"""Sparse rational function fields. """


from typing import Any, Dict
from functools import reduce

from operator import add, mul, lt, le, gt, ge

from sympy.core.compatibility import is_sequence
from sympy.core.expr import Expr
from sympy.core.mod import Mod
from sympy.core.numbers import Exp1
from sympy.core.singleton import S
from sympy.core.symbol import Symbol
from sympy.core.sympify import CantSympify, sympify
from sympy.functions.elementary.exponential import ExpBase
from sympy.polys.domains.domainelement import DomainElement
from sympy.polys.domains.fractionfield import FractionField
from sympy.polys.domains.polynomialring import PolynomialRing
from sympy.polys.constructor import construct_domain
from sympy.polys.orderings import lex
from sympy.polys.polyerrors import CoercionFailed
from sympy.polys.polyoptions import build_options
from sympy.polys.polyutils import _parallel_dict_from_expr
from sympy.polys.rings import PolyElement
from sympy.printing.defaults import DefaultPrinting
from sympy.utilities import public
from sympy.utilities.magic import pollute

@public
def field(symbols, domain, order=lex):
    """Construct new rational function field returning (field, x1, ..., xn). """
    _field = FracField(symbols, domain, order)
    return (_field,) + _field.gens

@public
def xfield(symbols, domain, order=lex):
    """Construct new rational function field returning (field, (x1, ..., xn)). """
    _field = FracField(symbols, domain, order)
    return (_field, _field.gens)

@public
def vfield(symbols, domain, order=lex):
    """Construct new rational function field and inject generators into global namespace. """
    _field = FracField(symbols, domain, order)
    pollute([ sym.name for sym in _field.symbols ], _field.gens)
    return _field

@public
def sfield(exprs, *symbols, **options):
    """Construct a field deriving generators and domain
    from options and input expressions.

    Parameters
    ==========

    exprs   : py:class:`~.Expr` or sequence of :py:class:`~.Expr` (sympifiable)
    symbols : sequence of :py:class:`~.Symbol`/:py:class:`~.Expr`
    options : keyword arguments understood by :py:class:`~.Options`

    Examples
    ========

    >>> from sympy.core import symbols
    >>> from sympy.functions import exp, log
    >>> from sympy.polys.fields import sfield

    >>> x = symbols("x")
    >>> K, f = sfield((x*log(x) + 4*x**2)*exp(1/x + log(x)/3)/x**2)
    >>> K
    Rational function field in x, exp(1/x), log(x), x**(1/3) over ZZ with lex order
    >>> f
    (4*x**2*(exp(1/x)) + x*(exp(1/x))*(log(x)))/((x**(1/3))**5)
    """
    single = False
    if not is_sequence(exprs):
        exprs, single = [exprs], True

    exprs = list(map(sympify, exprs))
    opt = build_options(symbols, options)
    numdens = []
    for expr in exprs:
        numdens.extend(expr.as_numer_denom())
    reps, opt = _parallel_dict_from_expr(numdens, opt)

    if opt.domain is None:
        # NOTE: this is inefficient because construct_domain() automatically
        # performs conversion to the target domain. It shouldn't do this.
        coeffs = sum([list(rep.values()) for rep in reps], [])
        opt.domain, _ = construct_domain(coeffs, opt=opt)

    _field = FracField(opt.gens, opt.domain, opt.order)
    fracs = []
    for i in range(0, len(reps), 2):
        fracs.append(_field(tuple(reps[i:i+2])))

    if single:
        return (_field, fracs[0])
    else:
        return (_field, fracs)

_field_cache = {}  # type: Dict[Any, Any]

class FracField(DefaultPrinting):
    """Multivariate distributed rational function field. """

    def __new__(cls, symbols, domain, order=lex):
        from sympy.polys.rings import PolyRing
        ring = PolyRing(symbols, domain, order)
        symbols = ring.symbols
        ngens = ring.ngens
        domain = ring.domain
        order = ring.order

        _hash_tuple = (cls.__name__, symbols, ngens, domain, order)
        obj = _field_cache.get(_hash_tuple)

        if obj is None:
            obj = object.__new__(cls)
            obj._hash_tuple = _hash_tuple
            obj._hash = hash(_hash_tuple)
            obj.ring = ring
            obj.dtype = type("FracElement", (FracElement,), {"field": obj})
            obj.symbols = symbols
            obj.ngens = ngens
            obj.domain = domain
            obj.order = order

            obj.zero = obj.dtype(ring.zero)
            obj.one = obj.dtype(ring.one)

            obj.gens = obj._gens()

            for symbol, generator in zip(obj.symbols, obj.gens):
                if isinstance(symbol, Symbol):
                    name = symbol.name

                    if not hasattr(obj, name):
                        setattr(obj, name, generator)

            _field_cache[_hash_tuple] = obj

        return obj

    def _gens(self):
        """Return a list of polynomial generators. """
        return tuple([ self.dtype(gen) for gen in self.ring.gens ])

    def __getnewargs__(self):
        return (self.symbols, self.domain, self.order)

    def __hash__(self):
        return self._hash

    def index(self, gen):
        if isinstance(gen, self.dtype):
            return self.ring.index(gen.to_poly())
        else:
            raise ValueError("expected a %s, got %s instead" % (self.dtype,gen))

    def __eq__(self, other):
        return isinstance(other, FracField) and \
            (self.symbols, self.ngens, self.domain, self.order) == \
            (other.symbols, other.ngens, other.domain, other.order)

    def __ne__(self, other):
        return not self == other

    def raw_new(self, numer, denom=None):
        return self.dtype(numer, denom)
    def new(self, numer, denom=None):
        if denom is None: denom = self.ring.one
        numer, denom = numer.cancel(denom)
        return self.raw_new(numer, denom)

    def domain_new(self, element):
        return self.domain.convert(element)

    def ground_new(self, element):
        try:
            return self.new(self.ring.ground_new(element))
        except CoercionFailed:
            domain = self.domain

            if not domain.is_Field and domain.has_assoc_Field:
                ring = self.ring
                ground_field = domain.get_field()
                element = ground_field.convert(element)
                numer = ring.ground_new(ground_field.numer(element))
                denom = ring.ground_new(ground_field.denom(element))
                return self.raw_new(numer, denom)
            else:
                raise

    def field_new(self, element):
        if isinstance(element, FracElement):
            if self == element.field:
                return element

            if isinstance(self.domain, FractionField) and \
                self.domain.field == element.field:
                return self.ground_new(element)
            elif isinstance(self.domain, PolynomialRing) and \
                self.domain.ring.to_field() == element.field:
                return self.ground_new(element)
            else:
                raise NotImplementedError("conversion")
        elif isinstance(element, PolyElement):
            denom, numer = element.clear_denoms()

            if isinstance(self.domain, PolynomialRing) and \
                numer.ring == self.domain.ring:
                numer = self.ring.ground_new(numer)
            elif isinstance(self.domain, FractionField) and \
                numer.ring == self.domain.field.to_ring():
                numer = self.ring.ground_new(numer)
            else:
                numer = numer.set_ring(self.ring)

            denom = self.ring.ground_new(denom)
            return self.raw_new(numer, denom)
        elif isinstance(element, tuple) and len(element) == 2:
            numer, denom = list(map(self.ring.ring_new, element))
            return self.new(numer, denom)
        elif isinstance(element, str):
            raise NotImplementedError("parsing")
        elif isinstance(element, Expr):
            return self.from_expr(element)
        else:
            return self.ground_new(element)

    __call__ = field_new

    def _rebuild_expr(self, expr, mapping):
        domain = self.domain
        powers = tuple((gen, gen.as_base_exp()) for gen in mapping.keys()
            if gen.is_Pow or isinstance(gen, ExpBase))

        def _rebuild(expr):
            generator = mapping.get(expr)

            if generator is not None:
                return generator
            elif expr.is_Add:
                return reduce(add, list(map(_rebuild, expr.args)))
            elif expr.is_Mul:
                return reduce(mul, list(map(_rebuild, expr.args)))
            elif expr.is_Pow or isinstance(expr, (ExpBase, Exp1)):
                b, e = expr.as_base_exp()
                # look for bg**eg whose integer power may be b**e
                for gen, (bg, eg) in powers:
                    if bg == b and Mod(e, eg) == 0:
                        return mapping.get(gen)**int(e/eg)
                if e.is_Integer and e is not S.One:
                    return _rebuild(b)**int(e)

            try:
                return domain.convert(expr)
            except CoercionFailed:
                if not domain.is_Field and domain.has_assoc_Field:
                    return domain.get_field().convert(expr)
                else:
                    raise

        return _rebuild(sympify(expr))

    def from_expr(self, expr):
        mapping = dict(list(zip(self.symbols, self.gens)))

        try:
            frac = self._rebuild_expr(expr, mapping)
        except CoercionFailed:
            raise ValueError("expected an expression convertible to a rational function in %s, got %s" % (self, expr))
        else:
            return self.field_new(frac)

    def to_domain(self):
        return FractionField(self)

    def to_ring(self):
        from sympy.polys.rings import PolyRing
        return PolyRing(self.symbols, self.domain, self.order)

class FracElement(DomainElement, DefaultPrinting, CantSympify):
    """Element of multivariate distributed rational function field. """

    def __init__(self, numer, denom=None):
        if denom is None:
            denom = self.field.ring.one
        elif not denom:
            raise ZeroDivisionError("zero denominator")

        self.numer = numer
        self.denom = denom

    def raw_new(f, numer, denom):
        return f.__class__(numer, denom)
    def new(f, numer, denom):
        return f.raw_new(*numer.cancel(denom))

    def to_poly(f):
        if f.denom != 1:
            raise ValueError("f.denom should be 1")
        return f.numer

    def parent(self):
        return self.field.to_domain()

    def __getnewargs__(self):
        return (self.field, self.numer, self.denom)

    _hash = None

    def __hash__(self):
        _hash = self._hash
        if _hash is None:
            self._hash = _hash = hash((self.field, self.numer, self.denom))
        return _hash

    def copy(self):
        return self.raw_new(self.numer.copy(), self.denom.copy())

    def set_field(self, new_field):
        if self.field == new_field:
            return self
        else:
            new_ring = new_field.ring
            numer = self.numer.set_ring(new_ring)
            denom = self.denom.set_ring(new_ring)
            return new_field.new(numer, denom)

    def as_expr(self, *symbols):
        return self.numer.as_expr(*symbols)/self.denom.as_expr(*symbols)

    def __eq__(f, g):
        if isinstance(g, FracElement) and f.field == g.field:
            return f.numer == g.numer and f.denom == g.denom
        else:
            return f.numer == g and f.denom == f.field.ring.one

    def __ne__(f, g):
        return not f == g

    def __bool__(f):
        return bool(f.numer)

    def sort_key(self):
        return (self.denom.sort_key(), self.numer.sort_key())

    def _cmp(f1, f2, op):
        if isinstance(f2, f1.field.dtype):
            return op(f1.sort_key(), f2.sort_key())
        else:
            return NotImplemented

    def __lt__(f1, f2):
        return f1._cmp(f2, lt)
    def __le__(f1, f2):
        return f1._cmp(f2, le)
    def __gt__(f1, f2):
        return f1._cmp(f2, gt)
    def __ge__(f1, f2):
        return f1._cmp(f2, ge)

    def __pos__(f):
        """Negate all coefficients in ``f``. """
        return f.raw_new(f.numer, f.denom)

    def __neg__(f):
        """Negate all coefficients in ``f``. """
        return f.raw_new(-f.numer, f.denom)

    def _extract_ground(self, element):
        domain = self.field.domain

        try:
            element = domain.convert(element)
        except CoercionFailed:
            if not domain.is_Field and domain.has_assoc_Field:
                ground_field = domain.get_field()

                try:
                    element = ground_field.convert(element)
                except CoercionFailed:
                    pass
                else:
                    return -1, ground_field.numer(element), ground_field.denom(element)

            return 0, None, None
        else:
            return 1, element, None

    def __add__(f, g):
        """Add rational functions ``f`` and ``g``. """
        field = f.field

        if not g:
            return f
        elif not f:
            return g
        elif isinstance(g, field.dtype):
            if f.denom == g.denom:
                return f.new(f.numer + g.numer, f.denom)
            else:
                return f.new(f.numer*g.denom + f.denom*g.numer, f.denom*g.denom)
        elif isinstance(g, field.ring.dtype):
            return f.new(f.numer + f.denom*g, f.denom)
        else:
            if isinstance(g, FracElement):
                if isinstance(field.domain, FractionField) and field.domain.field == g.field:
                    pass
                elif isinstance(g.field.domain, FractionField) and g.field.domain.field == field:
                    return g.__radd__(f)
                else:
                    return NotImplemented
            elif isinstance(g, PolyElement):
                if isinstance(field.domain, PolynomialRing) and field.domain.ring == g.ring:
                    pass
                else:
                    return g.__radd__(f)

        return f.__radd__(g)

    def __radd__(f, c):
        if isinstance(c, f.field.ring.dtype):
            return f.new(f.numer + f.denom*c, f.denom)

        op, g_numer, g_denom = f._extract_ground(c)

        if op == 1:
            return f.new(f.numer + f.denom*g_numer, f.denom)
        elif not op:
            return NotImplemented
        else:
            return f.new(f.numer*g_denom + f.denom*g_numer, f.denom*g_denom)

    def __sub__(f, g):
        """Subtract rational functions ``f`` and ``g``. """
        field = f.field

        if not g:
            return f
        elif not f:
            return -g
        elif isinstance(g, field.dtype):
            if f.denom == g.denom:
                return f.new(f.numer - g.numer, f.denom)
            else:
                return f.new(f.numer*g.denom - f.denom*g.numer, f.denom*g.denom)
        elif isinstance(g, field.ring.dtype):
            return f.new(f.numer - f.denom*g, f.denom)
        else:
            if isinstance(g, FracElement):
                if isinstance(field.domain, FractionField) and field.domain.field == g.field:
                    pass
                elif isinstance(g.field.domain, FractionField) and g.field.domain.field == field:
                    return g.__rsub__(f)
                else:
                    return NotImplemented
            elif isinstance(g, PolyElement):
                if isinstance(field.domain, PolynomialRing) and field.domain.ring == g.ring:
                    pass
                else:
                    return g.__rsub__(f)

        op, g_numer, g_denom = f._extract_ground(g)

        if op == 1:
            return f.new(f.numer - f.denom*g_numer, f.denom)
        elif not op:
            return NotImplemented
        else:
            return f.new(f.numer*g_denom - f.denom*g_numer, f.denom*g_denom)

    def __rsub__(f, c):
        if isinstance(c, f.field.ring.dtype):
            return f.new(-f.numer + f.denom*c, f.denom)

        op, g_numer, g_denom = f._extract_ground(c)

        if op == 1:
            return f.new(-f.numer + f.denom*g_numer, f.denom)
        elif not op:
            return NotImplemented
        else:
            return f.new(-f.numer*g_denom + f.denom*g_numer, f.denom*g_denom)

    def __mul__(f, g):
        """Multiply rational functions ``f`` and ``g``. """
        field = f.field

        if not f or not g:
            return field.zero
        elif isinstance(g, field.dtype):
            return f.new(f.numer*g.numer, f.denom*g.denom)
        elif isinstance(g, field.ring.dtype):
            return f.new(f.numer*g, f.denom)
        else:
            if isinstance(g, FracElement):
                if isinstance(field.domain, FractionField) and field.domain.field == g.field:
                    pass
                elif isinstance(g.field.domain, FractionField) and g.field.domain.field == field:
                    return g.__rmul__(f)
                else:
                    return NotImplemented
            elif isinstance(g, PolyElement):
                if isinstance(field.domain, PolynomialRing) and field.domain.ring == g.ring:
                    pass
                else:
                    return g.__rmul__(f)

        return f.__rmul__(g)

    def __rmul__(f, c):
        if isinstance(c, f.field.ring.dtype):
            return f.new(f.numer*c, f.denom)

        op, g_numer, g_denom = f._extract_ground(c)

        if op == 1:
            return f.new(f.numer*g_numer, f.denom)
        elif not op:
            return NotImplemented
        else:
            return f.new(f.numer*g_numer, f.denom*g_denom)

    def __truediv__(f, g):
        """Computes quotient of fractions ``f`` and ``g``. """
        field = f.field

        if not g:
            raise ZeroDivisionError
        elif isinstance(g, field.dtype):
            return f.new(f.numer*g.denom, f.denom*g.numer)
        elif isinstance(g, field.ring.dtype):
            return f.new(f.numer, f.denom*g)
        else:
            if isinstance(g, FracElement):
                if isinstance(field.domain, FractionField) and field.domain.field == g.field:
                    pass
                elif isinstance(g.field.domain, FractionField) and g.field.domain.field == field:
                    return g.__rtruediv__(f)
                else:
                    return NotImplemented
            elif isinstance(g, PolyElement):
                if isinstance(field.domain, PolynomialRing) and field.domain.ring == g.ring:
                    pass
                else:
                    return g.__rtruediv__(f)

        op, g_numer, g_denom = f._extract_ground(g)

        if op == 1:
            return f.new(f.numer, f.denom*g_numer)
        elif not op:
            return NotImplemented
        else:
            return f.new(f.numer*g_denom, f.denom*g_numer)

    def __rtruediv__(f, c):
        if not f:
            raise ZeroDivisionError
        elif isinstance(c, f.field.ring.dtype):
            return f.new(f.denom*c, f.numer)

        op, g_numer, g_denom = f._extract_ground(c)

        if op == 1:
            return f.new(f.denom*g_numer, f.numer)
        elif not op:
            return NotImplemented
        else:
            return f.new(f.denom*g_numer, f.numer*g_denom)

    def __pow__(f, n):
        """Raise ``f`` to a non-negative power ``n``. """
        if n >= 0:
            return f.raw_new(f.numer**n, f.denom**n)
        elif not f:
            raise ZeroDivisionError
        else:
            return f.raw_new(f.denom**-n, f.numer**-n)

    def diff(f, x):
        """Computes partial derivative in ``x``.

        Examples
        ========

        >>> from sympy.polys.fields import field
        >>> from sympy.polys.domains import ZZ

        >>> _, x, y, z = field("x,y,z", ZZ)
        >>> ((x**2 + y)/(z + 1)).diff(x)
        2*x/(z + 1)

        """
        x = x.to_poly()
        return f.new(f.numer.diff(x)*f.denom - f.numer*f.denom.diff(x), f.denom**2)

    def __call__(f, *values):
        if 0 < len(values) <= f.field.ngens:
            return f.evaluate(list(zip(f.field.gens, values)))
        else:
            raise ValueError("expected at least 1 and at most %s values, got %s" % (f.field.ngens, len(values)))

    def evaluate(f, x, a=None):
        if isinstance(x, list) and a is None:
            x = [ (X.to_poly(), a) for X, a in x ]
            numer, denom = f.numer.evaluate(x), f.denom.evaluate(x)
        else:
            x = x.to_poly()
            numer, denom = f.numer.evaluate(x, a), f.denom.evaluate(x, a)

        field = numer.ring.to_field()
        return field.new(numer, denom)

    def subs(f, x, a=None):
        if isinstance(x, list) and a is None:
            x = [ (X.to_poly(), a) for X, a in x ]
            numer, denom = f.numer.subs(x), f.denom.subs(x)
        else:
            x = x.to_poly()
            numer, denom = f.numer.subs(x, a), f.denom.subs(x, a)

        return f.new(numer, denom)

    def compose(f, x, a=None):
        raise NotImplementedError