result *= i;
      }
      return BigInteger.valueOf(result);
    }

    /*
     * We want each multiplication to have both sides with approximately the same number of digits.
     * Currently, we just divide the range in half.
     */
    int mid = (n1 + n2) >>> 1;
    return oldSlowFactorial(n1, mid).multiply(oldSlowFactorial(mid, n2));
  }

  @Benchmark
  int slowFactorial(int reps) {
    int tmp = 0;

{* ../../docs_src/sub_applications/tutorial001.py hl[3, 6:8] *}

### Sub-application { #sub-application }

Then, create your sub-application, and its *path operations*.

This sub-application is just another standard FastAPI application, but this is the one that will be "mounted":

{* ../../docs_src/sub_applications/tutorial001.py hl[11, 14:16] *}

### Mount the sub-application { #mount-the-sub-application }

/**
 * A {@link Collection} whose contents will never change, and which offers a few additional
 * guarantees detailed below.
 *
 * <p><b>Warning:</b> avoid <i>direct</i> usage of {@link ImmutableCollection} as a type (just as
 * with {@link Collection} itself). Prefer subtypes such as {@link ImmutableSet} or {@link
 * ImmutableList}, which have well-defined {@link #equals} semantics, thus avoiding a common source
 * of bugs and confusion.
 *

 * except that the method parameter must accept whatever condition-like object is passed into {@code
 * callAndAssertWaits} by the test.
 *
 * @param <L> the type of the lock-like object to be used
 * @author Justin T. Sampson
 */
@NullUnmarked
public final class TestThread<L> extends Thread implements TearDown {

  private static final long DUE_DILIGENCE_MILLIS = 100;
  private static final long TIMEOUT_MILLIS = 5000;

   *     unimplement<i>able</i> (for example, consider a {@code Converter<Chicken, ChickenNugget>}),
   *     then this is not logically a {@code Converter} at all, and should just implement {@link
   *     Function}.
   */
  @ForOverride
  protected abstract A doBackward(B b);

  // API (consumer-side) methods

  /**

<img src="/img/tutorial/path-params/image01.png">

/// check

Again, just with that same Python type declaration, **FastAPI** gives you automatic, interactive documentation (integrating Swagger UI).

Notice that the path parameter is declared to be an integer.

///

          cache-dependency-glob: |
            requirements**.txt
            pyproject.toml
      - name: Install docs extras
        run: uv pip install -r requirements-docs.txt
      # Install MkDocs Material Insiders here just to put it in the cache for the rest of the steps
      - name: Install Material for MkDocs Insiders
        if: ( github.event_name != 'pull_request' || github.secret_source == 'Actions' )

)

REM max direct memory
if NOT "%FESS_DIRECT_SIZE%" == "" (
set FESS_JAVA_OPTS=%FESS_JAVA_OPTS% -XX:MaxDirectMemorySize=%FESS_DIRECT_SIZE%
)

REM set to headless, just in case
set FESS_JAVA_OPTS=%FESS_JAVA_OPTS% -Djava.awt.headless=true

REM maximum # keep-alive connections to maintain at once
set FESS_JAVA_OPTS=%FESS_JAVA_OPTS% -Dhttp.maxConnections=20

If there is no associated issue, then write "N/A".
-->

#### Special notes for your reviewer:

#### Does this PR introduce a user-facing change?
<!--
If no, just write "NONE" in the release-note block below.
If yes, a release note is required:

   * on the fields of TimeoutFuture. This field is slightly different from the fields discussed
   * there: cancel() never reads this field, only writes to it. That makes the race here completely
   * harmless, rather than just 99.99% harmless.
   */
  @LazyInit private @Nullable List<@Nullable Present<V>> values;

  @SuppressWarnings("EmptyList") // ImmutableList doesn't support nullable element types
  CollectionFuture(