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This module implements Pauli algebra by subclassing Symbol. Only algebraic
properties of Pauli matrices are used (we don't use the Matrix class).

See the documentation to the class Pauli for examples.

References
==========

.. [1] https://en.wikipedia.org/wiki/Pauli_matrices
�)�Symbol�I�Mul�Pow�Add)�
TensorProduct�evaluate_pauli_productcCs||krdSdS)z�
    Returns 1 if ``i == j``, else 0.

    This is used in the multiplication of Pauli matrices.

    Examples
    ========

    >>> from sympy.physics.paulialgebra import delta
    >>> delta(1, 1)
    1
    >>> delta(2, 3)
    0
    �r�)�i�jr
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�</usr/lib/python3/dist-packages/sympy/physics/paulialgebra.py�deltasrcCs(|||fdvr	dS|||fdvrdSdS)ak
    Return 1 if i,j,k is equal to (1,2,3), (2,3,1), or (3,1,2);
    -1 if ``i``,``j``,``k`` is equal to (1,3,2), (3,2,1), or (2,1,3);
    else return 0.

    This is used in the multiplication of Pauli matrices.

    Examples
    ========

    >>> from sympy.physics.paulialgebra import epsilon
    >>> epsilon(1, 2, 3)
    1
    >>> epsilon(1, 3, 2)
    -1
    )�r	��)rrr	)rr	rr	))r	rr)rrr	)rr	r���rr
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    The class representing algebraic properties of Pauli matrices.

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

    The symbol used to display the Pauli matrices can be changed with an
    optional parameter ``label="sigma"``. Pauli matrices with different
    ``label`` attributes cannot multiply together.

    If the left multiplication of symbol or number with Pauli matrix is needed,
    please use parentheses  to separate Pauli and symbolic multiplication
    (for example: 2*I*(Pauli(3)*Pauli(2))).

    Another variant is to use evaluate_pauli_product function to evaluate
    the product of Pauli matrices and other symbols (with commutative
    multiply rules).

    See Also
    ========

    evaluate_pauli_product

    Examples
    ========

    >>> from sympy.physics.paulialgebra import Pauli
    >>> Pauli(1)
    sigma1
    >>> Pauli(1)*Pauli(2)
    I*sigma3
    >>> Pauli(1)*Pauli(1)
    1
    >>> Pauli(3)**4
    1
    >>> Pauli(1)*Pauli(2)*Pauli(3)
    I

    >>> from sympy.physics.paulialgebra import Pauli
    >>> Pauli(1, label="tau")
    tau1
    >>> Pauli(1)*Pauli(2, label="tau")
    sigma1*tau2
    >>> Pauli(1, label="tau")*Pauli(2, label="tau")
    I*tau3

    >>> from sympy import I
    >>> I*(Pauli(2)*Pauli(3))
    -sigma1

    >>> from sympy.physics.paulialgebra import evaluate_pauli_product
    >>> f = I*Pauli(2)*Pauli(3)
    >>> f
    I*sigma2*sigma3
    >>> evaluate_pauli_product(f)
    -sigma1
    �r�label�sigmacCs:|dvrtd��tj|d||fddd�}||_||_|S)NrzInvalid Pauli indexz%s%dFT)�commutative�	hermitian)�
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isinstancerrrrrr�super�__mul__)r!�otherrr�jlab�klab��	__class__r
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is_Integer�is_positiver%�__pow__�int)�b�er*r
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__module__�__qualname__�__doc__�	__slots__rr"r#r&r2�
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rcCs�|}|}t|t�rt|jdt�r|jdjr|jdSdSt|t�r-tdd�|jD��St|t�r<tdd�|jD��St|t�sC|S||k||k||k@Br�|}|��}d}d}d}|dD]M}t|t�rl||9}q`|j	s�t|t�r�t|jdt�r�|jdjr�||jd9}q`t|t�r�||tdd�|jD��}d}q`|||}d}q`||9}q`|d|||}||kr�|S||k||k||k@BsP|S)a�Help function to evaluate Pauli matrices product
    with symbolic objects.

    Parameters
    ==========

    arg: symbolic expression that contains Paulimatrices

    Examples
    ========

    >>> from sympy.physics.paulialgebra import Pauli, evaluate_pauli_product
    >>> from sympy import I
    >>> evaluate_pauli_product(I*Pauli(1)*Pauli(2))
    -sigma3

    >>> from sympy.abc import x
    >>> evaluate_pauli_product(x**2*Pauli(2)*Pauli(1))
    -I*x**2*sigma3
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