
    \h`                     $   S SK Jr  S SKJrJrJrJrJrJrJ	r	J
r
JrJr  S SKJr  S SKJr  SSKJrJr  SSKJrJr  SSKJr  SS	KJrJrJrJrJr  S S
KJ r   S SK!J"r"  / SQr#S r$\$=RJ                  \RH                  RJ                  -  sl%        S r&\&=RJ                  \RL                  RJ                  -  sl%        SS jr'SS jr(S r)\)=RJ                  \RR                  RJ                  -  sl%        SS jr*S r+S r,SS jr-\" S5      \-l.        S\-l/        g)    )reduce)
sympifydiffsincosMatrixsymbolsFunctionSSymbollinear_eq_to_matrix)	integratetrigsimp   )Vector_check_vector)CoordinateSym_check_frame)Dyadic)vprintvsprintvpprintvlatexinit_vprinting)iterable)	translate)crossdotexpresstime_derivativeouterkinematic_equationsget_motion_paramspartial_velocitydynamicsymbolsr   r   r   r   r   c                 V    [        U [        [        45      (       d  [        S5      eX-  $ )z7Cross product convenience wrapper for Vector.cross(): 
z$Cross product is between two vectors
isinstancer   r   	TypeErrorvec1vec2s     V/var/www/auris/envauris/lib/python3.13/site-packages/sympy/physics/vector/functions.pyr   r      s(    dVV,-->??;    c                 V    [        U [        [        45      (       d  [        S5      eX-  $ )z3Dot product convenience wrapper for Vector.dot(): 
z"Dot product is between two vectorsr(   r+   s     r.   r   r      s(    dVV,--<==;r/   Nc           
      x   [        U5        U S:X  a  U $ [        U [        5      (       a  U(       aZ  U R                   Vs/ s H  oDS   PM	     nn0 nU H#  nUR	                  UR                  U5      5        M%     U R                  U5      n [        / 5      nU R                   Hn  n	U	S   U:w  aS  UR                  U	S   5      U	S   -  n
[        R                  (       a  U
R                  S 5      n
U[        X4/5      -  nM_  U[        U	/5      -  nMp     U$ [        U [        5      (       a\  Uc  Un[        U5        [        S5      nU R                   H/  n	U[        U	S   XS9[        U	S   XS9[        U	S   X#S9-  -  -  nM1     U$ U(       a  [        5       n[        U 5      n U R                   HG  n[        U[        5      (       d  M  UR                   U:w  d  M,  UR#                  UR                   5        MI     0 nU H#  nUR	                  UR                  U5      5        M%     U R                  U5      $ U $ s  snf )a.  
Global function for 'express' functionality.

Re-expresses a Vector, scalar(sympyfiable) or Dyadic in given frame.

Refer to the local methods of Vector and Dyadic for details.
If 'variables' is True, then the coordinate variables (CoordinateSym
instances) of other frames present in the vector/scalar field or
dyadic expression are also substituted in terms of the base scalars of
this frame.

Parameters
==========

expr : Vector/Dyadic/scalar(sympyfiable)
    The expression to re-express in ReferenceFrame 'frame'

frame: ReferenceFrame
    The reference frame to express expr in

frame2 : ReferenceFrame
    The other frame required for re-expression(only for Dyadic expr)

variables : boolean
    Specifies whether to substitute the coordinate variables present
    in expr, in terms of those of frame

Examples
========

>>> from sympy.physics.vector import ReferenceFrame, outer, dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> N = ReferenceFrame('N')
>>> q = dynamicsymbols('q')
>>> B = N.orientnew('B', 'Axis', [q, N.z])
>>> d = outer(N.x, N.x)
>>> from sympy.physics.vector import express
>>> express(d, B, N)
cos(q)*(B.x|N.x) - sin(q)*(B.y|N.x)
>>> express(B.x, N)
cos(q)*N.x + sin(q)*N.y
>>> express(N[0], B, variables=True)
B_x*cos(q) - B_y*sin(q)

r   r   c                     [        U SS9$ )Nfu)methodr   )xs    r.   <lambda>express.<locals>.<lambda>m   s    *21T*Br/   	variables   )r   r)   r   argsupdatevariable_mapsubsdcmsimp	applyfuncr   r    setr   free_symbolsr   frameadd)exprrE   frame2r:   r6   
frame_list	subs_dictfoutvecvtempol	frame_sets                r.   r    r    (   s   ` qy$ *.3AB%J3I  !67  99Y'DAtu}yy11-;;>> +C DD&4-11&!+%  $>FVAYA'!A$;1Q4<1Q4=>? ?B  	 I4=D&&a//AGGu4DMM!''* ' I  !67 99Y''U 4s   H7c                 T   [         R                  n[        U5        US:X  a  U $ US-  S:w  d  US:  a  [        S5      e[	        U [
        5      (       a  / nU R                   Hy  nUS   U:X  a%  U[        US   USS9R                  U5      U4/-  nM1  U[        [        U/5      US   5      US   R                  U5      [        U/5      -  -   R                  -  nM{     [        U5      n[        XaUS-
  5      $ [	        U [        5      (       a  [        S5      nU R                   H_  nXuS   R                  U5      US   US   -  -  -  nXuS   [        US   U5      US   -  -  -  nXuS   US   [        US   U5      -  -  -  nMa     [        XqUS-
  5      $ [        [        XSS9X25      $ )a  
Calculate the time derivative of a vector/scalar field function
or dyadic expression in given frame.

References
==========

https://en.wikipedia.org/wiki/Rotating_reference_frame#Time_derivatives_in_the_two_frames

Parameters
==========

expr : Vector/Dyadic/sympifyable
    The expression whose time derivative is to be calculated

frame : ReferenceFrame
    The reference frame to calculate the time derivative in

order : integer
    The order of the derivative to be calculated

Examples
========

>>> from sympy.physics.vector import ReferenceFrame, dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> from sympy import Symbol
>>> q1 = Symbol('q1')
>>> u1 = dynamicsymbols('u1')
>>> N = ReferenceFrame('N')
>>> A = N.orientnew('A', 'Axis', [q1, N.x])
>>> v = u1 * N.x
>>> A.set_ang_vel(N, 10*A.x)
>>> from sympy.physics.vector import time_derivative
>>> time_derivative(v, N)
u1'*N.x
>>> time_derivative(u1*A[0], N)
N_x*u1'
>>> B = N.orientnew('B', 'Axis', [u1, N.z])
>>> from sympy.physics.vector import outer
>>> d = outer(N.x, N.x)
>>> time_derivative(d, B)
- u1'*(N.y|N.x) - u1'*(N.x|N.y)

r   r   z"Unsupported value of order enteredTr9   r;   )r&   _tr   
ValueErrorr)   r   r<   r    r   r!   
ang_vel_inr   )rG   rE   ordertoutlistrM   rL   rO   s           r.   r!   r!      s   ` 	AzqyA~=>>$Atu}WQqT5DAFFqI"$ % % OFA3K1>qT__U3faSkACDHDI  veai88$AYAQ499Q<1Q4!A$;/0BQ4?1Q47!A$>?@BQ41Q4/!A$">>?@B  r%!)44 GD48!CCr/   c                 d    [        U [        5      (       d  [        S5      eU R                  U5      $ )z6Outer product convenience wrapper for Vector.outer():
z$Outer product is between two Vectors)r)   r   r*   r"   r+   s     r.   r"   r"      s*    dF##>??::dr/   c           	      :   Sn[        [        U5      SS5      nUR                  5       n[        U [        [
        45      (       d  [        S5      e[        U 5      S:w  a  [        S5      e[        U[        [
        45      (       d  [        S5      eUS;   Ga  X4;  a  [        S	5      e[        U5      S:w  a  [        S
5      eU u  pVnXVs=:X  a  Us=:X  a  S:X  a  O  O[        R                  /S-  $ Uu  pn
U Vs/ s H  n[        U[        R                  5      PM     snu  pn[        U5      [        U	5      [        U
5      /u  nnn[        U5      [        U	5      [        U
5      /u  nnnUS:X  Ga  US:X  a3  XU-  UU-  -
  U-  -
  XU-  -
  UU-  -
  X* U-  UU-  -   U-  U-  -
  U-
  /$ US:X  a3  XU-  UU-  -
  U-  -
  XU-  -
  UU-  -
  X-
  U* U-  UU-  -   U-  U-  -
  /$ US:X  a3  X* U-  UU-  -   U-  -
  XU-  -
  UU-  -
  XU-  UU-  -
  U-  U-  -
  U-
  /$ US:X  a2  XU-  UU-  -   U-  -
  XU-  -   UU-  -
  XU-  UU-  -   U-  U-  -
  U-
  /$ US:X  a2  XU-  UU-  -   U-  -
  XU-  -
  UU-  -   XU-  UU-  -   U-  U-  -
  U-
  /$ US:X  a2  XU-  UU-  -   U-  -
  XU-  -
  UU-  -   X-
  UU-  UU-  -   U-  U-  -
  /$ US:X  a2  XU-  UU-  -   U-  -
  XU-  -
  UU-  -   X-
  UU-  UU-  -   U-  U-  -   /$ US:X  a3  X* U-  UU-  -   U-  -
  XU-  -
  UU-  -
  X-
  UU-  UU-  -
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  /$ US:X  a3  XU-  UU-  -
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  UU-  -
  X* U-  UU-  -   U-  U-  -
  U-
  /$ US:X  a2  XU-  UU-  -   U-  -
  XU-  -   UU-  -
  XU-  UU-  -   U-  U-  -   U-
  /$ US:X  a2  XU-  UU-  -   U-  -
  XU-  -
  UU-  -   XU-  UU-  -   U-  U-  -   U-
  /$ US:X  a3  X* U-  UU-  -   U-  -
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  U-
  /$ US:X  Ga  US:X  a0  X-
  Xo-  UU-  -   U-  U-  -
  XU-  -
  X-  -   XU-  UU-  -   U-  -
  /$ US:X  a0  XU-  X-  -   U-  U-  -
  U-
  XU-  -   UU-  -
  XU-  X-  -   U-  -
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  U-
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  Xo-  -   XU-  UU-  -   U-  -
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  U* U-  X-  -   U-  U-  -
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  XU-  X-  -   U-  -
  /$ US:X  a1  XU-  UU-  -   U-  U-  -   U-
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  /$ g!g!US:X  a  US:w  a  [        S5      e[        U5      S:w  a  [        S5      eUu  nnnn[!        U S/-   5      n[!        UU* UU/UUU* U/U* UUU/U* U* U* U//5      n[!        UUUU4 Vs/ s H  n[        U[        R                  5      PM     sn5      n[	        UR"                  SUR"                  -  UR"                  -  -
  5      $ [        S 5      es  snf s  snf )"a}  Gives equations relating the qdot's to u's for a rotation type.

Supply rotation type and order as in orient. Speeds are assumed to be
body-fixed; if we are defining the orientation of B in A using by rot_type,
the angular velocity of B in A is assumed to be in the form: speed[0]*B.x +
speed[1]*B.y + speed[2]*B.z

Parameters
==========

speeds : list of length 3
    The body fixed angular velocity measure numbers.
coords : list of length 3 or 4
    The coordinates used to define the orientation of the two frames.
rot_type : str
    The type of rotation used to create the equations. Body, Space, or
    Quaternion only
rot_order : str or int
    If applicable, the order of a series of rotations.

Examples
========

>>> from sympy.physics.vector import dynamicsymbols
>>> from sympy.physics.vector import kinematic_equations, vprint
>>> u1, u2, u3 = dynamicsymbols('u1 u2 u3')
>>> q1, q2, q3 = dynamicsymbols('q1 q2 q3')
>>> vprint(kinematic_equations([u1,u2,u3], [q1,q2,q3], 'body', '313'),
...     order=None)
[-(u1*sin(q3) + u2*cos(q3))/sin(q2) + q1', -u1*cos(q3) + u2*sin(q3) + q2', (u1*sin(q3) + u2*cos(q3))*cos(q2)/sin(q2) - u3 + q3']

)123231312132213321121131212232313323123 XYZxyz123123zNeed to supply speeds in a list   z"Need to supply 3 body-fixed speedsz$Need to supply coordinates in a list)bodyspacez Not an acceptable rotation orderz$Need 3 coordinates for body or spacer   rm   rZ   r[   r\   r]   r^   r_   r`   ra   rb   rc   rd   re   rn   
quaternionri   z)Cannot have rotation order for quaternion   z!Need 4 coordinates for quaterniong      ?z/Not an approved rotation type for this functionN)r   strlowerr)   listtupler*   lenrS   r   Zeror   r&   rR   r   r   r   T)speedscoordsrot_type	rot_orderapproved_ordersw1w2w3q1q2q3iq1dq2dq3ds1s2s3c1c2c3e0e1e2e3wEedotss                               r.   r#   r#      s,   FFO #i.(H=I~~HftUm,,9::
6{a<==ftUm,,>??$$+?@@v;!CDD
rQFF8A:
=CDVa!2!23VD#"gs2wB0
B"gs2wB0
BvE!Bwb0B662gI 938b2g#5";b"@@2EG GE!Bwb0B662gI 9H"r	BG(;r'AB'FFH HE!sRx"r'1R77BwJ :7R"W#4":R"??"DF FE!Bwb0B662gI 97R"W#4":R"??"DF FE!Bwb0B662gI 97R"W#4":R"??"DF FE!Bwb0B662gI 9HR"r'(9R'?"'DDF FE!Bwb0B662gI 9HR"r'(9R'?"'DDF FE!sRx"r'1R77BwJ :HR"r'(9R'?"'DDF FE!Bwb0B662gI 938b2g#5";b"@@2EG GE!Bwb0B662gI 97R"W#4":R"??"DF FE!Bwb0B662gI 97R"W#4":R"??"DF FE!sRx"r'1R77BwJ :7R"W#4":R"??"DF FwE!BGb2g$5#;b#@@#I CWC%&)"WrBw->",D&DF FE!Bw0B6;;b@#I C"WC%&)"Wrw->",D&DF FE!Bwb0B6;;b@#I CWC%&)"WrBw->",D&DF FE!RC"Hrw$6"#<r#AA3J D"WD%&)"Wrw->",D&DF FE!Bwb0B6;;b@#I CWC%&)S2XR-?2,E&EG GE!sRx"'1R7"<<rA3J D"WD%&)"Wrw->",D&DF FE!BGb2g$5#;b#@@#I CWC%&)"WrBw->",D&DF FE!BGbg$5#;b#@@#I C"WC%&)S2X-?2,E&EG GE!sRx"r'1R7"<<rA3J DWD%&)"WrBw->",D&DF FE!Bw0B6;;b@#I C"WC%&)"Wrw->",D&DF FE!Bwb0B6;;b@#I CWC%&)"WrBw->",D&DF FE!Bw0B6;;b@#I C"WC%&)S2X-?2,E&EG G "E J 
\	!?HIIv;!@AABB6QC< R"b"%bS"%S"b"%S2#sB') * RR<LM<LqQ 1 12<LMNEGGcACCi!##o-..JKKy Er Ns   8$^+$^c                 X   S n[        U 5        SU;   a  SnOSU;   a  SnOSn/ SQn[        U5       HW  u  pVXa;  a)  US:  a  [        S5      X'   M  [        R                  X'   M3  US:  a  [        X   5        MH  [        X   5      X'   MY     US:X  aO  U" US   US   [        R                  US	   U 5      S   nU" XqS
   [        R                  US   U 5      S   nUS   Xx4$ US:X  a#  U" US   US
   [        R                  US   U 5      $ [        US
   U 5      n[        Xp5      n	XUS
   4$ )a  
Returns the three motion parameters - (acceleration, velocity, and
position) as vectorial functions of time in the given frame.

If a higher order differential function is provided, the lower order
functions are used as boundary conditions. For example, given the
acceleration, the velocity and position parameters are taken as
boundary conditions.

The values of time at which the boundary conditions are specified
are taken from timevalue1(for position boundary condition) and
timevalue2(for velocity boundary condition).

If any of the boundary conditions are not provided, they are taken
to be zero by default (zero vectors, in case of vectorial inputs). If
the boundary conditions are also functions of time, they are converted
to constants by substituting the time values in the dynamicsymbols._t
time Symbol.

This function can also be used for calculating rotational motion
parameters. Have a look at the Parameters and Examples for more clarity.

Parameters
==========

frame : ReferenceFrame
    The frame to express the motion parameters in

acceleration : Vector
    Acceleration of the object/frame as a function of time

velocity : Vector
    Velocity as function of time or as boundary condition
    of velocity at time = timevalue1

position : Vector
    Velocity as function of time or as boundary condition
    of velocity at time = timevalue1

timevalue1 : sympyfiable
    Value of time for position boundary condition

timevalue2 : sympyfiable
    Value of time for velocity boundary condition

Examples
========

>>> from sympy.physics.vector import ReferenceFrame, get_motion_params, dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> from sympy import symbols
>>> R = ReferenceFrame('R')
>>> v1, v2, v3 = dynamicsymbols('v1 v2 v3')
>>> v = v1*R.x + v2*R.y + v3*R.z
>>> get_motion_params(R, position = v)
(v1''*R.x + v2''*R.y + v3''*R.z, v1'*R.x + v2'*R.y + v3'*R.z, v1*R.x + v2*R.y + v3*R.z)
>>> a, b, c = symbols('a b c')
>>> v = a*R.x + b*R.y + c*R.z
>>> get_motion_params(R, velocity = v)
(0, a*R.x + b*R.y + c*R.z, a*t*R.x + b*t*R.y + c*t*R.z)
>>> parameters = get_motion_params(R, acceleration = v)
>>> parameters[1]
a*t*R.x + b*t*R.y + c*t*R.z
>>> parameters[2]
a*t**2/2*R.x + b*t**2/2*R.y + c*t**2/2*R.z

c                 :   US:w  a
  [        XSS9nU [        S5      :X  a  SSU4$ [        X5      n[        X5      n[        S5      nX#U4nU HI  n	UR                  U	5      n
U	R                  U5      R	                  X#05      nU[        X5      U-   U	-  -  nMK     X`U4$ )z
Helper function for get_motion methods. Finds derivative of vectdiff
wrt variable, and its integral using the specified boundary condition
at value of variable = ordinate.
Returns a tuple of - (derivative, function and integral) wrt vectdiff

r   Tr9   )r    r   r!   r   r?   r   )vectdiff	conditionvariableordinaterE   	vectdiff1	vectdiff2	vectdiff0limsdim	function1abscissas               r.   _process_vector_differential7get_motion_params.<locals>._process_vector_differential  s     >	DAIvay q)$$H,	#H4	1I	H-C!c*Iwwy)../CDH )I4x?3FFI  Y//r/   accelerationr;   velocityr   r   )r   r   position	timevalue
timevalue1
timevalue2rl   r   r   r   )
r   	enumerater   r   rv   r   r   r&   rR   r!   )
rE   kwargsr   mode
conditionsr   r6   velposaccs
             r.   r$   r$     su   L0@ 	v	
;J*%?1u"1I	FF	U&)$	*FI & qy*6.+A+1*+=+9+<+<+1,+?H IJK +3z0B+9+<+<+1,+?HHIK ~&11	+F:,>,2:,>,:,=,=,2<,@%I 	I
 fZ0%8c)&,--r/   c                    [        U 5      (       d  [        S5      e[        U5      (       d  [        S5      e/ n[        U5      nU  H  nU Vs/ s H  n[        S5      PM     nnUR                   H_  u  px[        Xq5      u  p[        [        U5      5       H5  n
[        U5       H#  u  pXU
4   S:w  d  M  Xj==   XX4   -  -  ss'   M%     M7     Ma     UR                  U5        M     U$ s  snf )a7  Returns a list of partial velocities with respect to the provided
generalized speeds in the given reference frame for each of the supplied
velocity vectors.

The output is a list of lists. The outer list has a number of elements
equal to the number of supplied velocity vectors. The inner lists are, for
each velocity vector, the partial derivatives of that velocity vector with
respect to the generalized speeds supplied.

Parameters
==========

vel_vecs : iterable
    An iterable of velocity vectors (angular or linear).
gen_speeds : iterable
    An iterable of generalized speeds.
frame : ReferenceFrame
    The reference frame that the partial derivatives are going to be taken
    in.

Examples
========

>>> from sympy.physics.vector import Point, ReferenceFrame
>>> from sympy.physics.vector import dynamicsymbols
>>> from sympy.physics.vector import partial_velocity
>>> u = dynamicsymbols('u')
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> P.set_vel(N, u * N.x)
>>> vel_vecs = [P.vel(N)]
>>> gen_speeds = [u]
>>> partial_velocity(vel_vecs, gen_speeds, N)
[[N.x]]

z2Velocity vectors must be contained in an iterable.z3Generalized speeds must be contained in an iterabler   )
r   r*   rs   r   r<   r   rangeru   r   append)vel_vecs
gen_speedsrE   vec_partialsr   _partials
componentsrefmatr   r   	directions                r.   r%   r%     s    L HLMMJMNNLj!J'12z!F1Iz2"xxOJ(@FC3z?+&/nNC6{a' ysv;'>> '5 ,  ( 	H%   3s   C*c           
          [        U 4S[        0UD6n[        R                  n[	        U5      (       a-  U Vs/ s H  n[        [        U/U-  U" U5      5      PM      nnU$ [        [        U/U-  U" U5      5      $ s  snf )aI  Uses symbols and Function for functions of time.

Creates a SymPy UndefinedFunction, which is then initialized as a function
of a variable, the default being Symbol('t').

Parameters
==========

names : str
    Names of the dynamic symbols you want to create; works the same way as
    inputs to symbols
level : int
    Level of differentiation of the returned function; d/dt once of t,
    twice of t, etc.
assumptions :
    - real(bool) : This is used to set the dynamicsymbol as real,
                by default is False.
    - positive(bool) : This is used to set the dynamicsymbol as positive,
                by default is False.
    - commutative(bool) : This is used to set the commutative property of
                a dynamicsymbol, by default is True.
    - integer(bool) : This is used to set the dynamicsymbol as integer,
                by default is False.

Examples
========

>>> from sympy.physics.vector import dynamicsymbols
>>> from sympy import diff, Symbol
>>> q1 = dynamicsymbols('q1')
>>> q1
q1(t)
>>> q2 = dynamicsymbols('q2', real=True)
>>> q2.is_real
True
>>> q3 = dynamicsymbols('q3', positive=True)
>>> q3.is_positive
True
>>> q4, q5 = dynamicsymbols('q4,q5', commutative=False)
>>> bool(q4*q5 != q5*q4)
True
>>> q6 = dynamicsymbols('q6', integer=True)
>>> q6.is_integer
True
>>> diff(q1, Symbol('t'))
Derivative(q1(t), t)

cls)r	   r
   r&   rR   r   r   r   )nameslevelassumptionsessesrV   es         r.   r&   r&   O  sx    b E7x7;7EA:?@%QqcEk1Q40%@dQC%Kq22 As   %A;rV   ')NF)r   )ri   )r   )0	functoolsr   sympyr   r   r   r   r   r	   r
   r   r   r   sympy.integrals.integralsr   sympy.simplify.trigsimpr   vectorr   r   rE   r   r   dyadicr   printingr   r   r   r   r   sympy.utilities.iterablesr   sympy.utilities.miscr   __all__r   __doc__r   r    r!   r"   r#   r$   r%   r&   rR   _str r/   r.   <module>r      s    J J J / , ) .  F F . * %% % vzz!! !dNMD` %% %ULpO.d9x73t 3K  r/   