content: -- |
  -- Module:      Math.NumberTheory.Prefactored
  -- Copyright:   (c) 2017 Andrew Lelechenko
  -- Licence:     MIT
  -- Maintainer:  Andrew Lelechenko <andrew.lelechenko@gmail.com>
  --
  -- Type for numbers, accompanied by their factorisation.
  --
  
  {-# LANGUAGE TypeFamilies  #-}
  
  {-# OPTIONS_GHC -fno-warn-unused-imports #-}
  
  module Math.NumberTheory.Prefactored
    ( Prefactored(prefValue, prefFactors)
    , fromValue
    , fromFactors
    ) where
  
  import Prelude hiding ((^), gcd)
  import Control.Arrow
  import Data.Euclidean
  import Data.Semigroup
  import Data.Semiring (Semiring(..), Mul(..), (^))
  import qualified Data.Semiring as Semiring
  import Unsafe.Coerce
  
  import Math.NumberTheory.Euclidean.Coprimes
  import Math.NumberTheory.Primes
  import Math.NumberTheory.Primes.Types
  
  -- | A container for a number and its pairwise coprime (but not necessarily prime)
  -- factorisation.
  -- It is designed to preserve information about factors under multiplication.
  -- One can use this representation to speed up prime factorisation
  -- and computation of arithmetic functions.
  --
  -- For instance, let @p@ and @q@ be big primes:
  --
  -- >>> let p = 1000000000000000000000000000057 :: Integer
  -- >>> let q = 2000000000000000000000000000071 :: Integer
  --
  -- It would be difficult to compute the totient function
  -- of their product as is, because once we multiplied them
  -- the information of factors is lost and
  -- 'Math.NumberTheory.ArithmeticFunctions.totient' (@p@ * @q@)
  -- would take ages. Things become different if we simply
  -- change types of @p@ and @q@ to prefactored ones:
  --
  -- >>> let p = 1000000000000000000000000000057 :: Prefactored Integer
  -- >>> let q = 2000000000000000000000000000071 :: Prefactored Integer
  --
  -- Now the 'Math.NumberTheory.ArithmeticFunctions.totient' function
  -- can be computed instantly:
  --
  -- >>> import Math.NumberTheory.ArithmeticFunctions
  -- >>> prefValue $ totient (p^2 * q^3)
  -- 8000000000000000000000000001752000000000000000000000000151322000000000000000000000006445392000000000000000000000135513014000000000000000000001126361040
  -- >>> prefValue $ totient $ totient (p^2 * q^3)
  -- 2133305798262843681544648472180210822742702690942899511234131900112583590230336435053688694839034890779375223070157301188739881477320529552945446912000
  --
  -- Let us look under the hood:
  --
  -- >>> import Math.NumberTheory.ArithmeticFunctions
  -- >>> prefFactors $ totient (p^2 * q^3)
  -- Coprimes {unCoprimes = [(1000000000000000000000000000057,1),(41666666666666666666666666669,1),(2000000000000000000000000000071,2),(111111111111111111111111111115,1),(2,4),(3,3)]}
  -- >>> prefFactors $ totient $ totient (p^2 * q^3)
  -- Coprimes {unCoprimes = [(39521,1),(227098769,1),(22222222222222222222222222223,1),(2000000000000000000000000000071,1),(361696272343,1),(85331809838489,1),(6046667,1),(199937,1),(5,3),(41666666666666666666666666669,1),(2,22),(3,8)]}
  --
  -- Pairwise coprimality of factors is crucial, because it allows
  -- us to process them independently, possibly even
  -- in parallel or concurrent fashion.
  --
  -- Following invariant is guaranteed to hold:
  --
  -- > abs (prefValue x) = abs (product (map (uncurry (^)) (prefFactors x)))
  data Prefactored a = Prefactored
    { prefValue   :: a
      -- ^ Number itself.
    , prefFactors :: Coprimes a Word
      -- ^ List of pairwise coprime (but not necessarily prime) factors,
      -- accompanied by their multiplicities.
    } deriving (Eq, Show)
  
  -- | Create 'Prefactored' from a given number.
  --
  -- >>> fromValue 123
  -- Prefactored {prefValue = 123, prefFactors = Coprimes {unCoprimes = [(123,1)]}}
  fromValue :: (Eq a, GcdDomain a) => a -> Prefactored a
  fromValue a = Prefactored a (singleton a 1)
  
  -- | Create 'Prefactored' from a given list of pairwise coprime
  -- (but not necessarily prime) factors with multiplicities.
  --
  -- >>> fromFactors (splitIntoCoprimes [(140, 1), (165, 1)])
  -- Prefactored {prefValue = 23100, prefFactors = Coprimes {unCoprimes = [(28,1),(33,1),(5,2)]}}
  -- >>> fromFactors (splitIntoCoprimes [(140, 2), (165, 3)])
  -- Prefactored {prefValue = 88045650000, prefFactors = Coprimes {unCoprimes = [(28,2),(33,3),(5,5)]}}
  fromFactors :: Semiring a => Coprimes a Word -> Prefactored a
  fromFactors as = Prefactored (getMul $ foldMap (\(a, k) -> Mul $ a ^ k) (unCoprimes as)) as
  
  instance (Eq a, GcdDomain a) => Semiring (Prefactored a) where
    Prefactored v1 _ `plus` Prefactored v2 _
      = fromValue (v1 `plus` v2)
    Prefactored v1 f1 `times` Prefactored v2 f2
      = Prefactored (v1 `times` v2) (f1 <> f2)
    fromNatural n = fromValue (fromNatural n)
  
  instance (Eq a, Num a, GcdDomain a) => Num (Prefactored a) where
    Prefactored v1 _ + Prefactored v2 _
      = fromValue (v1 + v2)
    Prefactored v1 _ - Prefactored v2 _
      = fromValue (v1 - v2)
    Prefactored v1 f1 * Prefactored v2 f2
      = Prefactored (v1 * v2) (f1 <> f2)
    negate (Prefactored v f) = Prefactored (negate v) f
    abs (Prefactored v f)    = Prefactored (abs v) f
    signum (Prefactored v _) = Prefactored (signum v) mempty
    fromInteger n = fromValue (fromInteger n)
  
  instance (Eq a, GcdDomain a, UniqueFactorisation a) => UniqueFactorisation (Prefactored a) where
    factorise (Prefactored _ f)
      = concatMap (\(x, xm) -> map (\(p, k) -> (Prime $ fromValue $ unPrime p, k * xm)) (factorise x)) (unCoprimes f)
    isPrime (Prefactored _ f) = case unCoprimes f of
      [(n, 1)] -> Prime . fromValue . unPrime <$> isPrime n
      _        -> Nothing
  