# Source code for diofant.functions.special.tensor_functions

from ...core import Function, Integer, prod
from ...utilities import default_sort_key, has_dups

###############################################################################
# #################### Kronecker Delta, Levi-Civita etc. #################### #
###############################################################################

[docs]def Eijk(*args, **kwargs):
"""
Represent the Levi-Civita symbol.

This is just compatibility wrapper to LeviCivita().

========

diofant.functions.special.tensor_functions.LeviCivita

"""
return LeviCivita(*args, **kwargs)

[docs]def eval_levicivita(*args):
"""Evaluate Levi-Civita symbol."""
from .. import factorial
n = len(args)
return prod(
prod(args[j] - args[i] for j in range(i + 1, n))
/ factorial(i) for i in range(n))
# converting factorial(i) to int is slightly faster

[docs]class LeviCivita(Function):
"""Represent the Levi-Civita symbol.

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 diofant.abc import i, j
>>> 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

========

diofant.functions.special.tensor_functions.Eijk

"""

is_integer = True

@classmethod
def eval(cls, *args):
if all(isinstance(a, (int, Integer)) for a in args):
return eval_levicivita(*args)
if has_dups(args):
return Integer(0)

def doit(self, **hints):
return eval_levicivita(*self.args)

[docs]class KroneckerDelta(Function):
"""The discrete, or Kronecker, delta function.

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.

Parameters
==========

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

Examples
========

A simple example with integer indices::

>>> KroneckerDelta(1, 2)
0
>>> KroneckerDelta(3, 3)
1

Symbolic indices::

>>> from diofant.abc import i, j
>>> 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)

========

diofant.functions.special.tensor_functions.KroneckerDelta.eval
diofant.functions.special.delta_functions.DiracDelta

References
==========

* https://en.wikipedia.org/wiki/Kronecker_delta

"""

is_integer = True

[docs]    @classmethod
def eval(cls, i, j):
"""
Evaluates the discrete delta function.

Examples
========

>>> from diofant.abc import i, j

>>> 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)

"""
diff = i - j
if diff.is_zero:
return Integer(1)
elif diff.is_nonzero:
return Integer(0)

if i._assumptions.get("below_fermi") and \
j._assumptions.get("above_fermi"):
return Integer(0)
if j._assumptions.get("below_fermi") and \
i._assumptions.get("above_fermi"):
return Integer(0)
# to make KroneckerDelta canonical
# following lines will check if inputs are in order
# if not, will return KroneckerDelta with correct order
if i is not min(i, j, key=default_sort_key):
return cls(j, i)

def _eval_power(self, expt):
if expt.is_positive:
return self
if expt.is_negative and -expt != 1:
return 1/self

@property
def is_above_fermi(self):
"""
True if Delta can be non-zero above fermi

Examples
========

>>> 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

========

diofant.functions.special.tensor_functions.KroneckerDelta.is_below_fermi
diofant.functions.special.tensor_functions.KroneckerDelta.is_only_below_fermi
diofant.functions.special.tensor_functions.KroneckerDelta.is_only_above_fermi

"""
if self.args[0]._assumptions.get("below_fermi"):
return False
if self.args[1]._assumptions.get("below_fermi"):
return False
return True

@property
def is_below_fermi(self):
"""
True if Delta can be non-zero below fermi

Examples
========

>>> 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

========

diofant.functions.special.tensor_functions.KroneckerDelta.is_above_fermi
diofant.functions.special.tensor_functions.KroneckerDelta.is_only_above_fermi
diofant.functions.special.tensor_functions.KroneckerDelta.is_only_below_fermi

"""
if self.args[0]._assumptions.get("above_fermi"):
return False
if self.args[1]._assumptions.get("above_fermi"):
return False
return True

@property
def is_only_above_fermi(self):
"""
True if Delta is restricted to above fermi

Examples
========

>>> 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

========

diofant.functions.special.tensor_functions.KroneckerDelta.is_above_fermi
diofant.functions.special.tensor_functions.KroneckerDelta.is_below_fermi
diofant.functions.special.tensor_functions.KroneckerDelta.is_only_below_fermi

"""
return (self.args[0]._assumptions.get("above_fermi") or
self.args[1]._assumptions.get("above_fermi") or False)

@property
def is_only_below_fermi(self):
"""
True if Delta is restricted to below fermi

Examples
========

>>> 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

========

diofant.functions.special.tensor_functions.KroneckerDelta.is_above_fermi
diofant.functions.special.tensor_functions.KroneckerDelta.is_below_fermi
diofant.functions.special.tensor_functions.KroneckerDelta.is_only_above_fermi

"""
return (self.args[0]._assumptions.get("below_fermi") or
self.args[1]._assumptions.get("below_fermi") or False)

@property
def indices_contain_equal_information(self):
"""
Returns True if indices are either both above or below fermi.

Examples
========

>>> 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

"""
if (self.args[0]._assumptions.get("below_fermi") and
self.args[1]._assumptions.get("below_fermi")):
return True
if (self.args[0]._assumptions.get("above_fermi")
and self.args[1]._assumptions.get("above_fermi")):
return True

# if both indices are general we are True, else false
return self.is_below_fermi and self.is_above_fermi

@property
def preferred_index(self):
"""
Returns the index which is preferred to keep in the final expression.

level.  If indices contain same information, 'a' is preferred before
'b'.

Examples
========

>>> 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

========

diofant.functions.special.tensor_functions.KroneckerDelta.killable_index

"""
if self._get_preferred_index():
return self.args[1]
else:
return self.args[0]

@property
def killable_index(self):
"""
Returns the index which is preferred to substitute in the final
expression.

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

Examples
========

>>> 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

========

diofant.functions.special.tensor_functions.KroneckerDelta.preferred_index

"""
if self._get_preferred_index():
return self.args[0]
else:
return self.args[1]

def _get_preferred_index(self):
"""
Returns the index which is preferred to keep in the final expression.

level.  If indices contain same information, index 0 is returned.

"""
if not self.is_above_fermi:
if self.args[0]._assumptions.get("below_fermi"):
return 0
else:
return 1
elif not self.is_below_fermi:
if self.args[0]._assumptions.get("above_fermi"):
return 0
else:
return 1
else:
return 0

@staticmethod
def _latex_no_arg(printer):
return r'\delta'