -> x
act t
t (b -> b) -> (a -> b) -> a -> b
forall b c a. (b -> c) -> (a -> b) -> a -> c
. a -> b
p (a -> b) -> (a -> a) -> a -> b
forall b c a. (b -> c) -> (a -> b) -> a -> c
. s -> a -> a
forall s x. Act s x => s -> x -> x
act s
s

-- | Action of an opposite group using inverses.

instance {-# OVERLAPPABLE #-} ( Act g x, Group g ) => Act ( Dual g ) x where
  act :: Dual g -> x -> x
act ( Dual g
g ) = g -> x -> x
forall s x. Act s x => s -> x -> x
act ( g -> g
forall g. Group g => g -> g
inverse g
g )

-- | Action of a group on endomorphisms.

instance ( Group g, Act g a ) => Act g ( Endo a ) where
  act :: g -> Endo a -> Endo a
act g
g = ((a -> a) -> a -> a) -> Endo a -> Endo a
coerce ( (Dual g, g) -> (a -> a) -> a -> a
forall s x. Act s x => s -> x -> x
act ( g -> Dual g
forall a. a -> Dual a
Dual g
g, g
g ) :: ( a -> a ) -> ( a -> a ) )

-----------------------------------------------------------------


-- | A left __torsor__ consists of a /free/ and /transitive/ left action of a group on an inhabited type.

--

-- This precisely means that for any two terms @x@, @y@, there exists a /unique/ group element @g@ taking @x@ to @y@,

-- which is denoted @ y <-- x @ (or @ x --> y @, but the left-pointing arrow is more natural when working with left actions).

--

-- That is @ y <-- x @ is the /unique/ element satisfying:

--

-- @( y <-- x ) • x = y@

--

--

-- Note the order of composition of @<--@ and @-->@ with respect to @<>@:

--

-- > ( z <-- y ) <> ( y <-- x ) = z <-- x

--

-- > ( y --> z ) <> ( x --> y ) = x --> z

class ( Group g, Act g x ) => Torsor g x where
  {-# MINIMAL (-->) | (<--) #-}
  -- | Unique group element effecting the given transition

  (<--), (-->) :: x -> x -> g
  (-->) = (x -> x -> g) -> x -> x -> g
forall a b c. (a -> b -> c) -> b -> a -> c
flip x -> x -> g
forall g x. Torsor g x => x -> x -> g
(<--)
  (<--) = (x -> x -> g) -> x -> x -> g
forall a b c. (a -> b -> c) -> b -> a -> c
flip x -> x -> g
forall g x. Torsor g x => x -> x -> g
(-->)

infix 7 -->
infix 7 <--

-- | Any group is a torsor under its own natural left action.

instance Group g => Torsor g g where
  g
h <-- :: g -> g -> g
<-- g
g = g
h g -> g -> g
forall s. Semigroup s => s -> s -> s
<> g -> g
forall g. Group g => g -> g
inverse g
g

deriving via ( Sum a ) instance Num a => Torsor ( Sum a ) a

-- | Given

-- 

--  * \( g \in G \) acting on \( A \),

--  * \( B \) a torsor under \( H \),

--  * a map \( p \colon A \to B \),

--

-- this function returns the unique element \( h \in H \) making the following diagram commute:

--

-- <<img/intertwiner.svg>>

intertwiner :: forall h g a b. ( Act g a, Torsor h b ) => g -> ( a -> b ) -> a -> h
intertwiner :: g -> (a -> b) -> a -> h
intertwiner g
g a -> b
p a
a = a -> b
p a
a b -> b -> h
forall g x. Torsor g x => x -> x -> g
--> a -> b
p ( g
g g -> a -> a
forall s x. Act s x => s -> x -> x
• a
a )