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from sympy.concrete import Sum |
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from sympy.concrete.delta import deltaproduct as dp, deltasummation as ds, _extract_delta |
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from sympy.core import Eq, S, symbols, oo |
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from sympy.functions import KroneckerDelta as KD, Piecewise, piecewise_fold |
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from sympy.logic import And |
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from sympy.testing.pytest import raises |
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i, j, k, l, m = symbols("i j k l m", integer=True, finite=True) |
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x, y = symbols("x y", commutative=False) |
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def test_deltaproduct_trivial(): |
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assert dp(x, (j, 1, 0)) == 1 |
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assert dp(x, (j, 1, 3)) == x**3 |
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assert dp(x + y, (j, 1, 3)) == (x + y)**3 |
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assert dp(x*y, (j, 1, 3)) == (x*y)**3 |
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assert dp(KD(i, j), (k, 1, 3)) == KD(i, j) |
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assert dp(x*KD(i, j), (k, 1, 3)) == x**3*KD(i, j) |
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assert dp(x*y*KD(i, j), (k, 1, 3)) == (x*y)**3*KD(i, j) |
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def test_deltaproduct_basic(): |
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assert dp(KD(i, j), (j, 1, 3)) == 0 |
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assert dp(KD(i, j), (j, 1, 1)) == KD(i, 1) |
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assert dp(KD(i, j), (j, 2, 2)) == KD(i, 2) |
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assert dp(KD(i, j), (j, 3, 3)) == KD(i, 3) |
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assert dp(KD(i, j), (j, 1, k)) == KD(i, 1)*KD(k, 1) + KD(k, 0) |
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assert dp(KD(i, j), (j, k, 3)) == KD(i, 3)*KD(k, 3) + KD(k, 4) |
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assert dp(KD(i, j), (j, k, l)) == KD(i, l)*KD(k, l) + KD(k, l + 1) |
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def test_deltaproduct_mul_x_kd(): |
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assert dp(x*KD(i, j), (j, 1, 3)) == 0 |
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assert dp(x*KD(i, j), (j, 1, 1)) == x*KD(i, 1) |
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assert dp(x*KD(i, j), (j, 2, 2)) == x*KD(i, 2) |
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assert dp(x*KD(i, j), (j, 3, 3)) == x*KD(i, 3) |
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assert dp(x*KD(i, j), (j, 1, k)) == x*KD(i, 1)*KD(k, 1) + KD(k, 0) |
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assert dp(x*KD(i, j), (j, k, 3)) == x*KD(i, 3)*KD(k, 3) + KD(k, 4) |
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assert dp(x*KD(i, j), (j, k, l)) == x*KD(i, l)*KD(k, l) + KD(k, l + 1) |
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def test_deltaproduct_mul_add_x_y_kd(): |
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assert dp((x + y)*KD(i, j), (j, 1, 3)) == 0 |
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assert dp((x + y)*KD(i, j), (j, 1, 1)) == (x + y)*KD(i, 1) |
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assert dp((x + y)*KD(i, j), (j, 2, 2)) == (x + y)*KD(i, 2) |
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assert dp((x + y)*KD(i, j), (j, 3, 3)) == (x + y)*KD(i, 3) |
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assert dp((x + y)*KD(i, j), (j, 1, k)) == \ |
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(x + y)*KD(i, 1)*KD(k, 1) + KD(k, 0) |
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assert dp((x + y)*KD(i, j), (j, k, 3)) == \ |
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(x + y)*KD(i, 3)*KD(k, 3) + KD(k, 4) |
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assert dp((x + y)*KD(i, j), (j, k, l)) == \ |
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(x + y)*KD(i, l)*KD(k, l) + KD(k, l + 1) |
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def test_deltaproduct_add_kd_kd(): |
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assert dp(KD(i, k) + KD(j, k), (k, 1, 3)) == 0 |
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assert dp(KD(i, k) + KD(j, k), (k, 1, 1)) == KD(i, 1) + KD(j, 1) |
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assert dp(KD(i, k) + KD(j, k), (k, 2, 2)) == KD(i, 2) + KD(j, 2) |
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assert dp(KD(i, k) + KD(j, k), (k, 3, 3)) == KD(i, 3) + KD(j, 3) |
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assert dp(KD(i, k) + KD(j, k), (k, 1, l)) == KD(l, 0) + \ |
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KD(i, 1)*KD(l, 1) + KD(j, 1)*KD(l, 1) + \ |
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KD(i, 1)*KD(j, 2)*KD(l, 2) + KD(j, 1)*KD(i, 2)*KD(l, 2) |
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assert dp(KD(i, k) + KD(j, k), (k, l, 3)) == KD(l, 4) + \ |
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KD(i, 3)*KD(l, 3) + KD(j, 3)*KD(l, 3) + \ |
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KD(i, 2)*KD(j, 3)*KD(l, 2) + KD(i, 3)*KD(j, 2)*KD(l, 2) |
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assert dp(KD(i, k) + KD(j, k), (k, l, m)) == KD(l, m + 1) + \ |
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KD(i, m)*KD(l, m) + KD(j, m)*KD(l, m) + \ |
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KD(i, m)*KD(j, m - 1)*KD(l, m - 1) + KD(i, m - 1)*KD(j, m)*KD(l, m - 1) |
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def test_deltaproduct_mul_x_add_kd_kd(): |
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assert dp(x*(KD(i, k) + KD(j, k)), (k, 1, 3)) == 0 |
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assert dp(x*(KD(i, k) + KD(j, k)), (k, 1, 1)) == x*(KD(i, 1) + KD(j, 1)) |
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assert dp(x*(KD(i, k) + KD(j, k)), (k, 2, 2)) == x*(KD(i, 2) + KD(j, 2)) |
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assert dp(x*(KD(i, k) + KD(j, k)), (k, 3, 3)) == x*(KD(i, 3) + KD(j, 3)) |
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assert dp(x*(KD(i, k) + KD(j, k)), (k, 1, l)) == KD(l, 0) + \ |
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x*KD(i, 1)*KD(l, 1) + x*KD(j, 1)*KD(l, 1) + \ |
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x**2*KD(i, 1)*KD(j, 2)*KD(l, 2) + x**2*KD(j, 1)*KD(i, 2)*KD(l, 2) |
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assert dp(x*(KD(i, k) + KD(j, k)), (k, l, 3)) == KD(l, 4) + \ |
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x*KD(i, 3)*KD(l, 3) + x*KD(j, 3)*KD(l, 3) + \ |
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x**2*KD(i, 2)*KD(j, 3)*KD(l, 2) + x**2*KD(i, 3)*KD(j, 2)*KD(l, 2) |
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assert dp(x*(KD(i, k) + KD(j, k)), (k, l, m)) == KD(l, m + 1) + \ |
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x*KD(i, m)*KD(l, m) + x*KD(j, m)*KD(l, m) + \ |
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x**2*KD(i, m - 1)*KD(j, m)*KD(l, m - 1) + \ |
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x**2*KD(i, m)*KD(j, m - 1)*KD(l, m - 1) |
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def test_deltaproduct_mul_add_x_y_add_kd_kd(): |
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assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, 1, 3)) == 0 |
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assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, 1, 1)) == \ |
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(x + y)*(KD(i, 1) + KD(j, 1)) |
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assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, 2, 2)) == \ |
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(x + y)*(KD(i, 2) + KD(j, 2)) |
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assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, 3, 3)) == \ |
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(x + y)*(KD(i, 3) + KD(j, 3)) |
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assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, 1, l)) == KD(l, 0) + \ |
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(x + y)*KD(i, 1)*KD(l, 1) + (x + y)*KD(j, 1)*KD(l, 1) + \ |
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(x + y)**2*KD(i, 1)*KD(j, 2)*KD(l, 2) + \ |
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(x + y)**2*KD(j, 1)*KD(i, 2)*KD(l, 2) |
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assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, l, 3)) == KD(l, 4) + \ |
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(x + y)*KD(i, 3)*KD(l, 3) + (x + y)*KD(j, 3)*KD(l, 3) + \ |
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(x + y)**2*KD(i, 2)*KD(j, 3)*KD(l, 2) + \ |
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(x + y)**2*KD(i, 3)*KD(j, 2)*KD(l, 2) |
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assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, l, m)) == KD(l, m + 1) + \ |
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(x + y)*KD(i, m)*KD(l, m) + (x + y)*KD(j, m)*KD(l, m) + \ |
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(x + y)**2*KD(i, m - 1)*KD(j, m)*KD(l, m - 1) + \ |
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(x + y)**2*KD(i, m)*KD(j, m - 1)*KD(l, m - 1) |
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def test_deltaproduct_add_mul_x_y_mul_x_kd(): |
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assert dp(x*y + x*KD(i, j), (j, 1, 3)) == (x*y)**3 + \ |
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x*(x*y)**2*KD(i, 1) + (x*y)*x*(x*y)*KD(i, 2) + (x*y)**2*x*KD(i, 3) |
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assert dp(x*y + x*KD(i, j), (j, 1, 1)) == x*y + x*KD(i, 1) |
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assert dp(x*y + x*KD(i, j), (j, 2, 2)) == x*y + x*KD(i, 2) |
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assert dp(x*y + x*KD(i, j), (j, 3, 3)) == x*y + x*KD(i, 3) |
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assert dp(x*y + x*KD(i, j), (j, 1, k)) == \ |
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(x*y)**k + Piecewise( |
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((x*y)**(i - 1)*x*(x*y)**(k - i), And(1 <= i, i <= k)), |
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(0, True) |
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) |
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assert dp(x*y + x*KD(i, j), (j, k, 3)) == \ |
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(x*y)**(-k + 4) + Piecewise( |
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((x*y)**(i - k)*x*(x*y)**(3 - i), And(k <= i, i <= 3)), |
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(0, True) |
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) |
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assert dp(x*y + x*KD(i, j), (j, k, l)) == \ |
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(x*y)**(-k + l + 1) + Piecewise( |
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((x*y)**(i - k)*x*(x*y)**(l - i), And(k <= i, i <= l)), |
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(0, True) |
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) |
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def test_deltaproduct_mul_x_add_y_kd(): |
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assert dp(x*(y + KD(i, j)), (j, 1, 3)) == (x*y)**3 + \ |
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x*(x*y)**2*KD(i, 1) + (x*y)*x*(x*y)*KD(i, 2) + (x*y)**2*x*KD(i, 3) |
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assert dp(x*(y + KD(i, j)), (j, 1, 1)) == x*(y + KD(i, 1)) |
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assert dp(x*(y + KD(i, j)), (j, 2, 2)) == x*(y + KD(i, 2)) |
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assert dp(x*(y + KD(i, j)), (j, 3, 3)) == x*(y + KD(i, 3)) |
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assert dp(x*(y + KD(i, j)), (j, 1, k)) == \ |
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(x*y)**k + Piecewise( |
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((x*y)**(i - 1)*x*(x*y)**(k - i), And(1 <= i, i <= k)), |
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(0, True) |
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).expand() |
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assert dp(x*(y + KD(i, j)), (j, k, 3)) == \ |
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((x*y)**(-k + 4) + Piecewise( |
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((x*y)**(i - k)*x*(x*y)**(3 - i), And(k <= i, i <= 3)), |
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(0, True) |
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)).expand() |
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assert dp(x*(y + KD(i, j)), (j, k, l)) == \ |
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((x*y)**(-k + l + 1) + Piecewise( |
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((x*y)**(i - k)*x*(x*y)**(l - i), And(k <= i, i <= l)), |
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(0, True) |
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)).expand() |
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def test_deltaproduct_mul_x_add_y_twokd(): |
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assert dp(x*(y + 2*KD(i, j)), (j, 1, 3)) == (x*y)**3 + \ |
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2*x*(x*y)**2*KD(i, 1) + 2*x*y*x*x*y*KD(i, 2) + 2*(x*y)**2*x*KD(i, 3) |
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assert dp(x*(y + 2*KD(i, j)), (j, 1, 1)) == x*(y + 2*KD(i, 1)) |
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assert dp(x*(y + 2*KD(i, j)), (j, 2, 2)) == x*(y + 2*KD(i, 2)) |
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assert dp(x*(y + 2*KD(i, j)), (j, 3, 3)) == x*(y + 2*KD(i, 3)) |
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assert dp(x*(y + 2*KD(i, j)), (j, 1, k)) == \ |
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(x*y)**k + Piecewise( |
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(2*(x*y)**(i - 1)*x*(x*y)**(k - i), And(1 <= i, i <= k)), |
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(0, True) |
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).expand() |
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assert dp(x*(y + 2*KD(i, j)), (j, k, 3)) == \ |
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((x*y)**(-k + 4) + Piecewise( |
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(2*(x*y)**(i - k)*x*(x*y)**(3 - i), And(k <= i, i <= 3)), |
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(0, True) |
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)).expand() |
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assert dp(x*(y + 2*KD(i, j)), (j, k, l)) == \ |
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((x*y)**(-k + l + 1) + Piecewise( |
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(2*(x*y)**(i - k)*x*(x*y)**(l - i), And(k <= i, i <= l)), |
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(0, True) |
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)).expand() |
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def test_deltaproduct_mul_add_x_y_add_y_kd(): |
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assert dp((x + y)*(y + KD(i, j)), (j, 1, 3)) == ((x + y)*y)**3 + \ |
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(x + y)*((x + y)*y)**2*KD(i, 1) + \ |
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(x + y)*y*(x + y)**2*y*KD(i, 2) + \ |
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((x + y)*y)**2*(x + y)*KD(i, 3) |
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assert dp((x + y)*(y + KD(i, j)), (j, 1, 1)) == (x + y)*(y + KD(i, 1)) |
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assert dp((x + y)*(y + KD(i, j)), (j, 2, 2)) == (x + y)*(y + KD(i, 2)) |
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assert dp((x + y)*(y + KD(i, j)), (j, 3, 3)) == (x + y)*(y + KD(i, 3)) |
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assert dp((x + y)*(y + KD(i, j)), (j, 1, k)) == \ |
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((x + y)*y)**k + Piecewise( |
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(((x + y)*y)**(-1)*((x + y)*y)**i*(x + y)*((x + y)*y |
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)**k*((x + y)*y)**(-i), (i >= 1) & (i <= k)), (0, True)) |
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assert dp((x + y)*(y + KD(i, j)), (j, k, 3)) == ( |
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(x + y)*y)**4*((x + y)*y)**(-k) + Piecewise((((x + y)*y)**i*( |
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(x + y)*y)**(-k)*(x + y)*((x + y)*y)**3*((x + y)*y)**(-i), |
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(i >= k) & (i <= 3)), (0, True)) |
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assert dp((x + y)*(y + KD(i, j)), (j, k, l)) == \ |
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(x + y)*y*((x + y)*y)**l*((x + y)*y)**(-k) + Piecewise( |
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(((x + y)*y)**i*((x + y)*y)**(-k)*(x + y)*((x + y)*y |
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)**l*((x + y)*y)**(-i), (i >= k) & (i <= l)), (0, True)) |
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def test_deltaproduct_mul_add_x_kd_add_y_kd(): |
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assert dp((x + KD(i, k))*(y + KD(i, j)), (j, 1, 3)) == \ |
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KD(i, 1)*(KD(i, k) + x)*((KD(i, k) + x)*y)**2 + \ |
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KD(i, 2)*(KD(i, k) + x)*y*(KD(i, k) + x)**2*y + \ |
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KD(i, 3)*((KD(i, k) + x)*y)**2*(KD(i, k) + x) + \ |
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((KD(i, k) + x)*y)**3 |
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assert dp((x + KD(i, k))*(y + KD(i, j)), (j, 1, 1)) == \ |
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(x + KD(i, k))*(y + KD(i, 1)) |
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assert dp((x + KD(i, k))*(y + KD(i, j)), (j, 2, 2)) == \ |
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(x + KD(i, k))*(y + KD(i, 2)) |
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assert dp((x + KD(i, k))*(y + KD(i, j)), (j, 3, 3)) == \ |
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(x + KD(i, k))*(y + KD(i, 3)) |
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assert dp((x + KD(i, k))*(y + KD(i, j)), (j, 1, k)) == \ |
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((KD(i, k) + x)*y)**k + Piecewise( |
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(((KD(i, k) + x)*y)**(-1)*((KD(i, k) + x)*y)**i*(KD(i, k) + x |
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)*((KD(i, k) + x)*y)**k*((KD(i, k) + x)*y)**(-i), (i >= 1 |
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) & (i <= k)), (0, True)) |
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assert dp((x + KD(i, k))*(y + KD(i, j)), (j, k, 3)) == ( |
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(KD(i, k) + x)*y)**4*((KD(i, k) + x)*y)**(-k) + Piecewise( |
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(((KD(i, k) + x)*y)**i*((KD(i, k) + x)*y)**(-k)*(KD(i, k) |
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+ x)*((KD(i, k) + x)*y)**3*((KD(i, k) + x)*y)**(-i), |
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(i >= k) & (i <= 3)), (0, True)) |
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assert dp((x + KD(i, k))*(y + KD(i, j)), (j, k, l)) == ( |
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KD(i, k) + x)*y*((KD(i, k) + x)*y)**l*((KD(i, k) + x)*y |
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)**(-k) + Piecewise((((KD(i, k) + x)*y)**i*((KD(i, k) + x |
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)*y)**(-k)*(KD(i, k) + x)*((KD(i, k) + x)*y)**l*((KD(i, k) + x |
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)*y)**(-i), (i >= k) & (i <= l)), (0, True)) |
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def test_deltasummation_trivial(): |
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assert ds(x, (j, 1, 0)) == 0 |
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assert ds(x, (j, 1, 3)) == 3*x |
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assert ds(x + y, (j, 1, 3)) == 3*(x + y) |
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assert ds(x*y, (j, 1, 3)) == 3*x*y |
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assert ds(KD(i, j), (k, 1, 3)) == 3*KD(i, j) |
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assert ds(x*KD(i, j), (k, 1, 3)) == 3*x*KD(i, j) |
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assert ds(x*y*KD(i, j), (k, 1, 3)) == 3*x*y*KD(i, j) |
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def test_deltasummation_basic_numerical(): |
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n = symbols('n', integer=True, nonzero=True) |
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assert ds(KD(n, 0), (n, 1, 3)) == 0 |
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assert ds(KD(i**2, j**2), (j, -oo, oo)) == \ |
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Sum(KD(i**2, j**2), (j, -oo, oo)) |
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assert Piecewise((KD(i, k), And(1 <= i, i <= 3)), (0, True)) == \ |
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ds(KD(i, j)*KD(j, k), (j, 1, 3)) == \ |
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ds(KD(j, k)*KD(i, j), (j, 1, 3)) |
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assert ds(KD(i, k), (k, -oo, oo)) == 1 |
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assert ds(KD(i, k), (k, 0, oo)) == Piecewise((1, S.Zero <= i), (0, True)) |
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assert ds(KD(i, k), (k, 1, 3)) == \ |
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Piecewise((1, And(1 <= i, i <= 3)), (0, True)) |
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assert ds(k*KD(i, j)*KD(j, k), (k, -oo, oo)) == j*KD(i, j) |
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assert ds(j*KD(i, j), (j, -oo, oo)) == i |
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assert ds(i*KD(i, j), (i, -oo, oo)) == j |
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assert ds(x, (i, 1, 3)) == 3*x |
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assert ds((i + j)*KD(i, j), (j, -oo, oo)) == 2*i |
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def test_deltasummation_basic_symbolic(): |
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assert ds(KD(i, j), (j, 1, 3)) == \ |
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Piecewise((1, And(1 <= i, i <= 3)), (0, True)) |
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assert ds(KD(i, j), (j, 1, 1)) == Piecewise((1, Eq(i, 1)), (0, True)) |
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assert ds(KD(i, j), (j, 2, 2)) == Piecewise((1, Eq(i, 2)), (0, True)) |
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assert ds(KD(i, j), (j, 3, 3)) == Piecewise((1, Eq(i, 3)), (0, True)) |
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assert ds(KD(i, j), (j, 1, k)) == \ |
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Piecewise((1, And(1 <= i, i <= k)), (0, True)) |
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assert ds(KD(i, j), (j, k, 3)) == \ |
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Piecewise((1, And(k <= i, i <= 3)), (0, True)) |
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assert ds(KD(i, j), (j, k, l)) == \ |
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Piecewise((1, And(k <= i, i <= l)), (0, True)) |
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def test_deltasummation_mul_x_kd(): |
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assert ds(x*KD(i, j), (j, 1, 3)) == \ |
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Piecewise((x, And(1 <= i, i <= 3)), (0, True)) |
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assert ds(x*KD(i, j), (j, 1, 1)) == Piecewise((x, Eq(i, 1)), (0, True)) |
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assert ds(x*KD(i, j), (j, 2, 2)) == Piecewise((x, Eq(i, 2)), (0, True)) |
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assert ds(x*KD(i, j), (j, 3, 3)) == Piecewise((x, Eq(i, 3)), (0, True)) |
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assert ds(x*KD(i, j), (j, 1, k)) == \ |
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Piecewise((x, And(1 <= i, i <= k)), (0, True)) |
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assert ds(x*KD(i, j), (j, k, 3)) == \ |
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Piecewise((x, And(k <= i, i <= 3)), (0, True)) |
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assert ds(x*KD(i, j), (j, k, l)) == \ |
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Piecewise((x, And(k <= i, i <= l)), (0, True)) |
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def test_deltasummation_mul_add_x_y_kd(): |
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assert ds((x + y)*KD(i, j), (j, 1, 3)) == \ |
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Piecewise((x + y, And(1 <= i, i <= 3)), (0, True)) |
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assert ds((x + y)*KD(i, j), (j, 1, 1)) == \ |
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Piecewise((x + y, Eq(i, 1)), (0, True)) |
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assert ds((x + y)*KD(i, j), (j, 2, 2)) == \ |
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Piecewise((x + y, Eq(i, 2)), (0, True)) |
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assert ds((x + y)*KD(i, j), (j, 3, 3)) == \ |
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Piecewise((x + y, Eq(i, 3)), (0, True)) |
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assert ds((x + y)*KD(i, j), (j, 1, k)) == \ |
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Piecewise((x + y, And(1 <= i, i <= k)), (0, True)) |
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assert ds((x + y)*KD(i, j), (j, k, 3)) == \ |
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Piecewise((x + y, And(k <= i, i <= 3)), (0, True)) |
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assert ds((x + y)*KD(i, j), (j, k, l)) == \ |
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Piecewise((x + y, And(k <= i, i <= l)), (0, True)) |
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def test_deltasummation_add_kd_kd(): |
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assert ds(KD(i, k) + KD(j, k), (k, 1, 3)) == piecewise_fold( |
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Piecewise((1, And(1 <= i, i <= 3)), (0, True)) + |
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Piecewise((1, And(1 <= j, j <= 3)), (0, True))) |
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assert ds(KD(i, k) + KD(j, k), (k, 1, 1)) == piecewise_fold( |
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Piecewise((1, Eq(i, 1)), (0, True)) + |
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Piecewise((1, Eq(j, 1)), (0, True))) |
|
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assert ds(KD(i, k) + KD(j, k), (k, 2, 2)) == piecewise_fold( |
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Piecewise((1, Eq(i, 2)), (0, True)) + |
|
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Piecewise((1, Eq(j, 2)), (0, True))) |
|
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assert ds(KD(i, k) + KD(j, k), (k, 3, 3)) == piecewise_fold( |
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Piecewise((1, Eq(i, 3)), (0, True)) + |
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Piecewise((1, Eq(j, 3)), (0, True))) |
|
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assert ds(KD(i, k) + KD(j, k), (k, 1, l)) == piecewise_fold( |
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Piecewise((1, And(1 <= i, i <= l)), (0, True)) + |
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|
Piecewise((1, And(1 <= j, j <= l)), (0, True))) |
|
|
assert ds(KD(i, k) + KD(j, k), (k, l, 3)) == piecewise_fold( |
|
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Piecewise((1, And(l <= i, i <= 3)), (0, True)) + |
|
|
Piecewise((1, And(l <= j, j <= 3)), (0, True))) |
|
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assert ds(KD(i, k) + KD(j, k), (k, l, m)) == piecewise_fold( |
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|
Piecewise((1, And(l <= i, i <= m)), (0, True)) + |
|
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Piecewise((1, And(l <= j, j <= m)), (0, True))) |
|
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|
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def test_deltasummation_add_mul_x_kd_kd(): |
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assert ds(x*KD(i, k) + KD(j, k), (k, 1, 3)) == piecewise_fold( |
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|
Piecewise((x, And(1 <= i, i <= 3)), (0, True)) + |
|
|
Piecewise((1, And(1 <= j, j <= 3)), (0, True))) |
|
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assert ds(x*KD(i, k) + KD(j, k), (k, 1, 1)) == piecewise_fold( |
|
|
Piecewise((x, Eq(i, 1)), (0, True)) + |
|
|
Piecewise((1, Eq(j, 1)), (0, True))) |
|
|
assert ds(x*KD(i, k) + KD(j, k), (k, 2, 2)) == piecewise_fold( |
|
|
Piecewise((x, Eq(i, 2)), (0, True)) + |
|
|
Piecewise((1, Eq(j, 2)), (0, True))) |
|
|
assert ds(x*KD(i, k) + KD(j, k), (k, 3, 3)) == piecewise_fold( |
|
|
Piecewise((x, Eq(i, 3)), (0, True)) + |
|
|
Piecewise((1, Eq(j, 3)), (0, True))) |
|
|
assert ds(x*KD(i, k) + KD(j, k), (k, 1, l)) == piecewise_fold( |
|
|
Piecewise((x, And(1 <= i, i <= l)), (0, True)) + |
|
|
Piecewise((1, And(1 <= j, j <= l)), (0, True))) |
|
|
assert ds(x*KD(i, k) + KD(j, k), (k, l, 3)) == piecewise_fold( |
|
|
Piecewise((x, And(l <= i, i <= 3)), (0, True)) + |
|
|
Piecewise((1, And(l <= j, j <= 3)), (0, True))) |
|
|
assert ds(x*KD(i, k) + KD(j, k), (k, l, m)) == piecewise_fold( |
|
|
Piecewise((x, And(l <= i, i <= m)), (0, True)) + |
|
|
Piecewise((1, And(l <= j, j <= m)), (0, True))) |
|
|
|
|
|
|
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|
def test_deltasummation_mul_x_add_kd_kd(): |
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|
assert ds(x*(KD(i, k) + KD(j, k)), (k, 1, 3)) == piecewise_fold( |
|
|
Piecewise((x, And(1 <= i, i <= 3)), (0, True)) + |
|
|
Piecewise((x, And(1 <= j, j <= 3)), (0, True))) |
|
|
assert ds(x*(KD(i, k) + KD(j, k)), (k, 1, 1)) == piecewise_fold( |
|
|
Piecewise((x, Eq(i, 1)), (0, True)) + |
|
|
Piecewise((x, Eq(j, 1)), (0, True))) |
|
|
assert ds(x*(KD(i, k) + KD(j, k)), (k, 2, 2)) == piecewise_fold( |
|
|
Piecewise((x, Eq(i, 2)), (0, True)) + |
|
|
Piecewise((x, Eq(j, 2)), (0, True))) |
|
|
assert ds(x*(KD(i, k) + KD(j, k)), (k, 3, 3)) == piecewise_fold( |
|
|
Piecewise((x, Eq(i, 3)), (0, True)) + |
|
|
Piecewise((x, Eq(j, 3)), (0, True))) |
|
|
assert ds(x*(KD(i, k) + KD(j, k)), (k, 1, l)) == piecewise_fold( |
|
|
Piecewise((x, And(1 <= i, i <= l)), (0, True)) + |
|
|
Piecewise((x, And(1 <= j, j <= l)), (0, True))) |
|
|
assert ds(x*(KD(i, k) + KD(j, k)), (k, l, 3)) == piecewise_fold( |
|
|
Piecewise((x, And(l <= i, i <= 3)), (0, True)) + |
|
|
Piecewise((x, And(l <= j, j <= 3)), (0, True))) |
|
|
assert ds(x*(KD(i, k) + KD(j, k)), (k, l, m)) == piecewise_fold( |
|
|
Piecewise((x, And(l <= i, i <= m)), (0, True)) + |
|
|
Piecewise((x, And(l <= j, j <= m)), (0, True))) |
|
|
|
|
|
|
|
|
def test_deltasummation_mul_add_x_y_add_kd_kd(): |
|
|
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 1, 3)) == piecewise_fold( |
|
|
Piecewise((x + y, And(1 <= i, i <= 3)), (0, True)) + |
|
|
Piecewise((x + y, And(1 <= j, j <= 3)), (0, True))) |
|
|
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 1, 1)) == piecewise_fold( |
|
|
Piecewise((x + y, Eq(i, 1)), (0, True)) + |
|
|
Piecewise((x + y, Eq(j, 1)), (0, True))) |
|
|
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 2, 2)) == piecewise_fold( |
|
|
Piecewise((x + y, Eq(i, 2)), (0, True)) + |
|
|
Piecewise((x + y, Eq(j, 2)), (0, True))) |
|
|
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 3, 3)) == piecewise_fold( |
|
|
Piecewise((x + y, Eq(i, 3)), (0, True)) + |
|
|
Piecewise((x + y, Eq(j, 3)), (0, True))) |
|
|
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 1, l)) == piecewise_fold( |
|
|
Piecewise((x + y, And(1 <= i, i <= l)), (0, True)) + |
|
|
Piecewise((x + y, And(1 <= j, j <= l)), (0, True))) |
|
|
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, l, 3)) == piecewise_fold( |
|
|
Piecewise((x + y, And(l <= i, i <= 3)), (0, True)) + |
|
|
Piecewise((x + y, And(l <= j, j <= 3)), (0, True))) |
|
|
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, l, m)) == piecewise_fold( |
|
|
Piecewise((x + y, And(l <= i, i <= m)), (0, True)) + |
|
|
Piecewise((x + y, And(l <= j, j <= m)), (0, True))) |
|
|
|
|
|
|
|
|
def test_deltasummation_add_mul_x_y_mul_x_kd(): |
|
|
assert ds(x*y + x*KD(i, j), (j, 1, 3)) == \ |
|
|
Piecewise((3*x*y + x, And(1 <= i, i <= 3)), (3*x*y, True)) |
|
|
assert ds(x*y + x*KD(i, j), (j, 1, 1)) == \ |
|
|
Piecewise((x*y + x, Eq(i, 1)), (x*y, True)) |
|
|
assert ds(x*y + x*KD(i, j), (j, 2, 2)) == \ |
|
|
Piecewise((x*y + x, Eq(i, 2)), (x*y, True)) |
|
|
assert ds(x*y + x*KD(i, j), (j, 3, 3)) == \ |
|
|
Piecewise((x*y + x, Eq(i, 3)), (x*y, True)) |
|
|
assert ds(x*y + x*KD(i, j), (j, 1, k)) == \ |
|
|
Piecewise((k*x*y + x, And(1 <= i, i <= k)), (k*x*y, True)) |
|
|
assert ds(x*y + x*KD(i, j), (j, k, 3)) == \ |
|
|
Piecewise(((4 - k)*x*y + x, And(k <= i, i <= 3)), ((4 - k)*x*y, True)) |
|
|
assert ds(x*y + x*KD(i, j), (j, k, l)) == Piecewise( |
|
|
((l - k + 1)*x*y + x, And(k <= i, i <= l)), ((l - k + 1)*x*y, True)) |
|
|
|
|
|
|
|
|
def test_deltasummation_mul_x_add_y_kd(): |
|
|
assert ds(x*(y + KD(i, j)), (j, 1, 3)) == \ |
|
|
Piecewise((3*x*y + x, And(1 <= i, i <= 3)), (3*x*y, True)) |
|
|
assert ds(x*(y + KD(i, j)), (j, 1, 1)) == \ |
|
|
Piecewise((x*y + x, Eq(i, 1)), (x*y, True)) |
|
|
assert ds(x*(y + KD(i, j)), (j, 2, 2)) == \ |
|
|
Piecewise((x*y + x, Eq(i, 2)), (x*y, True)) |
|
|
assert ds(x*(y + KD(i, j)), (j, 3, 3)) == \ |
|
|
Piecewise((x*y + x, Eq(i, 3)), (x*y, True)) |
|
|
assert ds(x*(y + KD(i, j)), (j, 1, k)) == \ |
|
|
Piecewise((k*x*y + x, And(1 <= i, i <= k)), (k*x*y, True)) |
|
|
assert ds(x*(y + KD(i, j)), (j, k, 3)) == \ |
|
|
Piecewise(((4 - k)*x*y + x, And(k <= i, i <= 3)), ((4 - k)*x*y, True)) |
|
|
assert ds(x*(y + KD(i, j)), (j, k, l)) == Piecewise( |
|
|
((l - k + 1)*x*y + x, And(k <= i, i <= l)), ((l - k + 1)*x*y, True)) |
|
|
|
|
|
|
|
|
def test_deltasummation_mul_x_add_y_twokd(): |
|
|
assert ds(x*(y + 2*KD(i, j)), (j, 1, 3)) == \ |
|
|
Piecewise((3*x*y + 2*x, And(1 <= i, i <= 3)), (3*x*y, True)) |
|
|
assert ds(x*(y + 2*KD(i, j)), (j, 1, 1)) == \ |
|
|
Piecewise((x*y + 2*x, Eq(i, 1)), (x*y, True)) |
|
|
assert ds(x*(y + 2*KD(i, j)), (j, 2, 2)) == \ |
|
|
Piecewise((x*y + 2*x, Eq(i, 2)), (x*y, True)) |
|
|
assert ds(x*(y + 2*KD(i, j)), (j, 3, 3)) == \ |
|
|
Piecewise((x*y + 2*x, Eq(i, 3)), (x*y, True)) |
|
|
assert ds(x*(y + 2*KD(i, j)), (j, 1, k)) == \ |
|
|
Piecewise((k*x*y + 2*x, And(1 <= i, i <= k)), (k*x*y, True)) |
|
|
assert ds(x*(y + 2*KD(i, j)), (j, k, 3)) == Piecewise( |
|
|
((4 - k)*x*y + 2*x, And(k <= i, i <= 3)), ((4 - k)*x*y, True)) |
|
|
assert ds(x*(y + 2*KD(i, j)), (j, k, l)) == Piecewise( |
|
|
((l - k + 1)*x*y + 2*x, And(k <= i, i <= l)), ((l - k + 1)*x*y, True)) |
|
|
|
|
|
|
|
|
def test_deltasummation_mul_add_x_y_add_y_kd(): |
|
|
assert ds((x + y)*(y + KD(i, j)), (j, 1, 3)) == Piecewise( |
|
|
(3*(x + y)*y + x + y, And(1 <= i, i <= 3)), (3*(x + y)*y, True)) |
|
|
assert ds((x + y)*(y + KD(i, j)), (j, 1, 1)) == \ |
|
|
Piecewise(((x + y)*y + x + y, Eq(i, 1)), ((x + y)*y, True)) |
|
|
assert ds((x + y)*(y + KD(i, j)), (j, 2, 2)) == \ |
|
|
Piecewise(((x + y)*y + x + y, Eq(i, 2)), ((x + y)*y, True)) |
|
|
assert ds((x + y)*(y + KD(i, j)), (j, 3, 3)) == \ |
|
|
Piecewise(((x + y)*y + x + y, Eq(i, 3)), ((x + y)*y, True)) |
|
|
assert ds((x + y)*(y + KD(i, j)), (j, 1, k)) == Piecewise( |
|
|
(k*(x + y)*y + x + y, And(1 <= i, i <= k)), (k*(x + y)*y, True)) |
|
|
assert ds((x + y)*(y + KD(i, j)), (j, k, 3)) == Piecewise( |
|
|
((4 - k)*(x + y)*y + x + y, And(k <= i, i <= 3)), |
|
|
((4 - k)*(x + y)*y, True)) |
|
|
assert ds((x + y)*(y + KD(i, j)), (j, k, l)) == Piecewise( |
|
|
((l - k + 1)*(x + y)*y + x + y, And(k <= i, i <= l)), |
|
|
((l - k + 1)*(x + y)*y, True)) |
|
|
|
|
|
|
|
|
def test_deltasummation_mul_add_x_kd_add_y_kd(): |
|
|
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 1, 3)) == piecewise_fold( |
|
|
Piecewise((KD(i, k) + x, And(1 <= i, i <= 3)), (0, True)) + |
|
|
3*(KD(i, k) + x)*y) |
|
|
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 1, 1)) == piecewise_fold( |
|
|
Piecewise((KD(i, k) + x, Eq(i, 1)), (0, True)) + |
|
|
(KD(i, k) + x)*y) |
|
|
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 2, 2)) == piecewise_fold( |
|
|
Piecewise((KD(i, k) + x, Eq(i, 2)), (0, True)) + |
|
|
(KD(i, k) + x)*y) |
|
|
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 3, 3)) == piecewise_fold( |
|
|
Piecewise((KD(i, k) + x, Eq(i, 3)), (0, True)) + |
|
|
(KD(i, k) + x)*y) |
|
|
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 1, k)) == piecewise_fold( |
|
|
Piecewise((KD(i, k) + x, And(1 <= i, i <= k)), (0, True)) + |
|
|
k*(KD(i, k) + x)*y) |
|
|
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, k, 3)) == piecewise_fold( |
|
|
Piecewise((KD(i, k) + x, And(k <= i, i <= 3)), (0, True)) + |
|
|
(4 - k)*(KD(i, k) + x)*y) |
|
|
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, k, l)) == piecewise_fold( |
|
|
Piecewise((KD(i, k) + x, And(k <= i, i <= l)), (0, True)) + |
|
|
(l - k + 1)*(KD(i, k) + x)*y) |
|
|
|
|
|
|
|
|
def test_extract_delta(): |
|
|
raises(ValueError, lambda: _extract_delta(KD(i, j) + KD(k, l), i)) |
|
|
|
