code: def test_dummy_order_inner_outer_lines_VT1T1T1T1_AT():
      ii, jj = symbols("i j", below_fermi=True)
      aa, bb = symbols("a b", above_fermi=True)
      k, l = symbols("k l", below_fermi=True, cls=Dummy)
      c, d = symbols("c d", above_fermi=True, cls=Dummy)
  
      # Coupled-Cluster T2 terms with V*T1*T1*T1*T1
      # non-equivalent substitutions (change of sign)
      exprs = [
          # permut t <=> swapping external lines
          atv(k, l, c, d)
          * att(c, ii)
          * att(d, jj)
          * att(aa, k)
          * att(bb, l),
          atv(k, l, c, d)
          * att(c, jj)
          * att(d, ii)
          * att(aa, k)
          * att(bb, l),
          atv(k, l, c, d)
          * att(c, ii)
          * att(d, jj)
          * att(bb, k)
          * att(aa, l),
      ]
      for permut in exprs[1:]:
          assert substitute_dummies(exprs[0]) == -substitute_dummies(
              permut
          )
  
      # equivalent substitutions
      exprs = [
          atv(k, l, c, d)
          * att(c, ii)
          * att(d, jj)
          * att(aa, k)
          * att(bb, l),
          # permut t <=> swapping external lines
          atv(k, l, c, d)
          * att(c, jj)
          * att(d, ii)
          * att(bb, k)
          * att(aa, l),
      ]
      for permut in exprs[1:]:
          assert substitute_dummies(exprs[0]) == substitute_dummies(
              permut
          )
  
imports: from sympy.physics.secondquant import Dagger, Bd, VarBosonicBasis, BBra, B, BKet, FixedBosonicBasis, matrix_rep, apply_operators, InnerProduct, Commutator, KroneckerDelta, AnnihilateBoson, CreateBoson, BosonicOperator, F, Fd, FKet, BosonState, CreateFermion, AnnihilateFermion, evaluate_deltas, AntiSymmetricTensor, contraction, NO, wicks, PermutationOperator, simplify_index_permutations, _sort_anticommuting_fermions, _get_ordered_dummies, substitute_dummies, FockStateBosonKet, ContractionAppliesOnlyToFermions
  from sympy.concrete.summations import Sum
  from sympy.core.function import Function, expand
  from sympy.core.numbers import I, Rational
  from sympy.core.singleton import S
  from sympy.core.symbol import Dummy, Symbol, symbols
  from sympy.functions.elementary.miscellaneous import sqrt
  from sympy.printing.repr import srepr
  from sympy.simplify.simplify import simplify
  from sympy.testing.pytest import slow, raises
  from sympy.printing.latex import latex
label_desc: Use image node_id 47 for calling a global function with example usage: test_dummy_order_inner_outer_lines_VT1T1T1T1_AT() without return types