#!/usr/bin/python

# TBD: read about Factory Method, variable in docstring, sage documentation

"""
This script calculates signature functions for knots (cable sums).

The script can be run as a sage script from the terminal
or used in interactive mode.

A knot (cable sum) is encoded as a list where each element (also a list)
corresponds to a cable knot, e.g. a list
[[1, 3], [2], [-1, -2], [-3]] encodes
T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7).

To calculate the number of characters for which signature function vanish use
the function eval_cable_for_null_signature as shown below.

sage: eval_cable_for_null_signature([[1, 3], [2], [-1, -2], [-3]])

T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7)
Zero cases: 1
All cases: 1225
Zero theta combinations:
(0, 0, 0, 0)

sage:

The numbers given to the function eval_cable_for_null_signature are k-values for each
component/cable in a direct sum.

To calculate signature function for a knot and a theta value, use function
get_function_of_theta_for_sum (see help/docstring for details).

About notation:
Cables that we work with follow a schema:
    T(2, q_0; 2, q_1; 2, q_3) # -T(2, q_1; 2, q_3) #
            # T(2, q_2; 2, q_3) # -T(2, q_0; 2, q_2; 2, q_3)
In knot_formula each k[i] is related with some q_i value, where
q_i = 2*k[i] + 1.
So we can work in the following steps:
1) choose a schema/formula by changing the value of knot_formula
2) set each q_i all or choose range in which q_i should varry
3) choose vector v / theata vector.


"""

import os
import sys

import collections
import inspect
import itertools as it
import pandas as pd
import numpy as np
import re


class MySettings(object):
    def __init__(self):
        self.f_results = os.path.join(os.getcwd(), "results.out")

        # is the ratio restriction for values in k_vector taken into account
        # False flag is usefull to make quick script tests
        # self.only_slice_candidates = True
        self.only_slice_candidates = False

        # knot_formula is a schema for knots which signature function
        # will be calculated
        self.knot_formula = "[[k[0], k[1], k[3]], [-k[1], -k[3]], \
                             [k[2], k[3]], [-k[0], -k[2], -k[3]]]"


        # self.knot_formula = "[[k[0], k[1], k[2]], [k[3], k[4]], \
        #                      [-k[0], -k[3], -k[4]], [-k[1], -k[2]]]"
        # self.knot_formula = "[[k[0], k[1], k[2]], [k[3]],\
        #                          [-k[0], -k[1], -k[3]], [-k[2]]]"
        self.default_limit = 3

        self.verbose = True


class SignatureFunction(object):
    """
    This simple class encodes twisted and untwisted signature functions
    of knots. Since the signature function is entirely encoded by its signature
    jump, the class stores only information about signature jumps
    in a dictionary self.data.
    The dictionary stores data of the signature jump as a key/values pair,
    where the key is the argument at which the functions jumps
    and value encodes the value of the jump. Remember that we treat
    signature functions as defined on the interval [0,1).
    """
    def __init__(self, values=[]):
        # We will store data of signature jumps here.
        self.data = collections.defaultdict(int)
        # values contain initial data of singature jumps
        for jump_arg, jump in values:
            assert 0 <= jump_arg < 1, \
                "Signature function is defined on the interval [0, 1)."
            self.data[jump_arg] = jump

    def sum_of_absolute_values(self):
        return sum([abs(i) for i in self.data.values()])

    def double_cover(self):
        # to read values for t^2
        new_data = []
        for jump_arg, jump in self.data.items():
            new_data.append((mod_one(jump_arg/2), jump))
            new_data.append((mod_one(1/2 + jump_arg/2), jump))
        return SignatureFunction(new_data)

    def square_root(self):
        # to read values for t^(1/2)
        new_data = []
        for jump_arg, jump in self.data.items():
            if jump_arg < 1/2:
                new_data.append((2 * jump_arg, jump))
        return SignatureFunction(new_data)

    def get_signture_jump(self, t):
        return self.data.get(t, 0)

    def minus_square_root(self):
        # to read values for t^(1/2)
        new_data = []
        for jump_arg, jump in self.data.items():
            if jump_arg >= 1/2:
                new_data.append((mod_one(2 * jump_arg), jump))
        return SignatureFunction(new_data)

    def __lshift__(self, shift):
        # A shift of the signature functions corresponds to the rotation.
        return self.__rshift__(-shift)

    def __rshift__(self, shift):
        new_data = []
        for jump_arg, jump in self.data.items():
            new_data.append((mod_one(jump_arg + shift), jump))
        return SignatureFunction(new_data)

    def __neg__(self):
        # we can perform arithmetic operations on signature functions.
        new_data = []
        for jump_arg, jump in self.data.items():
            new_data.append(jump_arg, -jump)
        return SignatureFunction(new_data)

    def __add__(self, other):
        new_signature_function = SignatureFunction()
        new_data = collections.defaultdict(int)
        for jump_arg, jump in other.data.items():
            new_data[jump_arg] = jump + self.data.get(jump_arg, 0)
        for jump_arg, jump in self.data.items():
            if jump_arg not in new_data.keys():
                new_data[jump_arg] = self.data[jump_arg]
        new_signature_function.data = new_data
        return new_signature_function

    def __sub__(self, other):
        return self + other.__neg__()

    def __str__(self):
        return ''.join([str(jump_arg) + ": " + str(jump) + "\n"
                          for jump_arg, jump in sorted(self.data.items())])
    def value(self, arg):
      # Compute the value of the signature function at the point arg.
      # This requires summing all signature jumps that occur before arg.
      assert 0 <= arg and arg < 1
      val = 0
      for jump_arg, jump in self.data.items():
          if jump_arg < arg:
              val += 2 * jump
          elif jump_arg == arg:
              val += jump
      return val


def main(arg):
    try:
        new_limit = int(arg[1])
    except:
        new_limit = None
    search_for_large_signature_value(limit=new_limit)
    # search_for_null_signature_value(limit=new_limit)


def search_for_large_signature_value(knot_formula=None,
                                     limit=None, verbose=None):
    if limit is None:
        limit = config.default_limit
    if knot_formula is None:
        knot_formula = config.knot_formula
    if verbose is None:
        vebose = config.verbose

    k_vector_size = extract_max(knot_formula) + 1
    limit = max(limit, k_vector_size)
    print limit
    combinations = it.combinations(range(1, limit + 1), k_vector_size)
    for k in combinations:
            print
            print k
#  P = Primes()
# sage: P.unrank(0)
# 2
# sage: P.unrank(5)
# 13
# sage: P.unrank(42)
    return None
    with open(config.f_results, 'w') as f_results:
        for k in combinations:
            if verbose:
                print
                print k
            # TBD:  maybe the following condition or the function
            # get_shifted_combination should be redefined to a dynamic version
            if config.only_slice_candidates and k_vector_size == 5:
                k = get_shifted_combination(k)
            # print k
            knot_sum = eval(knot_formula)
            print "knot_sum"
            print knot_sum

            if is_trivial_combination(knot_sum):
                if verbose:
                    print "\nTrivial combination" + str(knot_sum)
                continue
            result = eval_cable_for_large_signature(knot_sum,
                                                    print_results=False)
            if result is not None:
                knot_description, null_comb, all_comb = result
                line = (str(k) + ", " + str(null_comb) + ", " +
                        str(all_comb) + "\n")
                f_results.write(line)

def eval_cable_for_large_signature(knot_sum, print_results=True, verbose=None):
    if verbose is None:
        verbose = config.verbose
    if verbose:
        print "\neval_cable_for_large_signature"
    if len(knot_sum) != 4:
        print "Wrong number of cable direct summands!"
        return None
    q = 2 * abs(knot_sum[-1][-1]) + 1
    print "q is " + str(q)

    f = get_function_of_theta_for_sum(*knot_sum, verbose=False)
    knot_description = get_knot_descrption(*knot_sum)
    all_combinations = get_number_of_combinations(*knot_sum)

    null_combinations = 0
    zero_theta_combinations = []

    ranges_list = [range(abs(knot[-1]) + 1) for knot in knot_sum]
    if verbose:
        print "eval_cable_for_large_signature - knot_description: "
        print knot_description
    for v_theta in it.product(*ranges_list):
        y = f(*v_theta).value(1/2)
        print y
        if abs(y) > 5 + np.count_nonzero(v_theta):
            print "hura hura"
        # == 0:
        #     zero_theta_combinations.append(v_theta)
        #     m = len([theta for theta in v_theta if theta != 0])
        #     null_combinations += 2^m
        # else:
        #     assert sum(v_theta) != 0

    if print_results:
        print
        print knot_description
        print "Zero cases: " + str(null_combinations)
        print "All cases: " + str(all_combinations)
        if zero_theta_combinations:
            print "Zero theta combinations: "
            for el in zero_theta_combinations:
                print el
    if null_combinations^2 >= all_combinations:
        return knot_description, null_combinations, all_combinations
    return None


def search_for_null_signature_value(knot_formula=None, limit=None):
    if limit is None:
        limit = config.default_limit
    if knot_formula is None:
        knot_formula = config.knot_formula

    k_vector_size = extract_max(knot_formula) + 1
    combinations = it.combinations_with_replacement(range(1, limit + 1),
                                                    k_vector_size)

    with open(config.f_results, 'w') as f_results:
        for k in combinations:
            # print
            # print k
            # TBD:  maybe the following condition or the function
            # get_shifted_combination should be redefined to a dynamic version
            if confi.only_slice_candidates and k_vector_size == 5:
                k = get_shifted_combination(k)
            # print k
            knot_sum = eval(knot_formula)

            if is_trivial_combination(knot_sum):
                continue
            result = search_for_thetas(knot_sum, print_results=False)
            if result is not None:
                knot_description, null_comb, all_comb = result
                line = (str(k) + ", " + str(null_comb) + ", " +
                        str(all_comb) + "\n")
                f_results.write(line)

def search_for_thetas(knot_sum, print_results=False, verbose=None):
    if verbose is None:
        vebose = confi.verbose
    if verbose:
        print "search_for_thetas"
    f = get_function_of_theta_for_sum(*knot_sum)
    knot_description = get_knot_descrption(*knot_sum)
    all_combinations = get_number_of_combinations(*knot_sum)

    large_value_combinations = 0
    good_theta_list = []

    ranges_list = [range(abs(knot[-1]) + 1) for knot in knot_sum]
    if verbose:
        print "knot_description:"
        print knot_description
    for v_theta in it.product(*ranges_list):
        if (v_theta[0]^2 - v_theta[1]^2 + v_theta[2]^2 - v_theta[3]^2) % q:
            print "ojojoj"
            print (v_theta[0]^2 - v_theta[1]^2 + v_theta[2]^2 - v_theta[3]^2) % q
            continue
        print f(*v_theta).value(-1)
        if f(*v_theta).value(-1):
            print "Hip hip hura"
            good_theta_list.append(v_theta)
            m = len([theta for theta in v_theta if theta != 0])
            large_value_combinations += 2^m
        else:
            print "smuteczek"
        # else:
        #     assert sum(v_theta) != 0

    if print_results:
        print
        print knot_description
        print "Zero cases: " + str(null_combinations)
        print "All cases: " + str(all_combinations)
        if zero_theta_combinations:
            print "Zero theta combinations: "
            for el in zero_theta_combinations:
                print el
    if null_combinations^2 >= all_combinations:
        return knot_description, null_combinations, all_combinations
    return None


def is_trivial_combination(knot_sum):
    # for now is applicable only for schema that are sums of 4 cables
    if len(knot_sum) == 4:
        oposit_to_first = [-k for k in knot_sum[0]]
        if oposit_to_first in knot_sum:
            return True
    return False


def get_shifted_combination(combination):
    # for now applicable only for schama
    # "[[k[0], k[1], k[2]], [k[3], k[4]],
    # [-k[0], -k[3], -k[4]], [-k[1], -k[2]]]"
    # shift the combination so that the knot can be a candidate for slice
    combination = [combination[0], 4 * combination[0] + combination[1],
         4 * (4 * combination[0] + combination[1]) + combination[2],
         4 * combination[0] + combination[3],
         4 * (4 * combination[0] + combination[3]) + combination[4]]
    return combination


def get_blanchfield_for_pattern(k_n, theta):
    if theta == 0:
        a = get_untwisted_signature_function(k_n)
        return a.square_root() + a.minus_square_root()

    results = []
    k = abs(k_n)
    ksi = 1/(2 * k + 1)

    # lambda_odd, i.e. (theta + e) % 2 != 0
    for e in range(1, k + 1):
        if (theta + e) % 2 != 0:
            results.append((e * ksi, 1 * sgn(k_n)))
            results.append((1 - e * ksi, -1 * sgn(k_n)))

    # lambda_even
    # print "normal"
    for e in range(1, theta):
        if (theta + e) % 2 == 0:
            results.append((e * ksi, 1 * sgn(k_n)))
            results.append((1 - e * ksi, -1 * sgn(k_n)))
    # print "reversed"
    for e in range(theta + 1, k + 1):
        if (theta + e) % 2 != 0:
            continue
        results.append((e * ksi, -1 * sgn(k_n)))
        results.append((1 - e * ksi, 1 * sgn(k_n)))
    return SignatureFunction(results)


def get_cable_signature_as_theta_function(*arg):
    def get_signture_function(theta):
        # TBD: another formula (for t^2) description

        k_n = abs(arg[-1])
        if theta > k_n:
            msg = "k for the pattern in the cable is " + str(arg[-1]) + \
                  ". Parameter theta should not be larger than abs(k)."
            raise ValueError(msg)

        # twisted part
        cable_signature = get_blanchfield_for_pattern(arg[-1], theta)

        # untwisted part
        for i, k in enumerate(arg[:-1][::-1]):
            ksi = 1/(2 * k_n + 1)
            power = 2^i
            a = get_untwisted_signature_function(k)
            shift = theta * ksi * power
            b = a >> shift
            c = a << shift
            for _ in range(i):
                b = b.double_cover()
                c = c.double_cover()
            cable_signature += b + c
        return cable_signature
    get_signture_function.__doc__ = get_signture_function_docsting
    return get_signture_function


def get_untwisted_signature_function(j):
    """This function returns the signature function of the T_{2,2k+1}
    torus knot."""
    k = abs(j)
    w = ([((2 * a + 1)/(4 * k + 2), -1 * sgn(j)) for a in range(k)] +
         [((2 * a + 1)/(4 * k + 2), 1 * sgn(j))
         for a in range(k + 1, 2 * k + 1)])
    return SignatureFunction(w)


def get_function_of_theta_for_sum(*arg, **key_args):
    if 'verbose' in key_args:
        verbose_default = key_args['verbose']
    else:
        verbose_default = confi.verbose
    def signature_function_for_sum(*thetas, **kwargs):
        verbose = verbose_default
        if 'verbose' in kwargs:
            verbose = kwargs['verbose']
        la = len(arg)
        lt = len(thetas)

        # call with no arguments
        if lt == 0:
            return signature_function_for_sum(*(la * [0]))

        if lt != la:
            msg = "This function takes exactly " + str(la) + \
                  " arguments or no argument at all (" + str(lt) + " given)."
            raise TypeError(msg)

        sf = SignatureFunction([(0, 0)])

        # for each cable in cable sum apply theta
        for i, knot in enumerate(arg):
            try:
                sf += (get_cable_signature_as_theta_function(*knot))(thetas[i])
            # in case wrong theata value was given
            except ValueError as e:
                print "ValueError: " + str(e.args[0]) +\
                      " Please change " + str(i + 1) + ". parameter."
                return None
        if verbose:
            print
            print str(thetas)
            print sf
        return sf
    signature_function_for_sum.__doc__ = signature_function_for_sum_docstring
    return signature_function_for_sum


def eval_cable_for_null_signature(knot_sum, print_results=True, verbose=None):
    # search for zero combinations
    if verbose is None:
        vebose = confi.verbose
    f = get_function_of_theta_for_sum(*knot_sum, verbose=False)
    knot_description = get_knot_descrption(*knot_sum)
    all_combinations = get_number_of_combinations(*knot_sum)

    null_combinations = 0
    zero_theta_combinations = []

    ranges_list = [range(abs(knot[-1]) + 1) for knot in knot_sum]
    if verbose:
        print
        print knot_description
    for v_theta in it.product(*ranges_list):
        if f(*v_theta, verbose=False).sum_of_absolute_values() == 0:
            zero_theta_combinations.append(v_theta)
            m = len([theta for theta in v_theta if theta != 0])
            null_combinations += 2^m
        # else:
        #     assert sum(v_theta) != 0

    if print_results:
        print
        print knot_description
        print "Zero cases: " + str(null_combinations)
        print "All cases: " + str(all_combinations)
        if zero_theta_combinations:
            print "Zero theta combinations: "
            for el in zero_theta_combinations:
                print el
    if null_combinations^2 >= all_combinations:
        return knot_description, null_combinations, all_combinations
    return None


def get_number_of_combinations(*arg):
    number_of_combinations = 1
    for knot in arg:
        number_of_combinations *= (2 * abs(knot[-1]) + 1)
    return number_of_combinations


def extract_max(string):
    numbers = re.findall('\d+', string)
    numbers = map(int, numbers)
    return max(numbers)


def mod_one(n):
    return n - floor(n)


def get_knot_descrption(*arg):
    description = ""
    for knot in arg:
        if knot[0] < 0:
            description += "-"
        description += "T("
        for k in knot:
            description += "2, " + str(2 * abs(k) + 1) + "; "
        description = description[:-2] + ") # "
    return description[:-3]

get_blanchfield_for_pattern.__doc__ = \
    """
    Arguments:
        k_n:    a number s.t. q_n = 2 * k_n + 1, where
                T(2, q_n) is a pattern knot for a single cable from a cable sum
        theta:  twist/character for the cable (value form v vector)
    Return:
        SignatureFunction created for twisted signature function
        for a given cable and theta/character
    Based on:
        Proposition 9.8. in Twisted Blanchfield Pairing
        (https://arxiv.org/pdf/1809.08791.pdf)
    """

get_number_of_combinations.__doc__ = \
    """
    Arguments:
        arbitrary number of lists of numbers, each list encodes a single cable
    Return:
        number of possible theta values combinations that could be applied
        for a given cable sum,
        i.e. the product of q_j for j = {1,.. n},
        where n is a number of direct components in the cable sum,
        and q_j is the last q parameter for the component (a single cable)
    """

get_knot_descrption.__doc__ = \
    """
    Arguments:
        arbitrary number of lists of numbers, each list encodes a single cable.
    Examples:
        sage: get_knot_descrption([1, 3], [2], [-1, -2], [-3])
        'T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7)'
    """

mod_one.__doc__ = \
    """
    Argument:
        a number
    Return:
        the fractional part of the argument
    Examples:
        sage: mod_one(9 + 3/4)
        3/4
        sage: mod_one(-9 + 3/4)
        3/4
        sage: mod_one(-3/4)
        1/4
    """

search_for_null_signature_value.__doc__ = \
    """
    This function calculates signature functions for knots constracted
    accordinga a schema for a cable sum. The schema is given as an argument
    or defined in the class MySettings.
    Results of calculations will be writen to a file and the stdout.
    limit is the upper bound for the first value in k_vector,
    i.e k[0] value in a cable sum, where q_0 = 2 * k[0] + 1.

    (the number of knots that will be constracted depends on limit value).
    For each knot/cable sum the function eval_cable_for_null_signature is called.
    eval_cable_for_null_signature calculetes the number of all possible thetas
    (characters) and the number of combinations for which signature function
    equeles zero. In case the first number is larger than squere of the second,
    eval_cable_for_null_signature returns None (i.e. the knot can not be slice).
    Data for knots that are candidates for slice knots are saved to a file.
    """

extract_max.__doc__ = \
    """
    Return:
        maximum of absolute values of numbers from given string
    Examples:
        sage: extract_max("([1, 3], [2], [-1, -2], [-10])")
        10
        sage: extract_max("3, 55, ewewe, -42, 3300, 50")
        3300
    """

eval_cable_for_null_signature.__doc__ = \
    """
    This function calculates all possible twisted signature functions for
    a knot that is given as an argument. The knot should be encoded as a list
    of its direct component. Each component schould be presented as a list
    of integers. This integers correspond to the k - values in each component/
    cable. If a component is a mirror image of a cable the minus sign should
    be written before each number for this component. For example:
    eval_cable_for_null_signature([[1, 8], [2], [-2, -8], [-2]])
    eval_cable_for_null_signature([[1, 2], [-1, -2]])

    sage: eval_cable_for_null_signature([[1, 3], [2], [-1, -2], [-3]])

    T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7)
    Zero cases: 1
    All cases: 1225
    Zero theta combinations:
    (0, 0, 0, 0)

    sage:
    The numbers given to the function eval_cable_for_null_signature are k-values for each
    component/cable in a direct sum.
    """

get_function_of_theta_for_sum.__doc__ = \
    """
    Function intended to construct signature function for a connected
    sum of multiple cables with varying theta parameter values.
    Accept arbitrary number of arguments (depending on number of cables in
    connected sum).
    Each argument should be given as list of integer representing
    k - parameters for a cable: parameters k_i (i=1,.., n-1) for satelit knots
    T(2, 2k_i + 1) and - the last one - k_n for a pattern knot T(2, 2k_n + 1).
    Returns a function that will take theta vector as an argument and return
    an object SignatureFunction.

    To calculate signature function for a cable sum and a theta values vector,
    use as below.

    sage: signature_function_generator = get_function_of_theta_for_sum(
                                             [1, 3], [2], [-1, -2], [-3])
    sage: sf = signature_function_generator(2, 1, 2, 2)
    sage: print sf
    0: 0
    5/42: 1
    1/7: 0
    1/5: -1
    7/30: -1
    2/5: 1
    3/7: 0
    13/30: -1
    19/42: -1
    23/42: 1
    17/30: 1
    4/7: 0
    3/5: -1
    23/30: 1
    4/5: 1
    6/7: 0
    37/42: -1

    Or like below.
    sage: print get_function_of_theta_for_sum([1, 3], [2], [-1, -2], [-3]
                                                )(2, 1, 2, 2)
    0: 0
    1/7: 0
    1/6: 0
    1/5: -1
    2/5: 1
    3/7: 0
    1/2: 0
    4/7: 0
    3/5: -1
    4/5: 1
    5/6: 0
    6/7: 0
    """

get_cable_signature_as_theta_function.__doc__ = \
    """
    Argument:
        n integers that encode a single cable, i.e.
        values of q_i for T(2,q_0; 2,q_1; ... 2, q_n)
    Return:
        a function that returns SignatureFunction for this single cable
        and a theta given as an argument
    """

get_signture_function_docsting = \
    """
    This function returns SignatureFunction for previously defined single
    cable T_(2, q) and a theta given as an argument.
    The cable was defined by calling function
    get_cable_signature_as_theta_function(*arg)
    with the cable description as an argument.
    It is an implementaion of the formula:
        Bl_theta(K'_(2, d)) =
            Bl_theta(T_2, d) + Bl(K')(ksi_l^(-theta) * t)
            + Bl(K')(ksi_l^theta * t)
    """

signature_function_for_sum_docstring = \
    """
    Arguments:

    Returns object of SignatureFunction class for a previously defined
    connected sum of len(arg) cables.
    Acept len(arg) arguments: for each cable one theta parameter.
    If call with no arguments, all theta parameters are set to be 0.
    """

main.__doc__ = \
    """
    This function is run if the script was called from the terminal.
    It calls another function, search_for_null_signature_value,
    to calculate signature functions for a schema
    of a cable sum defined in the class MySettings.
    Optionaly a parameter (a limit for k_0 value) can be given.
    Thought to be run for time consuming calculations.
    """

config = MySettings()
if __name__ == '__main__' and '__file__' in globals():
    # not called in interactive mode as __file__ is not defined
    main(sys.argv)


# def calculate_form(x, y, q4):
                                                                                                                                                #     x1, x2, x3, x4 = x
                                                                                                                                                #     y1, y2, y3, y4 = y
                                                                                                                                                #     form = (x1 * y1 - x2 * y2 + x3 * y3 - x4 * y4) % q_4
#     # TBD change for ring modulo q_4
#     return form
#
# def check_condition(v, q4):
#     if calculate_form(v, v, q4):
#         return False
#     return True
#
# def find_v(q4):
#     results = []
#     for i in range(q4):
#         for j in range(q4):
#             for k in range(q4):
#                 for m in range(q4):
#                     if check_condition([i, j, k, m], q_4):
#                         results.add(v)
#     return results
#
# def check_inequality(q, v):
#     a1, a2, a3, a4 = v
#     q1, q2, q3, q4 = q
#     pattern = [q1, q2, q4],[-q2, -q4],[q3, q4],[-q1, -q3, -q4]
#     signature_function_generator = get_function_of_theta_for_sum(pattern)
#     signature_function_for_sum = signature_function_generator(a1, a2, a3, a4)
#
#     # sigma_v = sigma(q4, a1) - s(a2) + s(a3) - s(a4)
#
#