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87
GHC/TypeLits/Induction.hs
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87
GHC/TypeLits/Induction.hs
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-- |
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-- Module : GHC.TypeLits.Induction
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-- Description : Induction for GHC TypeLits
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-- Copyright : (C) 2017 mniip
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-- License : MIT
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-- Maintainer : mniip@mniip.com
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-- Stability : experimental
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-- Portability : portable
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{-# LANGUAGE TypeOperators, KindSignatures, DataKinds, PolyKinds, GADTs, RankNTypes, StandaloneDeriving, InstanceSigs, ScopedTypeVariables #-}
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{-# OPTIONS_GHC -fno-warn-tabs #-}
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module GHC.TypeLits.Induction
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(
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-- * Natural number induction
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induceIsZero,
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inducePeano,
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induceTwosComp,
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induceBaseComp,
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-- * Positive number induction
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induceUnary,
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inducePosBinary,
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inducePosBase
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)
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where
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import Data.Proxy
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import GHC.TypeLits
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import GHC.TypeLits.Singletons
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-- | \((\forall n. n > 0 \to P(n)) \to P(0) \to \forall m. P(m)\)
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induceIsZero :: KnownNat m => (forall n. KnownNat n => p (1 + n)) -> p 0 -> p m
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induceIsZero = go natSingleton
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where
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go :: NatIsZero m -> (forall n. KnownNat n => p (1 + n)) -> p 0 -> p m
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go IsZero y z = z
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go IsNonZero y z = y
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-- | \((\forall n. P(n) \to P(n + 1)) \to P(0) \to \forall m. P(m)\)
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inducePeano :: KnownNat m => (forall n. KnownNat n => p n -> p (1 + n)) -> p 0 -> p m
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inducePeano = go natSingleton
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where
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go :: NatPeano m -> (forall n. KnownNat n => p n -> p (1 + n)) -> p 0 -> p m
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go PeanoZero f z = z
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go (PeanoSucc n) f z = f (go n f z)
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-- | \((\forall n. P(n) \to P(2n + 1)) \to \\ (\forall n. P(n) \to P(2n + 2)) \to \\ P(0) \to \\ \forall m. P(m)\)
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induceTwosComp :: KnownNat m => (forall n. KnownNat n => p n -> p (1 + 2 * n)) -> (forall n. KnownNat n => p n -> p (2 + 2 * n)) -> p 0 -> p m
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induceTwosComp = go natSingleton
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where
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go :: NatTwosComp m -> (forall n. KnownNat n => p n -> p (1 + 2 * n)) -> (forall n. KnownNat n => p n -> p (2 + 2 * n)) -> p 0 -> p m
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go TwosCompZero f g z = z
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go (TwosCompx2p1 n) f g z = f (go n f g z)
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go (TwosCompx2p2 n) f g z = g (go n f g z)
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-- | \(\forall b. (\prod_d Q(d)) \to \\ (\forall n. \forall d. d < b \to Q(d) \to P(n) \to P(bn + d + 1)) \to \\ \forall m. P(m)\)
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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
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induceBaseComp _ = go natSingleton
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where
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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
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go BaseCompZero q f z = z
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go (BaseCompxBp1p k n) q f z = f q (go n q f z)
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-- | \((\forall n. P(n) \to P(n + 1)) \to P(1) \to \forall m > 0. P(m)\)
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induceUnary :: KnownNat m => (forall n. KnownNat n => p n -> p (1 + n)) -> p 1 -> p (1 + m)
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induceUnary = go posSingleton
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where
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go :: Unary m -> (forall n. KnownNat n => p n -> p (1 + n)) -> p 1 -> p m
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go UnaryOne f o = o
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go (UnarySucc n) f o = f (go n f o)
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-- | \((\forall n. P(n) \to P(2n)) \to \\ (\forall n. P(n) \to P(2n + 1)) \to \\ P(1) \to \\ \forall m > 0. P(m)\)
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inducePosBinary :: KnownNat m => (forall n. KnownNat n => p n -> p (2 * n)) -> (forall n. KnownNat n => p n -> p (1 + 2 * n)) -> p 1 -> p (1 + m)
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inducePosBinary = go posSingleton
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where
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go :: PosBinary m -> (forall n. KnownNat n => p n -> p (2 * n)) -> (forall n. KnownNat n => p n -> p (1 + 2 * n)) -> p 1 -> p m
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go BinOne f g z = z
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go (Bin0 n) f g z = f (go n f g z)
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go (Bin1 n) f g z = g (go n f g z)
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-- | \(\forall b. (\prod_d Q(d)) \to \\ (\forall n. \forall d. d < b \to Q(d) \to P(n) \to P(bn + d)) \to \\ (\forall d. d > 0 \to d < b \to Q(d) \to P(d)) \to \\ \forall m > 0. P (m)\)
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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)
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inducePosBase _ = go posSingleton
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where
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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
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go (BaseLead n) q f g = g q
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go (BaseDigit k n) q f g = f q (go n q f g)
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184
GHC/TypeLits/Singletons.hs
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184
GHC/TypeLits/Singletons.hs
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-- |
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-- Module : GHC.TypeLits.Singletons
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-- Description : Singletons for GHC TypeLits
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-- Copyright : (C) 2017 mniip
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-- License : MIT
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-- Maintainer : mniip@mniip.com
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-- Stability : experimental
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-- Portability : portable
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{-# LANGUAGE TypeOperators, KindSignatures, DataKinds, PolyKinds, GADTs, RankNTypes, StandaloneDeriving, InstanceSigs, ScopedTypeVariables #-}
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{-# OPTIONS_GHC -fno-warn-tabs #-}
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module GHC.TypeLits.Singletons
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(
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-- * Natural number singletons
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NatSingleton(..),
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NatIsZero(..),
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NatPeano(..),
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NatTwosComp(..),
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NatBaseComp(..),
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-- * Positive number singleton
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PositiveSingleton(..),
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Unary(..),
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PosBinary(..),
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PosBase(..),
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ShowN(..)
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)
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where
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import GHC.TypeLits
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import Data.Type.Equality
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import Unsafe.Coerce
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import Data.Proxy
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-- | A class of singletons capable of representing any 'KnownNat' natural number.
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class NatSingleton (p :: Nat -> *) where
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natSingleton :: KnownNat n => p n
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-- | Auxiliary class for implementing 'Show' on higher-kinded singletons.
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class ShowN (p :: Nat -> *) where showsPrecN :: Int -> p n -> ShowS
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instance ShowN Proxy where showsPrecN = showsPrec
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instance NatSingleton Proxy where
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natSingleton = Proxy
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-- | A natural number is either 0 or 1 plus something.
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data NatIsZero (n :: Nat) where
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IsZero :: NatIsZero 0
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IsNonZero :: KnownNat n => NatIsZero (1 + n)
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deriving instance Show (NatIsZero n)
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instance ShowN NatIsZero where showsPrecN = showsPrec
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instance NatSingleton NatIsZero where
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natSingleton :: forall n. KnownNat n => NatIsZero n
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natSingleton = case natVal (Proxy :: Proxy n) of
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0 -> (unsafeCoerce :: NatIsZero 0 -> NatIsZero n) IsZero
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n -> case someNatVal (n - 1) of
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Just (SomeNat (p :: Proxy m)) -> (unsafeCoerce :: NatIsZero (1 + m) -> NatIsZero n) $ IsNonZero
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Nothing -> error $ "Malformed KnownNat instance: " ++ show n
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-- | A natural number is either 0 or a successor of another natural number.
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data NatPeano (n :: Nat) where
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PeanoZero :: NatPeano 0
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PeanoSucc :: KnownNat n => NatPeano n -> NatPeano (1 + n)
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deriving instance Show (NatPeano n)
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instance ShowN NatPeano where showsPrecN = showsPrec
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instance NatSingleton NatPeano where
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natSingleton :: forall n. KnownNat n => NatPeano n
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natSingleton = case natSingleton :: NatIsZero n of
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IsZero -> PeanoZero
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IsNonZero -> PeanoSucc natSingleton
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-- | A natural number is either 0, or twice another natural number plus 1 or 2.
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data NatTwosComp (n :: Nat) where
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TwosCompZero :: NatTwosComp 0
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TwosCompx2p1 :: KnownNat n => NatTwosComp n -> NatTwosComp (1 + 2 * n)
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TwosCompx2p2 :: KnownNat n => NatTwosComp n -> NatTwosComp (2 + 2 * n)
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deriving instance Show (NatTwosComp n)
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instance ShowN NatTwosComp where showsPrecN = showsPrec
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instance NatSingleton NatTwosComp where
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natSingleton :: forall n. KnownNat n => NatTwosComp n
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natSingleton = case natVal (Proxy :: Proxy n) of
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0 -> (unsafeCoerce :: NatTwosComp 0 -> NatTwosComp n) TwosCompZero
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n -> case someNatVal ((n - 1) `div` 2) of
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Just (SomeNat (p :: Proxy m)) -> if even n
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then (unsafeCoerce :: NatTwosComp (2 + 2 * m) -> NatTwosComp n) $ TwosCompx2p2 natSingleton
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else (unsafeCoerce :: NatTwosComp (1 + 2 * m) -> NatTwosComp n) $ TwosCompx2p1 natSingleton
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Nothing -> error $ "Malformed KnownNat instance: " ++ show n
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-- | A natural number is either 0, or @b@ times another natural number plus a digit in @[1, b]@.
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data NatBaseComp (p :: Nat -> *) (b :: Nat) (n :: Nat) where
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BaseCompZero :: NatBaseComp p b 0
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BaseCompxBp1p :: (KnownNat k, 1 + k <= b, KnownNat n) => p k -> NatBaseComp p b n -> NatBaseComp p b (1 + k + b * n)
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instance ShowN p => Show (NatBaseComp p b n) where
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showsPrec d BaseCompZero = showString "BaseCompZero"
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showsPrec d (BaseCompxBp1p a b) = showParen (d > 10) $ showString "BaseCompxBp1p " . showsPrecN 11 a . showString " " . showsPrec 11 b
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instance ShowN p => ShowN (NatBaseComp p b) where showsPrecN = showsPrec
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instance (KnownNat b, NatSingleton p) => NatSingleton (NatBaseComp p b) where
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natSingleton :: forall n. KnownNat n => NatBaseComp p b n
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natSingleton = case natVal (Proxy :: Proxy n) of
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0 -> (unsafeCoerce :: NatBaseComp p b 0 -> NatBaseComp p b n) BaseCompZero
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n -> case someNatVal ((n - 1) `div` base) of
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Just (SomeNat (p :: Proxy m)) -> case someNatVal ((n - 1) `mod` base) of
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Just (SomeNat (p :: Proxy k)) -> (unsafeCoerce :: NatBaseComp p b (1 + k + b * m) -> NatBaseComp p b n) $ case unsafeCoerce Refl :: (1 + k <=? b) :~: True of
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Refl -> BaseCompxBp1p (natSingleton :: p k) (natSingleton :: NatBaseComp p b m)
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Nothing -> error $ "Malformed KnownNat instance: " ++ show base
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Nothing -> error $ "Malformed KnownNat instance: " ++ show n
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where
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base = natVal (Proxy :: Proxy b)
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-- | A class of singletons capable of representing postive 'KnownNat' natural numbers.
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class PositiveSingleton (p :: Nat -> *) where
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posSingleton :: KnownNat n => p (1 + n)
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instance PositiveSingleton Proxy where
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posSingleton = Proxy
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-- | A positive number is either 1 or a successor of another positive number.
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data Unary (n :: Nat) where
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UnaryOne :: Unary 1
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UnarySucc :: KnownNat n => Unary n -> Unary (1 + n)
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deriving instance Show (Unary n)
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instance ShowN Unary where showsPrecN = showsPrec
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instance PositiveSingleton Unary where
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posSingleton :: forall n. KnownNat n => Unary (1 + n)
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posSingleton = case natSingleton :: NatIsZero n of
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IsZero -> UnaryOne
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IsNonZero -> UnarySucc posSingleton
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-- | A positive number is either 1, or twice another positive number plus 0 or 1.
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data PosBinary (n :: Nat) where
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BinOne :: PosBinary 1
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Bin0 :: KnownNat n => PosBinary n -> PosBinary (2 * n)
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Bin1 :: KnownNat n => PosBinary n -> PosBinary (1 + 2 * n)
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deriving instance Show (PosBinary n)
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instance ShowN PosBinary where showsPrecN = showsPrec
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instance PositiveSingleton PosBinary where
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posSingleton :: forall n. KnownNat n => PosBinary (1 + n)
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posSingleton = case natVal (Proxy :: Proxy n) of
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0 -> (unsafeCoerce :: PosBinary 1 -> PosBinary (1 + n)) BinOne
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n -> case someNatVal ((n - 1) `div` 2) of
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Just (SomeNat (p :: Proxy m)) -> case someNatVal (natVal (Proxy :: Proxy m) + 1) of
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Just (SomeNat (q :: Proxy l)) -> case unsafeCoerce Refl :: l :~: 1 + m of
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Refl -> if even n
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then (unsafeCoerce :: PosBinary (1 + 2 * l) -> PosBinary (1 + n)) $ Bin1 posSingleton
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else (unsafeCoerce :: PosBinary (2 * l) -> PosBinary (1 + n)) $ Bin0 posSingleton
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Nothing -> error $ "Malformed KnownNat instance: " ++ show n
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Nothing -> error $ "Malformed KnownNat instance: " ++ show n
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-- | A positive number is either a digit in @[1, b)@, or another positive number times @b@ plus a digit in @[0, b)@.
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data PosBase (p :: Nat -> *) (b :: Nat) (n :: Nat) where
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BaseLead :: (KnownNat k, 1 + k <= b, k ~ (1 + n)) => p k -> PosBase p b k
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BaseDigit :: (KnownNat k, 1 + k <= b, KnownNat n) => p k -> PosBase p b n -> PosBase p b (k + b * n)
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instance ShowN p => Show (PosBase p b n) where
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showsPrec d (BaseLead a) = showParen (d > 10) $ showString "BaseLead " . showsPrecN 11 a
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showsPrec d (BaseDigit a b) = showParen (d > 10) $ showString "BaseDigit " . showsPrecN 11 a . showString " " . showsPrec 11 b
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instance ShowN p => ShowN (PosBase p b) where showsPrecN = showsPrec
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instance (KnownNat b, NatSingleton p) => PositiveSingleton (PosBase p b) where
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posSingleton :: forall n. KnownNat n => PosBase p b (1 + n)
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posSingleton = case natVal (Proxy :: Proxy n) of
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n | n < base - 1 -> case someNatVal (n + 1) of
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Just (SomeNat (p :: Proxy k)) -> case unsafeCoerce Refl :: k :~: (1 + n) of
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Refl -> case unsafeCoerce Refl :: (1 + (1 + n) <=? b) :~: True of
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Refl -> BaseLead natSingleton
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Nothing -> error $ "Malformed KnownNat instance: " ++ show n
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n -> case someNatVal ((n - base + 1) `div` base) of
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Just (SomeNat (p :: Proxy m)) -> case someNatVal (natVal (Proxy :: Proxy m) + 1) of
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Just (SomeNat (q :: Proxy l)) -> case unsafeCoerce Refl :: l :~: 1 + m of
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Refl -> case someNatVal ((n - base + 1) `mod` base) of
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Just (SomeNat (p :: Proxy k)) -> (unsafeCoerce :: PosBase p b (k + b * l) -> PosBase p b (1 + n)) $ case unsafeCoerce Refl :: (1 + k <=? b) :~: True of
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Refl -> BaseDigit (natSingleton :: p k) (posSingleton :: PosBase p b l)
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Nothing -> error $ "Malformed KnownNat instance: " ++ show base
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Nothing -> error $ "Malformed KnownNat instance: " ++ show base
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Nothing -> error $ "Malformed KnownNat instance: " ++ show n
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where
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base = natVal (Proxy :: Proxy b)
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30
LICENSE
Normal file
30
LICENSE
Normal file
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|
@ -0,0 +1,30 @@
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|||
Copyright (c) 2017, mniip
|
||||
|
||||
All rights reserved.
|
||||
|
||||
Redistribution and use in source and binary forms, with or without
|
||||
modification, are permitted provided that the following conditions are met:
|
||||
|
||||
* Redistributions of source code must retain the above copyright
|
||||
notice, this list of conditions and the following disclaimer.
|
||||
|
||||
* Redistributions in binary form must reproduce the above
|
||||
copyright notice, this list of conditions and the following
|
||||
disclaimer in the documentation and/or other materials provided
|
||||
with the distribution.
|
||||
|
||||
* Neither the name of mniip nor the names of other
|
||||
contributors may be used to endorse or promote products derived
|
||||
from this software without specific prior written permission.
|
||||
|
||||
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
|
||||
"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
|
||||
LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
|
||||
A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
|
||||
OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
|
||||
SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
|
||||
LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
|
||||
DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
|
||||
THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
|
||||
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
|
||||
OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
|
||||
2
Setup.hs
Normal file
2
Setup.hs
Normal file
|
|
@ -0,0 +1,2 @@
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import Distribution.Simple
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||||
main = defaultMain
|
||||
17
singleton-typelits.cabal
Normal file
17
singleton-typelits.cabal
Normal file
|
|
@ -0,0 +1,17 @@
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name: singleton-typelits
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version: 0.0.0.0
|
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synopsis: Singletons and induction over GHC TypeLits
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description: Singletons and induction schemes over 'GHC.TypeLits.Nat'
|
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homepage: https://github.com/mniip/singleton-typelits
|
||||
license: BSD3
|
||||
license-file: LICENSE
|
||||
author: mniip
|
||||
maintainer: mniip@mniip.com
|
||||
category: Data
|
||||
build-type: Simple
|
||||
cabal-version: >=1.10
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|
||||
library
|
||||
exposed-modules: GHC.TypeLits.Singletons, GHC.TypeLits.Induction
|
||||
build-depends: base >=4.9 && <4.11
|
||||
default-language: Haskell2010
|
||||
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