Lists and Patterns
This chapter will focus on two common elements of programming in OCaml: lists and pattern matching. Both of these were discussed in Chapter 1, A Guided Tour, but we'll go into more depth here, presenting the two topics together and using one to help illustrate the other.
open Base;; [1;2;3];; >- : int list = [1; 2; 3]
1 :: (2 :: (3 :: )) ;; >- : int list = [1; 2; 3] 1 :: 2 :: 3 ::  ;; >- : int list = [1; 2; 3]
As you can see, the
:: operator is right-associative, which means that we can build up lists without parentheses. The empty list
 is used to terminate a list. Note that the empty list is polymorphic, meaning it can be used with elements of any type, as you can see here:
let empty = ;; >val empty : 'a list =  3 :: empty;; >- : int list =  "three" :: empty;; >- : string list = ["three"]
The way in which the
:: operator attaches elements to the front of a list reflects the fact that OCaml's lists are in fact singly linked lists. The figure below is a rough graphical representation of how the list
1 :: 2 :: 3 ::  is laid out as a data structure. The final arrow (from the box containing
3) points to the empty list.
:: essentially adds a new block to the proceding picture. Such a block contains two things: a reference to the data in that list element, and a reference to the remainder of the list. This is why
:: can extend a list without modifying it; extension allocates a new list element but does not change any of the existing ones, as you can see:
let l = 1 :: 2 :: 3 :: ;; >val l : int list = [1; 2; 3] let m = 0 :: l;; >val m : int list = [0; 1; 2; 3] l;; >- : int list = [1; 2; 3]
Using Patterns to Extract Data from a List
let rec sum l = match l with |  -> 0 | hd :: tl -> hd + sum tl ;; >val sum : int list -> int = <fun> sum [1;2;3];; >- : int = 6 sum ;; >- : int = 0
This code follows the convention of using
hd to represent the first element (or head) of the list, and
tl to represent the remainder (or tail).
match statement in
sum is really doing two things: first, it's acting as a case-analysis tool, breaking down the possibilities into a pattern-indexed list of cases. Second, it lets you name substructures within the data structure being matched. In this case, the variables
tl are bound by the pattern that defines the second case of the match statement. Variables that are bound in this way can be used in the expression to the right of the arrow for the pattern in question.
The fact that
match statements can be used to bind new variables can be a source of confusion. To see how, imagine we wanted to write a function that filtered out from a list all elements equal to a particular value. You might be tempted to write that code as follows, but when you do, the compiler will immediately warn you that something is wrong:
let rec drop_value l to_drop = match l with |  ->  | to_drop :: tl -> drop_value tl to_drop | hd :: tl -> hd :: drop_value tl to_drop ;; >Characters 106-114: >Warning 11: this match case is unused. >val drop_value : 'a list -> 'a -> 'a list = <fun>
Moreover, the function clearly does the wrong thing, filtering out all elements of the list rather than just those equal to the provided value, as you can see here:
drop_value [1;2;3] 2;; >- : int list = 
So, what's going on?
The key observation is that the appearance of
to_drop in the second case doesn't imply a check that the first element is equal to the value
to_drop that was passed in as an argument to
drop_value. Instead, it just causes a new variable
to_drop to be bound to whatever happens to be in the first element of the list, shadowing the earlier definition of
to_drop. The third case is unused because it is essentially the same pattern as we had in the second case.
A better way to write this code is not to use pattern matching for determining whether the first element is equal to
to_drop, but to instead use an ordinary
let rec drop_value l to_drop = match l with |  ->  | hd :: tl -> let new_tl = drop_value tl to_drop in if hd = to_drop then new_tl else hd :: new_tl ;; >val drop_value : int list -> int -> int list = <fun> drop_value [1;2;3] 2;; >- : int list = [1; 3]
If we wanted to drop a particular literal value, rather than a value that was passed in, we could do this using something like our original implementation of
let rec drop_zero l = match l with |  ->  | 0 :: tl -> drop_zero tl | hd :: tl -> hd :: drop_zero tl ;; >val drop_zero : int list -> int list = <fun> drop_zero [1;2;0;3];; >- : int list = [1; 2; 3]
Limitations (and Blessings) of Pattern Matching
The preceding example highlights an important fact about patterns, which is that they can't be used to express arbitrary conditions. Patterns can characterize the layout of a data structure and can even include literals, as in the
drop_zero example, but that's where they stop. A pattern can check if a list has two elements, but it can't check if the first two elements are equal to each other.
You can think of patterns as a specialized sublanguage that can express a limited (though still quite rich) set of conditions. The fact that the pattern language is limited turns out to be a good thing, making it possible to build better support for patterns in the compiler. In particular, both the efficiency of
match statements and the ability of the compiler to detect errors in matches depend on the constrained nature of patterns.
Naively, you might think that it would be necessary to check each case in a
match in sequence to figure out which one fires. If the cases of a match were guarded by arbitrary code, that would be the case. But OCaml is often able to generate machine code that jumps directly to the matched case based on an efficiently chosen set of runtime checks.
As an example, consider the following rather silly functions for incrementing an integer by one. The first is implemented with a
match statement, and the second with a sequence of
let plus_one_match x = match x with | 0 -> 1 | 1 -> 2 | 2 -> 3 | 3 -> 4 | 4 -> 5 | 5 -> 6 | _ -> x + 1 ;; >val plus_one_match : int -> int = <fun> let plus_one_if x = if x = 0 then 1 else if x = 1 then 2 else if x = 2 then 3 else if x = 3 then 4 else if x = 4 then 5 else if x = 5 then 6 else x + 1 ;; >val plus_one_if : int -> int = <fun>
Note the use of
_ in the above match. This is a wildcard pattern that matches any value, but without binding a variable name to the value in question.
If you benchmark these functions, you'll see that
plus_one_if is considerably slower than
plus_one_match, and the advantage gets larger as the number of cases increases. Here, we'll benchmark these functions using the
core_bench library, which can be installed by running
opam install core_bench from the command line.
#require "core_bench";; open Core_bench;; [ Bench.Test.create ~name:"plus_one_match" (fun () -> ignore (plus_one_match 10)) ; Bench.Test.create ~name:"plus_one_if" (fun () -> ignore (plus_one_if 10)) ] |> Bench.bench ;; >Estimated testing time 20s (2 benchmarks x 10s). Change using -quota SECS. >┌────────────────┬──────────┐ >│ Name │ Time/Run │ >├────────────────┼──────────┤ >│ plus_one_match │ 34.86ns │ >│ plus_one_if │ 54.89ns │ >└────────────────┴──────────┘ >- : unit = ()
Here's another, less artificial example. We can rewrite the
sum function we described earlier in the chapter using an
if statement rather than a match. We can then use the functions
tl_exn from the
List module to deconstruct the list, allowing us to implement the entire function without pattern matching:
let rec sum_if l = if List.is_empty l then 0 else List.hd_exn l + sum_if (List.tl_exn l) ;; >val sum_if : int list -> int = <fun>
Again, we can benchmark these to see the difference:
let numbers = List.range 0 1000 in [ Bench.Test.create ~name:"sum_if" (fun () -> ignore (sum_if numbers)) ; Bench.Test.create ~name:"sum" (fun () -> ignore (sum numbers)) ] |> Bench.bench ;; >Estimated testing time 20s (2 benchmarks x 10s). Change using -quota SECS. >┌────────┬──────────┐ >│ Name │ Time/Run │ >├────────┼──────────┤ >│ sum_if │ 62.00us │ >│ sum │ 17.99us │ >└────────┴──────────┘ >- : unit = ()
In this case, the
match-based implementation is many times faster than the
if-based implementation. The difference comes because we need to effectively do the same work multiple times, since each function we call has to reexamine the first element of the list to determine whether or not it's the empty cell. With a
match statement, this work happens exactly once per list element.
This is a more general phenomena: pattern matching is very efficient, and pattern matching code is usually a win over what you might write by hand.
The error-detecting capabilities of
match statements are if anything more important than their performance. We've already seen one example of OCaml's ability to find problems in a pattern match: in our broken implementation of
drop_value, OCaml warned us that the final case was redundant. There are no algorithms for determining if a predicate written in a general-purpose language is redundant, but it can be solved reliably in the context of patterns.
OCaml also checks
match statements for exhaustiveness. Consider what happens if we modify
drop_zero by deleting the handler for one of the cases. As you can see, the compiler will produce a warning that we've missed a case, along with an example of an unmatched pattern:
let rec drop_zero l = match l with |  ->  | 0 :: tl -> drop_zero tl ;; >Characters 24-78: >Warning 8: this pattern-matching is not exhaustive. >Here is an example of a case that is not matched: >1::_ >val drop_zero : int list -> 'a list = <fun>
Even for simple examples like this, exhaustiveness checks are pretty useful. But as we'll see in Chapter 6, Variants, they become yet more valuable as you get to more complicated examples, especially those involving user-defined types. In addition to catching outright errors, they act as a sort of refactoring tool, guiding you to the locations where you need to adapt your code to deal with changing types.
Using the List Module Effectively
We've so far written a fair amount of list-munging code using pattern matching and recursive functions. In real life, you're usually better off using the
List module, which is full of reusable functions that abstract out common patterns for computing with lists.
Let's work through a concrete example. We'll write a function
render_table that, given a list of column headers and a list of rows, prints them out in a well-formatted text table, as follows:
Stdio.print_endline (render_table ["language";"architect";"first release"] [ ["Lisp" ;"John McCarthy" ;"1958"] ; ["C" ;"Dennis Ritchie";"1969"] ; ["ML" ;"Robin Milner" ;"1973"] ; ["OCaml";"Xavier Leroy" ;"1996"] ; ]);; >| language | architect | first release | >|----------+----------------+---------------| >| Lisp | John McCarthy | 1958 | >| C | Dennis Ritchie | 1969 | >| ML | Robin Milner | 1973 | >| OCaml | Xavier Leroy | 1996 | >- : unit = ()
The first step is to write a function to compute the maximum width of each column of data. We can do this by converting the header and each row into a list of integer lengths, and then taking the element-wise max of those lists of lengths. Writing the code for all of this directly would be a bit of a chore, but we can do it quite concisely by making use of three functions from the
List.map ~f:String.length ["Hello"; "World!"];; >- : int list = [5; 6]
List.map2_exn ~f:Int.max [1;2;3] [3;2;1];; >- : int list = [3; 2; 3]
_exn is there because the function throws an exception if the lists are of mismatched length:
List.map2_exn ~f:Int.max [1;2;3] [3;2;1;0];; >Exception: (Invalid_argument "length mismatch in map2_exn: 3 <> 4 ").
List.fold is the most complicated of the three, taking three arguments: a list to process, an initial accumulator value, and a function for updating the accumulator.
List.fold walks over the list from left to right, updating the accumulator at each step and returning the final value of the accumulator when it's done. You can see some of this by looking at the type-signature for
List.fold;; >- : 'a list -> init:'accum -> f:('accum -> 'a -> 'accum) -> 'accum = <fun>
We can use
List.fold for something as simple as summing up a list:
List.fold ~init:0 ~f:(+) [1;2;3;4];; >- : int = 10
This example is particularly simple because the accumulator and the list elements are of the same type. But
fold is not limited to such cases. We can for example use
fold to reverse a list, in which case the accumulator is itself a list:
List.fold ~init: ~f:(fun list x -> x :: list) [1;2;3;4];; >- : int list = [4; 3; 2; 1]
Let's bring our three functions together to compute the maximum column widths:
let max_widths header rows = let lengths l = List.map ~f:String.length l in List.fold rows ~init:(lengths header) ~f:(fun acc row -> List.map2_exn ~f:Int.max acc (lengths row)) ;; >val max_widths : string list -> string list list -> int list = <fun>
List.map we define the function
lengths, which converts a list of strings to a list of integer lengths.
List.fold is then used to iterate over the rows, using
map2_exn to take the max of the accumulator with the lengths of the strings in each row of the table, with the accumulator initialized to the lengths of the header row.
Now that we know how to compute column widths, we can write the code to generate the line that separates the header from the rest of the text table. We'll do this in part by mapping
String.make over the lengths of the columns to generate a string of dashes of the appropriate length. We'll then join these sequences of dashes together using
String.concat, which concatenates a list of strings with an optional separator string, and
^, which is a pairwise string concatenation function, to add the delimiters on the outside:
let render_separator widths = let pieces = List.map widths ~f:(fun w -> String.make (w + 2) '-') in "|" ^ String.concat ~sep:"+" pieces ^ "|" ;; >val render_separator : int list -> string = <fun> render_separator [3;6;2];; >- : string = "|-----+--------+----|"
Performance of String.concat and ^
In the preceding code we’ve concatenated strings two different ways:
String.concat, which operates on lists of strings; and
^, which is a pairwise operator. You should avoid
^ for joining long numbers of strings, since it allocates a new string every time it runs. Thus, the following code
let s = "." ^ "." ^ "." ^ "." ^ "." ^ "." ^ ".";; >val s : string = "......."
will allocate strings of length 2, 3, 4, 5, 6 and 7, whereas this code
let s = String.concat [".";".";".";".";".";".";"."];; >val s : string = "......."
allocates one string of size 7, as well as a list of length 7. At these small sizes, the differences don't amount to much, but for assembling large strings, it can be a serious performance issue.
let pad s length = " " ^ s ^ String.make (length - String.length s + 1) ' ' ;; >val pad : string -> int -> string = <fun> pad "hello" 10;; >- : string = " hello "
We can render a row of data by merging together the padded strings. Again, we'll use
List.map2_exn for combining the list of data in the row with the list of widths:
let render_row row widths = let padded = List.map2_exn row widths ~f:pad in "|" ^ String.concat ~sep:"|" padded ^ "|" ;; >val render_row : string list -> int list -> string = <fun> render_row ["Hello";"World"] [10;15];; >- : string = "| Hello | World |"
Now we can bring this all together in a single function that renders the table:
let render_table header rows = let widths = max_widths header rows in String.concat ~sep:"\n" (render_row header widths :: render_separator widths :: List.map rows ~f:(fun row -> render_row row widths) ) ;; >val render_table : string list -> string list list -> string = <fun>
More Useful List Functions
The previous example we worked through touched on only three of the functions in
List. We won't cover the entire interface (for that you should look at the online docs), but a few more functions are useful enough to mention here.
Combining list elements with List.reduce
List.fold, which we described earlier, is a very general and powerful function. Sometimes, however, you want something simpler and easier to use. One such function is
List.reduce, which is essentially a specialized version of
List.fold that doesn't require an explicit starting value, and whose accumulator has to consume and produce values of the same type as the elements of the list it applies to.
Here's the type signature:
List.reduce;; >- : 'a list -> f:('a -> 'a -> 'a) -> 'a option = <fun>
reduce returns an optional result, returning
None when the input list is empty.
Now we can see
reduce in action:
List.reduce ~f:(+) [1;2;3;4;5];; >- : int option = Some 15 List.reduce ~f:(+) ;; >- : int option = None
Filtering with List.filter and List.filter_map
List.filter ~f:(fun x -> x % 2 = 0) [1;2;3;4;5];; >- : int list = [2; 4]
Note that the
mod used above is an infix operator, as described in Chapter 2, Variables And Functions.
Sometimes, you want to both transform and filter as part of the same computation. In that case,
List.filter_map is what you need. The function passed to
List.filter_map returns an optional value, and
List.filter_map drops all elements for which
None is returned.
Here's an example. The following function computes a list of file extensions from a list of files, piping the results through
List.dedup to remove duplicates. Note that this example uses
String.rsplit2 from the String module to split a string on the rightmost appearance of a given character:
let extensions filenames = List.filter_map filenames ~f:(fun fname -> match String.rsplit2 ~on:'.' fname with | None | Some ("",_) -> None | Some (_,ext) -> Some ext) |> List.dedup_and_sort ~compare:String.compare ;; >val extensions : string list -> string list = <fun> extensions ["foo.c"; "foo.ml"; "bar.ml"; "bar.mli"];; >- : string list = ["c"; "ml"; "mli"]
The preceding code is also an example of an Or pattern, which allows you to have multiple subpatterns within a larger pattern. In this case,
None | Some ("",_) is an Or pattern. As we'll see later, Or patterns can be nested anywhere within larger patterns.
Partitioning with List.partition_tf
Another useful operation that's closely related to filtering is partitioning. The function
List.partition_tf takes a list and a function for computing a Boolean condition on the list elements, and returns two lists. The
tf in the name is a mnemonic to remind the user that
true elements go to the first list and
false ones go to the second. Here's an example:
let is_ocaml_source s = match String.rsplit2 s ~on:'.' with | Some (_,("ml"|"mli")) -> true | _ -> false ;; >val is_ocaml_source : string -> bool = <fun> let (ml_files,other_files) = List.partition_tf ["foo.c"; "foo.ml"; "bar.ml"; "bar.mli"] ~f:is_ocaml_source;; >val ml_files : string list = ["foo.ml"; "bar.ml"; "bar.mli"] >val other_files : string list = ["foo.c"]
List.append [1;2;3] [4;5;6];; >- : int list = [1; 2; 3; 4; 5; 6]
@, an operator equivalent of
[1;2;3] @ [4;5;6];; >- : int list = [1; 2; 3; 4; 5; 6]
In addition, there is
List.concat, for concatenating a list of lists:
List.concat [[1;2];[3;4;5];;];; >- : int list = [1; 2; 3; 4; 5; 6]
Here's an example of using
List.concat along with
List.map to compute a recursive listing of a directory tree.
module Sys = Core.Sys module Filename = Core.Filename;; >module Sys = Core.Sys >module Filename = Core.Filename let rec ls_rec s = if Sys.is_file_exn ~follow_symlinks:true s then [s] else Sys.ls_dir s |> List.map ~f:(fun sub -> ls_rec (Filename.concat s sub)) |> List.concat ;; >val ls_rec : string -> string list = <fun>
Note that this example uses some functions from the
Filename modules from
Core for accessing the filesystem and dealing with filenames.
The preceding combination of
List.concat is common enough that there is a function
List.concat_map that combines these into one, more efficient operation:
let rec ls_rec s = if Sys.is_file_exn ~follow_symlinks:true s then [s] else Sys.ls_dir s |> List.concat_map ~f:(fun sub -> ls_rec (Filename.concat s sub)) ;; >val ls_rec : string -> string list = <fun>
The only way to compute the length of an OCaml list is to walk the list from beginning to end. As a result, computing the length of a list takes time linear in the size of the list. Here's a simple function for doing so:
let rec length = function |  -> 0 | _ :: tl -> 1 + length tl ;; >val length : 'a list -> int = <fun> length [1;2;3];; >- : int = 3
This looks simple enough, but you'll discover that this implementation runs into problems on very large lists, as we'll show in the following code:
let make_list n = List.init n ~f:(fun x -> x);; >val make_list : int -> int list = <fun> length (make_list 10);; >- : int = 10 length (make_list 10_000_000);; >Stack overflow during evaluation (looping recursion?).
The preceding example creates lists using
List.init, which takes an integer
n and a function
f and creates a list of length
n, where the data for each element is created by calling
f on the index of that element.
To understand where the error in the above example comes from, you need to learn a bit more about how function calls work. Typically, a function call needs some space to keep track of information associated with the call, such as the arguments passed to the function, or the location of the code that needs to start executing when the function call is complete. To allow for nested function calls, this information is typically organized in a stack, where a new stack frame is allocated for each nested function call, and then deallocated when the function call is complete.
And that's the problem with our call to
length: it tried to allocate 10 million stack frames, which exhausted the available stack space. Happily, there's a way around this problem. Consider the following alternative implementation:
let rec length_plus_n l n = match l with |  -> n | _ :: tl -> length_plus_n tl (n + 1) ;; >val length_plus_n : 'a list -> int -> int = <fun> let length l = length_plus_n l 0 ;; >val length : 'a list -> int = <fun> length [1;2;3;4];; >- : int = 4
This implementation depends on a helper function,
length_plus_n, that computes the length of a given list plus a given
n. In practice,
n acts as an accumulator in which the answer is built up, step by step. As a result, we can do the additions along the way rather than doing them as we unwind the nested sequence of function calls, as we did in our first implementation of
The advantage of this approach is that the recursive call in
length_plus_n is a tail call. We'll explain more precisely what it means to be a tail call shortly, but the reason it's important is that tail calls don't require the allocation of a new stack frame, due to what is called the tail-call optimization. A recursive function is said to be tail recursive if all of its recursive calls are tail calls.
length_plus_n is indeed tail recursive, and as a result,
length can take a long list as input without blowing the stack:
length (make_list 10_000_000);; >- : int = 10000000
So when is a call a tail call? Let's think about the situation where one function (the caller) invokes another (the callee). The invocation is considered a tail call when the caller doesn't do anything with the value returned by the callee except to return it. The tail-call optimization makes sense because, when a caller makes a tail call, the caller's stack frame need never be used again, and so you don't need to keep it around. Thus, instead of allocating a new stack frame for the callee, the compiler is free to reuse the caller's stack frame.
Tail recursion is important for more than just lists. Ordinary nontail recursive calls are reasonable when dealing with data structures like binary trees, where the depth of the tree is logarithmic in the size of your data. But when dealing with situations where the depth of the sequence of nested calls is on the order of the size of your data, tail recursion is usually the right approach.
Terser and Faster Patterns
Now that we know more about how lists and patterns work, let's consider how we can improve on an example from Chapter 1, Recursive List Functions: the function
destutter, which removes sequential duplicates from a list. Here's the implementation that was described earlier:
let rec destutter list = match list with |  ->  | [hd] -> [hd] | hd :: hd' :: tl -> if hd = hd' then destutter (hd' :: tl) else hd :: destutter (hd' :: tl) ;; >val destutter : int list -> int list = <fun>
We'll consider some ways of making this code more concise and more efficient.
First, let's consider efficiency. One problem with the
destutter code above is that it in some cases re-creates on the righthand side of the arrow a value that already existed on the lefthand side. Thus, the pattern
[hd] -> [hd] actually allocates a new list element, when really, it should be able to just return the list being matched. We can reduce allocation here by using an
as pattern, which allows us to declare a name for the thing matched by a pattern or subpattern. While we're at it, we'll use the
function keyword to eliminate the need for an explicit match:
let rec destutter = function |  as l -> l | [_] as l -> l | hd :: (hd' :: _ as tl) -> if hd = hd' then destutter tl else hd :: destutter tl ;; >val destutter : int list -> int list = <fun>
We can further collapse this by combining the first two cases into one, using an or pattern:
let rec destutter = function |  | [_] as l -> l | hd :: (hd' :: _ as tl) -> if hd = hd' then destutter tl else hd :: destutter tl ;; >val destutter : int list -> int list = <fun>
We can make the code slightly terser now by using a
when clause. A
when clause allows us to add an extra precondition to a pattern in the form of an arbitrary OCaml expression. In this case, we can use it to include the check on whether the first two elements are equal:
let rec destutter = function |  | [_] as l -> l | hd :: (hd' :: _ as tl) when hd = hd' -> destutter tl | hd :: tl -> hd :: destutter tl ;; >val destutter : int list -> int list = <fun>
when clauses have some downsides. As we noted earlier, the static checks associated with pattern matches rely on the fact that patterns are restricted in what they can express. Once we add the ability to add an arbitrary condition to a pattern, something is lost. In particular, the ability of the compiler to determine if a match is exhaustive, or if some case is redundant, is compromised.
Consider the following function, which takes a list of optional values, and returns the number of those values that are
Some. Because this implementation uses
when clauses, the compiler can't tell that the code is exhaustive:
let rec count_some list = match list with |  -> 0 | x :: tl when Option.is_none x -> count_some tl | x :: tl when Option.is_some x -> 1 + count_some tl ;; >Characters 28-161: >Warning 8: this pattern-matching is not exhaustive. >Here is an example of a case that is not matched: >_::_ >(However, some guarded clause may match this value.) >val count_some : 'a option list -> int = <fun>
Despite the warning, the function does work fine:
count_some [Some 3; None; Some 4];; >- : int = 2
If we add another redundant case without a
when clause, the compiler will stop complaining about exhaustiveness and won't produce a warning about the redundancy.
let rec count_some list = match list with |  -> 0 | x :: tl when Option.is_none x -> count_some tl | x :: tl when Option.is_some x -> 1 + count_some tl | x :: tl -> -1 (* unreachable *) ;; >val count_some : 'a option list -> int = <fun>
Probably a better approach is to simply drop the second
let rec count_some list = match list with |  -> 0 | x :: tl when Option.is_none x -> count_some tl | _ :: tl -> 1 + count_some tl ;; >val count_some : 'a option list -> int = <fun>
This is a little less clear, however, than the direct pattern-matching solution, where the meaning of each pattern is clearer on its own:
let rec count_some list = match list with |  -> 0 | None :: tl -> count_some tl | Some _ :: tl -> 1 + count_some tl ;; >val count_some : 'a option list -> int = <fun>
The takeaway from all of this is although
when clauses can be useful, we should prefer patterns wherever they are sufficient.
As a side note, the above implementation of
count_some is longer than necessary; even worse, it is not tail recursive. In real life, you would probably just use the
List.count function from
let count_some l = List.count ~f:Option.is_some l;; >val count_some : 'a option list -> int = <fun>