Safe Haskell	Safe
Language	Haskell2010

Data.Universe.Helpers

Contents

Building lists
Building cardinalities
Debugging

Synopsis

universeDef :: (Bounded a, Enum a) => [a]
interleave :: [[a]] -> [a]
diagonal :: [[a]] -> [a]
diagonals :: [[a]] -> [[a]]
(+++) :: [a] -> [a] -> [a]
cartesianProduct :: (a -> b -> c) -> [a] -> [b] -> [c]
(+*+) :: [a] -> [b] -> [(a, b)]
(<+*+>) :: [a -> b] -> [a] -> [b]
choices :: [[a]] -> [[a]]
retagWith :: (a -> b) -> Tagged a x -> Tagged b x
retag :: forall k1 k2 (s :: k1) b (t :: k2). Tagged s b -> Tagged t b
newtype Tagged (s :: k) b = Tagged {
- unTagged :: b
}
data Natural
unfairCartesianProduct :: (a -> b -> c) -> [a] -> [b] -> [c]
unfairChoices :: [[a]] -> [[a]]

Documentation

This module is for functions that are useful for writing instances, but not necessarily for using them (and hence are not exported by the main module to avoid cluttering up the namespace).

Building lists

universeDef :: (Bounded a, Enum a) => [a] Source #

For many types, the universe should be [minBound .. maxBound]; universeDef makes it easy to make such types an instance of Universe via the snippet

instance Universe Foo where universe = universeDef

interleave :: [[a]] -> [a] Source #

Fair n-way interleaving: given a finite number of (possibly infinite) lists, produce a single list such that whenever v has finite index in one of the input lists, v also has finite index in the output list. No list's elements occur more frequently (on average) than another's.

diagonal :: [[a]] -> [a] Source #

Unfair n-way interleaving: given a possibly infinite number of (possibly infinite) lists, produce a single list such that whenever v has finite index in an input list at finite index, v also has finite index in the output list. Elements from lists at lower index occur more frequently, but not exponentially so.

diagonals :: [[a]] -> [[a]] Source #

Like diagonal, but expose a tiny bit more (non-semantic) information: if you lay out the input list in two dimensions, each list in the result will be one of the diagonals of the input. In particular, each element of the output will be a list whose elements are each from a distinct input list.

(+++) :: [a] -> [a] -> [a] Source #

Fair 2-way interleaving.

cartesianProduct :: (a -> b -> c) -> [a] -> [b] -> [c] Source #

Slightly unfair 2-way Cartesian product: given two (possibly infinite) lists, produce a single list such that whenever v and w have finite indices in the input lists, (v,w) has finite index in the output list. Lower indices occur as the fst part of the tuple more frequently, but not exponentially so.

(+*+) :: [a] -> [b] -> [(a, b)] Source #

cartesianProduct (,)

(<+*+>) :: [a -> b] -> [a] -> [b] Source #

A +*+ with application.

cartesianProduct ($)

choices :: [[a]] -> [[a]] Source #

Slightly unfair n-way Cartesian product: given a finite number of (possibly infinite) lists, produce a single list such that whenever vi has finite index in list i for each i, [v1, ..., vn] has finite index in the output list.

Building cardinalities

These functions are handy for inheriting the definition of cardinality in a newtype instance. For example, one might write

newtype Foo = Foo Bar
instance Finite Foo where cardinality = retagWith Foo cardinality

retagWith :: (a -> b) -> Tagged a x -> Tagged b x Source #

retag :: forall k1 k2 (s :: k1) b (t :: k2). Tagged s b -> Tagged t b #

Some times you need to change the tag you have lying around. Idiomatic usage is to make a new combinator for the relationship between the tags that you want to enforce, and define that combinator using retag.

data Succ n
retagSucc :: Tagged n a -> Tagged (Succ n) a
retagSucc = retag

newtype Tagged (s :: k) b #

A Tagged s b value is a value b with an attached phantom type s. This can be used in place of the more traditional but less safe idiom of passing in an undefined value with the type, because unlike an (s -> b), a Tagged s b can't try to use the argument s as a real value.

Moreover, you don't have to rely on the compiler to inline away the extra argument, because the newtype is "free"

Tagged has kind k -> * -> * if the compiler supports PolyKinds, therefore there is an extra k showing in the instance haddocks that may cause confusion.

Constructors

Tagged
Fields unTagged :: b

Instances

Instances details

Bitraversable (Tagged :: Type -> Type -> Type)
Instance details Defined in Data.Tagged Methods bitraverse :: Applicative f => (a -> f c) -> (b -> f d) -> Tagged a b -> f (Tagged c d) #
Bifoldable (Tagged :: Type -> Type -> Type)
Instance details Defined in Data.Tagged Methods bifold :: Monoid m => Tagged m m -> m # bifoldMap :: Monoid m => (a -> m) -> (b -> m) -> Tagged a b -> m # bifoldr :: (a -> c -> c) -> (b -> c -> c) -> c -> Tagged a b -> c # bifoldl :: (c -> a -> c) -> (c -> b -> c) -> c -> Tagged a b -> c #
Bifunctor (Tagged :: Type -> Type -> Type)
Instance details Defined in Data.Tagged Methods bimap :: (a -> b) -> (c -> d) -> Tagged a c -> Tagged b d # first :: (a -> b) -> Tagged a c -> Tagged b c # second :: (b -> c) -> Tagged a b -> Tagged a c #
Eq2 (Tagged :: Type -> Type -> Type)
Instance details Defined in Data.Tagged Methods liftEq2 :: (a -> b -> Bool) -> (c -> d -> Bool) -> Tagged a c -> Tagged b d -> Bool #
Ord2 (Tagged :: Type -> Type -> Type)
Instance details Defined in Data.Tagged Methods liftCompare2 :: (a -> b -> Ordering) -> (c -> d -> Ordering) -> Tagged a c -> Tagged b d -> Ordering #
Read2 (Tagged :: Type -> Type -> Type)
Instance details Defined in Data.Tagged Methods liftReadsPrec2 :: (Int -> ReadS a) -> ReadS [a] -> (Int -> ReadS b) -> ReadS [b] -> Int -> ReadS (Tagged a b) # liftReadList2 :: (Int -> ReadS a) -> ReadS [a] -> (Int -> ReadS b) -> ReadS [b] -> ReadS [Tagged a b] # liftReadPrec2 :: ReadPrec a -> ReadPrec [a] -> ReadPrec b -> ReadPrec [b] -> ReadPrec (Tagged a b) # liftReadListPrec2 :: ReadPrec a -> ReadPrec [a] -> ReadPrec b -> ReadPrec [b] -> ReadPrec [Tagged a b] #
Show2 (Tagged :: Type -> Type -> Type)
Instance details Defined in Data.Tagged Methods liftShowsPrec2 :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> (Int -> b -> ShowS) -> ([b] -> ShowS) -> Int -> Tagged a b -> ShowS # liftShowList2 :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> (Int -> b -> ShowS) -> ([b] -> ShowS) -> [Tagged a b] -> ShowS #
Generic1 (Tagged s :: Type -> Type)
Instance details Defined in Data.Tagged Associated Types type Rep1 (Tagged s) :: k -> Type # Methods from1 :: forall (a :: k). Tagged s a -> Rep1 (Tagged s) a # to1 :: forall (a :: k). Rep1 (Tagged s) a -> Tagged s a #
Monad (Tagged s)
Instance details Defined in Data.Tagged Methods (>>=) :: Tagged s a -> (a -> Tagged s b) -> Tagged s b # (>>) :: Tagged s a -> Tagged s b -> Tagged s b # return :: a -> Tagged s a #
Functor (Tagged s)
Instance details Defined in Data.Tagged Methods fmap :: (a -> b) -> Tagged s a -> Tagged s b # (<$) :: a -> Tagged s b -> Tagged s a #
Applicative (Tagged s)
Instance details Defined in Data.Tagged Methods pure :: a -> Tagged s a # (<>) :: Tagged s (a -> b) -> Tagged s a -> Tagged s b # liftA2 :: (a -> b -> c) -> Tagged s a -> Tagged s b -> Tagged s c # (>) :: Tagged s a -> Tagged s b -> Tagged s b # (<*) :: Tagged s a -> Tagged s b -> Tagged s a #
Foldable (Tagged s)
Instance details Defined in Data.Tagged Methods fold :: Monoid m => Tagged s m -> m # foldMap :: Monoid m => (a -> m) -> Tagged s a -> m # foldMap' :: Monoid m => (a -> m) -> Tagged s a -> m # foldr :: (a -> b -> b) -> b -> Tagged s a -> b # foldr' :: (a -> b -> b) -> b -> Tagged s a -> b # foldl :: (b -> a -> b) -> b -> Tagged s a -> b # foldl' :: (b -> a -> b) -> b -> Tagged s a -> b # foldr1 :: (a -> a -> a) -> Tagged s a -> a # foldl1 :: (a -> a -> a) -> Tagged s a -> a # toList :: Tagged s a -> [a] # null :: Tagged s a -> Bool # length :: Tagged s a -> Int # elem :: Eq a => a -> Tagged s a -> Bool # maximum :: Ord a => Tagged s a -> a # minimum :: Ord a => Tagged s a -> a # sum :: Num a => Tagged s a -> a # product :: Num a => Tagged s a -> a #
Traversable (Tagged s)
Instance details Defined in Data.Tagged Methods traverse :: Applicative f => (a -> f b) -> Tagged s a -> f (Tagged s b) # sequenceA :: Applicative f => Tagged s (f a) -> f (Tagged s a) # mapM :: Monad m => (a -> m b) -> Tagged s a -> m (Tagged s b) # sequence :: Monad m => Tagged s (m a) -> m (Tagged s a) #
Eq1 (Tagged s)
Instance details Defined in Data.Tagged Methods liftEq :: (a -> b -> Bool) -> Tagged s a -> Tagged s b -> Bool #
Ord1 (Tagged s)
Instance details Defined in Data.Tagged Methods liftCompare :: (a -> b -> Ordering) -> Tagged s a -> Tagged s b -> Ordering #
Read1 (Tagged s)
Instance details Defined in Data.Tagged Methods liftReadsPrec :: (Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (Tagged s a) # liftReadList :: (Int -> ReadS a) -> ReadS [a] -> ReadS [Tagged s a] # liftReadPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec (Tagged s a) # liftReadListPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec [Tagged s a] #
Show1 (Tagged s)
Instance details Defined in Data.Tagged Methods liftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> Tagged s a -> ShowS # liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [Tagged s a] -> ShowS #
Bounded b => Bounded (Tagged s b)
Instance details Defined in Data.Tagged Methods minBound :: Tagged s b # maxBound :: Tagged s b #
Enum a => Enum (Tagged s a)
Instance details Defined in Data.Tagged Methods succ :: Tagged s a -> Tagged s a # pred :: Tagged s a -> Tagged s a # toEnum :: Int -> Tagged s a # fromEnum :: Tagged s a -> Int # enumFrom :: Tagged s a -> [Tagged s a] # enumFromThen :: Tagged s a -> Tagged s a -> [Tagged s a] # enumFromTo :: Tagged s a -> Tagged s a -> [Tagged s a] # enumFromThenTo :: Tagged s a -> Tagged s a -> Tagged s a -> [Tagged s a] #
Eq b => Eq (Tagged s b)
Instance details Defined in Data.Tagged Methods (==) :: Tagged s b -> Tagged s b -> Bool # (/=) :: Tagged s b -> Tagged s b -> Bool #
Floating a => Floating (Tagged s a)
Instance details Defined in Data.Tagged Methods pi :: Tagged s a # exp :: Tagged s a -> Tagged s a # log :: Tagged s a -> Tagged s a # sqrt :: Tagged s a -> Tagged s a # (**) :: Tagged s a -> Tagged s a -> Tagged s a # logBase :: Tagged s a -> Tagged s a -> Tagged s a # sin :: Tagged s a -> Tagged s a # cos :: Tagged s a -> Tagged s a # tan :: Tagged s a -> Tagged s a # asin :: Tagged s a -> Tagged s a # acos :: Tagged s a -> Tagged s a # atan :: Tagged s a -> Tagged s a # sinh :: Tagged s a -> Tagged s a # cosh :: Tagged s a -> Tagged s a # tanh :: Tagged s a -> Tagged s a # asinh :: Tagged s a -> Tagged s a # acosh :: Tagged s a -> Tagged s a # atanh :: Tagged s a -> Tagged s a # log1p :: Tagged s a -> Tagged s a # expm1 :: Tagged s a -> Tagged s a # log1pexp :: Tagged s a -> Tagged s a # log1mexp :: Tagged s a -> Tagged s a #
Fractional a => Fractional (Tagged s a)
Instance details Defined in Data.Tagged Methods (/) :: Tagged s a -> Tagged s a -> Tagged s a # recip :: Tagged s a -> Tagged s a # fromRational :: Rational -> Tagged s a #
Integral a => Integral (Tagged s a)
Instance details Defined in Data.Tagged Methods quot :: Tagged s a -> Tagged s a -> Tagged s a # rem :: Tagged s a -> Tagged s a -> Tagged s a # div :: Tagged s a -> Tagged s a -> Tagged s a # mod :: Tagged s a -> Tagged s a -> Tagged s a # quotRem :: Tagged s a -> Tagged s a -> (Tagged s a, Tagged s a) # divMod :: Tagged s a -> Tagged s a -> (Tagged s a, Tagged s a) # toInteger :: Tagged s a -> Integer #
(Data s, Data b) => Data (Tagged s b)
Instance details Defined in Data.Tagged Methods gfoldl :: (forall d b0. Data d => c (d -> b0) -> d -> c b0) -> (forall g. g -> c g) -> Tagged s b -> c (Tagged s b) # gunfold :: (forall b0 r. Data b0 => c (b0 -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Tagged s b) # toConstr :: Tagged s b -> Constr # dataTypeOf :: Tagged s b -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (Tagged s b)) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Tagged s b)) # gmapT :: (forall b0. Data b0 => b0 -> b0) -> Tagged s b -> Tagged s b # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Tagged s b -> r # gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Tagged s b -> r # gmapQ :: (forall d. Data d => d -> u) -> Tagged s b -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> Tagged s b -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> Tagged s b -> m (Tagged s b) # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Tagged s b -> m (Tagged s b) # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Tagged s b -> m (Tagged s b) #
Num a => Num (Tagged s a)
Instance details Defined in Data.Tagged Methods (+) :: Tagged s a -> Tagged s a -> Tagged s a # (-) :: Tagged s a -> Tagged s a -> Tagged s a # (*) :: Tagged s a -> Tagged s a -> Tagged s a # negate :: Tagged s a -> Tagged s a # abs :: Tagged s a -> Tagged s a # signum :: Tagged s a -> Tagged s a # fromInteger :: Integer -> Tagged s a #
Ord b => Ord (Tagged s b)
Instance details Defined in Data.Tagged Methods compare :: Tagged s b -> Tagged s b -> Ordering # (<) :: Tagged s b -> Tagged s b -> Bool # (<=) :: Tagged s b -> Tagged s b -> Bool # (>) :: Tagged s b -> Tagged s b -> Bool # (>=) :: Tagged s b -> Tagged s b -> Bool # max :: Tagged s b -> Tagged s b -> Tagged s b # min :: Tagged s b -> Tagged s b -> Tagged s b #
Read b => Read (Tagged s b)
Instance details Defined in Data.Tagged Methods readsPrec :: Int -> ReadS (Tagged s b) # readList :: ReadS [Tagged s b] # readPrec :: ReadPrec (Tagged s b) # readListPrec :: ReadPrec [Tagged s b] #
Real a => Real (Tagged s a)
Instance details Defined in Data.Tagged Methods toRational :: Tagged s a -> Rational #
RealFloat a => RealFloat (Tagged s a)
Instance details Defined in Data.Tagged Methods floatRadix :: Tagged s a -> Integer # floatDigits :: Tagged s a -> Int # floatRange :: Tagged s a -> (Int, Int) # decodeFloat :: Tagged s a -> (Integer, Int) # encodeFloat :: Integer -> Int -> Tagged s a # exponent :: Tagged s a -> Int # significand :: Tagged s a -> Tagged s a # scaleFloat :: Int -> Tagged s a -> Tagged s a # isNaN :: Tagged s a -> Bool # isInfinite :: Tagged s a -> Bool # isDenormalized :: Tagged s a -> Bool # isNegativeZero :: Tagged s a -> Bool # isIEEE :: Tagged s a -> Bool # atan2 :: Tagged s a -> Tagged s a -> Tagged s a #
RealFrac a => RealFrac (Tagged s a)
Instance details Defined in Data.Tagged Methods properFraction :: Integral b => Tagged s a -> (b, Tagged s a) # truncate :: Integral b => Tagged s a -> b # round :: Integral b => Tagged s a -> b # ceiling :: Integral b => Tagged s a -> b # floor :: Integral b => Tagged s a -> b #
Show b => Show (Tagged s b)
Instance details Defined in Data.Tagged Methods showsPrec :: Int -> Tagged s b -> ShowS # show :: Tagged s b -> String # showList :: [Tagged s b] -> ShowS #
Ix b => Ix (Tagged s b)
Instance details Defined in Data.Tagged Methods range :: (Tagged s b, Tagged s b) -> [Tagged s b] # index :: (Tagged s b, Tagged s b) -> Tagged s b -> Int # unsafeIndex :: (Tagged s b, Tagged s b) -> Tagged s b -> Int # inRange :: (Tagged s b, Tagged s b) -> Tagged s b -> Bool # rangeSize :: (Tagged s b, Tagged s b) -> Int # unsafeRangeSize :: (Tagged s b, Tagged s b) -> Int #
IsString a => IsString (Tagged s a)
Instance details Defined in Data.Tagged Methods fromString :: String -> Tagged s a #
Generic (Tagged s b)
Instance details Defined in Data.Tagged Associated Types type Rep (Tagged s b) :: Type -> Type # Methods from :: Tagged s b -> Rep (Tagged s b) x # to :: Rep (Tagged s b) x -> Tagged s b #
Semigroup a => Semigroup (Tagged s a)
Instance details Defined in Data.Tagged Methods (<>) :: Tagged s a -> Tagged s a -> Tagged s a # sconcat :: NonEmpty (Tagged s a) -> Tagged s a # stimes :: Integral b => b -> Tagged s a -> Tagged s a #
(Semigroup a, Monoid a) => Monoid (Tagged s a)
Instance details Defined in Data.Tagged Methods mempty :: Tagged s a # mappend :: Tagged s a -> Tagged s a -> Tagged s a # mconcat :: [Tagged s a] -> Tagged s a #
Storable a => Storable (Tagged s a)
Instance details Defined in Data.Tagged Methods sizeOf :: Tagged s a -> Int # alignment :: Tagged s a -> Int # peekElemOff :: Ptr (Tagged s a) -> Int -> IO (Tagged s a) # pokeElemOff :: Ptr (Tagged s a) -> Int -> Tagged s a -> IO () # peekByteOff :: Ptr b -> Int -> IO (Tagged s a) # pokeByteOff :: Ptr b -> Int -> Tagged s a -> IO () # peek :: Ptr (Tagged s a) -> IO (Tagged s a) # poke :: Ptr (Tagged s a) -> Tagged s a -> IO () #
Bits a => Bits (Tagged s a)
Instance details Defined in Data.Tagged Methods (.&.) :: Tagged s a -> Tagged s a -> Tagged s a # (.\|.) :: Tagged s a -> Tagged s a -> Tagged s a # xor :: Tagged s a -> Tagged s a -> Tagged s a # complement :: Tagged s a -> Tagged s a # shift :: Tagged s a -> Int -> Tagged s a # rotate :: Tagged s a -> Int -> Tagged s a # zeroBits :: Tagged s a # bit :: Int -> Tagged s a # setBit :: Tagged s a -> Int -> Tagged s a # clearBit :: Tagged s a -> Int -> Tagged s a # complementBit :: Tagged s a -> Int -> Tagged s a # testBit :: Tagged s a -> Int -> Bool # bitSizeMaybe :: Tagged s a -> Maybe Int # bitSize :: Tagged s a -> Int # isSigned :: Tagged s a -> Bool # shiftL :: Tagged s a -> Int -> Tagged s a # unsafeShiftL :: Tagged s a -> Int -> Tagged s a # shiftR :: Tagged s a -> Int -> Tagged s a # unsafeShiftR :: Tagged s a -> Int -> Tagged s a # rotateL :: Tagged s a -> Int -> Tagged s a # rotateR :: Tagged s a -> Int -> Tagged s a # popCount :: Tagged s a -> Int #
FiniteBits a => FiniteBits (Tagged s a)
Instance details Defined in Data.Tagged Methods finiteBitSize :: Tagged s a -> Int # countLeadingZeros :: Tagged s a -> Int # countTrailingZeros :: Tagged s a -> Int #
NFData b => NFData (Tagged s b)
Instance details Defined in Data.Tagged Methods rnf :: Tagged s b -> () #
Finite a => Finite (Tagged b a) Source #
Instance details Defined in Data.Universe.Class Methods universeF :: [Tagged b a] Source # cardinality :: Tagged (Tagged b a) Natural Source #
Universe a => Universe (Tagged b a) Source #
Instance details Defined in Data.Universe.Class Methods universe :: [Tagged b a] Source #
type Rep1 (Tagged s :: Type -> Type)
Instance details Defined in Data.Tagged type Rep1 (Tagged s :: Type -> Type) = D1 ('MetaData "Tagged" "Data.Tagged" "tagged-0.8.6.1-A3E8I7zg2qBFE9O5vQsg6C" 'True) (C1 ('MetaCons "Tagged" 'PrefixI 'True) (S1 ('MetaSel ('Just "unTagged") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) Par1))
type Rep (Tagged s b)
Instance details Defined in Data.Tagged type Rep (Tagged s b) = D1 ('MetaData "Tagged" "Data.Tagged" "tagged-0.8.6.1-A3E8I7zg2qBFE9O5vQsg6C" 'True) (C1 ('MetaCons "Tagged" 'PrefixI 'True) (S1 ('MetaSel ('Just "unTagged") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 b)))

data Natural #

Type representing arbitrary-precision non-negative integers.

>>> 2^100 :: Natural
1267650600228229401496703205376

Operations whose result would be negative throw (Underflow :: ArithException),

>>> -1 :: Natural
*** Exception: arithmetic underflow

Since: base-4.8.0.0

Instances

Instances details

Enum Natural	Since: base-4.8.0.0
Instance details Defined in GHC.Enum Methods succ :: Natural -> Natural # pred :: Natural -> Natural # toEnum :: Int -> Natural # fromEnum :: Natural -> Int # enumFrom :: Natural -> [Natural] # enumFromThen :: Natural -> Natural -> [Natural] # enumFromTo :: Natural -> Natural -> [Natural] # enumFromThenTo :: Natural -> Natural -> Natural -> [Natural] #
Eq Natural	Since: base-4.8.0.0
Instance details Defined in GHC.Natural Methods (==) :: Natural -> Natural -> Bool # (/=) :: Natural -> Natural -> Bool #
Integral Natural	Since: base-4.8.0.0
Instance details Defined in GHC.Real Methods quot :: Natural -> Natural -> Natural # rem :: Natural -> Natural -> Natural # div :: Natural -> Natural -> Natural # mod :: Natural -> Natural -> Natural # quotRem :: Natural -> Natural -> (Natural, Natural) # divMod :: Natural -> Natural -> (Natural, Natural) # toInteger :: Natural -> Integer #
Num Natural	Note that `Natural`'s `Num` instance isn't a ring: no element but 0 has an additive inverse. It is a semiring though. Since: base-4.8.0.0
Instance details Defined in GHC.Num Methods (+) :: Natural -> Natural -> Natural # (-) :: Natural -> Natural -> Natural # (*) :: Natural -> Natural -> Natural # negate :: Natural -> Natural # abs :: Natural -> Natural # signum :: Natural -> Natural # fromInteger :: Integer -> Natural #
Ord Natural	Since: base-4.8.0.0
Instance details Defined in GHC.Natural Methods compare :: Natural -> Natural -> Ordering # (<) :: Natural -> Natural -> Bool # (<=) :: Natural -> Natural -> Bool # (>) :: Natural -> Natural -> Bool # (>=) :: Natural -> Natural -> Bool # max :: Natural -> Natural -> Natural # min :: Natural -> Natural -> Natural #
Read Natural	Since: base-4.8.0.0
Instance details Defined in GHC.Read Methods readsPrec :: Int -> ReadS Natural # readList :: ReadS [Natural] # readPrec :: ReadPrec Natural # readListPrec :: ReadPrec [Natural] #
Real Natural	Since: base-4.8.0.0
Instance details Defined in GHC.Real Methods toRational :: Natural -> Rational #
Show Natural	Since: base-4.8.0.0
Instance details Defined in GHC.Show Methods showsPrec :: Int -> Natural -> ShowS # show :: Natural -> String # showList :: [Natural] -> ShowS #
Universe Natural Source #
Instance details Defined in Data.Universe.Class Methods universe :: [Natural] Source #

Debugging

These functions exist primarily as a specification to test against.

unfairCartesianProduct :: (a -> b -> c) -> [a] -> [b] -> [c] Source #

Very unfair 2-way Cartesian product: same guarantee as the slightly unfair one, except that lower indices may occur as the fst part of the tuple exponentially more frequently.

unfairChoices :: [[a]] -> [[a]] Source #

Very unfair n-way Cartesian product: same guarantee as the slightly unfair one, but not as good in the same sense that the very unfair 2-way product is worse than the slightly unfair 2-way product.