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437 lines
12 KiB
437 lines
12 KiB
"""
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Handlers related to order relations: positive, negative, etc.
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"""
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from sympy.assumptions import Q, ask
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from sympy.core import Add, Basic, Expr, Mul, Pow
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from sympy.core.logic import fuzzy_not, fuzzy_and, fuzzy_or
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from sympy.core.numbers import E, ImaginaryUnit, NaN, I, pi
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from sympy.functions import Abs, acos, acot, asin, atan, exp, factorial, log
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from sympy.matrices import Determinant, Trace
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from sympy.matrices.expressions.matexpr import MatrixElement
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from sympy.multipledispatch import MDNotImplementedError
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from ..predicates.order import (NegativePredicate, NonNegativePredicate,
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NonZeroPredicate, ZeroPredicate, NonPositivePredicate, PositivePredicate,
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ExtendedNegativePredicate, ExtendedNonNegativePredicate,
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ExtendedNonPositivePredicate, ExtendedNonZeroPredicate,
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ExtendedPositivePredicate,)
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# NegativePredicate
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def _NegativePredicate_number(expr, assumptions):
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r, i = expr.as_real_imag()
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# If the imaginary part can symbolically be shown to be zero then
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# we just evaluate the real part; otherwise we evaluate the imaginary
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# part to see if it actually evaluates to zero and if it does then
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# we make the comparison between the real part and zero.
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if not i:
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r = r.evalf(2)
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if r._prec != 1:
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return r < 0
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else:
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i = i.evalf(2)
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if i._prec != 1:
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if i != 0:
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return False
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r = r.evalf(2)
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if r._prec != 1:
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return r < 0
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@NegativePredicate.register(Basic)
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def _(expr, assumptions):
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if expr.is_number:
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return _NegativePredicate_number(expr, assumptions)
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@NegativePredicate.register(Expr)
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def _(expr, assumptions):
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ret = expr.is_negative
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if ret is None:
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raise MDNotImplementedError
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return ret
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@NegativePredicate.register(Add)
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def _(expr, assumptions):
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"""
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Positive + Positive -> Positive,
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Negative + Negative -> Negative
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"""
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if expr.is_number:
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return _NegativePredicate_number(expr, assumptions)
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r = ask(Q.real(expr), assumptions)
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if r is not True:
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return r
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nonpos = 0
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for arg in expr.args:
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if ask(Q.negative(arg), assumptions) is not True:
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if ask(Q.positive(arg), assumptions) is False:
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nonpos += 1
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else:
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break
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else:
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if nonpos < len(expr.args):
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return True
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@NegativePredicate.register(Mul)
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def _(expr, assumptions):
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if expr.is_number:
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return _NegativePredicate_number(expr, assumptions)
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result = None
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for arg in expr.args:
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if result is None:
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result = False
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if ask(Q.negative(arg), assumptions):
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result = not result
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elif ask(Q.positive(arg), assumptions):
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pass
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else:
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return
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return result
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@NegativePredicate.register(Pow)
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def _(expr, assumptions):
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"""
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Real ** Even -> NonNegative
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Real ** Odd -> same_as_base
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NonNegative ** Positive -> NonNegative
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"""
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if expr.base == E:
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# Exponential is always positive:
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if ask(Q.real(expr.exp), assumptions):
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return False
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return
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if expr.is_number:
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return _NegativePredicate_number(expr, assumptions)
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if ask(Q.real(expr.base), assumptions):
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if ask(Q.positive(expr.base), assumptions):
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if ask(Q.real(expr.exp), assumptions):
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return False
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if ask(Q.even(expr.exp), assumptions):
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return False
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if ask(Q.odd(expr.exp), assumptions):
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return ask(Q.negative(expr.base), assumptions)
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@NegativePredicate.register_many(Abs, ImaginaryUnit)
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def _(expr, assumptions):
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return False
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@NegativePredicate.register(exp)
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def _(expr, assumptions):
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if ask(Q.real(expr.exp), assumptions):
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return False
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raise MDNotImplementedError
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# NonNegativePredicate
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@NonNegativePredicate.register(Basic)
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def _(expr, assumptions):
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if expr.is_number:
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notnegative = fuzzy_not(_NegativePredicate_number(expr, assumptions))
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if notnegative:
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return ask(Q.real(expr), assumptions)
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else:
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return notnegative
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@NonNegativePredicate.register(Expr)
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def _(expr, assumptions):
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ret = expr.is_nonnegative
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if ret is None:
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raise MDNotImplementedError
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return ret
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# NonZeroPredicate
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@NonZeroPredicate.register(Expr)
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def _(expr, assumptions):
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ret = expr.is_nonzero
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if ret is None:
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raise MDNotImplementedError
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return ret
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@NonZeroPredicate.register(Basic)
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def _(expr, assumptions):
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if ask(Q.real(expr)) is False:
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return False
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if expr.is_number:
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# if there are no symbols just evalf
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i = expr.evalf(2)
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def nonz(i):
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if i._prec != 1:
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return i != 0
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return fuzzy_or(nonz(i) for i in i.as_real_imag())
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@NonZeroPredicate.register(Add)
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def _(expr, assumptions):
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if all(ask(Q.positive(x), assumptions) for x in expr.args) \
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or all(ask(Q.negative(x), assumptions) for x in expr.args):
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return True
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@NonZeroPredicate.register(Mul)
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def _(expr, assumptions):
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for arg in expr.args:
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result = ask(Q.nonzero(arg), assumptions)
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if result:
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continue
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return result
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return True
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@NonZeroPredicate.register(Pow)
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def _(expr, assumptions):
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return ask(Q.nonzero(expr.base), assumptions)
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@NonZeroPredicate.register(Abs)
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def _(expr, assumptions):
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return ask(Q.nonzero(expr.args[0]), assumptions)
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@NonZeroPredicate.register(NaN)
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def _(expr, assumptions):
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return None
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# ZeroPredicate
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@ZeroPredicate.register(Expr)
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def _(expr, assumptions):
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ret = expr.is_zero
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if ret is None:
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raise MDNotImplementedError
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return ret
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@ZeroPredicate.register(Basic)
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def _(expr, assumptions):
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return fuzzy_and([fuzzy_not(ask(Q.nonzero(expr), assumptions)),
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ask(Q.real(expr), assumptions)])
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@ZeroPredicate.register(Mul)
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def _(expr, assumptions):
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# TODO: This should be deducible from the nonzero handler
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return fuzzy_or(ask(Q.zero(arg), assumptions) for arg in expr.args)
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# NonPositivePredicate
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@NonPositivePredicate.register(Expr)
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def _(expr, assumptions):
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ret = expr.is_nonpositive
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if ret is None:
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raise MDNotImplementedError
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return ret
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@NonPositivePredicate.register(Basic)
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def _(expr, assumptions):
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if expr.is_number:
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notpositive = fuzzy_not(_PositivePredicate_number(expr, assumptions))
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if notpositive:
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return ask(Q.real(expr), assumptions)
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else:
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return notpositive
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# PositivePredicate
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def _PositivePredicate_number(expr, assumptions):
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r, i = expr.as_real_imag()
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# If the imaginary part can symbolically be shown to be zero then
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# we just evaluate the real part; otherwise we evaluate the imaginary
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# part to see if it actually evaluates to zero and if it does then
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# we make the comparison between the real part and zero.
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if not i:
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r = r.evalf(2)
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if r._prec != 1:
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return r > 0
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else:
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i = i.evalf(2)
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if i._prec != 1:
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if i != 0:
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return False
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r = r.evalf(2)
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if r._prec != 1:
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return r > 0
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@PositivePredicate.register(Expr)
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def _(expr, assumptions):
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ret = expr.is_positive
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if ret is None:
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raise MDNotImplementedError
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return ret
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@PositivePredicate.register(Basic)
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def _(expr, assumptions):
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if expr.is_number:
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return _PositivePredicate_number(expr, assumptions)
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@PositivePredicate.register(Mul)
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def _(expr, assumptions):
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if expr.is_number:
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return _PositivePredicate_number(expr, assumptions)
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result = True
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for arg in expr.args:
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if ask(Q.positive(arg), assumptions):
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continue
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elif ask(Q.negative(arg), assumptions):
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result = result ^ True
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else:
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return
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return result
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@PositivePredicate.register(Add)
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def _(expr, assumptions):
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if expr.is_number:
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return _PositivePredicate_number(expr, assumptions)
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r = ask(Q.real(expr), assumptions)
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if r is not True:
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return r
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nonneg = 0
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for arg in expr.args:
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if ask(Q.positive(arg), assumptions) is not True:
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if ask(Q.negative(arg), assumptions) is False:
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nonneg += 1
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else:
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break
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else:
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if nonneg < len(expr.args):
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return True
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@PositivePredicate.register(Pow)
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def _(expr, assumptions):
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if expr.base == E:
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if ask(Q.real(expr.exp), assumptions):
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return True
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if ask(Q.imaginary(expr.exp), assumptions):
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return ask(Q.even(expr.exp/(I*pi)), assumptions)
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return
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if expr.is_number:
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return _PositivePredicate_number(expr, assumptions)
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if ask(Q.positive(expr.base), assumptions):
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if ask(Q.real(expr.exp), assumptions):
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return True
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if ask(Q.negative(expr.base), assumptions):
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if ask(Q.even(expr.exp), assumptions):
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return True
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if ask(Q.odd(expr.exp), assumptions):
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return False
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@PositivePredicate.register(exp)
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def _(expr, assumptions):
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if ask(Q.real(expr.exp), assumptions):
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return True
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if ask(Q.imaginary(expr.exp), assumptions):
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return ask(Q.even(expr.exp/(I*pi)), assumptions)
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@PositivePredicate.register(log)
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def _(expr, assumptions):
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r = ask(Q.real(expr.args[0]), assumptions)
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if r is not True:
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return r
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if ask(Q.positive(expr.args[0] - 1), assumptions):
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return True
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if ask(Q.negative(expr.args[0] - 1), assumptions):
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return False
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@PositivePredicate.register(factorial)
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def _(expr, assumptions):
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x = expr.args[0]
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if ask(Q.integer(x) & Q.positive(x), assumptions):
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return True
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@PositivePredicate.register(ImaginaryUnit)
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def _(expr, assumptions):
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return False
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@PositivePredicate.register(Abs)
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def _(expr, assumptions):
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return ask(Q.nonzero(expr), assumptions)
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@PositivePredicate.register(Trace)
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def _(expr, assumptions):
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if ask(Q.positive_definite(expr.arg), assumptions):
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return True
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@PositivePredicate.register(Determinant)
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def _(expr, assumptions):
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if ask(Q.positive_definite(expr.arg), assumptions):
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return True
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@PositivePredicate.register(MatrixElement)
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def _(expr, assumptions):
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if (expr.i == expr.j
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and ask(Q.positive_definite(expr.parent), assumptions)):
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return True
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@PositivePredicate.register(atan)
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def _(expr, assumptions):
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return ask(Q.positive(expr.args[0]), assumptions)
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@PositivePredicate.register(asin)
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def _(expr, assumptions):
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x = expr.args[0]
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if ask(Q.positive(x) & Q.nonpositive(x - 1), assumptions):
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return True
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if ask(Q.negative(x) & Q.nonnegative(x + 1), assumptions):
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return False
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@PositivePredicate.register(acos)
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def _(expr, assumptions):
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x = expr.args[0]
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if ask(Q.nonpositive(x - 1) & Q.nonnegative(x + 1), assumptions):
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return True
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@PositivePredicate.register(acot)
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def _(expr, assumptions):
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return ask(Q.real(expr.args[0]), assumptions)
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@PositivePredicate.register(NaN)
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def _(expr, assumptions):
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return None
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# ExtendedNegativePredicate
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@ExtendedNegativePredicate.register(object)
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def _(expr, assumptions):
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return ask(Q.negative(expr) | Q.negative_infinite(expr), assumptions)
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# ExtendedPositivePredicate
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@ExtendedPositivePredicate.register(object)
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def _(expr, assumptions):
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return ask(Q.positive(expr) | Q.positive_infinite(expr), assumptions)
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# ExtendedNonZeroPredicate
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@ExtendedNonZeroPredicate.register(object)
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def _(expr, assumptions):
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return ask(
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Q.negative_infinite(expr) | Q.negative(expr) | Q.positive(expr) | Q.positive_infinite(expr),
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assumptions)
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# ExtendedNonPositivePredicate
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@ExtendedNonPositivePredicate.register(object)
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def _(expr, assumptions):
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return ask(
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Q.negative_infinite(expr) | Q.negative(expr) | Q.zero(expr),
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assumptions)
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# ExtendedNonNegativePredicate
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@ExtendedNonNegativePredicate.register(object)
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def _(expr, assumptions):
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return ask(
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Q.zero(expr) | Q.positive(expr) | Q.positive_infinite(expr),
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assumptions)
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