content: \begin{code}
  
  {-# LANGUAGE Trustworthy #-}
  {-# LANGUAGE CPP, NoImplicitPrelude, MagicHash #-}
  {-# OPTIONS_HADDOCK hide #-}
  
  -----------------------------------------------------------------------------
  -- |
  -- Module      :  GHC.List
  -- Copyright   :  (c) The University of Glasgow 1994-2002
  -- License     :  see libraries/base/LICENSE
  -- 
  -- Maintainer  :  cvs-ghc@haskell.org
  -- Stability   :  internal
  -- Portability :  non-portable (GHC Extensions)
  --
  -- The List data type and its operations
  --
  -----------------------------------------------------------------------------
  
  -- #hide
  module GHC.List (
     -- [] (..),          -- Not Haskell 98; built in syntax
  
     map, (++), filter, concat,
     head, last, tail, init, null, length, (!!),
     foldl, scanl, scanl1, foldr, foldr1, scanr, scanr1,
     iterate, repeat, replicate, cycle,
     take, drop, splitAt, takeWhile, dropWhile, span, break,
     reverse, and, or,
     any, all, elem, notElem, lookup,
     concatMap,
     zip, zip3, zipWith, zipWith3, unzip, unzip3,
     errorEmptyList,
  
  #ifndef USE_REPORT_PRELUDE
     -- non-standard, but hidden when creating the Prelude
     -- export list.
     takeUInt_append
  #endif
  
   ) where
  
  import Data.Maybe
  import GHC.Base
  import GHC.Num
  import Language.Haskell.Liquid.Prelude (liquidAssert, liquidError)
  
  infixl 9  !!
  infix  4 `elem`, `notElem`
  
  
  
  
  \end{code}
  
  %*********************************************************
  %*                                                      *
  \subsection{List-manipulation functions}
  %*                                                      *
  %*********************************************************
  
  \begin{code}
  -- | Extract the first element of a list, which must be non-empty.
  {-@ assert head         :: xs:{v: [a] | len(v) > 0} -> a @-}
  head                    :: [a] -> a
  head (x:_)              =  x
  head []                 =  errorEmptyList "head"
  
  badHead :: a
  badHead = error "errorEmptyList head" -- errorEmptyList "head"
  
  -- This rule is useful in cases like 
  --      head [y | (x,y) <- ps, x==t]
  {- RULES
  "head/build"    forall (g::forall b.(a->b->b)->b->b) .
                  head (build g) = g (\x _ -> x) badHead
  "head/augment"  forall xs (g::forall b. (a->b->b) -> b -> b) . 
                  head (augment g xs) = g (\x _ -> x) (head xs)
   -}
  
  -- | Extract the elements after the head of a list, which must be non-empty.
  {-@ assert tail         :: xs:{v: [a] | len(v) > 0} -> {v: [a] | len(v) = (len(xs) - 1)}  @-}
  tail                    :: [a] -> [a]
  tail (_:xs)             =  xs
  tail []                 =  liquidError "tail" -- errorEmptyList "tail"
  
  -- | Extract the last element of a list, which must be finite and non-empty.
  {-@ assert last         :: xs:{v: [a] | len(v) > 0} -> a @-}
  last                    :: [a] -> a
  #ifdef USE_REPORT_PRELUDE
  last [x]                =  x
  last (_:xs)             =  last xs
  last []                 =  liquidError "last" -- errorEmptyList "last"
  #else
  -- eliminate repeated cases
  last []                 =  liquidError "last" -- errorEmptyList "last"
  last (x:xs)             =  last' x xs
    where last' y []     = y
          last' _ (y:ys) = last' y ys
  #endif
  
  -- | Return all the elements of a list except the last one.
  -- The list must be non-empty.
  {-@ assert init         :: xs:{v: [a] | len(v) > 0} -> {v: [a] | len(v) = len(xs) - 1}  @-}
  init                    :: [a] -> [a]
  #ifdef USE_REPORT_PRELUDE
  init [x]                =  []
  init (x:xs)             =  x : init xs
  init []                 =  liquidError "init" -- errorEmptyList "init"
  #else
  -- eliminate repeated cases
  init []                 =  liquidError "init" --errorEmptyList "init"
  init (x:xs)             =  init' x xs
    where init' _ []     = []
          init' y (z:zs) = y : init' z zs
  #endif
  
  -- | Test whether a list is empty.
  {-@ assert null :: xs:[a] -> {v: Bool | (Prop(v) <=> len(xs) = 0) }  @-}
  null                    :: [a] -> Bool
  null []                 =  True
  null (_:_)              =  False
  
  -- | /O(n)/. 'length' returns the length of a finite list as an 'Int'.
  -- It is an instance of the more general 'Data.List.genericLength',
  -- the result type of which may be any kind of number.
  {-@ assert length :: xs:[a] -> {v: GHC.Types.Int | v = len(xs)}  @-}
  length                  :: [a] -> Int
  length l                =  len l 0#
    where
      --LIQUID FIXME: leaving the type signature causes this to compile to very strange core
      --LIQUID len :: [a] -> Int# -> Int
      len []     a# = I# a#
      len (_:xs) a# = len xs (a# +# 1#)
  
  -- | 'filter', applied to a predicate and a list, returns the list of
  -- those elements that satisfy the predicate; i.e.,
  --
  -- > filter p xs = [ x | x <- xs, p x]
  
  {-@ assert filter :: (a -> GHC.Types.Bool) -> xs:[a] -> {v: [a] | len(v) <= len(xs)} @-}
  filter :: (a -> Bool) -> [a] -> [a]
  filter _pred []    = []
  filter pred (x:xs)
    | pred x         = x : filter pred xs
    | otherwise      = filter pred xs
  
  {-# NOINLINE [0] filterFB #-}
  filterFB :: (a -> b -> b) -> (a -> Bool) -> a -> b -> b
  filterFB c p x r | p x       = x `c` r
                   | otherwise = r
  
  {- RULES
  "filter"     [~1] forall p xs.  filter p xs = build (\c n -> foldr (filterFB c p) n xs)
  "filterList" [1]  forall p.     foldr (filterFB (:) p) [] = filter p
  "filterFB"        forall c p q. filterFB (filterFB c p) q = filterFB c (\x -> q x && p x)
   -}
  
  -- Note the filterFB rule, which has p and q the "wrong way round" in the RHS.
  --     filterFB (filterFB c p) q a b
  --   = if q a then filterFB c p a b else b
  --   = if q a then (if p a then c a b else b) else b
  --   = if q a && p a then c a b else b
  --   = filterFB c (\x -> q x && p x) a b
  -- I originally wrote (\x -> p x && q x), which is wrong, and actually
  -- gave rise to a live bug report.  SLPJ.
  
  
  -- | 'foldl', applied to a binary operator, a starting value (typically
  -- the left-identity of the operator), and a list, reduces the list
  -- using the binary operator, from left to right:
  --
  -- > foldl f z [x1, x2, ..., xn] == (...((z `f` x1) `f` x2) `f`...) `f` xn
  --
  -- The list must be finite.
  
  -- We write foldl as a non-recursive thing, so that it
  -- can be inlined, and then (often) strictness-analysed,
  -- and hence the classic space leak on foldl (+) 0 xs
  
  foldl        :: (a -> b -> a) -> a -> [b] -> a
  foldl f z0 xs0 = lgo z0 xs0
               where
                  lgo z []     = z
                  lgo z (x:xs) = lgo (f z x) xs
  
  -- | 'scanl' is similar to 'foldl', but returns a list of successive
  -- reduced values from the left:
  --
  -- > scanl f z [x1, x2, ...] == [z, z `f` x1, (z `f` x1) `f` x2, ...]
  --
  -- Note that
  --
  -- > last (scanl f z xs) == foldl f z xs.
  {-@ assert scanl        :: (a -> b -> a) -> a -> xs:[b] -> {v: [a] | len(v) = 1 + len(xs) } @-}
  scanl                   :: (a -> b -> a) -> a -> [b] -> [a]
  scanl f q ls            =  q : (case ls of
                                  []   -> []
                                  x:xs -> scanl f (f q x) xs)
  
  -- | 'scanl1' is a variant of 'scanl' that has no starting value argument:
  --
  -- > scanl1 f [x1, x2, ...] == [x1, x1 `f` x2, ...]
  
  {-@ assert scanl1       :: (a -> a -> a) -> xs:{v: [a] | len(v) > 0} -> {v: [a] | len(v) = len(xs) } @-}
  scanl1                  :: (a -> a -> a) -> [a] -> [a]
  scanl1 f (x:xs)         =  scanl f x xs
  scanl1 _ []             =  []
  
  -- foldr, foldr1, scanr, and scanr1 are the right-to-left duals of the
  -- above functions.
  
  -- | 'foldr1' is a variant of 'foldr' that has no starting value argument,
  -- and thus must be applied to non-empty lists.
  
  {-@ assert foldr1       :: (a -> a -> a) -> xs:{v: [a] | len(v) > 0} -> a @-}
  foldr1                  :: (a -> a -> a) -> [a] -> a
  foldr1 _ [x]            =  x
  foldr1 f (x:xs@(_:_))   =  f x (foldr1 f xs)
  foldr1 _ []             =  liquidError "foldr1" -- errorEmptyList "foldr1"
  
  -- | 'scanr' is the right-to-left dual of 'scanl'.
  -- Note that
  --
  -- > head (scanr f z xs) == foldr f z xs.
  
  {-@ assert scanr        :: (a -> b -> b) -> b -> xs:[a] -> {v: [b] | len(v) = 1 + len(xs) } @-}
  scanr                   :: (a -> b -> b) -> b -> [a] -> [b]
  scanr _ q0 []           =  [q0]
  scanr f q0 (x:xs)       =  f x q : qs
                             where qs@(q:_) = scanr f q0 xs 
  
  -- | 'scanr1' is a variant of 'scanr' that has no starting value argument.
  
  {-@ assert scanr1       :: (a -> a -> a) -> xs:{v: [a] | len(v) > 0} -> {v: [a] | len(v) = len(xs) } @-}
  scanr1                  :: (a -> a -> a) -> [a] -> [a]
  scanr1 _ []             =  []
  scanr1 _ [x]            =  [x]
  scanr1 f (x:xs@(_:_))   =  f x q : qs
                             where qs@(q:_) = scanr1 f xs 
  
  -- | 'iterate' @f x@ returns an infinite list of repeated applications
  -- of @f@ to @x@:
  --
  -- > iterate f x == [x, f x, f (f x), ...]
  
  {-@ lazy GHC.List.iterate @-}
  {-@ iterate :: (a -> a) -> a -> [a] @-}
  iterate :: (a -> a) -> a -> [a]
  iterate f x =  x : iterate f (f x)
  
  {-@ lazy GHC.List.iterateFB @-}
  {-@ iterateFB :: (a -> b -> b) -> (a -> a) -> a -> b @-}
  iterateFB :: (a -> b -> b) -> (a -> a) -> a -> b
  iterateFB c f x = x `c` iterateFB c f (f x)
  
  
  {- RULES
  "iterate"    [~1] forall f x.   iterate f x = build (\c _n -> iterateFB c f x)
  "iterateFB"  [1]                iterateFB (:) = iterate
   -}
  
  
  -- | 'repeat' @x@ is an infinite list, with @x@ the value of every element.
  {- measure inf :: Int @-}
  {- invariant {v:Int | v < inf} @-}
  {- repeat :: a -> {v:[a] | (len v) = inf} @-}
  {-@ repeat :: a -> [a] @-}
  {-@ lazy GHC.List.repeat @-}
  repeat :: a -> [a]
  {-# INLINE [0] repeat #-}
  -- The pragma just gives the rules more chance to fire
  repeat x = xs where xs = x : xs
  
  {-# INLINE [0] repeatFB #-}     -- ditto
  {-@ lazy GHC.List.repeatFB @-}
  {-@ repeatFB :: (a -> b -> b) -> a -> b @-}
  repeatFB :: (a -> b -> b) -> a -> b
  repeatFB c x = xs where xs = x `c` xs
  
  
  {- RULES
  "repeat"    [~1] forall x. repeat x = build (\c _n -> repeatFB c x)
  "repeatFB"  [1]  repeatFB (:)       = repeat
   -}
  
  -- | 'replicate' @n x@ is a list of length @n@ with @x@ the value of
  -- every element.
  -- It is an instance of the more general 'Data.List.genericReplicate',
  -- in which @n@ may be of any integral type.
  {-# INLINE replicate #-}
  {-@ assert replicate    :: n:Nat -> x:a -> {v: [{v:a | v = x}] | len(v) = n} @-}
  replicate               :: Int -> a -> [a]
  --LIQUID replicate n x           =  take n (repeat x)
  replicate 0 _ = []
  replicate n x = x : replicate (n-1) x
  
  -- | 'cycle' ties a finite list into a circular one, or equivalently,
  -- the infinite repetition of the original list.  It is the identity
  -- on infinite lists.
  
  {-@ assert cycle        :: {v: [a] | len(v) > 0 } -> [a] @-}
  {-@ lazy cycle @-}
  cycle                   :: [a] -> [a]
  cycle []                = liquidError {- error -} "Prelude.cycle: empty list"
  cycle xs                = xs' where xs' = xs ++ xs'
  
  -- | 'takeWhile', applied to a predicate @p@ and a list @xs@, returns the
  -- longest prefix (possibly empty) of @xs@ of elements that satisfy @p@:
  --
  -- > takeWhile (< 3) [1,2,3,4,1,2,3,4] == [1,2]
  -- > takeWhile (< 9) [1,2,3] == [1,2,3]
  -- > takeWhile (< 0) [1,2,3] == []
  --
  
  {-@ assert takeWhile    :: (a -> Bool) -> xs:[a] -> {v: [a] | len(v) <= len(xs)} @-}
  takeWhile               :: (a -> Bool) -> [a] -> [a]
  takeWhile _ []          =  []
  takeWhile p (x:xs) 
              | p x       =  x : takeWhile p xs
              | otherwise =  []
  
  -- | 'dropWhile' @p xs@ returns the suffix remaining after 'takeWhile' @p xs@:
  --
  -- > dropWhile (< 3) [1,2,3,4,5,1,2,3] == [3,4,5,1,2,3]
  -- > dropWhile (< 9) [1,2,3] == []
  -- > dropWhile (< 0) [1,2,3] == [1,2,3]
  --
  
  {-@ assert dropWhile    :: (a -> Bool) -> xs:[a] -> {v: [a] | len(v) <= len(xs)} @-}
  dropWhile               :: (a -> Bool) -> [a] -> [a]
  dropWhile _ []          =  []
  dropWhile p xs@(x:xs')
              | p x       =  dropWhile p xs'
              | otherwise =  xs
  
  -- | 'take' @n@, applied to a list @xs@, returns the prefix of @xs@
  -- of length @n@, or @xs@ itself if @n > 'length' xs@:
  --
  -- > take 5 "Hello World!" == "Hello"
  -- > take 3 [1,2,3,4,5] == [1,2,3]
  -- > take 3 [1,2] == [1,2]
  -- > take 3 [] == []
  -- > take (-1) [1,2] == []
  -- > take 0 [1,2] == []
  --
  -- It is an instance of the more general 'Data.List.genericTake',
  -- in which @n@ may be of any integral type.
  
  
  {-@ take :: n:Int
           -> xs:[a] 
           -> {v:[a] | (if (n >=0) then ((len v) = ((len(xs) < n) ? len(xs):n)) else ((len v) = 0))} 
    @-}
  take                   :: Int -> [a] -> [a]
  
  -- | 'drop' @n xs@ returns the suffix of @xs@
  -- after the first @n@ elements, or @[]@ if @n > 'length' xs@:
  --
  -- > drop 6 "Hello World!" == "World!"
  -- > drop 3 [1,2,3,4,5] == [4,5]
  -- > drop 3 [1,2] == []
  -- > drop 3 [] == []
  -- > drop (-1) [1,2] == [1,2]
  -- > drop 0 [1,2] == [1,2]
  --
  -- It is an instance of the more general 'Data.List.genericDrop',
  -- in which @n@ may be of any integral type.
  {-@ drop  :: n: Int 
            -> xs:[a] 
            -> {v:[a] | (if (n >= 0) then (len(v) = ((len(xs) <  n) ? 0 : len(xs) - n)) else ((len v) = (len xs)))} @-}
  drop                   :: Int -> [a] -> [a]
  
  -- | 'splitAt' @n xs@ returns a tuple where first element is @xs@ prefix of
  -- length @n@ and second element is the remainder of the list:
  --
  -- > splitAt 6 "Hello World!" == ("Hello ","World!")
  -- > splitAt 3 [1,2,3,4,5] == ([1,2,3],[4,5])
  -- > splitAt 1 [1,2,3] == ([1],[2,3])
  -- > splitAt 3 [1,2,3] == ([1,2,3],[])
  -- > splitAt 4 [1,2,3] == ([1,2,3],[])
  -- > splitAt 0 [1,2,3] == ([],[1,2,3])
  -- > splitAt (-1) [1,2,3] == ([],[1,2,3])
  --
  -- It is equivalent to @('take' n xs, 'drop' n xs)@ when @n@ is not @_|_@
  -- (@splitAt _|_ xs = _|_@).
  -- 'splitAt' is an instance of the more general 'Data.List.genericSplitAt',
  -- in which @n@ may be of any integral type.
  -- Liquid: TODO
  {-@ splitAt :: n:Int -> x:[a] -> ({v:[a] | (if (n >= 0) then (Min (len v) (len x) n) else ((len v) = 0))},[a])<{\x1 x2 -> (len x2) = (len x) - (len x1)}> @-}
  splitAt                :: Int -> [a] -> ([a],[a])
  
  #ifdef USE_REPORT_PRELUDE
  take n _      | n <= 0 =  []
  take _ []              =  []
  take n (x:xs)          =  x : take (n-1) xs
  
  drop n xs     | n <= 0 =  xs
  drop _ []              =  []
  drop n (_:xs)          =  drop (n-1) xs
  
  splitAt n xs           =  (take n xs, drop n xs)
  
  #else /* hack away */
  {- RULES
  "take"     [~1] forall n xs . take n xs = takeFoldr n xs 
  "takeList"  [1] forall n xs . foldr (takeFB (:) []) (takeConst []) xs n = takeUInt n xs
   -}
  
  {-# INLINE takeFoldr #-}
  takeFoldr :: Int -> [a] -> [a]
  takeFoldr (I# n#) xs
    = build (\c nil -> if n# <=# 0# then nil else
                       foldr (takeFB c nil) (takeConst nil) xs n#)
  
  {-# NOINLINE [0] takeConst #-}
  -- just a version of const that doesn't get inlined too early, so we
  -- can spot it in rules.  Also we need a type sig due to the unboxed Int#.
  takeConst :: a -> Int# -> a
  takeConst x _ = x
  
  {-# NOINLINE [0] takeFB #-}
  takeFB :: (a -> b -> b) -> b -> a -> (Int# -> b) -> Int# -> b
  takeFB c n x xs m | m <=# 1#  = x `c` n
                    | otherwise = x `c` xs (m -# 1#)
  
  {-- INLINE [0] take #-}
  take (I# n#) xs = takeUInt n# xs
  
  -- The general code for take, below, checks n <= maxInt
  -- No need to check for maxInt overflow when specialised
  -- at type Int or Int# since the Int must be <= maxInt
  
  takeUInt :: Int# -> [b] -> [b]
  takeUInt n xs
    | n >=# 0#  =  take_unsafe_UInt n xs
    | otherwise =  []
  
  take_unsafe_UInt :: Int# -> [b] -> [b]
  take_unsafe_UInt 0#  _  = []
  take_unsafe_UInt m   ls =
    case ls of
      []     -> []
      (x:xs) -> x : take_unsafe_UInt (m -# 1#) xs
  
  takeUInt_append :: Int# -> [b] -> [b] -> [b]
  takeUInt_append n xs rs
    | n >=# 0#  =  take_unsafe_UInt_append n xs rs
    | otherwise =  []
  
  take_unsafe_UInt_append :: Int# -> [b] -> [b] -> [b]
  take_unsafe_UInt_append 0#  _ rs  = rs
  take_unsafe_UInt_append m  ls rs  =
    case ls of
      []     -> rs
      (x:xs) -> x : take_unsafe_UInt_append (m -# 1#) xs rs
  
  drop (I# n#) ls
    | n# <# 0#    = ls
    | otherwise   = drop# n# ls
      where
          drop# :: Int# -> [a] -> [a]
          drop# 0# xs      = xs
          drop# _  xs@[]   = xs
          drop# m# (_:xs)  = drop# (m# -# 1#) xs
  
  splitAt (I# n#) ls
    | n# <# 0#    = ([], ls)
    | otherwise   = splitAt# n# ls
      where
          splitAt# :: Int# -> [a] -> ([a], [a])
          splitAt# 0# xs     = ([], xs)
          splitAt# _  xs@[]  = (xs, xs)
          splitAt# m# (x:xs) = (x:xs', xs'')
            where
              (xs', xs'') = splitAt# (m# -# 1#) xs
  
  #endif /* USE_REPORT_PRELUDE */
  
  -- | 'span', applied to a predicate @p@ and a list @xs@, returns a tuple where
  -- first element is longest prefix (possibly empty) of @xs@ of elements that
  -- satisfy @p@ and second element is the remainder of the list:
  -- 
  -- > span (< 3) [1,2,3,4,1,2,3,4] == ([1,2],[3,4,1,2,3,4])
  -- > span (< 9) [1,2,3] == ([1,2,3],[])
  -- > span (< 0) [1,2,3] == ([],[1,2,3])
  -- 
  -- 'span' @p xs@ is equivalent to @('takeWhile' p xs, 'dropWhile' p xs)@
  -- Liquid: TODO
  {-@
  span    :: (a -> Bool) 
          -> xs:[a] 
          -> ({v:[a]|((len v)<=(len xs))}, {v:[a]|((len v)<=(len xs))})
  @-}
  span                    :: (a -> Bool) -> [a] -> ([a], [a])
  span _ xs@[]            =  (xs, xs)
  span p xs@(x:xs')
           | p x          =  let (ys,zs) = span p xs' in (x:ys,zs)
           | otherwise    =  ([],xs)
  
  -- | 'break', applied to a predicate @p@ and a list @xs@, returns a tuple where
  -- first element is longest prefix (possibly empty) of @xs@ of elements that
  -- /do not satisfy/ @p@ and second element is the remainder of the list:
  -- 
  -- > break (> 3) [1,2,3,4,1,2,3,4] == ([1,2,3],[4,1,2,3,4])
  -- > break (< 9) [1,2,3] == ([],[1,2,3])
  -- > break (> 9) [1,2,3] == ([1,2,3],[])
  --
  -- 'break' @p@ is equivalent to @'span' ('not' . p)@.
  -- liquid:TODO
  {-@ break :: (a -> Bool) -> xs:[a] -> ([a],[a])<{\x y -> (len xs) = (len x) + (len y)}> @-}
  break                   :: (a -> Bool) -> [a] -> ([a],[a])
  #ifdef USE_REPORT_PRELUDE
  break p                 =  span (not . p)
  #else
  -- HBC version (stolen)
  break _ xs@[]           =  (xs, xs)
  break p xs@(x:xs')
             | p x        =  ([],xs)
             | otherwise  =  let (ys,zs) = break p xs' in (x:ys,zs)
  #endif
  
  -- | 'reverse' @xs@ returns the elements of @xs@ in reverse order.
  -- @xs@ must be finite.
  {-@ assert reverse      :: xs:[a] -> {v: [a] | len(v) = len(xs)} @-}
  {-@ include <len.hquals> @-}
  reverse                 :: [a] -> [a]
  #ifdef USE_REPORT_PRELUDE
  reverse                 =  foldl (flip (:)) []
  #else
  reverse l =  rev l []
    where
      rev []     a = a
      rev (x:xs) a = rev xs (x:a)
  #endif
  
  -- | 'and' returns the conjunction of a Boolean list.  For the result to be
  -- 'True', the list must be finite; 'False', however, results from a 'False'
  -- value at a finite index of a finite or infinite list.
  and                     :: [Bool] -> Bool
  
  -- | 'or' returns the disjunction of a Boolean list.  For the result to be
  -- 'False', the list must be finite; 'True', however, results from a 'True'
  -- value at a finite index of a finite or infinite list.
  or                      :: [Bool] -> Bool
  #ifdef USE_REPORT_PRELUDE
  and                     =  foldr (&&) True
  or                      =  foldr (||) False
  #else
  and []          =  True
  and (x:xs)      =  x && and xs
  or []           =  False
  or (x:xs)       =  x || or xs
  
  {- RULES
  "and/build"     forall (g::forall b.(Bool->b->b)->b->b) . 
                  and (build g) = g (&&) True
  "or/build"      forall (g::forall b.(Bool->b->b)->b->b) . 
                  or (build g) = g (||) False
   -}
  #endif
  
  -- | Applied to a predicate and a list, 'any' determines if any element
  -- of the list satisfies the predicate.  For the result to be
  -- 'False', the list must be finite; 'True', however, results from a 'True'
  -- value for the predicate applied to an element at a finite index of a finite or infinite list.
  any                     :: (a -> Bool) -> [a] -> Bool
  
  -- | Applied to a predicate and a list, 'all' determines if all elements
  -- of the list satisfy the predicate. For the result to be
  -- 'True', the list must be finite; 'False', however, results from a 'False'
  -- value for the predicate applied to an element at a finite index of a finite or infinite list.
  all                     :: (a -> Bool) -> [a] -> Bool
  #ifdef USE_REPORT_PRELUDE
  any p                   =  or . map p
  all p                   =  and . map p
  #else
  any _ []        = False
  any p (x:xs)    = p x || any p xs
  
  all _ []        =  True
  all p (x:xs)    =  p x && all p xs
  {- RULES
  "any/build"     forall p (g::forall b.(a->b->b)->b->b) . 
                  any p (build g) = g ((||) . p) False
  "all/build"     forall p (g::forall b.(a->b->b)->b->b) . 
                  all p (build g) = g ((&&) . p) True
   -}
  #endif
  
  -- | 'elem' is the list membership predicate, usually written in infix form,
  -- e.g., @x \`elem\` xs@.  For the result to be
  -- 'False', the list must be finite; 'True', however, results from an element equal to @x@ found at a finite index of a finite or infinite list.
  elem                    :: (Eq a) => a -> [a] -> Bool
  
  -- | 'notElem' is the negation of 'elem'.
  notElem                 :: (Eq a) => a -> [a] -> Bool
  #ifdef USE_REPORT_PRELUDE
  elem x                  =  any (== x)
  notElem x               =  all (/= x)
  #else
  elem _ []       = False
  elem x (y:ys)   = x==y || elem x ys
  
  notElem _ []    =  True
  notElem x (y:ys)=  x /= y && notElem x ys
  #endif
  
  -- | 'lookup' @key assocs@ looks up a key in an association list.
  lookup                  :: (Eq a) => a -> [(a,b)] -> Maybe b
  lookup _key []          =  Nothing
  lookup  key ((x,y):xys)
      | key == x          =  Just y
      | otherwise         =  lookup key xys
  
  -- | Map a function over a list and concatenate the results.
  concatMap               :: (a -> [b]) -> [a] -> [b]
  concatMap f             =  foldr ((++) . f) []
  
  -- | Concatenate a list of lists.
  concat :: [[a]] -> [a]
  concat = foldr (++) []
  
  {- RULES
    "concat" forall xs. concat xs = build (\c n -> foldr (\x y -> foldr c y x) n xs)
  -- We don't bother to turn non-fusible applications of concat back into concat
   -}
  
  \end{code}
  
  
  \begin{code}
  -- | List index (subscript) operator, starting from 0.
  -- It is an instance of the more general 'Data.List.genericIndex',
  -- which takes an index of any integral type.
  
  {-@ assert GHC.List.!!         :: xs:[a] -> {v: Int | ((0 <= v) && (v < len(xs)))} -> a @-}
  (!!)                    :: [a] -> Int -> a
  #ifdef USE_REPORT_PRELUDE
  xs     !! n | n < 0 =  liquidError {- error -} "Prelude.!!: negative index"
  []     !! _         =  liquidError {- error -} "Prelude.!!: index too large"
  (x:_)  !! 0         =  x
  (_:xs) !! n         =  xs !! (n-1)
  #else
  -- HBC version (stolen), then unboxified
  -- The semantics is not quite the same for error conditions
  -- in the more efficient version.
  --
  xs !! (I# n0) | n0 <# 0#   =  liquidError {- error -} "Prelude.(!!): negative index\n"
                 | otherwise =  sub xs n0
                           where
                              sub :: [a] -> Int# -> a
                              sub []     _ = liquidError {- error -} "Prelude.(!!): index too large\n"
                              sub (y:ys) n = if n ==# 0#
                                             then y
                                             else sub ys (n -# 1#)
  #endif
  \end{code}
  
  
  %*********************************************************
  %*                                                      *
  \subsection{The zip family}
  %*                                                      *
  %*********************************************************
  
  \begin{code}
  foldr2 :: (a -> b -> c -> c) -> c -> [a] -> [b] -> c
  foldr2 _k z []    _ys    = z
  foldr2 _k z _xs   []     = z
  foldr2 k z (x:xs) (y:ys) = k x y (foldr2 k z xs ys)
  
  foldr2_left :: (a -> b -> c -> d) -> d -> a -> ([b] -> c) -> [b] -> d
  foldr2_left _k  z _x _r []     = z
  foldr2_left  k _z  x  r (y:ys) = k x y (r ys)
  
  foldr2_right :: (a -> b -> c -> d) -> d -> b -> ([a] -> c) -> [a] -> d
  foldr2_right _k z  _y _r []     = z
  foldr2_right  k _z  y  r (x:xs) = k x y (r xs)
  
  -- foldr2 k z xs ys = foldr (foldr2_left k z)  (\_ -> z) xs ys
  -- foldr2 k z xs ys = foldr (foldr2_right k z) (\_ -> z) ys xs
  {- RULES
  "foldr2/left"   forall k z ys (g::forall b.(a->b->b)->b->b) . 
                    foldr2 k z (build g) ys = g (foldr2_left  k z) (\_ -> z) ys
  
  "foldr2/right"  forall k z xs (g::forall b.(a->b->b)->b->b) . 
                    foldr2 k z xs (build g) = g (foldr2_right k z) (\_ -> z) xs
   -}
  \end{code}
  
  The foldr2/right rule isn't exactly right, because it changes
  the strictness of foldr2 (and thereby zip)
  
  E.g. main = print (null (zip nonobviousNil (build undefined)))
            where   nonobviousNil = f 3
                    f n = if n == 0 then [] else f (n-1)
  
  I'm going to leave it though.
  
  
  Zips for larger tuples are in the List module.
  
  \begin{code}
  ----------------------------------------------
  -- | 'zip' takes two lists and returns a list of corresponding pairs.
  -- If one input list is short, excess elements of the longer list are
  -- discarded.
  
  {-@ zip :: xs : [a] -> ys:[b] 
              -> {v : [(a, b)] | ((((len v) <= (len xs)) && ((len v) <= (len ys)))
              && (((len xs) = (len ys)) => ((len v) = (len xs))) )} @-}
  
  zip :: [a] -> [b] -> [(a,b)]
  zip (a:as) (b:bs) = (a,b) : zip as bs
  zip _      _      = []
  
  {-# INLINE [0] zipFB #-}
  zipFB :: ((a, b) -> c -> d) -> a -> b -> c -> d
  zipFB c = \x y r -> (x,y) `c` r
  
  {- RULES
  "zip"      [~1] forall xs ys. zip xs ys = build (\c n -> foldr2 (zipFB c) n xs ys)
  "zipList"  [1]  foldr2 (zipFB (:)) []   = zip
   -}
  \end{code}
  
  \begin{code}
  ----------------------------------------------
  -- | 'zip3' takes three lists and returns a list of triples, analogous to
  -- 'zip'.
  zip3 :: [a] -> [b] -> [c] -> [(a,b,c)]
  -- Specification
  -- zip3 =  zipWith3 (,,)
  zip3 (a:as) (b:bs) (c:cs) = (a,b,c) : zip3 as bs cs
  zip3 _      _      _      = []
  \end{code}
  
  
  -- The zipWith family generalises the zip family by zipping with the
  -- function given as the first argument, instead of a tupling function.
  
  \begin{code}
  ----------------------------------------------
  -- | 'zipWith' generalises 'zip' by zipping with the function given
  -- as the first argument, instead of a tupling function.
  -- For example, @'zipWith' (+)@ is applied to two lists to produce the
  -- list of corresponding sums.
  
  
  {-@ zipWith :: (a -> b -> c) 
              -> xs : [a] -> ys:[b] 
              -> {v : [c] | (((len v) <= (len xs)) && ((len v) <= (len ys)))} @-}
  zipWith :: (a->b->c) -> [a]->[b]->[c]
  zipWith f (a:as) (b:bs) = f a b : zipWith f as bs
  zipWith _ _      _      = []
  
  -- zipWithFB must have arity 2 since it gets two arguments in the "zipWith"
  -- rule; it might not get inlined otherwise
  {-# INLINE [0] zipWithFB #-}
  zipWithFB :: (a -> b -> c) -> (d -> e -> a) -> d -> e -> b -> c
  zipWithFB c f = \x y r -> (x `f` y) `c` r
  
  {- RULES
  "zipWith"       [~1] forall f xs ys.    zipWith f xs ys = build (\c n -> foldr2 (zipWithFB c f) n xs ys)
  "zipWithList"   [1]  forall f.  foldr2 (zipWithFB (:) f) [] = zipWith f
    -}
  \end{code}
  
  \begin{code}
  -- | The 'zipWith3' function takes a function which combines three
  -- elements, as well as three lists and returns a list of their point-wise
  -- combination, analogous to 'zipWith'.
  zipWith3                :: (a->b->c->d) -> [a]->[b]->[c]->[d]
  zipWith3 z (a:as) (b:bs) (c:cs)
                          =  z a b c : zipWith3 z as bs cs
  zipWith3 _ _ _ _        =  []
  
  -- | 'unzip' transforms a list of pairs into a list of first components
  -- and a list of second components.
  unzip    :: [(a,b)] -> ([a],[b])
  {-# INLINE unzip #-}
  unzip    =  foldr (\(a,b) ~(as,bs) -> (a:as,b:bs)) ([],[])
  
  -- | The 'unzip3' function takes a list of triples and returns three
  -- lists, analogous to 'unzip'.
  unzip3   :: [(a,b,c)] -> ([a],[b],[c])
  {-# INLINE unzip3 #-}
  unzip3   =  foldr (\(a,b,c) ~(as,bs,cs) -> (a:as,b:bs,c:cs))
                    ([],[],[])
  \end{code}
  
  
  %*********************************************************
  %*                                                      *
  \subsection{Error code}
  %*                                                      *
  %*********************************************************
  
  Common up near identical calls to `error' to reduce the number
  constant strings created when compiled:
  
  \begin{code}
  {-@ assert errorEmptyList :: {v: String | (0 = 1)} -> a @-}
  errorEmptyList :: String -> a
  errorEmptyList fun =
    liquidError {- error -} (prel_list_str ++ fun ++ ": empty list")
  
  prel_list_str :: String
  prel_list_str = "Prelude."
  \end{code}
  