{-# LANGUAGE AllowAmbiguousTypes #-}
{-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE LambdaCase #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TupleSections #-}
{-# LANGUAGE TypeApplications #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE TypeInType #-}
{-# LANGUAGE TypeOperators #-}
module Data.Type.Predicate.Logic (
Evident, Impossible
, type Not, decideNot
, type (&&&), decideAnd
, type (|||), decideOr, type (^||), type (||^)
, type (^^^), decideXor
, type (==>), proveImplies, Implies
, type (<==>), Equiv
, compImpl, explosion, atom
, complementation, doubleNegation, tripleNegation, negateTwice
, contrapositive, contrapositive'
, projAndFst, projAndSnd, injOrLeft, injOrRight
) where
import Data.Singletons
import Data.Singletons.Decide
import Data.Type.Predicate
import Data.Void
data (&&&) :: Predicate k -> Predicate k -> Predicate k
type instance Apply (p &&& q) a = (p @@ a, q @@ a)
infixr 3 &&&
instance (Decidable p, Decidable q) => Decidable (p &&& q) where
decide :: Sing a -> Decision ((p &&& q) @@ a)
decide (x :: Sing a) = Decision (p @@ a) -> Decision (q @@ a) -> Decision ((p &&& q) @@ a)
forall k1 (p :: k1 ~> *) (q :: k1 ~> *) (a :: k1).
Decision (p @@ a) -> Decision (q @@ a) -> Decision ((p &&& q) @@ a)
decideAnd @p @q @a (Sing a -> Decision (p @@ a)
forall k1 (p :: k1 ~> *). Decidable p => Decide p
decide @p Sing a
x) (Sing a -> Decision (q @@ a)
forall k1 (p :: k1 ~> *). Decidable p => Decide p
decide @q Sing a
x)
instance (Provable p, Provable q) => Provable (p &&& q) where
prove :: Sing a -> (p &&& q) @@ a
prove x :: Sing a
x = (Sing a -> p @@ a
forall k1 (p :: k1 ~> *). Provable p => Prove p
prove @p Sing a
x, Sing a -> q @@ a
forall k1 (p :: k1 ~> *). Provable p => Prove p
prove @q Sing a
x)
decideAnd
:: forall p q a. ()
=> Decision (p @@ a)
-> Decision (q @@ a)
-> Decision ((p &&& q) @@ a)
decideAnd :: Decision (p @@ a) -> Decision (q @@ a) -> Decision ((p &&& q) @@ a)
decideAnd = \case
Proved p :: p @@ a
p -> ((q @@ a) -> (p @@ a, q @@ a))
-> ((p @@ a, q @@ a) -> q @@ a)
-> Decision (q @@ a)
-> Decision (p @@ a, q @@ a)
forall a b. (a -> b) -> (b -> a) -> Decision a -> Decision b
mapDecision (p @@ a
p,) (p @@ a, q @@ a) -> q @@ a
forall a b. (a, b) -> b
snd
Disproved v :: Refuted (p @@ a)
v -> \_ -> Refuted (p @@ a, q @@ a) -> Decision ((p &&& q) @@ a)
forall a. Refuted a -> Decision a
Disproved (Refuted (p @@ a, q @@ a) -> Decision ((p &&& q) @@ a))
-> Refuted (p @@ a, q @@ a) -> Decision ((p &&& q) @@ a)
forall a b. (a -> b) -> a -> b
$ \(p :: p @@ a
p, _) -> Refuted (p @@ a)
v p @@ a
p
data (|||) :: Predicate k -> Predicate k -> Predicate k
type instance Apply (p ||| q) a = Either (p @@ a) (q @@ a)
infixr 2 |||
instance (Decidable p, Decidable q) => Decidable (p ||| q) where
decide :: Sing a -> Decision ((p ||| q) @@ a)
decide (x :: Sing a) = Decision (p @@ a) -> Decision (q @@ a) -> Decision ((p ||| q) @@ a)
forall k1 (p :: k1 ~> *) (q :: k1 ~> *) (a :: k1).
Decision (p @@ a) -> Decision (q @@ a) -> Decision ((p ||| q) @@ a)
decideOr @p @q @a (Sing a -> Decision (p @@ a)
forall k1 (p :: k1 ~> *). Decidable p => Decide p
decide @p Sing a
x) (Sing a -> Decision (q @@ a)
forall k1 (p :: k1 ~> *). Decidable p => Decide p
decide @q Sing a
x)
decideOr
:: forall p q a. ()
=> Decision (p @@ a)
-> Decision (q @@ a)
-> Decision ((p ||| q) @@ a)
decideOr :: Decision (p @@ a) -> Decision (q @@ a) -> Decision ((p ||| q) @@ a)
decideOr = \case
Proved p :: p @@ a
p -> \_ -> Either (p @@ a) (q @@ a) -> Decision ((p ||| q) @@ a)
forall a. a -> Decision a
Proved (Either (p @@ a) (q @@ a) -> Decision ((p ||| q) @@ a))
-> Either (p @@ a) (q @@ a) -> Decision ((p ||| q) @@ a)
forall a b. (a -> b) -> a -> b
$ (p @@ a) -> Either (p @@ a) (q @@ a)
forall a b. a -> Either a b
Left p @@ a
p
Disproved v :: Refuted (p @@ a)
v -> ((q @@ a) -> Either (p @@ a) (q @@ a))
-> (Either (p @@ a) (q @@ a) -> q @@ a)
-> Decision (q @@ a)
-> Decision (Either (p @@ a) (q @@ a))
forall a b. (a -> b) -> (b -> a) -> Decision a -> Decision b
mapDecision (q @@ a) -> Either (p @@ a) (q @@ a)
forall a b. b -> Either a b
Right (((p @@ a) -> q @@ a)
-> ((q @@ a) -> q @@ a) -> Either (p @@ a) (q @@ a) -> q @@ a
forall a c b. (a -> c) -> (b -> c) -> Either a b -> c
either (Void -> q @@ a
forall a. Void -> a
absurd (Void -> q @@ a) -> Refuted (p @@ a) -> (p @@ a) -> q @@ a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Refuted (p @@ a)
v) (q @@ a) -> q @@ a
forall a. a -> a
id)
type p ^|| q = p ||| Not p &&& q
type p ||^ q = p &&& Not q ||| q
type p ^^^ q = (p &&& Not q) ||| (Not p &&& q)
decideXor
:: forall p q a. ()
=> Decision (p @@ a)
-> Decision (q @@ a)
-> Decision ((p ^^^ q) @@ a)
decideXor :: Decision (p @@ a) -> Decision (q @@ a) -> Decision ((p ^^^ q) @@ a)
decideXor p :: Decision (p @@ a)
p q :: Decision (q @@ a)
q = Decision ((p &&& Not q) @@ a)
-> Decision ((Not p &&& q) @@ a) -> Decision ((p ^^^ q) @@ a)
forall k1 (p :: k1 ~> *) (q :: k1 ~> *) (a :: k1).
Decision (p @@ a) -> Decision (q @@ a) -> Decision ((p ||| q) @@ a)
decideOr @(p &&& Not q) @(Not p &&& q) @a
(Decision (p @@ a)
-> Decision (Not q @@ a) -> Decision ((p &&& Not q) @@ a)
forall k1 (p :: k1 ~> *) (q :: k1 ~> *) (a :: k1).
Decision (p @@ a) -> Decision (q @@ a) -> Decision ((p &&& q) @@ a)
decideAnd @p @(Not q) @a Decision (p @@ a)
p (Decision (q @@ a) -> Decision (Not q @@ a)
forall k1 (p :: k1 ~> *) (a :: k1).
Decision (p @@ a) -> Decision (Not p @@ a)
decideNot @q @a Decision (q @@ a)
q))
(Decision (Not p @@ a)
-> Decision (q @@ a) -> Decision ((Not p &&& q) @@ a)
forall k1 (p :: k1 ~> *) (q :: k1 ~> *) (a :: k1).
Decision (p @@ a) -> Decision (q @@ a) -> Decision ((p &&& q) @@ a)
decideAnd @(Not p) @q @a (Decision (p @@ a) -> Decision (Not p @@ a)
forall k1 (p :: k1 ~> *) (a :: k1).
Decision (p @@ a) -> Decision (Not p @@ a)
decideNot @p @a Decision (p @@ a)
p) Decision (q @@ a)
q)
data (==>) :: Predicate k -> Predicate k -> Predicate k
type instance Apply (p ==> q) a = p @@ a -> q @@ a
infixr 1 ==>
instance Decidable (Impossible ==> p) where
instance Provable (Impossible ==> p) where
prove :: Sing a -> (Impossible ==> p) @@ a
prove = Impossible --> p
forall k1 (p :: k1 ~> *). Impossible --> p
explosion @p
instance (Decidable (p ==> q), Decidable q) => Decidable (Not q ==> Not p) where
decide :: Sing a -> Decision ((Not q ==> Not p) @@ a)
decide x :: Sing a
x = case Sing a -> Decision ((p ==> q) @@ a)
forall k1 (p :: k1 ~> *). Decidable p => Decide p
decide @(p ==> q) Sing a
x of
Proved pq :: (p ==> q) @@ a
pq -> ((Apply q a -> Void) -> Apply p a -> Void)
-> Decision ((Not q ==> Not p) @@ a)
forall a. a -> Decision a
Proved (((Apply q a -> Void) -> Apply p a -> Void)
-> Decision ((Not q ==> Not p) @@ a))
-> ((Apply q a -> Void) -> Apply p a -> Void)
-> Decision ((Not q ==> Not p) @@ a)
forall a b. (a -> b) -> a -> b
$ \vq :: Apply q a -> Void
vq p :: Apply p a
p -> Apply q a -> Void
vq ((p ==> q) @@ a
Apply p a -> Apply q a
pq Apply p a
p)
Disproved vpq :: Refuted ((p ==> q) @@ a)
vpq -> case Sing a -> Decision (Apply q a)
forall k1 (p :: k1 ~> *). Decidable p => Decide p
decide @q Sing a
x of
Proved q :: Apply q a
q -> Refuted ((Apply q a -> Void) -> Apply p a -> Void)
-> Decision ((Not q ==> Not p) @@ a)
forall a. Refuted a -> Decision a
Disproved (Refuted ((Apply q a -> Void) -> Apply p a -> Void)
-> Decision ((Not q ==> Not p) @@ a))
-> Refuted ((Apply q a -> Void) -> Apply p a -> Void)
-> Decision ((Not q ==> Not p) @@ a)
forall a b. (a -> b) -> a -> b
$ \_ -> Refuted ((p ==> q) @@ a)
vpq (Apply q a -> Apply p a -> Apply q a
forall a b. a -> b -> a
const Apply q a
q)
Disproved vq :: Apply q a -> Void
vq -> Refuted ((Apply q a -> Void) -> Apply p a -> Void)
-> Decision ((Not q ==> Not p) @@ a)
forall a. Refuted a -> Decision a
Disproved (Refuted ((Apply q a -> Void) -> Apply p a -> Void)
-> Decision ((Not q ==> Not p) @@ a))
-> Refuted ((Apply q a -> Void) -> Apply p a -> Void)
-> Decision ((Not q ==> Not p) @@ a)
forall a b. (a -> b) -> a -> b
$ \vnpnq :: (Apply q a -> Void) -> Apply p a -> Void
vnpnq -> Refuted ((p ==> q) @@ a)
vpq (Void -> Apply q a
forall a. Void -> a
absurd (Void -> Apply q a)
-> (Apply p a -> Void) -> Apply p a -> Apply q a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (Apply q a -> Void) -> Apply p a -> Void
vnpnq Apply q a -> Void
vq)
instance Provable (p ==> q) => Provable (Not q ==> Not p) where
prove :: Sing a -> (Not q ==> Not p) @@ a
prove = (p --> q) -> Not q --> Not p
forall k (p :: k ~> *) (q :: k ~> *). (p --> q) -> Not q --> Not p
contrapositive @p @q (Provable (p ==> q) => Prove (p ==> q)
forall k1 (p :: k1 ~> *). Provable p => Prove p
prove @(p ==> q))
instance {-# OVERLAPPING #-} Decidable (p &&& q ==> p) where
instance {-# OVERLAPPING #-} Provable (p &&& q ==> p) where
prove :: Sing a -> ((p &&& q) ==> p) @@ a
prove = (p &&& q) --> p
forall k1 (p :: Predicate k1) (q :: Predicate k1). (p &&& q) --> p
projAndFst @p @q
instance {-# OVERLAPPING #-} Decidable (p &&& q ==> q) where
instance {-# OVERLAPPING #-} Provable (p &&& q ==> q) where
prove :: Sing a -> ((p &&& q) ==> q) @@ a
prove = (p &&& q) --> q
forall k1 (p :: Predicate k1) (q :: Predicate k1). (p &&& q) --> q
projAndSnd @p @q
instance {-# OVERLAPPING #-} Decidable (p &&& p ==> p) where
instance {-# OVERLAPPING #-} Provable (p &&& p ==> p) where
prove :: Sing a -> ((p &&& p) ==> p) @@ a
prove = (p &&& p) --> p
forall k1 (p :: Predicate k1) (q :: Predicate k1). (p &&& q) --> p
projAndFst @p @p
instance {-# OVERLAPPING #-} Decidable (p ==> p ||| q)
instance {-# OVERLAPPING #-} Provable (p ==> p ||| q) where
prove :: Sing a -> (p ==> (p ||| q)) @@ a
prove = p --> (p ||| q)
forall k (p :: k ~> *) (q :: k ~> *). p --> (p ||| q)
injOrLeft @p @q
instance {-# OVERLAPPING #-} Decidable (q ==> p ||| q)
instance {-# OVERLAPPING #-} Provable (q ==> p ||| q) where
prove :: Sing a -> (q ==> (p ||| q)) @@ a
prove = q --> (p ||| q)
forall k (p :: Predicate k) (q :: Predicate k). q --> (p ||| q)
injOrRight @p @q
instance {-# OVERLAPPING #-} Decidable (p ==> p ||| p)
instance {-# OVERLAPPING #-} Provable (p ==> p ||| p) where
prove :: Sing a -> (p ==> (p ||| p)) @@ a
prove = p --> (p ||| p)
forall k (p :: k ~> *) (q :: k ~> *). p --> (p ||| q)
injOrLeft @p @p
type Implies p q = Provable (p ==> q)
type Equiv p q = Provable (p <==> q)
proveImplies :: Prove q -> Prove (p ==> q)
proveImplies :: Prove q -> Prove (p ==> q)
proveImplies q :: Prove q
q x :: Sing a
x _ = Sing a -> q @@ a
Prove q
q Sing a
x
type (p <==> q) = p ==> q &&& q ==> p
infixr 1 <==>
explosion :: Impossible --> p
explosion :: Sing a -> (Impossible @@ a) -> p @@ a
explosion x :: Sing a
x v :: Impossible @@ a
v = Void -> p @@ a
forall a. Void -> a
absurd (Void -> p @@ a) -> Void -> p @@ a
forall a b. (a -> b) -> a -> b
$ Impossible @@ a
Sing a -> Void
v Sing a
x
atom :: p --> Evident
atom :: Sing a -> (p @@ a) -> Evident @@ a
atom = Sing a -> (p @@ a) -> Evident @@ a
forall a b. a -> b -> a
const
complementation :: forall p. (p &&& Not p) --> Impossible
complementation :: Sing a -> ((p &&& Not p) @@ a) -> Impossible @@ a
complementation _ (p, notP) _ = Apply p a -> Void
notP Apply p a
p
instance {-# OVERLAPPING #-} Provable (p &&& Not p ==> Impossible) where
prove :: Sing a -> ((p &&& Not p) ==> Impossible) @@ a
prove = (p &&& Not p) --> Impossible
forall k1 (p :: Predicate k1). (p &&& Not p) --> Impossible
complementation @p
contrapositive
:: (p --> q)
-> (Not q --> Not p)
contrapositive :: (p --> q) -> Not q --> Not p
contrapositive f :: p --> q
f x :: Sing a
x vQ :: Not q @@ a
vQ p :: Apply p a
p = Not q @@ a
Refuted (q @@ a)
vQ (Sing a -> Apply p a -> q @@ a
p --> q
f Sing a
x Apply p a
p)
contrapositive'
:: forall p q. Decidable q
=> (Not q --> Not p)
-> (p --> q)
contrapositive' :: (Not q --> Not p) -> p --> q
contrapositive' f :: Not q --> Not p
f x :: Sing a
x p :: p @@ a
p = Decision (Apply q a) -> Refuted (Refuted (Apply q a)) -> Apply q a
forall a. Decision a -> Refuted (Refuted a) -> a
elimDisproof (Sing a -> Decision (Apply q a)
forall k1 (p :: k1 ~> *). Decidable p => Decide p
decide @q Sing a
x) (Refuted (Refuted (Apply q a)) -> Apply q a)
-> Refuted (Refuted (Apply q a)) -> Apply q a
forall a b. (a -> b) -> a -> b
$ \vQ :: Refuted (Apply q a)
vQ ->
Sing a -> (Not q @@ a) -> Refuted (p @@ a)
Not q --> Not p
f Sing a
x Not q @@ a
Refuted (Apply q a)
vQ p @@ a
p
doubleNegation :: forall p. Decidable p => Not (Not p) --> p
doubleNegation :: Not (Not p) --> p
doubleNegation x :: Sing a
x vvP :: Not (Not p) @@ a
vvP = Decision (Apply p a) -> Refuted (Refuted (Apply p a)) -> Apply p a
forall a. Decision a -> Refuted (Refuted a) -> a
elimDisproof (Sing a -> Decision (Apply p a)
forall k1 (p :: k1 ~> *). Decidable p => Decide p
decide @p Sing a
x) (Refuted (Refuted (Apply p a)) -> Apply p a)
-> Refuted (Refuted (Apply p a)) -> Apply p a
forall a b. (a -> b) -> a -> b
$ \vP :: Refuted (Apply p a)
vP ->
Not (Not p) @@ a
Refuted (Refuted (Apply p a))
vvP Refuted (Apply p a)
vP
tripleNegation :: forall p. Not (Not (Not p)) --> Not p
tripleNegation :: Sing a -> (Not (Not (Not p)) @@ a) -> Not p @@ a
tripleNegation _ vvvP :: Not (Not (Not p)) @@ a
vvvP p :: Apply p a
p = Not (Not (Not p)) @@ a
(Refuted (Apply p a) -> Void) -> Void
vvvP ((Refuted (Apply p a) -> Void) -> Void)
-> (Refuted (Apply p a) -> Void) -> Void
forall a b. (a -> b) -> a -> b
$ \vP :: Refuted (Apply p a)
vP -> Refuted (Apply p a)
vP Apply p a
p
negateTwice :: p --> Not (Not p)
negateTwice :: Sing a -> (p @@ a) -> Not (Not p) @@ a
negateTwice _ p :: p @@ a
p vP :: (p @@ a) -> Void
vP = (p @@ a) -> Void
vP p @@ a
p
projAndFst :: (p &&& q) --> p
projAndFst :: Sing a -> ((p &&& q) @@ a) -> p @@ a
projAndFst _ = ((p &&& q) @@ a) -> p @@ a
forall a b. (a, b) -> a
fst
projAndSnd :: (p &&& q) --> q
projAndSnd :: Sing a -> ((p &&& q) @@ a) -> q @@ a
projAndSnd _ = ((p &&& q) @@ a) -> q @@ a
forall a b. (a, b) -> b
snd
injOrLeft :: forall p q. p --> (p ||| q)
injOrLeft :: Sing a -> (p @@ a) -> (p ||| q) @@ a
injOrLeft _ = (p @@ a) -> (p ||| q) @@ a
forall a b. a -> Either a b
Left
injOrRight :: forall p q. q --> (p ||| q)
injOrRight :: Sing a -> (q @@ a) -> (p ||| q) @@ a
injOrRight _ = (q @@ a) -> (p ||| q) @@ a
forall a b. b -> Either a b
Right