-----------------------------------------------------------------------------
-- |
-- Module     : Algebra.Graph.Relation.Reflexive
-- Copyright  : (c) Andrey Mokhov 2016-2022
-- License    : MIT (see the file LICENSE)
-- Maintainer : [email protected]
-- Stability  : experimental
--
-- An abstract implementation of reflexive binary relations. Use
-- "Algebra.Graph.Class" for polymorphic construction and manipulation.
-----------------------------------------------------------------------------
module Algebra.Graph.Relation.Reflexive (
    -- * Data structure
    ReflexiveRelation, fromRelation, toRelation
    ) where

import Algebra.Graph.Relation
import Control.DeepSeq
import Data.String

import qualified Algebra.Graph.Class as C

{-| The 'ReflexiveRelation' data type represents a /reflexive binary relation/
over a set of elements. Reflexive relations satisfy all laws of the
'Reflexive' type class and, in particular, the /self-loop/ axiom:

@'vertex' x == 'vertex' x * 'vertex' x@

The 'Show' instance produces reflexively closed expressions:

@show (1     :: ReflexiveRelation Int) == "edge 1 1"
show (1 * 2 :: ReflexiveRelation Int) == "edges [(1,1),(1,2),(2,2)]"@
-}
newtype ReflexiveRelation a = ReflexiveRelation { forall a. ReflexiveRelation a -> Relation a
fromReflexive :: Relation a }
    deriving (String -> ReflexiveRelation a
forall a. IsString a => String -> ReflexiveRelation a
forall a. (String -> a) -> IsString a
fromString :: String -> ReflexiveRelation a
$cfromString :: forall a. IsString a => String -> ReflexiveRelation a
IsString, ReflexiveRelation a -> ()
forall a. NFData a => ReflexiveRelation a -> ()
forall a. (a -> ()) -> NFData a
rnf :: ReflexiveRelation a -> ()
$crnf :: forall a. NFData a => ReflexiveRelation a -> ()
NFData, Integer -> ReflexiveRelation a
ReflexiveRelation a -> ReflexiveRelation a
ReflexiveRelation a -> ReflexiveRelation a -> ReflexiveRelation a
forall a. (Ord a, Num a) => Integer -> ReflexiveRelation a
forall a.
(Ord a, Num a) =>
ReflexiveRelation a -> ReflexiveRelation a
forall a.
(Ord a, Num a) =>
ReflexiveRelation a -> ReflexiveRelation a -> ReflexiveRelation a
forall a.
(a -> a -> a)
-> (a -> a -> a)
-> (a -> a -> a)
-> (a -> a)
-> (a -> a)
-> (a -> a)
-> (Integer -> a)
-> Num a
fromInteger :: Integer -> ReflexiveRelation a
$cfromInteger :: forall a. (Ord a, Num a) => Integer -> ReflexiveRelation a
signum :: ReflexiveRelation a -> ReflexiveRelation a
$csignum :: forall a.
(Ord a, Num a) =>
ReflexiveRelation a -> ReflexiveRelation a
abs :: ReflexiveRelation a -> ReflexiveRelation a
$cabs :: forall a.
(Ord a, Num a) =>
ReflexiveRelation a -> ReflexiveRelation a
negate :: ReflexiveRelation a -> ReflexiveRelation a
$cnegate :: forall a.
(Ord a, Num a) =>
ReflexiveRelation a -> ReflexiveRelation a
* :: ReflexiveRelation a -> ReflexiveRelation a -> ReflexiveRelation a
$c* :: forall a.
(Ord a, Num a) =>
ReflexiveRelation a -> ReflexiveRelation a -> ReflexiveRelation a
- :: ReflexiveRelation a -> ReflexiveRelation a -> ReflexiveRelation a
$c- :: forall a.
(Ord a, Num a) =>
ReflexiveRelation a -> ReflexiveRelation a -> ReflexiveRelation a
+ :: ReflexiveRelation a -> ReflexiveRelation a -> ReflexiveRelation a
$c+ :: forall a.
(Ord a, Num a) =>
ReflexiveRelation a -> ReflexiveRelation a -> ReflexiveRelation a
Num)

instance Ord a => Eq (ReflexiveRelation a) where
    ReflexiveRelation a
x == :: ReflexiveRelation a -> ReflexiveRelation a -> Bool
== ReflexiveRelation a
y = forall a. Ord a => ReflexiveRelation a -> Relation a
toRelation ReflexiveRelation a
x forall a. Eq a => a -> a -> Bool
== forall a. Ord a => ReflexiveRelation a -> Relation a
toRelation ReflexiveRelation a
y

instance Ord a => Ord (ReflexiveRelation a) where
    compare :: ReflexiveRelation a -> ReflexiveRelation a -> Ordering
compare ReflexiveRelation a
x ReflexiveRelation a
y = forall a. Ord a => a -> a -> Ordering
compare (forall a. Ord a => ReflexiveRelation a -> Relation a
toRelation ReflexiveRelation a
x) (forall a. Ord a => ReflexiveRelation a -> Relation a
toRelation ReflexiveRelation a
y)

instance (Ord a, Show a) => Show (ReflexiveRelation a) where
    show :: ReflexiveRelation a -> String
show = forall a. Show a => a -> String
show forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. Ord a => ReflexiveRelation a -> Relation a
toRelation

instance Ord a => C.Graph (ReflexiveRelation a) where
    type Vertex (ReflexiveRelation a) = a
    empty :: ReflexiveRelation a
empty       = forall a. Relation a -> ReflexiveRelation a
ReflexiveRelation forall a. Relation a
empty
    vertex :: Vertex (ReflexiveRelation a) -> ReflexiveRelation a
vertex      = forall a. Relation a -> ReflexiveRelation a
ReflexiveRelation forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. a -> Relation a
vertex
    overlay :: ReflexiveRelation a -> ReflexiveRelation a -> ReflexiveRelation a
overlay ReflexiveRelation a
x ReflexiveRelation a
y = forall a. Relation a -> ReflexiveRelation a
ReflexiveRelation forall a b. (a -> b) -> a -> b
$ forall a. ReflexiveRelation a -> Relation a
fromReflexive ReflexiveRelation a
x forall a. Ord a => Relation a -> Relation a -> Relation a
`overlay` forall a. ReflexiveRelation a -> Relation a
fromReflexive ReflexiveRelation a
y
    connect :: ReflexiveRelation a -> ReflexiveRelation a -> ReflexiveRelation a
connect ReflexiveRelation a
x ReflexiveRelation a
y = forall a. Relation a -> ReflexiveRelation a
ReflexiveRelation forall a b. (a -> b) -> a -> b
$ forall a. ReflexiveRelation a -> Relation a
fromReflexive ReflexiveRelation a
x forall a. Ord a => Relation a -> Relation a -> Relation a
`connect` forall a. ReflexiveRelation a -> Relation a
fromReflexive ReflexiveRelation a
y

instance Ord a => C.Reflexive (ReflexiveRelation a)

-- | Construct a reflexive relation from a 'Relation'.
-- Complexity: /O(1)/ time.
fromRelation :: Relation a -> ReflexiveRelation a
fromRelation :: forall a. Relation a -> ReflexiveRelation a
fromRelation = forall a. Relation a -> ReflexiveRelation a
ReflexiveRelation

-- | Extract the underlying relation.
-- Complexity: /O(n*log(m))/ time.
toRelation :: Ord a => ReflexiveRelation a -> Relation a
toRelation :: forall a. Ord a => ReflexiveRelation a -> Relation a
toRelation = forall a. Ord a => Relation a -> Relation a
reflexiveClosure forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. ReflexiveRelation a -> Relation a
fromReflexive