def test_issue_5526():
    assert reduce_inequalities(S(0) <= x + Integral(y**2, (y, 1, 3)) - 1, True, [x]) == \
        (-Integral(y**2, (y, 1, 3)) + 1 <= x)
    f = Function('f')
    e = Sum(f(x), (x, 1, 3))
    assert reduce_inequalities(S(0) <= x + e + y**2, True, [x]) == \
        (-y**2 - Sum(f(x), (x, 1, 3)) <= x)

def test_reduce_inequalities_errors():
    x = Symbol('x')
    y = Symbol('y')

    raises(NotImplementedError, lambda: reduce_inequalities(Ge(sin(x) + x, 1)))
    raises(NotImplementedError, lambda: reduce_inequalities(Ge(x**2*y + y, 1)))
    raises(NotImplementedError, lambda: reduce_inequalities(Ge(sqrt(2)*x, 1)))

def test_reduce_abs_inequalities():
    real = Q.real(x)

    assert reduce_inequalities(abs(x - 5) < 3, assume=real) == And(Gt(x, 2), Lt(x, 8))
    assert reduce_inequalities(abs(2*x + 3) >= 8, assume=real) == Or(Le(x, -S(11)/2), Ge(x, S(5)/2))
    assert reduce_inequalities(abs(x - 4) + abs(3*x - 5) < 7, assume=real) == And(Gt(x, S(1)/2), Lt(x, 4))
    assert reduce_inequalities(abs(x - 4) + abs(3*abs(x) - 5) < 7, assume=real) == Or(And(-2 < x, x < -1), And(S(1)/2 < x, x < 4))

    raises(NotImplementedError, "reduce_inequalities(abs(x - 5) < 3)")

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_reduce_abs_inequalities():
    real = Q.real(x)

    assert reduce_inequalities(
        abs(x - 5) < 3, assume=real) == And(Lt(2, x), Lt(x, 8))
    assert reduce_inequalities(
        abs(2*x + 3) >= 8, assume=real) == 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, assume=real) == And(Lt(S(1)/2, x), Lt(x, 4))
    assert reduce_inequalities(abs(x - 4) + abs(3*abs(x) - 5) < 7, assume=real) == Or(And(S(-2) < x, x < -1), And(S(1)/2 < x, x < 4))

    raises(NotImplementedError, lambda: reduce_inequalities(abs(x - 5) < 3))

def test_reduce_abs_inequalities():
    x = Symbol('x', real=True)

    assert reduce_inequalities(abs(x - 5) < 3) == And(Lt(2, x), Lt(x, 8))
    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))

    x = Symbol('x')
    raises(NotImplementedError, lambda: reduce_inequalities(abs(x - 5) < 3))

def test_issue_8235():
    x = Symbol('x', real=True)
    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**2 - 9
    sol = solve(eq >= 0)
    known_sol = Or(And(-oo < x, RootOf(x**8 + x**2 - 9, 1) <= x, x < oo), \
            And(-oo < x, x < oo, x <= RootOf(x**8 + x**2 - 9, 0)))
    assert sol == known_sol

def test_issue_8545():
    from sympy import refine
    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(refine(eq) < 0) == ans

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_reduce_inequalities_multivariate():
    x = Symbol('x')
    y = Symbol('y')

    assert reduce_inequalities([Ge(x**2, 1), Ge(y**2, 1)]) == \
        And(Eq(im(x), 0), Eq(im(y), 0), Or(And(Le(1, re(x)), Lt(re(x), oo)),
                                           And(Le(re(x), -1), Lt(-oo, re(x)))),
            Or(And(Le(1, re(y)), Lt(re(y), oo)), And(Le(re(y), -1), Lt(-oo, re(y)))))

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_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_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_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_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_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 _solve(f, *symbols, **flags):
    """Solves equations and systems of equations.

       Currently supported are univariate polynomial, transcendental
       equations, piecewise combinations thereof and systems of linear
       and polynomial equations.  Input is formed as a single expression
       or an equation,  or an iterable container in case of an equation
       system.  The type of output may vary and depends heavily on the
       input. For more details refer to more problem specific functions.

       By default all solutions are simplified to make the output more
       readable. If this is not the expected behavior (e.g., because of
       speed issues) set simplified=False in function arguments.

       To solve equations and systems of equations like recurrence relations
       or differential equations, use rsolve() or dsolve(), respectively.

       >>> from sympy import I, solve
       >>> from sympy.abc import x, y

       Solve a polynomial equation:

       >>> solve(x**4-1, x)
       [1, -1, -I, I]

       Solve a linear system:

       >>> solve((x+5*y-2, -3*x+6*y-15), x, y)
       {x: -3, y: 1}

    """

    def sympified_list(w):
        return map(sympify, iff(isinstance(w,(list, tuple, set)), w, [w]))
    # make f and symbols into lists of sympified quantities
    # keeping track of how f was passed since if it is a list
    # a dictionary of results will be returned.
    bare_f = not iterable(f)
    f, symbols = (sympified_list(w) for w in [f, symbols])

    for i, fi in enumerate(f):
        if isinstance(fi, Equality):
            f[i] = fi.lhs - fi.rhs
        elif isinstance(fi, Poly):
            f[i] = fi.as_expr()
        elif isinstance(fi, bool) or fi.is_Relational:
            return reduce_inequalities(f, assume=flags.get('assume'))

    if not symbols:
        #get symbols from equations or supply dummy symbols since
        #solve(3,x) returns []...though it seems that it should raise some sort of error TODO
        symbols = set([])
        for fi in f:
            symbols |= fi.free_symbols or set([Dummy('x')])
        symbols = list(symbols)
        symbols.sort(key=Basic.sort_key)

    if len(symbols) == 1:
        if isinstance(symbols[0], (list, tuple, set)):
            symbols = symbols[0]

    result = list()

    # Begin code handling for Function and Derivative instances
    # Basic idea:  store all the passed symbols in symbols_passed, check to see
    # if any of them are Function or Derivative types, if so, use a dummy
    # symbol in their place, and set symbol_swapped = True so that other parts
    # of the code can be aware of the swap.  Once all swapping is done, the
    # continue on with regular solving as usual, and swap back at the end of
    # the routine, so that whatever was passed in symbols is what is returned.
    symbols_new = []
    symbol_swapped = False

    symbols_passed = list(symbols)

    for i, s in enumerate(symbols):
        if s.is_Symbol:
            s_new = s
        elif s.is_Function:
            symbol_swapped = True
            s_new = Dummy('F%d' % i)
        elif s.is_Derivative:
            symbol_swapped = True
            s_new = Dummy('D%d' % i)
        else:
            raise TypeError('not a Symbol or a Function')
        symbols_new.append(s_new)

        if symbol_swapped:
            swap_back_dict = dict(zip(symbols_new, symbols))
    # End code for handling of Function and Derivative instances

    if bare_f:
        f = f[0]

        # Create a swap dictionary for storing the passed symbols to be solved
        # for, so that they may be swapped back.
        if symbol_swapped:
            swap_dict = zip(symbols, symbols_new)
            f = f.subs(swap_dict)
#.........这里部分代码省略.........

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

#.........这里部分代码省略.........
                    [x*exp(x/2), -x*exp(x/2)]

                Presently, assumptions aren't checked either when `solve()` input
                involves relationals or bools.

       See also:
          rsolve() for solving recurrence relationships
          dsolve() for solving differential equations

    """
    # make f and symbols into lists of sympified quantities
    # keeping track of how f was passed since if it is a list
    # a dictionary of results will be returned.
    ###########################################################################
    def sympified_list(w):
        return map(sympify, w if iterable(w) else [w])
    bare_f = not iterable(f)
    ordered_symbols = (symbols and
                       symbols[0] and
                       (isinstance(symbols[0], Symbol) or
                        is_sequence(symbols[0], include=GeneratorType)
                       )
                      )
    f, symbols = (sympified_list(w) for w in [f, symbols])

    # preprocess equation(s)
    ###########################################################################
    for i, fi in enumerate(f):
        if isinstance(fi, Equality):
            f[i] = fi.lhs - fi.rhs
        elif isinstance(fi, Poly):
            f[i] = fi.as_expr()
        elif isinstance(fi, bool) or fi.is_Relational:
            return reduce_inequalities(f, assume=flags.get('assume'))
        # Any embedded piecewise functions need to be brought out to the
        # top level so that the appropriate strategy gets selected.
        f[i] = piecewise_fold(f[i])

    # preprocess symbol(s)
    ###########################################################################
    if not symbols:
        # get symbols from equations or supply dummy symbols so solve(3) behaves
        # like solve(3, x).
        symbols = set([])
        for fi in f:
            symbols |= fi.free_symbols or set([Dummy()])
        ordered_symbols = False
    elif len(symbols) == 1 and iterable(symbols[0]):
        symbols = symbols[0]
    if not ordered_symbols:
        # we do this to make the results returned canonical in case f
        # contains a system of nonlinear equations; all other cases should
        # be unambiguous
        symbols = sorted(symbols, key=lambda i: i.sort_key())

    # we can solve for Function and Derivative instances by replacing them
    # with Dummy symbols
    symbols_new = []
    symbol_swapped = False
    symbols_passed = list(symbols)

    for i, s in enumerate(symbols):
        if s.is_Symbol:
            s_new = s
        elif s.is_Function:
            symbol_swapped = True

def test_reduce_inequalities_boolean():
    assert reduce_inequalities(
        [Eq(x**2, 0), True]) == And(Eq(re(x), 0), Eq(im(x), 0))
    assert reduce_inequalities([Eq(x**2, 0), False]) is False