"""
This module contains functions to:

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

    - 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
from sympy.core.numbers import I, Number, Rational, oo
from sympy.core.function import (Lambda, expand, expand_complex)
from sympy.core.relational import Eq
from sympy.simplify.simplify import simplify, fraction, trigsimp
from sympy.core.symbol import Symbol
from sympy.functions import (log, Abs, tan, cot, sin, cos, sec, csc, exp,
                             acos, asin, acsc, asec, arg,
                             piecewise_fold)
from sympy.functions.elementary.trigonometric import (TrigonometricFunction,
                                                      HyperbolicFunction)
from sympy.functions.elementary.miscellaneous import real_root
from sympy.sets import (FiniteSet, EmptySet, imageset, Interval, Intersection,
                        Union, ConditionSet, ImageSet, Complement)
from sympy.matrices import Matrix
from sympy.polys import (roots, Poly, degree, together, PolynomialError,
                         RootOf)
from sympy.solvers.solvers import checksol, denoms, unrad, _simple_dens
from sympy.solvers.polysys import solve_poly_system
from sympy.solvers.inequalities import solve_univariate_inequality
from sympy.utilities import filldedent
from sympy.calculus.util import periodicity
from sympy.core.compatibility import ordered, default_sort_key


def _invert(f_x, y, x, domain=S.Complexes):
    """
    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 their domain in which they are valid, through set
    operations. For instance, if ``y = Abs(x) - n``, is inverted
    in the real domain, then, the ``set_h`` doesn't simply return
    `{-n, n}`, as the nature of `n` is unknown; rather it will return:
    `Intersection([0, oo) {n}) U Intersection((-oo, 0], {-n})`

    By default, the complex domain is used but note that inverting even
    seemingly simple functions like ``exp(x)`` can give very different
    result in the complex domain than are 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 use 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((-oo, oo), {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 not f_x.has(x):
        raise ValueError("Inverse of constant function doesn't exist")
    y = sympify(y)
    if y.has(x):
        raise ValueError("y should be independent of x ")

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


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):
        pos = Interval(0, S.Infinity)
        neg = Interval(S.NegativeInfinity, 0)
        return _invert_real(f.args[0],
                    Union(imageset(Lambda(n, n), g_ys).intersect(pos),
                          imageset(Lambda(n, -n), g_ys).intersect(neg)), 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 numer == S.One or numer == - S.One:
                    return _invert_real(base, res, symbol)
                else:
                    if 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:
            return _invert_real(expo,
                imageset(Lambda(n, log(n)/log(base)), g_ys), 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:
            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, 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 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
        if A.get('finite', None) is None:
            A['finite'] = True
        A.setdefault('complex', True)
        A.setdefault('real', domain.is_subset(S.Reals))
        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):
        return _solve_as_poly(g, symbol, 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):
    """ 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, FiniteSet):
        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_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 free
            # variables because that makes the solution more complicated. For
            # example expand_complex(a) returns re(a) + I*im(a)
            if all([s.free_symbols == 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):
        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 {}
    if not pattern_match.get(p, S.Zero).is_zero:
        f_p, f_q, f_r = pattern_match[p], pattern_match[q], pattern_match[r]
        q_pos_cond = solve_univariate_inequality(f_q >= 0, symbol,
                                                 relational=False)
        q_neg_cond = solve_univariate_inequality(f_q < 0, symbol,
                                                 relational=False)

        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 {pi*(2*n + 1) - 2 | 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

            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
    f = together(f)
    if f.is_Mul:
        _, f = f.as_independent(symbol, as_Add=False)
    if f.is_Add:
        a, h = f.as_independent(symbol)
        m, h = h.as_independent(symbol, as_Add=False)
        f = a/m + h  # XXX condition `m != 0` should be added to soln
    f = piecewise_fold(f)

    # assign the solvers to use
    solver = lambda f, x, domain=domain: _solveset(f, x, domain)
    if domain.is_subset(S.Reals):
        inverter_func = invert_real
    else:
        inverter_func = invert_complex
    inverter = lambda f, rhs, symbol: inverter_func(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 f.is_Piecewise:
        dom = domain
        result = EmptySet()
        expr_set_pairs = f.as_expr_set_pairs()
        for (expr, in_set) in expr_set_pairs:
            if in_set.is_Relational:
                in_set = in_set.as_set()
            if in_set.is_Interval:
                dom -= in_set
            solns = solver(expr, symbol, in_set)
            result += solns
    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 += _solve_as_rational(equation, symbol, domain)
                else:
                    result += solver(equation, symbol)

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

    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 solveset(f, symbol=None, domain=S.Complexes):
    """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)
    (0, oo)

    """
    f = sympify(f)

    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))

    free_symbols = f.free_symbols

    if 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()
        else:
            raise ValueError(filldedent('''
                The independent variable must be specified for a
                multivariate equation.'''))
    elif not getattr(symbol, 'is_Symbol', False):
        raise ValueError('A Symbol must be given, not type %s: %s' %
            (type(symbol), symbol))

    if isinstance(f, Eq):
        from sympy.core import Add
        f = Add(f.lhs, - f.rhs, evaluate=False)
    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, relational=False) - _invalid_solutions(
            f, symbol, domain)
        except NotImplementedError:
            result = ConditionSet(symbol, f, domain)
        return result

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


def _invalid_solutions(f, symbol, domain):
    bad = S.EmptySet
    for d in denoms(f):
        bad += _solveset(d, symbol, domain, _check=False)
    return bad


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
            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])
    # initialise 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 initialised (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 to solve for should be given as input in all the
    cases either in an iterable or as comma separated arguments.
    This is done to maintain consistency in returning solutions
    in the form of variable input by the user.

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

    Returns
    =======

    A FiniteSet of ordered tuple of values of `symbols` for which
    the `system` has solution.

    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.

    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 under determined, the function
      will return parametric solution in terms of the given symbols.
      Free symbols in the system are returned as it is. For e.g. in the system
      below, `z` is returned as the solution for variable z, which means z is a
      free symbol, i.e. it can take arbitrary values.

    >>> 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)}

    * List of Equations as input

    >>> Eqns = [3*x + 2*y - z - 1, 2*x - 2*y + 4*z + 2, - x + S(1)/2*y - 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 not symbols:
        raise ValueError('Symbols must be given, for which solution of the '
                         'system is to be found.')

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

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

    if not sym:
        raise ValueError('Symbols or iterable of symbols must be given as '
                         'second argument, not type %s: %s' % (type(symbols[0]), symbols[0]))

    # 1). Augmented Matrix input Form
    if isinstance(system, Matrix):
        A, b = system[:, :-1], system[:, -1:]

    elif hasattr(system, '__iter__'):

        # 2). A & b as input Form
        if len(system) == 2 and system[0].is_Matrix:
            A, b = system[0], system[1]

        # 3). List of equations Form
        if not system[0].is_Matrix:
            A, b = linear_eq_to_matrix(system, symbols)

    else:
        raise ValueError("Invalid arguments")

    # Solve using Gauss-Jordan elimination
    try:
        sol, params, free_syms = A.gauss_jordan_solve(b, freevar=True)
    except ValueError:
        # No solution
        return EmptySet()

    # Replace free parameters with free symbols
    solution = []
    if params:
        for s in sol:
            for k, v in enumerate(params):
                s = s.xreplace({v: symbols[free_syms[k]]})
            solution.append(simplify(s))
    else:
        for s in sol:
            solution.append(simplify(s))

    # Return solutions
    solution = FiniteSet(tuple(solution))
    return solution


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

def _return_conditionset(eqs, symbols):
        # return conditionset
        condition_set = ConditionSet(
            FiniteSet(*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 intersecton.
                    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]

    try:
        sym = symbols[0].is_Symbol
    except AttributeError:
        sym = False
    except IndexError:
        msg = ('Symbols must be given, for which solution of the '
               'system is to be found.')
        raise IndexError(filldedent(msg))

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

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