text: """Tests for tools for solving inequalities and systems of inequalities. """
  
  from sympy import (And, Eq, FiniteSet, Ge, Gt, Interval, Le, Lt, Ne, oo,
                     Or, S, sin, cos, tan, sqrt, Symbol, Union, Integral, Sum,
                     Function, Poly, PurePoly, pi, root, log, exp, Dummy, Abs)
  from sympy.solvers.inequalities import (reduce_inequalities,
                                          solve_poly_inequality as psolve,
                                          reduce_rational_inequalities,
                                          solve_univariate_inequality as isolve,
                                          reduce_abs_inequality)
  from sympy.polys.rootoftools import rootof
  from sympy.solvers.solvers import solve
  from sympy.solvers.solveset import solveset
  from sympy.abc import x, y
  
  from sympy.utilities.pytest import raises, slow, XFAIL
  
  
  inf = oo.evalf()
  
  
  def test_solve_poly_inequality():
      assert psolve(Poly(0, x), '==') == [S.Reals]
      assert psolve(Poly(1, x), '==') == [S.EmptySet]
      assert psolve(PurePoly(x + 1, x), ">") == [Interval(-1, oo, True, False)]
  
  
  def test_reduce_poly_inequalities_real_interval():
      assert reduce_rational_inequalities(
          [[Eq(x**2, 0)]], x, relational=False) == FiniteSet(0)
      assert reduce_rational_inequalities(
          [[Le(x**2, 0)]], x, relational=False) == FiniteSet(0)
      assert reduce_rational_inequalities(
          [[Lt(x**2, 0)]], x, relational=False) == S.EmptySet
      assert reduce_rational_inequalities(
          [[Ge(x**2, 0)]], x, relational=False) == \
          S.Reals if x.is_real else Interval(-oo, oo)
      assert reduce_rational_inequalities(
          [[Gt(x**2, 0)]], x, relational=False) == \
          FiniteSet(0).complement(S.Reals)
      assert reduce_rational_inequalities(
          [[Ne(x**2, 0)]], x, relational=False) == \
          FiniteSet(0).complement(S.Reals)
  
      assert reduce_rational_inequalities(
          [[Eq(x**2, 1)]], x, relational=False) == FiniteSet(-1, 1)
      assert reduce_rational_inequalities(
          [[Le(x**2, 1)]], x, relational=False) == Interval(-1, 1)
      assert reduce_rational_inequalities(
          [[Lt(x**2, 1)]], x, relational=False) == Interval(-1, 1, True, True)
      assert reduce_rational_inequalities(
          [[Ge(x**2, 1)]], x, relational=False) == \
          Union(Interval(-oo, -1), Interval(1, oo))
      assert reduce_rational_inequalities(
          [[Gt(x**2, 1)]], x, relational=False) == \
          Interval(-1, 1).complement(S.Reals)
      assert reduce_rational_inequalities(
          [[Ne(x**2, 1)]], x, relational=False) == \
          FiniteSet(-1, 1).complement(S.Reals)
      assert reduce_rational_inequalities([[Eq(
          x**2, 1.0)]], x, relational=False) == FiniteSet(-1.0, 1.0).evalf()
      assert reduce_rational_inequalities(
          [[Le(x**2, 1.0)]], x, relational=False) == Interval(-1.0, 1.0)
      assert reduce_rational_inequalities([[Lt(
          x**2, 1.0)]], x, relational=False) == Interval(-1.0, 1.0, True, True)
      assert reduce_rational_inequalities(
          [[Ge(x**2, 1.0)]], x, relational=False) == \
          Union(Interval(-inf, -1.0), Interval(1.0, inf))
      assert reduce_rational_inequalities(
          [[Gt(x**2, 1.0)]], x, relational=False) == \
          Union(Interval(-inf, -1.0, right_open=True),
          Interval(1.0, inf, left_open=True))
      assert reduce_rational_inequalities([[Ne(
          x**2, 1.0)]], x, relational=False) == \
          FiniteSet(-1.0, 1.0).complement(S.Reals)
  
      s = sqrt(2)
  
      assert reduce_rational_inequalities([[Lt(
          x**2 - 1, 0), Gt(x**2 - 1, 0)]], x, relational=False) == S.EmptySet
      assert reduce_rational_inequalities([[Le(x**2 - 1, 0), Ge(
          x**2 - 1, 0)]], x, relational=False) == FiniteSet(-1, 1)
      assert reduce_rational_inequalities(
          [[Le(x**2 - 2, 0), Ge(x**2 - 1, 0)]], x, relational=False
          ) == Union(Interval(-s, -1, False, False), Interval(1, s, False, False))
      assert reduce_rational_inequalities(
          [[Le(x**2 - 2, 0), Gt(x**2 - 1, 0)]], x, relational=False
          ) == Union(Interval(-s, -1, False, True), Interval(1, s, True, False))
      assert reduce_rational_inequalities(
          [[Lt(x**2 - 2, 0), Ge(x**2 - 1, 0)]], x, relational=False
          ) == Union(Interval(-s, -1, True, False), Interval(1, s, False, True))
      assert reduce_rational_inequalities(
          [[Lt(x**2 - 2, 0), Gt(x**2 - 1, 0)]], x, relational=False
          ) == Union(Interval(-s, -1, True, True), Interval(1, s, True, True))
      assert reduce_rational_inequalities(
          [[Lt(x**2 - 2, 0), Ne(x**2 - 1, 0)]], x, relational=False
          ) == Union(Interval(-s, -1, True, True), Interval(-1, 1, True, True),
          Interval(1, s, True, True))
  
  
  def test_reduce_poly_inequalities_complex_relational():
      assert reduce_rational_inequalities(
          [[Eq(x**2, 0)]], x, relational=True) == Eq(x, 0)
      assert reduce_rational_inequalities(
          [[Le(x**2, 0)]], x, relational=True) == Eq(x, 0)
      assert reduce_rational_inequalities(
          [[Lt(x**2, 0)]], x, relational=True) == False
      assert reduce_rational_inequalities(
          [[Ge(x**2, 0)]], x, relational=True) == And(Lt(-oo, x), Lt(x, oo))
      assert reduce_rational_inequalities(
          [[Gt(x**2, 0)]], x, relational=True) == \
          And(Or(And(Lt(-oo, x), Lt(x, 0)), And(Lt(0, x), Lt(x, oo))))
      assert reduce_rational_inequalities(
          [[Ne(x**2, 0)]], x, relational=True) == \
          And(Or(And(Lt(-oo, x), Lt(x, 0)), And(Lt(0, x), Lt(x, oo))))
  
      for one in (S(1), S(1.0)):
          inf = one*oo
          assert reduce_rational_inequalities(
              [[Eq(x**2, one)]], x, relational=True) == \
              Or(Eq(x, -one), Eq(x, one))
          assert reduce_rational_inequalities(
              [[Le(x**2, one)]], x, relational=True) == \
              And(And(Le(-one, x), Le(x, one)))
          assert reduce_rational_inequalities(
              [[Lt(x**2, one)]], x, relational=True) == \
              And(And(Lt(-one, x), Lt(x, one)))
          assert reduce_rational_inequalities(
              [[Ge(x**2, one)]], x, relational=True) == \
              And(Or(And(Le(one, x), Lt(x, inf)), And(Le(x, -one), Lt(-inf, x))))
          assert reduce_rational_inequalities(
              [[Gt(x**2, one)]], x, relational=True) == \
              And(Or(And(Lt(-inf, x), Lt(x, -one)), And(Lt(one, x), Lt(x, inf))))
          assert reduce_rational_inequalities(
              [[Ne(x**2, one)]], x, relational=True) == \
              Or(And(Lt(-inf, x), Lt(x, -one)),
                 And(Lt(-one, x), Lt(x, one)),
                 And(Lt(one, x), Lt(x, inf)))
  
  
  def test_reduce_rational_inequalities_real_relational():
      assert reduce_rational_inequalities([], x) == False
      assert reduce_rational_inequalities(
          [[(x**2 + 3*x + 2)/(x**2 - 16) >= 0]], x, relational=False) == \
          Union(Interval.open(-oo, -4), Interval(-2, -1), Interval.open(4, oo))
  
      assert reduce_rational_inequalities(
          [[((-2*x - 10)*(3 - x))/((x**2 + 5)*(x - 2)**2) < 0]], x,
          relational=False) == \
          Union(Interval.open(-5, 2), Interval.open(2, 3))
  
      assert reduce_rational_inequalities([[(x + 1)/(x - 5) <= 0]], x,
          relational=False) == \
          Interval.Ropen(-1, 5)
  
      assert reduce_rational_inequalities([[(x**2 + 4*x + 3)/(x - 1) > 0]], x,
          relational=False) == \
          Union(Interval.open(-3, -1), Interval.open(1, oo))
  
      assert reduce_rational_inequalities([[(x**2 - 16)/(x - 1)**2 < 0]], x,
          relational=False) == \
          Union(Interval.open(-4, 1), Interval.open(1, 4))
  
      assert reduce_rational_inequalities([[(3*x + 1)/(x + 4) >= 1]], x,
          relational=False) == \
          Union(Interval.open(-oo, -4), Interval.Ropen(S(3)/2, oo))
  
      assert reduce_rational_inequalities([[(x - 8)/x <= 3 - x]], x,
          relational=False) == \
          Union(Interval.Lopen(-oo, -2), Interval.Lopen(0, 4))
  
      # issue sympy/sympy#10237
      assert reduce_rational_inequalities(
          [[x < oo, x >= 0, -oo < x]], x, relational=False) == Interval(0, oo)
  
  
  def test_reduce_abs_inequalities():
      e = abs(x - 5) < 3
      ans = And(Lt(2, x), Lt(x, 8))
      assert reduce_inequalities(e) == ans
      assert reduce_inequalities(e, x) == ans
      assert reduce_inequalities(abs(x - 5)) == Eq(x, 5)
      assert reduce_inequalities(
          abs(2*x + 3) >= 8) == Or(And(Le(S(5)/2, x), Lt(x, oo)),
          And(Le(x, -S(11)/2), Lt(-oo, x)))
      assert reduce_inequalities(abs(x - 4) + abs(
          3*x - 5) < 7) == And(Lt(S(1)/2, x), Lt(x, 4))
      assert reduce_inequalities(abs(x - 4) + abs(3*abs(x) - 5) < 7) == \
          Or(And(S(-2) < x, x < -1), And(S(1)/2 < x, x < 4))
  
      nr = Symbol('nr', real=False)
      raises(TypeError, lambda: reduce_inequalities(abs(nr - 5) < 3))
      assert reduce_inequalities(x < 3, symbols=[x, nr]) == And(-oo < x, x < 3)
  
  
  def test_reduce_inequalities_general():
      assert reduce_inequalities(Ge(sqrt(2)*x, 1)) == And(sqrt(2)/2 <= x, x < oo)
      assert reduce_inequalities(PurePoly(x + 1, x) > 0) == And(S(-1) < x, x < oo)
  
  
  def test_reduce_inequalities_boolean():
      assert reduce_inequalities(
          [Eq(x**2, 0), True]) == Eq(x, 0)
      assert reduce_inequalities([Eq(x**2, 0), False]) == False
      assert reduce_inequalities(x**2 >= 0) is S.true  # issue 10196
  
  
  def test_reduce_inequalities_multivariate():
      assert reduce_inequalities([Ge(x**2, 1), Ge(y**2, 1)]) == And(
          Or(And(Le(1, x), Lt(x, oo)), And(Le(x, -1), Lt(-oo, x))),
          Or(And(Le(1, y), Lt(y, oo)), And(Le(y, -1), Lt(-oo, y))))
  
  
  def test_reduce_inequalities_errors():
      raises(NotImplementedError, lambda: reduce_inequalities(Ge(sin(x) + x, 1)))
      raises(NotImplementedError, lambda: reduce_inequalities(Ge(x**2*y + y, 1)))
  
  
  def test_hacky_inequalities():
      assert reduce_inequalities(x + y < 1, symbols=[x]) == (x < 1 - y)
      assert reduce_inequalities(x + y >= 1, symbols=[x]) == (x >= 1 - y)
      assert reduce_inequalities(Eq(0, x - y), symbols=[x]) == Eq(x, y)
      assert reduce_inequalities(Ne(0, x - y), symbols=[x]) == Ne(x, y)
  
  
  def test_issue_6343():
      eq = -3*x**2/2 - 45*x/4 + S(33)/2 > 0
      assert reduce_inequalities(eq) == \
          And(x < -S(15)/4 + sqrt(401)/4, -sqrt(401)/4 - S(15)/4 < x)
  
  
  def test_issue_8235():
      assert reduce_inequalities(x**2 - 1 < 0) == \
          And(S(-1) < x, x < S(1))
      assert reduce_inequalities(x**2 - 1 <= 0) == \
          And(S(-1) <= x, x <= 1)
      assert reduce_inequalities(x**2 - 1 > 0) == \
          Or(And(-oo < x, x < -1), And(x < oo, S(1) < x))
      assert reduce_inequalities(x**2 - 1 >= 0) == \
          Or(And(-oo < x, x <= S(-1)), And(S(1) <= x, x < oo))
  
      eq = x**8 + x - 9  # we want CRootOf solns here
      sol = solve(eq >= 0)
      tru = Or(And(rootof(eq, 1) <= x, x < oo), And(-oo < x, x <= rootof(eq, 0)))
      assert sol == tru
  
      # recast vanilla as real
      assert solve(sqrt((-x + 1)**2) < 1) == And(S(0) < x, x < 2)
  
  
  def test_issue_5526():
      assert reduce_inequalities(S(0) <=
          x + Integral(y**2, (y, 1, 3)) - 1, [x]) == \
          (x >= -Integral(y**2, (y, 1, 3)) + 1)
      f = Function('f')
      e = Sum(f(x), (x, 1, 3))
      assert reduce_inequalities(S(0) <= x + e + y**2, [x]) == \
          (x >= -y**2 - Sum(f(x), (x, 1, 3)))
  
  
  def test_solve_univariate_inequality():
      assert isolve(x**2 >= 4, x, relational=False) == Union(Interval(-oo, -2),
          Interval(2, oo))
      assert isolve(x**2 >= 4, x) == Or(And(Le(2, x), Lt(x, oo)), And(Le(x, -2),
          Lt(-oo, x)))
      assert isolve((x - 1)*(x - 2)*(x - 3) >= 0, x, relational=False) == \
          Union(Interval(1, 2), Interval(3, oo))
      assert isolve((x - 1)*(x - 2)*(x - 3) >= 0, x) == \
          Or(And(Le(1, x), Le(x, 2)), And(Le(3, x), Lt(x, oo)))
      # issue 2785:
      assert isolve(x**3 - 2*x - 1 > 0, x, relational=False) == \
          Union(Interval(-1, -sqrt(5)/2 + S(1)/2, True, True),
                Interval(S(1)/2 + sqrt(5)/2, oo, True, True))
      # issue 2794:
      assert isolve(x**3 - x**2 + x - 1 > 0, x, relational=False) == \
          Interval(1, oo, True)
  
      # numerical testing in valid() is needed
      assert isolve(x**7 - x - 2 > 0, x) == \
          And(rootof(x**7 - x - 2, 0) < x, x < oo)
  
      # handle numerator and denominator; although these would be handled as
      # rational inequalities, these test confirm that the right thing is done
      # when the domain is EX (e.g. when 2 is replaced with sqrt(2))
      assert isolve(1/(x - 2) > 0, x) == And(S(2) < x, x < oo)
      den = ((x - 1)*(x - 2)).expand()
      assert isolve((x - 1)/den <= 0, x) == \
          Or(And(-oo < x, x < 1), And(S(1) < x, x < 2))
  
      n = Dummy('n')
      raises(NotImplementedError, lambda: isolve(Abs(x) <= n, x, relational=False))
  
  
  def test_trig_inequalities():
      # all the inequalities are solved in a periodic interval.
      assert isolve(sin(x) < S.Half, x, relational=False) == \
          Union(Interval(0, pi/6, False, True), Interval(5*pi/6, 2*pi, True, False))
      assert isolve(sin(x) > S.Half, x, relational=False) == \
          Interval(pi/6, 5*pi/6, True, True)
      assert isolve(cos(x) < S.Zero, x, relational=False) == \
          Interval(pi/2, 3*pi/2, True, True)
      assert isolve(cos(x) >= S.Zero, x, relational=False) == \
          Union(Interval(0, pi/2), Interval(3*pi/2, 2*pi))
  
      assert isolve(tan(x) < S.One, x, relational=False) == \
          Union(Interval.Ropen(0, pi/4), Interval.Lopen(pi/2, pi))
  
      assert isolve(sin(x) <= S.Zero, x, relational=False) == \
          Union(FiniteSet(S(0)), Interval(pi, 2*pi))
  
      assert isolve(sin(x) <= S(1), x, relational=False) == S.Reals
      assert isolve(cos(x) < S(-2), x, relational=False) == S.EmptySet
      assert isolve(sin(x) >= S(-1), x, relational=False) == S.Reals
      assert isolve(cos(x) > S(1), x, relational=False) == S.EmptySet
  
  
  def test_issue_9954():
      assert isolve(x**2 >= 0, x, relational=False) == S.Reals
      assert isolve(x**2 >= 0, x, relational=True) == S.Reals.as_relational(x)
      assert isolve(x**2 < 0, x, relational=False) == S.EmptySet
      assert isolve(x**2 < 0, x, relational=True) == S.EmptySet.as_relational(x)
  
  
  @XFAIL
  def test_slow_general_univariate():
      r = rootof(x**5 - x**2 + 1, 0)
      assert solve(sqrt(x) + 1/root(x, 3) > 1) == \
          Or(And(S(0) < x, x < r**6), And(r**6 < x, x < oo))
  
  
  def test_issue_8545():
      eq = 1 - x - abs(1 - x)
      ans = And(Lt(1, x), Lt(x, oo))
      assert reduce_abs_inequality(eq, '<', x) == ans
      eq = 1 - x - sqrt((1 - x)**2)
      assert reduce_inequalities(eq < 0) == ans
  
  
  def test_issue_8974():
      assert isolve(-oo < x, x) == And(-oo < x, x < oo)
      assert isolve(oo > x, x) == And(-oo < x, x < oo)
  
  
  def test_issue_10198():
      assert reduce_inequalities(
          -1 + 1/abs(1/x - 1) < 0) == Or(
          And(-oo < x, x < 0), And(S(0) < x, x < S(1)/2)
          )
      assert reduce_inequalities(abs(1/sqrt(x)) - 1, x) == Eq(x, 1)
      assert reduce_abs_inequality(-3 + 1/abs(1 - 1/x), '<', x) == \
          Or(And(-oo < x, x < 0),
          And(S(0) < x, x < S(3)/4), And(S(3)/2 < x, x < oo))
      raises(ValueError,lambda: reduce_abs_inequality(-3 + 1/abs(
          1 - 1/sqrt(x)), '<', x))
  
  
  def test_issue_10047():
      assert solve(sin(x) < 2) == And(-oo < x, x < oo)
  
  
  def test_issue_10268():
      assert solve(log(x) < 1000) == And(S(0) < x, x < exp(1000))
  
  
  @XFAIL
  def test_isolve_Sets():
      n = Dummy('n')
      assert isolve(Abs(x) <= n, x, relational=False) == \
          Piecewise((S.EmptySet, n < 0), (Interval(-n, n), True))
  
  
  def test_issue_10671_12466():
      assert solveset(sin(y), y, Interval(0, pi)) == FiniteSet(0, pi)
      i = Interval(1, 10)
      assert solveset((1/x).diff(x) < 0, x, i) == i
      assert solveset((log(x - 6)/x) <= 0, x, S.Reals) == \
          Interval.Lopen(6, 7)
  