{-# 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 :: forall a b. (Ix a, Ix b) => Relation a b -> Relation b a
mirror Relation a b
r =
let ((a
a,b
b),(a
c,b
d)) = forall a b. (Ix a, Ix b) => Relation a b -> ((a, b), (a, b))
bounds Relation a b
r
in 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)) forall a b. (a -> b) -> a -> b
$ do (a
x,b
y) <- forall a b. (Ix a, Ix b) => Relation a b -> [(a, b)]
indices Relation a b
r ; forall (m :: * -> *) a. Monad m => a -> m a
return ((b
y,a
x), Relation a b
rforall 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 :: forall a b. (Ix a, Ix b) => Relation a b -> Relation a b
complement Relation a b
r =
forall a b.
(Ix a, Ix b) =>
((a, b), (a, b)) -> [((a, b), Bit)] -> Relation a b
build (forall a b. (Ix a, Ix b) => Relation a b -> ((a, b), (a, b))
bounds Relation a b
r) forall a b. (a -> b) -> a -> b
$ do (a, b)
i <- forall a b. (Ix a, Ix b) => Relation a b -> [(a, b)]
indices Relation a b
r ; forall (m :: * -> *) a. Monad m => a -> m a
return ( (a, b)
i, forall b. Boolean b => b -> b
not forall a b. (a -> b) -> a -> b
$ Relation a b
rforall 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 :: forall a b.
(Ix a, Ix b) =>
Relation a b -> Relation a b -> Relation a b
difference Relation a b
r Relation a b
s =
forall a b.
(Ix a, Ix b) =>
Relation a b -> Relation a b -> Relation a b
intersection Relation a b
r forall a b. (a -> b) -> 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 :: forall a b.
(Ix a, Ix b) =>
Relation a b -> Relation a b -> Relation a b
union Relation a b
r Relation a b
s = forall a b.
(Ix a, Ix b) =>
((a, b), (a, b)) -> [((a, b), Bit)] -> Relation a b
build ( forall a b. (Ix a, Ix b) => Relation a b -> ((a, b), (a, b))
bounds Relation a b
r ) forall a b. (a -> b) -> a -> b
$ do
(a, b)
i <- forall a b. (Ix a, Ix b) => Relation a b -> [(a, b)]
indices Relation a b
r
forall (m :: * -> *) a. Monad m => a -> m a
return ((a, b)
i, forall b (t :: * -> *). (Boolean b, Foldable t) => t b -> b
or [ Relation a b
rforall a b. (Ix a, Ix b) => Relation a b -> (a, b) -> Bit
!(a, b)
i, Relation a b
sforall 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 :: forall a b c.
(Ix a, Ix b, Ix c) =>
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)) = 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)) = 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 forall a b.
(Ix a, Ix b) =>
((a, b), (a, b)) -> [((a, b), Bit)] -> Relation a b
build ((a, c), (a, c))
bnd forall a b. (a -> b) -> a -> b
$ do
i :: (a, c)
i@(a
x,c
z) <- forall a. Ix a => (a, a) -> [a]
range ((a, c), (a, c))
bnd
forall (m :: * -> *) a. Monad m => a -> m a
return ((a, c)
i, forall b (t :: * -> *). (Boolean b, Foldable t) => t b -> b
or forall a b. (a -> b) -> a -> b
$ do
b
y <- forall a. Ix a => (a, a) -> [a]
range ( b
al, b
ar )
forall (m :: * -> *) a. Monad m => a -> m a
return forall a b. (a -> b) -> a -> b
$ forall b (t :: * -> *). (Boolean b, Foldable t) => t b -> b
and [ Relation a b
aforall a b. (Ix a, Ix b) => Relation a b -> (a, b) -> Bit
!(a
x,b
y), Relation b c
bforall 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 :: forall a. Ix a => Int -> Relation a a -> Relation a a
power Int
0 Relation a a
r = forall a. Ix a => ((a, a), (a, a)) -> Relation a a
identity ( 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) = forall a. Integral a => a -> a -> (a, a)
divMod Int
e Int
2
s :: Relation a a
s = forall a. Ix a => Int -> Relation a a -> Relation a a
power Int
d Relation a a
r
s2 :: Relation a a
s2 = 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
_ -> 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 :: forall a b.
(Ix a, Ix b) =>
Relation a b -> Relation a b -> Relation a b
intersection Relation a b
r Relation a b
s = forall a b.
(Ix a, Ix b) =>
((a, b), (a, b)) -> [((a, b), Bit)] -> Relation a b
build ( forall a b. (Ix a, Ix b) => Relation a b -> ((a, b), (a, b))
bounds Relation a b
r ) forall a b. (a -> b) -> a -> b
$ do
(a, b)
i <- forall a b. (Ix a, Ix b) => Relation a b -> [(a, b)]
indices Relation a b
r
forall (m :: * -> *) a. Monad m => a -> m a
return ((a, b)
i, forall b (t :: * -> *). (Boolean b, Foldable t) => t b -> b
and [ Relation a b
rforall a b. (Ix a, Ix b) => Relation a b -> (a, b) -> Bit
!(a, b)
i, Relation a b
sforall 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 :: forall a. Ix a => Relation a a -> Relation a a
reflexive_closure Relation a a
t =
forall a b.
(Ix a, Ix b) =>
Relation a b -> Relation a b -> Relation a b
union Relation a a
t forall a b. (a -> b) -> a -> b
$ forall a. Ix a => ((a, a), (a, a)) -> Relation a a
identity forall a b. (a -> b) -> a -> b
$ 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 :: forall a. Ix a => Relation a a -> Relation a a
symmetric_closure Relation a a
r =
forall a b.
(Ix a, Ix b) =>
Relation a b -> Relation a b -> Relation a b
union Relation a a
r forall a b. (a -> b) -> a -> b
$ forall a b. (Ix a, Ix b) => Relation a b -> Relation b a
mirror Relation a a
r