-- Copyright (c) 2011, David Amos. All rights reserved.
{-# LANGUAGE NoMonomorphismRestriction, TupleSections #-}
-- |A module of simple utility functions which are used throughout the rest of the library
module Math.Core.Utils where
import Data.List as L
import qualified Data.Set as S
toSet = S.toList . S.fromList
sortDesc = L.sortBy (flip compare)
insertDesc = L.insertBy (flip compare)
-- |The set union of two ascending lists. If both inputs are strictly increasing, then the output is their union
-- and is strictly increasing. The code does not check that the lists are strictly increasing.
setUnionAsc :: Ord a => [a] -> [a] -> [a]
setUnionAsc (x:xs) (y:ys) =
case compare x y of
LT -> x : setUnionAsc xs (y:ys)
EQ -> x : setUnionAsc xs ys
GT -> y : setUnionAsc (x:xs) ys
setUnionAsc xs ys = xs ++ ys
setUnionDesc :: Ord a => [a] -> [a] -> [a]
setUnionDesc (x:xs) (y:ys) =
case compare x y of
GT -> x : setUnionDesc xs (y:ys)
EQ -> x : setUnionDesc xs ys
LT -> y : setUnionDesc (x:xs) ys
setUnionDesc xs ys = xs ++ ys
-- |The (multi-)set intersection of two ascending lists. If both inputs are strictly increasing,
-- then the output is the set intersection and is strictly increasing. If both inputs are weakly increasing,
-- then the output is the multiset intersection (with multiplicity), and is weakly increasing.
intersectAsc :: Ord a => [a] -> [a] -> [a]
intersectAsc (x:xs) (y:ys) =
case compare x y of
LT -> intersectAsc xs (y:ys)
EQ -> x : intersectAsc xs ys
GT -> intersectAsc (x:xs) ys
intersectAsc _ _ = []
-- |The multiset sum of two ascending lists. If xs and ys are ascending, then multisetSumAsc xs ys == sort (xs++ys).
-- The code does not check that the lists are ascending.
multisetSumAsc :: Ord a => [a] -> [a] -> [a]
multisetSumAsc (x:xs) (y:ys) =
case compare x y of
LT -> x : multisetSumAsc xs (y:ys)
EQ -> x : y : multisetSumAsc xs ys
GT -> y : multisetSumAsc (x:xs) ys
multisetSumAsc xs ys = xs ++ ys
-- |The multiset sum of two descending lists. If xs and ys are descending, then multisetSumDesc xs ys == sortDesc (xs++ys).
-- The code does not check that the lists are descending.
multisetSumDesc :: Ord a => [a] -> [a] -> [a]
multisetSumDesc (x:xs) (y:ys) =
case compare x y of
GT -> x : multisetSumDesc xs (y:ys)
EQ -> x : y : multisetSumDesc xs ys
LT -> y : multisetSumDesc (x:xs) ys
multisetSumDesc xs ys = xs ++ ys
-- |The multiset or set difference between two ascending lists. If xs and ys are ascending, then diffAsc xs ys == xs \\ ys,
-- and diffAsc is more efficient. If xs and ys are sets (that is, have no repetitions), then diffAsc xs ys is the set difference.
-- The code does not check that the lists are ascending.
diffAsc :: Ord a => [a] -> [a] -> [a]
diffAsc (x:xs) (y:ys) = case compare x y of
LT -> x : diffAsc xs (y:ys)
EQ -> diffAsc xs ys
GT -> diffAsc (x:xs) ys
diffAsc xs [] = xs
diffAsc [] _ = []
-- |The multiset or set difference between two descending lists. If xs and ys are descending, then diffDesc xs ys == xs \\ ys,
-- and diffDesc is more efficient. If xs and ys are sets (that is, have no repetitions), then diffDesc xs ys is the set difference.
-- The code does not check that the lists are descending.
diffDesc :: Ord a => [a] -> [a] -> [a]
diffDesc (x:xs) (y:ys) = case compare x y of
GT -> x : diffDesc xs (y:ys)
EQ -> diffDesc xs ys
LT -> diffDesc (x:xs) ys
diffDesc xs [] = xs
diffDesc [] _ = []
isSubsetAsc = isSubMultisetAsc
isSubMultisetAsc (x:xs) (y:ys) =
case compare x y of
LT -> False
EQ -> isSubMultisetAsc xs ys
GT -> isSubMultisetAsc (x:xs) ys
isSubMultisetAsc [] ys = True
isSubMultisetAsc xs [] = False
-- |Is the element in the ascending list?
--
-- With infinite lists, this can fail to terminate.
-- For example, elemAsc 1 [1/2,3/4,7/8..] would fail to terminate.
-- However, with a list of Integer, this will always terminate.
elemAsc :: Ord a => a -> [a] -> Bool
elemAsc x (y:ys) = case compare x y of
LT -> False
EQ -> True
GT -> elemAsc x ys
-- or x `elemAsc` ys = x `elem` takeWhile (<= x) ys
-- |Is the element not in the ascending list? (With infinite lists, this can fail to terminate.)
notElemAsc :: Ord a => a -> [a] -> Bool
notElemAsc x (y:ys) = case compare x y of
LT -> True
EQ -> False
GT -> notElemAsc x ys
-- From Conor McBride
-- http://stackoverflow.com/questions/12869097/splitting-list-into-a-list-of-possible-tuples/12872133#12872133
-- |Return all the ways to \"pick one and leave the others\" from a list
picks :: [a] -> [(a,[a])]
picks [] = []
picks (x:xs) = (x,xs) : [(y,x:ys) | (y,ys) <- picks xs]
pairs (x:xs) = map (x,) xs ++ pairs xs
pairs [] = []
ordpair x y | x < y = (x,y)
| otherwise = (y,x)
-- fold a comparison operator through a list
foldcmpl p xs = and $ zipWith p xs (tail xs)
-- foldcmpl p (x1:x2:xs) = p x1 x2 && foldcmpl p (x2:xs)
-- foldcmpl _ _ = True
-- foldcmpl _ [] = True
-- foldcmpl p xs = and $ zipWith p xs (tail xs)
isWeaklyIncreasing :: Ord t => [t] -> Bool
isWeaklyIncreasing = foldcmpl (<=)
isStrictlyIncreasing :: Ord t => [t] -> Bool
isStrictlyIncreasing = foldcmpl (<)
isWeaklyDecreasing :: Ord t => [t] -> Bool
isWeaklyDecreasing = foldcmpl (>=)
isStrictlyDecreasing :: Ord t => [t] -> Bool
isStrictlyDecreasing = foldcmpl (>)
-- for use with L.sortBy
cmpfst x y = compare (fst x) (fst y)
-- for use with L.groupBy
eqfst x y = (==) (fst x) (fst y)
fromBase b xs = foldl' (\n x -> n * b + x) 0 xs
-- |Given a set @xs@, represented as an ordered list, @powersetdfs xs@ returns the list of all subsets of xs, in lex order
powersetdfs :: [a] -> [[a]]
powersetdfs xs = map reverse $ dfs [ ([],xs) ]
where dfs ( (ls,rs) : nodes ) = ls : dfs (successors (ls,rs) ++ nodes)
dfs [] = []
successors (ls,rs) = [ (r:ls, rs') | r:rs' <- L.tails rs ]
-- |Given a set @xs@, represented as an ordered list, @powersetbfs xs@ returns the list of all subsets of xs, in shortlex order
powersetbfs :: [a] -> [[a]]
powersetbfs xs = map reverse $ bfs [ ([],xs) ]
where bfs ( (ls,rs) : nodes ) = ls : bfs ( nodes ++ successors (ls,rs) )
bfs [] = []
successors (ls,rs) = [ (r:ls, rs') | r:rs' <- L.tails rs ]
-- |Given a positive integer @k@, and a set @xs@, represented as a list,
-- @combinationsOf k xs@ returns all k-element subsets of xs.
-- The result will be in lex order, relative to the order of the xs.
combinationsOf :: Int -> [a] -> [[a]]
combinationsOf 0 _ = [[]]
combinationsOf _ [] = []
combinationsOf k (x:xs) | k > 0 = map (x:) (combinationsOf (k-1) xs) ++ combinationsOf k xs
-- |@choose n k@ is the number of ways of choosing k distinct elements from an n-set
choose :: (Integral a) => a -> a -> a
choose n k = product [n-k+1..n] `div` product [1..k]
-- |The class of finite sets
class FinSet x where
elts :: [x]
-- |A class representing algebraic structures having an inverse operation.
-- Note that in some cases not every element has an inverse.
class HasInverses a where
inverse :: a -> a
infix 8 ^-
-- |A trick: x^-1 returns the inverse of x
(^-) :: (Num a, HasInverses a, Integral b) => a -> b -> a
x ^- n = inverse x ^ n