{-# language FlexibleInstances, MultiParamTypeClasses #-}

module Ersatz.Relation.Op

( mirror
, union
, complement
, difference
, product, power
, intersection
, reflexive_closure
, symmetric_closure
)

where

import Ersatz.Relation.Data

import Prelude hiding ( and, or, not, product )
import Ersatz.Bit (and, or, not)

import Data.Ix

mirror :: ( Ix a , Ix b ) => Relation a b -> Relation b a
mirror :: Relation a b -> Relation b a
mirror Relation a b
r =
    let ((a
a,b
b),(a
c,b
d)) = Relation a b -> ((a, b), (a, b))
forall a b. (Ix a, Ix b) => Relation a b -> ((a, b), (a, b))
bounds Relation a b
r
    in  ((b, a), (b, a)) -> [((b, a), Bit)] -> Relation b a
forall a b.
(Ix a, Ix b) =>
((a, b), (a, b)) -> [((a, b), Bit)] -> Relation a b
build ((b
b,a
a),(b
d,a
c)) ([((b, a), Bit)] -> Relation b a)
-> [((b, a), Bit)] -> Relation b a
forall a b. (a -> b) -> a -> b
$ do (a
x,b
y) <- Relation a b -> [(a, b)]
forall a b. (Ix a, Ix b) => Relation a b -> [(a, b)]
indices Relation a b
r ; ((b, a), Bit) -> [((b, a), Bit)]
forall (m :: * -> *) a. Monad m => a -> m a
return ((b
y,a
x), Relation a b
rRelation a b -> (a, b) -> Bit
forall a b. (Ix a, Ix b) => Relation a b -> (a, b) -> Bit
!(a
x,b
y))

complement :: ( Ix a , Ix b ) => Relation a b -> Relation a b
complement :: Relation a b -> Relation a b
complement Relation a b
r =
    ((a, b), (a, b)) -> [((a, b), Bit)] -> Relation a b
forall a b.
(Ix a, Ix b) =>
((a, b), (a, b)) -> [((a, b), Bit)] -> Relation a b
build (Relation a b -> ((a, b), (a, b))
forall a b. (Ix a, Ix b) => Relation a b -> ((a, b), (a, b))
bounds Relation a b
r) ([((a, b), Bit)] -> Relation a b)
-> [((a, b), Bit)] -> Relation a b
forall a b. (a -> b) -> a -> b
$ do (a, b)
i <- Relation a b -> [(a, b)]
forall a b. (Ix a, Ix b) => Relation a b -> [(a, b)]
indices Relation a b
r ; ((a, b), Bit) -> [((a, b), Bit)]
forall (m :: * -> *) a. Monad m => a -> m a
return ( (a, b)
i, Bit -> Bit
forall b. Boolean b => b -> b
not (Bit -> Bit) -> Bit -> Bit
forall a b. (a -> b) -> a -> b
$ Relation a b
rRelation a b -> (a, b) -> Bit
forall a b. (Ix a, Ix b) => Relation a b -> (a, b) -> Bit
!(a, b)
i )

difference :: ( Ix a , Ix b )
        => Relation a b -> Relation a b ->  Relation a b
difference :: Relation a b -> Relation a b -> Relation a b
difference Relation a b
r Relation a b
s =
    Relation a b -> Relation a b -> Relation a b
forall a b.
(Ix a, Ix b) =>
Relation a b -> Relation a b -> Relation a b
intersection Relation a b
r (Relation a b -> Relation a b) -> Relation a b -> Relation a b
forall a b. (a -> b) -> a -> b
$ Relation a b -> Relation a b
forall a b. (Ix a, Ix b) => Relation a b -> Relation a b
complement Relation a b
s

union :: ( Ix a , Ix b )
        => Relation a b -> Relation a b ->  Relation a b
union :: Relation a b -> Relation a b -> Relation a b
union Relation a b
r Relation a b
s =  ((a, b), (a, b)) -> [((a, b), Bit)] -> Relation a b
forall a b.
(Ix a, Ix b) =>
((a, b), (a, b)) -> [((a, b), Bit)] -> Relation a b
build ( Relation a b -> ((a, b), (a, b))
forall a b. (Ix a, Ix b) => Relation a b -> ((a, b), (a, b))
bounds Relation a b
r ) ([((a, b), Bit)] -> Relation a b)
-> [((a, b), Bit)] -> Relation a b
forall a b. (a -> b) -> a -> b
$ do
    (a, b)
i <- Relation a b -> [(a, b)]
forall a b. (Ix a, Ix b) => Relation a b -> [(a, b)]
indices Relation a b
r
    ((a, b), Bit) -> [((a, b), Bit)]
forall (m :: * -> *) a. Monad m => a -> m a
return ((a, b)
i, [Bit] -> Bit
forall b (t :: * -> *). (Boolean b, Foldable t) => t b -> b
or [ Relation a b
rRelation a b -> (a, b) -> Bit
forall a b. (Ix a, Ix b) => Relation a b -> (a, b) -> Bit
!(a, b)
i, Relation a b
sRelation a b -> (a, b) -> Bit
forall a b. (Ix a, Ix b) => Relation a b -> (a, b) -> Bit
!(a, b)
i ] )

product :: ( Ix a , Ix b, Ix c )
        => Relation a b -> Relation b c ->  Relation a c
product :: Relation a b -> Relation b c -> Relation a c
product Relation a b
a Relation b c
b =
    let ((a
ao,b
al),(a
au,b
ar)) = Relation a b -> ((a, b), (a, b))
forall a b. (Ix a, Ix b) => Relation a b -> ((a, b), (a, b))
bounds Relation a b
a
        ((b
_ ,c
bl),(b
_ ,c
br)) = Relation b c -> ((b, c), (b, c))
forall a b. (Ix a, Ix b) => Relation a b -> ((a, b), (a, b))
bounds Relation b c
b
        bnd :: ((a, c), (a, c))
bnd = ((a
ao,c
bl),(a
au,c
br))
    in  ((a, c), (a, c)) -> [((a, c), Bit)] -> Relation a c
forall a b.
(Ix a, Ix b) =>
((a, b), (a, b)) -> [((a, b), Bit)] -> Relation a b
build ((a, c), (a, c))
bnd ([((a, c), Bit)] -> Relation a c)
-> [((a, c), Bit)] -> Relation a c
forall a b. (a -> b) -> a -> b
$ do
          i :: (a, c)
i@(a
x,c
z) <- ((a, c), (a, c)) -> [(a, c)]
forall a. Ix a => (a, a) -> [a]
range ((a, c), (a, c))
bnd
          ((a, c), Bit) -> [((a, c), Bit)]
forall (m :: * -> *) a. Monad m => a -> m a
return ((a, c)
i, [Bit] -> Bit
forall b (t :: * -> *). (Boolean b, Foldable t) => t b -> b
or ([Bit] -> Bit) -> [Bit] -> Bit
forall a b. (a -> b) -> a -> b
$ do
                b
y <- (b, b) -> [b]
forall a. Ix a => (a, a) -> [a]
range ( b
al, b
ar )
                Bit -> [Bit]
forall (m :: * -> *) a. Monad m => a -> m a
return (Bit -> [Bit]) -> Bit -> [Bit]
forall a b. (a -> b) -> a -> b
$ [Bit] -> Bit
forall b (t :: * -> *). (Boolean b, Foldable t) => t b -> b
and [ Relation a b
aRelation a b -> (a, b) -> Bit
forall a b. (Ix a, Ix b) => Relation a b -> (a, b) -> Bit
!(a
x,b
y), Relation b c
bRelation b c -> (b, c) -> Bit
forall a b. (Ix a, Ix b) => Relation a b -> (a, b) -> Bit
!(b
y,c
z) ]
                )

power  :: ( Ix a  )
        => Int -> Relation a a -> Relation a a
power :: Int -> Relation a a -> Relation a a
power Int
0 Relation a a
r = ((a, a), (a, a)) -> Relation a a
forall a. Ix a => ((a, a), (a, a)) -> Relation a a
identity ( Relation a a -> ((a, a), (a, a))
forall a b. (Ix a, Ix b) => Relation a b -> ((a, b), (a, b))
bounds Relation a a
r )
power Int
1 Relation a a
r = Relation a a
r
power Int
e Relation a a
r =
    let (Int
d,Int
m) = Int -> Int -> (Int, Int)
forall a. Integral a => a -> a -> (a, a)
divMod Int
e Int
2
        s :: Relation a a
s = Int -> Relation a a -> Relation a a
forall a. Ix a => Int -> Relation a a -> Relation a a
power Int
d Relation a a
r
        s2 :: Relation a a
s2 = Relation a a -> Relation a a -> Relation a a
forall a b c.
(Ix a, Ix b, Ix c) =>
Relation a b -> Relation b c -> Relation a c
product Relation a a
s Relation a a
s
    in  case Int
m of
        Int
0 -> Relation a a
s2
        Int
_ -> Relation a a -> Relation a a -> Relation a a
forall a b c.
(Ix a, Ix b, Ix c) =>
Relation a b -> Relation b c -> Relation a c
product Relation a a
s2 Relation a a
r

intersection :: ( Ix a , Ix b)
      => Relation a b -> Relation a b
      -> Relation a b
intersection :: Relation a b -> Relation a b -> Relation a b
intersection Relation a b
r Relation a b
s = ((a, b), (a, b)) -> [((a, b), Bit)] -> Relation a b
forall a b.
(Ix a, Ix b) =>
((a, b), (a, b)) -> [((a, b), Bit)] -> Relation a b
build ( Relation a b -> ((a, b), (a, b))
forall a b. (Ix a, Ix b) => Relation a b -> ((a, b), (a, b))
bounds Relation a b
r ) ([((a, b), Bit)] -> Relation a b)
-> [((a, b), Bit)] -> Relation a b
forall a b. (a -> b) -> a -> b
$ do
        (a, b)
i <- Relation a b -> [(a, b)]
forall a b. (Ix a, Ix b) => Relation a b -> [(a, b)]
indices Relation a b
r
        ((a, b), Bit) -> [((a, b), Bit)]
forall (m :: * -> *) a. Monad m => a -> m a
return ((a, b)
i, [Bit] -> Bit
forall b (t :: * -> *). (Boolean b, Foldable t) => t b -> b
and [ Relation a b
rRelation a b -> (a, b) -> Bit
forall a b. (Ix a, Ix b) => Relation a b -> (a, b) -> Bit
!(a, b)
i, Relation a b
sRelation a b -> (a, b) -> Bit
forall a b. (Ix a, Ix b) => Relation a b -> (a, b) -> Bit
!(a, b)
i ] )

reflexive_closure :: Ix a => Relation a a -> Relation a a
reflexive_closure :: Relation a a -> Relation a a
reflexive_closure Relation a a
t =
    Relation a a -> Relation a a -> Relation a a
forall a b.
(Ix a, Ix b) =>
Relation a b -> Relation a b -> Relation a b
union Relation a a
t (Relation a a -> Relation a a) -> Relation a a -> Relation a a
forall a b. (a -> b) -> a -> b
$ ((a, a), (a, a)) -> Relation a a
forall a. Ix a => ((a, a), (a, a)) -> Relation a a
identity (((a, a), (a, a)) -> Relation a a)
-> ((a, a), (a, a)) -> Relation a a
forall a b. (a -> b) -> a -> b
$ Relation a a -> ((a, a), (a, a))
forall a b. (Ix a, Ix b) => Relation a b -> ((a, b), (a, b))
bounds Relation a a
t

symmetric_closure :: Ix a => Relation a a -> Relation a a
symmetric_closure :: Relation a a -> Relation a a
symmetric_closure Relation a a
r =
    Relation a a -> Relation a a -> Relation a a
forall a b.
(Ix a, Ix b) =>
Relation a b -> Relation a b -> Relation a b
union Relation a a
r (Relation a a -> Relation a a) -> Relation a a -> Relation a a
forall a b. (a -> b) -> a -> b
$ Relation a a -> Relation a a
forall a b. (Ix a, Ix b) => Relation a b -> Relation b a
mirror Relation a a
r