'''simple docstring'''


from __future__ import annotations

from collections.abc import Sequence
from typing import Literal





def       lowerCAmelCase					(		UpperCamelCase__				:   str    ,					UpperCamelCase__				:   str       ):


	"""simple docstring"""
	__UpperCAmelCase							=			list(UpperCamelCase__       )
	__UpperCAmelCase							=			list(UpperCamelCase__       )
	__UpperCAmelCase							=			0
	for i in range(len(UpperCamelCase__       )       ):
		if lista[i] != lista[i]:
			count += 1
			__UpperCAmelCase							=			'''_'''
	if count > 1:
		return False
	else:
		return "".join(UpperCamelCase__       )





def       lowerCAmelCase					(		UpperCamelCase__				:   list[str]       ):


	"""simple docstring"""
	__UpperCAmelCase							=			[]
	while True:
		__UpperCAmelCase							=			['''$'''] * len(UpperCamelCase__       )
		__UpperCAmelCase							=			[]
		for i in range(len(UpperCamelCase__       )       ):
			for j in range(i + 1    ,					len(UpperCamelCase__       )       ):
				__UpperCAmelCase							=			compare_string(binary[i]    ,					binary[j]       )
				if k is False:
					__UpperCAmelCase							=			'''*'''
					__UpperCAmelCase							=			'''*'''
					temp.append('''X'''       )
		for i in range(len(UpperCamelCase__       )       ):
			if checka[i] == "$":
				pi.append(binary[i]       )
		if len(UpperCamelCase__       ) == 0:
			return pi
		__UpperCAmelCase							=			list(set(UpperCamelCase__       )       )





def       lowerCAmelCase					(		UpperCamelCase__				:   int    ,					UpperCamelCase__				:   Sequence[float]       ):


	"""simple docstring"""
	__UpperCAmelCase							=			[]
	for minterm in minterms:
		__UpperCAmelCase							=			''''''
		for _ in range(UpperCamelCase__       ):
			__UpperCAmelCase							=			str(minterm % 2       ) + string
			minterm //= 2
		temp.append(UpperCamelCase__       )
	return temp





def       lowerCAmelCase					(		UpperCamelCase__				:   str    ,					UpperCamelCase__				:   str    ,					UpperCamelCase__				:   int       ):


	"""simple docstring"""
	__UpperCAmelCase							=			list(UpperCamelCase__       )
	__UpperCAmelCase							=			list(UpperCamelCase__       )
	__UpperCAmelCase							=			0
	for i in range(len(UpperCamelCase__       )       ):
		if lista[i] != lista[i]:
			count_n += 1
	return count_n == count





def       lowerCAmelCase					(		UpperCamelCase__				:   list[list[int]]    ,					UpperCamelCase__				:   list[str]       ):


	"""simple docstring"""
	__UpperCAmelCase							=			[]
	__UpperCAmelCase							=			[0] * len(UpperCamelCase__       )
	for i in range(len(chart[0]       )       ):
		__UpperCAmelCase							=			0
		__UpperCAmelCase							=			-1
		for j in range(len(UpperCamelCase__       )       ):
			if chart[j][i] == 1:
				count += 1
				__UpperCAmelCase							=			j
		if count == 1:
			__UpperCAmelCase							=			1
	for i in range(len(UpperCamelCase__       )       ):
		if select[i] == 1:
			for j in range(len(chart[0]       )       ):
				if chart[i][j] == 1:
					for k in range(len(UpperCamelCase__       )       ):
						__UpperCAmelCase							=			0
			temp.append(prime_implicants[i]       )
	while True:
		__UpperCAmelCase							=			0
		__UpperCAmelCase							=			-1
		__UpperCAmelCase							=			0
		for i in range(len(UpperCamelCase__       )       ):
			__UpperCAmelCase							=			chart[i].count(1       )
			if count_n > max_n:
				__UpperCAmelCase							=			count_n
				__UpperCAmelCase							=			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(UpperCamelCase__       )       ):
					__UpperCAmelCase							=			0





def       lowerCAmelCase					(		UpperCamelCase__				:   list[str]    ,					UpperCamelCase__				:   list[str]       ):


	"""simple docstring"""
	__UpperCAmelCase							=			[[0 for x in range(len(UpperCamelCase__       )       )] for x in range(len(UpperCamelCase__       )       )]
	for i in range(len(UpperCamelCase__       )       ):
		__UpperCAmelCase							=			prime_implicants[i].count('''_'''       )
		for j in range(len(UpperCamelCase__       )       ):
			if is_for_table(prime_implicants[i]    ,					binary[j]    ,					UpperCamelCase__       ):
				__UpperCAmelCase							=			1

	return chart





def       lowerCAmelCase					(		):


	"""simple docstring"""
	__UpperCAmelCase							=			int(input('''Enter the no. of variables\n'''       )       )
	__UpperCAmelCase							=			[
	    float(UpperCamelCase__       )
	    for x in input(
	        '''Enter the decimal representation of Minterms \'Spaces Separated\'\n'''       ).split()
	]
	__UpperCAmelCase							=			decimal_to_binary(UpperCamelCase__    ,					UpperCamelCase__       )

	__UpperCAmelCase							=			check(UpperCamelCase__       )
	print('''Prime Implicants are:'''       )
	print(UpperCamelCase__       )
	__UpperCAmelCase							=			prime_implicant_chart(UpperCamelCase__    ,					UpperCamelCase__       )

	__UpperCAmelCase							=			selection(UpperCamelCase__    ,					UpperCamelCase__       )
	print('''Essential Prime Implicants are:'''       )
	print(UpperCamelCase__       )


if __name__ == "__main__":
					import doctest

					doctest.testmod()
					main()

'''simple docstring'''


from math import sqrt

import numpy as np
from sympy import symbols

# Coefficient
# Speed of light (m/s)
__lowerCAmelCase	:       str						       =	299_792_458

# Symbols
__lowerCAmelCase       ,	__lowerCAmelCase       ,	__lowerCAmelCase       ,	__lowerCAmelCase	:       Any						       =	symbols("ct x y z")





def       lowerCAmelCase					(		UpperCamelCase__				:   float       ):


	"""simple docstring"""
	if velocity > c:
		raise ValueError('''Speed must not exceed light speed 299,792,458 [m/s]!'''       )
	elif velocity < 1:
		# Usually the speed should be much higher than 1 (c order of magnitude)
		raise ValueError('''Speed must be greater than or equal to 1!'''       )

	return velocity / c





def       lowerCAmelCase					(		UpperCamelCase__				:   float       ):


	"""simple docstring"""
	return 1 / sqrt(1 - beta(UpperCamelCase__       ) ** 2       )





def       lowerCAmelCase					(		UpperCamelCase__				:   float       ):


	"""simple docstring"""
	return np.array(
	    [
	        [gamma(UpperCamelCase__       ), -gamma(UpperCamelCase__       ) * beta(UpperCamelCase__       ), 0, 0],
	        [-gamma(UpperCamelCase__       ) * beta(UpperCamelCase__       ), gamma(UpperCamelCase__       ), 0, 0],
	        [0, 0, 1, 0],
	        [0, 0, 0, 1],
	    ]       )





def       lowerCAmelCase					(		UpperCamelCase__				:   float    ,					UpperCamelCase__				:   np.ndarray | None = None       ):


	"""simple docstring"""
	# Ensure event is not empty
	if event is None:
		__UpperCAmelCase							=			np.array([ct, x, y, z]       )  # Symbolic four vector
	else:
		event[0] *= c  # x0 is ct (speed of light * time)

	return transformation_matrix(UpperCamelCase__       ) @ event


if __name__ == "__main__":
					import doctest

					doctest.testmod()

					# Example of symbolic vector:
					__lowerCAmelCase	:       Dict						       =	transform(29_979_245)
					print("Example of four vector: ")
					print(F"""ct' = {four_vector[0]}""")
					print(F"""x' = {four_vector[1]}""")
					print(F"""y' = {four_vector[2]}""")
					print(F"""z' = {four_vector[3]}""")

					# Substitute symbols with numerical values
					__lowerCAmelCase	:       Union[str, Any]						       =	{ct: c, x: 1, y: 1, z: 1}
					__lowerCAmelCase	:       Optional[int]						       =	[four_vector[i].subs(sub_dict) for i in range(4)]

					print(F"""\n{numerical_vector}""")
