def test_Sum_doit():
    assert Sum(n*Integral(a**2), (n, 0, 2)).doit() == a**3
    assert Sum(n*Integral(a**2), (n, 0, 2)).doit(deep=False) == \
        3*Integral(a**2)
    assert summation(n*Integral(a**2), (n, 0, 2)) == 3*Integral(a**2)

    # test nested sum evaluation
    s = Sum( Sum( Sum(2,(z,1,n+1)), (y,x+1,n)), (x,1,n))
    assert 0 == (s.doit() - n*(n+1)*(n-1)).factor()

    assert Sum(Sum(KroneckerDelta(m, n), (m, 1, 3)), (n, 1, 3)).doit() == 3
    assert Sum(Sum(KroneckerDelta(k, m), (m, 1, 3)), (n, 1, 3)).doit() == \
        3*Piecewise((1, And(S(1) <= k, k <= 3)), (0, True))
    assert Sum(f(n)*Sum(KroneckerDelta(m, n), (m, 0, oo)), (n, 1, 3)).doit() == \
        f(1) + f(2) + f(3)
    assert Sum(f(n)*Sum(KroneckerDelta(m, n), (m, 0, oo)), (n, 1, oo)).doit() == \
        Sum(Piecewise((f(n), n >= 0), (0, True)), (n, 1, oo))
    l = Symbol('l', integer=True, positive=True)
    assert Sum(f(l)*Sum(KroneckerDelta(m, l), (m, 0, oo)), (l, 1, oo)).doit() == \
        Sum(f(l), (l, 1, oo))

    # issue 2597
    nmax = symbols('N', integer=True, positive=True)
    pw = Piecewise((1, And(S(1) <= n, n <= nmax)), (0, True))
    assert Sum(pw, (n, 1, nmax)).doit() == Sum(pw, (n, 1, nmax))

def test_issue_6966():
    i, k, m = symbols('i k m', integer=True)
    z_i, q_i = symbols('z_i q_i')
    a_k = Sum(-q_i*z_i/k,(i,1,m))
    b_k = a_k.diff(z_i)
    assert isinstance(b_k, Sum)
    assert b_k == Sum(-q_i/k,(i,1,m))

def test_eval_diff():
    assert Sum(x, (x, 1, 2)).diff(x) == 0
    assert Sum(x * y, (x, 1, 2)).diff(x) == 0
    assert Sum(x * y, (y, 1, 2)).diff(x) == Sum(y, (y, 1, 2))
    e = Sum(x * y, (x, 1, a))
    assert e.diff(a) == Derivative(e, a)
    assert Sum(x * y, (x, 1, 3), (a, 2, 5)).diff(y) == Sum(x * y, (x, 1, 3), (a, 2, 5)).doit().diff(y) == 24

class TimeSum:
    def setup(self):
        self.expr = Sum(
            Ts(int32ASU2L, ((), ((0, 1, -I / 2), (1, 0, I / 2)), ()), ext12Pi, int12Pi)
            * Ts(int42ASU2L, ((), ((0, 1, -I / 2), (1, 0, I / 2)), ()), ext22Pi, int22Pi)
            * Ts(int52ASU2L, ((), ((0, 1, -I / 2), (1, 0, I / 2)), ()), int12Pi, ext32Pi)
            * Ts(int62ASU2L, ((), ((0, 1, -I / 2), (1, 0, I / 2)), ()), int22Pi, ext42Pi)
            * KroneckerDelta(int32ASU2L, int42ASU2L)
            * KroneckerDelta(int52ASU2L, int62ASU2L),
            (int12Pi, 1, 2),
            (int22Pi, 1, 2),
            (exgg12SU2L, intgg11SU2L, intgg11SU2L),
            (ext12Pi, ext11Pi, ext11Pi),
            (exgg22SU2L, intgg11SU2L, intgg11SU2L),
            (ext22Pi, ext31Pi, ext31Pi),
            (exgg32SU2L, intgg21SU2L, intgg21SU2L),
            (ext32Pi, ext21Pi, ext21Pi),
            (exgg42SU2L, intgg21SU2L, intgg21SU2L),
            (ext42Pi, ext41Pi, ext41Pi),
            (int32ASU2L, 0, 2),
            (int42ASU2L, 0, 2),
            (int52ASU2L, 0, 2),
            (int62ASU2L, 0, 2),
        )

    def time_doit(self):
        self.expr.doit()

def test_multiple_sums():
    s = Sum(i * x + j, (i, a, b), (j, c, d))

    l = lambdarepr(s)
    assert l == "(builtins.sum(i*x + j for i in range(a, b+1) for j in range(c, d+1)))"

    assert (lambdify((x, a, b, c, d), s)(2, 3, 4, 5, 6) ==
            s.subs([(x, 2), (a, 3), (b, 4), (c, 5), (d, 6)]).doit())

def test_sum__2():
    s = Sum(i * x, (i, a, b))
    l = lambdarepr(s)
    assert l == "(builtins.sum(i*x for i in range(a, b+1)))"

    args = x, a, b
    f = lambdify(args, s)
    v = 2, 3, 8
    assert f(*v) == s.subs(zip(args, v)).doit()

def test_issue_2787():
    n, k = symbols('n k', positive=True, integer=True)
    p = symbols('p', positive=True)
    binomial_dist = binomial(n, k)*p**k*(1 - p)**(n - k)
    s = Sum(binomial_dist*k, (k, 0, n))
    res = s.doit().simplify()
    assert res == Piecewise((n*p, And(Or(-n + 1 < 0, -n + 1 >= 0),
        Or(-n + 1 < 0, Ne(p/(p - 1), 1)), p*Abs(1/(p - 1)) <= 1)),
        (Sum(k*p**k*(-p + 1)**(-k)*(-p + 1)**n*binomial(n, k), (k, 0, n)), True))

def test_issue_2787():
    n, k = symbols('n k', positive=True, integer=True)
    p = symbols('p', positive=True)
    binomial_dist = binomial(n, k)*p**k*(1 - p)**(n - k)
    s = Sum(binomial_dist*k, (k, 0, n))
    res = s.doit().simplify()
    assert res == Piecewise(
        (n*p, p/Abs(p - 1) <= 1),
        ((-p + 1)**n*Sum(k*p**k*(-p + 1)**(-k)*binomial(n, k), (k, 0, n)),
        True))

def test_sum__1():
    # In each case, test eval() the lambdarepr() to make sure that
    # it evaluates to the same results as the symbolic expression
    s = Sum(x ** i, (i, a, b))
    l = lambdarepr(s)
    assert l == "(builtins.sum(x**i for i in range(a, b+1)))"

    args = x, a, b
    f = lambdify(args, s)
    v = 2, 3, 8
    assert f(*v) == s.subs(zip(args, v)).doit()

def test_multiple_sums():
    s = Sum(i * x + j, (i, a, b), (j, c, d))

    l = lambdarepr(s)
    assert l == "(builtins.sum(i*x + j for i in range(a, b+1) for j in range(c, d+1)))"

    args = x, a, b, c, d
    f = lambdify(args, s)
    vals = 2, 3, 4, 5, 6
    f_ref = s.subs(zip(args, vals)).doit()
    f_res = f(*vals)
    assert f_res == f_ref

def test_sho():
    n, m = symbols('n m')
    h_n = Bd(n)*B(n)*(n + Rational(1, 2))
    H = Sum(h_n, (n, 0, 5))
    o = H.doit(deep = False)
    b = FixedBosonicBasis(2, 6)
    m = matrix_rep(o, b)
    # We need to double check these energy values to make sure that they
    # are correct and have the proper degeneracies!
    diag = [1, 2, 3, 3, 4, 5, 4, 5, 6, 7, 5, 6, 7, 8, 9, 6, 7, 8, 9, 10, 11]
    for i in range(len(diag)):
        assert diag[i] == m[i, i]

def test_evalf_sum():
    assert Sum(n,(n,1,2)).evalf() == 3.
    assert Sum(n,(n,1,2)).doit().evalf() == 3.
    # the next test should return instantly
    assert Sum(1/n,(n,1,2)).evalf() == 1.5

    # issue 8219
    assert Sum(E/factorial(n), (n, 0, oo)).evalf() == (E*E).evalf()
    # issue 8254
    assert Sum(2**n*n/factorial(n), (n, 0, oo)).evalf() == (2*E*E).evalf()
    # issue 8411
    s = Sum(1/x**2, (x, 100, oo))
    assert s.n() == s.doit().n()

def test_arithmetic_sums():
    assert summation(1, (n, a, b)) == b - a + 1
    assert Sum(S.NaN, (n, a, b)) is S.NaN
    assert Sum(x, (n, a, a)).doit() == x
    assert Sum(x, (x, a, a)).doit() == a
    assert Sum(x, (n, 1, a)).doit() == a*x
    lo, hi = 1, 2
    s1 = Sum(n, (n, lo, hi))
    s2 = Sum(n, (n, hi, lo))
    assert s1 != s2
    assert s1.doit() == s2.doit() == 3
    lo, hi = x, x + 1
    s1 = Sum(n, (n, lo, hi))
    s2 = Sum(n, (n, hi, lo))
    assert s1 != s2
    assert s1.doit() == s2.doit() == 2*x + 1
    assert Sum(Integral(x, (x, 1, y)) + x, (x, 1, 2)).doit() == \
        y**2 + 2
    assert summation(1, (n, 1, 10)) == 10
    assert summation(2*n, (n, 0, 10**10)) == 100000000010000000000
    assert summation(4*n*m, (n, a, 1), (m, 1, d)).expand() == \
        2*d + 2*d**2 + a*d + a*d**2 - d*a**2 - a**2*d**2
    assert summation(cos(n), (n, -2, 1)) == cos(-2) + cos(-1) + cos(0) + cos(1)
    assert summation(cos(n), (n, x, x + 2)) == cos(x) + cos(x + 1) + cos(x + 2)
    assert isinstance(summation(cos(n), (n, x, x + S.Half)), Sum)

def test_issue_14640():
    i, n = symbols("i n", integer=True)
    a, b, c = symbols("a b c")

    assert Sum(a**-i/(a - b), (i, 0, n)).doit() == Sum(
        1/(a*a**i - a**i*b), (i, 0, n)).doit() == Piecewise(
            (n + 1, Eq(1/a, 1)),
            ((-a**(-n - 1) + 1)/(1 - 1/a), True))/(a - b)

    assert Sum((b*a**i - c*a**i)**-2, (i, 0, n)).doit() == Piecewise(
        (n + 1, Eq(a**(-2), 1)),
        ((-a**(-2*n - 2) + 1)/(1 - 1/a**2), True))/(b - c)**2

    s = Sum(i*(a**(n - i) - b**(n - i))/(a - b), (i, 0, n)).doit()
    assert not s.has(Sum)
    assert s.subs({a: 2, b: 3, n: 5}) == 122

def test_euler_maclaurin():
    # Exact polynomial sums with E-M
    def check_exact(f, a, b, m, n):
        A = Sum(f, (k, a, b))
        s, e = A.euler_maclaurin(m, n)
        assert (e == 0) and (s.expand() == A.doit())
    check_exact(k**4, a, b, 0, 2)
    check_exact(k**4 + 2*k, a, b, 1, 2)
    check_exact(k**4 + k**2, a, b, 1, 5)
    check_exact(k**5, 2, 6, 1, 2)
    check_exact(k**5, 2, 6, 1, 3)
    # Not exact
    assert Sum(k**6, (k, a, b)).euler_maclaurin(0, 2)[1] != 0
    # Numerical test
    for m, n in [(2, 4), (2, 20), (10, 20), (18, 20)]:
        A = Sum(1/k**3, (k, 1, oo))
        s, e = A.euler_maclaurin(m, n)
        assert abs((s - zeta(3)).evalf()) < e.evalf()

def test_sum():
    # In each case, test eval() the lambdarepr() to make sure that
    # it evaluates to the same results as the symbolic expression

    s = Sum(x ** i, (i, a, b))

    l = lambdarepr(s)
    assert l == "(builtins.sum(x**i for i in range(a, b+1)))"

    assert (lambdify((x, a, b), s)(2, 3, 8) ==
            s.subs([(x, 2), (a, 3), (b, 8)]).doit())

    s = Sum(i * x, (i, a, b))

    l = lambdarepr(s)
    assert l == "(builtins.sum(i*x for i in range(a, b+1)))"

    assert (lambdify((x, a, b), s)(2, 3, 8) ==
            s.subs([(x, 2), (a, 3), (b, 8)]).doit())

def test_Sum_doit():
    f = Function('f')
    assert Sum(n*Integral(a**2), (n, 0, 2)).doit() == a**3
    assert Sum(n*Integral(a**2), (n, 0, 2)).doit(deep=False) == \
        3*Integral(a**2)
    assert summation(n*Integral(a**2), (n, 0, 2)) == 3*Integral(a**2)

    # test nested sum evaluation
    s = Sum( Sum( Sum(2,(z,1,n+1)), (y,x+1,n)), (x,1,n))
    assert 0 == (s.doit() - n*(n+1)*(n-1)).factor()

    assert Sum(KroneckerDelta(m, n), (m, -oo, oo)).doit() == Piecewise((1, And(-oo < n, n < oo)), (0, True))
    assert Sum(x*KroneckerDelta(m, n), (m, -oo, oo)).doit() == Piecewise((x, And(-oo < n, n < oo)), (0, True))
    assert Sum(Sum(KroneckerDelta(m, n), (m, 1, 3)), (n, 1, 3)).doit() == 3
    assert Sum(Sum(KroneckerDelta(k, m), (m, 1, 3)), (n, 1, 3)).doit() == \
           3 * Piecewise((1, And(S(1) <= k, k <= 3)), (0, True))
    assert Sum(f(n) * Sum(KroneckerDelta(m, n), (m, 0, oo)), (n, 1, 3)).doit() == \
           f(1) + f(2) + f(3)
    assert Sum(f(n) * Sum(KroneckerDelta(m, n), (m, 0, oo)), (n, 1, oo)).doit() == \
           Sum(Piecewise((f(n), And(Le(0, n), n < oo)), (0, True)), (n, 1, oo))
    l = Symbol('l', integer=True, positive=True)
    assert Sum(f(l) * Sum(KroneckerDelta(m, l), (m, 0, oo)), (l, 1, oo)).doit() == \
           Sum(f(l), (l, 1, oo))

    # issue 2597
    nmax = symbols('N', integer=True, positive=True)
    pw = Piecewise((1, And(S(1) <= n, n <= nmax)), (0, True))
    assert Sum(pw, (n, 1, nmax)).doit() == Sum(pw, (n, 1, nmax))

    q, s = symbols('q, s')
    assert summation(1/n**(2*s), (n, 1, oo)) == Piecewise((zeta(2*s), 2*s > 1),
        (Sum(n**(-2*s), (n, 1, oo)), True))
    assert summation(1/(n+1)**s, (n, 0, oo)) == Piecewise((zeta(s), s > 1),
        (Sum((n + 1)**(-s), (n, 0, oo)), True))
    assert summation(1/(n+q)**s, (n, 0, oo)) == Piecewise(
        (zeta(s, q), And(q > 0, s > 1)),
        (Sum((n + q)**(-s), (n, 0, oo)), True))
    assert summation(1/(n+q)**s, (n, q, oo)) == Piecewise(
        (zeta(s, 2*q), And(2*q > 0, s > 1)),
        (Sum((n + q)**(-s), (n, q, oo)), True))
    assert summation(1/n**2, (n, 1, oo)) == zeta(2)
    assert summation(1/n**s, (n, 0, oo)) == Sum(n**(-s), (n, 0, oo))

def test_euler_maclaurin():
    # Exact polynomial sums with E-M
    def check_exact(f, a, b, m, n):
        A = Sum(f, (k, a, b))
        s, e = A.euler_maclaurin(m, n)
        assert (e == 0) and (s.expand() == A.doit())
    check_exact(k**4, a, b, 0, 2)
    check_exact(k**4 + 2*k, a, b, 1, 2)
    check_exact(k**4 + k**2, a, b, 1, 5)
    check_exact(k**5, 2, 6, 1, 2)
    check_exact(k**5, 2, 6, 1, 3)
    assert Sum(x-1, (x, 0, 2)).euler_maclaurin(m=30, n=30, eps=2**-15) == (0, 0)
    # Not exact
    assert Sum(k**6, (k, a, b)).euler_maclaurin(0, 2)[1] != 0
    # Numerical test
    for m, n in [(2, 4), (2, 20), (10, 20), (18, 20)]:
        A = Sum(1/k**3, (k, 1, oo))
        s, e = A.euler_maclaurin(m, n)
        assert abs((s - zeta(3)).evalf()) < e.evalf()

    raises(ValueError, lambda: Sum(1, (x, 0, 1), (k, 0, 1)).euler_maclaurin())

def test_conjugate_transpose():
    A, B = symbols("A B", commutative=False)
    p = Sum(A*B**n, (n, 1, 3))
    assert p.adjoint().doit() == p.doit().adjoint()
    assert p.conjugate().doit() == p.doit().conjugate()
    assert p.transpose().doit() == p.doit().transpose()

 def check_exact(f, a, b, m, n):
     A = Sum(f, (k, a, b))
     s, e = A.euler_maclaurin(m, n)
     assert (e == 0) and (s.expand() == A.doit())