docs: def formula_dual(input_formula: str) -> str:
      """ Returns the dual of the input formula.
  
      The dual operation on formulas in :math:`B^+(X)` is defined as:
      the dual :math:`\overline{θ}` of a formula :math:`θ` is obtained from θ by
      switching :math:`∧` and :math:`∨`, and
      by switching :math:`true` and :math:`false`.
  
      :param str input_formula: original string.
      :return: *(str)*, dual of input formula.
      """
      conversion_dictionary = {
          'and': 'or',
          'or': 'and',
          'True': 'False',
          'False': 'True'
      }
  
      return re.sub(
          '|'.join(re.escape(key) for key in conversion_dictionary.keys()),
          lambda k: conversion_dictionary[k.group(0)], input_formula)