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Make singleton witnesses more accurate and make induction axioms stronger

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This commit is contained in:
mniip 2018-07-11 21:31:53 +03:00
parent b3c104ebef
commit 617598f09f
3 changed files with 23 additions and 13 deletions
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26
GHC/TypeLits/Induction.hs
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@ -38,6 +38,7 @@ induceIsZero = go natSingleton
go IsNonZero y z = y
-- | \((\forall n. P(n) \to P(n + 1)) \to P(0) \to \forall m. P(m)\)
--
-- For example:
--
-- > inducePeano f z :: p 3 = f (f (f z))
@ -49,6 +50,7 @@ inducePeano = go natSingleton
go (PeanoSucc n) f z = f (go n f z)
-- | \((\forall n. P(n) \to P(2n + 1)) \to \\ (\forall n. P(n) \to P(2n + 2)) \to \\ P(0) \to \\ \forall m. P(m)\)
--
-- For example:
--
-- > induceTwosComp f1 f2 z :: p 12 = f2 (f1 (f2 z))
@ -61,17 +63,21 @@ induceTwosComp = go natSingleton
go (TwosCompx2p2 n) f g z = g (go n f g z)
-- | \(\forall b. (\prod_d Q(d)) \to \\ (\forall n. \forall d. d < b \to Q(d) \to P(n) \to P(bn + d + 1)) \to \\ \forall m. P(m)\)
--
-- For example:
--
-- > induceBaseComp (_ :: r 10) d f z :: p 123 = (d :: q 2) `f` ((d :: q 1) `f` ((d :: q 0) `f` z)
induceBaseComp :: forall r b q p m. (KnownNat b, KnownNat m) => r b -> (forall m. KnownNat m => q m) -> (forall k n. (KnownNat b, 1 + k <= b, KnownNat n) => q k -> p n -> p (1 + k + b * n)) -> p 0 -> p m
induceBaseComp _ = go natSingleton
-- > induceBaseComp (_ :: s q) (_ :: r 10) d f z :: p 123 = (d :: q 2) `f` ((d :: q 1) `f` ((d :: q 0) `f` z)
--
-- The @s q@ parameter is necessary because presently GHC is unable to unify @q@ under the equational constraint @1 + k <= b@.
induceBaseComp :: forall r b q p m c s. (KnownNat b, b ~ (1 + c), KnownNat m) => s q -> r b -> (forall k. (KnownNat k, 1 + k <= b) => q k) -> (forall k n. (KnownNat k, 1 + k <= b, KnownNat n) => q k -> p n -> p (1 + k + b * n)) -> p 0 -> p m
induceBaseComp _ _ = go natSingleton
where
go :: forall m. NatBaseComp Proxy b m -> (forall m. (KnownNat m, 1 + m <= b) => q m) -> (forall k n. (KnownNat b, 1 + k <= b, KnownNat n) => q k -> p n -> p (1 + k + b * n)) -> p 0 -> p m
go :: forall m. NatBaseComp Proxy b m -> (forall k. (KnownNat k, 1 + k <= b) => q k) -> (forall k n. (KnownNat k, 1 + k <= b, KnownNat n) => q k -> p n -> p (1 + k + b * n)) -> p 0 -> p m
go BaseCompZero q f z = z
go (BaseCompxBp1p k n) q f z = f q (go n q f z)
-- | \((\forall n. P(n) \to P(n + 1)) \to P(1) \to \forall m > 0. P(m)\)
--
-- For example:
--
-- > induceUnary f o :: p 5 = f (f (f (f o)))
@ -83,6 +89,7 @@ induceUnary = go posSingleton
go (UnarySucc n) f o = f (go n f o)
-- | \((\forall n. P(n) \to P(2n)) \to \\ (\forall n. P(n) \to P(2n + 1)) \to \\ P(1) \to \\ \forall m > 0. P(m)\)
--
-- For example:
--
-- > inducePosBinary f0 f1 o :: p 10 = f0 (f1 (f0 o))
@ -95,12 +102,15 @@ inducePosBinary = go posSingleton
go (Bin1 n) f g z = g (go n f g z)
-- | \(\forall b. (\prod_d Q(d)) \to \\ (\forall n. \forall d. d < b \to Q(d) \to P(n) \to P(bn + d)) \to \\ (\forall d. d > 0 \to d < b \to Q(d) \to P(d)) \to \\ \forall m > 0. P (m)\)
--
-- For example:
--
-- > inducePosBase (_ :: r 10) d f l :: p 123 = (d :: q 3) `f` ((d :: q 2) `f` l (d :: q 1))
inducePosBase :: forall r b q p m. (KnownNat b, KnownNat m) => r b -> (forall m. KnownNat m => q m) -> (forall k n. (KnownNat b, 1 + k <= b, KnownNat n) => q k -> p n -> p (k + b * n)) -> (forall k n. (KnownNat k, 1 + k <= b, k ~ (1 + n)) => q k -> p k) -> p (1 + m)
inducePosBase _ = go posSingleton
-- > inducePosBase (_ :: s q) (_ :: r 10) d f l :: p 123 = (d :: q 3) `f` ((d :: q 2) `f` l (d :: q 1))
--
-- The @s q@ parameter is necessary because presently GHC is unable to unify @q@ under the equational constraint @1 + k <= b@.
inducePosBase :: forall r b q p m c s. (KnownNat b, b ~ (2 + c), KnownNat m) => s q -> r b -> (forall k. (KnownNat k, 1 + k <= b) => q k) -> (forall k n. (KnownNat k, 1 + k <= b, KnownNat n) => q k -> p n -> p (k + b * n)) -> (forall k n. (KnownNat k, 1 + k <= b, k ~ (1 + n)) => q k -> p k) -> p (1 + m)
inducePosBase _ _ = go posSingleton
where
go :: forall m. PosBase Proxy b m -> (forall m. KnownNat m => q m) -> (forall k n. (KnownNat b, 1 + k <= b, KnownNat n) => q k -> p n -> p (k + b * n)) -> (forall k n. (KnownNat k, 1 + k <= b, k ~ (1 + n)) => q k -> p k) -> p m
go :: forall m. PosBase Proxy b m -> (forall k. (KnownNat k, 1 + k <= b) => q k) -> (forall k n. (KnownNat k, 1 + k <= b, KnownNat n) => q k -> p n -> p (k + b * n)) -> (forall k n. (KnownNat k, 1 + k <= b, k ~ (1 + n)) => q k -> p k) -> p m
go (BaseLead n) q f g = g q
go (BaseDigit k n) q f g = f q (go n q f g)

8
GHC/TypeLits/Singletons.hs
View file

@ -106,14 +106,14 @@ instance NatSingleton NatTwosComp where
--
-- > BaseCompxBp1p (natSingleton :: p 2) (BaseCompxBp1p (natSingleton :: p 1) (BaseCompxBp1p (natSingleton :: p 0) BaseCompZero)) :: NatBaseComp p 10 123
data NatBaseComp (p :: Nat -> *) (b :: Nat) (n :: Nat) where
BaseCompZero :: NatBaseComp p b 0
BaseCompxBp1p :: (KnownNat k, 1 + k <= b, KnownNat n) => p k -> NatBaseComp p b n -> NatBaseComp p b (1 + k + b * n)
BaseCompZero :: (KnownNat b, b ~ (1 + c)) => NatBaseComp p b 0
BaseCompxBp1p :: (KnownNat b, b ~ (1 + c), KnownNat k, 1 + k <= b, KnownNat n) => p k -> NatBaseComp p b n -> NatBaseComp p b (1 + k + b * n)
instance ShowN p => Show (NatBaseComp p b n) where
showsPrec d BaseCompZero = showString "BaseCompZero"
showsPrec d (BaseCompxBp1p a b) = showParen (d > 10) $ showString "BaseCompxBp1p " . showsPrecN 11 a . showString " " . showsPrec 11 b
instance ShowN p => ShowN (NatBaseComp p b) where showsPrecN = showsPrec
instance (KnownNat b, NatSingleton p) => NatSingleton (NatBaseComp p b) where
instance (KnownNat b, b ~ (1 + c), NatSingleton p) => NatSingleton (NatBaseComp p b) where
natSingleton :: forall n. KnownNat n => NatBaseComp p b n
natSingleton = case natVal (Proxy :: Proxy n) of
0 -> (unsafeCoerce :: NatBaseComp p b 0 -> NatBaseComp p b n) BaseCompZero
@ -188,7 +188,7 @@ instance ShowN p => Show (PosBase p b n) where
showsPrec d (BaseDigit a b) = showParen (d > 10) $ showString "BaseDigit " . showsPrecN 11 a . showString " " . showsPrec 11 b
instance ShowN p => ShowN (PosBase p b) where showsPrecN = showsPrec
instance (KnownNat b, NatSingleton p) => PositiveSingleton (PosBase p b) where
instance (KnownNat b, b ~ (2 + c), NatSingleton p) => PositiveSingleton (PosBase p b) where
posSingleton :: forall n. KnownNat n => PosBase p b (1 + n)
posSingleton = case natVal (Proxy :: Proxy n) of
n | n < base - 1 -> case someNatVal (n + 1) of

2
singleton-typelits.cabal
View file

@ -1,5 +1,5 @@
name: singleton-typelits
version: 0.0.0.1
version: 0.1.0.0
synopsis: Singletons and induction over GHC TypeLits
description:
Singletons and induction schemes over 'GHC.TypeLits.Nat'.