code: def _check_conditions(sigma, num_classes, orders):
      """Symbolic-numeric verification of conditions C5 and C6.
  
      The conditions on the beta function are verified by constructing the beta
      function symbolically, and then checking that its derivative (computed
      symbolically) is non-negative within the interval of conjectured monotonicity.
      The last check is performed numerically.
      """
  
      print("Checking conditions C5 and C6 for all orders.")
      sys.stdout.flush()
      conditions_hold = True
  
      for order in orders:
          cond5, cond6 = pate_ss.check_conditions(
              sigma, num_classes, order
          )
          conditions_hold &= cond5 and cond6
          if not cond5:
              print("Condition C5 does not hold for order =", order)
          elif not cond6:
              print("Condition C6 does not hold for order =", order)
  
      if conditions_hold:
          print("Conditions C5-C6 hold for all orders.")
      sys.stdout.flush()
      return conditions_hold
  
file_path: privacy/research/pate_2018/ICLR2018/smooth_sensitivity_table.py
imports: from __future__ import absolute_import
  from __future__ import division
  from __future__ import print_function
  import math
  import os
  import sys
  from absl import app
  from absl import flags
  import numpy
  import core
  import smooth_sensitivity
label_desc: Use image node_id 1 for calling a global function with example usage: _check_conditions(sigma, num_classes, orders) and returns: conditions_hold