// We already know that nodeU equals other.nodeU. Combined with the earlier statement,
        // and the transitive property of equality, this implies that nodeU equals nodeV.
        // If nodeU equals nodeV, condition1 == condition2, so checking condition1 is sufficient.
        return nodeV().equals(other.nodeV());
      }

    // Explicitly call `equals`; `assertEquals` might return fast
    assertTrue(foo1.equals(foo1));
    assertEquals(control("foo", 1), foo1);
    assertEquals(control("bar", 2), entry("bar", 2));
    assertFalse(control("foo", 1).equals(entry("foo", 2)));
    assertFalse(foo1.equals(control("bar", 1)));
    assertFalse(foo1.equals(new Object()));
    assertFalse(foo1.equals(null));
  }

  public void testEqualsNull() {

  public static Equivalence<Object> identity() {
    return Identity.INSTANCE;
  }

  static final class Equals extends Equivalence<Object> implements Serializable {

    static final Equals INSTANCE = new Equals();

    @Override
    protected boolean doEquivalent(Object a, Object b) {
      return a.equals(b);
    }

    @Override
    protected int doHash(Object o) {
      return o.hashCode();
    }

        // Different input
        assertFalse(item1.equals(item3));

        // Same id and input
        assertTrue(item1.equals(item4));

        // Null object
        assertFalse(item1.equals(null));

        // Different class
        assertFalse(item1.equals("word"));

        // Symmetry
        assertTrue(item1.equals(item2));
        assertTrue(item2.equals(item1));
    }

            // Test null vs null
            assertTrue(auth1.equals(new NtlmPasswordAuthenticator("domain", "user", (String) null)));

            // Test null vs empty
            assertFalse(auth1.equals(auth2));

            // Test null vs password
            assertFalse(auth1.equals(auth3));

            // Test empty vs password
            assertFalse(auth2.equals(auth3));

      assertTrue(item + " must not be Object#equals to null", !item.equals(null));
      assertTrue(
          item + " must not be Object#equals to an arbitrary object of another class",
          !item.equals(NotAnInstance.EQUAL_TO_NOTHING));
      assertTrue(item + " must be Object#equals to itself", item.equals(item));
      assertEquals(

      !"Connection".equals(fieldName, ignoreCase = true) &&
        !"Keep-Alive".equals(fieldName, ignoreCase = true) &&
        !"Proxy-Authenticate".equals(fieldName, ignoreCase = true) &&
        !"Proxy-Authorization".equals(fieldName, ignoreCase = true) &&
        !"TE".equals(fieldName, ignoreCase = true) &&
        !"Trailers".equals(fieldName, ignoreCase = true) &&

        // Act & Assert
        assertTrue(left.equals(right), "Blobs with equal content should be equal");
        assertTrue(right.equals(left), "Equality should be symmetric for identical content");
    }

    // Demonstrates observed equals() asymmetry when this is shorter than argument
    @Test
    @DisplayName("equals: this shorter but same prefix -> true (observed behavior)")

        assertTrue(item1.equals(item3));

        // Different input value
        assertFalse(item1.equals(item4));

        // Null and different class
        assertFalse(item1.equals(null));
        assertFalse(item1.equals("string"));
        assertFalse(item1.equals(new Object()));
    }

    @Test
    public void test_equals_withNewInput() {

      assertTrue(item + " must not be Object#equals to null", !item.equals(null));
      assertTrue(
          item + " must not be Object#equals to an arbitrary object of another class",
          !item.equals(NotAnInstance.EQUAL_TO_NOTHING));
      assertTrue(item + " must be Object#equals to itself", item.equals(item));
      assertEquals(