Ejemplo n.º 1
0
def _(expr, assumptions):
    return ask(Q.negative_infinite(expr)
               | Q.negative(expr)
               | Q.zero(expr)
               | Q.positive(expr)
               | Q.positive_infinite(expr),
            assumptions)
Ejemplo n.º 2
0
def refine_atan2(expr, assumptions):
    """
    Handler for the atan2 function

    Examples
    ========

    >>> from sympy import Symbol, Q, refine, atan2
    >>> from sympy.assumptions.refine import refine_atan2
    >>> from sympy.abc import x, y
    >>> refine_atan2(atan2(y,x), Q.real(y) & Q.positive(x))
    atan(y/x)
    >>> refine_atan2(atan2(y,x), Q.negative(y) & Q.negative(x))
    atan(y/x) - pi
    >>> refine_atan2(atan2(y,x), Q.positive(y) & Q.negative(x))
    atan(y/x) + pi
    >>> refine_atan2(atan2(y,x), Q.zero(y) & Q.negative(x))
    pi
    >>> refine_atan2(atan2(y,x), Q.positive(y) & Q.zero(x))
    pi/2
    >>> refine_atan2(atan2(y,x), Q.negative(y) & Q.zero(x))
    -pi/2
    >>> refine_atan2(atan2(y,x), Q.zero(y) & Q.zero(x))
    nan
    """
    from sympy.functions.elementary.trigonometric import atan
    from sympy.core import S

    y, x = expr.args
    if ask(Q.real(y) & Q.positive(x), assumptions):
        return atan(y / x)
    elif ask(Q.negative(y) & Q.negative(x), assumptions):
        return atan(y / x) - S.Pi
    elif ask(Q.positive(y) & Q.negative(x), assumptions):
        return atan(y / x) + S.Pi
    elif ask(Q.zero(y) & Q.negative(x), assumptions):
        return S.Pi
    elif ask(Q.positive(y) & Q.zero(x), assumptions):
        return S.Pi / 2
    elif ask(Q.negative(y) & Q.zero(x), assumptions):
        return -S.Pi / 2
    elif ask(Q.zero(y) & Q.zero(x), assumptions):
        return S.NaN
    else:
        return expr
Ejemplo n.º 3
0
def refine_atan2(expr, assumptions):
    """
    Handler for the atan2 function

    Examples
    ========

    >>> from sympy import Symbol, Q, refine, atan2
    >>> from sympy.assumptions.refine import refine_atan2
    >>> from sympy.abc import x, y
    >>> refine_atan2(atan2(y,x), Q.real(y) & Q.positive(x))
    atan(y/x)
    >>> refine_atan2(atan2(y,x), Q.negative(y) & Q.negative(x))
    atan(y/x) - pi
    >>> refine_atan2(atan2(y,x), Q.positive(y) & Q.negative(x))
    atan(y/x) + pi
    >>> refine_atan2(atan2(y,x), Q.zero(y) & Q.negative(x))
    pi
    >>> refine_atan2(atan2(y,x), Q.positive(y) & Q.zero(x))
    pi/2
    >>> refine_atan2(atan2(y,x), Q.negative(y) & Q.zero(x))
    -pi/2
    >>> refine_atan2(atan2(y,x), Q.zero(y) & Q.zero(x))
    nan
    """
    from sympy.functions.elementary.trigonometric import atan
    from sympy.core import S

    y, x = expr.args
    if ask(Q.real(y) & Q.positive(x), assumptions):
        return atan(y / x)
    elif ask(Q.negative(y) & Q.negative(x), assumptions):
        return atan(y / x) - S.Pi
    elif ask(Q.positive(y) & Q.negative(x), assumptions):
        return atan(y / x) + S.Pi
    elif ask(Q.zero(y) & Q.negative(x), assumptions):
        return S.Pi
    elif ask(Q.positive(y) & Q.zero(x), assumptions):
        return S.Pi / 2
    elif ask(Q.negative(y) & Q.zero(x), assumptions):
        return -S.Pi / 2
    elif ask(Q.zero(y) & Q.zero(x), assumptions):
        return S.NaN
    else:
        return expr
Ejemplo n.º 4
0
def refine_sign(expr, assumptions):
    """
    Handler for sign.

    Examples
    ========

    >>> from sympy.assumptions.refine import refine_sign
    >>> from sympy import Symbol, Q, sign, im
    >>> x = Symbol('x', real = True)
    >>> expr = sign(x)
    >>> refine_sign(expr, Q.positive(x) & Q.nonzero(x))
    1
    >>> refine_sign(expr, Q.negative(x) & Q.nonzero(x))
    -1
    >>> refine_sign(expr, Q.zero(x))
    0
    >>> y = Symbol('y', imaginary = True)
    >>> expr = sign(y)
    >>> refine_sign(expr, Q.positive(im(y)))
    I
    >>> refine_sign(expr, Q.negative(im(y)))
    -I
    """
    arg = expr.args[0]
    if ask(Q.zero(arg), assumptions):
        return S.Zero
    if ask(Q.real(arg)):
        if ask(Q.positive(arg), assumptions):
            return S.One
        if ask(Q.negative(arg), assumptions):
            return S.NegativeOne
    if ask(Q.imaginary(arg)):
        arg_re, arg_im = arg.as_real_imag()
        if ask(Q.positive(arg_im), assumptions):
            return S.ImaginaryUnit
        if ask(Q.negative(arg_im), assumptions):
            return -S.ImaginaryUnit
    return expr
Ejemplo n.º 5
0
def _(expr, assumptions):
    if isinstance(expr, MatrixBase):
        return None
    if ask(Q.zero(expr), assumptions):
        return True
    return ask(Q.imaginary(expr), assumptions)
Ejemplo n.º 6
0
 def Mul(expr, assumptions):
     # TODO: This should be deducible from the nonzero handler
     return fuzzy_or(ask(Q.zero(arg), assumptions) for arg in expr.args)
Ejemplo n.º 7
0
def _(expr, assumptions):
    # After complex -> finite fact is registered to new assumption system,
    # querying Q.infinite may be removed.
    if ask(Q.infinite(expr.args[0]), assumptions):
        return False
    return ask(~Q.zero(expr.args[0]), assumptions)
Ejemplo n.º 8
0
 def Mul(expr, assumptions):
     # TODO: This should be deducible from the nonzero handler
     return fuzzy_or(ask(Q.zero(arg), assumptions) for arg in expr.args)
Ejemplo n.º 9
0
def get_known_facts(x=None):
    """
    Facts between unary predicates.

    Parameters
    ==========

    x : Symbol, optional
        Placeholder symbol for unary facts. Default is ``Symbol('x')``.

    Returns
    =======

    fact : Known facts in conjugated normal form.

    """
    if x is None:
        x = Symbol('x')

    fact = And(
        # primitive predicates for extended real exclude each other.
        Exclusive(Q.negative_infinite(x), Q.negative(x), Q.zero(x),
                  Q.positive(x), Q.positive_infinite(x)),

        # build complex plane
        Exclusive(Q.real(x), Q.imaginary(x)),
        Implies(Q.real(x) | Q.imaginary(x), Q.complex(x)),

        # other subsets of complex
        Exclusive(Q.transcendental(x), Q.algebraic(x)),
        Equivalent(Q.real(x),
                   Q.rational(x) | Q.irrational(x)),
        Exclusive(Q.irrational(x), Q.rational(x)),
        Implies(Q.rational(x), Q.algebraic(x)),

        # integers
        Exclusive(Q.even(x), Q.odd(x)),
        Implies(Q.integer(x), Q.rational(x)),
        Implies(Q.zero(x), Q.even(x)),
        Exclusive(Q.composite(x), Q.prime(x)),
        Implies(Q.composite(x) | Q.prime(x),
                Q.integer(x) & Q.positive(x)),
        Implies(Q.even(x) & Q.positive(x) & ~Q.prime(x), Q.composite(x)),

        # hermitian and antihermitian
        Implies(Q.real(x), Q.hermitian(x)),
        Implies(Q.imaginary(x), Q.antihermitian(x)),
        Implies(Q.zero(x),
                Q.hermitian(x) | Q.antihermitian(x)),

        # define finity and infinity, and build extended real line
        Exclusive(Q.infinite(x), Q.finite(x)),
        Implies(Q.complex(x), Q.finite(x)),
        Implies(
            Q.negative_infinite(x) | Q.positive_infinite(x), Q.infinite(x)),

        # commutativity
        Implies(Q.finite(x) | Q.infinite(x), Q.commutative(x)),

        # matrices
        Implies(Q.orthogonal(x), Q.positive_definite(x)),
        Implies(Q.orthogonal(x), Q.unitary(x)),
        Implies(Q.unitary(x) & Q.real_elements(x), Q.orthogonal(x)),
        Implies(Q.unitary(x), Q.normal(x)),
        Implies(Q.unitary(x), Q.invertible(x)),
        Implies(Q.normal(x), Q.square(x)),
        Implies(Q.diagonal(x), Q.normal(x)),
        Implies(Q.positive_definite(x), Q.invertible(x)),
        Implies(Q.diagonal(x), Q.upper_triangular(x)),
        Implies(Q.diagonal(x), Q.lower_triangular(x)),
        Implies(Q.lower_triangular(x), Q.triangular(x)),
        Implies(Q.upper_triangular(x), Q.triangular(x)),
        Implies(Q.triangular(x),
                Q.upper_triangular(x) | Q.lower_triangular(x)),
        Implies(Q.upper_triangular(x) & Q.lower_triangular(x), Q.diagonal(x)),
        Implies(Q.diagonal(x), Q.symmetric(x)),
        Implies(Q.unit_triangular(x), Q.triangular(x)),
        Implies(Q.invertible(x), Q.fullrank(x)),
        Implies(Q.invertible(x), Q.square(x)),
        Implies(Q.symmetric(x), Q.square(x)),
        Implies(Q.fullrank(x) & Q.square(x), Q.invertible(x)),
        Equivalent(Q.invertible(x), ~Q.singular(x)),
        Implies(Q.integer_elements(x), Q.real_elements(x)),
        Implies(Q.real_elements(x), Q.complex_elements(x)),
    )
    return fact