```{-# LANGUAGE DeriveDataTypeable, FlexibleContexts, FlexibleInstances, MultiParamTypeClasses, UndecidableInstances #-}
module Data.Logic.Types.Propositional where

import Data.Generics (Data, Typeable)
import Data.Logic.Classes.Combine (Combinable(..), Combination(..), BinOp(..))
import Data.Logic.Classes.Constants (Constants(..), asBool)
import qualified Data.Logic.Classes.Formula as C
import Data.Logic.Classes.Literal (Literal(..))
import Data.Logic.Classes.Negate (Negatable(..))
import Data.Logic.Classes.Pretty (Pretty(pretty), HasFixity(..), topFixity)
import Data.Logic.Classes.Propositional (PropositionalFormula(..), prettyPropositional, fixityPropositional, foldAtomsPropositional, mapAtomsPropositional)

-- | The range of a formula is {True, False} when it has no free variables.
data Formula atom
= Combine (Combination (Formula atom))
| Atom atom
| T
| F
-- Note that a derived Eq instance is not going to tell us that
-- a&b is equal to b&a, let alone that ~(a&b) equals (~a)|(~b).
deriving (Eq,Ord,Data,Typeable)

instance Negatable (Formula atom) where
negatePrivate x = Combine ((:~:) x)
foldNegation normal inverted (Combine ((:~:) x)) = foldNegation inverted normal x
foldNegation normal _ x = normal x

instance (Ord atom) => Combinable (Formula atom) where
x .<=>. y = Combine (BinOp  x (:<=>:) y)
x .=>.  y = Combine (BinOp  x (:=>:)  y)
x .|.   y = Combine (BinOp  x (:|:)   y)
x .&.   y = Combine (BinOp  x (:&:)   y)

instance Constants (Formula atom) where
fromBool True = T
fromBool False = F
asBool T = Just True
asBool F = Just False
asBool _ = Nothing

instance (Pretty atom, HasFixity atom, Ord atom) => C.Formula (Formula atom) atom where
atomic = Atom
foldAtoms = foldAtomsPropositional
mapAtoms = mapAtomsPropositional

instance (Pretty atom, HasFixity atom, Ord atom) => Literal (Formula atom) atom where
foldLiteral neg tf at formula =
case formula of
Combine ((:~:) p) -> neg p
Combine _ -> error ("Unexpected literal: " ++ show (pretty formula))
Atom x -> at x
T -> tf True
F -> tf False

instance (C.Formula (Formula atom) atom, Pretty atom, HasFixity atom, Ord atom) => PropositionalFormula (Formula atom) atom where
foldPropositional co tf at formula =
case formula of
Combine x -> co x
Atom x -> at x
T -> tf True
F -> tf False

instance (Pretty atom, HasFixity atom, Ord atom) => Pretty (Formula atom) where
pretty = prettyPropositional pretty topFixity

instance (Pretty atom, HasFixity atom, Ord atom) => HasFixity (Formula atom) where
fixity = fixityPropositional
```