324 lines
9.2 KiB
Python
324 lines
9.2 KiB
Python
from collections import defaultdict
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from sympy.assumptions.ask import Q
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from sympy.core import (Add, Mul, Pow, Number, NumberSymbol, Symbol)
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from sympy.core.numbers import ImaginaryUnit
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from sympy.functions.elementary.complexes import Abs
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from sympy.logic.boolalg import (Equivalent, And, Or, Implies)
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from sympy.matrices.expressions import MatMul
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# APIs here may be subject to change
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### Helper functions ###
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def allargs(symbol, fact, expr):
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"""
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Apply all arguments of the expression to the fact structure.
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Parameters
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==========
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symbol : Symbol
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A placeholder symbol.
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fact : Boolean
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Resulting ``Boolean`` expression.
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expr : Expr
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Examples
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========
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>>> from sympy import Q
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>>> from sympy.assumptions.sathandlers import allargs
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>>> from sympy.abc import x, y
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>>> allargs(x, Q.negative(x) | Q.positive(x), x*y)
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(Q.negative(x) | Q.positive(x)) & (Q.negative(y) | Q.positive(y))
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"""
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return And(*[fact.subs(symbol, arg) for arg in expr.args])
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def anyarg(symbol, fact, expr):
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"""
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Apply any argument of the expression to the fact structure.
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Parameters
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==========
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symbol : Symbol
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A placeholder symbol.
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fact : Boolean
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Resulting ``Boolean`` expression.
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expr : Expr
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Examples
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========
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>>> from sympy import Q
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>>> from sympy.assumptions.sathandlers import anyarg
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>>> from sympy.abc import x, y
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>>> anyarg(x, Q.negative(x) & Q.positive(x), x*y)
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(Q.negative(x) & Q.positive(x)) | (Q.negative(y) & Q.positive(y))
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"""
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return Or(*[fact.subs(symbol, arg) for arg in expr.args])
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def exactlyonearg(symbol, fact, expr):
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"""
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Apply exactly one argument of the expression to the fact structure.
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Parameters
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==========
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symbol : Symbol
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A placeholder symbol.
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fact : Boolean
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Resulting ``Boolean`` expression.
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expr : Expr
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Examples
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========
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>>> from sympy import Q
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>>> from sympy.assumptions.sathandlers import exactlyonearg
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>>> from sympy.abc import x, y
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>>> exactlyonearg(x, Q.positive(x), x*y)
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(Q.positive(x) & ~Q.positive(y)) | (Q.positive(y) & ~Q.positive(x))
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"""
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pred_args = [fact.subs(symbol, arg) for arg in expr.args]
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res = Or(*[And(pred_args[i], *[~lit for lit in pred_args[:i] +
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pred_args[i+1:]]) for i in range(len(pred_args))])
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return res
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### Fact registry ###
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class ClassFactRegistry:
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"""
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Register handlers against classes.
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Explanation
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===========
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``register`` method registers the handler function for a class. Here,
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handler function should return a single fact. ``multiregister`` method
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registers the handler function for multiple classes. Here, handler function
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should return a container of multiple facts.
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``registry(expr)`` returns a set of facts for *expr*.
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Examples
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========
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Here, we register the facts for ``Abs``.
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>>> from sympy import Abs, Equivalent, Q
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>>> from sympy.assumptions.sathandlers import ClassFactRegistry
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>>> reg = ClassFactRegistry()
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>>> @reg.register(Abs)
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... def f1(expr):
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... return Q.nonnegative(expr)
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>>> @reg.register(Abs)
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... def f2(expr):
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... arg = expr.args[0]
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... return Equivalent(~Q.zero(arg), ~Q.zero(expr))
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Calling the registry with expression returns the defined facts for the
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expression.
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>>> from sympy.abc import x
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>>> reg(Abs(x))
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{Q.nonnegative(Abs(x)), Equivalent(~Q.zero(x), ~Q.zero(Abs(x)))}
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Multiple facts can be registered at once by ``multiregister`` method.
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>>> reg2 = ClassFactRegistry()
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>>> @reg2.multiregister(Abs)
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... def _(expr):
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... arg = expr.args[0]
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... return [Q.even(arg) >> Q.even(expr), Q.odd(arg) >> Q.odd(expr)]
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>>> reg2(Abs(x))
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{Implies(Q.even(x), Q.even(Abs(x))), Implies(Q.odd(x), Q.odd(Abs(x)))}
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"""
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def __init__(self):
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self.singlefacts = defaultdict(frozenset)
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self.multifacts = defaultdict(frozenset)
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def register(self, cls):
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def _(func):
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self.singlefacts[cls] |= {func}
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return func
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return _
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def multiregister(self, *classes):
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def _(func):
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for cls in classes:
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self.multifacts[cls] |= {func}
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return func
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return _
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def __getitem__(self, key):
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ret1 = self.singlefacts[key]
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for k in self.singlefacts:
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if issubclass(key, k):
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ret1 |= self.singlefacts[k]
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ret2 = self.multifacts[key]
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for k in self.multifacts:
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if issubclass(key, k):
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ret2 |= self.multifacts[k]
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return ret1, ret2
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def __call__(self, expr):
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ret = set()
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handlers1, handlers2 = self[type(expr)]
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for h in handlers1:
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ret.add(h(expr))
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for h in handlers2:
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ret.update(h(expr))
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return ret
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class_fact_registry = ClassFactRegistry()
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### Class fact registration ###
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x = Symbol('x')
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## Abs ##
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@class_fact_registry.multiregister(Abs)
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def _(expr):
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arg = expr.args[0]
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return [Q.nonnegative(expr),
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Equivalent(~Q.zero(arg), ~Q.zero(expr)),
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Q.even(arg) >> Q.even(expr),
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Q.odd(arg) >> Q.odd(expr),
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Q.integer(arg) >> Q.integer(expr),
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]
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### Add ##
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@class_fact_registry.multiregister(Add)
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def _(expr):
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return [allargs(x, Q.positive(x), expr) >> Q.positive(expr),
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allargs(x, Q.negative(x), expr) >> Q.negative(expr),
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allargs(x, Q.real(x), expr) >> Q.real(expr),
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allargs(x, Q.rational(x), expr) >> Q.rational(expr),
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allargs(x, Q.integer(x), expr) >> Q.integer(expr),
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exactlyonearg(x, ~Q.integer(x), expr) >> ~Q.integer(expr),
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]
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@class_fact_registry.register(Add)
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def _(expr):
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allargs_real = allargs(x, Q.real(x), expr)
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onearg_irrational = exactlyonearg(x, Q.irrational(x), expr)
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return Implies(allargs_real, Implies(onearg_irrational, Q.irrational(expr)))
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### Mul ###
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@class_fact_registry.multiregister(Mul)
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def _(expr):
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return [Equivalent(Q.zero(expr), anyarg(x, Q.zero(x), expr)),
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allargs(x, Q.positive(x), expr) >> Q.positive(expr),
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allargs(x, Q.real(x), expr) >> Q.real(expr),
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allargs(x, Q.rational(x), expr) >> Q.rational(expr),
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allargs(x, Q.integer(x), expr) >> Q.integer(expr),
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exactlyonearg(x, ~Q.rational(x), expr) >> ~Q.integer(expr),
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allargs(x, Q.commutative(x), expr) >> Q.commutative(expr),
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]
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@class_fact_registry.register(Mul)
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def _(expr):
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# Implicitly assumes Mul has more than one arg
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# Would be allargs(x, Q.prime(x) | Q.composite(x)) except 1 is composite
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# More advanced prime assumptions will require inequalities, as 1 provides
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# a corner case.
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allargs_prime = allargs(x, Q.prime(x), expr)
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return Implies(allargs_prime, ~Q.prime(expr))
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@class_fact_registry.register(Mul)
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def _(expr):
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# General Case: Odd number of imaginary args implies mul is imaginary(To be implemented)
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allargs_imag_or_real = allargs(x, Q.imaginary(x) | Q.real(x), expr)
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onearg_imaginary = exactlyonearg(x, Q.imaginary(x), expr)
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return Implies(allargs_imag_or_real, Implies(onearg_imaginary, Q.imaginary(expr)))
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@class_fact_registry.register(Mul)
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def _(expr):
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allargs_real = allargs(x, Q.real(x), expr)
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onearg_irrational = exactlyonearg(x, Q.irrational(x), expr)
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return Implies(allargs_real, Implies(onearg_irrational, Q.irrational(expr)))
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@class_fact_registry.register(Mul)
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def _(expr):
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# Including the integer qualification means we don't need to add any facts
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# for odd, since the assumptions already know that every integer is
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# exactly one of even or odd.
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allargs_integer = allargs(x, Q.integer(x), expr)
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anyarg_even = anyarg(x, Q.even(x), expr)
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return Implies(allargs_integer, Equivalent(anyarg_even, Q.even(expr)))
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### MatMul ###
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@class_fact_registry.register(MatMul)
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def _(expr):
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allargs_square = allargs(x, Q.square(x), expr)
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allargs_invertible = allargs(x, Q.invertible(x), expr)
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return Implies(allargs_square, Equivalent(Q.invertible(expr), allargs_invertible))
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### Pow ###
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@class_fact_registry.multiregister(Pow)
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def _(expr):
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base, exp = expr.base, expr.exp
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return [
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(Q.real(base) & Q.even(exp) & Q.nonnegative(exp)) >> Q.nonnegative(expr),
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(Q.nonnegative(base) & Q.odd(exp) & Q.nonnegative(exp)) >> Q.nonnegative(expr),
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(Q.nonpositive(base) & Q.odd(exp) & Q.nonnegative(exp)) >> Q.nonpositive(expr),
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Equivalent(Q.zero(expr), Q.zero(base) & Q.positive(exp))
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]
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### Numbers ###
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_old_assump_getters = {
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Q.positive: lambda o: o.is_positive,
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Q.zero: lambda o: o.is_zero,
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Q.negative: lambda o: o.is_negative,
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Q.rational: lambda o: o.is_rational,
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Q.irrational: lambda o: o.is_irrational,
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Q.even: lambda o: o.is_even,
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Q.odd: lambda o: o.is_odd,
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Q.imaginary: lambda o: o.is_imaginary,
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Q.prime: lambda o: o.is_prime,
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Q.composite: lambda o: o.is_composite,
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}
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@class_fact_registry.multiregister(Number, NumberSymbol, ImaginaryUnit)
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def _(expr):
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ret = []
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for p, getter in _old_assump_getters.items():
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pred = p(expr)
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prop = getter(expr)
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if prop is not None:
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ret.append(Equivalent(pred, prop))
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return ret
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