Data.Algebra.Boolean

Documentation

class Boolean b where

A class for boolean algebras. Instances of this class are expected to obey all the laws of boolean algebra).

Minimal complete definition: true or false, not or (<-->, false), || or &&.

Minimal complete definition

(false | true), (not | (<-->), false), ((||) | (&&))

Methods

true :: b

Truth value, defined as the top of the bounded lattice

false :: b

False value, defined as the bottom of the bounded lattice.

not :: b -> b

Logical negation.

(&&) :: b -> b -> b infixr 3

Logical conjunction. (infixr 3)

(||) :: b -> b -> b infixr 2

Logical inclusive disjunction. (infixr 2)

xor :: b -> b -> b infixr 1

Logical exclusive disjunction. (infixr 1)

(-->) :: b -> b -> b infixr 1

Logical implication. (infixr 1)

(<-->) :: b -> b -> b infixr 1

Logical biconditional. (infixr 1)

Instances

Boolean All

Could be done via `deriving via` from GHC8.6.1 onwards

Defined in Data.Algebra.Boolean

Methods

true :: All

false :: All

not :: All -> All

(&&) :: All -> All -> All

(||) :: All -> All -> All

xor :: All -> All -> All

(-->) :: All -> All -> All

(<-->) :: All -> All -> All

Boolean Any

Could be done via `deriving via` from GHC8.6.1 onwards

Defined in Data.Algebra.Boolean

Methods

true :: Any

false :: Any

not :: Any -> Any

(&&) :: Any -> Any -> Any

(||) :: Any -> Any -> Any

xor :: Any -> Any -> Any

(-->) :: Any -> Any -> Any

(<-->) :: Any -> Any -> Any

Boolean ()

The trivial boolean algebra

Defined in Data.Algebra.Boolean

Methods

true :: ()

false :: ()

not :: () -> ()

(&&) :: () -> () -> ()

(||) :: () -> () -> ()

xor :: () -> () -> ()

(-->) :: () -> () -> ()

(<-->) :: () -> () -> ()

Boolean Bool

Defined in Data.Algebra.Boolean

Methods

true :: Bool

false :: Bool

not :: Bool -> Bool

(&&) :: Bool -> Bool -> Bool

(||) :: Bool -> Bool -> Bool

xor :: Bool -> Bool -> Bool

(-->) :: Bool -> Bool -> Bool

(<-->) :: Bool -> Bool -> Bool

Boolean (Dual Bool)

Could be done via `deriving via` from GHC8.6.1 onwards

Defined in Data.Algebra.Boolean

Methods

true :: Dual Bool

false :: Dual Bool

not :: Dual Bool -> Dual Bool

(&&) :: Dual Bool -> Dual Bool -> Dual Bool

(||) :: Dual Bool -> Dual Bool -> Dual Bool

xor :: Dual Bool -> Dual Bool -> Dual Bool

(-->) :: Dual Bool -> Dual Bool -> Dual Bool

(<-->) :: Dual Bool -> Dual Bool -> Dual Bool

Boolean a => Boolean (Endo a)

Could be done via `deriving via` from GHC8.6.1 onwards

Defined in Data.Algebra.Boolean

Methods

true :: Endo a

false :: Endo a

not :: Endo a -> Endo a

(&&) :: Endo a -> Endo a -> Endo a

(||) :: Endo a -> Endo a -> Endo a

xor :: Endo a -> Endo a -> Endo a

(-->) :: Endo a -> Endo a -> Endo a

(<-->) :: Endo a -> Endo a -> Endo a

(Num a, Bits a) => Boolean (Bitwise a)

Defined in Data.Algebra.Boolean

Methods

true :: Bitwise a

false :: Bitwise a

not :: Bitwise a -> Bitwise a

(&&) :: Bitwise a -> Bitwise a -> Bitwise a

(||) :: Bitwise a -> Bitwise a -> Bitwise a

xor :: Bitwise a -> Bitwise a -> Bitwise a

(-->) :: Bitwise a -> Bitwise a -> Bitwise a

(<-->) :: Bitwise a -> Bitwise a -> Bitwise a

Boolean a => Boolean (Opp a)

Opposite boolean algebra: exchanges true and false, and and and or, etc

Defined in Data.Algebra.Boolean

Methods

true :: Opp a

false :: Opp a

not :: Opp a -> Opp a

(&&) :: Opp a -> Opp a -> Opp a

(||) :: Opp a -> Opp a -> Opp a

xor :: Opp a -> Opp a -> Opp a

(-->) :: Opp a -> Opp a -> Opp a

(<-->) :: Opp a -> Opp a -> Opp a

Boolean b => Boolean (a -> b)

Pointwise boolean algebra.

Defined in Data.Algebra.Boolean

Methods

true :: a -> b

false :: a -> b

not :: (a -> b) -> a -> b

(&&) :: (a -> b) -> (a -> b) -> a -> b

(||) :: (a -> b) -> (a -> b) -> a -> b

xor :: (a -> b) -> (a -> b) -> a -> b

(-->) :: (a -> b) -> (a -> b) -> a -> b

(<-->) :: (a -> b) -> (a -> b) -> a -> b

(Boolean x, Boolean y) => Boolean (x, y)

Defined in Data.Algebra.Boolean

Methods

true :: (x, y)

false :: (x, y)

not :: (x, y) -> (x, y)

(&&) :: (x, y) -> (x, y) -> (x, y)

(||) :: (x, y) -> (x, y) -> (x, y)

xor :: (x, y) -> (x, y) -> (x, y)

(-->) :: (x, y) -> (x, y) -> (x, y)

(<-->) :: (x, y) -> (x, y) -> (x, y)

(Boolean x, Boolean y, Boolean z) => Boolean (x, y, z)

Defined in Data.Algebra.Boolean

Methods

true :: (x, y, z)

false :: (x, y, z)

not :: (x, y, z) -> (x, y, z)

(&&) :: (x, y, z) -> (x, y, z) -> (x, y, z)

(||) :: (x, y, z) -> (x, y, z) -> (x, y, z)

xor :: (x, y, z) -> (x, y, z) -> (x, y, z)

(-->) :: (x, y, z) -> (x, y, z) -> (x, y, z)

(<-->) :: (x, y, z) -> (x, y, z) -> (x, y, z)

fromBool :: Boolean b => Bool -> b

Injection from Bool into a boolean algebra.

newtype Bitwise a

A newtype wrapper that derives a Boolean instance from any type that is both a Bits instance and a Num instance, such that boolean logic operations on the Bitwise wrapper correspond to bitwise logic operations on the inner type. It should be noted that false is defined as Bitwise 0 and true is defined as not false.

In addition, a number of other classes are automatically derived from the inner type. These classes were chosen on the basis that many other Bits instances defined in base are also instances of these classes.

Constructors

Bitwise

Fields

• getBits :: a

Instances

Data a => Data (Bitwise a)

Defined in Data.Algebra.Boolean

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Bitwise a -> c (Bitwise a)

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Bitwise a)

toConstr :: Bitwise a -> Constr

dataTypeOf :: Bitwise a -> DataType

dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (Bitwise a))

dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Bitwise a))

gmapT :: (forall b. Data b => b -> b) -> Bitwise a -> Bitwise a

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Bitwise a -> r

gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Bitwise a -> r

gmapQ :: (forall d. Data d => d -> u) -> Bitwise a -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> Bitwise a -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> Bitwise a -> m (Bitwise a)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Bitwise a -> m (Bitwise a)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Bitwise a -> m (Bitwise a)

Storable a => Storable (Bitwise a)

Defined in Data.Algebra.Boolean

Methods

sizeOf :: Bitwise a -> Int

alignment :: Bitwise a -> Int

peekElemOff :: Ptr (Bitwise a) -> Int -> IO (Bitwise a)

pokeElemOff :: Ptr (Bitwise a) -> Int -> Bitwise a -> IO ()

peekByteOff :: Ptr b -> Int -> IO (Bitwise a)

pokeByteOff :: Ptr b -> Int -> Bitwise a -> IO ()

peek :: Ptr (Bitwise a) -> IO (Bitwise a)

poke :: Ptr (Bitwise a) -> Bitwise a -> IO ()

Bits a => Bits (Bitwise a)

Defined in Data.Algebra.Boolean

Methods

(.&.) :: Bitwise a -> Bitwise a -> Bitwise a

(.|.) :: Bitwise a -> Bitwise a -> Bitwise a

xor :: Bitwise a -> Bitwise a -> Bitwise a

complement :: Bitwise a -> Bitwise a

shift :: Bitwise a -> Int -> Bitwise a

rotate :: Bitwise a -> Int -> Bitwise a

zeroBits :: Bitwise a

bit :: Int -> Bitwise a

setBit :: Bitwise a -> Int -> Bitwise a

clearBit :: Bitwise a -> Int -> Bitwise a

complementBit :: Bitwise a -> Int -> Bitwise a

testBit :: Bitwise a -> Int -> Bool

bitSizeMaybe :: Bitwise a -> Maybe Int

bitSize :: Bitwise a -> Int

isSigned :: Bitwise a -> Bool

shiftL :: Bitwise a -> Int -> Bitwise a

unsafeShiftL :: Bitwise a -> Int -> Bitwise a

shiftR :: Bitwise a -> Int -> Bitwise a

unsafeShiftR :: Bitwise a -> Int -> Bitwise a

rotateL :: Bitwise a -> Int -> Bitwise a

rotateR :: Bitwise a -> Int -> Bitwise a

popCount :: Bitwise a -> Int

Bounded a => Bounded (Bitwise a)

Defined in Data.Algebra.Boolean

Methods

minBound :: Bitwise a

maxBound :: Bitwise a

Enum a => Enum (Bitwise a)

Defined in Data.Algebra.Boolean

Methods

succ :: Bitwise a -> Bitwise a

pred :: Bitwise a -> Bitwise a

toEnum :: Int -> Bitwise a

fromEnum :: Bitwise a -> Int

enumFrom :: Bitwise a -> [Bitwise a]

enumFromThen :: Bitwise a -> Bitwise a -> [Bitwise a]

enumFromTo :: Bitwise a -> Bitwise a -> [Bitwise a]

enumFromThenTo :: Bitwise a -> Bitwise a -> Bitwise a -> [Bitwise a]

Ix a => Ix (Bitwise a)

Defined in Data.Algebra.Boolean

Methods

range :: (Bitwise a, Bitwise a) -> [Bitwise a]

index :: (Bitwise a, Bitwise a) -> Bitwise a -> Int

unsafeIndex :: (Bitwise a, Bitwise a) -> Bitwise a -> Int

inRange :: (Bitwise a, Bitwise a) -> Bitwise a -> Bool

rangeSize :: (Bitwise a, Bitwise a) -> Int

unsafeRangeSize :: (Bitwise a, Bitwise a) -> Int

Num a => Num (Bitwise a)

Defined in Data.Algebra.Boolean

Methods

(+) :: Bitwise a -> Bitwise a -> Bitwise a

(-) :: Bitwise a -> Bitwise a -> Bitwise a

(*) :: Bitwise a -> Bitwise a -> Bitwise a

negate :: Bitwise a -> Bitwise a

abs :: Bitwise a -> Bitwise a

signum :: Bitwise a -> Bitwise a

fromInteger :: Integer -> Bitwise a

Read a => Read (Bitwise a)

Defined in Data.Algebra.Boolean

Methods

readsPrec :: Int -> ReadS (Bitwise a)

readList :: ReadS [Bitwise a]

readPrec :: ReadPrec (Bitwise a)

readListPrec :: ReadPrec [Bitwise a]

Integral a => Integral (Bitwise a)

Defined in Data.Algebra.Boolean

Methods

quot :: Bitwise a -> Bitwise a -> Bitwise a

rem :: Bitwise a -> Bitwise a -> Bitwise a

div :: Bitwise a -> Bitwise a -> Bitwise a

mod :: Bitwise a -> Bitwise a -> Bitwise a

quotRem :: Bitwise a -> Bitwise a -> (Bitwise a, Bitwise a)

divMod :: Bitwise a -> Bitwise a -> (Bitwise a, Bitwise a)

toInteger :: Bitwise a -> Integer

Real a => Real (Bitwise a)

Defined in Data.Algebra.Boolean

Methods

toRational :: Bitwise a -> Rational

Show a => Show (Bitwise a)

Defined in Data.Algebra.Boolean

Methods

showsPrec :: Int -> Bitwise a -> ShowS

show :: Bitwise a -> String

showList :: [Bitwise a] -> ShowS

PrintfArg a => PrintfArg (Bitwise a)

Defined in Data.Algebra.Boolean

Methods

formatArg :: Bitwise a -> FieldFormatter

parseFormat :: Bitwise a -> ModifierParser

(Num a, Bits a) => Boolean (Bitwise a)

Defined in Data.Algebra.Boolean

Methods

true :: Bitwise a

false :: Bitwise a

not :: Bitwise a -> Bitwise a

(&&) :: Bitwise a -> Bitwise a -> Bitwise a

(||) :: Bitwise a -> Bitwise a -> Bitwise a

xor :: Bitwise a -> Bitwise a -> Bitwise a

(-->) :: Bitwise a -> Bitwise a -> Bitwise a

(<-->) :: Bitwise a -> Bitwise a -> Bitwise a

Eq a => Eq (Bitwise a)

Defined in Data.Algebra.Boolean

Methods

(==) :: Bitwise a -> Bitwise a -> Bool

(/=) :: Bitwise a -> Bitwise a -> Bool

Ord a => Ord (Bitwise a)

Defined in Data.Algebra.Boolean

Methods

compare :: Bitwise a -> Bitwise a -> Ordering

(<) :: Bitwise a -> Bitwise a -> Bool

(<=) :: Bitwise a -> Bitwise a -> Bool

(>) :: Bitwise a -> Bitwise a -> Bool

(>=) :: Bitwise a -> Bitwise a -> Bool

max :: Bitwise a -> Bitwise a -> Bitwise a

min :: Bitwise a -> Bitwise a -> Bitwise a

and :: (Boolean b, Foldable t) => t b -> b

The logical conjunction of several values.

or :: (Boolean b, Foldable t) => t b -> b

The logical disjunction of several values.

nand :: (Boolean b, Foldable t) => t b -> b

The negated logical conjunction of several values.

nand = not . and

nor :: (Boolean b, Foldable t) => t b -> b

The negated logical disjunction of several values.

nor = not . or

any :: (Boolean b, Foldable t) => (a -> b) -> t a -> b

The logical disjunction of the mapping of a function over several values.

all :: (Boolean b, Foldable t) => (a -> b) -> t a -> b

The logical conjunction of the mapping of a function over several values.

newtype Opp a

Constructors

Opp

Fields

• getOpp :: a

Instances

Show a => Show (Opp a)

Defined in Data.Algebra.Boolean

Methods

showsPrec :: Int -> Opp a -> ShowS

show :: Opp a -> String

showList :: [Opp a] -> ShowS

Boolean a => Boolean (Opp a)

Opposite boolean algebra: exchanges true and false, and and and or, etc

Defined in Data.Algebra.Boolean

Methods

true :: Opp a

false :: Opp a

not :: Opp a -> Opp a

(&&) :: Opp a -> Opp a -> Opp a

(||) :: Opp a -> Opp a -> Opp a

xor :: Opp a -> Opp a -> Opp a

(-->) :: Opp a -> Opp a -> Opp a

(<-->) :: Opp a -> Opp a -> Opp a

Eq a => Eq (Opp a)

Defined in Data.Algebra.Boolean

Methods

(==) :: Opp a -> Opp a -> Bool

(/=) :: Opp a -> Opp a -> Bool

Ord a => Ord (Opp a)

Defined in Data.Algebra.Boolean

Methods

compare :: Opp a -> Opp a -> Ordering

(<) :: Opp a -> Opp a -> Bool

(<=) :: Opp a -> Opp a -> Bool

(>) :: Opp a -> Opp a -> Bool

(>=) :: Opp a -> Opp a -> Bool

max :: Opp a -> Opp a -> Opp a

min :: Opp a -> Opp a -> Opp a

newtype AnyB b

A boolean algebra regarded as a monoid under disjunction

Constructors

AnyB

Fields

• getAnyB :: b

Instances

Boolean b => Monoid (AnyB b)

Defined in Data.Algebra.Boolean

Methods

mempty :: AnyB b

mappend :: AnyB b -> AnyB b -> AnyB b

mconcat :: [AnyB b] -> AnyB b

Boolean b => Semigroup (AnyB b)

Defined in Data.Algebra.Boolean

Methods

(<>) :: AnyB b -> AnyB b -> AnyB b

sconcat :: NonEmpty (AnyB b) -> AnyB b

stimes :: Integral b0 => b0 -> AnyB b -> AnyB b

Show b => Show (AnyB b)

Defined in Data.Algebra.Boolean

Methods

showsPrec :: Int -> AnyB b -> ShowS

show :: AnyB b -> String

showList :: [AnyB b] -> ShowS

Eq b => Eq (AnyB b)

Defined in Data.Algebra.Boolean

Methods

(==) :: AnyB b -> AnyB b -> Bool

(/=) :: AnyB b -> AnyB b -> Bool

Ord b => Ord (AnyB b)

Defined in Data.Algebra.Boolean

Methods

compare :: AnyB b -> AnyB b -> Ordering

(<) :: AnyB b -> AnyB b -> Bool

(<=) :: AnyB b -> AnyB b -> Bool

(>) :: AnyB b -> AnyB b -> Bool

(>=) :: AnyB b -> AnyB b -> Bool

max :: AnyB b -> AnyB b -> AnyB b

min :: AnyB b -> AnyB b -> AnyB b

newtype AllB b

A boolean algebra regarded as a monoid under conjunction

Constructors

AllB

Fields

• getAllB :: b

Instances

Boolean b => Monoid (AllB b)

Defined in Data.Algebra.Boolean

Methods

mempty :: AllB b

mappend :: AllB b -> AllB b -> AllB b

mconcat :: [AllB b] -> AllB b

Boolean b => Semigroup (AllB b)

Defined in Data.Algebra.Boolean

Methods

(<>) :: AllB b -> AllB b -> AllB b

sconcat :: NonEmpty (AllB b) -> AllB b

stimes :: Integral b0 => b0 -> AllB b -> AllB b

Show b => Show (AllB b)

Defined in Data.Algebra.Boolean

Methods

showsPrec :: Int -> AllB b -> ShowS

show :: AllB b -> String

showList :: [AllB b] -> ShowS

Eq b => Eq (AllB b)

Defined in Data.Algebra.Boolean

Methods

(==) :: AllB b -> AllB b -> Bool

(/=) :: AllB b -> AllB b -> Bool

Ord b => Ord (AllB b)

Defined in Data.Algebra.Boolean

Methods

compare :: AllB b -> AllB b -> Ordering

(<) :: AllB b -> AllB b -> Bool

(<=) :: AllB b -> AllB b -> Bool

(>) :: AllB b -> AllB b -> Bool

(>=) :: AllB b -> AllB b -> Bool

max :: AllB b -> AllB b -> AllB b

min :: AllB b -> AllB b -> AllB b

newtype XorB b

A boolean algebra regarded as a monoid under exclusive or

Constructors

XorB

Fields

• getXorB :: b

Instances

Boolean b => Monoid (XorB b)

Defined in Data.Algebra.Boolean

Methods

mempty :: XorB b

mappend :: XorB b -> XorB b -> XorB b

mconcat :: [XorB b] -> XorB b

Boolean b => Semigroup (XorB b)

Defined in Data.Algebra.Boolean

Methods

(<>) :: XorB b -> XorB b -> XorB b

sconcat :: NonEmpty (XorB b) -> XorB b

stimes :: Integral b0 => b0 -> XorB b -> XorB b

Show b => Show (XorB b)

Defined in Data.Algebra.Boolean

Methods

showsPrec :: Int -> XorB b -> ShowS

show :: XorB b -> String

showList :: [XorB b] -> ShowS

Eq b => Eq (XorB b)

Defined in Data.Algebra.Boolean

Methods

(==) :: XorB b -> XorB b -> Bool

(/=) :: XorB b -> XorB b -> Bool

Ord b => Ord (XorB b)

Defined in Data.Algebra.Boolean

Methods

compare :: XorB b -> XorB b -> Ordering

(<) :: XorB b -> XorB b -> Bool

(<=) :: XorB b -> XorB b -> Bool

(>) :: XorB b -> XorB b -> Bool

(>=) :: XorB b -> XorB b -> Bool

max :: XorB b -> XorB b -> XorB b

min :: XorB b -> XorB b -> XorB b

newtype EquivB b

A boolean algebra regarded as a monoid under equivalence

Constructors

EquivB

Fields

• getEquivB :: b

Instances

Boolean b => Monoid (EquivB b)

Defined in Data.Algebra.Boolean

Methods

mempty :: EquivB b

mappend :: EquivB b -> EquivB b -> EquivB b

mconcat :: [EquivB b] -> EquivB b

Boolean b => Semigroup (EquivB b)

Defined in Data.Algebra.Boolean

Methods

(<>) :: EquivB b -> EquivB b -> EquivB b

sconcat :: NonEmpty (EquivB b) -> EquivB b

stimes :: Integral b0 => b0 -> EquivB b -> EquivB b

Show b => Show (EquivB b)

Defined in Data.Algebra.Boolean

Methods

showsPrec :: Int -> EquivB b -> ShowS

show :: EquivB b -> String

showList :: [EquivB b] -> ShowS

Eq b => Eq (EquivB b)

Defined in Data.Algebra.Boolean

Methods

(==) :: EquivB b -> EquivB b -> Bool

(/=) :: EquivB b -> EquivB b -> Bool

Ord b => Ord (EquivB b)

Defined in Data.Algebra.Boolean

Methods

compare :: EquivB b -> EquivB b -> Ordering

(<) :: EquivB b -> EquivB b -> Bool

(<=) :: EquivB b -> EquivB b -> Bool

(>) :: EquivB b -> EquivB b -> Bool

(>=) :: EquivB b -> EquivB b -> Bool

max :: EquivB b -> EquivB b -> EquivB b

min :: EquivB b -> EquivB b -> EquivB b