1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
|
{-# LANGUAGE Trustworthy #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE NoStarIsType #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE ExistentialQuantification #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE NoImplicitPrelude #-}
{-# LANGUAGE MagicHash #-}
{-# LANGUAGE PolyKinds #-}
{-| This module is an internal GHC module. It declares the constants used
in the implementation of type-level natural numbers. The programmer interface
for working with type-level naturals should be defined in a separate library.
@since 4.10.0.0
-}
module GHC.TypeNats
( -- * Nat Kind
Nat -- declared in GHC.Types in package ghc-prim
-- * Linking type and value level
, KnownNat, natVal, natVal'
, SomeNat(..)
, someNatVal
, sameNat
-- * Functions on type literals
, type (<=), type (<=?), type (+), type (*), type (^), type (-)
, CmpNat
, Div, Mod, Log2
) where
import GHC.Base(Eq(..), Ord(..), Bool(True), Ordering(..), otherwise)
import GHC.Types( Nat )
import GHC.Natural(Natural)
import GHC.Show(Show(..))
import GHC.Read(Read(..))
import GHC.Prim(magicDict, Proxy#)
import Data.Maybe(Maybe(..))
import Data.Proxy (Proxy(..))
import Data.Type.Equality((:~:)(Refl))
import Unsafe.Coerce(unsafeCoerce)
--------------------------------------------------------------------------------
-- | This class gives the integer associated with a type-level natural.
-- There are instances of the class for every concrete literal: 0, 1, 2, etc.
--
-- @since 4.7.0.0
class KnownNat (n :: Nat) where
natSing :: SNat n
-- | @since 4.10.0.0
natVal :: forall n proxy. KnownNat n => proxy n -> Natural
natVal _ = case natSing :: SNat n of
SNat x -> x
-- | @since 4.10.0.0
natVal' :: forall n. KnownNat n => Proxy# n -> Natural
natVal' _ = case natSing :: SNat n of
SNat x -> x
-- | This type represents unknown type-level natural numbers.
--
-- @since 4.10.0.0
data SomeNat = forall n. KnownNat n => SomeNat (Proxy n)
-- | Convert an integer into an unknown type-level natural.
--
-- @since 4.10.0.0
someNatVal :: Natural -> SomeNat
someNatVal n = withSNat SomeNat (SNat n) Proxy
-- | @since 4.7.0.0
instance Eq SomeNat where
SomeNat x == SomeNat y = natVal x == natVal y
-- | @since 4.7.0.0
instance Ord SomeNat where
compare (SomeNat x) (SomeNat y) = compare (natVal x) (natVal y)
-- | @since 4.7.0.0
instance Show SomeNat where
showsPrec p (SomeNat x) = showsPrec p (natVal x)
-- | @since 4.7.0.0
instance Read SomeNat where
readsPrec p xs = do (a,ys) <- readsPrec p xs
[(someNatVal a, ys)]
--------------------------------------------------------------------------------
infix 4 <=?, <=
infixl 6 +, -
infixl 7 *, `Div`, `Mod`
infixr 8 ^
-- | Comparison of type-level naturals, as a constraint.
--
-- @since 4.7.0.0
type x <= y = (x <=? y) ~ 'True
-- | Comparison of type-level naturals, as a function.
--
-- @since 4.7.0.0
type family CmpNat (m :: Nat) (n :: Nat) :: Ordering
{- | Comparison of type-level naturals, as a function.
NOTE: The functionality for this function should be subsumed
by 'CmpNat', so this might go away in the future.
Please let us know, if you encounter discrepancies between the two. -}
type family (m :: Nat) <=? (n :: Nat) :: Bool
-- | Addition of type-level naturals.
--
-- @since 4.7.0.0
type family (m :: Nat) + (n :: Nat) :: Nat
-- | Multiplication of type-level naturals.
--
-- @since 4.7.0.0
type family (m :: Nat) * (n :: Nat) :: Nat
-- | Exponentiation of type-level naturals.
--
-- @since 4.7.0.0
type family (m :: Nat) ^ (n :: Nat) :: Nat
-- | Subtraction of type-level naturals.
--
-- @since 4.7.0.0
type family (m :: Nat) - (n :: Nat) :: Nat
-- | Division (round down) of natural numbers.
-- @Div x 0@ is undefined (i.e., it cannot be reduced).
--
-- @since 4.11.0.0
type family Div (m :: Nat) (n :: Nat) :: Nat
-- | Modulus of natural numbers.
-- @Mod x 0@ is undefined (i.e., it cannot be reduced).
--
-- @since 4.11.0.0
type family Mod (m :: Nat) (n :: Nat) :: Nat
-- | Log base 2 (round down) of natural numbers.
-- @Log 0@ is undefined (i.e., it cannot be reduced).
--
-- @since 4.11.0.0
type family Log2 (m :: Nat) :: Nat
--------------------------------------------------------------------------------
-- | We either get evidence that this function was instantiated with the
-- same type-level numbers, or 'Nothing'.
--
-- @since 4.7.0.0
sameNat :: (KnownNat a, KnownNat b) =>
Proxy a -> Proxy b -> Maybe (a :~: b)
sameNat x y
| natVal x == natVal y = Just (unsafeCoerce Refl)
| otherwise = Nothing
--------------------------------------------------------------------------------
-- PRIVATE:
newtype SNat (n :: Nat) = SNat Natural
data WrapN a b = WrapN (KnownNat a => Proxy a -> b)
-- See Note [magicDictId magic] in "basicType/MkId.hs"
withSNat :: (KnownNat a => Proxy a -> b)
-> SNat a -> Proxy a -> b
withSNat f x y = magicDict (WrapN f) x y
|
