Boolean.hs

{-# LANGUAGE TypeFamilies, GADTs, TypeOperators, ScopedTypeVariables, UndecidableInstances #-}
-----------------------------------------------------------------------------
-- |
-- Module      :  Data.Category.Boolean
-- License     :  BSD-style (see the file LICENSE)
--
-- Maintainer  :  sjoerd@w3future.com
-- Stability   :  experimental
-- Portability :  non-portable
--
-- /2/, or the Boolean category. 
-- It contains 2 objects, one for true and one for false.
-- It contains 3 arrows, 2 identity arrows and one from false to true.
-----------------------------------------------------------------------------
module Data.Category.Boolean where

import Prelude hiding ((.), id, Functor)

import Data.Category
import Data.Category.Functor
import Data.Category.NaturalTransformation
import Data.Category.Product
import Data.Category.Limit
import Data.Category.Monoidal
import Data.Category.CartesianClosed


data Fls
data Tru
  
data Boolean a b where
  Fls :: Boolean Fls Fls
  F2T :: Boolean Fls Tru
  Tru :: Boolean Tru Tru

-- | @Boolean@ is the category with true and false as objects, and an arrow from false to true.
instance Category Boolean where
  
  src Fls   = Fls
  src F2T   = Fls
  src Tru   = Tru
  
  tgt Fls   = Fls
  tgt F2T   = Tru
  tgt Tru   = Tru
  
  Fls . Fls = Fls
  F2T . Fls = F2T
  Tru . F2T = F2T
  Tru . Tru = Tru
  _      . _      = error "Other combinations should not type check"


-- | False is the initial object in the Boolean category.
instance HasInitialObject Boolean where
  type InitialObject Boolean = Fls
  initialObject = Fls
  initialize Fls = Fls
  initialize Tru = F2T
  initialize _   = error "Other values should not type check"
  
-- | True is the terminal object in the Boolean category.
instance HasTerminalObject Boolean where
  type TerminalObject Boolean = Tru
  terminalObject = Tru
  terminate Fls = F2T
  terminate Tru = Tru
  terminate _   = error "Other values should not type check"


type instance BinaryProduct Boolean Fls Fls = Fls
type instance BinaryProduct Boolean Fls Tru = Fls
type instance BinaryProduct Boolean Tru Fls = Fls
type instance BinaryProduct Boolean Tru Tru = Tru

-- | Conjunction is the binary product in the Boolean category.
instance HasBinaryProducts Boolean where 
  
  proj1 Fls Fls = Fls
  proj1 Fls Tru = Fls
  proj1 Tru Fls = F2T
  proj1 Tru Tru = Tru
  proj1 _   _   = error "Other combinations should not type check"
  proj2 Fls Fls = Fls
  proj2 Fls Tru = F2T
  proj2 Tru Fls = Fls
  proj2 Tru Tru = Tru
  proj2 _   _   = error "Other combinations should not type check"
    
  Fls &&& Fls = Fls
  Fls &&& F2T = Fls
  F2T &&& Fls = Fls
  F2T &&& F2T = F2T
  Tru &&& Tru = Tru
  _   &&& _   = error "Other combinations should not type check"


type instance BinaryCoproduct Boolean Fls Fls = Fls
type instance BinaryCoproduct Boolean Fls Tru = Tru
type instance BinaryCoproduct Boolean Tru Fls = Tru
type instance BinaryCoproduct Boolean Tru Tru = Tru

-- | Disjunction is the binary coproduct in the Boolean category.
instance HasBinaryCoproducts Boolean where 
  
  inj1 Fls Fls = Fls
  inj1 Fls Tru = F2T
  inj1 Tru Fls = Tru
  inj1 Tru Tru = Tru
  inj1 _   _   = error "Other combinations should not type check"
  inj2 Fls Fls = Fls
  inj2 Fls Tru = Tru
  inj2 Tru Fls = F2T
  inj2 Tru Tru = Tru
  inj2 _   _   = error "Other combinations should not type check"
    
  Fls ||| Fls = Fls
  F2T ||| F2T = F2T
  F2T ||| Tru = Tru
  Tru ||| F2T = Tru
  Tru ||| Tru = Tru
  _   ||| _   = error "Other combinations should not type check"


type instance Exponential Boolean Fls Fls = Tru
type instance Exponential Boolean Fls Tru = Tru
type instance Exponential Boolean Tru Fls = Fls
type instance Exponential Boolean Tru Tru = Tru

-- | Implication makes the Boolean category cartesian closed.
instance CartesianClosed Boolean where
  
  apply Fls Fls = Fls
  apply Fls Tru = F2T
  apply Tru Fls = Fls
  apply Tru Tru = Tru
  apply _   _   = error "Other combinations should not type check"
  
  tuple Fls Fls = F2T
  tuple Fls Tru = Tru
  tuple Tru Fls = Fls
  tuple Tru Tru = Tru
  tuple _   _   = error "Other combinations should not type check"
  
  Fls ^^^ Fls = Tru
  Fls ^^^ F2T = F2T
  Fls ^^^ Tru = Fls
  F2T ^^^ Fls = Tru
  F2T ^^^ F2T = F2T
  F2T ^^^ Tru = F2T
  Tru ^^^ Fls = Tru
  Tru ^^^ F2T = Tru
  Tru ^^^ Tru = Tru


trueProductMonoid :: MonoidObject (ProductFunctor Boolean) Tru
trueProductMonoid = MonoidObject Tru Tru

falseCoproductComonoid :: ComonoidObject (CoproductFunctor Boolean) Fls
falseCoproductComonoid = ComonoidObject Fls Fls

trueProductComonoid :: ComonoidObject (ProductFunctor Boolean) Tru
trueProductComonoid = ComonoidObject Tru Tru

falseCoproductMonoid :: MonoidObject (CoproductFunctor Boolean) Fls
falseCoproductMonoid = MonoidObject Fls Fls

trueCoproductMonoid :: MonoidObject (CoproductFunctor Boolean) Tru
trueCoproductMonoid = MonoidObject F2T Tru

falseProductComonoid :: ComonoidObject (ProductFunctor Boolean) Fls
falseProductComonoid = ComonoidObject F2T Fls


data NatAsFunctor f g where
  NatAsFunctor :: (Functor f, Functor g, Category c, Category d, Dom f ~ c, Cod f ~ d, Dom g ~ c, Cod g ~ d)
               => Nat (Dom f) (Cod f) f g -> NatAsFunctor f g
type instance Dom (NatAsFunctor f g) = Dom f :**: Boolean
type instance Cod (NatAsFunctor f g) = Cod f
type instance NatAsFunctor f g :% (a, Fls) = f :% a
type instance NatAsFunctor f g :% (a, Tru) = g :% a
-- | A natural transformation @Nat c d@ is isomorphic to a functor from @c :**: 2@ to @d@.
instance (Category (Dom f), Category (Cod f)) => Functor (NatAsFunctor f g) where
  NatAsFunctor (Nat f _ _) % (a :**: Fls) = f % a
  NatAsFunctor (Nat _ g _) % (a :**: Tru) = g % a
  NatAsFunctor n           % (a :**: F2T) = n ! a