Add a new element to the front/left of a sequence

 lcons x <x0,...,xn-1> = <x,x0,...,xn-1>

Axioms:

• lcons x xs = append (singleton x) xs

This function is always unambiguous.

Default running time: O( 1 )

Add a new element to the right/rear of a sequence

 rcons x <x0,...,xn-1> = <x0,...,xn-1,x>

Axioms:

• rcons x xs = append xs (singleton x)

This function is always unambiguous.

Default running time: O( n )

Convert a list into a sequence

 fromList [x0,...,xn-1] = <x0,...,xn-1>

Axioms:

• fromList xs = foldr lcons empty xs

This function is always unambiguous.

Default running time: O( n )

Create a sequence containing n copies of the given element. Return empty if n<0.

copy n x = <x,...,x>

Axioms:

• n > 0    ==> copy n x = cons x (copy (n-1) x)

• n <= 0   ==> copy n x = empty

This function is always unambiguous.

Default running time: O( n )

Separate a sequence into its first (leftmost) element and the remaining sequence. Calls fail if the sequence is empty.

Axioms:

• lview empty = fail

• lview (lcons x xs) = return (x,xs)

This function is always unambiguous.

Default running time: O( 1 )

Return the first element of a sequence. Signals an error if the sequence is empty.

Axioms:

• lhead empty = undefined

• lhead (lcons x xs) = x

This function is always unambiguous.

Default running time: O( 1 )

Returns the first element of a sequence. Calls fail if the sequence is empty.

Axioms:

• lheadM empty = fail

• lheadM (lcons x xs) = return x

This function is always unambiguous.

Default running time: O( 1 )

Delete the first element of the sequence. Signals error if sequence is empty.

Axioms:

• ltail empty = undefined

• ltail (lcons x xs) = xs

This function is always unambiguous.

Default running time: O( 1 )

Delete the first element of the sequence. Calls fail if the sequence is empty.

Axioms:

• ltailM empty = fail

• ltailM (lcons x xs) = return xs

This function is always unambiguous.

Default running time: O( 1 )

Separate a sequence into its last (rightmost) element and the remaining sequence. Calls fail if the sequence is empty.

Axioms:

• rview empty = fail

• rview (rcons x xs) = return (x,xs)

This function is always unambiguous.

Default running time: O( n )

Return the last (rightmost) element of the sequence. Signals error if sequence is empty.

Axioms:

• rhead empty = undefined

• rhead (rcons x xs) = x

This function is always unambiguous.

Default running time: O( n )

Returns the last element of the sequence. Calls fail if the sequence is empty.

Axioms:

• rheadM empty = fail

• rheadM (rcons x xs) = return x

This function is always unambiguous.

Default running time: O( n )

Delete the last (rightmost) element of the sequence. Signals an error if the sequence is empty.

Axioms:

• rtail empty = undefined

• rtail (rcons x xs) = xs

This function is always unambiguous.

Default running time: O( n )

Delete the last (rightmost) element of the sequence. Calls fail of the sequence is empty

Axioms:

• rtailM empty = fail

• rtailM (rcons x xs) = return xs

This function is always unambiguous.

Default running time: O( n )

Returns True if the sequence is empty and False otherwise.

 null <x0,...,xn-1> = (n==0)

Axioms:

• null xs = (size xs == 0)

This function is always unambiguous.

Default running time: O( 1 )

Returns the length of a sequence.

 size <x0,...,xn-1> = n

Axioms:

• size empty = 0

• size (lcons x xs) = 1 + size xs

This function is always unambiguous.

Default running time: O( n )

Convert a sequence to a list.

 toList <x0,...,xn-1> = [x0,...,xn-1]

Axioms:

• toList empty = []

• toList (lcons x xs) = x : toList xs

This function is always unambiguous.

Default running time: O( n )

Flatten a sequence of sequences into a simple sequence.

 concat xss = foldr append empty xss

Axioms:

• concat xss = foldr append empty xss

This function is always unambiguous.

Default running time: O( n + m ) where n is the length of the input sequence and m is length of the output sequence.

Reverse the order of a sequence

 reverse <x0,...,xn-1> = <xn-1,...,x0>

Axioms:

• reverse empty = empty

• reverse (lcons x xs) = rcons x (reverse xs)

This function is always unambiguous.

Default running time: O( n )

Reverse a sequence onto the front of another sequence.

 reverseOnto <x0,...,xn-1> <y0,...,ym-1> = <xn-1,...,x0,y0,...,ym-1>

Axioms:

• reverseOnto xs ys = append (reverse xs) ys

This function is always unambiguous.

Default running time: O( n1 )

Combine all the elements of a sequence into a single value, given a combining function and an initial value. The order in which the elements are applied to the combining function is unspecified. fold is one of the few ambiguous sequence functions.

Axioms:

• fold f c empty = c

• f is fold-commutative ==> fold f = foldr f = foldl f

fold f is unambiguous iff f is fold-commutative.

Default running type: O( t * n ) where t is the running tome of f.

A strict variant of fold. fold' is one of the few ambiguous sequence functions.

Axioms:

• forall a. f a _|_ = _|_ ==> fold f x xs = fold' f x xs

fold f is unambiguous iff f is fold-commutative.

Default running type: O( t * n ) where t is the running tome of f.

Combine all the elements of a non-empty sequence into a single value, given a combining function. Signals an error if the sequence is empty.

Axioms:

• f is fold-commutative ==> fold1 f = foldr1 f = foldl1 f

fold1 f is unambiguous iff f is fold-commutative.

Default running type: O( t * n ) where t is the running tome of f.

A strict variant of fold1.

Axioms:

• forall a. f a _|_ = _|_ ==> fold1' f xs = fold1 f xs

fold1' f is unambiguous iff f is fold-commutative.

Default running time: O( t * n ) where t is the running time of f

Combine all the elements of a sequence into a single value, given a combining function and an initial value. The function is applied with right nesting.

 foldr (%) c <x0,...,xn-1> = x0 % (x1 % ( ... % (xn-1 % c)))

Axioms:

• foldr f c empty = c

• foldr f c (lcons x xs) = f x (foldr f c xs)

This function is always unambiguous.

Default running time: O( t * n ) where t is the running time of f

Strict variant of foldr.

Axioms:

• forall a. f a _|_ = _|_ ==> foldr f x xs = foldr' f x xs

This function is always unambiguous.

Default running time: O( t * n ) where t is the running time of f

Combine all the elements of a sequence into a single value, given a combining function and an initial value. The function is applied with left nesting.

 foldl (%) c <x0,...,xn-1> = (((c % x0) % x1) % ... ) % xn-1

Axioms:

• foldl f c empty = c

• foldl f c (lcons x xs) = foldl f (f c x) xs

This function is always unambiguous.

Default running time: O( t * n ) where t is the running time of f

Strict variant of foldl.

Axioms:

• forall a. f _|_ a = _|_ ==> foldl f z xs = foldl' f z xs

This function is always unambiguous.

Default running time: O( t * n ) where t is the running time of f

Combine all the elements of a non-empty sequence into a single value, given a combining function. The function is applied with right nesting. Signals an error if the sequence is empty.

 foldr1 (+) <x0,...,xn-1>
   | n==0 = error "ModuleName.foldr1: empty sequence"
   | n>0  = x0 + (x1 + ... + xn-1)

Axioms:

• foldr1 f empty = undefined

• foldr1 f (rcons x xs) = foldr f x xs

This function is always unambiguous.

Default running time: O( t * n ) where t is the running time of f

Strict variant of foldr1.

Axioms:

• forall a. f a _|_ = _|_ ==> foldr1 f xs = foldr1' f xs

This function is always unambiguous.

Default running time: O( t * n ) where t is the running time of f

Combine all the elements of a non-empty sequence into a single value, given a combining function. The function is applied with left nesting. Signals an error if the sequence is empty.

 foldl1 (+) <x0,...,xn-1>
  | n==0 = error "ModuleName.foldl1: empty sequence"
  | n>0  = (x0 + x1) + ... + xn-1

Axioms:

• foldl1 f empty = undefined

• foldl1 f (lcons x xs) = foldl f x xs

This function is always unambiguous.

Default running time: O( t * n ) where t is the running time of f

Strict variant of foldl1.

Axioms:

• forall a. f _|_ a = _|_ ==> foldl1 f xs = foldl1' f xs

This function is always unambiguous.

Default running time: O( t * n ) where t is the running time of f

See reduce1 for additional notes.

 reducer f x xs = reduce1 f (cons x xs)

Axioms:

• reducer f c xs = foldr f c xs for associative f

reducer f is unambiguous iff f is an associative function.

Default running time: O( t * n ) where t is the running time of f

Strict variant of reducer.

See reduce1 for additional notes.

Axioms:

• forall a. f a _|_ = _|_ && forall a. f _|_ a = _|_ ==>
            reducer f x xs = reducer' f x xs

reducer' f is unambiguous iff f is an associative function.

Default running time: O( t * n ) where t is the running time of f

See reduce1 for additional notes.

 reducel f x xs = reduce1 f (rcons x xs)

Axioms:

• reducel f c xs = foldl f c xs for associative f

reducel f is unambiguous iff f is an associative function.

Default running time: O( t * n ) where t is the running time of f

Strict variant of reducel.

See reduce1 for additional notes.

Axioms:

• forall a. f a _|_ = _|_ && forall a. f _|_ a = _|_ ==>
            reducel f x xs = reducel' f x xs

reducel' f is unambiguous iff f is an associative function.

Default running time: O( t * n ) where t is the running time of f

A reduce is similar to a fold, but combines elements in a balanced fashion. The combining function should usually be associative. If the combining function is associative, the various reduce functions yield the same results as the corresponding folds.

What is meant by "in a balanced fashion"? We mean that reduce1 (%) <x0,x1,...,xn-1> equals some complete parenthesization of x0 % x1 % ... % xn-1 such that the nesting depth of parentheses is O( log n ). The precise shape of this parenthesization is unspecified.

 reduce1 f <x> = x
 reduce1 f <x0,...,xn-1> =
     f (reduce1 f <x0,...,xi>) (reduce1 f <xi+1,...,xn-1>)

for some i such that 0 <= i && i < n-1

Although the exact value of i is unspecified it tends toward n/2 so that the depth of calls to f is at most logarithmic.

Note that reduce* are some of the only sequence operations for which different implementations are permitted to yield different answers. Also note that a single implementation may choose different parenthisizations for different sequences, even if they are the same length. This will typically happen when the sequences were constructed differently.

The canonical applications of the reduce functions are algorithms like merge sort where:

 mergesort xs = reducer merge empty (map singleton xs)

Axioms:

• reduce1 f empty = undefined

• reduce1 f xs = foldr1 f xs = foldl1 f xs for associative f

reduce1 f is unambiguous iff f is an associative function.

Default running time: O( t * n ) where t is the running time of f

Strict variant of reduce1.

Axioms:

• forall a. f a _|_ = _|_ && forall a. f _|_ a = _|_ ==>
            reduce1 f xs = reduce1' f xs

reduce1' f is unambiguous iff f is an associative function.

Default running time: O( t * n ) where t is the running time of f

Extract a prefix of length i from the sequence. Return empty if i is negative, or the entire sequence if i is too large.

 take i xs = fst (splitAt i xs)

Axioms:

• i < 0        ==> take i xs = empty

• i > size xs  ==> take i xs = xs

• size xs == i ==> take i (append xs ys) = xs

This function is always unambiguous.

Default running time: O( i )

Delete a prefix of length i from a sequence. Return the entire sequence if i is negative, or empty if i is too large.

 drop i xs = snd (splitAt i xs)

Axioms:

• i < 0        ==> drop i xs = xs

• i > size xs  ==> drop i xs = empty

• size xs == i ==> drop i (append xs ys) = ys

This function is always unambiguous.

Default running time: O( i )

Split a sequence into a prefix of length i and the remaining sequence. Behaves the same as the corresponding calls to take and drop if i is negative or too large.

 splitAt i xs
  | i < 0  = (<>           , <x0,...,xn-1>)
  | i < n  = (<x0,...,xi-1>, <xi,...,xn-1>)
  | i >= n = (<x0,...,xn-1>, <>           )

Axioms:

• splitAt i xs = (take i xs,drop i xs)

This function is always unambiguous.

Default running time: O( i )

Extract a subsequence from a sequence. The integer arguments are "start index" and "length" NOT "start index" and "end index". Behaves the same as the corresponding calls to take and drop if the start index or length are negative or too large.

 subseq i len xs = take len (drop i xs)

Axioms:

• subseq i len xs = take len (drop i xs)

This function is always unambiguous.

Default running time: O( i + len )

Extract the elements of a sequence that satisfy the given predicate, retaining the relative ordering of elements from the original sequence.

 filter p xs = foldr pcons empty xs
      where pcons x xs = if p x then cons x xs else xs

Axioms:

• filter p empty = empty

• filter p (lcons x xs) = if p x 
         then lcons x (filter p xs)
         else filter p xs

This function is always unambiguous.

Default running time: O( t * n ) where t is the running time of p

Separate the elements of a sequence into those that satisfy the given predicate and those that do not, retaining the relative ordering of elements from the original sequence.

 partition p xs = (filter p xs, filter (not . p) xs)

Axioms:

• partition p xs = (filter p xs, filter (not . p) xs)

This function is always unambiguous.

Default running time: O( t * n ) where t is the running time of p

Extract the maximal prefix of elements satisfying the given predicate.

 takeWhile p xs = fst (splitWhile p xs)

Axioms:

• takeWhile p empty = empty

• takeWhile p (lcons x xs) = if p x
         then lcons x (takeWhile p xs)
         else empty

This function is always unambiguous.

Default running time: O( t * n ) where t is the running time of p

Delete the maximal prefix of elements satisfying the given predicate.

 dropWhile p xs = snd (splitWhile p xs)

Axioms:

• dropWhile p empty = empty

• dropWhile p (lcons x xs) = if p x
        then dropWhile p xs
        else lcons x xs

This function is always unambiguous.

Default running time: O( t * n ) where t is the running time of p

Split a sequence into the maximal prefix of elements satisfying the given predicate, and the remaining sequence.

 splitWhile p <x0,...,xn-1> = (<x0,...,xi-1>, <xi,...,xn-1>)
   where i = min j such that p xj (or n if no such j)

Axioms:

• splitWhile p xs = (takeWhile p xs,dropWhile p xs)

This function is always unambiguous.

Default running time: O( t * n ) where t is the running time of p

Test whether an index is valid for the given sequence. All indexes are 0 based.

 inBounds i <x0,...,xn-1> = (0 <= i && i < n)

Axioms:

• inBounds i xs = (0 <= i && i < size xs)

This function is always unambiguous.

Default running time: O( i )

Return the element at the given index. All indexes are 0 based. Signals error if the index out of bounds.

 lookup i xs@<x0,...,xn-1>
   | inBounds i xs = xi
   | otherwise = error "ModuleName.lookup: index out of bounds"

Axioms:

• not (inBounds i xs)  ==> lookup i xs = undefined

• size xs == i ==> lookup i (append xs (lcons x ys)) = x

This function is always unambiguous.

Default running time: O( i )

Return the element at the given index. All indexes are 0 based. Calls fail if the index is out of bounds.

 lookupM i xs@<x0,...,xn-1>
   | inBounds i xs = Just xi
   | otherwise = Nothing

Axioms:

• not (inBounds i xs) ==> lookupM i xs = fail

• size xs == i ==> lookupM i (append xs (lcons x ys)) = return x

This function is always unambiguous.

Default running time: O( i )

Return the element at the given index, or the default argument if the index is out of bounds. All indexes are 0 based.

 lookupWithDefault d i xs@<x0,...,xn-1>
   | inBounds i xs = xi
   | otherwise = d

Axioms:

• not (inBounds i xs) ==> lookupWithDefault d i xs = d

• size xs == i ==> lookupWithDefault d i (append xs (lcons x ys)) = x

This function is always unambiguous.

Default running time: O( i )

Replace the element at the given index, or return the original sequence if the index is out of bounds. All indexes are 0 based.

 update i y xs@<x0,...,xn-1>
   | inBounds i xs = <x0,...xi-1,y,xi+1,...,xn-1>
   | otherwise = xs

Axioms:

• not (inBounds i xs) ==> update i y xs = xs

• size xs == i ==> update i y (append xs (lcons x ys)) =
        append xs (lcons y ys)

This function is always unambiguous.

Default running time: O( i )

Apply a function to the element at the given index, or return the original sequence if the index is out of bounds. All indexes are 0 based.

 adjust f i xs@<x0,...,xn-1>
   | inBounds i xs = <x0,...xi-1,f xi,xi+1,...,xn-1>
   | otherwise = xs

Axioms:

• not (inBounds i xs) ==> adjust f i xs = xs

• size xs == i ==> adjust f i (append xs (lcons x ys)) =
        append xs (cons (f x) ys)

This function is always unambiguous.

Default running time: O( i + t ) where t is the running time of f

Like map, but include the index with each element. All indexes are 0 based.

 mapWithIndex f <x0,...,xn-1> = <f 0 x0,...,f (n-1) xn-1>

Axioms:

• mapWithIndex f empty = empty

• mapWithIndex f (rcons x xs) = rcons (f (size xs) x) (mapWithIndex f xs)

This function is always unambiguous.

Default running time: O( t * n ) where t is the running time of f

Like foldr, but include the index with each element. All indexes are 0 based.

 foldrWithIndex f c <x0,...,xn-1> = 
     f 0 x0 (f 1 x1 (... (f (n-1) xn-1 c)))

Axioms:

• foldrWithIndex f c empty = c

• foldrWithIndex f c (rcons x xs) =
        foldrWithIndex f (f (size xs) x c) xs

This function is always unambiguous.

Default running time: O( t * n ) where t is the running time of f

Strict variant of foldrWithIndex.

Axioms:

• forall i a. f i a _|_ = _|_ ==> foldrWithIndex f x xs = 
         foldrWithIndex' f x xs

This function is always unambiguous.

Default running time: O( t * n ) where t is the running time of f

Like foldl, but include the index with each element. All indexes are 0 based.

 foldlWithIndex f c <x0,...,xn-1> =
     f (...(f (f c 0 x0) 1 x1)...) (n-1) xn-1)

Axioms:

• foldlWithIndex f c empty = c

• foldlWithIndex f c (rcons x xs) =
        f (foldlWithIndex f c xs) (size xs) x

This function is always unambiguous.

Default running time: O( t * n ) where t is the running time of f

Strict variant of foldlWithIndex.

Axioms:

• forall i a. f _|_ i a = _|_ ==> foldlWithIndex f x xs = 
         foldlWithIndex' f x xs

This function is always unambiguous.

Default running time: O( t * n ) where t is the running time of f

Combine two sequences into a sequence of pairs. If the sequences are different lengths, the excess elements of the longer sequence is discarded.

 zip <x0,...,xn-1> <y0,...,ym-1> = <(x0,y0),...,(xj-1,yj-1)>
     where j = min {n,m}

Axioms:

• zip xs ys = zipWith (,) xs ys

This function is always unambiguous.

Default running time: O( min( n1, n2 ) )

Like zip, but combines three sequences into triples.

 zip3 <x0,...,xn-1> <y0,...,ym-1> <z0,...,zk-1> = 
      <(x0,y0,z0),...,(xj-1,yj-1,zj-1)>
    where j = min {n,m,k}

Axioms:

• zip3 xs ys zs = zipWith3 (,,) xs ys zs

This function is always unambiguous.

Default running time: O( min( n1, n2, n3 ) )

Combine two sequences into a single sequence by mapping a combining function across corresponding elements. If the sequences are of different lengths, the excess elements of the longer sequence are discarded.

 zipWith f xs ys = map (uncurry f) (zip xs ys)

Axioms:

• zipWith f (lcons x xs) (lcons y ys) =
       lcons (f x y) (zipWith f xs ys)

• (null xs || null ys) ==> zipWith xs ys = empty

This function is always unambiguous.

Default running time: O( t * min( n1, n2 ) ) where t is the running time of f

Like zipWith but for a three-place function and three sequences.

 zipWith3 f xs ys zs = map (uncurry f) (zip3 xs ys zs)

Axioms:

• zipWith3 (lcons x xs) (lcons y ys) (lcons z zs) =
        lcons (f x y z) (zipWith3 f xs ys zs)

This function is always unambiguous.

Default running time: O( t * min( n1, n2, n3 ) ) where t is the running time of f

Transpose a sequence of pairs into a pair of sequences.

 unzip xs = (map fst xs, map snd xs)

Axioms:

• unzip xys = unzipWith fst snd xys

This function is always unambiguous.

Default running time: O( n )

Transpose a sequence of triples into a triple of sequences

 unzip3 xs = (map fst3 xs, map snd3 xs, map thd3 xs)
    where fst3 (x,y,z) = x
          snd3 (x,y,z) = y
          thd3 (x,y,z) = z

Axioms:

• unzip3 xyzs = unzipWith3 fst3 snd3 thd3 xyzs

This function is always unambiguous.

Default running time: O( n )

Map two functions across every element of a sequence, yielding a pair of sequences

 unzipWith f g xs = (map f xs, map g xs)

Axioms:

• unzipWith f g xs = (map f xs, map g xs)

This function is always unambiguous.

Default running time: O( t * n ) where t is the maximum running time of f and g

Map three functions across every element of a sequence, yielding a triple of sequences.

 unzipWith3 f g h xs = (map f xs, map g xs, map h xs)

Axioms:

• unzipWith3 f g h xs = (map f xs,map g xs,map h xs)

This function is always unambiguous.

Default running time: O( t * n ) where t is the maximum running time of f, g, and h

Semanticly, this function is a partial identity function. If the datastructure is infinite in size or contains exceptions or non-termination in the structure itself, then strict will result in bottom. Operationally, this function walks the datastructure forcing any closures. Elements contained in the sequence are not forced.

Axioms:

• strict xs = xs OR strict xs = _|_

This function is always unambiguous.

Default running time: O( n )

Similar to strict, this function walks the datastructure forcing closures. However, strictWith will additionally apply the given function to the sequence elements, force the result using seq, and then ignore it. This function can be used to perform various levels of forcing on the sequence elements. In particular:

 strictWith id xs

will force the spine of the datastructure and reduce each element to WHNF.

Axioms:

• forall f :: a -> b, strictWith f xs = xs OR strictWith f xs = _|_

This function is always unambiguous.

Default running time: unbounded (forcing element closures can take arbitrairly long)

A method to facilitate unit testing. Returns True if the structural invariants of the implementation hold for the given sequence. If this function returns False, it represents a bug in the implementation.

The name of the module implementing s.