from sympy import (atan, Eq, exp, Function, log,
    Rational, sin, sqrt, Symbol, tan, symbols)

from sympy.solvers.ode import (classify_ode, checkinfsol, dsolve, infinitesimals)

from sympy.solvers.ode.subscheck import checkodesol

from sympy.testing.pytest import XFAIL


C1 = Symbol('C1')
x, y = symbols("x y")
f = Function('f')
xi = Function('xi')
eta = Function('eta')


def test_heuristic1():
    a, b, c, a4, a3, a2, a1, a0 = symbols("a b c a4 a3 a2 a1 a0")
    df = f(x).diff(x)
    eq = Eq(df, x**2*f(x))
    eq1 = f(x).diff(x) + a*f(x) - c*exp(b*x)
    eq2 = f(x).diff(x) + 2*x*f(x) - x*exp(-x**2)
    eq3 = (1 + 2*x)*df + 2 - 4*exp(-f(x))
    eq4 = f(x).diff(x) - (a4*x**4 + a3*x**3 + a2*x**2 + a1*x + a0)**Rational(-1, 2)
    eq5 = x**2*df - f(x) + x**2*exp(x - (1/x))
    eqlist = [eq, eq1, eq2, eq3, eq4, eq5]

    i = infinitesimals(eq, hint='abaco1_simple')
    assert i == [{eta(x, f(x)): exp(x**3/3), xi(x, f(x)): 0},
        {eta(x, f(x)): f(x), xi(x, f(x)): 0},
        {eta(x, f(x)): 0, xi(x, f(x)): x**(-2)}]
    i1 = infinitesimals(eq1, hint='abaco1_simple')
    assert i1 == [{eta(x, f(x)): exp(-a*x), xi(x, f(x)): 0}]
    i2 = infinitesimals(eq2, hint='abaco1_simple')
    assert i2 == [{eta(x, f(x)): exp(-x**2), xi(x, f(x)): 0}]
    i3 = infinitesimals(eq3, hint='abaco1_simple')
    assert i3 == [{eta(x, f(x)): 0, xi(x, f(x)): 2*x + 1},
        {eta(x, f(x)): 0, xi(x, f(x)): 1/(exp(f(x)) - 2)}]
    i4 = infinitesimals(eq4, hint='abaco1_simple')
    assert i4 == [{eta(x, f(x)): 1, xi(x, f(x)): 0},
        {eta(x, f(x)): 0,
        xi(x, f(x)): sqrt(a0 + a1*x + a2*x**2 + a3*x**3 + a4*x**4)}]
    i5 = infinitesimals(eq5, hint='abaco1_simple')
    assert i5 == [{xi(x, f(x)): 0, eta(x, f(x)): exp(-1/x)}]

    ilist = [i, i1, i2, i3, i4, i5]
    for eq, i in (zip(eqlist, ilist)):
        check = checkinfsol(eq, i)
        assert check[0]

    # This ODE can be solved by the Lie Group method, when there are
    # better assumptions
    eq6 = df - (f(x)/x)*(x*log(x**2/f(x)) + 2)
    i = infinitesimals(eq6, hint='abaco1_product')
    assert i == [{eta(x, f(x)): f(x)*exp(-x), xi(x, f(x)): 0}]
    assert checkinfsol(eq6, i)[0]

    eq7 = x*(f(x).diff(x)) + 1 - f(x)**2
    i = infinitesimals(eq7, hint='chi')
    assert checkinfsol(eq7, i)[0]


def test_heuristic3():
    a, b = symbols("a b")
    df = f(x).diff(x)

    eq = x**2*df + x*f(x) + f(x)**2 + x**2
    i = infinitesimals(eq, hint='bivariate')
    assert i == [{eta(x, f(x)): f(x), xi(x, f(x)): x}]
    assert checkinfsol(eq, i)[0]

    eq = x**2*(-f(x)**2 + df)- a*x**2*f(x) + 2 - a*x
    i = infinitesimals(eq, hint='bivariate')
    assert checkinfsol(eq, i)[0]


def test_heuristic_function_sum():
    eq = f(x).diff(x) - (3*(1 + x**2/f(x)**2)*atan(f(x)/x) + (1 - 2*f(x))/x +
       (1 - 3*f(x))*(x/f(x)**2))
    i = infinitesimals(eq, hint='function_sum')
    assert i == [{eta(x, f(x)): f(x)**(-2) + x**(-2), xi(x, f(x)): 0}]
    assert checkinfsol(eq, i)[0]


def test_heuristic_abaco2_similar():
    a, b = symbols("a b")
    F = Function('F')
    eq = f(x).diff(x) - F(a*x + b*f(x))
    i = infinitesimals(eq, hint='abaco2_similar')
    assert i == [{eta(x, f(x)): -a/b, xi(x, f(x)): 1}]
    assert checkinfsol(eq, i)[0]

    eq = f(x).diff(x) - (f(x)**2 / (sin(f(x) - x) - x**2 + 2*x*f(x)))
    i = infinitesimals(eq, hint='abaco2_similar')
    assert i == [{eta(x, f(x)): f(x)**2, xi(x, f(x)): f(x)**2}]
    assert checkinfsol(eq, i)[0]


def test_heuristic_abaco2_unique_unknown():

    a, b = symbols("a b")
    F = Function('F')
    eq = f(x).diff(x) - x**(a - 1)*(f(x)**(1 - b))*F(x**a/a + f(x)**b/b)
    i = infinitesimals(eq, hint='abaco2_unique_unknown')
    assert i == [{eta(x, f(x)): -f(x)*f(x)**(-b), xi(x, f(x)): x*x**(-a)}]
    assert checkinfsol(eq, i)[0]

    eq = f(x).diff(x) + tan(F(x**2 + f(x)**2) + atan(x/f(x)))
    i = infinitesimals(eq, hint='abaco2_unique_unknown')
    assert i == [{eta(x, f(x)): x, xi(x, f(x)): -f(x)}]
    assert checkinfsol(eq, i)[0]

    eq = (x*f(x).diff(x) + f(x) + 2*x)**2 -4*x*f(x) -4*x**2 -4*a
    i = infinitesimals(eq, hint='abaco2_unique_unknown')
    assert checkinfsol(eq, i)[0]


def test_heuristic_linear():
    a, b, m, n = symbols("a b m n")

    eq = x**(n*(m + 1) - m)*(f(x).diff(x)) - a*f(x)**n -b*x**(n*(m + 1))
    i = infinitesimals(eq, hint='linear')
    assert checkinfsol(eq, i)[0]


@XFAIL
def test_kamke():
    a, b, alpha, c = symbols("a b alpha c")
    eq = x**2*(a*f(x)**2+(f(x).diff(x))) + b*x**alpha + c
    i = infinitesimals(eq, hint='sum_function')  # XFAIL
    assert checkinfsol(eq, i)[0]


def test_user_infinitesimals():
    x = Symbol("x") # assuming x is real generates an error
    eq = x*(f(x).diff(x)) + 1 - f(x)**2
    sol = Eq(f(x), (C1 + x**2)/(C1 - x**2))
    infinitesimals = {'xi':sqrt(f(x) - 1)/sqrt(f(x) + 1), 'eta':0}
    assert dsolve(eq, hint='lie_group', **infinitesimals) == sol
    assert checkodesol(eq, sol) == (True, 0)


@XFAIL
def test_lie_group_issue15219():
    eqn = exp(f(x).diff(x)-f(x))
    assert 'lie_group' not in classify_ode(eqn, f(x))