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"""Test ideals.py code.""" |
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from sympy.polys import QQ, ilex |
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from sympy.abc import x, y, z |
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from sympy.testing.pytest import raises |
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def test_ideal_operations(): |
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R = QQ.old_poly_ring(x, y) |
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I = R.ideal(x) |
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J = R.ideal(y) |
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S = R.ideal(x*y) |
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T = R.ideal(x, y) |
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assert not (I == J) |
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assert I == I |
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assert I.union(J) == T |
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assert I + J == T |
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assert I + T == T |
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assert not I.subset(T) |
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assert T.subset(I) |
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assert I.product(J) == S |
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assert I*J == S |
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assert x*J == S |
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assert I*y == S |
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assert R.convert(x)*J == S |
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assert I*R.convert(y) == S |
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assert not I.is_zero() |
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assert not J.is_whole_ring() |
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assert R.ideal(x**2 + 1, x).is_whole_ring() |
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assert R.ideal() == R.ideal(0) |
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assert R.ideal().is_zero() |
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assert T.contains(x*y) |
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assert T.subset([x, y]) |
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assert T.in_terms_of_generators(x) == [R(1), R(0)] |
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assert T**0 == R.ideal(1) |
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assert T**1 == T |
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assert T**2 == R.ideal(x**2, y**2, x*y) |
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assert I**5 == R.ideal(x**5) |
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def test_exceptions(): |
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I = QQ.old_poly_ring(x).ideal(x) |
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J = QQ.old_poly_ring(y).ideal(1) |
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raises(ValueError, lambda: I.union(x)) |
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raises(ValueError, lambda: I + J) |
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raises(ValueError, lambda: I * J) |
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raises(ValueError, lambda: I.union(J)) |
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assert (I == J) is False |
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assert I != J |
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def test_nontriv_global(): |
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R = QQ.old_poly_ring(x, y, z) |
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def contains(I, f): |
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return R.ideal(*I).contains(f) |
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assert contains([x, y], x) |
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assert contains([x, y], x + y) |
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assert not contains([x, y], 1) |
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assert not contains([x, y], z) |
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assert contains([x**2 + y, x**2 + x], x - y) |
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assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x**2) |
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assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**3) |
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assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**4) |
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assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x*y**2) |
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assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**4 + y**3 + 2*z*y*x) |
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assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x*y*z) |
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assert contains([x, 1 + x + y, 5 - 7*y], 1) |
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assert contains( |
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[x**3 + y**3, y**3 + z**3, z**3 + x**3, x**2*y + x**2*z + y**2*z], |
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x**3) |
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assert not contains( |
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[x**3 + y**3, y**3 + z**3, z**3 + x**3, x**2*y + x**2*z + y**2*z], |
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x**2 + y**2) |
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assert not contains([x*(1 + x + y), y*(1 + z)], x) |
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assert not contains([x*(1 + x + y), y*(1 + z)], x + y) |
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def test_nontriv_local(): |
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R = QQ.old_poly_ring(x, y, z, order=ilex) |
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def contains(I, f): |
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return R.ideal(*I).contains(f) |
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assert contains([x, y], x) |
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assert contains([x, y], x + y) |
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assert not contains([x, y], 1) |
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assert not contains([x, y], z) |
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assert contains([x**2 + y, x**2 + x], x - y) |
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assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x**2) |
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assert contains([x*(1 + x + y), y*(1 + z)], x) |
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assert contains([x*(1 + x + y), y*(1 + z)], x + y) |
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def test_intersection(): |
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R = QQ.old_poly_ring(x, y, z) |
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assert R.ideal(x, y).intersect(R.ideal(y**2, z)) == R.ideal(y**2, y*z, x*z) |
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assert R.ideal(x, y).intersect(R.ideal()).is_zero() |
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R = QQ.old_poly_ring(x, y, z, order="ilex") |
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assert R.ideal(x, y).intersect(R.ideal(y**2 + y**2*z, z + z*x**3*y)) == \ |
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R.ideal(y**2, y*z, x*z) |
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def test_quotient(): |
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R = QQ.old_poly_ring(x, y, z) |
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assert R.ideal(x, y).quotient(R.ideal(y**2, z)) == R.ideal(x, y) |
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def test_reduction(): |
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from sympy.polys.distributedmodules import sdm_nf_buchberger_reduced |
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R = QQ.old_poly_ring(x, y) |
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I = R.ideal(x**5, y) |
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e = R.convert(x**3 + y**2) |
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assert I.reduce_element(e) == e |
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assert I.reduce_element(e, NF=sdm_nf_buchberger_reduced) == R.convert(x**3) |
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