module Sequence:sig
..end
The elements are computed on demand, possibly repeating work if they are demanded multiple times. A sequence can be built by unfolding from some initial state, which will in practice often be other containers.
Most functions constructing a sequence will not immediately compute any elements of the sequence. These functions will always return in O(1), but traversing the resulting sequence may be more expensive. The most they will do immediately is generate a new internal state and a new step function.
Functions that transform existing sequences sometimes have to reconstruct some suffix
of the input sequence, even if it is unmodified. For example, calling drop 1
will
return a sequence with a slightly larger state and whose elements all cost slightly
more to traverse. Because this is sometimes undesirable (for example, applying drop
1
n times will cost O(n) per element traversed in the result), there are also more
eager versions of many functions (whose names are suffixed with _eagerly
) that do
more work up front. A function has the _eagerly
suffix iff it matches both of these
conditions:
It might consume an element from an input t
before returning.
It only returns a t
(not paired with something else, not wrapped in an option
,
etc.). If it returns anything other than a t
and it has at least one t
input,
it's probably demanding elements from the input t
anyway.
Only *_exn
functions can raise exceptions, except if the function underlying the
sequence (the f
passed to unfold
) raises, in which case the exception will
cascade.
type +'a
t
type'a
sequence ='a t
include Container.S1
include Monad.S
val empty : 'a t
empty
is a sequence with no elements.val next : 'a t -> ('a * 'a t) option
next
returns the next element of a sequence and the next tail if the sequence is not
finished. It is the most primitive way to walk over a sequence.module Step:sig
..end
Step
describes the next step of the sequence construction.
val unfold_step : init:'s -> f:('s -> ('a, 's) Step.t) -> 'a t
unfold_step ~init ~f
constructs a sequence by giving an initial state init
and a
function f
explaining how to continue the next step from a given state.val unfold : init:'s -> f:('s -> ('a * 's) option) -> 'a t
unfold ~init f
is a simplified version of unfold_step
that does not allow
Skip
.val unfold_with : 'a t ->
init:'s -> f:('s -> 'a -> ('b, 's) Step.t) -> 'b t
unfold_with t ~init ~f
folds a state through the sequence t
to create a new
sequenceval unfold_with_and_finish : 'a t ->
init:'s_a ->
running_step:('s_a -> 'a -> ('b, 's_a) Step.t) ->
inner_finished:('s_a -> 's_b) ->
finishing_step:('s_b -> ('b, 's_b) Step.t) -> 'b t
unfold_with_and_finish t ~init ~running_step ~inner_finished ~finishing_step
folds a
state through the sequence t
to create a new sequence. The new sequence can
continue once t
has finished.val nth : 'a t -> int -> 'a option
val nth_exn : 'a t -> int -> 'a
val mapi : 'a t -> f:(int -> 'a -> 'b) -> 'b t
val filteri : 'a t -> f:(int -> 'a -> bool) -> 'a t
val filter : 'a t -> f:('a -> bool) -> 'a t
val merge : 'a t -> 'a t -> cmp:('a -> 'a -> int) -> 'a t
merge t1 t2 ~cmp
produces the interleaved elements of t1
and t2
, always picking
the smallest of the two available elements from t1
and t2
, according to cmp
.
When the two available elements are equal, the one from t1
is preferred.val hd : 'a t -> 'a option
val hd_exn : 'a t -> 'a
val tl : 'a t -> 'a t option
tl t
and tl_eagerly_exn t
immediately evaluate the first element of t
and return
the unevaluated tail.val tl_eagerly_exn : 'a t -> 'a t
val findi : 'a t -> f:(int -> 'a -> bool) -> (int * 'a) option
val find_exn : 'a t -> f:('a -> bool) -> 'a
find_exn t ~f
returns the first element of t
that satisfies f
. It raises if
there is no such element.val append : 'a t -> 'a t -> 'a t
append t1 t2
first produces the elements of t1
, then produces the elements of
t2
.val concat : 'a t t -> 'a t
concat tt
produces the elements of each inner sequence sequentially. If any inner
sequences are infinite, elements of subsequent inner sequences will not be reached.val concat_map : 'a t -> f:('a -> 'b t) -> 'b t
concat_map t ~f
is concat (map t ~f)
.val concat_mapi : 'a t -> f:(int -> 'a -> 'b t) -> 'b t
concat_mapi t ~f
is like concat_map, but passes the index as an argument.val interleave : 'a t t -> 'a t
interleave tt
produces each element of the inner sequences of tt
eventually, even
if any or all of the inner sequences are infinite. The elements of each inner
sequence are produced in order with respect to that inner sequence. The manner of
interleaving among the separate inner sequences is deterministic but unspecified.val zip : 'a t -> 'b t -> ('a * 'b) t
WARNING: Unlike List.zip
, this will not error out if the two input sequences are of
different lengths, because zip
may have already returned some elements by the time
this becomes apparent.
val zip_full : 'a t ->
'b t -> [ `Both of 'a * 'b | `Left of 'a | `Right of 'b ] t
zip_full
is like zip
, but if one sequence ends before the other, then it keeps
producing elements from the other sequence until it has ended as well.val iteri : 'a t -> f:(int -> 'a -> unit) -> unit
iteri
is just like iter
, but it also passes in the index of each element to
f
.val foldi : 'a t -> f:(int -> 'b -> 'a -> 'b) -> init:'b -> 'b
foldi
is just like fold
, but it also passes in the index of each element to
f
.val reduce_exn : 'a t -> f:('a -> 'a -> 'a) -> 'a
reduce_exn f [a1; ...; an]
is f (... (f (f a1 a2) a3) ...) an
. It fails on the
empty sequence.val reduce : 'a t -> f:('a -> 'a -> 'a) -> 'a option
val find_consecutive_duplicate : 'a t -> equal:('a -> 'a -> bool) -> ('a * 'a) option
find_consecutive_duplicate t ~equal
returns the first pair of consecutive elements
(a1, a2)
in t
such that equal a1 a2
. They are returned in the same order as
they appear in t
.val remove_consecutive_duplicates : 'a t -> equal:('a -> 'a -> bool) -> 'a t
val range : ?stride:int ->
?start:[ `exclusive | `inclusive ] ->
?stop:[ `exclusive | `inclusive ] -> int -> int -> int t
range ?stride ?start ?stop start_i stop_i
is the sequence of integers from start_i
to stop_i
, stepping by stride
. If stride
< 0 then we need start_i
> stop_i
for the result to be nonempty (or start_i
>= stop_i
in the case where both bounds
are inclusive).val init : int -> f:(int -> 'a) -> 'a t
init n ~f
is [(f 0); (f 1); ...; (f (n-1))]
. It is an error if n < 0
.val filter_map : 'a t -> f:('a -> 'b option) -> 'b t
filter_map t ~f
produce mapped elements of t
which are not None
.val filter_mapi : 'a t -> f:(int -> 'a -> 'b option) -> 'b t
filter_mapi
is just like filter_map
, but it also passes in the index of each
element to f
.val filter_opt : 'a option t -> 'a t
filter_opt t
produces the elements of t
which are not None
. filter_opt t
=
filter_map t ~f:ident
val sub : 'a t -> pos:int -> len:int -> 'a t
sub t ~pos ~len
is the len
-element subsequence of t
, starting at pos
. If the
sequence is shorter than pos + len
, it returns t[pos] ... t[l-1]
, where l
is
the length of the sequence.val take : 'a t -> int -> 'a t
take t n
produces the first n
elements of t
.val drop : 'a t -> int -> 'a t
drop t n
produces all elements of t
except the first n
elements. If there are
fewer than n
elements in t
, there is no error; the resulting sequence simply
produces no elements. Usually you will probably want to use drop_eagerly
because it
can be significantly cheaper.val drop_eagerly : 'a t -> int -> 'a t
drop_eagerly t n
immediately consumes the first n
elements of t
and returns the
unevaluated tail of t
.val take_while : 'a t -> f:('a -> bool) -> 'a t
take_while t ~f
produces the longest prefix of t
for which f
applied to each
element is true
.val drop_while : 'a t -> f:('a -> bool) -> 'a t
drop_while t ~f
produces the suffix of t
beginning with the first element of t
for which f
is false
. Usually you will probably want to use drop_while_option
because it can be significantly cheaper.val drop_while_option : 'a t -> f:('a -> bool) -> ('a * 'a t) option
drop_while_option t ~f
immediately consumes the elements from t
until the
predicate f
fails and returns the first element that failed along with the
unevaluated tail of t
. The first element is returned separately because the
alternatives would mean forcing the consumer to evaluate the first element again (if
the previous state of the sequence is returned) or take on extra cost for each element
(if the element is added to the final state of the sequence using shift_right
).val split_n_eagerly : 'a t -> int -> 'a t * 'a t
split_n_eagerly t n
immediately consumes the first n
elements of t
and returns
the consumed prefix, as a new stream, along with the unevaluated tail of t
.val shift_right : 'a t -> 'a -> 'a t
shift_right t a
produces a
and then produces each element of t
.val shift_right_with_list : 'a t -> 'a list -> 'a t
shift_right_with_list t l
produces the elements of l
, then produces the elements
of t
. It is better to call shift_right_with_list
with a list of size n than
shift_right
n times; the former will require O(1) work per element produced and the
later O(n) work per element produced.val shift_left : 'a t -> int -> 'a t
shift_left t n
is a synonym for drop t n
.module Infix:sig
..end
val cartesian_product : 'a t -> 'b t -> ('a * 'b) t
cartesian_product
with expensive or side-effecting functions. If the
second sequence is infinite, some values in the first sequence may not be reached.val interleaved_cartesian_product : 'a t -> 'b t -> ('a * 'b) t
interleaved_cartesian_product
with expensive or side-effecting functions.val intersperse : 'a t -> sep:'a -> 'a t
intersperse xs ~sep
produces sep
between adjacent elements of xs
.
e.g. intersperse [1;2;3] ~sep:0 = [1;0;2;0;3]
val cycle_list_exn : 'a list -> 'a t
cycle_list_exn xs
repeats the elements of xs
forever. If xs
is empty, it
raises.val repeat : 'a -> 'a t
repeat a
repeats a
forever.val singleton : 'a -> 'a t
singleton a
produces a
exactly once.val delayed_fold : 'a t ->
init:'s -> f:('s -> 'a -> k:('s -> 'r) -> 'r) -> finish:('s -> 'r) -> 'r
delayed_fold
allows to do an on-demand fold, while maintaining a state. This
function is sufficient to implement fold_m
in any monad.
let fold_m t ~init ~f =
let open M in
delayed_fold t ~init
~f:(fun s a ~k -> f s a >>= k)
~finish:return
It is possible to exit early by not calling k
in f
. It is also possible to call
k
multiple times. This results in the rest of the sequence being folded over
multiple times, independently.
val to_list : 'a t -> 'a list
val to_list_rev : 'a t -> 'a list
to_list_rev t
returns a list of the elements of t
, in reverse order. It is faster
than to_list
.val of_list : 'a list -> 'a t
val memoize : 'a t -> 'a t
memoize t
produces each element of t
, but also memoizes them so that if you
consume the same element multiple times it is only computed once. It's a non-eager
version of force_eagerly
.val force_eagerly : 'a t -> 'a t
force_eagerly t
precomputes the sequence. It is behaviorally equivalent to of_list
(to_list t)
, but may at some point have a more efficient implementation. It's an
eager version of memoize
.val bounded_length : 'a t -> at_most:int -> [ `Greater | `Is of int ]
bounded_length ~at_most t
returns `Is len
if len = length t <= at_most
, and
otherwise returns `Greater
. Walks through only as much of the sequence as
necessary. Always returns `Greater
if at_most < 0
.val length_is_bounded_by : ?min:int -> ?max:int -> 'a t -> bool
length_is_bounded_by ~min ~max t
returns true if min <= length t
and length t <=
max
When min
or max
are not provided, the check for that bound is omitted. Walks
through only as much of the sequence as necessary.Generator
is a monadic interface to generate sequences in a direct style, similar to
Python's generators.
Here are some examples:
open Generator
let rec traverse_list = function
| [] -> return ()
| x :: xs -> yield x >>= fun () -> traverse_list xs
let traverse_option = function
| None -> return ()
| Some x -> yield x
let traverse_array arr =
let n = Array.length arr in
let rec loop i =
if i >= n then return () else yield arr.(i) >>= fun () -> loop (i + 1)
in
loop 0
let rec traverse_bst = function
| Node.Empty -> return ()
| Node.Branch (left, value, right) ->
traverse_bst left >>= fun () ->
yield value >>= fun () ->
traverse_bst right
let sequence_of_list x = Generator.run (traverse_list x)
let sequence_of_option x = Generator.run (traverse_option x)
let sequence_of_array x = Generator.run (traverse_array x)
let sequence_of_bst x = Generator.run (traverse_bst x)
module Generator:sig
..end
val sexp_of_t : ('a -> Sexplib.Sexp.t) -> 'a t -> Sexplib.Sexp.t
val compare : ('a -> 'a -> int) -> 'a t -> 'a t -> int
empty
is a sequence with no elements.next
returns the next element of a sequence and the next tail if the sequence is not
finished. It is the most primitive way to walk over a sequence.Step
describes the next step of the sequence construction. Done
indicates the
sequence is finished. Skip
indicates the sequence continues with another state
without producing the next element yet. Yield
outputs an element and introduces a
new state.
Modifying 's
doesn't violate any *internal* invariants, but it may violate some
undocumented expectations. For example, one might expect that producing an element
from the same point in the sequence would always give the same value, but if the state
can mutate, that is not so.
unfold_step ~init ~f
constructs a sequence by giving an initial state init
and a
function f
explaining how to continue the next step from a given state.
unfold ~init f
is a simplified version of unfold_step
that does not allow
Skip
.
unfold_with t ~init ~f
folds a state through the sequence t
to create a new
sequence
unfold_with_and_finish t ~init ~running_step ~inner_finished ~finishing_step
folds a
state through the sequence t
to create a new sequence. The new sequence can
continue once t
has finished.
return the nth element
merge t1 t2 ~cmp
produces the interleaved elements of t1
and t2
, always picking
the smallest of the two available elements from t1
and t2
, according to cmp
.
When the two available elements are equal, the one from t1
is preferred.
tl t
and tl_eagerly_exn t
immediately evaluate the first element of t
and return
the unevaluated tail.
find_exn t ~f
returns the first element of t
that satisfies f
. It raises if
there is no such element.
append t1 t2
first produces the elements of t1
, then produces the elements of
t2
.
concat tt
produces the elements of each inner sequence sequentially. If any inner
sequences are infinite, elements of subsequent inner sequences will not be reached.
concat_map t ~f
is concat (map t ~f)
.
concat_mapi t ~f
is like concat_map, but passes the index as an argument.
interleave tt
produces each element of the inner sequences of tt
eventually, even
if any or all of the inner sequences are infinite. The elements of each inner
sequence are produced in order with respect to that inner sequence. The manner of
interleaving among the separate inner sequences is deterministic but unspecified.
Transforms a pair of sequences into a sequence of pairs. The length of the returned
sequence is the length of the shorter input. The remaining elements of the longer
input are discarded.
WARNING: Unlike List.zip
, this will not error out if the two input sequences are of
different lengths, because zip
may have already returned some elements by the time
this becomes apparent.
zip_full
is like zip
, but if one sequence ends before the other, then it keeps
producing elements from the other sequence until it has ended as well.
iteri
is just like iter
, but it also passes in the index of each element to
f
.
foldi
is just like fold
, but it also passes in the index of each element to
f
.
reduce_exn f [a1; ...; an]
is f (... (f (f a1 a2) a3) ...) an
. It fails on the
empty sequence.
find_consecutive_duplicate t ~equal
returns the first pair of consecutive elements
(a1, a2)
in t
such that equal a1 a2
. They are returned in the same order as
they appear in t
.
The same sequence with consecutive duplicates removed. The relative order of the
other elements is unaffected.
range ?stride ?start ?stop start_i stop_i
is the sequence of integers from start_i
to stop_i
, stepping by stride
. If stride
< 0 then we need start_i
> stop_i
for the result to be nonempty (or start_i
>= stop_i
in the case where both bounds
are inclusive).
default is 1
default is `inclusive
default is `exclusive
init n ~f
is [(f 0); (f 1); ...; (f (n-1))]
. It is an error if n < 0
.
filter_map t ~f
produce mapped elements of t
which are not None
.
filter_mapi
is just like filter_map
, but it also passes in the index of each
element to f
.
filter_opt t
produces the elements of t
which are not None
. filter_opt t
=
filter_map t ~f:ident
sub t ~pos ~len
is the len
-element subsequence of t
, starting at pos
. If the
sequence is shorter than pos + len
, it returns t[pos] ... t[l-1]
, where l
is
the length of the sequence.
take t n
produces the first n
elements of t
.
drop t n
produces all elements of t
except the first n
elements. If there are
fewer than n
elements in t
, there is no error; the resulting sequence simply
produces no elements. Usually you will probably want to use drop_eagerly
because it
can be significantly cheaper.
drop_eagerly t n
immediately consumes the first n
elements of t
and returns the
unevaluated tail of t
.
take_while t ~f
produces the longest prefix of t
for which f
applied to each
element is true
.
drop_while t ~f
produces the suffix of t
beginning with the first element of t
for which f
is false
. Usually you will probably want to use drop_while_option
because it can be significantly cheaper.
drop_while_option t ~f
immediately consumes the elements from t
until the
predicate f
fails and returns the first element that failed along with the
unevaluated tail of t
. The first element is returned separately because the
alternatives would mean forcing the consumer to evaluate the first element again (if
the previous state of the sequence is returned) or take on extra cost for each element
(if the element is added to the final state of the sequence using shift_right
).
split_n_eagerly t n
immediately consumes the first n
elements of t
and returns
the consumed prefix, as a new stream, along with the unevaluated tail of t
.
shift_right t a
produces a
and then produces each element of t
.
shift_right_with_list t l
produces the elements of l
, then produces the elements
of t
. It is better to call shift_right_with_list
with a list of size n than
shift_right
n times; the former will require O(1) work per element produced and the
later O(n) work per element produced.
shift_left t n
is a synonym for drop t n
.
Returns a sequence with all possible pairs. The stepper function of the second
sequence passed as argument may be applied to the same state multiple times, so be
careful using cartesian_product
with expensive or side-effecting functions. If the
second sequence is infinite, some values in the first sequence may not be reached.
Returns a sequence that eventually reaches every possible pair of elements of the
inputs, even if either or both are infinite. The step function of both inputs may be
applied to the same state repeatedly, so be careful using
interleaved_cartesian_product
with expensive or side-effecting functions.
intersperse xs ~sep
produces sep
between adjacent elements of xs
.
e.g. intersperse [1;2;3] ~sep:0 = [1;0;2;0;3]
cycle_list_exn xs
repeats the elements of xs
forever. If xs
is empty, it
raises.
repeat a
repeats a
forever.
singleton a
produces a
exactly once.
delayed_fold
allows to do an on-demand fold, while maintaining a state. This
function is sufficient to implement fold_m
in any monad.
let fold_m t ~init ~f =
let open M in
delayed_fold t ~init
~f:(fun s a ~k -> f s a >>= k)
~finish:return
It is possible to exit early by not calling k
in f
. It is also possible to call
k
multiple times. This results in the rest of the sequence being folded over
multiple times, independently.
k
stands for "continuation"
to_list_rev t
returns a list of the elements of t
, in reverse order. It is faster
than to_list
.
memoize t
produces each element of t
, but also memoizes them so that if you
consume the same element multiple times it is only computed once. It's a non-eager
version of force_eagerly
.
force_eagerly t
precomputes the sequence. It is behaviorally equivalent to of_list
(to_list t)
, but may at some point have a more efficient implementation. It's an
eager version of memoize
.
bounded_length ~at_most t
returns `Is len
if len = length t <= at_most
, and
otherwise returns `Greater
. Walks through only as much of the sequence as
necessary. Always returns `Greater
if at_most < 0
.
length_is_bounded_by ~min ~max t
returns true if min <= length t
and length t <=
max
When min
or max
are not provided, the check for that bound is omitted. Walks
through only as much of the sequence as necessary.
Generator
is a monadic interface to generate sequences in a direct style, similar to
Python's generators.
Here are some examples:
open Generator
let rec traverse_list = function
| [] -> return ()
| x :: xs -> yield x >>= fun () -> traverse_list xs
let traverse_option = function
| None -> return ()
| Some x -> yield x
let traverse_array arr =
let n = Array.length arr in
let rec loop i =
if i >= n then return () else yield arr.(i) >>= fun () -> loop (i + 1)
in
loop 0
let rec traverse_bst = function
| Node.Empty -> return ()
| Node.Branch (left, value, right) ->
traverse_bst left >>= fun () ->
yield value >>= fun () ->
traverse_bst right
let sequence_of_list x = Generator.run (traverse_list x)
let sequence_of_option x = Generator.run (traverse_option x)
let sequence_of_array x = Generator.run (traverse_array x)
let sequence_of_bst x = Generator.run (traverse_bst x)