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|
{-# OPTIONS_GHC -fno-warn-orphans #-}
{-# LANGUAGE DeriveDataTypeable #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE EmptyDataDecls #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE TemplateHaskell #-}
{-# LANGUAGE ScopedTypeVariables #-}
-- |
-- Module : TypeLevel.Number.Int
-- Copyright : Alexey Khudyakov
-- License : BSD3-style (see LICENSE)
--
-- Maintainer : Alexey Khudyakov <alexey.skladnoy@gmail.com>
-- Stability : unstable
-- Portability : unportable (GHC only)
--
-- Type level signed integer numbers are implemented using balanced
-- ternary encoding much in the same way as natural numbers.
--
-- Currently following operations are supported: Next, Prev, Add, Sub,
-- Mul.
module TypeLevel.Number.Int ( -- * Integer numbers
ZZ
, Dn
, D0
, D1
, IntT(..)
-- ** Lifting
, SomeInt
, withInt
-- * Template haskell utilities
, intT
, module TypeLevel.Number.Classes
) where
import Data.Typeable (Typeable)
import Language.Haskell.TH
import TypeLevel.Number.Classes
import TypeLevel.Number.Int.Types
import TypeLevel.Util
splitToTrits :: Integer -> [Int]
splitToTrits 0 = []
splitToTrits x | n == 0 = 0 : splitToTrits rest
| n == 1 = 1 : splitToTrits rest
| n == 2 = -1 : splitToTrits (rest + 1)
where
(rest,n) = divMod x 3
splitToTrits _ = error "Internal error"
-- | Generate type for integer number.
intT :: Integer -> TypeQ
intT = foldr appT (conT ''ZZ) . map con . splitToTrits
where
con (-1) = conT ''Dn
con 0 = conT ''D0
con 1 = conT ''D1
con x = error $ "Strange trit: " ++ show x
----------------------------------------------------------------
--
-- | Type class for type level integers. Only numbers without leading
-- zeroes are members of the class.
class IntT n where
-- | Convert natural number to integral value. It's not checked
-- whether value could be represented.
toInt :: Integral i => n -> i
instance IntT ZZ where toInt _ = 0
instance IntT (D1 ZZ) where toInt _ = 1
instance IntT (Dn ZZ) where toInt _ = -1
instance IntT (Dn n) => IntT (Dn (Dn n)) where toInt n = -1 + 3 * toInt' n
instance IntT (Dn n) => IntT (D0 (Dn n)) where toInt n = 0 + 3 * toInt' n
instance IntT (Dn n) => IntT (D1 (Dn n)) where toInt n = 1 + 3 * toInt' n
instance IntT (D0 n) => IntT (Dn (D0 n)) where toInt n = -1 + 3 * toInt' n
instance IntT (D0 n) => IntT (D0 (D0 n)) where toInt n = 0 + 3 * toInt' n
instance IntT (D0 n) => IntT (D1 (D0 n)) where toInt n = 1 + 3 * toInt' n
instance IntT (D1 n) => IntT (Dn (D1 n)) where toInt n = -1 + 3 * toInt' n
instance IntT (D1 n) => IntT (D0 (D1 n)) where toInt n = 0 + 3 * toInt' n
instance IntT (D1 n) => IntT (D1 (D1 n)) where toInt n = 1 + 3 * toInt' n
toInt' :: (IntT n, Integral i) => t n -> i
toInt' = toInt . cdr
instance Show ZZ where show _ = "[0:Z]"
instance IntT (Dn n) => Show (Dn n) where show n = "["++show (toInt n :: Integer)++":Z]"
instance IntT (D0 n) => Show (D0 n) where show n = "["++show (toInt n :: Integer)++":Z]"
instance IntT (D1 n) => Show (D1 n) where show n = "["++show (toInt n :: Integer)++":Z]"
-- | Some natural number
data SomeInt where
SomeInt :: IntT n => n -> SomeInt
deriving Typeable
instance Show SomeInt where
showsPrec d (SomeInt n) = showParen (d > 10) $
showString "withInt SomeInt " . shows (toInt n :: Integer)
-- | Apply function which could work with any 'Nat' value only know at runtime.
withInt :: forall i a. (Integral i) => (forall n. IntT n => n -> a) -> i -> a
withInt f i0
| i0 == 0 = f (undefined :: ZZ)
| otherwise = cont (fromIntegral i0) f f f
where
cont :: Integer -> (forall n m. (IntT n, n ~ Dn m) => n -> a)
-> (forall n m. (IntT n, n ~ D0 m) => n -> a)
-> (forall n m. (IntT n, n ~ D1 m) => n -> a) -> a
cont (-1) kN _ _ = kN (undefined :: Dn ZZ)
cont 1 _ _ k1 = k1 (undefined :: D1 ZZ)
cont i kN k0 k1 = cont i' kN' k0' k1'
where
(i',bit) = case divMod i 3 of
(x,2) -> (x+1,-1)
x -> x
kN' :: forall n m. (IntT n, n ~ Dn m) => n -> a
kN' _ | bit == -1 = kN (undefined :: Dn n)
| bit == 0 = k0 (undefined :: D0 n)
| otherwise = k1 (undefined :: D1 n)
k0' :: forall n m. (IntT n, n ~ D0 m) => n -> a
k0' _ | bit == -1 = kN (undefined :: Dn n)
| bit == 0 = k0 (undefined :: D0 n)
| otherwise = k1 (undefined :: D1 n)
k1' :: forall n m. (IntT n, n ~ D1 m) => n -> a
k1' _ | bit == -1 = kN (undefined :: Dn n)
| bit == 0 = k0 (undefined :: D0 n)
| otherwise = k1 (undefined :: D1 n)
----------------------------------------------------------------
-- Number normalization
type family AddBit n :: *
type instance AddBit ZZ = ZZ
type instance AddBit (Dn a) = D0 (Dn a)
type instance AddBit (D0 a) = D0 (D0 a)
type instance AddBit (D1 a) = D0 (D1 a)
type instance Normalized ZZ = ZZ
type instance Normalized (Dn n) = Dn (Normalized n)
type instance Normalized (D0 n) = AddBit (Normalized n)
type instance Normalized (D1 n) = D1 (Normalized n)
----------------------------------------------------------------
-- Next Number
type instance Next ZZ = D1 ZZ
type instance Next (Dn n) = Normalized (D0 n)
type instance Next (D0 n) = D1 n
type instance Next (D1 n) = Normalized (Dn (Next n))
----------------------------------------------------------------
-- Previous number
type instance Prev ZZ = Dn ZZ
type instance Prev (Dn n) = Normalized (D1 (Prev n))
type instance Prev (D0 n) = Dn n
type instance Prev (D1 n) = Normalized (D0 n)
----------------------------------------------------------------
-- Negate number
type instance Negate ZZ = ZZ
type instance Negate (Dn n) = D1 (Negate n)
type instance Negate (D0 n) = D0 (Negate n)
type instance Negate (D1 n) = Dn (Negate n)
----------------------------------------------------------------
-- Addition
-- Type class which actually implement addtition of natural numbers
type family Add' n m carry :: *
data CarryN
data Carry0
data Carry1
-- Special cases with ZZ
type instance Add' ZZ ZZ Carry0 = ZZ
type instance Add' ZZ (Dn n) Carry0 = (Dn n)
type instance Add' ZZ (D0 n) Carry0 = (D0 n)
type instance Add' ZZ (D1 n) Carry0 = (D1 n)
type instance Add' (Dn n) ZZ Carry0 = (Dn n)
type instance Add' (D0 n) ZZ Carry0 = (D0 n)
type instance Add' (D1 n) ZZ Carry0 = (D1 n)
--
type instance Add' ZZ ZZ CarryN = Dn ZZ
type instance Add' ZZ (Dn n) CarryN = Prev (Dn n)
type instance Add' ZZ (D0 n) CarryN = (Dn n)
type instance Add' ZZ (D1 n) CarryN = (D0 n)
type instance Add' (Dn n) ZZ CarryN = Prev (Dn n)
type instance Add' (D0 n) ZZ CarryN = (Dn n)
type instance Add' (D1 n) ZZ CarryN = (D0 n)
--
type instance Add' ZZ ZZ Carry1 = D1 ZZ
type instance Add' ZZ (Dn n) Carry1 = (D0 n)
type instance Add' ZZ (D0 n) Carry1 = (D1 n)
type instance Add' ZZ (D1 n) Carry1 = Next (D1 n)
type instance Add' (Dn n) ZZ Carry1 = (D0 n)
type instance Add' (D0 n) ZZ Carry1 = (D1 n)
type instance Add' (D1 n) ZZ Carry1 = Next (D1 n)
-- == General recursion ==
-- No carry
type instance Add' (Dn n) (Dn m) Carry0 = D1 (Add' n m CarryN)
type instance Add' (D0 n) (Dn m) Carry0 = Dn (Add' n m Carry0)
type instance Add' (D1 n) (Dn m) Carry0 = D0 (Add' n m Carry0)
--
type instance Add' (Dn n) (D0 m) Carry0 = Dn (Add' n m Carry0)
type instance Add' (D0 n) (D0 m) Carry0 = D0 (Add' n m Carry0)
type instance Add' (D1 n) (D0 m) Carry0 = D1 (Add' n m Carry0)
--
type instance Add' (Dn n) (D1 m) Carry0 = D0 (Add' n m Carry0)
type instance Add' (D0 n) (D1 m) Carry0 = D1 (Add' n m Carry0)
type instance Add' (D1 n) (D1 m) Carry0 = Dn (Add' n m Carry1)
-- Carry '-'
type instance Add' (Dn n) (Dn m) CarryN = D0 (Add' n m CarryN)
type instance Add' (D0 n) (Dn m) CarryN = D1 (Add' n m CarryN)
type instance Add' (D1 n) (Dn m) CarryN = Dn (Add' n m Carry0)
--
type instance Add' (Dn n) (D0 m) CarryN = D1 (Add' n m CarryN)
type instance Add' (D0 n) (D0 m) CarryN = Dn (Add' n m Carry0)
type instance Add' (D1 n) (D0 m) CarryN = D0 (Add' n m Carry0)
--
type instance Add' (Dn n) (D1 m) CarryN = Dn (Add' n m Carry0)
type instance Add' (D0 n) (D1 m) CarryN = D0 (Add' n m Carry0)
type instance Add' (D1 n) (D1 m) CarryN = D1 (Add' n m Carry0)
-- Carry '+'
type instance Add' (Dn n) (Dn m) Carry1 = Dn (Add' n m Carry0)
type instance Add' (D0 n) (Dn m) Carry1 = D0 (Add' n m Carry0)
type instance Add' (D1 n) (Dn m) Carry1 = D1 (Add' n m Carry0)
--
type instance Add' (Dn n) (D0 m) Carry1 = D0 (Add' n m Carry0)
type instance Add' (D0 n) (D0 m) Carry1 = D1 (Add' n m Carry0)
type instance Add' (D1 n) (D0 m) Carry1 = Dn (Add' n m Carry1)
--
type instance Add' (Dn n) (D1 m) Carry1 = D1 (Add' n m Carry0)
type instance Add' (D0 n) (D1 m) Carry1 = Dn (Add' n m Carry1)
type instance Add' (D1 n) (D1 m) Carry1 = D0 (Add' n m Carry1)
-- Instances for AddN
type instance Add ZZ ZZ = ZZ
type instance Add ZZ (Dn n) = Normalized (Dn n)
type instance Add ZZ (D0 n) = Normalized (D0 n)
type instance Add ZZ (D1 n) = Normalized (D1 n)
type instance Add (Dn n) ZZ = Normalized (Dn n)
type instance Add (D0 n) ZZ = Normalized (D0 n)
type instance Add (D1 n) ZZ = Normalized (D1 n)
--
type instance Add (Dn n) (Dn m) = Normalized (Add' (Dn n) (Dn m) Carry0)
type instance Add (D0 n) (Dn m) = Normalized (Add' (D0 n) (Dn m) Carry0)
type instance Add (D1 n) (Dn m) = Normalized (Add' (D1 n) (Dn m) Carry0)
--
type instance Add (Dn n) (D0 m) = Normalized (Add' (Dn n) (D0 m) Carry0)
type instance Add (D0 n) (D0 m) = Normalized (Add' (D0 n) (D0 m) Carry0)
type instance Add (D1 n) (D0 m) = Normalized (Add' (D1 n) (D0 m) Carry0)
--
type instance Add (Dn n) (D1 m) = Normalized (Add' (Dn n) (D1 m) Carry0)
type instance Add (D0 n) (D1 m) = Normalized (Add' (D0 n) (D1 m) Carry0)
type instance Add (D1 n) (D1 m) = Normalized (Add' (D1 n) (D1 m) Carry0)
----------------------------------------------------------------
-- Subtraction.
--
-- Subtraction is much easier since is ise defined using
-- addition and negation
type instance Sub ZZ ZZ = ZZ
type instance Sub ZZ (Dn n) = Negate (Dn n)
type instance Sub ZZ (D0 n) = Negate (D0 n)
type instance Sub ZZ (D1 n) = Negate (D1 n)
type instance Sub (Dn n) ZZ = (Dn n)
type instance Sub (D0 n) ZZ = (D0 n)
type instance Sub (D1 n) ZZ = (D1 n)
type instance Sub (Dn n) (Dn m) = Add (Dn n) (Negate (Dn m))
type instance Sub (D0 n) (Dn m) = Add (D0 n) (Negate (Dn m))
type instance Sub (D1 n) (Dn m) = Add (D1 n) (Negate (Dn m))
--
type instance Sub (Dn n) (D0 m) = Add (Dn n) (Negate (D0 m))
type instance Sub (D0 n) (D0 m) = Add (D0 n) (Negate (D0 m))
type instance Sub (D1 n) (D0 m) = Add (D1 n) (Negate (D0 m))
--
type instance Sub (Dn n) (D1 m) = Add (Dn n) (Negate (D1 m))
type instance Sub (D0 n) (D1 m) = Add (D0 n) (Negate (D1 m))
type instance Sub (D1 n) (D1 m) = Add (D1 n) (Negate (D1 m))
----------------------------------------------------------------
-- Multiplication
type instance Mul n ZZ = ZZ
type instance Mul n (Dn m) = Normalized (Add' (Negate n) (D0 (Mul n m)) Carry0)
type instance Mul n (D0 m) = Normalized (D0 (Mul n m))
type instance Mul n (D1 m) = Normalized (Add' n (D0 (Mul n m)) Carry0)
|
