/

Source code for sympy.core.mul

from collections import defaultdict
import operator

from .sympify import sympify
from .basic import Basic, C
from .singleton import S
from .operations import AssocOp
from .cache import cacheit
from .logic import fuzzy_not
from .compatibility import cmp_to_key
from functools import reduce

# internal marker to indicate:
#   "there are still non-commutative objects -- don't forget to process them"
class NC_Marker:
is_Order    = False
is_Mul      = False
is_Number   = False
is_Poly     = False

is_commutative = False

[docs]class Mul(AssocOp):

__slots__ = []

is_Mul = True

#identity = S.One
# cyclic import, so defined in numbers.py

# Key for sorting commutative args in canonical order
_args_sortkey = cmp_to_key(Basic.compare)

@classmethod
[docs]    def flatten(cls, seq):
"""Return commutative, noncommutative and order arguments by
combining related terms.

Notes
=====
* In an expression like a*b*c, python process this through sympy
as Mul(Mul(a, b), c). This can have undesirable consequences.

-  Sometimes terms are not combined as one would like:

>>> from sympy import Mul, sqrt
>>> from sympy.abc import x, y, z
>>> 2*(x + 1) # this is the 2-arg Mul behavior
2*x + 2
>>> y*(x + 1)*2
2*y*(x + 1)
>>> 2*(x + 1)*y # 2-arg result will be obtained first
y*(2*x + 2)
>>> Mul(2, x + 1, y) # all 3 args simultaneously processed
2*y*(x + 1)
>>> 2*((x + 1)*y) # parentheses can control this behavior
2*y*(x + 1)

Powers with compound bases may not find a single base to
combine with unless all arguments are processed at once.
Post-processing may be necessary in such cases.

>>> a = sqrt(x*sqrt(y))
>>> a**3
(x*sqrt(y))**(3/2)
>>> Mul(a,a,a)
(x*sqrt(y))**(3/2)
>>> a*a*a
x*sqrt(y)*sqrt(x*sqrt(y))
>>> _.subs(a.base, z).subs(z, a.base)
(x*sqrt(y))**(3/2)

-  If more than two terms are being multiplied then all the
previous terms will be re-processed for each new argument.
So if each of a, b and c were :class:Mul
expression, then a*b*c (or building up the product
with *=) will process all the arguments of a and
b twice: once when a*b is computed and again when
c is multiplied.

Using Mul(a, b, c) will process all arguments once.

* The results of Mul are cached according to arguments, so flatten
will only be called once for Mul(a, b, c). If you can
structure a calculation so the arguments are most likely to be
repeats then this can save time in computing the answer. For
example, say you had a Mul, M, that you wished to divide by d[i]
and multiply by n[i] and you suspect there are many repeats
in n. It would be better to compute M*n[i]/d[i] rather
than M/d[i]*n[i] since every time n[i] is a repeat, the
product, M*n[i] will be returned without flattening -- the
cached value will be returned. If you divide by the d[i]
first (and those are more unique than the n[i]) then that will
create a new Mul, M/d[i] the args of which will be traversed
again when it is multiplied by n[i].

This consideration is moot if the cache is turned off.

NB
--
The validity of the above notes depends on the implementation
details of Mul and flatten which may change at any time. Therefore,
you should only consider them when your code is highly performance
sensitive.

Removal of 1 from the sequence is already handled by AssocOp.__new__.
"""
rv = None
if len(seq) == 2:
a, b = seq
if b.is_Rational:
a, b = b, a
assert not a is S.One
if a and a.is_Rational:
r, b = b.as_coeff_Mul()
a *= r
if b.is_Mul:
bargs, nc = b.args_cnc()
rv = bargs, nc, None
if a is not S.One:
bargs.insert(0, a)

if a is S.One:
rv = [b], [], None
else:
bargs = [_keep_coeff(a, bi) for bi in Add.make_args(b)]
bargs.sort(key=hash)
ar = a*r
if ar:
bargs.insert(0, ar)
rv = bargs, [], None
if rv:
return rv

# apply associativity, separate commutative part of seq
c_part = []         # out: commutative factors
nc_part = []        # out: non-commutative factors

nc_seq = []

coeff = S.One       # standalone term
# e.g. 3 * ...

iu = []             # ImaginaryUnits, I

c_powers = []       # (base,exp)      n
# e.g. (x,n) for x

num_exp = []        # (num-base, exp)           y
# e.g.  (3, y)  for  ... * 3  * ...

neg1e = 0           # exponent on -1 extracted from Number-based Pow

pnum_rat = {}       # (num-base, Rat-exp)          1/2
# e.g.  (3, 1/2)  for  ... * 3     * ...

order_symbols = None

# --- PART 1 ---
#
# "collect powers and coeff":
#
# o coeff
# o c_powers
# o num_exp
# o neg1e
# o pnum_rat
#
# NOTE: this is optimized for all-objects-are-commutative case

for o in seq:
# O(x)
if o.is_Order:
o, order_symbols = o.as_expr_variables(order_symbols)

# Mul([...])
if o.is_Mul:
if o.is_commutative:
seq.extend(o.args)    # XXX zerocopy?

else:
# NCMul can have commutative parts as well
for q in o.args:
if q.is_commutative:
seq.append(q)
else:
nc_seq.append(q)

# append non-commutative marker, so we don't forget to
# process scheduled non-commutative objects
seq.append(NC_Marker)

continue

# 3
elif o.is_Number:
if o is S.NaN or coeff is S.ComplexInfinity and o is S.Zero:
# we know for sure the result will be nan
return [S.NaN], [], None
elif coeff.is_Number: # it could be zoo
coeff *= o
if coeff is S.NaN:
# we know for sure the result will be nan
return [S.NaN], [], None
continue

elif o is S.ComplexInfinity:
if not coeff:
# 0 * zoo = NaN
return [S.NaN], [], None
if coeff is S.ComplexInfinity:
# zoo * zoo = zoo
return [S.ComplexInfinity], [], None
coeff = S.ComplexInfinity
continue

elif o is S.ImaginaryUnit:
iu.append(o)
continue

elif o.is_commutative:
#      e
# o = b
b, e = o.as_base_exp()

#  y
# 3
if o.is_Pow and b.is_Number:

# get all the factors with numeric base so they can be
# combined below, but don't combine negatives unless
# the exponent is an integer
if e.is_Rational:
if e.is_Integer:
coeff *= Pow(b, e) # it is an unevaluated power
continue
elif e.is_negative:    # also a sign of an unevaluated power
seq.append(Pow(b, e))
continue
elif b.is_negative:
neg1e += e
b = -b
if b is not S.One:
pnum_rat.setdefault(b, []).append(e)
continue
elif b.is_positive or e.is_integer:
num_exp.append((b, e))
continue
c_powers.append((b, e))

# NON-COMMUTATIVE
# TODO: Make non-commutative exponents not combine automatically
else:
if o is not NC_Marker:
nc_seq.append(o)

# process nc_seq (if any)
while nc_seq:
o = nc_seq.pop(0)
if not nc_part:
nc_part.append(o)
continue

#                             b    c       b+c
# try to combine last terms: a  * a   ->  a
o1 = nc_part.pop()
b1,e1 = o1.as_base_exp()
b2,e2 = o.as_base_exp()
new_exp = e1 + e2
# Only allow powers to combine if the new exponent is
# not an Add. This allow things like a**2*b**3 == a**5
# if a.is_commutative == False, but prohibits
# a**x*a**y and x**a*x**b from combining (x,y commute).
o12 = b1 ** new_exp

# now o12 could be a commutative object
if o12.is_commutative:
seq.append(o12)
continue
else:
nc_seq.insert(0, o12)

else:
nc_part.append(o1)
nc_part.append(o)

# handle the ImaginaryUnits
if iu:
if len(iu) == 1:
c_powers.append((iu, S.One))
else:
# a product of I's has one of 4 values; select that value
# based on the length of iu:
# len(iu) % 4 of (0, 1, 2, 3) has a corresponding value of
#                (1, I,-1,-I)
niu = len(iu) % 4
if niu % 2:
c_powers.append((S.ImaginaryUnit, S.One))
if niu in (2, 3):
coeff = -coeff

# We do want a combined exponent if it would not be an Add, such as
#  y    2y     3y
# x  * x   -> x
# We determine if two exponents have the same term by using
# as_coeff_Mul.
#
# Unfortunately, this isn't smart enough to consider combining into
#  z - y    y
# x      * x  will be left alone.  This is because checking every possible
# combination can slow things down.

# gather exponents of common bases...
def _gather(c_powers):
new_c_powers = []
common_b = {} # b:e
for b, e in c_powers:
co = e.as_coeff_Mul()
common_b.setdefault(b, {}).setdefault(co, []).append(co)
for b, d in list(common_b.items()):
for di, li in list(d.items()):
for b, e in list(common_b.items()):
for t, c in list(e.items()):
new_c_powers.append((b, c*t))
return new_c_powers

# in c_powers
c_powers = _gather(c_powers)

# and in num_exp
num_exp = _gather(num_exp)

# --- PART 2 ---
#
# o process collected powers  (x**0 -> 1; x**1 -> x; otherwise Pow)
# o combine collected powers  (2**x * 3**x -> 6**x)
#   with numeric base

# ................................
# now we have:
# - coeff:
# - c_powers:    (b, e)
# - num_exp:     (2, e)
# - pnum_rat:    {(1/3, [1/3, 2/3, 1/4])}

#  0             1
# x  -> 1       x  -> x
for b, e in c_powers:
if e is S.One:
if b.is_Number:
coeff *= b
else:
c_part.append(b)
elif e is not S.Zero:
c_part.append(Pow(b, e))

#  x    x     x
# 2  * 3  -> 6
inv_exp_dict = {}   # exp:Mul(num-bases)     x    x
# e.g.  x:6  for  ... * 2  * 3  * ...
for b, e in num_exp:
inv_exp_dict.setdefault(e, []).append(b)
for e, b in list(inv_exp_dict.items()):
inv_exp_dict[e] = Mul(*b)
c_part.extend([Pow(b, e) for e, b in inv_exp_dict.items() if e])

# b, e -> e' = sum(e), b
# {(1/5, [1/3]), (1/2, [1/12, 1/4]} -> {(1/3, [1/5, 1/2])}
comb_e = {}
for b, e in pnum_rat.items():
del pnum_rat
# process them, reducing exponents to values less than 1
# and updating coeff if necessary else adding them to
# num_rat for further processing
num_rat = []
for e, b in comb_e.items():
b = Mul(*b)
if e.q == 1:
coeff *= Pow(b, e)
continue
if e.p > e.q:
e_i, ep = divmod(e.p, e.q)
coeff *= Pow(b, e_i)
e = Rational(ep, e.q)
num_rat.append((b, e))
del comb_e

# extract gcd of bases in num_rat
# 2**(1/3)*6**(1/4) -> 2**(1/3+1/4)*3**(1/4)
pnew = {}
i = 0 # steps through num_rat which may grow
while i < len(num_rat):
bi, ei = num_rat[i]
grow = []
for j in range(i + 1, len(num_rat)):
bj, ej = num_rat[j]
g = igcd(bi, bj)
if g != 1:
# 4**r1*6**r2 -> 2**(r1+r2)  *  2**r1 *  3**r2
# this might have a gcd with something else
e = ei + ej
if e.q == 1:
coeff *= Pow(g, e)
else:
if e.p > e.q:
e_i, ep = divmod(e.p, e.q) # change e in place
coeff *= Pow(g, e_i)
e = Rational(ep, e.q)
grow.append((g, e))
# update the jth item
num_rat[j] = (bj//g, ej)
# update bi that we are checking with
bi = bi//g
if bi is S.One:
break
if bi is not S.One:
obj = Pow(bi, ei)
if obj.is_Number:
coeff *= obj
else:
if obj.is_Mul: # sqrt(12) -> 2*sqrt(3)
c, obj = obj.args # expecting only 2 args
coeff *= c
assert obj.is_Pow
bi, ei = obj.args
pnew.setdefault(ei, []).append(bi)

num_rat.extend(grow)
i += 1

# combine bases of the new powers
for e, b in pnew.items():
pnew[e] = Mul(*b)

# see if there is a base with matching coefficient
# that the -1 can be joined with
if neg1e:
p = Pow(S.NegativeOne, neg1e)
if p.is_Number:
coeff *= p
else:
c, p = p.as_coeff_Mul()
coeff *= c
if p.is_Pow and p.base is S.NegativeOne:
neg1e = p.exp
for e, b in pnew.items():
if e == neg1e and b.is_positive:
pnew[e] = -b
break
else:
c_part.append(p)

# add all the pnew powers
c_part.extend([Pow(b, e) for e, b in pnew.items()])

# oo, -oo
if (coeff is S.Infinity) or (coeff is S.NegativeInfinity):
def _handle_for_oo(c_part, coeff_sign):
new_c_part = []
for t in c_part:
if t.is_positive:
continue
if t.is_negative:
coeff_sign *= -1
continue
new_c_part.append(t)
return new_c_part, coeff_sign
c_part, coeff_sign = _handle_for_oo(c_part, 1)
nc_part, coeff_sign = _handle_for_oo(nc_part, coeff_sign)
coeff *= coeff_sign

# zoo
if coeff is S.ComplexInfinity:
# zoo might be
#   unbounded_real + bounded_im
#   bounded_real + unbounded_im
#   unbounded_real + unbounded_im
# and non-zero real or imaginary will not change that status.
c_part = [c for c in c_part if not (c.is_nonzero and
c.is_real is not None)]
nc_part = [c for c in nc_part if not (c.is_nonzero and
c.is_real is not None)]

# 0
elif coeff is S.Zero:
# we know for sure the result will be 0
return [coeff], [], order_symbols

# order commutative part canonically
c_part.sort(key=cls._args_sortkey)

# current code expects coeff to be always in slot-0
if coeff is not S.One:
c_part.insert(0, coeff)

# we are done
if len(c_part)==2 and c_part.is_Number and c_part.is_Add:
# 2*(1+a) -> 2 + 2 * a
coeff = c_part
c_part = [Add(*[coeff*f for f in c_part.args])]

return c_part, nc_part, order_symbols

def _eval_power(b, e):

# don't break up NC terms: (A*B)**3 != A**3*B**3, it is A*B*A*B*A*B
coeff, b = b.as_coeff_Mul()
bc, bnc = b.args_cnc()

done = [Pow(Mul._from_args(bnc), e, evaluate=False) if bnc else S.One]

if e.is_Number:
if e.is_Integer:
# (a*b)**2 -> a**2 * b**2
return Mul(*([s**e for s in [coeff] + bc] + done))

if e.is_Rational or e.is_Float:
unk = []
nonneg = []
neg = []
iu = []
for bi in bc:
if bi.is_polar:
nonneg.append(bi)
elif bi.is_negative is not None:
if bi.is_negative:
neg.append(bi)
elif bi.is_nonnegative:
nonneg.append(bi)
elif bi is S.ImaginaryUnit:
iu.append(bi)
else:
unk.append(bi)
else:
unk.append(bi)

if iu:
niu = len(iu) % 4
i = S.ImaginaryUnit if niu % 2 else S.One
if niu in (2, 3):
coeff = -coeff
if i is S.ImaginaryUnit:
if unk or e.is_Float:
unk.append(i)
else:
if coeff.is_negative and e.is_Rational:
coeff = -coeff
ie = Rational(4*e.q - e.p, 2*e.q)
done.append(Pow(-1, ie))
else:
done.append(i**e)

if len(unk) == len(bc) or len(neg) == len(bc) == 1:
# if all terms were unknown there is nothing to pull
# out except maybe the coeff; if there is a single
# negative term, this is the base case which cannot
# be processed further
if coeff.is_negative:
coeff *= -1
bc = -bc
if coeff is S.One:
return None
return Mul(*([Pow(coeff, e), Pow(Mul(*bc), e)] + done))

# otherwise return the new expression expanding out the known
# terms; those that are not known can be expanded out with
# expand_power_base() but this will introduce a lot of
# "garbage" that is needed to keep one on the same branch as
# the unexpanded expression. The negatives are brought out
# with a negative sign added and a negative left behind in the
# unexpanded terms if there were an odd number of negatives.
if coeff.is_negative:
coeff = -coeff
neg.append(S.NegativeOne)
if neg:
neg = [-w for w in neg]
if len(neg) % 2:
unk.append(S.NegativeOne)

done.extend([Pow(s, e) for s in nonneg + neg + [coeff, Mul(*unk)]])
return Mul(*done)

if e.is_even and coeff.is_negative:
return Pow(-coeff, e)*Pow(b, e)

@classmethod
def class_key(cls):
return 3, 0, cls.__name__

def _eval_evalf(self, prec):
c, m = self.as_coeff_Mul()
if c is S.NegativeOne:
if m.is_Mul:
rv = -AssocOp._eval_evalf(m, prec)
else:
mnew = m._eval_evalf(prec)
if mnew is not None:
m = mnew
rv = -m
else:
rv = AssocOp._eval_evalf(self, prec)
if rv.is_number:
return rv.expand()
return rv

@cacheit
[docs]    def as_two_terms(self):
"""Return head and tail of self.

This is the most efficient way to get the head and tail of an
expression.

- if you want only the head, use self.args;
- if you want to process the arguments of the tail then use
self.as_coef_mul() which gives the head and a tuple containing
the arguments of the tail when treated as a Mul.
- if you want the coefficient when self is treated as an Add

>>> from sympy.abc import x, y
>>> (3*x*y).as_two_terms()
(3, x*y)
"""
args = self.args

if len(args) == 1:
return S.One, self
elif len(args) == 2:
return args

else:
return args, self._new_rawargs(*args[1:])

@cacheit
def as_coeff_mul(self, *deps):
if deps:
l1 = []
l2 = []
for f in self.args:
if f.has(*deps):
l2.append(f)
else:
l1.append(f)
return self._new_rawargs(*l1), tuple(l2)
args = self.args
if args.is_Rational:
return args, args[1:]
elif args is S.NegativeInfinity:
return S.NegativeOne, (-args,) + args[1:]
return S.One, args

[docs]    def as_coeff_Mul(self, rational=False):
"""Efficiently extract the coefficient of a product. """
coeff, args = self.args, self.args[1:]

if coeff.is_Number and not (rational and not coeff.is_Rational):
if len(args) == 1:
return coeff, args
else:
return coeff, self._new_rawargs(*args)
else:
return S.One, self

def as_real_imag(self, deep=True, **hints):
other = []
coeff = S(1)
for a in self.args:
if a.is_real:
coeff *= a
else:
other.append(a)
m = Mul(*other)
if hints.get('ignore') == m:
return None
else:
return (coeff*C.re(m), coeff*C.im(m))

@staticmethod
def _expandsums(sums):
"""
Helper function for _eval_expand_mul.

sums must be a list of instances of Basic.
"""

L = len(sums)
if L == 1:
return sums.args
terms = []
left = Mul._expandsums(sums[:L//2])
right = Mul._expandsums(sums[L//2:])

terms = [Mul(a, b) for a in left for b in right]

def _eval_expand_mul(self, **hints):
from sympy import fraction, expand_mul

# Handle things like 1/(x*(x + 1)), which are automatically converted
# to 1/x*1/(x + 1)
expr = self
n, d = fraction(expr)
if d.is_Mul:
expr = n/d._eval_expand_mul(**hints)
if not expr.is_Mul:
return expand_mul(expr, deep=False)

plain, sums, rewrite = [], [], False
for factor in expr.args:
sums.append(factor)
rewrite = True
else:
if factor.is_commutative:
plain.append(factor)
else:
sums.append(Basic(factor)) # Wrapper

if not rewrite:
return expr
else:
plain = Mul(*plain)
if sums:
terms = Mul._expandsums(sums)
args = []
for term in terms:
t = Mul(plain, term)
if t.is_Mul and any(a.is_Add for a in t.args):
t = t._eval_expand_mul()
args.append(t)
else:
return plain

def _eval_derivative(self, s):
terms = list(self.args)
factors = []
for i in range(len(terms)):
t = terms[i].diff(s)
if t is S.Zero:
continue
factors.append(Mul(*(terms[:i]+[t]+terms[i+1:])))

def _matches_simple(self, expr, repl_dict):
# handle (w*3).matches('x*5') -> {w: x*5/3}
coeff, terms = self.as_coeff_Mul()
terms = Mul.make_args(terms)
if len(terms) == 1:
newexpr = self.__class__._combine_inverse(expr, coeff)
return terms.matches(newexpr, repl_dict)
return

def matches(self, expr, repl_dict={}):
expr = sympify(expr)
if self.is_commutative and expr.is_commutative:
return AssocOp._matches_commutative(self, expr, repl_dict)
# todo for commutative parts, until then use the default matches method for non-commutative products
return self._matches(expr, repl_dict)

def _matches(self, expr, repl_dict={}):
# weed out negative one prefixes
sign = 1
a, b = self.as_two_terms()
if a is S.NegativeOne:
if b.is_Mul:
sign = -sign
else:
# the remainder, b, is not a Mul anymore
return b.matches(-expr, repl_dict)
expr = sympify(expr)
if expr.is_Mul and expr.args is S.NegativeOne:
expr = -expr; sign = -sign

if not expr.is_Mul:
# expr can only match if it matches b and a matches +/- 1
if len(self.args) == 2:
# quickly test for equality
if b == expr:
return a.matches(Rational(sign), repl_dict)
# do more expensive match
dd = b.matches(expr, repl_dict)
if dd == None:
return None
dd = a.matches(Rational(sign), dd)
return dd
return None

d = repl_dict.copy()

# weed out identical terms
pp = list(self.args)
ee = list(expr.args)
for p in self.args:
if p in expr.args:
ee.remove(p)
pp.remove(p)

# only one symbol left in pattern -> match the remaining expression
if len(pp) == 1 and isinstance(pp, C.Wild):
if len(ee) == 1:
d[pp] = sign * ee
else:
d[pp] = sign * expr.func(*ee)
return d

if len(ee) != len(pp):
return None

for p, e in zip(pp, ee):
d = p.xreplace(d).matches(e, d)
if d is None:
return None
return d

@staticmethod
def _combine_inverse(lhs, rhs):
"""
Returns lhs/rhs, but treats arguments like symbols, so things like
oo/oo return 1, instead of a nan.
"""
if lhs == rhs:
return S.One
def check(l, r):
if l.is_Float and r.is_comparable:
# if both objects are added to 0 they will share the same "normalization"
# and are more likely to compare the same. Since Add(foo, 0) will not allow
# the 0 to pass, we use __add__ directly.
return False
if check(lhs, rhs) or check(rhs, lhs):
return S.One
if lhs.is_Mul and rhs.is_Mul:
a = list(lhs.args)
b = 
for x in rhs.args:
if x in a:
a.remove(x)
else:
b.append(x)
return Mul(*a)/Mul(*b)
return lhs/rhs

def as_powers_dict(self):
d = defaultdict(list)
for term in self.args:
b, e = term.as_base_exp()
d[b].append(e)
for b, e in d.items():
if len(e) == 1:
e = e
else:
d[b] = e
return d

def as_numer_denom(self):
# don't use _from_args to rebuild the numerators and denominators
# as the order is not guaranteed to be the same once they have
# been separated from each other
numers, denoms = list(zip(*[f.as_numer_denom() for f in self.args]))
return Mul(*numers), Mul(*denoms)

def as_base_exp(self):
e1 = None
bases = []
nc = 0
for m in self.args:
b, e = m.as_base_exp()
if not b.is_commutative:
nc += 1
if e1 is None:
e1 = e
elif e != e1 or nc > 1:
return self, S.One
bases.append(b)
return Mul(*bases), e1

def _eval_is_polynomial(self, syms):
return all(term._eval_is_polynomial(syms) for term in self.args)

def _eval_is_rational_function(self, syms):
return all(term._eval_is_rational_function(syms) for term in self.args)

_eval_is_bounded = lambda self: self._eval_template_is_attr('is_bounded')
_eval_is_integer = lambda self: self._eval_template_is_attr('is_integer', when_multiple=None)
_eval_is_commutative = lambda self: self._eval_template_is_attr('is_commutative')

def _eval_is_polar(self):
has_polar = any(arg.is_polar for arg in self.args)
return has_polar and \
all(arg.is_polar or arg.is_positive for arg in self.args)

# I*I -> R,  I*I*I -> -I
def _eval_is_real(self):
im_count = 0
is_neither = False
for t in self.args:
if t.is_imaginary:
im_count += 1
continue
t_real = t.is_real
if t_real:
continue
elif t_real is False:
if is_neither:
return None
else:
is_neither = True
else:
return None
if is_neither:
return False

return (im_count % 2 == 0)

def _eval_is_imaginary(self):
im_count = 0
is_neither = False
for t in self.args:
if t.is_imaginary:
im_count += 1
continue
t_real = t.is_real
if t_real:
continue
elif t_real is False:
if is_neither:
return None
else:
is_neither = True
else:
return None
if is_neither:
return False

return (im_count % 2 == 1)

def _eval_is_hermitian(self):
nc_count = 0
im_count = 0
is_neither = False
for t in self.args:
if not t.is_commutative:
nc_count += 1
if nc_count > 1:
return None
if t.is_antihermitian:
im_count += 1
continue
t_real = t.is_hermitian
if t_real:
continue
elif t_real is False:
if is_neither:
return None
else:
is_neither = True
else:
return None
if is_neither:
return False

return (im_count % 2 == 0)

def _eval_is_antihermitian(self):
nc_count = 0
im_count = 0
is_neither = False
for t in self.args:
if not t.is_commutative:
nc_count += 1
if nc_count > 1:
return None
if t.is_antihermitian:
im_count += 1
continue
t_real = t.is_hermitian
if t_real:
continue
elif t_real is False:
if is_neither:
return None
else:
is_neither = True
else:
return None
if is_neither:
return False

return (im_count % 2 == 1)

def _eval_is_irrational(self):
for t in self.args:
a = t.is_irrational
if a:
others = list(self.args)
others.remove(t)
if all(x.is_rational is True for x in others):
return True
return None
if a is None:
return
return False

def _eval_is_zero(self):
zero = None
for a in self.args:
if a.is_zero:
zero = True
continue
bound = a.is_bounded
if not bound:
return bound
if zero:
return True

def _eval_is_positive(self):
"""Return True if self is positive, False if not, and None if it
cannot be determined.

This algorithm is non-recursive and works by keeping track of the
sign which changes when a negative or nonpositive is encountered.
Whether a nonpositive or nonnegative is seen is also tracked since
the presence of these makes it impossible to return True, but
possible to return False if the end result is nonpositive. e.g.

pos * neg * nonpositive -> pos or zero -> None is returned
pos * neg * nonnegative -> neg or zero -> False is returned
"""

sign = 1
saw_NON = False
for t in self.args:
if t.is_positive:
continue
elif t.is_negative:
sign = -sign
elif t.is_zero:
return False
elif t.is_nonpositive:
sign = -sign
saw_NON = True
elif t.is_nonnegative:
saw_NON = True
else:
return
if sign == 1 and saw_NON is False:
return True
if sign < 0:
return False

def _eval_is_negative(self):
"""Return True if self is negative, False if not, and None if it
cannot be determined.

This algorithm is non-recursive and works by keeping track of the
sign which changes when a negative or nonpositive is encountered.
Whether a nonpositive or nonnegative is seen is also tracked since
the presence of these makes it impossible to return True, but
possible to return False if the end result is nonnegative. e.g.

pos * neg * nonpositive -> pos or zero -> False is returned
pos * neg * nonnegative -> neg or zero -> None is returned
"""

sign = 1
saw_NON = False
for t in self.args:
if t.is_positive:
continue
elif t.is_negative:
sign = -sign
elif t.is_zero:
return False
elif t.is_nonpositive:
sign = -sign
saw_NON = True
elif t.is_nonnegative:
saw_NON = True
else:
return
if sign == -1 and saw_NON is False:
return True
if sign > 0:
return False

def _eval_is_odd(self):
is_integer = self.is_integer

if is_integer:
r = True
for t in self.args:
if t.is_even:
return False
if t.is_odd is None:
r = None
return r

# !integer -> !odd
elif is_integer == False:
return False

def _eval_is_even(self):
is_integer = self.is_integer

if is_integer:
return fuzzy_not(self._eval_is_odd())

elif is_integer == False:
return False

def _eval_subs(self, old, new):
from sympy import sign, Dummy, multiplicity
from sympy.simplify.simplify import powdenest, fraction

if not old.is_Mul:
return None

def base_exp(a):
# if I and -1 are in a Mul, they get both end up with
# a -1 base (see issue 3322); all we want here are the
# true Pow or exp separated into base and exponent
if a.is_Pow or a.func is C.exp:
return a.as_base_exp()
return a, S.One

def breakup(eq):
"""break up powers of eq when treated as a Mul:
b**(Rational*e) -> b**e, Rational
commutatives come back as a dictionary {b**e: Rational}
noncommutatives come back as a list [(b**e, Rational)]
"""

(c, nc) = (defaultdict(int), list())
for a in Mul.make_args(eq):
a = powdenest(a)
(b, e) = base_exp(a)
if e is not S.One:
(co, _) = e.as_coeff_mul()
b = Pow(b, e/co)
e = co
if a.is_commutative:
c[b] += e
else:
nc.append([b, e])
return (c, nc)

def rejoin(b, co):
"""
Put rational back with exponent; in general this is not ok, but
since we took it from the exponent for analysis, it's ok to put
it back.
"""

(b, e) = base_exp(b)
return Pow(b, e*co)

def ndiv(a, b):
"""if b divides a in an extractive way (like 1/4 divides 1/2
but not vice versa, and 2/5 does not divide 1/3) then return
the integer number of times it divides, else return 0.
"""

if not b.q % a.q or not a.q % b.q:
return int(a/b)
return 0

# give Muls in the denominator a chance to be changed (see issue 2552)
# rv will be the default return value
rv = None
n, d = fraction(self)
if d is not S.One:
self2 = n._subs(old, new)/d._subs(old, new)
if not self2.is_Mul:
return self2._subs(old, new)
if self2 != self:
self = rv = self2

# Now continue with regular substitution.

# handle the leading coefficient and use it to decide if anything
# should even be started; we always know where to find the Rational
# so it's a quick test

co_self = self.args
co_old = old.args
co_xmul = None
if co_old.is_Rational and co_self.is_Rational:
# if coeffs are the same there will be no updating to do
# below after breakup() step; so skip (and keep co_xmul=None)
if co_old != co_self:
co_xmul = co_self.extract_multiplicatively(co_old)
elif co_old.is_Rational:
return rv

# break self and old into factors

(c, nc) = breakup(self)
(old_c, old_nc) = breakup(old)

# update the coefficients if we had an extraction
# e.g. if co_self were 2*(3/35*x)**2 and co_old = 3/5
# then co_self in c is replaced by (3/5)**2 and co_residual
# is 2*(1/7)**2

if co_xmul and co_xmul.is_Rational:
n_old, d_old = co_old.as_numer_denom()
n_self, d_self = co_self.as_numer_denom()
def _multiplicity(p, n):
p = abs(p)
if p is S.One:
return S.Infinity
return multiplicity(p, abs(n))
mult = S(min(_multiplicity(n_old, n_self),
_multiplicity(d_old, d_self)))
c.pop(co_self)
c[co_old] = mult
co_residual = co_self/co_old**mult
else:
co_residual = 1

# do quick tests to see if we can't succeed

ok = True
if (
# more non-commutative terms
len(old_nc) > len(nc)):
ok = False
elif (
# more commutative terms
len(old_c) > len(c)):
ok = False
elif (
# unmatched non-commutative bases
set(_ for _ in  old_nc).difference(set(_ for _ in nc))):
ok = False
elif (
# unmatched commutative terms
set(old_c).difference(set(c))):
ok = False
elif (
# differences in sign
any(sign(c[b]) != sign(old_c[b]) for b in old_c)):
ok = False
if not ok:
return rv

if not old_c:
cdid = None
else:
rat = []
for (b, old_e) in list(old_c.items()):
c_e = c[b]
rat.append(ndiv(c_e, old_e))
if not rat[-1]:
return rv
cdid = min(rat)

if not old_nc:
ncdid = None
for i in range(len(nc)):
nc[i] = rejoin(*nc[i])
else:
ncdid = 0  # number of nc replacements we did
take = len(old_nc)  # how much to look at each time
limit = cdid or S.Infinity # max number that we can take
failed = []  # failed terms will need subs if other terms pass
i = 0
while limit and i + take <= len(nc):
hit = False

# the bases must be equivalent in succession, and
# the powers must be extractively compatible on the
# first and last factor but equal inbetween.

rat = []
for j in range(take):
if nc[i + j] != old_nc[j]:
break
elif j == 0:
rat.append(ndiv(nc[i + j], old_nc[j]))
elif j == take - 1:
rat.append(ndiv(nc[i + j], old_nc[j]))
elif nc[i + j] != old_nc[j]:
break
else:
rat.append(1)
j += 1
else:
ndo = min(rat)
if ndo:
if take == 1:
if cdid:
ndo = min(cdid, ndo)
nc[i] = Pow(new, ndo)*rejoin(nc[i],
nc[i] - ndo*old_nc)
else:
ndo = 1

# the left residual

l = rejoin(nc[i], nc[i] - ndo*
old_nc)

# eliminate all middle terms

mid = new

# the right residual (which may be the same as the middle if take == 2)

ir = i + take - 1
r = (nc[ir], nc[ir] - ndo*
old_nc[-1])
if r:
if i + take < len(nc):
nc[i:i + take] = [l*mid, r]
else:
r = rejoin(*r)
nc[i:i + take] = [l*mid*r]
else:

# there was nothing left on the right

nc[i:i + take] = [l*mid]

limit -= ndo
ncdid += ndo
hit = True
if not hit:

# do the subs on this failing factor

failed.append(i)
i += 1
else:

if not ncdid:
return rv

# although we didn't fail, certain nc terms may have
# failed so we rebuild them after attempting a partial
# subs on them

failed.extend(list(range(i, len(nc))))
for i in failed:
nc[i] = rejoin(*nc[i]).subs(old, new)

# rebuild the expression

if cdid is None:
do = ncdid
elif ncdid is None:
do = cdid
else:
do = min(ncdid, cdid)

margs = []
for b in c:
if b in old_c:

# calculate the new exponent

e = c[b] - old_c[b]*do
margs.append(rejoin(b, e))
else:
margs.append(rejoin(b.subs(old, new), c[b]))
if cdid and not ncdid:

# in case we are replacing commutative with non-commutative,
# we want the new term to come at the front just like the
# rest of this routine

margs = [Pow(new, cdid)] + margs
return co_residual*Mul(*margs)*Mul(*nc)

def _eval_nseries(self, x, n, logx):
from sympy import powsimp
terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
return powsimp(Mul(*terms).expand(), combine='exp', deep=True)

return Mul(*[t.as_leading_term(x) for t in self.args])

def _eval_conjugate(self):
return Mul(*[t.conjugate() for t in self.args])

def _eval_transpose(self):
return Mul(*[t.transpose() for t in self.args[::-1]])

return Mul(*[t.adjoint() for t in self.args[::-1]])

def _sage_(self):
s = 1
for x in self.args:
s *= x._sage_()
return s

"""Return the tuple (R, self/R) where R is the positive Rational
extracted from self.

Examples
========

>>> from sympy import sqrt
>>> (-3*sqrt(2)*(2 - 2*sqrt(2))).as_content_primitive()
(6, -sqrt(2)*(-sqrt(2) + 1))

See docstring of Expr.as_content_primitive for more examples.
"""

coef = S.One
args = []
for i, a in enumerate(self.args):
coef *= c
if p is not S.One:
args.append(p)
# don't use self._from_args here to reconstruct args
# since there may be identical args now that should be combined
# e.g. (2+2*x)*(3+3*x) should be (6, (1 + x)**2) not (6, (1+x)*(1+x))
return coef, Mul(*args)

[docs]    def as_ordered_factors(self, order=None):
"""Transform an expression into an ordered list of factors.

Examples
========

>>> from sympy import sin, cos
>>> from sympy.abc import x, y

>>> (2*x*y*sin(x)*cos(x)).as_ordered_factors()
[2, x, y, sin(x), cos(x)]

"""
cpart, ncpart = self.args_cnc()
cpart.sort(key=lambda expr: expr.sort_key(order=order))
return cpart + ncpart

@property
def _sorted_args(self):
return self.as_ordered_factors()

[docs]def prod(a, start=1):
"""Return product of elements of a. Start with int 1 so if only
ints are included then an int result is returned.

Examples
========

>>> from sympy import prod, S
>>> prod(list(range(3)))
0
>>> type(_) is int
True
>>> prod([S(2), 3])
6
>>> _.is_Integer
True

You can start the product at something other than 1:
>>> prod([1, 2], 3)
6

"""
return reduce(operator.mul, a, start)

def _keep_coeff(coeff, factors, clear=True):
"""Return coeff*factors unevaluated if necessary.

If clear is False, do not keep the coefficient as a factor
if it can be distributed on a single factor such that one or
more terms will still have integer coefficients.

Examples
========

>>> from sympy.core.mul import _keep_coeff
>>> from sympy.abc import x, y
>>> from sympy import S

>>> _keep_coeff(S.Half, x + 2)
(x + 2)/2
>>> _keep_coeff(S.Half, x + 2, clear=False)
x/2 + 1
>>> _keep_coeff(S.Half, (x + 2)*y, clear=False)
y*(x + 2)/2
"""

if not coeff.is_Number:
if factors.is_Number:
factors, coeff = coeff, factors
else:
return coeff*factors
if coeff == 1:
return factors
elif coeff == -1: # don't keep sign?
return -factors
if not clear and coeff.is_Rational and coeff.q != 1:
q = S(coeff.q)
for i in factors.args:
c, t = i.as_coeff_Mul()
r = c/q
if r == int(r):
return coeff*factors
return Mul._from_args((coeff, factors))
elif factors.is_Mul:
margs = list(factors.args)
if margs.is_Number:
margs *= coeff
if margs == 1:
margs.pop(0)
else:
margs.insert(0, coeff)
return Mul._from_args(margs)
else:
return coeff*factors

from .numbers import Rational, igcd
from .power import Pow