module Data.List.HT.Private where

import Data.List  as List  (find, transpose, unfoldr, isPrefixOf,
                            findIndices, foldl', mapAccumL, )
import Data.Maybe as Maybe (fromMaybe, catMaybes, isJust, mapMaybe, )
import Data.Maybe.HT       (toMaybe, )
import Control.Monad.HT    ((<=<), )
import Control.Monad       (guard, msum, mplus, liftM2, )
import Control.Applicative ((<$>), (<*>), )
import Data.Tuple.HT       (mapPair, mapFst, mapSnd, forcePair, swap, )

import qualified Control.Functor.HT as Func

import qualified Data.List.Key.Private   as Key
import qualified Data.List.Match.Private as Match
import qualified Data.List.Reverse.StrictElement as Rev

import Prelude hiding (unzip, break, span, )


-- $setup
-- >>> import qualified Test.QuickCheck as QC
-- >>> import Test.Utility (forAllPredicates)
-- >>> import Test.QuickCheck (NonNegative(NonNegative), Positive(Positive), NonEmptyList(NonEmpty))
-- >>> import qualified Data.List as List
-- >>> import Data.List (transpose)
-- >>> import Data.Maybe.HT (toMaybe)
-- >>> import Data.Maybe (mapMaybe, isNothing)
-- >>> import Data.Char (isLetter, toUpper)
-- >>> import Data.Eq.HT (equating)
-- >>> import Control.Monad (liftM2)
-- >>>
-- >>> divMaybe :: Int -> Int -> Maybe Int
-- >>> divMaybe m n = case divMod n m of (q,0) -> Just q; _ -> Nothing
-- >>>
-- >>> forAllMaybeFn :: (QC.Testable test) => ((Int -> Maybe Int) -> test) -> QC.Property
-- >>> forAllMaybeFn prop = QC.forAll (QC.choose (1,4)) $ prop . divMaybe


-- * Improved standard functions

{- |
This function is lazier than the one suggested in the Haskell 98 report.
It is @inits undefined = [] : undefined@,
in contrast to @Data.List.inits undefined = undefined@.
-}
{-
suggested in
<http://www.haskell.org/pipermail/libraries/2014-July/023291.html>
-}
inits :: [a] -> [[a]]
inits :: forall a. [a] -> [[a]]
inits = forall a b. (a -> b) -> [a] -> [b]
map forall a. [a] -> [a]
reverse forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall b a. (b -> a -> b) -> b -> [a] -> [b]
scanl (forall a b c. (a -> b -> c) -> b -> a -> c
flip (:)) []

{- |
As lazy as 'inits' but less efficient because of repeated 'map'.
-}
initsLazy :: [a] -> [[a]]
initsLazy :: forall a. [a] -> [[a]]
initsLazy [a]
xt =
   [] forall a. a -> [a] -> [a]
:
   case [a]
xt of
      [] -> []
      a
x:[a]
xs -> forall a b. (a -> b) -> [a] -> [b]
map (a
xforall a. a -> [a] -> [a]
:) (forall a. [a] -> [[a]]
initsLazy [a]
xs)

{- |
Suggested implementation in the Haskell 98 report.
It is not as lazy as possible.
-}
inits98 :: [a] -> [[a]]
inits98 :: forall a. [a] -> [[a]]
inits98 []     = [[]]
inits98 (a
x:[a]
xs) = [[]] forall a. [a] -> [a] -> [a]
++ forall a b. (a -> b) -> [a] -> [b]
map (a
xforall a. a -> [a] -> [a]
:) (forall a. [a] -> [[a]]
inits98 [a]
xs)

inits98' :: [a] -> [[a]]
inits98' :: forall a. [a] -> [[a]]
inits98' =
   forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr (\a
x [[a]]
prefixes -> [] forall a. a -> [a] -> [a]
: forall a b. (a -> b) -> [a] -> [b]
map (a
xforall a. a -> [a] -> [a]
:) [[a]]
prefixes) [[]]


{- |
This function is lazier than the one suggested in the Haskell 98 report.
It is @tails undefined = ([] : undefined) : undefined@,
in contrast to @Data.List.tails undefined = undefined@.
-}
tails :: [a] -> [[a]]
tails :: forall a. [a] -> [[a]]
tails [a]
xt =
   forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry (:) forall a b. (a -> b) -> a -> b
$
   case [a]
xt of
      [] -> ([],[])
      a
_:[a]
xs -> ([a]
xt, forall a. [a] -> [[a]]
tails [a]
xs)

tails' :: [a] -> [[a]]
tails' :: forall a. [a] -> [[a]]
tails' = forall a b. (a, b) -> a
fst forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. (a -> Bool) -> [a] -> ([a], [a])
breakAfter forall (t :: * -> *) a. Foldable t => t a -> Bool
null forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. (a -> a) -> a -> [a]
iterate forall a. [a] -> [a]
tail

tails98            :: [a] -> [[a]]
tails98 :: forall a. [a] -> [[a]]
tails98 []         = [[]]
tails98 xxs :: [a]
xxs@(a
_:[a]
xs) = [a]
xxs forall a. a -> [a] -> [a]
: forall a. [a] -> [[a]]
tails98 [a]
xs

{- |
This function compares adjacent elements of a list.
If two adjacent elements satisfy a relation then they are put into the same sublist.
Example:

>>> groupBy (<) "abcdebcdef"
["abcde","bcdef"]

In contrast to that 'Data.List.groupBy' compares
the head of each sublist with each candidate for this sublist.
This yields

>>> List.groupBy (<) "abcdebcdef"
["abcdebcdef"]

The second @'b'@ is compared with the leading @'a'@.
Thus it is put into the same sublist as @'a'@.

The sublists are never empty.
Thus the more precise result type would be @[(a,[a])]@.
-}
groupBy :: (a -> a -> Bool) -> [a] -> [[a]]
groupBy :: forall a. (a -> a -> Bool) -> [a] -> [[a]]
groupBy = forall a. (a -> a -> Bool) -> [a] -> [[a]]
Key.groupBy

group :: (Eq a) => [a] -> [[a]]
group :: forall a. Eq a => [a] -> [[a]]
group = forall a. (a -> a -> Bool) -> [a] -> [[a]]
groupBy forall a. Eq a => a -> a -> Bool
(==)


{- |
Like standard 'unzip' but more lazy.
It is @Data.List.unzip undefined == undefined@,
but @unzip undefined == (undefined, undefined)@.
-}
unzip :: [(a,b)] -> ([a],[b])
unzip :: forall a b. [(a, b)] -> ([a], [b])
unzip =
   forall a b. (a, b) -> (a, b)
forcePair forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr (\ (a
x,b
y) ~([a]
xs,[b]
ys) -> (a
xforall a. a -> [a] -> [a]
:[a]
xs,b
yforall a. a -> [a] -> [a]
:[b]
ys)) ([],[])


{- |
'Data.List.partition' of GHC 6.2.1 fails on infinite lists.
But this one does not.
-}
{-
The lazy pattern match @(y,z)@ is necessary
since otherwise it fails on infinite lists.
-}
partition :: (a -> Bool) -> [a] -> ([a], [a])
partition :: forall a. (a -> Bool) -> [a] -> ([a], [a])
partition a -> Bool
p =
   forall a b. (a, b) -> (a, b)
forcePair forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr
      (\a
x ~([a]
y,[a]
z) ->
         if a -> Bool
p a
x
           then (a
x forall a. a -> [a] -> [a]
: [a]
y, [a]
z)
           else ([a]
y, a
x forall a. a -> [a] -> [a]
: [a]
z))
      ([],[])

{- |
It is @Data.List.span f undefined = undefined@,
whereas @span f undefined = (undefined, undefined)@.
-}
span, break :: (a -> Bool) -> [a] -> ([a],[a])
span :: forall a. (a -> Bool) -> [a] -> ([a], [a])
span a -> Bool
p =
   let recourse :: [a] -> ([a], [a])
recourse [a]
xt =
          forall a b. (a, b) -> (a, b)
forcePair forall a b. (a -> b) -> a -> b
$
          forall a. a -> Maybe a -> a
fromMaybe ([],[a]
xt) forall a b. (a -> b) -> a -> b
$
          do (a
x,[a]
xs) <- forall a. [a] -> Maybe (a, [a])
viewL [a]
xt
             forall (f :: * -> *). Alternative f => Bool -> f ()
guard forall a b. (a -> b) -> a -> b
$ a -> Bool
p a
x
             forall (m :: * -> *) a. Monad m => a -> m a
return forall a b. (a -> b) -> a -> b
$ forall a c b. (a -> c) -> (a, b) -> (c, b)
mapFst (a
xforall a. a -> [a] -> [a]
:) forall a b. (a -> b) -> a -> b
$ [a] -> ([a], [a])
recourse [a]
xs
   in  [a] -> ([a], [a])
recourse

break :: forall a. (a -> Bool) -> [a] -> ([a], [a])
break a -> Bool
p =  forall a. (a -> Bool) -> [a] -> ([a], [a])
span (Bool -> Bool
not forall b c a. (b -> c) -> (a -> b) -> a -> c
. a -> Bool
p)



-- * Split

{- |
Split the list at the occurrences of a separator into sub-lists.
Remove the separators.
This is somehow a generalization of 'lines' and 'words'.
But note the differences:

>>> words "a  a"
["a","a"]
>>> chop (' '==) "a  a"
["a","","a"]

>>> lines "a\n\na"
["a","","a"]
>>> chop ('\n'==) "a\n\na"
["a","","a"]

>>> lines "a\n"
["a"]
>>> chop ('\n'==) "a\n"
["a",""]
-}
chop :: (a -> Bool) -> [a] -> [[a]]
chop :: forall a. (a -> Bool) -> [a] -> [[a]]
chop a -> Bool
p =
   forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry (:) forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr (\ a
x ~([a]
y,[[a]]
ys) -> if a -> Bool
p a
x then ([],[a]
yforall a. a -> [a] -> [a]
:[[a]]
ys) else ((a
xforall a. a -> [a] -> [a]
:[a]
y),[[a]]
ys) ) ([],[])

chop' :: (a -> Bool) -> [a] -> [[a]]
chop' :: forall a. (a -> Bool) -> [a] -> [[a]]
chop' a -> Bool
p =
   let recourse :: [a] -> [[a]]
recourse =
          forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry (:) forall b c a. (b -> c) -> (a -> b) -> a -> c
.
          forall b c a. (b -> c) -> (a, b) -> (a, c)
mapSnd (forall b a. b -> (a -> [a] -> b) -> [a] -> b
switchL [] (forall a b. a -> b -> a
const [a] -> [[a]]
recourse)) forall b c a. (b -> c) -> (a -> b) -> a -> c
.
          forall a. (a -> Bool) -> [a] -> ([a], [a])
break a -> Bool
p
   in  [a] -> [[a]]
recourse


chopAtRun :: (a -> Bool) -> [a] -> [[a]]
chopAtRun :: forall a. (a -> Bool) -> [a] -> [[a]]
chopAtRun a -> Bool
p =
   let recourse :: [a] -> [[a]]
recourse [] = [[]]
       recourse [a]
y =
          let ([a]
z,[a]
zs) = forall a. (a -> Bool) -> [a] -> ([a], [a])
break a -> Bool
p (forall a. (a -> Bool) -> [a] -> [a]
dropWhile a -> Bool
p [a]
y)
          in [a]
z forall a. a -> [a] -> [a]
: [a] -> [[a]]
recourse [a]
zs
   in  [a] -> [[a]]
recourse


{- |
Like 'break', but splits after the matching element.

prop> forAllPredicates $ \p xs -> uncurry (++) (breakAfter p xs) == xs
-}
breakAfter :: (a -> Bool) -> [a] -> ([a], [a])
breakAfter :: forall a. (a -> Bool) -> [a] -> ([a], [a])
breakAfter = forall a. (a -> Bool) -> [a] -> ([a], [a])
breakAfterRec

breakAfterRec :: (a -> Bool) -> [a] -> ([a], [a])
breakAfterRec :: forall a. (a -> Bool) -> [a] -> ([a], [a])
breakAfterRec a -> Bool
p =
   let recourse :: [a] -> ([a], [a])
recourse [] = ([],[])
       recourse (a
x:[a]
xs) =
          forall a c b. (a -> c) -> (a, b) -> (c, b)
mapFst (a
xforall a. a -> [a] -> [a]
:) forall a b. (a -> b) -> a -> b
$
          if a -> Bool
p a
x
            then ([],[a]
xs)
            else [a] -> ([a], [a])
recourse [a]
xs
   in  forall a b. (a, b) -> (a, b)
forcePair forall b c a. (b -> c) -> (a -> b) -> a -> c
. [a] -> ([a], [a])
recourse

{-
The use of 'foldr' might allow for fusion,
but unfortunately this simple implementation would copy the tail of the list.
-}
-- | prop> forAllPredicates $ \p xs -> breakAfterRec p xs == breakAfterFoldr p xs
breakAfterFoldr :: (a -> Bool) -> [a] -> ([a], [a])
breakAfterFoldr :: forall a. (a -> Bool) -> [a] -> ([a], [a])
breakAfterFoldr a -> Bool
p =
   forall a b. (a, b) -> (a, b)
forcePair forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr
      (\a
x ([a], [a])
yzs -> forall a c b. (a -> c) -> (a, b) -> (c, b)
mapFst (a
xforall a. a -> [a] -> [a]
:) forall a b. (a -> b) -> a -> b
$ if a -> Bool
p a
x then ([], forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry forall a. [a] -> [a] -> [a]
(++) ([a], [a])
yzs) else ([a], [a])
yzs)
      ([],[])

-- | prop> forAllPredicates $ \p xs -> breakAfterRec p xs == breakAfterBreak p xs
breakAfterBreak :: (a -> Bool) -> [a] -> ([a], [a])
breakAfterBreak :: forall a. (a -> Bool) -> [a] -> ([a], [a])
breakAfterBreak a -> Bool
p [a]
xs =
   case forall a. (a -> Bool) -> [a] -> ([a], [a])
break a -> Bool
p [a]
xs of
      ([a]
ys, []) -> ([a]
ys, [])
      ([a]
ys, a
z:[a]
zs) -> ([a]
ysforall a. [a] -> [a] -> [a]
++[a
z], [a]
zs)

-- | prop> forAllPredicates $ \p xs -> breakAfterRec p xs == breakAfterTakeUntil p xs
breakAfterTakeUntil :: (a -> Bool) -> [a] -> ([a], [a])
breakAfterTakeUntil :: forall a. (a -> Bool) -> [a] -> ([a], [a])
breakAfterTakeUntil a -> Bool
p [a]
xs =
   forall a b. (a, b) -> (a, b)
forcePair forall a b. (a -> b) -> a -> b
$
   (\[(a, [a])]
ys -> (forall a b. (a -> b) -> [a] -> [b]
map forall a b. (a, b) -> a
fst [(a, [a])]
ys, forall b a. b -> (a -> b) -> Maybe a -> b
maybe [] (forall a b. (a, b) -> b
snd forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a b. (a, b) -> b
snd) forall a b. (a -> b) -> a -> b
$ forall a. [a] -> Maybe ([a], a)
viewR [(a, [a])]
ys)) forall a b. (a -> b) -> a -> b
$
   forall a. (a -> Bool) -> [a] -> [a]
takeUntil (a -> Bool
p forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a b. (a, b) -> a
fst) forall a b. (a -> b) -> a -> b
$ forall a b. [a] -> [b] -> [(a, b)]
zip [a]
xs forall a b. (a -> b) -> a -> b
$ forall a. [a] -> [a]
tail forall a b. (a -> b) -> a -> b
$ forall a. [a] -> [[a]]
tails [a]
xs

{- |
Take all elements until one matches.
The matching element is returned, too.
This is the key difference to @takeWhile (not . p)@.
It holds:

prop> forAllPredicates $ \p xs -> takeUntil p xs == fst (breakAfter p xs)
-}
takeUntil :: (a -> Bool) -> [a] -> [a]
takeUntil :: forall a. (a -> Bool) -> [a] -> [a]
takeUntil a -> Bool
p = forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr (\a
x [a]
ys -> a
x forall a. a -> [a] -> [a]
: if a -> Bool
p a
x then [] else [a]
ys) []


{- |
Split the list after each occurence of a terminator.
Keep the terminator.
There is always a list for the part after the last terminator.
It may be empty.
See package @non-empty@ for more precise result type.

prop> forAllPredicates $ \p xs -> concat (segmentAfter p xs) == xs
prop> forAllPredicates $ \p xs -> length (filter p xs) == length (tail (segmentAfter p xs))
prop> forAllPredicates $ \p -> all (p . last) . init . segmentAfter p
prop> forAllPredicates $ \p -> all (all (not . p) . init) . init . segmentAfter p

This test captures both infinitely many groups and infinitely big groups:

prop> forAllPredicates $ \p x -> flip seq True . (!!100) . concat . segmentAfter p . cycle . (x:)
-}
segmentAfter :: (a -> Bool) -> [a] -> [[a]]
segmentAfter :: forall a. (a -> Bool) -> [a] -> [[a]]
segmentAfter a -> Bool
p =
   forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry (:) forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr
      (\a
x ~([a]
y,[[a]]
ys) ->
         forall a c b. (a -> c) -> (a, b) -> (c, b)
mapFst (a
xforall a. a -> [a] -> [a]
:) forall a b. (a -> b) -> a -> b
$
         if a -> Bool
p a
x then ([],[a]
yforall a. a -> [a] -> [a]
:[[a]]
ys) else ([a]
y,[[a]]
ys))
      ([],[])

segmentAfter' :: (a -> Bool) -> [a] -> [[a]]
segmentAfter' :: forall a. (a -> Bool) -> [a] -> [[a]]
segmentAfter' a -> Bool
p =
   forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr (\ a
x ~yt :: [[a]]
yt@([a]
y:[[a]]
ys) -> if a -> Bool
p a
x then [a
x]forall a. a -> [a] -> [a]
:[[a]]
yt else (a
xforall a. a -> [a] -> [a]
:[a]
y)forall a. a -> [a] -> [a]
:[[a]]
ys) [[]]

{- |
Split the list before each occurence of a leading character.
Keep these characters.
There is always a list for the part before the first leading character.
It may be empty.
See package @non-empty@ for more precise result type.

prop> forAllPredicates $ \p xs -> concat (segmentBefore p xs) == xs
prop> forAllPredicates $ \p xs -> length (filter p xs) == length (tail (segmentBefore p xs))
prop> forAllPredicates $ \p -> all (p . head) . tail . segmentBefore p
prop> forAllPredicates $ \p -> all (all (not . p) . tail) . tail . segmentBefore p
prop> forAllPredicates $ \p x -> flip seq True . (!!100) . concat . segmentBefore p . cycle . (x:)
-}
segmentBefore :: (a -> Bool) -> [a] -> [[a]]
segmentBefore :: forall a. (a -> Bool) -> [a] -> [[a]]
segmentBefore a -> Bool
p =
--   foldr (\ x ~(y:ys) -> (if p x then ([]:) else id) ((x:y):ys)) [[]]
   forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry (:) forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr
      (\ a
x ~([a]
y,[[a]]
ys) ->
         let xs :: [a]
xs = a
xforall a. a -> [a] -> [a]
:[a]
y
         in  if a -> Bool
p a
x then ([],[a]
xsforall a. a -> [a] -> [a]
:[[a]]
ys) else ([a]
xs,[[a]]
ys))
      ([],[])

-- | prop> forAllPredicates $ \p xs -> segmentBefore p xs == segmentBefore' p xs
segmentBefore' :: (a -> Bool) -> [a] -> [[a]]
segmentBefore' :: forall a. (a -> Bool) -> [a] -> [[a]]
segmentBefore' a -> Bool
p =
   forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry (:) forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   (\[[a]]
xst ->
      forall a. a -> Maybe a -> a
fromMaybe ([],[[a]]
xst) forall a b. (a -> b) -> a -> b
$ do
         ((a
x:[a]
xs):[[a]]
xss) <- forall a. a -> Maybe a
Just [[a]]
xst
         forall (f :: * -> *). Alternative f => Bool -> f ()
guard forall a b. (a -> b) -> a -> b
$ Bool -> Bool
not forall a b. (a -> b) -> a -> b
$ a -> Bool
p a
x
         forall (m :: * -> *) a. Monad m => a -> m a
return (a
xforall a. a -> [a] -> [a]
:[a]
xs, [[a]]
xss)) forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   forall a. (a -> a -> Bool) -> [a] -> [[a]]
groupBy (\a
_ a
x -> Bool -> Bool
not forall a b. (a -> b) -> a -> b
$ a -> Bool
p a
x)

-- | prop> forAllPredicates $ \p xs -> segmentBefore p xs == segmentBefore'' p xs
segmentBefore'' :: (a -> Bool) -> [a] -> [[a]]
segmentBefore'' :: forall a. (a -> Bool) -> [a] -> [[a]]
segmentBefore'' a -> Bool
p =
   (\[[a]]
xst ->
      case [[a]]
xst of
         ~([a]
xs:[[a]]
xss) ->
            forall a. [a] -> [a]
tail [a]
xs forall a. a -> [a] -> [a]
: [[a]]
xss) forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   forall a. (a -> a -> Bool) -> [a] -> [[a]]
groupBy (\a
_ a
x -> Bool -> Bool
not forall a b. (a -> b) -> a -> b
$ a -> Bool
p a
x) forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   (forall a. HasCallStack => [Char] -> a
error [Char]
"segmentBefore: dummy element" forall a. a -> [a] -> [a]
:)


{- |
>>> segmentBeforeJust (\c -> toMaybe (isLetter c) (toUpper c)) "123a5345b---"
("123",[('A',"5345"),('B',"---")])
-}
segmentBeforeJust ::
   (a -> Maybe b) ->
   [a] -> ([a], [(b, [a])])
segmentBeforeJust :: forall a b. (a -> Maybe b) -> [a] -> ([a], [(b, [a])])
segmentBeforeJust a -> Maybe b
f =
   forall a b. (a, b) -> (a, b)
forcePair forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr
      (\ a
x ~([a]
y,[(b, [a])]
ys) ->
         case a -> Maybe b
f a
x of
            Just b
b -> ([],(b
b,[a]
y)forall a. a -> [a] -> [a]
:[(b, [a])]
ys)
            Maybe b
Nothing -> (a
xforall a. a -> [a] -> [a]
:[a]
y,[(b, [a])]
ys))
      ([],[])

{- |
>>> segmentAfterJust (\c -> toMaybe (isLetter c) (toUpper c)) "123a5345b---"
([("123",'A'),("5345",'B')],"---")
-}
segmentAfterJust ::
   (a -> Maybe b) ->
   [a] -> ([([a], b)], [a])
segmentAfterJust :: forall a b. (a -> Maybe b) -> [a] -> ([([a], b)], [a])
segmentAfterJust a -> Maybe b
f =
   forall a b. (a, b) -> (b, a)
swap forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry (forall (t :: * -> *) s a b.
Traversable t =>
(s -> a -> (s, b)) -> s -> t a -> (s, t b)
mapAccumL (\[a]
as0 (b
b,[a]
as1) -> ([a]
as1, ([a]
as0,b
b)))) forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   forall a b. (a -> Maybe b) -> [a] -> ([a], [(b, [a])])
segmentBeforeJust a -> Maybe b
f


{- |
>>> segmentBeforeRight [Left 'a', Right LT, Right GT, Left 'b']
("a",[(LT,""),(GT,"b")])

prop> forAllMaybeFn $ \f xs -> segmentBeforeJust f xs == segmentBeforeRight (map (\x -> maybe (Left x) Right (f x)) xs)
-}
segmentBeforeRight ::
   [Either a b] -> ([a], [(b, [a])])
segmentBeforeRight :: forall a b. [Either a b] -> ([a], [(b, [a])])
segmentBeforeRight =
   forall a b. (a, b) -> (a, b)
forcePair forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr
      (\ Either a b
x ~([a]
y,[(b, [a])]
ys) ->
         case Either a b
x of
            Right b
b -> ([],(b
b,[a]
y)forall a. a -> [a] -> [a]
:[(b, [a])]
ys)
            Left a
a -> (a
aforall a. a -> [a] -> [a]
:[a]
y,[(b, [a])]
ys))
      ([],[])

{- |
>>> segmentAfterRight [Left 'a', Right LT, Right GT, Left 'b']
([("a",LT),("",GT)],"b")

prop> forAllMaybeFn $ \f xs -> segmentAfterJust f xs == segmentAfterRight (map (\x -> maybe (Left x) Right (f x)) xs)
-}
segmentAfterRight ::
   [Either a b] -> ([([a], b)], [a])
segmentAfterRight :: forall a b. [Either a b] -> ([([a], b)], [a])
segmentAfterRight =
   forall a b. (a, b) -> (b, a)
swap forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry (forall (t :: * -> *) s a b.
Traversable t =>
(s -> a -> (s, b)) -> s -> t a -> (s, t b)
mapAccumL (\[a]
as0 (b
b,[a]
as1) -> ([a]
as1, ([a]
as0,b
b)))) forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   forall a b. [Either a b] -> ([a], [(b, [a])])
segmentBeforeRight


-- cf. Matroid.hs
{- |
@removeEach xs@ represents a list of sublists of @xs@,
where each element of @xs@ is removed and
the removed element is separated.
It seems to be much simpler to achieve with
@zip xs (map (flip List.delete xs) xs)@,
but the implementation of 'removeEach' does not need the 'Eq' instance
and thus can also be used for lists of functions.

See also the proposal
 <http://www.haskell.org/pipermail/libraries/2008-February/009270.html>

>>> removeEach "abc"
[('a',"bc"),('b',"ac"),('c',"ab")]
>>> removeEach "a"
[('a',"")]
>>> removeEach ""
[]
-}
removeEach :: [a] -> [(a, [a])]
removeEach :: forall a. [a] -> [(a, [a])]
removeEach =
   forall a b. (a -> b) -> [a] -> [b]
map (\([a]
ys, a
pivot, [a]
zs) -> (a
pivot,[a]
ysforall a. [a] -> [a] -> [a]
++[a]
zs)) forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. [a] -> [([a], a, [a])]
splitEverywhere

{- |
>>> splitEverywhere "abc"
[("",'a',"bc"),("a",'b',"c"),("ab",'c',"")]
>>> splitEverywhere "a"
[("",'a',"")]
>>> splitEverywhere ""
[]
-}
splitEverywhere :: [a] -> [([a], a, [a])]
splitEverywhere :: forall a. [a] -> [([a], a, [a])]
splitEverywhere [a]
xs =
   forall a b. (a -> b) -> [a] -> [b]
map
      (\([a]
y, [a]
zs0) ->
         case [a]
zs0 of
            a
z:[a]
zs -> ([a]
y,a
z,[a]
zs)
            [] -> forall a. HasCallStack => [Char] -> a
error [Char]
"splitEverywhere: empty list")
      (forall a. [a] -> [a]
init (forall a b. [a] -> [b] -> [(a, b)]
zip (forall a. [a] -> [[a]]
inits [a]
xs) (forall a. [a] -> [[a]]
tails [a]
xs)))



--  * inspect ends of a list

{-# DEPRECATED splitLast "use viewR instead" #-}
{- |
It holds @splitLast xs == (init xs, last xs)@,
but 'splitLast' is more efficient
if the last element is accessed after the initial ones,
because it avoids memoizing list.

prop> \(NonEmpty xs) -> splitLast (xs::String)  ==  (init xs, last xs)
-}
splitLast :: [a] -> ([a], a)
splitLast :: forall a. [a] -> ([a], a)
splitLast [] = forall a. HasCallStack => [Char] -> a
error [Char]
"splitLast: empty list"
splitLast [a
x] = ([], a
x)
splitLast (a
x:[a]
xs) =
   let ([a]
xs', a
lastx) = forall a. [a] -> ([a], a)
splitLast [a]
xs in (a
xforall a. a -> [a] -> [a]
:[a]
xs', a
lastx)


{- |
Should be prefered to 'head' and 'tail'.
-}
{-# INLINE viewL #-}
viewL :: [a] -> Maybe (a, [a])
viewL :: forall a. [a] -> Maybe (a, [a])
viewL (a
x:[a]
xs) = forall a. a -> Maybe a
Just (a
x,[a]
xs)
viewL []     = forall a. Maybe a
Nothing

{- |
Should be prefered to 'init' and 'last'.

prop> \xs -> maybe True ((init xs, last xs) == ) (viewR (xs::String))
-}
viewR :: [a] -> Maybe ([a], a)
viewR :: forall a. [a] -> Maybe ([a], a)
viewR =
   forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr (\a
x -> forall a. a -> Maybe a
Just forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a b. (a, b) -> (a, b)
forcePair forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall b a. b -> (a -> b) -> Maybe a -> b
maybe ([],a
x) (forall a c b. (a -> c) -> (a, b) -> (c, b)
mapFst (a
xforall a. a -> [a] -> [a]
:))) forall a. Maybe a
Nothing

{- |
Should be prefered to 'head' and 'tail'.
-}
{-# INLINE switchL #-}
switchL :: b -> (a -> [a] -> b) -> [a] -> b
switchL :: forall b a. b -> (a -> [a] -> b) -> [a] -> b
switchL b
n a -> [a] -> b
_ [] = b
n
switchL b
_ a -> [a] -> b
j (a
x:[a]
xs) = a -> [a] -> b
j a
x [a]
xs

switchL' :: b -> (a -> [a] -> b) -> [a] -> b
switchL' :: forall b a. b -> (a -> [a] -> b) -> [a] -> b
switchL' b
n a -> [a] -> b
j =
   forall b a. b -> (a -> b) -> Maybe a -> b
maybe b
n (forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry a -> [a] -> b
j) forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. [a] -> Maybe (a, [a])
viewL

{- |
Should be prefered to 'init' and 'last'.

prop> \xs -> switchR True (\ixs lxs -> ixs == init xs && lxs == last xs) (xs::String)
-}
{-# INLINE switchR #-}
switchR :: b -> ([a] -> a -> b) -> [a] -> b
switchR :: forall b a. b -> ([a] -> a -> b) -> [a] -> b
switchR b
n [a] -> a -> b
j =
   forall b a. b -> (a -> b) -> Maybe a -> b
maybe b
n (forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry [a] -> a -> b
j) forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. [a] -> Maybe ([a], a)
viewR


-- * List processing starting at the end

{- |
@takeRev n@ is like @reverse . take n . reverse@
but it is lazy enough to work for infinite lists, too.

prop> \n xs -> takeRev n (xs::String) == reverse (take n (reverse xs))
-}
takeRev :: Int -> [a] -> [a]
takeRev :: forall a. Int -> [a] -> [a]
takeRev Int
n [a]
xs = forall b a. [b] -> [a] -> [a]
Match.drop (forall a. Int -> [a] -> [a]
drop Int
n [a]
xs) [a]
xs

{- |
@dropRev n@ is like @reverse . drop n . reverse@
but it is lazy enough to work for infinite lists, too.

prop> \n xs -> dropRev n (xs::String) == reverse (drop n (reverse xs))
-}
dropRev :: Int -> [a] -> [a]
dropRev :: forall a. Int -> [a] -> [a]
dropRev Int
n [a]
xs = forall b a. [b] -> [a] -> [a]
Match.take (forall a. Int -> [a] -> [a]
drop Int
n [a]
xs) [a]
xs

{- |
@splitAtRev n xs == (dropRev n xs, takeRev n xs)@.

prop> \n xs -> splitAtRev n (xs::String) == (dropRev n xs, takeRev n xs)
prop> \n xs -> (xs::String) == uncurry (++) (splitAtRev n xs)
-}
splitAtRev :: Int -> [a] -> ([a], [a])
splitAtRev :: forall a. Int -> [a] -> ([a], [a])
splitAtRev Int
n [a]
xs = forall b a. [b] -> [a] -> ([a], [a])
Match.splitAt (forall a. Int -> [a] -> [a]
drop Int
n [a]
xs) [a]
xs


-- * List processing with Maybe and Either

{- |
@maybePrefixOf xs ys@ is @Just zs@ if @xs@ is a prefix of @ys@,
where @zs@ is @ys@ without the prefix @xs@.
Otherwise it is @Nothing@.
It is the same as 'Data.List.stripPrefix'.

>>> maybePrefixOf "abc" "abcdef"
Just "def"
>>> maybePrefixOf "def" "abcdef"
Nothing
-}
maybePrefixOf :: Eq a => [a] -> [a] -> Maybe [a]
maybePrefixOf :: forall a. Eq a => [a] -> [a] -> Maybe [a]
maybePrefixOf (a
x:[a]
xs) (a
y:[a]
ys) = forall (f :: * -> *). Alternative f => Bool -> f ()
guard (a
xforall a. Eq a => a -> a -> Bool
==a
y) forall (m :: * -> *) a b. Monad m => m a -> m b -> m b
>> forall a. Eq a => [a] -> [a] -> Maybe [a]
maybePrefixOf [a]
xs [a]
ys
maybePrefixOf [] [a]
ys = forall a. a -> Maybe a
Just [a]
ys
maybePrefixOf [a]
_  [] = forall a. Maybe a
Nothing

{- |
>>> maybeSuffixOf "abc" "abcdef"
Nothing
>>> maybeSuffixOf "def" "abcdef"
Just "abc"
-}
maybeSuffixOf :: Eq a => [a] -> [a] -> Maybe [a]
maybeSuffixOf :: forall a. Eq a => [a] -> [a] -> Maybe [a]
maybeSuffixOf [a]
xs [a]
ys =
   forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap forall a. [a] -> [a]
reverse forall a b. (a -> b) -> a -> b
$ forall a. Eq a => [a] -> [a] -> Maybe [a]
maybePrefixOf (forall a. [a] -> [a]
reverse [a]
xs) (forall a. [a] -> [a]
reverse [a]
ys)


{- |
Partition a list into elements which evaluate to @Just@ or @Nothing@ by @f@.

prop> forAllMaybeFn $ \f xs -> partitionMaybe f xs == (mapMaybe f xs, filter (isNothing . f) xs)
prop> forAllPredicates $ \p xs -> partition p xs == partitionMaybe (\x -> toMaybe (p x) x) xs
-}
partitionMaybe :: (a -> Maybe b) -> [a] -> ([b], [a])
partitionMaybe :: forall a b. (a -> Maybe b) -> [a] -> ([b], [a])
partitionMaybe a -> Maybe b
f =
   forall a b. (a, b) -> (a, b)
forcePair forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr
      (\a
x -> forall b a. b -> (a -> b) -> Maybe a -> b
maybe (forall b c a. (b -> c) -> (a, b) -> (a, c)
mapSnd (a
xforall a. a -> [a] -> [a]
:)) (\b
y -> forall a c b. (a -> c) -> (a, b) -> (c, b)
mapFst (b
yforall a. a -> [a] -> [a]
:)) (a -> Maybe b
f a
x))
      ([],[])

{- |
This is the cousin of 'takeWhile'
analogously to 'catMaybes' being the cousin of 'filter'.

>>> takeWhileJust [Just 'a', Just 'b', Nothing, Just 'c']
"ab"

Example: Keep the heads of sublists until an empty list occurs.

>>> takeWhileJust $ map (fmap fst . viewL) ["abc","def","","xyz"]
"ad"

For consistency with 'takeWhile',
'partitionMaybe' and 'dropWhileNothing' it should have been:

> takeWhileJust_ :: (a -> Maybe b) -> a -> [b]

However, both variants are interchangeable:

> takeWhileJust_ f == takeWhileJust . map f
> takeWhileJust == takeWhileJust_ id
-}
takeWhileJust :: [Maybe a] -> [a]
takeWhileJust :: forall a. [Maybe a] -> [a]
takeWhileJust =
   forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr (\Maybe a
x [a]
acc -> forall b a. b -> (a -> b) -> Maybe a -> b
maybe [] (forall a. a -> [a] -> [a]
:[a]
acc) Maybe a
x) []

dropWhileNothing :: (a -> Maybe b) -> [a] -> Maybe (b, [a])
dropWhileNothing :: forall a b. (a -> Maybe b) -> [a] -> Maybe (b, [a])
dropWhileNothing a -> Maybe b
f =
   forall (t :: * -> *) (m :: * -> *) a.
(Foldable t, MonadPlus m) =>
t (m a) -> m a
msum forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a b. (a -> b) -> [a] -> [b]
map (forall (f :: * -> *) a c b.
Functor f =>
(a -> f c) -> (a, b) -> f (c, b)
Func.mapFst a -> Maybe b
f forall (m :: * -> *) b c a.
Monad m =>
(b -> m c) -> (a -> m b) -> a -> m c
<=< forall a. [a] -> Maybe (a, [a])
viewL) forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. [a] -> [[a]]
tails

-- | prop> forAllMaybeFn $ \f xs -> dropWhileNothing f xs == dropWhileNothingRec f xs
dropWhileNothingRec :: (a -> Maybe b) -> [a] -> Maybe (b, [a])
dropWhileNothingRec :: forall a b. (a -> Maybe b) -> [a] -> Maybe (b, [a])
dropWhileNothingRec a -> Maybe b
f =
   let go :: [a] -> Maybe (b, [a])
go [] = forall a. Maybe a
Nothing
       go (a
a:[a]
xs) = (forall a b c. (a -> b -> c) -> b -> a -> c
flip (,) [a]
xs forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> a -> Maybe b
f a
a) forall (m :: * -> *) a. MonadPlus m => m a -> m a -> m a
`mplus` [a] -> Maybe (b, [a])
go [a]
xs
   in  [a] -> Maybe (b, [a])
go

-- | prop> forAllMaybeFn $ \f xs -> snd (breakJust f xs) == dropWhileNothing f xs
breakJust :: (a -> Maybe b) -> [a] -> ([a], Maybe (b, [a]))
breakJust :: forall a b. (a -> Maybe b) -> [a] -> ([a], Maybe (b, [a]))
breakJust a -> Maybe b
f =
   let go :: [a] -> ([a], Maybe (b, [a]))
go [] = ([], forall a. Maybe a
Nothing)
       go (a
a:[a]
xs) =
         case a -> Maybe b
f a
a of
            Maybe b
Nothing -> forall a c b. (a -> c) -> (a, b) -> (c, b)
mapFst (a
aforall a. a -> [a] -> [a]
:) forall a b. (a -> b) -> a -> b
$ [a] -> ([a], Maybe (b, [a]))
go [a]
xs
            Just b
b -> ([], forall a. a -> Maybe a
Just (b
b, [a]
xs))
   in  [a] -> ([a], Maybe (b, [a]))
go

-- memory leak, because xs is hold all the time
-- | prop> forAllMaybeFn $ \f xs -> breakJust f xs == breakJustRemoveEach f xs
breakJustRemoveEach :: (a -> Maybe b) -> [a] -> ([a], Maybe (b, [a]))
breakJustRemoveEach :: forall a b. (a -> Maybe b) -> [a] -> ([a], Maybe (b, [a]))
breakJustRemoveEach a -> Maybe b
f [a]
xs =
   forall b a. b -> (a -> [a] -> b) -> [a] -> b
switchL ([a]
xs, forall a. Maybe a
Nothing) forall a b. a -> b -> a
const forall a b. (a -> b) -> a -> b
$
   forall a b. (a -> Maybe b) -> [a] -> [b]
mapMaybe (\([a]
ys,a
a,[a]
zs) -> (\b
b -> ([a]
ys, forall a. a -> Maybe a
Just (b
b,[a]
zs))) forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> a -> Maybe b
f a
a) forall a b. (a -> b) -> a -> b
$
   forall a. [a] -> [([a], a, [a])]
splitEverywhere [a]
xs

-- needs to apply 'f' twice at the end and uses partial functions
-- | prop> forAllMaybeFn $ \f xs -> breakJust f xs == breakJustPartial f xs
breakJustPartial :: (a -> Maybe b) -> [a] -> ([a], Maybe (b, [a]))
breakJustPartial :: forall a b. (a -> Maybe b) -> [a] -> ([a], Maybe (b, [a]))
breakJustPartial a -> Maybe b
f [a]
xs =
   let ([a]
ys,[a]
zs) = forall a. (a -> Bool) -> [a] -> ([a], [a])
break (forall a. Maybe a -> Bool
isJust forall b c a. (b -> c) -> (a -> b) -> a -> c
. a -> Maybe b
f) [a]
xs
   in  ([a]
ys,
        forall a c b. (a -> c) -> (a, b) -> (c, b)
mapFst (forall b a. b -> (a -> b) -> Maybe a -> b
maybe (forall a. HasCallStack => [Char] -> a
error [Char]
"breakJust: unexpected Nothing") forall a. a -> a
id forall b c a. (b -> c) -> (a -> b) -> a -> c
. a -> Maybe b
f) forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$>
            forall a. [a] -> Maybe (a, [a])
viewL [a]
zs)

spanJust :: (a -> Maybe b) -> [a] -> ([b], [a])
spanJust :: forall a b. (a -> Maybe b) -> [a] -> ([b], [a])
spanJust a -> Maybe b
f =
   let go :: [a] -> ([b], [a])
go [] = ([], [])
       go xt :: [a]
xt@(a
a:[a]
xs) =
         case a -> Maybe b
f a
a of
            Just b
b -> forall a c b. (a -> c) -> (a, b) -> (c, b)
mapFst (b
bforall a. a -> [a] -> [a]
:) forall a b. (a -> b) -> a -> b
$ [a] -> ([b], [a])
go [a]
xs
            Maybe b
Nothing -> ([], [a]
xt)
   in  [a] -> ([b], [a])
go


unzipEithers :: [Either a b] -> ([a], [b])
unzipEithers :: forall a b. [Either a b] -> ([a], [b])
unzipEithers =
   forall a b. (a, b) -> (a, b)
forcePair forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr (forall a c b. (a -> c) -> (b -> c) -> Either a b -> c
either (\a
x -> forall a c b. (a -> c) -> (a, b) -> (c, b)
mapFst (a
xforall a. a -> [a] -> [a]
:)) (\b
y -> forall b c a. (b -> c) -> (a, b) -> (a, c)
mapSnd (b
yforall a. a -> [a] -> [a]
:))) ([],[])


-- * Sieve and slice

{- | keep every k-th value from the list

>>> sieve 6 ['a'..'z']
"agmsy"
-}
sieve, sieve', sieve'', sieve''' :: Int -> [a] -> [a]
sieve :: forall a. Int -> [a] -> [a]
sieve Int
k =
   forall b a. (b -> Maybe (a, b)) -> b -> [a]
unfoldr (\[a]
xs -> forall a. Bool -> a -> Maybe a
toMaybe (Bool -> Bool
not (forall (t :: * -> *) a. Foldable t => t a -> Bool
null [a]
xs)) (forall a. [a] -> a
head [a]
xs, forall a. Int -> [a] -> [a]
drop Int
k [a]
xs))

-- | prop> \(Positive n) xs -> sieve n xs == sieve' n (xs::String)
sieve' :: forall a. Int -> [a] -> [a]
sieve' Int
k = forall a b. (a -> b) -> [a] -> [b]
map forall a. [a] -> a
head forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. Int -> [a] -> [[a]]
sliceVertical Int
k

-- | prop> \(Positive n) xs -> sieve n xs == sieve'' n (xs::String)
sieve'' :: forall a. Int -> [a] -> [a]
sieve'' Int
k [a]
x = forall a b. (a -> b) -> [a] -> [b]
map ([a]
xforall a. [a] -> Int -> a
!!) [Int
0,Int
k..(forall (t :: * -> *) a. Foldable t => t a -> Int
length [a]
xforall a. Num a => a -> a -> a
-Int
1)]

-- | prop> \(Positive n) xs -> sieve n xs == sieve''' n (xs::String)
sieve''' :: forall a. Int -> [a] -> [a]
sieve''' Int
k = forall a b. (a -> b) -> [a] -> [b]
map forall a. [a] -> a
head forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. (a -> Bool) -> [a] -> [a]
takeWhile (Bool -> Bool
not forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (t :: * -> *) a. Foldable t => t a -> Bool
null) forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. (a -> a) -> a -> [a]
iterate (forall a. Int -> [a] -> [a]
drop Int
k)


{-
sliceHorizontal is faster than sliceHorizontal' but consumes slightly more memory
(although it needs no swapping)
-}
{- |
>>> sliceHorizontal 6 ['a'..'z']
["agmsy","bhntz","ciou","djpv","ekqw","flrx"]

prop> \(NonEmpty xs) -> QC.forAll (QC.choose (1, length xs)) $ \n -> sliceHorizontal n xs == transpose (sliceVertical n (xs::String))
prop> \(NonEmpty xs) -> QC.forAll (QC.choose (1, length xs)) $ \n -> sliceVertical  n xs == transpose (sliceHorizontal n (xs::String))

The properties do not hold for empty lists because of:

>>> sliceHorizontal 4 ([]::[Int])
[[],[],[],[]]
-}
sliceHorizontal, sliceHorizontal', sliceHorizontal'', sliceHorizontal''' ::
   Int -> [a] -> [[a]]
sliceHorizontal :: forall a. Int -> [a] -> [[a]]
sliceHorizontal Int
n =
   forall a b. (a -> b) -> [a] -> [b]
map (forall a. Int -> [a] -> [a]
sieve Int
n) forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. Int -> [a] -> [a]
take Int
n forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. (a -> a) -> a -> [a]
iterate (forall a. Int -> [a] -> [a]
drop Int
1)

-- | prop> \(NonNegative n) xs -> sliceHorizontal n xs == sliceHorizontal' n (xs::String)
sliceHorizontal' :: forall a. Int -> [a] -> [[a]]
sliceHorizontal' Int
n =
   forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr (\a
x [[a]]
ys -> let y :: [a]
y = forall a. [a] -> a
last [[a]]
ys in forall b a. [b] -> [a] -> [a]
Match.take [[a]]
ys ((a
xforall a. a -> [a] -> [a]
:[a]
y)forall a. a -> [a] -> [a]
:[[a]]
ys)) (forall a. Int -> a -> [a]
replicate Int
n [])

-- | prop> \(Positive n) xs -> sliceHorizontal n xs == sliceHorizontal'' n (xs::String)
sliceHorizontal'' :: forall a. Int -> [a] -> [[a]]
sliceHorizontal'' Int
n =
   forall a. [a] -> [a]
reverse forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr (\a
x ~([a]
y:[[a]]
ys) -> [[a]]
ys forall a. [a] -> [a] -> [a]
++ [a
xforall a. a -> [a] -> [a]
:[a]
y]) (forall a. Int -> a -> [a]
replicate Int
n [])

sliceHorizontal''' :: forall a. Int -> [a] -> [[a]]
sliceHorizontal''' Int
n =
   forall a. Int -> [a] -> [a]
take Int
n forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. [[a]] -> [[a]]
transpose forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. (a -> Bool) -> [a] -> [a]
takeWhile (Bool -> Bool
not forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (t :: * -> *) a. Foldable t => t a -> Bool
null) forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. (a -> a) -> a -> [a]
iterate (forall a. Int -> [a] -> [a]
drop Int
n)


{- |
>>> sliceVertical 6 ['a'..'z']
["abcdef","ghijkl","mnopqr","stuvwx","yz"]
-}
sliceVertical, sliceVertical' :: Int -> [a] -> [[a]]
sliceVertical :: forall a. Int -> [a] -> [[a]]
sliceVertical Int
n =
   forall a b. (a -> b) -> [a] -> [b]
map (forall a. Int -> [a] -> [a]
take Int
n) forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. (a -> Bool) -> [a] -> [a]
takeWhile (Bool -> Bool
not forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (t :: * -> *) a. Foldable t => t a -> Bool
null) forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. (a -> a) -> a -> [a]
iterate (forall a. Int -> [a] -> [a]
drop Int
n)
      {- takeWhile must be performed before (map take)
         in order to handle (n==0) correctly -}

-- | prop> \(NonNegative n) xs -> equating (take 100000) (sliceVertical n xs) (sliceVertical' n (xs::String))
sliceVertical' :: forall a. Int -> [a] -> [[a]]
sliceVertical' Int
n =
   forall b a. (b -> Maybe (a, b)) -> b -> [a]
unfoldr (\[a]
x -> forall a. Bool -> a -> Maybe a
toMaybe (Bool -> Bool
not (forall (t :: * -> *) a. Foldable t => t a -> Bool
null [a]
x)) (forall a. Int -> [a] -> ([a], [a])
splitAt Int
n [a]
x))




-- * Search&replace

search :: (Eq a) => [a] -> [a] -> [Int]
search :: forall a. Eq a => [a] -> [a] -> [Int]
search [a]
sub [a]
str = forall a. (a -> Bool) -> [a] -> [Int]
findIndices (forall a. Eq a => [a] -> [a] -> Bool
isPrefixOf [a]
sub) (forall a. [a] -> [[a]]
tails [a]
str)

{- |
prop> \(NonEmpty xs) ys -> replace xs xs ys == (ys::String)
prop> \(NonEmpty xs) (NonEmpty ys) -> equating (take 1000) (replace xs ys (cycle xs)) (cycle (ys::String))
-}
replace :: Eq a => [a] -> [a] -> [a] -> [a]
replace :: forall a. Eq a => [a] -> [a] -> [a] -> [a]
replace [a]
src [a]
dst =
   let recourse :: [a] -> [a]
recourse [] = []
       recourse str :: [a]
str@(a
s:[a]
ss) =
          forall a. a -> Maybe a -> a
fromMaybe
             (a
s forall a. a -> [a] -> [a]
: [a] -> [a]
recourse [a]
ss)
             (forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap (([a]
dstforall a. [a] -> [a] -> [a]
++) forall b c a. (b -> c) -> (a -> b) -> a -> c
. [a] -> [a]
recourse) forall a b. (a -> b) -> a -> b
$
              forall a. Eq a => [a] -> [a] -> Maybe [a]
maybePrefixOf [a]
src [a]
str)
   in  [a] -> [a]
recourse

markSublists :: (Eq a) => [a] -> [a] -> [Maybe [a]]
markSublists :: forall a. Eq a => [a] -> [a] -> [Maybe [a]]
markSublists [a]
sub [a]
ys =
   let ~([a]
hd', [Maybe [a]]
rest') =
          forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr (\a
c ~([a]
hd, [Maybe [a]]
rest) ->
                   let xs :: [a]
xs = a
cforall a. a -> [a] -> [a]
:[a]
hd
                   in  case forall a. Eq a => [a] -> [a] -> Maybe [a]
maybePrefixOf [a]
sub [a]
xs of
                         Just [a]
suffix -> ([], forall a. Maybe a
Nothing forall a. a -> [a] -> [a]
: forall a. a -> Maybe a
Just [a]
suffix forall a. a -> [a] -> [a]
: [Maybe [a]]
rest)
                         Maybe [a]
Nothing -> ([a]
xs, [Maybe [a]]
rest)) ([],[]) [a]
ys
   in  forall a. a -> Maybe a
Just [a]
hd' forall a. a -> [a] -> [a]
: [Maybe [a]]
rest'

replace' :: (Eq a) => [a] -> [a] -> [a] -> [a]
replace' :: forall a. Eq a => [a] -> [a] -> [a] -> [a]
replace' [a]
src [a]
dst [a]
xs =
   forall (t :: * -> *) a b. Foldable t => (a -> [b]) -> t a -> [b]
concatMap (forall a. a -> Maybe a -> a
fromMaybe [a]
dst) (forall a. Eq a => [a] -> [a] -> [Maybe [a]]
markSublists [a]
src [a]
xs)

{- | This is slightly wrong, because it re-replaces things.
     That's also the reason for inefficiency:
        The replacing can go on only when subsequent replacements are finished.
     Thus this functiob fails on infinite lists. -}
replace'' :: (Eq a) => [a] -> [a] -> [a] -> [a]
replace'' :: forall a. Eq a => [a] -> [a] -> [a] -> [a]
replace'' [a]
src [a]
dst =
    forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr (\a
x [a]
xs -> let y :: [a]
y=a
xforall a. a -> [a] -> [a]
:[a]
xs
                    in  if forall a. Eq a => [a] -> [a] -> Bool
isPrefixOf [a]
src [a]
y
                          then [a]
dst forall a. [a] -> [a] -> [a]
++ forall a. Int -> [a] -> [a]
drop (forall (t :: * -> *) a. Foldable t => t a -> Int
length [a]
src) [a]
y
                          else [a]
y) []

{- |
prop \src dst xs -> replace src dst xs == multiReplace [(src,dst)] (xs::String)
-}
multiReplace :: Eq a => [([a], [a])] -> [a] -> [a]
multiReplace :: forall a. Eq a => [([a], [a])] -> [a] -> [a]
multiReplace [([a], [a])]
dict =
   let recourse :: [a] -> [a]
recourse [] = []
       recourse str :: [a]
str@(a
s:[a]
ss) =
          forall a. a -> Maybe a -> a
fromMaybe
             (a
s forall a. a -> [a] -> [a]
: [a] -> [a]
recourse [a]
ss)
             (forall (t :: * -> *) (m :: * -> *) a.
(Foldable t, MonadPlus m) =>
t (m a) -> m a
msum forall a b. (a -> b) -> a -> b
$
              forall a b. (a -> b) -> [a] -> [b]
map (\([a]
src,[a]
dst) ->
                      forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap (([a]
dstforall a. [a] -> [a] -> [a]
++) forall b c a. (b -> c) -> (a -> b) -> a -> c
. [a] -> [a]
recourse) forall a b. (a -> b) -> a -> b
$
                      forall a. Eq a => [a] -> [a] -> Maybe [a]
maybePrefixOf [a]
src [a]
str) [([a], [a])]
dict)
   in  [a] -> [a]
recourse

multiReplace' :: Eq a => [([a], [a])] -> [a] -> [a]
multiReplace' :: forall a. Eq a => [([a], [a])] -> [a] -> [a]
multiReplace' [([a], [a])]
dict =
   let recourse :: [a] -> [a]
recourse [] = []
       recourse str :: [a]
str@(a
s:[a]
ss) =
          forall b a. b -> (a -> b) -> Maybe a -> b
maybe
             (a
s forall a. a -> [a] -> [a]
: [a] -> [a]
recourse [a]
ss)
             (\([a]
src, [a]
dst) -> [a]
dst forall a. [a] -> [a] -> [a]
++ [a] -> [a]
recourse (forall b a. [b] -> [a] -> [a]
Match.drop [a]
src [a]
str))
             (forall (t :: * -> *) a. Foldable t => (a -> Bool) -> t a -> Maybe a
find (forall a b c. (a -> b -> c) -> b -> a -> c
flip forall a. Eq a => [a] -> [a] -> Bool
isPrefixOf [a]
str forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a b. (a, b) -> a
fst) [([a], [a])]
dict)
   in  [a] -> [a]
recourse


-- * Lists of lists

{- |
Transform

> [[00,01,02,...],          [[00],
>  [10,11,12,...],   -->     [10,01],
>  [20,21,22,...],           [20,11,02],
>  ...]                      ...]

With @concat . shear@ you can perform a Cantor diagonalization,
that is an enumeration of all elements of the sub-lists
where each element is reachable within a finite number of steps.
It is also useful for polynomial multiplication (convolution).
-}
shear :: [[a]] -> [[a]]
shear :: forall a. [[a]] -> [[a]]
shear =
   forall a b. (a -> b) -> [a] -> [b]
map forall a. [Maybe a] -> [a]
catMaybes forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   forall a. [[a]] -> [[a]]
shearTranspose forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   forall a. [[a]] -> [[Maybe a]]
transposeFill

transposeFill :: [[a]] -> [[Maybe a]]
transposeFill :: forall a. [[a]] -> [[Maybe a]]
transposeFill =
   forall b a. (b -> Maybe (a, b)) -> b -> [a]
unfoldr (\[[a]]
xs ->
      forall a. Bool -> a -> Maybe a
toMaybe (Bool -> Bool
not (forall (t :: * -> *) a. Foldable t => t a -> Bool
null [[a]]
xs))
         (forall b c a. (b -> c) -> (a, b) -> (a, c)
mapSnd (forall a. (a -> Bool) -> [a] -> [a]
Rev.dropWhile forall (t :: * -> *) a. Foldable t => t a -> Bool
null) forall a b. (a -> b) -> a -> b
$ forall a. [[a]] -> ([Maybe a], [[a]])
unzipCons [[a]]
xs))

unzipCons :: [[a]] -> ([Maybe a], [[a]])
unzipCons :: forall a. [[a]] -> ([Maybe a], [[a]])
unzipCons =
   forall a b. [(a, b)] -> ([a], [b])
unzip forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   forall a b. (a -> b) -> [a] -> [b]
map ((\Maybe (a, [a])
my -> (forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap forall a b. (a, b) -> a
fst Maybe (a, [a])
my, forall b a. b -> (a -> b) -> Maybe a -> b
maybe [] forall a b. (a, b) -> b
snd Maybe (a, [a])
my)) forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. [a] -> Maybe (a, [a])
viewL)

{- |
It's somehow inverse to zipCons,
but the difficult part is,
that a trailing empty list on the right side is suppressed.
-}
unzipConsSkew :: [[a]] -> ([Maybe a], [[a]])
unzipConsSkew :: forall a. [[a]] -> ([Maybe a], [[a]])
unzipConsSkew =
   let aux :: [a] -> [[a]] -> ([Maybe a], [[a]])
aux [] [] = ([],[])  -- one empty list at the end will be removed
       aux [a]
xs [[a]]
ys = forall b c a. (b -> c) -> (a, b) -> (a, c)
mapSnd ([a]
xsforall a. a -> [a] -> [a]
:) forall a b. (a -> b) -> a -> b
$ [[a]] -> ([Maybe a], [[a]])
prep [[a]]
ys
       prep :: [[a]] -> ([Maybe a], [[a]])
prep =
          forall a b. (a, b) -> (a, b)
forcePair forall b c a. (b -> c) -> (a -> b) -> a -> c
.
          forall b a. b -> (a -> [a] -> b) -> [a] -> b
switchL ([],[])
             (\[a]
y [[a]]
ys ->
                let my :: Maybe (a, [a])
my = forall a. [a] -> Maybe (a, [a])
viewL [a]
y
                in  forall a c b. (a -> c) -> (a, b) -> (c, b)
mapFst (forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap forall a b. (a, b) -> a
fst Maybe (a, [a])
my forall a. a -> [a] -> [a]
:) forall a b. (a -> b) -> a -> b
$
                    [a] -> [[a]] -> ([Maybe a], [[a]])
aux (forall b a. b -> (a -> b) -> Maybe a -> b
maybe [] forall a b. (a, b) -> b
snd Maybe (a, [a])
my) [[a]]
ys)
   in  forall a. [[a]] -> ([Maybe a], [[a]])
prep



shear' :: [[a]] -> [[a]]
shear' :: forall a. [[a]] -> [[a]]
shear' xs :: [[a]]
xs@([a]
_:[[a]]
_) =
   let ([a]
y:[[a]]
ys,[[a]]
zs) = forall a b. [(a, b)] -> ([a], [b])
unzip (forall a b. (a -> b) -> [a] -> [b]
map (forall a. Int -> [a] -> ([a], [a])
splitAt Int
1) [[a]]
xs)
       zipConc :: [[a]] -> [[a]] -> [[a]]
zipConc ([a]
a:[[a]]
as) ([a]
b:[[a]]
bs) = ([a]
aforall a. [a] -> [a] -> [a]
++[a]
b) forall a. a -> [a] -> [a]
: [[a]] -> [[a]] -> [[a]]
zipConc [[a]]
as [[a]]
bs
       zipConc [] [[a]]
bs = [[a]]
bs
       zipConc [[a]]
as [] = [[a]]
as
   in  [a]
y forall a. a -> [a] -> [a]
: forall {a}. [[a]] -> [[a]] -> [[a]]
zipConc [[a]]
ys (forall a. [[a]] -> [[a]]
shear' (forall a. (a -> Bool) -> [a] -> [a]
Rev.dropWhile forall (t :: * -> *) a. Foldable t => t a -> Bool
null [[a]]
zs))
              {- Dropping trailing empty lists is necessary,
                 otherwise finite lists are filled with empty lists. -}
shear' [] = []

{- |
Transform

> [[00,01,02,...],          [[00],
>  [10,11,12,...],   -->     [01,10],
>  [20,21,22,...],           [02,11,20],
>  ...]                      ...]

It's like 'shear' but the order of elements in the sub list is reversed.
Its implementation seems to be more efficient than that of 'shear'.
If the order does not matter, better choose 'shearTranspose'.

prop> \xs -> shearTranspose xs  ==  map reverse (shear (xs::[String]))
-}
shearTranspose :: [[a]] -> [[a]]
shearTranspose :: forall a. [[a]] -> [[a]]
shearTranspose =
   forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr forall a. [a] -> [[a]] -> [[a]]
zipConsSkew []

zipConsSkew :: [a] -> [[a]] -> [[a]]
zipConsSkew :: forall a. [a] -> [[a]] -> [[a]]
zipConsSkew [a]
xt [[a]]
yss =
   forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry (:) forall a b. (a -> b) -> a -> b
$
   case [a]
xt of
      a
x:[a]
xs -> ([a
x], forall a. [a] -> [[a]] -> [[a]]
zipCons [a]
xs [[a]]
yss)
      [] -> ([], [[a]]
yss)

{- |
zipCons is like @zipWith (:)@ but it keeps lists which are too long
This version works also for @zipCons something undefined@.
-}
zipCons :: [a] -> [[a]] -> [[a]]
zipCons :: forall a. [a] -> [[a]] -> [[a]]
zipCons (a
x:[a]
xs) [[a]]
yt =
   let ([a]
y,[[a]]
ys) = forall b a. b -> (a -> [a] -> b) -> [a] -> b
switchL ([],[]) (,) [[a]]
yt
   in  (a
xforall a. a -> [a] -> [a]
:[a]
y) forall a. a -> [a] -> [a]
: forall a. [a] -> [[a]] -> [[a]]
zipCons [a]
xs [[a]]
ys
zipCons [] [[a]]
ys = [[a]]
ys

-- | zipCons' is like @zipWith (:)@ but it keeps lists which are too long
zipCons' :: [a] -> [[a]] -> [[a]]
zipCons' :: forall a. [a] -> [[a]] -> [[a]]
zipCons' (a
x:[a]
xs) ([a]
y:[[a]]
ys) = (a
xforall a. a -> [a] -> [a]
:[a]
y) forall a. a -> [a] -> [a]
: forall a. [a] -> [[a]] -> [[a]]
zipCons' [a]
xs [[a]]
ys
zipCons' [] [[a]]
ys = [[a]]
ys
zipCons' [a]
xs [] = forall a b. (a -> b) -> [a] -> [b]
map (forall a. a -> [a] -> [a]
:[]) [a]
xs


{- |
Operate on each combination of elements of the first and the second list.
In contrast to the list instance of 'Monad.liftM2'
it holds the results in a list of lists.

prop> \xs ys -> let f x y = (x::Char,y::Int) in concat (outerProduct f xs ys)  ==  liftM2 f xs ys
-}
outerProduct :: (a -> b -> c) -> [a] -> [b] -> [[c]]
outerProduct :: forall a b c. (a -> b -> c) -> [a] -> [b] -> [[c]]
outerProduct a -> b -> c
f [a]
xs [b]
ys = forall a b. (a -> b) -> [a] -> [b]
map (forall a b c. (a -> b -> c) -> b -> a -> c
flip forall a b. (a -> b) -> [a] -> [b]
map [b]
ys forall b c a. (b -> c) -> (a -> b) -> a -> c
. a -> b -> c
f) [a]
xs



-- * Miscellaneous

{- |
Take while first predicate holds,
then continue taking while second predicate holds,
and so on.
-}
takeWhileMulti :: [a -> Bool] -> [a] -> [a]
takeWhileMulti :: forall a. [a -> Bool] -> [a] -> [a]
takeWhileMulti [] [a]
_  = []
takeWhileMulti [a -> Bool]
_  [] = []
takeWhileMulti aps :: [a -> Bool]
aps@(a -> Bool
p:[a -> Bool]
ps) axs :: [a]
axs@(a
x:[a]
xs) =
   if a -> Bool
p a
x
      then a
x forall a. a -> [a] -> [a]
: forall a. [a -> Bool] -> [a] -> [a]
takeWhileMulti [a -> Bool]
aps [a]
xs
      else forall a. [a -> Bool] -> [a] -> [a]
takeWhileMulti [a -> Bool]
ps [a]
axs

{- |
prop> \ys xs -> let ps = map (<=) ys in takeWhileMulti ps xs == takeWhileMulti' ps (xs::String)
-}
takeWhileMulti' :: [a -> Bool] -> [a] -> [a]
takeWhileMulti' :: forall a. [a -> Bool] -> [a] -> [a]
takeWhileMulti' [a -> Bool]
ps [a]
xs =
   forall (t :: * -> *) a b. Foldable t => (a -> [b]) -> t a -> [b]
concatMap forall a b. (a, b) -> a
fst (forall a. [a] -> [a]
tail
      (forall b a. (b -> a -> b) -> b -> [a] -> [b]
scanl (forall a b c. (a -> b -> c) -> b -> a -> c
flip forall a. (a -> Bool) -> [a] -> ([a], [a])
span forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a b. (a, b) -> b
snd) (forall a. HasCallStack => a
undefined,[a]
xs) [a -> Bool]
ps))

{-
Debug.QuickCheck.quickCheck (propTakeWhileMulti [(<0), (>0), odd, even, ((0::Int)==)])
-}

{- |
This is a combination of 'foldl'' and 'foldr'
in the sense of 'propFoldl'r'.
It is however more efficient
because it avoids storing the whole input list as a result of sharing.
-}
foldl'r, foldl'rStrict, foldl'rNaive ::
   (b -> a -> b) -> b -> (c -> d -> d) -> d -> [(a,c)] -> (b,d)
foldl'r :: forall b a c d.
(b -> a -> b) -> b -> (c -> d -> d) -> d -> [(a, c)] -> (b, d)
foldl'r b -> a -> b
f b
b0 c -> d -> d
g d
d0 =
--   (\(k,d1) -> (k b0, d1)) .
   forall a c b. (a -> c) -> (a, b) -> (c, b)
mapFst (forall a b. (a -> b) -> a -> b
$ b
b0) forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr (\(a
a,c
c) ~(b -> b
k,d
d) -> (\b
b -> b -> b
k forall a b. (a -> b) -> a -> b
$! b -> a -> b
f b
b a
a, c -> d -> d
g c
c d
d)) (forall a. a -> a
id,d
d0)

foldl'rStrict :: forall b a c d.
(b -> a -> b) -> b -> (c -> d -> d) -> d -> [(a, c)] -> (b, d)
foldl'rStrict b -> a -> b
f b
b0 c -> d -> d
g d
d0 =
   forall a c b. (a -> c) -> (a, b) -> (c, b)
mapFst (forall a b. (a -> b) -> a -> b
$ b
b0) forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr (\(a
a,c
c) ~(b -> b
k,d
d) -> ((,) forall a b. (a -> b) -> a -> b
$! (\b
b -> b -> b
k forall a b. (a -> b) -> a -> b
$! b -> a -> b
f b
b a
a)) forall a b. (a -> b) -> a -> b
$! c -> d -> d
g c
c d
d) (forall a. a -> a
id,d
d0)

foldl'rNaive :: forall b a c d.
(b -> a -> b) -> b -> (c -> d -> d) -> d -> [(a, c)] -> (b, d)
foldl'rNaive b -> a -> b
f b
b c -> d -> d
g d
d [(a, c)]
xs =
   forall a c b d. (a -> c, b -> d) -> (a, b) -> (c, d)
mapPair (forall (t :: * -> *) b a.
Foldable t =>
(b -> a -> b) -> b -> t a -> b
foldl' b -> a -> b
f b
b, forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr c -> d -> d
g d
d) forall a b. (a -> b) -> a -> b
$ forall a b. [(a, b)] -> ([a], [b])
unzip [(a, c)]
xs

propFoldl'r :: (Eq b, Eq d) =>
   (b -> a -> b) -> b -> (c -> d -> d) -> d -> [(a,c)] -> Bool
propFoldl'r :: forall b d a c.
(Eq b, Eq d) =>
(b -> a -> b) -> b -> (c -> d -> d) -> d -> [(a, c)] -> Bool
propFoldl'r b -> a -> b
f b
b c -> d -> d
g d
d [(a, c)]
xs =
   forall b a c d.
(b -> a -> b) -> b -> (c -> d -> d) -> d -> [(a, c)] -> (b, d)
foldl'r b -> a -> b
f b
b c -> d -> d
g d
d [(a, c)]
xs forall a. Eq a => a -> a -> Bool
== forall b a c d.
(b -> a -> b) -> b -> (c -> d -> d) -> d -> [(a, c)] -> (b, d)
foldl'rNaive b -> a -> b
f b
b c -> d -> d
g d
d [(a, c)]
xs

{-
The results in GHCi surprise:

*List.HT> mapSnd last $ foldl'rNaive (+) (0::Integer) (:) "" $ replicate 1000000 (1,'a')
(1000000,'a')
(0.44 secs, 141032856 bytes)

*List.HT> mapSnd last $ foldl'r (+) (0::Integer) (:) "" $ replicate 1000000 (1,'a')
(1000000,'a')
(2.64 secs, 237424948 bytes)
-}

{-
Debug.QuickCheck.quickCheck (\b d -> propFoldl'r (+) (b::Int) (++) (d::[Int]))
-}


{- |
>>> lengthAtLeast 0 ""
True
>>> lengthAtLeast 3 "ab"
False
>>> lengthAtLeast 3 "abc"
True
>>> lengthAtLeast 3 $ repeat 'a'
True
>>> lengthAtLeast 3 $ "abc" ++ undefined
True

prop> \n xs -> lengthAtLeast n (xs::String)  ==  (length xs >= n)
-}
lengthAtLeast :: Int -> [a] -> Bool
lengthAtLeast :: forall a. Int -> [a] -> Bool
lengthAtLeast Int
n =
   if Int
nforall a. Ord a => a -> a -> Bool
<=Int
0
     then forall a b. a -> b -> a
const Bool
True
     else Bool -> Bool
not forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (t :: * -> *) a. Foldable t => t a -> Bool
null forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. Int -> [a] -> [a]
drop (Int
nforall a. Num a => a -> a -> a
-Int
1)

{- |
>>> lengthAtMost 0 ""
True
>>> lengthAtMost 3 "ab"
True
>>> lengthAtMost 3 "abc"
True
>>> lengthAtMost 3 "abcd"
False
>>> lengthAtMost 3 $ repeat 'a'
False
>>> lengthAtMost 3 $ "abcd" ++ undefined
False

prop> \n xs -> lengthAtMost n (xs::String)  ==  (length xs <= n)
-}
lengthAtMost :: Int -> [a] -> Bool
lengthAtMost :: forall a. Int -> [a] -> Bool
lengthAtMost Int
n =
   if Int
nforall a. Ord a => a -> a -> Bool
<Int
0
     then forall a b. a -> b -> a
const Bool
False
     else forall (t :: * -> *) a. Foldable t => t a -> Bool
null forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. Int -> [a] -> [a]
drop Int
n

{- |
prop> \n xs -> lengthAtMost0 n (xs::String)  ==  (length xs <= n)
-}
lengthAtMost0 :: Int -> [a] -> Bool
lengthAtMost0 :: forall a. Int -> [a] -> Bool
lengthAtMost0 Int
n = (Int
nforall a. Ord a => a -> a -> Bool
>=) forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (t :: * -> *) a. Foldable t => t a -> Int
length forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. Int -> [a] -> [a]
take (Int
nforall a. Num a => a -> a -> a
+Int
1)

{-
Iterate until elements start to cycle.
This implementation is inspired by Elements of Programming
but I am still not satisfied
where the iteration actually stops.
-}
iterateUntilCycle :: (Eq a) => (a -> a) -> a -> [a]
iterateUntilCycle :: forall a. Eq a => (a -> a) -> a -> [a]
iterateUntilCycle a -> a
f a
a =
   let as :: [a]
as = forall a. (a -> a) -> a -> [a]
iterate a -> a
f a
a
   in  (a
aforall a. a -> [a] -> [a]
:) forall a b. (a -> b) -> a -> b
$ forall a b. (a -> b) -> [a] -> [b]
map forall a b. (a, b) -> a
fst forall a b. (a -> b) -> a -> b
$
       forall a. (a -> Bool) -> [a] -> [a]
takeWhile (forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry forall a. Eq a => a -> a -> Bool
(/=)) forall a b. (a -> b) -> a -> b
$
       forall a b. [a] -> [b] -> [(a, b)]
zip (forall a. [a] -> [a]
tail [a]
as) (forall (t :: * -> *) a b. Foldable t => (a -> [b]) -> t a -> [b]
concatMap (\a
ai->[a
ai,a
ai]) [a]
as)

{-
iterateUntilCycleQ :: (Eq a) => (a -> a) -> a -> [a]
iterateUntilCycleQ f a =
   let as = tail $ iterate f a
   in  (a:) $ map fst $
       takeWhile (uncurry (/=)) $
       zip as (downsample2 (tail as))
-}

iterateUntilCycleP :: (Eq a) => (a -> a) -> a -> [a]
iterateUntilCycleP :: forall a. Eq a => (a -> a) -> a -> [a]
iterateUntilCycleP a -> a
f a
a =
   let as :: [a]
as = forall a. (a -> a) -> a -> [a]
iterate a -> a
f a
a
   in  forall a b. (a -> b) -> [a] -> [b]
map forall a b. (a, b) -> a
fst forall a b. (a -> b) -> a -> b
$
       forall a. (a -> Bool) -> [a] -> [a]
takeWhile (\(a
a1,(a
a20,a
a21)) -> a
a1forall a. Eq a => a -> a -> Bool
/=a
a20 Bool -> Bool -> Bool
&& a
a1forall a. Eq a => a -> a -> Bool
/=a
a21) forall a b. (a -> b) -> a -> b
$
       forall a b. [a] -> [b] -> [(a, b)]
zip [a]
as (forall t. [t] -> [(t, t)]
pairs (forall a. [a] -> [a]
tail [a]
as))

pairs :: [t] -> [(t, t)]
pairs :: forall t. [t] -> [(t, t)]
pairs [] = []
pairs (t
_:[]) = forall a. HasCallStack => [Char] -> a
error [Char]
"pairs: odd number of elements"
pairs (t
x0:t
x1:[t]
xs) = (t
x0,t
x1) forall a. a -> [a] -> [a]
: forall t. [t] -> [(t, t)]
pairs [t]
xs


{- | rotate left -}
rotate, rotate', rotate'' :: Int -> [a] -> [a]
rotate :: forall a. Int -> [a] -> [a]
rotate Int
n [a]
x =
   forall b a. [b] -> [a] -> [a]
Match.take [a]
x (forall a. Int -> [a] -> [a]
drop (forall a. Integral a => a -> a -> a
mod Int
n (forall (t :: * -> *) a. Foldable t => t a -> Int
length [a]
x)) (forall a. [a] -> [a]
cycle [a]
x))

{- | more efficient implementation of rotate'

prop> \n (NonEmpty xs) -> rotate n xs == rotate' n (xs::String)
-}
rotate' :: forall a. Int -> [a] -> [a]
rotate' Int
n [a]
x =
   forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry (forall a b c. (a -> b -> c) -> b -> a -> c
flip forall a. [a] -> [a] -> [a]
(++))
           (forall a. Int -> [a] -> ([a], [a])
splitAt (forall a. Integral a => a -> a -> a
mod Int
n (forall (t :: * -> *) a. Foldable t => t a -> Int
length [a]
x)) [a]
x)

{- |
prop> \(NonNegative n) xs -> rotate n xs == rotate'' n (xs::String)
-}
rotate'' :: forall a. Int -> [a] -> [a]
rotate'' Int
n [a]
x =
   forall b a. [b] -> [a] -> [a]
Match.take [a]
x (forall a. Int -> [a] -> [a]
drop Int
n (forall a. [a] -> [a]
cycle [a]
x))

{- |
Given two lists that are ordered
(i.e. @p x y@ holds for subsequent @x@ and @y@)
'mergeBy' them into a list that is ordered, again.

>>> mergeBy (<=) "agh" "begz"
"abegghz"
-}
mergeBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]
mergeBy :: forall a. (a -> a -> Bool) -> [a] -> [a] -> [a]
mergeBy = forall a. (a -> a -> Bool) -> [a] -> [a] -> [a]
Key.mergeBy


{- |
>>> allEqual "aab"
False
>>> allEqual "aaa"
True
>>> allEqual "aa"
True
>>> allEqual "a"
True
>>> allEqual ""
True
-}
allEqual :: Eq a => [a] -> Bool
allEqual :: forall a. Eq a => [a] -> Bool
allEqual = forall (t :: * -> *). Foldable t => t Bool -> Bool
and forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a b. (a -> a -> b) -> [a] -> [b]
mapAdjacent forall a. Eq a => a -> a -> Bool
(==)

{- |
>>> isAscending "abc"
True
>>> isAscending "abb"
True
>>> isAscending "aba"
False
>>> isAscending "cba"
False
>>> isAscending "a"
True
>>> isAscending ""
True
-}
isAscending :: (Ord a) => [a] -> Bool
isAscending :: forall a. Ord a => [a] -> Bool
isAscending = forall (t :: * -> *). Foldable t => t Bool -> Bool
and forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. Ord a => [a] -> [Bool]
isAscendingLazy

isAscendingLazy :: (Ord a) => [a] -> [Bool]
isAscendingLazy :: forall a. Ord a => [a] -> [Bool]
isAscendingLazy = forall a b. (a -> a -> b) -> [a] -> [b]
mapAdjacent forall a. Ord a => a -> a -> Bool
(<=)

{- |
This function combines every pair of neighbour elements
in a list with a certain function.

>>> mapAdjacent (<=) ""
[]
>>> mapAdjacent (<=) "a"
[]
>>> mapAdjacent (<=) "aba"
[True,False]
>>> mapAdjacent (,) "abc"
[('a','b'),('b','c')]

prop> \x xs -> mapAdjacent subtract (scanl (+) x xs) == (xs::[Integer])
-}
mapAdjacent :: (a -> a -> b) -> [a] -> [b]
mapAdjacent :: forall a b. (a -> a -> b) -> [a] -> [b]
mapAdjacent a -> a -> b
f [a]
xs = forall a b c. (a -> b -> c) -> [a] -> [b] -> [c]
zipWith a -> a -> b
f [a]
xs (forall a. [a] -> [a]
tail [a]
xs)

{- |
<http://mail.haskell.org/libraries/2016-April/026912.html>

prop> \xs -> mapAdjacent (,) xs == mapAdjacentPointfree (,) (xs::String)
-}
mapAdjacentPointfree :: (a -> a -> b) -> [a] -> [b]
mapAdjacentPointfree :: forall a b. (a -> a -> b) -> [a] -> [b]
mapAdjacentPointfree a -> a -> b
f = forall a b c. (a -> b -> c) -> [a] -> [b] -> [c]
zipWith a -> a -> b
f forall (f :: * -> *) a b. Applicative f => f (a -> b) -> f a -> f b
<*> forall a. [a] -> [a]
tail


{- |
>>> let f x y z = [x,y]++show(z::Int) in mapAdjacent1 f 'a' [('b',1), ('c',2), ('d',3)]
["ab1","bc2","cd3"]
-}
mapAdjacent1 :: (a -> a -> b -> c) -> a -> [(a,b)] -> [c]
mapAdjacent1 :: forall a b c. (a -> a -> b -> c) -> a -> [(a, b)] -> [c]
mapAdjacent1 a -> a -> b -> c
f a
a [(a, b)]
xs =
   forall a b c. (a -> b -> c) -> [a] -> [b] -> [c]
zipWith (\a
a0 (a
a1,b
b) -> a -> a -> b -> c
f a
a0 a
a1 b
b) (a
a forall a. a -> [a] -> [a]
: forall a b. (a -> b) -> [a] -> [b]
map forall a b. (a, b) -> a
fst [(a, b)]
xs) [(a, b)]
xs


{- |
>>> equalWith (<=) "ab" "bb"
True
>>> equalWith (<=) "aa" "bbb"
False
>>> equalWith (==) "aa" "aaa"
False

prop> \as bs -> let f a b = abs (a-b) <= (10::Int) in equalWith f as bs ==  equalWithRec f as bs
prop> \as bs -> let f a b = abs (a-b) <= (10::Int) in equalWith f as bs ==  equalWithLiftM f as bs
-}
equalWith, equalWithLiftM, equalWithRec ::
   (a -> b -> Bool) -> [a] -> [b] -> Bool
equalWith :: forall a b. (a -> b -> Bool) -> [a] -> [b] -> Bool
equalWith a -> b -> Bool
f [a]
as [b]
bs =
   forall (t :: * -> *). Foldable t => t Bool -> Bool
and forall a b. (a -> b) -> a -> b
$
   forall a b c. (a -> b -> c) -> [a] -> [b] -> [c]
zipWith
      (\Maybe a
ma Maybe b
mb ->
         case (Maybe a
ma,Maybe b
mb) of
            (Just a
a, Just b
b) -> a -> b -> Bool
f a
a b
b
            (Maybe a
Nothing, Maybe b
Nothing) -> Bool
True
            (Maybe a, Maybe b)
_ -> Bool
False)
      (forall a b. (a -> b) -> [a] -> [b]
map forall a. a -> Maybe a
Just [a]
as forall a. [a] -> [a] -> [a]
++ [forall a. Maybe a
Nothing])
      (forall a b. (a -> b) -> [a] -> [b]
map forall a. a -> Maybe a
Just [b]
bs forall a. [a] -> [a] -> [a]
++ [forall a. Maybe a
Nothing])

equalWithLiftM :: forall a b. (a -> b -> Bool) -> [a] -> [b] -> Bool
equalWithLiftM a -> b -> Bool
f [a]
as [b]
bs =
   forall (t :: * -> *) a. Foldable t => (a -> Bool) -> t a -> Bool
all (forall a. a -> Maybe a
Just Bool
True forall a. Eq a => a -> a -> Bool
==) forall a b. (a -> b) -> a -> b
$
   forall a b c. (a -> b -> c) -> [a] -> [b] -> [c]
zipWith
      (\Maybe a
ma Maybe b
mb ->
         case (Maybe a
ma,Maybe b
mb) of
            (Maybe a
Nothing, Maybe b
Nothing) -> forall a. a -> Maybe a
Just Bool
True
            (Maybe a, Maybe b)
_ -> forall (m :: * -> *) a1 a2 r.
Monad m =>
(a1 -> a2 -> r) -> m a1 -> m a2 -> m r
liftM2 a -> b -> Bool
f Maybe a
ma Maybe b
mb)
      (forall a b. (a -> b) -> [a] -> [b]
map forall a. a -> Maybe a
Just [a]
as forall a. [a] -> [a] -> [a]
++ [forall a. Maybe a
Nothing])
      (forall a b. (a -> b) -> [a] -> [b]
map forall a. a -> Maybe a
Just [b]
bs forall a. [a] -> [a] -> [a]
++ [forall a. Maybe a
Nothing])

equalWithRec :: forall a b. (a -> b -> Bool) -> [a] -> [b] -> Bool
equalWithRec a -> b -> Bool
f =
   let go :: [a] -> [b] -> Bool
go (a
a:[a]
as) (b
b:[b]
bs) = a -> b -> Bool
f a
a b
b Bool -> Bool -> Bool
&& [a] -> [b] -> Bool
go [a]
as [b]
bs
       go [] [] = Bool
True
       go [a]
_ [b]
_ = Bool
False
   in [a] -> [b] -> Bool
go


{- |
Enumerate without Enum context.
For Enum equivalent to enumFrom.

>>> range 0 :: [Integer]
[]
>>> range 1 :: [Integer]
[0]
>>> range 8 :: [Integer]
[0,1,2,3,4,5,6,7]

prop> \(NonNegative n) -> length (range n :: [Integer]) == n
-}
range :: Num a => Int -> [a]
range :: forall a. Num a => Int -> [a]
range Int
n = forall a. Int -> [a] -> [a]
take Int
n (forall a. (a -> a) -> a -> [a]
iterate (forall a. Num a => a -> a -> a
+a
1) a
0)


{-# INLINE padLeft #-}
padLeft :: a -> Int -> [a] -> [a]
padLeft :: forall a. a -> Int -> [a] -> [a]
padLeft  a
c Int
n [a]
xs = forall a. Int -> a -> [a]
replicate (Int
n forall a. Num a => a -> a -> a
- forall (t :: * -> *) a. Foldable t => t a -> Int
length [a]
xs) a
c forall a. [a] -> [a] -> [a]
++ [a]
xs


{-# INLINE padRight #-}
padRight, padRight1 :: a -> Int -> [a] -> [a]
padRight :: forall a. a -> Int -> [a] -> [a]
padRight  a
c Int
n [a]
xs = forall a. Int -> [a] -> [a]
take Int
n forall a b. (a -> b) -> a -> b
$ [a]
xs forall a. [a] -> [a] -> [a]
++ forall a. a -> [a]
repeat a
c
padRight1 :: forall a. a -> Int -> [a] -> [a]
padRight1 a
c Int
n [a]
xs = [a]
xs forall a. [a] -> [a] -> [a]
++ forall a. Int -> a -> [a]
replicate (Int
n forall a. Num a => a -> a -> a
- forall (t :: * -> *) a. Foldable t => t a -> Int
length [a]
xs) a
c

{- |
For an associative operation @op@ this computes
   @iterateAssociative op a = iterate (op a) a@
but it is even faster than @map (powerAssociative op a a) [0..]@
since it shares temporary results.

The idea is:
From the list @map (powerAssociative op a a) [0,(2*n)..]@
we compute the list @map (powerAssociative op a a) [0,n..]@,
and iterate that until @n==1@.

prop> \x -> equating (take 1000) (List.iterate (x+) x) (iterateAssociative (+) (x::Integer))
-}
iterateAssociative :: (a -> a -> a) -> a -> [a]
iterateAssociative :: forall a. (a -> a -> a) -> a -> [a]
iterateAssociative a -> a -> a
op a
a =
   forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr (\a
pow [a]
xs -> a
pow forall a. a -> [a] -> [a]
: forall (t :: * -> *) a b. Foldable t => (a -> [b]) -> t a -> [b]
concatMap (\a
x -> [a
x, a -> a -> a
op a
x a
pow]) [a]
xs)
         forall a. HasCallStack => a
undefined (forall a. (a -> a) -> a -> [a]
iterate (\a
x -> a -> a -> a
op a
x a
x) a
a)

{- |
This is equal to 'iterateAssociative'.
The idea is the following:
The list we search is the fixpoint of the function:
"Square all elements of the list,
then spread it and fill the holes with successive numbers
of their left neighbour."
This also preserves log n applications per value.
However it has a space leak,
because for the value with index @n@
all elements starting at @div n 2@ must be kept.

prop> \x -> equating (take 1000) (List.iterate (x+) x) (iterateLeaky (+) (x::Integer))
-}
iterateLeaky :: (a -> a -> a) -> a -> [a]
iterateLeaky :: forall a. (a -> a -> a) -> a -> [a]
iterateLeaky a -> a -> a
op a
x =
   let merge :: [a] -> [a] -> [a]
merge (a
a:[a]
as) [a]
b = a
a forall a. a -> [a] -> [a]
: [a] -> [a] -> [a]
merge [a]
b [a]
as
       merge [a]
_ [a]
_ = forall a. HasCallStack => [Char] -> a
error [Char]
"iterateLeaky: an empty list cannot occur"
       sqrs :: [a]
sqrs = forall a b. (a -> b) -> [a] -> [b]
map (\a
y -> a -> a -> a
op a
y a
y) [a]
z
       z :: [a]
z = a
x forall a. a -> [a] -> [a]
: forall a. [a] -> [a] -> [a]
merge [a]
sqrs (forall a b. (a -> b) -> [a] -> [b]
map (a -> a -> a
op a
x) [a]
sqrs)
   in  [a]
z