from sympy import (Abs, exp, Expr, I, pi, Q, Rational, refine, S, sqrt, atan, atan2, nan, Symbol, re, im, sign, arg) 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 = "" @property 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_eval_refine(): from sympy.core.expr import Expr 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]