/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES

After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5 },
it remains to prove termination of the 158-rule system
{ 0 1 0 ⟶ 0 0 1 2 ,
  0 1 0 ⟶ 0 1 2 0 ,
  0 1 0 ⟶ 0 1 3 0 ,
  0 1 0 ⟶ 1 2 0 0 ,
  0 1 0 ⟶ 1 3 0 0 ,
  0 1 0 ⟶ 0 0 1 2 4 ,
  0 1 0 ⟶ 0 0 1 4 4 ,
  0 1 0 ⟶ 0 0 4 1 2 ,
  0 1 0 ⟶ 0 1 2 0 2 ,
  0 1 0 ⟶ 0 1 3 0 2 ,
  0 1 0 ⟶ 0 1 3 0 4 ,
  0 1 0 ⟶ 0 1 4 5 0 ,
  0 1 0 ⟶ 0 2 1 3 0 ,
  0 1 0 ⟶ 0 3 1 3 0 ,
  0 1 0 ⟶ 0 4 1 2 0 ,
  0 1 0 ⟶ 0 4 1 3 0 ,
  0 1 0 ⟶ 1 1 2 0 0 ,
  0 1 0 ⟶ 1 2 0 5 0 ,
  0 1 0 ⟶ 1 2 2 0 0 ,
  0 1 0 ⟶ 1 2 3 0 0 ,
  0 1 0 ⟶ 1 2 5 0 0 ,
  0 1 0 ⟶ 1 3 0 2 0 ,
  0 1 0 ⟶ 1 3 0 3 0 ,
  0 1 0 ⟶ 1 3 0 4 0 ,
  0 1 0 ⟶ 1 3 0 5 0 ,
  0 1 0 ⟶ 1 4 2 0 0 ,
  0 1 0 ⟶ 1 4 3 0 0 ,
  0 1 0 ⟶ 1 5 2 0 0 ,
  0 1 0 ⟶ 2 0 0 2 1 ,
  0 1 0 ⟶ 2 0 0 3 1 ,
  0 1 0 ⟶ 2 1 2 0 0 ,
  0 1 0 ⟶ 2 1 3 0 0 ,
  0 1 0 ⟶ 2 1 5 0 0 ,
  0 1 0 ⟶ 3 1 2 0 0 ,
  0 1 0 ⟶ 3 1 3 0 0 ,
  0 1 0 ⟶ 4 1 3 0 0 ,
  0 1 0 ⟶ 0 0 2 1 2 4 ,
  0 1 0 ⟶ 0 0 3 4 1 2 ,
  0 1 0 ⟶ 0 0 4 1 2 2 ,
  0 1 0 ⟶ 0 0 5 1 4 4 ,
  0 1 0 ⟶ 0 0 5 5 1 2 ,
  0 1 0 ⟶ 0 1 1 2 5 0 ,
  0 1 0 ⟶ 0 1 2 2 3 0 ,
  0 1 0 ⟶ 0 1 3 0 4 2 ,
  0 1 0 ⟶ 0 1 4 1 3 0 ,
  0 1 0 ⟶ 0 2 0 2 1 2 ,
  0 1 0 ⟶ 0 2 1 4 4 0 ,
  0 1 0 ⟶ 0 2 1 4 5 0 ,
  0 1 0 ⟶ 0 2 2 1 2 0 ,
  0 1 0 ⟶ 0 2 5 1 2 0 ,
  0 1 0 ⟶ 0 3 1 3 0 4 ,
  0 1 0 ⟶ 0 3 4 1 3 0 ,
  0 1 0 ⟶ 0 4 5 1 3 0 ,
  0 1 0 ⟶ 1 1 1 2 0 0 ,
  0 1 0 ⟶ 1 1 1 3 0 0 ,
  0 1 0 ⟶ 1 1 2 0 3 0 ,
  0 1 0 ⟶ 1 1 2 2 0 0 ,
  0 1 0 ⟶ 1 1 3 0 5 0 ,
  0 1 0 ⟶ 1 1 4 5 0 0 ,
  0 1 0 ⟶ 1 2 2 5 0 0 ,
  0 1 0 ⟶ 1 3 1 5 0 0 ,
  0 1 0 ⟶ 1 3 5 0 0 2 ,
  0 1 0 ⟶ 1 4 2 2 0 0 ,
  0 1 0 ⟶ 1 4 4 0 0 2 ,
  0 1 0 ⟶ 1 4 4 3 0 0 ,
  0 1 0 ⟶ 1 4 4 5 0 0 ,
  0 1 0 ⟶ 1 5 0 1 3 0 ,
  0 1 0 ⟶ 2 0 0 1 2 1 ,
  0 1 0 ⟶ 2 0 0 2 1 4 ,
  0 1 0 ⟶ 2 0 0 2 3 1 ,
  0 1 0 ⟶ 2 0 0 5 1 2 ,
  0 1 0 ⟶ 2 0 1 1 2 0 ,
  0 1 0 ⟶ 2 0 4 1 2 0 ,
  0 1 0 ⟶ 2 1 0 2 4 0 ,
  0 1 0 ⟶ 2 1 5 0 3 0 ,
  0 1 0 ⟶ 2 2 1 3 0 0 ,
  0 1 0 ⟶ 3 1 2 0 2 0 ,
  0 1 0 ⟶ 3 5 0 0 1 2 ,
  0 1 0 ⟶ 4 1 2 0 3 0 ,
  0 1 0 ⟶ 4 2 0 0 1 2 ,
  0 1 0 ⟶ 4 2 1 3 0 0 ,
  0 1 0 ⟶ 4 3 1 5 0 0 ,
  0 1 0 ⟶ 4 5 1 3 0 0 ,
  0 1 0 ⟶ 5 0 0 1 1 2 ,
  0 1 0 ⟶ 5 0 0 2 1 2 ,
  0 1 0 ⟶ 5 1 2 0 5 0 ,
  0 1 0 ⟶ 5 1 2 3 0 0 ,
  0 1 0 ⟶ 5 1 2 5 0 0 ,
  0 1 0 ⟶ 5 1 3 0 0 2 ,
  0 1 0 ⟶ 5 1 3 0 3 0 ,
  0 1 0 ⟶ 5 1 4 3 0 0 ,
  0 1 0 ⟶ 5 5 1 3 0 0 ,
  0 0 0 0 ⟶ 0 0 0 2 0 ,
  0 0 1 0 ⟶ 0 0 0 3 1 ,
  0 0 1 0 ⟶ 0 0 1 2 0 ,
  0 0 1 0 ⟶ 0 0 0 1 3 1 ,
  0 0 1 0 ⟶ 0 0 2 1 3 0 ,
  0 0 1 0 ⟶ 0 0 2 2 1 0 ,
  0 0 1 0 ⟶ 0 0 4 2 1 0 ,
  0 0 1 0 ⟶ 0 0 5 0 1 2 ,
  0 0 1 0 ⟶ 0 1 2 4 0 0 ,
  0 0 1 0 ⟶ 0 1 3 0 3 0 ,
  0 0 1 0 ⟶ 0 2 1 5 0 0 ,
  0 0 1 0 ⟶ 0 5 1 2 0 0 ,
  0 1 0 0 ⟶ 1 3 0 0 0 ,
  0 1 0 0 ⟶ 0 0 0 1 2 4 ,
  0 1 0 0 ⟶ 0 0 1 4 2 0 ,
  0 1 0 0 ⟶ 0 0 4 1 2 0 ,
  0 1 0 0 ⟶ 0 1 3 0 0 4 ,
  0 1 0 0 ⟶ 0 1 4 3 0 0 ,
  0 1 0 0 ⟶ 0 2 0 1 3 0 ,
  0 1 0 0 ⟶ 1 2 4 0 0 0 ,
  0 1 0 4 ⟶ 0 0 1 2 4 ,
  0 1 0 4 ⟶ 0 0 1 4 2 ,
  0 1 0 4 ⟶ 0 0 4 1 2 ,
  0 1 0 4 ⟶ 1 2 0 0 4 ,
  0 1 0 4 ⟶ 0 0 5 1 4 2 ,
  0 1 0 4 ⟶ 1 2 0 4 5 0 ,
  0 1 0 4 ⟶ 1 4 4 0 2 0 ,
  0 1 1 0 ⟶ 0 1 1 3 0 ,
  0 1 1 0 ⟶ 1 1 3 0 0 ,
  0 1 1 0 ⟶ 0 0 1 1 2 4 ,
  0 1 1 0 ⟶ 0 1 0 5 3 1 ,
  0 1 1 0 ⟶ 0 1 4 3 1 0 ,
  0 1 1 0 ⟶ 0 2 0 1 4 1 ,
  0 1 1 0 ⟶ 0 4 1 2 1 0 ,
  0 1 1 0 ⟶ 0 4 1 4 1 0 ,
  0 1 1 0 ⟶ 0 5 0 2 1 1 ,
  0 1 1 0 ⟶ 0 5 1 3 1 0 ,
  0 1 1 0 ⟶ 1 1 1 3 0 0 ,
  0 1 1 0 ⟶ 1 2 0 0 2 1 ,
  0 1 1 0 ⟶ 1 4 5 0 1 0 ,
  0 1 1 0 ⟶ 2 0 1 2 1 0 ,
  0 1 1 0 ⟶ 2 1 0 0 2 1 ,
  0 1 1 0 ⟶ 2 1 2 0 0 1 ,
  0 1 1 0 ⟶ 4 1 1 3 0 0 ,
  0 4 1 0 ⟶ 0 0 4 1 2 ,
  0 4 1 0 ⟶ 0 0 4 1 2 4 ,
  0 4 1 0 ⟶ 0 3 0 1 4 2 ,
  0 4 1 0 ⟶ 0 5 1 4 5 0 ,
  0 5 1 0 ⟶ 1 2 5 0 0 ,
  0 5 1 0 ⟶ 1 5 2 0 0 ,
  0 5 1 0 ⟶ 4 3 1 5 0 0 ,
  0 5 1 0 ⟶ 5 2 1 2 0 0 ,
  0 5 4 0 ⟶ 0 1 4 5 0 ,
  0 5 4 0 ⟶ 5 0 0 1 4 4 ,
  0 0 0 1 0 ⟶ 0 0 0 2 1 0 ,
  0 1 0 4 4 ⟶ 0 0 4 4 1 3 ,
  0 1 0 4 4 ⟶ 0 4 4 1 3 0 ,
  0 1 0 4 4 ⟶ 3 1 4 4 0 0 ,
  0 1 1 1 0 ⟶ 0 0 1 1 2 1 ,
  0 2 0 1 0 ⟶ 0 3 1 2 0 0 ,
  0 3 0 1 0 ⟶ 0 5 0 0 3 1 ,
  0 4 4 1 0 ⟶ 0 0 4 1 2 4 ,
  0 4 5 1 0 ⟶ 0 0 5 4 1 2 ,
  0 5 0 1 0 ⟶ 5 0 0 1 2 0 ,
  0 5 1 1 0 ⟶ 0 1 5 1 3 0 ,
  0 5 4 4 0 ⟶ 4 0 4 5 0 2 }

The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 5:

0 ↦ ⎛           ⎞
    ⎜ 1 0 1 0 0 ⎟
    ⎜ 0 1 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 ⎟
    ⎝           ⎠

1 ↦ ⎛           ⎞
    ⎜ 1 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 1 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎝           ⎠

2 ↦ ⎛           ⎞
    ⎜ 1 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎝           ⎠

3 ↦ ⎛           ⎞
    ⎜ 1 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎝           ⎠

4 ↦ ⎛           ⎞
    ⎜ 1 0 1 0 0 ⎟
    ⎜ 0 1 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎝           ⎠

5 ↦ ⎛           ⎞
    ⎜ 1 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 ⎟
    ⎜ 0 0 0 1 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎝           ⎠

After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5 },
it remains to prove termination of the 153-rule system
{ 0 1 0 ⟶ 0 0 1 2 ,
  0 1 0 ⟶ 0 1 2 0 ,
  0 1 0 ⟶ 0 1 3 0 ,
  0 1 0 ⟶ 1 2 0 0 ,
  0 1 0 ⟶ 1 3 0 0 ,
  0 1 0 ⟶ 0 0 1 2 4 ,
  0 1 0 ⟶ 0 0 1 4 4 ,
  0 1 0 ⟶ 0 0 4 1 2 ,
  0 1 0 ⟶ 0 1 2 0 2 ,
  0 1 0 ⟶ 0 1 3 0 2 ,
  0 1 0 ⟶ 0 1 3 0 4 ,
  0 1 0 ⟶ 0 1 4 5 0 ,
  0 1 0 ⟶ 0 2 1 3 0 ,
  0 1 0 ⟶ 0 3 1 3 0 ,
  0 1 0 ⟶ 0 4 1 2 0 ,
  0 1 0 ⟶ 0 4 1 3 0 ,
  0 1 0 ⟶ 1 1 2 0 0 ,
  0 1 0 ⟶ 1 2 0 5 0 ,
  0 1 0 ⟶ 1 2 2 0 0 ,
  0 1 0 ⟶ 1 2 3 0 0 ,
  0 1 0 ⟶ 1 2 5 0 0 ,
  0 1 0 ⟶ 1 3 0 2 0 ,
  0 1 0 ⟶ 1 3 0 3 0 ,
  0 1 0 ⟶ 1 3 0 4 0 ,
  0 1 0 ⟶ 1 3 0 5 0 ,
  0 1 0 ⟶ 1 4 2 0 0 ,
  0 1 0 ⟶ 1 4 3 0 0 ,
  0 1 0 ⟶ 1 5 2 0 0 ,
  0 1 0 ⟶ 2 0 0 2 1 ,
  0 1 0 ⟶ 2 0 0 3 1 ,
  0 1 0 ⟶ 2 1 2 0 0 ,
  0 1 0 ⟶ 2 1 3 0 0 ,
  0 1 0 ⟶ 2 1 5 0 0 ,
  0 1 0 ⟶ 3 1 2 0 0 ,
  0 1 0 ⟶ 3 1 3 0 0 ,
  0 1 0 ⟶ 4 1 3 0 0 ,
  0 1 0 ⟶ 0 0 2 1 2 4 ,
  0 1 0 ⟶ 0 0 3 4 1 2 ,
  0 1 0 ⟶ 0 0 4 1 2 2 ,
  0 1 0 ⟶ 0 0 5 1 4 4 ,
  0 1 0 ⟶ 0 0 5 5 1 2 ,
  0 1 0 ⟶ 0 1 1 2 5 0 ,
  0 1 0 ⟶ 0 1 2 2 3 0 ,
  0 1 0 ⟶ 0 1 3 0 4 2 ,
  0 1 0 ⟶ 0 1 4 1 3 0 ,
  0 1 0 ⟶ 0 2 0 2 1 2 ,
  0 1 0 ⟶ 0 2 1 4 4 0 ,
  0 1 0 ⟶ 0 2 1 4 5 0 ,
  0 1 0 ⟶ 0 2 2 1 2 0 ,
  0 1 0 ⟶ 0 2 5 1 2 0 ,
  0 1 0 ⟶ 0 3 1 3 0 4 ,
  0 1 0 ⟶ 0 3 4 1 3 0 ,
  0 1 0 ⟶ 0 4 5 1 3 0 ,
  0 1 0 ⟶ 1 1 1 2 0 0 ,
  0 1 0 ⟶ 1 1 1 3 0 0 ,
  0 1 0 ⟶ 1 1 2 0 3 0 ,
  0 1 0 ⟶ 1 1 2 2 0 0 ,
  0 1 0 ⟶ 1 1 3 0 5 0 ,
  0 1 0 ⟶ 1 1 4 5 0 0 ,
  0 1 0 ⟶ 1 2 2 5 0 0 ,
  0 1 0 ⟶ 1 3 1 5 0 0 ,
  0 1 0 ⟶ 1 3 5 0 0 2 ,
  0 1 0 ⟶ 1 4 2 2 0 0 ,
  0 1 0 ⟶ 1 4 4 0 0 2 ,
  0 1 0 ⟶ 1 4 4 3 0 0 ,
  0 1 0 ⟶ 1 4 4 5 0 0 ,
  0 1 0 ⟶ 1 5 0 1 3 0 ,
  0 1 0 ⟶ 2 0 0 1 2 1 ,
  0 1 0 ⟶ 2 0 0 2 1 4 ,
  0 1 0 ⟶ 2 0 0 2 3 1 ,
  0 1 0 ⟶ 2 0 0 5 1 2 ,
  0 1 0 ⟶ 2 0 1 1 2 0 ,
  0 1 0 ⟶ 2 0 4 1 2 0 ,
  0 1 0 ⟶ 2 1 0 2 4 0 ,
  0 1 0 ⟶ 2 1 5 0 3 0 ,
  0 1 0 ⟶ 2 2 1 3 0 0 ,
  0 1 0 ⟶ 3 1 2 0 2 0 ,
  0 1 0 ⟶ 3 5 0 0 1 2 ,
  0 1 0 ⟶ 4 1 2 0 3 0 ,
  0 1 0 ⟶ 4 2 0 0 1 2 ,
  0 1 0 ⟶ 4 2 1 3 0 0 ,
  0 1 0 ⟶ 4 3 1 5 0 0 ,
  0 1 0 ⟶ 4 5 1 3 0 0 ,
  0 1 0 ⟶ 5 0 0 1 1 2 ,
  0 1 0 ⟶ 5 0 0 2 1 2 ,
  0 1 0 ⟶ 5 1 2 0 5 0 ,
  0 1 0 ⟶ 5 1 2 3 0 0 ,
  0 1 0 ⟶ 5 1 2 5 0 0 ,
  0 1 0 ⟶ 5 1 3 0 0 2 ,
  0 1 0 ⟶ 5 1 3 0 3 0 ,
  0 1 0 ⟶ 5 1 4 3 0 0 ,
  0 1 0 ⟶ 5 5 1 3 0 0 ,
  0 0 0 0 ⟶ 0 0 0 2 0 ,
  0 0 1 0 ⟶ 0 0 0 3 1 ,
  0 0 1 0 ⟶ 0 0 1 2 0 ,
  0 0 1 0 ⟶ 0 0 0 1 3 1 ,
  0 0 1 0 ⟶ 0 0 2 1 3 0 ,
  0 0 1 0 ⟶ 0 0 2 2 1 0 ,
  0 0 1 0 ⟶ 0 0 4 2 1 0 ,
  0 0 1 0 ⟶ 0 0 5 0 1 2 ,
  0 0 1 0 ⟶ 0 1 2 4 0 0 ,
  0 0 1 0 ⟶ 0 1 3 0 3 0 ,
  0 0 1 0 ⟶ 0 2 1 5 0 0 ,
  0 0 1 0 ⟶ 0 5 1 2 0 0 ,
  0 1 0 0 ⟶ 1 3 0 0 0 ,
  0 1 0 0 ⟶ 0 0 0 1 2 4 ,
  0 1 0 0 ⟶ 0 0 1 4 2 0 ,
  0 1 0 0 ⟶ 0 0 4 1 2 0 ,
  0 1 0 0 ⟶ 0 1 3 0 0 4 ,
  0 1 0 0 ⟶ 0 1 4 3 0 0 ,
  0 1 0 0 ⟶ 0 2 0 1 3 0 ,
  0 1 0 0 ⟶ 1 2 4 0 0 0 ,
  0 1 0 4 ⟶ 0 0 1 2 4 ,
  0 1 0 4 ⟶ 0 0 1 4 2 ,
  0 1 0 4 ⟶ 0 0 4 1 2 ,
  0 1 0 4 ⟶ 1 2 0 0 4 ,
  0 1 0 4 ⟶ 0 0 5 1 4 2 ,
  0 1 0 4 ⟶ 1 2 0 4 5 0 ,
  0 1 0 4 ⟶ 1 4 4 0 2 0 ,
  0 1 1 0 ⟶ 0 1 1 3 0 ,
  0 1 1 0 ⟶ 1 1 3 0 0 ,
  0 1 1 0 ⟶ 0 0 1 1 2 4 ,
  0 1 1 0 ⟶ 0 1 0 5 3 1 ,
  0 1 1 0 ⟶ 0 1 4 3 1 0 ,
  0 1 1 0 ⟶ 0 2 0 1 4 1 ,
  0 1 1 0 ⟶ 0 4 1 2 1 0 ,
  0 1 1 0 ⟶ 0 4 1 4 1 0 ,
  0 1 1 0 ⟶ 0 5 0 2 1 1 ,
  0 1 1 0 ⟶ 0 5 1 3 1 0 ,
  0 1 1 0 ⟶ 1 1 1 3 0 0 ,
  0 1 1 0 ⟶ 1 2 0 0 2 1 ,
  0 1 1 0 ⟶ 1 4 5 0 1 0 ,
  0 1 1 0 ⟶ 2 0 1 2 1 0 ,
  0 1 1 0 ⟶ 2 1 0 0 2 1 ,
  0 1 1 0 ⟶ 2 1 2 0 0 1 ,
  0 1 1 0 ⟶ 4 1 1 3 0 0 ,
  0 4 1 0 ⟶ 0 0 4 1 2 ,
  0 4 1 0 ⟶ 0 0 4 1 2 4 ,
  0 4 1 0 ⟶ 0 3 0 1 4 2 ,
  0 4 1 0 ⟶ 0 5 1 4 5 0 ,
  0 5 4 0 ⟶ 0 1 4 5 0 ,
  0 5 4 0 ⟶ 5 0 0 1 4 4 ,
  0 0 0 1 0 ⟶ 0 0 0 2 1 0 ,
  0 1 0 4 4 ⟶ 0 0 4 4 1 3 ,
  0 1 0 4 4 ⟶ 0 4 4 1 3 0 ,
  0 1 0 4 4 ⟶ 3 1 4 4 0 0 ,
  0 1 1 1 0 ⟶ 0 0 1 1 2 1 ,
  0 2 0 1 0 ⟶ 0 3 1 2 0 0 ,
  0 3 0 1 0 ⟶ 0 5 0 0 3 1 ,
  0 4 4 1 0 ⟶ 0 0 4 1 2 4 ,
  0 5 0 1 0 ⟶ 5 0 0 1 2 0 ,
  0 5 1 1 0 ⟶ 0 1 5 1 3 0 ,
  0 5 4 4 0 ⟶ 4 0 4 5 0 2 }

The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 6:

0 ↦ ⎛             ⎞
    ⎜ 1 0 1 0 0 0 ⎟
    ⎜ 0 1 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 0 ⎟
    ⎝             ⎠

1 ↦ ⎛             ⎞
    ⎜ 1 0 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 1 0 ⎟
    ⎜ 0 0 0 0 0 1 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎝             ⎠

2 ↦ ⎛             ⎞
    ⎜ 1 0 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎝             ⎠

3 ↦ ⎛             ⎞
    ⎜ 1 0 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎝             ⎠

4 ↦ ⎛             ⎞
    ⎜ 1 0 1 0 0 0 ⎟
    ⎜ 0 1 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎝             ⎠

5 ↦ ⎛             ⎞
    ⎜ 1 0 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 0 ⎟
    ⎜ 0 0 0 1 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎝             ⎠

After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5 },
it remains to prove termination of the 152-rule system
{ 0 1 0 ⟶ 0 0 1 2 ,
  0 1 0 ⟶ 0 1 2 0 ,
  0 1 0 ⟶ 0 1 3 0 ,
  0 1 0 ⟶ 1 2 0 0 ,
  0 1 0 ⟶ 1 3 0 0 ,
  0 1 0 ⟶ 0 0 1 2 4 ,
  0 1 0 ⟶ 0 0 1 4 4 ,
  0 1 0 ⟶ 0 0 4 1 2 ,
  0 1 0 ⟶ 0 1 2 0 2 ,
  0 1 0 ⟶ 0 1 3 0 2 ,
  0 1 0 ⟶ 0 1 3 0 4 ,
  0 1 0 ⟶ 0 1 4 5 0 ,
  0 1 0 ⟶ 0 2 1 3 0 ,
  0 1 0 ⟶ 0 3 1 3 0 ,
  0 1 0 ⟶ 0 4 1 2 0 ,
  0 1 0 ⟶ 0 4 1 3 0 ,
  0 1 0 ⟶ 1 1 2 0 0 ,
  0 1 0 ⟶ 1 2 0 5 0 ,
  0 1 0 ⟶ 1 2 2 0 0 ,
  0 1 0 ⟶ 1 2 3 0 0 ,
  0 1 0 ⟶ 1 2 5 0 0 ,
  0 1 0 ⟶ 1 3 0 2 0 ,
  0 1 0 ⟶ 1 3 0 3 0 ,
  0 1 0 ⟶ 1 3 0 4 0 ,
  0 1 0 ⟶ 1 3 0 5 0 ,
  0 1 0 ⟶ 1 4 2 0 0 ,
  0 1 0 ⟶ 1 4 3 0 0 ,
  0 1 0 ⟶ 1 5 2 0 0 ,
  0 1 0 ⟶ 2 0 0 2 1 ,
  0 1 0 ⟶ 2 0 0 3 1 ,
  0 1 0 ⟶ 2 1 2 0 0 ,
  0 1 0 ⟶ 2 1 3 0 0 ,
  0 1 0 ⟶ 2 1 5 0 0 ,
  0 1 0 ⟶ 3 1 2 0 0 ,
  0 1 0 ⟶ 3 1 3 0 0 ,
  0 1 0 ⟶ 4 1 3 0 0 ,
  0 1 0 ⟶ 0 0 2 1 2 4 ,
  0 1 0 ⟶ 0 0 3 4 1 2 ,
  0 1 0 ⟶ 0 0 4 1 2 2 ,
  0 1 0 ⟶ 0 0 5 1 4 4 ,
  0 1 0 ⟶ 0 0 5 5 1 2 ,
  0 1 0 ⟶ 0 1 1 2 5 0 ,
  0 1 0 ⟶ 0 1 2 2 3 0 ,
  0 1 0 ⟶ 0 1 3 0 4 2 ,
  0 1 0 ⟶ 0 1 4 1 3 0 ,
  0 1 0 ⟶ 0 2 0 2 1 2 ,
  0 1 0 ⟶ 0 2 1 4 4 0 ,
  0 1 0 ⟶ 0 2 1 4 5 0 ,
  0 1 0 ⟶ 0 2 2 1 2 0 ,
  0 1 0 ⟶ 0 2 5 1 2 0 ,
  0 1 0 ⟶ 0 3 1 3 0 4 ,
  0 1 0 ⟶ 0 3 4 1 3 0 ,
  0 1 0 ⟶ 0 4 5 1 3 0 ,
  0 1 0 ⟶ 1 1 1 2 0 0 ,
  0 1 0 ⟶ 1 1 1 3 0 0 ,
  0 1 0 ⟶ 1 1 2 0 3 0 ,
  0 1 0 ⟶ 1 1 2 2 0 0 ,
  0 1 0 ⟶ 1 1 3 0 5 0 ,
  0 1 0 ⟶ 1 1 4 5 0 0 ,
  0 1 0 ⟶ 1 2 2 5 0 0 ,
  0 1 0 ⟶ 1 3 1 5 0 0 ,
  0 1 0 ⟶ 1 3 5 0 0 2 ,
  0 1 0 ⟶ 1 4 2 2 0 0 ,
  0 1 0 ⟶ 1 4 4 0 0 2 ,
  0 1 0 ⟶ 1 4 4 3 0 0 ,
  0 1 0 ⟶ 1 4 4 5 0 0 ,
  0 1 0 ⟶ 1 5 0 1 3 0 ,
  0 1 0 ⟶ 2 0 0 1 2 1 ,
  0 1 0 ⟶ 2 0 0 2 1 4 ,
  0 1 0 ⟶ 2 0 0 2 3 1 ,
  0 1 0 ⟶ 2 0 0 5 1 2 ,
  0 1 0 ⟶ 2 0 1 1 2 0 ,
  0 1 0 ⟶ 2 0 4 1 2 0 ,
  0 1 0 ⟶ 2 1 0 2 4 0 ,
  0 1 0 ⟶ 2 1 5 0 3 0 ,
  0 1 0 ⟶ 2 2 1 3 0 0 ,
  0 1 0 ⟶ 3 1 2 0 2 0 ,
  0 1 0 ⟶ 3 5 0 0 1 2 ,
  0 1 0 ⟶ 4 1 2 0 3 0 ,
  0 1 0 ⟶ 4 2 0 0 1 2 ,
  0 1 0 ⟶ 4 2 1 3 0 0 ,
  0 1 0 ⟶ 4 3 1 5 0 0 ,
  0 1 0 ⟶ 4 5 1 3 0 0 ,
  0 1 0 ⟶ 5 0 0 1 1 2 ,
  0 1 0 ⟶ 5 0 0 2 1 2 ,
  0 1 0 ⟶ 5 1 2 0 5 0 ,
  0 1 0 ⟶ 5 1 2 3 0 0 ,
  0 1 0 ⟶ 5 1 2 5 0 0 ,
  0 1 0 ⟶ 5 1 3 0 0 2 ,
  0 1 0 ⟶ 5 1 3 0 3 0 ,
  0 1 0 ⟶ 5 1 4 3 0 0 ,
  0 1 0 ⟶ 5 5 1 3 0 0 ,
  0 0 0 0 ⟶ 0 0 0 2 0 ,
  0 0 1 0 ⟶ 0 0 0 3 1 ,
  0 0 1 0 ⟶ 0 0 1 2 0 ,
  0 0 1 0 ⟶ 0 0 0 1 3 1 ,
  0 0 1 0 ⟶ 0 0 2 1 3 0 ,
  0 0 1 0 ⟶ 0 0 2 2 1 0 ,
  0 0 1 0 ⟶ 0 0 4 2 1 0 ,
  0 0 1 0 ⟶ 0 0 5 0 1 2 ,
  0 0 1 0 ⟶ 0 1 2 4 0 0 ,
  0 0 1 0 ⟶ 0 1 3 0 3 0 ,
  0 0 1 0 ⟶ 0 2 1 5 0 0 ,
  0 0 1 0 ⟶ 0 5 1 2 0 0 ,
  0 1 0 0 ⟶ 1 3 0 0 0 ,
  0 1 0 0 ⟶ 0 0 0 1 2 4 ,
  0 1 0 0 ⟶ 0 0 1 4 2 0 ,
  0 1 0 0 ⟶ 0 0 4 1 2 0 ,
  0 1 0 0 ⟶ 0 1 3 0 0 4 ,
  0 1 0 0 ⟶ 0 1 4 3 0 0 ,
  0 1 0 0 ⟶ 0 2 0 1 3 0 ,
  0 1 0 0 ⟶ 1 2 4 0 0 0 ,
  0 1 0 4 ⟶ 0 0 1 2 4 ,
  0 1 0 4 ⟶ 0 0 1 4 2 ,
  0 1 0 4 ⟶ 0 0 4 1 2 ,
  0 1 0 4 ⟶ 1 2 0 0 4 ,
  0 1 0 4 ⟶ 0 0 5 1 4 2 ,
  0 1 0 4 ⟶ 1 2 0 4 5 0 ,
  0 1 0 4 ⟶ 1 4 4 0 2 0 ,
  0 1 1 0 ⟶ 0 1 1 3 0 ,
  0 1 1 0 ⟶ 1 1 3 0 0 ,
  0 1 1 0 ⟶ 0 0 1 1 2 4 ,
  0 1 1 0 ⟶ 0 1 0 5 3 1 ,
  0 1 1 0 ⟶ 0 1 4 3 1 0 ,
  0 1 1 0 ⟶ 0 2 0 1 4 1 ,
  0 1 1 0 ⟶ 0 4 1 2 1 0 ,
  0 1 1 0 ⟶ 0 4 1 4 1 0 ,
  0 1 1 0 ⟶ 0 5 0 2 1 1 ,
  0 1 1 0 ⟶ 0 5 1 3 1 0 ,
  0 1 1 0 ⟶ 1 1 1 3 0 0 ,
  0 1 1 0 ⟶ 1 2 0 0 2 1 ,
  0 1 1 0 ⟶ 1 4 5 0 1 0 ,
  0 1 1 0 ⟶ 2 0 1 2 1 0 ,
  0 1 1 0 ⟶ 2 1 0 0 2 1 ,
  0 1 1 0 ⟶ 2 1 2 0 0 1 ,
  0 1 1 0 ⟶ 4 1 1 3 0 0 ,
  0 4 1 0 ⟶ 0 0 4 1 2 ,
  0 4 1 0 ⟶ 0 0 4 1 2 4 ,
  0 4 1 0 ⟶ 0 3 0 1 4 2 ,
  0 4 1 0 ⟶ 0 5 1 4 5 0 ,
  0 5 4 0 ⟶ 0 1 4 5 0 ,
  0 5 4 0 ⟶ 5 0 0 1 4 4 ,
  0 0 0 1 0 ⟶ 0 0 0 2 1 0 ,
  0 1 0 4 4 ⟶ 0 0 4 4 1 3 ,
  0 1 0 4 4 ⟶ 0 4 4 1 3 0 ,
  0 1 0 4 4 ⟶ 3 1 4 4 0 0 ,
  0 1 1 1 0 ⟶ 0 0 1 1 2 1 ,
  0 2 0 1 0 ⟶ 0 3 1 2 0 0 ,
  0 3 0 1 0 ⟶ 0 5 0 0 3 1 ,
  0 4 4 1 0 ⟶ 0 0 4 1 2 4 ,
  0 5 0 1 0 ⟶ 5 0 0 1 2 0 ,
  0 5 4 4 0 ⟶ 4 0 4 5 0 2 }

The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 5:

0 ↦ ⎛           ⎞
    ⎜ 1 0 1 0 0 ⎟
    ⎜ 0 1 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 1 1 1 0 ⎟
    ⎝           ⎠

1 ↦ ⎛           ⎞
    ⎜ 1 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 ⎟
    ⎜ 0 0 0 1 0 ⎟
    ⎜ 0 0 0 0 1 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎝           ⎠

2 ↦ ⎛           ⎞
    ⎜ 1 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎝           ⎠

3 ↦ ⎛           ⎞
    ⎜ 1 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎝           ⎠

4 ↦ ⎛           ⎞
    ⎜ 1 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 ⎟
    ⎜ 0 0 1 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎝           ⎠

5 ↦ ⎛           ⎞
    ⎜ 1 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎝           ⎠

After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5 },
it remains to prove termination of the 135-rule system
{ 0 1 0 ⟶ 0 0 1 2 ,
  0 1 0 ⟶ 0 1 2 0 ,
  0 1 0 ⟶ 0 1 3 0 ,
  0 1 0 ⟶ 1 2 0 0 ,
  0 1 0 ⟶ 1 3 0 0 ,
  0 1 0 ⟶ 0 0 1 2 4 ,
  0 1 0 ⟶ 0 0 1 4 4 ,
  0 1 0 ⟶ 0 0 4 1 2 ,
  0 1 0 ⟶ 0 1 2 0 2 ,
  0 1 0 ⟶ 0 1 3 0 2 ,
  0 1 0 ⟶ 0 1 3 0 4 ,
  0 1 0 ⟶ 0 1 4 5 0 ,
  0 1 0 ⟶ 0 2 1 3 0 ,
  0 1 0 ⟶ 0 3 1 3 0 ,
  0 1 0 ⟶ 0 4 1 2 0 ,
  0 1 0 ⟶ 0 4 1 3 0 ,
  0 1 0 ⟶ 1 1 2 0 0 ,
  0 1 0 ⟶ 1 2 0 5 0 ,
  0 1 0 ⟶ 1 2 2 0 0 ,
  0 1 0 ⟶ 1 2 3 0 0 ,
  0 1 0 ⟶ 1 2 5 0 0 ,
  0 1 0 ⟶ 1 3 0 2 0 ,
  0 1 0 ⟶ 1 3 0 3 0 ,
  0 1 0 ⟶ 1 3 0 4 0 ,
  0 1 0 ⟶ 1 3 0 5 0 ,
  0 1 0 ⟶ 1 4 2 0 0 ,
  0 1 0 ⟶ 1 4 3 0 0 ,
  0 1 0 ⟶ 1 5 2 0 0 ,
  0 1 0 ⟶ 2 0 0 2 1 ,
  0 1 0 ⟶ 2 0 0 3 1 ,
  0 1 0 ⟶ 2 1 2 0 0 ,
  0 1 0 ⟶ 2 1 3 0 0 ,
  0 1 0 ⟶ 2 1 5 0 0 ,
  0 1 0 ⟶ 3 1 2 0 0 ,
  0 1 0 ⟶ 3 1 3 0 0 ,
  0 1 0 ⟶ 4 1 3 0 0 ,
  0 1 0 ⟶ 0 0 2 1 2 4 ,
  0 1 0 ⟶ 0 0 3 4 1 2 ,
  0 1 0 ⟶ 0 0 4 1 2 2 ,
  0 1 0 ⟶ 0 0 5 1 4 4 ,
  0 1 0 ⟶ 0 0 5 5 1 2 ,
  0 1 0 ⟶ 0 1 1 2 5 0 ,
  0 1 0 ⟶ 0 1 2 2 3 0 ,
  0 1 0 ⟶ 0 1 3 0 4 2 ,
  0 1 0 ⟶ 0 1 4 1 3 0 ,
  0 1 0 ⟶ 0 2 0 2 1 2 ,
  0 1 0 ⟶ 0 2 1 4 4 0 ,
  0 1 0 ⟶ 0 2 1 4 5 0 ,
  0 1 0 ⟶ 0 2 2 1 2 0 ,
  0 1 0 ⟶ 0 2 5 1 2 0 ,
  0 1 0 ⟶ 0 3 1 3 0 4 ,
  0 1 0 ⟶ 0 3 4 1 3 0 ,
  0 1 0 ⟶ 0 4 5 1 3 0 ,
  0 1 0 ⟶ 1 1 1 2 0 0 ,
  0 1 0 ⟶ 1 1 1 3 0 0 ,
  0 1 0 ⟶ 1 1 2 0 3 0 ,
  0 1 0 ⟶ 1 1 2 2 0 0 ,
  0 1 0 ⟶ 1 1 3 0 5 0 ,
  0 1 0 ⟶ 1 1 4 5 0 0 ,
  0 1 0 ⟶ 1 2 2 5 0 0 ,
  0 1 0 ⟶ 1 3 1 5 0 0 ,
  0 1 0 ⟶ 1 3 5 0 0 2 ,
  0 1 0 ⟶ 1 4 2 2 0 0 ,
  0 1 0 ⟶ 1 4 4 0 0 2 ,
  0 1 0 ⟶ 1 4 4 3 0 0 ,
  0 1 0 ⟶ 1 4 4 5 0 0 ,
  0 1 0 ⟶ 1 5 0 1 3 0 ,
  0 1 0 ⟶ 2 0 0 1 2 1 ,
  0 1 0 ⟶ 2 0 0 2 1 4 ,
  0 1 0 ⟶ 2 0 0 2 3 1 ,
  0 1 0 ⟶ 2 0 0 5 1 2 ,
  0 1 0 ⟶ 2 0 1 1 2 0 ,
  0 1 0 ⟶ 2 0 4 1 2 0 ,
  0 1 0 ⟶ 2 1 0 2 4 0 ,
  0 1 0 ⟶ 2 1 5 0 3 0 ,
  0 1 0 ⟶ 2 2 1 3 0 0 ,
  0 1 0 ⟶ 3 1 2 0 2 0 ,
  0 1 0 ⟶ 3 5 0 0 1 2 ,
  0 1 0 ⟶ 4 1 2 0 3 0 ,
  0 1 0 ⟶ 4 2 0 0 1 2 ,
  0 1 0 ⟶ 4 2 1 3 0 0 ,
  0 1 0 ⟶ 4 3 1 5 0 0 ,
  0 1 0 ⟶ 4 5 1 3 0 0 ,
  0 1 0 ⟶ 5 0 0 1 1 2 ,
  0 1 0 ⟶ 5 0 0 2 1 2 ,
  0 1 0 ⟶ 5 1 2 0 5 0 ,
  0 1 0 ⟶ 5 1 2 3 0 0 ,
  0 1 0 ⟶ 5 1 2 5 0 0 ,
  0 1 0 ⟶ 5 1 3 0 0 2 ,
  0 1 0 ⟶ 5 1 3 0 3 0 ,
  0 1 0 ⟶ 5 1 4 3 0 0 ,
  0 1 0 ⟶ 5 5 1 3 0 0 ,
  0 0 0 0 ⟶ 0 0 0 2 0 ,
  0 0 1 0 ⟶ 0 0 0 3 1 ,
  0 0 1 0 ⟶ 0 0 1 2 0 ,
  0 0 1 0 ⟶ 0 0 0 1 3 1 ,
  0 0 1 0 ⟶ 0 0 2 1 3 0 ,
  0 0 1 0 ⟶ 0 0 2 2 1 0 ,
  0 0 1 0 ⟶ 0 0 4 2 1 0 ,
  0 0 1 0 ⟶ 0 0 5 0 1 2 ,
  0 0 1 0 ⟶ 0 1 2 4 0 0 ,
  0 0 1 0 ⟶ 0 1 3 0 3 0 ,
  0 0 1 0 ⟶ 0 2 1 5 0 0 ,
  0 0 1 0 ⟶ 0 5 1 2 0 0 ,
  0 1 0 0 ⟶ 1 3 0 0 0 ,
  0 1 0 0 ⟶ 0 0 0 1 2 4 ,
  0 1 0 0 ⟶ 0 0 1 4 2 0 ,
  0 1 0 0 ⟶ 0 0 4 1 2 0 ,
  0 1 0 0 ⟶ 0 1 3 0 0 4 ,
  0 1 0 0 ⟶ 0 1 4 3 0 0 ,
  0 1 0 0 ⟶ 0 2 0 1 3 0 ,
  0 1 0 0 ⟶ 1 2 4 0 0 0 ,
  0 1 0 4 ⟶ 0 0 1 2 4 ,
  0 1 0 4 ⟶ 0 0 1 4 2 ,
  0 1 0 4 ⟶ 0 0 4 1 2 ,
  0 1 0 4 ⟶ 1 2 0 0 4 ,
  0 1 0 4 ⟶ 0 0 5 1 4 2 ,
  0 1 0 4 ⟶ 1 2 0 4 5 0 ,
  0 1 0 4 ⟶ 1 4 4 0 2 0 ,
  0 4 1 0 ⟶ 0 0 4 1 2 ,
  0 4 1 0 ⟶ 0 0 4 1 2 4 ,
  0 4 1 0 ⟶ 0 3 0 1 4 2 ,
  0 4 1 0 ⟶ 0 5 1 4 5 0 ,
  0 5 4 0 ⟶ 0 1 4 5 0 ,
  0 5 4 0 ⟶ 5 0 0 1 4 4 ,
  0 0 0 1 0 ⟶ 0 0 0 2 1 0 ,
  0 1 0 4 4 ⟶ 0 0 4 4 1 3 ,
  0 1 0 4 4 ⟶ 0 4 4 1 3 0 ,
  0 1 0 4 4 ⟶ 3 1 4 4 0 0 ,
  0 1 1 1 0 ⟶ 0 0 1 1 2 1 ,
  0 2 0 1 0 ⟶ 0 3 1 2 0 0 ,
  0 3 0 1 0 ⟶ 0 5 0 0 3 1 ,
  0 4 4 1 0 ⟶ 0 0 4 1 2 4 ,
  0 5 0 1 0 ⟶ 5 0 0 1 2 0 ,
  0 5 4 4 0 ⟶ 4 0 4 5 0 2 }

The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 4:

0 ↦ ⎛         ⎞
    ⎜ 1 0 1 0 ⎟
    ⎜ 0 1 0 0 ⎟
    ⎜ 0 0 0 0 ⎟
    ⎜ 0 1 0 0 ⎟
    ⎝         ⎠

1 ↦ ⎛         ⎞
    ⎜ 1 0 0 0 ⎟
    ⎜ 0 1 0 0 ⎟
    ⎜ 0 0 0 1 ⎟
    ⎜ 0 0 0 0 ⎟
    ⎝         ⎠

2 ↦ ⎛         ⎞
    ⎜ 1 0 0 0 ⎟
    ⎜ 0 1 0 0 ⎟
    ⎜ 0 0 0 0 ⎟
    ⎜ 0 0 0 0 ⎟
    ⎝         ⎠

3 ↦ ⎛         ⎞
    ⎜ 1 0 0 0 ⎟
    ⎜ 0 1 0 0 ⎟
    ⎜ 0 0 0 0 ⎟
    ⎜ 0 0 0 0 ⎟
    ⎝         ⎠

4 ↦ ⎛         ⎞
    ⎜ 1 0 1 0 ⎟
    ⎜ 0 1 0 0 ⎟
    ⎜ 0 0 0 0 ⎟
    ⎜ 0 0 0 0 ⎟
    ⎝         ⎠

5 ↦ ⎛         ⎞
    ⎜ 1 0 0 0 ⎟
    ⎜ 0 1 0 0 ⎟
    ⎜ 0 0 0 0 ⎟
    ⎜ 0 0 0 0 ⎟
    ⎝         ⎠

After renaming modulo the bijection { 0 ↦ 0, 2 ↦ 1, 5 ↦ 2, 4 ↦ 3, 1 ↦ 4 },
it remains to prove termination of the 5-rule system
{ 0 0 0 0 ⟶ 0 0 0 1 0 ,
  0 2 3 0 ⟶ 0 4 3 2 0 ,
  0 2 3 0 ⟶ 2 0 0 4 3 3 ,
  0 4 4 4 0 ⟶ 0 0 4 4 1 4 ,
  0 2 3 3 0 ⟶ 3 0 3 2 0 1 }

The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 6:

0 ↦ ⎛             ⎞
    ⎜ 1 0 1 0 0 0 ⎟
    ⎜ 0 1 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 0 ⎟
    ⎝             ⎠

1 ↦ ⎛             ⎞
    ⎜ 1 0 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎝             ⎠

2 ↦ ⎛             ⎞
    ⎜ 1 0 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎝             ⎠

3 ↦ ⎛             ⎞
    ⎜ 1 0 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎝             ⎠

4 ↦ ⎛             ⎞
    ⎜ 1 0 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 0 ⎟
    ⎜ 0 0 0 1 0 0 ⎟
    ⎜ 0 0 0 0 1 0 ⎟
    ⎜ 0 0 0 0 0 1 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎝             ⎠

After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4 },
it remains to prove termination of the 4-rule system
{ 0 0 0 0 ⟶ 0 0 0 1 0 ,
  0 2 3 0 ⟶ 0 4 3 2 0 ,
  0 2 3 0 ⟶ 2 0 0 4 3 3 ,
  0 2 3 3 0 ⟶ 3 0 3 2 0 1 }

The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 5:

0 ↦ ⎛           ⎞
    ⎜ 1 0 1 0 0 ⎟
    ⎜ 0 1 0 0 0 ⎟
    ⎜ 0 0 0 1 0 ⎟
    ⎜ 0 0 0 0 1 ⎟
    ⎜ 0 1 0 0 0 ⎟
    ⎝           ⎠

1 ↦ ⎛           ⎞
    ⎜ 1 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎝           ⎠

2 ↦ ⎛           ⎞
    ⎜ 1 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎝           ⎠

3 ↦ ⎛           ⎞
    ⎜ 1 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎝           ⎠

4 ↦ ⎛           ⎞
    ⎜ 1 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎝           ⎠

After renaming modulo the bijection { 0 ↦ 0, 2 ↦ 1, 3 ↦ 2, 4 ↦ 3, 1 ↦ 4 },
it remains to prove termination of the 3-rule system
{ 0 1 2 0 ⟶ 0 3 2 1 0 ,
  0 1 2 0 ⟶ 1 0 0 3 2 2 ,
  0 1 2 2 0 ⟶ 2 0 2 1 0 4 }

The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 6:

0 ↦ ⎛             ⎞
    ⎜ 1 0 1 0 0 0 ⎟
    ⎜ 0 1 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 1 0 0 0 ⎟
    ⎜ 0 1 0 0 0 0 ⎟
    ⎝             ⎠

1 ↦ ⎛             ⎞
    ⎜ 1 0 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 0 ⎟
    ⎜ 0 0 0 1 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎝             ⎠

2 ↦ ⎛             ⎞
    ⎜ 1 0 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 1 0 ⎟
    ⎜ 0 0 0 0 0 1 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎝             ⎠

3 ↦ ⎛             ⎞
    ⎜ 1 0 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎝             ⎠

4 ↦ ⎛             ⎞
    ⎜ 1 0 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 0 ⎟
    ⎝             ⎠

After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3 },
it remains to prove termination of the 2-rule system
{ 0 1 2 0 ⟶ 0 3 2 1 0 ,
  0 1 2 0 ⟶ 1 0 0 3 2 2 }

The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 5:

0 ↦ ⎛           ⎞
    ⎜ 1 0 1 0 0 ⎟
    ⎜ 0 1 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 ⎟
    ⎝           ⎠

1 ↦ ⎛           ⎞
    ⎜ 1 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 ⎟
    ⎜ 0 0 0 1 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎝           ⎠

2 ↦ ⎛           ⎞
    ⎜ 1 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 1 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎝           ⎠

3 ↦ ⎛           ⎞
    ⎜ 1 0 0 0 0 ⎟
    ⎜ 0 1 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎜ 0 0 0 0 0 ⎟
    ⎝           ⎠

After renaming modulo the bijection {  },
it remains to prove termination of the 0-rule system
{ }

The system is trivially terminating.