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| from sympy.assumptions.ask import Q | |
| from sympy.assumptions.refine import refine | |
| from sympy.core.expr import Expr | |
| from sympy.core.numbers import (I, Rational, nan, pi) | |
| from sympy.core.singleton import S | |
| from sympy.core.symbol import Symbol | |
| from sympy.functions.elementary.complexes import (Abs, arg, im, re, sign) | |
| from sympy.functions.elementary.exponential import exp | |
| from sympy.functions.elementary.miscellaneous import sqrt | |
| from sympy.functions.elementary.trigonometric import (atan, atan2) | |
| from sympy.abc import w, x, y, z | |
| from sympy.core.relational import Eq, Ne | |
| from sympy.functions.elementary.piecewise import Piecewise | |
| from sympy.matrices.expressions.matexpr import MatrixSymbol | |
| def test_Abs(): | |
| assert refine(Abs(x), Q.positive(x)) == x | |
| assert refine(1 + Abs(x), Q.positive(x)) == 1 + x | |
| assert refine(Abs(x), Q.negative(x)) == -x | |
| assert refine(1 + Abs(x), Q.negative(x)) == 1 - x | |
| assert refine(Abs(x**2)) != x**2 | |
| assert refine(Abs(x**2), Q.real(x)) == x**2 | |
| def test_pow1(): | |
| assert refine((-1)**x, Q.even(x)) == 1 | |
| assert refine((-1)**x, Q.odd(x)) == -1 | |
| assert refine((-2)**x, Q.even(x)) == 2**x | |
| # nested powers | |
| assert refine(sqrt(x**2)) != Abs(x) | |
| assert refine(sqrt(x**2), Q.complex(x)) != Abs(x) | |
| assert refine(sqrt(x**2), Q.real(x)) == Abs(x) | |
| assert refine(sqrt(x**2), Q.positive(x)) == x | |
| assert refine((x**3)**Rational(1, 3)) != x | |
| assert refine((x**3)**Rational(1, 3), Q.real(x)) != x | |
| assert refine((x**3)**Rational(1, 3), Q.positive(x)) == x | |
| assert refine(sqrt(1/x), Q.real(x)) != 1/sqrt(x) | |
| assert refine(sqrt(1/x), Q.positive(x)) == 1/sqrt(x) | |
| # powers of (-1) | |
| assert refine((-1)**(x + y), Q.even(x)) == (-1)**y | |
| assert refine((-1)**(x + y + z), Q.odd(x) & Q.odd(z)) == (-1)**y | |
| assert refine((-1)**(x + y + 1), Q.odd(x)) == (-1)**y | |
| assert refine((-1)**(x + y + 2), Q.odd(x)) == (-1)**(y + 1) | |
| assert refine((-1)**(x + 3)) == (-1)**(x + 1) | |
| # continuation | |
| assert refine((-1)**((-1)**x/2 - S.Half), Q.integer(x)) == (-1)**x | |
| assert refine((-1)**((-1)**x/2 + S.Half), Q.integer(x)) == (-1)**(x + 1) | |
| assert refine((-1)**((-1)**x/2 + 5*S.Half), Q.integer(x)) == (-1)**(x + 1) | |
| def test_pow2(): | |
| assert refine((-1)**((-1)**x/2 - 7*S.Half), Q.integer(x)) == (-1)**(x + 1) | |
| assert refine((-1)**((-1)**x/2 - 9*S.Half), Q.integer(x)) == (-1)**x | |
| # powers of Abs | |
| assert refine(Abs(x)**2, Q.real(x)) == x**2 | |
| assert refine(Abs(x)**3, Q.real(x)) == Abs(x)**3 | |
| assert refine(Abs(x)**2) == Abs(x)**2 | |
| def test_exp(): | |
| x = Symbol('x', integer=True) | |
| assert refine(exp(pi*I*2*x)) == 1 | |
| assert refine(exp(pi*I*2*(x + S.Half))) == -1 | |
| assert refine(exp(pi*I*2*(x + Rational(1, 4)))) == I | |
| assert refine(exp(pi*I*2*(x + Rational(3, 4)))) == -I | |
| def test_Piecewise(): | |
| assert refine(Piecewise((1, x < 0), (3, True)), (x < 0)) == 1 | |
| assert refine(Piecewise((1, x < 0), (3, True)), ~(x < 0)) == 3 | |
| assert refine(Piecewise((1, x < 0), (3, True)), (y < 0)) == \ | |
| Piecewise((1, x < 0), (3, True)) | |
| assert refine(Piecewise((1, x > 0), (3, True)), (x > 0)) == 1 | |
| assert refine(Piecewise((1, x > 0), (3, True)), ~(x > 0)) == 3 | |
| assert refine(Piecewise((1, x > 0), (3, True)), (y > 0)) == \ | |
| Piecewise((1, x > 0), (3, True)) | |
| assert refine(Piecewise((1, x <= 0), (3, True)), (x <= 0)) == 1 | |
| assert refine(Piecewise((1, x <= 0), (3, True)), ~(x <= 0)) == 3 | |
| assert refine(Piecewise((1, x <= 0), (3, True)), (y <= 0)) == \ | |
| Piecewise((1, x <= 0), (3, True)) | |
| assert refine(Piecewise((1, x >= 0), (3, True)), (x >= 0)) == 1 | |
| assert refine(Piecewise((1, x >= 0), (3, True)), ~(x >= 0)) == 3 | |
| assert refine(Piecewise((1, x >= 0), (3, True)), (y >= 0)) == \ | |
| Piecewise((1, x >= 0), (3, True)) | |
| assert refine(Piecewise((1, Eq(x, 0)), (3, True)), (Eq(x, 0)))\ | |
| == 1 | |
| assert refine(Piecewise((1, Eq(x, 0)), (3, True)), (Eq(0, x)))\ | |
| == 1 | |
| assert refine(Piecewise((1, Eq(x, 0)), (3, True)), ~(Eq(x, 0)))\ | |
| == 3 | |
| assert refine(Piecewise((1, Eq(x, 0)), (3, True)), ~(Eq(0, x)))\ | |
| == 3 | |
| assert refine(Piecewise((1, Eq(x, 0)), (3, True)), (Eq(y, 0)))\ | |
| == Piecewise((1, Eq(x, 0)), (3, True)) | |
| assert refine(Piecewise((1, Ne(x, 0)), (3, True)), (Ne(x, 0)))\ | |
| == 1 | |
| assert refine(Piecewise((1, Ne(x, 0)), (3, True)), ~(Ne(x, 0)))\ | |
| == 3 | |
| assert refine(Piecewise((1, Ne(x, 0)), (3, True)), (Ne(y, 0)))\ | |
| == Piecewise((1, Ne(x, 0)), (3, True)) | |
| def test_atan2(): | |
| assert refine(atan2(y, x), Q.real(y) & Q.positive(x)) == atan(y/x) | |
| assert refine(atan2(y, x), Q.negative(y) & Q.positive(x)) == atan(y/x) | |
| assert refine(atan2(y, x), Q.negative(y) & Q.negative(x)) == atan(y/x) - pi | |
| assert refine(atan2(y, x), Q.positive(y) & Q.negative(x)) == atan(y/x) + pi | |
| assert refine(atan2(y, x), Q.zero(y) & Q.negative(x)) == pi | |
| assert refine(atan2(y, x), Q.positive(y) & Q.zero(x)) == pi/2 | |
| assert refine(atan2(y, x), Q.negative(y) & Q.zero(x)) == -pi/2 | |
| assert refine(atan2(y, x), Q.zero(y) & Q.zero(x)) is nan | |
| def test_re(): | |
| assert refine(re(x), Q.real(x)) == x | |
| assert refine(re(x), Q.imaginary(x)) is S.Zero | |
| assert refine(re(x+y), Q.real(x) & Q.real(y)) == x + y | |
| assert refine(re(x+y), Q.real(x) & Q.imaginary(y)) == x | |
| assert refine(re(x*y), Q.real(x) & Q.real(y)) == x * y | |
| assert refine(re(x*y), Q.real(x) & Q.imaginary(y)) == 0 | |
| assert refine(re(x*y*z), Q.real(x) & Q.real(y) & Q.real(z)) == x * y * z | |
| def test_im(): | |
| assert refine(im(x), Q.imaginary(x)) == -I*x | |
| assert refine(im(x), Q.real(x)) is S.Zero | |
| assert refine(im(x+y), Q.imaginary(x) & Q.imaginary(y)) == -I*x - I*y | |
| assert refine(im(x+y), Q.real(x) & Q.imaginary(y)) == -I*y | |
| assert refine(im(x*y), Q.imaginary(x) & Q.real(y)) == -I*x*y | |
| assert refine(im(x*y), Q.imaginary(x) & Q.imaginary(y)) == 0 | |
| assert refine(im(1/x), Q.imaginary(x)) == -I/x | |
| assert refine(im(x*y*z), Q.imaginary(x) & Q.imaginary(y) | |
| & Q.imaginary(z)) == -I*x*y*z | |
| def test_complex(): | |
| assert refine(re(1/(x + I*y)), Q.real(x) & Q.real(y)) == \ | |
| x/(x**2 + y**2) | |
| assert refine(im(1/(x + I*y)), Q.real(x) & Q.real(y)) == \ | |
| -y/(x**2 + y**2) | |
| assert refine(re((w + I*x) * (y + I*z)), Q.real(w) & Q.real(x) & Q.real(y) | |
| & Q.real(z)) == w*y - x*z | |
| assert refine(im((w + I*x) * (y + I*z)), Q.real(w) & Q.real(x) & Q.real(y) | |
| & Q.real(z)) == w*z + x*y | |
| def test_sign(): | |
| x = Symbol('x', real = True) | |
| assert refine(sign(x), Q.positive(x)) == 1 | |
| assert refine(sign(x), Q.negative(x)) == -1 | |
| assert refine(sign(x), Q.zero(x)) == 0 | |
| assert refine(sign(x), True) == sign(x) | |
| assert refine(sign(Abs(x)), Q.nonzero(x)) == 1 | |
| x = Symbol('x', imaginary=True) | |
| assert refine(sign(x), Q.positive(im(x))) == S.ImaginaryUnit | |
| assert refine(sign(x), Q.negative(im(x))) == -S.ImaginaryUnit | |
| assert refine(sign(x), True) == sign(x) | |
| x = Symbol('x', complex=True) | |
| assert refine(sign(x), Q.zero(x)) == 0 | |
| def test_arg(): | |
| x = Symbol('x', complex = True) | |
| assert refine(arg(x), Q.positive(x)) == 0 | |
| assert refine(arg(x), Q.negative(x)) == pi | |
| def test_func_args(): | |
| class MyClass(Expr): | |
| # A class with nontrivial .func | |
| def __init__(self, *args): | |
| self.my_member = "" | |
| def func(self): | |
| def my_func(*args): | |
| obj = MyClass(*args) | |
| obj.my_member = self.my_member | |
| return obj | |
| return my_func | |
| x = MyClass() | |
| x.my_member = "A very important value" | |
| assert x.my_member == refine(x).my_member | |
| def test_issue_refine_9384(): | |
| assert refine(Piecewise((1, x < 0), (0, True)), Q.positive(x)) == 0 | |
| assert refine(Piecewise((1, x < 0), (0, True)), Q.negative(x)) == 1 | |
| assert refine(Piecewise((1, x > 0), (0, True)), Q.positive(x)) == 1 | |
| assert refine(Piecewise((1, x > 0), (0, True)), Q.negative(x)) == 0 | |
| def test_eval_refine(): | |
| class MockExpr(Expr): | |
| def _eval_refine(self, assumptions): | |
| return True | |
| mock_obj = MockExpr() | |
| assert refine(mock_obj) | |
| def test_refine_issue_12724(): | |
| expr1 = refine(Abs(x * y), Q.positive(x)) | |
| expr2 = refine(Abs(x * y * z), Q.positive(x)) | |
| assert expr1 == x * Abs(y) | |
| assert expr2 == x * Abs(y * z) | |
| y1 = Symbol('y1', real = True) | |
| expr3 = refine(Abs(x * y1**2 * z), Q.positive(x)) | |
| assert expr3 == x * y1**2 * Abs(z) | |
| def test_matrixelement(): | |
| x = MatrixSymbol('x', 3, 3) | |
| i = Symbol('i', positive = True) | |
| j = Symbol('j', positive = True) | |
| assert refine(x[0, 1], Q.symmetric(x)) == x[0, 1] | |
| assert refine(x[1, 0], Q.symmetric(x)) == x[0, 1] | |
| assert refine(x[i, j], Q.symmetric(x)) == x[j, i] | |
| assert refine(x[j, i], Q.symmetric(x)) == x[j, i] | |