text: from sympy import (Symbol, Rational, Order, C, exp, ln, log, O, var, nan, pi,
      S, Integral, sin, conjugate, expand, transpose)
  from sympy.utilities.pytest import XFAIL, raises
  from sympy.abc import w, x, y, z
  
  def test_caching_bug():
      #needs to be a first test, so that all caches are clean
      #cache it
      e = O(w)
      #and test that this won't raise an exception
      f = O(w**(-1/x/log(3)*log(5)), w)
  
  def test_simple_1():
      o = Rational(0)
      assert Order(2*x) == Order(x)
      assert Order(x)*3 == Order(x)
      assert -28*Order(x) == Order(x)
      assert Order(Order(x)) == Order(x)
      assert Order(Order(x), y) == Order(Order(x), x, y)
      assert Order(-23) == Order(1)
      assert Order(exp(x)) == Order(1,x)
      assert Order(exp(1/x)).expr == exp(1/x)
      assert Order(x*exp(1/x)).expr == x*exp(1/x)
      assert Order(x**(o/3)).expr == x**(o/3)
      assert Order(x**(5*o/3)).expr == x**(5*o/3)
      assert Order(x**2 + x + y, x) == O(1, x)
      assert Order(x**2 + x + y, y) == O(1, y)
      raises(NotImplementedError, lambda: Order(x, 2 - x))
  
  def test_simple_2():
      assert Order(2*x)*x == Order(x**2)
      assert Order(2*x)/x == Order(1,x)
      assert Order(2*x)*x*exp(1/x) == Order(x**2*exp(1/x))
      assert (Order(2*x)*x*exp(1/x)/ln(x)**3).expr == x**2*exp(1/x)*ln(x)**-3
  
  def test_simple_3():
      assert Order(x)+x == Order(x)
      assert Order(x)+2 == 2+Order(x)
      assert Order(x)+x**2 == Order(x)
      assert Order(x)+1/x == 1/x+Order(x)
      assert Order(1/x)+1/x**2 == 1/x**2+Order(1/x)
      assert Order(x)+exp(1/x) == Order(x)+exp(1/x)
  
  def test_simple_4():
      assert Order(x)**2 == Order(x**2)
      assert Order(x**3)**-2 == Order(x**-6)
  
  def test_simple_5():
      assert Order(x)+Order(x**2) == Order(x)
      assert Order(x)+Order(x**-2) == Order(x**-2)
      assert Order(x)+Order(1/x) == Order(1/x)
  
  def test_simple_6():
      assert Order(x)-Order(x) == Order(x)
      assert Order(x)+Order(1) == Order(1)
      assert Order(x)+Order(x**2) == Order(x)
      assert Order(1/x)+Order(1) == Order(1/x)
      assert Order(x)+Order(exp(1/x)) == Order(exp(1/x))
      assert Order(x**3)+Order(exp(2/x)) == Order(exp(2/x))
      assert Order(x**-3)+Order(exp(2/x)) == Order(exp(2/x))
  
  def test_simple_7():
      assert 1+O(1) == O(1)
      assert 2+O(1) == O(1)
      assert x+O(1) == O(1)
      assert 1/x+O(1) == 1/x+O(1)
  
  def test_as_expr_variables():
      assert Order(x).as_expr_variables(None) == (x, (x,))
      assert Order(x).as_expr_variables((x,)) == (x, (x,))
      assert Order(y).as_expr_variables((x,)) == (y, (x, y))
      assert Order(y).as_expr_variables((x, y)) == (y, (x, y))
  
  def test_contains_0():
      assert Order(1,x).contains(Order(1,x))
      assert Order(1,x).contains(Order(1))
      assert Order(1).contains(Order(1,x))
  
  def test_contains_1():
      assert Order(x).contains(Order(x))
      assert Order(x).contains(Order(x**2))
      assert not Order(x**2).contains(Order(x))
      assert not Order(x).contains(Order(1/x))
      assert not Order(1/x).contains(Order(exp(1/x)))
      assert not Order(x).contains(Order(exp(1/x)))
      assert Order(1/x).contains(Order(x))
      assert Order(exp(1/x)).contains(Order(x))
      assert Order(exp(1/x)).contains(Order(1/x))
      assert Order(exp(1/x)).contains(Order(exp(1/x)))
      assert Order(exp(2/x)).contains(Order(exp(1/x)))
      assert not Order(exp(1/x)).contains(Order(exp(2/x)))
  
  def test_contains_2():
      assert Order(x).contains(Order(y)) is None
      assert Order(x).contains(Order(y*x))
      assert Order(y*x).contains(Order(x))
      assert Order(y).contains(Order(x*y))
      assert Order(x).contains(Order(y**2*x))
  
  def test_contains_3():
      assert Order(x*y**2).contains(Order(x**2*y)) is None
      assert Order(x**2*y).contains(Order(x*y**2)) is None
  
  def test_add_1():
      assert Order(x+x) == Order(x)
      assert Order(3*x-2*x**2) == Order(x)
      assert Order(1+x) == Order(1,x)
      assert Order(1+1/x) == Order(1/x)
      assert Order(ln(x)+1/ln(x)) == Order(ln(x))
      assert Order(exp(1/x)+x) == Order(exp(1/x))
      assert Order(exp(1/x)+1/x**20) == Order(exp(1/x))
  
  def test_ln_args():
      assert O(log(x)) + O(log(2*x)) == O(log(x))
      assert O(log(x)) + O(log(x**3)) == O(log(x))
      assert O(log(x*y)) + O(log(x)+log(y)) == O(log(x*y))
  
  def test_multivar_0():
      assert Order(x*y).expr == x*y
      assert Order(x*y**2).expr == x*y**2
      assert Order(x*y,x).expr == x
      assert Order(x*y**2,y).expr == y**2
      assert Order(x*y*z).expr == x*y*z
      assert Order(x/y).expr == x/y
      assert Order(x*exp(1/y)).expr == x*exp(1/y)
      assert Order(exp(x)*exp(1/y)).expr == exp(1/y)
  
  def test_multivar_0a():
      assert Order(exp(1/x)*exp(1/y)).expr == exp(1/x + 1/y)
  
  def test_multivar_1():
      assert Order(x+y).expr == x+y
      assert Order(x+2*y).expr == x+y
      assert (Order(x+y)+x).expr == (x+y)
      assert (Order(x+y)+x**2) == Order(x+y)
      assert (Order(x+y)+1/x) == 1/x+Order(x+y)
      assert Order(x**2+y*x).expr == x**2+y*x
  
  def test_multivar_2():
      assert Order(x**2*y+y**2*x,x,y).expr == x**2*y+y**2*x
  
  def test_multivar_mul_1():
      assert Order(x+y)*x == Order(x**2+y*x,x,y)
  
  def test_multivar_3():
      assert (Order(x)+Order(y)).args in [
              (Order(x), Order(y)),
              (Order(y), Order(x))]
      assert Order(x)+Order(y)+Order(x+y) == Order(x+y)
      assert (Order(x**2*y)+Order(y**2*x)).args in [
              (Order(x*y**2), Order(y*x**2)),
              (Order(y*x**2), Order(x*y**2))]
      assert (Order(x**2*y)+Order(y*x)) == Order(x*y)
  
  def test_issue369():
      x = Symbol('x')
      y = Symbol('y', negative=True)
      z = Symbol('z', complex=True)
  
      # check that Order does not modify assumptions about symbols
      Order(x)
      Order(y)
      Order(z)
  
      assert x.is_positive == None
      assert y.is_positive == False
      assert z.is_positive == None
  
      assert x.is_infinitesimal == None
      assert y.is_infinitesimal == None
      assert z.is_infinitesimal == None
  
  def test_leading_order():
      assert (x+1+1/x**5).extract_leading_order(x) == ((1/x**5, O(1/x**5)),)
      assert (1+1/x).extract_leading_order(x) == ((1/x, O(1/x)),)
      assert (1+x).extract_leading_order(x) == ((1, O(1, x)),)
      assert (1+x**2).extract_leading_order(x) == ((1, O(1, x)),)
      assert (2+x**2).extract_leading_order(x) == ((2, O(1, x)),)
      assert (x+x**2).extract_leading_order(x) == ((x, O(x)),)
  
  def test_leading_order2():
      assert set((2+pi+x**2).extract_leading_order(x)) == set(((pi, O(1, x)),
              (S(2), O(1, x))))
      assert set((2*x+pi*x+x**2).extract_leading_order(x)) == set(((2*x, O(x)),
              (x*pi, O(x))))
  
  def test_order_leadterm():
      assert O(x**2)._eval_as_leading_term(x) == O(x**2)
  
  def test_order_symbols():
      e = x*y*sin(x)*Integral(x, (x, 1, 2))
      assert O(e) == O(x**2*y, x, y)
      assert O(e, x) == O(x**2)
  
  def test_nan():
      assert O(nan) == nan
      assert not O(x).contains(nan)
  
  def test_O1():
      assert O(1, x) * x == O(x)
      assert O(1, y) * x == O(1, y)
  
  def test_getn():
      # other lines are tested incidentally by the suite
      assert O(x).getn() == 1
      assert O(x/log(x)).getn() == 1
      assert O(x**2/log(x)**2).getn() == 2
      assert O(x*log(x)).getn() == 1
      raises(NotImplementedError, lambda: (O(x) + O(y)).getn())
  
  def test_diff():
      assert O(x**2).diff(x) == O(x)
  
  def test_getO():
      assert (x).getO() is None
      assert (x).removeO() == x
      assert (O(x)).getO() == O(x)
      assert (O(x)).removeO() == 0
      assert (z + O(x) + O(y)).getO() == O(x) + O(y)
      assert (z + O(x) + O(y)).removeO() == z
      raises(NotImplementedError, lambda: (O(x)+O(y)).getn())
  
  def test_leading_term():
      from sympy import digamma
      assert O(1/digamma(1/x)) == O(1/log(x))
  
  def test_eval():
      y = Symbol('y')
      from sympy import Basic
      assert Order(x).subs(Order(x), 1) == 1
      assert Order(x).subs(x, y) == Order(y)
      assert Order(x).subs(y, x) == Order(x)
      assert Order(x).subs(x, x + y) == Order(x + y)
      assert (O(1)**x).is_Pow
  
  def test_oseries():
      assert Order(x).oseries(x) == Order(x)
  
  @XFAIL
  def test_issue_1180():
      a, b = symbols('a b')
      assert O(a+b,a,b)+O(1,a,b) == O(1, a, b)
  
  @XFAIL
  def test_issue_1756():
      x = Symbol('x')
      f = Function('f')
      g = Function('g')
      assert 1/O(1) != O(1)
      assert 1/O(x) != O(1/x)
      assert 1/O(f(x)) != O(1/x)
  
  def test_order_conjugate_transpose():
      x = Symbol('x', real=True)
      y = Symbol('y', imaginary=True)
      assert conjugate(Order(x)) == Order(conjugate(x))
      assert conjugate(Order(y)) == Order(conjugate(y))
      assert conjugate(Order(x**2)) == Order(conjugate(x)**2)
      assert conjugate(Order(y**2)) == Order(conjugate(y)**2)
      assert transpose(Order(x)) == Order(transpose(x))
      assert transpose(Order(y)) == Order(transpose(y))
      assert transpose(Order(x**2)) == Order(transpose(x)**2)
      assert transpose(Order(y**2)) == Order(transpose(y)**2)
  
  def test_order_noncommutative():
      A = Symbol('A', commutative=False)
      x = Symbol('x')
      assert Order(A + A*x, x) == Order(1, x)
      assert (A + A*x)*Order(x) == Order(x)
      assert (A*x)*Order(x) == Order(x**2, x)
      assert expand((1 + Order(x))*A*A*x) == A*A*x + Order(x**2, x)
      assert expand((A*A + Order(x))*x) == A*A*x + Order(x**2, x)
      assert expand((A + Order(x))*A*x) == A*A*x + Order(x**2, x)
  