class_docstr: Properties
  ==========
  
  [1] `0 \in \Omega` and `\tau(1) = \epsilon`
  [2] `\alpha^x = \beta \Leftrightarrow \beta^{x^{-1}} = \alpha`
  [3] If `\alpha^x = \beta`, then `H \tau(\alpha)x = H \tau(\beta)`
  [4] `\forall \alpha \in \Omega, 1^{\tau(\alpha)} = \alpha`
  
  References
  ==========
  
  .. [1] Holt, D., Eick, B., O'Brien, E.
         "Handbook of Computational Group Theory"
  
  .. [2] John J. Cannon; Lucien A. Dimino; George Havas; Jane M. Watson
         Mathematics of Computation, Vol. 27, No. 123. (Jul., 1973), pp. 463-490.
         "Implementation and Analysis of the Todd-Coxeter Algorithm"
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docstr: A variation of ``process_deductions``, this calls ``scan_check``
  wherever ``process_deductions`` calls ``scan``, described on Pg. [1].
  
  See Also
  ========
  
  process_deductions
docstr_tok: ["A","variation","of","``","process_deductions","``",",","this","calls","``","scan_check","``","wherever","``","process_deductions","``","calls","``","scan","``",",","described","on","Pg",".","[","1","]",".","See","Also","========","process_deductions"]
code: def process_deductions_check(self, R_c_x, R_c_x_inv):
      """
      A variation of ``process_deductions``, this calls ``scan_check``
      wherever ``process_deductions`` calls ``scan``, described on Pg. [1].
  
      See Also
      ========
  
      process_deductions
  
      """
      table = self.table
      while len(self.deduction_stack) > 0:
          alpha, x = self.deduction_stack.pop()
          for w in R_c_x:
              if not self.scan_check(alpha, w):
                  return False
          beta = table[alpha][self.A_dict[x]]
          if beta is not None:
              for w in R_c_x_inv:
                  if not self.scan_check(beta, w):
                      return False
      return True
  
imports: from sympy.combinatorics.free_groups import free_group
  from sympy.printing.defaults import DefaultPrinting
  from itertools import chain, product
  from bisect import bisect_left
label_desc: Use image node_id 20 for calling the CosetTable obj's underlying member method code with example usage: obj.process_deductions_check(R_c_x, R_c_x_inv) and returns: True, False, False