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 }

  }

  @CollectionSize.Require(SEVERAL)
  @MapFeature.Require(SUPPORTS_REMOVE)
  /*
   * SetMultimap.asMap essentially returns a Map<K, Set<V>>; we just can't declare it that way.
   * Thus, calls like asMap().values().remove(someSet) are safe because they are comparing a set to
   * a collection of other sets.
   */
  @SuppressWarnings("CollectionUndefinedEquality")

/**
 * 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.
 *

    }
  }

  /**
   * Moves a file from one path to another. This method can rename a file and/or move it to a
   * different directory. In either case {@code to} must be the target path for the file itself; not
   * just the new name for the file or the path to the new parent directory.
   *
   * <p><b>{@link java.nio.file.Path} equivalent:</b> {@link java.nio.file.Files#move}.
   *
   * @param from the source file

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(

        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;

          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' )

			keys = append(keys, foundKey)
			plSize += len(foundKey)
		} else {
			keys = append(keys, []byte(newKey))
			plSize += len(newKey)
			found = true
		}
	}
	// If not found, just return.
	if !found {
		return false
	}

	// Reserialize...
	x.serialize(plSize, keys, vals)
	return true
}

// remove will remove one or more keys.
// Returns true if any key was found.