Add IntegralGCD interface
This interface allows numeric operations to be total while keeping things (relatively) safe.
This commit is contained in:
parent
412cfd2b5b
commit
2e6f56dfde
35
src/Data/IntegralGCD.idr
Normal file
35
src/Data/IntegralGCD.idr
Normal file
|
@ -0,0 +1,35 @@
|
|||
module Data.IntegralGCD
|
||||
|
||||
%default total
|
||||
|
||||
|
||||
||| An interface for integer types that it is possible to take the GCD
|
||||
||| (greatest common denominator) of.
|
||||
|||
|
||||
||| This interface exists for totality purposes: Euclid's algorithm is
|
||||
||| possible to implement for any type that satisfies `Integral`, but it
|
||||
||| is not provably total unless that implementation satisfies some properties.
|
||||
|||
|
||||
||| Implementing this interface asserts to the compiler that Euclid's algorithm
|
||||
||| is total on this type.
|
||||
public export
|
||||
interface (Eq a, Integral a) => IntegralGCD a where
|
||||
gcd : a -> a -> a
|
||||
gcd x y =
|
||||
if x == 0 then y
|
||||
else if y == 0 then x
|
||||
else assert_total $ gcd y (x `mod` y)
|
||||
|
||||
|
||||
export IntegralGCD Integer where
|
||||
|
||||
export IntegralGCD Int where
|
||||
export IntegralGCD Int8 where
|
||||
export IntegralGCD Int16 where
|
||||
export IntegralGCD Int32 where
|
||||
export IntegralGCD Int64 where
|
||||
|
||||
export IntegralGCD Bits8 where
|
||||
export IntegralGCD Bits16 where
|
||||
export IntegralGCD Bits32 where
|
||||
export IntegralGCD Bits64 where
|
|
@ -1,9 +1,12 @@
|
|||
module Data.Ratio
|
||||
|
||||
infix 10 //
|
||||
import Data.Maybe
|
||||
import Data.So
|
||||
import Data.IntegralGCD
|
||||
|
||||
%default total
|
||||
|
||||
|
||||
||| Ratio types, represented by a numerator and denominator of type `a`.
|
||||
|||
|
||||
||| Most numeric operations require an instance `Integral a` in order to
|
||||
|
@ -22,58 +25,61 @@ Rational : Type
|
|||
Rational = Ratio Integer
|
||||
|
||||
|
||||
-- This function is almost always total; after all, it's a fairly standard
|
||||
-- implementation of Euclid's algorithm. Unfortunately, we can't _guarantee_
|
||||
-- it's total without knowing exactly what the implementation of `Integral a` is,
|
||||
-- so using an `assert_total` here would be potentially unsafe. The only option
|
||||
-- is to mark this function, and by extention `reduce`, as non-total.
|
||||
covering
|
||||
gcd : (Eq a, Integral a) => a -> a -> a
|
||||
gcd x y =
|
||||
if x == 0 then y
|
||||
else if y == 0 then x
|
||||
else gcd y (x `mod` y)
|
||||
|
||||
||| Reduce a ratio by dividing both components by their common factor. Most
|
||||
||| operations automatically reduce the returned ratio, so explicitly calling
|
||||
||| this function is usually unnecessary.
|
||||
export covering
|
||||
reduce : (Eq a, Integral a) => Ratio a -> Ratio a
|
||||
export
|
||||
reduce : IntegralGCD a => Ratio a -> Ratio a
|
||||
reduce (MkRatio n d) =
|
||||
let g = gcd n d in MkRatio (n `div` g) (d `div` g)
|
||||
|
||||
|
||||
||| Construct a ratio of two values, returning `Nothing` if the denominator is
|
||||
||| zero.
|
||||
export
|
||||
mkRatioMaybe : IntegralGCD a => (n, d : a) -> Maybe (Ratio a)
|
||||
mkRatioMaybe n d = toMaybe (d /= 0) (reduce $ MkRatio n d)
|
||||
|
||||
infix 9 //
|
||||
|
||||
||| Construct a ratio of two values.
|
||||
export partial
|
||||
(//) : (Eq a, Integral a) => a -> a -> Ratio a
|
||||
n // d = case d /= 0 of
|
||||
True => reduce $ MkRatio n d
|
||||
export
|
||||
(//) : IntegralGCD a => (n, d : a) -> {auto 0 ok : So (d /= 0)} -> Ratio a
|
||||
(//) n d = reduce $ MkRatio n d
|
||||
|
||||
|
||||
||| Round a ratio down to a positive integer.
|
||||
export
|
||||
floor : Integral a => Ratio a -> a
|
||||
floor (MkRatio n d) = n `div` d
|
||||
|
||||
|
||||
namespace Ratio
|
||||
||| Return the numerator of the fraction in reduced form.
|
||||
||| Return the numerator of the ratio in reduced form.
|
||||
export %inline
|
||||
numer : Ratio a -> a
|
||||
numer = nm
|
||||
|
||||
||| Return the numerator of the fraction in reduced form.
|
||||
||| Return the numerator of the ratio in reduced form.
|
||||
export %inline
|
||||
(.numer) : Ratio a -> a
|
||||
(.numer) = nm
|
||||
|
||||
||| Return the denominator of the fraction in reduced form.
|
||||
||| Return the denominator of the ratio in reduced form.
|
||||
||| This value is guaranteed to always be positive.
|
||||
export %inline
|
||||
denom : Ratio a -> a
|
||||
denom = dn
|
||||
|
||||
||| Return the denominator of the fraction in reduced form.
|
||||
||| Return the denominator of the ratio in reduced form.
|
||||
||| This value is guaranteed to always be positive.
|
||||
export %inline
|
||||
(.denom) : Ratio a -> a
|
||||
(.denom) = dn
|
||||
|
||||
|
||||
|
||||
|
||||
export
|
||||
Eq a => Eq (Ratio a) where
|
||||
MkRatio n d == MkRatio m b = n == m && d == b
|
||||
|
@ -94,23 +100,23 @@ Show a => Show (Ratio a) where
|
|||
let p' = User 10
|
||||
in showParens (p >= p') (showPrec p' n ++ " // " ++ showPrec p' d)
|
||||
|
||||
export covering
|
||||
(Eq a, Integral a) => Num (Ratio a) where
|
||||
export
|
||||
IntegralGCD a => Num (Ratio a) where
|
||||
MkRatio n d + MkRatio m b = reduce $ MkRatio (n*b + m*d) (d*b)
|
||||
MkRatio n d * MkRatio m b = reduce $ MkRatio (n*m) (d*b)
|
||||
fromInteger x = MkRatio (fromInteger x) 1
|
||||
|
||||
export covering
|
||||
(Eq a, Integral a, Neg a) => Neg (Ratio a) where
|
||||
export
|
||||
(IntegralGCD a, Neg a) => Neg (Ratio a) where
|
||||
negate (MkRatio n d) = MkRatio (-n) d
|
||||
MkRatio n d - MkRatio m b = reduce $ MkRatio (n*b - m*d) (d*b)
|
||||
|
||||
export
|
||||
(Eq a, Integral a, Abs a) => Abs (Ratio a) where
|
||||
(IntegralGCD a, Abs a) => Abs (Ratio a) where
|
||||
abs (MkRatio n d) = MkRatio (abs n) (abs d)
|
||||
|
||||
export
|
||||
(Eq a, Integral a) => Fractional (Ratio a) where
|
||||
IntegralGCD a => Fractional (Ratio a) where
|
||||
recip (MkRatio n d) = case n /= 0 of True => MkRatio d n
|
||||
MkRatio n d / MkRatio m b = case m /= 0 of
|
||||
True => reduce $ MkRatio (n*b) (m*d)
|
||||
|
|
Loading…
Reference in a new issue