263 lines
8.2 KiB
Python
263 lines
8.2 KiB
Python
from sympy.core.singleton import S
|
|
from sympy.sets.sets import Set
|
|
from sympy.calculus.singularities import singularities
|
|
from sympy.core import Expr, Add
|
|
from sympy.core.function import Lambda, FunctionClass, diff, expand_mul
|
|
from sympy.core.numbers import Float, oo
|
|
from sympy.core.symbol import Dummy, symbols, Wild
|
|
from sympy.functions.elementary.exponential import exp, log
|
|
from sympy.functions.elementary.miscellaneous import Min, Max
|
|
from sympy.logic.boolalg import true
|
|
from sympy.multipledispatch import Dispatcher
|
|
from sympy.sets import (imageset, Interval, FiniteSet, Union, ImageSet,
|
|
Intersection, Range, Complement)
|
|
from sympy.sets.sets import EmptySet, is_function_invertible_in_set
|
|
from sympy.sets.fancysets import Integers, Naturals, Reals
|
|
from sympy.functions.elementary.exponential import match_real_imag
|
|
|
|
|
|
_x, _y = symbols("x y")
|
|
|
|
FunctionUnion = (FunctionClass, Lambda)
|
|
|
|
_set_function = Dispatcher('_set_function')
|
|
|
|
|
|
@_set_function.register(FunctionClass, Set)
|
|
def _(f, x):
|
|
return None
|
|
|
|
@_set_function.register(FunctionUnion, FiniteSet)
|
|
def _(f, x):
|
|
return FiniteSet(*map(f, x))
|
|
|
|
@_set_function.register(Lambda, Interval)
|
|
def _(f, x):
|
|
from sympy.solvers.solveset import solveset
|
|
from sympy.series import limit
|
|
# TODO: handle functions with infinitely many solutions (eg, sin, tan)
|
|
# TODO: handle multivariate functions
|
|
|
|
expr = f.expr
|
|
if len(expr.free_symbols) > 1 or len(f.variables) != 1:
|
|
return
|
|
var = f.variables[0]
|
|
if not var.is_real:
|
|
if expr.subs(var, Dummy(real=True)).is_real is False:
|
|
return
|
|
|
|
if expr.is_Piecewise:
|
|
result = S.EmptySet
|
|
domain_set = x
|
|
for (p_expr, p_cond) in expr.args:
|
|
if p_cond is true:
|
|
intrvl = domain_set
|
|
else:
|
|
intrvl = p_cond.as_set()
|
|
intrvl = Intersection(domain_set, intrvl)
|
|
|
|
if p_expr.is_Number:
|
|
image = FiniteSet(p_expr)
|
|
else:
|
|
image = imageset(Lambda(var, p_expr), intrvl)
|
|
result = Union(result, image)
|
|
|
|
# remove the part which has been `imaged`
|
|
domain_set = Complement(domain_set, intrvl)
|
|
if domain_set is S.EmptySet:
|
|
break
|
|
return result
|
|
|
|
if not x.start.is_comparable or not x.end.is_comparable:
|
|
return
|
|
|
|
try:
|
|
from sympy.polys.polyutils import _nsort
|
|
sing = list(singularities(expr, var, x))
|
|
if len(sing) > 1:
|
|
sing = _nsort(sing)
|
|
except NotImplementedError:
|
|
return
|
|
|
|
if x.left_open:
|
|
_start = limit(expr, var, x.start, dir="+")
|
|
elif x.start not in sing:
|
|
_start = f(x.start)
|
|
if x.right_open:
|
|
_end = limit(expr, var, x.end, dir="-")
|
|
elif x.end not in sing:
|
|
_end = f(x.end)
|
|
|
|
if len(sing) == 0:
|
|
soln_expr = solveset(diff(expr, var), var)
|
|
if not (isinstance(soln_expr, FiniteSet)
|
|
or soln_expr is S.EmptySet):
|
|
return
|
|
solns = list(soln_expr)
|
|
|
|
extr = [_start, _end] + [f(i) for i in solns
|
|
if i.is_real and i in x]
|
|
start, end = Min(*extr), Max(*extr)
|
|
|
|
left_open, right_open = False, False
|
|
if _start <= _end:
|
|
# the minimum or maximum value can occur simultaneously
|
|
# on both the edge of the interval and in some interior
|
|
# point
|
|
if start == _start and start not in solns:
|
|
left_open = x.left_open
|
|
if end == _end and end not in solns:
|
|
right_open = x.right_open
|
|
else:
|
|
if start == _end and start not in solns:
|
|
left_open = x.right_open
|
|
if end == _start and end not in solns:
|
|
right_open = x.left_open
|
|
|
|
return Interval(start, end, left_open, right_open)
|
|
else:
|
|
return imageset(f, Interval(x.start, sing[0],
|
|
x.left_open, True)) + \
|
|
Union(*[imageset(f, Interval(sing[i], sing[i + 1], True, True))
|
|
for i in range(0, len(sing) - 1)]) + \
|
|
imageset(f, Interval(sing[-1], x.end, True, x.right_open))
|
|
|
|
@_set_function.register(FunctionClass, Interval)
|
|
def _(f, x):
|
|
if f == exp:
|
|
return Interval(exp(x.start), exp(x.end), x.left_open, x.right_open)
|
|
elif f == log:
|
|
return Interval(log(x.start), log(x.end), x.left_open, x.right_open)
|
|
return ImageSet(Lambda(_x, f(_x)), x)
|
|
|
|
@_set_function.register(FunctionUnion, Union)
|
|
def _(f, x):
|
|
return Union(*(imageset(f, arg) for arg in x.args))
|
|
|
|
@_set_function.register(FunctionUnion, Intersection)
|
|
def _(f, x):
|
|
# If the function is invertible, intersect the maps of the sets.
|
|
if is_function_invertible_in_set(f, x):
|
|
return Intersection(*(imageset(f, arg) for arg in x.args))
|
|
else:
|
|
return ImageSet(Lambda(_x, f(_x)), x)
|
|
|
|
@_set_function.register(FunctionUnion, EmptySet)
|
|
def _(f, x):
|
|
return x
|
|
|
|
@_set_function.register(FunctionUnion, Set)
|
|
def _(f, x):
|
|
return ImageSet(Lambda(_x, f(_x)), x)
|
|
|
|
@_set_function.register(FunctionUnion, Range)
|
|
def _(f, self):
|
|
if not self:
|
|
return S.EmptySet
|
|
if not isinstance(f.expr, Expr):
|
|
return
|
|
if self.size == 1:
|
|
return FiniteSet(f(self[0]))
|
|
if f is S.IdentityFunction:
|
|
return self
|
|
|
|
x = f.variables[0]
|
|
expr = f.expr
|
|
# handle f that is linear in f's variable
|
|
if x not in expr.free_symbols or x in expr.diff(x).free_symbols:
|
|
return
|
|
if self.start.is_finite:
|
|
F = f(self.step*x + self.start) # for i in range(len(self))
|
|
else:
|
|
F = f(-self.step*x + self[-1])
|
|
F = expand_mul(F)
|
|
if F != expr:
|
|
return imageset(x, F, Range(self.size))
|
|
|
|
@_set_function.register(FunctionUnion, Integers)
|
|
def _(f, self):
|
|
expr = f.expr
|
|
if not isinstance(expr, Expr):
|
|
return
|
|
|
|
n = f.variables[0]
|
|
if expr == abs(n):
|
|
return S.Naturals0
|
|
|
|
# f(x) + c and f(-x) + c cover the same integers
|
|
# so choose the form that has the fewest negatives
|
|
c = f(0)
|
|
fx = f(n) - c
|
|
f_x = f(-n) - c
|
|
neg_count = lambda e: sum(_.could_extract_minus_sign()
|
|
for _ in Add.make_args(e))
|
|
if neg_count(f_x) < neg_count(fx):
|
|
expr = f_x + c
|
|
|
|
a = Wild('a', exclude=[n])
|
|
b = Wild('b', exclude=[n])
|
|
match = expr.match(a*n + b)
|
|
if match and match[a] and (
|
|
not match[a].atoms(Float) and
|
|
not match[b].atoms(Float)):
|
|
# canonical shift
|
|
a, b = match[a], match[b]
|
|
if a in [1, -1]:
|
|
# drop integer addends in b
|
|
nonint = []
|
|
for bi in Add.make_args(b):
|
|
if not bi.is_integer:
|
|
nonint.append(bi)
|
|
b = Add(*nonint)
|
|
if b.is_number and a.is_real:
|
|
# avoid Mod for complex numbers, #11391
|
|
br, bi = match_real_imag(b)
|
|
if br and br.is_comparable and a.is_comparable:
|
|
br %= a
|
|
b = br + S.ImaginaryUnit*bi
|
|
elif b.is_number and a.is_imaginary:
|
|
br, bi = match_real_imag(b)
|
|
ai = a/S.ImaginaryUnit
|
|
if bi and bi.is_comparable and ai.is_comparable:
|
|
bi %= ai
|
|
b = br + S.ImaginaryUnit*bi
|
|
expr = a*n + b
|
|
|
|
if expr != f.expr:
|
|
return ImageSet(Lambda(n, expr), S.Integers)
|
|
|
|
|
|
@_set_function.register(FunctionUnion, Naturals)
|
|
def _(f, self):
|
|
expr = f.expr
|
|
if not isinstance(expr, Expr):
|
|
return
|
|
|
|
x = f.variables[0]
|
|
if not expr.free_symbols - {x}:
|
|
if expr == abs(x):
|
|
if self is S.Naturals:
|
|
return self
|
|
return S.Naturals0
|
|
step = expr.coeff(x)
|
|
c = expr.subs(x, 0)
|
|
if c.is_Integer and step.is_Integer and expr == step*x + c:
|
|
if self is S.Naturals:
|
|
c += step
|
|
if step > 0:
|
|
if step == 1:
|
|
if c == 0:
|
|
return S.Naturals0
|
|
elif c == 1:
|
|
return S.Naturals
|
|
return Range(c, oo, step)
|
|
return Range(c, -oo, step)
|
|
|
|
|
|
@_set_function.register(FunctionUnion, Reals)
|
|
def _(f, self):
|
|
expr = f.expr
|
|
if not isinstance(expr, Expr):
|
|
return
|
|
return _set_function(f, Interval(-oo, oo))
|