Data.PartialOrd

Partial orders

Comparisons in partial orders

data PartialOrdering

A data type representing relationships between two objects in a poset: they can be related (by EQ', LT' or GT'; like EQ, LT or GT), or unrelated (NT').

Constructors

EQ'

LT'

GT'

NT'

Instances

Monoid PartialOrdering

Defined in Data.PartialOrd

Methods

mempty :: PartialOrdering

mappend :: PartialOrdering -> PartialOrdering -> PartialOrdering

mconcat :: [PartialOrdering] -> PartialOrdering

Semigroup PartialOrdering

A comparison (less than or equal, greater than or equal) holds if and only if it does on both arguments.

Defined in Data.PartialOrd

Methods

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

sconcat :: NonEmpty PartialOrdering -> PartialOrdering

stimes :: Integral b => b -> PartialOrdering -> PartialOrdering

Show PartialOrdering

Defined in Data.PartialOrd

Methods

showsPrec :: Int -> PartialOrdering -> ShowS

show :: PartialOrdering -> String

showList :: [PartialOrdering] -> ShowS

Eq PartialOrdering

Defined in Data.PartialOrd

Methods

(==) :: PartialOrdering -> PartialOrdering -> Bool

(/=) :: PartialOrdering -> PartialOrdering -> Bool

fromOrd :: Ordering -> PartialOrdering

Convert an ordering into a partial ordering

toMaybeOrd :: PartialOrdering -> Maybe Ordering

Convert a partial ordering to an ordering

fromMaybeOrd :: Maybe Ordering -> PartialOrdering

Convert an ordering into a partial ordering

fromLeqGeq :: Bool -> Bool -> PartialOrdering

Convert from leq and geq to a partial ordering

fromCompare :: Ord a => a -> a -> PartialOrdering

Lift a compare to a compare`

Partial orderings

class PartialOrd a where

A typeclass expressing partially ordered types: any two elements are related by a PartialOrdering.

Minimal complete definition

compare' | leq

Methods

compare' :: a -> a -> PartialOrdering

leq :: a -> a -> Bool

geq :: a -> a -> Bool

Instances

PartialOrd IntSet

Defined in Data.PartialOrd

Methods

compare' :: IntSet -> IntSet -> PartialOrdering

leq :: IntSet -> IntSet -> Bool

geq :: IntSet -> IntSet -> Bool

PartialOrd Integer

It's hard to imagine another sensible instance

Defined in Data.PartialOrd

Methods

compare' :: Integer -> Integer -> PartialOrdering

leq :: Integer -> Integer -> Bool

geq :: Integer -> Integer -> Bool

PartialOrd ()

Defined in Data.PartialOrd

Methods

compare' :: () -> () -> PartialOrdering

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

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

PartialOrd Int

It's hard to imagine another sensible instance

Defined in Data.PartialOrd

Methods

compare' :: Int -> Int -> PartialOrdering

leq :: Int -> Int -> Bool

geq :: Int -> Int -> Bool

Ord a => PartialOrd (Set a)

Defined in Data.PartialOrd

Methods

compare' :: Set a -> Set a -> PartialOrdering

leq :: Set a -> Set a -> Bool

geq :: Set a -> Set a -> Bool

Eq a => PartialOrd (Discrete a)

Defined in Data.PartialOrd

Methods

compare' :: Discrete a -> Discrete a -> PartialOrdering

leq :: Discrete a -> Discrete a -> Bool

geq :: Discrete a -> Discrete a -> Bool

Ord a => PartialOrd (FullyOrd a)

Defined in Data.PartialOrd

Methods

compare' :: FullyOrd a -> FullyOrd a -> PartialOrdering

leq :: FullyOrd a -> FullyOrd a -> Bool

geq :: FullyOrd a -> FullyOrd a -> Bool

Eq a => PartialOrd (Infix a)

Defined in Data.PartialOrd

Methods

compare' :: Infix a -> Infix a -> PartialOrdering

leq :: Infix a -> Infix a -> Bool

geq :: Infix a -> Infix a -> Bool

Eq a => PartialOrd (Prefix a)

Defined in Data.PartialOrd

Methods

compare' :: Prefix a -> Prefix a -> PartialOrdering

leq :: Prefix a -> Prefix a -> Bool

geq :: Prefix a -> Prefix a -> Bool

Eq a => PartialOrd (Subseq a)

Defined in Data.PartialOrd

Methods

compare' :: Subseq a -> Subseq a -> PartialOrdering

leq :: Subseq a -> Subseq a -> Bool

geq :: Subseq a -> Subseq a -> Bool

Eq a => PartialOrd (Suffix a)

Defined in Data.PartialOrd

Methods

compare' :: Suffix a -> Suffix a -> PartialOrdering

leq :: Suffix a -> Suffix a -> Bool

geq :: Suffix a -> Suffix a -> Bool

(PartialOrd a, PartialOrd b) => PartialOrd (a, b)

This is equivalent to

  compare' (a,b) (c,d) = compare' a c <> compare' b d

but may be more efficient: if compare' a1 a2 is LT' or GT' we seek less information about b1 and b2 (and this can be faster).

Defined in Data.PartialOrd

Methods

compare' :: (a, b) -> (a, b) -> PartialOrdering

leq :: (a, b) -> (a, b) -> Bool

geq :: (a, b) -> (a, b) -> Bool

(PartialOrd a, PartialOrd b, PartialOrd c) => PartialOrd (a, b, c)

Defined in Data.PartialOrd

Methods

compare' :: (a, b, c) -> (a, b, c) -> PartialOrdering

leq :: (a, b, c) -> (a, b, c) -> Bool

geq :: (a, b, c) -> (a, b, c) -> Bool

(PartialOrd a, PartialOrd b, PartialOrd c, PartialOrd d) => PartialOrd (a, b, c, d)

Defined in Data.PartialOrd

Methods

compare' :: (a, b, c, d) -> (a, b, c, d) -> PartialOrdering

leq :: (a, b, c, d) -> (a, b, c, d) -> Bool

geq :: (a, b, c, d) -> (a, b, c, d) -> Bool

(PartialOrd a, PartialOrd b, PartialOrd c, PartialOrd d, PartialOrd e) => PartialOrd (a, b, c, d, e)

Defined in Data.PartialOrd

Methods

compare' :: (a, b, c, d, e) -> (a, b, c, d, e) -> PartialOrdering

leq :: (a, b, c, d, e) -> (a, b, c, d, e) -> Bool

geq :: (a, b, c, d, e) -> (a, b, c, d, e) -> Bool

comparable :: PartialOrd a => a -> a -> Bool

Are they LT', EQ', GT'

Special partial orderings

newtype FullyOrd a

A helper type for constructing partial orderings from total orderings (using deriving via)

Constructors

FullyOrd

Fields

• getOrd :: a

Instances

Show a => Show (FullyOrd a)

Defined in Data.PartialOrd

Methods

showsPrec :: Int -> FullyOrd a -> ShowS

show :: FullyOrd a -> String

showList :: [FullyOrd a] -> ShowS

Eq a => Eq (FullyOrd a)

Defined in Data.PartialOrd

Methods

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

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

Ord a => Ord (FullyOrd a)

Defined in Data.PartialOrd

Methods

compare :: FullyOrd a -> FullyOrd a -> Ordering

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

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

(>) :: FullyOrd a -> FullyOrd a -> Bool

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

max :: FullyOrd a -> FullyOrd a -> FullyOrd a

min :: FullyOrd a -> FullyOrd a -> FullyOrd a

Ord a => PartialOrd (FullyOrd a)

Defined in Data.PartialOrd

Methods

compare' :: FullyOrd a -> FullyOrd a -> PartialOrdering

leq :: FullyOrd a -> FullyOrd a -> Bool

geq :: FullyOrd a -> FullyOrd a -> Bool

newtype Discrete a

A helper type for constructing partial orderings where everything is equal or incomparable.

Constructors

Discrete

Fields

• getDiscrete :: a

Instances

Show a => Show (Discrete a)

Defined in Data.PartialOrd

Methods

showsPrec :: Int -> Discrete a -> ShowS

show :: Discrete a -> String

showList :: [Discrete a] -> ShowS

Eq a => Eq (Discrete a)

Defined in Data.PartialOrd

Methods

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

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

Eq a => PartialOrd (Discrete a)

Defined in Data.PartialOrd

Methods

compare' :: Discrete a -> Discrete a -> PartialOrdering

leq :: Discrete a -> Discrete a -> Bool

geq :: Discrete a -> Discrete a -> Bool

Maxima and minima

newtype Maxima a

Sets of incomparable elements, with a monoidal structure obtained by taking the maximal ones.

Unfortunately, we need a full ordering for these to work (since they use sets), though we don't assume this ordering has any compatibility with the partial order. The monoid structures are most efficient with pre-reduced sets as the left-hand argument.

Constructors

Maxima

Fields

• maxSet :: Set a

Instances

(Ord a, PartialOrd a) => Monoid (Maxima a)

Defined in Data.PartialOrd

Methods

mempty :: Maxima a

mappend :: Maxima a -> Maxima a -> Maxima a

mconcat :: [Maxima a] -> Maxima a

(Ord a, PartialOrd a) => Semigroup (Maxima a)

Defined in Data.PartialOrd

Methods

(<>) :: Maxima a -> Maxima a -> Maxima a

sconcat :: NonEmpty (Maxima a) -> Maxima a

stimes :: Integral b => b -> Maxima a -> Maxima a

maxima :: (Ord a, PartialOrd a) => [a] -> [a]

Find the maxima of a list (passing it through the machinery above)

newtype Minima a

As above, but with minima

Constructors

Minima

Fields

• minSet :: Set a

Instances

(Ord a, PartialOrd a) => Monoid (Minima a)

Defined in Data.PartialOrd

Methods

mempty :: Minima a

mappend :: Minima a -> Minima a -> Minima a

mconcat :: [Minima a] -> Minima a

(Ord a, PartialOrd a) => Semigroup (Minima a)

Defined in Data.PartialOrd

Methods

(<>) :: Minima a -> Minima a -> Minima a

sconcat :: NonEmpty (Minima a) -> Minima a

stimes :: Integral b => b -> Minima a -> Minima a

minima :: (Ord a, PartialOrd a) => [a] -> [a]

Find the minima of a list (passing it through the machinery above)

Partial orders on lists

newtype Infix a

Lists partially ordered by infix inclusion

Constructors

Infix

Fields

• unInfix :: [a]

Instances

Show a => Show (Infix a)

Defined in Data.PartialOrd

Methods

showsPrec :: Int -> Infix a -> ShowS

show :: Infix a -> String

showList :: [Infix a] -> ShowS

Eq a => Eq (Infix a)

Defined in Data.PartialOrd

Methods

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

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

Eq a => PartialOrd (Infix a)

Defined in Data.PartialOrd

Methods

compare' :: Infix a -> Infix a -> PartialOrdering

leq :: Infix a -> Infix a -> Bool

geq :: Infix a -> Infix a -> Bool

newtype Prefix a

Lists partially ordered by prefix inclusion

Constructors

Prefix

Fields

• unPrefix :: [a]

Instances

Show a => Show (Prefix a)

Defined in Data.PartialOrd

Methods

showsPrec :: Int -> Prefix a -> ShowS

show :: Prefix a -> String

showList :: [Prefix a] -> ShowS

Eq a => Eq (Prefix a)

Defined in Data.PartialOrd

Methods

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

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

Eq a => PartialOrd (Prefix a)

Defined in Data.PartialOrd

Methods

compare' :: Prefix a -> Prefix a -> PartialOrdering

leq :: Prefix a -> Prefix a -> Bool

geq :: Prefix a -> Prefix a -> Bool

newtype Suffix a

Lists partially ordered by suffix inclusion

Constructors

Suffix

Fields

• unSuffix :: [a]

Instances

Show a => Show (Suffix a)

Defined in Data.PartialOrd

Methods

showsPrec :: Int -> Suffix a -> ShowS

show :: Suffix a -> String

showList :: [Suffix a] -> ShowS

Eq a => Eq (Suffix a)

Defined in Data.PartialOrd

Methods

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

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

Eq a => PartialOrd (Suffix a)

Defined in Data.PartialOrd

Methods

compare' :: Suffix a -> Suffix a -> PartialOrdering

leq :: Suffix a -> Suffix a -> Bool

geq :: Suffix a -> Suffix a -> Bool

newtype Subseq a

Lists partially ordered by the subsequence relation

Constructors

Subseq

Fields

• unSubseq :: [a]

Instances

Show a => Show (Subseq a)

Defined in Data.PartialOrd

Methods

showsPrec :: Int -> Subseq a -> ShowS

show :: Subseq a -> String

showList :: [Subseq a] -> ShowS

Eq a => Eq (Subseq a)

Defined in Data.PartialOrd

Methods

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

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

Eq a => PartialOrd (Subseq a)

Defined in Data.PartialOrd

Methods

compare' :: Subseq a -> Subseq a -> PartialOrdering

leq :: Subseq a -> Subseq a -> Bool

geq :: Subseq a -> Subseq a -> Bool