Copyright | Copyright (c) 1998-1999 Chris Okasaki |
---|---|

License | MIT; see COPYRIGHT file for terms and conditions |

Maintainer | robdockins AT fastmail DOT fm |

Stability | stable |

Portability | GHC, Hugs (MPTC and FD) |

Safe Haskell | Safe |

Language | Haskell2010 |

The sequence abstraction is usually viewed as a hierarchy of ADTs including lists, queues, deques, catenable lists, etc. However, such a hierarchy is based on efficiency rather than functionality. For example, a list supports all the operations that a deque supports, even though some of the operations may be inefficient. Hence, in Edison, all sequence data structures are defined as instances of the single Sequence class:

class (Functor s, MonadPlus s) => Sequence s

All sequences are also instances of `Functor`

, `Monad`

, and `MonadPlus`

.
In addition, all sequences are expected to be instances of `Eq`

, `Show`

,
and `Read`

, although this is not enforced.

We follow the naming convention that every module implementing sequences
defines a type constructor named `Seq`

.

For each method the "default" complexity is listed. Individual implementations may differ for some methods. The documentation for each implementation will list those methods for which the running time differs from these.

A description of each Sequence function appears below. In most cases psudeocode is also provided. Obviously, the psudeocode is illustrative only.

Sequences are represented in psudecode between angle brackets:

<x0,x1,x2...,xn-1>

Such that `x0`

is at the left (front) of the sequence and
`xn-1`

is at the right (rear) of the sequence.

- map :: Sequence s => (a -> b) -> s a -> s b
- singleton :: Sequence s => a -> s a
- concatMap :: Sequence s => (a -> s b) -> s a -> s b
- empty :: Sequence s => s a
- append :: Sequence s => s a -> s a -> s a
- class (Functor s, MonadPlus s) => Sequence s where
- lcons :: a -> s a -> s a
- rcons :: a -> s a -> s a
- fromList :: [a] -> s a
- copy :: Int -> a -> s a
- lview :: Monad m => s a -> m (a, s a)
- lhead :: s a -> a
- lheadM :: Monad m => s a -> m a
- ltail :: s a -> s a
- ltailM :: Monad m => s a -> m (s a)
- rview :: Monad m => s a -> m (a, s a)
- rhead :: s a -> a
- rheadM :: Monad m => s a -> m a
- rtail :: s a -> s a
- rtailM :: Monad m => s a -> m (s a)
- null :: s a -> Bool
- size :: s a -> Int
- toList :: s a -> [a]
- concat :: s (s a) -> s a
- reverse :: s a -> s a
- reverseOnto :: s a -> s a -> s a
- fold :: (a -> b -> b) -> b -> s a -> b
- fold' :: (a -> b -> b) -> b -> s a -> b
- fold1 :: (a -> a -> a) -> s a -> a
- fold1' :: (a -> a -> a) -> s a -> a
- foldr :: (a -> b -> b) -> b -> s a -> b
- foldr' :: (a -> b -> b) -> b -> s a -> b
- foldl :: (b -> a -> b) -> b -> s a -> b
- foldl' :: (b -> a -> b) -> b -> s a -> b
- foldr1 :: (a -> a -> a) -> s a -> a
- foldr1' :: (a -> a -> a) -> s a -> a
- foldl1 :: (a -> a -> a) -> s a -> a
- foldl1' :: (a -> a -> a) -> s a -> a
- reducer :: (a -> a -> a) -> a -> s a -> a
- reducer' :: (a -> a -> a) -> a -> s a -> a
- reducel :: (a -> a -> a) -> a -> s a -> a
- reducel' :: (a -> a -> a) -> a -> s a -> a
- reduce1 :: (a -> a -> a) -> s a -> a
- reduce1' :: (a -> a -> a) -> s a -> a
- take :: Int -> s a -> s a
- drop :: Int -> s a -> s a
- splitAt :: Int -> s a -> (s a, s a)
- subseq :: Int -> Int -> s a -> s a
- filter :: (a -> Bool) -> s a -> s a
- partition :: (a -> Bool) -> s a -> (s a, s a)
- takeWhile :: (a -> Bool) -> s a -> s a
- dropWhile :: (a -> Bool) -> s a -> s a
- splitWhile :: (a -> Bool) -> s a -> (s a, s a)
- inBounds :: Int -> s a -> Bool
- lookup :: Int -> s a -> a
- lookupM :: Monad m => Int -> s a -> m a
- lookupWithDefault :: a -> Int -> s a -> a
- update :: Int -> a -> s a -> s a
- adjust :: (a -> a) -> Int -> s a -> s a
- mapWithIndex :: (Int -> a -> b) -> s a -> s b
- foldrWithIndex :: (Int -> a -> b -> b) -> b -> s a -> b
- foldrWithIndex' :: (Int -> a -> b -> b) -> b -> s a -> b
- foldlWithIndex :: (b -> Int -> a -> b) -> b -> s a -> b
- foldlWithIndex' :: (b -> Int -> a -> b) -> b -> s a -> b
- zip :: s a -> s b -> s (a, b)
- zip3 :: s a -> s b -> s c -> s (a, b, c)
- zipWith :: (a -> b -> c) -> s a -> s b -> s c
- zipWith3 :: (a -> b -> c -> d) -> s a -> s b -> s c -> s d
- unzip :: s (a, b) -> (s a, s b)
- unzip3 :: s (a, b, c) -> (s a, s b, s c)
- unzipWith :: (a -> b) -> (a -> c) -> s a -> (s b, s c)
- unzipWith3 :: (a -> b) -> (a -> c) -> (a -> d) -> s a -> (s b, s c, s d)
- strict :: s a -> s a
- strictWith :: (a -> b) -> s a -> s a
- structuralInvariant :: s a -> Bool
- instanceName :: s a -> String

# Superclass aliases

## Functor aliases

map :: Sequence s => (a -> b) -> s a -> s b Source

Return the result of applying a function to
every element of a sequence. Identical
to `fmap`

from `Functor`

.

map f <x0,...,xn-1> = <f x0,...,f xn-1>

*Axioms:*

map f empty = empty

map f (lcons x xs) = lcons (f x) (map f xs)

This function is always *unambiguous*.

Default running time: `O( t * n )`

where `t`

is the running time of `f`

## Monad aliases

singleton :: Sequence s => a -> s a Source

Create a singleton sequence. Identical to `return`

from `Monad`

.

singleton x = <x>

*Axioms:*

singleton x = lcons x empty = rcons x empty

This function is always *unambiguous*.

Default running time: `O( 1 )`

concatMap :: Sequence s => (a -> s b) -> s a -> s b Source

Apply a sequence-producing function to every element
of a sequence and flatten the result. `concatMap`

is the bind `(>>=)`

operation of from `Monad`

with the
arguments in the reverse order.

concatMap f xs = concat (map f xs)

*Axioms:*

concatMap f xs = concat (map f xs)

This function is always *unambiguous*.

Default running time: `O( t * n + m )`

where `n`

is the length of the input sequence, `m`

is the
length of the output sequence, and `t`

is the running time of `f`

## MonadPlus aliases

empty :: Sequence s => s a Source

The empty sequence. Identical to `mzero`

from `MonadPlus`

.

empty = <>

This function is always *unambiguous*.

Default running time: `O( 1 )`

append :: Sequence s => s a -> s a -> s a Source

Append two sequence, with the first argument on the left
and the second argument on the right. Identical to `mplus`

from `MonadPlus`

.

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

*Axioms:*

append xs ys = foldr lcons ys xs

This function is always *unambiguous*.

Default running time: `O( n1 )`

# The Sequence class

class (Functor s, MonadPlus s) => Sequence s where Source

The `Sequence`

class defines an interface for datatypes which
implement sequences. A description for each function is
given below.

lcons :: a -> s a -> s a Source

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 )`

rcons :: a -> s a -> s a Source

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 )`

copy :: Int -> a -> s a Source

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 )`

lview :: Monad m => s a -> m (a, s a) Source

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 )`

lheadM :: Monad m => s a -> m a Source

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 )`

ltailM :: Monad m => s a -> m (s a) Source

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 )`

rview :: Monad m => s a -> m (a, s a) Source

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 )`

rheadM :: Monad m => s a -> m a Source

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 )`

rtailM :: Monad m => s a -> m (s a) Source

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 )`

concat :: s (s a) -> s a Source

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 )`

reverseOnto :: s a -> s a -> s a Source

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 )`

fold :: (a -> b -> b) -> b -> s a -> b Source

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`

.

fold' :: (a -> b -> b) -> b -> s a -> b Source

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`

.

fold1 :: (a -> a -> a) -> s a -> a Source

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`

.

fold1' :: (a -> a -> a) -> s a -> a Source

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`

foldr :: (a -> b -> b) -> b -> s a -> b Source

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`

foldr' :: (a -> b -> b) -> b -> s a -> b Source

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`

foldl :: (b -> a -> b) -> b -> s a -> b Source

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`

foldl' :: (b -> a -> b) -> b -> s a -> b Source

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`

foldr1 :: (a -> a -> a) -> s a -> a Source

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`

foldr1' :: (a -> a -> a) -> s a -> a Source

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`

foldl1 :: (a -> a -> a) -> s a -> a Source

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`

foldl1' :: (a -> a -> a) -> s a -> a Source

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`

reducer :: (a -> a -> a) -> a -> s a -> a Source

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`

reducer' :: (a -> a -> a) -> a -> s a -> a Source

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`

reducel :: (a -> a -> a) -> a -> s a -> a Source

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`

reducel' :: (a -> a -> a) -> a -> s a -> a Source

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`

reduce1 :: (a -> a -> a) -> s a -> a Source

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`

reduce1' :: (a -> a -> a) -> s a -> a Source

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`

take :: Int -> s a -> s a Source

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 )`

drop :: Int -> s a -> s a Source

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 )`

splitAt :: Int -> s a -> (s a, s a) Source

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 )`

subseq :: Int -> Int -> s a -> s a Source

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 )`

filter :: (a -> Bool) -> s a -> s a Source

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`

partition :: (a -> Bool) -> s a -> (s a, s a) Source

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`

takeWhile :: (a -> Bool) -> s a -> s a Source

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`

dropWhile :: (a -> Bool) -> s a -> s a Source

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`

splitWhile :: (a -> Bool) -> s a -> (s a, s a) Source

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`

inBounds :: Int -> s a -> Bool Source

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 )`

lookup :: Int -> s a -> a Source

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 )`

lookupM :: Monad m => Int -> s a -> m a Source

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 )`

lookupWithDefault :: a -> Int -> s a -> a Source

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 )`

update :: Int -> a -> s a -> s a Source

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 )`

adjust :: (a -> a) -> Int -> s a -> s a Source

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`

mapWithIndex :: (Int -> a -> b) -> s a -> s b Source

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`

foldrWithIndex :: (Int -> a -> b -> b) -> b -> s a -> b Source

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`

foldrWithIndex' :: (Int -> a -> b -> b) -> b -> s a -> b Source

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`

foldlWithIndex :: (b -> Int -> a -> b) -> b -> s a -> b Source

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`

foldlWithIndex' :: (b -> Int -> a -> b) -> b -> s a -> b Source

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`

zip :: s a -> s b -> s (a, b) Source

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 ) )`

zip3 :: s a -> s b -> s c -> s (a, b, c) Source

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 ) )`

zipWith :: (a -> b -> c) -> s a -> s b -> s c Source

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`

zipWith3 :: (a -> b -> c -> d) -> s a -> s b -> s c -> s d Source

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`

unzip :: s (a, b) -> (s a, s b) Source

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 )`

unzip3 :: s (a, b, c) -> (s a, s b, s c) Source

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 )`

unzipWith :: (a -> b) -> (a -> c) -> s a -> (s b, s c) Source

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`

unzipWith3 :: (a -> b) -> (a -> c) -> (a -> d) -> s a -> (s b, s c, s d) Source

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 )`

strictWith :: (a -> b) -> s a -> s a Source

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)

structuralInvariant :: s a -> Bool Source

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.

instanceName :: s a -> String Source

The name of the module implementing s.