Module Base.Sequence

A sequence of elements that can be produced one at a time, on demand, normally with no sharing.

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:

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
val sexp_of_t : ('a -> Sexplib0.Sexp.t) -> 'a t -> Sexplib0.Sexp.t
type 'a sequence := 'a t
val equal : ('a -> 'a -> bool) -> 'a t -> 'a t -> bool
val compare : ('a -> 'a -> int) -> 'a t -> 'a t -> int
include Indexed_container.S1 with type 'a t := 'a t
include Container.S1 with type 'a t := 'a t
val mem : 'a t -> 'a -> equal:('a -> 'a -> bool) -> bool

Checks whether the provided element is there, using equal.

val length : 'a t -> int
val is_empty : 'a t -> bool
val iter : 'a t -> f:('a -> unit) -> unit
val fold : 'a t -> init:'acc -> f:('acc -> 'a -> 'acc) -> 'acc

fold t ~init ~f returns f (... f (f (f init e1) e2) e3 ...) en, where e1..en are the elements of t

val fold_result : 'a t -> init:'acc -> f:('acc -> 'a -> ('acc, 'e) Result.t) -> ('acc, 'e) Result.t

fold_result t ~init ~f is a short-circuiting version of fold that runs in the Result monad. If f returns an Error _, that value is returned without any additional invocations of f.

val fold_until : 'a t -> init:'acc -> f:('acc -> 'a -> ('acc, 'final) Container.Continue_or_stop.t) -> finish:('acc -> 'final) -> 'final

fold_until t ~init ~f ~finish is a short-circuiting version of fold. If f returns Stop _ the computation ceases and results in that value. If f returns Continue _, the fold will proceed. If f never returns Stop _, the final result is computed by finish.

Example:

type maybe_negative =
  | Found_negative of int
  | All_nonnegative of { sum : int }

(** [first_neg_or_sum list] returns the first negative number in [list], if any,
    otherwise returns the sum of the list. *)
let first_neg_or_sum =
  List.fold_until ~init:0
    ~f:(fun sum x ->
      if x < 0
      then Stop (Found_negative x)
      else Continue (sum + x))
    ~finish:(fun sum -> All_nonnegative { sum })
;;

let x = first_neg_or_sum [1; 2; 3; 4; 5]
val x : maybe_negative = All_nonnegative {sum = 15}

let y = first_neg_or_sum [1; 2; -3; 4; 5]
val y : maybe_negative = Found_negative -3
val exists : 'a t -> f:('a -> bool) -> bool

Returns true if and only if there exists an element for which the provided function evaluates to true. This is a short-circuiting operation.

val for_all : 'a t -> f:('a -> bool) -> bool

Returns true if and only if the provided function evaluates to true for all elements. This is a short-circuiting operation.

val count : 'a t -> f:('a -> bool) -> int

Returns the number of elements for which the provided function evaluates to true.

val sum : (module Container.Summable with type t = 'sum) -> 'a t -> f:('a -> 'sum) -> 'sum

Returns the sum of f i for all i in the container.

val find : 'a t -> f:('a -> bool) -> 'a option

Returns as an option the first element for which f evaluates to true.

val find_map : 'a t -> f:('a -> 'b option) -> 'b option

Returns the first evaluation of f that returns Some, and returns None if there is no such element.

val to_list : 'a t -> 'a list
val to_array : 'a t -> 'a array
val min_elt : 'a t -> compare:('a -> 'a -> int) -> 'a option

Returns a minimum (resp maximum) element from the collection using the provided compare function, or None if the collection is empty. In case of a tie, the first element encountered while traversing the collection is returned. The implementation uses fold so it has the same complexity as fold.

val max_elt : 'a t -> compare:('a -> 'a -> int) -> 'a option

These are all like their equivalents in Container except that an index starting at 0 is added as the first argument to f.

val foldi : 'a t -> init:_ -> f:(int -> _ -> 'a -> _) -> _
val iteri : 'a t -> f:(int -> 'a -> unit) -> unit
val existsi : 'a t -> f:(int -> 'a -> bool) -> bool
val counti : 'a t -> f:(int -> 'a -> bool) -> int
val findi : 'a t -> f:(int -> 'a -> bool) -> (int * 'a) option
val find_mapi : 'a t -> f:(int -> 'a -> 'b option) -> 'b option
include Monad.S with type 'a t := 'a t
val (>>=) : 'a t -> ('a -> 'b t) -> 'b t

t >>= f returns a computation that sequences the computations represented by two monad elements. The resulting computation first does t to yield a value v, and then runs the computation returned by f v.

val (>>|) : 'a t -> ('a -> 'b) -> 'b t

t >>| f is t >>= (fun a -> return (f a)).

module Monad_infix : sig ... end
val bind : 'a t -> f:('a -> 'b t) -> 'b t

bind t ~f = t >>= f

val return : 'a -> 'a t

return v returns the (trivial) computation that returns v.

val map : 'a t -> f:('a -> 'b) -> 'b t

map t ~f is t >>| f.

val join : 'a t t -> 'a t

join t is t >>= (fun t' -> t').

val ignore_m : 'a t -> unit t

ignore_m t is map t ~f:(fun _ -> ()). ignore_m used to be called ignore, but we decided that was a bad name, because it shadowed the widely used Stdlib.ignore. Some monads still do let ignore = ignore_m for historical reasons.

val all : 'a t list -> 'a list t
val all_unit : unit t list -> unit t

Like all, but ensures that every monadic value in the list produces a unit value, all of which are discarded rather than being collected into a list.

module Let_syntax : sig ... end

These are convenient to have in scope when programming with a monad:

val empty : _ 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.

module Step : sig ... end

A 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.

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 sequence

val 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 t to create a new sequence (like unfold_with t ~init ~f:running_step), and then continues the new sequence by unfolding the final state (like unfold_step ~init:(inner_finished final_state) ~f:finishing_step).

val nth : 'a t -> int -> 'a option

Returns the nth element.

val nth_exn : 'a t -> int -> 'a
val folding_map : 'a t -> init:'acc -> f:('acc -> 'a -> 'acc * 'b) -> 'b t

folding_map is a version of map that threads an accumulator through calls to f.

val folding_mapi : 'a t -> init:'acc -> f:(int -> 'acc -> 'a -> 'acc * 'b) -> 'b t
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_deduped_and_sorted : 'a t -> 'a t -> compare:('a -> 'a -> int) -> 'a t

If t1 and t2 are each sorted without duplicates, merge_deduped_and_sorted t1 t2 ~compare merges t1 and t2 into a sorted sequence without duplicates. Whenever identical elements are found in both t1 and t2, the one from t1 is used and the one from t2 is discarded. The behavior is undefined if the inputs aren't sorted or contain duplicates.

val merge : 'a t -> 'a t -> compare:('a -> 'a -> int) -> 'a t
  • deprecated [since 2021-07] For identical behavior, use [Sequence.merge_deduped_and_sorted], but consider using [Sequence.merge_sorted] instead.
val merge_sorted : 'a t -> 'a t -> compare:('a -> 'a -> int) -> 'a t

If t1 and t2 are each sorted, merge_sorted t1 t2 ~compare merges t1 and t2 into a sorted sequence. Whenever identical elements are found in both t1 and t2, the one from t1 is used first. The behavior is undefined if the inputs aren't sorted.

module Merge_with_duplicates_element : sig ... end
val merge_with_duplicates : 'a t -> 'b t -> compare:('a -> 'b -> int) -> ('a, 'b) Merge_with_duplicates_element.t t

merge_with_duplicates_element t1 t2 ~compare is like merge, except that for each element it indicates which input(s) the element comes from, using Merge_with_duplicates_element.

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 evaluates the first element of t and returns the unevaluated tail.

val tl_eagerly_exn : 'a t -> 'a t
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 for_alli : 'a t -> f:(int -> 'a -> bool) -> bool

Like for_all, but passes the index as an argument.

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 round_robin : 'a t list -> 'a t

round_robin list is like interleave (of_list list), except that the manner of interleaving among the inner sequences is guaranteed to be round-robin. The input sequences may be of different lengths; an empty sequence is dropped from subsequent rounds of interleaving.

val zip : 'a t -> 'b t -> ('a * 'b) t

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.

val zip_full : 'a t -> 'b t -> [ `Left of 'a | `Both of 'a * 'b | `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 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 group : 'a t -> break:('a -> 'a -> bool) -> 'a list t

group l ~break returns a sequence of lists (i.e., groups) whose concatenation is equal to the original sequence. Each group is broken where break returns true on a pair of successive elements.

Example:

group ~break:(<>) (of_list ['M';'i';'s';'s';'i';'s';'s';'i';'p';'p';'i']) ->

of_list [['M'];['i'];['s';'s'];['i'];['s';'s'];['i'];['p';'p'];['i']]
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

The same sequence with consecutive duplicates removed. The relative order of the other elements is unaffected.

val range : ?stride:int -> ?start:[ `inclusive | `exclusive ] -> ?stop:[ `inclusive | `exclusive ] -> 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:Fn.id.

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 : 'a t -> int -> 'a list * 'a t

split_n t n immediately consumes the first n elements of t and returns the consumed prefix, as a list, along with the unevaluated tail of t.

val chunks_exn : 'a t -> int -> 'a list t

chunks_exn t n produces lists of elements of t, up to n elements at a time. The last list may contain fewer than n elements. No list contains zero elements. If n is not positive, it raises.

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 latter 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

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.

val interleaved_cartesian_product : 'a t -> 'b t -> ('a * 'b) t

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.

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.

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.

Note that delayed_fold, when targeting JavaScript, can result in stack overflow as JavaScript doesn't generally have tail call optimization.

val fold_m : bind:('acc_m -> f:('acc -> 'acc_m) -> 'acc_m) -> return:('acc -> 'acc_m) -> 'elt t -> init:'acc -> f:('acc -> 'elt -> 'acc_m) -> 'acc_m

fold_m is a monad-friendly version of fold. Supply it with the monad's return and bind, and it will chain them through the computation.

val iter_m : bind:('unit_m -> f:(unit -> 'unit_m) -> 'unit_m) -> return:(unit -> 'unit_m) -> 'elt t -> f:('elt -> 'unit_m) -> 'unit_m

iter_m is a monad-friendly version of iter. Supply it with the monad's return and bind, and it will chain them through the computation.

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 of_lazy : 'a t Lazy.t -> 'a t

of_lazy t_lazy produces a sequence that forces t_lazy the first time it needs to compute an element.

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 : _ t -> at_most:int -> [ `Is of int | `Greater ]

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 -> _ 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.

val of_seq : 'a Seq.t -> 'a t
val to_seq : 'a t -> 'a Seq.t

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
module Expert : sig ... end

The functions in Expert expose internal structure which is normally meant to be hidden. For example, at least when f is purely functional, it is not intended for client code to distinguish between