+
    ‹°³j ã                  óˆ  € ^ RI Ht ^ RIHtHt ^ RIHt ^RIHt ^RI	H
t
 ^RIHt ^RIHt ^RIHt ^R	IHtHtHtHtHt ^R
IHtHtHtHt ^RIHt ^RIHtH t  ^RI!H"t"H#t# ^ RI$H%t% ^ RI&H't' ^ RI(H)t) ^ RI*H+t+  ! R R]4      t,]+! R4      t-]-P]                  ]/]/3],4       ^RI.H0t0 ^RI1H2t2H3t3 ^RI4H5t5H6t6 ^RI7H8t8H9t9H:t: R# )é    )Úannotations)ÚCallableÚTYPE_CHECKING)Úproduct)Ú_sympify)Úcacheit)ÚS)ÚExpr)ÚPrecisionExhausted)Úexpand_complexÚexpand_multinomialÚ
expand_mulÚ_mexpandÚ	PoleError)Ú
fuzzy_boolÚ	fuzzy_notÚ	fuzzy_andÚfuzzy_or)Úglobal_parameters)Úis_gtÚis_lt)Ú
NumberKindÚUndefinedKind)Úsift)Úsympy_deprecation_warning)Úas_int)Ú
Dispatcherc                  óþ  a € ] tR t^tRtRtR?t]'       d   ]R R l4       t	]R R l4       t
]R R l4       t]R	 4       t]R@R
 R ll4       tRAR lt]R 4       tR tR tR tR tR tR tR tR tR tR tR tR tR tR tR t R t!R t"R t#R  t$R! t%R" t&R# t'R$ t(R% t)R& t*RBR' lt+R( t,R) t-R* t.R+ t/R, t0R- t1R. t2R/ t3R0 t4R1 t5RCR2 lt6RDR3 lt7R4 t8]R5 4       t9V 3R6 lt:R7 t;R8 t<R9 t=R: t>RER; lt?R< t@R= tAR>tBV ;tC# )FÚPowañ  
Defines the expression x**y as "x raised to a power y"

.. deprecated:: 1.7

   Using arguments that aren't subclasses of :class:`~.Expr` in core
   operators (:class:`~.Mul`, :class:`~.Add`, and :class:`~.Pow`) is
   deprecated. See :ref:`non-expr-args-deprecated` for details.

Singleton definitions involving (0, 1, -1, oo, -oo, I, -I):

+--------------+---------+-----------------------------------------------+
| expr         | value   | reason                                        |
+==============+=========+===============================================+
| z**0         | 1       | Although arguments over 0**0 exist, see [2].  |
+--------------+---------+-----------------------------------------------+
| z**1         | z       |                                               |
+--------------+---------+-----------------------------------------------+
| (-oo)**(-1)  | 0       |                                               |
+--------------+---------+-----------------------------------------------+
| (-1)**-1     | -1      |                                               |
+--------------+---------+-----------------------------------------------+
| S.Zero**-1   | zoo     | This is not strictly true, as 0**-1 may be    |
|              |         | undefined, but is convenient in some contexts |
|              |         | where the base is assumed to be positive.     |
+--------------+---------+-----------------------------------------------+
| 1**-1        | 1       |                                               |
+--------------+---------+-----------------------------------------------+
| oo**-1       | 0       |                                               |
+--------------+---------+-----------------------------------------------+
| 0**oo        | 0       | Because for all complex numbers z near        |
|              |         | 0, z**oo -> 0.                                |
+--------------+---------+-----------------------------------------------+
| 0**-oo       | zoo     | This is not strictly true, as 0**oo may be    |
|              |         | oscillating between positive and negative     |
|              |         | values or rotating in the complex plane.      |
|              |         | It is convenient, however, when the base      |
|              |         | is positive.                                  |
+--------------+---------+-----------------------------------------------+
| 1**oo        | nan     | Because there are various cases where         |
| 1**-oo       |         | lim(x(t),t)=1, lim(y(t),t)=oo (or -oo),       |
|              |         | but lim( x(t)**y(t), t) != 1.  See [3].       |
+--------------+---------+-----------------------------------------------+
| b**zoo       | nan     | Because b**z has no limit as z -> zoo         |
+--------------+---------+-----------------------------------------------+
| (-1)**oo     | nan     | Because of oscillations in the limit.         |
| (-1)**(-oo)  |         |                                               |
+--------------+---------+-----------------------------------------------+
| oo**oo       | oo      |                                               |
+--------------+---------+-----------------------------------------------+
| oo**-oo      | 0       |                                               |
+--------------+---------+-----------------------------------------------+
| (-oo)**oo    | nan     |                                               |
| (-oo)**-oo   |         |                                               |
+--------------+---------+-----------------------------------------------+
| oo**I        | nan     | oo**e could probably be best thought of as    |
| (-oo)**I     |         | the limit of x**e for real x as x tends to    |
|              |         | oo. If e is I, then the limit does not exist  |
|              |         | and nan is used to indicate that.             |
+--------------+---------+-----------------------------------------------+
| oo**(1+I)    | zoo     | If the real part of e is positive, then the   |
| (-oo)**(1+I) |         | limit of abs(x**e) is oo. So the limit value  |
|              |         | is zoo.                                       |
+--------------+---------+-----------------------------------------------+
| oo**(-1+I)   | 0       | If the real part of e is negative, then the   |
| -oo**(-1+I)  |         | limit is 0.                                   |
+--------------+---------+-----------------------------------------------+

Because symbolic computations are more flexible than floating point
calculations and we prefer to never return an incorrect answer,
we choose not to conform to all IEEE 754 conventions.  This helps
us avoid extra test-case code in the calculation of limits.

See Also
========

sympy.core.numbers.Infinity
sympy.core.numbers.NegativeInfinity
sympy.core.numbers.NaN

References
==========

.. [1] https://en.wikipedia.org/wiki/Exponentiation
.. [2] https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero
.. [3] https://en.wikipedia.org/wiki/Indeterminate_forms

Tc               ó   € V ^8„  d   QhRR/# )é   Úreturnztuple[Expr, Expr]© )Úformats   "ÚQ/home/luca/screen-checking/.venv/lib/python3.14/site-packages/sympy/core/power.pyÚ__annotate__ÚPow.__annotate__v   s   € ÷ 	ñ 	Ð+ñ 	ó    c                	ó   € R # ©Nr#   ©Úselfs   &r%   ÚargsÚPow.argsu   s   € ár(   c               ó   € V ^8„  d   QhRR/# ©r!   r"   r
   r#   )r$   s   "r%   r&   r'   z   s   € ÷ ñ �dñ r(   c                	ó(   € V P                   ^ ,          # ©r   ©r-   r+   s   &r%   ÚbaseÚPow.basey   ó   € à�y‰y˜�|Ðr(   c               ó   € V ^8„  d   QhRR/# r0   r#   )r$   s   "r%   r&   r'   ~   s   € ÷ ñ �Tñ r(   c                	ó(   € V P                   ^,          # ©é   r3   r+   s   &r%   ÚexpÚPow.exp}   r6   r(   c                	óv   € V P                   P                  \        J d   V P                  P                  # \        # r*   )r;   Úkindr   r4   r   r+   s   &r%   r>   ÚPow.kind�   s&   € à�8‰8�=‰=œJÓ&Ø—9‘9—>‘>Ð!ä Ð r(   c               ó$   € V ^8„  d   QhRRRRRR/# )r!   ÚbzExpr | complexÚer"   r
   r#   )r$   s   "r%   r&   r'   ‰   s&   € ÷ `ñ `˜ð `¨>ð `ÈTñ `r(   c                	ó0  € Vf   \         P                  p\        V4      p\        V4      p^RIHp \        WF4      '       g   \        WV4      '       d   \        R4      hWE3 FB  p\        V\        4      '       d   K  \        R\        V4      P                  : R2RR^R7       KD  	  V'       Ed‚   V\        P                  J d   \        P                  # V\        P                  J dã   \        V\        P                   4      '       d   \        P                  # \        V\        P"                  4      '       d1   \%        V\        P                   4      '       d   \        P&                  # \%        V\        P"                  4      '       dC   VP(                  '       d   \        P                  # VP(                  RJ d   \        P                  # V\        P&                  J d   \        P                   # V\        P                   J d   V# VR8X  d   V'       g   \        P                  # VP*                  P                  R	8X  dL   V\        P,                  8X  d6   ^ R
IHp V! \3        WEP4                  4      \3        WEP6                  4      4      # M¸VP8                  '       d   VP:                  '       g   VP<                  '       dƒ   VP>                  '       d   VP@                  '       g   VPB                  '       dM   VPE                  4       '       d7   VPF                  '       d   V) pM VPH                  '       d   \3        V) V4      ) # \        P                  WE39   d   \        P                  # V\        P                   J d<   \K        V4      PL                  '       d   \        P                  # \        P                   # ^ RI'H(p	 VPR                  '       EgL   V\        P,                  JEd7   \        WI4      '       Eg%   ^RI*H+p
 ^ RI'H,p ^ RI-H.p V
! VRR7      P_                  4       w  rÞV! V4      w  pp\        VV4      '       d6   VP`                  ^ ,          V8X  d   \        P,                  Wß,          ,          # VPb                  '       d•   ^ RI2H3pH4p V! V! V4      4      pVPB                  '       dm   V'       de   VV! V
! VRR7      ) 4      V\        Pj                  ,          \        Pl                  ,          ,           8X  d   \        P,                  Wß,          ,          # VPo                  V4      pVe   V# \        Pp                  ! WV4      pV Ps                  V4      p\        V\2        4      '       g   V# VPt                  ;'       d    VPt                  Vn:        V# )N)Ú
Relationalz Relational cannot be used in Powzf
    Using non-Expr arguments in Pow is deprecated (in this case, one of the
    arguments is of type zf).

    If you really did intend to construct a power with this base, use the **
    operator instead.z1.7znon-expr-args-deprecated)Údeprecated_since_versionÚactive_deprecations_targetÚ
stacklevelFÚAccumulationBounds©ÚAccumBounds)Ú	exp_polar)Úfactor_terms©Úlog)Úfraction)Úsign)rP   Úiméÿÿÿÿ);r   Úevaluater   Ú
relationalrD   Ú
isinstanceÚ	TypeErrorr
   r   ÚtypeÚ__name__r	   ÚComplexInfinityÚNaNÚInfinityr   ÚOneÚNegativeOner   ÚZeroÚ	is_finiteÚ	__class__ÚExp1Ú!sympy.calculus.accumulationboundsrJ   r   ÚminÚmaxÚ	is_SymbolÚ
is_integerÚ
is_IntegerÚ	is_numberÚis_MulÚ	is_NumberÚcould_extract_minus_signÚis_evenÚis_oddÚabsÚis_infiniteÚ&sympy.functions.elementary.exponentialrK   Úis_AtomÚ	exprtoolsrL   rN   Úsympy.simplify.radsimprO   Úas_coeff_Mulr-   Úis_AddÚ$sympy.functions.elementary.complexesrP   rQ   ÚImaginaryUnitÚPiÚ_eval_powerÚ__new__Ú _exec_constructor_postprocessorsÚis_commutative)ÚclsrA   rB   rS   r4   r;   rD   ÚargrJ   rK   rL   rN   rO   ÚcÚexÚnumÚdenrP   rQ   ÚsÚobjs   &&&&                 r%   rz   ÚPow.__new__ˆ   sÃ  € àÒÜ(×1Ñ1ˆHä˜‹{ˆÜ�q‹kˆõ 	+Ü�d×'Ò'¬:°c×+FÒ+FÜÐ>Ó?Ð?ð “;ˆCÜ˜c¤4×(Ô(Ü)ðä˜s›)×,Ñ,Ñ/ð 0ðð .3Ø/IØ ÷
ñ ÷ ˆ8Ø”a×'Ñ'Ó'Ü—u‘u�Ø”a—j‘jÓ Ü˜œqŸu™u×%Ò%ÜŸ:™:Ð%Ü˜œqŸ}™}×-Ò-´%¸¼a¿e¹e×2DÒ2DÜŸ6™6�MÜ˜œqŸ}™}×-Ò-Ø—~—~�~Ü ×0Ñ0Ð0Ø—~‘~¨Ó.Ü Ÿu™u˜Ø”a—f‘f‹}Ü—u‘u�ØœŸ™“Ø�Ø˜”§4Ü×(Ñ(Ð(Ø—‘×'Ñ'Ð+?Ô?Øœ1Ÿ6™6”>ÝMÙ&¤s¨4·±Ó'9¼3¸tÇWÁWÓ;MÓNÐNð "ð —-—-�- C§N§N N°c·n·n°nØŸ>Ÿ>˜>¨d¯k¯k¨k¸T¿^¿^¸^Ø×7Ñ7×9Ò9Ø—;—;�;Ø ˜5‘DØ—Z—Z�ZÜ   s›OÐ+Ð+Ü�u‰u˜˜Ô#Ü—u‘u�ØœŸ™“Ü�s“8×'×'Ð'ÜŸ5™5�LÜ—u‘u�õ MØ—{—{‘{ t´1·6±6Ô'9Ä*ÈT×B]ÓB]Ý7ÝJÝ?Ù(¨°5Ô9×FÑFÓH‘E�AÙ'¨›|‘H�C˜Ü! # s×+Ò+°·±¸µ¸tÔ0CÜ Ÿv™v¨­�Ð.ØŸŸ˜ßQÙ ¡ D£›N˜ØŸ;Ÿ;˜;¯1°Ù #¡\°$¸UÔ%CÐ$CÓ DÀqÌÏÉÕGXÔYZ×Y]ÑY]ÕG]Õ ]ô2^ä#$§6¡6¨A­E¥?Ð2à×&Ñ& sÓ+�Ø’?Ø�JÜ�lŠl˜3 cÓ*ˆØ×2Ñ2°3Ó7ˆÜ˜#œs×#Ò#ØˆJØ"×1Ñ1×HÐH°c×6HÑ6HˆÔØˆ
r(   c                	óT   € V P                   \        P                  8X  d	   ^ RIHp V# R# )r   rM   N)r4   r	   ra   rp   rN   )r,   ÚargindexrN   s   && r%   ÚinverseÚPow.inverseë   s   € Ø�9‰9œŸ™ÔÝBØˆJÙr(   c                	ó    € ^^V P                   3# )é   )rX   ©r}   s   &r%   Ú	class_keyÚPow.class_keyñ   s   € à�!�S—\‘\Ð!Ð!r(   c                	óX  € ^ RI HpHp V P                  4       w  rEV! VP	                  V4      V4      '       dr   VP                  4       '       dZ   V! VP                  V4      V4      '       d   \        V) V4      # V! VP                  V4      V4      '       d   \        V) V4      ) # R# R# R# )r   )ÚaskÚQN)	Úsympy.assumptions.askr�   r‘   Úas_base_expÚintegerrk   Úevenr   Úodd)r,   Úassumptionsr�   r‘   rA   rB   s   &&    r%   Ú_eval_refineÚPow._eval_refineõ   sŽ   € ß0Ø×ÑÓ!‰ˆÙˆq�y‰y˜‹|˜[×)Ò)¨a×.HÑ.H×.JÒ.JÙ�1—6‘6˜!“9˜k×*Ò*Ü˜A˜2˜q“zÐ!Ù�Q—U‘U˜1“X˜{×+Ò+Ü˜Q˜B ›
�{Ð"ñ ,ñ /KÑ)r(   c                	ó<  € V P                  4       w  r#V\        P                  J d   W#,          V,          # R pVP                  '       d   ^pEMVP                  '       d   ^pEMVP
                  Eeú   ^ RIHpHpH	pH
p ^ RIHp	Hp
 ^ RIHp R pR pVP
                  '       Ed   VR8X  dk   V! V4      '       d\   VP                   RJ d0   \        P"                  V,          \%        V) W1,          4      ,          # VP                   RJ d   \%        W!) 4      # MgVP&                  '       dV   VP
                  '       d   \)        V4      pVP*                  '       d'   \)        V! V4      4      \        P,                  ,          p\)        V4      ^8  R8X  g   V^8X  d   ^pEMÐVP.                  '       d   ^pEMºV! V4      P.                  '       d   \)        V4      ^8  R8X  d   ^pEM‹V! V4      '       d»   V	! ^\        P0                  ,          \        P,                  ,          V,          V! \        P2                  W5! V4      ,          ^\        P0                  ,          ,          ,
          4      ,          4      pVP
                  '       d$   V! V! V4      V,
          4      ^ 8X  d
   V! V4      pMÅR pMÂ V	! ^\        P,                  ,          \        P0                  ,          V,          V! \        P2                  V! W:! V4      ,          4      ^,          \        P0                  ,          ,
          4      ,          4      pVP
                  '       d$   V! V! V4      V,
          4      ^ 8X  d
   V! V4      pMR p Ve   V\%        W#V,          4      ,          # R #   \4         d    R p L.i ; i)	N)r~   rQ   ÚrerP   ©r;   rN   )Úfloorc                óŒ   € \        V RR4      ^8X  d   R# V P                  4       w  rVP                  '       d   V^8X  d   R# R# R# )zJReturn True if the exponent has a literal 2 as the
denominator, else None.ÚqNT)ÚgetattrÚas_numer_denomrf   )rB   ÚnÚds   &  r%   Ú_halfÚPow._eval_power.<locals>._half  sA   € ô ˜1˜c 4Ó(¨AÔ-ÙØ×'Ñ'Ó)‘�Ø—<—<�< A¨¤FÙñ %+‘<r(   c                óz   €  V P                  ^RR7      pVP                  '       d   V# R#   \         d     R# i ; i)zHReturn ``e`` evaluated to a Number with 2 significant
digits, else None.T©ÚstrictN)Úevalfrj   r   )rB   Úrvs   & r%   Ú_n2ÚPow._eval_power.<locals>._n2  s>   € ðØŸ™ ¨4˜Ó0�BØ—|—|�|Ø!˜	ñ $øä)ô Úðús   ‚$+ §+ «:¹:TFrR   )r“   r	   rZ   rf   Úis_polarÚis_extended_realrv   r~   rQ   r›   rP   rp   r;   rN   Ú#sympy.functions.elementary.integersr�   Úis_negativer]   r   rl   rn   Úis_imaginaryrw   Úis_extended_nonnegativerx   ÚHalfr   )r,   ÚexptrA   rB   rƒ   r~   rQ   r›   rP   r;   rN   r�   r¤   r«   s   &&            r%   ry   ÚPow._eval_powerþ   sŸ  € Ø×ÑÓ!‰ˆØ”—‘‹:Ø•D˜4•<ÐàˆØ�?�?ˆ?ØŠAØ�Z�ZˆZØŠAØ×ÑÓ+ßNÓNßGÝAò òð ×!×!Ñ!ð ˜”7á˜T—{’{ØŸ=™=¨DÓ0Ü#$§=¡=°$Õ#6´s¸A¸2¸q½v³Õ#FÐFØŸ]™]¨eÓ3Ü#& q¨%£=Ð0øØ—Y—Y�YØ×)×)Ð)Ü ›F˜Ø—~—~�~Ü¡ 1£›J¤q§¡Õ6˜ä˜“F˜Q‘J 4Ô'¨1°¬6Ø’AØ×.×.Ð.Ø’AÙ˜“U×2×2Ð2¼¸A»À¹
ÀtÔ7KØ’AÙ˜4—[’[Ù˜AœaŸd™d�F¤1§?¡?Õ2°4Õ7¹ÜŸ™  3 q£6¥¨1¬Q¯T©T­6Õ!2Õ2ó94õ 4ó 5�Aà×)×)Ð)©c±$°q³'¸Aµ+Ó.>À!Ô.CÙ  ›G™à ˜øð

Ù˜AœaŸo™oÕ-¬a¯d©dÕ2°4Õ7ÙœaŸf™f¡r¨!¨C°«F­(£|°A¥~´a·d±dÕ':Õ:Ó;õ<ó =�Að ×)×)Ð)©c±$°q³'¸Aµ+Ó.>À!Ô.CÙ  ›G™à ™ð Š=Ø”S˜˜d�F“^Õ#Ð#ñ øô *ô Ø’Aðús   Ê+B=N Í)N ÎNÎNc                ól  € V P                   V P                  r2VP                  '       Ed
   VP                  '       Edõ   VP                  '       d   W!,          ^ 8X  d   \        P
                  # ^ RIHp VP                  '       dÆ   VP                  '       d´   VP                  '       d¢   \        V4      \        V4      \        V4      rvpVP                  4       pV^P8:  dV   Wh8¼  dP   VP                  4       ^,          V8¼  d4   \        V! V4      4      p	\        \        WYWi,          ,           V4      4      # \        \        WVV4      4      # ^RIHp
 \        V\         4      '       dA   VP                  '       d/   VP"                  '       d   V
! W!4      pV
! \!        W#RR7      V4      # \        V\         4      '       du   VP                  '       da   VP"                  '       dM   \        V4      P                  4       pV^P8:  d+   V! V4      p	Wš! W94      ,           pV
! \!        W#RR7      V4      # R# R# R# R# R# R# )aÏ  A dispatched function to compute `b^e \bmod q`, dispatched
by ``Mod``.

Notes
=====

Algorithms:

1. For unevaluated integer power, use built-in ``pow`` function
with 3 arguments, if powers are not too large wrt base.

2. For very large powers, use totient reduction if $e \ge \log(m)$.
Bound on m, is for safe factorization memory wise i.e. $m^{1/4}$.
For pollard-rho to be faster than built-in pow $\log(e) > m^{1/4}$
check is added.

3. For any unevaluated power found in `b` or `e`, the step 2
will be recursed down to the base and the exponent
such that the $b \bmod q$ becomes the new base and
$\phi(q) + e \bmod \phi(q)$ becomes the new exponent, and then
the computation for the reduced expression can be done.
)Útotient)ÚModF©rS   N)r4   r;   rf   Úis_positiver	   r^   Ú%sympy.functions.combinatorial.numbersr·   rg   ÚintÚ
bit_lengthÚIntegerÚpowÚmodr¸   rU   r   rh   )r,   rŸ   r4   r;   r·   rA   rB   ÚmÚmbÚphir¸   r½   s   &&          r%   Ú	_eval_ModÚPow._eval_ModR  s‡  € ð0 —I‘I˜tŸx™xˆcà�>�>‰>˜cŸoŸo™oØ�|�|ˆ| ¥¨A¤Ü—v‘v�åEà��ˆ 3§>§> >°a·l·l°lÜ˜d›)¤S¨£X¬s°1«v�a�Ø—\‘\“^�Ø˜”8 ¤¨A¯L©L«N¸AÕ,=ÀÔ,BÜ™g a›j›/�CÜ"¤3 q°µ­+°qÓ#9Ó:Ð:Üœs 1¨›|Ó,Ð,å ä˜$¤×$Ò$¨¯¯¨¸T¿^¿^¸^Ù˜4“|�Ùœ3˜t°5Ô9¸1Ó=Ð=ä˜#œs×#Ò#¨¯¯¨¸3¿=¿=¸=Ü  ›V×.Ñ.Ó0�
ð  Ô#Ù! !›*�CØ  C£Õ-�CÙœs 4°uÔ=¸qÓAÐAñ $ñ <I©Ñ#ñ) .‰>r(   c                	ó¦   € V P                   P                  '       d5   V P                   P                  '       d   V P                  P                  # R # R # r*   )r;   rf   rº   r4   rl   r+   s   &r%   Ú_eval_is_evenÚPow._eval_is_evenŠ  s9   € Ø�8‰8××Ð 4§8¡8×#7×#7Ð#7Ø—9‘9×$Ñ$Ð$ñ $8Ñr(   c                	óT   € \         P                  V 4      pVR J d   V P                  # V# ©T)r   Ú_eval_is_extended_negativer_   )r,   Úext_negs   & r%   Ú_eval_is_negativeÚPow._eval_is_negativeŽ  s(   € Ü×0Ñ0°Ó6ˆØ�d‹?Ø—>‘>Ð!Øˆr(   c                	óF  € V P                   V P                  8X  d!   V P                   P                  '       d   R # R# V P                   P                  '       d!   V P                  P                  '       d   R # R# V P                   P
                  '       d?   V P                  P                  '       d   R # V P                  P                  '       d   R# R# V P                   P                  '       d5   V P                  P                  '       d   V P                  P                  # R# V P                   P                  '       d!   V P                  P                  '       d   R# R# V P                   P                  '       d¨   V P                  P                  '       dL   V P                  ^,          pVP                  '       d   R # VP                  '       d   VP                  RJ d   R# V P                  P                  '       d#   ^ RIHp V! V P                   4      P                  # R# R# )TFrM   N)r4   r;   r²   rº   Úis_realÚis_extended_negativerl   rm   Úis_zeror®   Úis_extended_nonpositiver±   rf   rp   rN   )r,   rÁ   rN   s   &  r%   Ú_eval_is_extended_positiveÚPow._eval_is_extended_positive”  se  € Ø�9‰9˜Ÿ™Ô Ø�y‰y×0×0Ð0Ùñ 1à�Y‰Y×"×"Ð"Ø�x‰x××ÐÙñ  à�Y‰Y×+×+Ð+Ø�x‰x××ÐÙØ�x‰x��ˆÙñ à�Y‰Y××ÐØ�x‰x×(×(Ð(Ø—x‘x×'Ñ'Ð'ñ )à�Y‰Y×.×.Ð.Ø�x‰x��ˆÙñ à�Y‰Y×#×#Ð#Ø�x‰x×"×"Ð"Ø—H‘H˜q•L�Ø—9—9�9ÙØ—<—<�< A§I¡I°Ó$6Ù Ø�x‰x×$×$Ð$ÝFÙ˜4Ÿ9™9“~×2Ñ2Ð2ñ %ñ $r(   c                	óú  € V P                   \        P                  J d;   V P                  P                  '       g   V P                  P
                  '       d   R # V P                  P                  '       d[   V P                   P                  '       d   V P                  P                  '       d   R# V P                   P                  '       d   R # R# V P                  P                  '       d!   V P                   P
                  '       d   R # R# V P                  P                  '       d!   V P                   P
                  '       d   R # R# V P                  P                  '       d!   V P                   P                  '       d   R # R# V P                  P                  '       d!   V P                   P                  '       d   R # R# V P                  P
                  '       d!   V P                   P                  '       d   R # R# R# ©FTN)r;   r	   r³   r4   Ú
is_complexr®   rÑ   rm   r_   rl   Úis_extended_positiverÒ   r²   rÓ   r+   s   &r%   rË   ÚPow._eval_is_extended_negative±  sH  € Ø�8‰8”q—v‘vÓØ�y‰y×#×#Ð# t§y¡y×'A×'AÐ'AÙØ�9‰9×)×)Ð)Ø�x‰x��ˆ 4§9¡9×#6×#6Ð#6ÙØ�x‰x××ÐÙñ  à�Y‰Y×+×+Ð+Ø�x‰x×(×(Ð(Ùñ )à�Y‰Y××ÐØ�x‰x×(×(Ð(Ùñ )à�Y‰Y×.×.Ð.Ø�x‰x×/×/Ð/Ùñ 0à�Y‰Y×.×.Ð.Ø�x‰x××ÐÙñ  à�Y‰Y×'×'Ð'Ø�x‰x××ÐÙñ  ñ (r(   c                	óv  € V P                   P                  '       d?   V P                  P                  '       d   R # V P                  P                  '       d   R# R# V P                   \
        P                  8X  d   V P                  \
        P                  J # V P                   P                  RJ EdM   V P                   P                  '       d   V P                  P                  '       d   R# V P                  P                  '       d   V P                   P                  # V P                  P                  '       d   R# V P                  P                  '       d¥   V P                  P                  '       d‡   ^\        V P                   4      ,
          P                  '       d   V P                  P                  # ^\        V P                   4      ,
          P                  '       d   V P                  P                  # R# R# R# V P                   P                  '       d!   V P                  P                  '       d   R# R# R# ©TFN)r4   rÒ   r;   rÙ   rÓ   r	   ra   ÚNegativeInfinityr_   r°   ro   Úis_nonnegativer®   rn   rÑ   r+   s   &r%   Ú_eval_is_zeroÚPow._eval_is_zeroÊ  st  € Ø�9‰9××ÐØ�x‰x×,×,Ð,ÙØ—‘×1×1Ð1Ùñ 2à�Y‰Yœ!Ÿ&™&Ô Ø—8‘8œq×1Ñ1Ð1Ð1Ø�Y‰Y×Ñ %Ô'Ø�y‰y×"×"Ð" t§x¡x×'9×'9Ð'9ÙØ—‘×%×%Ð%Ø—y‘y×,Ñ,Ð,Ø—‘×(×(Ð(ÙØ—‘×%×%Ð%¨$¯(©(×*C×*CÐ*CØœ˜DŸI™I›Õ&×<×<Ð<ØŸ8™8×8Ñ8Ð8Øœ#˜dŸi™i›.Õ(×>×>Ð>ØŸ8™8×8Ñ8Ð8ñ ?ñ +DÑ%ð
 �Y‰Y× × Ð  T§X¡X×%9×%9Ð%9áñ &:Ñ r(   c                	ó  € V P                   w  rVP                  '       d%   VP                  R J d   VP                  '       d   R # VP                  '       dO   VP                  '       d=   V\        P
                  J d   R# VP                  '       g   VP                  '       d   R# VP                  '       d}   VP                  '       dk   VP                  '       g   VP                  '       dG   \        V^,
          P                  4      '       d%   \        V^,           P                  4      '       d   R # VP                  '       d9   VP                  '       d'   V P                  ! V P                   !  pVP                  # VP                  '       d.   VP                  '       d   V^,
          P                  '       d   R # VP                  '       d2   VP                  '       d   V^,           P                  '       d   R # R# R# R# r×   )r-   Úis_rationalrf   rº   r	   r]   rÞ   r°   r_   r   rÒ   rj   Úfuncrg   )r,   rA   rB   Úchecks   &   r%   Ú_eval_is_integerÚPow._eval_is_integerâ  s,  € Ø�y‰y‰ˆØ�=�=ˆ=Ø�|‰|˜uÓ$¨¯¯¨ÙØ�<�<ˆ<˜AŸLŸL˜LØ”A—M‘MÓ!ÙØ××Ð 1§=§= =ÙØ�<�<ˆ<˜AŸMŸM˜M¨q¯{¯{¨{¸a¿l¿l¸lÜ˜!˜a�%Ÿ™×)Ò)¬i¸¸Q½¿¹×.HÒ.HÙØ�;�;ˆ;˜1Ÿ;Ÿ;˜;Ø—I’I˜tŸy™yÑ)ˆEØ×#Ñ#Ð#Ø�=�=ˆ=˜QŸ]Ÿ]˜]°°Aµ×/B×/BÐ/BÙØ�=�=ˆ=˜QŸ]Ÿ]˜]°°Aµ×/B×/BÐ/BÙñ 0C™]‰=r(   c                	óH  € V P                   \        P                  J d‚   V P                  P                  '       d   R # V P                  P
                  '       dH   ^\        P                  ,          V P                  ,          \        P                  ,          P                  # ^ RI	H
pHp V P                   P                  pVfÝ   V P                   P                  V8X  d=   V P                   P                  P
                  '       d   V P                  P
                  # V P                   P                  \        8X  de   V P                   P                   \        P                  J d=   V P                   P                  P
                  '       d   V P                  P
                  # R# V P                  P                  pVf   R# V'       Ed   V'       Ed   V P                   P                  '       d   R # V P                   P                  '       d   V P                  P                  '       d   R # V P                  P                  '       d   V P                   P                   '       d   R # V P                  P                  '       d   V P                  P"                  '       d   R # V P                   P$                  '       d   V P                  P&                  '       d   R# V'       db   V P                  P$                  '       dF   V P                   P(                  RJ d,   \        V P                   V P                  ) 4      P                  # V P                   P
                  pV P                  P
                  pV'       Edg   V P                  P                  '       d?   V P                  P                  '       d   R # V P                  P*                  '       d   R# EMV'       d%   V! V P                   4      P
                  '       d   R # V P                  P,                  '       dr   V P                  P/                  4       w  rxV'       dM   VP0                  '       d;   \3        V P                   V,          V P                   V,          RR7      P                  # MSV P                   \        P                  ) \        P                  39   d$   V P                  ^,          P                  RJ d   R# V'       Ed   V'       Ed   V P                   \        P4                  J d   R # V P                  P7                  \        P                  4      pV'       d¿   V P                   P8                  '       df   VP8                  '       dT   V P                   P:                  '       d8   V P                   ^,
          P:                  '       d   VP:                  '       d   R# Wq! V P                   4      ,          \        P                  ,          P                  p	V	e   V	# VRJ d¦   V'       dœ   \=        V P                  \>        4      '       d   V P                  P@                  ^8X  d   R# ^ RI!H"p
 V
! V P                   4      V P                  ,          \        P                  ,          pVPF                  '       d   VP                  # R# R# R# )T)rN   r;   NFr¹   ©r~   )$r4   r	   ra   r;   r®   r±   rw   rx   rl   rp   rN   rã   r   rÙ   r²   rf   Úis_extended_nonzerorÞ   rÑ   Úis_RationalrÒ   rm   ru   Úas_coeff_Addrg   ÚMulr]   Úcoeffrâ   Ú
is_nonzerorU   ÚRationalÚprv   r~   rØ   )r,   rN   r;   Úreal_bÚreal_eÚim_bÚim_er   ÚaÚokr~   Úis   &           r%   Ú_eval_is_extended_realÚPow._eval_is_extended_real÷  sÿ  € Ø�9‰9œŸ™ÓØ�x‰x×(×(Ð(ÙØ—‘×&×&Ð&Øœ!Ÿ/™/Õ)¨$¯(©(Õ2´1·4±4Õ7×@Ñ@Ð@çCØ—‘×+Ñ+ˆØŠ>Ø�y‰y�~‰~ Ô$¨¯©¯©×)C×)CÐ)CØ—x‘x×,Ñ,Ð,Ø�y‰y�~‰~¤Ô$¨¯©¯©¼1¿6¹6Ó)AÀdÇiÁiÇmÁm×F`×F`ÐF`Ø—x‘x×,Ñ,Ð,ÙØ—‘×*Ñ*ˆØŠ>Ùßˆ6—f�fØ�y‰y×-×-Ð-ÙØ—‘×2×2Ð2°t·x±x×7W×7WÐ7WÙØ—‘×$×$Ð$¨¯©×)F×)FÐ)FÙØ—‘×$×$Ð$¨¯©×)@×)@Ð)@ÙØ—‘×/×/Ð/Ø—8‘8×'×'Ð'Ù ß�d—h‘h×3×3Ð3¸¿	¹	×8IÑ8IÈUÓ8RÜ�t—y‘y 4§8¡8 )Ó,×=Ñ=Ð=Ø�y‰y×%Ñ%ˆØ�x‰x×$Ñ$ˆßˆ4Ø�x‰x×"×"Ð"Ø—8‘8×#×#Ð#ÙØ—X‘X—_—_�_Ù ñ %ç™#˜dŸi™i›.×5×5Ð5ÙØ—‘——�Ø—x‘x×,Ñ,Ó.‘�ß˜ŸŸ˜ÜØŸ	™	 1� d§i¡i°¥l¸UôDßDTÑDTðUøà—‘¤§¡Ð/´·±ÐAÔAØ—H‘H˜Q•J×*Ñ*¨eÓ3Ù ßˆ6—d�dØ�y‰yœAŸM™MÓ)ÙØ—‘—‘œqŸ™Ó/ˆAßØ—9‘9×(×(Ð(¨Q¯]¯]¨]Ø—y‘y×+×+Ð+°·±¸Qµ×0J×0JÐ0JÈqÏ|Ï|È|Ù$Ø˜˜DŸI™I›Õ&¤q§t¡tÕ+×7Ñ7�Ø’>Ø�Ià�U‹?ŸvÜ˜$Ÿ(™(¤H×-Ò-°$·(±(·*±*À´/ÙÝ@Ù�D—I‘I“˜tŸx™xÕ'¬¯©Õ,ˆAØ�|�|ˆ|Ø—|‘|Ð#ñ ñ  &‰?r(   c                	ó~  € V P                   \        P                  8X  d6   \        V P                  P
                  V P                  P                  .4      # \        ;QJ d&    R  V P                   4       F  '       d   K   RM	  RM! R  V P                   4       4      '       d   V P                  4       '       d   R# R# R# )c              3  ó8   "  € T F  qP                   x € K  	  R # 5ir*   )rØ   )Ú.0rõ   s   & r%   Ú	<genexpr>Ú'Pow._eval_is_complex.<locals>.<genexpr>B  s   é € Ð/¡Y �|Ž|£Yùó   ‚FTN)
r4   r	   ra   r   r;   rØ   rÑ   Úallr-   Ú_eval_is_finiter+   s   &r%   Ú_eval_is_complexÚPow._eval_is_complex=  sy   € à�9‰9œŸ™ÔÜ˜TŸX™X×0Ñ0°$·(±(×2OÑ2OÐPÓQÐQç‹3Ñ/ T§Y¢YÓ/�3�3Š3Ñ/ T§Y¢YÓ/×/Ò/°D×4HÑ4H×4JÒ4JÙñ 5KÑ/r(   c                	ó  € V P                   P                  R J d   R # V P                   P                  '       d;   V P                  P                  '       d   V P                  P
                  pVe   V# R# V P                   \        P                  8X  dh   ^V P                  ,          \        P                  \        P                  ,          ,          pVP                  '       d   R # VP
                  '       d   R# R# V P                  P                  '       d)   ^ RIHp V! V P                   4      P                  pVe   R # V P                   P                  '       d¶   V P                  P                  '       dš   V P                   P                  '       d   R # V P                  P                  pV'       g   V# V P                  P                  '       d   R # ^V P                  ,          P                  pV'       d   V P                   P                   # V# V P                   P                  R J dZ   ^ RIHp V! V P                   4      V P                  ,          \        P                  ,          p^V,          P
                  p	V	e   V	# R# R# )FNTrM   rè   )r4   r|   r±   r;   rf   rm   r	   ra   rx   rw   rl   rp   rN   r®   rº   râ   r°   rv   r~   )
r,   r–   ÚfrN   ÚimlogÚratÚhalfr~   r÷   Úisodds
   &         r%   Ú_eval_is_imaginaryÚPow._eval_is_imaginaryE  s§  € Ø�9‰9×#Ñ# uÓ,Ùà�9‰9×!×!Ð!Ø�x‰x×"×"Ð"Ø—h‘h—o‘o�Ø’?Ø�JÙà�9‰9œŸ™ÔØ�D—H‘H•¤§¡¤Q§_¡_Õ 4Õ5ˆAà�y�yˆyÙà�x�xˆxÙÙà�8‰8× × Ð ÝBÙ˜Ÿ	™	“N×/Ñ/ˆEØÒ Ùà�9‰9×%×%Ð%¨$¯(©(×*C×*CÐ*CØ�y‰y×$×$Ð$Ùà—h‘h×*Ñ*�ßØ�JØ—8‘8×&×&Ð&Ù à˜dŸh™h�J×2Ñ2�DßØ#Ÿy™y×4Ñ4Ð4Ø�Kà�9‰9×%Ñ%¨Ó.Ý@Ù�D—I‘I“˜tŸx™xÕ'¬¯©Õ,ˆAØ�q•S—L‘LˆEØÒ Ø�ñ !ñ	 /r(   c                	óZ  € V P                   P                  '       d�   V P                   P                  '       d   V P                  P                  # V P                   P
                  '       d   V P                  P                  '       d   R # V P                  \        P                  J d   R # R# R# ©TN)r;   rf   rº   r4   rm   rÞ   r	   r]   r+   s   &r%   Ú_eval_is_oddÚPow._eval_is_oddv  sq   € Ø�8‰8××ÐØ�x‰x×#×#Ð#Ø—y‘y×'Ñ'Ð'Ø—‘×(×(Ð(¨T¯Y©Y×-=×-=Ð-=ÙØ—‘œaŸm™mÓ+Ùñ ,ñ r(   c                	ó  € V P                   P                  '       dY   V P                  P                  '       d   R # V P                  P                  '       g   V P                  P
                  '       d   R# V P                  P                  pVf   R# V P                   P                  pVf   R# V'       dP   V'       dF   V P                   P                  '       g&   \        V P                  P                  4      '       d   R# R# R# R# r×   )	r;   r°   r4   rÒ   ro   rî   r_   rÞ   r   )r,   Úc1Úc2s   &  r%   r  ÚPow._eval_is_finite  s®   € Ø�8‰8××ÐØ�y‰y× × Ð ÙØ�y‰y×$×$Ð$¨¯	©	×(<×(<Ð(<ÙØ�Y‰Y× Ñ ˆØŠ:ÙØ�X‰X×ÑˆØŠ:Ùß—"Ø�x‰x×&×&Ð&¬)°D·I±I×4EÑ4E×*FÒ*FÙñ +Gñ ‰2r(   c                óÈ   € V P                   P                  '       dF   V P                  P                  '       d(   V P                  ^,
          P                  '       d   R# R# R# R# )z=
An integer raised to the n(>=2)-th power cannot be a prime.
FN)r4   rf   r;   rº   r+   s   &r%   Ú_eval_is_primeÚPow._eval_is_prime�  sE   € ð �9‰9××Ð D§H¡H×$7×$7Ð$7¸T¿X¹XÈ½\×<V×<VÐ<VÙñ =WÑ$7Ñr(   c                óÌ  € V P                   P                  '       dÈ   V P                  P                  '       dª   V P                   ^,
          P                  '       d$   V P                  ^,
          P                  '       g\   V P                   ^,           P                  '       d?   V P                  P                  '       d!   V P                  P
                  '       d   R# R# R# R# R# R# )zC
A power is composite if both base and exponent are greater than 1
TN)r4   rf   r;   rº   r°   rl   r+   s   &r%   Ú_eval_is_compositeÚPow._eval_is_composite–  s”   € ð �I‰I× × Ð  T§X¡X×%8×%8Ð%8Ø�i‰i˜!�m×(×(Ð(¨d¯h©h¸­l×-G×-GÐ-GØ�Y‰Y˜�]×'×'Ð'¨D¯H©H×,@×,@Ð,@ÀTÇXÁX×EU×EUÐEUÙñ FVÑ,@Ñ'ñ &9Ñ r(   c                	ó.   € V P                   P                  # r*   )r4   r­   r+   s   &r%   Ú_eval_is_polarÚPow._eval_is_polarŸ  s   € Ø�y‰y×!Ñ!Ð!r(   c                	ó¶	  € ^ RI Hp \        V P                  V4      '       dj   V P                  P                  W4      pV P                  P                  W4      p\        WS4      '       d   VP                  V4      # V P                  WE4      # ^ RIHpH	p R pWP                  8X  g&   W8X  dŠ   V P                  \        P                  8X  dk   VP                  '       d8   \        V\        4      '       d"   V! V P                  P                  W4      4      # W P                  P                  W4      ,          # \        WP                  4      '       dV   V P                  VP                  8X  d;   V! V P                  VP                  4      p	V	P                  '       d   \!        W)4      # \        WP                  4      '       Edò   V P                  VP                  8X  EdÖ   V P                  P"                  RJ d�   V P                  P%                  \&        RR7      p
VP                  P%                  \&        RR7      pV! W«V4      w  rÍpV'       d8   V P                  W-4      pVe!   \)        V\!        VP                  V4      4      pV# EM,VP                  p. p. pVP+                  4       pV P                  P,                   Fž  pVP                  W4      pVP+                  4       p
V! W«V4      w  rÍpV'       d/   VP/                  W-,          4       Ve   VP/                  V4       Kf  VP0                  '       g   VP2                  '       g    R# VP/                  V4       K   	  V'       dM   \5        V!  pTP/                  V^8w  d   \!        V P                  VRR7      MV P                  4       \)        V!  # \        W4      '       g2   VP6                  '       Ed   VP                  \        P                  J dã   V P                  P8                  '       dÅ   V P                  P:                  '       d§   VP                  P%                  \&        RR7      p
V P                  V! V P                  4      ,          P%                  \&        RR7      pV! W«V4      w  rÍpV'       d8   V P                  W-4      pVe!   \)        V\!        VP                  V4      4      pV# R# R# R# R# R# )r   rI   rœ   c                óì  € V w  r4Vw  rVWF8X  dé   VP                   '       d   W5,          p \        VRR7       RpW‡R3# \        V\        4      '       g   V3p\        ;QJ d    R V 4       F  '       d   K   RM	  RM! R V 4       4      '       g   R#  \        \        V4      \        V4      4      w  r{V^ 8  d#   V^ 8w  d   V^,          pV\        V4      ,          pV^ 8X  d   RpM\        V.VO5!  pRW|3# R#   \         d_    TP                  4       w  ršT	P                  ;'       d    T
P
                  ;'       g!    T	P                  ;'       d    T
P                  p EL(i ; i  \         d     L{i ; i)aš  Return (bool, pow, remainder_pow) where, if bool is True, then the
exponent of Pow `old` will combine with `pow` so the substitution
is valid, otherwise bool will be False.

For noncommutative objects, `pow` will be an integer, and a factor
`Pow(old.base, remainder_pow)` needs to be included. If there is
no such factor, None is returned. For commutative objects,
remainder_pow is always None.

cti are the coefficient and terms of an exponent of self or old
In this _eval_subs routine a change like (b**(2*x)).subs(b**x, y)
will give y**2 since (b**x)**2 == b**(2*x); if that equality does
not hold then the substitution should not occur so `bool` will be
False.

Fr§   TNc              3  ó8   "  € T F  qP                   x € K  	  R # 5ir*   )rf   )rü   Úterms   & r%   rý   Ú1Pow._eval_subs.<locals>._check.<locals>.<genexpr>Ò  s   é € ÐB¹6°4Ÿž»6ùrÿ   )FNN)r|   r   Ú
ValueErrorr“   rº   rÐ   rÞ   rU   Útupler   Údivmodrì   )Úct1Úct2ÚoldÚcoeff1Úterms1Úcoeff2Úterms2r¿   ÚcombinesrA   rB   Ú	remainderÚremainder_pows   &&&          r%   Ú_checkÚPow._eval_subs.<locals>._check®  s]  € ð" !‰NˆFØ ‰NˆFØÔØ×%×%Ð%à �-�CðhÜ˜s¨5Õ1Ø#'˜ð $¨$Ð.Ð.ô & f¬e×4Ò4Ø"( ˜ß›3ÑB¹6ÓBŸ3Ÿ3š3ÑB¹6ÓB×BÒBØ0Ð0ðä)/´°v³ÄÀvÃÓ)O™˜Ø œ7 y°A¤~Ø 1�H˜CØ%¬°«Õ7˜Ià$¨œ>Ø,0™Mä,/°	Ð,C¸FÓ,C˜Mà# SÐ7Ð7ð
 %Ð$øô= &ô hØ"Ÿ™Ó0™˜à#$§=¡=×#>Ð#>°Q·Y±Y×#gÐ#gÀ!×BRÑBR×BgÐBgÐWX×WgÑWg›ðhûô4 &ô áðús6   ªC9 ÂA#E% Ã9/E"Ä)E"Ä=E"ÅE"Å!E"Å%E3Å2E3F)Úas_AddNr¹   )rb   rJ   rU   r;   r4   ÚsubsÚ__rpow__rã   rp   rN   r	   ra   Úis_Functionr   Ú_subsrj   r   ru   Úas_independentÚSymbolrì   Úas_coeff_mulr-   Úappendr|   rf   ÚAddÚis_Powr®   rº   )r,   r'  ÚnewrJ   rA   rB   r;   rN   r/  Úlr%  r&  rö   r¿   r.  ÚresultÚoargÚnew_lÚo_alrõ   ÚnewaÚexpos   &&&                   r%   Ú
_eval_subsÚPow._eval_subs¢  s•  € ÝAä�d—h‘h ×,Ò,Ø—	‘	—‘˜sÓ(ˆAØ—‘—‘˜cÓ'ˆAÜ˜!×)Ò)Ø—z‘z !“}Ð$Ø—9‘9˜Q“?Ð"çCò8	%ðt —)‘)Ô ¤
¨t¯y©y¼A¿F¹FÔ/BØ��ˆ¤:¨c´8×#<Ò#<Ù˜4Ÿ8™8Ÿ>™>¨#Ó3Ó4Ð4àŸH™HŸN™N¨3Ó4Õ4Ð4ô �cŸ9™9×%Ò%¨$¯(©(°c·g±gÔ*=Ù�D—I‘I˜sŸx™xÓ(ˆAØ�{�{ˆ{Ü˜3“{Ð"ä�cŸ9™9×%Ó%¨$¯)©)°s·x±xÕ*?Ø�x‰x�‰ %Ó'Ø—h‘h×-Ñ-¬f¸UÐ-ÓC�Ø—g‘g×,Ñ,¬V¸EÐ,ÓB�Ù)/°¸#Ó)>Ñ&�˜ßà!ŸY™Y sÓ0�FØ$Ò0Ü!$ V¬S°·±¸=Ó-IÓ!J˜Ø!�Mñ ð —w‘w�Ø�Ø�Ø×'Ñ'Ó)�ØŸ™Ÿœ�AØŸ7™7 3Ó,�DØ×+Ñ+Ó-�CÙ-3°C¸cÓ-BÑ*�B˜]ßØŸ™ S¥XÔ.Ø(Ò4Ø ŸK™K¨Ô6Ù Ø ×/×/Ð/¸¿¿¸ò Ø—K‘K Ö%ñ '÷ Ü ™:�DØ—L‘LÈÐQRÌ¤ T§Y¡Y°¸uÕ!EÐX\×XaÑXaÔbÜ ™;Ð&ä�s× Ò  S§Z§Z¡Z°C·H±HÄÇÁÓ4FÈTÏXÉX×Mf×MfÐMfÐko×ktÑkt÷  lA÷  lAð  lAØ—'‘'×(Ñ(¬¸Ð(Ó>ˆCØ—8‘8™C §	¡	›NÕ*×:Ñ:Ü˜uð ;ó &ˆCá%+¨C°cÓ%:Ñ"ˆB�]ßØŸ™ 3Ó,�Ø Ò,Ü  ¬¨S¯X©X°}Ó)EÓF�FØ�ñ	 ñ lAÑMfÑ4F¡Zr(   c                ó¼   € V P                   w  rVP                  '       d;   VP                  ^8X  d*   VP                  ^8w  d   \	        VP                  4      V) 3# W3# )az  Return base and exp of self.

Explanation
===========

If base a Rational less than 1, then return 1/Rational, -exp.
If this extra processing is not needed, the base and exp
properties will give the raw arguments.

Examples
========

>>> from sympy import Pow, S
>>> p = Pow(S.Half, 2, evaluate=False)
>>> p.as_base_exp()
(2, -2)
>>> p.args
(1/2, 2)
>>> p.base, p.exp
(1/2, 2)

)r-   rê   rð   rŸ   r¾   )r,   rA   rB   s   &  r%   r“   ÚPow.as_base_exp#  sH   € ð. �y‰y‰ˆØ�=�=ˆ=˜QŸS™S AœX¨!¯#©#°¬(Ü˜1Ÿ3™3“< ! Ð#Ð#Øˆtˆr(   c                	óf  € ^ RI Hp V P                  P                  V P                  P
                  r2V'       d$   V! V P                  4      V P                  ,          # V'       d$   V P                  V! V P                  4      ,          # VRJ d$   VRJ d   \        V 4      pW@8w  d	   V! V4      # R# R# R# )r   )ÚadjointFN)rv   rI  r;   rf   r4   rº   r   )r,   rI  r÷   rð   Úexpandeds   &    r%   Ú_eval_adjointÚPow._eval_adjoint?  sŠ   € Ý@Ø�x‰x×"Ñ" D§I¡I×$9Ñ$9ˆ1ßÙ˜4Ÿ9™9Ó% t§x¡xÕ/Ð/ßØ—9‘9™g d§h¡hÓ/Õ/Ð/Ø�‹:˜!˜u›*Ü% dÓ+ˆHØÔÙ˜xÓ(Ð(ñ  ñ %‰:r(   c                	ó†  € ^ RI Hp V P                  P                  V P                  P
                  r2V'       d$   V! V P                  4      V P                  ,          # V'       d$   V P                  V! V P                  4      ,          # VRJ d    VRJ d   \        V 4      pW@8w  d	   V! V4      # V P                  '       d   V # R# )r   )Ú	conjugateFN)rv   rN  r;   rf   r4   rº   r   r®   )r,   r   r÷   rð   rJ  s   &    r%   Ú_eval_conjugateÚPow._eval_conjugateK  s‘   € ÝGØ�x‰x×"Ñ" D§I¡I×$9Ñ$9ˆ1ßÙ�T—Y‘Y“< §¡Õ)Ð)ßØ—9‘9™a §¡›kÕ)Ð)Ø�‹:˜!˜u›*Ü% dÓ+ˆHØÔÙ˜“{Ð"Ø× × Ð ØˆKñ !r(   c                	óD  € ^ RI Hp V P                  \        P                  8X  d9   V P                  \        P                  V P                  P                  4       4      # V P                  P                  V P                  P                  ;'       g    V P                  P                  r2V'       d   V P                  V P                  ,          # V'       d$   V! V P                  4      V P                  ,          # VRJ d$   VRJ d   \        V 4      pW@8w  d	   V! V4      # R# R# R# )r   )Ú	transposeFN)rv   rR  r4   r	   ra   rã   r;   rf   rØ   ro   r   )r,   rR  r÷   rð   rJ  s   &    r%   Ú_eval_transposeÚPow._eval_transposeY  sÉ   € ÝBØ�9‰9œŸ™ÔØ—9‘9œQŸV™V T§X¡X×%7Ñ%7Ó%9Ó:Ð:Ø�x‰x×"Ñ" T§Y¡Y×%9Ñ%9×%RÐ%R¸T¿Y¹Y×=RÑ=Rˆ1ßØ—9‘9˜dŸh™hÕ&Ð&ßÙ˜TŸY™YÓ'¨¯©Õ1Ð1Ø�‹:˜!˜u›*Ü% dÓ+ˆHØÔÙ  Ó*Ð*ñ  ñ %‰:r(   c           	     óP  € V P                   pV P                  pV\        P                  8X  d]   ^ RIHp \        W44      '       dF   VP                  '       d4   ^ RIH	p V! V P                  W#P                  4      .VP                  O5!  # VP                  '       Ed    VP                  RR4      '       g'   VP                  RJ g   VP!                  4       '       dÂ   VP                  '       d2   \#        VP$                   Uu. uF  q`P                  W&4      NK  	  up!  # VP                  '       dm   \'        VP$                  R RR7      w  rxV'       dJ   \#        V Uu. uF  q`P                  W&4      NK  	  up!  V\(        P*                  ! V4      ,          ,          # V # u upi u upi )za**(n + m) -> a**n*a**m)ÚSum)ÚProductÚforceFc                ó   € V P                   # r*   ©r|   ©Úxs   &r%   Ú<lambda>Ú,Pow._eval_expand_power_exp.<locals>.<lambda>u  s
   € ¨q×/?Ò/?r(   T©Úbinary)r4   r;   r	   ra   Úsympy.concrete.summationsrV  rU   r|   Úsympy.concrete.productsrW  rã   ÚfunctionÚlimitsru   ÚgetrÒ   Ú_all_nonneg_or_nonpposrì   r-   r   r:  Ú
_from_args)	r,   ÚhintsrA   rB   rV  rW  r\  r   Úncs	   &,       r%   Ú_eval_expand_power_expÚPow._eval_expand_power_expg  s,  € à�I‰IˆØ�H‰HˆØ”—‘Œ;Ý5Ü˜!×!Ò! a×&6×&6Ð&6Ý;Ù˜tŸy™y¨¯J©JÓ7ÐC¸!¿(¹(ÓCÐCØ�8�8‰8˜Ÿ™ 7¨E×2Ò2Ø—	‘	˜UÓ" a×&>Ñ&>×&@Ò&@Ø××ÐÜ°a·f²fÓ=±f°ŸY™Y qž_±fÑ=Ñ>Ð>Ø××ÐÜ˜QŸV™VÑ%?ÈÔM‘�ßÜ¹!Ó <¹!°Q§¡¨1¦¹!Ñ <ñ ØœSŸ^š^¨BÓ/Õ/õ0ð 0àˆùò >ùò !=s   Ä FÅF#c                ó�
  € VP                  RR4      pV P                  pV P                  pVP                  '       g   V # VP	                  RR7      w  rVV'       dé   V Uu. uF)  p\        VR4      '       d   VP                  ! R/ VB MTNK+  	  ppVP                  '       du   VP                  '       d   \        Wd,          !  pM1\        VRRR1,           Uu. uF  qwR,          NK  	  upV) ,          !  pV'       d   V\        V!  V,          ,          pV# V'       g   V P                  \        V!  VRR7      # \        V!  .p\        VR RR	7      w  ršR
 p\        W«4      pVR,          pWœR,          ,          p	VR,          pV\        P                  ,          pV'       Ed   \        P                  p\        V4      ^,          pV^ 8X  d   MîV^8X  d   V	P                  V4       MÕV^8X  d`   V'       d8   VP!                  4       ) pV\        P"                  Jd   VP                  V4       M�VP                  \        P$                  4       MoV'       d8   VP!                  4       ) pV\        P"                  Jd   VP                  V4       MVP                  \        P$                  4       V	P                  V4       ?V'       g   VP&                  '       d   WÞ,           V	,           pTp	EMtVP                  '       d   Q h\        V4      ^8”  d¦   \        P"                  pV	'       g2   V^ ,          P(                  '       d   VVP!                  ^ 4      ,          p\        V4      ^,          '       d   V) pV F  pVP                  V) 4       K  	  V\        P"                  Jd   V	P                  V4       M V'       dˆ   V	'       d€   V^ ,          P(                  '       dU   V^ ,          \        P$                  Jd:   V	P                  \        P$                  4       VP                  V^ ,          ) 4       M#V	P+                  V4       MV	P+                  V4       ?TpW–,          p	\        P"                  pV'       d•   VP,                  '       dS   \        VR RR	7      w  pp\        V Uu. uF,  q0P                  VP                  ! VP.                  !  V4      NK.  	  up!  pT\        V Uu. uF  q0P                  W4RR7      NK  	  up!  ,          pV	'       d"   W€P                  \        V	!  VRR7      ,          pV# u upi u upi u upi u upi )z(a*b)**n -> a**n * b**nrX  F)Úsplit_1Ú_eval_expand_power_baseNr¹   c                ó   € V P                   R J # )F)r®   r[  s   &r%   r]  Ú-Pow._eval_expand_power_base.<locals>.<lambda>�  s   € °!×2DÑ2DÈÑ2Mr(   Tr_  c                ó¬   € V \         P                  J d   \         P                  # V P                  pV'       d   R # Vf   \        V P                  4      # R# r  )r	   rw   r­   r   r²   )r\  Úpolars   & r%   ÚpredÚ)Pow._eval_expand_power_base.<locals>.predŸ  sB   € Ø”A—O‘OÓ#Ü—‘Ð&Ø—J‘JˆEßÙØŠ}Ü! !×";Ñ";Ó<Ð<ñ r(   c                ó’   € V P                   ;'       d5    V P                  P                  ;'       d    V P                  P                  # r*   )r;  r;   rê   r4   rh   r[  s   &r%   r]  rp  í  s;   € °A·H±H÷ 5;ð 5;Ø—E‘E×%Ñ%÷5;ð 5;Ø*+¯&©&×*:Ñ*:ð5;r(   r#   rR   )re  r4   r;   ri   Úargs_cncÚhasattrrn  rg   rº   rì   rã   r   r	   rw   Úlenr9  Úpopr\   r]   rf   rj   Úextendrê   r-   )r,   rh  rX  rA   rB   Úcargsri  r÷   rª   ÚotherÚ
maybe_realrs  ÚsiftedÚnonnegÚnegÚimagÚIÚnonnÚor¢   Únpows   &,                   r%   rn  ÚPow._eval_expand_power_base{  s  € à—	‘	˜' 5Ó)ˆà�I‰IˆØ�H‰HˆØ�x�xˆxØˆKà—J‘J u�JÓ-‰	ˆ÷
 ñ óá�Aô ˜1Ð7×8Ò8ð ×+Ò+Ñ4¨eÒ4Ø>?ò@áð ð ð �|�|ˆ|Ø—=—=�=Ü˜b�d™‘Bä¨b±°2°®hÓ7©h¨ "Ÿu˜u©hÑ7¸¸Õ:Ñ;�BßØœ#˜u™+ q�.Õ(�BØ�	çØ—y‘y¤ b¡¨1°u�yÓ=Ð=ä�r‘(�ˆBô ! Ñ(MØôÑˆò	=ô �jÓ'ˆØ˜•ˆØ˜•ÕˆØ�U�mˆØ”a—o‘oÕ&ˆßˆ4Ü—‘ˆAÜ�D“	˜A•ˆAØ�AŒvØØ�a”Ø—‘˜Q•Ø�a”ßØŸG™G›I˜:�DØ¤1§5¡5Ó(ØŸ™ dÔ+øà—J‘JœqŸ}™}Õ-çØŸG™G›I˜:�DØ¤1§5¡5Ó(ØŸ™ dÔ+øà—J‘JœqŸ}™}Ô-Ø—‘˜Q”Ø÷ �A—L—L�Là•L 5Õ(ˆEØŠEð —|—|�|Ð#Ð#ô �3‹x˜!Œ|Ü—E‘E�ß  Q¥×!1×!1Ð!1Ø˜Ÿ™ ›•O�AÜ�s“8˜a—<”<Ø˜�AÛ�AØ—M‘M 1 "Ö%ñ àœAŸE™E“>Ø—L‘L ”OøßŸØ�q•6×#×#Ð#¨¨A­´a·m±mÓ(CØ—L‘L¤§¡Ô/Ø—M‘M 3 q¥6 'Õ*à—L‘L Õ%à—‘˜SÔ!ØàˆEØ�KˆEä�U‰UˆßØ�}�}ˆ}Ü" 5ñ +;àô!‘��eô Á$ÓGÁ$¸QŸ9™9 Q§V¢V¨Q¯V©V¡_°aÖ8Á$ÑGÑH�Ø”#ÁÓGÁ¸AŸ	™	 !°˜	Ö7ÁÑGÑHÕHˆBßØ—)‘)œC ™K¨°U�)Ó;Õ;ˆBØˆ	ùòUùò 8ùò| HùÚGs   Á/T4ÃT9Ò"2T>Ó#U
c           	     ó®  € V P                   w  r#T pVP                  '       Ed.   VP                  ^ 8”  Ed   VP                  '       Ed	   VP                  '       gº   \        VP                  VP                  ,          4      pV'       g   V# V P                  W#V,
          4      . rFV P                  W%4      pVP                  '       d   VP                  4       p\        P                  ! V4       F  pVP                  W†,          4       K  	  \        V!  # \        V4      pVP                  '       EdF   . . r©VP                    F9  pVP                  '       d   V	P                  V4       K(  V
P                  V4       K;  	  V	'       d‚   \        V
!  p\        V	!  pV^8X  d(   \!        WÅ,          RR7      W\,          V,          ,           # \!        WÅ^,
          ,          RR7      p\#        WÎ,          RR7      W^,          V,          ,           # VP$                  '       Ed4   VP'                  4       w  rûVP                  '       Ed   VP                  '       Edü   VP                  '       gµ   VP                  '       gh   V P                  VP                  VP                  ,          V4      pVP                  VP                  ,          VP                  VP                  ,          r¿M‰V P                  VP                  V4      pVP                  VP                  V,          r¿MNVP                  '       g;   V P                  VP                  V4      pWûP                  ,          VP                  r¿M^p\        V4      \        V4      ^^ 3w  rûppV'       dv   V^,          '       d8   VV,          VV,          ,
          VV,          VV,          ,           ppV^,          pWÿ,          W»,          ,
          ^V,          V,          r¿V^,          pK}  \(        P*                  pV^8X  d   VVV,          ,           # \        V4      V,          VV,          V,          ,           # T
p^ RIHp ^ RIHp V! \5        V4      V4      pV! V.VO5!  # V^8X  d>   \        VP                    UUu. uF  qÂP                    F  qìV,          NK  	  K  	  upp!  # W%^,
          ,          P                  4       pVP                  '       d?   \        VP                    UUu. uF  pVP                    F  pWÎ,          NK  	  K   	  upp!  # \        VP                    Uu. uF  qÌV,          NK  	  up!  # VP                  '       do   VP                  ^ 8  d^   VP                  '       dL   \7        VP                  4      VP                  8”  d(   ^V P                  W#) 4      P                  4       ,          # VP                  '       dâ   VP8                  '       dÐ   VP;                  RR4      '       g'   VP<                  RJ g   VP?                  4       '       d’   . . ppVP                    FH  pVP8                  '       d#   VP                  V P                  W(4      4       K7  VP                  V4       KJ  	  \A        VV P                  V\        PB                  ! V4      4      .,           !  # V# u uppi u uppi u upi )zA(a + b + ..)**n -> a**n + n*a**(n-1)*b + .., n is nonzero integerF©Údeep)Úmultinomial_coefficients)Úbasic_from_dictrX  )"r-   rê   rð   ru   rg   r¾   rŸ   rã   r;  Ú_eval_expand_multinomialr:  Ú	make_argsr9  r¼   r|   Úis_Orderr   r   rh   Úas_real_imagr	   rw   Úsympy.ntheory.multinomialrŠ  Úsympy.polys.polyutilsr‹  rx  rn   rj   re  rÒ   rf  rì   rg  )r,   rh  r4   r;   r>  r¢   ÚradicalÚexpanded_base_nr   Úorder_termsÚother_termsrA   r  r„  Úgrõ   Úkr   r£   r‚  rð   rŠ  r‹  Úexpansion_dictÚmultirí   Útails   &,                         r%   rŒ  ÚPow._eval_expand_multinomialö  sá  € ð —I‘I‰	ˆØˆà�?�?‰?˜sŸu™u q�y¨T¯[¯[©[Ø—>—>�>Ü˜CŸE™E S§U¡U�NÓ+�çØ!�Mà&*§i¡i°¸AµgÓ&>À˜Và&*§i¡i°Ó&8�OØ&×-×-Ð-à+×DÑDÓFð (ä #§¢¨oÖ >˜ØŸ™ d¥lÖ3ñ !?ô  ™<Ð'ä�C“ˆAà×"×"Ñ"Ø+-¨r˜[àŸœ�AØ—z—z�zØ#×*Ñ*¨1Ö-à#×*Ñ*¨1Ö-ñ	 #÷ ä˜[Ñ)�AÜ˜[Ñ)�Aà˜A”vÜ1°!µ$¸UÔCÀaÅcÈ!ÅeÕKÐKä.¨q°qµ5­zÀÔF˜Ü)¨!­#°EÔ:¸Q½SÀ½UÕBÐBà—>—>‘>ð  ×,Ñ,Ó.‘D�Aà—}—}‘}¨¯¯©Ø Ÿ|Ÿ|˜|Ø#$§<§< <Ø$(§I¡I¨a¯c©c°A·C±C­i¸Ó$; Ø'(§s¡s¨1¯3©3¥w°·±°A·C±Cµ¡1à$(§I¡I¨a¯c©c°1Ó$5 Ø'(§s¡s¨A¯C©C°­E¡1Ø!"§§ Ø $§	¡	¨!¯#©#¨qÓ 1˜AØ#$§S¡S¥5¨!¯#©#™qà !˜Aä%(¨£V¬S°«V°Q¸Ð%9™
˜˜a çØ  1ŸuœuØ'(¨¥s¨Q¨q­S¥y°!°Aµ#¸¸!½µ) 1 Ø ! Q¥ Ø#$¥3¨­¥9¨a°­c°!­e˜qØ !�GšAäŸO™O˜à œ6Ø#$ q¨¥s¥7˜Nä#*¨1£:¨a¥<°!°Aµ#°aµ%Õ#7Ð7à�õ OÝAÙ!9¼#¸a»&À!Ó!D�ñ ' ~Ð:¸Ó:Ð:à˜”6Ü¨d¯iªiÔ K©i¨ÇÄ¸A 1§ Á¡©iÒ KÑLÐLà!¨¥E�]×DÑDÓF�EØ—|—|�|Ü"°$·)²)ô %1±)¨QØ%*§Z¤Z ð &'§S SÙ%/ñ &)±)ò %1ñ  2ð 2ô  #°d·i²iÓ$@±i° u§W W±iÑ$@ÑAÐAØ�o�oˆo #§%¡%¨!¤)°··°Ü�C—E‘E“
˜SŸU™UÔ"Ø�t—y‘y  tÓ,×EÑEÓGÕGÐGØ�Z�ZˆZ˜DŸNŸN˜N°·	±	¸'À5×0IÒ0IØ—‘ Ó%¨×)CÑ)C×)EÒ)Eð ˜b�4ˆEØŸœ�Ø—>—>�>Ø—L‘L §¡¨4Ó!6Ö7à—K‘K Ö%ñ	 !ô
 ˜ $§)¡)¨D´#·.².ÀÓ2FÓ"GÐ!HÕHÑJÐJàˆMùó3 !Lùó%1ùò %As   Ò#[
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     	óÆ  € V P                   P                  '       Ed2   ^ RIHp V P                   pV P                  P                  VR7      w  rVV'       g   V \        P                  3# \        R\        R7      w  rxV^ 8¼  dm   VP                  '       dE   VP                  '       d3   \        V P                  V,          4      p	W�8w  d   V	P                  4       # V! Wx,           V,          4      p	M¦V^,          V^,          ,           p
WZ,          V) V
,          reVP                  '       dW   VP                  '       dE   \        WV\        P                  ,          ,           V) ,          4      p	W�8w  d   V	P                  4       # V! Wx,           V) ,          4      p	V	P                  4        Uu. uF#  q»^ ,          ^,          ^,          '       d   K!  VNK%  	  pp\        V UUUu. uF#  w  w  rÞqÿW},          ,          WŽ,          ,          NK%  	  uppp!  pV	P                  4        Uu. uF"  q»^ ,          ^,          ^,          ^8X  g   K   VNK$  	  pp\        V UUUu. uF#  w  w  rÞqÿW},          ,          WŽ,          ,          NK%  	  uppp!  pV	P                  4        Uu. uF"  q»^ ,          ^,          ^,          ^8X  g   K   VNK$  	  pp\        V UUUu. uF#  w  w  rÞqÿW},          ,          WŽ,          ,          NK%  	  uppp!  pVP                  WuV\        P                  V,          /4      VP                  WuW†/4      VP                  WuW†) /4      ,           3# ^ RIHpHpHp V P                   P(                  '       EdN   V P                  P                  VR7      w  rVVP*                  '       dƒ   V P                   \        P,                  J de   VP.                  '       d   V \        P                  3# VP0                  '       d/   \        P                  V P                  ) V P                   ,          3# V P3                  V P3                  V^4      V P3                  V^4      ,           \        P,                  4      pV! We4      pV P3                  WÀP                   4      VV P                   ,          ppVV! V4      ,          VV! V4      ,          3# V P                  \        P4                  J d   ^ RIH p V P                   P                  4       w  rVV'       d'   VP8                  ! V3/ VB pVP8                  ! V3/ VB pV! V4      V! V4      ppV! V4      V,          V! V4      V,          3# ^ RIHpHp V'       dA   RVR	&   V P8                  ! V3/ VB pVPA                  R
4      V8X  d   R# V! V4      V! V4      3# V! V 4      V! V 4      3# u upi u upppi u upi u upppi u upi u upppi )r   )Úpolyrˆ  za brŒ   )Úatan2ÚcosÚsin©r;   )rQ   r›   FÚcomplexÚignoreN)!r;   rg   Úsympy.polys.polytoolsr�  r4   r�  r	   r^   ÚsymbolsÚDummyrj   r   rw   Útermsr:  r2  Ú(sympy.functions.elementary.trigonometricrž  rŸ  r   rê   rÒ   r³   r²   rÓ   rã   ra   rp   Úexpandrv   rQ   r›   re  )r,   r‰  rh  r�  r;   Úre_erô   rõ   rA   ÚexprÚmagr÷   ÚrÚaaÚbbÚccÚre_partÚim_part1Úim_part3rž  rŸ  r   ÚtÚrpÚtpr   rƒ   rQ   r›   rJ  s   &&,                           r%   r�  ÚPow.as_real_imagp  sc  € Ø�8‰8××ÑÝ2à—(‘(ˆCØŸ™×/Ñ/°TÐ/Ó:‰JˆDßØœQŸV™V�|Ð#Ü˜5¤eÔ,‰DˆAØ�aŒxØ—>—>�> d§n§n nä-¨d¯i©i¸­nÓ=�DØ”|Ø#×0Ñ0Ó2Ð2áØ•U˜S•Ló"‘ð ˜A•g  a¥Õ'�Ø!�X¨ u¨S¥y�dØ—>—>�> d§n§n nä-¨t¼1¿?¹?Õ6JÕ/JÈcÈTÕ.QÓR�DØ”|Ø#×0Ñ0Ó2Ð2á˜Q�U c T�MÓ*�ð !ŸJ™JœLÓ<™L�q°!µ°Qµ¸!·±—�™LˆAÐ<Ü¹qÕA¹q©|©x°¸˜q�u�H Q¥UŸN˜N¹qÓAÑBˆGà ŸJ™JœLÓ=™L�q¨a­D°­G°a­K¸1Ñ,<—�™LˆAÐ=ÜÁÕBÁ±±°¸" ¥�X a¥eŸ^˜^ÁÓBÑCˆHØ ŸJ™JœLÓ=™L�q¨a­D°­G°a­K¸1Ñ,<—�™LˆAÐ=ÜÁÕBÁ±±°¸" ¥�X a¥eŸ^˜^ÁÓBÑCˆHà—L‘L !¨1¬a¯o©o¸dÕ.BÐ!CÓDØ�M‰M˜1 AÐ,Ó-°·±¸qÈÈ5Ð>QÓ0RÕRðTð T÷ 	MÑLà�8‰8××ÑØŸ™×/Ñ/°TÐ/Ó:‰JˆDà�|�|ˆ| §¡¬A¯F©FÓ 2Ø×/×/Ð/Ø¤§¡˜<Ð'Ø×/×/Ð/ÜŸ6™6 T§Y¡Y J°·±Õ#9Ð9Ð9ð
 —	‘	˜$Ÿ)™) D¨!Ó,¨t¯y©y¸¸qÓ/AÕAÄ1Ç6Á6ÓJˆAá�dÓ!ˆAà—Y‘Y˜q§(¡(Ó+¨Q¨t¯x©x­Z�ˆBà‘c˜"“g•:˜r¡# b£'�zÐ)Ð)Ø�Y‰Yœ!Ÿ&™&Ó ÝBØŸ™×.Ñ.Ó0‰JˆDßØ—{’{ 4Ñ1¨5Ñ1�Ø—{’{ 4Ñ1¨5Ñ1�Ù�t“9™c $›iˆqˆAÙ�t“9˜Q•;¡ D£	¨!¥Ð+Ð+çCßØ#(��iÑ àŸ;š; tÑ5¨uÑ5�Ø—9‘9˜XÓ&¨(Ô2Ùá˜x›L©"¨X«,Ð7Ð7á˜$“x¡ D£Ð)Ð)ùòg =ùÜAùâ=ùÜBùÚ=ùÜBs6   Æ&V?Ç	V?Ç)WÈWÉ WÉ)WÊWÊ7WË
)Wc                	ó  € ^ RI Hp V P                  P                  V4      pV P                  P                  V4      pWV! V P                  4      ,          W0P                  ,          V P                  ,          ,           ,          # )r   rM   )rp   rN   r4   Údiffr;   )r,   rƒ   rN   ÚdbaseÚdexps   &&   r%   Ú_eval_derivativeÚPow._eval_derivativeÃ  sT   € Ý>Ø—	‘	—‘˜qÓ!ˆØ�x‰x�}‰}˜QÓˆØ™c $§)¡)›nÕ,¨u·x±xÕ/?ÀÇ	Á	Õ/IÕIÕJÐJr(   c                	óL  € V P                  4       w  r#V\        P                  8X  d*   ^ RIHp V! V P                  RR7      P                  V4      # VP                  V4      pVP                  '       g   VP                  V4      pVP                  '       d   VP                  '       dm   VP                  RJ d]   VP                  4       W"P                  4       ,          P                  V4      ,          pV) pV P                  W#4      P                  4       # V P                  W#4      # )r   r¡  Fr¹   )r“   r	   ra   rp   r;   Ú_eval_evalfÚ_evalfrg   r°   rh   r®   rN  rã   r©  )r,   Úprecr4   r;   Úexp_functions   &&   r%   r¿  ÚPow._eval_evalfÉ  sÓ   € Ø×$Ñ$Ó&‰	ˆØ”1—6‘6Œ>åRÙ §¡°5Ô9×EÑEÀdÓKÐKØ�{‰{˜4Ó ˆØ�~�~ˆ~Ø—*‘*˜TÓ"ˆCØ�?�?ˆ?˜tŸ~Ÿ~˜~°$×2GÑ2GÈ5Ó2PØ—>‘>Ó# t¯n©nÓ.>Õ'>×&FÑ&FÀtÓ&LÕLˆDØ�$ˆCØ—9‘9˜TÓ'×.Ñ.Ó0Ð0Ø�y‰y˜Ó#Ð#r(   c                	ó<  € V P                   P                  ! V!  '       d   R # V P                  P                  ! V!  '       dZ   \        V P                  P	                  V4      ;'       d.    V P                   P
                  ;'       d    V P                   ^ 8¬  4      # R# ©FT)r;   Úhasr4   ÚboolÚ_eval_is_polynomialrg   ©r,   Úsymss   &&r%   rÈ  ÚPow._eval_is_polynomialØ  sz   € Ø�8‰8�<Š<˜×ÐÙà�9‰9�=Š=˜$×ÐÜ˜Ÿ	™	×5Ñ5°dÓ;÷ 8ð 8Ø—‘×#Ñ#÷8ð 8Ø)-¯©°Q©ó9ð 9ñ r(   c                	óZ  € V P                   P                  '       dc   V P                  P                  '       dG   \	        \        V P                   P                  V P                  P                  .4      4      '       d   R # V P                  ! V P                  4       !  pVP                  '       g   VP                  # VP                  4       w  r#VP                  '       d   VP                  '       d   R# VP                  '       di   VP                  '       d9   \	        VP                  4      '       g   VP                  '       d   R # W#8X  d   R # MVP                  '       d   VP                  # V\        P                  J d+   VP                  '       d   VP                   '       d   R# R# R# R# rÜ   )r;   rf   r4   râ   r   r   r°   rÒ   rã   r“   r;  rê   rÞ   Úis_irrationalr	   ra   rî   )r,   rð   rA   rB   s   &   r%   Ú_eval_is_rationalÚPow._eval_is_rationalâ  s  € ð �H‰H××Ð D§I¡I×$9×$9Ð$9Üœi¨¯©×)=Ñ)=¸t¿y¹y×?PÑ?PÐ(QÓR×SÒSÙØ�IŠI�t×'Ñ'Ó)Ñ*ˆØ�x�xˆxØ—=‘=Ð Ø�}‰}‹‰ˆØ�=�=ˆ=˜QŸ]Ÿ]˜]ñ Ø�<�<ˆ<Ø�}�}ˆ}Ü˜QŸY™Y×'Ò'¨1×+;×+;Ð+;ÙØ”6Ùð à——�Ø—y‘yÐ Ø”—‘‹;Ø�}�}ˆ} §§ Ùñ ".‰}ñ r(   c                	ó  € R  pV P                   P                  '       g   V! V P                   4      '       d   R# V P                   \        P                  J dû   V P                  ! V P
                  !  pVP                  V P                  8X  dº   V P                  P                  '       dœ   V P                  P                  '       d   R# V P                  \        P                  ,          P                  '       d   R# V P                  \        P                  \        P                  ,          ,          P                  '       d   R# R# R# VP                  # V P                  P                  '       dÏ   V P                   P                  RJ d   V P                  P                  # V P                   P                  RJ dQ   V P                  P                  '       d   V P                   P                  # V P                   P                  '       d   R# V P                  P                  '       d   V P                   P                  # R# V P                   P                  '       d³   V P                  P                  '       d•   \        V P                   P                  4      '       d"   \        V! V P                   4      4      '       g7   V P                   P                  RJ g   V P                   P                  '       d   V P                  P                  # R# R# R# )c                óN   €  V ^,
          P                   #   \         d     R# i ; i)r:   F)rÒ   r"  )r«  s   &r%   Ú_is_oneÚ'Pow._eval_is_algebraic.<locals>._is_oneþ  s)   € ðØ˜q�×)Ñ)Ð)øÜô âðús   ‚ •$£$TFN)r4   rÒ   r	   ra   rã   r-   r;   rî   Úis_algebraicrx   râ   rw   rº   r   rf   rÍ  )r,   rÒ  rƒ   s   &  r%   Ú_eval_is_algebraicÚPow._eval_is_algebraicý  sù  € ò	ð �9‰9××Ð¡¨¯	©	× 2Ò 2ÙØ�Y‰Yœ!Ÿ&™&Ó Ø—	’	˜4Ÿ9™9Ñ%ˆAØ�v‰v˜Ÿ™Ô"Ø—8‘8×&×&Ð&Ø—x‘x×,×,Ð,Ù$ØŸ(™(¤1§4¡4�-×4×4Ð4Ù$ØŸ(™(¤A§O¡O´A·D±DÕ$8Õ9×F×FÐFÙ#ñ Gñ 'ð —~‘~Ð%Ø�X‰X×!×!Ð!Ø�y‰y×%Ñ%¨Ó.Ø—x‘x×'Ñ'Ð'Ø�y‰y× Ñ  EÓ)Ø—8‘8×&×&Ð&ØŸ9™9×1Ñ1Ð1Ø—Y‘Y×+×+Ð+ÙØ�x‰x×#×#Ð#Ø—y‘y×-Ñ-Ð-ñ $à�Y‰Y×#×#Ð#¨¯©×(=×(=Ð(=Ü˜4Ÿ9™9×,Ñ,×-Ò-Ü™g d§i¡iÓ0×1Ò1Ø—9‘9×'Ñ'¨5Ó0Ø—9‘9×*×*Ð*Ø—x‘x×+Ñ+Ð+ñ +ñ	 )>Ñ#r(   c                	óü   € V P                   P                  ! V!  '       d   R # V P                  P                  ! V!  '       d:   V P                  P                  V4      ;'       d    V P                   P                  # R# rÅ  )r;   rÆ  r4   Ú_eval_is_rational_functionrg   rÉ  s   &&r%   rØ  ÚPow._eval_is_rational_function$  s^   € Ø�8‰8�<Š<˜×ÐÙà�9‰9�=Š=˜$×ÐØ—9‘9×7Ñ7¸Ó=÷ $ð $Ø—‘×#Ñ#ð$ñ r(   c                	óü  € V P                   P                  W4      pV P                  P                  pV'       d   V# V P                  P                  W4      pVR J d   V'       d   R # R# Vf   R# V P                   P	                  W4      pVP
                  pV'       d   R pM \        VP                  \        V4      34      pVR J d   V# Vf   R# V'       g   V# V P                  P	                  W4      P                  # )FN)	r4   Ú_eval_is_meromorphicr;   rg   r2  rÒ   r   r_   r   )	r,   r\  rõ   Ú
base_meromÚexp_integerÚ	exp_meromrA   Úb_zeroÚlog_defineds	   &&&      r%   rÛ  ÚPow._eval_is_meromorphic.  sÚ   € ð —Y‘Y×3Ñ3°AÓ9ˆ
Ø—h‘h×)Ñ)ˆßØÐà—H‘H×1Ñ1°!Ó7ˆ	Ø˜Ó÷ &�5Ð/¨4Ð/ØÒÙà�I‰I�N‰N˜1Ó ˆð —‘ˆßØ‰Kä# Q§[¡[´)¸FÓ2CÐ$DÓEˆKà˜%ÓØÐØÒ ÙçØÐà�x‰x�}‰}˜QÓ"×,Ñ,Ð,r(   c                	óü   € V P                   P                  ! V!  '       d   R # V P                  P                  ! V!  '       d:   V P                  P                  V4      ;'       d    V P                   P                  # R# rÅ  )r;   rÆ  r4   Ú_eval_is_algebraic_exprrê   rÉ  s   &&r%   rã  ÚPow._eval_is_algebraic_exprS  s^   € Ø�8‰8�<Š<˜×ÐÙà�9‰9�=Š=˜$×ÐØ—9‘9×4Ñ4°TÓ:÷ %ð %Ø—‘×$Ñ$ð%ñ r(   c                	ó8  € ^ RI HpHp VP                  '       g/   VP	                  V4      '       g   VP	                  V4      '       d	   W,          # VP	                  \
        4      pVP	                  \
        4      '       dW   \        P                  '       d*   \        \        P                  V! V4      V,          VR7      # V! V! V4      V,          VR7      # ^ RIHpHp V! V! V! V4      4      \        P                  V! V4      ,          ,           V,          4      # )r   rœ   r¹   )r~   ÚAbs)rp   r;   rN   rÒ   rÆ  r7  r   Ú
exp_is_powr   r	   ra   rv   r~   ræ  rw   )	r,   r4   rC  Úkwargsr;   rN   rS   r~   ræ  s	   &&&,     r%   Ú_eval_rewrite_as_expÚPow._eval_rewrite_as_exp]  s¹   € ßCà�<�<ˆ<˜4Ÿ8™8 CŸ=š=¨D¯H©H°S¯MªMØ•:Ðà—8‘8œFÓ#ˆà�8‰8”F×Òô !×+×+Ð+Üœ1Ÿ6™6¡3 t£9¨T¥>¸HÔEÐEá™3˜t›9 T�>°HÔ=Ð=÷ FÙ™™C ›I›¬¯©¹¸T»Õ)BÕBÀDÕHÓIÐIr(   c                	ón  € V P                   '       g   V \        P                  3# V P                  4       w  rVP	                  4       w  r4VP
                  pVP                  '       d+   V'       g#   VP                  '       g   VP                  4       pVP                  pVP                  '       g   V'       g   Tp\        P                  pVP                  pV'       d   V) V) rCMVf   V'       g   Tp\        P                  pV'       d   YCrCV) pVP                  '       dv   V\        P                  J d'   V\        P                  Jd   W0P                  WB4      3# V\        P                  Jd(   V\        P                  J d   V P                  W24      V3# V P                  W24      V P                  WB4      3# r*   )r|   r	   r\   r“   r¡   r°   ri   rº   rk   rf   r®   Úis_nonpositivero   rã   )r,   r4   r;   r¢   r£   Úneg_expÚint_expÚdnonposs   &       r%   r¡   ÚPow.as_numer_denomq  sB  € Ø×"×"Ð"ØœŸ™�;ÐØ×$Ñ$Ó&‰	ˆØ×"Ñ"Ó$‰ˆð —/‘/ˆØ�:�:ˆ:Ÿg¨c¯o¯o¨oØ×2Ñ2Ó4ˆGØ—.‘.ˆð ×"×"Ð"§gØˆAÜ—‘ˆAØ×"Ñ"ˆßØ�2˜�r‰qØŠ_§WØˆAÜ—‘ˆAßØˆqØ�$ˆCØ�?�?ˆ?Ø”A—E‘E‹z˜a¤q§u¡u›nØŸ)™) AÓ+Ð+Ð+ØœŸ™‹~ !¤q§u¡u£*Ø—y‘y Ó(¨!Ð+Ð+Ø�y‰y˜Ó  $§)¡)¨AÓ"3Ð3Ð3r(   c                	ó  € \        V4      pVf   / pV\        P                  J d1   V P                  P	                  \        P
                  V4      pVe   V# \        V\        4      '       g   R # VP                  4       w  rVV P                  4       w  rxVP                  '       dl   VP                  '       dZ   V'       dR   VP                  '       d    VP	                  WVV,          ,          V4      # VP	                  V^V,          ,          V4      # VP                  4       pV P                  P	                  WT4      pVf   R # V P                  P                  V4      P	                  Wd4      pVf   \        P                  ! WV4      # V# r*   )r   r	   r\   r;   Úmatchesr^   rU   r
   r“   re   rg   râ   Úcopyr4   Úxreplace)	r,   r«  Ú	repl_dictr'  r£   rA   rB   ÚsbÚses	   &&&&     r%   rò  ÚPow.matches”  s)  € Ü˜‹~ˆØÒØˆIð ”1—5‘5‹=Ø—‘× Ñ ¤§¡¨Ó3ˆAØŠ}Ø�ô ˜$¤×%Ò%Ùà×ÑÓ!‰ˆð ×!Ñ!Ó#‰ˆØ�<�<ˆ<˜BŸMŸM˜M¯dØ�}�}ˆ}Ø—z‘z !¨¥d¥)¨YÓ7Ð7Ø—:‘:˜d Q r¥T�l¨IÓ6Ð6à�N‰NÓˆØ�I‰I×Ñ˜aÓ#ˆØŠ9Ùà�H‰H×Ñ˜aÓ ×(Ñ(¨Ó.ˆØŠ9Ü—<’< ¨IÓ6Ð6Øˆr(   c                	óp  a2€ ^ RI HpHp ^ RIHp ^ RIHp ^ RIHp	 V P                  \        P                  J dþ   V P                  P                  WVR7      p
V
P                  '       d
   ^V
,           # V! V
P                  4       V^ 4      pV\        P                  J d   V! W,          V4      # V\        P                   J d   V # W«,
          pV! V4      ;rÞ\#        ^V4       F-  pWìV,          ,          pVP                  WVR7      pWÞ,          pK/  	  WØ! WÂ,          V4      ,          p^ RIHp V! VRRR	7      # ^ R
IHp ^RIHp V! V RR7      P/                  4       p V P1                  4       w  ppVP2                  ! V!  '       d   \5        4       hVP3                  V4      '       d'   V! VV! V4      ,          4      P7                  WW4R7      # Ves   VP3                  V4      '       d\   ^RIHp \=        RVV.R7      w  ppVP?                  V! VVV,          ,          4      V! V4      VV,          ,           4      pVV,          p TP                  4       p ^ RI H!p VP3                  V\        PD                  4      '       d   Ve   \G        4       hVPI                  V4      w  ppVP3                  V4      '       d   ^ RI)H*p V! V4      PW                  4       pVPX                  '       g€   VPZ                  '       d   VP\                  '       g\   W P_                  WVR7      8X  dE   V! VV! V4      ,          4      P7                  WW4R7      pVV! VV! V4      ,          4      8X  d   V # V# VPa                  WR7      p\c        V4      V,
          PW                  4       pVV,          pVPZ                  '       g   \K        4       hVVV,          ,
          o2S2P3                  \d        4      '       d	   V	! V4      o2S2Pf                  '       d   V! VVV,          ,          V4      # VPX                  '       d)   VV,          pVV 8w  d   VV! W,          V4      ,          pV# R p V23R lp! VPI                  WR7      w  pp"TPh                  '       d?   T"\        Pj                  8X  d*   TP?                  R R 4      pTPI                  YR7      w  pp"T"Pl                  '       g¢   TPo                  4       pTPX                  '       d
   TT,          # TPI                  YR7      w  pp"T"Pl                  '       gP   TT,
          T,          Pq                  4       pTPI                  YR7      w  pp"T"Pl                  '       g   \K        4       h^ RI9H:p# TP7                  TT#! S24      Y4R7      P                  4       p$/ p%\v        Px                  ! T$4       F8  pT ! Yá4      w  p&p'T%P{                  T'\        Pj                  4      T&,           T%T'&   K:  	  \        P|                  p(\        Pj                  \        P|                  /p)T%p*^ RI?H@p+HAp, T(T",          S2,
          Pf                  '       dz   T,! TT(4      T+! T(4      ,          p-T* F;  pT)P{                  T\        Pj                  4      T-T*T,          ,          ,           T)T&   K=  	  T!! T*T%4      p*T(\        P|                  ,          p(K™  ^ RIBHCp. TPˆ                  '       gÛ   TPX                  '       dÉ   TPf                  '       d·   TT,
          P‹                  Y4      p/T.! T/4      Pf                  '       d*   T ! TT,          RRT,          ,          ,          T4      w  p0p1MpT.! T/4      PX                  '       d2   T ! T! TT! T4      ,          4      Pa                  YTR7      T4      w  p0p1M'T ! TT,          T4      w  p0p1MT ! TT,          T4      w  p0p1\        Pj                  pT) F1  p'T'T1,           pTT)T',          T0,          TT,          ,          ,          pK3  	  TPˆ                  '       dC   TPl                  '       d1   TT",          T,
          PŒ                  '       d   T\c        T 4      8X  g    TT! Y,          T4      ,          pT# T#   \F        \J        \4        3 dƒ    TP7                  T\M        ^T4      Y4R7      P                  4       pTP3                  \        PN                  \        PP                  4      '       d   \K        4       hTPI                  T4      w  pp ELzi ; i  \F        \J        3 dP    T! TTS2,          ,          T^ 4      ^ 8X  d(   TT,          TTT,          ,          T,          ,           u # \K        4       hi ; i  \J         d*    T! TT! T4      ,          4      P7                  YY4R7      u # i ; i)r   rœ   )Úlimit)ÚOrder©Úsympify)r¢   Úlogx)ÚpowsimpTr;   )r‰  Úcombine)Ú	powdenest)Ú_illegal)rX  )r¢   rþ  Úcdir)ÚWildzc, ex)r}   Úexclude)Ú	polygamma)Ú
logcombine©rþ  r  ©rþ  c                ód  € \         P                  \         P                  r2\        P                  ! V 4       FP  pVP                  V4      '       d/   VP                  4       w  rSWQ8w  d    V P                  V4      u # KH  W$,          pKR  	  W#3#   \         d    T \         P                  3u u # i ; ir*   )	r	   r\   r^   rì   r�  rÆ  r“   Úleadtermr"  )r   r\  rí   r;   Úfactorr4   s   &&    r%   Ú	coeff_expÚ$Pow._eval_nseries.<locals>.coeff_exp  s‘   € ÜŸ™¤§¡�3ÜŸ-š-¨Ö-�Ø—:‘:˜a—=’=Ø &× 2Ñ 2Ó 4‘I�DØ”yð0Ø#'§=¡=°Ó#3Ò3ñ !ð •O’Eñ .ð �:Ðøô	  *ô 0Ø#'¬¯© <Ô/ð0ús   Á)BÂB/Â.B/c                óÎ   <€ / p\        W4       FR  w  r4W4,           pVS8  g   K  VP                  V\        P                  4      W,          W,          ,          ,           W%&   KT  	  V# r*   )r   re  r	   r^   )Úd1Úd2ÚresÚe1Úe2r€   Úmaxpows   &&    €r%   ÚmulÚPow._eval_nseries.<locals>.mul"  sP   ø€ ØˆCÜ! "ž/‘�Ø•W�Ø˜–;Ø!Ÿg™g b¬!¯&©&Ó1°BµF¸2½6µMÕA�C“Gñ *ð ˆJr(   c                ó   € V P                   # r*   )Úis_Floatr[  s   &r%   r]  Ú#Pow._eval_nseries.<locals>.<lambda>6  s   €  A§J¢Jr(   c                ó   € \        V 4      # r*   )rï   r[  s   &r%   r]  r  6  s   € ¼(À1¼+r(   )Úceiling)Ú	factorialÚff©rQ   rR   éþÿÿÿ)Grp   r;   rN   Úsympy.series.limitsrú  Úsympy.series.orderrû  Úsympy.core.sympifyrý  r4   r	   ra   ÚnseriesrŽ  ÚremoveOrÝ   r[   ÚrangeÚsympy.simplify.powsimprÿ  r  Únumbersr  Útrigsimpr“   rÆ  r   Ú_eval_nseriesÚsymbolr  r¥  ÚreplaceÚ'sympy.functions.special.gamma_functionsr  Ú
EulerGammar"  r  ÚNotImplementedErrorrd   rZ   rY   Úsympy.simplify.simplifyr  ÚcancelrÒ   rh   rÐ   Ú_eval_as_leading_termÚas_leading_termr   r7  r°   r  r^   rº   Úsimplifyr©  r¯   r  r:  r�  re  r\   Ú(sympy.functions.combinatorial.factorialsr  r  rv   rQ   rf   Údirrì  )3r,   r\  r¢   rþ  r  r;   rN   rú  rû  rý  Úe_seriesÚe0r´  Ú
exp_seriesr   r÷   rÿ  r  r  rA   rB   r  r   r€   r  Ú_rÁ   r  r  r  r–  r­  r  r  r£   r  ÚgpolyÚgtermsÚco1r  r—  r§  Útkr  r  rí   rQ   ÚndirÚincoÚinexr  s3   &&&&&                                             @r%   r*  ÚPow._eval_nseries¶  s‡  ø€ ÷ 	DÝ-Ý,Ý.Ø�9‰9œŸ™ÓØ—x‘x×'Ñ'¨°TÐ'Ó:ˆHØ× × Ð Ø˜8•|Ð#Ù�x×'Ñ'Ó)¨1¨aÓ0ˆBØ”Q×'Ñ'Ó'Ù˜Q�T 1“~Ð%Ø”Q—Z‘ZÓØ�Ø•ˆAÙ # B£Ð'ˆJä˜1˜a–[�Ø˜!�•�Ø—|‘| A°�|Ó6�ØÕ"’
ñ !ð ˜% ¥ a›.Õ(ˆJÝ6Ù˜:¨D¸%Ô@Ð@Ý4Ý%Ù˜ TÔ*×3Ñ3Ó5ˆØ×ÑÓ!‰ˆˆ1à�5Š5�(×ÐÜ“+Ðà�5‰5��8Š8Ù�q™˜Q›•x“=×.Ñ.¨q¸DÐ.ÓLÐLàÒ §¡ c§
¢
Ý$Ü˜G¨¸°sÔ;‰EˆAˆrØ—	‘	™#˜a  2¥�g›,©¨A«°°DµÕ(8Ó9ˆAØ�a•4ˆDà�I‰I‹Kˆð		!ÝIØ�u‰u�Y¤§¡×-Ò-°$Ò2BÜ “lÐ"Ø—:‘:˜a“=‰DˆAˆqð �5‰5��:Š:Ý:Ù˜1“×$Ñ$Ó&ˆAà—	—	�	˜QŸ[Ÿ[˜[¨Q¯Y¯Y¨YØ×1Ñ1°!ÀTÐ1ÓJÔJÙ˜!™C ›F�(“m×1Ñ1°!¸tÐ1ÓO�Ø™#˜a¡ A£�h›-Ô'Ø�KØ�
à×Ñ˜aÐÓ+ˆÜ�a‹[˜1�_×$Ñ$Ó&ˆØˆa�CˆØ�{�{ˆ{Ü%Ó'Ð'Ø�Q�q•S•ˆØ�:‰:”f×ÒÙ˜Q“ZˆFà××ÐÙ˜˜Q˜q�S� 1Ó%Ð%à�9�9ˆ9Ø�1•ˆAØ�DŒyØ‘U˜1�4 “^Õ#�ØˆHò	õ	ð	,Ø—:‘:˜a�:Ó+‰DˆAˆqð �:�:ˆ:˜!œqŸv™vœ+ð —	‘	Ñ.Ñ0EÓFˆAØ—:‘:˜a�:Ó+‰DˆAˆqØ�}�}ˆ}Ø—
‘
“ˆAØ�y�yˆyØ˜!•t�Ø—:‘:˜a�:Ó+‰DˆAˆqØ—=—=�=Ø˜!•e˜Q•Y×&Ñ&Ó(�Ø—z‘z !�zÓ/‘��1Ø—}—}�}Ü-Ó/Ð/å?Ø—‘ ¡W¨V£_¸4�ÓK×SÑSÓUˆØˆä—M’M %Ö(ˆDÙ Ó(‰GˆC�ØŸ™ B¬¯©Ó/°#Õ5ˆF�2‹Jñ )ô �E‰EˆÜ—‘œŸ™�ˆØˆçJà��s�V�|×(×(Ð(Ù�q˜!“H™Y q›\Õ)ˆEÛ�Ø!ŸI™I b¬!¯&©&Ó1°E¸"¸R½&µLÕ@��b“	ñ á�R˜“ˆBØ”—‘�JŠAå;à�|�|ˆ| §	§	 	¨a¯m¯m¨mØ˜•E—;‘;˜qÓ'ˆDÙ�$‹x×#×#Ð#Ù& q¨!¥t¨R°2°aµ4­LÕ'8¸!Ó<‘
�‘dÙ�D“×!×!Ð!Ù&¡s¨1©S°«V­8£}×'DÑ'DÀQÐX\Ð'DÓ']Ð_`Óa‘
�‘dá& q¨!¥t¨QÓ/‘
�‘dá" 1 a¥4¨Ó+‰JˆD�$Ü�f‰fˆãˆBØ�d•ˆBØ�5˜•9˜T•> ! b¥'Õ)Õ)ŠCñ ð ——� §§ °A°aµC¸!µG×3K×3KÐ3KØ”x “~Ô%ðQØ‘u˜Q�T 1“~Õ%�ð ˆ
ˆsˆ
øô Ô/´Ð;ô 	!Ø—‘ ¤S¨¨A£Y°T�ÓE×MÑMÓOˆAØ�u‰u”Q—U‘UœA×-Ñ-×.Ò.Ü)Ó+Ð+Ø—:‘:˜a“=‰DˆA‹qð		!ûôv Ô/Ð0ô 	,Ù�Q�q˜&•y•[ ! QÓ'¨1Ô,à˜!•t˜a  1¥�f Q�h•Ò&ä)Ó+Ð+ð	,ûôD 'ô QÙ˜1™S ›V�8“}×2Ñ2°1ÀÐ2ÓPÒPðQús>   É+b É>"b Ñd á)f âBdädäAe>å3e>æ1f5æ4f5c                	ó:  € ^ RI HpHp V P                  pV P                  pV P                  \        P
                  J d„   VP                  WR7      pVP                  V^ 4      p	V	\        P                  J d   VP                  V^ 4      p	V	P                  RJ d   \        P
                  V	,          # \        RV ,          4      hVP                  V4      '       d(   V! We! V4      ,          4      p
V
P                  WVR7      # ^ RIHp  VP                  WVR7      pTP                  '       gÎ   TP                   '       d¼   TP                  T4      '       g¥   Y|,
          P#                  Y4      pT! T4      P                   '       d'   T P%                  YÆ4      RRT,          ,          ,          # T! T4      P&                  '       d8   T! T4      P)                  YTR7      pTP                  RJ d   T! Yn,          4      # T P%                  YÆ4      #   \         d    T u # i ; i)	r   rœ   r	  FzCannot expand %s around 0r  r  rR   r   )rp   r;   rN   r4   r	   ra   r3  r2  rZ   rú  ro   r   rÆ  rv   rQ   rf   r°   r6  rã   rÒ   r2  )r,   r\  rþ  r  r;   rN   rB   rA   r~   Úarg0ÚltrQ   r  r?  Úlog_leadterms   &&&&           r%   r2  ÚPow._eval_as_leading_termr  s¦  € ßCØ�H‰HˆØ�I‰IˆØ�9‰9œŸ™ÓØ×#Ñ# AÐ#Ó1ˆCØ—8‘8˜A˜q“>ˆDØ”q—u‘u‹}Ø—y‘y  A“�Ø×Ñ 5Ó(Ü—v‘v˜t•|Ð#ÜÐ7¸4Õ@ÓAÐAØ�U‰U�1�XŠXÙ�Q˜˜Q›•Z“ˆBØ×%Ñ% a¸Ð%Ó>Ð>å?ðØ×%Ñ% a¸Ð%Ó>�ð —<—<�< A§M§M M¸!¿%¹%À¿(º(Ø�—{‘{ 1Ó+�Ù�d“8×'×'Ð'ð  Ÿ9™9 Q›?¨b°B°qµD­\Õ9Ð9Ù˜“X×%×%Ð%Ù#& q£6×#?Ñ#?ÀÐSWÐ#?Ó#X�LØ#×/Ñ/°5Ó8Ù" 1¥>Ó2Ð2Ø—9‘9˜Q“?Ð"øô ô Ø’ðús   ÄH
 È
HÈHc                	ó`   € ^ RI Hp V! V P                  V4      V P                  W!4      ,          # )r   )Úbinomial)r5  rI  r;   rã   )r,   r¢   r\  Úprevious_termsrI  s   &&&* r%   Ú_taylor_termÚPow._taylor_term”  s#   € åEÙ˜Ÿ™ !Ó$ t§y¡y°£Õ6Ð6r(   c                	óZ  <€ V P                   \        P                  Jd   \        SV `  ! W.VO5!  # V^ 8  d   \        P
                  # V^ 8X  d   \        P                  # ^RIHp V! V4      pV'       d   VR,          pVe   WR,          V,          # ^ RIH	p W!,          V! V4      ,          # )r   rü  )r  rR   )
r4   r	   ra   ÚsuperÚtaylor_termr^   r\   rý  r5  r  )r,   r¢   r\  rJ  rý  rð   r  r`   s   &&&*   €r%   rO  ÚPow.taylor_term™  s†   ø€ Ø�9‰9œAŸF™FÓ"Ü‘7Ò& qÐ=¨nÓ=Ð=ØˆqŒ5Ü—6‘6ˆMØ�Œ6Ü—5‘5ˆLÝ$Ù�A‹JˆßØ˜rÕ"ˆAØŠ}Ø•u˜q•yÐ ÝFØ�t‘I˜a“LÕ Ð r(   c                	óV  € V P                   \        P                  J d‹   ^ RIHp V! \        P
                  V P                  ,          \        P                  ^,          ,           4      \        P
                  V! \        P
                  V P                  ,          4      ,          ,
          # R# )r   )r   N)r4   r	   ra   r¨  r   rw   r;   rx   )r,   r4   r;   rh  r   s   &&&, r%   Ú_eval_rewrite_as_sinÚPow._eval_rewrite_as_sin©  se   € Ø�9‰9œŸ™ÓÝDÙ”q—‘ t§x¡xÕ/´!·$±$°qµ&Õ8Ó9¼A¿O¹OÉCÔPQ×P_ÑP_Ð`d×`hÑ`hÕPhÓLiÕ<iÕiÐiñ r(   c                	óV  € V P                   \        P                  J d‹   ^ RIHp V! \        P
                  V P                  ,          4      \        P
                  V! \        P
                  V P                  ,          \        P                  ^,          ,           4      ,          ,           # R# )r   )rŸ  N)r4   r	   ra   r¨  rŸ  rw   r;   rx   )r,   r4   r;   rh  rŸ  s   &&&, r%   Ú_eval_rewrite_as_cosÚPow._eval_rewrite_as_cos®  sg   € Ø�9‰9œŸ™ÓÝDÙ”q—‘ t§x¡xÕ/Ó0´1·?±?Á3ÄqÇÁÐW[×W_ÑW_ÕG_Ôbc×bfÑbfÐghÕbhÕGhÓCiÕ3iÕiÐiñ r(   c                	óØ   € V P                   \        P                  J dL   ^ RIHp ^V! V P
                  ^,          4      ,           ^V! V P
                  ^,          4      ,
          ,          # R# )r   )ÚtanhN)r4   r	   ra   Ú%sympy.functions.elementary.hyperbolicrX  r;   )r,   r4   r;   rh  rX  s   &&&, r%   Ú_eval_rewrite_as_tanhÚPow._eval_rewrite_as_tanh³  sI   € Ø�9‰9œŸ™ÓÝBØ™˜TŸX™X a�ZÓ(Õ(¨1©t°D·H±H¸QµJÓ/?Õ+?Õ@Ð@ñ r(   c                	ó  € ^ RI HpHp V\        P                  Jd   R# VP
                  '       dÐ   VP                  \        P                  \        P                  ,          4      pV'       d’   VP                  '       d~   V! \        P                  V,          4      V! \        P                  V,          4      r‡\        Wu4      '       g2   \        W„4      '       g   V\        P                  V,          ,           # R# R# R# R# R# )r   )r   rŸ  N)r¨  r   rŸ  r	   ra   ri   rí   rx   rw   rh   rU   )	r,   r4   r;   rè  r   rŸ  rí   ÚcosineÚsines	   &&&,     r%   Ú_eval_rewrite_as_sqrtÚPow._eval_rewrite_as_sqrt¸  s    € ßEØ”q—v‘vÓÙØ�:�:ˆ:Ø—I‘IœaŸd™d¤Q§_¡_Õ4Ó5ˆEß˜ŸŸ˜Ù"¤1§4¡4¨¥:›±´A·D±D¸µJ³˜Ü! &×.Ò.´zÀ4×7MÒ7MØ!¤A§O¡O°DÕ$8Õ8Ð8ñ 8NÑ.ñ )‰uñ r(   c           
     ó   € V P                  4       w  r4\        VP                  WR7      !  pVP                  WR7      w  rVVP                  '       dâ   VP	                  4       w  rxVP                  '       d¾   V\
        P                  8w  d©   WW,          p	V P                  W94      p
\
        P                  pV
P                  '       g4   \        V	P                  V	P                  4      w  rËV P                  W<4      p
W P                  V\        WXWµ,          V	P                  ,          ,           4      4      3# \        WV4      pVP                  '       d�   VP                  '       d~   VP                  WR7      w  rxV P                  Wt4      P                  4       w  r­VP                  4       w  rÞV\
        P                  J g   Wä8X  d   W P                  \        WØ4      V4      3# \
        P                  V P                  W44      3# )aî  Return the tuple (R, self/R) where R is the positive Rational
extracted from self.

Examples
========

>>> from sympy import sqrt
>>> sqrt(4 + 4*sqrt(2)).as_content_primitive()
(2, sqrt(1 + sqrt(2)))
>>> sqrt(3 + 3*sqrt(2)).as_content_primitive()
(1, sqrt(3)*sqrt(1 + sqrt(2)))

>>> from sympy import expand_power_base, powsimp, Mul
>>> from sympy.abc import x, y

>>> ((2*x + 2)**2).as_content_primitive()
(4, (x + 1)**2)
>>> (4**((1 + y)/2)).as_content_primitive()
(2, 4**(y/2))
>>> (3**((1 + y)/2)).as_content_primitive()
(1, 3**((y + 1)/2))
>>> (3**((5 + y)/2)).as_content_primitive()
(9, 3**((y + 1)/2))
>>> eq = 3**(2 + 2*x)
>>> powsimp(eq) == eq
True
>>> eq.as_content_primitive()
(9, 3**(2*x))
>>> powsimp(Mul(*_))
3**(2*x + 2)

>>> eq = (2 + 2*x)**y
>>> s = expand_power_base(eq); s.is_Mul, s
(False, (2*x + 2)**y)
>>> eq.as_content_primitive()
(1, (2*(x + 1))**y)
>>> s = expand_power_base(_[1]); s.is_Mul, s
(True, 2**y*(x + 1)**y)

See docstring of Expr.as_content_primitive for more examples.
)r’  Úclear)r“   Ú_keep_coeffÚas_content_primitiverê   rë   r	   r^   rã   r$  rð   rŸ   ri   rt   r\   )r,   r’  rb  rA   rB   ÚceÚpeÚhr´  Úcehr   r­  ÚicehrÁ   Úmes   &&&            r%   rd  ÚPow.as_content_primitiveÃ  sv  € ðV ×ÑÓ!‰ˆÜ˜×/Ñ/¸Ð/ÓMÑNˆØ×'Ñ'°Ð'ÓE‰ˆØ�=�=ˆ=ð —?‘?Ó$‰DˆAØ�}�}ˆ} ¤a§f¡f¤Ø•d�Ø—I‘I˜aÓ%�Ü—F‘F�Ø—}—}�}Ü$ S§U¡U¨C¯E©EÓ2‘G�DØŸ	™	 !Ó*�AØŸ)™) A¤{°2¸1½4ÀÇÁ½:µ~Ó'FÓGÐGÐGÜ˜Óˆà�=�=ˆ=˜QŸXŸX˜XØ×)Ñ)°'Ð)ÓG‰DˆAØ—9‘9˜Q“?×/Ñ/Ó1‰DˆAØ—M‘M“O‰EˆAØ”A—E‘E‹z˜RœWð Ÿ)™)¤K°Ó$5°qÓ9Ð9Ð9Ü�u‰u�d—i‘i “oÐ%Ð%r(   c                	ó®  € T pVP                  R R4      '       d   VP                  4       pVP                  4       w  rEVP                  ^ 4      pV'       d   WE,          pWs8w  d   VP	                  4       # VP                  ! V!  pVP                  ! V!  p	V	'       d%   V'       d   R# VP                  ^ 4      pVRJ d   R# MV	f   R# VP                  ^ 4      # )r4  TFN)re  r4  r“   ÚequalsÚis_constant)
r,   ÚwrtÚflagsr«  rA   rB   Úbzr<  ÚeconÚbcons
   &*,       r%   rn  ÚPow.is_constant  s³   € ØˆØ�9‰9�Z ×&Ò&Ø—=‘=“?ˆDØ×ÑÓ!‰ˆØ�X‰X�a‹[ˆßØ•$ˆCØŒ{Ø—‘Ó(Ð(Ø�}Š}˜cÑ"ˆØ�}Š}˜cÑ"ˆßßÙØ—‘˜!“ˆBØ�U‹{Ùð àŠ\Ùà�x‰x˜‹{Ðr(   c                	óì   € V P                   w  r4VP                  V4      '       dO   VP                  V4      '       g6   VP                  WV,           4      pW5V,
          ,          ^,
          V ,          # R# R# )r:   N)r-   rÆ  r2  )r,   r¢   ÚsteprA   rB   Únew_es   &&&   r%   Ú_eval_difference_deltaÚPow._eval_difference_delta+  sU   € Ø�y‰y‰ˆØ�5‰5��8Š8˜AŸE™E !ŸHšHØ—F‘F˜1 $�hÓ'ˆEØ �	•N QÕ&¨$Õ.Ð.ñ %‰8r(   r#   rZ  r*   r9   rÊ   )NFr2   rÅ  )DrX   Ú
__module__Ú__qualname__Ú__firstlineno__Ú__doc__r;  Ú	__slots__r   Úpropertyr-   r4   r;   r>   r   rz   rˆ   Úclassmethodr�   r˜   ry   rÄ   rÇ   rÍ   rÔ   rË   rß   rå   rø   r  r
  r  r  r  r  r  rD  r“   rK  rO  rS  rj  rn  rŒ  r�  r¼  r¿  rÈ  rÎ  rÕ  rØ  rÛ  rã  ré  r¡   rò  r*  r2  rK  rO  rR  rU  rZ  r_  rd  rn  rx  Ú__static_attributes__Ú__classcell__)r`   s   @r%   r   r      s¸  ø† ñWðp €Fà#€Içà	ô	ó 
ð	ð ôó ðð ôó ðð ñ!ó ð!ð ö`ó ð`ôDð ñ"ó ð"ò#òR$òh6Bòp%òò3ò:ò2ò0ò*D$òLò/òbòò òò"òòBò8
)òò+òò(yòvxôtQ*òfKò$òòò6%,òNò#-òJòJò(!4ôF ôDzòx #ðD ñ7ó ð7õ!ò jò
jò
Aò
	9ôO&òb÷./ð /r(   r   Úpower)r:  )r¾   rï   )rì   rc  )r7  r¦  r¥  N);Ú
__future__r   Útypingr   r   Ú	itertoolsr   rý  r   Úcacher   Ú	singletonr	   r«  r
   r©   r   rc  r   r   r   r   r   Úlogicr   r   r   r   Ú
parametersr   rT   r   r   r>   r   r   Úsympy.utilities.iterablesr   Úsympy.utilities.exceptionsr   Úsympy.utilities.miscr   Úsympy.multipledispatchr   r   rƒ  ÚaddÚobjectr:  r(  r¾   rï   r  rì   rc  r+  r7  r¦  r¥  r#   r(   r%   Ú<module>r‘     s   ðÝ "ß *Ý å Ý Ý Ý Ý %÷%õ %ç =Ó =Ý )ß $ß +Ý *Ý @Ý 'Ý -ôY/ˆ$ô Y/ñv8 	�7Ó€Ø ‡	�	ˆ6�6Ð
˜CÔ  å ß &ß !ß *Ó *r(   