{-# OPTIONS_GHC -Wno-partial-type-signatures #-}
{-# LANGUAGE PartialTypeSignatures #-}
{-# LANGUAGE TemplateHaskell #-}
{-|
    Module      :  Numeric.MixedType.Mul
    Description :  Bottom-up typed multiplication with ring laws
    Copyright   :  (c) Michal Konecny
    License     :  BSD3

    Maintainer  :  mikkonecny@gmail.com
    Stability   :  experimental
    Portability :  portable

-}

module Numeric.MixedTypes.Mul
(
  -- ** Multiplication
  CanMul, CanMulAsymmetric(..), CanMulBy, CanMulSameType
  , (*), product
  -- ** Tests
  , specCanMul, specCanMulNotMixed, specCanMulSameType
)
where

import Utils.TH.DeclForTypes

import Numeric.MixedTypes.PreludeHiding
import qualified Prelude as P
import Text.Printf

import qualified Data.List as List

import Test.Hspec
import Test.QuickCheck

import qualified Numeric.CollectErrors as CN
import Numeric.CollectErrors ( CN )

import Numeric.MixedTypes.Literals
import Numeric.MixedTypes.Bool
import Numeric.MixedTypes.Eq
-- import Numeric.MixedTypes.MinMaxAbs
import Numeric.MixedTypes.AddSub
import Numeric.MixedTypes.Reduce

{---- Multiplication -----}

type CanMul t1 t2 =
  (CanMulAsymmetric t1 t2, CanMulAsymmetric t2 t1,
   MulType t1 t2 ~ MulType t2 t1)

{-|
  A replacement for Prelude's `P.*`.  If @t1 = t2@ and @Num t1@,
  then one can use the default implementation to mirror Prelude's @*@.
-}
class CanMulAsymmetric t1 t2 where
  type MulType t1 t2
  type MulType t1 t2 = t1 -- default
  mul :: t1 -> t2 -> MulType t1 t2
  default mul :: (MulType t1 t2 ~ t1, t1~t2, P.Num t1) => t1 -> t2 -> MulType t1 t2
  mul = t1 -> t2 -> MulType t1 t2
forall a. Num a => a -> a -> a
(P.*)

infixl 7  *

(*) :: (CanMulAsymmetric t1 t2) => t1 -> t2 -> MulType t1 t2
* :: t1 -> t2 -> MulType t1 t2
(*) = t1 -> t2 -> MulType t1 t2
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
mul

type CanMulBy t1 t2 =
  (CanMul t1 t2, MulType t1 t2 ~ t1)
type CanMulSameType t =
  CanMulBy t t

product :: (CanMulSameType t, ConvertibleExactly Integer t) => [t] -> t
product :: [t] -> t
product [t]
xs = (t -> t -> t) -> t -> [t] -> t
forall (t :: * -> *) b a.
Foldable t =>
(b -> a -> b) -> b -> t a -> b
List.foldl' t -> t -> t
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
mul (Integer -> t
forall t1 t2. ConvertibleExactly t1 t2 => t1 -> t2
convertExactly Integer
1) [t]
xs

{-|
  HSpec properties that each implementation of CanMul should satisfy.
 -}
specCanMul ::
  _ => T t1 -> T t2 -> T t3 -> Spec
specCanMul :: T t1 -> T t2 -> T t3 -> Spec
specCanMul (T String
typeName1 :: T t1) (T String
typeName2 :: T t2) (T String
typeName3 :: T t3) =
  String -> Spec -> Spec
forall a. HasCallStack => String -> SpecWith a -> SpecWith a
describe (String -> String -> String -> String -> String -> String
forall r. PrintfType r => String -> r
printf String
"CanMul %s %s, CanMul %s %s" String
typeName1 String
typeName2 String
typeName2 String
typeName3) (Spec -> Spec) -> Spec -> Spec
forall a b. (a -> b) -> a -> b
$ do
    String -> Property -> SpecWith (Arg Property)
forall a.
(HasCallStack, Example a) =>
String -> a -> SpecWith (Arg a)
it String
"absorbs 1" (Property -> SpecWith (Arg Property))
-> Property -> SpecWith (Arg Property)
forall a b. (a -> b) -> a -> b
$ do
      (t1 -> Property) -> Property
forall prop. Testable prop => prop -> Property
property ((t1 -> Property) -> Property) -> (t1 -> Property) -> Property
forall a b. (a -> b) -> a -> b
$ \ (t1
x :: t1) -> let one :: t2
one = (Integer -> t2
forall t1 t2. ConvertibleExactly t1 t2 => t1 -> t2
convertExactly Integer
1 :: t2) in (t1
x t1 -> t2 -> MulType t1 t2
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
* t2
one) MulType t1 t2 -> t1 -> Property
forall a b.
(HasEqCertainlyAsymmetric a b, Show a, Show b) =>
a -> b -> Property
?==?$ t1
x
    String -> Property -> SpecWith (Arg Property)
forall a.
(HasCallStack, Example a) =>
String -> a -> SpecWith (Arg a)
it String
"is commutative" (Property -> SpecWith (Arg Property))
-> Property -> SpecWith (Arg Property)
forall a b. (a -> b) -> a -> b
$ do
      (t1 -> t2 -> Property) -> Property
forall prop. Testable prop => prop -> Property
property ((t1 -> t2 -> Property) -> Property)
-> (t1 -> t2 -> Property) -> Property
forall a b. (a -> b) -> a -> b
$ \ (t1
x :: t1) (t2
y :: t2) -> (t1
x t1 -> t2 -> MulType t1 t2
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
* t2
y) MulType t1 t2 -> MulType t2 t1 -> Property
forall a b.
(HasEqCertainlyAsymmetric a b, Show a, Show b) =>
a -> b -> Property
?==?$ (t2
y t2 -> t1 -> MulType t2 t1
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
* t1
x)
    String -> Property -> SpecWith (Arg Property)
forall a.
(HasCallStack, Example a) =>
String -> a -> SpecWith (Arg a)
it String
"is associative" (Property -> SpecWith (Arg Property))
-> Property -> SpecWith (Arg Property)
forall a b. (a -> b) -> a -> b
$ do
      (t1 -> t2 -> t3 -> Property) -> Property
forall prop. Testable prop => prop -> Property
property ((t1 -> t2 -> t3 -> Property) -> Property)
-> (t1 -> t2 -> t3 -> Property) -> Property
forall a b. (a -> b) -> a -> b
$ \ (t1
x :: t1) (t2
y :: t2) (t3
z :: t3) ->
                      (t1
x t1 -> MulType t2 t3 -> MulType t1 (MulType t2 t3)
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
* (t2
y t2 -> t3 -> MulType t2 t3
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
* t3
z)) MulType t1 (MulType t2 t3)
-> MulType (MulType t1 t2) t3 -> Property
forall a b.
(HasEqCertainlyAsymmetric a b, Show a, Show b) =>
a -> b -> Property
?==?$ ((t1
x t1 -> t2 -> MulType t1 t2
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
* t2
y) MulType t1 t2 -> t3 -> MulType (MulType t1 t2) t3
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
* t3
z)
    String -> Property -> SpecWith (Arg Property)
forall a.
(HasCallStack, Example a) =>
String -> a -> SpecWith (Arg a)
it String
"distributes over addition" (Property -> SpecWith (Arg Property))
-> Property -> SpecWith (Arg Property)
forall a b. (a -> b) -> a -> b
$ do
      (t1 -> t2 -> t3 -> Property) -> Property
forall prop. Testable prop => prop -> Property
property ((t1 -> t2 -> t3 -> Property) -> Property)
-> (t1 -> t2 -> t3 -> Property) -> Property
forall a b. (a -> b) -> a -> b
$ \ (t1
x :: t1) (t2
y :: t2) (t3
z :: t3) ->
                      (t1
x t1 -> AddType t2 t3 -> MulType t1 (AddType t2 t3)
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
* (t2
y t2 -> t3 -> AddType t2 t3
forall t1 t2. CanAddAsymmetric t1 t2 => t1 -> t2 -> AddType t1 t2
+ t3
z)) MulType t1 (AddType t2 t3)
-> AddType (MulType t1 t2) (MulType t1 t3) -> Property
forall a b.
(HasEqCertainlyAsymmetric a b, Show a, Show b) =>
a -> b -> Property
?==?$ (t1
x t1 -> t2 -> MulType t1 t2
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
* t2
y) MulType t1 t2
-> MulType t1 t3 -> AddType (MulType t1 t2) (MulType t1 t3)
forall t1 t2. CanAddAsymmetric t1 t2 => t1 -> t2 -> AddType t1 t2
+ (t1
x t1 -> t3 -> MulType t1 t3
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
* t3
z)
  where
  infix 4 ?==?$
  (?==?$) :: (HasEqCertainlyAsymmetric a b, Show a, Show b) => a -> b -> Property
  ?==?$ :: a -> b -> Property
(?==?$) = String -> (a -> b -> Bool) -> a -> b -> Property
forall prop a b.
(Testable prop, Show a, Show b) =>
String -> (a -> b -> prop) -> a -> b -> Property
printArgsIfFails2 String
"?==?" a -> b -> Bool
forall a b. HasEqCertainlyAsymmetric a b => a -> b -> Bool
(?==?)

{-|
  HSpec properties that each implementation of CanMul should satisfy.
 -}
specCanMulNotMixed ::
  _ => T t -> Spec
specCanMulNotMixed :: T t -> Spec
specCanMulNotMixed (T t
t :: T t) = T t -> T t -> T t -> Spec
forall t1 t2 t3.
(Arbitrary t1, Arbitrary t2, Arbitrary t3,
 HasEqAsymmetric (MulType t1 t2) t1,
 HasEqAsymmetric (MulType t1 t2) (MulType t2 t1),
 HasEqAsymmetric
   (MulType t1 (MulType t2 t3)) (MulType (MulType t1 t2) t3),
 HasEqAsymmetric
   (MulType t1 (AddType t2 t3))
   (AddType (MulType t1 t2) (MulType t1 t3)),
 CanTestCertainly (EqCompareType (MulType t1 t2) t1),
 CanTestCertainly (EqCompareType (MulType t1 t2) (MulType t2 t1)),
 CanTestCertainly
   (EqCompareType
      (MulType t1 (MulType t2 t3)) (MulType (MulType t1 t2) t3)),
 CanTestCertainly
   (EqCompareType
      (MulType t1 (AddType t2 t3))
      (AddType (MulType t1 t2) (MulType t1 t3))),
 Show t1, Show (MulType t1 t2), Show t2, Show (MulType t2 t1),
 Show t3, Show (MulType t1 (MulType t2 t3)),
 Show (MulType (MulType t1 t2) t3),
 Show (MulType t1 (AddType t2 t3)),
 Show (AddType (MulType t1 t2) (MulType t1 t3)),
 CanAddAsymmetric t2 t3,
 CanAddAsymmetric (MulType t1 t2) (MulType t1 t3),
 CanMulAsymmetric t2 t1, CanMulAsymmetric t2 t3,
 CanMulAsymmetric (MulType t1 t2) t3, CanMulAsymmetric t1 t2,
 CanMulAsymmetric t1 (MulType t2 t3),
 CanMulAsymmetric t1 (AddType t2 t3), CanMulAsymmetric t1 t3,
 ConvertibleExactly Integer t2) =>
T t1 -> T t2 -> T t3 -> Spec
specCanMul T t
t T t
t T t
t

{-|
  HSpec properties that each implementation of CanMulSameType should satisfy.
 -}
specCanMulSameType ::
  (Show t, ConvertibleExactly Integer t,
   CanTestCertainly (EqCompareType t t), HasEqAsymmetric t t,
   CanMulAsymmetric t t, MulType t t ~ t)
   =>
   T t -> Spec
specCanMulSameType :: T t -> Spec
specCanMulSameType (T String
typeName :: T t) =
  String -> Spec -> Spec
forall a. HasCallStack => String -> SpecWith a -> SpecWith a
describe (String -> String -> String
forall r. PrintfType r => String -> r
printf String
"CanMulSameType %s" String
typeName) (Spec -> Spec) -> Spec -> Spec
forall a b. (a -> b) -> a -> b
$ do
    String -> Property -> SpecWith (Arg Property)
forall a.
(HasCallStack, Example a) =>
String -> a -> SpecWith (Arg a)
it String
"has product working over integers" (Property -> SpecWith (Arg Property))
-> Property -> SpecWith (Arg Property)
forall a b. (a -> b) -> a -> b
$ do
      ([Integer] -> Property) -> Property
forall prop. Testable prop => prop -> Property
property (([Integer] -> Property) -> Property)
-> ([Integer] -> Property) -> Property
forall a b. (a -> b) -> a -> b
$ \ ([Integer]
xsi :: [Integer]) ->
        ([t] -> t
forall t.
(CanMulSameType t, ConvertibleExactly Integer t) =>
[t] -> t
product ([t] -> t) -> [t] -> t
forall a b. (a -> b) -> a -> b
$ ((Integer -> t) -> [Integer] -> [t]
forall a b. (a -> b) -> [a] -> [b]
map Integer -> t
forall t1 t2. ConvertibleExactly t1 t2 => t1 -> t2
convertExactly [Integer]
xsi :: [t])) t -> t -> Property
forall a b.
(HasEqCertainlyAsymmetric a b, Show a, Show b) =>
a -> b -> Property
?==?$ (Integer -> t
forall t1 t2. ConvertibleExactly t1 t2 => t1 -> t2
convertExactly ([Integer] -> Integer
forall t.
(CanMulSameType t, ConvertibleExactly Integer t) =>
[t] -> t
product [Integer]
xsi) :: t)
    String -> Property -> SpecWith (Arg Property)
forall a.
(HasCallStack, Example a) =>
String -> a -> SpecWith (Arg a)
it String
"has product [] = 1" (Property -> SpecWith (Arg Property))
-> Property -> SpecWith (Arg Property)
forall a b. (a -> b) -> a -> b
$ do
        ([t] -> t
forall t.
(CanMulSameType t, ConvertibleExactly Integer t) =>
[t] -> t
product ([] :: [t])) t -> t -> Property
forall a b.
(HasEqCertainlyAsymmetric a b, Show a, Show b) =>
a -> b -> Property
?==?$ (Integer -> t
forall t1 t2. ConvertibleExactly t1 t2 => t1 -> t2
convertExactly Integer
1 :: t)
  where
  infix 4 ?==?$
  (?==?$) :: (HasEqCertainlyAsymmetric a b, Show a, Show b) => a -> b -> Property
  ?==?$ :: a -> b -> Property
(?==?$) = String -> (a -> b -> Bool) -> a -> b -> Property
forall prop a b.
(Testable prop, Show a, Show b) =>
String -> (a -> b -> prop) -> a -> b -> Property
printArgsIfFails2 String
"?==?" a -> b -> Bool
forall a b. HasEqCertainlyAsymmetric a b => a -> b -> Bool
(?==?)

instance CanMulAsymmetric Int Int where
  type MulType Int Int = Integer -- do not risk overflow
  mul :: Int -> Int -> MulType Int Int
mul Int
a Int
b = (Int -> Integer
forall t. CanBeInteger t => t -> Integer
integer Int
a) Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
P.* (Int -> Integer
forall t. CanBeInteger t => t -> Integer
integer Int
b)
instance CanMulAsymmetric Integer Integer
instance CanMulAsymmetric Rational Rational
instance CanMulAsymmetric Double Double

instance CanMulAsymmetric Int Integer where
  type MulType Int Integer = Integer
  mul :: Int -> Integer -> MulType Int Integer
mul = (Integer -> Integer -> Integer) -> Int -> Integer -> Integer
forall a b c.
ConvertibleExactly a b =>
(b -> b -> c) -> a -> b -> c
convertFirst Integer -> Integer -> Integer
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
mul
instance CanMulAsymmetric Integer Int where
  type MulType Integer Int = Integer
  mul :: Integer -> Int -> MulType Integer Int
mul = (Integer -> Integer -> Integer) -> Integer -> Int -> Integer
forall b a c.
ConvertibleExactly b a =>
(a -> a -> c) -> a -> b -> c
convertSecond Integer -> Integer -> Integer
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
mul

instance CanMulAsymmetric Int Rational where
  type MulType Int Rational = Rational
  mul :: Int -> Rational -> MulType Int Rational
mul = (Rational -> Rational -> Rational) -> Int -> Rational -> Rational
forall a b c.
ConvertibleExactly a b =>
(b -> b -> c) -> a -> b -> c
convertFirst Rational -> Rational -> Rational
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
mul
instance CanMulAsymmetric Rational Int where
  type MulType Rational Int = Rational
  mul :: Rational -> Int -> MulType Rational Int
mul = (Rational -> Rational -> Rational) -> Rational -> Int -> Rational
forall b a c.
ConvertibleExactly b a =>
(a -> a -> c) -> a -> b -> c
convertSecond Rational -> Rational -> Rational
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
mul

instance CanMulAsymmetric Integer Rational where
  type MulType Integer Rational = Rational
  mul :: Integer -> Rational -> MulType Integer Rational
mul = (Rational -> Rational -> Rational)
-> Integer -> Rational -> Rational
forall a b c.
ConvertibleExactly a b =>
(b -> b -> c) -> a -> b -> c
convertFirst Rational -> Rational -> Rational
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
mul
instance CanMulAsymmetric Rational Integer where
  type MulType Rational Integer = Rational
  mul :: Rational -> Integer -> MulType Rational Integer
mul = (Rational -> Rational -> Rational)
-> Rational -> Integer -> Rational
forall b a c.
ConvertibleExactly b a =>
(a -> a -> c) -> a -> b -> c
convertSecond Rational -> Rational -> Rational
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
mul

instance CanMulAsymmetric Int Double where
  type MulType Int Double = Double
  mul :: Int -> Double -> MulType Int Double
mul Int
n Double
d = Double -> Double -> MulType Double Double
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
mul (Int -> Double
forall t. CanBeDouble t => t -> Double
double Int
n) Double
d
instance CanMulAsymmetric Double Int where
  type MulType Double Int = Double
  mul :: Double -> Int -> MulType Double Int
mul Double
d Int
n = Double -> Double -> MulType Double Double
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
mul Double
d (Int -> Double
forall t. CanBeDouble t => t -> Double
double Int
n)

instance CanMulAsymmetric Integer Double where
  type MulType Integer Double = Double
  mul :: Integer -> Double -> MulType Integer Double
mul Integer
n Double
d = Double -> Double -> MulType Double Double
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
mul (Integer -> Double
forall t. CanBeDouble t => t -> Double
double Integer
n) Double
d
instance CanMulAsymmetric Double Integer where
  type MulType Double Integer = Double
  mul :: Double -> Integer -> MulType Double Integer
mul Double
d Integer
n = Double -> Double -> MulType Double Double
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
mul Double
d (Integer -> Double
forall t. CanBeDouble t => t -> Double
double Integer
n)

instance CanMulAsymmetric Rational Double where
  type MulType Rational Double = Double
  mul :: Rational -> Double -> MulType Rational Double
mul Rational
n Double
d = Double -> Double -> MulType Double Double
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
mul (Rational -> Double
forall t. CanBeDouble t => t -> Double
double Rational
n) Double
d
instance CanMulAsymmetric Double Rational where
  type MulType Double Rational = Double
  mul :: Double -> Rational -> MulType Double Rational
mul Double
d Rational
n = Double -> Double -> MulType Double Double
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
mul Double
d (Rational -> Double
forall t. CanBeDouble t => t -> Double
double Rational
n)

instance (CanMulAsymmetric a b) => CanMulAsymmetric [a] [b] where
  type MulType [a] [b] = [MulType a b]
  mul :: [a] -> [b] -> MulType [a] [b]
mul (a
x:[a]
xs) (b
y:[b]
ys) = (a -> b -> MulType a b
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
mul a
x b
y) MulType a b -> [MulType a b] -> [MulType a b]
forall a. a -> [a] -> [a]
: ([a] -> [b] -> MulType [a] [b]
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
mul [a]
xs [b]
ys)
  mul [a]
_ [b]
_ = []

instance (CanMulAsymmetric a b) => CanMulAsymmetric (Maybe a) (Maybe b) where
  type MulType (Maybe a) (Maybe b) = Maybe (MulType a b)
  mul :: Maybe a -> Maybe b -> MulType (Maybe a) (Maybe b)
mul (Just a
x) (Just b
y) = MulType a b -> Maybe (MulType a b)
forall a. a -> Maybe a
Just (a -> b -> MulType a b
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
mul a
x b
y)
  mul Maybe a
_ Maybe b
_ = MulType (Maybe a) (Maybe b)
forall a. Maybe a
Nothing

instance
  (CanMulAsymmetric a b, CanGiveUpIfVeryInaccurate (MulType a b))
  =>
  CanMulAsymmetric (CN a) (CN b)
  where
  type MulType (CN a) (CN b) = CN (MulType a b)
  mul :: CN a -> CN b -> MulType (CN a) (CN b)
mul CN a
a CN b
b = CN (MulType a b) -> CN (MulType a b)
forall t. CanGiveUpIfVeryInaccurate t => CN t -> CN t
giveUpIfVeryInaccurate (CN (MulType a b) -> CN (MulType a b))
-> CN (MulType a b) -> CN (MulType a b)
forall a b. (a -> b) -> a -> b
$ (a -> b -> MulType a b) -> CN a -> CN b -> CN (MulType a b)
forall es a b c.
Monoid es =>
(a -> b -> c)
-> CollectErrors es a -> CollectErrors es b -> CollectErrors es c
CN.lift2 a -> b -> MulType a b
forall t1 t2. CanMulAsymmetric t1 t2 => t1 -> t2 -> MulType t1 t2
mul CN a
a CN b
b

$(declForTypes
  [[t| Integer |], [t| Int |], [t| Rational |], [t| Double |]]
  (\ t -> [d|

    instance
      (CanMulAsymmetric $t b, CanGiveUpIfVeryInaccurate (MulType $t b))
      =>
      CanMulAsymmetric $t (CN b)
      where
      type MulType $t (CN b) = CN (MulType $t b)
      mul a b = giveUpIfVeryInaccurate $ CN.liftT1 mul a b

    instance
      (CanMulAsymmetric a $t, CanGiveUpIfVeryInaccurate (MulType a $t))
      =>
      CanMulAsymmetric (CN a) $t
      where
      type MulType (CN a) $t = CN (MulType a $t)
      mul a b = giveUpIfVeryInaccurate $ CN.lift1T mul a b
  |]))