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|
{-# OPTIONS_HADDOCK show-extensions #-}
{-# LANGUAGE AutoDeriveTypeable #-}
{-# LANGUAGE CPP #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE UndecidableInstances #-}
#if __GLASGOW_HASKELL__ > 805
{-# LANGUAGE NoStarIsType #-}
#endif
{- |
Copyright : Copyright (C) 2006-2015 Bjorn Buckwalter
License : BSD3
Maintainer : bjorn.buckwalter@gmail.com
Stability : Stable
Portability: GHC only
= Summary
Type-level integers for GHC 7.8+.
We provide type level arithmetic operations. We also provide term-level arithmetic operations on proxys,
and conversion from the type level to the term level.
= Planned Obsolesence
We commit this package to hackage in sure and certain hope of the coming of glorious GHC integer type literals,
when the sea shall give up her dead, and this package shall be rendered unto obsolescence.
-}
module Numeric.NumType.DK.Integers
(
-- * Type-Level Integers
type TypeInt(..),
-- * Type-level Arithmetic
Pred, Succ, Negate, Abs, Signum,
type (+), type (-), type (*), type (/), type (^),
-- * Arithmetic on Proxies
pred, succ, negate, abs, signum,
(+), (-), (*), (/), (^),
-- * Convenience Synonyms for Proxies
zero,
pos1, pos2, pos3, pos4, pos5, pos6, pos7, pos8, pos9,
neg1, neg2, neg3, neg4, neg5, neg6, neg7, neg8, neg9,
-- * Conversion from Types to Terms
KnownTypeInt(..)
)
where
import Data.Proxy
import Prelude hiding ((+), (-), (*), (/), (^), pred, succ, negate, abs, signum)
import qualified Prelude
#if MIN_VERSION_base(4, 8, 0)
-- Use @Nat@s from @GHC.TypeLits@.
import qualified GHC.TypeLits as TN
type Z = 0 -- GHC.TypeLits
type N1 = 1 -- GHC.TypeLits
#else
-- Use custom @Typeable@ @Nat@s.
import qualified Numeric.NumType.DK.Naturals as TN
type Z = 'TN.Z
type N1 = 'TN.S 'TN.Z -- TypeInt.DK.Nats
#endif
-- Use the same fixity for operators as the Prelude.
infixr 8 ^
infixl 7 *, /
infixl 6 +, -
-- Natural numbers
-- ===============
type family NatPred (n::TN.Nat) :: TN.Nat where NatPred n = n TN.- N1
type family NatSucc (n::TN.Nat) :: TN.Nat where NatSucc n = n TN.+ N1
-- Integers
-- ========
data TypeInt = Neg10Minus TN.Nat -- 10, 11, 12, 13, ...
| Neg9
| Neg8
| Neg7
| Neg6
| Neg5
| Neg4
| Neg3
| Neg2
| Neg1
| Zero -- 0
| Pos1
| Pos2
| Pos3
| Pos4
| Pos5
| Pos6
| Pos7
| Pos8
| Pos9
| Pos10Plus TN.Nat -- -10, -11, -12, -13, ...
-- Unary operations
-- ----------------
type family Pred (i::TypeInt) :: TypeInt where
Pred ('Neg10Minus n) = 'Neg10Minus (NatSucc n)
Pred 'Neg9 = 'Neg10Minus Z
Pred 'Neg8 = 'Neg9
Pred 'Neg7 = 'Neg8
Pred 'Neg6 = 'Neg7
Pred 'Neg5 = 'Neg6
Pred 'Neg4 = 'Neg5
Pred 'Neg3 = 'Neg4
Pred 'Neg2 = 'Neg3
Pred 'Neg1 = 'Neg2
Pred 'Zero = 'Neg1
Pred 'Pos1 = 'Zero
Pred 'Pos2 = 'Pos1
Pred 'Pos3 = 'Pos2
Pred 'Pos4 = 'Pos3
Pred 'Pos5 = 'Pos4
Pred 'Pos6 = 'Pos5
Pred 'Pos7 = 'Pos6
Pred 'Pos8 = 'Pos7
Pred 'Pos9 = 'Pos8
Pred ('Pos10Plus Z) = 'Pos9
Pred ('Pos10Plus n) = 'Pos10Plus (NatPred n)
type family Succ (i::TypeInt) :: TypeInt where
Succ ('Neg10Minus Z) = 'Neg9
Succ ('Neg10Minus n) = 'Neg10Minus (NatPred n)
Succ 'Neg9 = 'Neg8
Succ 'Neg8 = 'Neg7
Succ 'Neg7 = 'Neg6
Succ 'Neg6 = 'Neg5
Succ 'Neg5 = 'Neg4
Succ 'Neg4 = 'Neg3
Succ 'Neg3 = 'Neg2
Succ 'Neg2 = 'Neg1
Succ 'Neg1 = 'Zero
Succ 'Zero = 'Pos1
Succ 'Pos1 = 'Pos2
Succ 'Pos2 = 'Pos3
Succ 'Pos3 = 'Pos4
Succ 'Pos4 = 'Pos5
Succ 'Pos5 = 'Pos6
Succ 'Pos6 = 'Pos7
Succ 'Pos7 = 'Pos8
Succ 'Pos8 = 'Pos9
Succ 'Pos9 = 'Pos10Plus Z
Succ ('Pos10Plus n) = 'Pos10Plus (NatSucc n)
-- | TypeInt negation.
type family Negate (i::TypeInt) :: TypeInt where
Negate ('Neg10Minus n) = 'Pos10Plus n
Negate 'Neg9 = 'Pos9
Negate 'Neg8 = 'Pos8
Negate 'Neg7 = 'Pos7
Negate 'Neg6 = 'Pos6
Negate 'Neg5 = 'Pos5
Negate 'Neg4 = 'Pos4
Negate 'Neg3 = 'Pos3
Negate 'Neg2 = 'Pos2
Negate 'Neg1 = 'Pos1
Negate 'Zero = 'Zero
Negate 'Pos1 = 'Neg1
Negate 'Pos2 = 'Neg2
Negate 'Pos3 = 'Neg3
Negate 'Pos4 = 'Neg4
Negate 'Pos5 = 'Neg5
Negate 'Pos6 = 'Neg6
Negate 'Pos7 = 'Neg7
Negate 'Pos8 = 'Neg8
Negate 'Pos9 = 'Neg9
Negate ('Pos10Plus n) = 'Neg10Minus n
-- | Absolute value.
type family Abs (i::TypeInt) :: TypeInt where
Abs ('Neg10Minus n) = 'Pos10Plus n
Abs 'Neg9 = 'Pos9
Abs 'Neg8 = 'Pos8
Abs 'Neg7 = 'Pos7
Abs 'Neg6 = 'Pos6
Abs 'Neg5 = 'Pos5
Abs 'Neg4 = 'Pos4
Abs 'Neg3 = 'Pos3
Abs 'Neg2 = 'Pos2
Abs 'Neg1 = 'Pos1
Abs i = i
-- | Signum.
type family Signum (i::TypeInt) :: TypeInt where
Signum ('Neg10Minus n) = 'Neg1
Signum 'Neg9 = 'Neg1
Signum 'Neg8 = 'Neg1
Signum 'Neg7 = 'Neg1
Signum 'Neg6 = 'Neg1
Signum 'Neg5 = 'Neg1
Signum 'Neg4 = 'Neg1
Signum 'Neg3 = 'Neg1
Signum 'Neg2 = 'Neg1
Signum 'Neg1 = 'Neg1
Signum 'Zero = 'Zero
Signum i = 'Pos1
-- Binary operations
-- -----------------
-- | TypeInt addition.
type family (i::TypeInt) + (i'::TypeInt) :: TypeInt where
'Zero + i = i
i + 'Neg10Minus n = Pred i + Succ ('Neg10Minus n)
i + 'Neg9 = Pred i + 'Neg8
i + 'Neg8 = Pred i + 'Neg7
i + 'Neg7 = Pred i + 'Neg6
i + 'Neg6 = Pred i + 'Neg5
i + 'Neg5 = Pred i + 'Neg4
i + 'Neg4 = Pred i + 'Neg3
i + 'Neg3 = Pred i + 'Neg2
i + 'Neg2 = Pred i + 'Neg1
i + 'Neg1 = Pred i
i + 'Zero = i
i + i' = Succ i + Pred i' -- i + Pos
-- | TypeInt subtraction.
type family (i::TypeInt) - (i'::TypeInt) :: TypeInt where
i - i' = i + Negate i'
-- | TypeInt multiplication.
type family (i::TypeInt) * (i'::TypeInt) :: TypeInt
where
'Zero * i = 'Zero
i * 'Zero = 'Zero
i * 'Pos1 = i
i * 'Pos2 = i + i
i * 'Pos3 = i + i + i
i * 'Pos4 = i + i + i + i
i * 'Pos5 = i + i + i + i + i
i * 'Pos6 = i + i + i + i + i + i
i * 'Pos7 = i + i + i + i + i + i + i
i * 'Pos8 = i + i + i + i + i + i + i + i
i * 'Pos9 = i + i + i + i + i + i + i + i + i
i * 'Pos10Plus n = i + i * Pred ('Pos10Plus n)
i * i' = Negate (i * Negate i')
-- | TypeInt exponentiation.
type family (i::TypeInt) ^ (i'::TypeInt) :: TypeInt
where
i ^ 'Zero = 'Pos1
i ^ 'Pos1 = i
i ^ 'Pos2 = i * i
i ^ 'Pos3 = i * i * i
i ^ 'Pos4 = i * i * i * i
i ^ 'Pos5 = i * i * i * i * i
i ^ 'Pos6 = i * i * i * i * i * i
i ^ 'Pos7 = i * i * i * i * i * i * i
i ^ 'Pos8 = i * i * i * i * i * i * i * i
i ^ 'Pos9 = i * i * i * i * i * i * i * i * i
i ^ 'Pos10Plus n = i * i ^ Pred ('Pos10Plus n)
-- | TypeInt division.
type family (i::TypeInt) / (i'::TypeInt) :: TypeInt
where
i / 'Pos1 = i
i / 'Neg1 = Negate i
-- @Zero / n = Zero@ would allow division by zero.
-- @i / i = Pos1@ would allow division by zero.
'Zero / ('Neg10Minus n) = 'Zero
'Zero / 'Neg9 = 'Zero
'Zero / 'Neg8 = 'Zero
'Zero / 'Neg7 = 'Zero
'Zero / 'Neg6 = 'Zero
'Zero / 'Neg5 = 'Zero
'Zero / 'Neg4 = 'Zero
'Zero / 'Neg3 = 'Zero
'Zero / 'Neg2 = 'Zero
'Zero / 'Pos2 = 'Zero
'Zero / 'Pos3 = 'Zero
'Zero / 'Pos4 = 'Zero
'Zero / 'Pos5 = 'Zero
'Zero / 'Pos6 = 'Zero
'Zero / 'Pos7 = 'Zero
'Zero / 'Pos8 = 'Zero
'Zero / 'Pos9 = 'Zero
'Zero / ('Pos10Plus n) = 'Zero
'Neg2 / 'Neg2 = 'Pos1
'Neg3 / 'Neg3 = 'Pos1
'Neg4 / 'Neg4 = 'Pos1
'Neg5 / 'Neg5 = 'Pos1
'Neg6 / 'Neg6 = 'Pos1
'Neg7 / 'Neg7 = 'Pos1
'Neg8 / 'Neg8 = 'Pos1
'Neg9 / 'Neg9 = 'Pos1
'Neg10Minus n / 'Neg10Minus n = 'Pos1
'Neg2 / 'Pos2 = 'Neg1
'Neg3 / 'Pos3 = 'Neg1
'Neg4 / 'Pos4 = 'Neg1
'Neg5 / 'Pos5 = 'Neg1
'Neg6 / 'Pos6 = 'Neg1
'Neg7 / 'Pos7 = 'Neg1
'Neg8 / 'Pos8 = 'Neg1
'Neg9 / 'Pos9 = 'Neg1
'Neg10Minus n / 'Pos10Plus n = 'Neg1
'Pos2 / 'Neg2 = 'Neg1
'Pos3 / 'Neg3 = 'Neg1
'Pos4 / 'Neg4 = 'Neg1
'Pos5 / 'Neg5 = 'Neg1
'Pos6 / 'Neg6 = 'Neg1
'Pos7 / 'Neg7 = 'Neg1
'Pos8 / 'Neg8 = 'Neg1
'Pos9 / 'Neg9 = 'Neg1
'Pos10Plus n / 'Neg10Minus n = 'Neg1
'Pos2 / 'Pos2 = 'Pos1
'Pos3 / 'Pos3 = 'Pos1
'Pos4 / 'Pos4 = 'Pos1
'Pos5 / 'Pos5 = 'Pos1
'Pos6 / 'Pos6 = 'Pos1
'Pos7 / 'Pos7 = 'Pos1
'Pos8 / 'Pos8 = 'Pos1
'Pos9 / 'Pos9 = 'Pos1
'Pos10Plus n / 'Pos10Plus n = 'Pos1
'Neg4 / 'Neg2 = 'Pos2
'Neg6 / 'Neg2 = 'Pos3
'Neg8 / 'Neg2 = 'Pos4
'Neg6 / 'Neg3 = 'Pos2
'Neg9 / 'Neg3 = 'Pos3
'Neg8 / 'Neg4 = 'Pos2
'Neg10Minus n / i = ('Neg10Minus n + Abs i) / i - Signum i
'Neg4 / 'Pos2 = 'Neg2
'Neg6 / 'Pos2 = 'Neg3
'Neg8 / 'Pos2 = 'Neg4
'Neg6 / 'Pos3 = 'Neg2
'Neg9 / 'Pos3 = 'Neg3
'Neg8 / 'Pos4 = 'Neg2
'Pos4 / 'Neg2 = 'Neg2
'Pos6 / 'Neg2 = 'Neg3
'Pos8 / 'Neg2 = 'Neg4
'Pos6 / 'Neg3 = 'Neg2
'Pos9 / 'Neg3 = 'Neg3
'Pos8 / 'Neg4 = 'Neg2
'Pos4 / 'Pos2 = 'Pos2
'Pos6 / 'Pos2 = 'Pos3
'Pos8 / 'Pos2 = 'Pos4
'Pos6 / 'Pos3 = 'Pos2
'Pos9 / 'Pos3 = 'Pos3
'Pos8 / 'Pos4 = 'Pos2
'Pos10Plus n / i = ('Pos10Plus n - Abs i) / i + Signum i
-- Term level
-- ==========
-- Term level operators
-- --------------------
pred :: Proxy i -> Proxy (Pred i); pred _ = Proxy
succ :: Proxy i -> Proxy (Succ i); succ _ = Proxy
negate :: Proxy i -> Proxy (Negate i); negate _ = Proxy
abs :: Proxy i -> Proxy (Abs i); abs _ = Proxy
signum :: Proxy i -> Proxy (Signum i); signum _ = Proxy
(+) :: Proxy i -> Proxy i' -> Proxy (i + i'); _ + _ = Proxy
(-) :: Proxy i -> Proxy i' -> Proxy (i - i'); _ - _ = Proxy
(*) :: Proxy i -> Proxy i' -> Proxy (i * i'); _ * _ = Proxy
(/) :: Proxy i -> Proxy i' -> Proxy (i / i'); _ / _ = Proxy
(^) :: Proxy i -> Proxy i' -> Proxy (i ^ i'); _ ^ _ = Proxy
-- Term level TypeNats for convenience
-- -----------------------------------
neg9 :: Proxy 'Neg9
neg9 = Proxy :: Proxy 'Neg9
neg8 :: Proxy 'Neg8
neg8 = Proxy :: Proxy 'Neg8
neg7 :: Proxy 'Neg7
neg7 = Proxy :: Proxy 'Neg7
neg6 :: Proxy 'Neg6
neg6 = Proxy :: Proxy 'Neg6
neg5 :: Proxy 'Neg5
neg5 = Proxy :: Proxy 'Neg5
neg4 :: Proxy 'Neg4
neg4 = Proxy :: Proxy 'Neg4
neg3 :: Proxy 'Neg3
neg3 = Proxy :: Proxy 'Neg3
neg2 :: Proxy 'Neg2
neg2 = Proxy :: Proxy 'Neg2
neg1 :: Proxy 'Neg1
neg1 = Proxy :: Proxy 'Neg1
zero :: Proxy 'Zero
zero = Proxy :: Proxy 'Zero
pos1 :: Proxy 'Pos1
pos1 = Proxy :: Proxy 'Pos1
pos2 :: Proxy 'Pos2
pos2 = Proxy :: Proxy 'Pos2
pos3 :: Proxy 'Pos3
pos3 = Proxy :: Proxy 'Pos3
pos4 :: Proxy 'Pos4
pos4 = Proxy :: Proxy 'Pos4
pos5 :: Proxy 'Pos5
pos5 = Proxy :: Proxy 'Pos5
pos6 :: Proxy 'Pos6
pos6 = Proxy :: Proxy 'Pos6
pos7 :: Proxy 'Pos7
pos7 = Proxy :: Proxy 'Pos7
pos8 :: Proxy 'Pos8
pos8 = Proxy :: Proxy 'Pos8
pos9 :: Proxy 'Pos9
pos9 = Proxy :: Proxy 'Pos9
-- Reification
-- -----------
-- | Conversion to a @Num@.
class KnownTypeInt (i::TypeInt) where toNum :: Num a => Proxy i -> a
instance KnownTypeInt (Succ ('Neg10Minus n)) => KnownTypeInt ('Neg10Minus n)
where toNum = (Prelude.- 1) . toNum . succ
instance KnownTypeInt 'Neg9 where toNum _ = -9
instance KnownTypeInt 'Neg8 where toNum _ = -8
instance KnownTypeInt 'Neg7 where toNum _ = -7
instance KnownTypeInt 'Neg6 where toNum _ = -6
instance KnownTypeInt 'Neg5 where toNum _ = -5
instance KnownTypeInt 'Neg4 where toNum _ = -4
instance KnownTypeInt 'Neg3 where toNum _ = -3
instance KnownTypeInt 'Neg2 where toNum _ = -2
instance KnownTypeInt 'Neg1 where toNum _ = -1
instance KnownTypeInt 'Zero where toNum _ = 0
instance KnownTypeInt 'Pos1 where toNum _ = 1
instance KnownTypeInt 'Pos2 where toNum _ = 2
instance KnownTypeInt 'Pos3 where toNum _ = 3
instance KnownTypeInt 'Pos4 where toNum _ = 4
instance KnownTypeInt 'Pos5 where toNum _ = 5
instance KnownTypeInt 'Pos6 where toNum _ = 6
instance KnownTypeInt 'Pos7 where toNum _ = 7
instance KnownTypeInt 'Pos8 where toNum _ = 8
instance KnownTypeInt 'Pos9 where toNum _ = 9
instance KnownTypeInt (Pred ('Pos10Plus n)) => KnownTypeInt ('Pos10Plus n)
where toNum = (Prelude.+ 1) . toNum . pred
|
