```-- |
-- Module:      Math.NumberTheory.SmoothNumbers
-- Copyright:   (c) 2018 Frederick Schneider
-- Licence:     MIT
-- Maintainer:  Frederick Schneider <frederick.schneider2011@gmail.com>
--
-- A <https://en.wikipedia.org/wiki/Smooth_number smooth number>
-- is an integer, which can be represented as a product of powers of elements
-- from a given set (smooth basis). E. g., 48 = 3 * 4 * 4 is smooth
-- over a set {3, 4}, and 24 is not.
--

{-# LANGUAGE FlexibleContexts    #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TypeApplications    #-}

module Math.NumberTheory.SmoothNumbers
( -- * Create a smooth basis
SmoothBasis
, fromSet
, fromList
, fromSmoothUpperBound
-- * Generate smooth numbers
, smoothOver
, smoothOver'
, smoothOverInRange
, smoothOverInRangeBF

-- * Verify if a number is smooth
, isSmooth
) where

import Prelude hiding (div, mod, gcd)
import Data.Bits (Bits)
import Data.Coerce
import Data.List (nub)
import Data.Semiring (isZero)
import qualified Data.Set as S

import qualified Math.NumberTheory.Euclidean as E
import Math.NumberTheory.Primes

-- | An abstract representation of a smooth basis.
-- It consists of a set of numbers ≥2.
newtype SmoothBasis a = SmoothBasis { unSmoothBasis :: [a] } deriving (Eq, Show)

-- | Build a 'SmoothBasis' from a set of numbers ≥2.
--
-- >>> import qualified Data.Set as Set
-- >>> fromSet (Set.fromList [2, 3])
-- Just (SmoothBasis {unSmoothBasis = [2,3]})
-- >>> fromSet (Set.fromList [2, 4])
-- Just (SmoothBasis {unSmoothBasis = [2,4]})
-- >>> fromSet (Set.fromList [1, 3]) -- should be >= 2
-- Nothing
fromSet :: (Eq a, E.GcdDomain a) => S.Set a -> Maybe (SmoothBasis a)
fromSet s = if isValid l then Just (SmoothBasis l) else Nothing where l = S.elems s
{-# DEPRECATED fromSet "Use 'fromList' instead " #-}

-- | Build a 'SmoothBasis' from a list of numbers ≥2.
--
-- >>> fromList [2, 3]
-- Just (SmoothBasis {unSmoothBasis = [2,3]})
-- >>> fromList [2, 2]
-- Just (SmoothBasis {unSmoothBasis = [2]})
-- >>> fromList [2, 4]
-- Just (SmoothBasis {unSmoothBasis = [2,4]})
-- >>> fromList [1, 3] -- should be >= 2
-- Nothing
fromList :: (Eq a, E.GcdDomain a) => [a] -> Maybe (SmoothBasis a)
fromList l = if isValid l' then Just (SmoothBasis l') else Nothing
where
l' = nub l

-- | Build a 'SmoothBasis' from a list of primes below given bound.
--
-- >>> fromSmoothUpperBound 10
-- Just (SmoothBasis {unSmoothBasis = [2,3,5,7]})
-- >>> fromSmoothUpperBound 1
-- Nothing
fromSmoothUpperBound
:: (Integral a, Enum (Prime a), Bits a, UniqueFactorisation a)
=> a
-> Maybe (SmoothBasis a)
fromSmoothUpperBound n
| n < 2     = Nothing
| otherwise = Just \$ SmoothBasis \$ map unPrime [nextPrime 2 .. precPrime n]
{-# DEPRECATED fromSmoothUpperBound "Use 'fromList' with an appropriate list of primes instead " #-}

-- | Helper used by @smoothOver@ (@Integral@ constraint) and @smoothOver'@
-- (@Euclidean@ constraint) Since the typeclass constraint is just
-- @Num@, it receives a @norm@ comparison function for the generated smooth
-- numbers.
-- This function relies on the fact that for any element of a smooth basis @p@
-- and any @a@ it is true that @norm (a * p) > norm a@.
-- This condition is not checked.
smoothOver'
:: forall a b. (Eq a, Num a, Ord b)
=> (a -> b)
-> SmoothBasis a
-> [a]
smoothOver' norm pl =
foldr
(\p l -> mergeListLists \$ iterate (map (* p)) l)
[1]
(nub \$ unSmoothBasis pl)
where
{-# INLINE mergeListLists #-}
mergeListLists :: [[a]] -> [a]
mergeListLists = foldr go1 []
where
go1 :: [a] -> [a] -> [a]
go1 []    b = b
go1 (h:t) b = h:(go2 t b)

go2 :: [a] -> [a] -> [a]
go2 a [] = a
go2 [] b = b
go2 a@(ah:at) b@(bh:bt)
| norm bh < norm ah = bh : (go2 a bt)
| abs ah == abs bh  = ah : (go2 at bt)
| otherwise         = ah : (go2 at b)

-- | Generate an infinite ascending list of
-- <https://en.wikipedia.org/wiki/Smooth_number smooth numbers>
-- over a given smooth basis.
--
-- >>> import Data.Maybe
-- >>> take 10 (smoothOver (fromJust (fromList [2, 5])))
-- [1,2,4,5,8,10,16,20,25,32]
smoothOver :: (Ord a, Num a) => SmoothBasis a -> [a]
smoothOver = smoothOver' abs

-- | Generate an ascending list of
-- <https://en.wikipedia.org/wiki/Smooth_number smooth numbers>
-- over a given smooth basis in a given range.
--
-- It may appear inefficient
-- for short, but distant ranges;
-- consider using 'smoothOverInRangeBF' in such cases.
--
-- >>> import Data.Maybe
-- >>> smoothOverInRange (fromJust (fromList [2, 5])) 100 200
-- [100,125,128,160,200]
smoothOverInRange :: (Ord a, Num a) => SmoothBasis a -> a -> a -> [a]
smoothOverInRange s lo hi
= takeWhile (<= hi)
\$ dropWhile (< lo)
\$ smoothOver s
{-# DEPRECATED smoothOverInRange "Use 'smoothOver' instead" #-}

-- | Generate an ascending list of
-- <https://en.wikipedia.org/wiki/Smooth_number smooth numbers>
-- over a given smooth basis in a given range.
--
-- It is inefficient
-- for large or starting near 0 ranges;
-- consider using 'smoothOverInRange' in such cases.
--
-- Suffix BF stands for the brute force algorithm, involving a lot of divisions.
--
-- >>> import Data.Maybe
-- >>> smoothOverInRangeBF (fromJust (fromList [2, 5])) 100 200
-- [100,125,128,160,200]
smoothOverInRangeBF
:: (Eq a, Enum a, E.GcdDomain a)
=> SmoothBasis a
-> a
-> a
-> [a]
smoothOverInRangeBF prs lo hi
= coerce
\$ filter (isSmooth prs)
\$ coerce [lo..hi]
{-# DEPRECATED smoothOverInRangeBF "Use filtering by 'isSmooth' instead" #-}

isValid :: (Eq a, E.GcdDomain a) => [a] -> Bool
isValid [] = False
isValid xs = all (\x -> not (isZero x) && not (E.isUnit x)) xs

-- | @isSmooth@ checks if a given number is smooth under a certain @SmoothBasis@.
-- Does not check if the @SmoothBasis@ is valid.
isSmooth :: (Eq a, E.GcdDomain a) => SmoothBasis a -> a -> Bool
isSmooth prs x = not (isZero x) && go (unSmoothBasis prs) x
where
go :: (Eq a, E.GcdDomain a) => [a] -> a -> Bool
go [] n = E.isUnit n
go pps@(p:ps) n = case n `E.divide` p of
Nothing -> go ps n
Just q  -> go pps q || go ps n
```