streamly-0.7.2: src/Streamly/Prelude.hs
{-# LANGUAGE CPP #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE FlexibleContexts #-}
#if __GLASGOW_HASKELL__ >= 800
{-# OPTIONS_GHC -Wno-orphans #-}
#endif
#include "inline.hs"
-- |
-- Module : Streamly.Prelude
-- Copyright : (c) 2017 Harendra Kumar
--
-- License : BSD3
-- Maintainer : streamly@composewell.com
-- Stability : experimental
-- Portability : GHC
--
-- This module is designed to be imported qualified:
--
-- @
-- import qualified Streamly.Prelude as S
-- @
--
-- Functions with the suffix @M@ are general functions that work on monadic
-- arguments. The corresponding functions without the suffix @M@ work on pure
-- arguments and can in general be derived from their monadic versions but are
-- provided for convenience and for consistency with other pure APIs in the
-- @base@ package.
--
-- In many cases, short definitions of the combinators are provided in the
-- documentation for illustration. The actual implementation may differ for
-- performance reasons.
--
-- Functions having a 'MonadAsync' constraint work concurrently when used with
-- appropriate stream type combinator. Please be careful to not use 'parallely'
-- with infinite streams.
--
-- Deconstruction and folds accept a 'SerialT' type instead of a polymorphic
-- type to ensure that streams always have a concrete monomorphic type by
-- default, reducing type errors. In case you want to use any other type of
-- stream you can use one of the type combinators provided in the "Streamly"
-- module to convert the stream type.
module Streamly.Prelude
(
-- * Construction
-- ** Primitives
-- | Primitives to construct a stream from pure values or monadic actions.
-- All other stream construction and generation combinators described later
-- can be expressed in terms of these primitives. However, the special
-- versions provided in this module can be much more efficient in most
-- cases. Users can create custom combinators using these primitives.
nil
, cons
, (.:)
, consM
, (|:)
-- ** From Values
-- | Generate a monadic stream from a seed value or values.
, yield
, yieldM
, repeat
, repeatM
, replicate
, replicateM
-- Note: Using enumeration functions e.g. 'Prelude.enumFromThen' turns out
-- to be slightly faster than the idioms like @[from, then..]@.
--
-- ** Enumeration
-- | We can use the 'Enum' type class to enumerate a type producing a list
-- and then convert it to a stream:
--
-- @
-- 'fromList' $ 'Prelude.enumFromThen' from then
-- @
--
-- However, this is not particularly efficient.
-- The 'Enumerable' type class provides corresponding functions that
-- generate a stream instead of a list, efficiently.
, Enumerable (..)
, enumerate
, enumerateTo
-- ** From Generators
-- | Generate a monadic stream from a seed value and a generator function.
, unfoldr
, unfoldrM
, unfold
, iterate
, iterateM
, fromIndices
, fromIndicesM
-- ** From Containers
-- | Convert an input structure, container or source into a stream. All of
-- these can be expressed in terms of primitives.
, fromList
, fromListM
, fromFoldable
, fromFoldableM
-- * Elimination
-- ** Deconstruction
-- | It is easy to express all the folds in terms of the 'uncons' primitive,
-- however the specific implementations provided later are generally more
-- efficient.
--
, uncons
, tail
, init
-- ** Folding
-- | In imperative terms a fold can be considered as a loop over the stream
-- that reduces the stream to a single value.
-- Left and right folds use a fold function @f@ and an identity element @z@
-- (@zero@) to recursively deconstruct a structure and then combine and reduce
-- the values or transform and reconstruct a new container.
--
-- In general, a right fold is suitable for transforming and reconstructing a
-- right associated structure (e.g. cons lists and streamly streams) and a left
-- fold is suitable for reducing a right associated structure. The behavior of
-- right and left folds are described in detail in the individual fold's
-- documentation. To illustrate the two folds for cons lists:
--
-- > foldr :: (a -> b -> b) -> b -> [a] -> b
-- > foldr f z [] = z
-- > foldr f z (x:xs) = x `f` foldr f z xs
-- >
-- > foldl :: (b -> a -> b) -> b -> [a] -> b
-- > foldl f z [] = z
-- > foldl f z (x:xs) = foldl f (z `f` x) xs
--
-- @foldr@ is conceptually equivalent to:
--
-- > foldr f z [] = z
-- > foldr f z [x] = f x z
-- > foldr f z xs = foldr f (foldr f z (tail xs)) [head xs]
--
-- @foldl@ is conceptually equivalent to:
--
-- > foldl f z [] = z
-- > foldl f z [x] = f z x
-- > foldl f z xs = foldl f (foldl f z (init xs)) [last xs]
--
-- Left and right folds are duals of each other.
--
-- @
-- foldr f z xs = foldl (flip f) z (reverse xs)
-- foldl f z xs = foldr (flip f) z (reverse xs)
-- @
--
-- More generally:
--
-- @
-- foldr f z xs = foldl g id xs z where g k x = k . f x
-- foldl f z xs = foldr g id xs z where g x k = k . flip f x
-- @
--
-- As a general rule, foldr cannot have state and foldl cannot have control.
-- NOTE: Folds are inherently serial as each step needs to use the result of
-- the previous step. However, it is possible to fold parts of the stream in
-- parallel and then combine the results using a monoid.
-- ** Right Folds
-- $rightfolds
, foldrM
, foldr
-- ** Left Folds
-- $leftfolds
, foldl'
, foldl1'
, foldlM'
-- ** Composable Left Folds
-- $runningfolds
, fold
-- ** Full Folds
-- | Folds that are guaranteed to evaluate the whole stream.
-- -- ** To Summary (Full Folds)
-- -- | Folds that summarize the stream to a single value.
, drain
, last
, length
, sum
, product
-- -- ** To Summary (Maybe) (Full Folds)
-- -- | Folds that summarize a non-empty stream to a 'Just' value and return
-- 'Nothing' for an empty stream.
, maximumBy
, maximum
, minimumBy
, minimum
, the
-- ** Lazy Folds
--
-- | Folds that generate a lazy structure. Note that the generated
-- structure may not be lazy if the underlying monad is strict.
-- -- ** To Containers (Full Folds)
-- -- | Convert or divert a stream into an output structure, container or
-- sink.
, toList
-- ** Partial Folds
-- | Folds that may terminate before evaluating the whole stream. These
-- folds strictly evaluate the stream until the result is determined.
-- -- ** To Elements (Partial Folds)
, drainN
, drainWhile
-- -- | Folds that extract selected elements of a stream or their properties.
, (!!)
, head
, findM
, find
, lookup
, findIndex
, elemIndex
-- -- ** To Boolean (Partial Folds)
-- -- | Folds that summarize the stream to a boolean value.
, null
, elem
, notElem
, all
, any
, and
, or
-- ** Multi-Stream folds
, eqBy
, cmpBy
, isPrefixOf
, isSubsequenceOf
-- trimming sequences
, stripPrefix
-- * Transformation
-- ** Mapping
-- | In imperative terms a map operation can be considered as a loop over
-- the stream that transforms the stream into another stream by performing
-- an operation on each element of the stream.
--
-- 'map' is the least powerful transformation operation with strictest
-- guarantees. A map, (1) is a stateless loop which means that no state is
-- allowed to be carried from one iteration to another, therefore,
-- operations on different elements are guaranteed to not affect each
-- other, (2) is a strictly one-to-one transformation of stream elements
-- which means it guarantees that no elements can be added or removed from
-- the stream, it can merely transform them.
, map
, sequence
, mapM
-- ** Special Maps
, mapM_
, trace
, tap
-- ** Scanning
--
-- | A scan is more powerful than map. While a 'map' is a stateless loop, a
-- @scan@ is a stateful loop which means that a state can be shared across
-- all the loop iterations, therefore, future iterations can be impacted by
-- the state changes made by the past iterations. A scan yields the state
-- of the loop after each iteration. Like a map, a @postscan@ or @prescan@
-- does not add or remove elements in the stream, it just transforms them.
-- However, a @scan@ adds one extra element to the stream.
--
-- A left associative scan, also known as a prefix sum, can be thought of
-- as a stream transformation consisting of left folds of all prefixes of a
-- stream. Another way of thinking about it is that it streams all the
-- intermediate values of the accumulator while applying a left fold on the
-- input stream. A right associative scan, on the other hand, can be
-- thought of as a stream consisting of right folds of all the suffixes of
-- a stream.
--
-- The following equations hold for lists:
--
-- > scanl f z xs == map (foldl f z) $ inits xs
-- > scanr f z xs == map (foldr f z) $ tails xs
--
-- @
-- > scanl (+) 0 [1,2,3,4]
-- 0 = 0
-- 0 + 1 = 1
-- 0 + 1 + 2 = 3
-- 0 + 1 + 2 + 3 = 6
-- 0 + 1 + 2 + 3 + 4 = 10
--
-- > scanr (+) 0 [1,2,3,4]
-- 1 + 2 + 3 + 4 + 0 = 10
-- 2 + 3 + 4 + 0 = 9
-- 3 + 4 + 0 = 7
-- 4 + 0 = 4
-- 0 = 0
-- @
--
-- Left and right scans are duals:
--
-- > scanr f z xs == reverse $ scanl (flip f) z (reverse xs)
-- > scanl f z xs == reverse $ scanr (flip f) z (reverse xs)
--
-- A scan is a stateful map i.e. a combination of map and fold:
--
-- > map f xs = tail $ scanl (\_ x -> f x) z xs
-- > map f xs = reverse $ head $ scanr (\_ x -> f x) z xs
--
-- > foldl f z xs = last $ scanl f z xs
-- > foldr f z xs = head $ scanr f z xs
-- ** Left scans
, scanl'
, scanlM'
, postscanl'
, postscanlM'
, scanl1'
, scanl1M'
-- ** Scan Using Fold
, scan
, postscan
-- ** Filtering
-- | Remove some elements from the stream based on a predicate. In
-- imperative terms a filter over a stream corresponds to a loop with a
-- @continue@ clause for the cases when the predicate fails.
, filter
, filterM
-- ** Mapping Filters
-- | Mapping along with filtering
, mapMaybe
, mapMaybeM
-- ** Deleting Elements
-- | Deleting elements is a special case of de-interleaving streams.
, deleteBy
, uniq
-- ** Inserting Elements
-- | Inserting elements is a special case of interleaving/merging streams.
, insertBy
, intersperseM
, intersperse
-- ** Indexing
-- | Indexing can be considered as a special type of zipping where we zip a
-- stream with an index stream.
, indexed
, indexedR
-- ** Reordering Elements
, reverse
-- ** Trimming
-- | Take or remove elements from one or both ends of a stream.
, take
, takeWhile
, takeWhileM
, drop
, dropWhile
, dropWhileM
-- -- ** Breaking
, chunksOf
, intervalsOf
-- ** Searching
-- | Finding the presence or location of an element, a sequence of elements
-- or another stream within a stream.
-- -- ** Searching Elements
, findIndices
, elemIndices
-- ** Splitting
-- | In general we can express splitting in terms of parser combinators.
-- These are some common use functions for convenience and efficiency.
-- While parsers can fail these functions are designed to transform a
-- stream without failure with a predefined behavior for all cases.
--
-- In general, there are three possible ways of combining stream segments
-- with a separator. The separator could be prefixed to each segment,
-- suffixed to each segment, or it could be infixed between segments.
-- 'intersperse' and 'intercalate' operations are examples of infixed
-- combining whereas 'unlines' is an example of suffixed combining. When we
-- split a stream with separators we can split in three different ways,
-- each being an opposite of the three ways of combining.
--
-- Splitting may keep the separator or drop it. Depending on how we split,
-- the separator may be kept attached to the stream segments in prefix or
-- suffix position or as a separate element in infix position. Combinators
-- like 'splitOn' that use @On@ in their names drop the separator and
-- combinators that use 'With' in their names keep the separator. When a
-- segment is missing it is considered as empty, therefore, we never
-- encounter an error in parsing.
-- -- ** Splitting By Elements
, splitOn
, splitOnSuffix
, splitWithSuffix
, wordsBy -- strip, compact and split
-- ** Grouping
-- | Splitting a stream by combining multiple contiguous elements into
-- groups using some criterion.
, groups
, groupsBy
, groupsByRolling
-- * Combining Streams
-- | New streams can be constructed by appending, merging or zipping
-- existing streams.
-- ** Appending
-- | Streams form a 'Semigroup' and a 'Monoid' under the append
-- operation. Appending can be considered as a generalization of the `cons`
-- operation to consing a stream to a stream.
--
-- @
--
-- -------Stream m a------|-------Stream m a------|=>----Stream m a---
--
-- @
--
-- @
-- >> S.toList $ S.fromList [1,2] \<> S.fromList [3,4]
-- [1,2,3,4]
-- >> S.toList $ fold $ [S.fromList [1,2], S.fromList [3,4]]
-- [1,2,3,4]
-- @
-- ** Merging
-- | Streams form a commutative semigroup under the merge
-- operation. Merging can be considered as a generalization of inserting an
-- element in a stream to interleaving a stream with another stream.
--
-- @
--
-- -------Stream m a------|
-- |=>----Stream m a---
-- -------Stream m a------|
-- @
--
-- , merge
, mergeBy
, mergeByM
, mergeAsyncBy
, mergeAsyncByM
-- ** Zipping
-- |
-- @
--
-- -------Stream m a------|
-- |=>----Stream m c---
-- -------Stream m b------|
-- @
--
, zipWith
, zipWithM
, zipAsyncWith
, zipAsyncWithM
{-
-- ** Folding Containers of Streams
-- | These are variants of standard 'Foldable' fold functions that use a
-- polymorphic stream sum operation (e.g. 'async' or 'wSerial') to fold a
-- finite container of streams. Note that these are just special cases of
-- the more general 'concatMapWith' operation.
--
, foldMapWith
, forEachWith
, foldWith
-}
-- ** Folding Streams of Streams
-- | Stream operations like map and filter represent loop processing in
-- imperative programming terms. Similarly, the imperative concept of
-- nested loops are represented by streams of streams. The 'concatMap'
-- operation represents nested looping.
-- A 'concatMap' operation loops over the input stream and then for each
-- element of the input stream generates another stream and then loops over
-- that inner stream as well producing effects and generating a single
-- output stream.
-- The 'Monad' instances of different stream types provide a more
-- convenient way of writing nested loops. Note that the monad bind
-- operation is just @flip concatMap@.
--
-- One dimension loops are just a special case of nested loops. For
-- example, 'concatMap' can degenerate to a simple map operation:
--
-- > map f m = S.concatMap (\x -> S.yield (f x)) m
--
-- Similarly, 'concatMap' can perform filtering by mapping an element to a
-- 'nil' stream:
--
-- > filter p m = S.concatMap (\x -> if p x then S.yield x else S.nil) m
--
, concatMapWith
, concatMap
, concatMapM
, concatUnfold
-- * Exceptions
, before
, after
, bracket
, onException
, finally
, handle
-- * Deprecated
, once
, each
, scanx
, foldx
, foldxM
, foldr1
, runStream
, runN
, runWhile
, fromHandle
, toHandle
)
where
import Prelude
hiding (filter, drop, dropWhile, take, takeWhile, zipWith, foldr,
foldl, map, mapM, mapM_, sequence, all, any, sum, product, elem,
notElem, maximum, minimum, head, last, tail, length, null,
reverse, iterate, init, and, or, lookup, foldr1, (!!),
scanl, scanl1, repeat, replicate, concatMap, span)
import Streamly.Internal.Prelude
-- $rightfolds
--
-- Let's take a closer look at the @foldr@ definition for lists, as given
-- earlier:
--
-- @
-- foldr f z (x:xs) = x \`f` foldr f z xs
-- @
--
-- @foldr@ invokes the fold step function @f@ as @f x (foldr f z xs)@. At each
-- invocation of @f@, @foldr@ gives us the next element in the input container
-- @x@ and a recursive expression @foldr f z xs@ representing the yet unbuilt
-- (lazy thunk) part of the output.
--
-- When @f x xs@ is lazy in @xs@ it can consume the input one element at a time
-- in FIFO order to build a lazy output expression. For example,
--
-- > f x remaining = show x : remaining
--
-- @take 2 $ foldr f [] (1:2:undefined)@ would consume the input lazily on
-- demand, consuming only first two elements and resulting in ["1", "2"]. @f@
-- can terminate recursion by not evaluating the @remaining@ part:
--
-- > f 2 remaining = show 2 : []
-- > f x remaining = show x : remaining
--
-- @f@ would terminate recursion whenever it sees element @2@ in the input.
-- Therefore, @take 2 $ foldr f [] (1:2:undefined)@ would work without any
-- problem.
--
-- On the other hand, if @f a b@ is strict in @b@ it would end up consuming the
-- whole input right away and expanding the recursive expression @b@ (i.e.
-- @foldr f z xs@) fully before it yields an output expression, resulting in
-- the following /right associated expression/:
--
-- @
-- foldr f z xs == x1 \`f` (x2 \`f` ...(xn \`f` z))
-- @
--
-- For example,
--
-- > f x remaining = x + remaining
--
-- With this definition, @foldr f 0 [1..1000]@, would recurse completely until
-- it reaches the terminating case @... `f` (1000 `f` 0)@, and then
-- start reducing the whole expression from right to left, therefore, consuming
-- the input elements in LIFO order. Thus, such an evaluation would require
-- memory proportional to the size of input. Try out @foldr (+) 0 (map (\\x ->
-- trace (show x) x) [1..10])@.
--
-- Notice the order of the arguments to the step function @f a b@. It follows
-- the order of @a@ and @b@ in the right associative recursive expression
-- generated by expanding @a \`f` b@.
--
-- A right fold is a pull fold, the step function is the puller, it can pull
-- more data from the input container by using its second argument in the
-- output expression or terminate pulling by not using it. As a corollary:
--
-- 1. a step function which is lazy in its second argument (usually functions
-- or constructors that build a lazy structure e.g. @(:)@) can pull lazily on
-- demand.
-- 2. a step function strict in its second argument (usually reducers e.g.
-- (+)) would end up pulling all of its input and buffer it in memory before
-- potentially reducing it.
--
-- A right fold is suitable for lazy reconstructions e.g. transformation,
-- mapping, filtering of /right associated input structures/ (e.g. cons lists).
-- Whereas a left fold is suitable for reductions (e.g. summing a stream of
-- numbers) of right associated structures. Note that these roles will reverse
-- for left associated structures (e.g. snoc lists). Most of our observations
-- here assume right associated structures, lists being the canonical example.
--
-- 1. A lazy FIFO style pull using a right fold allows pulling a potentially
-- /infinite/ input stream lazily, perform transformations on it, and
-- reconstruct a new structure without having to buffer the whole structure. In
-- contrast, a left fold would buffer the entire structure before the
-- reconstructed structure can be consumed.
-- 2. Even if buffering the entire input structure is ok, we need to keep in
-- mind that a right fold reconstructs structures in a FIFO style, whereas a
-- left fold reconstructs in a LIFO style, thereby reversing the order of
-- elements..
-- 3. A right fold has termination control and therefore can terminate early
-- without going through the entire input, a left fold cannot terminate
-- without consuming all of its input. For example, a right fold
-- implementation of 'or' can terminate as soon as it finds the first 'True'
-- element, whereas a left fold would necessarily go through the entire input
-- irrespective of that.
-- 4. Reduction (e.g. using (+) on a stream of numbers) using a right fold
-- occurs in a LIFO style, which means that the entire input gets buffered
-- before reduction starts. Whereas with a strict left fold reductions occur
-- incrementally in FIFO style. Therefore, a strict left fold is more suitable
-- for reductions.
--
-- $leftfolds
--
-- Note that the observations below about the behavior of a left fold assume
-- that we are working on a right associated structure like cons lists and
-- streamly streams. If we are working on a left associated structure (e.g.
-- snoc lists) the roles of right and left folds would reverse.
--
-- Let's take a closer look at the @foldl@ definition for lists given above:
--
-- @
-- foldl f z (x:xs) = foldl f (z \`f` x) xs
-- @
--
-- @foldl@ calls itself recursively, in each call it invokes @f@ as @f z x@
-- providing it with the result accumulated till now @z@ (the state) and the
-- next element from the input container. First call to @f@ is supplied with
-- the initial value of the accumulator @z@ and each subsequent call uses the
-- output of the previous call to @f z x@.
--
-- >> foldl' (+) 0 [1,2,3]
-- > 6
--
-- The recursive call at the head of the output expression is bound to be
-- evaluated until recursion terminates, therefore, a left fold always
-- /consumes the whole input container/. The following would result in an
-- error, even though the fold is not using the values at all:
--
-- >> foldl' (\_ _ -> 0) 0 (1:undefined)
-- > *** Exception: Prelude.undefined
--
-- As @foldl@ recurses, it builds the left associated expression shown below.
-- Notice, the order of the arguments to the step function @f b a@. It follows
-- the left associative recursive expression generated by expanding @b \`f` a@.
--
-- @
-- foldl f z xs == (((z \`f` x1) \`f` x2) ...) \`f` xn
-- @
--
--
-- The strict left fold @foldl'@ forces the reduction of its argument @z \`f`
-- x@ before using it, therefore it never builds the whole expression in
-- memory. Thus, @z \`f` x1@ would get reduced to @z1@ and then @z1 \`f` x2@
-- would get reduced to @z2@ and so on, incrementally reducing the expression
-- from left to right as it recurses, consuming the input in FIFO order. Try
-- out @foldl' (+) 0 (map (\\x -> trace (show x) x) [1..10])@ to see how it
-- works. For example:
--
-- @
-- > S.foldl' (+) 0 $ S.fromList [1,2,3,4]
-- 10
-- @
--
-- @
-- 0 + 1 = 1
-- 1 + 2 = 3
-- 3 + 3 = 6
-- 6 + 4 = 10
-- @
--
-- However, @foldl'@ evaluates the accumulator only to WHNF. It may further
-- help if the step function uses a strict data structure as accumulator to
-- improve performance and to keep the expression fully reduced at all times
-- during the fold.
--
-- A left fold can also build a new structure instead of reducing one if a
-- constructor is used as a fold step. However, it may not be very useful
-- because it will consume the whole input and construct the new structure in
-- memory before we can consume it. Thus the whole structure gets buffered in
-- memory. When the list constructor is used it would build a new list in
-- reverse (LIFO) order:
--
-- @
-- > S.foldl' (flip (:)) [] $ S.fromList [1,2,3,4]
-- [4,3,2,1]
-- @
--
-- A left fold is a push fold. The producer pushes its contents to the step
-- function of the fold. The step function therefore has no control to stop the
-- input, it can only discard it if it does not need it. We can also consider a
-- left fold as a state machine where the state is store in the accumulator,
-- the state can be modified based on new inputs that are pushed to the fold.
--
-- In general, a strict left fold is a reducing fold, whereas a right fold is a
-- constructing fold. A strict left fold reduces in a FIFO order whereas it
-- constructs in a LIFO order, and vice-versa for the right fold. See the
-- documentation of 'foldrM' for a discussion on where a left or right fold is
-- suitable.
--
-- To perform a left fold lazily without having to consume all the input one
-- can use @scanl@ to stream the intermediate results of the fold and consume
-- the resulting stream lazily.
--
-- $runningfolds
--
-- "Streamly.Data.Fold" module defines composable left folds which can be combined
-- together in many interesting ways. Those folds can be run using 'fold'.
-- The following two ways of folding are equivalent in functionality and
-- performance,
--
-- >>> S.fold FL.sum (S.enumerateFromTo 1 100)
-- 5050
-- >>> S.sum (S.enumerateFromTo 1 100)
-- 5050
--
-- However, left folds cannot terminate early even if it does not need to
-- consume more input to determine the result. Therefore, the performance is
-- equivalent only for full folds like 'sum' and 'length'. For partial folds
-- like 'head' or 'any' the the folds defined in this module may be much more
-- efficient because they are implemented as right folds that terminate as soon
-- as we get the result. Note that when a full fold is composed with a partial
-- fold in parallel the performance is not impacted as we anyway have to
-- consume the whole stream due to the full fold.
--
-- >>> S.head (1 `S.cons` undefined)
-- Just 1
-- >>> S.fold FL.head (1 `S.cons` undefined)
-- *** Exception: Prelude.undefined
--
-- However, we can wrap the fold in a scan to convert it into a lazy stream of
-- fold steps. We can then terminate the stream whenever we want. For example,
--
-- >>> S.toList $ S.take 1 $ S.scan FL.head (1 `S.cons` undefined)
-- [Nothing]
--
-- The following example extracts the input stream up to a point where the
-- running average of elements is no more than 10:
--
-- @
-- > S.toList
-- $ S.map (fromJust . fst)
-- $ S.takeWhile (\\(_,x) -> x <= 10)
-- $ S.postscan ((,) \<$> FL.last \<*> avg) (S.enumerateFromTo 1.0 100.0)
-- @
-- @
-- [1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0,9.0,10.0,11.0,12.0,13.0,14.0,15.0,16.0,17.0,18.0,19.0]
-- @