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env-llmeval/lib/python3.10/site-packages/sympy/concrete/tests/test_delta.py

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

env-llmeval/lib/python3.10/site-packages/sympy/concrete/tests/test_gosper.py

@@ -0,0 +1,204 @@
+"""Tests for Gosper's algorithm for hypergeometric summation. """
+
+from sympy.core.numbers import (Rational, pi)
+from sympy.core.singleton import S
+from sympy.core.symbol import Symbol
+from sympy.functions.combinatorial.factorials import (binomial, factorial)
+from sympy.functions.elementary.exponential import (exp, log)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.special.gamma_functions import gamma
+from sympy.polys.polytools import Poly
+from sympy.simplify.simplify import simplify
+from sympy.concrete.gosper import gosper_normal, gosper_sum, gosper_term
+from sympy.abc import a, b, j, k, m, n, r, x
+
+
+def test_gosper_normal():
+    eq = 4*n + 5, 2*(4*n + 1)*(2*n + 3), n
+    assert gosper_normal(*eq) == \
+        (Poly(Rational(1, 4), n), Poly(n + Rational(3, 2)), Poly(n + Rational(1, 4)))
+    assert gosper_normal(*eq, polys=False) == \
+        (Rational(1, 4), n + Rational(3, 2), n + Rational(1, 4))
+
+
+def test_gosper_term():
+    assert gosper_term((4*k + 1)*factorial(
+        k)/factorial(2*k + 1), k) == (-k - S.Half)/(k + Rational(1, 4))
+
+
+def test_gosper_sum():
+    assert gosper_sum(1, (k, 0, n)) == 1 + n
+    assert gosper_sum(k, (k, 0, n)) == n*(1 + n)/2
+    assert gosper_sum(k**2, (k, 0, n)) == n*(1 + n)*(1 + 2*n)/6
+    assert gosper_sum(k**3, (k, 0, n)) == n**2*(1 + n)**2/4
+
+    assert gosper_sum(2**k, (k, 0, n)) == 2*2**n - 1
+
+    assert gosper_sum(factorial(k), (k, 0, n)) is None
+    assert gosper_sum(binomial(n, k), (k, 0, n)) is None
+
+    assert gosper_sum(factorial(k)/k**2, (k, 0, n)) is None
+    assert gosper_sum((k - 3)*factorial(k), (k, 0, n)) is None
+
+    assert gosper_sum(k*factorial(k), k) == factorial(k)
+    assert gosper_sum(
+        k*factorial(k), (k, 0, n)) == n*factorial(n) + factorial(n) - 1
+
+    assert gosper_sum((-1)**k*binomial(n, k), (k, 0, n)) == 0
+    assert gosper_sum((
+        -1)**k*binomial(n, k), (k, 0, m)) == -(-1)**m*(m - n)*binomial(n, m)/n
+
+    assert gosper_sum((4*k + 1)*factorial(k)/factorial(2*k + 1), (k, 0, n)) == \
+        (2*factorial(2*n + 1) - factorial(n))/factorial(2*n + 1)
+
+    # issue 6033:
+    assert gosper_sum(
+        n*(n + a + b)*a**n*b**n/(factorial(n + a)*factorial(n + b)), \
+        (n, 0, m)).simplify() == -exp(m*log(a) + m*log(b))*gamma(a + 1) \
+        *gamma(b + 1)/(gamma(a)*gamma(b)*gamma(a + m + 1)*gamma(b + m + 1)) \
+        + 1/(gamma(a)*gamma(b))
+
+
+def test_gosper_sum_indefinite():
+    assert gosper_sum(k, k) == k*(k - 1)/2
+    assert gosper_sum(k**2, k) == k*(k - 1)*(2*k - 1)/6
+
+    assert gosper_sum(1/(k*(k + 1)), k) == -1/k
+    assert gosper_sum(-(27*k**4 + 158*k**3 + 430*k**2 + 678*k + 445)*gamma(2*k
+                      + 4)/(3*(3*k + 7)*gamma(3*k + 6)), k) == \
+        (3*k + 5)*(k**2 + 2*k + 5)*gamma(2*k + 4)/gamma(3*k + 6)
+
+
+def test_gosper_sum_parametric():
+    assert gosper_sum(binomial(S.Half, m - j + 1)*binomial(S.Half, m + j), (j, 1, n)) == \
+        n*(1 + m - n)*(-1 + 2*m + 2*n)*binomial(S.Half, 1 + m - n)* \
+        binomial(S.Half, m + n)/(m*(1 + 2*m))
+
+
+def test_gosper_sum_algebraic():
+    assert gosper_sum(
+        n**2 + sqrt(2), (n, 0, m)) == (m + 1)*(2*m**2 + m + 6*sqrt(2))/6
+
+
+def test_gosper_sum_iterated():
+    f1 = binomial(2*k, k)/4**k
+    f2 = (1 + 2*n)*binomial(2*n, n)/4**n
+    f3 = (1 + 2*n)*(3 + 2*n)*binomial(2*n, n)/(3*4**n)
+    f4 = (1 + 2*n)*(3 + 2*n)*(5 + 2*n)*binomial(2*n, n)/(15*4**n)
+    f5 = (1 + 2*n)*(3 + 2*n)*(5 + 2*n)*(7 + 2*n)*binomial(2*n, n)/(105*4**n)
+
+    assert gosper_sum(f1, (k, 0, n)) == f2
+    assert gosper_sum(f2, (n, 0, n)) == f3
+    assert gosper_sum(f3, (n, 0, n)) == f4
+    assert gosper_sum(f4, (n, 0, n)) == f5
+
+# the AeqB tests test expressions given in
+# www.math.upenn.edu/~wilf/AeqB.pdf
+
+
+def test_gosper_sum_AeqB_part1():
+    f1a = n**4
+    f1b = n**3*2**n
+    f1c = 1/(n**2 + sqrt(5)*n - 1)
+    f1d = n**4*4**n/binomial(2*n, n)
+    f1e = factorial(3*n)/(factorial(n)*factorial(n + 1)*factorial(n + 2)*27**n)
+    f1f = binomial(2*n, n)**2/((n + 1)*4**(2*n))
+    f1g = (4*n - 1)*binomial(2*n, n)**2/((2*n - 1)**2*4**(2*n))
+    f1h = n*factorial(n - S.Half)**2/factorial(n + 1)**2
+
+    g1a = m*(m + 1)*(2*m + 1)*(3*m**2 + 3*m - 1)/30
+    g1b = 26 + 2**(m + 1)*(m**3 - 3*m**2 + 9*m - 13)
+    g1c = (m + 1)*(m*(m**2 - 7*m + 3)*sqrt(5) - (
+        3*m**3 - 7*m**2 + 19*m - 6))/(2*m**3*sqrt(5) + m**4 + 5*m**2 - 1)/6
+    g1d = Rational(-2, 231) + 2*4**m*(m + 1)*(63*m**4 + 112*m**3 + 18*m**2 -
+             22*m + 3)/(693*binomial(2*m, m))
+    g1e = Rational(-9, 2) + (81*m**2 + 261*m + 200)*factorial(
+        3*m + 2)/(40*27**m*factorial(m)*factorial(m + 1)*factorial(m + 2))
+    g1f = (2*m + 1)**2*binomial(2*m, m)**2/(4**(2*m)*(m + 1))
+    g1g = -binomial(2*m, m)**2/4**(2*m)
+    g1h = 4*pi -(2*m + 1)**2*(3*m + 4)*factorial(m - S.Half)**2/factorial(m + 1)**2
+
+    g = gosper_sum(f1a, (n, 0, m))
+    assert g is not None and simplify(g - g1a) == 0
+    g = gosper_sum(f1b, (n, 0, m))
+    assert g is not None and simplify(g - g1b) == 0
+    g = gosper_sum(f1c, (n, 0, m))
+    assert g is not None and simplify(g - g1c) == 0
+    g = gosper_sum(f1d, (n, 0, m))
+    assert g is not None and simplify(g - g1d) == 0
+    g = gosper_sum(f1e, (n, 0, m))
+    assert g is not None and simplify(g - g1e) == 0
+    g = gosper_sum(f1f, (n, 0, m))
+    assert g is not None and simplify(g - g1f) == 0
+    g = gosper_sum(f1g, (n, 0, m))
+    assert g is not None and simplify(g - g1g) == 0
+    g = gosper_sum(f1h, (n, 0, m))
+    # need to call rewrite(gamma) here because we have terms involving
+    # factorial(1/2)
+    assert g is not None and simplify(g - g1h).rewrite(gamma) == 0
+
+
+def test_gosper_sum_AeqB_part2():
+    f2a = n**2*a**n
+    f2b = (n - r/2)*binomial(r, n)
+    f2c = factorial(n - 1)**2/(factorial(n - x)*factorial(n + x))
+
+    g2a = -a*(a + 1)/(a - 1)**3 + a**(
+        m + 1)*(a**2*m**2 - 2*a*m**2 + m**2 - 2*a*m + 2*m + a + 1)/(a - 1)**3
+    g2b = (m - r)*binomial(r, m)/2
+    ff = factorial(1 - x)*factorial(1 + x)
+    g2c = 1/ff*(
+        1 - 1/x**2) + factorial(m)**2/(x**2*factorial(m - x)*factorial(m + x))
+
+    g = gosper_sum(f2a, (n, 0, m))
+    assert g is not None and simplify(g - g2a) == 0
+    g = gosper_sum(f2b, (n, 0, m))
+    assert g is not None and simplify(g - g2b) == 0
+    g = gosper_sum(f2c, (n, 1, m))
+    assert g is not None and simplify(g - g2c) == 0
+
+
+def test_gosper_nan():
+    a = Symbol('a', positive=True)
+    b = Symbol('b', positive=True)
+    n = Symbol('n', integer=True)
+    m = Symbol('m', integer=True)
+    f2d = n*(n + a + b)*a**n*b**n/(factorial(n + a)*factorial(n + b))
+    g2d = 1/(factorial(a - 1)*factorial(
+        b - 1)) - a**(m + 1)*b**(m + 1)/(factorial(a + m)*factorial(b + m))
+    g = gosper_sum(f2d, (n, 0, m))
+    assert simplify(g - g2d) == 0
+
+
+def test_gosper_sum_AeqB_part3():
+    f3a = 1/n**4
+    f3b = (6*n + 3)/(4*n**4 + 8*n**3 + 8*n**2 + 4*n + 3)
+    f3c = 2**n*(n**2 - 2*n - 1)/(n**2*(n + 1)**2)
+    f3d = n**2*4**n/((n + 1)*(n + 2))
+    f3e = 2**n/(n + 1)
+    f3f = 4*(n - 1)*(n**2 - 2*n - 1)/(n**2*(n + 1)**2*(n - 2)**2*(n - 3)**2)
+    f3g = (n**4 - 14*n**2 - 24*n - 9)*2**n/(n**2*(n + 1)**2*(n + 2)**2*
+           (n + 3)**2)
+
+    # g3a -> no closed form
+    g3b = m*(m + 2)/(2*m**2 + 4*m + 3)
+    g3c = 2**m/m**2 - 2
+    g3d = Rational(2, 3) + 4**(m + 1)*(m - 1)/(m + 2)/3
+    # g3e -> no closed form
+    g3f = -(Rational(-1, 16) + 1/((m - 2)**2*(m + 1)**2))  # the AeqB key is wrong
+    g3g = Rational(-2, 9) + 2**(m + 1)/((m + 1)**2*(m + 3)**2)
+
+    g = gosper_sum(f3a, (n, 1, m))
+    assert g is None
+    g = gosper_sum(f3b, (n, 1, m))
+    assert g is not None and simplify(g - g3b) == 0
+    g = gosper_sum(f3c, (n, 1, m - 1))
+    assert g is not None and simplify(g - g3c) == 0
+    g = gosper_sum(f3d, (n, 1, m))
+    assert g is not None and simplify(g - g3d) == 0
+    g = gosper_sum(f3e, (n, 0, m - 1))
+    assert g is None
+    g = gosper_sum(f3f, (n, 4, m))
+    assert g is not None and simplify(g - g3f) == 0
+    g = gosper_sum(f3g, (n, 1, m))
+    assert g is not None and simplify(g - g3g) == 0

env-llmeval/lib/python3.10/site-packages/sympy/concrete/tests/test_guess.py

@@ -0,0 +1,82 @@
+from sympy.concrete.guess import (
+            find_simple_recurrence_vector,
+            find_simple_recurrence,
+            rationalize,
+            guess_generating_function_rational,
+            guess_generating_function,
+            guess
+        )
+from sympy.concrete.products import Product
+from sympy.core.function import Function
+from sympy.core.numbers import Rational
+from sympy.core.singleton import S
+from sympy.core.symbol import (Symbol, symbols)
+from sympy.core.sympify import sympify
+from sympy.functions.combinatorial.factorials import (RisingFactorial, factorial)
+from sympy.functions.combinatorial.numbers import fibonacci
+from sympy.functions.elementary.exponential import exp
+
+
+def test_find_simple_recurrence_vector():
+    assert find_simple_recurrence_vector(
+            [fibonacci(k) for k in range(12)]) == [1, -1, -1]
+
+
+def test_find_simple_recurrence():
+    a = Function('a')
+    n = Symbol('n')
+    assert find_simple_recurrence([fibonacci(k) for k in range(12)]) == (
+        -a(n) - a(n + 1) + a(n + 2))
+
+    f = Function('a')
+    i = Symbol('n')
+    a = [1, 1, 1]
+    for k in range(15): a.append(5*a[-1]-3*a[-2]+8*a[-3])
+    assert find_simple_recurrence(a, A=f, N=i) == (
+        -8*f(i) + 3*f(i + 1) - 5*f(i + 2) + f(i + 3))
+    assert find_simple_recurrence([0, 2, 15, 74, 12, 3, 0,
+                                    1, 2, 85, 4, 5, 63]) == 0
+
+
+def test_rationalize():
+    from mpmath import cos, pi, mpf
+    assert rationalize(cos(pi/3)) == S.Half
+    assert rationalize(mpf("0.333333333333333")) == Rational(1, 3)
+    assert rationalize(mpf("-0.333333333333333")) == Rational(-1, 3)
+    assert rationalize(pi, maxcoeff = 250) == Rational(355, 113)
+
+
+def test_guess_generating_function_rational():
+    x = Symbol('x')
+    assert guess_generating_function_rational([fibonacci(k)
+        for k in range(5, 15)]) == ((3*x + 5)/(-x**2 - x + 1))
+
+
+def test_guess_generating_function():
+    x = Symbol('x')
+    assert guess_generating_function([fibonacci(k)
+        for k in range(5, 15)])['ogf'] == ((3*x + 5)/(-x**2 - x + 1))
+    assert guess_generating_function(
+        [1, 2, 5, 14, 41, 124, 383, 1200, 3799, 12122, 38919])['ogf'] == (
+        (1/(x**4 + 2*x**2 - 4*x + 1))**S.Half)
+    assert guess_generating_function(sympify(
+       "[3/2, 11/2, 0, -121/2, -363/2, 121, 4719/2, 11495/2, -8712, -178717/2]")
+       )['ogf'] == (x + Rational(3, 2))/(11*x**2 - 3*x + 1)
+    assert guess_generating_function([factorial(k) for k in range(12)],
+       types=['egf'])['egf'] == 1/(-x + 1)
+    assert guess_generating_function([k+1 for k in range(12)],
+       types=['egf']) == {'egf': (x + 1)*exp(x), 'lgdegf': (x + 2)/(x + 1)}
+
+
+def test_guess():
+    i0, i1 = symbols('i0 i1')
+    assert guess([1, 2, 6, 24, 120], evaluate=False) == [Product(i1 + 1, (i1, 1, i0 - 1))]
+    assert guess([1, 2, 6, 24, 120]) == [RisingFactorial(2, i0 - 1)]
+    assert guess([1, 2, 7, 42, 429, 7436, 218348, 10850216], niter=4) == [
+        2**(i0 - 1)*(Rational(27, 16))**(i0**2/2 - 3*i0/2 +
+        1)*Product(RisingFactorial(Rational(5, 3), i1 - 1)*RisingFactorial(Rational(7, 3), i1
+        - 1)/(RisingFactorial(Rational(3, 2), i1 - 1)*RisingFactorial(Rational(5, 2), i1 -
+        1)), (i1, 1, i0 - 1))]
+    assert guess([1, 0, 2]) == []
+    x, y = symbols('x y')
+    assert guess([1, 2, 6, 24, 120], variables=[x, y]) == [RisingFactorial(2, x - 1)]

env-llmeval/lib/python3.10/site-packages/sympy/concrete/tests/test_products.py

@@ -0,0 +1,410 @@
+from sympy.concrete.products import (Product, product)
+from sympy.concrete.summations import Sum
+from sympy.core.function import (Derivative, Function, diff)
+from sympy.core.numbers import (Rational, oo, pi)
+from sympy.core.singleton import S
+from sympy.core.symbol import (Dummy, Symbol, symbols)
+from sympy.functions.combinatorial.factorials import (rf, factorial)
+from sympy.functions.elementary.exponential import (exp, log)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.functions.special.tensor_functions import KroneckerDelta
+from sympy.simplify.combsimp import combsimp
+from sympy.simplify.simplify import simplify
+from sympy.testing.pytest import raises
+
+a, k, n, m, x = symbols('a,k,n,m,x', integer=True)
+f = Function('f')
+
+
+def test_karr_convention():
+    # Test the Karr product convention that we want to hold.
+    # See his paper "Summation in Finite Terms" for a detailed
+    # reasoning why we really want exactly this definition.
+    # The convention is described for sums on page 309 and
+    # essentially in section 1.4, definition 3. For products
+    # we can find in analogy:
+    #
+    # \prod_{m <= i < n} f(i) 'has the obvious meaning'      for m < n
+    # \prod_{m <= i < n} f(i) = 0                            for m = n
+    # \prod_{m <= i < n} f(i) = 1 / \prod_{n <= i < m} f(i)  for m > n
+    #
+    # It is important to note that he defines all products with
+    # the upper limit being *exclusive*.
+    # In contrast, SymPy and the usual mathematical notation has:
+    #
+    # prod_{i = a}^b f(i) = f(a) * f(a+1) * ... * f(b-1) * f(b)
+    #
+    # with the upper limit *inclusive*. So translating between
+    # the two we find that:
+    #
+    # \prod_{m <= i < n} f(i) = \prod_{i = m}^{n-1} f(i)
+    #
+    # where we intentionally used two different ways to typeset the
+    # products and its limits.
+
+    i = Symbol("i", integer=True)
+    k = Symbol("k", integer=True)
+    j = Symbol("j", integer=True, positive=True)
+
+    # A simple example with a concrete factors and symbolic limits.
+
+    # The normal product: m = k and n = k + j and therefore m < n:
+    m = k
+    n = k + j
+
+    a = m
+    b = n - 1
+    S1 = Product(i**2, (i, a, b)).doit()
+
+    # The reversed product: m = k + j and n = k and therefore m > n:
+    m = k + j
+    n = k
+
+    a = m
+    b = n - 1
+    S2 = Product(i**2, (i, a, b)).doit()
+
+    assert S1 * S2 == 1
+
+    # Test the empty product: m = k and n = k and therefore m = n:
+    m = k
+    n = k
+
+    a = m
+    b = n - 1
+    Sz = Product(i**2, (i, a, b)).doit()
+
+    assert Sz == 1
+
+    # Another example this time with an unspecified factor and
+    # numeric limits. (We can not do both tests in the same example.)
+    f = Function("f")
+
+    # The normal product with m < n:
+    m = 2
+    n = 11
+
+    a = m
+    b = n - 1
+    S1 = Product(f(i), (i, a, b)).doit()
+
+    # The reversed product with m > n:
+    m = 11
+    n = 2
+
+    a = m
+    b = n - 1
+    S2 = Product(f(i), (i, a, b)).doit()
+
+    assert simplify(S1 * S2) == 1
+
+    # Test the empty product with m = n:
+    m = 5
+    n = 5
+
+    a = m
+    b = n - 1
+    Sz = Product(f(i), (i, a, b)).doit()
+
+    assert Sz == 1
+
+
+def test_karr_proposition_2a():
+    # Test Karr, page 309, proposition 2, part a
+    i, u, v = symbols('i u v', integer=True)
+
+    def test_the_product(m, n):
+        # g
+        g = i**3 + 2*i**2 - 3*i
+        # f = Delta g
+        f = simplify(g.subs(i, i+1) / g)
+        # The product
+        a = m
+        b = n - 1
+        P = Product(f, (i, a, b)).doit()
+        # Test if Product_{m <= i < n} f(i) = g(n) / g(m)
+        assert combsimp(P / (g.subs(i, n) / g.subs(i, m))) == 1
+
+    # m < n
+    test_the_product(u, u + v)
+    # m = n
+    test_the_product(u, u)
+    # m > n
+    test_the_product(u + v, u)
+
+
+def test_karr_proposition_2b():
+    # Test Karr, page 309, proposition 2, part b
+    i, u, v, w = symbols('i u v w', integer=True)
+
+    def test_the_product(l, n, m):
+        # Productmand
+        s = i**3
+        # First product
+        a = l
+        b = n - 1
+        S1 = Product(s, (i, a, b)).doit()
+        # Second product
+        a = l
+        b = m - 1
+        S2 = Product(s, (i, a, b)).doit()
+        # Third product
+        a = m
+        b = n - 1
+        S3 = Product(s, (i, a, b)).doit()
+        # Test if S1 = S2 * S3 as required
+        assert combsimp(S1 / (S2 * S3)) == 1
+
+    # l < m < n
+    test_the_product(u, u + v, u + v + w)
+    # l < m = n
+    test_the_product(u, u + v, u + v)
+    # l < m > n
+    test_the_product(u, u + v + w, v)
+    # l = m < n
+    test_the_product(u, u, u + v)
+    # l = m = n
+    test_the_product(u, u, u)
+    # l = m > n
+    test_the_product(u + v, u + v, u)
+    # l > m < n
+    test_the_product(u + v, u, u + w)
+    # l > m = n
+    test_the_product(u + v, u, u)
+    # l > m > n
+    test_the_product(u + v + w, u + v, u)
+
+
+def test_simple_products():
+    assert product(2, (k, a, n)) == 2**(n - a + 1)
+    assert product(k, (k, 1, n)) == factorial(n)
+    assert product(k**3, (k, 1, n)) == factorial(n)**3
+
+    assert product(k + 1, (k, 0, n - 1)) == factorial(n)
+    assert product(k + 1, (k, a, n - 1)) == rf(1 + a, n - a)
+
+    assert product(cos(k), (k, 0, 5)) == cos(1)*cos(2)*cos(3)*cos(4)*cos(5)
+    assert product(cos(k), (k, 3, 5)) == cos(3)*cos(4)*cos(5)
+    assert product(cos(k), (k, 1, Rational(5, 2))) != cos(1)*cos(2)
+
+    assert isinstance(product(k**k, (k, 1, n)), Product)
+
+    assert Product(x**k, (k, 1, n)).variables == [k]
+
+    raises(ValueError, lambda: Product(n))
+    raises(ValueError, lambda: Product(n, k))
+    raises(ValueError, lambda: Product(n, k, 1))
+    raises(ValueError, lambda: Product(n, k, 1, 10))
+    raises(ValueError, lambda: Product(n, (k, 1)))
+
+    assert product(1, (n, 1, oo)) == 1  # issue 8301
+    assert product(2, (n, 1, oo)) is oo
+    assert product(-1, (n, 1, oo)).func is Product
+
+
+def test_multiple_products():
+    assert product(x, (n, 1, k), (k, 1, m)) == x**(m**2/2 + m/2)
+    assert product(f(n), (
+        n, 1, m), (m, 1, k)) == Product(f(n), (n, 1, m), (m, 1, k)).doit()
+    assert Product(f(n), (m, 1, k), (n, 1, k)).doit() == \
+        Product(Product(f(n), (m, 1, k)), (n, 1, k)).doit() == \
+        product(f(n), (m, 1, k), (n, 1, k)) == \
+        product(product(f(n), (m, 1, k)), (n, 1, k)) == \
+        Product(f(n)**k, (n, 1, k))
+    assert Product(
+        x, (x, 1, k), (k, 1, n)).doit() == Product(factorial(k), (k, 1, n))
+
+    assert Product(x**k, (n, 1, k), (k, 1, m)).variables == [n, k]
+
+
+def test_rational_products():
+    assert product(1 + 1/k, (k, 1, n)) == rf(2, n)/factorial(n)
+
+
+def test_special_products():
+    # Wallis product
+    assert product((4*k)**2 / (4*k**2 - 1), (k, 1, n)) == \
+        4**n*factorial(n)**2/rf(S.Half, n)/rf(Rational(3, 2), n)
+
+    # Euler's product formula for sin
+    assert product(1 + a/k**2, (k, 1, n)) == \
+        rf(1 - sqrt(-a), n)*rf(1 + sqrt(-a), n)/factorial(n)**2
+
+
+def test__eval_product():
+    from sympy.abc import i, n
+    # issue 4809
+    a = Function('a')
+    assert product(2*a(i), (i, 1, n)) == 2**n * Product(a(i), (i, 1, n))
+    # issue 4810
+    assert product(2**i, (i, 1, n)) == 2**(n*(n + 1)/2)
+    k, m = symbols('k m', integer=True)
+    assert product(2**i, (i, k, m)) == 2**(-k**2/2 + k/2 + m**2/2 + m/2)
+    n = Symbol('n', negative=True, integer=True)
+    p = Symbol('p', positive=True, integer=True)
+    assert product(2**i, (i, n, p)) == 2**(-n**2/2 + n/2 + p**2/2 + p/2)
+    assert product(2**i, (i, p, n)) == 2**(n**2/2 + n/2 - p**2/2 + p/2)
+
+
+def test_product_pow():
+    # issue 4817
+    assert product(2**f(k), (k, 1, n)) == 2**Sum(f(k), (k, 1, n))
+    assert product(2**(2*f(k)), (k, 1, n)) == 2**Sum(2*f(k), (k, 1, n))
+
+
+def test_infinite_product():
+    # issue 5737
+    assert isinstance(Product(2**(1/factorial(n)), (n, 0, oo)), Product)
+
+
+def test_conjugate_transpose():
+    p = Product(x**k, (k, 1, 3))
+    assert p.adjoint().doit() == p.doit().adjoint()
+    assert p.conjugate().doit() == p.doit().conjugate()
+    assert p.transpose().doit() == p.doit().transpose()
+
+    A, B = symbols("A B", commutative=False)
+    p = Product(A*B**k, (k, 1, 3))
+    assert p.adjoint().doit() == p.doit().adjoint()
+    assert p.conjugate().doit() == p.doit().conjugate()
+    assert p.transpose().doit() == p.doit().transpose()
+
+    p = Product(B**k*A, (k, 1, 3))
+    assert p.adjoint().doit() == p.doit().adjoint()
+    assert p.conjugate().doit() == p.doit().conjugate()
+    assert p.transpose().doit() == p.doit().transpose()
+
+
+def test_simplify_prod():
+    y, t, b, c, v, d = symbols('y, t, b, c, v, d', integer = True)
+
+    _simplify = lambda e: simplify(e, doit=False)
+    assert _simplify(Product(x*y, (x, n, m), (y, a, k)) * \
+        Product(y, (x, n, m), (y, a, k))) == \
+            Product(x*y**2, (x, n, m), (y, a, k))
+    assert _simplify(3 * y* Product(x, (x, n, m)) * Product(x, (x, m + 1, a))) \
+        == 3 * y * Product(x, (x, n, a))
+    assert _simplify(Product(x, (x, k + 1, a)) * Product(x, (x, n, k))) == \
+        Product(x, (x, n, a))
+    assert _simplify(Product(x, (x, k + 1, a)) * Product(x + 1, (x, n, k))) == \
+        Product(x, (x, k + 1, a)) * Product(x + 1, (x, n, k))
+    assert _simplify(Product(x, (t, a, b)) * Product(y, (t, a, b)) * \
+        Product(x, (t, b+1, c))) == Product(x*y, (t, a, b)) * \
+            Product(x, (t, b+1, c))
+    assert _simplify(Product(x, (t, a, b)) * Product(x, (t, b+1, c)) * \
+        Product(y, (t, a, b))) == Product(x*y, (t, a, b)) * \
+            Product(x, (t, b+1, c))
+    assert _simplify(Product(sin(t)**2 + cos(t)**2 + 1, (t, a, b))) == \
+        Product(2, (t, a, b))
+    assert _simplify(Product(sin(t)**2 + cos(t)**2 - 1, (t, a, b))) == \
+           Product(0, (t, a, b))
+    assert _simplify(Product(v*Product(sin(t)**2 + cos(t)**2, (t, a, b)),
+                             (v, c, d))) == Product(v*Product(1, (t, a, b)), (v, c, d))
+
+
+def test_change_index():
+    b, y, c, d, z = symbols('b, y, c, d, z', integer = True)
+
+    assert Product(x, (x, a, b)).change_index(x, x + 1, y) == \
+        Product(y - 1, (y, a + 1, b + 1))
+    assert Product(x**2, (x, a, b)).change_index(x, x - 1) == \
+        Product((x + 1)**2, (x, a - 1, b - 1))
+    assert Product(x**2, (x, a, b)).change_index(x, -x, y) == \
+        Product((-y)**2, (y, -b, -a))
+    assert Product(x, (x, a, b)).change_index(x, -x - 1) == \
+        Product(-x - 1, (x, - b - 1, -a - 1))
+    assert Product(x*y, (x, a, b), (y, c, d)).change_index(x, x - 1, z) == \
+        Product((z + 1)*y, (z, a - 1, b - 1), (y, c, d))
+
+
+def test_reorder():
+    b, y, c, d, z = symbols('b, y, c, d, z', integer = True)
+
+    assert Product(x*y, (x, a, b), (y, c, d)).reorder((0, 1)) == \
+        Product(x*y, (y, c, d), (x, a, b))
+    assert Product(x, (x, a, b), (x, c, d)).reorder((0, 1)) == \
+        Product(x, (x, c, d), (x, a, b))
+    assert Product(x*y + z, (x, a, b), (z, m, n), (y, c, d)).reorder(\
+        (2, 0), (0, 1)) == Product(x*y + z, (z, m, n), (y, c, d), (x, a, b))
+    assert Product(x*y*z, (x, a, b), (y, c, d), (z, m, n)).reorder(\
+        (0, 1), (1, 2), (0, 2)) == \
+        Product(x*y*z, (x, a, b), (z, m, n), (y, c, d))
+    assert Product(x*y*z, (x, a, b), (y, c, d), (z, m, n)).reorder(\
+        (x, y), (y, z), (x, z)) == \
+        Product(x*y*z, (x, a, b), (z, m, n), (y, c, d))
+    assert Product(x*y, (x, a, b), (y, c, d)).reorder((x, 1)) == \
+        Product(x*y, (y, c, d), (x, a, b))
+    assert Product(x*y, (x, a, b), (y, c, d)).reorder((y, x)) == \
+        Product(x*y, (y, c, d), (x, a, b))
+
+
+def test_Product_is_convergent():
+    assert Product(1/n**2, (n, 1, oo)).is_convergent() is S.false
+    assert Product(exp(1/n**2), (n, 1, oo)).is_convergent() is S.true
+    assert Product(1/n, (n, 1, oo)).is_convergent() is S.false
+    assert Product(1 + 1/n, (n, 1, oo)).is_convergent() is S.false
+    assert Product(1 + 1/n**2, (n, 1, oo)).is_convergent() is S.true
+
+
+def test_reverse_order():
+    x, y, a, b, c, d= symbols('x, y, a, b, c, d', integer = True)
+
+    assert Product(x, (x, 0, 3)).reverse_order(0) == Product(1/x, (x, 4, -1))
+    assert Product(x*y, (x, 1, 5), (y, 0, 6)).reverse_order(0, 1) == \
+           Product(x*y, (x, 6, 0), (y, 7, -1))
+    assert Product(x, (x, 1, 2)).reverse_order(0) == Product(1/x, (x, 3, 0))
+    assert Product(x, (x, 1, 3)).reverse_order(0) == Product(1/x, (x, 4, 0))
+    assert Product(x, (x, 1, a)).reverse_order(0) == Product(1/x, (x, a + 1, 0))
+    assert Product(x, (x, a, 5)).reverse_order(0) == Product(1/x, (x, 6, a - 1))
+    assert Product(x, (x, a + 1, a + 5)).reverse_order(0) == \
+           Product(1/x, (x, a + 6, a))
+    assert Product(x, (x, a + 1, a + 2)).reverse_order(0) == \
+           Product(1/x, (x, a + 3, a))
+    assert Product(x, (x, a + 1, a + 1)).reverse_order(0) == \
+           Product(1/x, (x, a + 2, a))
+    assert Product(x, (x, a, b)).reverse_order(0) == Product(1/x, (x, b + 1, a - 1))
+    assert Product(x, (x, a, b)).reverse_order(x) == Product(1/x, (x, b + 1, a - 1))
+    assert Product(x*y, (x, a, b), (y, 2, 5)).reverse_order(x, 1) == \
+           Product(x*y, (x, b + 1, a - 1), (y, 6, 1))
+    assert Product(x*y, (x, a, b), (y, 2, 5)).reverse_order(y, x) == \
+           Product(x*y, (x, b + 1, a - 1), (y, 6, 1))
+
+
+def test_issue_9983():
+    n = Symbol('n', integer=True, positive=True)
+    p = Product(1 + 1/n**Rational(2, 3), (n, 1, oo))
+    assert p.is_convergent() is S.false
+    assert product(1 + 1/n**Rational(2, 3), (n, 1, oo)) == p.doit()
+
+
+def test_issue_13546():
+    n = Symbol('n')
+    k = Symbol('k')
+    p = Product(n + 1 / 2**k, (k, 0, n-1)).doit()
+    assert p.subs(n, 2).doit() == Rational(15, 2)
+
+
+def test_issue_14036():
+    a, n = symbols('a n')
+    assert product(1 - a**2 / (n*pi)**2, [n, 1, oo]) != 0
+
+
+def test_rewrite_Sum():
+    assert Product(1 - S.Half**2/k**2, (k, 1, oo)).rewrite(Sum) == \
+        exp(Sum(log(1 - 1/(4*k**2)), (k, 1, oo)))
+
+
+def test_KroneckerDelta_Product():
+    y = Symbol('y')
+    assert Product(x*KroneckerDelta(x, y), (x, 0, 1)).doit() == 0
+
+
+def test_issue_20848():
+    _i = Dummy('i')
+    t, y, z = symbols('t y z')
+    assert diff(Product(x, (y, 1, z)), x).as_dummy() == Sum(Product(x, (y, 1, _i - 1))*Product(x, (y, _i + 1, z)), (_i, 1, z)).as_dummy()
+    assert diff(Product(x, (y, 1, z)), x).doit() == x**(z - 1)*z
+    assert diff(Product(x, (y, x, z)), x) == Derivative(Product(x, (y, x, z)), x)
+    assert diff(Product(t, (x, 1, z)), x) == S(0)
+    assert Product(sin(n*x), (n, -1, 1)).diff(x).doit() == S(0)

env-llmeval/lib/python3.10/site-packages/sympy/concrete/tests/test_sums_products.py

@@ -0,0 +1,1646 @@
+from math import prod
+
+from sympy.concrete.expr_with_intlimits import ReorderError
+from sympy.concrete.products import (Product, product)
+from sympy.concrete.summations import (Sum, summation, telescopic,
+     eval_sum_residue, _dummy_with_inherited_properties_concrete)
+from sympy.core.function import (Derivative, Function)
+from sympy.core import (Catalan, EulerGamma)
+from sympy.core.facts import InconsistentAssumptions
+from sympy.core.mod import Mod
+from sympy.core.numbers import (E, I, Rational, nan, oo, pi)
+from sympy.core.relational import Eq
+from sympy.core.numbers import Float
+from sympy.core.singleton import S
+from sympy.core.symbol import (Dummy, Symbol, symbols)
+from sympy.core.sympify import sympify
+from sympy.functions.combinatorial.factorials import (rf, binomial, factorial)
+from sympy.functions.combinatorial.numbers import harmonic
+from sympy.functions.elementary.complexes import Abs
+from sympy.functions.elementary.exponential import (exp, log)
+from sympy.functions.elementary.hyperbolic import (sinh, tanh)
+from sympy.functions.elementary.integers import floor
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.functions.special.gamma_functions import (gamma, lowergamma)
+from sympy.functions.special.tensor_functions import KroneckerDelta
+from sympy.functions.special.zeta_functions import zeta
+from sympy.integrals.integrals import Integral
+from sympy.logic.boolalg import And, Or
+from sympy.matrices.expressions.matexpr import MatrixSymbol
+from sympy.matrices.expressions.special import Identity
+from sympy.matrices import (Matrix, SparseMatrix,
+    ImmutableDenseMatrix, ImmutableSparseMatrix, diag)
+from sympy.sets.fancysets import Range
+from sympy.sets.sets import Interval
+from sympy.simplify.combsimp import combsimp
+from sympy.simplify.simplify import simplify
+from sympy.tensor.indexed import (Idx, Indexed, IndexedBase)
+from sympy.testing.pytest import XFAIL, raises, slow
+from sympy.abc import a, b, c, d, k, m, x, y, z
+
+n = Symbol('n', integer=True)
+f, g = symbols('f g', cls=Function)
+
+def test_karr_convention():
+    # Test the Karr summation convention that we want to hold.
+    # See his paper "Summation in Finite Terms" for a detailed
+    # reasoning why we really want exactly this definition.
+    # The convention is described on page 309 and essentially
+    # in section 1.4, definition 3:
+    #
+    # \sum_{m <= i < n} f(i) 'has the obvious meaning'   for m < n
+    # \sum_{m <= i < n} f(i) = 0                         for m = n
+    # \sum_{m <= i < n} f(i) = - \sum_{n <= i < m} f(i)  for m > n
+    #
+    # It is important to note that he defines all sums with
+    # the upper limit being *exclusive*.
+    # In contrast, SymPy and the usual mathematical notation has:
+    #
+    # sum_{i = a}^b f(i) = f(a) + f(a+1) + ... + f(b-1) + f(b)
+    #
+    # with the upper limit *inclusive*. So translating between
+    # the two we find that:
+    #
+    # \sum_{m <= i < n} f(i) = \sum_{i = m}^{n-1} f(i)
+    #
+    # where we intentionally used two different ways to typeset the
+    # sum and its limits.
+
+    i = Symbol("i", integer=True)
+    k = Symbol("k", integer=True)
+    j = Symbol("j", integer=True)
+
+    # A simple example with a concrete summand and symbolic limits.
+
+    # The normal sum: m = k and n = k + j and therefore m < n:
+    m = k
+    n = k + j
+
+    a = m
+    b = n - 1
+    S1 = Sum(i**2, (i, a, b)).doit()
+
+    # The reversed sum: m = k + j and n = k and therefore m > n:
+    m = k + j
+    n = k
+
+    a = m
+    b = n - 1
+    S2 = Sum(i**2, (i, a, b)).doit()
+
+    assert simplify(S1 + S2) == 0
+
+    # Test the empty sum: m = k and n = k and therefore m = n:
+    m = k
+    n = k
+
+    a = m
+    b = n - 1
+    Sz = Sum(i**2, (i, a, b)).doit()
+
+    assert Sz == 0
+
+    # Another example this time with an unspecified summand and
+    # numeric limits. (We can not do both tests in the same example.)
+
+    # The normal sum with m < n:
+    m = 2
+    n = 11
+
+    a = m
+    b = n - 1
+    S1 = Sum(f(i), (i, a, b)).doit()
+
+    # The reversed sum with m > n:
+    m = 11
+    n = 2
+
+    a = m
+    b = n - 1
+    S2 = Sum(f(i), (i, a, b)).doit()
+
+    assert simplify(S1 + S2) == 0
+
+    # Test the empty sum with m = n:
+    m = 5
+    n = 5
+
+    a = m
+    b = n - 1
+    Sz = Sum(f(i), (i, a, b)).doit()
+
+    assert Sz == 0
+
+    e = Piecewise((exp(-i), Mod(i, 2) > 0), (0, True))
+    s = Sum(e, (i, 0, 11))
+    assert s.n(3) == s.doit().n(3)
+
+
+def test_karr_proposition_2a():
+    # Test Karr, page 309, proposition 2, part a
+    i = Symbol("i", integer=True)
+    u = Symbol("u", integer=True)
+    v = Symbol("v", integer=True)
+
+    def test_the_sum(m, n):
+        # g
+        g = i**3 + 2*i**2 - 3*i
+        # f = Delta g
+        f = simplify(g.subs(i, i+1) - g)
+        # The sum
+        a = m
+        b = n - 1
+        S = Sum(f, (i, a, b)).doit()
+        # Test if Sum_{m <= i < n} f(i) = g(n) - g(m)
+        assert simplify(S - (g.subs(i, n) - g.subs(i, m))) == 0
+
+    # m < n
+    test_the_sum(u,   u+v)
+    # m = n
+    test_the_sum(u,   u  )
+    # m > n
+    test_the_sum(u+v, u  )
+
+
+def test_karr_proposition_2b():
+    # Test Karr, page 309, proposition 2, part b
+    i = Symbol("i", integer=True)
+    u = Symbol("u", integer=True)
+    v = Symbol("v", integer=True)
+    w = Symbol("w", integer=True)
+
+    def test_the_sum(l, n, m):
+        # Summand
+        s = i**3
+        # First sum
+        a = l
+        b = n - 1
+        S1 = Sum(s, (i, a, b)).doit()
+        # Second sum
+        a = l
+        b = m - 1
+        S2 = Sum(s, (i, a, b)).doit()
+        # Third sum
+        a = m
+        b = n - 1
+        S3 = Sum(s, (i, a, b)).doit()
+        # Test if S1 = S2 + S3 as required
+        assert S1 - (S2 + S3) == 0
+
+    # l < m < n
+    test_the_sum(u,     u+v,   u+v+w)
+    # l < m = n
+    test_the_sum(u,     u+v,   u+v  )
+    # l < m > n
+    test_the_sum(u,     u+v+w, v    )
+    # l = m < n
+    test_the_sum(u,     u,     u+v  )
+    # l = m = n
+    test_the_sum(u,     u,     u    )
+    # l = m > n
+    test_the_sum(u+v,   u+v,   u    )
+    # l > m < n
+    test_the_sum(u+v,   u,     u+w  )
+    # l > m = n
+    test_the_sum(u+v,   u,     u    )
+    # l > m > n
+    test_the_sum(u+v+w, u+v,   u    )
+
+
+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
+    assert Sum(x, (x, Range(1, 11))).doit() == 55
+    assert Sum(x, (x, Range(1, 11, 2))).doit() == 25
+    assert Sum(x, (x, Range(1, 10, 2))) == Sum(x, (x, Range(9, 0, -2)))
+    lo, hi = 1, 2
+    s1 = Sum(n, (n, lo, hi))
+    s2 = Sum(n, (n, hi, lo))
+    assert s1 != s2
+    assert s1.doit() == 3 and s2.doit() == 0
+    lo, hi = x, x + 1
+    s1 = Sum(n, (n, lo, hi))
+    s2 = Sum(n, (n, hi, lo))
+    assert s1 != s2
+    assert s1.doit() == 2*x + 1 and s2.doit() == 0
+    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)
+    assert summation(k, (k, 0, oo)) is oo
+    assert summation(k, (k, Range(1, 11))) == 55
+
+
+def test_polynomial_sums():
+    assert summation(n**2, (n, 3, 8)) == 199
+    assert summation(n, (n, a, b)) == \
+        ((a + b)*(b - a + 1)/2).expand()
+    assert summation(n**2, (n, 1, b)) == \
+        ((2*b**3 + 3*b**2 + b)/6).expand()
+    assert summation(n**3, (n, 1, b)) == \
+        ((b**4 + 2*b**3 + b**2)/4).expand()
+    assert summation(n**6, (n, 1, b)) == \
+        ((6*b**7 + 21*b**6 + 21*b**5 - 7*b**3 + b)/42).expand()
+
+
+def test_geometric_sums():
+    assert summation(pi**n, (n, 0, b)) == (1 - pi**(b + 1)) / (1 - pi)
+    assert summation(2 * 3**n, (n, 0, b)) == 3**(b + 1) - 1
+    assert summation(S.Half**n, (n, 1, oo)) == 1
+    assert summation(2**n, (n, 0, b)) == 2**(b + 1) - 1
+    assert summation(2**n, (n, 1, oo)) is oo
+    assert summation(2**(-n), (n, 1, oo)) == 1
+    assert summation(3**(-n), (n, 4, oo)) == Rational(1, 54)
+    assert summation(2**(-4*n + 3), (n, 1, oo)) == Rational(8, 15)
+    assert summation(2**(n + 1), (n, 1, b)).expand() == 4*(2**b - 1)
+
+    # issue 6664:
+    assert summation(x**n, (n, 0, oo)) == \
+        Piecewise((1/(-x + 1), Abs(x) < 1), (Sum(x**n, (n, 0, oo)), True))
+
+    assert summation(-2**n, (n, 0, oo)) is -oo
+    assert summation(I**n, (n, 0, oo)) == Sum(I**n, (n, 0, oo))
+
+    # issue 6802:
+    assert summation((-1)**(2*x + 2), (x, 0, n)) == n + 1
+    assert summation((-2)**(2*x + 2), (x, 0, n)) == 4*4**(n + 1)/S(3) - Rational(4, 3)
+    assert summation((-1)**x, (x, 0, n)) == -(-1)**(n + 1)/S(2) + S.Half
+    assert summation(y**x, (x, a, b)) == \
+        Piecewise((-a + b + 1, Eq(y, 1)), ((y**a - y**(b + 1))/(-y + 1), True))
+    assert summation((-2)**(y*x + 2), (x, 0, n)) == \
+        4*Piecewise((n + 1, Eq((-2)**y, 1)),
+                    ((-(-2)**(y*(n + 1)) + 1)/(-(-2)**y + 1), True))
+
+    # issue 8251:
+    assert summation((1/(n + 1)**2)*n**2, (n, 0, oo)) is oo
+
+    #issue 9908:
+    assert Sum(1/(n**3 - 1), (n, -oo, -2)).doit() == summation(1/(n**3 - 1), (n, -oo, -2))
+
+    #issue 11642:
+    result = Sum(0.5**n, (n, 1, oo)).doit()
+    assert result == 1.0
+    assert result.is_Float
+
+    result = Sum(0.25**n, (n, 1, oo)).doit()
+    assert result == 1/3.
+    assert result.is_Float
+
+    result = Sum(0.99999**n, (n, 1, oo)).doit()
+    assert result == 99999.0
+    assert result.is_Float
+
+    result = Sum(S.Half**n, (n, 1, oo)).doit()
+    assert result == 1
+    assert not result.is_Float
+
+    result = Sum(Rational(3, 5)**n, (n, 1, oo)).doit()
+    assert result == Rational(3, 2)
+    assert not result.is_Float
+
+    assert Sum(1.0**n, (n, 1, oo)).doit() is oo
+    assert Sum(2.43**n, (n, 1, oo)).doit() is oo
+
+    # Issue 13979
+    i, k, q = symbols('i k q', integer=True)
+    result = summation(
+        exp(-2*I*pi*k*i/n) * exp(2*I*pi*q*i/n) / n, (i, 0, n - 1)
+    )
+    assert result.simplify() == Piecewise(
+            (1, Eq(exp(-2*I*pi*(k - q)/n), 1)), (0, True)
+    )
+
+    #Issue 23491
+    assert Sum(1/(n**2 + 1), (n, 1, oo)).doit() == S(-1)/2 + pi/(2*tanh(pi))
+
+def test_harmonic_sums():
+    assert summation(1/k, (k, 0, n)) == Sum(1/k, (k, 0, n))
+    assert summation(1/k, (k, 1, n)) == harmonic(n)
+    assert summation(n/k, (k, 1, n)) == n*harmonic(n)
+    assert summation(1/k, (k, 5, n)) == harmonic(n) - harmonic(4)
+
+
+def test_composite_sums():
+    f = S.Half*(7 - 6*n + Rational(1, 7)*n**3)
+    s = summation(f, (n, a, b))
+    assert not isinstance(s, Sum)
+    A = 0
+    for i in range(-3, 5):
+        A += f.subs(n, i)
+    B = s.subs(a, -3).subs(b, 4)
+    assert A == B
+
+
+def test_hypergeometric_sums():
+    assert summation(
+        binomial(2*k, k)/4**k, (k, 0, n)) == (1 + 2*n)*binomial(2*n, n)/4**n
+    assert summation(binomial(2*k, k)/5**k, (k, -oo, oo)) == sqrt(5)
+
+
+def test_other_sums():
+    f = m**2 + m*exp(m)
+    g = 3*exp(Rational(3, 2))/2 + exp(S.Half)/2 - exp(Rational(-1, 2))/2 - 3*exp(Rational(-3, 2))/2 + 5
+
+    assert summation(f, (m, Rational(-3, 2), Rational(3, 2))) == g
+    assert summation(f, (m, -1.5, 1.5)).evalf().epsilon_eq(g.evalf(), 1e-10)
+
+fac = factorial
+
+
+def NS(e, n=15, **options):
+    return str(sympify(e).evalf(n, **options))
+
+
+def test_evalf_fast_series():
+    # Euler transformed series for sqrt(1+x)
+    assert NS(Sum(
+        fac(2*n + 1)/fac(n)**2/2**(3*n + 1), (n, 0, oo)), 100) == NS(sqrt(2), 100)
+
+    # Some series for exp(1)
+    estr = NS(E, 100)
+    assert NS(Sum(1/fac(n), (n, 0, oo)), 100) == estr
+    assert NS(1/Sum((1 - 2*n)/fac(2*n), (n, 0, oo)), 100) == estr
+    assert NS(Sum((2*n + 1)/fac(2*n), (n, 0, oo)), 100) == estr
+    assert NS(Sum((4*n + 3)/2**(2*n + 1)/fac(2*n + 1), (n, 0, oo))**2, 100) == estr
+
+    pistr = NS(pi, 100)
+    # Ramanujan series for pi
+    assert NS(9801/sqrt(8)/Sum(fac(
+        4*n)*(1103 + 26390*n)/fac(n)**4/396**(4*n), (n, 0, oo)), 100) == pistr
+    assert NS(1/Sum(
+        binomial(2*n, n)**3 * (42*n + 5)/2**(12*n + 4), (n, 0, oo)), 100) == pistr
+    # Machin's formula for pi
+    assert NS(16*Sum((-1)**n/(2*n + 1)/5**(2*n + 1), (n, 0, oo)) -
+        4*Sum((-1)**n/(2*n + 1)/239**(2*n + 1), (n, 0, oo)), 100) == pistr
+
+    # Apery's constant
+    astr = NS(zeta(3), 100)
+    P = 126392*n**5 + 412708*n**4 + 531578*n**3 + 336367*n**2 + 104000* \
+        n + 12463
+    assert NS(Sum((-1)**n * P / 24 * (fac(2*n + 1)*fac(2*n)*fac(
+        n))**3 / fac(3*n + 2) / fac(4*n + 3)**3, (n, 0, oo)), 100) == astr
+    assert NS(Sum((-1)**n * (205*n**2 + 250*n + 77)/64 * fac(n)**10 /
+              fac(2*n + 1)**5, (n, 0, oo)), 100) == astr
+
+
+def test_evalf_fast_series_issue_4021():
+    # Catalan's constant
+    assert NS(Sum((-1)**(n - 1)*2**(8*n)*(40*n**2 - 24*n + 3)*fac(2*n)**3*
+        fac(n)**2/n**3/(2*n - 1)/fac(4*n)**2, (n, 1, oo))/64, 100) == \
+        NS(Catalan, 100)
+    astr = NS(zeta(3), 100)
+    assert NS(5*Sum(
+        (-1)**(n - 1)*fac(n)**2 / n**3 / fac(2*n), (n, 1, oo))/2, 100) == astr
+    assert NS(Sum((-1)**(n - 1)*(56*n**2 - 32*n + 5) / (2*n - 1)**2 * fac(n - 1)
+              **3 / fac(3*n), (n, 1, oo))/4, 100) == astr
+
+
+def test_evalf_slow_series():
+    assert NS(Sum((-1)**n / n, (n, 1, oo)), 15) == NS(-log(2), 15)
+    assert NS(Sum((-1)**n / n, (n, 1, oo)), 50) == NS(-log(2), 50)
+    assert NS(Sum(1/n**2, (n, 1, oo)), 15) == NS(pi**2/6, 15)
+    assert NS(Sum(1/n**2, (n, 1, oo)), 100) == NS(pi**2/6, 100)
+    assert NS(Sum(1/n**2, (n, 1, oo)), 500) == NS(pi**2/6, 500)
+    assert NS(Sum((-1)**n / (2*n + 1)**3, (n, 0, oo)), 15) == NS(pi**3/32, 15)
+    assert NS(Sum((-1)**n / (2*n + 1)**3, (n, 0, oo)), 50) == NS(pi**3/32, 50)
+
+
+def test_evalf_oo_to_oo():
+    # There used to be an error in certain cases
+    # Does not evaluate, but at least do not throw an error
+    # Evaluates symbolically to 0, which is not correct
+    assert Sum(1/(n**2+1), (n, -oo, oo)).evalf() == Sum(1/(n**2+1), (n, -oo, oo))
+    # This evaluates if from 1 to oo and symbolically
+    assert Sum(1/(factorial(abs(n))), (n, -oo, -1)).evalf() == Sum(1/(factorial(abs(n))), (n, -oo, -1))
+
+
+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 mi, ni in [(2, 4), (2, 20), (10, 20), (18, 20)]:
+        A = Sum(1/k**3, (k, 1, oo))
+        s, e = A.euler_maclaurin(mi, ni)
+        assert abs((s - zeta(3)).evalf()) < e.evalf()
+
+    raises(ValueError, lambda: Sum(1, (x, 0, 1), (k, 0, 1)).euler_maclaurin())
+
+
+@slow
+def test_evalf_euler_maclaurin():
+    assert NS(Sum(1/k**k, (k, 1, oo)), 15) == '1.29128599706266'
+    assert NS(Sum(1/k**k, (k, 1, oo)),
+              50) == '1.2912859970626635404072825905956005414986193682745'
+    assert NS(Sum(1/k - log(1 + 1/k), (k, 1, oo)), 15) == NS(EulerGamma, 15)
+    assert NS(Sum(1/k - log(1 + 1/k), (k, 1, oo)), 50) == NS(EulerGamma, 50)
+    assert NS(Sum(log(k)/k**2, (k, 1, oo)), 15) == '0.937548254315844'
+    assert NS(Sum(log(k)/k**2, (k, 1, oo)),
+              50) == '0.93754825431584375370257409456786497789786028861483'
+    assert NS(Sum(1/k, (k, 1000000, 2000000)), 15) == '0.693147930560008'
+    assert NS(Sum(1/k, (k, 1000000, 2000000)),
+              50) == '0.69314793056000780941723211364567656807940638436025'
+
+
+def test_evalf_symbolic():
+    # issue 6328
+    expr = Sum(f(x), (x, 1, 3)) + Sum(g(x), (x, 1, 3))
+    assert expr.evalf() == expr
+
+
+def test_evalf_issue_3273():
+    assert Sum(0, (k, 1, oo)).evalf() == 0
+
+
+def test_simple_products():
+    assert Product(S.NaN, (x, 1, 3)) is S.NaN
+    assert product(S.NaN, (x, 1, 3)) is S.NaN
+    assert Product(x, (n, a, a)).doit() == x
+    assert Product(x, (x, a, a)).doit() == a
+    assert Product(x, (y, 1, a)).doit() == x**a
+
+    lo, hi = 1, 2
+    s1 = Product(n, (n, lo, hi))
+    s2 = Product(n, (n, hi, lo))
+    assert s1 != s2
+    # This IS correct according to Karr product convention
+    assert s1.doit() == 2
+    assert s2.doit() == 1
+
+    lo, hi = x, x + 1
+    s1 = Product(n, (n, lo, hi))
+    s2 = Product(n, (n, hi, lo))
+    s3 = 1 / Product(n, (n, hi + 1, lo - 1))
+    assert s1 != s2
+    # This IS correct according to Karr product convention
+    assert s1.doit() == x*(x + 1)
+    assert s2.doit() == 1
+    assert s3.doit() == x*(x + 1)
+
+    assert Product(Integral(2*x, (x, 1, y)) + 2*x, (x, 1, 2)).doit() == \
+        (y**2 + 1)*(y**2 + 3)
+    assert product(2, (n, a, b)) == 2**(b - a + 1)
+    assert product(n, (n, 1, b)) == factorial(b)
+    assert product(n**3, (n, 1, b)) == factorial(b)**3
+    assert product(3**(2 + n), (n, a, b)) \
+        == 3**(2*(1 - a + b) + b/2 + (b**2)/2 + a/2 - (a**2)/2)
+    assert product(cos(n), (n, 3, 5)) == cos(3)*cos(4)*cos(5)
+    assert product(cos(n), (n, x, x + 2)) == cos(x)*cos(x + 1)*cos(x + 2)
+    assert isinstance(product(cos(n), (n, x, x + S.Half)), Product)
+    # If Product managed to evaluate this one, it most likely got it wrong!
+    assert isinstance(Product(n**n, (n, 1, b)), Product)
+
+
+def test_rational_products():
+    assert combsimp(product(1 + 1/n, (n, a, b))) == (1 + b)/a
+    assert combsimp(product(n + 1, (n, a, b))) == gamma(2 + b)/gamma(1 + a)
+    assert combsimp(product((n + 1)/(n - 1), (n, a, b))) == b*(1 + b)/(a*(a - 1))
+    assert combsimp(product(n/(n + 1)/(n + 2), (n, a, b))) == \
+        a*gamma(a + 2)/(b + 1)/gamma(b + 3)
+    assert combsimp(product(n*(n + 1)/(n - 1)/(n - 2), (n, a, b))) == \
+        b**2*(b - 1)*(1 + b)/(a - 1)**2/(a*(a - 2))
+
+
+def test_wallis_product():
+    # Wallis product, given in two different forms to ensure that Product
+    # can factor simple rational expressions
+    A = Product(4*n**2 / (4*n**2 - 1), (n, 1, b))
+    B = Product((2*n)*(2*n)/(2*n - 1)/(2*n + 1), (n, 1, b))
+    R = pi*gamma(b + 1)**2/(2*gamma(b + S.Half)*gamma(b + Rational(3, 2)))
+    assert simplify(A.doit()) == R
+    assert simplify(B.doit()) == R
+    # This one should eventually also be doable (Euler's product formula for sin)
+    # assert Product(1+x/n**2, (n, 1, b)) == ...
+
+
+def test_telescopic_sums():
+    #checks also input 2 of comment 1 issue 4127
+    assert Sum(1/k - 1/(k + 1), (k, 1, n)).doit() == 1 - 1/(1 + n)
+    assert Sum(
+        f(k) - f(k + 2), (k, m, n)).doit() == -f(1 + n) - f(2 + n) + f(m) + f(1 + m)
+    assert Sum(cos(k) - cos(k + 3), (k, 1, n)).doit() == -cos(1 + n) - \
+        cos(2 + n) - cos(3 + n) + cos(1) + cos(2) + cos(3)
+
+    # dummy variable shouldn't matter
+    assert telescopic(1/m, -m/(1 + m), (m, n - 1, n)) == \
+        telescopic(1/k, -k/(1 + k), (k, n - 1, n))
+
+    assert Sum(1/x/(x - 1), (x, a, b)).doit() == 1/(a - 1) - 1/b
+    eq = 1/((5*n + 2)*(5*(n + 1) + 2))
+    assert Sum(eq, (n, 0, oo)).doit() == S(1)/10
+    nz = symbols('nz', nonzero=True)
+    v = Sum(eq.subs(5, nz), (n, 0, oo)).doit()
+    assert v.subs(nz, 5).simplify() == S(1)/10
+    # check that apart is being used in non-symbolic case
+    s = Sum(eq, (n, 0, k)).doit()
+    v = Sum(eq, (n, 0, 10**100)).doit()
+    assert v == s.subs(k, 10**100)
+
+
+def test_sum_reconstruct():
+    s = Sum(n**2, (n, -1, 1))
+    assert s == Sum(*s.args)
+    raises(ValueError, lambda: Sum(x, x))
+    raises(ValueError, lambda: Sum(x, (x, 1)))
+
+
+def test_limit_subs():
+    for F in (Sum, Product, Integral):
+        assert F(a*exp(a), (a, -2, 2)) == F(a*exp(a), (a, -b, b)).subs(b, 2)
+        assert F(a, (a, F(b, (b, 1, 2)), 4)).subs(F(b, (b, 1, 2)), c) == \
+            F(a, (a, c, 4))
+        assert F(x, (x, 1, x + y)).subs(x, 1) == F(x, (x, 1, y + 1))
+
+
+def test_function_subs():
+    S = Sum(x*f(y),(x,0,oo),(y,0,oo))
+    assert S.subs(f(y),y) == Sum(x*y,(x,0,oo),(y,0,oo))
+    assert S.subs(f(x),x) == S
+    raises(ValueError, lambda: S.subs(f(y),x+y) )
+    S = Sum(x*log(y),(x,0,oo),(y,0,oo))
+    assert S.subs(log(y),y) == S
+    S = Sum(x*f(y),(x,0,oo),(y,0,oo))
+    assert S.subs(f(y),y) == Sum(x*y,(x,0,oo),(y,0,oo))
+
+
+def test_equality():
+    # if this fails remove special handling below
+    raises(ValueError, lambda: Sum(x, x))
+    r = symbols('x', real=True)
+    for F in (Sum, Product, Integral):
+        try:
+            assert F(x, x) != F(y, y)
+            assert F(x, (x, 1, 2)) != F(x, x)
+            assert F(x, (x, x)) != F(x, x)  # or else they print the same
+            assert F(1, x) != F(1, y)
+        except ValueError:
+            pass
+        assert F(a, (x, 1, 2)) != F(a, (x, 1, 3))  # diff limit
+        assert F(a, (x, 1, x)) != F(a, (y, 1, y))
+        assert F(a, (x, 1, 2)) != F(b, (x, 1, 2))  # diff expression
+        assert F(x, (x, 1, 2)) != F(r, (r, 1, 2))  # diff assumptions
+        assert F(1, (x, 1, x)) != F(1, (y, 1, x))  # only dummy is diff
+        assert F(1, (x, 1, x)).dummy_eq(F(1, (y, 1, x)))
+
+    # issue 5265
+    assert Sum(x, (x, 1, x)).subs(x, a) == Sum(x, (x, 1, a))
+
+
+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()
+
+    # Integer assumes finite
+    assert Sum(KroneckerDelta(x, y), (x, -oo, oo)).doit() == Piecewise((1, And(-oo < y, y < oo)), (0, True))
+    assert Sum(KroneckerDelta(m, n), (m, -oo, oo)).doit() == 1
+    assert Sum(m*KroneckerDelta(x, y), (x, -oo, oo)).doit() == Piecewise((m, And(-oo < y, y < oo)), (0, True))
+    assert Sum(x*KroneckerDelta(m, n), (m, -oo, oo)).doit() == x
+    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(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(f(n), (n, 1, oo))
+
+    # issue 2597
+    nmax = symbols('N', integer=True, positive=True)
+    pw = Piecewise((1, And(1 <= n, n <= nmax)), (0, True))
+    assert Sum(pw, (n, 1, nmax)).doit() == Sum(Piecewise((1, nmax >= n),
+                    (0, True)), (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_Product_doit():
+    assert Product(n*Integral(a**2), (n, 1, 3)).doit() == 2 * a**9 / 9
+    assert Product(n*Integral(a**2), (n, 1, 3)).doit(deep=False) == \
+        6*Integral(a**2)**3
+    assert product(n*Integral(a**2), (n, 1, 3)) == 6*Integral(a**2)**3
+
+
+def test_Sum_interface():
+    assert isinstance(Sum(0, (n, 0, 2)), Sum)
+    assert Sum(nan, (n, 0, 2)) is nan
+    assert Sum(nan, (n, 0, oo)) is nan
+    assert Sum(0, (n, 0, 2)).doit() == 0
+    assert isinstance(Sum(0, (n, 0, oo)), Sum)
+    assert Sum(0, (n, 0, oo)).doit() == 0
+    raises(ValueError, lambda: Sum(1))
+    raises(ValueError, lambda: summation(1))
+
+
+def test_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).doit() == \
+        Sum(x*y, (x, 1, 3), (a, 2, 5)).doit().diff(y) == 24
+    assert Sum(x, (x, 1, 2)).diff(y) == 0
+
+
+def test_hypersum():
+    assert simplify(summation(x**n/fac(n), (n, 1, oo))) == -1 + exp(x)
+    assert summation((-1)**n * x**(2*n) / fac(2*n), (n, 0, oo)) == cos(x)
+    assert simplify(summation((-1)**n*x**(2*n + 1) /
+        factorial(2*n + 1), (n, 3, oo))) == -x + sin(x) + x**3/6 - x**5/120
+
+    assert summation(1/(n + 2)**3, (n, 1, oo)) == Rational(-9, 8) + zeta(3)
+    assert summation(1/n**4, (n, 1, oo)) == pi**4/90
+
+    s = summation(x**n*n, (n, -oo, 0))
+    assert s.is_Piecewise
+    assert s.args[0].args[0] == -1/(x*(1 - 1/x)**2)
+    assert s.args[0].args[1] == (abs(1/x) < 1)
+
+    m = Symbol('n', integer=True, positive=True)
+    assert summation(binomial(m, k), (k, 0, m)) == 2**m
+
+
+def test_issue_4170():
+    assert summation(1/factorial(k), (k, 0, oo)) == E
+
+
+def test_is_commutative():
+    from sympy.physics.secondquant import NO, F, Fd
+    m = Symbol('m', commutative=False)
+    for f in (Sum, Product, Integral):
+        assert f(z, (z, 1, 1)).is_commutative is True
+        assert f(z*y, (z, 1, 6)).is_commutative is True
+        assert f(m*x, (x, 1, 2)).is_commutative is False
+
+        assert f(NO(Fd(x)*F(y))*z, (z, 1, 2)).is_commutative is False
+
+
+def test_is_zero():
+    for func in [Sum, Product]:
+        assert func(0, (x, 1, 1)).is_zero is True
+        assert func(x, (x, 1, 1)).is_zero is None
+
+    assert Sum(0, (x, 1, 0)).is_zero is True
+    assert Product(0, (x, 1, 0)).is_zero is False
+
+
+def test_is_number():
+    # is number should not rely on evaluation or assumptions,
+    # it should be equivalent to `not foo.free_symbols`
+    assert Sum(1, (x, 1, 1)).is_number is True
+    assert Sum(1, (x, 1, x)).is_number is False
+    assert Sum(0, (x, y, z)).is_number is False
+    assert Sum(x, (y, 1, 2)).is_number is False
+    assert Sum(x, (y, 1, 1)).is_number is False
+    assert Sum(x, (x, 1, 2)).is_number is True
+    assert Sum(x*y, (x, 1, 2), (y, 1, 3)).is_number is True
+
+    assert Product(2, (x, 1, 1)).is_number is True
+    assert Product(2, (x, 1, y)).is_number is False
+    assert Product(0, (x, y, z)).is_number is False
+    assert Product(1, (x, y, z)).is_number is False
+    assert Product(x, (y, 1, x)).is_number is False
+    assert Product(x, (y, 1, 2)).is_number is False
+    assert Product(x, (y, 1, 1)).is_number is False
+    assert Product(x, (x, 1, 2)).is_number is True
+
+
+def test_free_symbols():
+    for func in [Sum, Product]:
+        assert func(1, (x, 1, 2)).free_symbols == set()
+        assert func(0, (x, 1, y)).free_symbols == {y}
+        assert func(2, (x, 1, y)).free_symbols == {y}
+        assert func(x, (x, 1, 2)).free_symbols == set()
+        assert func(x, (x, 1, y)).free_symbols == {y}
+        assert func(x, (y, 1, y)).free_symbols == {x, y}
+        assert func(x, (y, 1, 2)).free_symbols == {x}
+        assert func(x, (y, 1, 1)).free_symbols == {x}
+        assert func(x, (y, 1, z)).free_symbols == {x, z}
+        assert func(x, (x, 1, y), (y, 1, 2)).free_symbols == set()
+        assert func(x, (x, 1, y), (y, 1, z)).free_symbols == {z}
+        assert func(x, (x, 1, y), (y, 1, y)).free_symbols == {y}
+        assert func(x, (y, 1, y), (y, 1, z)).free_symbols == {x, z}
+    assert Sum(1, (x, 1, y)).free_symbols == {y}
+    # free_symbols answers whether the object *as written* has free symbols,
+    # not whether the evaluated expression has free symbols
+    assert Product(1, (x, 1, y)).free_symbols == {y}
+    # don't count free symbols that are not independent of integration
+    # variable(s)
+    assert func(f(x), (f(x), 1, 2)).free_symbols == set()
+    assert func(f(x), (f(x), 1, x)).free_symbols == {x}
+    assert func(f(x), (f(x), 1, y)).free_symbols == {y}
+    assert func(f(x), (z, 1, y)).free_symbols == {x, y}
+
+
+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()
+
+    p = Sum(B**n*A, (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 test_noncommutativity_honoured():
+    A, B = symbols("A B", commutative=False)
+    M = symbols('M', integer=True, positive=True)
+    p = Sum(A*B**n, (n, 1, M))
+    assert p.doit() == A*Piecewise((M, Eq(B, 1)),
+                                   ((B - B**(M + 1))*(1 - B)**(-1), True))
+
+    p = Sum(B**n*A, (n, 1, M))
+    assert p.doit() == Piecewise((M, Eq(B, 1)),
+                                 ((B - B**(M + 1))*(1 - B)**(-1), True))*A
+
+    p = Sum(B**n*A*B**n, (n, 1, M))
+    assert p.doit() == p
+
+
+def test_issue_4171():
+    assert summation(factorial(2*k + 1)/factorial(2*k), (k, 0, oo)) is oo
+    assert summation(2*k + 1, (k, 0, oo)) is oo
+
+
+def test_issue_6273():
+    assert Sum(x, (x, 1, n)).n(2, subs={n: 1}) == Float(1, 2)
+
+
+def test_issue_6274():
+    assert Sum(x, (x, 1, 0)).doit() == 0
+    assert NS(Sum(x, (x, 1, 0))) == '0'
+    assert Sum(n, (n, 10, 5)).doit() == -30
+    assert NS(Sum(n, (n, 10, 5))) == '-30.0000000000000'
+
+
+def test_simplify_sum():
+    y, t, v = symbols('y, t, v')
+
+    _simplify = lambda e: simplify(e, doit=False)
+    assert _simplify(Sum(x*y, (x, n, m), (y, a, k)) + \
+        Sum(y, (x, n, m), (y, a, k))) == Sum(y * (x + 1), (x, n, m), (y, a, k))
+    assert _simplify(Sum(x, (x, n, m)) + Sum(x, (x, m + 1, a))) == \
+        Sum(x, (x, n, a))
+    assert _simplify(Sum(x, (x, k + 1, a)) + Sum(x, (x, n, k))) == \
+        Sum(x, (x, n, a))
+    assert _simplify(Sum(x, (x, k + 1, a)) + Sum(x + 1, (x, n, k))) == \
+        Sum(x, (x, n, a)) + Sum(1, (x, n, k))
+    assert _simplify(Sum(x, (x, 0, 3)) * 3 + 3 * Sum(x, (x, 4, 6)) + \
+        4 * Sum(z, (z, 0, 1))) == 4*Sum(z, (z, 0, 1)) + 3*Sum(x, (x, 0, 6))
+    assert _simplify(3*Sum(x**2, (x, a, b)) + Sum(x, (x, a, b))) == \
+        Sum(x*(3*x + 1), (x, a, b))
+    assert _simplify(Sum(x**3, (x, n, k)) * 3 + 3 * Sum(x, (x, n, k)) + \
+        4 * y * Sum(z, (z, n, k))) + 1 == \
+            4*y*Sum(z, (z, n, k)) + 3*Sum(x**3 + x, (x, n, k)) + 1
+    assert _simplify(Sum(x, (x, a, b)) + 1 + Sum(x, (x, b + 1, c))) == \
+        1 + Sum(x, (x, a, c))
+    assert _simplify(Sum(x, (t, a, b)) + Sum(y, (t, a, b)) + \
+        Sum(x, (t, b+1, c))) == x * Sum(1, (t, a, c)) + y * Sum(1, (t, a, b))
+    assert _simplify(Sum(x, (t, a, b)) + Sum(x, (t, b+1, c)) + \
+        Sum(y, (t, a, b))) == x * Sum(1, (t, a, c)) + y * Sum(1, (t, a, b))
+    assert _simplify(Sum(x, (t, a, b)) + 2 * Sum(x, (t, b+1, c))) == \
+        _simplify(Sum(x, (t, a, b)) + Sum(x, (t, b+1, c)) + Sum(x, (t, b+1, c)))
+    assert _simplify(Sum(x, (x, a, b))*Sum(x**2, (x, a, b))) == \
+        Sum(x, (x, a, b)) * Sum(x**2, (x, a, b))
+    assert _simplify(Sum(x, (t, a, b)) + Sum(y, (t, a, b)) + Sum(z, (t, a, b))) \
+        == (x + y + z) * Sum(1, (t, a, b))          # issue 8596
+    assert _simplify(Sum(x, (t, a, b)) + Sum(y, (t, a, b)) + Sum(z, (t, a, b)) + \
+        Sum(v, (t, a, b))) == (x + y + z + v) * Sum(1, (t, a, b))  # issue 8596
+    assert _simplify(Sum(x * y, (x, a, b)) / (3 * y)) == \
+        (Sum(x, (x, a, b)) / 3)
+    assert _simplify(Sum(f(x) * y * z, (x, a, b)) / (y * z)) \
+        == Sum(f(x), (x, a, b))
+    assert _simplify(Sum(c * x, (x, a, b)) - c * Sum(x, (x, a, b))) == 0
+    assert _simplify(c * (Sum(x, (x, a, b))  + y)) == c * (y + Sum(x, (x, a, b)))
+    assert _simplify(c * (Sum(x, (x, a, b)) + y * Sum(x, (x, a, b)))) == \
+        c * (y + 1) * Sum(x, (x, a, b))
+    assert _simplify(Sum(Sum(c * x, (x, a, b)), (y, a, b))) == \
+                c * Sum(x, (x, a, b), (y, a, b))
+    assert _simplify(Sum((3 + y) * Sum(c * x, (x, a, b)), (y, a, b))) == \
+                c * Sum((3 + y), (y, a, b)) * Sum(x, (x, a, b))
+    assert _simplify(Sum((3 + t) * Sum(c * t, (x, a, b)), (y, a, b))) == \
+                c*t*(t + 3)*Sum(1, (x, a, b))*Sum(1, (y, a, b))
+    assert _simplify(Sum(Sum(d * t, (x, a, b - 1)) + \
+                Sum(d * t, (x, b, c)), (t, a, b))) == \
+                    d * Sum(1, (x, a, c)) * Sum(t, (t, a, b))
+    assert _simplify(Sum(sin(t)**2 + cos(t)**2 + 1, (t, a, b))) == \
+        2 * Sum(1, (t, a, b))
+
+
+def test_change_index():
+    b, v, w = symbols('b, v, w', integer = True)
+
+    assert Sum(x, (x, a, b)).change_index(x, x + 1, y) == \
+        Sum(y - 1, (y, a + 1, b + 1))
+    assert Sum(x**2, (x, a, b)).change_index( x, x - 1) == \
+        Sum((x+1)**2, (x, a - 1, b - 1))
+    assert Sum(x**2, (x, a, b)).change_index( x, -x, y) == \
+        Sum((-y)**2, (y, -b, -a))
+    assert Sum(x, (x, a, b)).change_index( x, -x - 1) == \
+        Sum(-x - 1, (x, -b - 1, -a - 1))
+    assert Sum(x*y, (x, a, b), (y, c, d)).change_index( x, x - 1, z) == \
+        Sum((z + 1)*y, (z, a - 1, b - 1), (y, c, d))
+    assert Sum(x, (x, a, b)).change_index( x, x + v) == \
+        Sum(-v + x, (x, a + v, b + v))
+    assert Sum(x, (x, a, b)).change_index( x, -x - v) == \
+        Sum(-v - x, (x, -b - v, -a - v))
+    assert Sum(x, (x, a, b)).change_index(x, w*x, v) == \
+        Sum(v/w, (v, b*w, a*w))
+    raises(ValueError, lambda: Sum(x, (x, a, b)).change_index(x, 2*x))
+
+
+def test_reorder():
+    b, y, c, d, z = symbols('b, y, c, d, z', integer = True)
+
+    assert Sum(x*y, (x, a, b), (y, c, d)).reorder((0, 1)) == \
+        Sum(x*y, (y, c, d), (x, a, b))
+    assert Sum(x, (x, a, b), (x, c, d)).reorder((0, 1)) == \
+        Sum(x, (x, c, d), (x, a, b))
+    assert Sum(x*y + z, (x, a, b), (z, m, n), (y, c, d)).reorder(\
+        (2, 0), (0, 1)) == Sum(x*y + z, (z, m, n), (y, c, d), (x, a, b))
+    assert Sum(x*y*z, (x, a, b), (y, c, d), (z, m, n)).reorder(\
+        (0, 1), (1, 2), (0, 2)) == Sum(x*y*z, (x, a, b), (z, m, n), (y, c, d))
+    assert Sum(x*y*z, (x, a, b), (y, c, d), (z, m, n)).reorder(\
+        (x, y), (y, z), (x, z)) == Sum(x*y*z, (x, a, b), (z, m, n), (y, c, d))
+    assert Sum(x*y, (x, a, b), (y, c, d)).reorder((x, 1)) == \
+        Sum(x*y, (y, c, d), (x, a, b))
+    assert Sum(x*y, (x, a, b), (y, c, d)).reorder((y, x)) == \
+        Sum(x*y, (y, c, d), (x, a, b))
+
+
+def test_reverse_order():
+    assert Sum(x, (x, 0, 3)).reverse_order(0) == Sum(-x, (x, 4, -1))
+    assert Sum(x*y, (x, 1, 5), (y, 0, 6)).reverse_order(0, 1) == \
+           Sum(x*y, (x, 6, 0), (y, 7, -1))
+    assert Sum(x, (x, 1, 2)).reverse_order(0) == Sum(-x, (x, 3, 0))
+    assert Sum(x, (x, 1, 3)).reverse_order(0) == Sum(-x, (x, 4, 0))
+    assert Sum(x, (x, 1, a)).reverse_order(0) == Sum(-x, (x, a + 1, 0))
+    assert Sum(x, (x, a, 5)).reverse_order(0) == Sum(-x, (x, 6, a - 1))
+    assert Sum(x, (x, a + 1, a + 5)).reverse_order(0) == \
+                         Sum(-x, (x, a + 6, a))
+    assert Sum(x, (x, a + 1, a + 2)).reverse_order(0) == \
+           Sum(-x, (x, a + 3, a))
+    assert Sum(x, (x, a + 1, a + 1)).reverse_order(0) == \
+           Sum(-x, (x, a + 2, a))
+    assert Sum(x, (x, a, b)).reverse_order(0) == Sum(-x, (x, b + 1, a - 1))
+    assert Sum(x, (x, a, b)).reverse_order(x) == Sum(-x, (x, b + 1, a - 1))
+    assert Sum(x*y, (x, a, b), (y, 2, 5)).reverse_order(x, 1) == \
+        Sum(x*y, (x, b + 1, a - 1), (y, 6, 1))
+    assert Sum(x*y, (x, a, b), (y, 2, 5)).reverse_order(y, x) == \
+        Sum(x*y, (x, b + 1, a - 1), (y, 6, 1))
+
+
+def test_issue_7097():
+    assert sum(x**n/n for n in range(1, 401)) == summation(x**n/n, (n, 1, 400))
+
+
+def test_factor_expand_subs():
+    # test factoring
+    assert Sum(4 * x, (x, 1, y)).factor() == 4 * Sum(x, (x, 1, y))
+    assert Sum(x * a, (x, 1, y)).factor() == a * Sum(x, (x, 1, y))
+    assert Sum(4 * x * a, (x, 1, y)).factor() == 4 * a * Sum(x, (x, 1, y))
+    assert Sum(4 * x * y, (x, 1, y)).factor() == 4 * y * Sum(x, (x, 1, y))
+
+    # test expand
+    _x = Symbol('x', zero=False)
+    assert Sum(x+1,(x,1,y)).expand() == Sum(x,(x,1,y)) + Sum(1,(x,1,y))
+    assert Sum(x+a*x**2,(x,1,y)).expand() == Sum(x,(x,1,y)) + Sum(a*x**2,(x,1,y))
+    assert Sum(_x**(n + 1)*(n + 1), (n, -1, oo)).expand() \
+        == Sum(n*_x*_x**n + _x*_x**n, (n, -1, oo))
+    assert Sum(x**(n + 1)*(n + 1), (n, -1, oo)).expand(power_exp=False) \
+        == Sum(n*x**(n + 1) + x**(n + 1), (n, -1, oo))
+    assert Sum(x**(n + 1)*(n + 1), (n, -1, oo)).expand(force=True) \
+           == Sum(x*x**n, (n, -1, oo)) + Sum(n*x*x**n, (n, -1, oo))
+    assert Sum(a*n+a*n**2,(n,0,4)).expand() \
+        == Sum(a*n,(n,0,4)) + Sum(a*n**2,(n,0,4))
+    assert Sum(_x**a*_x**n,(x,0,3)) \
+        == Sum(_x**(a+n),(x,0,3)).expand(power_exp=True)
+    _a, _n = symbols('a n', positive=True)
+    assert Sum(x**(_a+_n),(x,0,3)).expand(power_exp=True) \
+        == Sum(x**_a*x**_n, (x, 0, 3))
+    assert Sum(x**(_a-_n),(x,0,3)).expand(power_exp=True) \
+        == Sum(x**(_a-_n),(x,0,3)).expand(power_exp=False)
+
+    # test subs
+    assert Sum(1/(1+a*x**2),(x,0,3)).subs([(a,3)]) == Sum(1/(1+3*x**2),(x,0,3))
+    assert Sum(x*y,(x,0,y),(y,0,x)).subs([(x,3)]) == Sum(x*y,(x,0,y),(y,0,3))
+    assert Sum(x,(x,1,10)).subs([(x,y-2)]) == Sum(x,(x,1,10))
+    assert Sum(1/x,(x,1,10)).subs([(x,(3+n)**3)]) == Sum(1/x,(x,1,10))
+    assert Sum(1/x,(x,1,10)).subs([(x,3*x-2)]) == Sum(1/x,(x,1,10))
+
+
+def test_distribution_over_equality():
+    assert Product(Eq(x*2, f(x)), (x, 1, 3)).doit() == Eq(48, f(1)*f(2)*f(3))
+    assert Sum(Eq(f(x), x**2), (x, 0, y)) == \
+        Eq(Sum(f(x), (x, 0, y)), Sum(x**2, (x, 0, y)))
+
+
+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()
+    ans = Piecewise(
+        (n*p, x),
+        (Sum(k*p**k*binomial(n, k)*(1 - p)**(n - k), (k, 0, n)),
+        True)).subs(x, (Eq(n, 1) | (n > 1)) & (p/Abs(p - 1) <= 1))
+    ans2 = Piecewise(
+        (n*p, x),
+        (factorial(n)*Sum(p**k*(1 - p)**(-k + n)/
+        (factorial(-k + n)*factorial(k - 1)), (k, 0, n)),
+        True)).subs(x, (Eq(n, 1) | (n > 1)) & (p/Abs(p - 1) <= 1))
+    assert res in [ans, ans2]  # XXX system dependent
+    # Issue #17165: make sure that another simplify does not complicate
+    # the result by much. Why didn't first simplify replace
+    # Eq(n, 1) | (n > 1) with True?
+    assert res.simplify().count_ops() <= res.count_ops() + 2
+
+
+def test_issue_4668():
+    assert summation(1/n, (n, 2, oo)) is oo
+
+
+def test_matrix_sum():
+    A = Matrix([[0, 1], [n, 0]])
+
+    result = Sum(A, (n, 0, 3)).doit()
+    assert result == Matrix([[0, 4], [6, 0]])
+    assert result.__class__ == ImmutableDenseMatrix
+
+    A = SparseMatrix([[0, 1], [n, 0]])
+
+    result = Sum(A, (n, 0, 3)).doit()
+    assert result.__class__ == ImmutableSparseMatrix
+
+
+def test_failing_matrix_sum():
+    n = Symbol('n')
+    # TODO Implement matrix geometric series summation.
+    A = Matrix([[0, 1, 0], [-1, 0, 0], [0, 0, 0]])
+    assert Sum(A ** n, (n, 1, 4)).doit() == \
+        Matrix([[0, 0, 0], [0, 0, 0], [0, 0, 0]])
+    # issue sympy/sympy#16989
+    assert summation(A**n, (n, 1, 1)) == A
+
+
+def test_indexed_idx_sum():
+    i = symbols('i', cls=Idx)
+    r = Indexed('r', i)
+    assert Sum(r, (i, 0, 3)).doit() == sum([r.xreplace({i: j}) for j in range(4)])
+    assert Product(r, (i, 0, 3)).doit() == prod([r.xreplace({i: j}) for j in range(4)])
+
+    j = symbols('j', integer=True)
+    assert Sum(r, (i, j, j+2)).doit() == sum([r.xreplace({i: j+k}) for k in range(3)])
+    assert Product(r, (i, j, j+2)).doit() == prod([r.xreplace({i: j+k}) for k in range(3)])
+
+    k = Idx('k', range=(1, 3))
+    A = IndexedBase('A')
+    assert Sum(A[k], k).doit() == sum([A[Idx(j, (1, 3))] for j in range(1, 4)])
+    assert Product(A[k], k).doit() == prod([A[Idx(j, (1, 3))] for j in range(1, 4)])
+
+    raises(ValueError, lambda: Sum(A[k], (k, 1, 4)))
+    raises(ValueError, lambda: Sum(A[k], (k, 0, 3)))
+    raises(ValueError, lambda: Sum(A[k], (k, 2, oo)))
+
+    raises(ValueError, lambda: Product(A[k], (k, 1, 4)))
+    raises(ValueError, lambda: Product(A[k], (k, 0, 3)))
+    raises(ValueError, lambda: Product(A[k], (k, 2, oo)))
+
+
+@slow
+def test_is_convergent():
+    # divergence tests --
+    assert Sum(n/(2*n + 1), (n, 1, oo)).is_convergent() is S.false
+    assert Sum(factorial(n)/5**n, (n, 1, oo)).is_convergent() is S.false
+    assert Sum(3**(-2*n - 1)*n**n, (n, 1, oo)).is_convergent() is S.false
+    assert Sum((-1)**n*n, (n, 3, oo)).is_convergent() is S.false
+    assert Sum((-1)**n, (n, 1, oo)).is_convergent() is S.false
+    assert Sum(log(1/n), (n, 2, oo)).is_convergent() is S.false
+
+    # Raabe's test --
+    assert Sum(Product((3*m),(m,1,n))/Product((3*m+4),(m,1,n)),(n,1,oo)).is_convergent() is S.true
+
+    # root test --
+    assert Sum((-12)**n/n, (n, 1, oo)).is_convergent() is S.false
+
+    # integral test --
+
+    # p-series test --
+    assert Sum(1/(n**2 + 1), (n, 1, oo)).is_convergent() is S.true
+    assert Sum(1/n**Rational(6, 5), (n, 1, oo)).is_convergent() is S.true
+    assert Sum(2/(n*sqrt(n - 1)), (n, 2, oo)).is_convergent() is S.true
+    assert Sum(1/(sqrt(n)*sqrt(n)), (n, 2, oo)).is_convergent() is S.false
+    assert Sum(factorial(n) / factorial(n+2), (n, 1, oo)).is_convergent() is S.true
+    assert Sum(rf(5,n)/rf(7,n),(n,1,oo)).is_convergent() is S.true
+    assert Sum((rf(1, n)*rf(2, n))/(rf(3, n)*factorial(n)),(n,1,oo)).is_convergent() is S.false
+
+    # comparison test --
+    assert Sum(1/(n + log(n)), (n, 1, oo)).is_convergent() is S.false
+    assert Sum(1/(n**2*log(n)), (n, 2, oo)).is_convergent() is S.true
+    assert Sum(1/(n*log(n)), (n, 2, oo)).is_convergent() is S.false
+    assert Sum(2/(n*log(n)*log(log(n))**2), (n, 5, oo)).is_convergent() is S.true
+    assert Sum(2/(n*log(n)**2), (n, 2, oo)).is_convergent() is S.true
+    assert Sum((n - 1)/(n**2*log(n)**3), (n, 2, oo)).is_convergent() is S.true
+    assert Sum(1/(n*log(n)*log(log(n))), (n, 5, oo)).is_convergent() is S.false
+    assert Sum((n - 1)/(n*log(n)**3), (n, 3, oo)).is_convergent() is S.false
+    assert Sum(2/(n**2*log(n)), (n, 2, oo)).is_convergent() is S.true
+    assert Sum(1/(n*sqrt(log(n))*log(log(n))), (n, 100, oo)).is_convergent() is S.false
+    assert Sum(log(log(n))/(n*log(n)**2), (n, 100, oo)).is_convergent() is S.true
+    assert Sum(log(n)/n**2, (n, 5, oo)).is_convergent() is S.true
+
+    # alternating series tests --
+    assert Sum((-1)**(n - 1)/(n**2 - 1), (n, 3, oo)).is_convergent() is S.true
+
+    # with -negativeInfinite Limits
+    assert Sum(1/(n**2 + 1), (n, -oo, 1)).is_convergent() is S.true
+    assert Sum(1/(n - 1), (n, -oo, -1)).is_convergent() is S.false
+    assert Sum(1/(n**2 - 1), (n, -oo, -5)).is_convergent() is S.true
+    assert Sum(1/(n**2 - 1), (n, -oo, 2)).is_convergent() is S.true
+    assert Sum(1/(n**2 - 1), (n, -oo, oo)).is_convergent() is S.true
+
+    # piecewise functions
+    f = Piecewise((n**(-2), n <= 1), (n**2, n > 1))
+    assert Sum(f, (n, 1, oo)).is_convergent() is S.false
+    assert Sum(f, (n, -oo, oo)).is_convergent() is S.false
+    assert Sum(f, (n, 1, 100)).is_convergent() is S.true
+    #assert Sum(f, (n, -oo, 1)).is_convergent() is S.true
+
+    # integral test
+
+    assert Sum(log(n)/n**3, (n, 1, oo)).is_convergent() is S.true
+    assert Sum(-log(n)/n**3, (n, 1, oo)).is_convergent() is S.true
+    # the following function has maxima located at (x, y) =
+    # (1.2, 0.43), (3.0, -0.25) and (6.8, 0.050)
+    eq = (x - 2)*(x**2 - 6*x + 4)*exp(-x)
+    assert Sum(eq, (x, 1, oo)).is_convergent() is S.true
+    assert Sum(eq, (x, 1, 2)).is_convergent() is S.true
+    assert Sum(1/(x**3), (x, 1, oo)).is_convergent() is S.true
+    assert Sum(1/(x**S.Half), (x, 1, oo)).is_convergent() is S.false
+
+    # issue 19545
+    assert Sum(1/n - 3/(3*n +2), (n, 1, oo)).is_convergent() is S.true
+
+    # issue 19836
+    assert Sum(4/(n + 2) - 5/(n + 1) + 1/n,(n, 7, oo)).is_convergent() is S.true
+
+
+def test_is_absolutely_convergent():
+    assert Sum((-1)**n, (n, 1, oo)).is_absolutely_convergent() is S.false
+    assert Sum((-1)**n/n**2, (n, 1, oo)).is_absolutely_convergent() is S.true
+
+
+@XFAIL
+def test_convergent_failing():
+    # dirichlet tests
+    assert Sum(sin(n)/n, (n, 1, oo)).is_convergent() is S.true
+    assert Sum(sin(2*n)/n, (n, 1, oo)).is_convergent() is S.true
+
+
+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_issue_10156():
+    cx = Sum(2*y**2*x, (x, 1,3))
+    e = 2*y*Sum(2*cx*x**2, (x, 1, 9))
+    assert e.factor() == \
+        8*y**3*Sum(x, (x, 1, 3))*Sum(x**2, (x, 1, 9))
+
+
+def test_issue_10973():
+    assert Sum((-n + (n**3 + 1)**(S(1)/3))/log(n), (n, 1, oo)).is_convergent() is S.true
+
+
+def test_issue_14129():
+    x = Symbol('x', zero=False)
+    assert Sum( k*x**k, (k, 0, n-1)).doit() == \
+        Piecewise((n**2/2 - n/2, Eq(x, 1)), ((n*x*x**n -
+            n*x**n - x*x**n + x)/(x - 1)**2, True))
+    assert Sum( x**k, (k, 0, n-1)).doit() == \
+        Piecewise((n, Eq(x, 1)), ((-x**n + 1)/(-x + 1), True))
+    assert Sum( k*(x/y+x)**k, (k, 0, n-1)).doit() == \
+        Piecewise((n*(n - 1)/2, Eq(x, y/(y + 1))),
+        (x*(y + 1)*(n*x*y*(x + x/y)**(n - 1) +
+        n*x*(x + x/y)**(n - 1) - n*y*(x + x/y)**(n - 1) -
+        x*y*(x + x/y)**(n - 1) - x*(x + x/y)**(n - 1) + y)/
+        (x*y + x - y)**2, True))
+
+
+def test_issue_14112():
+    assert Sum((-1)**n/sqrt(n), (n, 1, oo)).is_absolutely_convergent() is S.false
+    assert Sum((-1)**(2*n)/n, (n, 1, oo)).is_convergent() is S.false
+    assert Sum((-2)**n + (-3)**n, (n, 1, oo)).is_convergent() is S.false
+
+
+def test_issue_14219():
+    A = diag(0, 2, -3)
+    res = diag(1, 15, -20)
+    assert Sum(A**n, (n, 0, 3)).doit() == res
+
+
+def test_sin_times_absolutely_convergent():
+    assert Sum(sin(n) / n**3, (n, 1, oo)).is_convergent() is S.true
+    assert Sum(sin(n) * log(n) / n**3, (n, 1, oo)).is_convergent() is S.true
+
+
+def test_issue_14111():
+    assert Sum(1/log(log(n)), (n, 22, oo)).is_convergent() is S.false
+
+
+def test_issue_14484():
+    assert Sum(sin(n)/log(log(n)), (n, 22, oo)).is_convergent() is S.false
+
+
+def test_issue_14640():
+    i, n = symbols("i n", integer=True)
+    a, b, c = symbols("a b c", zero=False)
+
+    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_issue_15943():
+    s = Sum(binomial(n, k)*factorial(n - k), (k, 0, n)).doit().rewrite(gamma)
+    assert s == -E*(n + 1)*gamma(n + 1)*lowergamma(n + 1, 1)/gamma(n + 2
+        ) + E*gamma(n + 1)
+    assert s.simplify() == E*(factorial(n) - lowergamma(n + 1, 1))
+
+
+def test_Sum_dummy_eq():
+    assert not Sum(x, (x, a, b)).dummy_eq(1)
+    assert not Sum(x, (x, a, b)).dummy_eq(Sum(x, (x, a, b), (a, 1, 2)))
+    assert not Sum(x, (x, a, b)).dummy_eq(Sum(x, (x, a, c)))
+    assert Sum(x, (x, a, b)).dummy_eq(Sum(x, (x, a, b)))
+    d = Dummy()
+    assert Sum(x, (x, a, d)).dummy_eq(Sum(x, (x, a, c)), c)
+    assert not Sum(x, (x, a, d)).dummy_eq(Sum(x, (x, a, c)))
+    assert Sum(x, (x, a, c)).dummy_eq(Sum(y, (y, a, c)))
+    assert Sum(x, (x, a, d)).dummy_eq(Sum(y, (y, a, c)), c)
+    assert not Sum(x, (x, a, d)).dummy_eq(Sum(y, (y, a, c)))
+
+
+def test_issue_15852():
+    assert summation(x**y*y, (y, -oo, oo)).doit() == Sum(x**y*y, (y, -oo, oo))
+
+
+def test_exceptions():
+    S = Sum(x, (x, a, b))
+    raises(ValueError, lambda: S.change_index(x, x**2, y))
+    S = Sum(x, (x, a, b), (x, 1, 4))
+    raises(ValueError, lambda: S.index(x))
+    S = Sum(x, (x, a, b), (y, 1, 4))
+    raises(ValueError, lambda: S.reorder([x]))
+    S = Sum(x, (x, y, b), (y, 1, 4))
+    raises(ReorderError, lambda: S.reorder_limit(0, 1))
+    S = Sum(x*y, (x, a, b), (y, 1, 4))
+    raises(NotImplementedError, lambda: S.is_convergent())
+
+
+def test_sumproducts_assumptions():
+    M = Symbol('M', integer=True, positive=True)
+
+    m = Symbol('m', integer=True)
+    for func in [Sum, Product]:
+        assert func(m, (m, -M, M)).is_positive is None
+        assert func(m, (m, -M, M)).is_nonpositive is None
+        assert func(m, (m, -M, M)).is_negative is None
+        assert func(m, (m, -M, M)).is_nonnegative is None
+        assert func(m, (m, -M, M)).is_finite is True
+
+    m = Symbol('m', integer=True, nonnegative=True)
+    for func in [Sum, Product]:
+        assert func(m, (m, 0, M)).is_positive is None
+        assert func(m, (m, 0, M)).is_nonpositive is None
+        assert func(m, (m, 0, M)).is_negative is False
+        assert func(m, (m, 0, M)).is_nonnegative is True
+        assert func(m, (m, 0, M)).is_finite is True
+
+    m = Symbol('m', integer=True, positive=True)
+    for func in [Sum, Product]:
+        assert func(m, (m, 1, M)).is_positive is True
+        assert func(m, (m, 1, M)).is_nonpositive is False
+        assert func(m, (m, 1, M)).is_negative is False
+        assert func(m, (m, 1, M)).is_nonnegative is True
+        assert func(m, (m, 1, M)).is_finite is True
+
+    m = Symbol('m', integer=True, negative=True)
+    assert Sum(m, (m, -M, -1)).is_positive is False
+    assert Sum(m, (m, -M, -1)).is_nonpositive is True
+    assert Sum(m, (m, -M, -1)).is_negative is True
+    assert Sum(m, (m, -M, -1)).is_nonnegative is False
+    assert Sum(m, (m, -M, -1)).is_finite is True
+    assert Product(m, (m, -M, -1)).is_positive is None
+    assert Product(m, (m, -M, -1)).is_nonpositive is None
+    assert Product(m, (m, -M, -1)).is_negative is None
+    assert Product(m, (m, -M, -1)).is_nonnegative is None
+    assert Product(m, (m, -M, -1)).is_finite is True
+
+    m = Symbol('m', integer=True, nonpositive=True)
+    assert Sum(m, (m, -M, 0)).is_positive is False
+    assert Sum(m, (m, -M, 0)).is_nonpositive is True
+    assert Sum(m, (m, -M, 0)).is_negative is None
+    assert Sum(m, (m, -M, 0)).is_nonnegative is None
+    assert Sum(m, (m, -M, 0)).is_finite is True
+    assert Product(m, (m, -M, 0)).is_positive is None
+    assert Product(m, (m, -M, 0)).is_nonpositive is None
+    assert Product(m, (m, -M, 0)).is_negative is None
+    assert Product(m, (m, -M, 0)).is_nonnegative is None
+    assert Product(m, (m, -M, 0)).is_finite is True
+
+    m = Symbol('m', integer=True)
+    assert Sum(2, (m, 0, oo)).is_positive is None
+    assert Sum(2, (m, 0, oo)).is_nonpositive is None
+    assert Sum(2, (m, 0, oo)).is_negative is None
+    assert Sum(2, (m, 0, oo)).is_nonnegative is None
+    assert Sum(2, (m, 0, oo)).is_finite is None
+
+    assert Product(2, (m, 0, oo)).is_positive is None
+    assert Product(2, (m, 0, oo)).is_nonpositive is None
+    assert Product(2, (m, 0, oo)).is_negative is False
+    assert Product(2, (m, 0, oo)).is_nonnegative is None
+    assert Product(2, (m, 0, oo)).is_finite is None
+
+    assert Product(0, (x, M, M-1)).is_positive is True
+    assert Product(0, (x, M, M-1)).is_finite is True
+
+
+def test_expand_with_assumptions():
+    M = Symbol('M', integer=True, positive=True)
+    x = Symbol('x', positive=True)
+    m = Symbol('m', nonnegative=True)
+    assert log(Product(x**m, (m, 0, M))).expand() == Sum(m*log(x), (m, 0, M))
+    assert log(Product(exp(x**m), (m, 0, M))).expand() == Sum(x**m, (m, 0, M))
+    assert log(Product(x**m, (m, 0, M))).rewrite(Sum).expand() == Sum(m*log(x), (m, 0, M))
+    assert log(Product(exp(x**m), (m, 0, M))).rewrite(Sum).expand() == Sum(x**m, (m, 0, M))
+
+    n = Symbol('n', nonnegative=True)
+    i, j = symbols('i,j', positive=True, integer=True)
+    x, y = symbols('x,y', positive=True)
+    assert log(Product(x**i*y**j, (i, 1, n), (j, 1, m))).expand() \
+        == Sum(i*log(x) + j*log(y), (i, 1, n), (j, 1, m))
+
+    m = Symbol('m', nonnegative=True, integer=True)
+    s = Sum(x**m, (m, 0, M))
+    s_as_product = s.rewrite(Product)
+    assert s_as_product.has(Product)
+    assert s_as_product == log(Product(exp(x**m), (m, 0, M)))
+    assert s_as_product.expand() == s
+    s5 = s.subs(M, 5)
+    s5_as_product = s5.rewrite(Product)
+    assert s5_as_product.has(Product)
+    assert s5_as_product.doit().expand() == s5.doit()
+
+
+def test_has_finite_limits():
+    x = Symbol('x')
+    assert Sum(1, (x, 1, 9)).has_finite_limits is True
+    assert Sum(1, (x, 1, oo)).has_finite_limits is False
+    M = Symbol('M')
+    assert Sum(1, (x, 1, M)).has_finite_limits is None
+    M = Symbol('M', positive=True)
+    assert Sum(1, (x, 1, M)).has_finite_limits is True
+    x = Symbol('x', positive=True)
+    M = Symbol('M')
+    assert Sum(1, (x, 1, M)).has_finite_limits is True
+
+    assert Sum(1, (x, 1, M), (y, -oo, oo)).has_finite_limits is False
+
+def test_has_reversed_limits():
+    assert Sum(1, (x, 1, 1)).has_reversed_limits is False
+    assert Sum(1, (x, 1, 9)).has_reversed_limits is False
+    assert Sum(1, (x, 1, -9)).has_reversed_limits is True
+    assert Sum(1, (x, 1, 0)).has_reversed_limits is True
+    assert Sum(1, (x, 1, oo)).has_reversed_limits is False
+    M = Symbol('M')
+    assert Sum(1, (x, 1, M)).has_reversed_limits is None
+    M = Symbol('M', positive=True, integer=True)
+    assert Sum(1, (x, 1, M)).has_reversed_limits is False
+    assert Sum(1, (x, 1, M), (y, -oo, oo)).has_reversed_limits is False
+    M = Symbol('M', negative=True)
+    assert Sum(1, (x, 1, M)).has_reversed_limits is True
+
+    assert Sum(1, (x, 1, M), (y, -oo, oo)).has_reversed_limits is True
+    assert Sum(1, (x, oo, oo)).has_reversed_limits is None
+
+
+def test_has_empty_sequence():
+    assert Sum(1, (x, 1, 1)).has_empty_sequence is False
+    assert Sum(1, (x, 1, 9)).has_empty_sequence is False
+    assert Sum(1, (x, 1, -9)).has_empty_sequence is False
+    assert Sum(1, (x, 1, 0)).has_empty_sequence is True
+    assert Sum(1, (x, y, y - 1)).has_empty_sequence is True
+    assert Sum(1, (x, 3, 2), (y, -oo, oo)).has_empty_sequence is True
+    assert Sum(1, (y, -oo, oo), (x, 3, 2)).has_empty_sequence is True
+    assert Sum(1, (x, oo, oo)).has_empty_sequence is False
+
+
+def test_empty_sequence():
+    assert Product(x*y, (x, -oo, oo), (y, 1, 0)).doit() == 1
+    assert Product(x*y, (y, 1, 0), (x, -oo, oo)).doit() == 1
+    assert Sum(x, (x, -oo, oo), (y, 1, 0)).doit() == 0
+    assert Sum(x, (y, 1, 0), (x, -oo, oo)).doit() == 0
+
+
+def test_issue_8016():
+    k = Symbol('k', integer=True)
+    n, m = symbols('n, m', integer=True, positive=True)
+    s = Sum(binomial(m, k)*binomial(m, n - k)*(-1)**k, (k, 0, n))
+    assert s.doit().simplify() == \
+        cos(pi*n/2)*gamma(m + 1)/gamma(n/2 + 1)/gamma(m - n/2 + 1)
+
+
+def test_issue_14313():
+    assert Sum(S.Half**floor(n/2), (n, 1, oo)).is_convergent()
+
+
+def test_issue_14563():
+    # The assertion was failing due to no assumptions methods in Sums and Product
+    assert 1 % Sum(1, (x, 0, 1)) == 1
+
+
+def test_issue_16735():
+    assert Sum(5**n/gamma(n+1), (n, 1, oo)).is_convergent() is S.true
+
+
+def test_issue_14871():
+    assert Sum((Rational(1, 10))**n*rf(0, n)/factorial(n), (n, 0, oo)).rewrite(factorial).doit() == 1
+
+
+def test_issue_17165():
+    n = symbols("n", integer=True)
+    x = symbols('x')
+    s = (x*Sum(x**n, (n, -1, oo)))
+    ssimp = s.doit().simplify()
+
+    assert ssimp == Piecewise((-1/(x - 1), (x > -1) & (x < 1)),
+                              (x*Sum(x**n, (n, -1, oo)), True)), ssimp
+    assert ssimp.simplify() == ssimp
+
+
+def test_issue_19379():
+    assert Sum(factorial(n)/factorial(n + 2), (n, 1, oo)).is_convergent() is S.true
+
+
+def test_issue_20777():
+    assert Sum(exp(x*sin(n/m)), (n, 1, m)).doit() == Sum(exp(x*sin(n/m)), (n, 1, m))
+
+
+def test__dummy_with_inherited_properties_concrete():
+    x = Symbol('x')
+
+    from sympy.core.containers import Tuple
+    d = _dummy_with_inherited_properties_concrete(Tuple(x, 0, 5))
+    assert d.is_real
+    assert d.is_integer
+    assert d.is_nonnegative
+    assert d.is_extended_nonnegative
+
+    d = _dummy_with_inherited_properties_concrete(Tuple(x, 1, 9))
+    assert d.is_real
+    assert d.is_integer
+    assert d.is_positive
+    assert d.is_odd is None
+
+    d = _dummy_with_inherited_properties_concrete(Tuple(x, -5, 5))
+    assert d.is_real
+    assert d.is_integer
+    assert d.is_positive is None
+    assert d.is_extended_nonnegative is None
+    assert d.is_odd is None
+
+    d = _dummy_with_inherited_properties_concrete(Tuple(x, -1.5, 1.5))
+    assert d.is_real
+    assert d.is_integer is None
+    assert d.is_positive is None
+    assert d.is_extended_nonnegative is None
+
+    N = Symbol('N', integer=True, positive=True)
+    d = _dummy_with_inherited_properties_concrete(Tuple(x, 2, N))
+    assert d.is_real
+    assert d.is_positive
+    assert d.is_integer
+
+    # Return None if no assumptions are added
+    N = Symbol('N', integer=True, positive=True)
+    d = _dummy_with_inherited_properties_concrete(Tuple(N, 2, 4))
+    assert d is None
+
+    x = Symbol('x', negative=True)
+    raises(InconsistentAssumptions,
+           lambda: _dummy_with_inherited_properties_concrete(Tuple(x, 1, 5)))
+
+
+def test_matrixsymbol_summation_numerical_limits():
+    A = MatrixSymbol('A', 3, 3)
+    n = Symbol('n', integer=True)
+
+    assert Sum(A**n, (n, 0, 2)).doit() == Identity(3) + A + A**2
+    assert Sum(A, (n, 0, 2)).doit() == 3*A
+    assert Sum(n*A, (n, 0, 2)).doit() == 3*A
+
+    B = Matrix([[0, n, 0], [-1, 0, 0], [0, 0, 2]])
+    ans = Matrix([[0, 6, 0], [-4, 0, 0], [0, 0, 8]]) + 4*A
+    assert Sum(A+B, (n, 0, 3)).doit() == ans
+    ans = A*Matrix([[0, 6, 0], [-4, 0, 0], [0, 0, 8]])
+    assert Sum(A*B, (n, 0, 3)).doit() == ans
+
+    ans = (A**2*Matrix([[-2, 0, 0], [0,-2, 0], [0, 0, 4]]) +
+           A**3*Matrix([[0, -9, 0], [3, 0, 0], [0, 0, 8]]) +
+           A*Matrix([[0, 1, 0], [-1, 0, 0], [0, 0, 2]]))
+    assert Sum(A**n*B**n, (n, 1, 3)).doit() == ans
+
+
+def test_issue_21651():
+    i = Symbol('i')
+    a = Sum(floor(2*2**(-i)), (i, S.One, 2))
+    assert a.doit() == S.One
+
+
+@XFAIL
+def test_matrixsymbol_summation_symbolic_limits():
+    N = Symbol('N', integer=True, positive=True)
+
+    A = MatrixSymbol('A', 3, 3)
+    n = Symbol('n', integer=True)
+    assert Sum(A, (n, 0, N)).doit() == (N+1)*A
+    assert Sum(n*A, (n, 0, N)).doit() == (N**2/2+N/2)*A
+
+
+def test_summation_by_residues():
+    x = Symbol('x')
+
+    # Examples from Nakhle H. Asmar, Loukas Grafakos,
+    # Complex Analysis with Applications
+    assert eval_sum_residue(1 / (x**2 + 1), (x, -oo, oo)) == pi/tanh(pi)
+    assert eval_sum_residue(1 / x**6, (x, S(1), oo)) == pi**6/945
+    assert eval_sum_residue(1 / (x**2 + 9), (x, -oo, oo)) == pi/(3*tanh(3*pi))
+    assert eval_sum_residue(1 / (x**2 + 1)**2, (x, -oo, oo)).cancel() == \
+        (-pi**2*tanh(pi)**2 + pi*tanh(pi) + pi**2)/(2*tanh(pi)**2)
+    assert eval_sum_residue(x**2 / (x**2 + 1)**2, (x, -oo, oo)).cancel() == \
+        (-pi**2 + pi*tanh(pi) + pi**2*tanh(pi)**2)/(2*tanh(pi)**2)
+    assert eval_sum_residue(1 / (4*x**2 - 1), (x, -oo, oo)) == 0
+    assert eval_sum_residue(x**2 / (x**2 - S(1)/4)**2, (x, -oo, oo)) == pi**2/2
+    assert eval_sum_residue(1 / (4*x**2 - 1)**2, (x, -oo, oo)) == pi**2/8
+    assert eval_sum_residue(1 / ((x - S(1)/2)**2 + 1), (x, -oo, oo)) == pi*tanh(pi)
+    assert eval_sum_residue(1 / x**2, (x, S(1), oo)) == pi**2/6
+    assert eval_sum_residue(1 / x**4, (x, S(1), oo)) == pi**4/90
+    assert eval_sum_residue(1 / x**2 / (x**2 + 4), (x, S(1), oo)) == \
+        -pi*(-pi/12 - 1/(16*pi) + 1/(8*tanh(2*pi)))/2
+
+    # Some examples made from 1 / (x**2 + 1)
+    assert eval_sum_residue(1 / (x**2 + 1), (x, S(0), oo)) == \
+        S(1)/2 + pi/(2*tanh(pi))
+    assert eval_sum_residue(1 / (x**2 + 1), (x, S(1), oo)) == \
+        -S(1)/2 + pi/(2*tanh(pi))
+    assert eval_sum_residue(1 / (x**2 + 1), (x, S(-1), oo)) == \
+        1 + pi/(2*tanh(pi))
+    assert eval_sum_residue((-1)**x / (x**2 + 1), (x, -oo, oo)) == \
+        pi/sinh(pi)
+    assert eval_sum_residue((-1)**x / (x**2 + 1), (x, S(0), oo)) == \
+        pi/(2*sinh(pi)) + S(1)/2
+    assert eval_sum_residue((-1)**x / (x**2 + 1), (x, S(1), oo)) == \
+        -S(1)/2 + pi/(2*sinh(pi))
+    assert eval_sum_residue((-1)**x / (x**2 + 1), (x, S(-1), oo)) == \
+        pi/(2*sinh(pi))
+
+    # Some examples made from shifting of 1 / (x**2 + 1)
+    assert eval_sum_residue(1 / (x**2 + 2*x + 2), (x, S(-1), oo)) == S(1)/2 + pi/(2*tanh(pi))
+    assert eval_sum_residue(1 / (x**2 + 4*x + 5), (x, S(-2), oo)) == S(1)/2 + pi/(2*tanh(pi))
+    assert eval_sum_residue(1 / (x**2 - 2*x + 2), (x, S(1), oo)) == S(1)/2 + pi/(2*tanh(pi))
+    assert eval_sum_residue(1 / (x**2 - 4*x + 5), (x, S(2), oo)) == S(1)/2 + pi/(2*tanh(pi))
+    assert eval_sum_residue((-1)**x * -1 / (x**2 + 2*x + 2), (x, S(-1), oo)) ==  S(1)/2 + pi/(2*sinh(pi))
+    assert eval_sum_residue((-1)**x * -1 / (x**2 -2*x + 2), (x, S(1), oo)) == S(1)/2 + pi/(2*sinh(pi))
+
+    # Some examples made from 1 / x**2
+    assert eval_sum_residue(1 / x**2, (x, S(2), oo)) == -1 + pi**2/6
+    assert eval_sum_residue(1 / x**2, (x, S(3), oo)) == -S(5)/4 + pi**2/6
+    assert eval_sum_residue((-1)**x / x**2, (x, S(1), oo)) == -pi**2/12
+    assert eval_sum_residue((-1)**x / x**2, (x, S(2), oo)) == 1 - pi**2/12
+
+
+@slow
+def test_summation_by_residues_failing():
+    x = Symbol('x')
+
+    # Failing because of the bug in residue computation
+    assert eval_sum_residue(x**2 / (x**4 + 1), (x, S(1), oo))
+    assert eval_sum_residue(1 / ((x - 1)*(x - 2) + 1), (x, -oo, oo)) != 0
+
+
+def test_process_limits():
+    from sympy.concrete.expr_with_limits import _process_limits
+
+    # these should be (x, Range(3)) not Range(3)
+    raises(ValueError, lambda: _process_limits(
+        Range(3), discrete=True))
+    raises(ValueError, lambda: _process_limits(
+        Range(3), discrete=False))
+    # these should be (x, union) not union
+    # (but then we would get a TypeError because we don't
+    # handle non-contiguous sets: see below use of `union`)
+    union = Or(x < 1, x > 3).as_set()
+    raises(ValueError, lambda: _process_limits(
+        union, discrete=True))
+    raises(ValueError, lambda: _process_limits(
+        union, discrete=False))
+
+    # error not triggered if not needed
+    assert _process_limits((x, 1, 2)) == ([(x, 1, 2)], 1)
+
+    # this equivalence is used to detect Reals in _process_limits
+    assert isinstance(S.Reals, Interval)
+
+    C = Integral  # continuous limits
+    assert C(x, x >= 5) == C(x, (x, 5, oo))
+    assert C(x, x < 3) == C(x, (x, -oo, 3))
+    ans = C(x, (x, 0, 3))
+    assert C(x, And(x >= 0, x < 3)) == ans
+    assert C(x, (x, Interval.Ropen(0, 3))) == ans
+    raises(TypeError, lambda: C(x, (x, Range(3))))
+
+    # discrete limits
+    for D in (Sum, Product):
+        r, ans = Range(3, 10, 2), D(2*x + 3, (x, 0, 3))
+        assert D(x, (x, r)) == ans
+        assert D(x, (x, r.reversed)) == ans
+        r, ans = Range(3, oo, 2), D(2*x + 3, (x, 0, oo))
+        assert D(x, (x, r)) == ans
+        assert D(x, (x, r.reversed)) == ans
+        r, ans = Range(-oo, 5, 2), D(3 - 2*x, (x, 0, oo))
+        assert D(x, (x, r)) == ans
+        assert D(x, (x, r.reversed)) == ans
+        raises(TypeError, lambda: D(x, x > 0))
+        raises(ValueError, lambda: D(x, Interval(1, 3)))
+        raises(NotImplementedError, lambda: D(x, (x, union)))
+
+
+def test_pr_22677():
+    b = Symbol('b', integer=True, positive=True)
+    assert Sum(1/x**2,(x, 0, b)).doit() == Sum(x**(-2), (x, 0, b))
+    assert Sum(1/(x - b)**2,(x, 0, b-1)).doit() == Sum(
+        (-b + x)**(-2), (x, 0, b - 1))
+
+
+def test_issue_23952():
+    p, q = symbols("p q", real=True, nonnegative=True)
+    k1, k2 = symbols("k1 k2", integer=True, nonnegative=True)
+    n = Symbol("n", integer=True, positive=True)
+    expr = Sum(abs(k1 - k2)*p**k1 *(1 - q)**(n - k2),
+        (k1, 0, n), (k2, 0, n))
+    assert expr.subs(p,0).subs(q,1).subs(n, 3).doit() == 3

env-llmeval/lib/python3.10/site-packages/sympy/printing/aesaracode.py

@@ -0,0 +1,539 @@
+from __future__ import annotations
+from typing import Any
+
+from sympy.external import import_module
+from sympy.printing.printer import Printer
+from sympy.utilities.iterables import is_sequence
+import sympy
+from functools import partial
+
+
+aesara = import_module('aesara')
+
+if aesara:
+    aes = aesara.scalar
+    aet = aesara.tensor
+    from aesara.tensor import nlinalg
+    from aesara.tensor.elemwise import Elemwise
+    from aesara.tensor.elemwise import DimShuffle
+
+    # `true_divide` replaced `true_div` in Aesara 2.8.11 (released 2023) to
+    # match NumPy
+    # XXX: Remove this when not needed to support older versions.
+    true_divide = getattr(aet, 'true_divide', None)
+    if true_divide is None:
+        true_divide = aet.true_div
+
+    mapping = {
+            sympy.Add: aet.add,
+            sympy.Mul: aet.mul,
+            sympy.Abs: aet.abs,
+            sympy.sign: aet.sgn,
+            sympy.ceiling: aet.ceil,
+            sympy.floor: aet.floor,
+            sympy.log: aet.log,
+            sympy.exp: aet.exp,
+            sympy.sqrt: aet.sqrt,
+            sympy.cos: aet.cos,
+            sympy.acos: aet.arccos,
+            sympy.sin: aet.sin,
+            sympy.asin: aet.arcsin,
+            sympy.tan: aet.tan,
+            sympy.atan: aet.arctan,
+            sympy.atan2: aet.arctan2,
+            sympy.cosh: aet.cosh,
+            sympy.acosh: aet.arccosh,
+            sympy.sinh: aet.sinh,
+            sympy.asinh: aet.arcsinh,
+            sympy.tanh: aet.tanh,
+            sympy.atanh: aet.arctanh,
+            sympy.re: aet.real,
+            sympy.im: aet.imag,
+            sympy.arg: aet.angle,
+            sympy.erf: aet.erf,
+            sympy.gamma: aet.gamma,
+            sympy.loggamma: aet.gammaln,
+            sympy.Pow: aet.pow,
+            sympy.Eq: aet.eq,
+            sympy.StrictGreaterThan: aet.gt,
+            sympy.StrictLessThan: aet.lt,
+            sympy.LessThan: aet.le,
+            sympy.GreaterThan: aet.ge,
+            sympy.And: aet.bitwise_and,  # bitwise
+            sympy.Or: aet.bitwise_or,  # bitwise
+            sympy.Not: aet.invert,  # bitwise
+            sympy.Xor: aet.bitwise_xor,  # bitwise
+            sympy.Max: aet.maximum,  # Sympy accept >2 inputs, Aesara only 2
+            sympy.Min: aet.minimum,  # Sympy accept >2 inputs, Aesara only 2
+            sympy.conjugate: aet.conj,
+            sympy.core.numbers.ImaginaryUnit: lambda:aet.complex(0,1),
+            # Matrices
+            sympy.MatAdd: Elemwise(aes.add),
+            sympy.HadamardProduct: Elemwise(aes.mul),
+            sympy.Trace: nlinalg.trace,
+            sympy.Determinant : nlinalg.det,
+            sympy.Inverse: nlinalg.matrix_inverse,
+            sympy.Transpose: DimShuffle((False, False), [1, 0]),
+    }
+
+
+class AesaraPrinter(Printer):
+    """ Code printer which creates Aesara symbolic expression graphs.
+
+    Parameters
+    ==========
+
+    cache : dict
+        Cache dictionary to use. If None (default) will use
+        the global cache. To create a printer which does not depend on or alter
+        global state pass an empty dictionary. Note: the dictionary is not
+        copied on initialization of the printer and will be updated in-place,
+        so using the same dict object when creating multiple printers or making
+        multiple calls to :func:`.aesara_code` or :func:`.aesara_function` means
+        the cache is shared between all these applications.
+
+    Attributes
+    ==========
+
+    cache : dict
+        A cache of Aesara variables which have been created for SymPy
+        symbol-like objects (e.g. :class:`sympy.core.symbol.Symbol` or
+        :class:`sympy.matrices.expressions.MatrixSymbol`). This is used to
+        ensure that all references to a given symbol in an expression (or
+        multiple expressions) are printed as the same Aesara variable, which is
+        created only once. Symbols are differentiated only by name and type. The
+        format of the cache's contents should be considered opaque to the user.
+    """
+    printmethod = "_aesara"
+
+    def __init__(self, *args, **kwargs):
+        self.cache = kwargs.pop('cache', {})
+        super().__init__(*args, **kwargs)
+
+    def _get_key(self, s, name=None, dtype=None, broadcastable=None):
+        """ Get the cache key for a SymPy object.
+
+        Parameters
+        ==========
+
+        s : sympy.core.basic.Basic
+            SymPy object to get key for.
+
+        name : str
+            Name of object, if it does not have a ``name`` attribute.
+        """
+
+        if name is None:
+            name = s.name
+
+        return (name, type(s), s.args, dtype, broadcastable)
+
+    def _get_or_create(self, s, name=None, dtype=None, broadcastable=None):
+        """
+        Get the Aesara variable for a SymPy symbol from the cache, or create it
+        if it does not exist.
+        """
+
+        # Defaults
+        if name is None:
+            name = s.name
+        if dtype is None:
+            dtype = 'floatX'
+        if broadcastable is None:
+            broadcastable = ()
+
+        key = self._get_key(s, name, dtype=dtype, broadcastable=broadcastable)
+
+        if key in self.cache:
+            return self.cache[key]
+
+        value = aet.tensor(name=name, dtype=dtype, broadcastable=broadcastable)
+        self.cache[key] = value
+        return value
+
+    def _print_Symbol(self, s, **kwargs):
+        dtype = kwargs.get('dtypes', {}).get(s)
+        bc = kwargs.get('broadcastables', {}).get(s)
+        return self._get_or_create(s, dtype=dtype, broadcastable=bc)
+
+    def _print_AppliedUndef(self, s, **kwargs):
+        name = str(type(s)) + '_' + str(s.args[0])
+        dtype = kwargs.get('dtypes', {}).get(s)
+        bc = kwargs.get('broadcastables', {}).get(s)
+        return self._get_or_create(s, name=name, dtype=dtype, broadcastable=bc)
+
+    def _print_Basic(self, expr, **kwargs):
+        op = mapping[type(expr)]
+        children = [self._print(arg, **kwargs) for arg in expr.args]
+        return op(*children)
+
+    def _print_Number(self, n, **kwargs):
+        # Integers already taken care of below, interpret as float
+        return float(n.evalf())
+
+    def _print_MatrixSymbol(self, X, **kwargs):
+        dtype = kwargs.get('dtypes', {}).get(X)
+        return self._get_or_create(X, dtype=dtype, broadcastable=(None, None))
+
+    def _print_DenseMatrix(self, X, **kwargs):
+        if not hasattr(aet, 'stacklists'):
+            raise NotImplementedError(
+               "Matrix translation not yet supported in this version of Aesara")
+
+        return aet.stacklists([
+            [self._print(arg, **kwargs) for arg in L]
+            for L in X.tolist()
+        ])
+
+    _print_ImmutableMatrix = _print_ImmutableDenseMatrix = _print_DenseMatrix
+
+    def _print_MatMul(self, expr, **kwargs):
+        children = [self._print(arg, **kwargs) for arg in expr.args]
+        result = children[0]
+        for child in children[1:]:
+            result = aet.dot(result, child)
+        return result
+
+    def _print_MatPow(self, expr, **kwargs):
+        children = [self._print(arg, **kwargs) for arg in expr.args]
+        result = 1
+        if isinstance(children[1], int) and children[1] > 0:
+            for i in range(children[1]):
+                result = aet.dot(result, children[0])
+        else:
+            raise NotImplementedError('''Only non-negative integer
+           powers of matrices can be handled by Aesara at the moment''')
+        return result
+
+    def _print_MatrixSlice(self, expr, **kwargs):
+        parent = self._print(expr.parent, **kwargs)
+        rowslice = self._print(slice(*expr.rowslice), **kwargs)
+        colslice = self._print(slice(*expr.colslice), **kwargs)
+        return parent[rowslice, colslice]
+
+    def _print_BlockMatrix(self, expr, **kwargs):
+        nrows, ncols = expr.blocks.shape
+        blocks = [[self._print(expr.blocks[r, c], **kwargs)
+                        for c in range(ncols)]
+                        for r in range(nrows)]
+        return aet.join(0, *[aet.join(1, *row) for row in blocks])
+
+
+    def _print_slice(self, expr, **kwargs):
+        return slice(*[self._print(i, **kwargs)
+                        if isinstance(i, sympy.Basic) else i
+                        for i in (expr.start, expr.stop, expr.step)])
+
+    def _print_Pi(self, expr, **kwargs):
+        return 3.141592653589793
+
+    def _print_Piecewise(self, expr, **kwargs):
+        import numpy as np
+        e, cond = expr.args[0].args  # First condition and corresponding value
+
+        # Print conditional expression and value for first condition
+        p_cond = self._print(cond, **kwargs)
+        p_e = self._print(e, **kwargs)
+
+        # One condition only
+        if len(expr.args) == 1:
+            # Return value if condition else NaN
+            return aet.switch(p_cond, p_e, np.nan)
+
+        # Return value_1 if condition_1 else evaluate remaining conditions
+        p_remaining = self._print(sympy.Piecewise(*expr.args[1:]), **kwargs)
+        return aet.switch(p_cond, p_e, p_remaining)
+
+    def _print_Rational(self, expr, **kwargs):
+        return true_divide(self._print(expr.p, **kwargs),
+                           self._print(expr.q, **kwargs))
+
+    def _print_Integer(self, expr, **kwargs):
+        return expr.p
+
+    def _print_factorial(self, expr, **kwargs):
+        return self._print(sympy.gamma(expr.args[0] + 1), **kwargs)
+
+    def _print_Derivative(self, deriv, **kwargs):
+        from aesara.gradient import Rop
+
+        rv = self._print(deriv.expr, **kwargs)
+        for var in deriv.variables:
+            var = self._print(var, **kwargs)
+            rv = Rop(rv, var, aet.ones_like(var))
+        return rv
+
+    def emptyPrinter(self, expr):
+        return expr
+
+    def doprint(self, expr, dtypes=None, broadcastables=None):
+        """ Convert a SymPy expression to a Aesara graph variable.
+
+        The ``dtypes`` and ``broadcastables`` arguments are used to specify the
+        data type, dimension, and broadcasting behavior of the Aesara variables
+        corresponding to the free symbols in ``expr``. Each is a mapping from
+        SymPy symbols to the value of the corresponding argument to
+        ``aesara.tensor.var.TensorVariable``.
+
+        See the corresponding `documentation page`__ for more information on
+        broadcasting in Aesara.
+
+        .. __: https://aesara.readthedocs.io/en/latest/tutorial/broadcasting.html
+
+        Parameters
+        ==========
+
+        expr : sympy.core.expr.Expr
+            SymPy expression to print.
+
+        dtypes : dict
+            Mapping from SymPy symbols to Aesara datatypes to use when creating
+            new Aesara variables for those symbols. Corresponds to the ``dtype``
+            argument to ``aesara.tensor.var.TensorVariable``. Defaults to ``'floatX'``
+            for symbols not included in the mapping.
+
+        broadcastables : dict
+            Mapping from SymPy symbols to the value of the ``broadcastable``
+            argument to ``aesara.tensor.var.TensorVariable`` to use when creating Aesara
+            variables for those symbols. Defaults to the empty tuple for symbols
+            not included in the mapping (resulting in a scalar).
+
+        Returns
+        =======
+
+        aesara.graph.basic.Variable
+            A variable corresponding to the expression's value in a Aesara
+            symbolic expression graph.
+
+        """
+        if dtypes is None:
+            dtypes = {}
+        if broadcastables is None:
+            broadcastables = {}
+
+        return self._print(expr, dtypes=dtypes, broadcastables=broadcastables)
+
+
+global_cache: dict[Any, Any] = {}
+
+
+def aesara_code(expr, cache=None, **kwargs):
+    """
+    Convert a SymPy expression into a Aesara graph variable.
+
+    Parameters
+    ==========
+
+    expr : sympy.core.expr.Expr
+        SymPy expression object to convert.
+
+    cache : dict
+        Cached Aesara variables (see :class:`AesaraPrinter.cache
+        <AesaraPrinter>`). Defaults to the module-level global cache.
+
+    dtypes : dict
+        Passed to :meth:`.AesaraPrinter.doprint`.
+
+    broadcastables : dict
+        Passed to :meth:`.AesaraPrinter.doprint`.
+
+    Returns
+    =======
+
+    aesara.graph.basic.Variable
+        A variable corresponding to the expression's value in a Aesara symbolic
+        expression graph.
+
+    """
+    if not aesara:
+        raise ImportError("aesara is required for aesara_code")
+
+    if cache is None:
+        cache = global_cache
+
+    return AesaraPrinter(cache=cache, settings={}).doprint(expr, **kwargs)
+
+
+def dim_handling(inputs, dim=None, dims=None, broadcastables=None):
+    r"""
+    Get value of ``broadcastables`` argument to :func:`.aesara_code` from
+    keyword arguments to :func:`.aesara_function`.
+
+    Included for backwards compatibility.
+
+    Parameters
+    ==========
+
+    inputs
+        Sequence of input symbols.
+
+    dim : int
+        Common number of dimensions for all inputs. Overrides other arguments
+        if given.
+
+    dims : dict
+        Mapping from input symbols to number of dimensions. Overrides
+        ``broadcastables`` argument if given.
+
+    broadcastables : dict
+        Explicit value of ``broadcastables`` argument to
+        :meth:`.AesaraPrinter.doprint`. If not None function will return this value unchanged.
+
+    Returns
+    =======
+    dict
+        Dictionary mapping elements of ``inputs`` to their "broadcastable"
+        values (tuple of ``bool``\ s).
+    """
+    if dim is not None:
+        return {s: (False,) * dim for s in inputs}
+
+    if dims is not None:
+        maxdim = max(dims.values())
+        return {
+            s: (False,) * d + (True,) * (maxdim - d)
+            for s, d in dims.items()
+        }
+
+    if broadcastables is not None:
+        return broadcastables
+
+    return {}
+
+
+def aesara_function(inputs, outputs, scalar=False, *,
+        dim=None, dims=None, broadcastables=None, **kwargs):
+    """
+    Create a Aesara function from SymPy expressions.
+
+    The inputs and outputs are converted to Aesara variables using
+    :func:`.aesara_code` and then passed to ``aesara.function``.
+
+    Parameters
+    ==========
+
+    inputs
+        Sequence of symbols which constitute the inputs of the function.
+
+    outputs
+        Sequence of expressions which constitute the outputs(s) of the
+        function. The free symbols of each expression must be a subset of
+        ``inputs``.
+
+    scalar : bool
+        Convert 0-dimensional arrays in output to scalars. This will return a
+        Python wrapper function around the Aesara function object.
+
+    cache : dict
+        Cached Aesara variables (see :class:`AesaraPrinter.cache
+        <AesaraPrinter>`). Defaults to the module-level global cache.
+
+    dtypes : dict
+        Passed to :meth:`.AesaraPrinter.doprint`.
+
+    broadcastables : dict
+        Passed to :meth:`.AesaraPrinter.doprint`.
+
+    dims : dict
+        Alternative to ``broadcastables`` argument. Mapping from elements of
+        ``inputs`` to integers indicating the dimension of their associated
+        arrays/tensors. Overrides ``broadcastables`` argument if given.
+
+    dim : int
+        Another alternative to the ``broadcastables`` argument. Common number of
+        dimensions to use for all arrays/tensors.
+        ``aesara_function([x, y], [...], dim=2)`` is equivalent to using
+        ``broadcastables={x: (False, False), y: (False, False)}``.
+
+    Returns
+    =======
+    callable
+        A callable object which takes values of ``inputs`` as positional
+        arguments and returns an output array for each of the expressions
+        in ``outputs``. If ``outputs`` is a single expression the function will
+        return a Numpy array, if it is a list of multiple expressions the
+        function will return a list of arrays. See description of the ``squeeze``
+        argument above for the behavior when a single output is passed in a list.
+        The returned object will either be an instance of
+        ``aesara.compile.function.types.Function`` or a Python wrapper
+        function around one. In both cases, the returned value will have a
+        ``aesara_function`` attribute which points to the return value of
+        ``aesara.function``.
+
+    Examples
+    ========
+
+    >>> from sympy.abc import x, y, z
+    >>> from sympy.printing.aesaracode import aesara_function
+
+    A simple function with one input and one output:
+
+    >>> f1 = aesara_function([x], [x**2 - 1], scalar=True)
+    >>> f1(3)
+    8.0
+
+    A function with multiple inputs and one output:
+
+    >>> f2 = aesara_function([x, y, z], [(x**z + y**z)**(1/z)], scalar=True)
+    >>> f2(3, 4, 2)
+    5.0
+
+    A function with multiple inputs and multiple outputs:
+
+    >>> f3 = aesara_function([x, y], [x**2 + y**2, x**2 - y**2], scalar=True)
+    >>> f3(2, 3)
+    [13.0, -5.0]
+
+    See also
+    ========
+
+    dim_handling
+
+    """
+    if not aesara:
+        raise ImportError("Aesara is required for aesara_function")
+
+    # Pop off non-aesara keyword args
+    cache = kwargs.pop('cache', {})
+    dtypes = kwargs.pop('dtypes', {})
+
+    broadcastables = dim_handling(
+        inputs, dim=dim, dims=dims, broadcastables=broadcastables,
+    )
+
+    # Print inputs/outputs
+    code = partial(aesara_code, cache=cache, dtypes=dtypes,
+                   broadcastables=broadcastables)
+    tinputs = list(map(code, inputs))
+    toutputs = list(map(code, outputs))
+
+    #fix constant expressions as variables
+    toutputs = [output if isinstance(output, aesara.graph.basic.Variable) else aet.as_tensor_variable(output) for output in toutputs]
+
+    if len(toutputs) == 1:
+        toutputs = toutputs[0]
+
+    # Compile aesara func
+    func = aesara.function(tinputs, toutputs, **kwargs)
+
+    is_0d = [len(o.variable.broadcastable) == 0 for o in func.outputs]
+
+    # No wrapper required
+    if not scalar or not any(is_0d):
+        func.aesara_function = func
+        return func
+
+    # Create wrapper to convert 0-dimensional outputs to scalars
+    def wrapper(*args):
+        out = func(*args)
+        # out can be array(1.0) or [array(1.0), array(2.0)]
+
+        if is_sequence(out):
+            return [o[()] if is_0d[i] else o for i, o in enumerate(out)]
+        else:
+            return out[()]
+
+    wrapper.__wrapped__ = func
+    wrapper.__doc__ = func.__doc__
+    wrapper.aesara_function = func
+    return wrapper

env-llmeval/lib/python3.10/site-packages/sympy/printing/c.py

@@ -0,0 +1,747 @@
+"""
+C code printer
+
+The C89CodePrinter & C99CodePrinter converts single SymPy expressions into
+single C expressions, using the functions defined in math.h where possible.
+
+A complete code generator, which uses ccode extensively, can be found in
+sympy.utilities.codegen. The codegen module can be used to generate complete
+source code files that are compilable without further modifications.
+
+
+"""
+
+from __future__ import annotations
+from typing import Any
+
+from functools import wraps
+from itertools import chain
+
+from sympy.core import S
+from sympy.core.numbers import equal_valued
+from sympy.codegen.ast import (
+    Assignment, Pointer, Variable, Declaration, Type,
+    real, complex_, integer, bool_, float32, float64, float80,
+    complex64, complex128, intc, value_const, pointer_const,
+    int8, int16, int32, int64, uint8, uint16, uint32, uint64, untyped,
+    none
+)
+from sympy.printing.codeprinter import CodePrinter, requires
+from sympy.printing.precedence import precedence, PRECEDENCE
+from sympy.sets.fancysets import Range
+
+# These are defined in the other file so we can avoid importing sympy.codegen
+# from the top-level 'import sympy'. Export them here as well.
+from sympy.printing.codeprinter import ccode, print_ccode # noqa:F401
+
+# dictionary mapping SymPy function to (argument_conditions, C_function).
+# Used in C89CodePrinter._print_Function(self)
+known_functions_C89 = {
+    "Abs": [(lambda x: not x.is_integer, "fabs"), (lambda x: x.is_integer, "abs")],
+    "sin": "sin",
+    "cos": "cos",
+    "tan": "tan",
+    "asin": "asin",
+    "acos": "acos",
+    "atan": "atan",
+    "atan2": "atan2",
+    "exp": "exp",
+    "log": "log",
+    "sinh": "sinh",
+    "cosh": "cosh",
+    "tanh": "tanh",
+    "floor": "floor",
+    "ceiling": "ceil",
+    "sqrt": "sqrt", # To enable automatic rewrites
+}
+
+known_functions_C99 = dict(known_functions_C89, **{
+    'exp2': 'exp2',
+    'expm1': 'expm1',
+    'log10': 'log10',
+    'log2': 'log2',
+    'log1p': 'log1p',
+    'Cbrt': 'cbrt',
+    'hypot': 'hypot',
+    'fma': 'fma',
+    'loggamma': 'lgamma',
+    'erfc': 'erfc',
+    'Max': 'fmax',
+    'Min': 'fmin',
+    "asinh": "asinh",
+    "acosh": "acosh",
+    "atanh": "atanh",
+    "erf": "erf",
+    "gamma": "tgamma",
+})
+
+# These are the core reserved words in the C language. Taken from:
+# https://en.cppreference.com/w/c/keyword
+
+reserved_words = [
+    'auto', 'break', 'case', 'char', 'const', 'continue', 'default', 'do',
+    'double', 'else', 'enum', 'extern', 'float', 'for', 'goto', 'if', 'int',
+    'long', 'register', 'return', 'short', 'signed', 'sizeof', 'static',
+    'struct', 'entry',  # never standardized, we'll leave it here anyway
+    'switch', 'typedef', 'union', 'unsigned', 'void', 'volatile', 'while'
+]
+
+reserved_words_c99 = ['inline', 'restrict']
+
+def get_math_macros():
+    """ Returns a dictionary with math-related macros from math.h/cmath
+
+    Note that these macros are not strictly required by the C/C++-standard.
+    For MSVC they are enabled by defining "_USE_MATH_DEFINES" (preferably
+    via a compilation flag).
+
+    Returns
+    =======
+
+    Dictionary mapping SymPy expressions to strings (macro names)
+
+    """
+    from sympy.codegen.cfunctions import log2, Sqrt
+    from sympy.functions.elementary.exponential import log
+    from sympy.functions.elementary.miscellaneous import sqrt
+
+    return {
+        S.Exp1: 'M_E',
+        log2(S.Exp1): 'M_LOG2E',
+        1/log(2): 'M_LOG2E',
+        log(2): 'M_LN2',
+        log(10): 'M_LN10',
+        S.Pi: 'M_PI',
+        S.Pi/2: 'M_PI_2',
+        S.Pi/4: 'M_PI_4',
+        1/S.Pi: 'M_1_PI',
+        2/S.Pi: 'M_2_PI',
+        2/sqrt(S.Pi): 'M_2_SQRTPI',
+        2/Sqrt(S.Pi): 'M_2_SQRTPI',
+        sqrt(2): 'M_SQRT2',
+        Sqrt(2): 'M_SQRT2',
+        1/sqrt(2): 'M_SQRT1_2',
+        1/Sqrt(2): 'M_SQRT1_2'
+    }
+
+
+def _as_macro_if_defined(meth):
+    """ Decorator for printer methods
+
+    When a Printer's method is decorated using this decorator the expressions printed
+    will first be looked for in the attribute ``math_macros``, and if present it will
+    print the macro name in ``math_macros`` followed by a type suffix for the type
+    ``real``. e.g. printing ``sympy.pi`` would print ``M_PIl`` if real is mapped to float80.
+
+    """
+    @wraps(meth)
+    def _meth_wrapper(self, expr, **kwargs):
+        if expr in self.math_macros:
+            return '%s%s' % (self.math_macros[expr], self._get_math_macro_suffix(real))
+        else:
+            return meth(self, expr, **kwargs)
+
+    return _meth_wrapper
+
+
+class C89CodePrinter(CodePrinter):
+    """A printer to convert Python expressions to strings of C code"""
+    printmethod = "_ccode"
+    language = "C"
+    standard = "C89"
+    reserved_words = set(reserved_words)
+
+    _default_settings: dict[str, Any] = {
+        'order': None,
+        'full_prec': 'auto',
+        'precision': 17,
+        'user_functions': {},
+        'human': True,
+        'allow_unknown_functions': False,
+        'contract': True,
+        'dereference': set(),
+        'error_on_reserved': False,
+        'reserved_word_suffix': '_',
+    }
+
+    type_aliases = {
+        real: float64,
+        complex_: complex128,
+        integer: intc
+    }
+
+    type_mappings: dict[Type, Any] = {
+        real: 'double',
+        intc: 'int',
+        float32: 'float',
+        float64: 'double',
+        integer: 'int',
+        bool_: 'bool',
+        int8: 'int8_t',
+        int16: 'int16_t',
+        int32: 'int32_t',
+        int64: 'int64_t',
+        uint8: 'int8_t',
+        uint16: 'int16_t',
+        uint32: 'int32_t',
+        uint64: 'int64_t',
+    }
+
+    type_headers = {
+        bool_: {'stdbool.h'},
+        int8: {'stdint.h'},
+        int16: {'stdint.h'},
+        int32: {'stdint.h'},
+        int64: {'stdint.h'},
+        uint8: {'stdint.h'},
+        uint16: {'stdint.h'},
+        uint32: {'stdint.h'},
+        uint64: {'stdint.h'},
+    }
+
+    # Macros needed to be defined when using a Type
+    type_macros: dict[Type, tuple[str, ...]] = {}
+
+    type_func_suffixes = {
+        float32: 'f',
+        float64: '',
+        float80: 'l'
+    }
+
+    type_literal_suffixes = {
+        float32: 'F',
+        float64: '',
+        float80: 'L'
+    }
+
+    type_math_macro_suffixes = {
+        float80: 'l'
+    }
+
+    math_macros = None
+
+    _ns = ''  # namespace, C++ uses 'std::'
+    # known_functions-dict to copy
+    _kf: dict[str, Any] = known_functions_C89
+
+    def __init__(self, settings=None):
+        settings = settings or {}
+        if self.math_macros is None:
+            self.math_macros = settings.pop('math_macros', get_math_macros())
+        self.type_aliases = dict(chain(self.type_aliases.items(),
+                                       settings.pop('type_aliases', {}).items()))
+        self.type_mappings = dict(chain(self.type_mappings.items(),
+                                        settings.pop('type_mappings', {}).items()))
+        self.type_headers = dict(chain(self.type_headers.items(),
+                                       settings.pop('type_headers', {}).items()))
+        self.type_macros = dict(chain(self.type_macros.items(),
+                                       settings.pop('type_macros', {}).items()))
+        self.type_func_suffixes = dict(chain(self.type_func_suffixes.items(),
+                                        settings.pop('type_func_suffixes', {}).items()))
+        self.type_literal_suffixes = dict(chain(self.type_literal_suffixes.items(),
+                                        settings.pop('type_literal_suffixes', {}).items()))
+        self.type_math_macro_suffixes = dict(chain(self.type_math_macro_suffixes.items(),
+                                        settings.pop('type_math_macro_suffixes', {}).items()))
+        super().__init__(settings)
+        self.known_functions = dict(self._kf, **settings.get('user_functions', {}))
+        self._dereference = set(settings.get('dereference', []))
+        self.headers = set()
+        self.libraries = set()
+        self.macros = set()
+
+    def _rate_index_position(self, p):
+        return p*5
+
+    def _get_statement(self, codestring):
+        """ Get code string as a statement - i.e. ending with a semicolon. """
+        return codestring if codestring.endswith(';') else codestring + ';'
+
+    def _get_comment(self, text):
+        return "/* {} */".format(text)
+
+    def _declare_number_const(self, name, value):
+        type_ = self.type_aliases[real]
+        var = Variable(name, type=type_, value=value.evalf(type_.decimal_dig), attrs={value_const})
+        decl = Declaration(var)
+        return self._get_statement(self._print(decl))
+
+    def _format_code(self, lines):
+        return self.indent_code(lines)
+
+    def _traverse_matrix_indices(self, mat):
+        rows, cols = mat.shape
+        return ((i, j) for i in range(rows) for j in range(cols))
+
+    @_as_macro_if_defined
+    def _print_Mul(self, expr, **kwargs):
+        return super()._print_Mul(expr, **kwargs)
+
+    @_as_macro_if_defined
+    def _print_Pow(self, expr):
+        if "Pow" in self.known_functions:
+            return self._print_Function(expr)
+        PREC = precedence(expr)
+        suffix = self._get_func_suffix(real)
+        if equal_valued(expr.exp, -1):
+            literal_suffix = self._get_literal_suffix(real)
+            return '1.0%s/%s' % (literal_suffix, self.parenthesize(expr.base, PREC))
+        elif equal_valued(expr.exp, 0.5):
+            return '%ssqrt%s(%s)' % (self._ns, suffix, self._print(expr.base))
+        elif expr.exp == S.One/3 and self.standard != 'C89':
+            return '%scbrt%s(%s)' % (self._ns, suffix, self._print(expr.base))
+        else:
+            return '%spow%s(%s, %s)' % (self._ns, suffix, self._print(expr.base),
+                                   self._print(expr.exp))
+
+    def _print_Mod(self, expr):
+        num, den = expr.args
+        if num.is_integer and den.is_integer:
+            PREC = precedence(expr)
+            snum, sden = [self.parenthesize(arg, PREC) for arg in expr.args]
+            # % is remainder (same sign as numerator), not modulo (same sign as
+            # denominator), in C. Hence, % only works as modulo if both numbers
+            # have the same sign
+            if (num.is_nonnegative and den.is_nonnegative or
+                num.is_nonpositive and den.is_nonpositive):
+                return f"{snum} % {sden}"
+            return f"(({snum} % {sden}) + {sden}) % {sden}"
+        # Not guaranteed integer
+        return self._print_math_func(expr, known='fmod')
+
+    def _print_Rational(self, expr):
+        p, q = int(expr.p), int(expr.q)
+        suffix = self._get_literal_suffix(real)
+        return '%d.0%s/%d.0%s' % (p, suffix, q, suffix)
+
+    def _print_Indexed(self, expr):
+        # calculate index for 1d array
+        offset = getattr(expr.base, 'offset', S.Zero)
+        strides = getattr(expr.base, 'strides', None)
+        indices = expr.indices
+
+        if strides is None or isinstance(strides, str):
+            dims = expr.shape
+            shift = S.One
+            temp = ()
+            if strides == 'C' or strides is None:
+                traversal = reversed(range(expr.rank))
+                indices = indices[::-1]
+            elif strides == 'F':
+                traversal = range(expr.rank)
+
+            for i in traversal:
+                temp += (shift,)
+                shift *= dims[i]
+            strides = temp
+        flat_index = sum([x[0]*x[1] for x in zip(indices, strides)]) + offset
+        return "%s[%s]" % (self._print(expr.base.label),
+                           self._print(flat_index))
+
+    def _print_Idx(self, expr):
+        return self._print(expr.label)
+
+    @_as_macro_if_defined
+    def _print_NumberSymbol(self, expr):
+        return super()._print_NumberSymbol(expr)
+
+    def _print_Infinity(self, expr):
+        return 'HUGE_VAL'
+
+    def _print_NegativeInfinity(self, expr):
+        return '-HUGE_VAL'
+
+    def _print_Piecewise(self, expr):
+        if expr.args[-1].cond != True:
+            # We need the last conditional to be a True, otherwise the resulting
+            # function may not return a result.
+            raise ValueError("All Piecewise expressions must contain an "
+                             "(expr, True) statement to be used as a default "
+                             "condition. Without one, the generated "
+                             "expression may not evaluate to anything under "
+                             "some condition.")
+        lines = []
+        if expr.has(Assignment):
+            for i, (e, c) in enumerate(expr.args):
+                if i == 0:
+                    lines.append("if (%s) {" % self._print(c))
+                elif i == len(expr.args) - 1 and c == True:
+                    lines.append("else {")
+                else:
+                    lines.append("else if (%s) {" % self._print(c))
+                code0 = self._print(e)
+                lines.append(code0)
+                lines.append("}")
+            return "\n".join(lines)
+        else:
+            # The piecewise was used in an expression, need to do inline
+            # operators. This has the downside that inline operators will
+            # not work for statements that span multiple lines (Matrix or
+            # Indexed expressions).
+            ecpairs = ["((%s) ? (\n%s\n)\n" % (self._print(c),
+                                               self._print(e))
+                    for e, c in expr.args[:-1]]
+            last_line = ": (\n%s\n)" % self._print(expr.args[-1].expr)
+            return ": ".join(ecpairs) + last_line + " ".join([")"*len(ecpairs)])
+
+    def _print_ITE(self, expr):
+        from sympy.functions import Piecewise
+        return self._print(expr.rewrite(Piecewise, deep=False))
+
+    def _print_MatrixElement(self, expr):
+        return "{}[{}]".format(self.parenthesize(expr.parent, PRECEDENCE["Atom"],
+            strict=True), expr.j + expr.i*expr.parent.shape[1])
+
+    def _print_Symbol(self, expr):
+        name = super()._print_Symbol(expr)
+        if expr in self._settings['dereference']:
+            return '(*{})'.format(name)
+        else:
+            return name
+
+    def _print_Relational(self, expr):
+        lhs_code = self._print(expr.lhs)
+        rhs_code = self._print(expr.rhs)
+        op = expr.rel_op
+        return "{} {} {}".format(lhs_code, op, rhs_code)
+
+    def _print_For(self, expr):
+        target = self._print(expr.target)
+        if isinstance(expr.iterable, Range):
+            start, stop, step = expr.iterable.args
+        else:
+            raise NotImplementedError("Only iterable currently supported is Range")
+        body = self._print(expr.body)
+        return ('for ({target} = {start}; {target} < {stop}; {target} += '
+                '{step}) {{\n{body}\n}}').format(target=target, start=start,
+                stop=stop, step=step, body=body)
+
+    def _print_sign(self, func):
+        return '((({0}) > 0) - (({0}) < 0))'.format(self._print(func.args[0]))
+
+    def _print_Max(self, expr):
+        if "Max" in self.known_functions:
+            return self._print_Function(expr)
+        def inner_print_max(args): # The more natural abstraction of creating
+            if len(args) == 1:     # and printing smaller Max objects is slow
+                return self._print(args[0]) # when there are many arguments.
+            half = len(args) // 2
+            return "((%(a)s > %(b)s) ? %(a)s : %(b)s)" % {
+                'a': inner_print_max(args[:half]),
+                'b': inner_print_max(args[half:])
+            }
+        return inner_print_max(expr.args)
+
+    def _print_Min(self, expr):
+        if "Min" in self.known_functions:
+            return self._print_Function(expr)
+        def inner_print_min(args): # The more natural abstraction of creating
+            if len(args) == 1:     # and printing smaller Min objects is slow
+                return self._print(args[0]) # when there are many arguments.
+            half = len(args) // 2
+            return "((%(a)s < %(b)s) ? %(a)s : %(b)s)" % {
+                'a': inner_print_min(args[:half]),
+                'b': inner_print_min(args[half:])
+            }
+        return inner_print_min(expr.args)
+
+    def indent_code(self, code):
+        """Accepts a string of code or a list of code lines"""
+
+        if isinstance(code, str):
+            code_lines = self.indent_code(code.splitlines(True))
+            return ''.join(code_lines)
+
+        tab = "   "
+        inc_token = ('{', '(', '{\n', '(\n')
+        dec_token = ('}', ')')
+
+        code = [line.lstrip(' \t') for line in code]
+
+        increase = [int(any(map(line.endswith, inc_token))) for line in code]
+        decrease = [int(any(map(line.startswith, dec_token))) for line in code]
+
+        pretty = []
+        level = 0
+        for n, line in enumerate(code):
+            if line in ('', '\n'):
+                pretty.append(line)
+                continue
+            level -= decrease[n]
+            pretty.append("%s%s" % (tab*level, line))
+            level += increase[n]
+        return pretty
+
+    def _get_func_suffix(self, type_):
+        return self.type_func_suffixes[self.type_aliases.get(type_, type_)]
+
+    def _get_literal_suffix(self, type_):
+        return self.type_literal_suffixes[self.type_aliases.get(type_, type_)]
+
+    def _get_math_macro_suffix(self, type_):
+        alias = self.type_aliases.get(type_, type_)
+        dflt = self.type_math_macro_suffixes.get(alias, '')
+        return self.type_math_macro_suffixes.get(type_, dflt)
+
+    def _print_Tuple(self, expr):
+        return '{'+', '.join(self._print(e) for e in expr)+'}'
+
+    _print_List = _print_Tuple
+
+    def _print_Type(self, type_):
+        self.headers.update(self.type_headers.get(type_, set()))
+        self.macros.update(self.type_macros.get(type_, set()))
+        return self._print(self.type_mappings.get(type_, type_.name))
+
+    def _print_Declaration(self, decl):
+        from sympy.codegen.cnodes import restrict
+        var = decl.variable
+        val = var.value
+        if var.type == untyped:
+            raise ValueError("C does not support untyped variables")
+
+        if isinstance(var, Pointer):
+            result = '{vc}{t} *{pc} {r}{s}'.format(
+                vc='const ' if value_const in var.attrs else '',
+                t=self._print(var.type),
+                pc=' const' if pointer_const in var.attrs else '',
+                r='restrict ' if restrict in var.attrs else '',
+                s=self._print(var.symbol)
+            )
+        elif isinstance(var, Variable):
+            result = '{vc}{t} {s}'.format(
+                vc='const ' if value_const in var.attrs else '',
+                t=self._print(var.type),
+                s=self._print(var.symbol)
+            )
+        else:
+            raise NotImplementedError("Unknown type of var: %s" % type(var))
+        if val != None: # Must be "!= None", cannot be "is not None"
+            result += ' = %s' % self._print(val)
+        return result
+
+    def _print_Float(self, flt):
+        type_ = self.type_aliases.get(real, real)
+        self.macros.update(self.type_macros.get(type_, set()))
+        suffix = self._get_literal_suffix(type_)
+        num = str(flt.evalf(type_.decimal_dig))
+        if 'e' not in num and '.' not in num:
+            num += '.0'
+        num_parts = num.split('e')
+        num_parts[0] = num_parts[0].rstrip('0')
+        if num_parts[0].endswith('.'):
+            num_parts[0] += '0'
+        return 'e'.join(num_parts) + suffix
+
+    @requires(headers={'stdbool.h'})
+    def _print_BooleanTrue(self, expr):
+        return 'true'
+
+    @requires(headers={'stdbool.h'})
+    def _print_BooleanFalse(self, expr):
+        return 'false'
+
+    def _print_Element(self, elem):
+        if elem.strides == None: # Must be "== None", cannot be "is None"
+            if elem.offset != None: # Must be "!= None", cannot be "is not None"
+                raise ValueError("Expected strides when offset is given")
+            idxs = ']['.join((self._print(arg) for arg in elem.indices))
+        else:
+            global_idx = sum([i*s for i, s in zip(elem.indices, elem.strides)])
+            if elem.offset != None: # Must be "!= None", cannot be "is not None"
+                global_idx += elem.offset
+            idxs = self._print(global_idx)
+
+        return "{symb}[{idxs}]".format(
+            symb=self._print(elem.symbol),
+            idxs=idxs
+        )
+
+    def _print_CodeBlock(self, expr):
+        """ Elements of code blocks printed as statements. """
+        return '\n'.join([self._get_statement(self._print(i)) for i in expr.args])
+
+    def _print_While(self, expr):
+        return 'while ({condition}) {{\n{body}\n}}'.format(**expr.kwargs(
+            apply=lambda arg: self._print(arg)))
+
+    def _print_Scope(self, expr):
+        return '{\n%s\n}' % self._print_CodeBlock(expr.body)
+
+    @requires(headers={'stdio.h'})
+    def _print_Print(self, expr):
+        return 'printf({fmt}, {pargs})'.format(
+            fmt=self._print(expr.format_string),
+            pargs=', '.join((self._print(arg) for arg in expr.print_args))
+        )
+
+    def _print_FunctionPrototype(self, expr):
+        pars = ', '.join((self._print(Declaration(arg)) for arg in expr.parameters))
+        return "%s %s(%s)" % (
+            tuple((self._print(arg) for arg in (expr.return_type, expr.name))) + (pars,)
+        )
+
+    def _print_FunctionDefinition(self, expr):
+        return "%s%s" % (self._print_FunctionPrototype(expr),
+                         self._print_Scope(expr))
+
+    def _print_Return(self, expr):
+        arg, = expr.args
+        return 'return %s' % self._print(arg)
+
+    def _print_CommaOperator(self, expr):
+        return '(%s)' % ', '.join((self._print(arg) for arg in expr.args))
+
+    def _print_Label(self, expr):
+        if expr.body == none:
+            return '%s:' % str(expr.name)
+        if len(expr.body.args) == 1:
+            return '%s:\n%s' % (str(expr.name), self._print_CodeBlock(expr.body))
+        return '%s:\n{\n%s\n}' % (str(expr.name), self._print_CodeBlock(expr.body))
+
+    def _print_goto(self, expr):
+        return 'goto %s' % expr.label.name
+
+    def _print_PreIncrement(self, expr):
+        arg, = expr.args
+        return '++(%s)' % self._print(arg)
+
+    def _print_PostIncrement(self, expr):
+        arg, = expr.args
+        return '(%s)++' % self._print(arg)
+
+    def _print_PreDecrement(self, expr):
+        arg, = expr.args
+        return '--(%s)' % self._print(arg)
+
+    def _print_PostDecrement(self, expr):
+        arg, = expr.args
+        return '(%s)--' % self._print(arg)
+
+    def _print_struct(self, expr):
+        return "%(keyword)s %(name)s {\n%(lines)s}" % {
+            "keyword": expr.__class__.__name__, "name": expr.name, "lines": ';\n'.join(
+                [self._print(decl) for decl in expr.declarations] + [''])
+        }
+
+    def _print_BreakToken(self, _):
+        return 'break'
+
+    def _print_ContinueToken(self, _):
+        return 'continue'
+
+    _print_union = _print_struct
+
+class C99CodePrinter(C89CodePrinter):
+    standard = 'C99'
+    reserved_words = set(reserved_words + reserved_words_c99)
+    type_mappings=dict(chain(C89CodePrinter.type_mappings.items(), {
+        complex64: 'float complex',
+        complex128: 'double complex',
+    }.items()))
+    type_headers = dict(chain(C89CodePrinter.type_headers.items(), {
+        complex64: {'complex.h'},
+        complex128: {'complex.h'}
+    }.items()))
+
+    # known_functions-dict to copy
+    _kf: dict[str, Any] = known_functions_C99
+
+    # functions with versions with 'f' and 'l' suffixes:
+    _prec_funcs = ('fabs fmod remainder remquo fma fmax fmin fdim nan exp exp2'
+                   ' expm1 log log10 log2 log1p pow sqrt cbrt hypot sin cos tan'
+                   ' asin acos atan atan2 sinh cosh tanh asinh acosh atanh erf'
+                   ' erfc tgamma lgamma ceil floor trunc round nearbyint rint'
+                   ' frexp ldexp modf scalbn ilogb logb nextafter copysign').split()
+
+    def _print_Infinity(self, expr):
+        return 'INFINITY'
+
+    def _print_NegativeInfinity(self, expr):
+        return '-INFINITY'
+
+    def _print_NaN(self, expr):
+        return 'NAN'
+
+    # tgamma was already covered by 'known_functions' dict
+
+    @requires(headers={'math.h'}, libraries={'m'})
+    @_as_macro_if_defined
+    def _print_math_func(self, expr, nest=False, known=None):
+        if known is None:
+            known = self.known_functions[expr.__class__.__name__]
+        if not isinstance(known, str):
+            for cb, name in known:
+                if cb(*expr.args):
+                    known = name
+                    break
+            else:
+                raise ValueError("No matching printer")
+        try:
+            return known(self, *expr.args)
+        except TypeError:
+            suffix = self._get_func_suffix(real) if self._ns + known in self._prec_funcs else ''
+
+        if nest:
+            args = self._print(expr.args[0])
+            if len(expr.args) > 1:
+                paren_pile = ''
+                for curr_arg in expr.args[1:-1]:
+                    paren_pile += ')'
+                    args += ', {ns}{name}{suffix}({next}'.format(
+                        ns=self._ns,
+                        name=known,
+                        suffix=suffix,
+                        next = self._print(curr_arg)
+                    )
+                args += ', %s%s' % (
+                    self._print(expr.func(expr.args[-1])),
+                    paren_pile
+                )
+        else:
+            args = ', '.join((self._print(arg) for arg in expr.args))
+        return '{ns}{name}{suffix}({args})'.format(
+            ns=self._ns,
+            name=known,
+            suffix=suffix,
+            args=args
+        )
+
+    def _print_Max(self, expr):
+        return self._print_math_func(expr, nest=True)
+
+    def _print_Min(self, expr):
+        return self._print_math_func(expr, nest=True)
+
+    def _get_loop_opening_ending(self, indices):
+        open_lines = []
+        close_lines = []
+        loopstart = "for (int %(var)s=%(start)s; %(var)s<%(end)s; %(var)s++){"  # C99
+        for i in indices:
+            # C arrays start at 0 and end at dimension-1
+            open_lines.append(loopstart % {
+                'var': self._print(i.label),
+                'start': self._print(i.lower),
+                'end': self._print(i.upper + 1)})
+            close_lines.append("}")
+        return open_lines, close_lines
+
+
+for k in ('Abs Sqrt exp exp2 expm1 log log10 log2 log1p Cbrt hypot fma'
+          ' loggamma sin cos tan asin acos atan atan2 sinh cosh tanh asinh acosh '
+          'atanh erf erfc loggamma gamma ceiling floor').split():
+    setattr(C99CodePrinter, '_print_%s' % k, C99CodePrinter._print_math_func)
+
+
+class C11CodePrinter(C99CodePrinter):
+
+    @requires(headers={'stdalign.h'})
+    def _print_alignof(self, expr):
+        arg, = expr.args
+        return 'alignof(%s)' % self._print(arg)
+
+
+c_code_printers = {
+    'c89': C89CodePrinter,
+    'c99': C99CodePrinter,
+    'c11': C11CodePrinter
+}

env-llmeval/lib/python3.10/site-packages/sympy/printing/conventions.py

@@ -0,0 +1,88 @@
+"""
+A few practical conventions common to all printers.
+"""
+
+import re
+
+from collections.abc import Iterable
+from sympy.core.function import Derivative
+
+_name_with_digits_p = re.compile(r'^([^\W\d_]+)(\d+)$', re.U)
+
+
+def split_super_sub(text):
+    """Split a symbol name into a name, superscripts and subscripts
+
+    The first part of the symbol name is considered to be its actual
+    'name', followed by super- and subscripts. Each superscript is
+    preceded with a "^" character or by "__". Each subscript is preceded
+    by a "_" character.  The three return values are the actual name, a
+    list with superscripts and a list with subscripts.
+
+    Examples
+    ========
+
+    >>> from sympy.printing.conventions import split_super_sub
+    >>> split_super_sub('a_x^1')
+    ('a', ['1'], ['x'])
+    >>> split_super_sub('var_sub1__sup_sub2')
+    ('var', ['sup'], ['sub1', 'sub2'])
+
+    """
+    if not text:
+        return text, [], []
+
+    pos = 0
+    name = None
+    supers = []
+    subs = []
+    while pos < len(text):
+        start = pos + 1
+        if text[pos:pos + 2] == "__":
+            start += 1
+        pos_hat = text.find("^", start)
+        if pos_hat < 0:
+            pos_hat = len(text)
+        pos_usc = text.find("_", start)
+        if pos_usc < 0:
+            pos_usc = len(text)
+        pos_next = min(pos_hat, pos_usc)
+        part = text[pos:pos_next]
+        pos = pos_next
+        if name is None:
+            name = part
+        elif part.startswith("^"):
+            supers.append(part[1:])
+        elif part.startswith("__"):
+            supers.append(part[2:])
+        elif part.startswith("_"):
+            subs.append(part[1:])
+        else:
+            raise RuntimeError("This should never happen.")
+
+    # Make a little exception when a name ends with digits, i.e. treat them
+    # as a subscript too.
+    m = _name_with_digits_p.match(name)
+    if m:
+        name, sub = m.groups()
+        subs.insert(0, sub)
+
+    return name, supers, subs
+
+
+def requires_partial(expr):
+    """Return whether a partial derivative symbol is required for printing
+
+    This requires checking how many free variables there are,
+    filtering out the ones that are integers. Some expressions do not have
+    free variables. In that case, check its variable list explicitly to
+    get the context of the expression.
+    """
+
+    if isinstance(expr, Derivative):
+        return requires_partial(expr.expr)
+
+    if not isinstance(expr.free_symbols, Iterable):
+        return len(set(expr.variables)) > 1
+
+    return sum(not s.is_integer for s in expr.free_symbols) > 1

env-llmeval/lib/python3.10/site-packages/sympy/printing/dot.py

@@ -0,0 +1,294 @@
+from sympy.core.basic import Basic
+from sympy.core.expr import Expr
+from sympy.core.symbol import Symbol
+from sympy.core.numbers import Integer, Rational, Float
+from sympy.printing.repr import srepr
+
+__all__ = ['dotprint']
+
+default_styles = (
+    (Basic, {'color': 'blue', 'shape': 'ellipse'}),
+    (Expr,  {'color': 'black'})
+)
+
+slotClasses = (Symbol, Integer, Rational, Float)
+def purestr(x, with_args=False):
+    """A string that follows ```obj = type(obj)(*obj.args)``` exactly.
+
+    Parameters
+    ==========
+
+    with_args : boolean, optional
+        If ``True``, there will be a second argument for the return
+        value, which is a tuple containing ``purestr`` applied to each
+        of the subnodes.
+
+        If ``False``, there will not be a second argument for the
+        return.
+
+        Default is ``False``
+
+    Examples
+    ========
+
+    >>> from sympy import Float, Symbol, MatrixSymbol
+    >>> from sympy import Integer # noqa: F401
+    >>> from sympy.core.symbol import Str # noqa: F401
+    >>> from sympy.printing.dot import purestr
+
+    Applying ``purestr`` for basic symbolic object:
+    >>> code = purestr(Symbol('x'))
+    >>> code
+    "Symbol('x')"
+    >>> eval(code) == Symbol('x')
+    True
+
+    For basic numeric object:
+    >>> purestr(Float(2))
+    "Float('2.0', precision=53)"
+
+    For matrix symbol:
+    >>> code = purestr(MatrixSymbol('x', 2, 2))
+    >>> code
+    "MatrixSymbol(Str('x'), Integer(2), Integer(2))"
+    >>> eval(code) == MatrixSymbol('x', 2, 2)
+    True
+
+    With ``with_args=True``:
+    >>> purestr(Float(2), with_args=True)
+    ("Float('2.0', precision=53)", ())
+    >>> purestr(MatrixSymbol('x', 2, 2), with_args=True)
+    ("MatrixSymbol(Str('x'), Integer(2), Integer(2))",
+     ("Str('x')", 'Integer(2)', 'Integer(2)'))
+    """
+    sargs = ()
+    if not isinstance(x, Basic):
+        rv = str(x)
+    elif not x.args:
+        rv = srepr(x)
+    else:
+        args = x.args
+        sargs = tuple(map(purestr, args))
+        rv = "%s(%s)"%(type(x).__name__, ', '.join(sargs))
+    if with_args:
+        rv = rv, sargs
+    return rv
+
+
+def styleof(expr, styles=default_styles):
+    """ Merge style dictionaries in order
+
+    Examples
+    ========
+
+    >>> from sympy import Symbol, Basic, Expr, S
+    >>> from sympy.printing.dot import styleof
+    >>> styles = [(Basic, {'color': 'blue', 'shape': 'ellipse'}),
+    ...           (Expr,  {'color': 'black'})]
+
+    >>> styleof(Basic(S(1)), styles)
+    {'color': 'blue', 'shape': 'ellipse'}
+
+    >>> x = Symbol('x')
+    >>> styleof(x + 1, styles)  # this is an Expr
+    {'color': 'black', 'shape': 'ellipse'}
+    """
+    style = {}
+    for typ, sty in styles:
+        if isinstance(expr, typ):
+            style.update(sty)
+    return style
+
+
+def attrprint(d, delimiter=', '):
+    """ Print a dictionary of attributes
+
+    Examples
+    ========
+
+    >>> from sympy.printing.dot import attrprint
+    >>> print(attrprint({'color': 'blue', 'shape': 'ellipse'}))
+    "color"="blue", "shape"="ellipse"
+    """
+    return delimiter.join('"%s"="%s"'%item for item in sorted(d.items()))
+
+
+def dotnode(expr, styles=default_styles, labelfunc=str, pos=(), repeat=True):
+    """ String defining a node
+
+    Examples
+    ========
+
+    >>> from sympy.printing.dot import dotnode
+    >>> from sympy.abc import x
+    >>> print(dotnode(x))
+    "Symbol('x')_()" ["color"="black", "label"="x", "shape"="ellipse"];
+    """
+    style = styleof(expr, styles)
+
+    if isinstance(expr, Basic) and not expr.is_Atom:
+        label = str(expr.__class__.__name__)
+    else:
+        label = labelfunc(expr)
+    style['label'] = label
+    expr_str = purestr(expr)
+    if repeat:
+        expr_str += '_%s' % str(pos)
+    return '"%s" [%s];' % (expr_str, attrprint(style))
+
+
+def dotedges(expr, atom=lambda x: not isinstance(x, Basic), pos=(), repeat=True):
+    """ List of strings for all expr->expr.arg pairs
+
+    See the docstring of dotprint for explanations of the options.
+
+    Examples
+    ========
+
+    >>> from sympy.printing.dot import dotedges
+    >>> from sympy.abc import x
+    >>> for e in dotedges(x+2):
+    ...     print(e)
+    "Add(Integer(2), Symbol('x'))_()" -> "Integer(2)_(0,)";
+    "Add(Integer(2), Symbol('x'))_()" -> "Symbol('x')_(1,)";
+    """
+    if atom(expr):
+        return []
+    else:
+        expr_str, arg_strs = purestr(expr, with_args=True)
+        if repeat:
+            expr_str += '_%s' % str(pos)
+            arg_strs = ['%s_%s' % (a, str(pos + (i,)))
+                for i, a in enumerate(arg_strs)]
+        return ['"%s" -> "%s";' % (expr_str, a) for a in arg_strs]
+
+template = \
+"""digraph{
+
+# Graph style
+%(graphstyle)s
+
+#########
+# Nodes #
+#########
+
+%(nodes)s
+
+#########
+# Edges #
+#########
+
+%(edges)s
+}"""
+
+_graphstyle = {'rankdir': 'TD', 'ordering': 'out'}
+
+def dotprint(expr,
+    styles=default_styles, atom=lambda x: not isinstance(x, Basic),
+    maxdepth=None, repeat=True, labelfunc=str, **kwargs):
+    """DOT description of a SymPy expression tree
+
+    Parameters
+    ==========
+
+    styles : list of lists composed of (Class, mapping), optional
+        Styles for different classes.
+
+        The default is
+
+        .. code-block:: python
+
+            (
+                (Basic, {'color': 'blue', 'shape': 'ellipse'}),
+                (Expr,  {'color': 'black'})
+            )
+
+    atom : function, optional
+        Function used to determine if an arg is an atom.
+
+        A good choice is ``lambda x: not x.args``.
+
+        The default is ``lambda x: not isinstance(x, Basic)``.
+
+    maxdepth : integer, optional
+        The maximum depth.
+
+        The default is ``None``, meaning no limit.
+
+    repeat : boolean, optional
+        Whether to use different nodes for common subexpressions.
+
+        The default is ``True``.
+
+        For example, for ``x + x*y`` with ``repeat=True``, it will have
+        two nodes for ``x``; with ``repeat=False``, it will have one
+        node.
+
+        .. warning::
+            Even if a node appears twice in the same object like ``x`` in
+            ``Pow(x, x)``, it will still only appear once.
+            Hence, with ``repeat=False``, the number of arrows out of an
+            object might not equal the number of args it has.
+
+    labelfunc : function, optional
+        A function to create a label for a given leaf node.
+
+        The default is ``str``.
+
+        Another good option is ``srepr``.
+
+        For example with ``str``, the leaf nodes of ``x + 1`` are labeled,
+        ``x`` and ``1``.  With ``srepr``, they are labeled ``Symbol('x')``
+        and ``Integer(1)``.
+
+    **kwargs : optional
+        Additional keyword arguments are included as styles for the graph.
+
+    Examples
+    ========
+
+    >>> from sympy import dotprint
+    >>> from sympy.abc import x
+    >>> print(dotprint(x+2)) # doctest: +NORMALIZE_WHITESPACE
+    digraph{
+    <BLANKLINE>
+    # Graph style
+    "ordering"="out"
+    "rankdir"="TD"
+    <BLANKLINE>
+    #########
+    # Nodes #
+    #########
+    <BLANKLINE>
+    "Add(Integer(2), Symbol('x'))_()" ["color"="black", "label"="Add", "shape"="ellipse"];
+    "Integer(2)_(0,)" ["color"="black", "label"="2", "shape"="ellipse"];
+    "Symbol('x')_(1,)" ["color"="black", "label"="x", "shape"="ellipse"];
+    <BLANKLINE>
+    #########
+    # Edges #
+    #########
+    <BLANKLINE>
+    "Add(Integer(2), Symbol('x'))_()" -> "Integer(2)_(0,)";
+    "Add(Integer(2), Symbol('x'))_()" -> "Symbol('x')_(1,)";
+    }
+
+    """
+    # repeat works by adding a signature tuple to the end of each node for its
+    # position in the graph. For example, for expr = Add(x, Pow(x, 2)), the x in the
+    # Pow will have the tuple (1, 0), meaning it is expr.args[1].args[0].
+    graphstyle = _graphstyle.copy()
+    graphstyle.update(kwargs)
+
+    nodes = []
+    edges = []
+    def traverse(e, depth, pos=()):
+        nodes.append(dotnode(e, styles, labelfunc=labelfunc, pos=pos, repeat=repeat))
+        if maxdepth and depth >= maxdepth:
+            return
+        edges.extend(dotedges(e, atom=atom, pos=pos, repeat=repeat))
+        [traverse(arg, depth+1, pos + (i,)) for i, arg in enumerate(e.args) if not atom(arg)]
+    traverse(expr, 0)
+
+    return template%{'graphstyle': attrprint(graphstyle, delimiter='\n'),
+                     'nodes': '\n'.join(nodes),
+                     'edges': '\n'.join(edges)}

env-llmeval/lib/python3.10/site-packages/sympy/printing/precedence.py

@@ -0,0 +1,177 @@
+"""A module providing information about the necessity of brackets"""
+
+
+# Default precedence values for some basic types
+PRECEDENCE = {
+    "Lambda": 1,
+    "Xor": 10,
+    "Or": 20,
+    "And": 30,
+    "Relational": 35,
+    "Add": 40,
+    "Mul": 50,
+    "Pow": 60,
+    "Func": 70,
+    "Not": 100,
+    "Atom": 1000,
+    "BitwiseOr": 36,
+    "BitwiseXor": 37,
+    "BitwiseAnd": 38
+}
+
+# A dictionary assigning precedence values to certain classes. These values are
+# treated like they were inherited, so not every single class has to be named
+# here.
+# Do not use this with printers other than StrPrinter
+PRECEDENCE_VALUES = {
+    "Equivalent": PRECEDENCE["Xor"],
+    "Xor": PRECEDENCE["Xor"],
+    "Implies": PRECEDENCE["Xor"],
+    "Or": PRECEDENCE["Or"],
+    "And": PRECEDENCE["And"],
+    "Add": PRECEDENCE["Add"],
+    "Pow": PRECEDENCE["Pow"],
+    "Relational": PRECEDENCE["Relational"],
+    "Sub": PRECEDENCE["Add"],
+    "Not": PRECEDENCE["Not"],
+    "Function" : PRECEDENCE["Func"],
+    "NegativeInfinity": PRECEDENCE["Add"],
+    "MatAdd": PRECEDENCE["Add"],
+    "MatPow": PRECEDENCE["Pow"],
+    "MatrixSolve": PRECEDENCE["Mul"],
+    "Mod": PRECEDENCE["Mul"],
+    "TensAdd": PRECEDENCE["Add"],
+    # As soon as `TensMul` is a subclass of `Mul`, remove this:
+    "TensMul": PRECEDENCE["Mul"],
+    "HadamardProduct": PRECEDENCE["Mul"],
+    "HadamardPower": PRECEDENCE["Pow"],
+    "KroneckerProduct": PRECEDENCE["Mul"],
+    "Equality": PRECEDENCE["Mul"],
+    "Unequality": PRECEDENCE["Mul"],
+}
+
+# Sometimes it's not enough to assign a fixed precedence value to a
+# class. Then a function can be inserted in this dictionary that takes
+# an instance of this class as argument and returns the appropriate
+# precedence value.
+
+# Precedence functions
+
+
+def precedence_Mul(item):
+    if item.could_extract_minus_sign():
+        return PRECEDENCE["Add"]
+    return PRECEDENCE["Mul"]
+
+
+def precedence_Rational(item):
+    if item.p < 0:
+        return PRECEDENCE["Add"]
+    return PRECEDENCE["Mul"]
+
+
+def precedence_Integer(item):
+    if item.p < 0:
+        return PRECEDENCE["Add"]
+    return PRECEDENCE["Atom"]
+
+
+def precedence_Float(item):
+    if item < 0:
+        return PRECEDENCE["Add"]
+    return PRECEDENCE["Atom"]
+
+
+def precedence_PolyElement(item):
+    if item.is_generator:
+        return PRECEDENCE["Atom"]
+    elif item.is_ground:
+        return precedence(item.coeff(1))
+    elif item.is_term:
+        return PRECEDENCE["Mul"]
+    else:
+        return PRECEDENCE["Add"]
+
+
+def precedence_FracElement(item):
+    if item.denom == 1:
+        return precedence_PolyElement(item.numer)
+    else:
+        return PRECEDENCE["Mul"]
+
+
+def precedence_UnevaluatedExpr(item):
+    return precedence(item.args[0]) - 0.5
+
+
+PRECEDENCE_FUNCTIONS = {
+    "Integer": precedence_Integer,
+    "Mul": precedence_Mul,
+    "Rational": precedence_Rational,
+    "Float": precedence_Float,
+    "PolyElement": precedence_PolyElement,
+    "FracElement": precedence_FracElement,
+    "UnevaluatedExpr": precedence_UnevaluatedExpr,
+}
+
+
+def precedence(item):
+    """Returns the precedence of a given object.
+
+    This is the precedence for StrPrinter.
+    """
+    if hasattr(item, "precedence"):
+        return item.precedence
+    try:
+        mro = item.__class__.__mro__
+    except AttributeError:
+        return PRECEDENCE["Atom"]
+    for i in mro:
+        n = i.__name__
+        if n in PRECEDENCE_FUNCTIONS:
+            return PRECEDENCE_FUNCTIONS[n](item)
+        elif n in PRECEDENCE_VALUES:
+            return PRECEDENCE_VALUES[n]
+    return PRECEDENCE["Atom"]
+
+
+PRECEDENCE_TRADITIONAL = PRECEDENCE.copy()
+PRECEDENCE_TRADITIONAL['Integral'] = PRECEDENCE["Mul"]
+PRECEDENCE_TRADITIONAL['Sum'] = PRECEDENCE["Mul"]
+PRECEDENCE_TRADITIONAL['Product'] = PRECEDENCE["Mul"]
+PRECEDENCE_TRADITIONAL['Limit'] = PRECEDENCE["Mul"]
+PRECEDENCE_TRADITIONAL['Derivative'] = PRECEDENCE["Mul"]
+PRECEDENCE_TRADITIONAL['TensorProduct'] = PRECEDENCE["Mul"]
+PRECEDENCE_TRADITIONAL['Transpose'] = PRECEDENCE["Pow"]
+PRECEDENCE_TRADITIONAL['Adjoint'] = PRECEDENCE["Pow"]
+PRECEDENCE_TRADITIONAL['Dot'] = PRECEDENCE["Mul"] - 1
+PRECEDENCE_TRADITIONAL['Cross'] = PRECEDENCE["Mul"] - 1
+PRECEDENCE_TRADITIONAL['Gradient'] = PRECEDENCE["Mul"] - 1
+PRECEDENCE_TRADITIONAL['Divergence'] = PRECEDENCE["Mul"] - 1
+PRECEDENCE_TRADITIONAL['Curl'] = PRECEDENCE["Mul"] - 1
+PRECEDENCE_TRADITIONAL['Laplacian'] = PRECEDENCE["Mul"] - 1
+PRECEDENCE_TRADITIONAL['Union'] = PRECEDENCE['Xor']
+PRECEDENCE_TRADITIONAL['Intersection'] = PRECEDENCE['Xor']
+PRECEDENCE_TRADITIONAL['Complement'] = PRECEDENCE['Xor']
+PRECEDENCE_TRADITIONAL['SymmetricDifference'] = PRECEDENCE['Xor']
+PRECEDENCE_TRADITIONAL['ProductSet'] = PRECEDENCE['Xor']
+
+
+def precedence_traditional(item):
+    """Returns the precedence of a given object according to the
+    traditional rules of mathematics.
+
+    This is the precedence for the LaTeX and pretty printer.
+    """
+    # Integral, Sum, Product, Limit have the precedence of Mul in LaTeX,
+    # the precedence of Atom for other printers:
+    from sympy.core.expr import UnevaluatedExpr
+
+    if isinstance(item, UnevaluatedExpr):
+        return precedence_traditional(item.args[0])
+
+    n = item.__class__.__name__
+    if n in PRECEDENCE_TRADITIONAL:
+        return PRECEDENCE_TRADITIONAL[n]
+
+    return precedence(item)

env-llmeval/lib/python3.10/site-packages/sympy/printing/rcode.py

@@ -0,0 +1,410 @@
+"""
+R code printer
+
+The RCodePrinter converts single SymPy expressions into single R expressions,
+using the functions defined in math.h where possible.
+
+
+
+"""
+
+from __future__ import annotations
+from typing import Any
+
+from sympy.core.numbers import equal_valued
+from sympy.printing.codeprinter import CodePrinter
+from sympy.printing.precedence import precedence, PRECEDENCE
+from sympy.sets.fancysets import Range
+
+# dictionary mapping SymPy function to (argument_conditions, C_function).
+# Used in RCodePrinter._print_Function(self)
+known_functions = {
+    #"Abs": [(lambda x: not x.is_integer, "fabs")],
+    "Abs": "abs",
+    "sin": "sin",
+    "cos": "cos",
+    "tan": "tan",
+    "asin": "asin",
+    "acos": "acos",
+    "atan": "atan",
+    "atan2": "atan2",
+    "exp": "exp",
+    "log": "log",
+    "erf": "erf",
+    "sinh": "sinh",
+    "cosh": "cosh",
+    "tanh": "tanh",
+    "asinh": "asinh",
+    "acosh": "acosh",
+    "atanh": "atanh",
+    "floor": "floor",
+    "ceiling": "ceiling",
+    "sign": "sign",
+    "Max": "max",
+    "Min": "min",
+    "factorial": "factorial",
+    "gamma": "gamma",
+    "digamma": "digamma",
+    "trigamma": "trigamma",
+    "beta": "beta",
+    "sqrt": "sqrt",  # To enable automatic rewrite
+}
+
+# These are the core reserved words in the R language. Taken from:
+# https://cran.r-project.org/doc/manuals/r-release/R-lang.html#Reserved-words
+
+reserved_words = ['if',
+                  'else',
+                  'repeat',
+                  'while',
+                  'function',
+                  'for',
+                  'in',
+                  'next',
+                  'break',
+                  'TRUE',
+                  'FALSE',
+                  'NULL',
+                  'Inf',
+                  'NaN',
+                  'NA',
+                  'NA_integer_',
+                  'NA_real_',
+                  'NA_complex_',
+                  'NA_character_',
+                  'volatile']
+
+
+class RCodePrinter(CodePrinter):
+    """A printer to convert SymPy expressions to strings of R code"""
+    printmethod = "_rcode"
+    language = "R"
+
+    _default_settings: dict[str, Any] = {
+        'order': None,
+        'full_prec': 'auto',
+        'precision': 15,
+        'user_functions': {},
+        'human': True,
+        'contract': True,
+        'dereference': set(),
+        'error_on_reserved': False,
+        'reserved_word_suffix': '_',
+    }
+    _operators = {
+       'and': '&',
+        'or': '|',
+       'not': '!',
+    }
+
+    _relationals: dict[str, str] = {}
+
+    def __init__(self, settings={}):
+        CodePrinter.__init__(self, settings)
+        self.known_functions = dict(known_functions)
+        userfuncs = settings.get('user_functions', {})
+        self.known_functions.update(userfuncs)
+        self._dereference = set(settings.get('dereference', []))
+        self.reserved_words = set(reserved_words)
+
+    def _rate_index_position(self, p):
+        return p*5
+
+    def _get_statement(self, codestring):
+        return "%s;" % codestring
+
+    def _get_comment(self, text):
+        return "// {}".format(text)
+
+    def _declare_number_const(self, name, value):
+        return "{} = {};".format(name, value)
+
+    def _format_code(self, lines):
+        return self.indent_code(lines)
+
+    def _traverse_matrix_indices(self, mat):
+        rows, cols = mat.shape
+        return ((i, j) for i in range(rows) for j in range(cols))
+
+    def _get_loop_opening_ending(self, indices):
+        """Returns a tuple (open_lines, close_lines) containing lists of codelines
+        """
+        open_lines = []
+        close_lines = []
+        loopstart = "for (%(var)s in %(start)s:%(end)s){"
+        for i in indices:
+            # R arrays start at 1 and end at dimension
+            open_lines.append(loopstart % {
+                'var': self._print(i.label),
+                'start': self._print(i.lower+1),
+                'end': self._print(i.upper + 1)})
+            close_lines.append("}")
+        return open_lines, close_lines
+
+    def _print_Pow(self, expr):
+        if "Pow" in self.known_functions:
+            return self._print_Function(expr)
+        PREC = precedence(expr)
+        if equal_valued(expr.exp, -1):
+            return '1.0/%s' % (self.parenthesize(expr.base, PREC))
+        elif equal_valued(expr.exp, 0.5):
+            return 'sqrt(%s)' % self._print(expr.base)
+        else:
+            return '%s^%s' % (self.parenthesize(expr.base, PREC),
+                                 self.parenthesize(expr.exp, PREC))
+
+
+    def _print_Rational(self, expr):
+        p, q = int(expr.p), int(expr.q)
+        return '%d.0/%d.0' % (p, q)
+
+    def _print_Indexed(self, expr):
+        inds = [ self._print(i) for i in expr.indices ]
+        return "%s[%s]" % (self._print(expr.base.label), ", ".join(inds))
+
+    def _print_Idx(self, expr):
+        return self._print(expr.label)
+
+    def _print_Exp1(self, expr):
+        return "exp(1)"
+
+    def _print_Pi(self, expr):
+        return 'pi'
+
+    def _print_Infinity(self, expr):
+        return 'Inf'
+
+    def _print_NegativeInfinity(self, expr):
+        return '-Inf'
+
+    def _print_Assignment(self, expr):
+        from sympy.codegen.ast import Assignment
+
+        from sympy.matrices.expressions.matexpr import MatrixSymbol
+        from sympy.tensor.indexed import IndexedBase
+        lhs = expr.lhs
+        rhs = expr.rhs
+        # We special case assignments that take multiple lines
+        #if isinstance(expr.rhs, Piecewise):
+        #    from sympy.functions.elementary.piecewise import Piecewise
+        #    # Here we modify Piecewise so each expression is now
+        #    # an Assignment, and then continue on the print.
+        #    expressions = []
+        #    conditions = []
+        #    for (e, c) in rhs.args:
+        #        expressions.append(Assignment(lhs, e))
+        #        conditions.append(c)
+        #    temp = Piecewise(*zip(expressions, conditions))
+        #    return self._print(temp)
+        #elif isinstance(lhs, MatrixSymbol):
+        if isinstance(lhs, MatrixSymbol):
+            # Here we form an Assignment for each element in the array,
+            # printing each one.
+            lines = []
+            for (i, j) in self._traverse_matrix_indices(lhs):
+                temp = Assignment(lhs[i, j], rhs[i, j])
+                code0 = self._print(temp)
+                lines.append(code0)
+            return "\n".join(lines)
+        elif self._settings["contract"] and (lhs.has(IndexedBase) or
+                rhs.has(IndexedBase)):
+            # Here we check if there is looping to be done, and if so
+            # print the required loops.
+            return self._doprint_loops(rhs, lhs)
+        else:
+            lhs_code = self._print(lhs)
+            rhs_code = self._print(rhs)
+            return self._get_statement("%s = %s" % (lhs_code, rhs_code))
+
+    def _print_Piecewise(self, expr):
+        # This method is called only for inline if constructs
+        # Top level piecewise is handled in doprint()
+        if expr.args[-1].cond == True:
+            last_line = "%s" % self._print(expr.args[-1].expr)
+        else:
+            last_line = "ifelse(%s,%s,NA)" % (self._print(expr.args[-1].cond), self._print(expr.args[-1].expr))
+        code=last_line
+        for e, c in reversed(expr.args[:-1]):
+            code= "ifelse(%s,%s," % (self._print(c), self._print(e))+code+")"
+        return(code)
+
+    def _print_ITE(self, expr):
+        from sympy.functions import Piecewise
+        return self._print(expr.rewrite(Piecewise))
+
+    def _print_MatrixElement(self, expr):
+        return "{}[{}]".format(self.parenthesize(expr.parent, PRECEDENCE["Atom"],
+            strict=True), expr.j + expr.i*expr.parent.shape[1])
+
+    def _print_Symbol(self, expr):
+        name = super()._print_Symbol(expr)
+        if expr in self._dereference:
+            return '(*{})'.format(name)
+        else:
+            return name
+
+    def _print_Relational(self, expr):
+        lhs_code = self._print(expr.lhs)
+        rhs_code = self._print(expr.rhs)
+        op = expr.rel_op
+        return "{} {} {}".format(lhs_code, op, rhs_code)
+
+    def _print_AugmentedAssignment(self, expr):
+        lhs_code = self._print(expr.lhs)
+        op = expr.op
+        rhs_code = self._print(expr.rhs)
+        return "{} {} {};".format(lhs_code, op, rhs_code)
+
+    def _print_For(self, expr):
+        target = self._print(expr.target)
+        if isinstance(expr.iterable, Range):
+            start, stop, step = expr.iterable.args
+        else:
+            raise NotImplementedError("Only iterable currently supported is Range")
+        body = self._print(expr.body)
+        return 'for({target} in seq(from={start}, to={stop}, by={step}){{\n{body}\n}}'.format(target=target, start=start,
+                stop=stop-1, step=step, body=body)
+
+
+    def indent_code(self, code):
+        """Accepts a string of code or a list of code lines"""
+
+        if isinstance(code, str):
+            code_lines = self.indent_code(code.splitlines(True))
+            return ''.join(code_lines)
+
+        tab = "   "
+        inc_token = ('{', '(', '{\n', '(\n')
+        dec_token = ('}', ')')
+
+        code = [ line.lstrip(' \t') for line in code ]
+
+        increase = [ int(any(map(line.endswith, inc_token))) for line in code ]
+        decrease = [ int(any(map(line.startswith, dec_token)))
+                     for line in code ]
+
+        pretty = []
+        level = 0
+        for n, line in enumerate(code):
+            if line in ('', '\n'):
+                pretty.append(line)
+                continue
+            level -= decrease[n]
+            pretty.append("%s%s" % (tab*level, line))
+            level += increase[n]
+        return pretty
+
+
+def rcode(expr, assign_to=None, **settings):
+    """Converts an expr to a string of r code
+
+    Parameters
+    ==========
+
+    expr : Expr
+        A SymPy expression to be converted.
+    assign_to : optional
+        When given, the argument is used as the name of the variable to which
+        the expression is assigned. Can be a string, ``Symbol``,
+        ``MatrixSymbol``, or ``Indexed`` type. This is helpful in case of
+        line-wrapping, or for expressions that generate multi-line statements.
+    precision : integer, optional
+        The precision for numbers such as pi [default=15].
+    user_functions : dict, optional
+        A dictionary where the keys are string representations of either
+        ``FunctionClass`` or ``UndefinedFunction`` instances and the values
+        are their desired R string representations. Alternatively, the
+        dictionary value can be a list of tuples i.e. [(argument_test,
+        rfunction_string)] or [(argument_test, rfunction_formater)]. See below
+        for examples.
+    human : bool, optional
+        If True, the result is a single string that may contain some constant
+        declarations for the number symbols. If False, the same information is
+        returned in a tuple of (symbols_to_declare, not_supported_functions,
+        code_text). [default=True].
+    contract: bool, optional
+        If True, ``Indexed`` instances are assumed to obey tensor contraction
+        rules and the corresponding nested loops over indices are generated.
+        Setting contract=False will not generate loops, instead the user is
+        responsible to provide values for the indices in the code.
+        [default=True].
+
+    Examples
+    ========
+
+    >>> from sympy import rcode, symbols, Rational, sin, ceiling, Abs, Function
+    >>> x, tau = symbols("x, tau")
+    >>> rcode((2*tau)**Rational(7, 2))
+    '8*sqrt(2)*tau^(7.0/2.0)'
+    >>> rcode(sin(x), assign_to="s")
+    's = sin(x);'
+
+    Simple custom printing can be defined for certain types by passing a
+    dictionary of {"type" : "function"} to the ``user_functions`` kwarg.
+    Alternatively, the dictionary value can be a list of tuples i.e.
+    [(argument_test, cfunction_string)].
+
+    >>> custom_functions = {
+    ...   "ceiling": "CEIL",
+    ...   "Abs": [(lambda x: not x.is_integer, "fabs"),
+    ...           (lambda x: x.is_integer, "ABS")],
+    ...   "func": "f"
+    ... }
+    >>> func = Function('func')
+    >>> rcode(func(Abs(x) + ceiling(x)), user_functions=custom_functions)
+    'f(fabs(x) + CEIL(x))'
+
+    or if the R-function takes a subset of the original arguments:
+
+    >>> rcode(2**x + 3**x, user_functions={'Pow': [
+    ...   (lambda b, e: b == 2, lambda b, e: 'exp2(%s)' % e),
+    ...   (lambda b, e: b != 2, 'pow')]})
+    'exp2(x) + pow(3, x)'
+
+    ``Piecewise`` expressions are converted into conditionals. If an
+    ``assign_to`` variable is provided an if statement is created, otherwise
+    the ternary operator is used. Note that if the ``Piecewise`` lacks a
+    default term, represented by ``(expr, True)`` then an error will be thrown.
+    This is to prevent generating an expression that may not evaluate to
+    anything.
+
+    >>> from sympy import Piecewise
+    >>> expr = Piecewise((x + 1, x > 0), (x, True))
+    >>> print(rcode(expr, assign_to=tau))
+    tau = ifelse(x > 0,x + 1,x);
+
+    Support for loops is provided through ``Indexed`` types. With
+    ``contract=True`` these expressions will be turned into loops, whereas
+    ``contract=False`` will just print the assignment expression that should be
+    looped over:
+
+    >>> from sympy import Eq, IndexedBase, Idx
+    >>> len_y = 5
+    >>> y = IndexedBase('y', shape=(len_y,))
+    >>> t = IndexedBase('t', shape=(len_y,))
+    >>> Dy = IndexedBase('Dy', shape=(len_y-1,))
+    >>> i = Idx('i', len_y-1)
+    >>> e=Eq(Dy[i], (y[i+1]-y[i])/(t[i+1]-t[i]))
+    >>> rcode(e.rhs, assign_to=e.lhs, contract=False)
+    'Dy[i] = (y[i + 1] - y[i])/(t[i + 1] - t[i]);'
+
+    Matrices are also supported, but a ``MatrixSymbol`` of the same dimensions
+    must be provided to ``assign_to``. Note that any expression that can be
+    generated normally can also exist inside a Matrix:
+
+    >>> from sympy import Matrix, MatrixSymbol
+    >>> mat = Matrix([x**2, Piecewise((x + 1, x > 0), (x, True)), sin(x)])
+    >>> A = MatrixSymbol('A', 3, 1)
+    >>> print(rcode(mat, A))
+    A[0] = x^2;
+    A[1] = ifelse(x > 0,x + 1,x);
+    A[2] = sin(x);
+
+    """
+
+    return RCodePrinter(settings).doprint(expr, assign_to)
+
+
+def print_rcode(expr, **settings):
+    """Prints R representation of the given expression."""
+    print(rcode(expr, **settings))

env-llmeval/lib/python3.10/site-packages/sympy/printing/rust.py

@@ -0,0 +1,625 @@
+"""
+Rust code printer
+
+The `RustCodePrinter` converts SymPy expressions into Rust expressions.
+
+A complete code generator, which uses `rust_code` extensively, can be found
+in `sympy.utilities.codegen`. The `codegen` module can be used to generate
+complete source code files.
+
+"""
+
+# Possible Improvement
+#
+# * make sure we follow Rust Style Guidelines_
+# * make use of pattern matching
+# * better support for reference
+# * generate generic code and use trait to make sure they have specific methods
+# * use crates_ to get more math support
+#     - num_
+#         + BigInt_, BigUint_
+#         + Complex_
+#         + Rational64_, Rational32_, BigRational_
+#
+# .. _crates: https://crates.io/
+# .. _Guidelines: https://github.com/rust-lang/rust/tree/master/src/doc/style
+# .. _num: http://rust-num.github.io/num/num/
+# .. _BigInt: http://rust-num.github.io/num/num/bigint/struct.BigInt.html
+# .. _BigUint: http://rust-num.github.io/num/num/bigint/struct.BigUint.html
+# .. _Complex: http://rust-num.github.io/num/num/complex/struct.Complex.html
+# .. _Rational32: http://rust-num.github.io/num/num/rational/type.Rational32.html
+# .. _Rational64: http://rust-num.github.io/num/num/rational/type.Rational64.html
+# .. _BigRational: http://rust-num.github.io/num/num/rational/type.BigRational.html
+
+from __future__ import annotations
+from typing import Any
+
+from sympy.core import S, Rational, Float, Lambda
+from sympy.core.numbers import equal_valued
+from sympy.printing.codeprinter import CodePrinter
+
+# Rust's methods for integer and float can be found at here :
+#
+# * `Rust - Primitive Type f64 <https://doc.rust-lang.org/std/primitive.f64.html>`_
+# * `Rust - Primitive Type i64 <https://doc.rust-lang.org/std/primitive.i64.html>`_
+#
+# Function Style :
+#
+# 1. args[0].func(args[1:]), method with arguments
+# 2. args[0].func(), method without arguments
+# 3. args[1].func(), method without arguments (e.g. (e, x) => x.exp())
+# 4. func(args), function with arguments
+
+# dictionary mapping SymPy function to (argument_conditions, Rust_function).
+# Used in RustCodePrinter._print_Function(self)
+
+# f64 method in Rust
+known_functions = {
+    # "": "is_nan",
+    # "": "is_infinite",
+    # "": "is_finite",
+    # "": "is_normal",
+    # "": "classify",
+    "floor": "floor",
+    "ceiling": "ceil",
+    # "": "round",
+    # "": "trunc",
+    # "": "fract",
+    "Abs": "abs",
+    "sign": "signum",
+    # "": "is_sign_positive",
+    # "": "is_sign_negative",
+    # "": "mul_add",
+    "Pow": [(lambda base, exp: equal_valued(exp, -1), "recip", 2),   # 1.0/x
+            (lambda base, exp: equal_valued(exp, 0.5), "sqrt", 2),   # x ** 0.5
+            (lambda base, exp: equal_valued(exp, -0.5), "sqrt().recip", 2),   # 1/(x ** 0.5)
+            (lambda base, exp: exp == Rational(1, 3), "cbrt", 2),    # x ** (1/3)
+            (lambda base, exp: equal_valued(base, 2), "exp2", 3),    # 2 ** x
+            (lambda base, exp: exp.is_integer, "powi", 1),           # x ** y, for i32
+            (lambda base, exp: not exp.is_integer, "powf", 1)],      # x ** y, for f64
+    "exp": [(lambda exp: True, "exp", 2)],   # e ** x
+    "log": "ln",
+    # "": "log",          # number.log(base)
+    # "": "log2",
+    # "": "log10",
+    # "": "to_degrees",
+    # "": "to_radians",
+    "Max": "max",
+    "Min": "min",
+    # "": "hypot",        # (x**2 + y**2) ** 0.5
+    "sin": "sin",
+    "cos": "cos",
+    "tan": "tan",
+    "asin": "asin",
+    "acos": "acos",
+    "atan": "atan",
+    "atan2": "atan2",
+    # "": "sin_cos",
+    # "": "exp_m1",       # e ** x - 1
+    # "": "ln_1p",        # ln(1 + x)
+    "sinh": "sinh",
+    "cosh": "cosh",
+    "tanh": "tanh",
+    "asinh": "asinh",
+    "acosh": "acosh",
+    "atanh": "atanh",
+    "sqrt": "sqrt",  # To enable automatic rewrites
+}
+
+# i64 method in Rust
+# known_functions_i64 = {
+#     "": "min_value",
+#     "": "max_value",
+#     "": "from_str_radix",
+#     "": "count_ones",
+#     "": "count_zeros",
+#     "": "leading_zeros",
+#     "": "trainling_zeros",
+#     "": "rotate_left",
+#     "": "rotate_right",
+#     "": "swap_bytes",
+#     "": "from_be",
+#     "": "from_le",
+#     "": "to_be",    # to big endian
+#     "": "to_le",    # to little endian
+#     "": "checked_add",
+#     "": "checked_sub",
+#     "": "checked_mul",
+#     "": "checked_div",
+#     "": "checked_rem",
+#     "": "checked_neg",
+#     "": "checked_shl",
+#     "": "checked_shr",
+#     "": "checked_abs",
+#     "": "saturating_add",
+#     "": "saturating_sub",
+#     "": "saturating_mul",
+#     "": "wrapping_add",
+#     "": "wrapping_sub",
+#     "": "wrapping_mul",
+#     "": "wrapping_div",
+#     "": "wrapping_rem",
+#     "": "wrapping_neg",
+#     "": "wrapping_shl",
+#     "": "wrapping_shr",
+#     "": "wrapping_abs",
+#     "": "overflowing_add",
+#     "": "overflowing_sub",
+#     "": "overflowing_mul",
+#     "": "overflowing_div",
+#     "": "overflowing_rem",
+#     "": "overflowing_neg",
+#     "": "overflowing_shl",
+#     "": "overflowing_shr",
+#     "": "overflowing_abs",
+#     "Pow": "pow",
+#     "Abs": "abs",
+#     "sign": "signum",
+#     "": "is_positive",
+#     "": "is_negnative",
+# }
+
+# These are the core reserved words in the Rust language. Taken from:
+# http://doc.rust-lang.org/grammar.html#keywords
+
+reserved_words = ['abstract',
+                  'alignof',
+                  'as',
+                  'become',
+                  'box',
+                  'break',
+                  'const',
+                  'continue',
+                  'crate',
+                  'do',
+                  'else',
+                  'enum',
+                  'extern',
+                  'false',
+                  'final',
+                  'fn',
+                  'for',
+                  'if',
+                  'impl',
+                  'in',
+                  'let',
+                  'loop',
+                  'macro',
+                  'match',
+                  'mod',
+                  'move',
+                  'mut',
+                  'offsetof',
+                  'override',
+                  'priv',
+                  'proc',
+                  'pub',
+                  'pure',
+                  'ref',
+                  'return',
+                  'Self',
+                  'self',
+                  'sizeof',
+                  'static',
+                  'struct',
+                  'super',
+                  'trait',
+                  'true',
+                  'type',
+                  'typeof',
+                  'unsafe',
+                  'unsized',
+                  'use',
+                  'virtual',
+                  'where',
+                  'while',
+                  'yield']
+
+
+class RustCodePrinter(CodePrinter):
+    """A printer to convert SymPy expressions to strings of Rust code"""
+    printmethod = "_rust_code"
+    language = "Rust"
+
+    _default_settings: dict[str, Any] = {
+        'order': None,
+        'full_prec': 'auto',
+        'precision': 17,
+        'user_functions': {},
+        'human': True,
+        'contract': True,
+        'dereference': set(),
+        'error_on_reserved': False,
+        'reserved_word_suffix': '_',
+        'inline': False,
+    }
+
+    def __init__(self, settings={}):
+        CodePrinter.__init__(self, settings)
+        self.known_functions = dict(known_functions)
+        userfuncs = settings.get('user_functions', {})
+        self.known_functions.update(userfuncs)
+        self._dereference = set(settings.get('dereference', []))
+        self.reserved_words = set(reserved_words)
+
+    def _rate_index_position(self, p):
+        return p*5
+
+    def _get_statement(self, codestring):
+        return "%s;" % codestring
+
+    def _get_comment(self, text):
+        return "// %s" % text
+
+    def _declare_number_const(self, name, value):
+        return "const %s: f64 = %s;" % (name, value)
+
+    def _format_code(self, lines):
+        return self.indent_code(lines)
+
+    def _traverse_matrix_indices(self, mat):
+        rows, cols = mat.shape
+        return ((i, j) for i in range(rows) for j in range(cols))
+
+    def _get_loop_opening_ending(self, indices):
+        open_lines = []
+        close_lines = []
+        loopstart = "for %(var)s in %(start)s..%(end)s {"
+        for i in indices:
+            # Rust arrays start at 0 and end at dimension-1
+            open_lines.append(loopstart % {
+                'var': self._print(i),
+                'start': self._print(i.lower),
+                'end': self._print(i.upper + 1)})
+            close_lines.append("}")
+        return open_lines, close_lines
+
+    def _print_caller_var(self, expr):
+        if len(expr.args) > 1:
+            # for something like `sin(x + y + z)`,
+            # make sure we can get '(x + y + z).sin()'
+            # instead of 'x + y + z.sin()'
+            return '(' + self._print(expr) + ')'
+        elif expr.is_number:
+            return self._print(expr, _type=True)
+        else:
+            return self._print(expr)
+
+    def _print_Function(self, expr):
+        """
+        basic function for printing `Function`
+
+        Function Style :
+
+        1. args[0].func(args[1:]), method with arguments
+        2. args[0].func(), method without arguments
+        3. args[1].func(), method without arguments (e.g. (e, x) => x.exp())
+        4. func(args), function with arguments
+        """
+
+        if expr.func.__name__ in self.known_functions:
+            cond_func = self.known_functions[expr.func.__name__]
+            func = None
+            style = 1
+            if isinstance(cond_func, str):
+                func = cond_func
+            else:
+                for cond, func, style in cond_func:
+                    if cond(*expr.args):
+                        break
+            if func is not None:
+                if style == 1:
+                    ret = "%(var)s.%(method)s(%(args)s)" % {
+                        'var': self._print_caller_var(expr.args[0]),
+                        'method': func,
+                        'args': self.stringify(expr.args[1:], ", ") if len(expr.args) > 1 else ''
+                    }
+                elif style == 2:
+                    ret = "%(var)s.%(method)s()" % {
+                        'var': self._print_caller_var(expr.args[0]),
+                        'method': func,
+                    }
+                elif style == 3:
+                    ret = "%(var)s.%(method)s()" % {
+                        'var': self._print_caller_var(expr.args[1]),
+                        'method': func,
+                    }
+                else:
+                    ret = "%(func)s(%(args)s)" % {
+                        'func': func,
+                        'args': self.stringify(expr.args, ", "),
+                    }
+                return ret
+        elif hasattr(expr, '_imp_') and isinstance(expr._imp_, Lambda):
+            # inlined function
+            return self._print(expr._imp_(*expr.args))
+        elif expr.func.__name__ in self._rewriteable_functions:
+            # Simple rewrite to supported function possible
+            target_f, required_fs = self._rewriteable_functions[expr.func.__name__]
+            if self._can_print(target_f) and all(self._can_print(f) for f in required_fs):
+                return self._print(expr.rewrite(target_f))
+        else:
+            return self._print_not_supported(expr)
+
+    def _print_Pow(self, expr):
+        if expr.base.is_integer and not expr.exp.is_integer:
+            expr = type(expr)(Float(expr.base), expr.exp)
+            return self._print(expr)
+        return self._print_Function(expr)
+
+    def _print_Float(self, expr, _type=False):
+        ret = super()._print_Float(expr)
+        if _type:
+            return ret + '_f64'
+        else:
+            return ret
+
+    def _print_Integer(self, expr, _type=False):
+        ret = super()._print_Integer(expr)
+        if _type:
+            return ret + '_i32'
+        else:
+            return ret
+
+    def _print_Rational(self, expr):
+        p, q = int(expr.p), int(expr.q)
+        return '%d_f64/%d.0' % (p, q)
+
+    def _print_Relational(self, expr):
+        lhs_code = self._print(expr.lhs)
+        rhs_code = self._print(expr.rhs)
+        op = expr.rel_op
+        return "{} {} {}".format(lhs_code, op, rhs_code)
+
+    def _print_Indexed(self, expr):
+        # calculate index for 1d array
+        dims = expr.shape
+        elem = S.Zero
+        offset = S.One
+        for i in reversed(range(expr.rank)):
+            elem += expr.indices[i]*offset
+            offset *= dims[i]
+        return "%s[%s]" % (self._print(expr.base.label), self._print(elem))
+
+    def _print_Idx(self, expr):
+        return expr.label.name
+
+    def _print_Dummy(self, expr):
+        return expr.name
+
+    def _print_Exp1(self, expr, _type=False):
+        return "E"
+
+    def _print_Pi(self, expr, _type=False):
+        return 'PI'
+
+    def _print_Infinity(self, expr, _type=False):
+        return 'INFINITY'
+
+    def _print_NegativeInfinity(self, expr, _type=False):
+        return 'NEG_INFINITY'
+
+    def _print_BooleanTrue(self, expr, _type=False):
+        return "true"
+
+    def _print_BooleanFalse(self, expr, _type=False):
+        return "false"
+
+    def _print_bool(self, expr, _type=False):
+        return str(expr).lower()
+
+    def _print_NaN(self, expr, _type=False):
+        return "NAN"
+
+    def _print_Piecewise(self, expr):
+        if expr.args[-1].cond != True:
+            # We need the last conditional to be a True, otherwise the resulting
+            # function may not return a result.
+            raise ValueError("All Piecewise expressions must contain an "
+                             "(expr, True) statement to be used as a default "
+                             "condition. Without one, the generated "
+                             "expression may not evaluate to anything under "
+                             "some condition.")
+        lines = []
+
+        for i, (e, c) in enumerate(expr.args):
+            if i == 0:
+                lines.append("if (%s) {" % self._print(c))
+            elif i == len(expr.args) - 1 and c == True:
+                lines[-1] += " else {"
+            else:
+                lines[-1] += " else if (%s) {" % self._print(c)
+            code0 = self._print(e)
+            lines.append(code0)
+            lines.append("}")
+
+        if self._settings['inline']:
+            return " ".join(lines)
+        else:
+            return "\n".join(lines)
+
+    def _print_ITE(self, expr):
+        from sympy.functions import Piecewise
+        return self._print(expr.rewrite(Piecewise, deep=False))
+
+    def _print_MatrixBase(self, A):
+        if A.cols == 1:
+            return "[%s]" % ", ".join(self._print(a) for a in A)
+        else:
+            raise ValueError("Full Matrix Support in Rust need Crates (https://crates.io/keywords/matrix).")
+
+    def _print_SparseRepMatrix(self, mat):
+        # do not allow sparse matrices to be made dense
+        return self._print_not_supported(mat)
+
+    def _print_MatrixElement(self, expr):
+        return "%s[%s]" % (expr.parent,
+                           expr.j + expr.i*expr.parent.shape[1])
+
+    def _print_Symbol(self, expr):
+
+        name = super()._print_Symbol(expr)
+
+        if expr in self._dereference:
+            return '(*%s)' % name
+        else:
+            return name
+
+    def _print_Assignment(self, expr):
+        from sympy.tensor.indexed import IndexedBase
+        lhs = expr.lhs
+        rhs = expr.rhs
+        if self._settings["contract"] and (lhs.has(IndexedBase) or
+                rhs.has(IndexedBase)):
+            # Here we check if there is looping to be done, and if so
+            # print the required loops.
+            return self._doprint_loops(rhs, lhs)
+        else:
+            lhs_code = self._print(lhs)
+            rhs_code = self._print(rhs)
+            return self._get_statement("%s = %s" % (lhs_code, rhs_code))
+
+    def indent_code(self, code):
+        """Accepts a string of code or a list of code lines"""
+
+        if isinstance(code, str):
+            code_lines = self.indent_code(code.splitlines(True))
+            return ''.join(code_lines)
+
+        tab = "    "
+        inc_token = ('{', '(', '{\n', '(\n')
+        dec_token = ('}', ')')
+
+        code = [ line.lstrip(' \t') for line in code ]
+
+        increase = [ int(any(map(line.endswith, inc_token))) for line in code ]
+        decrease = [ int(any(map(line.startswith, dec_token)))
+                     for line in code ]
+
+        pretty = []
+        level = 0
+        for n, line in enumerate(code):
+            if line in ('', '\n'):
+                pretty.append(line)
+                continue
+            level -= decrease[n]
+            pretty.append("%s%s" % (tab*level, line))
+            level += increase[n]
+        return pretty
+
+
+def rust_code(expr, assign_to=None, **settings):
+    """Converts an expr to a string of Rust code
+
+    Parameters
+    ==========
+
+    expr : Expr
+        A SymPy expression to be converted.
+    assign_to : optional
+        When given, the argument is used as the name of the variable to which
+        the expression is assigned. Can be a string, ``Symbol``,
+        ``MatrixSymbol``, or ``Indexed`` type. This is helpful in case of
+        line-wrapping, or for expressions that generate multi-line statements.
+    precision : integer, optional
+        The precision for numbers such as pi [default=15].
+    user_functions : dict, optional
+        A dictionary where the keys are string representations of either
+        ``FunctionClass`` or ``UndefinedFunction`` instances and the values
+        are their desired C string representations. Alternatively, the
+        dictionary value can be a list of tuples i.e. [(argument_test,
+        cfunction_string)].  See below for examples.
+    dereference : iterable, optional
+        An iterable of symbols that should be dereferenced in the printed code
+        expression. These would be values passed by address to the function.
+        For example, if ``dereference=[a]``, the resulting code would print
+        ``(*a)`` instead of ``a``.
+    human : bool, optional
+        If True, the result is a single string that may contain some constant
+        declarations for the number symbols. If False, the same information is
+        returned in a tuple of (symbols_to_declare, not_supported_functions,
+        code_text). [default=True].
+    contract: bool, optional
+        If True, ``Indexed`` instances are assumed to obey tensor contraction
+        rules and the corresponding nested loops over indices are generated.
+        Setting contract=False will not generate loops, instead the user is
+        responsible to provide values for the indices in the code.
+        [default=True].
+
+    Examples
+    ========
+
+    >>> from sympy import rust_code, symbols, Rational, sin, ceiling, Abs, Function
+    >>> x, tau = symbols("x, tau")
+    >>> rust_code((2*tau)**Rational(7, 2))
+    '8*1.4142135623731*tau.powf(7_f64/2.0)'
+    >>> rust_code(sin(x), assign_to="s")
+    's = x.sin();'
+
+    Simple custom printing can be defined for certain types by passing a
+    dictionary of {"type" : "function"} to the ``user_functions`` kwarg.
+    Alternatively, the dictionary value can be a list of tuples i.e.
+    [(argument_test, cfunction_string)].
+
+    >>> custom_functions = {
+    ...   "ceiling": "CEIL",
+    ...   "Abs": [(lambda x: not x.is_integer, "fabs", 4),
+    ...           (lambda x: x.is_integer, "ABS", 4)],
+    ...   "func": "f"
+    ... }
+    >>> func = Function('func')
+    >>> rust_code(func(Abs(x) + ceiling(x)), user_functions=custom_functions)
+    '(fabs(x) + x.CEIL()).f()'
+
+    ``Piecewise`` expressions are converted into conditionals. If an
+    ``assign_to`` variable is provided an if statement is created, otherwise
+    the ternary operator is used. Note that if the ``Piecewise`` lacks a
+    default term, represented by ``(expr, True)`` then an error will be thrown.
+    This is to prevent generating an expression that may not evaluate to
+    anything.
+
+    >>> from sympy import Piecewise
+    >>> expr = Piecewise((x + 1, x > 0), (x, True))
+    >>> print(rust_code(expr, tau))
+    tau = if (x > 0) {
+        x + 1
+    } else {
+        x
+    };
+
+    Support for loops is provided through ``Indexed`` types. With
+    ``contract=True`` these expressions will be turned into loops, whereas
+    ``contract=False`` will just print the assignment expression that should be
+    looped over:
+
+    >>> from sympy import Eq, IndexedBase, Idx
+    >>> len_y = 5
+    >>> y = IndexedBase('y', shape=(len_y,))
+    >>> t = IndexedBase('t', shape=(len_y,))
+    >>> Dy = IndexedBase('Dy', shape=(len_y-1,))
+    >>> i = Idx('i', len_y-1)
+    >>> e=Eq(Dy[i], (y[i+1]-y[i])/(t[i+1]-t[i]))
+    >>> rust_code(e.rhs, assign_to=e.lhs, contract=False)
+    'Dy[i] = (y[i + 1] - y[i])/(t[i + 1] - t[i]);'
+
+    Matrices are also supported, but a ``MatrixSymbol`` of the same dimensions
+    must be provided to ``assign_to``. Note that any expression that can be
+    generated normally can also exist inside a Matrix:
+
+    >>> from sympy import Matrix, MatrixSymbol
+    >>> mat = Matrix([x**2, Piecewise((x + 1, x > 0), (x, True)), sin(x)])
+    >>> A = MatrixSymbol('A', 3, 1)
+    >>> print(rust_code(mat, A))
+    A = [x.powi(2), if (x > 0) {
+        x + 1
+    } else {
+        x
+    }, x.sin()];
+    """
+
+    return RustCodePrinter(settings).doprint(expr, assign_to)
+
+
+def print_rust_code(expr, **settings):
+    """Prints Rust representation of the given expression."""
+    print(rust_code(expr, **settings))

env-llmeval/lib/python3.10/site-packages/sympy/printing/smtlib.py

@@ -0,0 +1,526 @@
+import typing
+
+import sympy
+from sympy.core import Add, Mul
+from sympy.core import Symbol, Expr, Float, Rational, Integer, Basic
+from sympy.core.function import UndefinedFunction, Function
+from sympy.core.relational import Relational, Unequality, Equality, LessThan, GreaterThan, StrictLessThan, StrictGreaterThan
+from sympy.functions.elementary.complexes import Abs
+from sympy.functions.elementary.exponential import exp, log, Pow
+from sympy.functions.elementary.hyperbolic import sinh, cosh, tanh
+from sympy.functions.elementary.miscellaneous import Min, Max
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.functions.elementary.trigonometric import sin, cos, tan, asin, acos, atan, atan2
+from sympy.logic.boolalg import And, Or, Xor, Implies, Boolean
+from sympy.logic.boolalg import BooleanTrue, BooleanFalse, BooleanFunction, Not, ITE
+from sympy.printing.printer import Printer
+from sympy.sets import Interval
+
+
+class SMTLibPrinter(Printer):
+    printmethod = "_smtlib"
+
+    # based on dReal, an automated reasoning tool for solving problems that can be encoded as first-order logic formulas over the real numbers.
+    # dReal's special strength is in handling problems that involve a wide range of nonlinear real functions.
+    _default_settings: dict = {
+        'precision': None,
+        'known_types': {
+            bool: 'Bool',
+            int: 'Int',
+            float: 'Real'
+        },
+        'known_constants': {
+            # pi: 'MY_VARIABLE_PI_DECLARED_ELSEWHERE',
+        },
+        'known_functions': {
+            Add: '+',
+            Mul: '*',
+
+            Equality: '=',
+            LessThan: '<=',
+            GreaterThan: '>=',
+            StrictLessThan: '<',
+            StrictGreaterThan: '>',
+
+            exp: 'exp',
+            log: 'log',
+            Abs: 'abs',
+            sin: 'sin',
+            cos: 'cos',
+            tan: 'tan',
+            asin: 'arcsin',
+            acos: 'arccos',
+            atan: 'arctan',
+            atan2: 'arctan2',
+            sinh: 'sinh',
+            cosh: 'cosh',
+            tanh: 'tanh',
+            Min: 'min',
+            Max: 'max',
+            Pow: 'pow',
+
+            And: 'and',
+            Or: 'or',
+            Xor: 'xor',
+            Not: 'not',
+            ITE: 'ite',
+            Implies: '=>',
+        }
+    }
+
+    symbol_table: dict
+
+    def __init__(self, settings: typing.Optional[dict] = None,
+                 symbol_table=None):
+        settings = settings or {}
+        self.symbol_table = symbol_table or {}
+        Printer.__init__(self, settings)
+        self._precision = self._settings['precision']
+        self._known_types = dict(self._settings['known_types'])
+        self._known_constants = dict(self._settings['known_constants'])
+        self._known_functions = dict(self._settings['known_functions'])
+
+        for _ in self._known_types.values(): assert self._is_legal_name(_)
+        for _ in self._known_constants.values(): assert self._is_legal_name(_)
+        # for _ in self._known_functions.values(): assert self._is_legal_name(_)  # +, *, <, >, etc.
+
+    def _is_legal_name(self, s: str):
+        if not s: return False
+        if s[0].isnumeric(): return False
+        return all(_.isalnum() or _ == '_' for _ in s)
+
+    def _s_expr(self, op: str, args: typing.Union[list, tuple]) -> str:
+        args_str = ' '.join(
+            a if isinstance(a, str)
+            else self._print(a)
+            for a in args
+        )
+        return f'({op} {args_str})'
+
+    def _print_Function(self, e):
+        if e in self._known_functions:
+            op = self._known_functions[e]
+        elif type(e) in self._known_functions:
+            op = self._known_functions[type(e)]
+        elif type(type(e)) == UndefinedFunction:
+            op = e.name
+        else:
+            op = self._known_functions[e]  # throw KeyError
+
+        return self._s_expr(op, e.args)
+
+    def _print_Relational(self, e: Relational):
+        return self._print_Function(e)
+
+    def _print_BooleanFunction(self, e: BooleanFunction):
+        return self._print_Function(e)
+
+    def _print_Expr(self, e: Expr):
+        return self._print_Function(e)
+
+    def _print_Unequality(self, e: Unequality):
+        if type(e) in self._known_functions:
+            return self._print_Relational(e)  # default
+        else:
+            eq_op = self._known_functions[Equality]
+            not_op = self._known_functions[Not]
+            return self._s_expr(not_op, [self._s_expr(eq_op, e.args)])
+
+    def _print_Piecewise(self, e: Piecewise):
+        def _print_Piecewise_recursive(args: typing.Union[list, tuple]):
+            e, c = args[0]
+            if len(args) == 1:
+                assert (c is True) or isinstance(c, BooleanTrue)
+                return self._print(e)
+            else:
+                ite = self._known_functions[ITE]
+                return self._s_expr(ite, [
+                    c, e, _print_Piecewise_recursive(args[1:])
+                ])
+
+        return _print_Piecewise_recursive(e.args)
+
+    def _print_Interval(self, e: Interval):
+        if e.start.is_infinite and e.end.is_infinite:
+            return ''
+        elif e.start.is_infinite != e.end.is_infinite:
+            raise ValueError(f'One-sided intervals (`{e}`) are not supported in SMT.')
+        else:
+            return f'[{e.start}, {e.end}]'
+
+    # todo: Sympy does not support quantifiers yet as of 2022, but quantifiers can be handy in SMT.
+    # For now, users can extend this class and build in their own quantifier support.
+    # See `test_quantifier_extensions()` in test_smtlib.py for an example of how this might look.
+
+    # def _print_ForAll(self, e: ForAll):
+    #     return self._s('forall', [
+    #         self._s('', [
+    #             self._s(sym.name, [self._type_name(sym), Interval(start, end)])
+    #             for sym, start, end in e.limits
+    #         ]),
+    #         e.function
+    #     ])
+
+    def _print_BooleanTrue(self, x: BooleanTrue):
+        return 'true'
+
+    def _print_BooleanFalse(self, x: BooleanFalse):
+        return 'false'
+
+    def _print_Float(self, x: Float):
+        f = x.evalf(self._precision) if self._precision else x.evalf()
+        return str(f).rstrip('0')
+
+    def _print_float(self, x: float):
+        return str(x)
+
+    def _print_Rational(self, x: Rational):
+        return self._s_expr('/', [x.p, x.q])
+
+    def _print_Integer(self, x: Integer):
+        assert x.q == 1
+        return str(x.p)
+
+    def _print_int(self, x: int):
+        return str(x)
+
+    def _print_Symbol(self, x: Symbol):
+        assert self._is_legal_name(x.name)
+        return x.name
+
+    def _print_NumberSymbol(self, x):
+        name = self._known_constants.get(x)
+        return name if name else self._print_Float(x)
+
+    def _print_UndefinedFunction(self, x):
+        assert self._is_legal_name(x.name)
+        return x.name
+
+    def _print_Exp1(self, x):
+        return (
+            self._print_Function(exp(1, evaluate=False))
+            if exp in self._known_functions else
+            self._print_NumberSymbol(x)
+        )
+
+    def emptyPrinter(self, expr):
+        raise NotImplementedError(f'Cannot convert `{repr(expr)}` of type `{type(expr)}` to SMT.')
+
+
+def smtlib_code(
+    expr,
+    auto_assert=True, auto_declare=True,
+    precision=None,
+    symbol_table=None,
+    known_types=None, known_constants=None, known_functions=None,
+    prefix_expressions=None, suffix_expressions=None,
+    log_warn=None
+):
+    r"""Converts ``expr`` to a string of smtlib code.
+
+    Parameters
+    ==========
+
+    expr : Expr | List[Expr]
+        A SymPy expression or system to be converted.
+    auto_assert : bool, optional
+        If false, do not modify expr and produce only the S-Expression equivalent of expr.
+        If true, assume expr is a system and assert each boolean element.
+    auto_declare : bool, optional
+        If false, do not produce declarations for the symbols used in expr.
+        If true, prepend all necessary declarations for variables used in expr based on symbol_table.
+    precision : integer, optional
+        The ``evalf(..)`` precision for numbers such as pi.
+    symbol_table : dict, optional
+        A dictionary where keys are ``Symbol`` or ``Function`` instances and values are their Python type i.e. ``bool``, ``int``, ``float``, or ``Callable[...]``.
+        If incomplete, an attempt will be made to infer types from ``expr``.
+    known_types: dict, optional
+        A dictionary where keys are ``bool``, ``int``, ``float`` etc. and values are their corresponding SMT type names.
+        If not given, a partial listing compatible with several solvers will be used.
+    known_functions : dict, optional
+        A dictionary where keys are ``Function``, ``Relational``, ``BooleanFunction``, or ``Expr`` instances and values are their SMT string representations.
+        If not given, a partial listing optimized for dReal solver (but compatible with others) will be used.
+    known_constants: dict, optional
+        A dictionary where keys are ``NumberSymbol`` instances and values are their SMT variable names.
+        When using this feature, extra caution must be taken to avoid naming collisions between user symbols and listed constants.
+        If not given, constants will be expanded inline i.e. ``3.14159`` instead of ``MY_SMT_VARIABLE_FOR_PI``.
+    prefix_expressions: list, optional
+        A list of lists of ``str`` and/or expressions to convert into SMTLib and prefix to the output.
+    suffix_expressions: list, optional
+        A list of lists of ``str`` and/or expressions to convert into SMTLib and postfix to the output.
+    log_warn: lambda function, optional
+        A function to record all warnings during potentially risky operations.
+        Soundness is a core value in SMT solving, so it is good to log all assumptions made.
+
+    Examples
+    ========
+    >>> from sympy import smtlib_code, symbols, sin, Eq
+    >>> x = symbols('x')
+    >>> smtlib_code(sin(x).series(x).removeO(), log_warn=print)
+    Could not infer type of `x`. Defaulting to float.
+    Non-Boolean expression `x**5/120 - x**3/6 + x` will not be asserted. Converting to SMTLib verbatim.
+    '(declare-const x Real)\n(+ x (* (/ -1 6) (pow x 3)) (* (/ 1 120) (pow x 5)))'
+
+    >>> from sympy import Rational
+    >>> x, y, tau = symbols("x, y, tau")
+    >>> smtlib_code((2*tau)**Rational(7, 2), log_warn=print)
+    Could not infer type of `tau`. Defaulting to float.
+    Non-Boolean expression `8*sqrt(2)*tau**(7/2)` will not be asserted. Converting to SMTLib verbatim.
+    '(declare-const tau Real)\n(* 8 (pow 2 (/ 1 2)) (pow tau (/ 7 2)))'
+
+    ``Piecewise`` expressions are implemented with ``ite`` expressions by default.
+    Note that if the ``Piecewise`` lacks a default term, represented by
+    ``(expr, True)`` then an error will be thrown.  This is to prevent
+    generating an expression that may not evaluate to anything.
+
+    >>> from sympy import Piecewise
+    >>> pw = Piecewise((x + 1, x > 0), (x, True))
+    >>> smtlib_code(Eq(pw, 3), symbol_table={x: float}, log_warn=print)
+    '(declare-const x Real)\n(assert (= (ite (> x 0) (+ 1 x) x) 3))'
+
+    Custom printing can be defined for certain types by passing a dictionary of
+    PythonType : "SMT Name" to the ``known_types``, ``known_constants``, and ``known_functions`` kwargs.
+
+    >>> from typing import Callable
+    >>> from sympy import Function, Add
+    >>> f = Function('f')
+    >>> g = Function('g')
+    >>> smt_builtin_funcs = {  # functions our SMT solver will understand
+    ...   f: "existing_smtlib_fcn",
+    ...   Add: "sum",
+    ... }
+    >>> user_def_funcs = {  # functions defined by the user must have their types specified explicitly
+    ...   g: Callable[[int], float],
+    ... }
+    >>> smtlib_code(f(x) + g(x), symbol_table=user_def_funcs, known_functions=smt_builtin_funcs, log_warn=print)
+    Non-Boolean expression `f(x) + g(x)` will not be asserted. Converting to SMTLib verbatim.
+    '(declare-const x Int)\n(declare-fun g (Int) Real)\n(sum (existing_smtlib_fcn x) (g x))'
+    """
+    log_warn = log_warn or (lambda _: None)
+
+    if not isinstance(expr, list): expr = [expr]
+    expr = [
+        sympy.sympify(_, strict=True, evaluate=False, convert_xor=False)
+        for _ in expr
+    ]
+
+    if not symbol_table: symbol_table = {}
+    symbol_table = _auto_infer_smtlib_types(
+        *expr, symbol_table=symbol_table
+    )
+    # See [FALLBACK RULES]
+    # Need SMTLibPrinter to populate known_functions and known_constants first.
+
+    settings = {}
+    if precision: settings['precision'] = precision
+    del precision
+
+    if known_types: settings['known_types'] = known_types
+    del known_types
+
+    if known_functions: settings['known_functions'] = known_functions
+    del known_functions
+
+    if known_constants: settings['known_constants'] = known_constants
+    del known_constants
+
+    if not prefix_expressions: prefix_expressions = []
+    if not suffix_expressions: suffix_expressions = []
+
+    p = SMTLibPrinter(settings, symbol_table)
+    del symbol_table
+
+    # [FALLBACK RULES]
+    for e in expr:
+        for sym in e.atoms(Symbol, Function):
+            if (
+                sym.is_Symbol and
+                sym not in p._known_constants and
+                sym not in p.symbol_table
+            ):
+                log_warn(f"Could not infer type of `{sym}`. Defaulting to float.")
+                p.symbol_table[sym] = float
+            if (
+                sym.is_Function and
+                type(sym) not in p._known_functions and
+                type(sym) not in p.symbol_table and
+                not sym.is_Piecewise
+            ): raise TypeError(
+                f"Unknown type of undefined function `{sym}`. "
+                f"Must be mapped to ``str`` in known_functions or mapped to ``Callable[..]`` in symbol_table."
+            )
+
+    declarations = []
+    if auto_declare:
+        constants = {sym.name: sym for e in expr for sym in e.free_symbols
+                     if sym not in p._known_constants}
+        functions = {fnc.name: fnc for e in expr for fnc in e.atoms(Function)
+                     if type(fnc) not in p._known_functions and not fnc.is_Piecewise}
+        declarations = \
+            [
+                _auto_declare_smtlib(sym, p, log_warn)
+                for sym in constants.values()
+            ] + [
+                _auto_declare_smtlib(fnc, p, log_warn)
+                for fnc in functions.values()
+            ]
+        declarations = [decl for decl in declarations if decl]
+
+    if auto_assert:
+        expr = [_auto_assert_smtlib(e, p, log_warn) for e in expr]
+
+    # return SMTLibPrinter().doprint(expr)
+    return '\n'.join([
+        # ';; PREFIX EXPRESSIONS',
+        *[
+            e if isinstance(e, str) else p.doprint(e)
+            for e in prefix_expressions
+        ],
+
+        # ';; DECLARATIONS',
+        *sorted(e for e in declarations),
+
+        # ';; EXPRESSIONS',
+        *[
+            e if isinstance(e, str) else p.doprint(e)
+            for e in expr
+        ],
+
+        # ';; SUFFIX EXPRESSIONS',
+        *[
+            e if isinstance(e, str) else p.doprint(e)
+            for e in suffix_expressions
+        ],
+    ])
+
+
+def _auto_declare_smtlib(sym: typing.Union[Symbol, Function], p: SMTLibPrinter, log_warn: typing.Callable[[str], None]):
+    if sym.is_Symbol:
+        type_signature = p.symbol_table[sym]
+        assert isinstance(type_signature, type)
+        type_signature = p._known_types[type_signature]
+        return p._s_expr('declare-const', [sym, type_signature])
+
+    elif sym.is_Function:
+        type_signature = p.symbol_table[type(sym)]
+        assert callable(type_signature)
+        type_signature = [p._known_types[_] for _ in type_signature.__args__]
+        assert len(type_signature) > 0
+        params_signature = f"({' '.join(type_signature[:-1])})"
+        return_signature = type_signature[-1]
+        return p._s_expr('declare-fun', [type(sym), params_signature, return_signature])
+
+    else:
+        log_warn(f"Non-Symbol/Function `{sym}` will not be declared.")
+        return None
+
+
+def _auto_assert_smtlib(e: Expr, p: SMTLibPrinter, log_warn: typing.Callable[[str], None]):
+    if isinstance(e, Boolean) or (
+        e in p.symbol_table and p.symbol_table[e] == bool
+    ) or (
+        e.is_Function and
+        type(e) in p.symbol_table and
+        p.symbol_table[type(e)].__args__[-1] == bool
+    ):
+        return p._s_expr('assert', [e])
+    else:
+        log_warn(f"Non-Boolean expression `{e}` will not be asserted. Converting to SMTLib verbatim.")
+        return e
+
+
+def _auto_infer_smtlib_types(
+    *exprs: Basic,
+    symbol_table: typing.Optional[dict] = None
+) -> dict:
+    # [TYPE INFERENCE RULES]
+    # X is alone in an expr => X is bool
+    # X in BooleanFunction.args => X is bool
+    # X matches to a bool param of a symbol_table function => X is bool
+    # X matches to an int param of a symbol_table function => X is int
+    # X.is_integer => X is int
+    # X == Y, where X is T => Y is T
+
+    # [FALLBACK RULES]
+    # see _auto_declare_smtlib(..)
+    # X is not bool and X is not int and X is Symbol => X is float
+    # else (e.g. X is Function) => error. must be specified explicitly.
+
+    _symbols = dict(symbol_table) if symbol_table else {}
+
+    def safe_update(syms: set, inf):
+        for s in syms:
+            assert s.is_Symbol
+            if (old_type := _symbols.setdefault(s, inf)) != inf:
+                raise TypeError(f"Could not infer type of `{s}`. Apparently both `{old_type}` and `{inf}`?")
+
+    # EXPLICIT TYPES
+    safe_update({
+        e
+        for e in exprs
+        if e.is_Symbol
+    }, bool)
+
+    safe_update({
+        symbol
+        for e in exprs
+        for boolfunc in e.atoms(BooleanFunction)
+        for symbol in boolfunc.args
+        if symbol.is_Symbol
+    }, bool)
+
+    safe_update({
+        symbol
+        for e in exprs
+        for boolfunc in e.atoms(Function)
+        if type(boolfunc) in _symbols
+        for symbol, param in zip(boolfunc.args, _symbols[type(boolfunc)].__args__)
+        if symbol.is_Symbol and param == bool
+    }, bool)
+
+    safe_update({
+        symbol
+        for e in exprs
+        for intfunc in e.atoms(Function)
+        if type(intfunc) in _symbols
+        for symbol, param in zip(intfunc.args, _symbols[type(intfunc)].__args__)
+        if symbol.is_Symbol and param == int
+    }, int)
+
+    safe_update({
+        symbol
+        for e in exprs
+        for symbol in e.atoms(Symbol)
+        if symbol.is_integer
+    }, int)
+
+    safe_update({
+        symbol
+        for e in exprs
+        for symbol in e.atoms(Symbol)
+        if symbol.is_real and not symbol.is_integer
+    }, float)
+
+    # EQUALITY RELATION RULE
+    rels = [rel for expr in exprs for rel in expr.atoms(Equality)]
+    rels = [
+               (rel.lhs, rel.rhs) for rel in rels if rel.lhs.is_Symbol
+           ] + [
+               (rel.rhs, rel.lhs) for rel in rels if rel.rhs.is_Symbol
+           ]
+    for infer, reltd in rels:
+        inference = (
+            _symbols[infer] if infer in _symbols else
+            _symbols[reltd] if reltd in _symbols else
+
+            _symbols[type(reltd)].__args__[-1]
+            if reltd.is_Function and type(reltd) in _symbols else
+
+            bool if reltd.is_Boolean else
+            int if reltd.is_integer or reltd.is_Integer else
+            float if reltd.is_real else
+            None
+        )
+        if inference: safe_update({infer}, inference)
+
+    return _symbols

env-llmeval/lib/python3.10/site-packages/sympy/printing/tableform.py

@@ -0,0 +1,366 @@
+from sympy.core.containers import Tuple
+from sympy.core.singleton import S
+from sympy.core.symbol import Symbol
+from sympy.core.sympify import SympifyError
+
+from types import FunctionType
+
+
+class TableForm:
+    r"""
+    Create a nice table representation of data.
+
+    Examples
+    ========
+
+    >>> from sympy import TableForm
+    >>> t = TableForm([[5, 7], [4, 2], [10, 3]])
+    >>> print(t)
+    5  7
+    4  2
+    10 3
+
+    You can use the SymPy's printing system to produce tables in any
+    format (ascii, latex, html, ...).
+
+    >>> print(t.as_latex())
+    \begin{tabular}{l l}
+    $5$ & $7$ \\
+    $4$ & $2$ \\
+    $10$ & $3$ \\
+    \end{tabular}
+
+    """
+
+    def __init__(self, data, **kwarg):
+        """
+        Creates a TableForm.
+
+        Parameters:
+
+            data ...
+                            2D data to be put into the table; data can be
+                            given as a Matrix
+
+            headings ...
+                            gives the labels for rows and columns:
+
+                            Can be a single argument that applies to both
+                            dimensions:
+
+                                - None ... no labels
+                                - "automatic" ... labels are 1, 2, 3, ...
+
+                            Can be a list of labels for rows and columns:
+                            The labels for each dimension can be given
+                            as None, "automatic", or [l1, l2, ...] e.g.
+                            ["automatic", None] will number the rows
+
+                            [default: None]
+
+            alignments ...
+                            alignment of the columns with:
+
+                                - "left" or "<"
+                                - "center" or "^"
+                                - "right" or ">"
+
+                            When given as a single value, the value is used for
+                            all columns. The row headings (if given) will be
+                            right justified unless an explicit alignment is
+                            given for it and all other columns.
+
+                            [default: "left"]
+
+            formats ...
+                            a list of format strings or functions that accept
+                            3 arguments (entry, row number, col number) and
+                            return a string for the table entry. (If a function
+                            returns None then the _print method will be used.)
+
+            wipe_zeros ...
+                            Do not show zeros in the table.
+
+                            [default: True]
+
+            pad ...
+                            the string to use to indicate a missing value (e.g.
+                            elements that are None or those that are missing
+                            from the end of a row (i.e. any row that is shorter
+                            than the rest is assumed to have missing values).
+                            When None, nothing will be shown for values that
+                            are missing from the end of a row; values that are
+                            None, however, will be shown.
+
+                            [default: None]
+
+        Examples
+        ========
+
+        >>> from sympy import TableForm, Symbol
+        >>> TableForm([[5, 7], [4, 2], [10, 3]])
+        5  7
+        4  2
+        10 3
+        >>> TableForm([list('.'*i) for i in range(1, 4)], headings='automatic')
+          | 1 2 3
+        ---------
+        1 | .
+        2 | . .
+        3 | . . .
+        >>> TableForm([[Symbol('.'*(j if not i%2 else 1)) for i in range(3)]
+        ...            for j in range(4)], alignments='rcl')
+            .
+          . . .
+         .. . ..
+        ... . ...
+        """
+        from sympy.matrices.dense import Matrix
+
+        # We only support 2D data. Check the consistency:
+        if isinstance(data, Matrix):
+            data = data.tolist()
+        _h = len(data)
+
+        # fill out any short lines
+        pad = kwarg.get('pad', None)
+        ok_None = False
+        if pad is None:
+            pad = " "
+            ok_None = True
+        pad = Symbol(pad)
+        _w = max(len(line) for line in data)
+        for i, line in enumerate(data):
+            if len(line) != _w:
+                line.extend([pad]*(_w - len(line)))
+            for j, lj in enumerate(line):
+                if lj is None:
+                    if not ok_None:
+                        lj = pad
+                else:
+                    try:
+                        lj = S(lj)
+                    except SympifyError:
+                        lj = Symbol(str(lj))
+                line[j] = lj
+            data[i] = line
+        _lines = Tuple(*[Tuple(*d) for d in data])
+
+        headings = kwarg.get("headings", [None, None])
+        if headings == "automatic":
+            _headings = [range(1, _h + 1), range(1, _w + 1)]
+        else:
+            h1, h2 = headings
+            if h1 == "automatic":
+                h1 = range(1, _h + 1)
+            if h2 == "automatic":
+                h2 = range(1, _w + 1)
+            _headings = [h1, h2]
+
+        allow = ('l', 'r', 'c')
+        alignments = kwarg.get("alignments", "l")
+
+        def _std_align(a):
+            a = a.strip().lower()
+            if len(a) > 1:
+                return {'left': 'l', 'right': 'r', 'center': 'c'}.get(a, a)
+            else:
+                return {'<': 'l', '>': 'r', '^': 'c'}.get(a, a)
+        std_align = _std_align(alignments)
+        if std_align in allow:
+            _alignments = [std_align]*_w
+        else:
+            _alignments = []
+            for a in alignments:
+                std_align = _std_align(a)
+                _alignments.append(std_align)
+                if std_align not in ('l', 'r', 'c'):
+                    raise ValueError('alignment "%s" unrecognized' %
+                        alignments)
+        if _headings[0] and len(_alignments) == _w + 1:
+            _head_align = _alignments[0]
+            _alignments = _alignments[1:]
+        else:
+            _head_align = 'r'
+        if len(_alignments) != _w:
+            raise ValueError(
+                'wrong number of alignments: expected %s but got %s' %
+                (_w, len(_alignments)))
+
+        _column_formats = kwarg.get("formats", [None]*_w)
+
+        _wipe_zeros = kwarg.get("wipe_zeros", True)
+
+        self._w = _w
+        self._h = _h
+        self._lines = _lines
+        self._headings = _headings
+        self._head_align = _head_align
+        self._alignments = _alignments
+        self._column_formats = _column_formats
+        self._wipe_zeros = _wipe_zeros
+
+    def __repr__(self):
+        from .str import sstr
+        return sstr(self, order=None)
+
+    def __str__(self):
+        from .str import sstr
+        return sstr(self, order=None)
+
+    def as_matrix(self):
+        """Returns the data of the table in Matrix form.
+
+        Examples
+        ========
+
+        >>> from sympy import TableForm
+        >>> t = TableForm([[5, 7], [4, 2], [10, 3]], headings='automatic')
+        >>> t
+          | 1  2
+        --------
+        1 | 5  7
+        2 | 4  2
+        3 | 10 3
+        >>> t.as_matrix()
+        Matrix([
+        [ 5, 7],
+        [ 4, 2],
+        [10, 3]])
+        """
+        from sympy.matrices.dense import Matrix
+        return Matrix(self._lines)
+
+    def as_str(self):
+        # XXX obsolete ?
+        return str(self)
+
+    def as_latex(self):
+        from .latex import latex
+        return latex(self)
+
+    def _sympystr(self, p):
+        """
+        Returns the string representation of 'self'.
+
+        Examples
+        ========
+
+        >>> from sympy import TableForm
+        >>> t = TableForm([[5, 7], [4, 2], [10, 3]])
+        >>> s = t.as_str()
+
+        """
+        column_widths = [0] * self._w
+        lines = []
+        for line in self._lines:
+            new_line = []
+            for i in range(self._w):
+                # Format the item somehow if needed:
+                s = str(line[i])
+                if self._wipe_zeros and (s == "0"):
+                    s = " "
+                w = len(s)
+                if w > column_widths[i]:
+                    column_widths[i] = w
+                new_line.append(s)
+            lines.append(new_line)
+
+        # Check heading:
+        if self._headings[0]:
+            self._headings[0] = [str(x) for x in self._headings[0]]
+            _head_width = max([len(x) for x in self._headings[0]])
+
+        if self._headings[1]:
+            new_line = []
+            for i in range(self._w):
+                # Format the item somehow if needed:
+                s = str(self._headings[1][i])
+                w = len(s)
+                if w > column_widths[i]:
+                    column_widths[i] = w
+                new_line.append(s)
+            self._headings[1] = new_line
+
+        format_str = []
+
+        def _align(align, w):
+            return '%%%s%ss' % (
+                ("-" if align == "l" else ""),
+                str(w))
+        format_str = [_align(align, w) for align, w in
+                      zip(self._alignments, column_widths)]
+        if self._headings[0]:
+            format_str.insert(0, _align(self._head_align, _head_width))
+            format_str.insert(1, '|')
+        format_str = ' '.join(format_str) + '\n'
+
+        s = []
+        if self._headings[1]:
+            d = self._headings[1]
+            if self._headings[0]:
+                d = [""] + d
+            first_line = format_str % tuple(d)
+            s.append(first_line)
+            s.append("-" * (len(first_line) - 1) + "\n")
+        for i, line in enumerate(lines):
+            d = [l if self._alignments[j] != 'c' else
+                 l.center(column_widths[j]) for j, l in enumerate(line)]
+            if self._headings[0]:
+                l = self._headings[0][i]
+                l = (l if self._head_align != 'c' else
+                     l.center(_head_width))
+                d = [l] + d
+            s.append(format_str % tuple(d))
+        return ''.join(s)[:-1]  # don't include trailing newline
+
+    def _latex(self, printer):
+        """
+        Returns the string representation of 'self'.
+        """
+        # Check heading:
+        if self._headings[1]:
+            new_line = []
+            for i in range(self._w):
+                # Format the item somehow if needed:
+                new_line.append(str(self._headings[1][i]))
+            self._headings[1] = new_line
+
+        alignments = []
+        if self._headings[0]:
+            self._headings[0] = [str(x) for x in self._headings[0]]
+            alignments = [self._head_align]
+        alignments.extend(self._alignments)
+
+        s = r"\begin{tabular}{" + " ".join(alignments) + "}\n"
+
+        if self._headings[1]:
+            d = self._headings[1]
+            if self._headings[0]:
+                d = [""] + d
+            first_line = " & ".join(d) + r" \\" + "\n"
+            s += first_line
+            s += r"\hline" + "\n"
+        for i, line in enumerate(self._lines):
+            d = []
+            for j, x in enumerate(line):
+                if self._wipe_zeros and (x in (0, "0")):
+                    d.append(" ")
+                    continue
+                f = self._column_formats[j]
+                if f:
+                    if isinstance(f, FunctionType):
+                        v = f(x, i, j)
+                        if v is None:
+                            v = printer._print(x)
+                    else:
+                        v = f % x
+                    d.append(v)
+                else:
+                    v = printer._print(x)
+                    d.append("$%s$" % v)
+            if self._headings[0]:
+                d = [self._headings[0][i]] + d
+            s += " & ".join(d) + r" \\" + "\n"
+        s += r"\end{tabular}"
+        return s

env-llmeval/lib/python3.10/site-packages/sympy/printing/tensorflow.py

@@ -0,0 +1,216 @@
+from sympy.external.importtools import version_tuple
+from collections.abc import Iterable
+
+from sympy.core.mul import Mul
+from sympy.core.singleton import S
+from sympy.codegen.cfunctions import Sqrt
+from sympy.external import import_module
+from sympy.printing.precedence import PRECEDENCE
+from sympy.printing.pycode import AbstractPythonCodePrinter, ArrayPrinter
+import sympy
+
+tensorflow = import_module('tensorflow')
+
+class TensorflowPrinter(ArrayPrinter, AbstractPythonCodePrinter):
+    """
+    Tensorflow printer which handles vectorized piecewise functions,
+    logical operators, max/min, and relational operators.
+    """
+    printmethod = "_tensorflowcode"
+
+    mapping = {
+        sympy.Abs: "tensorflow.math.abs",
+        sympy.sign: "tensorflow.math.sign",
+
+        # XXX May raise error for ints.
+        sympy.ceiling: "tensorflow.math.ceil",
+        sympy.floor: "tensorflow.math.floor",
+        sympy.log: "tensorflow.math.log",
+        sympy.exp: "tensorflow.math.exp",
+        Sqrt: "tensorflow.math.sqrt",
+        sympy.cos: "tensorflow.math.cos",
+        sympy.acos: "tensorflow.math.acos",
+        sympy.sin: "tensorflow.math.sin",
+        sympy.asin: "tensorflow.math.asin",
+        sympy.tan: "tensorflow.math.tan",
+        sympy.atan: "tensorflow.math.atan",
+        sympy.atan2: "tensorflow.math.atan2",
+        # XXX Also may give NaN for complex results.
+        sympy.cosh: "tensorflow.math.cosh",
+        sympy.acosh: "tensorflow.math.acosh",
+        sympy.sinh: "tensorflow.math.sinh",
+        sympy.asinh: "tensorflow.math.asinh",
+        sympy.tanh: "tensorflow.math.tanh",
+        sympy.atanh: "tensorflow.math.atanh",
+
+        sympy.re: "tensorflow.math.real",
+        sympy.im: "tensorflow.math.imag",
+        sympy.arg: "tensorflow.math.angle",
+
+        # XXX May raise error for ints and complexes
+        sympy.erf: "tensorflow.math.erf",
+        sympy.loggamma: "tensorflow.math.lgamma",
+
+        sympy.Eq: "tensorflow.math.equal",
+        sympy.Ne: "tensorflow.math.not_equal",
+        sympy.StrictGreaterThan: "tensorflow.math.greater",
+        sympy.StrictLessThan: "tensorflow.math.less",
+        sympy.LessThan: "tensorflow.math.less_equal",
+        sympy.GreaterThan: "tensorflow.math.greater_equal",
+
+        sympy.And: "tensorflow.math.logical_and",
+        sympy.Or: "tensorflow.math.logical_or",
+        sympy.Not: "tensorflow.math.logical_not",
+        sympy.Max: "tensorflow.math.maximum",
+        sympy.Min: "tensorflow.math.minimum",
+
+        # Matrices
+        sympy.MatAdd: "tensorflow.math.add",
+        sympy.HadamardProduct: "tensorflow.math.multiply",
+        sympy.Trace: "tensorflow.linalg.trace",
+
+        # XXX May raise error for integer matrices.
+        sympy.Determinant : "tensorflow.linalg.det",
+    }
+
+    _default_settings = dict(
+        AbstractPythonCodePrinter._default_settings,
+        tensorflow_version=None
+    )
+
+    def __init__(self, settings=None):
+        super().__init__(settings)
+
+        version = self._settings['tensorflow_version']
+        if version is None and tensorflow:
+            version = tensorflow.__version__
+        self.tensorflow_version = version
+
+    def _print_Function(self, expr):
+        op = self.mapping.get(type(expr), None)
+        if op is None:
+            return super()._print_Basic(expr)
+        children = [self._print(arg) for arg in expr.args]
+        if len(children) == 1:
+            return "%s(%s)" % (
+                self._module_format(op),
+                children[0]
+            )
+        else:
+            return self._expand_fold_binary_op(op, children)
+
+    _print_Expr = _print_Function
+    _print_Application = _print_Function
+    _print_MatrixExpr = _print_Function
+    # TODO: a better class structure would avoid this mess:
+    _print_Relational = _print_Function
+    _print_Not = _print_Function
+    _print_And = _print_Function
+    _print_Or = _print_Function
+    _print_HadamardProduct = _print_Function
+    _print_Trace = _print_Function
+    _print_Determinant = _print_Function
+
+    def _print_Inverse(self, expr):
+        op = self._module_format('tensorflow.linalg.inv')
+        return "{}({})".format(op, self._print(expr.arg))
+
+    def _print_Transpose(self, expr):
+        version = self.tensorflow_version
+        if version and version_tuple(version) < version_tuple('1.14'):
+            op = self._module_format('tensorflow.matrix_transpose')
+        else:
+            op = self._module_format('tensorflow.linalg.matrix_transpose')
+        return "{}({})".format(op, self._print(expr.arg))
+
+    def _print_Derivative(self, expr):
+        variables = expr.variables
+        if any(isinstance(i, Iterable) for i in variables):
+            raise NotImplementedError("derivation by multiple variables is not supported")
+        def unfold(expr, args):
+            if not args:
+                return self._print(expr)
+            return "%s(%s, %s)[0]" % (
+                    self._module_format("tensorflow.gradients"),
+                    unfold(expr, args[:-1]),
+                    self._print(args[-1]),
+                )
+        return unfold(expr.expr, variables)
+
+    def _print_Piecewise(self, expr):
+        version = self.tensorflow_version
+        if version and version_tuple(version) < version_tuple('1.0'):
+            tensorflow_piecewise = "tensorflow.select"
+        else:
+            tensorflow_piecewise = "tensorflow.where"
+
+        from sympy.functions.elementary.piecewise import Piecewise
+        e, cond = expr.args[0].args
+        if len(expr.args) == 1:
+            return '{}({}, {}, {})'.format(
+                self._module_format(tensorflow_piecewise),
+                self._print(cond),
+                self._print(e),
+                0)
+
+        return '{}({}, {}, {})'.format(
+            self._module_format(tensorflow_piecewise),
+            self._print(cond),
+            self._print(e),
+            self._print(Piecewise(*expr.args[1:])))
+
+    def _print_Pow(self, expr):
+        # XXX May raise error for
+        # int**float or int**complex or float**complex
+        base, exp = expr.args
+        if expr.exp == S.Half:
+            return "{}({})".format(
+                self._module_format("tensorflow.math.sqrt"), self._print(base))
+        return "{}({}, {})".format(
+            self._module_format("tensorflow.math.pow"),
+            self._print(base), self._print(exp))
+
+    def _print_MatrixBase(self, expr):
+        tensorflow_f = "tensorflow.Variable" if expr.free_symbols else "tensorflow.constant"
+        data = "["+", ".join(["["+", ".join([self._print(j) for j in i])+"]" for i in expr.tolist()])+"]"
+        return "%s(%s)" % (
+            self._module_format(tensorflow_f),
+            data,
+        )
+
+    def _print_MatMul(self, expr):
+        from sympy.matrices.expressions import MatrixExpr
+        mat_args = [arg for arg in expr.args if isinstance(arg, MatrixExpr)]
+        args = [arg for arg in expr.args if arg not in mat_args]
+        if args:
+            return "%s*%s" % (
+                self.parenthesize(Mul.fromiter(args), PRECEDENCE["Mul"]),
+                self._expand_fold_binary_op(
+                    "tensorflow.linalg.matmul", mat_args)
+            )
+        else:
+            return self._expand_fold_binary_op(
+                "tensorflow.linalg.matmul", mat_args)
+
+    def _print_MatPow(self, expr):
+        return self._expand_fold_binary_op(
+            "tensorflow.linalg.matmul", [expr.base]*expr.exp)
+
+    def _print_CodeBlock(self, expr):
+        # TODO: is this necessary?
+        ret = []
+        for subexpr in expr.args:
+            ret.append(self._print(subexpr))
+        return "\n".join(ret)
+
+    _module = "tensorflow"
+    _einsum = "linalg.einsum"
+    _add = "math.add"
+    _transpose = "transpose"
+    _ones = "ones"
+    _zeros = "zeros"
+
+
+def tensorflow_code(expr, **settings):
+    printer = TensorflowPrinter(settings)
+    return printer.doprint(expr)

env-llmeval/lib/python3.10/site-packages/sympy/utilities/_compilation/__init__.py

@@ -0,0 +1,22 @@
+""" This sub-module is private, i.e. external code should not depend on it.
+
+These functions are used by tests run as part of continuous integration.
+Once the implementation is mature (it should support the major
+platforms: Windows, OS X & Linux) it may become official API which
+ may be relied upon by downstream libraries. Until then API may break
+without prior notice.
+
+TODO:
+- (optionally) clean up after tempfile.mkdtemp()
+- cross-platform testing
+- caching of compiler choice and intermediate files
+
+"""
+
+from .compilation import compile_link_import_strings, compile_run_strings
+from .availability import has_fortran, has_c, has_cxx
+
+__all__ = [
+    'compile_link_import_strings', 'compile_run_strings',
+    'has_fortran', 'has_c', 'has_cxx',
+]

env-llmeval/lib/python3.10/site-packages/sympy/utilities/_compilation/availability.py

@@ -0,0 +1,77 @@
+import os
+from .compilation import compile_run_strings
+from .util import CompilerNotFoundError
+
+def has_fortran():
+    if not hasattr(has_fortran, 'result'):
+        try:
+            (stdout, stderr), info = compile_run_strings(
+                [('main.f90', (
+                    'program foo\n'
+                    'print *, "hello world"\n'
+                    'end program'
+                ))], clean=True
+            )
+        except CompilerNotFoundError:
+            has_fortran.result = False
+            if os.environ.get('SYMPY_STRICT_COMPILER_CHECKS', '0') == '1':
+                raise
+        else:
+            if info['exit_status'] != os.EX_OK or 'hello world' not in stdout:
+                if os.environ.get('SYMPY_STRICT_COMPILER_CHECKS', '0') == '1':
+                    raise ValueError("Failed to compile test program:\n%s\n%s\n" % (stdout, stderr))
+                has_fortran.result = False
+            else:
+                has_fortran.result = True
+    return has_fortran.result
+
+
+def has_c():
+    if not hasattr(has_c, 'result'):
+        try:
+            (stdout, stderr), info = compile_run_strings(
+                [('main.c', (
+                    '#include <stdio.h>\n'
+                    'int main(){\n'
+                    'printf("hello world\\n");\n'
+                    'return 0;\n'
+                    '}'
+                ))], clean=True
+            )
+        except CompilerNotFoundError:
+            has_c.result = False
+            if os.environ.get('SYMPY_STRICT_COMPILER_CHECKS', '0') == '1':
+                raise
+        else:
+            if info['exit_status'] != os.EX_OK or 'hello world' not in stdout:
+                if os.environ.get('SYMPY_STRICT_COMPILER_CHECKS', '0') == '1':
+                    raise ValueError("Failed to compile test program:\n%s\n%s\n" % (stdout, stderr))
+                has_c.result = False
+            else:
+                has_c.result = True
+    return has_c.result
+
+
+def has_cxx():
+    if not hasattr(has_cxx, 'result'):
+        try:
+            (stdout, stderr), info = compile_run_strings(
+                [('main.cxx', (
+                    '#include <iostream>\n'
+                    'int main(){\n'
+                    'std::cout << "hello world" << std::endl;\n'
+                    '}'
+                ))], clean=True
+            )
+        except CompilerNotFoundError:
+            has_cxx.result = False
+            if os.environ.get('SYMPY_STRICT_COMPILER_CHECKS', '0') == '1':
+                raise
+        else:
+            if info['exit_status'] != os.EX_OK or 'hello world' not in stdout:
+                if os.environ.get('SYMPY_STRICT_COMPILER_CHECKS', '0') == '1':
+                    raise ValueError("Failed to compile test program:\n%s\n%s\n" % (stdout, stderr))
+                has_cxx.result = False
+            else:
+                has_cxx.result = True
+    return has_cxx.result

env-llmeval/lib/python3.10/site-packages/sympy/utilities/_compilation/compilation.py

@@ -0,0 +1,648 @@
+import glob
+import os
+import shutil
+import subprocess
+import sys
+import tempfile
+import warnings
+from sysconfig import get_config_var, get_config_vars, get_path
+
+from .runners import (
+    CCompilerRunner,
+    CppCompilerRunner,
+    FortranCompilerRunner
+)
+from .util import (
+    get_abspath, make_dirs, copy, Glob, ArbitraryDepthGlob,
+    glob_at_depth, import_module_from_file, pyx_is_cplus,
+    sha256_of_string, sha256_of_file, CompileError
+)
+
+if os.name == 'posix':
+    objext = '.o'
+elif os.name == 'nt':
+    objext = '.obj'
+else:
+    warnings.warn("Unknown os.name: {}".format(os.name))
+    objext = '.o'
+
+
+def compile_sources(files, Runner=None, destdir=None, cwd=None, keep_dir_struct=False,
+                    per_file_kwargs=None, **kwargs):
+    """ Compile source code files to object files.
+
+    Parameters
+    ==========
+
+    files : iterable of str
+        Paths to source files, if ``cwd`` is given, the paths are taken as relative.
+    Runner: CompilerRunner subclass (optional)
+        Could be e.g. ``FortranCompilerRunner``. Will be inferred from filename
+        extensions if missing.
+    destdir: str
+        Output directory, if cwd is given, the path is taken as relative.
+    cwd: str
+        Working directory. Specify to have compiler run in other directory.
+        also used as root of relative paths.
+    keep_dir_struct: bool
+        Reproduce directory structure in `destdir`. default: ``False``
+    per_file_kwargs: dict
+        Dict mapping instances in ``files`` to keyword arguments.
+    \\*\\*kwargs: dict
+        Default keyword arguments to pass to ``Runner``.
+
+    """
+    _per_file_kwargs = {}
+
+    if per_file_kwargs is not None:
+        for k, v in per_file_kwargs.items():
+            if isinstance(k, Glob):
+                for path in glob.glob(k.pathname):
+                    _per_file_kwargs[path] = v
+            elif isinstance(k, ArbitraryDepthGlob):
+                for path in glob_at_depth(k.filename, cwd):
+                    _per_file_kwargs[path] = v
+            else:
+                _per_file_kwargs[k] = v
+
+    # Set up destination directory
+    destdir = destdir or '.'
+    if not os.path.isdir(destdir):
+        if os.path.exists(destdir):
+            raise OSError("{} is not a directory".format(destdir))
+        else:
+            make_dirs(destdir)
+    if cwd is None:
+        cwd = '.'
+        for f in files:
+            copy(f, destdir, only_update=True, dest_is_dir=True)
+
+    # Compile files and return list of paths to the objects
+    dstpaths = []
+    for f in files:
+        if keep_dir_struct:
+            name, ext = os.path.splitext(f)
+        else:
+            name, ext = os.path.splitext(os.path.basename(f))
+        file_kwargs = kwargs.copy()
+        file_kwargs.update(_per_file_kwargs.get(f, {}))
+        dstpaths.append(src2obj(f, Runner, cwd=cwd, **file_kwargs))
+    return dstpaths
+
+
+def get_mixed_fort_c_linker(vendor=None, cplus=False, cwd=None):
+    vendor = vendor or os.environ.get('SYMPY_COMPILER_VENDOR', 'gnu')
+
+    if vendor.lower() == 'intel':
+        if cplus:
+            return (FortranCompilerRunner,
+                    {'flags': ['-nofor_main', '-cxxlib']}, vendor)
+        else:
+            return (FortranCompilerRunner,
+                    {'flags': ['-nofor_main']}, vendor)
+    elif vendor.lower() == 'gnu' or 'llvm':
+        if cplus:
+            return (CppCompilerRunner,
+                    {'lib_options': ['fortran']}, vendor)
+        else:
+            return (FortranCompilerRunner,
+                    {}, vendor)
+    else:
+        raise ValueError("No vendor found.")
+
+
+def link(obj_files, out_file=None, shared=False, Runner=None,
+         cwd=None, cplus=False, fort=False, **kwargs):
+    """ Link object files.
+
+    Parameters
+    ==========
+
+    obj_files: iterable of str
+        Paths to object files.
+    out_file: str (optional)
+        Path to executable/shared library, if ``None`` it will be
+        deduced from the last item in obj_files.
+    shared: bool
+        Generate a shared library?
+    Runner: CompilerRunner subclass (optional)
+        If not given the ``cplus`` and ``fort`` flags will be inspected
+        (fallback is the C compiler).
+    cwd: str
+        Path to the root of relative paths and working directory for compiler.
+    cplus: bool
+        C++ objects? default: ``False``.
+    fort: bool
+        Fortran objects? default: ``False``.
+    \\*\\*kwargs: dict
+        Keyword arguments passed to ``Runner``.
+
+    Returns
+    =======
+
+    The absolute path to the generated shared object / executable.
+
+    """
+    if out_file is None:
+        out_file, ext = os.path.splitext(os.path.basename(obj_files[-1]))
+        if shared:
+            out_file += get_config_var('EXT_SUFFIX')
+
+    if not Runner:
+        if fort:
+            Runner, extra_kwargs, vendor = \
+                get_mixed_fort_c_linker(
+                    vendor=kwargs.get('vendor', None),
+                    cplus=cplus,
+                    cwd=cwd,
+                )
+            for k, v in extra_kwargs.items():
+                if k in kwargs:
+                    kwargs[k].expand(v)
+                else:
+                    kwargs[k] = v
+        else:
+            if cplus:
+                Runner = CppCompilerRunner
+            else:
+                Runner = CCompilerRunner
+
+    flags = kwargs.pop('flags', [])
+    if shared:
+        if '-shared' not in flags:
+            flags.append('-shared')
+    run_linker = kwargs.pop('run_linker', True)
+    if not run_linker:
+        raise ValueError("run_linker was set to False (nonsensical).")
+
+    out_file = get_abspath(out_file, cwd=cwd)
+    runner = Runner(obj_files, out_file, flags, cwd=cwd, **kwargs)
+    runner.run()
+    return out_file
+
+
+def link_py_so(obj_files, so_file=None, cwd=None, libraries=None,
+               cplus=False, fort=False, **kwargs):
+    """ Link Python extension module (shared object) for importing
+
+    Parameters
+    ==========
+
+    obj_files: iterable of str
+        Paths to object files to be linked.
+    so_file: str
+        Name (path) of shared object file to create. If not specified it will
+        have the basname of the last object file in `obj_files` but with the
+        extension '.so' (Unix).
+    cwd: path string
+        Root of relative paths and working directory of linker.
+    libraries: iterable of strings
+        Libraries to link against, e.g. ['m'].
+    cplus: bool
+        Any C++ objects? default: ``False``.
+    fort: bool
+        Any Fortran objects? default: ``False``.
+    kwargs**: dict
+        Keyword arguments passed to ``link(...)``.
+
+    Returns
+    =======
+
+    Absolute path to the generate shared object.
+    """
+    libraries = libraries or []
+
+    include_dirs = kwargs.pop('include_dirs', [])
+    library_dirs = kwargs.pop('library_dirs', [])
+
+    # Add Python include and library directories
+    # PY_LDFLAGS does not available on all python implementations
+    # e.g. when with pypy, so it's LDFLAGS we need to use
+    if sys.platform == "win32":
+        warnings.warn("Windows not yet supported.")
+    elif sys.platform == 'darwin':
+        cfgDict = get_config_vars()
+        kwargs['linkline'] = kwargs.get('linkline', []) + [cfgDict['LDFLAGS']]
+        library_dirs += [cfgDict['LIBDIR']]
+
+        # In macOS, linker needs to compile frameworks
+        # e.g. "-framework CoreFoundation"
+        is_framework = False
+        for opt in cfgDict['LIBS'].split():
+            if is_framework:
+                kwargs['linkline'] = kwargs.get('linkline', []) + ['-framework', opt]
+                is_framework = False
+            elif opt.startswith('-l'):
+                libraries.append(opt[2:])
+            elif opt.startswith('-framework'):
+                is_framework = True
+        # The python library is not included in LIBS
+        libfile = cfgDict['LIBRARY']
+        libname = ".".join(libfile.split('.')[:-1])[3:]
+        libraries.append(libname)
+
+    elif sys.platform[:3] == 'aix':
+        # Don't use the default code below
+        pass
+    else:
+        if get_config_var('Py_ENABLE_SHARED'):
+            cfgDict = get_config_vars()
+            kwargs['linkline'] = kwargs.get('linkline', []) + [cfgDict['LDFLAGS']]
+            library_dirs += [cfgDict['LIBDIR']]
+            for opt in cfgDict['BLDLIBRARY'].split():
+                if opt.startswith('-l'):
+                    libraries += [opt[2:]]
+        else:
+            pass
+
+    flags = kwargs.pop('flags', [])
+    needed_flags = ('-pthread',)
+    for flag in needed_flags:
+        if flag not in flags:
+            flags.append(flag)
+
+    return link(obj_files, shared=True, flags=flags, cwd=cwd,
+                cplus=cplus, fort=fort, include_dirs=include_dirs,
+                libraries=libraries, library_dirs=library_dirs, **kwargs)
+
+
+def simple_cythonize(src, destdir=None, cwd=None, **cy_kwargs):
+    """ Generates a C file from a Cython source file.
+
+    Parameters
+    ==========
+
+    src: str
+        Path to Cython source.
+    destdir: str (optional)
+        Path to output directory (default: '.').
+    cwd: path string (optional)
+        Root of relative paths (default: '.').
+    **cy_kwargs:
+        Second argument passed to cy_compile. Generates a .cpp file if ``cplus=True`` in ``cy_kwargs``,
+        else a .c file.
+    """
+    from Cython.Compiler.Main import (
+        default_options, CompilationOptions
+    )
+    from Cython.Compiler.Main import compile as cy_compile
+
+    assert src.lower().endswith('.pyx') or src.lower().endswith('.py')
+    cwd = cwd or '.'
+    destdir = destdir or '.'
+
+    ext = '.cpp' if cy_kwargs.get('cplus', False) else '.c'
+    c_name = os.path.splitext(os.path.basename(src))[0] + ext
+
+    dstfile = os.path.join(destdir, c_name)
+
+    if cwd:
+        ori_dir = os.getcwd()
+    else:
+        ori_dir = '.'
+    os.chdir(cwd)
+    try:
+        cy_options = CompilationOptions(default_options)
+        cy_options.__dict__.update(cy_kwargs)
+        # Set language_level if not set by cy_kwargs
+        # as not setting it is deprecated
+        if 'language_level' not in cy_kwargs:
+            cy_options.__dict__['language_level'] = 3
+        cy_result = cy_compile([src], cy_options)
+        if cy_result.num_errors > 0:
+            raise ValueError("Cython compilation failed.")
+
+        # Move generated C file to destination
+        # In macOS, the generated C file is in the same directory as the source
+        # but the /var is a symlink to /private/var, so we need to use realpath
+        if os.path.realpath(os.path.dirname(src)) != os.path.realpath(destdir):
+            if os.path.exists(dstfile):
+                os.unlink(dstfile)
+            shutil.move(os.path.join(os.path.dirname(src), c_name), destdir)
+    finally:
+        os.chdir(ori_dir)
+    return dstfile
+
+
+extension_mapping = {
+    '.c': (CCompilerRunner, None),
+    '.cpp': (CppCompilerRunner, None),
+    '.cxx': (CppCompilerRunner, None),
+    '.f': (FortranCompilerRunner, None),
+    '.for': (FortranCompilerRunner, None),
+    '.ftn': (FortranCompilerRunner, None),
+    '.f90': (FortranCompilerRunner, None),  # ifort only knows about .f90
+    '.f95': (FortranCompilerRunner, 'f95'),
+    '.f03': (FortranCompilerRunner, 'f2003'),
+    '.f08': (FortranCompilerRunner, 'f2008'),
+}
+
+
+def src2obj(srcpath, Runner=None, objpath=None, cwd=None, inc_py=False, **kwargs):
+    """ Compiles a source code file to an object file.
+
+    Files ending with '.pyx' assumed to be cython files and
+    are dispatched to pyx2obj.
+
+    Parameters
+    ==========
+
+    srcpath: str
+        Path to source file.
+    Runner: CompilerRunner subclass (optional)
+        If ``None``: deduced from extension of srcpath.
+    objpath : str (optional)
+        Path to generated object. If ``None``: deduced from ``srcpath``.
+    cwd: str (optional)
+        Working directory and root of relative paths. If ``None``: current dir.
+    inc_py: bool
+        Add Python include path to kwarg "include_dirs". Default: False
+    \\*\\*kwargs: dict
+        keyword arguments passed to Runner or pyx2obj
+
+    """
+    name, ext = os.path.splitext(os.path.basename(srcpath))
+    if objpath is None:
+        if os.path.isabs(srcpath):
+            objpath = '.'
+        else:
+            objpath = os.path.dirname(srcpath)
+            objpath = objpath or '.'  # avoid objpath == ''
+
+    if os.path.isdir(objpath):
+        objpath = os.path.join(objpath, name + objext)
+
+    include_dirs = kwargs.pop('include_dirs', [])
+    if inc_py:
+        py_inc_dir = get_path('include')
+        if py_inc_dir not in include_dirs:
+            include_dirs.append(py_inc_dir)
+
+    if ext.lower() == '.pyx':
+        return pyx2obj(srcpath, objpath=objpath, include_dirs=include_dirs, cwd=cwd,
+                       **kwargs)
+
+    if Runner is None:
+        Runner, std = extension_mapping[ext.lower()]
+        if 'std' not in kwargs:
+            kwargs['std'] = std
+
+    flags = kwargs.pop('flags', [])
+    needed_flags = ('-fPIC',)
+    for flag in needed_flags:
+        if flag not in flags:
+            flags.append(flag)
+
+    # src2obj implies not running the linker...
+    run_linker = kwargs.pop('run_linker', False)
+    if run_linker:
+        raise CompileError("src2obj called with run_linker=True")
+
+    runner = Runner([srcpath], objpath, include_dirs=include_dirs,
+                    run_linker=run_linker, cwd=cwd, flags=flags, **kwargs)
+    runner.run()
+    return objpath
+
+
+def pyx2obj(pyxpath, objpath=None, destdir=None, cwd=None,
+            include_dirs=None, cy_kwargs=None, cplus=None, **kwargs):
+    """
+    Convenience function
+
+    If cwd is specified, pyxpath and dst are taken to be relative
+    If only_update is set to `True` the modification time is checked
+    and compilation is only run if the source is newer than the
+    destination
+
+    Parameters
+    ==========
+
+    pyxpath: str
+        Path to Cython source file.
+    objpath: str (optional)
+        Path to object file to generate.
+    destdir: str (optional)
+        Directory to put generated C file. When ``None``: directory of ``objpath``.
+    cwd: str (optional)
+        Working directory and root of relative paths.
+    include_dirs: iterable of path strings (optional)
+        Passed onto src2obj and via cy_kwargs['include_path']
+        to simple_cythonize.
+    cy_kwargs: dict (optional)
+        Keyword arguments passed onto `simple_cythonize`
+    cplus: bool (optional)
+        Indicate whether C++ is used. default: auto-detect using ``.util.pyx_is_cplus``.
+    compile_kwargs: dict
+        keyword arguments passed onto src2obj
+
+    Returns
+    =======
+
+    Absolute path of generated object file.
+
+    """
+    assert pyxpath.endswith('.pyx')
+    cwd = cwd or '.'
+    objpath = objpath or '.'
+    destdir = destdir or os.path.dirname(objpath)
+
+    abs_objpath = get_abspath(objpath, cwd=cwd)
+
+    if os.path.isdir(abs_objpath):
+        pyx_fname = os.path.basename(pyxpath)
+        name, ext = os.path.splitext(pyx_fname)
+        objpath = os.path.join(objpath, name + objext)
+
+    cy_kwargs = cy_kwargs or {}
+    cy_kwargs['output_dir'] = cwd
+    if cplus is None:
+        cplus = pyx_is_cplus(pyxpath)
+    cy_kwargs['cplus'] = cplus
+
+    interm_c_file = simple_cythonize(pyxpath, destdir=destdir, cwd=cwd, **cy_kwargs)
+
+    include_dirs = include_dirs or []
+    flags = kwargs.pop('flags', [])
+    needed_flags = ('-fwrapv', '-pthread', '-fPIC')
+    for flag in needed_flags:
+        if flag not in flags:
+            flags.append(flag)
+
+    options = kwargs.pop('options', [])
+
+    if kwargs.pop('strict_aliasing', False):
+        raise CompileError("Cython requires strict aliasing to be disabled.")
+
+    # Let's be explicit about standard
+    if cplus:
+        std = kwargs.pop('std', 'c++98')
+    else:
+        std = kwargs.pop('std', 'c99')
+
+    return src2obj(interm_c_file, objpath=objpath, cwd=cwd,
+                   include_dirs=include_dirs, flags=flags, std=std,
+                   options=options, inc_py=True, strict_aliasing=False,
+                   **kwargs)
+
+
+def _any_X(srcs, cls):
+    for src in srcs:
+        name, ext = os.path.splitext(src)
+        key = ext.lower()
+        if key in extension_mapping:
+            if extension_mapping[key][0] == cls:
+                return True
+    return False
+
+
+def any_fortran_src(srcs):
+    return _any_X(srcs, FortranCompilerRunner)
+
+
+def any_cplus_src(srcs):
+    return _any_X(srcs, CppCompilerRunner)
+
+
+def compile_link_import_py_ext(sources, extname=None, build_dir='.', compile_kwargs=None,
+                               link_kwargs=None):
+    """ Compiles sources to a shared object (Python extension) and imports it
+
+    Sources in ``sources`` which is imported. If shared object is newer than the sources, they
+    are not recompiled but instead it is imported.
+
+    Parameters
+    ==========
+
+    sources : string
+        List of paths to sources.
+    extname : string
+        Name of extension (default: ``None``).
+        If ``None``: taken from the last file in ``sources`` without extension.
+    build_dir: str
+        Path to directory in which objects files etc. are generated.
+    compile_kwargs: dict
+        keyword arguments passed to ``compile_sources``
+    link_kwargs: dict
+        keyword arguments passed to ``link_py_so``
+
+    Returns
+    =======
+
+    The imported module from of the Python extension.
+    """
+    if extname is None:
+        extname = os.path.splitext(os.path.basename(sources[-1]))[0]
+
+    compile_kwargs = compile_kwargs or {}
+    link_kwargs = link_kwargs or {}
+
+    try:
+        mod = import_module_from_file(os.path.join(build_dir, extname), sources)
+    except ImportError:
+        objs = compile_sources(list(map(get_abspath, sources)), destdir=build_dir,
+                               cwd=build_dir, **compile_kwargs)
+        so = link_py_so(objs, cwd=build_dir, fort=any_fortran_src(sources),
+                        cplus=any_cplus_src(sources), **link_kwargs)
+        mod = import_module_from_file(so)
+    return mod
+
+
+def _write_sources_to_build_dir(sources, build_dir):
+    build_dir = build_dir or tempfile.mkdtemp()
+    if not os.path.isdir(build_dir):
+        raise OSError("Non-existent directory: ", build_dir)
+
+    source_files = []
+    for name, src in sources:
+        dest = os.path.join(build_dir, name)
+        differs = True
+        sha256_in_mem = sha256_of_string(src.encode('utf-8')).hexdigest()
+        if os.path.exists(dest):
+            if os.path.exists(dest + '.sha256'):
+                with open(dest + '.sha256') as fh:
+                    sha256_on_disk = fh.read()
+            else:
+                sha256_on_disk = sha256_of_file(dest).hexdigest()
+
+            differs = sha256_on_disk != sha256_in_mem
+        if differs:
+            with open(dest, 'wt') as fh:
+                fh.write(src)
+            with open(dest + '.sha256', 'wt') as fh:
+                fh.write(sha256_in_mem)
+        source_files.append(dest)
+    return source_files, build_dir
+
+
+def compile_link_import_strings(sources, build_dir=None, **kwargs):
+    """ Compiles, links and imports extension module from source.
+
+    Parameters
+    ==========
+
+    sources : iterable of name/source pair tuples
+    build_dir : string (default: None)
+        Path. ``None`` implies use a temporary directory.
+    **kwargs:
+        Keyword arguments passed onto `compile_link_import_py_ext`.
+
+    Returns
+    =======
+
+    mod : module
+        The compiled and imported extension module.
+    info : dict
+        Containing ``build_dir`` as 'build_dir'.
+
+    """
+    source_files, build_dir = _write_sources_to_build_dir(sources, build_dir)
+    mod = compile_link_import_py_ext(source_files, build_dir=build_dir, **kwargs)
+    info = {"build_dir": build_dir}
+    return mod, info
+
+
+def compile_run_strings(sources, build_dir=None, clean=False, compile_kwargs=None, link_kwargs=None):
+    """ Compiles, links and runs a program built from sources.
+
+    Parameters
+    ==========
+
+    sources : iterable of name/source pair tuples
+    build_dir : string (default: None)
+        Path. ``None`` implies use a temporary directory.
+    clean : bool
+        Whether to remove build_dir after use. This will only have an
+        effect if ``build_dir`` is ``None`` (which creates a temporary directory).
+        Passing ``clean == True`` and ``build_dir != None`` raises a ``ValueError``.
+        This will also set ``build_dir`` in returned info dictionary to ``None``.
+    compile_kwargs: dict
+        Keyword arguments passed onto ``compile_sources``
+    link_kwargs: dict
+        Keyword arguments passed onto ``link``
+
+    Returns
+    =======
+
+    (stdout, stderr): pair of strings
+    info: dict
+        Containing exit status as 'exit_status' and ``build_dir`` as 'build_dir'
+
+    """
+    if clean and build_dir is not None:
+        raise ValueError("Automatic removal of build_dir is only available for temporary directory.")
+    try:
+        source_files, build_dir = _write_sources_to_build_dir(sources, build_dir)
+        objs = compile_sources(list(map(get_abspath, source_files)), destdir=build_dir,
+                               cwd=build_dir, **(compile_kwargs or {}))
+        prog = link(objs, cwd=build_dir,
+                    fort=any_fortran_src(source_files),
+                    cplus=any_cplus_src(source_files), **(link_kwargs or {}))
+        p = subprocess.Popen([prog], stdout=subprocess.PIPE, stderr=subprocess.PIPE)
+        exit_status = p.wait()
+        stdout, stderr = [txt.decode('utf-8') for txt in p.communicate()]
+    finally:
+        if clean and os.path.isdir(build_dir):
+            shutil.rmtree(build_dir)
+            build_dir = None
+    info = {"exit_status": exit_status, "build_dir": build_dir}
+    return (stdout, stderr), info

env-llmeval/lib/python3.10/site-packages/sympy/utilities/_compilation/tests/test_compilation.py

@@ -0,0 +1,62 @@
+import shutil
+from sympy.external import import_module
+from sympy.testing.pytest import skip
+
+from sympy.utilities._compilation.compilation import compile_link_import_strings
+
+numpy = import_module('numpy')
+cython = import_module('cython')
+
+_sources1 = [
+    ('sigmoid.c', r"""
+#include <math.h>
+
+void sigmoid(int n, const double * const restrict in,
+             double * const restrict out, double lim){
+    for (int i=0; i<n; ++i){
+        const double x = in[i];
+        out[i] = x*pow(pow(x/lim, 8)+1, -1./8.);
+    }
+}
+"""),
+    ('_sigmoid.pyx', r"""
+import numpy as np
+cimport numpy as cnp
+
+cdef extern void c_sigmoid "sigmoid" (int, const double * const,
+                                      double * const, double)
+
+def sigmoid(double [:] inp, double lim=350.0):
+    cdef cnp.ndarray[cnp.float64_t, ndim=1] out = np.empty(
+        inp.size, dtype=np.float64)
+    c_sigmoid(inp.size, &inp[0], &out[0], lim)
+    return out
+""")
+]
+
+
+def npy(data, lim=350.0):
+    return data/((data/lim)**8+1)**(1/8.)
+
+
+def test_compile_link_import_strings():
+    if not numpy:
+        skip("numpy not installed.")
+    if not cython:
+        skip("cython not installed.")
+
+    from sympy.utilities._compilation import has_c
+    if not has_c():
+        skip("No C compiler found.")
+
+    compile_kw = {"std": 'c99', "include_dirs": [numpy.get_include()]}
+    info = None
+    try:
+        mod, info = compile_link_import_strings(_sources1, compile_kwargs=compile_kw)
+        data = numpy.random.random(1024*1024*8)  # 64 MB of RAM needed..
+        res_mod = mod.sigmoid(data)
+        res_npy = npy(data)
+        assert numpy.allclose(res_mod, res_npy)
+    finally:
+        if info and info['build_dir']:
+            shutil.rmtree(info['build_dir'])

env-llmeval/lib/python3.10/site-packages/sympy/utilities/_compilation/util.py

@@ -0,0 +1,287 @@
+from collections import namedtuple
+from hashlib import sha256
+import os
+import shutil
+import sys
+import fnmatch
+
+from sympy.testing.pytest import XFAIL
+
+
+def may_xfail(func):
+    if sys.platform.lower() == 'darwin' or os.name == 'nt':
+        # sympy.utilities._compilation needs more testing on Windows and macOS
+        # once those two platforms are reliably supported this xfail decorator
+        # may be removed.
+        return XFAIL(func)
+    else:
+        return func
+
+
+class CompilerNotFoundError(FileNotFoundError):
+    pass
+
+
+class CompileError (Exception):
+    """Failure to compile one or more C/C++ source files."""
+
+
+def get_abspath(path, cwd='.'):
+    """ Returns the absolute path.
+
+    Parameters
+    ==========
+
+    path : str
+        (relative) path.
+    cwd : str
+        Path to root of relative path.
+    """
+    if os.path.isabs(path):
+        return path
+    else:
+        if not os.path.isabs(cwd):
+            cwd = os.path.abspath(cwd)
+        return os.path.abspath(
+            os.path.join(cwd, path)
+        )
+
+
+def make_dirs(path):
+    """ Create directories (equivalent of ``mkdir -p``). """
+    if path[-1] == '/':
+        parent = os.path.dirname(path[:-1])
+    else:
+        parent = os.path.dirname(path)
+
+    if len(parent) > 0:
+        if not os.path.exists(parent):
+            make_dirs(parent)
+
+    if not os.path.exists(path):
+        os.mkdir(path, 0o777)
+    else:
+        assert os.path.isdir(path)
+
+
+def copy(src, dst, only_update=False, copystat=True, cwd=None,
+         dest_is_dir=False, create_dest_dirs=False):
+    """ Variation of ``shutil.copy`` with extra options.
+
+    Parameters
+    ==========
+
+    src : str
+        Path to source file.
+    dst : str
+        Path to destination.
+    only_update : bool
+        Only copy if source is newer than destination
+        (returns None if it was newer), default: ``False``.
+    copystat : bool
+        See ``shutil.copystat``. default: ``True``.
+    cwd : str
+        Path to working directory (root of relative paths).
+    dest_is_dir : bool
+        Ensures that dst is treated as a directory. default: ``False``
+    create_dest_dirs : bool
+        Creates directories if needed.
+
+    Returns
+    =======
+
+    Path to the copied file.
+
+    """
+    if cwd:  # Handle working directory
+        if not os.path.isabs(src):
+            src = os.path.join(cwd, src)
+        if not os.path.isabs(dst):
+            dst = os.path.join(cwd, dst)
+
+    if not os.path.exists(src):  # Make sure source file extists
+        raise FileNotFoundError("Source: `{}` does not exist".format(src))
+
+    # We accept both (re)naming destination file _or_
+    # passing a (possible non-existent) destination directory
+    if dest_is_dir:
+        if not dst[-1] == '/':
+            dst = dst+'/'
+    else:
+        if os.path.exists(dst) and os.path.isdir(dst):
+            dest_is_dir = True
+
+    if dest_is_dir:
+        dest_dir = dst
+        dest_fname = os.path.basename(src)
+        dst = os.path.join(dest_dir, dest_fname)
+    else:
+        dest_dir = os.path.dirname(dst)
+
+    if not os.path.exists(dest_dir):
+        if create_dest_dirs:
+            make_dirs(dest_dir)
+        else:
+            raise FileNotFoundError("You must create directory first.")
+
+    if only_update:
+        # This function is not defined:
+        # XXX: This branch is clearly not tested!
+        if not missing_or_other_newer(dst, src): # noqa
+            return
+
+    if os.path.islink(dst):
+        dst = os.path.abspath(os.path.realpath(dst), cwd=cwd)
+
+    shutil.copy(src, dst)
+    if copystat:
+        shutil.copystat(src, dst)
+
+    return dst
+
+Glob = namedtuple('Glob', 'pathname')
+ArbitraryDepthGlob = namedtuple('ArbitraryDepthGlob', 'filename')
+
+def glob_at_depth(filename_glob, cwd=None):
+    if cwd is not None:
+        cwd = '.'
+    globbed = []
+    for root, dirs, filenames in os.walk(cwd):
+        for fn in filenames:
+            # This is not tested:
+            if fnmatch.fnmatch(fn, filename_glob):
+                globbed.append(os.path.join(root, fn))
+    return globbed
+
+def sha256_of_file(path, nblocks=128):
+    """ Computes the SHA256 hash of a file.
+
+    Parameters
+    ==========
+
+    path : string
+        Path to file to compute hash of.
+    nblocks : int
+        Number of blocks to read per iteration.
+
+    Returns
+    =======
+
+    hashlib sha256 hash object. Use ``.digest()`` or ``.hexdigest()``
+    on returned object to get binary or hex encoded string.
+    """
+    sh = sha256()
+    with open(path, 'rb') as f:
+        for chunk in iter(lambda: f.read(nblocks*sh.block_size), b''):
+            sh.update(chunk)
+    return sh
+
+
+def sha256_of_string(string):
+    """ Computes the SHA256 hash of a string. """
+    sh = sha256()
+    sh.update(string)
+    return sh
+
+
+def pyx_is_cplus(path):
+    """
+    Inspect a Cython source file (.pyx) and look for comment line like:
+
+    # distutils: language = c++
+
+    Returns True if such a file is present in the file, else False.
+    """
+    with open(path) as fh:
+        for line in fh:
+            if line.startswith('#') and '=' in line:
+                splitted = line.split('=')
+                if len(splitted) != 2:
+                    continue
+                lhs, rhs = splitted
+                if lhs.strip().split()[-1].lower() == 'language' and \
+                       rhs.strip().split()[0].lower() == 'c++':
+                            return True
+    return False
+
+def import_module_from_file(filename, only_if_newer_than=None):
+    """ Imports Python extension (from shared object file)
+
+    Provide a list of paths in `only_if_newer_than` to check
+    timestamps of dependencies. import_ raises an ImportError
+    if any is newer.
+
+    Word of warning: The OS may cache shared objects which makes
+    reimporting same path of an shared object file very problematic.
+
+    It will not detect the new time stamp, nor new checksum, but will
+    instead silently use old module. Use unique names for this reason.
+
+    Parameters
+    ==========
+
+    filename : str
+        Path to shared object.
+    only_if_newer_than : iterable of strings
+        Paths to dependencies of the shared object.
+
+    Raises
+    ======
+
+    ``ImportError`` if any of the files specified in ``only_if_newer_than`` are newer
+    than the file given by filename.
+    """
+    path, name = os.path.split(filename)
+    name, ext = os.path.splitext(name)
+    name = name.split('.')[0]
+    if sys.version_info[0] == 2:
+        from imp import find_module, load_module
+        fobj, filename, data = find_module(name, [path])
+        if only_if_newer_than:
+            for dep in only_if_newer_than:
+                if os.path.getmtime(filename) < os.path.getmtime(dep):
+                    raise ImportError("{} is newer than {}".format(dep, filename))
+        mod = load_module(name, fobj, filename, data)
+    else:
+        import importlib.util
+        spec = importlib.util.spec_from_file_location(name, filename)
+        if spec is None:
+            raise ImportError("Failed to import: '%s'" % filename)
+        mod = importlib.util.module_from_spec(spec)
+        spec.loader.exec_module(mod)
+    return mod
+
+
+def find_binary_of_command(candidates):
+    """ Finds binary first matching name among candidates.
+
+    Calls ``which`` from shutils for provided candidates and returns
+    first hit.
+
+    Parameters
+    ==========
+
+    candidates : iterable of str
+        Names of candidate commands
+
+    Raises
+    ======
+
+    CompilerNotFoundError if no candidates match.
+    """
+    from shutil import which
+    for c in candidates:
+        binary_path = which(c)
+        if c and binary_path:
+            return c, binary_path
+
+    raise CompilerNotFoundError('No binary located for candidates: {}'.format(candidates))
+
+
+def unique_list(l):
+    """ Uniquify a list (skip duplicate items). """
+    result = []
+    for x in l:
+        if x not in result:
+            result.append(x)
+    return result

env-llmeval/lib/python3.10/site-packages/sympy/utilities/mathml/__init__.py

@@ -0,0 +1,79 @@
+"""Module with some functions for MathML, like transforming MathML
+content in MathML presentation.
+
+To use this module, you will need lxml.
+"""
+
+from sympy.utilities.pkgdata import get_resource
+from sympy.utilities.decorator import doctest_depends_on
+
+
+__doctest_requires__ = {('apply_xsl', 'c2p'): ['lxml']}
+
+
+def add_mathml_headers(s):
+    return """<math xmlns:mml="http://www.w3.org/1998/Math/MathML"
+      xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
+      xsi:schemaLocation="http://www.w3.org/1998/Math/MathML
+        http://www.w3.org/Math/XMLSchema/mathml2/mathml2.xsd">""" + s + "</math>"
+
+
+@doctest_depends_on(modules=('lxml',))
+def apply_xsl(mml, xsl):
+    """Apply a xsl to a MathML string.
+
+    Parameters
+    ==========
+
+    mml
+        A string with MathML code.
+    xsl
+        A string representing a path to a xsl (xml stylesheet) file.
+        This file name is relative to the PYTHONPATH.
+
+    Examples
+    ========
+
+    >>> from sympy.utilities.mathml import apply_xsl
+    >>> xsl = 'mathml/data/simple_mmlctop.xsl'
+    >>> mml = '<apply> <plus/> <ci>a</ci> <ci>b</ci> </apply>'
+    >>> res = apply_xsl(mml,xsl)
+    >>> ''.join(res.splitlines())
+    '<?xml version="1.0"?><mrow xmlns="http://www.w3.org/1998/Math/MathML">  <mi>a</mi>  <mo> + </mo>  <mi>b</mi></mrow>'
+    """
+    from lxml import etree
+
+    parser = etree.XMLParser(resolve_entities=False)
+    ac = etree.XSLTAccessControl.DENY_ALL
+
+    s = etree.XML(get_resource(xsl).read(), parser=parser)
+    transform = etree.XSLT(s, access_control=ac)
+    doc = etree.XML(mml, parser=parser)
+    result = transform(doc)
+    s = str(result)
+    return s
+
+
+@doctest_depends_on(modules=('lxml',))
+def c2p(mml, simple=False):
+    """Transforms a document in MathML content (like the one that sympy produces)
+    in one document in MathML presentation, more suitable for printing, and more
+    widely accepted
+
+    Examples
+    ========
+
+    >>> from sympy.utilities.mathml import c2p
+    >>> mml = '<apply> <exp/> <cn>2</cn> </apply>'
+    >>> c2p(mml,simple=True) != c2p(mml,simple=False)
+    True
+
+    """
+
+    if not mml.startswith('<math'):
+        mml = add_mathml_headers(mml)
+
+    if simple:
+        return apply_xsl(mml, 'mathml/data/simple_mmlctop.xsl')
+
+    return apply_xsl(mml, 'mathml/data/mmlctop.xsl')

env-llmeval/lib/python3.10/site-packages/sympy/utilities/pytest.py

@@ -0,0 +1,12 @@
+"""
+.. deprecated:: 1.6
+
+   sympy.utilities.pytest has been renamed to sympy.testing.pytest.
+"""
+from sympy.utilities.exceptions import sympy_deprecation_warning
+
+sympy_deprecation_warning("The sympy.utilities.pytest submodule is deprecated. Use sympy.testing.pytest instead.",
+    deprecated_since_version="1.6",
+    active_deprecations_target="deprecated-sympy-utilities-submodules")
+
+from sympy.testing.pytest import *  # noqa:F401