





from __future__ import annotations

from collections.abc import Sequence
from typing import Literal



def     _lowerCAmelCase		(  _lowerCAmelCase  ,_lowerCAmelCase     ):
   '''simple docstring'''


   A_     :				str			      =	list(__A     )
   A_     :				Optional[Any]			      =	list(__A     )
   A_     :				str			      =	0
   for i in range(len(__A     )     ):
      if lista[i] != lista[i]:
         count += 1
         A_     :				Union[str, Any]			      =	'''_'''
   if count > 1:
      return False
   else:
      return "".join(__A     )



def     _lowerCAmelCase		(  _lowerCAmelCase     ):
   '''simple docstring'''


   A_     :				int			      =	[]
   while True:
      A_     :				Optional[int]			      =	['''$'''] * len(__A     )
      A_     :				int			      =	[]
      for i in range(len(__A     )     ):
         for j in range(i + 1  ,len(__A     )     ):
            A_     :				str			      =	compare_string(binary[i]  ,binary[j]     )
            if k is False:
               A_     :				List[str]			      =	'''*'''
               A_     :				Any			      =	'''*'''
               temp.append("""X"""     )
      for i in range(len(__A     )     ):
         if checka[i] == "$":
            pi.append(binary[i]     )
      if len(__A     ) == 0:
         return pi
      A_     :				Union[str, Any]			      =	list(set(__A     )     )



def     _lowerCAmelCase		(  _lowerCAmelCase  ,_lowerCAmelCase     ):
   '''simple docstring'''


   A_     :				str			      =	[]
   for minterm in minterms:
      A_     :				Dict			      =	''''''
      for _ in range(__A     ):
         A_     :				Union[str, Any]			      =	str(minterm % 2     ) + string
         minterm //= 2
      temp.append(__A     )
   return temp



def     _lowerCAmelCase		(  _lowerCAmelCase  ,_lowerCAmelCase  ,_lowerCAmelCase     ):
   '''simple docstring'''


   A_     :				Dict			      =	list(__A     )
   A_     :				Tuple			      =	list(__A     )
   A_     :				int			      =	0
   for i in range(len(__A     )     ):
      if lista[i] != lista[i]:
         count_n += 1
   return count_n == count



def     _lowerCAmelCase		(  _lowerCAmelCase  ,_lowerCAmelCase     ):
   '''simple docstring'''


   A_     :				Optional[Any]			      =	[]
   A_     :				List[str]			      =	[0] * len(__A     )
   for i in range(len(chart[0]     )     ):
      A_     :				int			      =	0
      A_     :				List[Any]			      =	-1
      for j in range(len(__A     )     ):
         if chart[j][i] == 1:
            count += 1
            A_     :				List[Any]			      =	j
      if count == 1:
         A_     :				int			      =	1
   for i in range(len(__A     )     ):
      if select[i] == 1:
         for j in range(len(chart[0]     )     ):
            if chart[i][j] == 1:
               for k in range(len(__A     )     ):
                  A_     :				Any			      =	0
         temp.append(prime_implicants[i]     )
   while True:
      A_     :				Any			      =	0
      A_     :				Any			      =	-1
      A_     :				List[str]			      =	0
      for i in range(len(__A     )     ):
         A_     :				List[Any]			      =	chart[i].count(1     )
         if count_n > max_n:
            A_     :				Dict			      =	count_n
            A_     :				Union[str, Any]			      =	i

      if max_n == 0:
         return temp

      temp.append(prime_implicants[rem]     )

      for i in range(len(chart[0]     )     ):
         if chart[rem][i] == 1:
            for j in range(len(__A     )     ):
               A_     :				int			      =	0



def     _lowerCAmelCase		(  _lowerCAmelCase  ,_lowerCAmelCase     ):
   '''simple docstring'''


   A_     :				List[Any]			      =	[[0 for x in range(len(__A     )     )] for x in range(len(__A     )     )]
   for i in range(len(__A     )     ):
      A_     :				Union[str, Any]			      =	prime_implicants[i].count("""_"""     )
      for j in range(len(__A     )     ):
         if is_for_table(prime_implicants[i]  ,binary[j]  ,__A     ):
            A_     :				Union[str, Any]			      =	1

   return chart



def     _lowerCAmelCase		(  ):
   '''simple docstring'''


   A_     :				Union[str, Any]			      =	int(input("""Enter the no. of variables\n"""     )     )
   A_     :				Dict			      =	[
       float(__A     )
       for x in input(
           """Enter the decimal representation of Minterms \'Spaces Separated\'\n"""     ).split()
   ]
   A_     :				int			      =	decimal_to_binary(__A  ,__A     )

   A_     :				Tuple			      =	check(__A     )
   print("""Prime Implicants are:"""     )
   print(__A     )
   A_     :				Optional[Any]			      =	prime_implicant_chart(__A  ,__A     )

   A_     :				Tuple			      =	selection(__A  ,__A     )
   print("""Essential Prime Implicants are:"""     )
   print(__A     )


if __name__ == "__main__":
      import doctest

      doctest.testmod()
      main()










from __future__ import annotations

import numpy as np
from numpy import floataa
from numpy.typing import NDArray



def     _lowerCAmelCase		(  _lowerCAmelCase  ,_lowerCAmelCase  ,_lowerCAmelCase  ,_lowerCAmelCase  ,):
   '''simple docstring'''


   A_      ,				A_     :				int			      =	coefficient_matrix.shape
   A_      ,				A_     :				Tuple			      =	constant_matrix.shape

   if rowsa != colsa:
      A_     :				int			      =	f"""Coefficient matrix dimensions must be nxn but received {rowsa}x{colsa}"""
      raise ValueError(_lowerCAmelCase     )

   if colsa != 1:
      A_     :				List[Any]			      =	f"""Constant matrix must be nx1 but received {rowsa}x{colsa}"""
      raise ValueError(_lowerCAmelCase     )

   if rowsa != rowsa:
      A_     :				str			      =	(
          """Coefficient and constant matrices dimensions must be nxn and nx1 but """
          f"""received {rowsa}x{colsa} and {rowsa}x{colsa}"""
      )
      raise ValueError(_lowerCAmelCase     )

   if len(_lowerCAmelCase     ) != rowsa:
      A_     :				Any			      =	(
          """Number of initial values must be equal to number of rows in coefficient """
          f"""matrix but received {len(_lowerCAmelCase     )} and {rowsa}"""
      )
      raise ValueError(_lowerCAmelCase     )

   if iterations <= 0:
      raise ValueError("""Iterations must be at least 1"""     )

   A_     :				NDArray[floataa]			      =	np.concatenate(
       (coefficient_matrix, constant_matrix)  ,axis=1     )

   A_      ,				A_     :				str			      =	table.shape

   strictly_diagonally_dominant(_lowerCAmelCase     )

   # Iterates the whole matrix for given number of times
   for _ in range(_lowerCAmelCase     ):
      A_     :				Union[str, Any]			      =	[]
      for row in range(_lowerCAmelCase     ):
         A_     :				str			      =	0
         for col in range(_lowerCAmelCase     ):
            if col == row:
               A_     :				Optional[Any]			      =	table[row][col]
            elif col == cols - 1:
               A_     :				List[str]			      =	table[row][col]
            else:
               temp += (-1) * table[row][col] * init_val[col]
         A_     :				str			      =	(temp + val) / denom
         new_val.append(_lowerCAmelCase     )
      A_     :				List[str]			      =	new_val

   return [float(_lowerCAmelCase     ) for i in new_val]



def     _lowerCAmelCase		(  _lowerCAmelCase     ):
   '''simple docstring'''


   A_      ,				A_     :				str			      =	table.shape

   A_     :				Any			      =	True

   for i in range(0  ,_lowerCAmelCase     ):
      A_     :				Optional[Any]			      =	0
      for j in range(0  ,cols - 1     ):
         if i == j:
            continue
         else:
            total += table[i][j]

      if table[i][i] <= total:
         raise ValueError("""Coefficient matrix is not strictly diagonally dominant"""     )

   return is_diagonally_dominant


# Test Cases
if __name__ == "__main__":
      import doctest

      doctest.testmod()






