
    g/                         d dl mZ d dlmZmZ d dlmZ d dlmZ d dl	m
Z
 d dlmZ d dlmZ d dlmZ d d	lmZ d d
lmZ d Zd Z G d de          Z G d de          ZdS )    )prod)SInteger)Function)	fuzzy_not)Ne)default_sort_key)
SYMPY_INTS)	factorial)	Piecewise)has_dupsc                      t          | i |S )z
    Represent the Levi-Civita symbol.

    This is a compatibility wrapper to ``LeviCivita()``.

    See Also
    ========

    LeviCivita

    )
LeviCivita)argskwargss     d/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/functions/special/tensor_functions.pyEijkr      s     t&v&&&    c                  v     t                     t           fdt                    D                       S )zEvaluate Levi-Civita symbol.c              3      K   | ]At          fd t          dz             D                       t                    z  V  BdS )c              3   :   K   | ]}|                  z
  V  d S N ).0jr   is     r   	<genexpr>z,eval_levicivita.<locals>.<genexpr>.<genexpr>%   s0      881T!WtAw888888r      N)r   ranger   )r   r   r   ns    @r   r   z"eval_levicivita.<locals>.<genexpr>$   sr       * * 	88888a!eQ88888
A,,	* * * * * *r   )lenr   r   )r   r    s   `@r   eval_levicivitar"   !   sR    D		A * * * * * %a* * * * * *r   c                   2    e Zd ZdZdZed             Zd ZdS )r   aU  
    Represent the Levi-Civita symbol.

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

    For even permutations of indices it returns 1, for odd permutations -1, and
    for everything else (a repeated index) it returns 0.

    Thus it represents an alternating pseudotensor.

    Examples
    ========

    >>> from sympy import LeviCivita
    >>> from sympy.abc import i, j, k
    >>> LeviCivita(1, 2, 3)
    1
    >>> LeviCivita(1, 3, 2)
    -1
    >>> LeviCivita(1, 2, 2)
    0
    >>> LeviCivita(i, j, k)
    LeviCivita(i, j, k)
    >>> LeviCivita(i, j, i)
    0

    See Also
    ========

    Eijk

    Tc                     t          d |D                       r	t          | S t          |          rt          j        S d S )Nc              3   N   K   | ] }t          |t          t          f          V  !d S r   )
isinstancer
   r   )r   as     r   r   z"LeviCivita.eval.<locals>.<genexpr>Q   s1      BBz!j'233BBBBBBr   )allr"   r   r   Zero)clsr   s     r   evalzLeviCivita.evalO   sK    BBTBBBBB 	*"D))D>> 	6M	 	r   c                     t          | j         S r   )r"   r   )selfhintss     r   doitzLeviCivita.doitV   s    	**r   N)__name__
__module____qualname____doc__
is_integerclassmethodr+   r/   r   r   r   r   r   *   sN           D J  [+ + + + +r   r   c                      e Zd ZdZdZedd            Zed             Zd Z	ed             Z
ed             Zed	             Zed
             Zed             Zed             Zed             Zd Zed             Zd ZdS )KroneckerDeltaa  
    The discrete, or Kronecker, delta function.

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

    A function that takes in two integers $i$ and $j$. It returns $0$ if $i$
    and $j$ are not equal, or it returns $1$ if $i$ and $j$ are equal.

    Examples
    ========

    An example with integer indices:

        >>> from sympy import KroneckerDelta
        >>> KroneckerDelta(1, 2)
        0
        >>> KroneckerDelta(3, 3)
        1

    Symbolic indices:

        >>> from sympy.abc import i, j, k
        >>> KroneckerDelta(i, j)
        KroneckerDelta(i, j)
        >>> KroneckerDelta(i, i)
        1
        >>> KroneckerDelta(i, i + 1)
        0
        >>> KroneckerDelta(i, i + 1 + k)
        KroneckerDelta(i, i + k + 1)

    Parameters
    ==========

    i : Number, Symbol
        The first index of the delta function.
    j : Number, Symbol
        The second index of the delta function.

    See Also
    ========

    eval
    DiracDelta

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Kronecker_delta

    TNc                    |i|\  }}||z
  dk    dk    rt           j        S ||z
  dk    dk    rt           j        S ||z
  dk     dk    rt           j        S ||z
  dk     dk    rt           j        S ||z
  }|j        rt           j        S t	          |j                  rt           j        S |j                            d          r&|j                            d          rt           j        S |j                            d          r&|j                            d          rt           j        S t          |          t          |          k     r|r | |||          S  | ||          S dS )a  
        Evaluates the discrete delta function.

        Examples
        ========

        >>> from sympy import KroneckerDelta
        >>> from sympy.abc import i, j, k

        >>> KroneckerDelta(i, j)
        KroneckerDelta(i, j)
        >>> KroneckerDelta(i, i)
        1
        >>> KroneckerDelta(i, i + 1)
        0
        >>> KroneckerDelta(i, i + 1 + k)
        KroneckerDelta(i, i + k + 1)

        # indirect doctest

        Nr   Tbelow_fermiabove_fermi)r   r)   is_zeroOner   assumptions0getr	   )r*   r   r   delta_rangedinfdsupdiffs          r   r+   zKroneckerDelta.eval   sn   0 "$JD$q1%%vq1%%vq1%%vq1%%v1u< 	5Lt|$$ 	6M>m,, 	""=11	6M>m,, 	""=11	6M A!1!!4!444 !s1a---s1ayy 	 54r   c                 P    t          | j                  dk    r| j        d         S d S )N   )r!   r   r-   s    r   r?   zKroneckerDelta.delta_range   s)    ty>>A9Q< r   c                 P    |j         r| S |j        r|t          j        urd| z  S d S d S )Nr   )is_positiveis_negativer   NegativeOne)r-   expts     r   _eval_powerzKroneckerDelta._eval_power   sB     	K 	AM 9 9T6M	 	 9 9r   c                     | j         d         j                            d          rdS | j         d         j                            d          rdS dS )aG  
        True if Delta can be non-zero above fermi.

        Examples
        ========

        >>> from sympy import KroneckerDelta, Symbol
        >>> a = Symbol('a', above_fermi=True)
        >>> i = Symbol('i', below_fermi=True)
        >>> p = Symbol('p')
        >>> q = Symbol('q')
        >>> KroneckerDelta(p, a).is_above_fermi
        True
        >>> KroneckerDelta(p, i).is_above_fermi
        False
        >>> KroneckerDelta(p, q).is_above_fermi
        True

        See Also
        ========

        is_below_fermi, is_only_below_fermi, is_only_above_fermi

        r   r9   Fr   Tr   r=   r>   rE   s    r   is_above_fermizKroneckerDelta.is_above_fermi   Q    4 9Q<$((77 	59Q<$((77 	5tr   c                     | j         d         j                            d          rdS | j         d         j                            d          rdS dS )aG  
        True if Delta can be non-zero below fermi.

        Examples
        ========

        >>> from sympy import KroneckerDelta, Symbol
        >>> a = Symbol('a', above_fermi=True)
        >>> i = Symbol('i', below_fermi=True)
        >>> p = Symbol('p')
        >>> q = Symbol('q')
        >>> KroneckerDelta(p, a).is_below_fermi
        False
        >>> KroneckerDelta(p, i).is_below_fermi
        True
        >>> KroneckerDelta(p, q).is_below_fermi
        True

        See Also
        ========

        is_above_fermi, is_only_above_fermi, is_only_below_fermi

        r   r:   Fr   TrM   rE   s    r   is_below_fermizKroneckerDelta.is_below_fermi   rO   r   c                     | j         d         j                            d          p&| j         d         j                            d          pdS )aS  
        True if Delta is restricted to above fermi.

        Examples
        ========

        >>> from sympy import KroneckerDelta, Symbol
        >>> a = Symbol('a', above_fermi=True)
        >>> i = Symbol('i', below_fermi=True)
        >>> p = Symbol('p')
        >>> q = Symbol('q')
        >>> KroneckerDelta(p, a).is_only_above_fermi
        True
        >>> KroneckerDelta(p, q).is_only_above_fermi
        False
        >>> KroneckerDelta(p, i).is_only_above_fermi
        False

        See Also
        ========

        is_above_fermi, is_below_fermi, is_only_below_fermi

        r   r:   r   FrM   rE   s    r   is_only_above_fermiz"KroneckerDelta.is_only_above_fermi  L    4 1*..}== =	!)--m<< 	r   c                     | j         d         j                            d          p&| j         d         j                            d          pdS )aS  
        True if Delta is restricted to below fermi.

        Examples
        ========

        >>> from sympy import KroneckerDelta, Symbol
        >>> a = Symbol('a', above_fermi=True)
        >>> i = Symbol('i', below_fermi=True)
        >>> p = Symbol('p')
        >>> q = Symbol('q')
        >>> KroneckerDelta(p, i).is_only_below_fermi
        True
        >>> KroneckerDelta(p, q).is_only_below_fermi
        False
        >>> KroneckerDelta(p, a).is_only_below_fermi
        False

        See Also
        ========

        is_above_fermi, is_below_fermi, is_only_above_fermi

        r   r9   r   FrM   rE   s    r   is_only_below_fermiz"KroneckerDelta.is_only_below_fermi4  rT   r   c                 N   | j         d         j                            d          r'| j         d         j                            d          rdS | j         d         j                            d          r'| j         d         j                            d          rdS | j        o| j        S )a0  
        Returns True if indices are either both above or below fermi.

        Examples
        ========

        >>> from sympy import KroneckerDelta, Symbol
        >>> a = Symbol('a', above_fermi=True)
        >>> i = Symbol('i', below_fermi=True)
        >>> p = Symbol('p')
        >>> q = Symbol('q')
        >>> KroneckerDelta(p, q).indices_contain_equal_information
        True
        >>> KroneckerDelta(p, q+1).indices_contain_equal_information
        True
        >>> KroneckerDelta(i, p).indices_contain_equal_information
        False

        r   r9   r   Tr:   )r   r=   r>   rQ   rN   rE   s    r   !indices_contain_equal_informationz0KroneckerDelta.indices_contain_equal_informationS  s    * IaL%))-88 		!)--m<<	4IaL%))-88 	IaL-11-@@	4 ":t'::r   c                 ^    |                                  r| j        d         S | j        d         S )a  
        Returns the index which is preferred to keep in the final expression.

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

        The preferred index is the index with more information regarding fermi
        level. If indices contain the same information, 'a' is preferred before
        'b'.

        Examples
        ========

        >>> from sympy import KroneckerDelta, Symbol
        >>> a = Symbol('a', above_fermi=True)
        >>> i = Symbol('i', below_fermi=True)
        >>> j = Symbol('j', below_fermi=True)
        >>> p = Symbol('p')
        >>> KroneckerDelta(p, i).preferred_index
        i
        >>> KroneckerDelta(p, a).preferred_index
        a
        >>> KroneckerDelta(i, j).preferred_index
        i

        See Also
        ========

        killable_index

        r   r   _get_preferred_indexr   rE   s    r   preferred_indexzKroneckerDelta.preferred_indexr  s1    B $$&& 	 9Q<9Q<r   c                 ^    |                                  r| j        d         S | j        d         S )a)  
        Returns the index which is preferred to substitute in the final
        expression.

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

        The index to substitute is the index with less information regarding
        fermi level. If indices contain the same information, 'a' is preferred
        before 'b'.

        Examples
        ========

        >>> from sympy import KroneckerDelta, Symbol
        >>> a = Symbol('a', above_fermi=True)
        >>> i = Symbol('i', below_fermi=True)
        >>> j = Symbol('j', below_fermi=True)
        >>> p = Symbol('p')
        >>> KroneckerDelta(p, i).killable_index
        p
        >>> KroneckerDelta(p, a).killable_index
        p
        >>> KroneckerDelta(i, j).killable_index
        j

        See Also
        ========

        preferred_index

        r   r   rZ   rE   s    r   killable_indexzKroneckerDelta.killable_index  s1    D $$&& 	 9Q<9Q<r   c                     | j         s)| j        d         j                            d          rdS dS | j        s)| j        d         j                            d          rdS dS dS )z
        Returns the index which is preferred to keep in the final expression.

        The preferred index is the index with more information regarding fermi
        level. If indices contain the same information, index 0 is returned.

        r   r9   r   r:   )rN   r   r=   r>   rQ   rE   s    r   r[   z#KroneckerDelta._get_preferred_index  sq     " 	y|(,,];; qq$ 	y|(,,];; qq1r   c                      | j         dd         S )Nr   rD   )r   rE   s    r   indiceszKroneckerDelta.indices  s    y1~r   c                 L    |\  }}t          dt          ||          fd          S )Nr   )r   T)r   r   )r-   r   r   r   r   s        r   _eval_rewrite_as_Piecewisez)KroneckerDelta._eval_rewrite_as_Piecewise  s'    1!R1XX	222r   r   )r0   r1   r2   r3   r4   r5   r+   propertyr?   rK   rN   rQ   rS   rV   rX   r\   r^   r[   ra   rc   r   r   r   r7   r7   Z   sg       3 3j J5! 5! 5! [5!n     X      X>   X>   X<   X< ; ; X;< #  #  X# J $  $  X$ L  *   X3 3 3 3 3r   r7   N)mathr   
sympy.corer   r   sympy.core.functionr   sympy.core.logicr   sympy.core.relationalr   sympy.core.sortingr	   sympy.external.gmpyr
   (sympy.functions.combinatorial.factorialsr   $sympy.functions.elementary.piecewiser   sympy.utilities.iterablesr   r   r"   r   r7   r   r   r   <module>ro      sF         ! ! ! ! ! ! ! ! ( ( ( ( ( ( & & & & & & $ $ $ $ $ $ / / / / / / * * * * * * > > > > > > : : : : : : . . . . . .' ' '* * *-+ -+ -+ -+ -+ -+ -+ -+`@3 @3 @3 @3 @3X @3 @3 @3 @3 @3r   