# Module Sequence

    module Sequence: sig .. end

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:

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

A `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 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 the sequence `t` to create a new sequence. The new sequence can continue once `t` has finished.

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

return the nth element

    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

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 -> [ `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

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

    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

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. 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.
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.

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)