MLPY/Lib/site-packages/sympy/sets/tests/test_conditionset.py

from sympy.core.expr import unchanged
from sympy.sets import (ConditionSet, Intersection, FiniteSet,
    EmptySet, Union, Contains, ImageSet)
from sympy.sets.sets import SetKind
from sympy.core.function import (Function, Lambda)
from sympy.core.mod import Mod
from sympy.core.kind import NumberKind
from sympy.core.numbers import (oo, pi)
from sympy.core.relational import (Eq, Ne)
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.complexes import Abs
from sympy.functions.elementary.trigonometric import (asin, sin)
from sympy.logic.boolalg import And
from sympy.matrices.dense import Matrix
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.sets.sets import Interval
from sympy.testing.pytest import raises, warns_deprecated_sympy


w = Symbol('w')
x = Symbol('x')
y = Symbol('y')
z = Symbol('z')
f = Function('f')


def test_CondSet():
    sin_sols_principal = ConditionSet(x, Eq(sin(x), 0),
                                      Interval(0, 2*pi, False, True))
    assert pi in sin_sols_principal
    assert pi/2 not in sin_sols_principal
    assert 3*pi not in sin_sols_principal
    assert oo not in sin_sols_principal
    assert 5 in ConditionSet(x, x**2 > 4, S.Reals)
    assert 1 not in ConditionSet(x, x**2 > 4, S.Reals)
    # in this case, 0 is not part of the base set so
    # it can't be in any subset selected by the condition
    assert 0 not in ConditionSet(x, y > 5, Interval(1, 7))
    # since 'in' requires a true/false, the following raises
    # an error because the given value provides no information
    # for the condition to evaluate (since the condition does
    # not depend on the dummy symbol): the result is `y > 5`.
    # In this case, ConditionSet is just acting like
    # Piecewise((Interval(1, 7), y > 5), (S.EmptySet, True)).
    raises(TypeError, lambda: 6 in ConditionSet(x, y > 5,
        Interval(1, 7)))

    X = MatrixSymbol('X', 2, 2)
    matrix_set = ConditionSet(X, Eq(X*Matrix([[1, 1], [1, 1]]), X))
    Y = Matrix([[0, 0], [0, 0]])
    assert matrix_set.contains(Y).doit() is S.true
    Z = Matrix([[1, 2], [3, 4]])
    assert matrix_set.contains(Z).doit() is S.false

    assert isinstance(ConditionSet(x, x < 1, {x, y}).base_set,
        FiniteSet)
    raises(TypeError, lambda: ConditionSet(x, x + 1, {x, y}))
    raises(TypeError, lambda: ConditionSet(x, x, 1))

    I = S.Integers
    U = S.UniversalSet
    C = ConditionSet
    assert C(x, False, I) is S.EmptySet
    assert C(x, True, I) is I
    assert C(x, x < 1, C(x, x < 2, I)
        ) == C(x, (x < 1) & (x < 2), I)
    assert C(y, y < 1, C(x, y < 2, I)
        ) == C(x, (x < 1) & (y < 2), I), C(y, y < 1, C(x, y < 2, I))
    assert C(y, y < 1, C(x, x < 2, I)
        ) == C(y, (y < 1) & (y < 2), I)
    assert C(y, y < 1, C(x, y < x, I)
        ) == C(x, (x < 1) & (y < x), I)
    assert unchanged(C, y, x < 1, C(x, y < x, I))
    assert ConditionSet(x, x < 1).base_set is U
    # arg checking is not done at instantiation but this
    # will raise an error when containment is tested
    assert ConditionSet((x,), x < 1).base_set is U

    c = ConditionSet((x, y), x < y, I**2)
    assert (1, 2) in c
    assert (1, pi) not in c

    raises(TypeError, lambda: C(x, x > 1, C((x, y), x > 1, I**2)))
    # signature mismatch since only 3 args are accepted
    raises(TypeError, lambda: C((x, y), x + y < 2, U, U))


def test_CondSet_intersect():
    input_conditionset = ConditionSet(x, x**2 > 4, Interval(1, 4, False,
        False))
    other_domain = Interval(0, 3, False, False)
    output_conditionset = ConditionSet(x, x**2 > 4, Interval(
        1, 3, False, False))
    assert Intersection(input_conditionset, other_domain
        ) == output_conditionset


def test_issue_9849():
    assert ConditionSet(x, Eq(x, x), S.Naturals
        ) is S.Naturals
    assert ConditionSet(x, Eq(Abs(sin(x)), -1), S.Naturals
        ) == S.EmptySet


def test_simplified_FiniteSet_in_CondSet():
    assert ConditionSet(x, And(x < 1, x > -3), FiniteSet(0, 1, 2)
        ) == FiniteSet(0)
    assert ConditionSet(x, x < 0, FiniteSet(0, 1, 2)) == EmptySet
    assert ConditionSet(x, And(x < -3), EmptySet) == EmptySet
    y = Symbol('y')
    assert (ConditionSet(x, And(x > 0), FiniteSet(-1, 0, 1, y)) ==
        Union(FiniteSet(1), ConditionSet(x, And(x > 0), FiniteSet(y))))
    assert (ConditionSet(x, Eq(Mod(x, 3), 1), FiniteSet(1, 4, 2, y)) ==
        Union(FiniteSet(1, 4), ConditionSet(x, Eq(Mod(x, 3), 1),
        FiniteSet(y))))


def test_free_symbols():
    assert ConditionSet(x, Eq(y, 0), FiniteSet(z)
        ).free_symbols == {y, z}
    assert ConditionSet(x, Eq(x, 0), FiniteSet(z)
        ).free_symbols == {z}
    assert ConditionSet(x, Eq(x, 0), FiniteSet(x, z)
        ).free_symbols == {x, z}
    assert ConditionSet(x, Eq(x, 0), ImageSet(Lambda(y, y**2),
        S.Integers)).free_symbols == set()


def test_bound_symbols():
    assert ConditionSet(x, Eq(y, 0), FiniteSet(z)
        ).bound_symbols == [x]
    assert ConditionSet(x, Eq(x, 0), FiniteSet(x, y)
        ).bound_symbols == [x]
    assert ConditionSet(x, x < 10, ImageSet(Lambda(y, y**2), S.Integers)
        ).bound_symbols == [x]
    assert ConditionSet(x, x < 10, ConditionSet(y, y > 1, S.Integers)
        ).bound_symbols == [x]


def test_as_dummy():
    _0, _1 = symbols('_0 _1')
    assert ConditionSet(x, x < 1, Interval(y, oo)
        ).as_dummy() == ConditionSet(_0, _0 < 1, Interval(y, oo))
    assert ConditionSet(x, x < 1, Interval(x, oo)
        ).as_dummy() == ConditionSet(_0, _0 < 1, Interval(x, oo))
    assert ConditionSet(x, x < 1, ImageSet(Lambda(y, y**2), S.Integers)
        ).as_dummy() == ConditionSet(
            _0, _0 < 1, ImageSet(Lambda(_0, _0**2), S.Integers))
    e = ConditionSet((x, y), x <= y, S.Reals**2)
    assert e.bound_symbols == [x, y]
    assert e.as_dummy() == ConditionSet((_0, _1), _0 <= _1, S.Reals**2)
    assert e.as_dummy() == ConditionSet((y, x), y <= x, S.Reals**2
        ).as_dummy()


def test_subs_CondSet():
    s = FiniteSet(z, y)
    c = ConditionSet(x, x < 2, s)
    assert c.subs(x, y) == c
    assert c.subs(z, y) == ConditionSet(x, x < 2, FiniteSet(y))
    assert c.xreplace({x: y}) == ConditionSet(y, y < 2, s)

    assert ConditionSet(x, x < y, s
        ).subs(y, w) == ConditionSet(x, x < w, s.subs(y, w))
    # if the user uses assumptions that cause the condition
    # to evaluate, that can't be helped from SymPy's end
    n = Symbol('n', negative=True)
    assert ConditionSet(n, 0 < n, S.Integers) is S.EmptySet
    p = Symbol('p', positive=True)
    assert ConditionSet(n, n < y, S.Integers
        ).subs(n, x) == ConditionSet(n, n < y, S.Integers)
    raises(ValueError, lambda: ConditionSet(
        x + 1, x < 1, S.Integers))
    assert ConditionSet(
        p, n < x, Interval(-5, 5)).subs(x, p) == Interval(-5, 5), ConditionSet(
        p, n < x, Interval(-5, 5)).subs(x, p)
    assert ConditionSet(
        n, n < x, Interval(-oo, 0)).subs(x, p
        ) == Interval(-oo, 0)

    assert ConditionSet(f(x), f(x) < 1, {w, z}
        ).subs(f(x), y) == ConditionSet(f(x), f(x) < 1, {w, z})

    # issue 17341
    k = Symbol('k')
    img1 = ImageSet(Lambda(k, 2*k*pi + asin(y)), S.Integers)
    img2 = ImageSet(Lambda(k, 2*k*pi + asin(S.One/3)), S.Integers)
    assert ConditionSet(x, Contains(
        y, Interval(-1,1)), img1).subs(y, S.One/3).dummy_eq(img2)

    assert (0, 1) in ConditionSet((x, y), x + y < 3, S.Integers**2)

    raises(TypeError, lambda: ConditionSet(n, n < -10, Interval(0, 10)))


def test_subs_CondSet_tebr():
    with warns_deprecated_sympy():
        assert ConditionSet((x, y), {x + 1, x + y}, S.Reals**2) == \
            ConditionSet((x, y), Eq(x + 1, 0) & Eq(x + y, 0), S.Reals**2)


def test_dummy_eq():
    C = ConditionSet
    I = S.Integers
    c = C(x, x < 1, I)
    assert c.dummy_eq(C(y, y < 1, I))
    assert c.dummy_eq(1) == False
    assert c.dummy_eq(C(x, x < 1, S.Reals)) == False

    c1 = ConditionSet((x, y), Eq(x + 1, 0) & Eq(x + y, 0), S.Reals**2)
    c2 = ConditionSet((x, y), Eq(x + 1, 0) & Eq(x + y, 0), S.Reals**2)
    c3 = ConditionSet((x, y), Eq(x + 1, 0) & Eq(x + y, 0), S.Complexes**2)
    assert c1.dummy_eq(c2)
    assert c1.dummy_eq(c3) is False
    assert c.dummy_eq(c1) is False
    assert c1.dummy_eq(c) is False

    # issue 19496
    m = Symbol('m')
    n = Symbol('n')
    a = Symbol('a')
    d1 = ImageSet(Lambda(m, m*pi), S.Integers)
    d2 = ImageSet(Lambda(n, n*pi), S.Integers)
    c1 = ConditionSet(x, Ne(a, 0), d1)
    c2 = ConditionSet(x, Ne(a, 0), d2)
    assert c1.dummy_eq(c2)


def test_contains():
    assert 6 in ConditionSet(x, x > 5, Interval(1, 7))
    assert (8 in ConditionSet(x, y > 5, Interval(1, 7))) is False
    # `in` should give True or False; in this case there is not
    # enough information for that result
    raises(TypeError,
        lambda: 6 in ConditionSet(x, y > 5, Interval(1, 7)))
    # here, there is enough information but the comparison is
    # not defined
    raises(TypeError, lambda: 0 in ConditionSet(x, 1/x >= 0, S.Reals))
    assert ConditionSet(x, y > 5, Interval(1, 7)
        ).contains(6) == (y > 5)
    assert ConditionSet(x, y > 5, Interval(1, 7)
        ).contains(8) is S.false
    assert ConditionSet(x, y > 5, Interval(1, 7)
        ).contains(w) == And(Contains(w, Interval(1, 7)), y > 5)
    # This returns an unevaluated Contains object
    # because 1/0 should not be defined for 1 and 0 in the context of
    # reals.
    assert ConditionSet(x, 1/x >= 0, S.Reals).contains(0) == \
        Contains(0, ConditionSet(x, 1/x >= 0, S.Reals), evaluate=False)
    c = ConditionSet((x, y), x + y > 1, S.Integers**2)
    assert not c.contains(1)
    assert c.contains((2, 1))
    assert not c.contains((0, 1))
    c = ConditionSet((w, (x, y)), w + x + y > 1, S.Integers*S.Integers**2)
    assert not c.contains(1)
    assert not c.contains((1, 2))
    assert not c.contains(((1, 2), 3))
    assert not c.contains(((1, 2), (3, 4)))
    assert c.contains((1, (3, 4)))


def test_as_relational():
    assert ConditionSet((x, y), x > 1, S.Integers**2).as_relational((x, y)
        ) == (x > 1) & Contains(x, S.Integers) & Contains(y, S.Integers)
    assert ConditionSet(x, x > 1, S.Integers).as_relational(x
        ) == Contains(x, S.Integers) & (x > 1)


def test_flatten():
    """Tests whether there is basic denesting functionality"""
    inner = ConditionSet(x, sin(x) + x > 0)
    outer = ConditionSet(x, Contains(x, inner), S.Reals)
    assert outer == ConditionSet(x, sin(x) + x > 0, S.Reals)

    inner = ConditionSet(y, sin(y) + y > 0)
    outer = ConditionSet(x, Contains(y, inner), S.Reals)
    assert outer != ConditionSet(x, sin(x) + x > 0, S.Reals)

    inner = ConditionSet(x, sin(x) + x > 0).intersect(Interval(-1, 1))
    outer = ConditionSet(x, Contains(x, inner), S.Reals)
    assert outer == ConditionSet(x, sin(x) + x > 0, Interval(-1, 1))


def test_duplicate():
    from sympy.core.function import BadSignatureError
    # test coverage for line 95 in conditionset.py, check for duplicates in symbols
    dup = symbols('a,a')
    raises(BadSignatureError, lambda: ConditionSet(dup, x < 0))


def test_SetKind_ConditionSet():
    assert ConditionSet(x, Eq(sin(x), 0), Interval(0, 2*pi)).kind is SetKind(NumberKind)
    assert ConditionSet(x, x < 0).kind is SetKind(NumberKind)