module Numeric.MixedTypes.Ring
(
CanAddSubMulBy, Ring, CertainlyEqRing, OrderedRing, OrderedCertainlyRing
, CanMul, CanMulAsymmetric(..), CanMulBy, CanMulSameType
, (*), product
, specCanMul, specCanMulNotMixed, specCanMulSameType, CanMulX
, CanPow(..), CanPowBy
, (^), (^^), (**)
, powUsingMul
, specCanPow, CanPowX
)
where
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 Numeric.MixedTypes.Literals
import Numeric.MixedTypes.Bool
import Numeric.MixedTypes.Eq
import Numeric.MixedTypes.Ord
import Numeric.MixedTypes.AddSub
type CanAddSubMulBy t s =
(CanAddThis t s, CanSubThis t s, CanMulBy t s)
type Ring t =
(CanNegSameType t, CanAddSameType t, CanSubSameType t, CanMulSameType t,
CanPowBy t Integer, CanPowBy t Int,
HasEq t t,
HasEq t Integer, CanAddSubMulBy t Integer,
CanSub Integer t, SubType Integer t ~ t,
HasEq t Int, CanAddSubMulBy t Int,
CanSub Int t, SubType Int t ~ t,
ConvertibleExactly Integer t)
type CertainlyEqRing t =
(Ring t, HasEqCertainly t t, HasEqCertainly t Int, HasEqCertainly t Integer)
type OrderedRing t =
(Ring t, HasOrder t t, HasOrder t Int, HasOrder t Integer)
type OrderedCertainlyRing t =
(CertainlyEqRing t, HasOrderCertainly t t, HasOrderCertainly t Int, HasOrderCertainly t Integer,
CanTestPosNeg t)
type CanMul t1 t2 =
(CanMulAsymmetric t1 t2, CanMulAsymmetric t2 t1,
MulType t1 t2 ~ MulType t2 t1)
class CanMulAsymmetric t1 t2 where
type MulType t1 t2
type MulType t1 t2 = t1
mul :: t1 -> t2 -> MulType t1 t2
default mul :: (MulType t1 t2 ~ t1, t1~t2, P.Num t1) => t1 -> t1 -> t1
mul = (P.*)
infixl 8 ^, ^^
infixl 7 *
(*) :: (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 xs = List.foldl' mul (convertExactly 1) xs
type CanMulX t1 t2 =
(CanMul t1 t2,
Show t1, Arbitrary t1,
Show t2, Arbitrary t2,
Show (MulType t1 t2),
HasEqCertainly t1 (MulType t1 t2),
HasEqCertainly t2 (MulType t1 t2),
HasEqCertainly (MulType t1 t2) (MulType t1 t2),
HasOrderCertainly t1 (MulType t1 t2),
HasOrderCertainly t2 (MulType t1 t2),
HasOrderCertainly (MulType t1 t2) (MulType t1 t2))
specCanMul ::
(CanMulX t1 t2,
CanMulX t1 t3,
CanMulX t2 t3,
CanMulX t1 (MulType t2 t3),
CanMulX (MulType t1 t2) t3,
HasEqCertainly (MulType t1 (MulType t2 t3)) (MulType (MulType t1 t2) t3),
CanAdd t2 t3,
CanMulX t1 (AddType t2 t3),
CanAddX (MulType t1 t2) (MulType t1 t3),
HasEqCertainly (MulType t1 (AddType t2 t3)) (AddType (MulType t1 t2) (MulType t1 t3)),
ConvertibleExactly Integer t2)
=>
T t1 -> T t2 -> T t3 -> Spec
specCanMul (T typeName1 :: T t1) (T typeName2 :: T t2) (T typeName3 :: T t3) =
describe (printf "CanMul %s %s, CanMul %s %s" typeName1 typeName2 typeName2 typeName3) $ do
it "absorbs 1" $ do
property $ \ (x :: t1) -> let one = (convertExactly 1 :: t2) in (x * one) ?==?$ x
it "is commutative" $ do
property $ \ (x :: t1) (y :: t2) -> (x * y) ?==?$ (y * x)
it "is associative" $ do
property $ \ (x :: t1) (y :: t2) (z :: t3) ->
(x * (y * z)) ?==?$ ((x * y) * z)
it "distributes over addition" $ do
property $ \ (x :: t1) (y :: t2) (z :: t3) ->
(x * (y + z)) ?==?$ (x * y) + (x * z)
where
infix 4 ?==?$
(?==?$) :: (HasEqCertainlyAsymmetric a b, Show a, Show b) => a -> b -> Property
(?==?$) = printArgsIfFails2 "?==?" (?==?)
specCanMulNotMixed ::
(CanMulX t t,
CanMulX t (MulType t t),
HasEqCertainly (MulType (MulType t t) t) (MulType t (MulType t t)),
CanAdd t t,
CanMulX t (AddType t t),
CanAddX (MulType t t) (MulType t t),
HasEqCertainly (MulType t (AddType t t)) (AddType (MulType t t) (MulType t t)),
ConvertibleExactly Integer t)
=>
T t -> Spec
specCanMulNotMixed t = specCanMul t t t
specCanMulSameType ::
(ConvertibleExactly Integer t, Show t,
HasEqCertainly t t, CanMulSameType t)
=>
T t -> Spec
specCanMulSameType (T typeName :: T t) =
describe (printf "CanMulSameType %s" typeName) $ do
it "has product working over integers" $ do
property $ \ (xsi :: [Integer]) ->
(product $ (map convertExactly xsi :: [t])) ?==?$ (convertExactly (product xsi) :: t)
it "has product [] = 1" $ do
(product ([] :: [t])) ?==?$ (convertExactly 1 :: t)
where
infix 4 ?==?$
(?==?$) :: (HasEqCertainlyAsymmetric a b, Show a, Show b) => a -> b -> Property
(?==?$) = printArgsIfFails2 "?==?" (?==?)
instance CanMulAsymmetric Int Int where
type MulType Int Int = Integer
mul a b = (integer a) P.* (integer b)
instance CanMulAsymmetric Integer Integer
instance CanMulAsymmetric Rational Rational
instance CanMulAsymmetric Double Double
instance CanMulAsymmetric Int Integer where
type MulType Int Integer = Integer
mul = convertFirst mul
instance CanMulAsymmetric Integer Int where
type MulType Integer Int = Integer
mul = convertSecond mul
instance CanMulAsymmetric Int Rational where
type MulType Int Rational = Rational
mul = convertFirst mul
instance CanMulAsymmetric Rational Int where
type MulType Rational Int = Rational
mul = convertSecond mul
instance CanMulAsymmetric Integer Rational where
type MulType Integer Rational = Rational
mul = convertFirst mul
instance CanMulAsymmetric Rational Integer where
type MulType Rational Integer = Rational
mul = convertSecond mul
instance CanMulAsymmetric Int Double where
type MulType Int Double = Double
mul = convertFirst mul
instance CanMulAsymmetric Double Int where
type MulType Double Int = Double
mul = convertSecond mul
instance CanMulAsymmetric Integer Double where
type MulType Integer Double = Double
mul = convertFirst mul
instance CanMulAsymmetric Double Integer where
type MulType Double Integer = Double
mul = convertSecond mul
instance CanMulAsymmetric Rational Double where
type MulType Rational Double = Double
mul = convertFirst mul
instance CanMulAsymmetric Double Rational where
type MulType Double Rational = Double
mul = convertSecond mul
instance (CanMulAsymmetric a b) => CanMulAsymmetric [a] [b] where
type MulType [a] [b] = [MulType a b]
mul (x:xs) (y:ys) = (mul x y) : (mul xs ys)
mul _ _ = []
instance (CanMulAsymmetric a b) => CanMulAsymmetric (Maybe a) (Maybe b) where
type MulType (Maybe a) (Maybe b) = Maybe (MulType a b)
mul (Just x) (Just y) = Just (mul x y)
mul _ _ = Nothing
class CanPow t1 t2 where
type PowType t1 t2
type PowType t1 t2 = t1
pow :: t1 -> t2 -> PowType t1 t2
default pow :: (PowType t1 t2 ~ t1, P.Num t1, P.Integral t2) => t1 -> t2 -> t1
pow = (P.^)
powUsingMul ::
(CanBeInteger e,
CanMulSameType t, ConvertibleExactly Integer t)
=>
t -> e -> t
powUsingMul x nPre
| n < 0 = error $ "powUsingMul is not defined for negative exponent " ++ show n
| n == 0 = convertExactly 1
| otherwise = aux n
where
n = integer nPre
aux m
| m == 1 = x
| even m =
let s = aux (m `div` 2) in s * s
| otherwise =
let s = aux ((m1) `div` 2) in x * s * s
(^) :: (CanPow t1 t2) => t1 -> t2 -> PowType t1 t2
(^) = pow
(^^) :: (CanPow t1 t2) => t1 -> t2 -> PowType t1 t2
(^^) = (^)
(**) :: (CanPow t1 t2) => t1 -> t2 -> (PowType t1 t2)
(**) = (^)
type CanPowBy t1 t2 =
(CanPow t1 t2, PowType t1 t2 ~ t1)
type CanPowX t1 t2 =
(CanPow t1 t2,
Show t1, Arbitrary t1,
Show t2, Arbitrary t2,
Show (PowType t1 t2))
specCanPow ::
(CanPowX t1 t2,
HasEqCertainly t1 (PowType t1 t2),
ConvertibleExactly Integer t1,
ConvertibleExactly Integer t2,
CanTestPosNeg t2,
CanAdd t2 Integer,
CanMulX t1 (PowType t1 t2),
CanPowX t1 (AddType t2 Integer),
HasEqCertainly (MulType t1 (PowType t1 t2)) (PowType t1 (AddType t2 Integer)))
=>
T t1 -> T t2 -> Spec
specCanPow (T typeName1 :: T t1) (T typeName2 :: T t2) =
describe (printf "CanPow %s %s" typeName1 typeName2) $ do
it "x^0 = 1" $ do
property $ \ (x :: t1) ->
let one = (convertExactly 1 :: t1) in
let z = (convertExactly 0 :: t2) in
(x ^ z) ?==?$ one
it "x^1 = x" $ do
property $ \ (x :: t1) ->
let one = (convertExactly 1 :: t2) in
(x ^ one) ?==?$ x
it "x^(y+1) = x*x^y" $ do
property $ \ (x :: t1) (y :: t2) ->
(isCertainlyNonNegative y) ==>
x * (x ^ y) ?==?$ (x ^ (y + 1))
where
infix 4 ?==?$
(?==?$) :: (HasEqCertainlyAsymmetric a b, Show a, Show b) => a -> b -> Property
(?==?$) = printArgsIfFails2 "?==?" (?==?)
instance CanPow Integer Integer
instance CanPow Integer Int
instance CanPow Int Integer where
type PowType Int Integer = Integer
pow x n = (integer x) P.^ n
instance CanPow Int Int where
type PowType Int Int = Integer
pow x n = (integer x) P.^ n
instance CanPow Rational Int where pow = (P.^^)
instance CanPow Rational Integer where pow = (P.^^)
instance CanPow Double Int where pow = (P.^^)
instance CanPow Double Integer where pow = (P.^^)
instance (CanPow a b) => CanPow (Maybe a) (Maybe b) where
type PowType (Maybe a) (Maybe b) = Maybe (PowType a b)
pow (Just x) (Just y) = Just (pow x y)
pow _ _ = Nothing