Spaces:
Sleeping
Sleeping
File size: 9,418 Bytes
6a86ad5 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 |
from collections import defaultdict
from sympy.assumptions.ask import Q
from sympy.core import (Add, Mul, Pow, Number, NumberSymbol, Symbol)
from sympy.core.numbers import ImaginaryUnit
from sympy.functions.elementary.complexes import Abs
from sympy.logic.boolalg import (Equivalent, And, Or, Implies)
from sympy.matrices.expressions import MatMul
# APIs here may be subject to change
### Helper functions ###
def allargs(symbol, fact, expr):
"""
Apply all arguments of the expression to the fact structure.
Parameters
==========
symbol : Symbol
A placeholder symbol.
fact : Boolean
Resulting ``Boolean`` expression.
expr : Expr
Examples
========
>>> from sympy import Q
>>> from sympy.assumptions.sathandlers import allargs
>>> from sympy.abc import x, y
>>> allargs(x, Q.negative(x) | Q.positive(x), x*y)
(Q.negative(x) | Q.positive(x)) & (Q.negative(y) | Q.positive(y))
"""
return And(*[fact.subs(symbol, arg) for arg in expr.args])
def anyarg(symbol, fact, expr):
"""
Apply any argument of the expression to the fact structure.
Parameters
==========
symbol : Symbol
A placeholder symbol.
fact : Boolean
Resulting ``Boolean`` expression.
expr : Expr
Examples
========
>>> from sympy import Q
>>> from sympy.assumptions.sathandlers import anyarg
>>> from sympy.abc import x, y
>>> anyarg(x, Q.negative(x) & Q.positive(x), x*y)
(Q.negative(x) & Q.positive(x)) | (Q.negative(y) & Q.positive(y))
"""
return Or(*[fact.subs(symbol, arg) for arg in expr.args])
def exactlyonearg(symbol, fact, expr):
"""
Apply exactly one argument of the expression to the fact structure.
Parameters
==========
symbol : Symbol
A placeholder symbol.
fact : Boolean
Resulting ``Boolean`` expression.
expr : Expr
Examples
========
>>> from sympy import Q
>>> from sympy.assumptions.sathandlers import exactlyonearg
>>> from sympy.abc import x, y
>>> exactlyonearg(x, Q.positive(x), x*y)
(Q.positive(x) & ~Q.positive(y)) | (Q.positive(y) & ~Q.positive(x))
"""
pred_args = [fact.subs(symbol, arg) for arg in expr.args]
res = Or(*[And(pred_args[i], *[~lit for lit in pred_args[:i] +
pred_args[i+1:]]) for i in range(len(pred_args))])
return res
### Fact registry ###
class ClassFactRegistry:
"""
Register handlers against classes.
Explanation
===========
``register`` method registers the handler function for a class. Here,
handler function should return a single fact. ``multiregister`` method
registers the handler function for multiple classes. Here, handler function
should return a container of multiple facts.
``registry(expr)`` returns a set of facts for *expr*.
Examples
========
Here, we register the facts for ``Abs``.
>>> from sympy import Abs, Equivalent, Q
>>> from sympy.assumptions.sathandlers import ClassFactRegistry
>>> reg = ClassFactRegistry()
>>> @reg.register(Abs)
... def f1(expr):
... return Q.nonnegative(expr)
>>> @reg.register(Abs)
... def f2(expr):
... arg = expr.args[0]
... return Equivalent(~Q.zero(arg), ~Q.zero(expr))
Calling the registry with expression returns the defined facts for the
expression.
>>> from sympy.abc import x
>>> reg(Abs(x))
{Q.nonnegative(Abs(x)), Equivalent(~Q.zero(x), ~Q.zero(Abs(x)))}
Multiple facts can be registered at once by ``multiregister`` method.
>>> reg2 = ClassFactRegistry()
>>> @reg2.multiregister(Abs)
... def _(expr):
... arg = expr.args[0]
... return [Q.even(arg) >> Q.even(expr), Q.odd(arg) >> Q.odd(expr)]
>>> reg2(Abs(x))
{Implies(Q.even(x), Q.even(Abs(x))), Implies(Q.odd(x), Q.odd(Abs(x)))}
"""
def __init__(self):
self.singlefacts = defaultdict(frozenset)
self.multifacts = defaultdict(frozenset)
def register(self, cls):
def _(func):
self.singlefacts[cls] |= {func}
return func
return _
def multiregister(self, *classes):
def _(func):
for cls in classes:
self.multifacts[cls] |= {func}
return func
return _
def __getitem__(self, key):
ret1 = self.singlefacts[key]
for k in self.singlefacts:
if issubclass(key, k):
ret1 |= self.singlefacts[k]
ret2 = self.multifacts[key]
for k in self.multifacts:
if issubclass(key, k):
ret2 |= self.multifacts[k]
return ret1, ret2
def __call__(self, expr):
ret = set()
handlers1, handlers2 = self[type(expr)]
ret.update(h(expr) for h in handlers1)
for h in handlers2:
ret.update(h(expr))
return ret
class_fact_registry = ClassFactRegistry()
### Class fact registration ###
x = Symbol('x')
## Abs ##
@class_fact_registry.multiregister(Abs)
def _(expr):
arg = expr.args[0]
return [Q.nonnegative(expr),
Equivalent(~Q.zero(arg), ~Q.zero(expr)),
Q.even(arg) >> Q.even(expr),
Q.odd(arg) >> Q.odd(expr),
Q.integer(arg) >> Q.integer(expr),
]
### Add ##
@class_fact_registry.multiregister(Add)
def _(expr):
return [allargs(x, Q.positive(x), expr) >> Q.positive(expr),
allargs(x, Q.negative(x), expr) >> Q.negative(expr),
allargs(x, Q.real(x), expr) >> Q.real(expr),
allargs(x, Q.rational(x), expr) >> Q.rational(expr),
allargs(x, Q.integer(x), expr) >> Q.integer(expr),
exactlyonearg(x, ~Q.integer(x), expr) >> ~Q.integer(expr),
]
@class_fact_registry.register(Add)
def _(expr):
allargs_real = allargs(x, Q.real(x), expr)
onearg_irrational = exactlyonearg(x, Q.irrational(x), expr)
return Implies(allargs_real, Implies(onearg_irrational, Q.irrational(expr)))
### Mul ###
@class_fact_registry.multiregister(Mul)
def _(expr):
return [Equivalent(Q.zero(expr), anyarg(x, Q.zero(x), expr)),
allargs(x, Q.positive(x), expr) >> Q.positive(expr),
allargs(x, Q.real(x), expr) >> Q.real(expr),
allargs(x, Q.rational(x), expr) >> Q.rational(expr),
allargs(x, Q.integer(x), expr) >> Q.integer(expr),
exactlyonearg(x, ~Q.rational(x), expr) >> ~Q.integer(expr),
allargs(x, Q.commutative(x), expr) >> Q.commutative(expr),
]
@class_fact_registry.register(Mul)
def _(expr):
# Implicitly assumes Mul has more than one arg
# Would be allargs(x, Q.prime(x) | Q.composite(x)) except 1 is composite
# More advanced prime assumptions will require inequalities, as 1 provides
# a corner case.
allargs_prime = allargs(x, Q.prime(x), expr)
return Implies(allargs_prime, ~Q.prime(expr))
@class_fact_registry.register(Mul)
def _(expr):
# General Case: Odd number of imaginary args implies mul is imaginary(To be implemented)
allargs_imag_or_real = allargs(x, Q.imaginary(x) | Q.real(x), expr)
onearg_imaginary = exactlyonearg(x, Q.imaginary(x), expr)
return Implies(allargs_imag_or_real, Implies(onearg_imaginary, Q.imaginary(expr)))
@class_fact_registry.register(Mul)
def _(expr):
allargs_real = allargs(x, Q.real(x), expr)
onearg_irrational = exactlyonearg(x, Q.irrational(x), expr)
return Implies(allargs_real, Implies(onearg_irrational, Q.irrational(expr)))
@class_fact_registry.register(Mul)
def _(expr):
# Including the integer qualification means we don't need to add any facts
# for odd, since the assumptions already know that every integer is
# exactly one of even or odd.
allargs_integer = allargs(x, Q.integer(x), expr)
anyarg_even = anyarg(x, Q.even(x), expr)
return Implies(allargs_integer, Equivalent(anyarg_even, Q.even(expr)))
### MatMul ###
@class_fact_registry.register(MatMul)
def _(expr):
allargs_square = allargs(x, Q.square(x), expr)
allargs_invertible = allargs(x, Q.invertible(x), expr)
return Implies(allargs_square, Equivalent(Q.invertible(expr), allargs_invertible))
### Pow ###
@class_fact_registry.multiregister(Pow)
def _(expr):
base, exp = expr.base, expr.exp
return [
(Q.real(base) & Q.even(exp) & Q.nonnegative(exp)) >> Q.nonnegative(expr),
(Q.nonnegative(base) & Q.odd(exp) & Q.nonnegative(exp)) >> Q.nonnegative(expr),
(Q.nonpositive(base) & Q.odd(exp) & Q.nonnegative(exp)) >> Q.nonpositive(expr),
Equivalent(Q.zero(expr), Q.zero(base) & Q.positive(exp))
]
### Numbers ###
_old_assump_getters = {
Q.positive: lambda o: o.is_positive,
Q.zero: lambda o: o.is_zero,
Q.negative: lambda o: o.is_negative,
Q.rational: lambda o: o.is_rational,
Q.irrational: lambda o: o.is_irrational,
Q.even: lambda o: o.is_even,
Q.odd: lambda o: o.is_odd,
Q.imaginary: lambda o: o.is_imaginary,
Q.prime: lambda o: o.is_prime,
Q.composite: lambda o: o.is_composite,
}
@class_fact_registry.multiregister(Number, NumberSymbol, ImaginaryUnit)
def _(expr):
ret = []
for p, getter in _old_assump_getters.items():
pred = p(expr)
prop = getter(expr)
if prop is not None:
ret.append(Equivalent(pred, prop))
return ret
|