



from __future__ import annotations

import math
def 						UpperCamelCase      (  snake_case__			:       list			,	snake_case__			:       list      )      ->	list:







   if len(snake_case_      ) != 2 or len(a[0]      ) != 2 or len(snake_case_      ) != 2 or len(b[0]      ) != 2:
      raise Exception('Matrices are not 2x2'      )
   UpperCamelCase       :    Tuple		      =      [
       [a[0][0] * b[0][0] + a[0][1] * b[1][0], a[0][0] * b[0][1] + a[0][1] * b[1][1]],
       [a[1][0] * b[0][0] + a[1][1] * b[1][0], a[1][0] * b[0][1] + a[1][1] * b[1][1]],
   ]
   return new_matrix
def 						UpperCamelCase      (  snake_case__			:       list			,	snake_case__			:       list      )      ->	Union[str, Any]:
   return [
       [matrix_a[row][col] + matrix_b[row][col] for col in range(len(matrix_a[row]      )      )]
       for row in range(len(snake_case_      )      )
   ]
def 						UpperCamelCase      (  snake_case__			:       list			,	snake_case__			:       list      )      ->	int:
   return [
       [matrix_a[row][col] - matrix_b[row][col] for col in range(len(matrix_a[row]      )      )]
       for row in range(len(snake_case_      )      )
   ]
def 						UpperCamelCase      (  snake_case__			:       list      )      ->	tuple[list, list, list, list]:







   if len(snake_case_      ) % 2 != 0 or len(a[0]      ) % 2 != 0:
      raise Exception('Odd matrices are not supported!'      )

   UpperCamelCase       :    Dict		      =      len(snake_case_      )
   UpperCamelCase       :    List[str]		      =      matrix_length // 2

   UpperCamelCase       :    Any		      =      [[a[i][j] for j in range(snake_case_			,	snake_case_      )] for i in range(snake_case_      )]
   UpperCamelCase       :    int		      =      [
       [a[i][j] for j in range(snake_case_			,	snake_case_      )] for i in range(snake_case_			,	snake_case_      )
   ]

   UpperCamelCase       :    str		      =      [[a[i][j] for j in range(snake_case_      )] for i in range(snake_case_      )]
   UpperCamelCase       :    int		      =      [[a[i][j] for j in range(snake_case_      )] for i in range(snake_case_			,	snake_case_      )]

   return top_left, top_right, bot_left, bot_right
def 						UpperCamelCase      (  snake_case__			:       list      )      ->	tuple[int, int]:
   return len(snake_case_      ), len(matrix[0]      )
def 						UpperCamelCase      (  snake_case__			:       list      )      ->	None:
   print('\n'.join(str(snake_case_      ) for line in matrix      )      )
def 						UpperCamelCase      (  snake_case__			:       list			,	snake_case__			:       list      )      ->	list:







   if matrix_dimensions(snake_case_      ) == (2, 2):
      return default_matrix_multiplication(snake_case_			,	snake_case_      )

   UpperCamelCase       :    Dict		      =      split_matrix(snake_case_      )
   UpperCamelCase       :    List[Any]		      =      split_matrix(snake_case_      )

   UpperCamelCase       :    Optional[int]		      =      actual_strassen(snake_case_			,	matrix_subtraction(snake_case_			,	snake_case_      )      )
   UpperCamelCase       :    Any		      =      actual_strassen(matrix_addition(snake_case_			,	snake_case_      )			,	snake_case_      )
   UpperCamelCase       :    Optional[int]		      =      actual_strassen(matrix_addition(snake_case_			,	snake_case_      )			,	snake_case_      )
   UpperCamelCase       :    List[Any]		      =      actual_strassen(snake_case_			,	matrix_subtraction(snake_case_			,	snake_case_      )      )
   UpperCamelCase       :    Union[str, Any]		      =      actual_strassen(matrix_addition(snake_case_			,	snake_case_      )			,	matrix_addition(snake_case_			,	snake_case_      )      )
   UpperCamelCase       :    List[str]		      =      actual_strassen(matrix_subtraction(snake_case_			,	snake_case_      )			,	matrix_addition(snake_case_			,	snake_case_      )      )
   UpperCamelCase       :    Any		      =      actual_strassen(matrix_subtraction(snake_case_			,	snake_case_      )			,	matrix_addition(snake_case_			,	snake_case_      )      )

   UpperCamelCase       :    Dict		      =      matrix_addition(matrix_subtraction(matrix_addition(snake_case_			,	snake_case_      )			,	snake_case_      )			,	snake_case_      )
   UpperCamelCase       :    Optional[Any]		      =      matrix_addition(snake_case_			,	snake_case_      )
   UpperCamelCase       :    Any		      =      matrix_addition(snake_case_			,	snake_case_      )
   UpperCamelCase       :    str		      =      matrix_subtraction(matrix_subtraction(matrix_addition(snake_case_			,	snake_case_      )			,	snake_case_      )			,	snake_case_      )

   # construct the new matrix from our 4 quadrants
   UpperCamelCase       :    Optional[int]		      =      []
   for i in range(len(snake_case_      )      ):
      new_matrix.append(top_left[i] + top_right[i]      )
   for i in range(len(snake_case_      )      ):
      new_matrix.append(bot_left[i] + bot_right[i]      )
   return new_matrix
def 						UpperCamelCase      (  snake_case__			:       list			,	snake_case__			:       list      )      ->	list:







   if matrix_dimensions(snake_case_      )[1] != matrix_dimensions(snake_case_      )[0]:
      UpperCamelCase       :    Optional[int]		      =      (
          '''Unable to multiply these matrices, please check the dimensions.\n'''
          F"""Matrix A: {matrixa}\n"""
          F"""Matrix B: {matrixa}"""
      )
      raise Exception(snake_case_      )
   UpperCamelCase       :    List[str]		      =      matrix_dimensions(snake_case_      )
   UpperCamelCase       :    List[Any]		      =      matrix_dimensions(snake_case_      )

   if dimensiona[0] == dimensiona[1] and dimensiona[0] == dimensiona[1]:
      return [matrixa, matrixa]

   UpperCamelCase       :    Tuple		      =      max(*snake_case_			,	*snake_case_      )
   UpperCamelCase       :    Tuple		      =      int(math.pow(2			,	math.ceil(math.loga(snake_case_      )      )      )      )
   UpperCamelCase       :    str		      =      matrixa
   UpperCamelCase       :    Union[str, Any]		      =      matrixa

   # Adding zeros to the matrices so that the arrays dimensions are the same and also
   # power of 2
   for i in range(0			,	snake_case_      ):
      if i < dimensiona[0]:
         for _ in range(dimensiona[1]			,	snake_case_      ):
            new_matrixa[i].append(0      )
      else:
         new_matrixa.append([0] * maxim      )
      if i < dimensiona[0]:
         for _ in range(dimensiona[1]			,	snake_case_      ):
            new_matrixa[i].append(0      )
      else:
         new_matrixa.append([0] * maxim      )

   UpperCamelCase       :    List[str]		      =      actual_strassen(snake_case_			,	snake_case_      )

   # Removing the additional zeros
   for i in range(0			,	snake_case_      ):
      if i < dimensiona[0]:
         for _ in range(dimensiona[1]			,	snake_case_      ):
            final_matrix[i].pop()
      else:
         final_matrix.pop()
   return final_matrix


if __name__ == "__main__":
    __UpperCAmelCase			  =			[
        [2, 3, 4, 5],
        [6, 4, 3, 1],
        [2, 3, 6, 7],
        [3, 1, 2, 4],
        [2, 3, 4, 5],
        [6, 4, 3, 1],
        [2, 3, 6, 7],
        [3, 1, 2, 4],
        [2, 3, 4, 5],
        [6, 2, 3, 1],
    ]
    __UpperCAmelCase			  =			[[0, 2, 1, 1], [16, 2, 3, 3], [2, 2, 7, 7], [13, 11, 22, 4]]
    print(strassen(matrixa, matrixa))










"""simple docstring"""



def 						__lowercase	(  snake_case_    :    int = 2000000		)  ->int:



    '''simple docstring'''







    __A       :		List[Any]							=	[0 for i in range(n + 1		)]
    __A       :		str							=	1
    __A       :		Dict							=	1

    for i in range(2							,int(n**0.5		) + 1		):
        if primality_list[i] == 0:
            for j in range(i * i							,n + 1							,snake_case_		):
                __A       :		List[Any]							=	1
    __A       :		Union[str, Any]							=	0
    for i in range(snake_case_		):
        if primality_list[i] == 0:
            sum_of_primes += i
    return sum_of_primes


if __name__ == "__main__":
       print(f'''{solution() = }''')




