Vehicle-cpp/hkpc/software/sympy/assumptions/tests/test_refine.py

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 = ""

        @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_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]