{-# LANGUAGE DerivingStrategies #-}
{-# LANGUAGE UndecidableInstances #-}

{- |
This module provides the data-type 'Relation' for encoding binary relations as SMT-problems.

==== __Example__

@
problem :: MonadSMT s m => StateT s m (Relation Int Int)
    setLogic \"QF_LIA\"

    r <- relation ((2,1), (6,5))

    forM_ (image r \<$\> domain r) (assert . exactly \@IntSort 1)
    forM_ (preimage r \<$\> codomain r) (assert . exactly \@IntSort 1)

    assertMaybe $ do
      item <- r^?ix (3,3)
      return $ item === true

    return r
@
-}
module Language.Hasmtlib.Type.Relation
(
  -- * Type
  Relation(..)

  -- * Construction
, relation, symmetric_relation, build, buildFrom, buildFromM, identity

  -- * Accessors
, (!?), (!)
, bounds, indices, elems, assocs
, domain, codomain, image, preimage

  -- * Pretty printing
, table
)
where

import Prelude hiding (and, (&&), any)
import Language.Hasmtlib.Type.MonadSMT
import Language.Hasmtlib.Type.SMTSort
import Language.Hasmtlib.Type.Expr
import Language.Hasmtlib.Boolean
import Language.Hasmtlib.Codec
import Data.Coerce
import Data.Array (Array, Ix(..))
import Data.Maybe
import qualified Data.Array as A
import Control.Monad
import Control.Lens hiding (indices)

-- | @Relation a b@ represents a binary relation \(R \subseteq A \times B \),
-- where the domain \(A\) is a finite subset of the type @a@,
-- and the codomain \(B\) is a finite subset of the type @b@.
--
-- A relation is stored internally as @Array (a,b) Expr BoolSort@,
-- so @a@ and @b@ have to be instances of 'Ix',
-- and both \(A\) and \(B\) are intervals.
newtype Relation a b = Relation (Array (a, b) (Expr BoolSort))

instance (Ix a, Ix b) => Codec (Relation a b) where
  type Decoded (Relation a b) = Array (a, b) Bool
  decode :: Solution -> Relation a b -> Maybe (Decoded (Relation a b))
decode Solution
s (Relation Array (a, b) (Expr 'BoolSort)
x) = Solution
-> Array (a, b) (Expr 'BoolSort)
-> Maybe (Decoded (Array (a, b) (Expr 'BoolSort)))
forall a. Codec a => Solution -> a -> Maybe (Decoded a)
decode Solution
s Array (a, b) (Expr 'BoolSort)
x
  encode :: Decoded (Relation a b) -> Relation a b
encode Decoded (Relation a b)
x = Array (a, b) (Expr 'BoolSort) -> Relation a b
forall a b. Array (a, b) (Expr 'BoolSort) -> Relation a b
Relation (Array (a, b) (Expr 'BoolSort) -> Relation a b)
-> Array (a, b) (Expr 'BoolSort) -> Relation a b
forall a b. (a -> b) -> a -> b
$ Decoded (Array (a, b) (Expr 'BoolSort))
-> Array (a, b) (Expr 'BoolSort)
forall a. Codec a => Decoded a -> a
encode Decoded (Array (a, b) (Expr 'BoolSort))
Decoded (Relation a b)
x

instance (Ix a, Ix b, a ~ c, b ~ d) => Each (Relation a b) (Relation c d) (Expr BoolSort) (Expr BoolSort) where
  each :: Traversal
  (Relation a b) (Relation c d) (Expr 'BoolSort) (Expr 'BoolSort)
each Expr 'BoolSort -> f (Expr 'BoolSort)
f (Relation Array (a, b) (Expr 'BoolSort)
arr) = Array (c, d) (Expr 'BoolSort) -> Relation c d
forall a b. Coercible a b => a -> b
coerce (Array (c, d) (Expr 'BoolSort) -> Relation c d)
-> f (Array (c, d) (Expr 'BoolSort)) -> f (Relation c d)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> (Expr 'BoolSort -> f (Expr 'BoolSort))
-> Array (a, b) (Expr 'BoolSort)
-> f (Array (c, d) (Expr 'BoolSort))
forall s t a b. Each s t a b => Traversal s t a b
Traversal
  (Array (a, b) (Expr 'BoolSort))
  (Array (c, d) (Expr 'BoolSort))
  (Expr 'BoolSort)
  (Expr 'BoolSort)
each Expr 'BoolSort -> f (Expr 'BoolSort)
f Array (a, b) (Expr 'BoolSort)
arr
  {-# INLINE each #-}

type instance Index (Relation a b) = (a,b)
type instance IxValue (Relation a b) = Expr BoolSort
instance (Ix a, Ix b) => Ixed (Relation a b) where
  ix :: Index (Relation a b)
-> Traversal' (Relation a b) (IxValue (Relation a b))
ix Index (Relation a b)
i IxValue (Relation a b) -> f (IxValue (Relation a b))
f (Relation Array (a, b) (Expr 'BoolSort)
arr) = Array (a, b) (Expr 'BoolSort) -> Relation a b
forall a b. Coercible a b => a -> b
coerce (Array (a, b) (Expr 'BoolSort) -> Relation a b)
-> f (Array (a, b) (Expr 'BoolSort)) -> f (Relation a b)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> Index (Array (a, b) (Expr 'BoolSort))
-> Traversal'
     (Array (a, b) (Expr 'BoolSort))
     (IxValue (Array (a, b) (Expr 'BoolSort)))
forall m. Ixed m => Index m -> Traversal' m (IxValue m)
ix Index (Array (a, b) (Expr 'BoolSort))
Index (Relation a b)
i IxValue (Array (a, b) (Expr 'BoolSort))
-> f (IxValue (Array (a, b) (Expr 'BoolSort)))
IxValue (Relation a b) -> f (IxValue (Relation a b))
f Array (a, b) (Expr 'BoolSort)
arr
  {-# INLINE ix #-}

-- | @relation ((amin,bmin),(amax,mbax))@ constructs an indeterminate relation \(R \subseteq A \times B \)
-- where \(A\) is @{amin .. amax}@ and \(B\) is @{bmin .. bmax}@.
relation :: (Ix a, Ix b, MonadSMT s m) =>
  ((a,b),(a,b))
  -> m (Relation a b)
relation :: forall a b s (m :: * -> *).
(Ix a, Ix b, MonadSMT s m) =>
((a, b), (a, b)) -> m (Relation a b)
relation ((a, b), (a, b))
bnd = do
    [((a, b), Expr 'BoolSort)]
pairs <- [m ((a, b), Expr 'BoolSort)] -> m [((a, b), Expr 'BoolSort)]
forall (t :: * -> *) (m :: * -> *) a.
(Traversable t, Monad m) =>
t (m a) -> m (t a)
forall (m :: * -> *) a. Monad m => [m a] -> m [a]
sequence ([m ((a, b), Expr 'BoolSort)] -> m [((a, b), Expr 'BoolSort)])
-> [m ((a, b), Expr 'BoolSort)] -> m [((a, b), Expr 'BoolSort)]
forall a b. (a -> b) -> a -> b
$ do
        (a, b)
p <- ((a, b), (a, b)) -> [(a, b)]
forall a. Ix a => (a, a) -> [a]
A.range ((a, b), (a, b))
bnd
        m ((a, b), Expr 'BoolSort) -> [m ((a, b), Expr 'BoolSort)]
forall a. a -> [a]
forall (m :: * -> *) a. Monad m => a -> m a
return (m ((a, b), Expr 'BoolSort) -> [m ((a, b), Expr 'BoolSort)])
-> m ((a, b), Expr 'BoolSort) -> [m ((a, b), Expr 'BoolSort)]
forall a b. (a -> b) -> a -> b
$ do
            Expr 'BoolSort
x <- m (Expr 'BoolSort)
forall (t :: SMTSort) s (m :: * -> *).
(KnownSMTSort t, MonadSMT s m) =>
m (Expr t)
var
            ((a, b), Expr 'BoolSort) -> m ((a, b), Expr 'BoolSort)
forall a. a -> m a
forall (m :: * -> *) a. Monad m => a -> m a
return ((a, b)
p, Expr 'BoolSort
x)
    Relation a b -> m (Relation a b)
forall a. a -> m a
forall (m :: * -> *) a. Monad m => a -> m a
return (Relation a b -> m (Relation a b))
-> Relation a b -> m (Relation a b)
forall a b. (a -> b) -> a -> b
$ ((a, b), (a, b)) -> [((a, b), Expr 'BoolSort)] -> Relation a b
forall a b.
(Ix a, Ix b) =>
((a, b), (a, b)) -> [((a, b), Expr 'BoolSort)] -> Relation a b
build ((a, b), (a, b))
bnd [((a, b), Expr 'BoolSort)]
pairs

-- | Constructs an indeterminate relation \(R \subseteq B \times B \)
-- that is symmetric, i.e., \(\forall x, y \in B: ((x,y) \in R) \rightarrow ((y,x) \in R) \).
symmetric_relation ::
  (MonadSMT s m, Ix b) =>
  ((b, b), (b, b)) -- ^ Since a symmetric relation must be homogeneous, the domain must equal the codomain.
                   -- Therefore, given bounds @((p,q),(r,s))@, it must hold that @p=q@ and @r=s@.
  -> m (Relation b b)
symmetric_relation :: forall s (m :: * -> *) b.
(MonadSMT s m, Ix b) =>
((b, b), (b, b)) -> m (Relation b b)
symmetric_relation ((b, b), (b, b))
bnd = do
    [[((b, b), Expr 'BoolSort)]]
pairs <- [m [((b, b), Expr 'BoolSort)]] -> m [[((b, b), Expr 'BoolSort)]]
forall (t :: * -> *) (m :: * -> *) a.
(Traversable t, Monad m) =>
t (m a) -> m (t a)
forall (m :: * -> *) a. Monad m => [m a] -> m [a]
sequence ([m [((b, b), Expr 'BoolSort)]] -> m [[((b, b), Expr 'BoolSort)]])
-> [m [((b, b), Expr 'BoolSort)]] -> m [[((b, b), Expr 'BoolSort)]]
forall a b. (a -> b) -> a -> b
$ do
        (b
p,b
q) <- ((b, b), (b, b)) -> [(b, b)]
forall a. Ix a => (a, a) -> [a]
A.range ((b, b), (b, b))
bnd
        Bool -> [()]
forall (f :: * -> *). Alternative f => Bool -> f ()
guard (Bool -> [()]) -> Bool -> [()]
forall a b. (a -> b) -> a -> b
$ b
p b -> b -> Bool
forall a. Ord a => a -> a -> Bool
<= b
q
        m [((b, b), Expr 'BoolSort)] -> [m [((b, b), Expr 'BoolSort)]]
forall a. a -> [a]
forall (m :: * -> *) a. Monad m => a -> m a
return (m [((b, b), Expr 'BoolSort)] -> [m [((b, b), Expr 'BoolSort)]])
-> m [((b, b), Expr 'BoolSort)] -> [m [((b, b), Expr 'BoolSort)]]
forall a b. (a -> b) -> a -> b
$ do
            Expr 'BoolSort
x <- m (Expr 'BoolSort)
forall (t :: SMTSort) s (m :: * -> *).
(KnownSMTSort t, MonadSMT s m) =>
m (Expr t)
var
            [((b, b), Expr 'BoolSort)] -> m [((b, b), Expr 'BoolSort)]
forall a. a -> m a
forall (m :: * -> *) a. Monad m => a -> m a
return ([((b, b), Expr 'BoolSort)] -> m [((b, b), Expr 'BoolSort)])
-> [((b, b), Expr 'BoolSort)] -> m [((b, b), Expr 'BoolSort)]
forall a b. (a -> b) -> a -> b
$   ((b
p,b
q), Expr 'BoolSort
x)
                   ((b, b), Expr 'BoolSort)
-> [((b, b), Expr 'BoolSort)] -> [((b, b), Expr 'BoolSort)]
forall a. a -> [a] -> [a]
: [ ((b
q,b
p), Expr 'BoolSort
x) | b
p b -> b -> Bool
forall a. Eq a => a -> a -> Bool
/= b
q ]
    Relation b b -> m (Relation b b)
forall a. a -> m a
forall (m :: * -> *) a. Monad m => a -> m a
return (Relation b b -> m (Relation b b))
-> Relation b b -> m (Relation b b)
forall a b. (a -> b) -> a -> b
$ ((b, b), (b, b)) -> [((b, b), Expr 'BoolSort)] -> Relation b b
forall a b.
(Ix a, Ix b) =>
((a, b), (a, b)) -> [((a, b), Expr 'BoolSort)] -> Relation a b
build ((b, b), (b, b))
bnd ([((b, b), Expr 'BoolSort)] -> Relation b b)
-> [((b, b), Expr 'BoolSort)] -> Relation b b
forall a b. (a -> b) -> a -> b
$ [[((b, b), Expr 'BoolSort)]] -> [((b, b), Expr 'BoolSort)]
forall (t :: * -> *) a. Foldable t => t [a] -> [a]
concat [[((b, b), Expr 'BoolSort)]]
pairs

-- | Constructs a relation \(R \subseteq A \times B \) from a list.
build :: (Ix a, Ix b)
      => ((a,b),(a,b))
      -> [((a,b), Expr BoolSort)] -- ^ A list of tuples, where the first element represents an element
                           -- \((x,y) \in A \times B \) and the second element is a positive 'Expr' 'BoolSort'
                           -- if \((x,y) \in R \), or a negative 'Expr' 'BoolSort' if \((x,y) \notin R \).
      -> Relation a b
build :: forall a b.
(Ix a, Ix b) =>
((a, b), (a, b)) -> [((a, b), Expr 'BoolSort)] -> Relation a b
build ((a, b), (a, b))
bnd [((a, b), Expr 'BoolSort)]
pairs = Array (a, b) (Expr 'BoolSort) -> Relation a b
forall a b. Array (a, b) (Expr 'BoolSort) -> Relation a b
Relation (Array (a, b) (Expr 'BoolSort) -> Relation a b)
-> Array (a, b) (Expr 'BoolSort) -> Relation a b
forall a b. (a -> b) -> a -> b
$ ((a, b), (a, b))
-> [((a, b), Expr 'BoolSort)] -> Array (a, b) (Expr 'BoolSort)
forall i e. Ix i => (i, i) -> [(i, e)] -> Array i e
A.array ((a, b), (a, b))
bnd [((a, b), Expr 'BoolSort)]
pairs

-- | Constructs a relation \(R \subseteq A \times B \) from a function.
buildFrom :: (Ix a, Ix b)
          => ((a,b),(a,b))
          -> ((a,b) -> Expr BoolSort) -- ^ A function that assigns a 'Expr' 'BoolSort'-value to each element \((x,y) \in A \times B \).
          -> Relation a b
buildFrom :: forall a b.
(Ix a, Ix b) =>
((a, b), (a, b)) -> ((a, b) -> Expr 'BoolSort) -> Relation a b
buildFrom ((a, b), (a, b))
bnd (a, b) -> Expr 'BoolSort
p = ((a, b), (a, b)) -> [((a, b), Expr 'BoolSort)] -> Relation a b
forall a b.
(Ix a, Ix b) =>
((a, b), (a, b)) -> [((a, b), Expr 'BoolSort)] -> Relation a b
build ((a, b), (a, b))
bnd ([((a, b), Expr 'BoolSort)] -> Relation a b)
-> [((a, b), Expr 'BoolSort)] -> Relation a b
forall a b. (a -> b) -> a -> b
$ (((a, b) -> ((a, b), Expr 'BoolSort))
 -> [(a, b)] -> [((a, b), Expr 'BoolSort)])
-> [(a, b)]
-> ((a, b) -> ((a, b), Expr 'BoolSort))
-> [((a, b), Expr 'BoolSort)]
forall a b c. (a -> b -> c) -> b -> a -> c
flip ((a, b) -> ((a, b), Expr 'BoolSort))
-> [(a, b)] -> [((a, b), Expr 'BoolSort)]
forall a b. (a -> b) -> [a] -> [b]
map (((a, b), (a, b)) -> [(a, b)]
forall a. Ix a => (a, a) -> [a]
A.range ((a, b), (a, b))
bnd) (((a, b) -> ((a, b), Expr 'BoolSort))
 -> [((a, b), Expr 'BoolSort)])
-> ((a, b) -> ((a, b), Expr 'BoolSort))
-> [((a, b), Expr 'BoolSort)]
forall a b. (a -> b) -> a -> b
$ \ (a, b)
i -> ((a, b)
i, (a, b) -> Expr 'BoolSort
p (a, b)
i)

-- | Constructs an indeterminate relation \(R \subseteq A \times B\) from a function.
buildFromM :: (Ix a, Ix b, MonadSMT s m)
          => ((a,b),(a,b))
          -> ((a,b) -> m (Expr BoolSort))
          -> m (Relation a b)
buildFromM :: forall a b s (m :: * -> *).
(Ix a, Ix b, MonadSMT s m) =>
((a, b), (a, b))
-> ((a, b) -> m (Expr 'BoolSort)) -> m (Relation a b)
buildFromM ((a, b), (a, b))
bnd (a, b) -> m (Expr 'BoolSort)
p = do
    [((a, b), Expr 'BoolSort)]
pairs <- [m ((a, b), Expr 'BoolSort)] -> m [((a, b), Expr 'BoolSort)]
forall (t :: * -> *) (m :: * -> *) a.
(Traversable t, Monad m) =>
t (m a) -> m (t a)
forall (m :: * -> *) a. Monad m => [m a] -> m [a]
sequence ([m ((a, b), Expr 'BoolSort)] -> m [((a, b), Expr 'BoolSort)])
-> [m ((a, b), Expr 'BoolSort)] -> m [((a, b), Expr 'BoolSort)]
forall a b. (a -> b) -> a -> b
$ do
        (a, b)
i <- ((a, b), (a, b)) -> [(a, b)]
forall a. Ix a => (a, a) -> [a]
A.range ((a, b), (a, b))
bnd
        m ((a, b), Expr 'BoolSort) -> [m ((a, b), Expr 'BoolSort)]
forall a. a -> [a]
forall (m :: * -> *) a. Monad m => a -> m a
return (m ((a, b), Expr 'BoolSort) -> [m ((a, b), Expr 'BoolSort)])
-> m ((a, b), Expr 'BoolSort) -> [m ((a, b), Expr 'BoolSort)]
forall a b. (a -> b) -> a -> b
$ do
            Expr 'BoolSort
x <- (a, b) -> m (Expr 'BoolSort)
p (a, b)
i
            ((a, b), Expr 'BoolSort) -> m ((a, b), Expr 'BoolSort)
forall a. a -> m a
forall (m :: * -> *) a. Monad m => a -> m a
return ((a, b)
i, Expr 'BoolSort
x)
    Relation a b -> m (Relation a b)
forall a. a -> m a
forall (m :: * -> *) a. Monad m => a -> m a
return (Relation a b -> m (Relation a b))
-> Relation a b -> m (Relation a b)
forall a b. (a -> b) -> a -> b
$ ((a, b), (a, b)) -> [((a, b), Expr 'BoolSort)] -> Relation a b
forall a b.
(Ix a, Ix b) =>
((a, b), (a, b)) -> [((a, b), Expr 'BoolSort)] -> Relation a b
build ((a, b), (a, b))
bnd [((a, b), Expr 'BoolSort)]
pairs

-- | Constructs the identity relation \(I = \{ (x,x) ~|~ x \in A \} \subseteq A \times A\).
identity :: (Ix a)
         => ((a,a),(a,a)) -- ^ Since the identity relation is homogeneous, the domain must equal the codomain.
                          -- Therefore, given bounds @((p,q),(r,s))@, it must hold that @p=q@ and @r=s@.
         -> Relation a a
identity :: forall a. Ix a => ((a, a), (a, a)) -> Relation a a
identity ((a
a,a
b),(a
c,a
d))
    | (a
a,a
c) (a, a) -> (a, a) -> Bool
forall a. Eq a => a -> a -> Bool
== (a
b,a
d) = ((a, a), (a, a)) -> ((a, a) -> Expr 'BoolSort) -> Relation a a
forall a b.
(Ix a, Ix b) =>
((a, b), (a, b)) -> ((a, b) -> Expr 'BoolSort) -> Relation a b
buildFrom ((a
a,a
b),(a
c,a
d)) (\ (a
i,a
j) -> Bool -> Expr 'BoolSort
forall b. Boolean b => Bool -> b
bool (Bool -> Expr 'BoolSort) -> Bool -> Expr 'BoolSort
forall a b. (a -> b) -> a -> b
$ a
i a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
j)
    | Bool
otherwise      = [Char] -> Relation a a
forall a. HasCallStack => [Char] -> a
error [Char]
"The domain must equal the codomain!"

-- | The bounds of the array that correspond to the matrix representation of the given relation.
bounds :: Relation a b -> ((a,b),(a,b))
bounds :: forall a b. Relation a b -> ((a, b), (a, b))
bounds (Relation Array (a, b) (Expr 'BoolSort)
r) = Array (a, b) (Expr 'BoolSort) -> ((a, b), (a, b))
forall i e. Array i e -> (i, i)
A.bounds Array (a, b) (Expr 'BoolSort)
r
{-# INLINE bounds #-}

-- | The list of indices, where each index represents an element \((x,y) \in A \times B \)
-- that may be contained in the given relation \(R \subseteq A \times B \).
indices :: (Ix a, Ix b) => Relation a b -> [(a, b)]
indices :: forall a b. (Ix a, Ix b) => Relation a b -> [(a, b)]
indices (Relation Array (a, b) (Expr 'BoolSort)
r) = Array (a, b) (Expr 'BoolSort) -> [(a, b)]
forall i e. Ix i => Array i e -> [i]
A.indices Array (a, b) (Expr 'BoolSort)
r
{-# INLINE indices #-}

-- | The list of tuples for the given relation \(R \subseteq A \times B \),
-- where the first element represents an element \((x,y) \in A \times B \)
-- and the second element indicates via a 'Expr' 'BoolSort' , if \((x,y) \in R \) or not.
assocs :: (Ix a, Ix b) => Relation a b -> [((a, b), Expr BoolSort)]
assocs :: forall a b.
(Ix a, Ix b) =>
Relation a b -> [((a, b), Expr 'BoolSort)]
assocs (Relation Array (a, b) (Expr 'BoolSort)
r) = Array (a, b) (Expr 'BoolSort) -> [((a, b), Expr 'BoolSort)]
forall i e. Ix i => Array i e -> [(i, e)]
A.assocs Array (a, b) (Expr 'BoolSort)
r
{-# INLINE assocs #-}

-- | The list of elements of the array
-- that correspond to the matrix representation of the given relation.
elems :: Relation a b -> [Expr BoolSort]
elems :: forall a b. Relation a b -> [Expr 'BoolSort]
elems (Relation Array (a, b) (Expr 'BoolSort)
r) = Array (a, b) (Expr 'BoolSort) -> [Expr 'BoolSort]
forall i e. Array i e -> [e]
A.elems Array (a, b) (Expr 'BoolSort)
r
{-# INLINE elems #-}

-- | 'Maybe' ('Expr' 'BoolSort') for a given element \((x,y) \in A \times B \)
-- and a given relation \(R \subseteq A \times B \) that indicates
-- if \((x,y) \in R \) or not.
--
-- 'Just' if given element is in 'bounds' of the relation.
-- 'Nothing' otherwise.
(!?) :: (Ix a, Ix b) => Relation a b -> (a, b) -> Maybe (Expr BoolSort)
Relation Array (a, b) (Expr 'BoolSort)
r !? :: forall a b.
(Ix a, Ix b) =>
Relation a b -> (a, b) -> Maybe (Expr 'BoolSort)
!? (a, b)
p = Array (a, b) (Expr 'BoolSort)
rArray (a, b) (Expr 'BoolSort)
-> Getting
     (First (Expr 'BoolSort))
     (Array (a, b) (Expr 'BoolSort))
     (Expr 'BoolSort)
-> Maybe (Expr 'BoolSort)
forall s a. s -> Getting (First a) s a -> Maybe a
^?Index (Array (a, b) (Expr 'BoolSort))
-> Traversal'
     (Array (a, b) (Expr 'BoolSort))
     (IxValue (Array (a, b) (Expr 'BoolSort)))
forall m. Ixed m => Index m -> Traversal' m (IxValue m)
ix (a, b)
Index (Array (a, b) (Expr 'BoolSort))
p
{-# INLINE (!?) #-}

-- | Unsafe version of '(!?)'.
-- Produces an array-indexing-error if given element is not within the 'bounds' of the relation.
(!) :: (Ix a, Ix b) => Relation a b -> (a, b) -> Expr BoolSort
Relation Array (a, b) (Expr 'BoolSort)
r ! :: forall a b.
(Ix a, Ix b) =>
Relation a b -> (a, b) -> Expr 'BoolSort
! (a, b)
p = Array (a, b) (Expr 'BoolSort)
r Array (a, b) (Expr 'BoolSort) -> (a, b) -> Expr 'BoolSort
forall i e. Ix i => Array i e -> i -> e
A.! (a, b)
p
{-# INLINE (!) #-}

-- | The domain \(A\) of a relation \(R \subseteq A \times B\).
domain :: Ix a => Relation a b -> [a]
domain :: forall a b. Ix a => Relation a b -> [a]
domain Relation a b
r =
  let ((a
x,b
_),(a
x',b
_)) = Relation a b -> ((a, b), (a, b))
forall a b. Relation a b -> ((a, b), (a, b))
bounds Relation a b
r
  in (a, a) -> [a]
forall a. Ix a => (a, a) -> [a]
A.range (a
x,a
x')
{-# INLINE domain #-}

-- | The codomain \(B\) of a relation \(R \subseteq A \times B\).
codomain :: Ix b => Relation a b -> [b]
codomain :: forall b a. Ix b => Relation a b -> [b]
codomain Relation a b
r =
  let ((a
_,b
y),(a
_,b
y')) = Relation a b -> ((a, b), (a, b))
forall a b. Relation a b -> ((a, b), (a, b))
bounds Relation a b
r
  in (b, b) -> [b]
forall a. Ix a => (a, a) -> [a]
A.range (b
y,b
y')
{-# INLINE codomain #-}

-- | Returns a list of 'Expr' 'BoolSort' indicating whether the projection on
-- given element \( x \in A \) holds for every element in the codomain:
--
-- \( \{ (x,y) \in R \mid y \in codomain(R) \} \)
image :: (Ix a, Ix b) => Relation a b -> a -> [Expr BoolSort]
image :: forall a b. (Ix a, Ix b) => Relation a b -> a -> [Expr 'BoolSort]
image Relation a b
r a
x = (b -> Maybe (Expr 'BoolSort)) -> [b] -> [Expr 'BoolSort]
forall a b. (a -> Maybe b) -> [a] -> [b]
mapMaybe ((Relation a b
r !?) ((a, b) -> Maybe (Expr 'BoolSort))
-> (b -> (a, b)) -> b -> Maybe (Expr 'BoolSort)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (a
x,)) (Relation a b -> [b]
forall b a. Ix b => Relation a b -> [b]
codomain Relation a b
r)
{-# INLINE image #-}

-- | Returns a list of 'Expr' 'BoolSort' indicating whether the projection on
-- given element \( y \in B \) holds for every element in the domain:
--
-- \( \{ (x,y) \in R \mid x \in domain(R) \} \)
preimage :: (Ix a, Ix b) => Relation a b -> b -> [Expr BoolSort]
preimage :: forall a b. (Ix a, Ix b) => Relation a b -> b -> [Expr 'BoolSort]
preimage Relation a b
r b
y = (a -> Maybe (Expr 'BoolSort)) -> [a] -> [Expr 'BoolSort]
forall a b. (a -> Maybe b) -> [a] -> [b]
mapMaybe ((Relation a b
r !?) ((a, b) -> Maybe (Expr 'BoolSort))
-> (a -> (a, b)) -> a -> Maybe (Expr 'BoolSort)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (,b
y)) (Relation a b -> [a]
forall a b. Ix a => Relation a b -> [a]
domain Relation a b
r)
{-# INLINE preimage #-}

-- | Print a satisfying assignment from an SMT solver, where the assignment is interpreted as a relation.
-- @putStrLn $ table \</assignment/\>@ corresponds to the matrix representation of this relation.
table :: (Ix a, Ix b)
      => Array (a,b) Bool -> String
table :: forall a b. (Ix a, Ix b) => Array (a, b) Bool -> [Char]
table Array (a, b) Bool
r = [[Char]] -> [Char]
unlines ([[Char]] -> [Char]) -> [[Char]] -> [Char]
forall a b. (a -> b) -> a -> b
$ do
    let ((a
a,b
b),(a
c,b
d)) = Array (a, b) Bool -> ((a, b), (a, b))
forall i e. Array i e -> (i, i)
A.bounds Array (a, b) Bool
r
    a
x <- (a, a) -> [a]
forall a. Ix a => (a, a) -> [a]
A.range (a
a,a
c)
    [Char] -> [[Char]]
forall a. a -> [a]
forall (m :: * -> *) a. Monad m => a -> m a
return ([Char] -> [[Char]]) -> [Char] -> [[Char]]
forall a b. (a -> b) -> a -> b
$ [[Char]] -> [Char]
unwords ([[Char]] -> [Char]) -> [[Char]] -> [Char]
forall a b. (a -> b) -> a -> b
$ do
        b
y <- (b, b) -> [b]
forall a. Ix a => (a, a) -> [a]
A.range (b
b,b
d)
        [Char] -> [[Char]]
forall a. a -> [a]
forall (m :: * -> *) a. Monad m => a -> m a
return ([Char] -> [[Char]]) -> [Char] -> [[Char]]
forall a b. (a -> b) -> a -> b
$ if Array (a, b) Bool
r Array (a, b) Bool -> (a, b) -> Bool
forall i e. Ix i => Array i e -> i -> e
A.! (a
x,b
y) then [Char]
"*" else [Char]
"."