388 lines
8.7 KiB
Python
388 lines
8.7 KiB
Python
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from sympy.assumptions import Predicate
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from sympy.multipledispatch import Dispatcher
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class IntegerPredicate(Predicate):
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"""
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Integer predicate.
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Explanation
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===========
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``Q.integer(x)`` is true iff ``x`` belongs to the set of integer
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numbers.
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Examples
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========
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>>> from sympy import Q, ask, S
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>>> ask(Q.integer(5))
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True
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>>> ask(Q.integer(S(1)/2))
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False
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Integer
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"""
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name = 'integer'
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handler = Dispatcher(
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"IntegerHandler",
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doc=("Handler for Q.integer.\n\n"
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"Test that an expression belongs to the field of integer numbers.")
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)
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class RationalPredicate(Predicate):
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"""
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Rational number predicate.
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Explanation
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===========
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``Q.rational(x)`` is true iff ``x`` belongs to the set of
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rational numbers.
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Examples
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========
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>>> from sympy import ask, Q, pi, S
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>>> ask(Q.rational(0))
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True
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>>> ask(Q.rational(S(1)/2))
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True
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>>> ask(Q.rational(pi))
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False
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Rational_number
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"""
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name = 'rational'
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handler = Dispatcher(
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"RationalHandler",
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doc=("Handler for Q.rational.\n\n"
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"Test that an expression belongs to the field of rational numbers.")
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)
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class IrrationalPredicate(Predicate):
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"""
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Irrational number predicate.
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Explanation
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===========
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``Q.irrational(x)`` is true iff ``x`` is any real number that
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cannot be expressed as a ratio of integers.
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Examples
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========
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>>> from sympy import ask, Q, pi, S, I
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>>> ask(Q.irrational(0))
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False
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>>> ask(Q.irrational(S(1)/2))
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False
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>>> ask(Q.irrational(pi))
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True
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>>> ask(Q.irrational(I))
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False
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Irrational_number
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"""
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name = 'irrational'
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handler = Dispatcher(
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"IrrationalHandler",
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doc=("Handler for Q.irrational.\n\n"
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"Test that an expression is irrational numbers.")
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)
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class RealPredicate(Predicate):
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r"""
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Real number predicate.
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Explanation
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===========
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``Q.real(x)`` is true iff ``x`` is a real number, i.e., it is in the
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interval `(-\infty, \infty)`. Note that, in particular the
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infinities are not real. Use ``Q.extended_real`` if you want to
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consider those as well.
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A few important facts about reals:
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- Every real number is positive, negative, or zero. Furthermore,
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because these sets are pairwise disjoint, each real number is
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exactly one of those three.
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- Every real number is also complex.
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- Every real number is finite.
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- Every real number is either rational or irrational.
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- Every real number is either algebraic or transcendental.
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- The facts ``Q.negative``, ``Q.zero``, ``Q.positive``,
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``Q.nonnegative``, ``Q.nonpositive``, ``Q.nonzero``,
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``Q.integer``, ``Q.rational``, and ``Q.irrational`` all imply
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``Q.real``, as do all facts that imply those facts.
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- The facts ``Q.algebraic``, and ``Q.transcendental`` do not imply
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``Q.real``; they imply ``Q.complex``. An algebraic or
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transcendental number may or may not be real.
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- The "non" facts (i.e., ``Q.nonnegative``, ``Q.nonzero``,
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``Q.nonpositive`` and ``Q.noninteger``) are not equivalent to
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not the fact, but rather, not the fact *and* ``Q.real``.
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For example, ``Q.nonnegative`` means ``~Q.negative & Q.real``.
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So for example, ``I`` is not nonnegative, nonzero, or
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nonpositive.
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Examples
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========
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>>> from sympy import Q, ask, symbols
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>>> x = symbols('x')
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>>> ask(Q.real(x), Q.positive(x))
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True
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>>> ask(Q.real(0))
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True
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Real_number
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"""
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name = 'real'
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handler = Dispatcher(
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"RealHandler",
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doc=("Handler for Q.real.\n\n"
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"Test that an expression belongs to the field of real numbers.")
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)
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class ExtendedRealPredicate(Predicate):
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r"""
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Extended real predicate.
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Explanation
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===========
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``Q.extended_real(x)`` is true iff ``x`` is a real number or
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`\{-\infty, \infty\}`.
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See documentation of ``Q.real`` for more information about related
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facts.
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Examples
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========
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>>> from sympy import ask, Q, oo, I
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>>> ask(Q.extended_real(1))
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True
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>>> ask(Q.extended_real(I))
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False
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>>> ask(Q.extended_real(oo))
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True
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"""
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name = 'extended_real'
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handler = Dispatcher(
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"ExtendedRealHandler",
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doc=("Handler for Q.extended_real.\n\n"
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"Test that an expression belongs to the field of extended real\n"
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"numbers, that is real numbers union {Infinity, -Infinity}.")
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)
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class HermitianPredicate(Predicate):
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"""
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Hermitian predicate.
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Explanation
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===========
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``ask(Q.hermitian(x))`` is true iff ``x`` belongs to the set of
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Hermitian operators.
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References
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==========
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.. [1] https://mathworld.wolfram.com/HermitianOperator.html
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"""
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# TODO: Add examples
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name = 'hermitian'
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handler = Dispatcher(
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"HermitianHandler",
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doc=("Handler for Q.hermitian.\n\n"
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"Test that an expression belongs to the field of Hermitian operators.")
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)
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class ComplexPredicate(Predicate):
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"""
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Complex number predicate.
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Explanation
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===========
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``Q.complex(x)`` is true iff ``x`` belongs to the set of complex
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numbers. Note that every complex number is finite.
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Examples
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========
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>>> from sympy import Q, Symbol, ask, I, oo
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>>> x = Symbol('x')
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>>> ask(Q.complex(0))
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True
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>>> ask(Q.complex(2 + 3*I))
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True
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>>> ask(Q.complex(oo))
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False
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Complex_number
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"""
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name = 'complex'
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handler = Dispatcher(
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"ComplexHandler",
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doc=("Handler for Q.complex.\n\n"
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"Test that an expression belongs to the field of complex numbers.")
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)
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class ImaginaryPredicate(Predicate):
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"""
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Imaginary number predicate.
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Explanation
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===========
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``Q.imaginary(x)`` is true iff ``x`` can be written as a real
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number multiplied by the imaginary unit ``I``. Please note that ``0``
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is not considered to be an imaginary number.
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Examples
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========
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>>> from sympy import Q, ask, I
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>>> ask(Q.imaginary(3*I))
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True
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>>> ask(Q.imaginary(2 + 3*I))
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False
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>>> ask(Q.imaginary(0))
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False
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Imaginary_number
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"""
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name = 'imaginary'
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handler = Dispatcher(
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"ImaginaryHandler",
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doc=("Handler for Q.imaginary.\n\n"
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"Test that an expression belongs to the field of imaginary numbers,\n"
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"that is, numbers in the form x*I, where x is real.")
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)
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class AntihermitianPredicate(Predicate):
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"""
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Antihermitian predicate.
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Explanation
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===========
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``Q.antihermitian(x)`` is true iff ``x`` belongs to the field of
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antihermitian operators, i.e., operators in the form ``x*I``, where
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``x`` is Hermitian.
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References
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==========
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.. [1] https://mathworld.wolfram.com/HermitianOperator.html
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"""
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# TODO: Add examples
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name = 'antihermitian'
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handler = Dispatcher(
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"AntiHermitianHandler",
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doc=("Handler for Q.antihermitian.\n\n"
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"Test that an expression belongs to the field of anti-Hermitian\n"
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"operators, that is, operators in the form x*I, where x is Hermitian.")
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)
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class AlgebraicPredicate(Predicate):
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r"""
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Algebraic number predicate.
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Explanation
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===========
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``Q.algebraic(x)`` is true iff ``x`` belongs to the set of
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algebraic numbers. ``x`` is algebraic if there is some polynomial
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in ``p(x)\in \mathbb\{Q\}[x]`` such that ``p(x) = 0``.
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Examples
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========
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>>> from sympy import ask, Q, sqrt, I, pi
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>>> ask(Q.algebraic(sqrt(2)))
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True
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>>> ask(Q.algebraic(I))
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True
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>>> ask(Q.algebraic(pi))
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False
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Algebraic_number
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"""
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name = 'algebraic'
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AlgebraicHandler = Dispatcher(
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"AlgebraicHandler",
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doc="""Handler for Q.algebraic key."""
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)
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class TranscendentalPredicate(Predicate):
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"""
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Transcedental number predicate.
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Explanation
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===========
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``Q.transcendental(x)`` is true iff ``x`` belongs to the set of
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transcendental numbers. A transcendental number is a real
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or complex number that is not algebraic.
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"""
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# TODO: Add examples
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name = 'transcendental'
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handler = Dispatcher(
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"Transcendental",
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doc="""Handler for Q.transcendental key."""
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)
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