code: def test_simplify_prod():
      y, t, b, c, v, d = symbols("y, t, b, c, v, d", integer=True)
  
      _simplify = lambda e: simplify(e, doit=False)
      assert _simplify(
          Product(x * y, (x, n, m), (y, a, k))
          * Product(y, (x, n, m), (y, a, k))
      ) == Product(x * y**2, (x, n, m), (y, a, k))
      assert _simplify(
          3 * y * Product(x, (x, n, m)) * Product(x, (x, m + 1, a))
      ) == 3 * y * Product(x, (x, n, a))
      assert _simplify(
          Product(x, (x, k + 1, a)) * Product(x, (x, n, k))
      ) == Product(x, (x, n, a))
      assert _simplify(
          Product(x, (x, k + 1, a)) * Product(x + 1, (x, n, k))
      ) == Product(x, (x, k + 1, a)) * Product(x + 1, (x, n, k))
      assert _simplify(
          Product(x, (t, a, b))
          * Product(y, (t, a, b))
          * Product(x, (t, b + 1, c))
      ) == Product(x * y, (t, a, b)) * Product(x, (t, b + 1, c))
      assert _simplify(
          Product(x, (t, a, b))
          * Product(x, (t, b + 1, c))
          * Product(y, (t, a, b))
      ) == Product(x * y, (t, a, b)) * Product(x, (t, b + 1, c))
      assert _simplify(
          Product(sin(t) ** 2 + cos(t) ** 2 + 1, (t, a, b))
      ) == Product(2, (t, a, b))
      assert _simplify(
          Product(sin(t) ** 2 + cos(t) ** 2 - 1, (t, a, b))
      ) == Product(0, (t, a, b))
      assert _simplify(
          Product(
              v * Product(sin(t) ** 2 + cos(t) ** 2, (t, a, b)),
              (v, c, d),
          )
      ) == Product(v * Product(1, (t, a, b)), (v, c, d))
  
imports: from sympy.concrete.products import Product, product
  from sympy.concrete.summations import Sum
  from sympy.core.function import Derivative, Function, diff
  from sympy.core.numbers import Rational, oo, pi
  from sympy.core.singleton import S
  from sympy.core.symbol import Dummy, Symbol, symbols
  from sympy.functions.combinatorial.factorials import rf, factorial
  from sympy.functions.elementary.exponential import exp, log
  from sympy.functions.elementary.miscellaneous import sqrt
  from sympy.functions.elementary.trigonometric import cos, sin
  from sympy.functions.special.tensor_functions import KroneckerDelta
  from sympy.simplify.combsimp import combsimp
  from sympy.simplify.simplify import simplify
  from sympy.testing.pytest import raises
label_desc: Use image node_id 12 for calling a global function with example usage: test_simplify_prod() without return types