text: """Tests for useful utilities for higher level polynomial classes. """
  
  from sympy import S, Integer, sin, cos, sqrt, symbols, pi, Eq, Integral, exp
  from sympy.utilities.pytest import raises
  
  from sympy.polys.polyutils import (
      _nsort,
      _sort_gens,
      _unify_gens,
      _analyze_gens,
      _sort_factors,
      parallel_dict_from_expr,
      dict_from_expr,
  )
  
  from sympy.polys.polyerrors import (
      GeneratorsNeeded,
      PolynomialError,
  )
  
  from sympy.polys.domains import ZZ
  
  x, y, z, p, q, r, s, t, u, v, w = symbols('x,y,z,p,q,r,s,t,u,v,w')
  A, B = symbols('A,B', commutative=False)
  
  
  def test__nsort():
      # issue 6137
      r = S('''[3/2 + sqrt(-14/3 - 2*(-415/216 + 13*I/12)**(1/3) - 4/sqrt(-7/3 +
      61/(18*(-415/216 + 13*I/12)**(1/3)) + 2*(-415/216 + 13*I/12)**(1/3)) -
      61/(18*(-415/216 + 13*I/12)**(1/3)))/2 - sqrt(-7/3 + 61/(18*(-415/216
      + 13*I/12)**(1/3)) + 2*(-415/216 + 13*I/12)**(1/3))/2, 3/2 - sqrt(-7/3
      + 61/(18*(-415/216 + 13*I/12)**(1/3)) + 2*(-415/216 +
      13*I/12)**(1/3))/2 - sqrt(-14/3 - 2*(-415/216 + 13*I/12)**(1/3) -
      4/sqrt(-7/3 + 61/(18*(-415/216 + 13*I/12)**(1/3)) + 2*(-415/216 +
      13*I/12)**(1/3)) - 61/(18*(-415/216 + 13*I/12)**(1/3)))/2, 3/2 +
      sqrt(-14/3 - 2*(-415/216 + 13*I/12)**(1/3) + 4/sqrt(-7/3 +
      61/(18*(-415/216 + 13*I/12)**(1/3)) + 2*(-415/216 + 13*I/12)**(1/3)) -
      61/(18*(-415/216 + 13*I/12)**(1/3)))/2 + sqrt(-7/3 + 61/(18*(-415/216
      + 13*I/12)**(1/3)) + 2*(-415/216 + 13*I/12)**(1/3))/2, 3/2 + sqrt(-7/3
      + 61/(18*(-415/216 + 13*I/12)**(1/3)) + 2*(-415/216 +
      13*I/12)**(1/3))/2 - sqrt(-14/3 - 2*(-415/216 + 13*I/12)**(1/3) +
      4/sqrt(-7/3 + 61/(18*(-415/216 + 13*I/12)**(1/3)) + 2*(-415/216 +
      13*I/12)**(1/3)) - 61/(18*(-415/216 + 13*I/12)**(1/3)))/2]''')
      ans = [r[1], r[0], r[-1], r[-2]]
      assert _nsort(r) == ans
      assert len(_nsort(r, separated=True)[0]) == 0
      b, c, a = exp(-1000), exp(-999), exp(-1001)
      assert _nsort((b, c, a)) == [a, b, c]
  
  
  def test__sort_gens():
      assert _sort_gens([]) == ()
  
      assert _sort_gens([x]) == (x,)
      assert _sort_gens([p]) == (p,)
      assert _sort_gens([q]) == (q,)
  
      assert _sort_gens([x, p]) == (x, p)
      assert _sort_gens([p, x]) == (x, p)
      assert _sort_gens([q, p]) == (p, q)
  
      assert _sort_gens([q, p, x]) == (x, p, q)
  
      assert _sort_gens([x, p, q], wrt=x) == (x, p, q)
      assert _sort_gens([x, p, q], wrt=p) == (p, x, q)
      assert _sort_gens([x, p, q], wrt=q) == (q, x, p)
  
      assert _sort_gens([x, p, q], wrt='x') == (x, p, q)
      assert _sort_gens([x, p, q], wrt='p') == (p, x, q)
      assert _sort_gens([x, p, q], wrt='q') == (q, x, p)
  
      assert _sort_gens([x, p, q], wrt='x,q') == (x, q, p)
      assert _sort_gens([x, p, q], wrt='q,x') == (q, x, p)
      assert _sort_gens([x, p, q], wrt='p,q') == (p, q, x)
      assert _sort_gens([x, p, q], wrt='q,p') == (q, p, x)
  
      assert _sort_gens([x, p, q], wrt='x, q') == (x, q, p)
      assert _sort_gens([x, p, q], wrt='q, x') == (q, x, p)
      assert _sort_gens([x, p, q], wrt='p, q') == (p, q, x)
      assert _sort_gens([x, p, q], wrt='q, p') == (q, p, x)
  
      assert _sort_gens([x, p, q], wrt=[x, 'q']) == (x, q, p)
      assert _sort_gens([x, p, q], wrt=[q, 'x']) == (q, x, p)
      assert _sort_gens([x, p, q], wrt=[p, 'q']) == (p, q, x)
      assert _sort_gens([x, p, q], wrt=[q, 'p']) == (q, p, x)
  
      assert _sort_gens([x, p, q], wrt=['x', 'q']) == (x, q, p)
      assert _sort_gens([x, p, q], wrt=['q', 'x']) == (q, x, p)
      assert _sort_gens([x, p, q], wrt=['p', 'q']) == (p, q, x)
      assert _sort_gens([x, p, q], wrt=['q', 'p']) == (q, p, x)
  
      assert _sort_gens([x, p, q], sort='x > p > q') == (x, p, q)
      assert _sort_gens([x, p, q], sort='p > x > q') == (p, x, q)
      assert _sort_gens([x, p, q], sort='p > q > x') == (p, q, x)
  
      assert _sort_gens([x, p, q], wrt='x', sort='q > p') == (x, q, p)
      assert _sort_gens([x, p, q], wrt='p', sort='q > x') == (p, q, x)
      assert _sort_gens([x, p, q], wrt='q', sort='p > x') == (q, p, x)
  
      X = symbols('x0,x1,x2,x10,x11,x12,x20,x21,x22')
  
      assert _sort_gens(X) == X
  
  
  def test__unify_gens():
      assert _unify_gens([], []) == ()
  
      assert _unify_gens([x], [x]) == (x,)
      assert _unify_gens([y], [y]) == (y,)
  
      assert _unify_gens([x, y], [x]) == (x, y)
      assert _unify_gens([x], [x, y]) == (x, y)
  
      assert _unify_gens([x, y], [x, y]) == (x, y)
      assert _unify_gens([y, x], [y, x]) == (y, x)
  
      assert _unify_gens([x], [y]) == (x, y)
      assert _unify_gens([y], [x]) == (y, x)
  
      assert _unify_gens([x], [y, x]) == (y, x)
      assert _unify_gens([y, x], [x]) == (y, x)
  
      assert _unify_gens([x, y, z], [x, y, z]) == (x, y, z)
      assert _unify_gens([z, y, x], [x, y, z]) == (z, y, x)
      assert _unify_gens([x, y, z], [z, y, x]) == (x, y, z)
      assert _unify_gens([z, y, x], [z, y, x]) == (z, y, x)
  
      assert _unify_gens([x, y, z], [t, x, p, q, z]) == (t, x, y, p, q, z)
  
  
  def test__analyze_gens():
      assert _analyze_gens((x, y, z)) == (x, y, z)
      assert _analyze_gens([x, y, z]) == (x, y, z)
  
      assert _analyze_gens(([x, y, z],)) == (x, y, z)
      assert _analyze_gens(((x, y, z),)) == (x, y, z)
  
  
  def test__sort_factors():
      assert _sort_factors([], multiple=True) == []
      assert _sort_factors([], multiple=False) == []
  
      F = [[1, 2, 3], [1, 2], [1]]
      G = [[1], [1, 2], [1, 2, 3]]
  
      assert _sort_factors(F, multiple=False) == G
  
      F = [[1, 2], [1, 2, 3], [1, 2], [1]]
      G = [[1], [1, 2], [1, 2], [1, 2, 3]]
  
      assert _sort_factors(F, multiple=False) == G
  
      F = [[2, 2], [1, 2, 3], [1, 2], [1]]
      G = [[1], [1, 2], [2, 2], [1, 2, 3]]
  
      assert _sort_factors(F, multiple=False) == G
  
      F = [([1, 2, 3], 1), ([1, 2], 1), ([1], 1)]
      G = [([1], 1), ([1, 2], 1), ([1, 2, 3], 1)]
  
      assert _sort_factors(F, multiple=True) == G
  
      F = [([1, 2], 1), ([1, 2, 3], 1), ([1, 2], 1), ([1], 1)]
      G = [([1], 1), ([1, 2], 1), ([1, 2], 1), ([1, 2, 3], 1)]
  
      assert _sort_factors(F, multiple=True) == G
  
      F = [([2, 2], 1), ([1, 2, 3], 1), ([1, 2], 1), ([1], 1)]
      G = [([1], 1), ([1, 2], 1), ([2, 2], 1), ([1, 2, 3], 1)]
  
      assert _sort_factors(F, multiple=True) == G
  
      F = [([2, 2], 1), ([1, 2, 3], 1), ([1, 2], 2), ([1], 1)]
      G = [([1], 1), ([2, 2], 1), ([1, 2], 2), ([1, 2, 3], 1)]
  
      assert _sort_factors(F, multiple=True) == G
  
  
  def test__dict_from_expr_if_gens():
      assert dict_from_expr(
          Integer(17), gens=(x,)) == ({(0,): Integer(17)}, (x,))
      assert dict_from_expr(
          Integer(17), gens=(x, y)) == ({(0, 0): Integer(17)}, (x, y))
      assert dict_from_expr(
          Integer(17), gens=(x, y, z)) == ({(0, 0, 0): Integer(17)}, (x, y, z))
  
      assert dict_from_expr(
          Integer(-17), gens=(x,)) == ({(0,): Integer(-17)}, (x,))
      assert dict_from_expr(
          Integer(-17), gens=(x, y)) == ({(0, 0): Integer(-17)}, (x, y))
      assert dict_from_expr(Integer(
          -17), gens=(x, y, z)) == ({(0, 0, 0): Integer(-17)}, (x, y, z))
  
      assert dict_from_expr(
          Integer(17)*x, gens=(x,)) == ({(1,): Integer(17)}, (x,))
      assert dict_from_expr(
          Integer(17)*x, gens=(x, y)) == ({(1, 0): Integer(17)}, (x, y))
      assert dict_from_expr(Integer(
          17)*x, gens=(x, y, z)) == ({(1, 0, 0): Integer(17)}, (x, y, z))
  
      assert dict_from_expr(
          Integer(17)*x**7, gens=(x,)) == ({(7,): Integer(17)}, (x,))
      assert dict_from_expr(
          Integer(17)*x**7*y, gens=(x, y)) == ({(7, 1): Integer(17)}, (x, y))
      assert dict_from_expr(Integer(17)*x**7*y*z**12, gens=(
          x, y, z)) == ({(7, 1, 12): Integer(17)}, (x, y, z))
  
      assert dict_from_expr(x + 2*y + 3*z, gens=(x,)) == \
          ({(1,): Integer(1), (0,): 2*y + 3*z}, (x,))
      assert dict_from_expr(x + 2*y + 3*z, gens=(x, y)) == \
          ({(1, 0): Integer(1), (0, 1): Integer(2), (0, 0): 3*z}, (x, y))
      assert dict_from_expr(x + 2*y + 3*z, gens=(x, y, z)) == \
          ({(1, 0, 0): Integer(
              1), (0, 1, 0): Integer(2), (0, 0, 1): Integer(3)}, (x, y, z))
  
      assert dict_from_expr(x*y + 2*x*z + 3*y*z, gens=(x,)) == \
          ({(1,): y + 2*z, (0,): 3*y*z}, (x,))
      assert dict_from_expr(x*y + 2*x*z + 3*y*z, gens=(x, y)) == \
          ({(1, 1): Integer(1), (1, 0): 2*z, (0, 1): 3*z}, (x, y))
      assert dict_from_expr(x*y + 2*x*z + 3*y*z, gens=(x, y, z)) == \
          ({(1, 1, 0): Integer(
              1), (1, 0, 1): Integer(2), (0, 1, 1): Integer(3)}, (x, y, z))
  
      assert dict_from_expr(2**y*x, gens=(x,)) == ({(1,): 2**y}, (x,))
      assert dict_from_expr(Integral(x, (x, 1, 2)) + x) == (
          {(0, 1): 1, (1, 0): 1}, (x, Integral(x, (x, 1, 2))))
      raises(PolynomialError, lambda: dict_from_expr(2**y*x, gens=(x, y)))
  
  
  def test__dict_from_expr_no_gens():
      raises(GeneratorsNeeded, lambda: dict_from_expr(Integer(17)))
  
      assert dict_from_expr(x) == ({(1,): Integer(1)}, (x,))
      assert dict_from_expr(y) == ({(1,): Integer(1)}, (y,))
  
      assert dict_from_expr(x*y) == ({(1, 1): Integer(1)}, (x, y))
      assert dict_from_expr(
          x + y) == ({(1, 0): Integer(1), (0, 1): Integer(1)}, (x, y))
  
      assert dict_from_expr(sqrt(2)) == ({(1,): Integer(1)}, (sqrt(2),))
      raises(GeneratorsNeeded, lambda: dict_from_expr(sqrt(2), greedy=False))
  
      assert dict_from_expr(x*y, domain=ZZ[x]) == ({(1,): x}, (y,))
      assert dict_from_expr(x*y, domain=ZZ[y]) == ({(1,): y}, (x,))
  
      assert dict_from_expr(3*sqrt(
          2)*pi*x*y, extension=None) == ({(1, 1, 1, 1): 3}, (x, y, pi, sqrt(2)))
      assert dict_from_expr(3*sqrt(
          2)*pi*x*y, extension=True) == ({(1, 1, 1): 3*sqrt(2)}, (x, y, pi))
  
      assert dict_from_expr(3*sqrt(
          2)*pi*x*y, extension=True) == ({(1, 1, 1): 3*sqrt(2)}, (x, y, pi))
  
      f = cos(x)*sin(x) + cos(x)*sin(y) + cos(y)*sin(x) + cos(y)*sin(y)
  
      assert dict_from_expr(f) == ({(0, 1, 0, 1): 1, (0, 1, 1, 0): 1,
          (1, 0, 0, 1): 1, (1, 0, 1, 0): 1}, (cos(x), cos(y), sin(x), sin(y)))
  
  
  def test__parallel_dict_from_expr_if_gens():
      assert parallel_dict_from_expr([x + 2*y + 3*z, Integer(7)], gens=(x,)) == \
          ([{(1,): Integer(1), (0,): 2*y + 3*z}, {(0,): Integer(7)}], (x,))
  
  
  def test__parallel_dict_from_expr_no_gens():
      assert parallel_dict_from_expr([x*y, Integer(3)]) == \
          ([{(1, 1): Integer(1)}, {(0, 0): Integer(3)}], (x, y))
      assert parallel_dict_from_expr([x*y, 2*z, Integer(3)]) == \
          ([{(1, 1, 0): Integer(
              1)}, {(0, 0, 1): Integer(2)}, {(0, 0, 0): Integer(3)}], (x, y, z))
  
  
  def test_parallel_dict_from_expr():
      assert parallel_dict_from_expr([Eq(x, 1), Eq(
          x**2, 2)]) == ([{(0,): -Integer(1), (1,): Integer(1)},
                          {(0,): -Integer(2), (2,): Integer(1)}], (x,))
      raises(PolynomialError, lambda: parallel_dict_from_expr([A*B - B*A]))
  
  
  def test_dict_from_expr():
      assert dict_from_expr(Eq(x, 1)) == \
          ({(0,): -Integer(1), (1,): Integer(1)}, (x,))
      raises(PolynomialError, lambda: dict_from_expr(A*B - B*A))
  