"""
This module contains functions to:

    - solve a single equation for a single variable, in any domain either real or complex.

    - solve a single transcendental equation for a single variable in any domain either real or complex.
      (currently supports solving in real domain only)

    - solve a system of linear equations with N variables and M equations.

    - solve a system of Non Linear Equations with N variables and M equations
"""
from __future__ import print_function, division

from sympy.core.sympify import sympify
from sympy.core import (S, Pow, Dummy, pi, Expr, Wild, Mul, Equality,
                        Add)
from sympy.core.containers import Tuple
from sympy.core.facts import InconsistentAssumptions
from sympy.core.numbers import I, Number, Rational, oo
from sympy.core.function import (Lambda, expand_complex, AppliedUndef,
                                expand_log)
from sympy.core.relational import Eq, Ne
from sympy.core.symbol import Symbol
from sympy.simplify.simplify import simplify, fraction, trigsimp
from sympy.simplify import powdenest, logcombine
from sympy.functions import (log, Abs, tan, cot, sin, cos, sec, csc, exp,
                             acos, asin, acsc, asec, arg,
                             piecewise_fold, Piecewise)
from sympy.functions.elementary.trigonometric import (TrigonometricFunction,
                                                      HyperbolicFunction)
from sympy.functions.elementary.miscellaneous import real_root
from sympy.logic.boolalg import And
from sympy.sets import (FiniteSet, EmptySet, imageset, Interval, Intersection,
                        Union, ConditionSet, ImageSet, Complement, Contains)
from sympy.sets.sets import Set
from sympy.matrices import Matrix
from sympy.polys import (roots, Poly, degree, together, PolynomialError,
                         RootOf, factor)
from sympy.solvers.solvers import (checksol, denoms, unrad,
    _simple_dens, recast_to_symbols)
from sympy.solvers.polysys import solve_poly_system
from sympy.solvers.inequalities import solve_univariate_inequality
from sympy.utilities import filldedent
from sympy.utilities.iterables import numbered_symbols
from sympy.calculus.util import periodicity, continuous_domain
from sympy.core.compatibility import ordered, default_sort_key, is_sequence

from types import GeneratorType


def _masked(f, *atoms):
    """Return ``f``, with all objects given by ``atoms`` replaced with
    Dummy symbols, ``d``, and the list of replacements, ``(d, e)``,
    where ``e`` is an object of type given by ``atoms`` in which
    any other instances of atoms have been recursively replaced with
    Dummy symbols, too. The tuples are ordered so that if they are
    applied in sequence, the origin ``f`` will be restored.

    Examples
    ========

    >>> from sympy import cos
    >>> from sympy.abc import x
    >>> from sympy.solvers.solveset import _masked

    >>> f = cos(cos(x) + 1)
    >>> f, reps = _masked(cos(1 + cos(x)), cos)
    >>> f
    _a1
    >>> reps
    [(_a1, cos(_a0 + 1)), (_a0, cos(x))]
    >>> for d, e in reps:
    ...     f = f.xreplace({d: e})
    >>> f
    cos(cos(x) + 1)
    """
    sym = numbered_symbols('a', cls=Dummy, real=True)
    mask = []
    for a in ordered(f.atoms(*atoms)):
        for i in mask:
            a = a.replace(*i)
        mask.append((a, next(sym)))
    for i, (o, n) in enumerate(mask):
        f = f.replace(o, n)
        mask[i] = (n, o)
    mask = list(reversed(mask))
    return f, mask


def _invert(f_x, y, x, domain=S.Complexes):
    r"""
    Reduce the complex valued equation ``f(x) = y`` to a set of equations
    ``{g(x) = h_1(y), g(x) = h_2(y), ..., g(x) = h_n(y) }`` where ``g(x)`` is
    a simpler function than ``f(x)``.  The return value is a tuple ``(g(x),
    set_h)``, where ``g(x)`` is a function of ``x`` and ``set_h`` is
    the set of function ``{h_1(y), h_2(y), ..., h_n(y)}``.
    Here, ``y`` is not necessarily a symbol.

    The ``set_h`` contains the functions, along with the information
    about the domain in which they are valid, through set
    operations. For instance, if ``y = Abs(x) - n`` is inverted
    in the real domain, then ``set_h`` is not simply
    `{-n, n}` as the nature of `n` is unknown; rather, it is:
    `Intersection([0, oo) {n}) U Intersection((-oo, 0], {-n})`

    By default, the complex domain is used which means that inverting even
    seemingly simple functions like ``exp(x)`` will give very different
    results from those obtained in the real domain.
    (In the case of ``exp(x)``, the inversion via ``log`` is multi-valued
    in the complex domain, having infinitely many branches.)

    If you are working with real values only (or you are not sure which
    function to use) you should probably set the domain to
    ``S.Reals`` (or use `invert\_real` which does that automatically).


    Examples
    ========

    >>> from sympy.solvers.solveset import invert_complex, invert_real
    >>> from sympy.abc import x, y
    >>> from sympy import exp, log

    When does exp(x) == y?

    >>> invert_complex(exp(x), y, x)
    (x, ImageSet(Lambda(_n, I*(2*_n*pi + arg(y)) + log(Abs(y))), Integers))
    >>> invert_real(exp(x), y, x)
    (x, Intersection(Reals, {log(y)}))

    When does exp(x) == 1?

    >>> invert_complex(exp(x), 1, x)
    (x, ImageSet(Lambda(_n, 2*_n*I*pi), Integers))
    >>> invert_real(exp(x), 1, x)
    (x, {0})

    See Also
    ========
    invert_real, invert_complex
    """
    x = sympify(x)
    if not x.is_Symbol:
        raise ValueError("x must be a symbol")
    f_x = sympify(f_x)
    if x not in f_x.free_symbols:
        raise ValueError("Inverse of constant function doesn't exist")
    y = sympify(y)
    if x in y.free_symbols:
        raise ValueError("y should be independent of x ")

    if domain.is_subset(S.Reals):
        x1, s = _invert_real(f_x, FiniteSet(y), x)
    else:
        x1, s = _invert_complex(f_x, FiniteSet(y), x)

    if not isinstance(s, FiniteSet) or x1 != x:
        return x1, s

    return x1, s.intersection(domain)


invert_complex = _invert


def invert_real(f_x, y, x, domain=S.Reals):
    """
    Inverts a real-valued function. Same as _invert, but sets
    the domain to ``S.Reals`` before inverting.
    """
    return _invert(f_x, y, x, domain)


def _invert_real(f, g_ys, symbol):
    """Helper function for _invert."""

    if f == symbol:
        return (f, g_ys)

    n = Dummy('n', real=True)

    if hasattr(f, 'inverse') and not isinstance(f, (
            TrigonometricFunction,
            HyperbolicFunction,
            )):
        if len(f.args) > 1:
            raise ValueError("Only functions with one argument are supported.")
        return _invert_real(f.args[0],
                            imageset(Lambda(n, f.inverse()(n)), g_ys),
                            symbol)

    if isinstance(f, Abs):
        return _invert_abs(f.args[0], g_ys, symbol)

    if f.is_Add:
        # f = g + h
        g, h = f.as_independent(symbol)
        if g is not S.Zero:
            return _invert_real(h, imageset(Lambda(n, n - g), g_ys), symbol)

    if f.is_Mul:
        # f = g*h
        g, h = f.as_independent(symbol)

        if g is not S.One:
            return _invert_real(h, imageset(Lambda(n, n/g), g_ys), symbol)

    if f.is_Pow:
        base, expo = f.args
        base_has_sym = base.has(symbol)
        expo_has_sym = expo.has(symbol)

        if not expo_has_sym:
            res = imageset(Lambda(n, real_root(n, expo)), g_ys)
            if expo.is_rational:
                numer, denom = expo.as_numer_denom()
                if denom % 2 == 0:
                    base_positive = solveset(base >= 0, symbol, S.Reals)
                    res = imageset(Lambda(n, real_root(n, expo)
                        ), g_ys.intersect(
                        Interval.Ropen(S.Zero, S.Infinity)))
                    _inv, _set = _invert_real(base, res, symbol)
                    return (_inv, _set.intersect(base_positive))

                elif numer % 2 == 0:
                        n = Dummy('n')
                        neg_res = imageset(Lambda(n, -n), res)
                        return _invert_real(base, res + neg_res, symbol)

                else:
                    return _invert_real(base, res, symbol)
            else:
                if not base.is_positive:
                    raise ValueError("x**w where w is irrational is not "
                                     "defined for negative x")
                return _invert_real(base, res, symbol)

        if not base_has_sym:
            rhs = g_ys.args[0]
            if base.is_positive:
                return _invert_real(expo,
                    imageset(Lambda(n, log(n, base, evaluate=False)), g_ys), symbol)
            elif base.is_negative:
                from sympy.core.power import integer_log
                s, b = integer_log(rhs, base)
                if b:
                    return _invert_real(expo, FiniteSet(s), symbol)
                else:
                    return _invert_real(expo, S.EmptySet, symbol)
            elif base.is_zero:
                one = Eq(rhs, 1)
                if one == S.true:
                    # special case: 0**x - 1
                    return _invert_real(expo, FiniteSet(0), symbol)
                elif one == S.false:
                    return _invert_real(expo, S.EmptySet, symbol)


    if isinstance(f, TrigonometricFunction):
        if isinstance(g_ys, FiniteSet):
            def inv(trig):
                if isinstance(f, (sin, csc)):
                    F = asin if isinstance(f, sin) else acsc
                    return (lambda a: n*pi + (-1)**n*F(a),)
                if isinstance(f, (cos, sec)):
                    F = acos if isinstance(f, cos) else asec
                    return (
                        lambda a: 2*n*pi + F(a),
                        lambda a: 2*n*pi - F(a),)
                if isinstance(f, (tan, cot)):
                    return (lambda a: n*pi + f.inverse()(a),)

            n = Dummy('n', integer=True)
            invs = S.EmptySet
            for L in inv(f):
                invs += Union(*[imageset(Lambda(n, L(g)), S.Integers) for g in g_ys])
            return _invert_real(f.args[0], invs, symbol)

    return (f, g_ys)


def _invert_complex(f, g_ys, symbol):
    """Helper function for _invert."""

    if f == symbol:
        return (f, g_ys)

    n = Dummy('n')

    if f.is_Add:
        # f = g + h
        g, h = f.as_independent(symbol)
        if g is not S.Zero:
            return _invert_complex(h, imageset(Lambda(n, n - g), g_ys), symbol)

    if f.is_Mul:
        # f = g*h
        g, h = f.as_independent(symbol)

        if g is not S.One:
            if g in set([S.NegativeInfinity, S.ComplexInfinity, S.Infinity]):
                return (h, S.EmptySet)
            return _invert_complex(h, imageset(Lambda(n, n/g), g_ys), symbol)

    if hasattr(f, 'inverse') and \
       not isinstance(f, TrigonometricFunction) and \
       not isinstance(f, HyperbolicFunction) and \
       not isinstance(f, exp):
        if len(f.args) > 1:
            raise ValueError("Only functions with one argument are supported.")
        return _invert_complex(f.args[0],
                               imageset(Lambda(n, f.inverse()(n)), g_ys), symbol)

    if isinstance(f, exp):
        if isinstance(g_ys, FiniteSet):
            exp_invs = Union(*[imageset(Lambda(n, I*(2*n*pi + arg(g_y)) +
                                               log(Abs(g_y))), S.Integers)
                               for g_y in g_ys if g_y != 0])
            return _invert_complex(f.args[0], exp_invs, symbol)

    return (f, g_ys)


def _invert_abs(f, g_ys, symbol):
    """Helper function for inverting absolute value functions.

    Returns the complete result of inverting an absolute value
    function along with the conditions which must also be satisfied.

    If it is certain that all these conditions are met, a `FiniteSet`
    of all possible solutions is returned. If any condition cannot be
    satisfied, an `EmptySet` is returned. Otherwise, a `ConditionSet`
    of the solutions, with all the required conditions specified, is
    returned.

    """
    if not g_ys.is_FiniteSet:
        # this could be used for FiniteSet, but the
        # results are more compact if they aren't, e.g.
        # ConditionSet(x, Contains(n, Interval(0, oo)), {-n, n}) vs
        # Union(Intersection(Interval(0, oo), {n}), Intersection(Interval(-oo, 0), {-n}))
        # for the solution of abs(x) - n
        pos = Intersection(g_ys, Interval(0, S.Infinity))
        parg = _invert_real(f, pos, symbol)
        narg = _invert_real(-f, pos, symbol)
        if parg[0] != narg[0]:
            raise NotImplementedError
        return parg[0], Union(narg[1], parg[1])

    # check conditions: all these must be true. If any are unknown
    # then return them as conditions which must be satisfied
    unknown = []
    for a in g_ys.args:
        ok = a.is_nonnegative if a.is_Number else a.is_positive
        if ok is None:
            unknown.append(a)
        elif not ok:
            return symbol, S.EmptySet
    if unknown:
        conditions = And(*[Contains(i, Interval(0, oo))
            for i in unknown])
    else:
        conditions = True
    n = Dummy('n', real=True)
    # this is slightly different than above: instead of solving
    # +/-f on positive values, here we solve for f on +/- g_ys
    g_x, values = _invert_real(f, Union(
        imageset(Lambda(n, n), g_ys),
        imageset(Lambda(n, -n), g_ys)), symbol)
    return g_x, ConditionSet(g_x, conditions, values)


def domain_check(f, symbol, p):
    """Returns False if point p is infinite or any subexpression of f
    is infinite or becomes so after replacing symbol with p. If none of
    these conditions is met then True will be returned.

    Examples
    ========

    >>> from sympy import Mul, oo
    >>> from sympy.abc import x
    >>> from sympy.solvers.solveset import domain_check
    >>> g = 1/(1 + (1/(x + 1))**2)
    >>> domain_check(g, x, -1)
    False
    >>> domain_check(x**2, x, 0)
    True
    >>> domain_check(1/x, x, oo)
    False

    * The function relies on the assumption that the original form
      of the equation has not been changed by automatic simplification.

    >>> domain_check(x/x, x, 0) # x/x is automatically simplified to 1
    True

    * To deal with automatic evaluations use evaluate=False:

    >>> domain_check(Mul(x, 1/x, evaluate=False), x, 0)
    False
    """
    f, p = sympify(f), sympify(p)
    if p.is_infinite:
        return False
    return _domain_check(f, symbol, p)


def _domain_check(f, symbol, p):
    # helper for domain check
    if f.is_Atom and f.is_finite:
        return True
    elif f.subs(symbol, p).is_infinite:
        return False
    else:
        return all([_domain_check(g, symbol, p)
                    for g in f.args])


def _is_finite_with_finite_vars(f, domain=S.Complexes):
    """
    Return True if the given expression is finite. For symbols that
    don't assign a value for `complex` and/or `real`, the domain will
    be used to assign a value; symbols that don't assign a value
    for `finite` will be made finite. All other assumptions are
    left unmodified.
    """
    def assumptions(s):
        A = s.assumptions0
        A.setdefault('finite', A.get('finite', True))
        if domain.is_subset(S.Reals):
            # if this gets set it will make complex=True, too
            A.setdefault('real', True)
        else:
            # don't change 'real' because being complex implies
            # nothing about being real
            A.setdefault('complex', True)
        return A

    reps = {s: Dummy(**assumptions(s)) for s in f.free_symbols}
    return f.xreplace(reps).is_finite


def _is_function_class_equation(func_class, f, symbol):
    """ Tests whether the equation is an equation of the given function class.

    The given equation belongs to the given function class if it is
    comprised of functions of the function class which are multiplied by
    or added to expressions independent of the symbol. In addition, the
    arguments of all such functions must be linear in the symbol as well.

    Examples
    ========

    >>> from sympy.solvers.solveset import _is_function_class_equation
    >>> from sympy import tan, sin, tanh, sinh, exp
    >>> from sympy.abc import x
    >>> from sympy.functions.elementary.trigonometric import (TrigonometricFunction,
    ... HyperbolicFunction)
    >>> _is_function_class_equation(TrigonometricFunction, exp(x) + tan(x), x)
    False
    >>> _is_function_class_equation(TrigonometricFunction, tan(x) + sin(x), x)
    True
    >>> _is_function_class_equation(TrigonometricFunction, tan(x**2), x)
    False
    >>> _is_function_class_equation(TrigonometricFunction, tan(x + 2), x)
    True
    >>> _is_function_class_equation(HyperbolicFunction, tanh(x) + sinh(x), x)
    True
    """
    if f.is_Mul or f.is_Add:
        return all(_is_function_class_equation(func_class, arg, symbol)
                   for arg in f.args)

    if f.is_Pow:
        if not f.exp.has(symbol):
            return _is_function_class_equation(func_class, f.base, symbol)
        else:
            return False

    if not f.has(symbol):
        return True

    if isinstance(f, func_class):
        try:
            g = Poly(f.args[0], symbol)
            return g.degree() <= 1
        except PolynomialError:
            return False
    else:
        return False


def _solve_as_rational(f, symbol, domain):
    """ solve rational functions"""
    f = together(f, deep=True)
    g, h = fraction(f)
    if not h.has(symbol):
        try:
            return _solve_as_poly(g, symbol, domain)
        except NotImplementedError:
            # The polynomial formed from g could end up having
            # coefficients in a ring over which finding roots
            # isn't implemented yet, e.g. ZZ[a] for some symbol a
            return ConditionSet(symbol, Eq(f, 0), domain)
    else:
        valid_solns = _solveset(g, symbol, domain)
        invalid_solns = _solveset(h, symbol, domain)
        return valid_solns - invalid_solns


def _solve_trig(f, symbol, domain):
    """Function to call other helpers to solve trigonometric equations """
    sol1 = sol = None
    try:
        sol1 = _solve_trig1(f, symbol, domain)
    except BaseException as error:
        pass
    if sol1 is None or isinstance(sol1, ConditionSet):
        try:
            sol = _solve_trig2(f, symbol, domain)
        except BaseException as error:
            sol = sol1
        if isinstance(sol1, ConditionSet) and isinstance(sol, ConditionSet):
            if sol1.count_ops() < sol.count_ops():
                sol = sol1
    else:
        sol = sol1
    if sol is None:
        raise NotImplementedError(filldedent('''
            Solution to this kind of trigonometric equations
            is yet to be implemented'''))
    return sol


def _solve_trig1(f, symbol, domain):
    """Primary Helper to solve trigonometric equations """
    f = trigsimp(f)
    f_original = f
    f = f.rewrite(exp)
    f = together(f)
    g, h = fraction(f)
    y = Dummy('y')
    g, h = g.expand(), h.expand()
    g, h = g.subs(exp(I*symbol), y), h.subs(exp(I*symbol), y)
    if g.has(symbol) or h.has(symbol):
        return ConditionSet(symbol, Eq(f, 0), S.Reals)

    solns = solveset_complex(g, y) - solveset_complex(h, y)
    if isinstance(solns, ConditionSet):
        raise NotImplementedError

    if isinstance(solns, FiniteSet):
        if any(isinstance(s, RootOf) for s in solns):
            raise NotImplementedError
        result = Union(*[invert_complex(exp(I*symbol), s, symbol)[1]
                       for s in solns])
        return Intersection(result, domain)
    elif solns is S.EmptySet:
        return S.EmptySet
    else:
        return ConditionSet(symbol, Eq(f_original, 0), S.Reals)


def _solve_trig2(f, symbol, domain):
    """Secondary helper to solve trigonometric equations,
    called when first helper fails """
    from sympy import ilcm, igcd, expand_trig, degree, simplify
    f = trigsimp(f)
    f_original = f
    trig_functions = f.atoms(sin, cos, tan, sec, cot, csc)
    trig_arguments = [e.args[0] for e in trig_functions]
    denominators = []
    numerators = []

    for ar in trig_arguments:
        try:
            poly_ar = Poly(ar, symbol)

        except ValueError:
            raise ValueError("give up, we can't solve if this is not a polynomial in x")
        if poly_ar.degree() > 1:  # degree >1 still bad
            raise ValueError("degree of variable inside polynomial should not exceed one")
        if poly_ar.degree() == 0:  # degree 0, don't care
            continue
        c = poly_ar.all_coeffs()[0]   # got the coefficient of 'symbol'
        numerators.append(Rational(c).p)
        denominators.append(Rational(c).q)

    x = Dummy('x')

    # ilcm() and igcd() require more than one argument
    if len(numerators) > 1:
        mu = Rational(2)*ilcm(*denominators)/igcd(*numerators)
    else:
        assert len(numerators) == 1
        mu = Rational(2)*denominators[0]/numerators[0]

    f = f.subs(symbol, mu*x)
    f = f.rewrite(tan)
    f = expand_trig(f)
    f = together(f)

    g, h = fraction(f)
    y = Dummy('y')
    g, h = g.expand(), h.expand()
    g, h = g.subs(tan(x), y), h.subs(tan(x), y)

    if g.has(x) or h.has(x):
        return ConditionSet(symbol, Eq(f_original, 0), domain)
    solns = solveset(g, y, S.Reals) - solveset(h, y, S.Reals)

    if isinstance(solns, FiniteSet):
        result = Union(*[invert_real(tan(symbol/mu), s, symbol)[1]
                       for s in solns])
        dsol = invert_real(tan(symbol/mu), oo, symbol)[1]
        if degree(h) > degree(g):                   # If degree(denom)>degree(num) then there
            result = Union(result, dsol)            # would be another sol at Lim(denom-->oo)
        return Intersection(result, domain)
    elif solns is S.EmptySet:
        return S.EmptySet
    else:
        return ConditionSet(symbol, Eq(f_original, 0), S.Reals)


def _solve_as_poly(f, symbol, domain=S.Complexes):
    """
    Solve the equation using polynomial techniques if it already is a
    polynomial equation or, with a change of variables, can be made so.
    """
    result = None
    if f.is_polynomial(symbol):
        solns = roots(f, symbol, cubics=True, quartics=True,
                      quintics=True, domain='EX')
        num_roots = sum(solns.values())
        if degree(f, symbol) <= num_roots:
            result = FiniteSet(*solns.keys())
        else:
            poly = Poly(f, symbol)
            solns = poly.all_roots()
            if poly.degree() <= len(solns):
                result = FiniteSet(*solns)
            else:
                result = ConditionSet(symbol, Eq(f, 0), domain)
    else:
        poly = Poly(f)
        if poly is None:
            result = ConditionSet(symbol, Eq(f, 0), domain)
        gens = [g for g in poly.gens if g.has(symbol)]

        if len(gens) == 1:
            poly = Poly(poly, gens[0])
            gen = poly.gen
            deg = poly.degree()
            poly = Poly(poly.as_expr(), poly.gen, composite=True)
            poly_solns = FiniteSet(*roots(poly, cubics=True, quartics=True,
                                          quintics=True).keys())

            if len(poly_solns) < deg:
                result = ConditionSet(symbol, Eq(f, 0), domain)

            if gen != symbol:
                y = Dummy('y')
                inverter = invert_real if domain.is_subset(S.Reals) else invert_complex
                lhs, rhs_s = inverter(gen, y, symbol)
                if lhs == symbol:
                    result = Union(*[rhs_s.subs(y, s) for s in poly_solns])
                else:
                    result = ConditionSet(symbol, Eq(f, 0), domain)
        else:
            result = ConditionSet(symbol, Eq(f, 0), domain)

    if result is not None:
        if isinstance(result, FiniteSet):
            # this is to simplify solutions like -sqrt(-I) to sqrt(2)/2
            # - sqrt(2)*I/2. We are not expanding for solution with symbols
            # or undefined functions because that makes the solution more complicated.
            # For example, expand_complex(a) returns re(a) + I*im(a)
            if all([s.atoms(Symbol, AppliedUndef) == set() and not isinstance(s, RootOf)
                    for s in result]):
                s = Dummy('s')
                result = imageset(Lambda(s, expand_complex(s)), result)
        if isinstance(result, FiniteSet):
            result = result.intersection(domain)
        return result
    else:
        return ConditionSet(symbol, Eq(f, 0), domain)


def _has_rational_power(expr, symbol):
    """
    Returns (bool, den) where bool is True if the term has a
    non-integer rational power and den is the denominator of the
    expression's exponent.

    Examples
    ========

    >>> from sympy.solvers.solveset import _has_rational_power
    >>> from sympy import sqrt
    >>> from sympy.abc import x
    >>> _has_rational_power(sqrt(x), x)
    (True, 2)
    >>> _has_rational_power(x**2, x)
    (False, 1)
    """
    a, p, q = Wild('a'), Wild('p'), Wild('q')
    pattern_match = expr.match(a*p**q) or {}
    if pattern_match.get(a, S.Zero) is S.Zero:
        return (False, S.One)
    elif p not in pattern_match.keys():
        return (False, S.One)
    elif isinstance(pattern_match[q], Rational) \
            and pattern_match[p].has(symbol):
        if not pattern_match[q].q == S.One:
            return (True, pattern_match[q].q)

    if not isinstance(pattern_match[a], Pow) \
            or isinstance(pattern_match[a], Mul):
        return (False, S.One)
    else:
        return _has_rational_power(pattern_match[a], symbol)


def _solve_radical(f, symbol, solveset_solver):
    """ Helper function to solve equations with radicals """
    eq, cov = unrad(f)
    if not cov:
        result = solveset_solver(eq, symbol) - \
            Union(*[solveset_solver(g, symbol) for g in denoms(f, symbol)])
    else:
        y, yeq = cov
        if not solveset_solver(y - I, y):
            yreal = Dummy('yreal', real=True)
            yeq = yeq.xreplace({y: yreal})
            eq = eq.xreplace({y: yreal})
            y = yreal
        g_y_s = solveset_solver(yeq, symbol)
        f_y_sols = solveset_solver(eq, y)
        result = Union(*[imageset(Lambda(y, g_y), f_y_sols)
                         for g_y in g_y_s])

    if isinstance(result, Complement) or isinstance(result,ConditionSet):
        solution_set = result
    else:
        f_set = []  # solutions for FiniteSet
        c_set = []  # solutions for ConditionSet
        for s in result:
            if checksol(f, symbol, s):
                f_set.append(s)
            else:
                c_set.append(s)
        solution_set = FiniteSet(*f_set) + ConditionSet(symbol, Eq(f, 0), FiniteSet(*c_set))

    return solution_set


def _solve_abs(f, symbol, domain):
    """ Helper function to solve equation involving absolute value function """
    if not domain.is_subset(S.Reals):
        raise ValueError(filldedent('''
            Absolute values cannot be inverted in the
            complex domain.'''))
    p, q, r = Wild('p'), Wild('q'), Wild('r')
    pattern_match = f.match(p*Abs(q) + r) or {}
    f_p, f_q, f_r = [pattern_match.get(i, S.Zero) for i in (p, q, r)]

    if not (f_p.is_zero or f_q.is_zero):
        domain = continuous_domain(f_q, symbol, domain)
        q_pos_cond = solve_univariate_inequality(f_q >= 0, symbol,
                                                 relational=False, domain=domain, continuous=True)
        q_neg_cond = q_pos_cond.complement(domain)

        sols_q_pos = solveset_real(f_p*f_q + f_r,
                                           symbol).intersect(q_pos_cond)
        sols_q_neg = solveset_real(f_p*(-f_q) + f_r,
                                           symbol).intersect(q_neg_cond)
        return Union(sols_q_pos, sols_q_neg)
    else:
        return ConditionSet(symbol, Eq(f, 0), domain)


def solve_decomposition(f, symbol, domain):
    """
    Function to solve equations via the principle of "Decomposition
    and Rewriting".

    Examples
    ========
    >>> from sympy import exp, sin, Symbol, pprint, S
    >>> from sympy.solvers.solveset import solve_decomposition as sd
    >>> x = Symbol('x')
    >>> f1 = exp(2*x) - 3*exp(x) + 2
    >>> sd(f1, x, S.Reals)
    {0, log(2)}
    >>> f2 = sin(x)**2 + 2*sin(x) + 1
    >>> pprint(sd(f2, x, S.Reals), use_unicode=False)
              3*pi
    {2*n*pi + ---- | n in Integers}
               2
    >>> f3 = sin(x + 2)
    >>> pprint(sd(f3, x, S.Reals), use_unicode=False)
    {2*n*pi - 2 | n in Integers} U {2*n*pi - 2 + pi | n in Integers}

    """
    from sympy.solvers.decompogen import decompogen
    from sympy.calculus.util import function_range
    # decompose the given function
    g_s = decompogen(f, symbol)
    # `y_s` represents the set of values for which the function `g` is to be
    # solved.
    # `solutions` represent the solutions of the equations `g = y_s` or
    # `g = 0` depending on the type of `y_s`.
    # As we are interested in solving the equation: f = 0
    y_s = FiniteSet(0)
    for g in g_s:
        frange = function_range(g, symbol, domain)
        y_s = Intersection(frange, y_s)
        result = S.EmptySet
        if isinstance(y_s, FiniteSet):
            for y in y_s:
                solutions = solveset(Eq(g, y), symbol, domain)
                if not isinstance(solutions, ConditionSet):
                    result += solutions

        else:
            if isinstance(y_s, ImageSet):
                iter_iset = (y_s,)

            elif isinstance(y_s, Union):
                iter_iset = y_s.args

            elif isinstance(y_s, EmptySet):
                # y_s is not in the range of g in g_s, so no solution exists
                #in the given domain
                return y_s

            for iset in iter_iset:
                new_solutions = solveset(Eq(iset.lamda.expr, g), symbol, domain)
                dummy_var = tuple(iset.lamda.expr.free_symbols)[0]
                base_set = iset.base_set
                if isinstance(new_solutions, FiniteSet):
                    new_exprs = new_solutions

                elif isinstance(new_solutions, Intersection):
                    if isinstance(new_solutions.args[1], FiniteSet):
                        new_exprs = new_solutions.args[1]

                for new_expr in new_exprs:
                    result += ImageSet(Lambda(dummy_var, new_expr), base_set)

        if result is S.EmptySet:
            return ConditionSet(symbol, Eq(f, 0), domain)

        y_s = result

    return y_s


def _solveset(f, symbol, domain, _check=False):
    """Helper for solveset to return a result from an expression
    that has already been sympify'ed and is known to contain the
    given symbol."""
    # _check controls whether the answer is checked or not
    from sympy.simplify.simplify import signsimp

    orig_f = f
    if f.is_Mul:
        coeff, f = f.as_independent(symbol, as_Add=False)
        if coeff in set([S.ComplexInfinity, S.NegativeInfinity, S.Infinity]):
            f = together(orig_f)
    elif f.is_Add:
        a, h = f.as_independent(symbol)
        m, h = h.as_independent(symbol, as_Add=False)
        if m not in set([S.ComplexInfinity, S.Zero, S.Infinity,
                              S.NegativeInfinity]):
            f = a/m + h  # XXX condition `m != 0` should be added to soln

    # assign the solvers to use
    solver = lambda f, x, domain=domain: _solveset(f, x, domain)
    inverter = lambda f, rhs, symbol: _invert(f, rhs, symbol, domain)

    result = EmptySet()

    if f.expand().is_zero:
        return domain
    elif not f.has(symbol):
        return EmptySet()
    elif f.is_Mul and all(_is_finite_with_finite_vars(m, domain)
            for m in f.args):
        # if f(x) and g(x) are both finite we can say that the solution of
        # f(x)*g(x) == 0 is same as Union(f(x) == 0, g(x) == 0) is not true in
        # general. g(x) can grow to infinitely large for the values where
        # f(x) == 0. To be sure that we are not silently allowing any
        # wrong solutions we are using this technique only if both f and g are
        # finite for a finite input.
        result = Union(*[solver(m, symbol) for m in f.args])
    elif _is_function_class_equation(TrigonometricFunction, f, symbol) or \
            _is_function_class_equation(HyperbolicFunction, f, symbol):
        result = _solve_trig(f, symbol, domain)
    elif isinstance(f, arg):
        a = f.args[0]
        result = solveset_real(a > 0, symbol)
    elif f.is_Piecewise:
        result = EmptySet()
        expr_set_pairs = f.as_expr_set_pairs(domain)
        for (expr, in_set) in expr_set_pairs:
            if in_set.is_Relational:
                in_set = in_set.as_set()
            solns = solver(expr, symbol, in_set)
            result += solns
    elif isinstance(f, Eq):
        result = solver(Add(f.lhs, - f.rhs, evaluate=False), symbol, domain)

    elif f.is_Relational:
        if not domain.is_subset(S.Reals):
            raise NotImplementedError(filldedent('''
                Inequalities in the complex domain are
                not supported. Try the real domain by
                setting domain=S.Reals'''))
        try:
            result = solve_univariate_inequality(
            f, symbol, domain=domain, relational=False)
        except NotImplementedError:
            result = ConditionSet(symbol, f, domain)
        return result
    else:
        lhs, rhs_s = inverter(f, 0, symbol)
        if lhs == symbol:
            # do some very minimal simplification since
            # repeated inversion may have left the result
            # in a state that other solvers (e.g. poly)
            # would have simplified; this is done here
            # rather than in the inverter since here it
            # is only done once whereas there it would
            # be repeated for each step of the inversion
            if isinstance(rhs_s, FiniteSet):
                rhs_s = FiniteSet(*[Mul(*
                    signsimp(i).as_content_primitive())
                    for i in rhs_s])
            result = rhs_s

        elif isinstance(rhs_s, FiniteSet):
            for equation in [lhs - rhs for rhs in rhs_s]:
                if equation == f:
                    if any(_has_rational_power(g, symbol)[0]
                           for g in equation.args) or _has_rational_power(
                           equation, symbol)[0]:
                        result += _solve_radical(equation,
                                                 symbol,
                                                 solver)
                    elif equation.has(Abs):
                        result += _solve_abs(f, symbol, domain)
                    else:
                        result_rational = _solve_as_rational(equation, symbol, domain)
                        if isinstance(result_rational, ConditionSet):
                            # may be a transcendental type equation
                            result += _transolve(equation, symbol, domain)
                        else:
                            result += result_rational
                else:
                    result += solver(equation, symbol)

        elif rhs_s is not S.EmptySet:
            result = ConditionSet(symbol, Eq(f, 0), domain)

    if isinstance(result, ConditionSet):
        num, den = f.as_numer_denom()
        if den.has(symbol):
            _result = _solveset(num, symbol, domain)
            if not isinstance(_result, ConditionSet):
                singularities = _solveset(den, symbol, domain)
                result = _result - singularities

    if _check:
        if isinstance(result, ConditionSet):
            # it wasn't solved or has enumerated all conditions
            # -- leave it alone
            return result

        # whittle away all but the symbol-containing core
        # to use this for testing
        fx = orig_f.as_independent(symbol, as_Add=True)[1]
        fx = fx.as_independent(symbol, as_Add=False)[1]

        if isinstance(result, FiniteSet):
            # check the result for invalid solutions
            result = FiniteSet(*[s for s in result
                      if isinstance(s, RootOf)
                      or domain_check(fx, symbol, s)])

    return result


def _term_factors(f):
    """
    Iterator to get the factors of all terms present
    in the given equation.

    Parameters
    ==========
    f : Expr
        Equation that needs to be addressed

    Returns
    =======
    Factors of all terms present in the equation.

    Examples
    ========

    >>> from sympy import symbols
    >>> from sympy.solvers.solveset import _term_factors
    >>> x = symbols('x')
    >>> list(_term_factors(-2 - x**2 + x*(x + 1)))
    [-2, -1, x**2, x, x + 1]
    """
    for add_arg in Add.make_args(f):
        for mul_arg in Mul.make_args(add_arg):
            yield mul_arg


def _solve_exponential(lhs, rhs, symbol, domain):
    r"""
    Helper function for solving (supported) exponential equations.

    Exponential equations are the sum of (currently) at most
    two terms with one or both of them having a power with a
    symbol-dependent exponent.

    For example

    .. math:: 5^{2x + 3} - 5^{3x - 1}

    .. math:: 4^{5 - 9x} - e^{2 - x}

    Parameters
    ==========

    lhs, rhs : Expr
        The exponential equation to be solved, `lhs = rhs`

    symbol : Symbol
        The variable in which the equation is solved

    domain : Set
        A set over which the equation is solved.

    Returns
    =======

    A set of solutions satisfying the given equation.
    A `ConditionSet` if the equation is unsolvable or
    if the assumptions are not properly defined, in that case
    a different style of `ConditionSet` is returned having the
    solution(s) of the equation with the desired assumptions.

    Examples
    ========

    >>> from sympy.solvers.solveset import _solve_exponential as solve_expo
    >>> from sympy import symbols, S
    >>> x = symbols('x', real=True)
    >>> a, b = symbols('a b')
    >>> solve_expo(2**x + 3**x - 5**x, 0, x, S.Reals)  # not solvable
    ConditionSet(x, Eq(2**x + 3**x - 5**x, 0), Reals)
    >>> solve_expo(a**x - b**x, 0, x, S.Reals)  # solvable but incorrect assumptions
    ConditionSet(x, (a > 0) & (b > 0), {0})
    >>> solve_expo(3**(2*x) - 2**(x + 3), 0, x, S.Reals)
    {-3*log(2)/(-2*log(3) + log(2))}
    >>> solve_expo(2**x - 4**x, 0, x, S.Reals)
    {0}

    * Proof of correctness of the method

    The logarithm function is the inverse of the exponential function.
    The defining relation between exponentiation and logarithm is:

    .. math:: {log_b x} = y \enspace if \enspace b^y = x

    Therefore if we are given an equation with exponent terms, we can
    convert every term to its corresponding log form. This is achieved by
    taking logarithms and expanding the equation using log identities
    so that it can easily be handled by `solveset`.

    For example:

    .. math:: 3^{2x} = 2^{x + 3}

    Taking log both sides will reduce the equation to

    .. math:: (2x)*log(3) = (x + 3)*log(2)

    This form can be easily handed by `solveset`.
    """
    unsolved_result = ConditionSet(symbol, Eq(lhs - rhs), domain)
    newlhs = powdenest(lhs)
    if lhs != newlhs:
        # it may also be advantageous to factor the new expr
        return _solveset(factor(newlhs - rhs), symbol, domain)  # try again with _solveset

    if not (isinstance(lhs, Add) and len(lhs.args) == 2):
        # solving for the sum of more than two powers is possible
        # but not yet implemented
        return unsolved_result

    if rhs != 0:
        return unsolved_result

    a, b = list(ordered(lhs.args))
    a_term = a.as_independent(symbol)[1]
    b_term = b.as_independent(symbol)[1]

    a_base, a_exp = a_term.base, a_term.exp
    b_base, b_exp = b_term.base, b_term.exp

    from sympy.functions.elementary.complexes import im

    if domain.is_subset(S.Reals):
        conditions = And(
            a_base > 0,
            b_base > 0,
            Eq(im(a_exp), 0),
            Eq(im(b_exp), 0))
    else:
        conditions = And(
            Ne(a_base, 0),
            Ne(b_base, 0))

    L, R = map(lambda i: expand_log(log(i), force=True), (a, -b))
    solutions = _solveset(L - R, symbol, domain)

    return ConditionSet(symbol, conditions, solutions)


def _is_exponential(f, symbol):
    r"""
    Return True if one or more terms contain symbol only in exponents,
    else False.

    Parameters
    ==========

    f : Expr
        The equation to be checked

    symbol : Symbol
        The variable in which the equation is checked

    Examples
    ========

    >>> from sympy import symbols, cos, exp
    >>> from sympy.solvers.solveset import _is_exponential as check
    >>> x, y = symbols('x y')
    >>> check(y, y)
    False
    >>> check(x**y - 1, y)
    True
    >>> check(x**y*2**y - 1, y)
    True
    >>> check(exp(x + 3) + 3**x, x)
    True
    >>> check(cos(2**x), x)
    False

    * Philosophy behind the helper

    The function extracts each term of the equation and checks if it is
    of exponential form w.r.t `symbol`.
    """
    rv = False
    for expr_arg in _term_factors(f):
        if symbol not in expr_arg.free_symbols:
            continue
        if (isinstance(expr_arg, Pow) and
           symbol not in expr_arg.base.free_symbols or
           isinstance(expr_arg, exp)):
            rv = True  # symbol in exponent
        else:
            return False  # dependent on symbol in non-exponential way
    return rv


def _solve_logarithm(lhs, rhs, symbol, domain):
    r"""
    Helper to solve logarithmic equations which are reducible
    to a single instance of `log`.

    Logarithmic equations are (currently) the equations that contains
    `log` terms which can be reduced to a single log term or
    a constant using various logarithmic identities.

    For example:

    .. math:: log(x) + log(x - 4)

    can be reduced to:

    .. math:: log(x(x - 4))

    Parameters
    ==========

    lhs, rhs : Expr
        The logarithmic equation to be solved, `lhs = rhs`

    symbol : Symbol
        The variable in which the equation is solved

    domain : Set
        A set over which the equation is solved.

    Returns
    =======

    A set of solutions satisfying the given equation.
    A `ConditionSet` if the equation is unsolvable.

    Examples
    ========

    >>> from sympy import symbols, log, S
    >>> from sympy.solvers.solveset import _solve_logarithm as solve_log
    >>> x = symbols('x')
    >>> f = log(x - 3) + log(x + 3)
    >>> solve_log(f, 0, x, S.Reals)
    {-sqrt(10), sqrt(10)}

    * Proof of correctness

    A logarithm is another way to write exponent and is defined by

    .. math:: {log_b x} = y \enspace if \enspace b^y = x

    When one side of the equation contains a single logarithm, the
    equation can be solved by rewriting the equation as an equivalent
    exponential equation as defined above. But if one side contains
    more than one logarithm, we need to use the properties of logarithm
    to condense it into a single logarithm.

    Take for example

    .. math:: log(2*x) - 15 = 0

    contains single logarithm, therefore we can directly rewrite it to
    exponential form as

    .. math:: x = exp(15)/2

    But if the equation has more than one logarithm as

    .. math:: log(x - 3) + log(x + 3) = 0

    we use log identities to convert into a reduced form

    Using,

    .. math:: log(a) + log(b) = log(a*b)

    therefore the equation becomes,

    .. math:: log((x - 3)*(x  + 3))

    This equation contains one logarithm and can be solved by rewriting
    to exponents.
    """
    new_lhs = logcombine(lhs, force=True)
    new_f = new_lhs - rhs

    return _solveset(new_f, symbol, domain)


def _is_logarithmic(f, symbol):
    r"""
    Return True if the equation is in the form
    `a*log(f(x)) + b*log(g(x)) + ... + c` else False.

    Parameters
    ==========

    f : Expr
        The equation to be checked

    symbol : Symbol
        The variable in which the equation is checked

    Returns
    =======

    `True` if the equation is logarithmic otherwise `False`.

    Examples
    ========

    >>> from sympy import symbols, tan, log
    >>> from sympy.solvers.solveset import _is_logarithmic as check
    >>> x, y = symbols('x y')
    >>> check(log(x + 2) - log(x + 3), x)
    True
    >>> check(tan(log(2*x)), x)
    False
    >>> check(x*log(x), x)
    False
    >>> check(x + log(x), x)
    False
    >>> check(y + log(x), x)
    True

    * Philosophy behind the helper

    The function extracts each term and checks whether it is
    logarithmic w.r.t `symbol`.
    """
    rv = False
    for term in Add.make_args(f):
        saw_log = False
        for term_arg in Mul.make_args(term):
            if symbol not in term_arg.free_symbols:
                continue
            if isinstance(term_arg, log):
                if saw_log:
                    return False  # more than one log in term
                saw_log = True
            else:
                return False  # dependent on symbol in non-log way
        if saw_log:
            rv = True
    return rv


def _transolve(f, symbol, domain):
    r"""
    Function to solve transcendental equations. It is a helper to
    `solveset` and should be used internally. `\_transolve`
    currently supports the following class of equations:

        - Exponential equations
        - Logarithmic equations

    Parameters
    ==========

    f : Any transcendental equation that needs to be solved.
        This needs to be an expression, which is assumed
        to be equal to 0.

    symbol : The variable for which the equation is solved.
        This needs to be of class `Symbol`.

    domain : A set over which the equation is solved.
        This needs to be of class `Set`.

    Returns
    =======

    Set
        A set of values for `symbol` for which `f` is equal to
        zero. An `EmptySet` is returned if `f` does not have solutions
        in respective domain. A `ConditionSet` is returned as unsolved
        object if algorithms to evaluate complete solution are not
        yet implemented.

    How to use `\_transolve`
    ========================

    `\_transolve` should not be used as an independent function, because
    it assumes that the equation (`f`) and the `symbol` comes from
    `solveset` and might have undergone a few modification(s).
    To use `\_transolve` as an independent function the equation (`f`)
    and the `symbol` should be passed as they would have been by
    `solveset`.

    Examples
    ========

    >>> from sympy.solvers.solveset import _transolve as transolve
    >>> from sympy.solvers.solvers import _tsolve as tsolve
    >>> from sympy import symbols, S, pprint
    >>> x = symbols('x', real=True) # assumption added
    >>> transolve(5**(x - 3) - 3**(2*x + 1), x, S.Reals)
    {-(log(3) + 3*log(5))/(-log(5) + 2*log(3))}

    How `\_transolve` works
    =======================

    `\_transolve` uses two types of helper functions to solve equations
    of a particular class:

    Identifying helpers: To determine whether a given equation
    belongs to a certain class of equation or not. Returns either
    True or False.

    Solving helpers: Once an equation is identified, a corresponding
    helper either solves the equation or returns a form of the equation
    that `solveset` might better be able to handle.

    * Philosophy behind the module

    The purpose of `\_transolve` is to take equations which are not
    already polynomial in their generator(s) and to either recast them
    as such through a valid transformation or to solve them outright.
    A pair of helper functions for each class of supported
    transcendental functions are employed for this purpose. One
    identifies the transcendental form of an equation and the other
    either solves it or recasts it into a tractable form that can be
    solved by  `solveset`.
    For example, an equation in the form `a*b**f(x) - c*d**g(x) = 0`
    can be transformed to
    `log(a) + f(x)*log(b) - log(c) - g(x)*log(d) = 0`
    (under certain assumptions) and this can be solved with `solveset`
    if `f(x)` and `g(x)` are in polynomial form.

    How `\_transolve` is better than `\_tsolve`
    ===========================================

    1) Better output

    `\_transolve` provides expressions in a more simplified form.

    Consider a simple exponential equation

    >>> f = 3**(2*x) - 2**(x + 3)
    >>> pprint(transolve(f, x, S.Reals), use_unicode=False)
        -3*log(2)
    {------------------}
     -2*log(3) + log(2)
    >>> pprint(tsolve(f, x), use_unicode=False)
         /   3     \
         | --------|
         | log(2/9)|
    [-log\2         /]

    2) Extensible

    The API of `\_transolve` is designed such that it is easily
    extensible, i.e. the code that solves a given class of
    equations is encapsulated in a helper and not mixed in with
    the code of \_transolve itself.

    3) Modular

    `\_transolve` is designed to be modular i.e, for every class of
    equation a separate helper for identification and solving is
    implemented. This makes it easy to change or modify any of the
    method implemented directly in the helpers without interfering
    with the actual structure of the API.

    4) Faster Computation

    Solving equation via `\_transolve` is much faster as compared to
    `\_tsolve`. In `solve`, attempts are made computing every possibility
    to get the solutions. This series of attempts makes solving a bit
    slow. In `\_transolve`, computation begins only after a particular
    type of equation is identified.

    How to add new class of equations
    =================================

    Adding a new class of equation solver is a three-step procedure:

    - Identify the type of the equations

      Determine the type of the class of equations to which they belong:
      it could be of `Add`, `Pow`, etc. types. Separate internal functions
      are used for each type. Write identification and solving helpers
      and use them from within the routine for the given type of equation
      (after adding it, if necessary). Something like:

      .. code-block:: python

        def add_type(lhs, rhs, x):
            ....
            if _is_exponential(lhs, x):
                new_eq = _solve_exponential(lhs, rhs, x)
        ....
        rhs, lhs = eq.as_independent(x)
        if lhs.is_Add:
            result = add_type(lhs, rhs, x)

    - Define the identification helper.

    - Define the solving helper.

    Apart from this, a few other things needs to be taken care while
    adding an equation solver:

    - Naming conventions:
      Name of the *identification* *helper* should be as
      `\_is\_class` where `class` will be the name or abbreviation
      of the class of equation. The *solving* *helper* will be named as
      `\_solve\_class`.
      For example: for exponential equations it becomes
      `\_is\_exponential` and `\_solve\_expo`.
    - The `identifying helpers`should take two input parameters,
      the equation to be checked and the variable for which a solution
      is being sought, while `solving helpers` would require an additional
      domain parameter.
    - Be sure to consider corner cases.
    - Add tests for each helper.
    - Add a docstring to your helper that describes the method
      implemented.
      The documentation of the helpers should identify:

      - the purpose of the helper,
      - the method used to identify and solve the equation,
      - a proof of correctness
      - the return values of the helpers
    """

    def add_type(lhs, rhs, symbol, domain):
        """
        Helper for `_transolve` to handle equations of
        `Add` type, i.e. equations taking the form as
        `a*f(x) + b*g(x) + .... = c`.
        For example: 4**x + 8**x = 0
        """
        result = ConditionSet(symbol, Eq(lhs - rhs, 0), domain)

        # check if it is exponential type equation
        if _is_exponential(lhs, symbol):
            result = _solve_exponential(lhs, rhs, symbol, domain)
        # check if it is logarithmic type equation
        elif _is_logarithmic(lhs, symbol):
            result = _solve_logarithm(lhs, rhs, symbol, domain)

        return result

    result = ConditionSet(symbol, Eq(f, 0), domain)

    # invert_complex handles the call to the desired inverter based
    # on the domain specified.
    lhs, rhs_s = invert_complex(f, 0, symbol, domain)

    if isinstance(rhs_s, FiniteSet):
        assert (len(rhs_s.args)) == 1
        rhs = rhs_s.args[0]

        if lhs.is_Add:
            result = add_type(lhs, rhs, symbol, domain)
    else:
        result = rhs_s

    return result


def solveset(f, symbol=None, domain=S.Complexes):
    r"""Solves a given inequality or equation with set as output

    Parameters
    ==========

    f : Expr or a relational.
        The target equation or inequality
    symbol : Symbol
        The variable for which the equation is solved
    domain : Set
        The domain over which the equation is solved

    Returns
    =======

    Set
        A set of values for `symbol` for which `f` is True or is equal to
        zero. An `EmptySet` is returned if `f` is False or nonzero.
        A `ConditionSet` is returned as unsolved object if algorithms
        to evaluate complete solution are not yet implemented.

    `solveset` claims to be complete in the solution set that it returns.

    Raises
    ======

    NotImplementedError
        The algorithms to solve inequalities in complex domain  are
        not yet implemented.
    ValueError
        The input is not valid.
    RuntimeError
        It is a bug, please report to the github issue tracker.


    Notes
    =====

    Python interprets 0 and 1 as False and True, respectively, but
    in this function they refer to solutions of an expression. So 0 and 1
    return the Domain and EmptySet, respectively, while True and False
    return the opposite (as they are assumed to be solutions of relational
    expressions).


    See Also
    ========

    solveset_real: solver for real domain
    solveset_complex: solver for complex domain

    Examples
    ========

    >>> from sympy import exp, sin, Symbol, pprint, S
    >>> from sympy.solvers.solveset import solveset, solveset_real

    * The default domain is complex. Not specifying a domain will lead
      to the solving of the equation in the complex domain (and this
      is not affected by the assumptions on the symbol):

    >>> x = Symbol('x')
    >>> pprint(solveset(exp(x) - 1, x), use_unicode=False)
    {2*n*I*pi | n in Integers}

    >>> x = Symbol('x', real=True)
    >>> pprint(solveset(exp(x) - 1, x), use_unicode=False)
    {2*n*I*pi | n in Integers}

    * If you want to use `solveset` to solve the equation in the
      real domain, provide a real domain. (Using `solveset\_real`
      does this automatically.)

    >>> R = S.Reals
    >>> x = Symbol('x')
    >>> solveset(exp(x) - 1, x, R)
    {0}
    >>> solveset_real(exp(x) - 1, x)
    {0}

    The solution is mostly unaffected by assumptions on the symbol,
    but there may be some slight difference:

    >>> pprint(solveset(sin(x)/x,x), use_unicode=False)
    ({2*n*pi | n in Integers} \ {0}) U ({2*n*pi + pi | n in Integers} \ {0})

    >>> p = Symbol('p', positive=True)
    >>> pprint(solveset(sin(p)/p, p), use_unicode=False)
    {2*n*pi | n in Integers} U {2*n*pi + pi | n in Integers}

    * Inequalities can be solved over the real domain only. Use of a complex
      domain leads to a NotImplementedError.

    >>> solveset(exp(x) > 1, x, R)
    Interval.open(0, oo)

    """
    f = sympify(f)
    symbol = sympify(symbol)

    if f is S.true:
        return domain

    if f is S.false:
        return S.EmptySet

    if not isinstance(f, (Expr, Number)):
        raise ValueError("%s is not a valid SymPy expression" % f)

    if not isinstance(symbol, Expr) and  symbol is not None:
        raise ValueError("%s is not a valid SymPy symbol" % symbol)

    if not isinstance(domain, Set):
        raise ValueError("%s is not a valid domain" %(domain))

    free_symbols = f.free_symbols

    if symbol is None and not free_symbols:
        b = Eq(f, 0)
        if b is S.true:
            return domain
        elif b is S.false:
            return S.EmptySet
        else:
            raise NotImplementedError(filldedent('''
                relationship between value and 0 is unknown: %s''' % b))

    if symbol is None:
        if len(free_symbols) == 1:
            symbol = free_symbols.pop()
        elif free_symbols:
            raise ValueError(filldedent('''
                The independent variable must be specified for a
                multivariate equation.'''))
    elif not isinstance(symbol, Symbol):
        f, s, swap = recast_to_symbols([f], [symbol])
        # the xreplace will be needed if a ConditionSet is returned
        return solveset(f[0], s[0], domain).xreplace(swap)

    if domain.is_subset(S.Reals):
        if not symbol.is_real:
            assumptions = symbol.assumptions0
            assumptions['real'] = True
            try:
                r = Dummy('r', **assumptions)
                return solveset(f.xreplace({symbol: r}), r, domain
                    ).xreplace({r: symbol})
            except InconsistentAssumptions:
                pass
    # Abs has its own handling method which avoids the
    # rewriting property that the first piece of abs(x)
    # is for x >= 0 and the 2nd piece for x < 0 -- solutions
    # can look better if the 2nd condition is x <= 0. Since
    # the solution is a set, duplication of results is not
    # an issue, e.g. {y, -y} when y is 0 will be {0}
    f, mask = _masked(f, Abs)
    f = f.rewrite(Piecewise) # everything that's not an Abs
    for d, e in mask:
        # everything *in* an Abs
        e = e.func(e.args[0].rewrite(Piecewise))
        f = f.xreplace({d: e})
    f = piecewise_fold(f)

    return _solveset(f, symbol, domain, _check=True)


def solveset_real(f, symbol):
    return solveset(f, symbol, S.Reals)


def solveset_complex(f, symbol):
    return solveset(f, symbol, S.Complexes)


def solvify(f, symbol, domain):
    """Solves an equation using solveset and returns the solution in accordance
    with the `solve` output API.

    Returns
    =======

    We classify the output based on the type of solution returned by `solveset`.

    Solution    |    Output
    ----------------------------------------
    FiniteSet   | list

    ImageSet,   | list (if `f` is periodic)
    Union       |

    EmptySet    | empty list

    Others      | None


    Raises
    ======

    NotImplementedError
        A ConditionSet is the input.

    Examples
    ========

    >>> from sympy.solvers.solveset import solvify, solveset
    >>> from sympy.abc import x
    >>> from sympy import S, tan, sin, exp
    >>> solvify(x**2 - 9, x, S.Reals)
    [-3, 3]
    >>> solvify(sin(x) - 1, x, S.Reals)
    [pi/2]
    >>> solvify(tan(x), x, S.Reals)
    [0]
    >>> solvify(exp(x) - 1, x, S.Complexes)

    >>> solvify(exp(x) - 1, x, S.Reals)
    [0]

    """
    solution_set = solveset(f, symbol, domain)
    result = None
    if solution_set is S.EmptySet:
        result = []

    elif isinstance(solution_set, ConditionSet):
        raise NotImplementedError('solveset is unable to solve this equation.')

    elif isinstance(solution_set, FiniteSet):
        result = list(solution_set)

    else:
        period = periodicity(f, symbol)
        if period is not None:
            solutions = S.EmptySet
            iter_solutions = ()
            if isinstance(solution_set, ImageSet):
                iter_solutions = (solution_set,)
            elif isinstance(solution_set, Union):
                if all(isinstance(i, ImageSet) for i in solution_set.args):
                    iter_solutions = solution_set.args

            for solution in iter_solutions:
                solutions += solution.intersect(Interval(0, period, False, True))

            if isinstance(solutions, FiniteSet):
                result = list(solutions)

        else:
            solution = solution_set.intersect(domain)
            if isinstance(solution, FiniteSet):
                result += solution

    return result


###############################################################################
################################ LINSOLVE #####################################
###############################################################################


def linear_eq_to_matrix(equations, *symbols):
    r"""
    Converts a given System of Equations into Matrix form.
    Here `equations` must be a linear system of equations in
    `symbols`. The order of symbols in input `symbols` will
    determine the order of coefficients in the returned
    Matrix.

    The Matrix form corresponds to the augmented matrix form.
    For example:

    .. math:: 4x + 2y + 3z  = 1
    .. math:: 3x +  y +  z  = -6
    .. math:: 2x + 4y + 9z  = 2

    This system would return `A` & `b` as given below:

    ::

         [ 4  2  3 ]          [ 1 ]
     A = [ 3  1  1 ]   b  =   [-6 ]
         [ 2  4  9 ]          [ 2 ]

    Examples
    ========

    >>> from sympy import linear_eq_to_matrix, symbols
    >>> x, y, z = symbols('x, y, z')
    >>> eqns = [x + 2*y + 3*z - 1, 3*x + y + z + 6, 2*x + 4*y + 9*z - 2]
    >>> A, b = linear_eq_to_matrix(eqns, [x, y, z])
    >>> A
    Matrix([
    [1, 2, 3],
    [3, 1, 1],
    [2, 4, 9]])
    >>> b
    Matrix([
    [ 1],
    [-6],
    [ 2]])
    >>> eqns = [x + z - 1, y + z, x - y]
    >>> A, b = linear_eq_to_matrix(eqns, [x, y, z])
    >>> A
    Matrix([
    [1,  0, 1],
    [0,  1, 1],
    [1, -1, 0]])
    >>> b
    Matrix([
    [1],
    [0],
    [0]])

    * Symbolic coefficients are also supported

    >>> a, b, c, d, e, f = symbols('a, b, c, d, e, f')
    >>> eqns = [a*x + b*y - c, d*x + e*y - f]
    >>> A, B = linear_eq_to_matrix(eqns, x, y)
    >>> A
    Matrix([
    [a, b],
    [d, e]])
    >>> B
    Matrix([
    [c],
    [f]])

    """

    if not symbols:
        raise ValueError('Symbols must be given, for which coefficients \
                         are to be found.')

    if hasattr(symbols[0], '__iter__'):
        symbols = symbols[0]

    M = Matrix([symbols])
    # initialize Matrix with symbols + 1 columns
    M = M.col_insert(len(symbols), Matrix([1]))
    row_no = 1

    for equation in equations:
        f = sympify(equation)
        if isinstance(f, Equality):
            f = f.lhs - f.rhs

        # Extract coeff of symbols
        coeff_list = []
        for symbol in symbols:
            coeff_list.append(f.coeff(symbol))

        # append constant term (term free from symbols)
        coeff_list.append(-f.as_coeff_add(*symbols)[0])

        # insert equations coeff's into rows
        M = M.row_insert(row_no, Matrix([coeff_list]))
        row_no += 1

    # delete the initialized (Ist) trivial row
    M.row_del(0)
    A, b = M[:, :-1], M[:, -1:]
    return A, b


def linsolve(system, *symbols):
    r"""
    Solve system of N linear equations with M variables, which
    means both under - and overdetermined systems are supported.
    The possible number of solutions is zero, one or infinite.
    Zero solutions throws a ValueError, where as infinite
    solutions are represented parametrically in terms of given
    symbols. For unique solution a FiniteSet of ordered tuple
    is returned.

    All Standard input formats are supported:
    For the given set of Equations, the respective input types
    are given below:

    .. math:: 3x + 2y -   z = 1
    .. math:: 2x - 2y + 4z = -2
    .. math:: 2x -   y + 2z = 0

    * Augmented Matrix Form, `system` given below:

    ::

              [3   2  -1  1]
     system = [2  -2   4 -2]
              [2  -1   2  0]

    * List Of Equations Form

    `system  =  [3x + 2y - z - 1, 2x - 2y + 4z + 2, 2x - y + 2z]`

    * Input A & b Matrix Form (from Ax = b) are given as below:

    ::

         [3   2  -1 ]         [  1 ]
     A = [2  -2   4 ]    b =  [ -2 ]
         [2  -1   2 ]         [  0 ]

    `system = (A, b)`

    Symbols can always be passed but are actually only needed
    when 1) a system of equations is being passed and 2) the
    system is passed as an underdetermined matrix and one wants
    to control the name of the free variables in the result.
    An error is raised if no symbols are used for case 1, but if
    no symbols are provided for case 2, internally generated symbols
    will be provided. When providing symbols for case 2, there should
    be at least as many symbols are there are columns in matrix A.

    The algorithm used here is Gauss-Jordan elimination, which
    results, after elimination, in a row echelon form matrix.

    Returns
    =======

    A FiniteSet containing an ordered tuple of values for the
    unknowns for which the `system` has a solution. (Wrapping
    the tuple in FiniteSet is used to maintain a consistent
    output format throughout solveset.)

    Returns EmptySet(), if the linear system is inconsistent.

    Raises
    ======

    ValueError
        The input is not valid.
        The symbols are not given.

    Examples
    ========

    >>> from sympy import Matrix, S, linsolve, symbols
    >>> x, y, z = symbols("x, y, z")
    >>> A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 10]])
    >>> b = Matrix([3, 6, 9])
    >>> A
    Matrix([
    [1, 2,  3],
    [4, 5,  6],
    [7, 8, 10]])
    >>> b
    Matrix([
    [3],
    [6],
    [9]])
    >>> linsolve((A, b), [x, y, z])
    {(-1, 2, 0)}

    * Parametric Solution: In case the system is underdetermined, the
      function will return a parametric solution in terms of the given
      symbols. Those that are free will be returned unchanged. e.g. in
      the system below, `z` is returned as the solution for variable z;
      it can take on any value.

    >>> A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
    >>> b = Matrix([3, 6, 9])
    >>> linsolve((A, b), x, y, z)
    {(z - 1, -2*z + 2, z)}

    If no symbols are given, internally generated symbols will be used.
    The `tau0` in the 3rd position indicates (as before) that the 3rd
    variable -- whatever it's named -- can take on any value:

    >>> linsolve((A, b))
    {(tau0 - 1, -2*tau0 + 2, tau0)}

    * List of Equations as input

    >>> Eqns = [3*x + 2*y - z - 1, 2*x - 2*y + 4*z + 2, - x + y/2 - z]
    >>> linsolve(Eqns, x, y, z)
    {(1, -2, -2)}

    * Augmented Matrix as input

    >>> aug = Matrix([[2, 1, 3, 1], [2, 6, 8, 3], [6, 8, 18, 5]])
    >>> aug
    Matrix([
    [2, 1,  3, 1],
    [2, 6,  8, 3],
    [6, 8, 18, 5]])
    >>> linsolve(aug, x, y, z)
    {(3/10, 2/5, 0)}

    * Solve for symbolic coefficients

    >>> a, b, c, d, e, f = symbols('a, b, c, d, e, f')
    >>> eqns = [a*x + b*y - c, d*x + e*y - f]
    >>> linsolve(eqns, x, y)
    {((-b*f + c*e)/(a*e - b*d), (a*f - c*d)/(a*e - b*d))}

    * A degenerate system returns solution as set of given
      symbols.

    >>> system = Matrix(([0, 0, 0], [0, 0, 0], [0, 0, 0]))
    >>> linsolve(system, x, y)
    {(x, y)}

    * For an empty system linsolve returns empty set

    >>> linsolve([], x)
    EmptySet()

    """
    if not system:
        return S.EmptySet

    # If second argument is an iterable
    if symbols and hasattr(symbols[0], '__iter__'):
        symbols = symbols[0]
    sym_gen = isinstance(symbols, GeneratorType)

    swap = {}
    b = None  # if we don't get b the input was bad
    syms_needed_msg = None

    # unpack system

    if hasattr(system, '__iter__'):

        # 1). (A, b)
        if len(system) == 2 and isinstance(system[0], Matrix):
            A, b = system

        # 2). (eq1, eq2, ...)
        if not isinstance(system[0], Matrix):
            if sym_gen or not symbols:
                raise ValueError(filldedent('''
                    When passing a system of equations, the explicit
                    symbols for which a solution is being sought must
                    be given as a sequence, too.
                '''))
            system = list(system)
            for i, eq in enumerate(system):
                try:
                    # since we are checking it, we might as well take the
                    # expanded expr that it will give
                    system[i] = Poly(eq, symbols).as_expr()
                    if any (degree(eq, sym) > 1 for sym in symbols):
                        raise PolynomialError
                except PolynomialError:
                    raise ValueError(filldedent('''
                        %s contains non-linear terms in the
                        variables to be evaluated
                    ''') % eq)
            system, symbols, swap = recast_to_symbols(system, symbols)
            A, b = linear_eq_to_matrix(system, symbols)
            syms_needed_msg = 'free symbols in the equations provided'

    elif isinstance(system, Matrix) and not (
            symbols and not isinstance(symbols, GeneratorType) and
            isinstance(symbols[0], Matrix)):
        # 3). A augmented with b
        A, b = system[:, :-1], system[:, -1:]

    if b is None:
        raise ValueError("Invalid arguments")

    syms_needed_msg  = syms_needed_msg or 'columns of A'

    if sym_gen:
        symbols = [next(symbols) for i in range(A.cols)]
        if any(set(symbols) & (A.free_symbols | b.free_symbols)):
            raise ValueError(filldedent('''
                At least one of the symbols provided
                already appears in the system to be solved.
                One way to avoid this is to use Dummy symbols in
                the generator, e.g. numbered_symbols('%s', cls=Dummy)
            ''' % symbols[0].name.rstrip('1234567890')))

    try:
        solution, params, free_syms = A.gauss_jordan_solve(b, freevar=True)
    except ValueError:
        # No solution
        return S.EmptySet

    # Replace free parameters with free symbols
    if params:
        if not symbols:
            symbols = [_ for _ in params]
            # re-use the parameters but put them in order
            # params       [x, y, z]
            # free_symbols [2, 0, 4]
            # idx          [1, 0, 2]
            idx = list(zip(*sorted(zip(free_syms, range(len(free_syms))))))[1]
            # simultaneous replacements {y: x, x: y, z: z}
            replace_dict = dict(zip(symbols, [symbols[i] for i in idx]))
        elif len(symbols) >= A.cols:
            replace_dict = {v: symbols[free_syms[k]] for k, v in enumerate(params)}
        else:
            raise IndexError(filldedent('''
                the number of symbols passed should have a length
                equal to the number of %s.
                ''' % syms_needed_msg))
        solution = [sol.xreplace(replace_dict) for sol in solution]

    solution = [simplify(sol).xreplace(swap) for sol in solution]
    return FiniteSet(tuple(solution))



##############################################################################
# ------------------------------nonlinsolve ---------------------------------#
##############################################################################

def _return_conditionset(eqs, symbols):
        # return conditionset
        condition_set = ConditionSet(
            Tuple(*symbols),
            FiniteSet(*eqs),
            S.Complexes)
        return condition_set


def substitution(system, symbols, result=[{}], known_symbols=[],
                 exclude=[], all_symbols=None):
    r"""
     Solves the `system` using substitution method. It is used in
     `nonlinsolve`. This will be called from `nonlinsolve` when any
     equation(s) is non polynomial equation.

    Parameters
    ==========

    system : list of equations
        The target system of equations
    symbols : list of symbols to be solved.
        The variable(s) for which the system is solved
    known_symbols : list of solved symbols
        Values are known for these variable(s)
    result : An empty list or list of dict
        If No symbol values is known then empty list otherwise
        symbol as keys and corresponding value in dict.
    exclude : Set of expression.
        Mostly denominator expression(s) of the equations of the system.
        Final solution should not satisfy these expressions.
    all_symbols : known_symbols + symbols(unsolved).

    Returns
    =======

    A FiniteSet of ordered tuple of values of `all_symbols` for which the
    `system` has solution. Order of values in the tuple is same as symbols
    present in the parameter `all_symbols`. If parameter `all_symbols` is None
    then same as symbols present in the parameter `symbols`.

    Please note that general FiniteSet is unordered, the solution returned
    here is not simply a FiniteSet of solutions, rather it is a FiniteSet of
    ordered tuple, i.e. the first & only argument to FiniteSet is a tuple of
    solutions, which is ordered, & hence the returned solution is ordered.

    Also note that solution could also have been returned as an ordered tuple,
    FiniteSet is just a wrapper `{}` around the tuple. It has no other
    significance except for the fact it is just used to maintain a consistent
    output format throughout the solveset.

    Raises
    ======

    ValueError
        The input is not valid.
        The symbols are not given.
    AttributeError
        The input symbols are not `Symbol` type.

    Examples
    ========

    >>> from sympy.core.symbol import symbols
    >>> x, y = symbols('x, y', real=True)
    >>> from sympy.solvers.solveset import substitution
    >>> substitution([x + y], [x], [{y: 1}], [y], set([]), [x, y])
    {(-1, 1)}

    * when you want soln should not satisfy eq `x + 1 = 0`

    >>> substitution([x + y], [x], [{y: 1}], [y], set([x + 1]), [y, x])
    EmptySet()
    >>> substitution([x + y], [x], [{y: 1}], [y], set([x - 1]), [y, x])
    {(1, -1)}
    >>> substitution([x + y - 1, y - x**2 + 5], [x, y])
    {(-3, 4), (2, -1)}

    * Returns both real and complex solution

    >>> x, y, z = symbols('x, y, z')
    >>> from sympy import exp, sin
    >>> substitution([exp(x) - sin(y), y**2 - 4], [x, y])
    {(log(sin(2)), 2), (ImageSet(Lambda(_n, I*(2*_n*pi + pi) +
        log(sin(2))), Integers), -2), (ImageSet(Lambda(_n, 2*_n*I*pi +
        Mod(log(sin(2)), 2*I*pi)), Integers), 2)}

    >>> eqs = [z**2 + exp(2*x) - sin(y), -3 + exp(-y)]
    >>> substitution(eqs, [y, z])
    {(-log(3), -sqrt(-exp(2*x) - sin(log(3)))),
    (-log(3), sqrt(-exp(2*x) - sin(log(3)))),
    (ImageSet(Lambda(_n, 2*_n*I*pi + Mod(-log(3), 2*I*pi)), Integers),
    ImageSet(Lambda(_n, -sqrt(-exp(2*x) + sin(2*_n*I*pi +
    Mod(-log(3), 2*I*pi)))), Integers)),
    (ImageSet(Lambda(_n, 2*_n*I*pi + Mod(-log(3), 2*I*pi)), Integers),
    ImageSet(Lambda(_n, sqrt(-exp(2*x) + sin(2*_n*I*pi +
        Mod(-log(3), 2*I*pi)))), Integers))}

    """

    from sympy import Complement
    from sympy.core.compatibility import is_sequence

    if not system:
        return S.EmptySet

    if not symbols:
        msg = ('Symbols must be given, for which solution of the '
               'system is to be found.')
        raise ValueError(filldedent(msg))

    if not is_sequence(symbols):
        msg = ('symbols should be given as a sequence, e.g. a list.'
               'Not type %s: %s')
        raise TypeError(filldedent(msg % (type(symbols), symbols)))

    try:
        sym = symbols[0].is_Symbol
    except AttributeError:
        sym = False

    if not sym:
        msg = ('Iterable of symbols must be given as '
               'second argument, not type %s: %s')
        raise ValueError(filldedent(msg % (type(symbols[0]), symbols[0])))

    # By default `all_symbols` will be same as `symbols`
    if all_symbols is None:
        all_symbols = symbols

    old_result = result
    # storing complements and intersection for particular symbol
    complements = {}
    intersections = {}

    # when total_solveset_call is equals to total_conditionset
    # means solvest fail to solve all the eq.
    total_conditionset = -1
    total_solveset_call = -1

    def _unsolved_syms(eq, sort=False):
        """Returns the unsolved symbol present
        in the equation `eq`.
        """
        free = eq.free_symbols
        unsolved = (free - set(known_symbols)) & set(all_symbols)
        if sort:
            unsolved = list(unsolved)
            unsolved.sort(key=default_sort_key)
        return unsolved
    # end of _unsolved_syms()

    # sort such that equation with the fewest potential symbols is first.
    # means eq with less number of variable first in the list.
    eqs_in_better_order = list(
        ordered(system, lambda _: len(_unsolved_syms(_))))

    def add_intersection_complement(result, sym_set, **flags):
        # If solveset have returned some intersection/complement
        # for any symbol. It will be added in final solution.
        final_result = []
        for res in result:
            res_copy = res
            for key_res, value_res in res.items():
                # Intersection/complement is in Interval or Set.
                intersection_true = flags.get('Intersection', True)
                complements_true = flags.get('Complement', True)
                for key_sym, value_sym in sym_set.items():
                    if key_sym == key_res:
                        if intersection_true:
                            # testcase is not added for this line(intersection)
                            new_value = \
                                Intersection(FiniteSet(value_res), value_sym)
                            if new_value is not S.EmptySet:
                                res_copy[key_res] = new_value
                        if complements_true:
                            new_value = \
                                Complement(FiniteSet(value_res), value_sym)
                            if new_value is not S.EmptySet:
                                res_copy[key_res] = new_value
            final_result.append(res_copy)
        return final_result
    # end of def add_intersection_complement()

    def _extract_main_soln(sol, soln_imageset):
            """separate the Complements, Intersections, ImageSet lambda expr
            and it's base_set.
            """
            # if there is union, then need to check
            # Complement, Intersection, Imageset.
            # Order should not be changed.
            if isinstance(sol, Complement):
                # extract solution and complement
                complements[sym] = sol.args[1]
                sol = sol.args[0]
                # complement will be added at the end
                # using `add_intersection_complement` method
            if isinstance(sol, Intersection):
                # Interval/Set will be at 0th index always
                if sol.args[0] != Interval(-oo, oo):
                    # sometimes solveset returns soln
                    # with intersection `S.Reals`, to confirm that
                    # soln is in `domain=S.Reals` or not. We don't consider
                    # that intersection.
                    intersections[sym] = sol.args[0]
                sol = sol.args[1]
            # after intersection and complement Imageset should
            # be checked.
            if isinstance(sol, ImageSet):
                soln_imagest = sol
                expr2 = sol.lamda.expr
                sol = FiniteSet(expr2)
                soln_imageset[expr2] = soln_imagest

            # if there is union of Imageset or other in soln.
            # no testcase is written for this if block
            if isinstance(sol, Union):
                sol_args = sol.args
                sol = S.EmptySet
                # We need in sequence so append finteset elements
                # and then imageset or other.
                for sol_arg2 in sol_args:
                    if isinstance(sol_arg2, FiniteSet):
                        sol += sol_arg2
                    else:
                        # ImageSet, Intersection, complement then
                        # append them directly
                        sol += FiniteSet(sol_arg2)

            if not isinstance(sol, FiniteSet):
                sol = FiniteSet(sol)
            return sol, soln_imageset
    # end of def _extract_main_soln()

    # helper function for _append_new_soln
    def _check_exclude(rnew, imgset_yes):
        rnew_ = rnew
        if imgset_yes:
            # replace all dummy variables (Imageset lambda variables)
            # with zero before `checksol`. Considering fundamental soln
            # for `checksol`.
            rnew_copy = rnew.copy()
            dummy_n = imgset_yes[0]
            for key_res, value_res in rnew_copy.items():
                rnew_copy[key_res] = value_res.subs(dummy_n, 0)
            rnew_ = rnew_copy
        # satisfy_exclude == true if it satisfies the expr of `exclude` list.
        try:
            # something like : `Mod(-log(3), 2*I*pi)` can't be
            # simplified right now, so `checksol` returns `TypeError`.
            # when this issue is fixed this try block should be
            # removed. Mod(-log(3), 2*I*pi) == -log(3)
            satisfy_exclude = any(
                checksol(d, rnew_) for d in exclude)
        except TypeError:
            satisfy_exclude = None
        return satisfy_exclude
    # end of def _check_exclude()

    # helper function for _append_new_soln
    def _restore_imgset(rnew, original_imageset, newresult):
        restore_sym = set(rnew.keys()) & \
            set(original_imageset.keys())
        for key_sym in restore_sym:
            img = original_imageset[key_sym]
            rnew[key_sym] = img
        if rnew not in newresult:
            newresult.append(rnew)
    # end of def _restore_imgset()

    def _append_eq(eq, result, res, delete_soln, n=None):
        u = Dummy('u')
        if n:
            eq = eq.subs(n, 0)
        satisfy = checksol(u, u, eq, minimal=True)
        if satisfy is False:
            delete_soln = True
            res = {}
        else:
            result.append(res)
        return result, res, delete_soln

    def _append_new_soln(rnew, sym, sol, imgset_yes, soln_imageset,
                         original_imageset, newresult, eq=None):
        """If `rnew` (A dict <symbol: soln>) contains valid soln
        append it to `newresult` list.
        `imgset_yes` is (base, dummy_var) if there was imageset in previously
         calculated result(otherwise empty tuple). `original_imageset` is dict
         of imageset expr and imageset from this result.
        `soln_imageset` dict of imageset expr and imageset of new soln.
        """
        satisfy_exclude = _check_exclude(rnew, imgset_yes)
        delete_soln = False
        # soln should not satisfy expr present in `exclude` list.
        if not satisfy_exclude:
            local_n = None
            # if it is imageset
            if imgset_yes:
                local_n = imgset_yes[0]
                base = imgset_yes[1]
                if sym and sol:
                    # when `sym` and `sol` is `None` means no new
                    # soln. In that case we will append rnew directly after
                    # substituting original imagesets in rnew values if present
                    # (second last line of this function using _restore_imgset)
                    dummy_list = list(sol.atoms(Dummy))
                    # use one dummy `n` which is in
                    # previous imageset
                    local_n_list = [
                        local_n for i in range(
                            0, len(dummy_list))]

                    dummy_zip = zip(dummy_list, local_n_list)
                    lam = Lambda(local_n, sol.subs(dummy_zip))
                    rnew[sym] = ImageSet(lam, base)
                if eq is not None:
                    newresult, rnew, delete_soln = _append_eq(
                        eq, newresult, rnew, delete_soln, local_n)
            elif eq is not None:
                newresult, rnew, delete_soln = _append_eq(
                    eq, newresult, rnew, delete_soln)
            elif soln_imageset:
                rnew[sym] = soln_imageset[sol]
                # restore original imageset
                _restore_imgset(rnew, original_imageset, newresult)
            else:
                newresult.append(rnew)
        elif satisfy_exclude:
            delete_soln = True
            rnew = {}
        _restore_imgset(rnew, original_imageset, newresult)
        return newresult, delete_soln
    # end of def _append_new_soln()

    def _new_order_result(result, eq):
        # separate first, second priority. `res` that makes `eq` value equals
        # to zero, should be used first then other result(second priority).
        # If it is not done then we may miss some soln.
        first_priority = []
        second_priority = []
        for res in result:
            if not any(isinstance(val, ImageSet) for val in res.values()):
                if eq.subs(res) == 0:
                    first_priority.append(res)
                else:
                    second_priority.append(res)
        if first_priority or second_priority:
            return first_priority + second_priority
        return result

    def _solve_using_known_values(result, solver):
        """Solves the system using already known solution
        (result contains the dict <symbol: value>).
        solver is `solveset_complex` or `solveset_real`.
        """
        # stores imageset <expr: imageset(Lambda(n, expr), base)>.
        soln_imageset = {}
        total_solvest_call = 0
        total_conditionst = 0

        # sort such that equation with the fewest potential symbols is first.
        # means eq with less variable first
        for index, eq in enumerate(eqs_in_better_order):
            newresult = []
            original_imageset = {}
            # if imageset expr is used to solve other symbol
            imgset_yes = False
            result = _new_order_result(result, eq)
            for res in result:
                got_symbol = set()  # symbols solved in one iteration
                if soln_imageset:
                    # find the imageset and use its expr.
                    for key_res, value_res in res.items():
                        if isinstance(value_res, ImageSet):
                            res[key_res] = value_res.lamda.expr
                            original_imageset[key_res] = value_res
                            dummy_n = value_res.lamda.expr.atoms(Dummy).pop()
                            base = value_res.base_set
                            imgset_yes = (dummy_n, base)
                # update eq with everything that is known so far
                eq2 = eq.subs(res)
                unsolved_syms = _unsolved_syms(eq2, sort=True)
                if not unsolved_syms:
                    if res:
                        newresult, delete_res = _append_new_soln(
                            res, None, None, imgset_yes, soln_imageset,
                            original_imageset, newresult, eq2)
                        if delete_res:
                            # `delete_res` is true, means substituting `res` in
                            # eq2 doesn't return `zero` or deleting the `res`
                            # (a soln) since it staisfies expr of `exclude`
                            # list.
                            result.remove(res)
                    continue  # skip as it's independent of desired symbols
                depen = eq2.as_independent(unsolved_syms)[0]
                if depen.has(Abs) and solver == solveset_complex:
                    # Absolute values cannot be inverted in the
                    # complex domain
                    continue
                soln_imageset = {}
                for sym in unsolved_syms:
                    not_solvable = False
                    try:
                        soln = solver(eq2, sym)
                        total_solvest_call += 1
                        soln_new = S.EmptySet
                        if isinstance(soln, Complement):
                            # separate solution and complement
                            complements[sym] = soln.args[1]
                            soln = soln.args[0]
                            # complement will be added at the end
                        if isinstance(soln, Intersection):
                            # Interval will be at 0th index always
                            if soln.args[0] != Interval(-oo, oo):
                                # sometimes solveset returns soln
                                # with intersection S.Reals, to confirm that
                                # soln is in domain=S.Reals
                                intersections[sym] = soln.args[0]
                            soln_new += soln.args[1]
                        soln = soln_new if soln_new else soln
                        if index > 0 and solver == solveset_real:
                            # one symbol's real soln , another symbol may have
                            # corresponding complex soln.
                            if not isinstance(soln, (ImageSet, ConditionSet)):
                                soln += solveset_complex(eq2, sym)
                    except NotImplementedError:
                        # If sovleset is not able to solve equation `eq2`. Next
                        # time we may get soln using next equation `eq2`
                        continue
                    if isinstance(soln, ConditionSet):
                            soln = S.EmptySet
                            # don't do `continue` we may get soln
                            # in terms of other symbol(s)
                            not_solvable = True
                            total_conditionst += 1

                    if soln is not S.EmptySet:
                        soln, soln_imageset = _extract_main_soln(
                            soln, soln_imageset)

                    for sol in soln:
                        # sol is not a `Union` since we checked it
                        # before this loop
                        sol, soln_imageset = _extract_main_soln(
                            sol, soln_imageset)
                        sol = set(sol).pop()
                        free = sol.free_symbols
                        if got_symbol and any([
                            ss in free for ss in got_symbol
                        ]):
                            # sol depends on previously solved symbols
                            # then continue
                            continue
                        rnew = res.copy()
                        # put each solution in res and append the new  result
                        # in the new result list (solution for symbol `s`)
                        # along with old results.
                        for k, v in res.items():
                            if isinstance(v, Expr):
                                # if any unsolved symbol is present
                                # Then subs known value
                                rnew[k] = v.subs(sym, sol)
                        # and add this new solution
                        if soln_imageset:
                            # replace all lambda variables with 0.
                            imgst = soln_imageset[sol]
                            rnew[sym] = imgst.lamda(
                                *[0 for i in range(0, len(
                                    imgst.lamda.variables))])
                        else:
                            rnew[sym] = sol
                        newresult, delete_res = _append_new_soln(
                            rnew, sym, sol, imgset_yes, soln_imageset,
                            original_imageset, newresult)
                        if delete_res:
                            # deleting the `res` (a soln) since it staisfies
                            # eq of `exclude` list
                            result.remove(res)
                    # solution got for sym
                    if not not_solvable:
                        got_symbol.add(sym)
            # next time use this new soln
            if newresult:
                result = newresult
        return result, total_solvest_call, total_conditionst
    # end def _solve_using_know_values()

    new_result_real, solve_call1, cnd_call1 = _solve_using_known_values(
        old_result, solveset_real)
    new_result_complex, solve_call2, cnd_call2 = _solve_using_known_values(
        old_result, solveset_complex)

    # when `total_solveset_call` is equals to `total_conditionset`
    # means solvest fails to solve all the eq.
    # return conditionset in this case
    total_conditionset += (cnd_call1 + cnd_call2)
    total_solveset_call += (solve_call1 + solve_call2)

    if total_conditionset == total_solveset_call and total_solveset_call != -1:
        return _return_conditionset(eqs_in_better_order, all_symbols)

    # overall result
    result = new_result_real + new_result_complex

    result_all_variables = []
    result_infinite = []
    for res in result:
        if not res:
            # means {None : None}
            continue
        # If length < len(all_symbols) means infinite soln.
        # Some or all the soln is dependent on 1 symbol.
        # eg. {x: y+2} then final soln {x: y+2, y: y}
        if len(res) < len(all_symbols):
            solved_symbols = res.keys()
            unsolved = list(filter(
                lambda x: x not in solved_symbols, all_symbols))
            for unsolved_sym in unsolved:
                res[unsolved_sym] = unsolved_sym
            result_infinite.append(res)
        if res not in result_all_variables:
            result_all_variables.append(res)

    if result_infinite:
        # we have general soln
        # eg : [{x: -1, y : 1}, {x : -y , y: y}] then
        # return [{x : -y, y : y}]
        result_all_variables = result_infinite
    if intersections and complements:
        # no testcase is added for this block
        result_all_variables = add_intersection_complement(
            result_all_variables, intersections,
            Intersection=True, Complement=True)
    elif intersections:
        result_all_variables = add_intersection_complement(
            result_all_variables, intersections, Intersection=True)
    elif complements:
        result_all_variables = add_intersection_complement(
            result_all_variables, complements, Complement=True)

    # convert to ordered tuple
    result = S.EmptySet
    for r in result_all_variables:
        temp = [r[symb] for symb in all_symbols]
        result += FiniteSet(tuple(temp))
    return result
# end of def substitution()


def _solveset_work(system, symbols):
        soln = solveset(system[0], symbols[0])
        if isinstance(soln, FiniteSet):
            _soln = FiniteSet(*[tuple((s,)) for s in soln])
            return _soln
        else:
            return FiniteSet(tuple(FiniteSet(soln)))


def _handle_positive_dimensional(polys, symbols, denominators):
    from sympy.polys.polytools import groebner
    # substitution method where new system is groebner basis of the system
    _symbols = list(symbols)
    _symbols.sort(key=default_sort_key)
    basis = groebner(polys, _symbols, polys=True)
    new_system = []
    for poly_eq in basis:
        new_system.append(poly_eq.as_expr())
    result = [{}]
    result = substitution(
        new_system, symbols, result, [],
        denominators)
    return result
# end of def _handle_positive_dimensional()


def _handle_zero_dimensional(polys, symbols, system):
    # solve 0 dimensional poly system using `solve_poly_system`
    result = solve_poly_system(polys, *symbols)
    # May be some extra soln is added because
    # we used `unrad` in `_separate_poly_nonpoly`, so
    # need to check and remove if it is not a soln.
    result_update = S.EmptySet
    for res in result:
        dict_sym_value = dict(list(zip(symbols, res)))
        if all(checksol(eq, dict_sym_value) for eq in system):
            result_update += FiniteSet(res)
    return result_update
# end of def _handle_zero_dimensional()


def _separate_poly_nonpoly(system, symbols):
    polys = []
    polys_expr = []
    nonpolys = []
    denominators = set()
    poly = None
    for eq in system:
        # Store denom expression if it contains symbol
        denominators.update(_simple_dens(eq, symbols))
        # try to remove sqrt and rational power
        without_radicals = unrad(simplify(eq))
        if without_radicals:
            eq_unrad, cov = without_radicals
            if not cov:
                eq = eq_unrad
        if isinstance(eq, Expr):
            eq = eq.as_numer_denom()[0]
            poly = eq.as_poly(*symbols, extension=True)
        elif simplify(eq).is_number:
            continue
        if poly is not None:
            polys.append(poly)
            polys_expr.append(poly.as_expr())
        else:
            nonpolys.append(eq)
    return polys, polys_expr, nonpolys, denominators
# end of def _separate_poly_nonpoly()


def nonlinsolve(system, *symbols):
    r"""
    Solve system of N non linear equations with M variables, which means both
    under and overdetermined systems are supported. Positive dimensional
    system is also supported (A system with infinitely many solutions is said
    to be positive-dimensional). In Positive dimensional system solution will
    be dependent on at least one symbol. Returns both real solution
    and complex solution(If system have). The possible number of solutions
    is zero, one or infinite.

    Parameters
    ==========

    system : list of equations
        The target system of equations
    symbols : list of Symbols
        symbols should be given as a sequence eg. list

    Returns
    =======

    A FiniteSet of ordered tuple of values of `symbols` for which the `system`
    has solution. Order of values in the tuple is same as symbols present in
    the parameter `symbols`.

    Please note that general FiniteSet is unordered, the solution returned
    here is not simply a FiniteSet of solutions, rather it is a FiniteSet of
    ordered tuple, i.e. the first & only argument to FiniteSet is a tuple of
    solutions, which is ordered, & hence the returned solution is ordered.

    Also note that solution could also have been returned as an ordered tuple,
    FiniteSet is just a wrapper `{}` around the tuple. It has no other
    significance except for the fact it is just used to maintain a consistent
    output format throughout the solveset.

    For the given set of Equations, the respective input types
    are given below:

    .. math:: x*y - 1 = 0
    .. math:: 4*x**2 + y**2 - 5 = 0

    `system  = [x*y - 1, 4*x**2 + y**2 - 5]`
    `symbols = [x, y]`

    Raises
    ======

    ValueError
        The input is not valid.
        The symbols are not given.
    AttributeError
        The input symbols are not `Symbol` type.

    Examples
    ========

    >>> from sympy.core.symbol import symbols
    >>> from sympy.solvers.solveset import nonlinsolve
    >>> x, y, z = symbols('x, y, z', real=True)
    >>> nonlinsolve([x*y - 1, 4*x**2 + y**2 - 5], [x, y])
    {(-1, -1), (-1/2, -2), (1/2, 2), (1, 1)}

    1. Positive dimensional system and complements:

    >>> from sympy import pprint
    >>> from sympy.polys.polytools import is_zero_dimensional
    >>> a, b, c, d = symbols('a, b, c, d', real=True)
    >>> eq1 =  a + b + c + d
    >>> eq2 = a*b + b*c + c*d + d*a
    >>> eq3 = a*b*c + b*c*d + c*d*a + d*a*b
    >>> eq4 = a*b*c*d - 1
    >>> system = [eq1, eq2, eq3, eq4]
    >>> is_zero_dimensional(system)
    False
    >>> pprint(nonlinsolve(system, [a, b, c, d]), use_unicode=False)
      -1       1               1      -1
    {(---, -d, -, {d} \ {0}), (-, -d, ---, {d} \ {0})}
       d       d               d       d
    >>> nonlinsolve([(x+y)**2 - 4, x + y - 2], [x, y])
    {(-y + 2, y)}

    2. If some of the equations are non polynomial equation then `nonlinsolve`
    will call `substitution` function and returns real and complex solutions,
    if present.

    >>> from sympy import exp, sin
    >>> nonlinsolve([exp(x) - sin(y), y**2 - 4], [x, y])
    {(log(sin(2)), 2), (ImageSet(Lambda(_n, I*(2*_n*pi + pi) +
        log(sin(2))), Integers), -2), (ImageSet(Lambda(_n, 2*_n*I*pi +
        Mod(log(sin(2)), 2*I*pi)), Integers), 2)}

    3. If system is Non linear polynomial zero dimensional then it returns
    both solution (real and complex solutions, if present using
    `solve_poly_system`):

    >>> from sympy import sqrt
    >>> nonlinsolve([x**2 - 2*y**2 -2, x*y - 2], [x, y])
    {(-2, -1), (2, 1), (-sqrt(2)*I, sqrt(2)*I), (sqrt(2)*I, -sqrt(2)*I)}

    4. `nonlinsolve` can solve some linear(zero or positive dimensional)
    system (because it is using `groebner` function to get the
    groebner basis and then `substitution` function basis as the new `system`).
    But it is not recommended to solve linear system using `nonlinsolve`,
    because `linsolve` is better for all kind of linear system.

    >>> nonlinsolve([x + 2*y -z - 3, x - y - 4*z + 9 , y + z - 4], [x, y, z])
    {(3*z - 5, -z + 4, z)}

    5. System having polynomial equations and only real solution is present
    (will be solved using `solve_poly_system`):

    >>> e1 = sqrt(x**2 + y**2) - 10
    >>> e2 = sqrt(y**2 + (-x + 10)**2) - 3
    >>> nonlinsolve((e1, e2), (x, y))
    {(191/20, -3*sqrt(391)/20), (191/20, 3*sqrt(391)/20)}
    >>> nonlinsolve([x**2 + 2/y - 2, x + y - 3], [x, y])
    {(1, 2), (1 + sqrt(5), -sqrt(5) + 2), (-sqrt(5) + 1, 2 + sqrt(5))}
    >>> nonlinsolve([x**2 + 2/y - 2, x + y - 3], [y, x])
    {(2, 1), (2 + sqrt(5), -sqrt(5) + 1), (-sqrt(5) + 2, 1 + sqrt(5))}

    6. It is better to use symbols instead of Trigonometric Function or
    Function (e.g. replace `sin(x)` with symbol, replace `f(x)` with symbol
    and so on. Get soln from `nonlinsolve` and then using `solveset` get
    the value of `x`)

    How nonlinsolve is better than old solver `_solve_system` :
    ===========================================================

    1. A positive dimensional system solver : nonlinsolve can return
    solution for positive dimensional system. It finds the
    Groebner Basis of the positive dimensional system(calling it as
    basis) then we can start solving equation(having least number of
    variable first in the basis) using solveset and substituting that
    solved solutions into other equation(of basis) to get solution in
    terms of minimum variables. Here the important thing is how we
    are substituting the known values and in which equations.

    2. Real and Complex both solutions : nonlinsolve returns both real
    and complex solution. If all the equations in the system are polynomial
    then using `solve_poly_system` both real and complex solution is returned.
    If all the equations in the system are not polynomial equation then goes to
    `substitution` method with this polynomial and non polynomial equation(s),
    to solve for unsolved variables. Here to solve for particular variable
    solveset_real and solveset_complex is used. For both real and complex
    solution function `_solve_using_know_values` is used inside `substitution`
    function.(`substitution` function will be called when there is any non
    polynomial equation(s) is present). When solution is valid then add its
    general solution in the final result.

    3. Complement and Intersection will be added if any : nonlinsolve maintains
    dict for complements and Intersections. If solveset find complements or/and
    Intersection with any Interval or set during the execution of
    `substitution` function ,then complement or/and Intersection for that
    variable is added before returning final solution.

    """
    from sympy.polys.polytools import is_zero_dimensional

    if not system:
        return S.EmptySet

    if not symbols:
        msg = ('Symbols must be given, for which solution of the '
               'system is to be found.')
        raise ValueError(filldedent(msg))

    if hasattr(symbols[0], '__iter__'):
        symbols = symbols[0]

    if not is_sequence(symbols) or not symbols:
        msg = ('Symbols must be given, for which solution of the '
               'system is to be found.')
        raise IndexError(filldedent(msg))

    system, symbols, swap = recast_to_symbols(system, symbols)
    if swap:
        soln = nonlinsolve(system, symbols)
        return FiniteSet(*[tuple(i.xreplace(swap) for i in s) for s in soln])

    if len(system) == 1 and len(symbols) == 1:
        return _solveset_work(system, symbols)

    # main code of def nonlinsolve() starts from here
    polys, polys_expr, nonpolys, denominators = _separate_poly_nonpoly(
        system, symbols)

    if len(symbols) == len(polys):
        # If all the equations in the system are poly
        if is_zero_dimensional(polys, symbols):
            # finite number of soln (Zero dimensional system)
            try:
                return _handle_zero_dimensional(polys, symbols, system)
            except NotImplementedError:
                # Right now it doesn't fail for any polynomial system of
                # equation. If `solve_poly_system` fails then `substitution`
                # method will handle it.
                result = substitution(
                    polys_expr, symbols, exclude=denominators)
                return result

        # positive dimensional system
        return _handle_positive_dimensional(polys, symbols, denominators)

    else:
        # If all the equations are not polynomial.
        # Use `substitution` method for the system
        result = substitution(
            polys_expr + nonpolys, symbols, exclude=denominators)
        return result
