text: from sympy import (Symbol, Rational, Order, exp, ln, log, nan, oo, O, pi, I,
      S, Integral, sin, cos, sqrt, conjugate, expand, transpose, symbols,
      Function, Add)
  from sympy.utilities.pytest import 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
      O(w**(-1/x/log(3)*log(5)), w)
  
  
  def test_free_symbols():
      assert Order(1).free_symbols == set()
      assert Order(x).free_symbols == {x}
      assert Order(1, x).free_symbols == {x}
      assert Order(x*y).free_symbols == {x, y}
      assert Order(x, x, y).free_symbols == {x, y}
  
  
  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(ValueError, lambda: Order(exp(x), x, x))
      raises(TypeError, 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)
  
  
  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_simple_8():
      assert O(sqrt(-x)) == O(sqrt(x))
      assert O(x**2*sqrt(x)) == O(x**(S(5)/2))
      assert O(x**3*sqrt(-(-x)**3)) == O(x**(S(9)/2))
      assert O(x**(S(3)/2)*sqrt((-x)**3)) == O(x**3)
      assert O(x*(-2*x)**(I/2)) == O(x*(-x)**(I/2))
  
  
  def test_as_expr_variables():
      assert Order(x).as_expr_variables(None) == (x, ((x, 0),))
      assert Order(x).as_expr_variables((((x, 0),))) == (x, ((x, 0),))
      assert Order(y).as_expr_variables(((x, 0),)) == (y, ((x, 0), (y, 0)))
      assert Order(y).as_expr_variables(((x, 0), (y, 0))) == (y, ((x, 0), (y, 0)))
  
  
  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)) is False
  
  
  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_contains_4():
      assert Order(sin(1/x**2)).contains(Order(cos(1/x**2))) is None
      assert Order(cos(1/x**2)).contains(Order(sin(1/x**2))) is None
  
  
  def test_contains():
      assert Order(1, x) not in Order(1)
      assert Order(1) in Order(1, x)
      raises(TypeError, lambda: Order(x*y**2) in Order(x**2*y))
  
  
  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_issue_3468():
      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 is None
      assert y.is_positive is False
      assert z.is_positive is 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():
      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, (x, -y))
      assert (O(1)**x).is_Pow
  
  
  def test_issue_4279():
      a, b = symbols('a b')
      assert O(a, a, b) + O(1, a, b) == O(1, a, b)
      assert O(b, a, b) + O(1, a, b) == O(1, a, b)
      assert O(a + b, a, b) + O(1, a, b) == O(1, a, b)
      assert O(1, a, b) + O(a, a, b) == O(1, a, b)
      assert O(1, a, b) + O(b, a, b) == O(1, a, b)
      assert O(1, a, b) + O(a + b, a, b) == O(1, a, b)
  
  
  def test_issue_4855():
      assert 1/O(1) != O(1)
      assert 1/O(x) != O(1/x)
      assert 1/O(x, (x, oo)) != O(1/x, (x, oo))
  
      f = Function('f')
      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)
      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)
  
  
  def test_issue_6753():
      assert (1 + x**2)**10000*O(x) == O(x)
  
  
  def test_order_at_infinity():
      assert Order(1 + x, (x, oo)) == Order(x, (x, oo))
      assert Order(3*x, (x, oo)) == Order(x, (x, oo))
      assert Order(x, (x, oo))*3 == Order(x, (x, oo))
      assert -28*Order(x, (x, oo)) == Order(x, (x, oo))
      assert Order(Order(x, (x, oo)), (x, oo)) == Order(x, (x, oo))
      assert Order(Order(x, (x, oo)), (y, oo)) == Order(x, (x, oo), (y, oo))
      assert Order(3, (x, oo)) == Order(1, (x, oo))
      assert Order(x**2 + x + y, (x, oo)) == O(x**2, (x, oo))
      assert Order(x**2 + x + y, (y, oo)) == O(y, (y, oo))
  
      assert Order(2*x, (x, oo))*x == Order(x**2, (x, oo))
      assert Order(2*x, (x, oo))/x == Order(1, (x, oo))
      assert Order(2*x, (x, oo))*x*exp(1/x) == Order(x**2*exp(1/x), (x, oo))
      assert Order(2*x, (x, oo))*x*exp(1/x)/ln(x)**3 == Order(x**2*exp(1/x)*ln(x)**-3, (x, oo))
  
      assert Order(x, (x, oo)) + 1/x == 1/x + Order(x, (x, oo)) == Order(x, (x, oo))
      assert Order(x, (x, oo)) + 1 == 1 + Order(x, (x, oo)) == Order(x, (x, oo))
      assert Order(x, (x, oo)) + x == x + Order(x, (x, oo)) == Order(x, (x, oo))
      assert Order(x, (x, oo)) + x**2 == x**2 + Order(x, (x, oo))
      assert Order(1/x, (x, oo)) + 1/x**2 == 1/x**2 + Order(1/x, (x, oo)) == Order(1/x, (x, oo))
      assert Order(x, (x, oo)) + exp(1/x) == exp(1/x) + Order(x, (x, oo))
  
      assert Order(x, (x, oo))**2 == Order(x**2, (x, oo))
  
      assert Order(x, (x, oo)) + Order(x**2, (x, oo)) == Order(x**2, (x, oo))
      assert Order(x, (x, oo)) + Order(x**-2, (x, oo)) == Order(x, (x, oo))
      assert Order(x, (x, oo)) + Order(1/x, (x, oo)) == Order(x, (x, oo))
  
      assert Order(x, (x, oo)) - Order(x, (x, oo)) == Order(x, (x, oo))
      assert Order(x, (x, oo)) + Order(1, (x, oo)) == Order(x, (x, oo))
      assert Order(x, (x, oo)) + Order(x**2, (x, oo)) == Order(x**2, (x, oo))
      assert Order(1/x, (x, oo)) + Order(1, (x, oo)) == Order(1, (x, oo))
      assert Order(x, (x, oo)) + Order(exp(1/x), (x, oo)) == Order(x, (x, oo))
      assert Order(x**3, (x, oo)) + Order(exp(2/x), (x, oo)) == Order(x**3, (x, oo))
      assert Order(x**-3, (x, oo)) + Order(exp(2/x), (x, oo)) == Order(exp(2/x), (x, oo))
  
      # issue 7207
      assert Order(exp(x), (x, oo)).expr == Order(2*exp(x), (x, oo)).expr == exp(x)
      assert Order(y**x, (x, oo)).expr == Order(2*y**x, (x, oo)).expr == exp(log(y)*x)
  
  
  def test_mixing_order_at_zero_and_infinity():
      assert (Order(x, (x, 0)) + Order(x, (x, oo))).is_Add
      assert Order(x, (x, 0)) + Order(x, (x, oo)) == Order(x, (x, oo)) + Order(x, (x, 0))
      assert Order(Order(x, (x, oo))) == Order(x, (x, oo))
  
      # not supported (yet)
      raises(NotImplementedError, lambda: Order(x, (x, 0))*Order(x, (x, oo)))
      raises(NotImplementedError, lambda: Order(x, (x, oo))*Order(x, (x, 0)))
      raises(NotImplementedError, lambda: Order(Order(x, (x, oo)), y))
      raises(NotImplementedError, lambda: Order(Order(x), (x, oo)))
  
  
  def test_order_at_some_point():
      assert Order(x, (x, 1)) == Order(1, (x, 1))
      assert Order(2*x - 2, (x, 1)) == Order(x - 1, (x, 1))
      assert Order(-x + 1, (x, 1)) == Order(x - 1, (x, 1))
      assert Order(x - 1, (x, 1))**2 == Order((x - 1)**2, (x, 1))
      assert Order(x - 2, (x, 2)) - O(x - 2, (x, 2)) == Order(x - 2, (x, 2))
  
  
  def test_order_subs_limits():
      # issue 3333
      assert (1 + Order(x)).subs(x, 1/x) == 1 + Order(1/x, (x, oo))
      assert (1 + Order(x)).limit(x, 0) == 1
      # issue 5769
      assert ((x + Order(x**2))/x).limit(x, 0) == 1
  
      assert Order(x**2).subs(x, y - 1) == Order((y - 1)**2, (y, 1))
      assert Order(10*x**2, (x, 2)).subs(x, y - 1) == Order(1, (y, 3))
  
  
  def test_issue_9192():
      assert O(1)*O(1) == O(1)
      assert O(1)**O(1) == O(1)
  
  def test_performance_of_adding_order():
      l = list(x**i for i in range(1000))
      l.append(O(x**1001))
      assert Add(*l).subs(x,1) == O(1)
  