def _monotonic_sign(self):
    """Return the value closest to 0 that ``self`` may have if all symbols
    are signed and the result is uniformly the same sign for all values of symbols.
    If a symbol is only signed but not known to be an
    integer or the result is 0 then a symbol representative of the sign of self
    will be returned. Otherwise, None is returned if a) the sign could be positive
    or negative or b) self is not in one of the following forms:

    - L(x, y, ...) + A: a function linear in all symbols x, y, ... with an
      additive constant; if A is zero then the function can be a monomial whose
      sign is monotonic over the range of the variables, e.g. (x + 1)**3 if x is
      nonnegative.
    - A/L(x, y, ...) + B: the inverse of a function linear in all symbols x, y, ...
      that does not have a sign change from positive to negative for any set
      of values for the variables.
    - M(x, y, ...) + A: a monomial M whose factors are all signed and a constant, A.
    - A/M(x, y, ...) + B: the inverse of a monomial and constants A and B.
    - P(x): a univariate polynomial

    Examples
    ========

    >>> from sympy.core.exprtools import _monotonic_sign as F
    >>> from sympy import Dummy, S
    >>> nn = Dummy(integer=True, nonnegative=True)
    >>> p = Dummy(integer=True, positive=True)
    >>> p2 = Dummy(integer=True, positive=True)
    >>> F(nn + 1)
    1
    >>> F(p - 1)
    _nneg
    >>> F(nn*p + 1)
    1
    >>> F(p2*p + 1)
    2
    >>> F(nn - 1)  # could be negative, zero or positive
    """
    if not self.is_real:
        return

    if (-self).is_Symbol:
        rv = _monotonic_sign(-self)
        return rv if rv is None else -rv

    if not self.is_Add and self.as_numer_denom()[1].is_number:
        s = self
        if s.is_prime:
            if s.is_odd:
                return S(3)
            else:
                return S(2)
        elif s.is_positive:
            if s.is_even:
                return S(2)
            elif s.is_integer:
                return S.One
            else:
                return _eps
        elif s.is_negative:
            if s.is_even:
                return S(-2)
            elif s.is_integer:
                return S.NegativeOne
            else:
                return -_eps
        if s.is_zero or s.is_nonpositive or s.is_nonnegative:
            return S.Zero
        return None

    # univariate polynomial
    free = self.free_symbols
    if len(free) == 1:
        if self.is_polynomial():
            from sympy.polys.polytools import real_roots
            from sympy.polys.polyroots import roots
            from sympy.polys.polyerrors import PolynomialError
            x = free.pop()
            x0 = _monotonic_sign(x)
            if x0 == _eps or x0 == -_eps:
                x0 = S.Zero
            if x0 is not None:
                d = self.diff(x)
                if d.is_number:
                    roots = []
                else:
                    try:
                        roots = real_roots(d)
                    except (PolynomialError, NotImplementedError):
                        roots = [r for r in roots(d, x) if r.is_real]
                y = self.subs(x, x0)
                if x.is_nonnegative and all(x0 <= r for r in roots):
                    if y.is_nonnegative and d.is_positive:
                        if y:
                            return y if y.is_positive else Dummy('pos', positive=True)
                        else:
                            return Dummy('nneg', nonnegative=True)
                    if y.is_nonpositive and d.is_negative:
                        if y:
                            return y if y.is_negative else Dummy('neg', negative=True)
                        else:
                            return Dummy('npos', nonpositive=True)
                elif x.is_nonpositive and all(r >= x0 for r in roots):
                    if y.is_nonnegative and d.is_negative:
                        if y:
                            return Dummy('pos', positive=True)
                        else:
                            return Dummy('nneg', nonnegative=True)
                    if y.is_nonpositive and d.is_positive:
                        if y:
                            return Dummy('neg', negative=True)
                        else:
                            return Dummy('npos', nonpositive=True)
        else:
            n, d = self.as_numer_denom()
            den = None
            if n.is_number:
                den = _monotonic_sign(d)
            elif not d.is_number:
                if _monotonic_sign(n) is not None:
                    den = _monotonic_sign(d)
            if den is not None and (den.is_positive or den.is_negative):
                v = n*den
                if v.is_positive:
                    return Dummy('pos', positive=True)
                elif v.is_nonnegative:
                    return Dummy('nneg', nonnegative=True)
                elif v.is_negative:
                    return Dummy('neg', negative=True)
                elif v.is_nonpositive:
                    return Dummy('npos', nonpositive=True)
        return None

    # multivariate
    c, a = self.as_coeff_Add()
    v = None
    if not a.is_polynomial():
        # F/A or A/F where A is a number and F is a signed, rational monomial
        n, d = a.as_numer_denom()
        if not (n.is_number or d.is_number):
            return
        if (
                a.is_Mul or a.is_Pow) and \
                a.is_rational and \
                all(p.exp.is_Integer for p in a.atoms(Pow) if p.is_Pow) and \
                (a.is_positive or a.is_negative):
            v = S(1)
            for ai in Mul.make_args(a):
                if ai.is_number:
                    v *= ai
                    continue
                reps = {}
                for x in ai.free_symbols:
                    reps[x] = _monotonic_sign(x)
                    if reps[x] is None:
                        return
                v *= ai.subs(reps)
    elif c:
        # signed linear expression
        if not any(p for p in a.atoms(Pow) if not p.is_number) and (a.is_nonpositive or a.is_nonnegative):
            free = list(a.free_symbols)
            p = {}
            for i in free:
                v = _monotonic_sign(i)
                if v is None:
                    return
                p[i] = v or (_eps if i.is_nonnegative else -_eps)
            v = a.xreplace(p)
    if v is not None:
        rv = v + c
        if v.is_nonnegative and rv.is_positive:
            return rv.subs(_eps, 0)
        if v.is_nonpositive and rv.is_negative:
            return rv.subs(_eps, 0)
dataset/ETHPy150Open sympy/sympy/sympy/core/exprtools.py/_monotonic_sign/swapped operands of `<=`