from sympy.multipledispatch import dispatch, Dispatcher
from sympy.core import Basic, Expr, Function, Add, Mul, Pow, Dummy, Integer
from sympy import Min, Max, Set, sympify, Lambda, symbols, exp, log, S, oo
from sympy.sets import (imageset, Interval, FiniteSet, Union, ImageSet,
    ProductSet, EmptySet, Intersection)
from sympy.core.function import FunctionClass
from sympy.logic.boolalg import And, Or, Not, true, false


_x, _y = symbols("x y")


@dispatch(Basic, Basic)
def _set_mul(x, y):
    return None

@dispatch(Set, Set)
def _set_mul(x, y):
    return None

@dispatch(Expr, Expr)
def _set_mul(x, y):
    return x*y

@dispatch(Interval, Interval)
def _set_mul(x, y):
    """
    Multiplications in interval arithmetic
    https://en.wikipedia.org/wiki/Interval_arithmetic
    """
    # TODO: some intervals containing 0 and oo will fail as 0*oo returns nan.
    comvals = (
        (x.start * y.start, bool(x.left_open or y.left_open)),
        (x.start * y.end, bool(x.left_open or y.right_open)),
        (x.end * y.start, bool(x.right_open or y.left_open)),
        (x.end * y.end, bool(x.right_open or y.right_open)),
    )
    # TODO: handle symbolic intervals
    minval, minopen = min(comvals)
    maxval, maxopen = max(comvals)
    return Interval(
        minval,
        maxval,
        minopen,
        maxopen
    )
    return SetExpr(Interval(start, end))


@dispatch(Basic, Basic)
def _set_div(x, y):
    return None

@dispatch(Expr, Expr)
def _set_div(x, y):
    return x/y

@dispatch(Set, Set)
def _set_div(x, y):
    return None

@dispatch(Interval, Interval)
def _set_div(x, y):
    """
    Divisions in interval arithmetic
    https://en.wikipedia.org/wiki/Interval_arithmetic
    """
    from sympy.sets.setexpr import set_mul
    from sympy import oo
    if (y.start*y.end).is_negative:
        return Interval(-oo, oo)
    if y.start == 0:
        s2 = oo
    else:
        s2 = 1/y.start
    if y.end == 0:
        s1 = -oo
    else:
        s1 = 1/y.end
    return set_mul(x, Interval(s1, s2, y.right_open, y.left_open))