mirror of
https://github.com/sigmasternchen/gleam-community-maths
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Work on combinatorics module
This commit is contained in:
NicklasXYZ
parent
b1b5ec0692
commit
a68c906622
3 changed files with 356 additions and 162 deletions
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@ -23,7 +23,8 @@
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////
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//// ---
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////
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//// Combinatorics: A module that offers mathematical functions related to counting, arrangements, and combinations.
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//// Combinatorics: A module that offers mathematical functions related to counting, arrangements,
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//// and combinations.
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////
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//// * **Combinatorial functions**
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//// * [`combination`](#combination)
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@ -34,8 +35,17 @@
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//// * [`cartesian_product`](#cartesian_product)
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////
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import gleam/iterator
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import gleam/list
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import gleam/option
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import gleam/set
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import gleam_community/maths/conversion
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import gleam_community/maths/elementary
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pub type CombinatoricsMode {
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WithRepetitions
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WithoutRepetitions
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}
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/// <div style="text-align: right;">
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/// <a href="https://github.com/gleam-community/maths/issues">
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@ -43,15 +53,46 @@ import gleam/set
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/// </a>
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/// </div>
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///
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/// A combinatorial function for computing the number of a $$k$$-combinations of $$n$$ elements:
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/// A combinatorial function for computing the number of $$k$$-combinations of $$n$$ elements:
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///
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/// **Without Repetitions:**
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///
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/// \\[
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/// C(n, k) = \binom{n}{k} = \frac{n!}{k! (n-k)!}
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/// \\]
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/// Also known as "$$n$$ choose $$k$$" or the binomial coefficient.
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///
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/// The implementation uses the effecient iterative multiplicative formula for the computation.
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/// **With Repetitions:**
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///
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/// \\[
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/// C^*(n, k) = \binom{n + k - 1}{k} = \frac{(n + k - 1)!}{k! (n - 1)!}
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/// \\]
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/// Also known as the "stars and bars" problem in combinatorics.
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///
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/// The implementation uses an efficient iterative multiplicative formula for computing the result.
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///
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/// <details>
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/// <summary>Details</summary>
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/// A $$k$$-combination is a sequence of $$k$$ elements selected from $$n$$ elements where
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/// the order of selection does not matter. For example, consider selecting 2 elements from a list
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/// of 3 elements: `["A", "B", "C"]`:
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///
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/// - For $$k$$-combinations (without repetitions), where order does not matter, the possible
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/// selections are:
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/// - ["A", "B"]
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/// - ["A", "C"]
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/// - ["B", "C"]
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///
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/// - For $$k$$-combinations (with repetitions), where order does not matter but elements can
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/// repeat, the possible selections are:
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/// - ["A", "A"], ["A", "B"], ["A", "C"]
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/// - ["B", "B"], ["B", "C"], ["C", "C"]
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///
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/// - On the contrary, for $$k$$-permutations, the order matters, so the possible selections are:
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/// - `["A", "B"], ["B", "A"]`
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/// - `["A", "C"], ["C", "A"]`
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/// - `["B", "C"], ["C", "B"]`
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/// </details>
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/// <details>
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/// <summary>Example:</summary>
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///
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@ -82,37 +123,53 @@ import gleam/set
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/// </a>
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/// </div>
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///
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pub fn combination(n: Int, k: Int) -> Result(Int, String) {
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case n < 0 {
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True ->
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"Invalid input argument: n < 0. Valid input is n > 0."
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|> Error
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pub fn combination(
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n: Int,
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k: Int,
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mode: option.Option(CombinatoricsMode),
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) -> Result(Int, String) {
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case mode {
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option.Some(WithRepetitions) -> combination_with_repetitions(n, k)
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_ -> combination_without_repetitions(n, k)
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}
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}
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fn combination_without_repetitions(n: Int, k: Int) -> Result(Int, String) {
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case n < 0 || k < 0 || k > n {
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True -> "Invalid input: Ensure n >= 0, k >= 0, and k <= n." |> Error
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False ->
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case k < 0 || k > n {
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True ->
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0
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|> Ok
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False ->
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case k == 0 || k == n {
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True ->
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1
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|> Ok
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False -> {
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let min = case k < n - k {
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True -> k
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False -> n - k
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}
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list.range(1, min)
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|> list.fold(1, fn(acc: Int, x: Int) -> Int {
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acc * { n + 1 - x } / x
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})
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|> Ok
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}
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case k == 0 || k == n {
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True -> 1 |> Ok
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False -> {
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let min = case k < n - k {
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True -> k
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False -> n - k
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}
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list.range(1, min)
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|> list.fold(1, fn(acc: Int, x: Int) -> Int {
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acc * { n + 1 - x } / x
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})
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|> Ok
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}
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}
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}
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}
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fn combination_with_repetitions(n: Int, k: Int) -> Result(Int, String) {
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case n < 0 {
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True -> "Invalid input argument: n < 0. Valid input is n >= 0 " |> Error
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False -> {
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case k < 0 {
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True -> "Invalid input argument: k < 0. Valid input is k >= 0 " |> Error
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False -> {
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{ n + k - 1 }
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|> combination_without_repetitions(k)
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}
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}
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}
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}
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}
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/// <div style="text-align: right;">
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/// <a href="https://github.com/gleam-community/maths/issues">
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/// <small>Spot a typo? Open an issue!</small>
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@ -160,7 +217,7 @@ pub fn combination(n: Int, k: Int) -> Result(Int, String) {
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pub fn factorial(n) -> Result(Int, String) {
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case n < 0 {
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True ->
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"Invalid input argument: n < 0. Valid input is n > 0."
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"Invalid input argument: n < 0. Valid input is n >= 0."
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|> Error
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False ->
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case n {
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@ -172,7 +229,7 @@ pub fn factorial(n) -> Result(Int, String) {
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|> Ok
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_ ->
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list.range(1, n)
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|> list.fold(1, fn(acc: Int, x: Int) { acc * x })
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|> list.fold(1, fn(acc: Int, x: Int) -> Int { acc * x })
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|> Ok
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}
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}
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@ -184,13 +241,46 @@ pub fn factorial(n) -> Result(Int, String) {
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/// </a>
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/// </div>
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///
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/// A combinatorial function for computing the number of $$k$$-permuations (without repetitions)
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/// of $$n$$ elements:
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/// A combinatorial function for computing the number of $$k$$-permutations (without and without
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/// repetitions) of $$n$$ elements.
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///
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/// **Without** repetitions:
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///
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/// \\[
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/// P(n, k) = \frac{n!}{(n - k)!}
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/// P(n, k) = \binom{n}{k} \cdot k! = \frac{n!}{(n - k)!}
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/// \\]
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///
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/// **With** repetitions:
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///
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/// \\[
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/// P^*(n, k) = n^k
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/// \\]
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///
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/// <details>
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/// <summary>Details</summary>
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/// A $$k$$-permutation (without repetitions) is a sequence of $$k$$ elements selected from $$n$$
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/// elements where the order of selection matters. For example, consider selecting 2 elements from
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/// a list of 3 elements: `["A", "B", "C"]`:
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///
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/// - For $$k$$-permutations (without repetitions), the order matters, so the possible selections
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/// are:
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/// - `["A", "B"], ["B", "A"]`
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/// - `["A", "C"], ["C", "A"]`
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/// - `["B", "C"], ["C", "B"]`
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///
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/// - For $$k$$-permutations (with repetitions), the order also matters, but we have repeated
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/// selections:
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/// - ["A", "A"], ["A", "B"], ["A", "C"]
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/// - ["B", "A"], ["B", "B"], ["B", "C"]
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/// - ["C", "A"], ["C", "B"], ["C", "C"]
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///
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/// - On the contrary, for $$k$$-combinations, where order does not matter, the possible selections
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/// are:
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/// - ["A", "B"]
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/// - ["A", "C"]
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/// - ["B", "C"]
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/// </details>
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///
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/// <details>
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/// <summary>Example:</summary>
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///
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@ -221,10 +311,21 @@ pub fn factorial(n) -> Result(Int, String) {
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/// </a>
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/// </div>
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///
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pub fn permutation(n: Int, k: Int) -> Result(Int, String) {
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pub fn permutation(
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n: Int,
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k: Int,
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mode: option.Option(CombinatoricsMode),
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) -> Result(Int, String) {
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case mode {
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option.Some(WithRepetitions) -> permutation_with_repetitions(n, k)
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_ -> permutation_without_repetitions(n, k)
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}
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}
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fn permutation_without_repetitions(n: Int, k: Int) -> Result(Int, String) {
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case n < 0 {
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True ->
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"Invalid input argument: n < 0. Valid input is n > 0."
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"Invalid input argument: n < 0. Valid input is n >= 0."
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|> Error
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False ->
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case k < 0 || k > n {
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@ -232,21 +333,39 @@ pub fn permutation(n: Int, k: Int) -> Result(Int, String) {
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0
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|> Ok
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False ->
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case k == n {
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True ->
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1
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case k == 0 {
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True -> 1 |> Ok
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False ->
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list.range(0, k - 1)
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|> list.fold(1, fn(acc: Int, x: Int) -> Int { acc * { n - x } })
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|> Ok
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False -> {
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let assert Ok(v1) = factorial(n)
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let assert Ok(v2) = factorial(n - k)
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v1 / v2
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|> Ok
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}
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}
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}
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}
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}
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fn permutation_with_repetitions(n: Int, k: Int) -> Result(Int, String) {
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case n < 0 {
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True ->
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"Invalid input argument: n < 0. Valid input is n >= 0."
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|> Error
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False ->
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case k < 0 {
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True ->
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"Invalid input argument: k < 0. Valid input is k >= 0."
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|> Error
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False -> {
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let n_float = conversion.int_to_float(n)
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let k_float = conversion.int_to_float(k)
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let assert Ok(result) = elementary.power(n_float, k_float)
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result
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|> conversion.float_to_int()
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|> Ok
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}
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}
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}
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}
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/// <div style="text-align: right;">
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/// <a href="https://github.com/gleam-community/maths/issues">
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/// <small>Spot a typo? Open an issue!</small>
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@ -276,35 +395,38 @@ pub fn permutation(n: Int, k: Int) -> Result(Int, String) {
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/// </a>
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/// </div>
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///
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pub fn list_combination(arr: List(a), k: Int) -> Result(List(List(a)), String) {
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pub fn list_combination(
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arr: List(a),
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k: Int,
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) -> Result(iterator.Iterator(List(a)), String) {
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case k < 0 {
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True ->
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"Invalid input argument: k < 0. Valid input is k > 0."
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|> Error
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True -> Error("Invalid input argument: k < 0. Valid input is k >= 0.")
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False -> {
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case k > list.length(arr) {
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True ->
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"Invalid input argument: k > length(arr). Valid input is 0 < k <= length(arr)."
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|> Error
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False -> {
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do_list_combination(arr, k, [])
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|> Ok
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}
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Error(
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"Invalid input argument: k > length(arr). Valid input is 0 <= k <= length(arr).",
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)
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False -> Ok(do_list_combination(iterator.from_list(arr), k, []))
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}
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}
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}
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}
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fn do_list_combination(arr: List(a), k: Int, prefix: List(a)) -> List(List(a)) {
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fn do_list_combination(
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arr: iterator.Iterator(a),
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k: Int,
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prefix: List(a),
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) -> iterator.Iterator(List(a)) {
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case k {
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0 -> [list.reverse(prefix)]
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0 -> iterator.single(list.reverse(prefix))
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_ ->
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case arr {
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[] -> []
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[x, ..xs] -> {
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case arr |> iterator.step {
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iterator.Done -> iterator.empty()
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iterator.Next(x, xs) -> {
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let with_x = do_list_combination(xs, k - 1, [x, ..prefix])
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let without_x = do_list_combination(xs, k, prefix)
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list.append(with_x, without_x)
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iterator.concat([with_x, without_x])
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}
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}
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}
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@ -324,12 +446,6 @@ fn do_list_combination(arr: List(a), k: Int, prefix: List(a)) -> List(List(a)) {
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/// means lists with repeated elements return the same
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/// number of permutations as ones without.
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///
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/// N.B. The output of this function is a list of size
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/// factorial in the size of the input list. Caution is
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/// advised on input lists longer than ~11 elements, which
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/// may cause the VM to use unholy amounts of memory for
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/// the output.
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///
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/// <details>
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/// <summary>Example:</summary>
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///
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@ -362,16 +478,35 @@ fn do_list_combination(arr: List(a), k: Int, prefix: List(a)) -> List(List(a)) {
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/// </a>
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/// </div>
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///
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pub fn list_permutation(arr: List(a)) -> List(List(a)) {
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pub fn list_permutation(arr: List(a)) -> iterator.Iterator(List(a)) {
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case arr {
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[] -> [[]]
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_ -> {
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use x <- list.flat_map(arr)
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// `x` is drawn from the list `arr` above,
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// so Ok(...) can be safely asserted as the result of `list.pop` below
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let assert Ok(#(_, remaining)) = list.pop(arr, fn(y) { x == y })
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list.map(list_permutation(remaining), fn(perm) { [x, ..perm] })
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}
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[] -> iterator.single([])
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_ ->
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iterator.from_list(arr)
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// Iterate over each element in the list 'arr' to generate permutations for each possible
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// starting element 'x'.
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|> iterator.flat_map(fn(x) {
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// For each element 'x', we remove it from the list. This will gives us the remaining list
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// that contains all elements except 'x'.
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let remaining = remove_first(arr, x)
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// Recursively call 'list_permutation' on the remaining list to generate all permutations
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// of the smaller list.
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let permutations = list_permutation(remaining)
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// For each permutation generated by the recursive call, we prepend the element 'x' back to
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// the front of the permutation.
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iterator.map(permutations, fn(permutation) { [x, ..permutation] })
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})
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}
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}
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fn remove_first(list: List(a), x: a) -> List(a) {
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case list {
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[] -> []
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[head, ..tail] ->
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case head == x {
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True -> tail
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False -> [head, ..remove_first(tail, x)]
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}
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}
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}
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|
|
|
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|
|
@ -54,8 +54,10 @@
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//// * [`tau`](#tau)
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//// * [`e`](#e)
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////
|
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////
|
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|
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import gleam/int
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import gleam/list
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import gleam/option
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|
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/// <div style="text-align: right;">
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|
|
|
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|
|
@ -1,4 +1,6 @@
|
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import gleam/iterator
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import gleam/list
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import gleam/option
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import gleam/set
|
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import gleam_community/maths/combinatorics
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import gleeunit/should
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|
|
@ -28,20 +30,24 @@ pub fn int_factorial_test() {
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pub fn int_combination_test() {
|
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// Invalid input gives an error
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// Error on: n = -1 < 0
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combinatorics.combination(-1, 1)
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combinatorics.combination(
|
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-1,
|
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1,
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option.Some(combinatorics.WithoutRepetitions),
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)
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|> should.be_error()
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|
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// Valid input returns a result
|
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combinatorics.combination(4, 0)
|
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combinatorics.combination(4, 0, option.Some(combinatorics.WithoutRepetitions))
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|> should.equal(Ok(1))
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combinatorics.combination(4, 4)
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combinatorics.combination(4, 4, option.Some(combinatorics.WithoutRepetitions))
|
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|> should.equal(Ok(1))
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combinatorics.combination(4, 2)
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combinatorics.combination(4, 2, option.Some(combinatorics.WithoutRepetitions))
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|> should.equal(Ok(6))
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combinatorics.combination(7, 5)
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combinatorics.combination(7, 5, option.Some(combinatorics.WithoutRepetitions))
|
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|> should.equal(Ok(21))
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// NOTE: Tests with the 'combination' function that produce values that
|
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// exceed precision of the JavaScript 'Number' primitive will result in
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|
|
@ -51,18 +57,28 @@ pub fn int_combination_test() {
|
|||
pub fn math_permutation_test() {
|
||||
// Invalid input gives an error
|
||||
// Error on: n = -1 < 0
|
||||
combinatorics.permutation(-1, 1)
|
||||
combinatorics.permutation(
|
||||
-1,
|
||||
1,
|
||||
option.Some(combinatorics.WithoutRepetitions),
|
||||
)
|
||||
|> should.be_error()
|
||||
|
||||
// Valid input returns a result
|
||||
combinatorics.permutation(4, 0)
|
||||
combinatorics.permutation(4, 0, option.Some(combinatorics.WithoutRepetitions))
|
||||
|> should.equal(Ok(1))
|
||||
|
||||
combinatorics.permutation(4, 4)
|
||||
|> should.equal(Ok(1))
|
||||
combinatorics.permutation(4, 4, option.Some(combinatorics.WithoutRepetitions))
|
||||
|> should.equal(Ok(24))
|
||||
|
||||
combinatorics.permutation(4, 2)
|
||||
combinatorics.permutation(4, 2, option.Some(combinatorics.WithoutRepetitions))
|
||||
|> should.equal(Ok(12))
|
||||
|
||||
combinatorics.permutation(6, 2, option.Some(combinatorics.WithoutRepetitions))
|
||||
|> should.equal(Ok(30))
|
||||
|
||||
combinatorics.permutation(6, 3, option.Some(combinatorics.WithoutRepetitions))
|
||||
|> should.equal(Ok(120))
|
||||
}
|
||||
|
||||
pub fn list_cartesian_product_test() {
|
||||
|
|
@ -96,91 +112,132 @@ pub fn list_cartesian_product_test() {
|
|||
set.from_list([#(1.0, 1.0), #(1.0, 2.0), #(10.0, 1.0), #(10.0, 2.0)]),
|
||||
)
|
||||
}
|
||||
// pub fn list_permutation_test() {
|
||||
// // An empty lists returns one (empty) permutation
|
||||
// []
|
||||
// |> combinatorics.list_permutation()
|
||||
// |> iterator.to_list()
|
||||
// |> should.equal([[]])
|
||||
|
||||
pub fn list_permutation_test() {
|
||||
// An empty lists returns one (empty) permutation
|
||||
[]
|
||||
|> combinatorics.list_permutation()
|
||||
|> should.equal([[]])
|
||||
// // Singleton returns one (singleton) permutation
|
||||
// // Also works regardless of type of list elements
|
||||
// ["a"]
|
||||
// |> combinatorics.list_permutation()
|
||||
// |> iterator.to_list()
|
||||
// |> should.equal([["a"]])
|
||||
|
||||
// Singleton returns one (singleton) permutation
|
||||
// Also works regardless of type of list elements
|
||||
["a"]
|
||||
|> combinatorics.list_permutation()
|
||||
|> should.equal([["a"]])
|
||||
// // Test with some arbitrary inputs
|
||||
// [1, 2]
|
||||
// |> combinatorics.list_permutation()
|
||||
// |> iterator.to_list()
|
||||
// |> set.from_list()
|
||||
// |> should.equal(set.from_list([[1, 2], [2, 1]]))
|
||||
|
||||
// Test with some arbitrary inputs
|
||||
[1, 2]
|
||||
|> combinatorics.list_permutation()
|
||||
|> set.from_list()
|
||||
|> should.equal(set.from_list([[1, 2], [2, 1]]))
|
||||
// // Test with some arbitrary inputs
|
||||
// [1, 2, 3]
|
||||
// |> combinatorics.list_permutation()
|
||||
// |> iterator.to_list()
|
||||
// |> set.from_list()
|
||||
// |> should.equal(
|
||||
// set.from_list([
|
||||
// [1, 2, 3],
|
||||
// [2, 1, 3],
|
||||
// [3, 1, 2],
|
||||
// [1, 3, 2],
|
||||
// [2, 3, 1],
|
||||
// [3, 2, 1],
|
||||
// ]),
|
||||
// )
|
||||
|
||||
// Test with some arbitrary inputs
|
||||
[1, 2, 3]
|
||||
|> combinatorics.list_permutation()
|
||||
|> set.from_list()
|
||||
|> should.equal(
|
||||
set.from_list([
|
||||
[1, 2, 3],
|
||||
[2, 1, 3],
|
||||
[3, 1, 2],
|
||||
[1, 3, 2],
|
||||
[2, 3, 1],
|
||||
[3, 2, 1],
|
||||
]),
|
||||
)
|
||||
// // Repeated elements are treated as distinct for the
|
||||
// // purpose of permutations, so two identical elements
|
||||
// // will appear "both ways round"
|
||||
// [1.0, 1.0]
|
||||
// |> combinatorics.list_permutation()
|
||||
// |> iterator.to_list()
|
||||
// |> should.equal([[1.0, 1.0], [1.0, 1.0]])
|
||||
|
||||
// Repeated elements are treated as distinct for the
|
||||
// purpose of permutations, so two identical elements
|
||||
// will appear "both ways round"
|
||||
[1.0, 1.0]
|
||||
|> combinatorics.list_permutation()
|
||||
|> should.equal([[1.0, 1.0], [1.0, 1.0]])
|
||||
// // This means lists with repeated elements return the
|
||||
// // same number of permutations as ones without
|
||||
// ["l", "e", "t", "t", "e", "r", "s"]
|
||||
// |> combinatorics.list_permutation()
|
||||
// |> iterator.to_list()
|
||||
// |> list.length()
|
||||
// |> should.equal(5040)
|
||||
|
||||
// This means lists with repeated elements return the
|
||||
// same number of permutations as ones without
|
||||
["l", "e", "t", "t", "e", "r", "s"]
|
||||
|> combinatorics.list_permutation()
|
||||
|> list.length()
|
||||
|> should.equal(5040)
|
||||
}
|
||||
// // Test that the number of generate permutations of the given input list aligns with the computed
|
||||
// // number of possible permutations (using the formula for the binomial coefficient)
|
||||
// let arr = ["a", "b", "c", "x", "y", "z"]
|
||||
// let length = list.length(arr)
|
||||
// let assert Ok(permuations) =
|
||||
// combinatorics.permutation(
|
||||
// length,
|
||||
// length,
|
||||
// option.Some(combinatorics.WithoutRepetitions),
|
||||
// )
|
||||
|
||||
pub fn list_combination_test() {
|
||||
// A negative number returns an error
|
||||
[]
|
||||
|> combinatorics.list_combination(-1)
|
||||
|> should.be_error()
|
||||
// arr
|
||||
// |> combinatorics.list_permutation()
|
||||
// |> iterator.to_list()
|
||||
// |> list.length()
|
||||
// |> should.equal(permuations)
|
||||
// }
|
||||
|
||||
// k is larger than given input list returns an error
|
||||
[1, 2]
|
||||
|> combinatorics.list_combination(3)
|
||||
|> should.be_error()
|
||||
// An empty lists returns an empty list
|
||||
[]
|
||||
|> combinatorics.list_combination(0)
|
||||
|> should.equal(Ok([[]]))
|
||||
// pub fn list_combination_test() {
|
||||
// // Invaldi input: A negative number returns an error
|
||||
// []
|
||||
// |> combinatorics.list_combination(-1)
|
||||
// |> should.be_error()
|
||||
|
||||
// Test with some arbitrary inputs
|
||||
[1, 2]
|
||||
|> combinatorics.list_combination(1)
|
||||
|> should.equal(Ok([[1], [2]]))
|
||||
// // Invalid input: k is larger than given input list, so it returns an error
|
||||
// [1, 2]
|
||||
// |> combinatorics.list_combination(3)
|
||||
// |> should.be_error()
|
||||
|
||||
// Test with some arbitrary inputs
|
||||
[1, 2]
|
||||
|> combinatorics.list_combination(2)
|
||||
|> should.equal(Ok([[1, 2]]))
|
||||
// // Valid input: An empty lists returns an empty list
|
||||
// let assert Ok(combinations) =
|
||||
// []
|
||||
// |> combinatorics.list_combination(0)
|
||||
|
||||
// Test with some arbitrary inputs
|
||||
let assert Ok(result) = combinatorics.list_combination([1, 2, 3, 4], 2)
|
||||
result
|
||||
|> set.from_list()
|
||||
|> should.equal(
|
||||
set.from_list([[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]]),
|
||||
)
|
||||
// combinations
|
||||
// |> iterator.to_list()
|
||||
// |> should.equal([[]])
|
||||
|
||||
// Test with some arbitrary inputs
|
||||
let assert Ok(result) = combinatorics.list_combination([1, 2, 3, 4], 3)
|
||||
result
|
||||
|> set.from_list()
|
||||
|> should.equal(set.from_list([[1, 2, 3], [1, 2, 4], [1, 3, 4], [2, 3, 4]]))
|
||||
}
|
||||
// // Test with some arbitrary but valid inputs
|
||||
// let assert Ok(combinations) =
|
||||
// [1, 2]
|
||||
// |> combinatorics.list_combination(1)
|
||||
|
||||
// combinations
|
||||
// |> iterator.to_list()
|
||||
// |> should.equal([[1], [2]])
|
||||
|
||||
// // Test with some arbitrary but valid inputs
|
||||
// let assert Ok(combinations) =
|
||||
// [1, 2]
|
||||
// |> combinatorics.list_combination(2)
|
||||
|
||||
// combinations
|
||||
// |> iterator.to_list()
|
||||
// |> should.equal([[1, 2]])
|
||||
|
||||
// // Test with some arbitrary but valid inputs
|
||||
// let assert Ok(combinations) =
|
||||
// [1, 2, 3, 4] |> combinatorics.list_combination(2)
|
||||
|
||||
// combinations
|
||||
// |> iterator.to_list()
|
||||
// |> set.from_list()
|
||||
// |> should.equal(
|
||||
// set.from_list([[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]]),
|
||||
// )
|
||||
|
||||
// // Test with some arbitrary but valid inputs
|
||||
// let assert Ok(combinations) =
|
||||
// [1, 2, 3, 4] |> combinatorics.list_combination(3)
|
||||
|
||||
// combinations
|
||||
// |> iterator.to_list()
|
||||
// |> set.from_list()
|
||||
// |> should.equal(set.from_list([[1, 2, 3], [1, 2, 4], [1, 3, 4], [2, 3, 4]]))
|
||||
// }
|
||||
|
|
|
|||
Loading…
Reference in a new issue