acts (empty) → 0.1.0.0

raw patch · 9 files changed

+2789/−0 lines, 9 filesdep +actsdep +basedep +deepseq

Dependencies added: acts, base, deepseq, finite-typelits, generic-data

Files

• acts.cabal +95/−0
• changelog.md +5/−0
• examples/Acts/Examples/MusicalIntervals.hs +332/−0
• img/coerce.svg +723/−0
• img/intertwiner.svg +480/−0
• img/transport.svg +518/−0
• src/Data/Act.hs +210/−0
• src/Data/Group.hs +251/−0
• src/Data/Group/Cyclic.hs +175/−0

@@ -0,0 +1,95 @@+cabal-version:  2.4
+name:           acts
+version:        0.1.0.0
+synopsis:       Semigroup actions, groups, and torsors.
+category:       Algebra, Math
+license:        BSD-3-Clause
+build-type:     Simple
+author:         Sam Derbyshire
+maintainer:     Sam Derbyshire
+homepage:       https://github.com/sheaf/acts
+bug-reports:    https://github.com/sheaf/acts/issues
+extra-source-files:
+    changelog.md
+extra-doc-files:
+    img/*.svg
+description:
+  Acts and torsors model types which can be transformed under the action of another type.
+  .
+  A prototypical example is affine space, which has an action by translation:
+  given any two points in affine space, there is a unique translation that brings one to the other.
+  .
+  This can be useful in a library keeping track of time: on top of needing to keep track
+  of units, one also needs to distinguish between absolute time (time stamps) and relative time
+  (time differences).
+  The operations one expects in this situation are:
+  .
+  * Addition and subtraction of time differences: time differences form a (commutative) group.
+  * Translation of an absolute time by a time difference: there is an action of relative time on absolute time.
+  * Given two absolute times, one can obtain the time difference between them: absolute time is a torsor under relative time.
+  .
+  This library provides a convenient framework which helps to avoid mixing up these two different notions.
+  .
+  .
+  A fleshed out example is available at "Acts.Examples.MusicalIntervals",
+  which showcases the use of actions and torsors in the context of musical intervals and harmony.
+  It also demonstrates common usage patterns of this library, such as how to automatically derive instances.
+  .
+  See also the [project readme](https://github.com/sheaf/acts),
+  which includes a simple example with 2D affine space.
+
+source-repository head
+  type: git
+  location: git://github.com/sheaf/acts.git
+
+common common
+
+  build-depends: 
+      base
+        >= 4.12 && < 4.15
+    , deepseq
+        ^>= 1.4.4.0
+    , finite-typelits
+        ^>= 0.1.4.2
+    , generic-data
+        >= 0.6.0.1 && < 0.7.0.0
+
+  default-language:
+      Haskell2010
+
+  ghc-options:
+      -O2
+      -Wall
+      -Wcompat
+      -fwarn-missing-local-signatures
+      -fwarn-incomplete-uni-patterns
+      -fwarn-missing-deriving-strategies
+
+
+library
+
+  import:
+      common
+
+  hs-source-dirs:
+      src
+
+  exposed-modules:
+      Data.Act
+    , Data.Group
+    , Data.Group.Cyclic
+
+
+library acts-examples
+
+  import:
+      common
+
+  hs-source-dirs:
+      examples
+
+  exposed-modules:
+      Acts.Examples.MusicalIntervals
+
+  build-depends:
+      acts

@@ -0,0 +1,5 @@+# Changelog for package `acts`
+
+## 0.1.0.0 ( February 2020 )
+
+* Initial release.

@@ -0,0 +1,332 @@+{-# LANGUAGE
+    DataKinds
+  , DeriveGeneric
+  , DerivingVia
+  , MultiParamTypeClasses
+  , PatternSynonyms
+  , TupleSections
+  , TypeApplications
+  , ViewPatterns
+#-}
+
+{-# OPTIONS_HADDOCK ignore-exports #-}
+
+{-|
+Module: Acts.MusicalIntervals
+
+Illustrative usage of 'Group', 'Act' and 'Torsor': manipulation of musical intervals.
+
+
+The musical distance between two musical notes is a musical interval.
+
+Intervals can be compounded and inverted, so they form a 'Group'.
+
+Notes can be translated by a given interval, which is an 'Act' of intervals on notes.
+
+There's a unique musical interval taking any note to any other given one, so notes are a torsor under intervals.
+
+
+This functionality is useful in providing enharmonically correct voicings of chords.
+-}
+
+module Acts.Examples.MusicalIntervals where
+
+-- base
+import Data.Monoid
+  ( Sum(..) )
+
+-- acts
+import Data.Act
+import Data.Group
+import Data.Group.Cyclic
+
+-----------------------------------------------------------------
+-- * Musical notes
+--
+-- $notenames
+-- We begin by defining note names, which are acted upon by the cyclic group of order 7.
+
+-- | Musical note names.
+--
+-- The enumeration starts with @C@ to conform with scientific pitch notation.
+data NoteName = C | D | E | F | G | A | B
+  deriving stock ( Eq, Ord, Show, Enum, Bounded )
+  deriving ( Act ( C 7 ), Torsor ( C 7 ) )
+    via CyclicEnum NoteName
+
+-- $deriving1
+-- In this case we used @DerivingVia@ to derive the action of @C 7@,
+-- using the 'CyclicEnum' newtype created for this exact purpose.
+
+-- | Alterations, i.e. sharps and flats.
+--
+-- Pattern synonyms such as 'Sharp' and 'Flat' are also bundled.
+newtype Alteration = Alteration { getAlteration :: Int }
+  deriving ( Semigroup, Monoid, Group )
+    via Sum Int
+
+-- $deriving2
+-- Note the use of @DerivingVia@ to transfer algebraic operations from @Sum Int@.
+-- 
+-- For non-newtypes, one can use generics, for example:
+--
+-- > data Klein4 = Klein4 ( C 2 ) ( C 2 )
+-- >   deriving stock Generic
+-- >   deriving ( Semigroup, Monoid, Group )
+-- >     via Generically Klein4
+--
+-- This uses the 'Generically' newtype from the @generic-data@ library.
+
+-- | Note names such as @A4@ or @C#6@: note name, alteration, and octave (scientific pitch notation).
+data Note = Note { name :: NoteName, alteration :: Alteration, octave :: Int }
+
+-----------------------------------------------------------------
+-- * Musical intervals
+--
+-- $intervals
+-- An interval is represented as a number of scale steps to take (relative to the major scale),
+-- together with an additional alteration to apply.
+--
+-- For instance, a major third is two steps up (diatonic steps relative to the root in a major scale):
+--
+-- > > Steps ( Sum 2 ) Natural
+-- > major 3rd up
+--
+-- A minor sixth is 5 steps up, and then a flat:
+--
+-- > > Steps ( Sum 5 ) Flat
+-- > minor 6th up
+--
+-- The smart constructor 'Interval' is also provided that is more intuitive to use:
+--
+-- > > Interval 3 Natural
+-- > major 3rd up
+--
+-- > > Interval 7 Flat
+-- > minor 7th up
+--
+-- Note that the @Semigroup@/@Group@ operations on intervals are __not__ the obvious ones, e.g.:
+--
+-- > > Steps ( Sum 2 ) Natural
+-- > major 3rd up
+--
+-- > > Steps ( Sum (-2) ) Natural
+-- > minor 3rd down
+--
+-- > > inverse ( Steps ( Sum 2 ) Natural )
+-- > Steps ( Sum (-2) ) Flat
+-- > major 3rd down
+
+-- | Musical interval: steps (relative to the root in a major scale) and additional alteration.
+data Interval = Steps { intervalSteps :: Sum Int, intervalAlteration :: Alteration }
+
+-- | Compute the number of semitones in an interval, using the reference of the C major scale.
+semitones :: Interval -> Int
+semitones ival = case act ival ( Note C Natural 0 ) of
+  Note n a o -> 12 * o + getAlteration a + majorValue
+    where
+      majorValue = let i = fromEnum n in 2 * i - fromEnum ( i >= 3 )
+
+-- $interval_operations
+-- To define algebraic operations on intervals,
+-- we use an equivariant bijection to the product group @( Sum Int, Sum Int )@.
+--
+-- Note that @( Sum Int, Sum Int )@ is automatically a 'Semigroup', 'Monoid' and 'Group'
+-- using the product structure.
+
+-- | Forward part of the bijection.
+straighten :: Interval -> ( Sum Int, Sum Int )
+straighten ival@( Steps steps _ ) = ( steps, Sum $ semitones ival )
+-- | Back part of the bijection.
+twist :: ( Sum Int, Sum Int ) -> Interval
+twist ( i, Sum a ) = Steps i ( Alteration ( semitones ( Steps i mempty ) ) --> Alteration a )
+
+instance Semigroup Interval where
+  iv1 <> iv2 = twist ( straighten iv1 <> straighten iv2 )
+instance Monoid Interval where
+  mempty = Steps mempty mempty
+instance Group Interval where
+  inverse = twist . inverse . straighten
+
+-- | Intervallically correct action of intervals on notes.
+--
+--  * minor third up from @C@: @Eb@
+--  * minor third up from @A@: @C@.
+instance Act Interval Note where
+  act ( Steps ( Sum steps ) a ) ( Note C a' o ) = Note ( act ( Cyclic @7 r ) C ) ( a <> a' ) ( q + o )
+    where
+      ( q, r ) = steps `divMod` 7
+  act ival note = act ( ival <> ( Note C Natural 0 --> note ) ) ( Note C Natural 0 )
+
+-- | Computes the interval between two notes.
+--
+-- > > Note C Natural 5 --> Note A Natural 4
+-- > minor 3rd down
+--
+-- > > Note E Flat 4 --> Note A Natural 5
+-- > augmented 11th up
+instance Torsor Interval Note where
+  Note C a o --> Note n a' o' = Steps ( Sum $ fromEnum n + 7 * (o' - o) ) ( a --> a' )
+  note1 --> note2 = ( Note C Natural 0 --> note1 :: Interval ) --> ( Note C Natural 0 --> note2 :: Interval )
+
+-----------------------------------------------------------------
+-- * Illustration of the functionality
+
+-- ** Chords
+
+-- | Major triad: major third, perfect fifth.
+majorTriad :: [ Interval ]
+majorTriad = [ mempty, Interval 3 Natural, Interval 5 Natural ]
+
+-- | Diminished seventh chord: minor third, diminished fifth, diminished seventh.
+diminished7th :: [ Interval ]
+diminished7th = [ mempty, Interval 3 Flat, Interval 5 Flat, Interval 7 DoubleFlat ]
+
+-- | Minor 11th chord (Kenny Barron voicing).
+minor11th :: [ Interval ]
+minor11th = [ mempty, Interval 5 Natural, Interval 9 Natural
+            , Interval 10 Flat, Interval 14 Flat, Interval 18 Natural
+            ]
+
+-- $chords
+-- Example chords:
+--
+-- > > majorTriad <&> ( • Note C Natural 4 )
+-- > [C4,E4,G4]
+-- 
+-- > > diminished7th <&> ( • Note G Sharp 3 )
+-- > [G#3,B3,D4,F4]
+--
+-- > > minor11th <&> ( • Note D Natural 3 )
+-- > [D3,A3,E4,F4,C5,G5]
+
+-- ** Scales
+
+-- | Modes of C major.
+mode :: NoteName -> [ Interval ]
+mode root =
+  map
+    ( \ ( n, i ) -> Note root Natural 0 --> Note n Natural i )
+    ( map ( , 0 ) [ root .. maxBound ] ++ map ( , 1 ) [ minBound .. root ] )
+
+-- | Phrygian scale.
+phrygian :: [ Interval ]
+phrygian = mode E
+
+-- | Lydian scale.
+lydian :: [ Interval ]
+lydian = mode F
+
+-- | Whole tone scale.
+wholeTone :: [ Interval ]
+wholeTone = scanl (<>) mempty
+  [ Interval 2 Natural, Interval 2 Natural, Interval 2 Natural, Interval 3 DoubleFlat, Interval 2 Natural ]
+
+-- $scales
+-- Example scales:
+--
+-- > > phrygian <&> ( • Note E Natural 3 )
+-- > [E3,F3,G3,A3,B3,C4,D4,E4]
+--
+-- > > phrygian <&> ( • Note C Sharp 3 )
+-- > [C#3,D3,E3,F#3,G#3,A3,B3,C#4]
+--
+-- > > lydian <&> ( • Note C Natural 4 )
+-- > [C4,D4,E4,F#4,G4,A4,B4,C5]
+--
+-- > > wholeTone <&> ( • Note G Natural 5 )
+-- > [G5,A5,B5,C#6,Eb6,F6]
+
+---------------------------------------------------
+-- * Helper code
+-- $end
+-- End of main example code.
+--
+-- Follows: helper code for reading/showing musical notes and intervals.
+
+pattern Natural :: Alteration
+pattern Natural = Alteration 0
+pattern Flat :: Alteration
+pattern Flat = Alteration (-1)
+pattern DoubleFlat :: Alteration
+pattern DoubleFlat = Alteration (-2)
+pattern Sharp :: Alteration
+pattern Sharp = Alteration 1
+pattern DoubleSharp :: Alteration
+pattern DoubleSharp = Alteration 2
+
+instance Show Alteration where
+  show ( Alteration i ) = replicate ( abs i ) accidental
+    where
+      accidental :: Char
+      accidental
+        | i >= 0 = '#'
+        | otherwise = 'b'
+
+instance Show Note where
+  show ( Note n alt oct ) = show n <> show alt <> show oct
+
+pattern Interval :: Int -> Alteration -> Interval
+pattern Interval i a <-
+  ( ( \ ( Steps ( Sum steps ) alt ) -> ( if steps >= 0 then steps + 1 else steps - 1, alt ) )
+  -> (i, a)
+  )
+  where
+    Interval i a = if i < 0 then Steps ( Sum (i+1) ) a else Steps ( Sum (i-1) ) a
+
+instance Show Interval where
+  show ival@( Steps ( Sum i ) _ )
+    | i == 0 || i == 7 || i == -7
+    , let
+        ivalName = case compare i 0 of
+          LT -> "octave down"
+          GT -> "octave up"
+          EQ -> "unison"
+    = if quality ival == "perfect"
+      then ivalName
+      else quality ival <> " " <> ivalName
+    | i < 0
+    = quality ( inverse ival ) <> " " <> showOrdinal (-i+1) <> " down"
+    | otherwise
+    = quality ival <> " " <> showOrdinal (i+1) <> " up"
+
+quality :: Interval -> String
+quality ( Steps ( Sum i ) ( Alteration a ) )
+  | ( i `mod` 7 ) `elem` [ 0, 3, 4 ]
+  = case a of
+      0 -> "perfect"
+      _ -> 
+        if a > 0
+        then multiplicity   a  <> "augmented"
+        else multiplicity (-a) <> "diminished"
+  | otherwise
+  = case a of
+      0    -> "major"
+      (-1) -> "minor"
+      _    -> 
+        if a > 0
+        then multiplicity   a    <> "augmented"
+        else multiplicity (-a-1) <> "diminished"
+
+showOrdinal :: Int -> String
+showOrdinal i
+  | i < 0
+  = "-" <> showOrdinal ( abs i )
+  | i `mod` 10 == 1 && i `mod` 100 /= 11
+  = show i <> "st"
+  | i `mod` 10 == 2 && i `mod` 100 /= 12
+  = show i <> "nd"
+  | i `mod` 10 == 3 && i `mod` 100 /= 13
+  = show i <> "rd"
+  | otherwise
+  = show i <> "th"
+
+multiplicity :: Int -> String
+multiplicity 1 = ""
+multiplicity 2 = "doubly "
+multiplicity 3 = "triply "
+multiplicity 4 = "quadruply "
+multiplicity 5 = "quintuply "
+multiplicity 6 = "sextuply "
+multiplicity 7 = "heptuply "
+multiplicity n = show n <> "-tuply "

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@@ -0,0 +1,210 @@+{-# 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
+  (•) = act
+  act = (•)
+
+infixr 5 •
+infixr 5 `act`
+
+-- | Transport an act:
+--
+-- <<img/transport.svg>>
+transportAction :: ( a -> b ) -> ( b -> a ) -> ( g -> b -> b ) -> ( g -> a -> a )
+transportAction to from actBy g = from . actBy g . to
+
+-- | Natural left action of a semigroup on itself.
+instance Semigroup s => Act s s where
+  (•) = (<>)
+
+-- | Trivial act of a semigroup on any type (acting by the identity).
+newtype Trivial a = Trivial { getTrivial :: a }
+  deriving stock   ( Show, Read, Data, Generic, Generic1 )
+  deriving newtype ( Eq, Ord, Enum, Bounded, NFData )
+instance Semigroup s => Act s ( Trivial a ) where
+  act _ = 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 _ = id
+instance ( Act s1 x1, Act s2 x2 )
+      => Act ( s1, s2 ) ( x1,x2 ) where
+  act ( s1, s2 ) ( x1, x2 ) =
+    ( act s1 x1, act s2 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 ) =
+    ( act s1 x1, act s2 x2, act s3 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 ) =
+    ( act s1 x1, act s2 x2, act s3 x3, act s4 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 ) =
+    ( act s1 x1, act s2 x2, act s3 x3, act s4 x4, act s5 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 = coerce ( fmap ( act 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 ) = coerce ( ( . act 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 ) p = act t . p . act s
+
+-- | Action of an opposite group using inverses.
+instance {-# OVERLAPPABLE #-} ( Act g x, Group g ) => Act ( Dual g ) x where
+  act ( Dual g ) = act ( inverse g )
+
+-- | Action of a group on endomorphisms.
+instance ( Group g, Act g a ) => Act g ( Endo a ) where
+  act g = coerce ( act ( Dual 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
+  (-->) = flip (<--)
+  (<--) = flip (-->)
+
+infix 7 -->
+infix 7 <--
+
+-- | Any group is a torsor under its own natural left action.
+instance Group g => Torsor g g where
+  h <-- g = h <> inverse 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 p a = p a --> p ( g • a )

@@ -0,0 +1,251 @@+{-# LANGUAGE
+    DataKinds
+  , DeriveAnyClass
+  , DeriveDataTypeable
+  , DeriveGeneric
+  , DerivingVia
+  , GeneralizedNewtypeDeriving
+  , KindSignatures
+  , ScopedTypeVariables
+  , StandaloneDeriving
+  , TypeApplications
+  , TypeOperators
+  , UndecidableInstances
+#-}
+
+{-|
+Module: Data.Group
+
+A 'Group' is a 'Monoid' for which the monoid operation can be undone.
+
+That is, \( G \) is a group if each \( g \in G \) has an inverse element \( g^{ -1 } \) such that
+
+\[ g^{ -1 } < \! > g = \text{mempty} = g < \! > g^{ -1 } \]
+
+Such inverses are necessarily unique.
+
+
+In Haskell, groups are mostly useful to describe objects possessing certain symmetries (such as translation or rotation).
+
+To automatically derive 'Group' instances, you can:
+
+- Use @DerivingVia@ to coerce an existing instance:
+
+> > newtype Seconds   = Seconds { getSeconds :: Double }
+> > newtype TimeDelta = TimeDelta { timeDeltaInSeconds :: Seconds }
+> >   deriving ( Semigroup, Monoid, Group )
+> >     via Sum Double
+
+- Use 'Generic' and 'Generic.Data.Generically':
+
+> > data MyRecord
+> >   = MyRecord
+> >   { field1 :: Sum Double
+> >   , field2 :: Product Double
+> >   , field3 :: Ap [] ( Sum Int, Sum Int )
+> >   }
+> >   deriving Generic
+> >   deriving ( Semigroup, Monoid, Group )
+> >     via Generically MyRecord
+-}
+
+
+module Data.Group
+  ( Group(..)
+  , Isom(..)
+  )
+  where
+
+-- base
+import Control.Monad.ST
+  ( ST )
+import Data.Data
+  ( Data )
+import Data.Functor.Const
+  ( Const(..) )
+import Data.Functor.Contravariant
+  ( Op(..) )
+import Data.Functor.Identity
+  ( Identity(..) )
+import Data.Monoid
+  ( Ap(..), Sum(..), Product(..) )
+import Data.Ord
+  ( Down(..) )
+import Data.Semigroup
+  ( Semigroup(..), Dual(..) )
+import Data.Proxy
+  ( Proxy(..) )
+import GHC.Generics
+  ( Generic, Generic1
+  , U1(..), Rec1(..), M1(..), K1(..), Par1(..), (:*:)(..)
+  , V1, (:+:)
+  )
+import qualified GHC.Generics as Generic
+  ( Generic(..) )
+import GHC.TypeLits
+  ( TypeError, ErrorMessage(Text) )
+
+-- deepseq
+import Control.DeepSeq
+  ( NFData )
+
+-- generic-data
+import Generic.Data
+  ( Generically(..) )
+
+-----------------------------------------------------------------------
+
+-- | A 'Group' is a 'Monoid' with inverses:
+--
+-- * @ inverse g <> g = g <> inverse g = mempty @
+--
+-- * @ inverse (g <> h) = inverse h <> inverse g @
+class Monoid g => Group g where
+  {-# MINIMAL inverse | gtimes #-}
+  -- | Group inversion anti-homomorphism.
+  inverse :: g -> g
+  inverse = gtimes ( (-1) :: Int )
+
+  -- | Take the @n@-th power of an element.
+  gtimes :: Integral n => n -> g -> g
+  gtimes n = case compare n 0 of
+    EQ -> const mempty
+    GT -> stimes n
+    LT -> stimes ( negate n ) . inverse
+
+-----------------------------------------------------------------------
+-- Instances.
+
+-- | Trivial group.
+instance Group () where
+  inverse  _ = ()
+  gtimes _ _ = ()
+
+-- | Additive groups (via 'Num').
+instance Num a => Group ( Sum a ) where
+  inverse  ( Sum a ) = Sum ( negate a )
+  gtimes n ( Sum a ) = Sum ( fromIntegral n * a )
+
+-- | Multiplicative group (via 'Num').
+instance Fractional a => Group ( Product a ) where
+  inverse  ( Product a ) = Product ( recip a )
+  gtimes n ( Product a ) = Product ( a ^^ toInteger n )
+
+-- | Opposite group.
+instance Group a => Group ( Dual a ) where
+  inverse  ( Dual a ) = Dual ( inverse a )
+  gtimes n ( Dual a ) = Dual ( gtimes n a )
+
+-- | Lifting group operations through an applicative functor.
+instance ( Group a, Applicative f ) => Group ( Ap f a ) where
+  inverse  = fmap inverse
+  gtimes n = fmap ( gtimes n )
+
+deriving via Ap ((->) r) a instance Group a => Group ( r  ->  a )
+deriving via Ap IO       a instance Group a => Group ( IO     a )
+deriving via Ap (ST s)   a instance Group a => Group ( ST s   a )
+
+deriving newtype instance Group a => Group ( Down     a )
+deriving newtype instance Group a => Group ( Identity a )
+deriving newtype instance Group a => Group ( Const a b )
+deriving newtype instance Group a => Group ( Op    a b )
+
+instance Group ( Proxy p ) where
+  inverse  _ = Proxy
+  gtimes _ _ = Proxy
+
+instance ( Group g1, Group g2 )
+      => Group ( g1, g2 ) where
+  inverse  ( g1, g2 ) =
+    ( inverse g1, inverse g2 )
+  gtimes n ( g1, g2 ) =
+    ( gtimes n g1, gtimes n g2 )
+
+instance ( Group g1, Group g2, Group g3 )
+      => Group ( g1, g2, g3 ) where
+  inverse  ( g1, g2, g3 ) =
+    ( inverse g1, inverse g2, inverse g3 )
+  gtimes n ( g1, g2, g3 ) =
+    ( gtimes n g1, gtimes n g2, gtimes n g3 )
+
+instance ( Group g1, Group g2, Group g3, Group g4 )
+      => Group ( g1, g2, g3, g4 ) where
+  inverse  ( g1, g2, g3, g4 ) =
+    ( inverse g1, inverse g2, inverse g3, inverse g4 )
+  gtimes n ( g1, g2, g3, g4 ) =
+    ( gtimes n g1, gtimes n g2, gtimes n g3, gtimes n g4 )
+
+instance ( Group g1, Group g2, Group g3, Group g4, Group g5 )
+      => Group ( g1, g2, g3, g4, g5 ) where
+  inverse  ( g1, g2, g3, g4, g5 ) =
+    ( inverse g1, inverse g2, inverse g3, inverse g4, inverse g5 )
+  gtimes n ( g1, g2, g3, g4, g5 ) =
+    ( gtimes n g1, gtimes n g2, gtimes n g3, gtimes n g4, gtimes n g5 )
+
+infix 7 :|:
+-- | Data type to keep track of a pair of inverse elements.
+data Isom a = (:|:) { to :: a, from :: Dual a }
+  deriving stock    ( Show, Read, Data, Generic, Generic1 )
+  deriving anyclass NFData
+instance Semigroup a => Semigroup ( Isom a ) where
+  ( p1 :|: q1 ) <> ( p2 :|: q2 ) = ( p1 <> p2 ) :|: ( q1 <> q2 )
+instance Monoid a => Monoid ( Isom a ) where
+  mempty = mempty :|: mempty
+instance Monoid a => Group ( Isom a ) where
+  inverse ( p :|: q ) = getDual q :|: Dual p
+
+-- Generics.
+
+instance Group ( U1 p ) where
+  inverse  _ = U1
+  gtimes _ _ = U1
+
+deriving newtype instance Group ( f p ) => Group ( Rec1 f p )
+deriving newtype instance Group ( f p ) => Group ( M1 i c f p )
+deriving newtype instance Group g       => Group ( K1 i g p )
+deriving newtype instance Group g       => Group ( Par1 g )
+
+instance ( Group ( f1 p ), Group ( f2 p ) ) => Group ( (f1 :*: f2) p ) where
+  inverse  ( g1 :*: g2 ) = ( inverse  g1 :*: inverse  g2 )
+  gtimes n ( g1 :*: g2 ) = ( gtimes n g1 :*: gtimes n g2 )
+
+instance
+  ( Generic g
+  , Monoid  ( Generic.Rep g () )
+  , GGroup  ( Generic.Rep g )
+  )
+  => Group ( Generically g ) where
+  inverse  = Generically . Generic.to . ginverse  . Generic.from . unGenerically
+  gtimes n = Generically . Generic.to . ggtimes n . Generic.from . unGenerically
+
+-- | Type class used for deriving 'Group' instances generically.
+class GGroup f where
+  ginverse :: f p -> f p
+  ggtimes  :: Integral n => n -> f p -> f p
+
+instance GGroup U1 where
+  ginverse  _ = U1
+  ggtimes _ _ = U1
+
+deriving newtype instance GGroup f => GGroup ( Rec1 f )
+deriving newtype instance GGroup f => GGroup ( M1 i c f )
+
+instance Group g => GGroup ( K1 i g ) where
+  ginverse  ( K1 g ) = K1 ( inverse  g )
+  ggtimes n ( K1 g ) = K1 ( gtimes n g )
+
+instance ( GGroup f1, GGroup f2 ) => GGroup ( f1 :*: f2 ) where
+  ginverse  ( g1 :*: g2 ) = ( ginverse  g1 :*: ginverse  g2 )
+  ggtimes n ( g1 :*: g2 ) = ( ggtimes n g1 :*: ggtimes n g2 )
+
+instance
+  TypeError ( 'Text "No 'Group' instance for empty generic representation." )
+  => GGroup V1 where
+  ginverse  _ = error "unreachable"
+  ggtimes _ _ = error "unreachable"
+
+instance
+  TypeError ( 'Text "No 'Group' instance for generic sum type." )
+  => GGroup ( f1 :+: f2 ) where
+  ginverse  _ = error "unreachable"
+  ggtimes _ _ = error "unreachable"

@@ -0,0 +1,175 @@+{-# LANGUAGE
+    DataKinds
+  , DeriveDataTypeable
+  , DeriveGeneric
+  , DerivingVia
+  , FlexibleContexts
+  , FlexibleInstances
+  , GADTs
+  , GeneralizedNewtypeDeriving
+  , MultiParamTypeClasses
+  , PatternSynonyms
+  , PolyKinds
+  , ScopedTypeVariables
+  , StandaloneDeriving
+  , TypeApplications
+  , TypeFamilies
+  , TypeOperators
+  , ViewPatterns
+#-}
+
+{-|
+Module: Data.Group.Cyclic
+
+Cyclic groups: integers modulo @n@ (clock arithmetic).
+-}
+
+module Data.Group.Cyclic
+  ( Cyclic(Cyclic), getCyclic
+  , C, Z
+  , CyclicEnum(..)
+  , pattern Involution, involution
+  , rootOfUnity
+  )
+  where
+
+-- base
+import Data.Coerce
+  ( coerce )
+import Data.Complex
+  ( Complex(..), conjugate, mkPolar )
+import Data.Data
+  ( Data )
+import Data.Monoid
+  ( Sum(..), Product(..) )
+import Data.Proxy
+  ( Proxy(..) )
+import GHC.Generics
+  ( Generic, Generic1 )
+import GHC.TypeNats
+  ( Nat, KnownNat, natVal
+  , type (<=)
+  )
+
+-- deepseq
+import Control.DeepSeq
+  ( NFData )
+
+-- finite-typelits
+import Data.Finite
+  ( Finite, getFinite )
+
+-- acts
+import Data.Act
+  ( Act(..), Torsor(..) )
+import Data.Group
+  ( Group(..) )
+
+-----------------------------------------------------------------
+
+-- | Cyclic group of order @n@: integers with addition modulo @n@.
+newtype Cyclic n = MkCyclic { runCyclic :: Finite n }
+  deriving stock   ( Show, Generic, Generic1 )
+  deriving newtype ( Eq, Ord, Enum, Bounded, NFData )
+deriving via ( Sum ( Finite n ) ) instance KnownNat n => Semigroup ( Cyclic n ) 
+deriving via ( Sum ( Finite n ) ) instance ( KnownNat n, 1 <= n ) => Monoid ( Cyclic n )
+deriving via ( Sum ( Finite n ) ) instance ( KnownNat n, 1 <= n ) => Group  ( Cyclic n )
+
+{-# COMPLETE Cyclic #-}
+-- | Smart pattern and constructor for elements of cyclic groups.
+pattern Cyclic :: forall n. KnownNat n => Int -> Cyclic n
+pattern Cyclic i <- ( fromIntegral . getFinite . runCyclic -> i )
+  where
+    Cyclic i = MkCyclic ( fromIntegral ( i `mod` ( fromIntegral ( natVal ( Proxy @n ) ) ) ) )
+
+-- | Obtain a representative in the range \( [0, n[ \).
+getCyclic :: forall n. KnownNat n => Cyclic n -> Int
+getCyclic ( Cyclic i ) = i
+
+-- | Synonym for finite cyclic group.
+type C ( n :: Nat ) = Cyclic n
+-- | Synonym for infinite cyclic group.
+type Z = Sum Int
+
+instance KnownNat n => Act ( Cyclic n ) Int where
+  act ( Cyclic f ) j
+    | r + i >= n
+    = ( i - n ) + j
+    | otherwise
+    = i + j
+    where
+      i, n, r :: Int
+      i = fromIntegral f
+      n = fromIntegral ( natVal ( Proxy @n ) )
+      r = j `mod` n
+
+-- | Nontrivial element of cyclic group of order 2.
+pattern Involution :: Cyclic 2
+pattern Involution = Cyclic 1
+
+-- | Act by an involution.
+involution :: Act ( Cyclic 2 ) x => x -> x
+involution = act Involution
+
+instance Act ( Cyclic 2 ) Bool where
+  act Involution = not
+  act _          = id
+
+instance Num i => Act ( Cyclic 2 ) ( Sum i ) where
+  act Involution = coerce ( negate :: i -> i )
+  act _          = id
+
+instance Fractional i => Act ( Cyclic 2 ) ( Product i ) where
+  act Involution = coerce ( recip :: i -> i )
+  act _          = id
+
+instance Num a => Act ( Cyclic 2 ) ( Complex a ) where
+  act Involution = conjugate
+  act _          = id
+
+-- | Natural complex representations of finite cyclic groups as roots of unity.
+rootOfUnity :: forall a n. ( KnownNat n, Floating a ) => Cyclic n -> Complex a
+rootOfUnity ( Cyclic f ) = mkPolar 1 ( 2 * pi * i / n )
+  where
+    i, n :: a
+    i = fromIntegral f
+    n = fromIntegral ( natVal ( Proxy @n ) )
+
+-- | Newtype for cycling through elements in a finite enumeration.
+--
+-- > data ABCD = A | B | C | D 
+-- >   deriving stock ( Enum, Bounded )
+-- >   deriving ( Act ( Cyclic 4 ), Torsor ( Cyclic 4 ) )
+-- >     via CyclicEnum ABCD
+-- 
+-- > > act ( Cyclic 2 ) C
+-- > A
+--
+-- > > act ( Cyclic (-1) ) A
+-- > D
+--
+-- > > ( C --> B :: Cyclic 4 )
+-- > Cyclic 3
+--
+-- __Warning__
+-- It is unfortunately not checked that the size of the group
+-- matches the size of the finite enumeration.
+-- Please manually ensure this condition.
+newtype CyclicEnum a = CyclicEnum { getCyclicEnum :: a }
+  deriving stock   ( Show, Data, Generic, Generic1 )
+  deriving newtype ( Eq, Ord, Enum, Bounded, NFData )
+
+instance ( Enum a, Bounded a, KnownNat n ) => Act ( Cyclic n ) ( CyclicEnum a ) where
+  act ( Cyclic f ) a = toEnum j
+    where
+      b_min, b_max, i, j :: Int
+      b_min = fromEnum ( minBound @a )
+      b_max = fromEnum ( maxBound @a )
+      i = fromIntegral f
+      j = b_min + ( fromEnum a + i - b_min ) `mod` ( 1 + b_max - b_min )
+      -- Assumes n ~ ( 1 + b_max - b_min ).
+instance ( Enum a, Bounded a, KnownNat n, 1 <= n ) => Torsor ( Cyclic n ) ( CyclicEnum a ) where
+  a --> b = Cyclic . fromIntegral . ( `mod` n ) $ fromEnum b - fromEnum a
+    where
+      n :: Int
+      n = fromIntegral ( natVal ( Proxy @n ) )