Correct exclusive reductions docs: the initial value is not emitted

Sources/AsyncAlgorithms/AsyncExclusiveReductionsSequence.swift

@@ -14,15 +14,18 @@ extension AsyncSequence {
   /// Returns an asynchronous sequence containing the accumulated results of combining the
   /// elements of the asynchronous sequence using the given closure.
   ///
-  /// This can be seen as applying the reduce function to each element and
-  /// providing the initial value followed by these results as an asynchronous sequence.
+  /// Starting from the initial value, this combines each element of the base
+  /// asynchronous sequence with the running result and emits each intermediate
+  /// result. The initial value itself is not emitted, so the first element of
+  /// the returned sequence is the result of combining the initial value with
+  /// the first element of the base sequence. If the base sequence is empty, the
+  /// returned sequence is also empty.
   ///
   /// - Parameters:
   ///   - initial: The value to use as the initial value.
   ///   - transform: A closure that combines the previously-reduced result and
   ///     the next element in the receiving asynchronous sequence, which it returns.
-  /// - Returns: An asynchronous sequence of the initial value followed by the reduced
-  ///   elements.
+  /// - Returns: An asynchronous sequence of the reduced elements.
   @available(AsyncAlgorithms 1.0, *)
   @inlinable
   public func reductions<Result>(
@@ -37,16 +40,19 @@ extension AsyncSequence {
   /// Returns an asynchronous sequence containing the accumulated results of combining the
   /// elements of the asynchronous sequence using the given closure.
   ///
-  /// This can be seen as applying the reduce function to each element and
-  /// providing the initial value followed by these results as an asynchronous sequence.
+  /// Starting from the initial value, this combines each element of the base
+  /// asynchronous sequence with the running result and emits each intermediate
+  /// result. The initial value itself is not emitted, so the first element of
+  /// the returned sequence is the result of combining the initial value with
+  /// the first element of the base sequence. If the base sequence is empty, the
+  /// returned sequence is also empty.
   ///
   /// - Parameters:
   ///   - initial: The value to use as the initial value.
   ///   - transform: A closure that combines the previously-reduced result and
   ///     the next element in the receiving asynchronous sequence, mutating the
   ///     previous result instead of returning a value.
-  /// - Returns: An asynchronous sequence of the initial value followed by the reduced
-  ///   elements.
+  /// - Returns: An asynchronous sequence of the reduced elements.
   @available(AsyncAlgorithms 1.0, *)
   @inlinable
   public func reductions<Result>(

Sources/AsyncAlgorithms/AsyncThrowingExclusiveReductionsSequence.swift

@@ -14,16 +14,19 @@ extension AsyncSequence {
   /// Returns an asynchronous sequence containing the accumulated results of combining the
   /// elements of the asynchronous sequence using the given error-throwing closure.
   ///
-  /// This can be seen as applying the reduce function to each element and
-  /// providing the initial value followed by these results as an asynchronous sequence.
+  /// Starting from the initial value, this combines each element of the base
+  /// asynchronous sequence with the running result and emits each intermediate
+  /// result. The initial value itself is not emitted, so the first element of
+  /// the returned sequence is the result of combining the initial value with
+  /// the first element of the base sequence. If the base sequence is empty, the
+  /// returned sequence is also empty.
   ///
   /// - Parameters:
   ///   - initial: The value to use as the initial value.
   ///   - transform: A closure that combines the previously reduced result and
   ///     the next element in the receiving asynchronous sequence and returns
   ///     the result. If the closure throws an error, the sequence throws.
-  /// - Returns: An asynchronous sequence of the initial value followed by the reduced
-  ///   elements.
+  /// - Returns: An asynchronous sequence of the reduced elements.
   @available(AsyncAlgorithms 1.0, *)
   @inlinable
   public func reductions<Result>(
@@ -38,17 +41,20 @@ extension AsyncSequence {
   /// Returns an asynchronous sequence containing the accumulated results of combining the
   /// elements of the asynchronous sequence using the given error-throwing closure.
   ///
-  /// This can be seen as applying the reduce function to each element and
-  /// providing the initial value followed by these results as an asynchronous sequence.
+  /// Starting from the initial value, this combines each element of the base
+  /// asynchronous sequence with the running result and emits each intermediate
+  /// result. The initial value itself is not emitted, so the first element of
+  /// the returned sequence is the result of combining the initial value with
+  /// the first element of the base sequence. If the base sequence is empty, the
+  /// returned sequence is also empty.
   ///
   /// - Parameters:
   ///   - initial: The value to use as the initial value.
   ///   - transform: A closure that combines the previously reduced result and
   ///     the next element in the receiving asynchronous sequence, mutating the
   ///     previous result instead of returning a value. If the closure throws an
   ///     error, the sequence throws.
-  /// - Returns: An asynchronous sequence of the initial value followed by the reduced
-  ///   elements.
+  /// - Returns: An asynchronous sequence of the reduced elements.
   @available(AsyncAlgorithms 1.0, *)
   @inlinable
   public func reductions<Result>(

Tests/AsyncAlgorithmsTests/TestReductions.swift

@@ -27,6 +27,34 @@ final class TestReductions: XCTestCase {
     XCTAssertNil(pastEnd)
   }
 
+  func test_reductions_empty() async {
+    let sequence = [Int]().async.reductions("") { partial, value in
+      partial + "\(value)"
+    }
+    var iterator = sequence.makeAsyncIterator()
+    var collected = [String]()
+    while let item = await iterator.next() {
+      collected.append(item)
+    }
+    XCTAssertEqual(collected, [])
+    let pastEnd = await iterator.next()
+    XCTAssertNil(pastEnd)
+  }
+
+  func test_throwing_reductions_empty() async throws {
+    let sequence = [Int]().async.reductions("") { (partial, value) throws -> String in
+      partial + "\(value)"
+    }
+    var iterator = sequence.makeAsyncIterator()
+    var collected = [String]()
+    while let item = try await iterator.next() {
+      collected.append(item)
+    }
+    XCTAssertEqual(collected, [])
+    let pastEnd = try await iterator.next()
+    XCTAssertNil(pastEnd)
+  }
+
   func test_inclusive_reductions() async {
     let sequence = [1, 2, 3, 4].async.reductions { $0 + $1 }
     var iterator = sequence.makeAsyncIterator()