/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- 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 50-rule system { 0 0 1 ⟶ 0 2 3 0 1 , 0 0 1 ⟶ 0 4 0 5 4 1 , 0 0 1 ⟶ 2 1 0 0 3 4 , 0 0 1 ⟶ 4 0 5 4 0 1 , 0 1 0 ⟶ 0 0 2 1 2 , 0 1 0 ⟶ 1 0 0 5 4 , 0 1 0 ⟶ 0 0 2 5 4 1 , 0 1 1 ⟶ 1 0 3 4 1 , 0 1 1 ⟶ 5 0 3 4 1 1 , 5 0 1 ⟶ 0 5 4 1 , 5 0 1 ⟶ 2 5 4 0 1 , 5 0 1 ⟶ 5 0 2 1 2 , 5 0 1 ⟶ 0 1 4 5 4 4 , 5 0 1 ⟶ 0 5 4 1 4 4 , 5 0 1 ⟶ 5 0 4 3 0 1 , 5 1 0 ⟶ 5 0 2 2 1 , 5 1 0 ⟶ 5 0 5 4 1 , 5 1 0 ⟶ 0 5 0 2 2 1 , 5 1 0 ⟶ 1 4 0 5 2 3 , 5 1 0 ⟶ 1 5 0 4 4 2 , 5 1 0 ⟶ 4 4 1 0 4 5 , 5 1 1 ⟶ 1 1 5 4 , 5 1 1 ⟶ 5 4 1 1 , 5 1 1 ⟶ 1 5 3 4 1 , 5 1 1 ⟶ 1 1 4 5 4 4 , 5 1 1 ⟶ 3 5 2 3 1 1 , 5 1 1 ⟶ 4 1 2 1 5 4 , 0 1 3 0 ⟶ 0 2 0 2 1 3 , 0 1 5 0 ⟶ 0 0 5 4 1 5 , 0 1 5 0 ⟶ 0 5 4 2 1 0 , 0 3 0 1 ⟶ 0 0 4 1 3 0 , 0 3 1 0 ⟶ 0 0 2 3 1 , 0 3 1 1 ⟶ 5 1 1 0 3 4 , 5 0 1 0 ⟶ 5 0 0 4 1 3 , 5 1 2 0 ⟶ 1 4 0 5 4 2 , 5 1 2 0 ⟶ 5 0 4 2 2 1 , 5 1 4 0 ⟶ 1 5 4 0 2 3 , 5 1 4 0 ⟶ 4 5 2 1 3 0 , 5 1 5 1 ⟶ 5 4 1 5 1 , 5 3 0 1 ⟶ 0 1 5 2 3 , 5 3 1 0 ⟶ 1 4 3 5 0 , 5 3 1 0 ⟶ 1 5 0 4 3 , 5 3 1 0 ⟶ 5 4 3 1 0 , 5 3 1 0 ⟶ 1 3 0 4 3 5 , 5 3 1 1 ⟶ 1 1 5 3 3 4 , 0 1 2 5 0 ⟶ 1 5 4 0 2 0 , 0 1 4 2 0 ⟶ 1 0 4 2 3 0 , 1 4 5 1 0 ⟶ 5 4 2 1 1 0 , 5 0 1 4 0 ⟶ 1 4 5 4 0 0 , 5 5 1 0 0 ⟶ 5 5 0 4 1 0 } The system was reversed. After renaming modulo the bijection { 1 ↦ 0, 0 ↦ 1, 3 ↦ 2, 2 ↦ 3, 4 ↦ 4, 5 ↦ 5 }, it remains to prove termination of the 50-rule system { 0 1 1 ⟶ 0 1 2 3 1 , 0 1 1 ⟶ 0 4 5 1 4 1 , 0 1 1 ⟶ 4 2 1 1 0 3 , 0 1 1 ⟶ 0 1 4 5 1 4 , 1 0 1 ⟶ 3 0 3 1 1 , 1 0 1 ⟶ 4 5 1 1 0 , 1 0 1 ⟶ 0 4 5 3 1 1 , 0 0 1 ⟶ 0 4 2 1 0 , 0 0 1 ⟶ 0 0 4 2 1 5 , 0 1 5 ⟶ 0 4 5 1 , 0 1 5 ⟶ 0 1 4 5 3 , 0 1 5 ⟶ 3 0 3 1 5 , 0 1 5 ⟶ 4 4 5 4 0 1 , 0 1 5 ⟶ 4 4 0 4 5 1 , 0 1 5 ⟶ 0 1 2 4 1 5 , 1 0 5 ⟶ 0 3 3 1 5 , 1 0 5 ⟶ 0 4 5 1 5 , 1 0 5 ⟶ 0 3 3 1 5 1 , 1 0 5 ⟶ 2 3 5 1 4 0 , 1 0 5 ⟶ 3 4 4 1 5 0 , 1 0 5 ⟶ 5 4 1 0 4 4 , 0 0 5 ⟶ 4 5 0 0 , 0 0 5 ⟶ 0 0 4 5 , 0 0 5 ⟶ 0 4 2 5 0 , 0 0 5 ⟶ 4 4 5 4 0 0 , 0 0 5 ⟶ 0 0 2 3 5 2 , 0 0 5 ⟶ 4 5 0 3 0 4 , 1 2 0 1 ⟶ 2 0 3 1 3 1 , 1 5 0 1 ⟶ 5 0 4 5 1 1 , 1 5 0 1 ⟶ 1 0 3 4 5 1 , 0 1 2 1 ⟶ 1 2 0 4 1 1 , 1 0 2 1 ⟶ 0 2 3 1 1 , 0 0 2 1 ⟶ 4 2 1 0 0 5 , 1 0 1 5 ⟶ 2 0 4 1 1 5 , 1 3 0 5 ⟶ 3 4 5 1 4 0 , 1 3 0 5 ⟶ 0 3 3 4 1 5 , 1 4 0 5 ⟶ 2 3 1 4 5 0 , 1 4 0 5 ⟶ 1 2 0 3 5 4 , 0 5 0 5 ⟶ 0 5 0 4 5 , 0 1 2 5 ⟶ 2 3 5 0 1 , 1 0 2 5 ⟶ 1 5 2 4 0 , 1 0 2 5 ⟶ 2 4 1 5 0 , 1 0 2 5 ⟶ 1 0 2 4 5 , 1 0 2 5 ⟶ 5 2 4 1 2 0 , 0 0 2 5 ⟶ 4 2 2 5 0 0 , 1 5 3 0 1 ⟶ 1 3 1 4 5 0 , 1 3 4 0 1 ⟶ 1 2 3 4 1 0 , 1 0 5 4 0 ⟶ 1 0 0 3 4 5 , 1 4 0 1 5 ⟶ 1 1 4 5 4 0 , 1 1 0 5 5 ⟶ 1 0 4 1 5 5 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (0,↑) ↦ 0, (1,↓) ↦ 1, (2,↓) ↦ 2, (3,↓) ↦ 3, (1,↑) ↦ 4, (4,↓) ↦ 5, (5,↓) ↦ 6, (0,↓) ↦ 7 }, it remains to prove termination of the 151-rule system { 0 1 1 ⟶ 0 1 2 3 1 , 0 1 1 ⟶ 4 2 3 1 , 0 1 1 ⟶ 0 5 6 1 5 1 , 0 1 1 ⟶ 4 5 1 , 0 1 1 ⟶ 4 1 7 3 , 0 1 1 ⟶ 4 7 3 , 0 1 1 ⟶ 0 3 , 0 1 1 ⟶ 0 1 5 6 1 5 , 0 1 1 ⟶ 4 5 6 1 5 , 0 1 1 ⟶ 4 5 , 4 7 1 ⟶ 0 3 1 1 , 4 7 1 ⟶ 4 1 , 4 7 1 ⟶ 4 1 7 , 4 7 1 ⟶ 4 7 , 4 7 1 ⟶ 0 , 4 7 1 ⟶ 0 5 6 3 1 1 , 0 7 1 ⟶ 0 5 2 1 7 , 0 7 1 ⟶ 4 7 , 0 7 1 ⟶ 0 , 0 7 1 ⟶ 0 7 5 2 1 6 , 0 7 1 ⟶ 0 5 2 1 6 , 0 7 1 ⟶ 4 6 , 0 1 6 ⟶ 0 5 6 1 , 0 1 6 ⟶ 4 , 0 1 6 ⟶ 0 1 5 6 3 , 0 1 6 ⟶ 4 5 6 3 , 0 1 6 ⟶ 0 3 1 6 , 0 1 6 ⟶ 0 1 , 0 1 6 ⟶ 0 1 2 5 1 6 , 0 1 6 ⟶ 4 2 5 1 6 , 4 7 6 ⟶ 0 3 3 1 6 , 4 7 6 ⟶ 4 6 , 4 7 6 ⟶ 0 5 6 1 6 , 4 7 6 ⟶ 0 3 3 1 6 1 , 4 7 6 ⟶ 4 6 1 , 4 7 6 ⟶ 4 , 4 7 6 ⟶ 4 5 7 , 4 7 6 ⟶ 0 , 4 7 6 ⟶ 4 6 7 , 4 7 6 ⟶ 4 7 5 5 , 4 7 6 ⟶ 0 5 5 , 0 7 6 ⟶ 0 7 , 0 7 6 ⟶ 0 , 0 7 6 ⟶ 0 7 5 6 , 0 7 6 ⟶ 0 5 6 , 0 7 6 ⟶ 0 5 2 6 7 , 0 7 6 ⟶ 0 7 2 3 6 2 , 0 7 6 ⟶ 0 2 3 6 2 , 0 7 6 ⟶ 0 3 7 5 , 0 7 6 ⟶ 0 5 , 4 2 7 1 ⟶ 0 3 1 3 1 , 4 2 7 1 ⟶ 4 3 1 , 4 6 7 1 ⟶ 0 5 6 1 1 , 4 6 7 1 ⟶ 4 1 , 4 6 7 1 ⟶ 4 7 3 5 6 1 , 4 6 7 1 ⟶ 0 3 5 6 1 , 0 1 2 1 ⟶ 4 2 7 5 1 1 , 0 1 2 1 ⟶ 0 5 1 1 , 0 1 2 1 ⟶ 4 1 , 4 7 2 1 ⟶ 0 2 3 1 1 , 4 7 2 1 ⟶ 4 1 , 0 7 2 1 ⟶ 4 7 7 6 , 0 7 2 1 ⟶ 0 7 6 , 0 7 2 1 ⟶ 0 6 , 4 7 1 6 ⟶ 0 5 1 1 6 , 4 7 1 6 ⟶ 4 1 6 , 4 3 7 6 ⟶ 4 5 7 , 4 3 7 6 ⟶ 0 , 4 3 7 6 ⟶ 0 3 3 5 1 6 , 4 3 7 6 ⟶ 4 6 , 4 5 7 6 ⟶ 4 5 6 7 , 4 5 7 6 ⟶ 0 , 4 5 7 6 ⟶ 4 2 7 3 6 5 , 4 5 7 6 ⟶ 0 3 6 5 , 0 6 7 6 ⟶ 0 6 7 5 6 , 0 6 7 6 ⟶ 0 5 6 , 0 1 2 6 ⟶ 0 1 , 0 1 2 6 ⟶ 4 , 4 7 2 6 ⟶ 4 6 2 5 7 , 4 7 2 6 ⟶ 0 , 4 7 2 6 ⟶ 4 6 7 , 4 7 2 6 ⟶ 4 7 2 5 6 , 4 7 2 6 ⟶ 0 2 5 6 , 4 7 2 6 ⟶ 4 2 7 , 0 7 2 6 ⟶ 0 7 , 0 7 2 6 ⟶ 0 , 4 6 3 7 1 ⟶ 4 3 1 5 6 7 , 4 6 3 7 1 ⟶ 4 5 6 7 , 4 6 3 7 1 ⟶ 0 , 4 3 5 7 1 ⟶ 4 2 3 5 1 7 , 4 3 5 7 1 ⟶ 4 7 , 4 3 5 7 1 ⟶ 0 , 4 7 6 5 7 ⟶ 4 7 7 3 5 6 , 4 7 6 5 7 ⟶ 0 7 3 5 6 , 4 7 6 5 7 ⟶ 0 3 5 6 , 4 5 7 1 6 ⟶ 4 1 5 6 5 7 , 4 5 7 1 6 ⟶ 4 5 6 5 7 , 4 5 7 1 6 ⟶ 0 , 4 1 7 6 6 ⟶ 4 7 5 1 6 6 , 4 1 7 6 6 ⟶ 0 5 1 6 6 , 4 1 7 6 6 ⟶ 4 6 6 , 7 1 1 →= 7 1 2 3 1 , 7 1 1 →= 7 5 6 1 5 1 , 7 1 1 →= 5 2 1 1 7 3 , 7 1 1 →= 7 1 5 6 1 5 , 1 7 1 →= 3 7 3 1 1 , 1 7 1 →= 5 6 1 1 7 , 1 7 1 →= 7 5 6 3 1 1 , 7 7 1 →= 7 5 2 1 7 , 7 7 1 →= 7 7 5 2 1 6 , 7 1 6 →= 7 5 6 1 , 7 1 6 →= 7 1 5 6 3 , 7 1 6 →= 3 7 3 1 6 , 7 1 6 →= 5 5 6 5 7 1 , 7 1 6 →= 5 5 7 5 6 1 , 7 1 6 →= 7 1 2 5 1 6 , 1 7 6 →= 7 3 3 1 6 , 1 7 6 →= 7 5 6 1 6 , 1 7 6 →= 7 3 3 1 6 1 , 1 7 6 →= 2 3 6 1 5 7 , 1 7 6 →= 3 5 5 1 6 7 , 1 7 6 →= 6 5 1 7 5 5 , 7 7 6 →= 5 6 7 7 , 7 7 6 →= 7 7 5 6 , 7 7 6 →= 7 5 2 6 7 , 7 7 6 →= 5 5 6 5 7 7 , 7 7 6 →= 7 7 2 3 6 2 , 7 7 6 →= 5 6 7 3 7 5 , 1 2 7 1 →= 2 7 3 1 3 1 , 1 6 7 1 →= 6 7 5 6 1 1 , 1 6 7 1 →= 1 7 3 5 6 1 , 7 1 2 1 →= 1 2 7 5 1 1 , 1 7 2 1 →= 7 2 3 1 1 , 7 7 2 1 →= 5 2 1 7 7 6 , 1 7 1 6 →= 2 7 5 1 1 6 , 1 3 7 6 →= 3 5 6 1 5 7 , 1 3 7 6 →= 7 3 3 5 1 6 , 1 5 7 6 →= 2 3 1 5 6 7 , 1 5 7 6 →= 1 2 7 3 6 5 , 7 6 7 6 →= 7 6 7 5 6 , 7 1 2 6 →= 2 3 6 7 1 , 1 7 2 6 →= 1 6 2 5 7 , 1 7 2 6 →= 2 5 1 6 7 , 1 7 2 6 →= 1 7 2 5 6 , 1 7 2 6 →= 6 2 5 1 2 7 , 7 7 2 6 →= 5 2 2 6 7 7 , 1 6 3 7 1 →= 1 3 1 5 6 7 , 1 3 5 7 1 →= 1 2 3 5 1 7 , 1 7 6 5 7 →= 1 7 7 3 5 6 , 1 5 7 1 6 →= 1 1 5 6 5 7 , 1 1 7 6 6 →= 1 7 5 1 6 6 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 5 ↦ 4, 6 ↦ 5, 4 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 120-rule system { 0 1 1 ⟶ 0 1 2 3 1 , 0 1 1 ⟶ 0 4 5 1 4 1 , 0 1 1 ⟶ 6 1 7 3 , 0 1 1 ⟶ 6 7 3 , 0 1 1 ⟶ 0 3 , 0 1 1 ⟶ 0 1 4 5 1 4 , 6 7 1 ⟶ 0 3 1 1 , 6 7 1 ⟶ 6 1 7 , 6 7 1 ⟶ 6 7 , 6 7 1 ⟶ 0 , 6 7 1 ⟶ 0 4 5 3 1 1 , 0 7 1 ⟶ 0 4 2 1 7 , 0 7 1 ⟶ 0 7 4 2 1 5 , 0 1 5 ⟶ 0 4 5 1 , 0 1 5 ⟶ 0 1 4 5 3 , 0 1 5 ⟶ 0 3 1 5 , 0 1 5 ⟶ 0 1 , 0 1 5 ⟶ 0 1 2 4 1 5 , 6 7 5 ⟶ 0 3 3 1 5 , 6 7 5 ⟶ 0 4 5 1 5 , 6 7 5 ⟶ 0 3 3 1 5 1 , 6 7 5 ⟶ 6 4 7 , 6 7 5 ⟶ 0 , 6 7 5 ⟶ 6 5 7 , 6 7 5 ⟶ 6 7 4 4 , 6 7 5 ⟶ 0 4 4 , 0 7 5 ⟶ 0 7 , 0 7 5 ⟶ 0 7 4 5 , 0 7 5 ⟶ 0 4 2 5 7 , 0 7 5 ⟶ 0 7 2 3 5 2 , 0 7 5 ⟶ 0 3 7 4 , 6 2 7 1 ⟶ 0 3 1 3 1 , 6 5 7 1 ⟶ 0 4 5 1 1 , 6 5 7 1 ⟶ 6 7 3 4 5 1 , 6 5 7 1 ⟶ 0 3 4 5 1 , 0 1 2 1 ⟶ 6 2 7 4 1 1 , 0 1 2 1 ⟶ 0 4 1 1 , 6 7 2 1 ⟶ 0 2 3 1 1 , 0 7 2 1 ⟶ 6 7 7 5 , 0 7 2 1 ⟶ 0 7 5 , 6 7 1 5 ⟶ 0 4 1 1 5 , 6 3 7 5 ⟶ 6 4 7 , 6 3 7 5 ⟶ 0 , 6 3 7 5 ⟶ 0 3 3 4 1 5 , 6 4 7 5 ⟶ 6 4 5 7 , 6 4 7 5 ⟶ 0 , 6 4 7 5 ⟶ 6 2 7 3 5 4 , 6 4 7 5 ⟶ 0 3 5 4 , 0 5 7 5 ⟶ 0 5 7 4 5 , 0 1 2 5 ⟶ 0 1 , 6 7 2 5 ⟶ 6 5 2 4 7 , 6 7 2 5 ⟶ 0 , 6 7 2 5 ⟶ 6 5 7 , 6 7 2 5 ⟶ 6 7 2 4 5 , 6 7 2 5 ⟶ 0 2 4 5 , 6 7 2 5 ⟶ 6 2 7 , 0 7 2 5 ⟶ 0 7 , 6 5 3 7 1 ⟶ 6 3 1 4 5 7 , 6 5 3 7 1 ⟶ 6 4 5 7 , 6 5 3 7 1 ⟶ 0 , 6 3 4 7 1 ⟶ 6 2 3 4 1 7 , 6 3 4 7 1 ⟶ 6 7 , 6 3 4 7 1 ⟶ 0 , 6 7 5 4 7 ⟶ 6 7 7 3 4 5 , 6 7 5 4 7 ⟶ 0 7 3 4 5 , 6 4 7 1 5 ⟶ 6 1 4 5 4 7 , 6 4 7 1 5 ⟶ 6 4 5 4 7 , 6 4 7 1 5 ⟶ 0 , 6 1 7 5 5 ⟶ 6 7 4 1 5 5 , 6 1 7 5 5 ⟶ 0 4 1 5 5 , 7 1 1 →= 7 1 2 3 1 , 7 1 1 →= 7 4 5 1 4 1 , 7 1 1 →= 4 2 1 1 7 3 , 7 1 1 →= 7 1 4 5 1 4 , 1 7 1 →= 3 7 3 1 1 , 1 7 1 →= 4 5 1 1 7 , 1 7 1 →= 7 4 5 3 1 1 , 7 7 1 →= 7 4 2 1 7 , 7 7 1 →= 7 7 4 2 1 5 , 7 1 5 →= 7 4 5 1 , 7 1 5 →= 7 1 4 5 3 , 7 1 5 →= 3 7 3 1 5 , 7 1 5 →= 4 4 5 4 7 1 , 7 1 5 →= 4 4 7 4 5 1 , 7 1 5 →= 7 1 2 4 1 5 , 1 7 5 →= 7 3 3 1 5 , 1 7 5 →= 7 4 5 1 5 , 1 7 5 →= 7 3 3 1 5 1 , 1 7 5 →= 2 3 5 1 4 7 , 1 7 5 →= 3 4 4 1 5 7 , 1 7 5 →= 5 4 1 7 4 4 , 7 7 5 →= 4 5 7 7 , 7 7 5 →= 7 7 4 5 , 7 7 5 →= 7 4 2 5 7 , 7 7 5 →= 4 4 5 4 7 7 , 7 7 5 →= 7 7 2 3 5 2 , 7 7 5 →= 4 5 7 3 7 4 , 1 2 7 1 →= 2 7 3 1 3 1 , 1 5 7 1 →= 5 7 4 5 1 1 , 1 5 7 1 →= 1 7 3 4 5 1 , 7 1 2 1 →= 1 2 7 4 1 1 , 1 7 2 1 →= 7 2 3 1 1 , 7 7 2 1 →= 4 2 1 7 7 5 , 1 7 1 5 →= 2 7 4 1 1 5 , 1 3 7 5 →= 3 4 5 1 4 7 , 1 3 7 5 →= 7 3 3 4 1 5 , 1 4 7 5 →= 2 3 1 4 5 7 , 1 4 7 5 →= 1 2 7 3 5 4 , 7 5 7 5 →= 7 5 7 4 5 , 7 1 2 5 →= 2 3 5 7 1 , 1 7 2 5 →= 1 5 2 4 7 , 1 7 2 5 →= 2 4 1 5 7 , 1 7 2 5 →= 1 7 2 4 5 , 1 7 2 5 →= 5 2 4 1 2 7 , 7 7 2 5 →= 4 2 2 5 7 7 , 1 5 3 7 1 →= 1 3 1 4 5 7 , 1 3 4 7 1 →= 1 2 3 4 1 7 , 1 7 5 4 7 →= 1 7 7 3 4 5 , 1 4 7 1 5 →= 1 1 4 5 4 7 , 1 1 7 5 5 →= 1 7 4 1 5 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 6: 0 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 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 1 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 1 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 ⎟ ⎝ ⎠ 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 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 1 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 117-rule system { 0 1 1 ⟶ 0 1 2 3 1 , 0 1 1 ⟶ 0 4 5 1 4 1 , 0 1 1 ⟶ 6 1 7 3 , 0 1 1 ⟶ 6 7 3 , 0 1 1 ⟶ 0 3 , 0 1 1 ⟶ 0 1 4 5 1 4 , 6 7 1 ⟶ 0 3 1 1 , 6 7 1 ⟶ 6 1 7 , 6 7 1 ⟶ 6 7 , 6 7 1 ⟶ 0 , 6 7 1 ⟶ 0 4 5 3 1 1 , 0 7 1 ⟶ 0 4 2 1 7 , 0 7 1 ⟶ 0 7 4 2 1 5 , 0 1 5 ⟶ 0 4 5 1 , 0 1 5 ⟶ 0 1 4 5 3 , 0 1 5 ⟶ 0 3 1 5 , 0 1 5 ⟶ 0 1 , 0 1 5 ⟶ 0 1 2 4 1 5 , 6 7 5 ⟶ 0 3 3 1 5 , 6 7 5 ⟶ 0 4 5 1 5 , 6 7 5 ⟶ 0 3 3 1 5 1 , 6 7 5 ⟶ 6 4 7 , 6 7 5 ⟶ 0 , 6 7 5 ⟶ 6 5 7 , 6 7 5 ⟶ 6 7 4 4 , 6 7 5 ⟶ 0 4 4 , 0 7 5 ⟶ 0 7 , 0 7 5 ⟶ 0 7 4 5 , 0 7 5 ⟶ 0 4 2 5 7 , 0 7 5 ⟶ 0 7 2 3 5 2 , 0 7 5 ⟶ 0 3 7 4 , 6 2 7 1 ⟶ 0 3 1 3 1 , 6 5 7 1 ⟶ 0 4 5 1 1 , 6 5 7 1 ⟶ 6 7 3 4 5 1 , 6 5 7 1 ⟶ 0 3 4 5 1 , 0 1 2 1 ⟶ 6 2 7 4 1 1 , 0 1 2 1 ⟶ 0 4 1 1 , 6 7 2 1 ⟶ 0 2 3 1 1 , 0 7 2 1 ⟶ 6 7 7 5 , 0 7 2 1 ⟶ 0 7 5 , 6 7 1 5 ⟶ 0 4 1 1 5 , 6 3 7 5 ⟶ 6 4 7 , 6 3 7 5 ⟶ 0 , 6 3 7 5 ⟶ 0 3 3 4 1 5 , 6 4 7 5 ⟶ 6 4 5 7 , 6 4 7 5 ⟶ 0 , 6 4 7 5 ⟶ 6 2 7 3 5 4 , 6 4 7 5 ⟶ 0 3 5 4 , 0 5 7 5 ⟶ 0 5 7 4 5 , 0 1 2 5 ⟶ 0 1 , 6 7 2 5 ⟶ 6 5 2 4 7 , 6 7 2 5 ⟶ 0 , 6 7 2 5 ⟶ 6 5 7 , 6 7 2 5 ⟶ 6 7 2 4 5 , 6 7 2 5 ⟶ 0 2 4 5 , 6 7 2 5 ⟶ 6 2 7 , 0 7 2 5 ⟶ 0 7 , 6 3 4 7 1 ⟶ 6 2 3 4 1 7 , 6 3 4 7 1 ⟶ 6 7 , 6 3 4 7 1 ⟶ 0 , 6 7 5 4 7 ⟶ 6 7 7 3 4 5 , 6 7 5 4 7 ⟶ 0 7 3 4 5 , 6 4 7 1 5 ⟶ 6 1 4 5 4 7 , 6 4 7 1 5 ⟶ 6 4 5 4 7 , 6 4 7 1 5 ⟶ 0 , 6 1 7 5 5 ⟶ 6 7 4 1 5 5 , 6 1 7 5 5 ⟶ 0 4 1 5 5 , 7 1 1 →= 7 1 2 3 1 , 7 1 1 →= 7 4 5 1 4 1 , 7 1 1 →= 4 2 1 1 7 3 , 7 1 1 →= 7 1 4 5 1 4 , 1 7 1 →= 3 7 3 1 1 , 1 7 1 →= 4 5 1 1 7 , 1 7 1 →= 7 4 5 3 1 1 , 7 7 1 →= 7 4 2 1 7 , 7 7 1 →= 7 7 4 2 1 5 , 7 1 5 →= 7 4 5 1 , 7 1 5 →= 7 1 4 5 3 , 7 1 5 →= 3 7 3 1 5 , 7 1 5 →= 4 4 5 4 7 1 , 7 1 5 →= 4 4 7 4 5 1 , 7 1 5 →= 7 1 2 4 1 5 , 1 7 5 →= 7 3 3 1 5 , 1 7 5 →= 7 4 5 1 5 , 1 7 5 →= 7 3 3 1 5 1 , 1 7 5 →= 2 3 5 1 4 7 , 1 7 5 →= 3 4 4 1 5 7 , 1 7 5 →= 5 4 1 7 4 4 , 7 7 5 →= 4 5 7 7 , 7 7 5 →= 7 7 4 5 , 7 7 5 →= 7 4 2 5 7 , 7 7 5 →= 4 4 5 4 7 7 , 7 7 5 →= 7 7 2 3 5 2 , 7 7 5 →= 4 5 7 3 7 4 , 1 2 7 1 →= 2 7 3 1 3 1 , 1 5 7 1 →= 5 7 4 5 1 1 , 1 5 7 1 →= 1 7 3 4 5 1 , 7 1 2 1 →= 1 2 7 4 1 1 , 1 7 2 1 →= 7 2 3 1 1 , 7 7 2 1 →= 4 2 1 7 7 5 , 1 7 1 5 →= 2 7 4 1 1 5 , 1 3 7 5 →= 3 4 5 1 4 7 , 1 3 7 5 →= 7 3 3 4 1 5 , 1 4 7 5 →= 2 3 1 4 5 7 , 1 4 7 5 →= 1 2 7 3 5 4 , 7 5 7 5 →= 7 5 7 4 5 , 7 1 2 5 →= 2 3 5 7 1 , 1 7 2 5 →= 1 5 2 4 7 , 1 7 2 5 →= 2 4 1 5 7 , 1 7 2 5 →= 1 7 2 4 5 , 1 7 2 5 →= 5 2 4 1 2 7 , 7 7 2 5 →= 4 2 2 5 7 7 , 1 5 3 7 1 →= 1 3 1 4 5 7 , 1 3 4 7 1 →= 1 2 3 4 1 7 , 1 7 5 4 7 →= 1 7 7 3 4 5 , 1 4 7 1 5 →= 1 1 4 5 4 7 , 1 1 7 5 5 →= 1 7 4 1 5 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 6: 0 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 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 1 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 1 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 1 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 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 1 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 114-rule system { 0 1 1 ⟶ 0 1 2 3 1 , 0 1 1 ⟶ 0 4 5 1 4 1 , 0 1 1 ⟶ 6 1 7 3 , 0 1 1 ⟶ 6 7 3 , 0 1 1 ⟶ 0 3 , 0 1 1 ⟶ 0 1 4 5 1 4 , 6 7 1 ⟶ 0 3 1 1 , 6 7 1 ⟶ 6 1 7 , 6 7 1 ⟶ 6 7 , 6 7 1 ⟶ 0 , 6 7 1 ⟶ 0 4 5 3 1 1 , 0 7 1 ⟶ 0 4 2 1 7 , 0 7 1 ⟶ 0 7 4 2 1 5 , 0 1 5 ⟶ 0 4 5 1 , 0 1 5 ⟶ 0 1 4 5 3 , 0 1 5 ⟶ 0 3 1 5 , 0 1 5 ⟶ 0 1 , 0 1 5 ⟶ 0 1 2 4 1 5 , 6 7 5 ⟶ 0 3 3 1 5 , 6 7 5 ⟶ 0 4 5 1 5 , 6 7 5 ⟶ 0 3 3 1 5 1 , 6 7 5 ⟶ 6 4 7 , 6 7 5 ⟶ 0 , 6 7 5 ⟶ 6 5 7 , 6 7 5 ⟶ 6 7 4 4 , 6 7 5 ⟶ 0 4 4 , 0 7 5 ⟶ 0 7 , 0 7 5 ⟶ 0 7 4 5 , 0 7 5 ⟶ 0 4 2 5 7 , 0 7 5 ⟶ 0 7 2 3 5 2 , 0 7 5 ⟶ 0 3 7 4 , 6 2 7 1 ⟶ 0 3 1 3 1 , 6 5 7 1 ⟶ 0 4 5 1 1 , 6 5 7 1 ⟶ 6 7 3 4 5 1 , 6 5 7 1 ⟶ 0 3 4 5 1 , 0 1 2 1 ⟶ 6 2 7 4 1 1 , 0 1 2 1 ⟶ 0 4 1 1 , 6 7 2 1 ⟶ 0 2 3 1 1 , 0 7 2 1 ⟶ 6 7 7 5 , 0 7 2 1 ⟶ 0 7 5 , 6 7 1 5 ⟶ 0 4 1 1 5 , 6 3 7 5 ⟶ 6 4 7 , 6 3 7 5 ⟶ 0 , 6 3 7 5 ⟶ 0 3 3 4 1 5 , 6 4 7 5 ⟶ 6 4 5 7 , 6 4 7 5 ⟶ 0 , 6 4 7 5 ⟶ 6 2 7 3 5 4 , 6 4 7 5 ⟶ 0 3 5 4 , 0 5 7 5 ⟶ 0 5 7 4 5 , 0 1 2 5 ⟶ 0 1 , 6 7 2 5 ⟶ 6 5 2 4 7 , 6 7 2 5 ⟶ 0 , 6 7 2 5 ⟶ 6 5 7 , 6 7 2 5 ⟶ 6 7 2 4 5 , 6 7 2 5 ⟶ 0 2 4 5 , 6 7 2 5 ⟶ 6 2 7 , 0 7 2 5 ⟶ 0 7 , 6 7 5 4 7 ⟶ 6 7 7 3 4 5 , 6 7 5 4 7 ⟶ 0 7 3 4 5 , 6 4 7 1 5 ⟶ 6 1 4 5 4 7 , 6 4 7 1 5 ⟶ 6 4 5 4 7 , 6 4 7 1 5 ⟶ 0 , 6 1 7 5 5 ⟶ 6 7 4 1 5 5 , 6 1 7 5 5 ⟶ 0 4 1 5 5 , 7 1 1 →= 7 1 2 3 1 , 7 1 1 →= 7 4 5 1 4 1 , 7 1 1 →= 4 2 1 1 7 3 , 7 1 1 →= 7 1 4 5 1 4 , 1 7 1 →= 3 7 3 1 1 , 1 7 1 →= 4 5 1 1 7 , 1 7 1 →= 7 4 5 3 1 1 , 7 7 1 →= 7 4 2 1 7 , 7 7 1 →= 7 7 4 2 1 5 , 7 1 5 →= 7 4 5 1 , 7 1 5 →= 7 1 4 5 3 , 7 1 5 →= 3 7 3 1 5 , 7 1 5 →= 4 4 5 4 7 1 , 7 1 5 →= 4 4 7 4 5 1 , 7 1 5 →= 7 1 2 4 1 5 , 1 7 5 →= 7 3 3 1 5 , 1 7 5 →= 7 4 5 1 5 , 1 7 5 →= 7 3 3 1 5 1 , 1 7 5 →= 2 3 5 1 4 7 , 1 7 5 →= 3 4 4 1 5 7 , 1 7 5 →= 5 4 1 7 4 4 , 7 7 5 →= 4 5 7 7 , 7 7 5 →= 7 7 4 5 , 7 7 5 →= 7 4 2 5 7 , 7 7 5 →= 4 4 5 4 7 7 , 7 7 5 →= 7 7 2 3 5 2 , 7 7 5 →= 4 5 7 3 7 4 , 1 2 7 1 →= 2 7 3 1 3 1 , 1 5 7 1 →= 5 7 4 5 1 1 , 1 5 7 1 →= 1 7 3 4 5 1 , 7 1 2 1 →= 1 2 7 4 1 1 , 1 7 2 1 →= 7 2 3 1 1 , 7 7 2 1 →= 4 2 1 7 7 5 , 1 7 1 5 →= 2 7 4 1 1 5 , 1 3 7 5 →= 3 4 5 1 4 7 , 1 3 7 5 →= 7 3 3 4 1 5 , 1 4 7 5 →= 2 3 1 4 5 7 , 1 4 7 5 →= 1 2 7 3 5 4 , 7 5 7 5 →= 7 5 7 4 5 , 7 1 2 5 →= 2 3 5 7 1 , 1 7 2 5 →= 1 5 2 4 7 , 1 7 2 5 →= 2 4 1 5 7 , 1 7 2 5 →= 1 7 2 4 5 , 1 7 2 5 →= 5 2 4 1 2 7 , 7 7 2 5 →= 4 2 2 5 7 7 , 1 5 3 7 1 →= 1 3 1 4 5 7 , 1 3 4 7 1 →= 1 2 3 4 1 7 , 1 7 5 4 7 →= 1 7 7 3 4 5 , 1 4 7 1 5 →= 1 1 4 5 4 7 , 1 1 7 5 5 →= 1 7 4 1 5 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 5: 0 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 ⎟ ⎜ 0 1 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 ⎟ ⎝ ⎠ 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 ⎟ ⎝ ⎠ 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 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 113-rule system { 0 1 1 ⟶ 0 1 2 3 1 , 0 1 1 ⟶ 0 4 5 1 4 1 , 0 1 1 ⟶ 6 1 7 3 , 0 1 1 ⟶ 6 7 3 , 0 1 1 ⟶ 0 3 , 0 1 1 ⟶ 0 1 4 5 1 4 , 6 7 1 ⟶ 0 3 1 1 , 6 7 1 ⟶ 6 1 7 , 6 7 1 ⟶ 6 7 , 6 7 1 ⟶ 0 , 6 7 1 ⟶ 0 4 5 3 1 1 , 0 7 1 ⟶ 0 4 2 1 7 , 0 7 1 ⟶ 0 7 4 2 1 5 , 0 1 5 ⟶ 0 4 5 1 , 0 1 5 ⟶ 0 1 4 5 3 , 0 1 5 ⟶ 0 3 1 5 , 0 1 5 ⟶ 0 1 , 0 1 5 ⟶ 0 1 2 4 1 5 , 6 7 5 ⟶ 0 3 3 1 5 , 6 7 5 ⟶ 0 4 5 1 5 , 6 7 5 ⟶ 0 3 3 1 5 1 , 6 7 5 ⟶ 6 4 7 , 6 7 5 ⟶ 0 , 6 7 5 ⟶ 6 5 7 , 6 7 5 ⟶ 6 7 4 4 , 6 7 5 ⟶ 0 4 4 , 0 7 5 ⟶ 0 7 , 0 7 5 ⟶ 0 7 4 5 , 0 7 5 ⟶ 0 4 2 5 7 , 0 7 5 ⟶ 0 7 2 3 5 2 , 0 7 5 ⟶ 0 3 7 4 , 6 2 7 1 ⟶ 0 3 1 3 1 , 6 5 7 1 ⟶ 0 4 5 1 1 , 6 5 7 1 ⟶ 6 7 3 4 5 1 , 6 5 7 1 ⟶ 0 3 4 5 1 , 0 1 2 1 ⟶ 6 2 7 4 1 1 , 0 1 2 1 ⟶ 0 4 1 1 , 0 7 2 1 ⟶ 6 7 7 5 , 0 7 2 1 ⟶ 0 7 5 , 6 7 1 5 ⟶ 0 4 1 1 5 , 6 3 7 5 ⟶ 6 4 7 , 6 3 7 5 ⟶ 0 , 6 3 7 5 ⟶ 0 3 3 4 1 5 , 6 4 7 5 ⟶ 6 4 5 7 , 6 4 7 5 ⟶ 0 , 6 4 7 5 ⟶ 6 2 7 3 5 4 , 6 4 7 5 ⟶ 0 3 5 4 , 0 5 7 5 ⟶ 0 5 7 4 5 , 0 1 2 5 ⟶ 0 1 , 6 7 2 5 ⟶ 6 5 2 4 7 , 6 7 2 5 ⟶ 0 , 6 7 2 5 ⟶ 6 5 7 , 6 7 2 5 ⟶ 6 7 2 4 5 , 6 7 2 5 ⟶ 0 2 4 5 , 6 7 2 5 ⟶ 6 2 7 , 0 7 2 5 ⟶ 0 7 , 6 7 5 4 7 ⟶ 6 7 7 3 4 5 , 6 7 5 4 7 ⟶ 0 7 3 4 5 , 6 4 7 1 5 ⟶ 6 1 4 5 4 7 , 6 4 7 1 5 ⟶ 6 4 5 4 7 , 6 4 7 1 5 ⟶ 0 , 6 1 7 5 5 ⟶ 6 7 4 1 5 5 , 6 1 7 5 5 ⟶ 0 4 1 5 5 , 7 1 1 →= 7 1 2 3 1 , 7 1 1 →= 7 4 5 1 4 1 , 7 1 1 →= 4 2 1 1 7 3 , 7 1 1 →= 7 1 4 5 1 4 , 1 7 1 →= 3 7 3 1 1 , 1 7 1 →= 4 5 1 1 7 , 1 7 1 →= 7 4 5 3 1 1 , 7 7 1 →= 7 4 2 1 7 , 7 7 1 →= 7 7 4 2 1 5 , 7 1 5 →= 7 4 5 1 , 7 1 5 →= 7 1 4 5 3 , 7 1 5 →= 3 7 3 1 5 , 7 1 5 →= 4 4 5 4 7 1 , 7 1 5 →= 4 4 7 4 5 1 , 7 1 5 →= 7 1 2 4 1 5 , 1 7 5 →= 7 3 3 1 5 , 1 7 5 →= 7 4 5 1 5 , 1 7 5 →= 7 3 3 1 5 1 , 1 7 5 →= 2 3 5 1 4 7 , 1 7 5 →= 3 4 4 1 5 7 , 1 7 5 →= 5 4 1 7 4 4 , 7 7 5 →= 4 5 7 7 , 7 7 5 →= 7 7 4 5 , 7 7 5 →= 7 4 2 5 7 , 7 7 5 →= 4 4 5 4 7 7 , 7 7 5 →= 7 7 2 3 5 2 , 7 7 5 →= 4 5 7 3 7 4 , 1 2 7 1 →= 2 7 3 1 3 1 , 1 5 7 1 →= 5 7 4 5 1 1 , 1 5 7 1 →= 1 7 3 4 5 1 , 7 1 2 1 →= 1 2 7 4 1 1 , 1 7 2 1 →= 7 2 3 1 1 , 7 7 2 1 →= 4 2 1 7 7 5 , 1 7 1 5 →= 2 7 4 1 1 5 , 1 3 7 5 →= 3 4 5 1 4 7 , 1 3 7 5 →= 7 3 3 4 1 5 , 1 4 7 5 →= 2 3 1 4 5 7 , 1 4 7 5 →= 1 2 7 3 5 4 , 7 5 7 5 →= 7 5 7 4 5 , 7 1 2 5 →= 2 3 5 7 1 , 1 7 2 5 →= 1 5 2 4 7 , 1 7 2 5 →= 2 4 1 5 7 , 1 7 2 5 →= 1 7 2 4 5 , 1 7 2 5 →= 5 2 4 1 2 7 , 7 7 2 5 →= 4 2 2 5 7 7 , 1 5 3 7 1 →= 1 3 1 4 5 7 , 1 3 4 7 1 →= 1 2 3 4 1 7 , 1 7 5 4 7 →= 1 7 7 3 4 5 , 1 4 7 1 5 →= 1 1 4 5 4 7 , 1 1 7 5 5 →= 1 7 4 1 5 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 4: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 1 0 1 ⎟ ⎝ ⎠ 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 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 1 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 1 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎜ 0 0 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 107-rule system { 0 1 1 ⟶ 0 1 2 3 1 , 0 1 1 ⟶ 0 4 5 1 4 1 , 0 1 1 ⟶ 6 1 7 3 , 0 1 1 ⟶ 6 7 3 , 0 1 1 ⟶ 0 3 , 0 1 1 ⟶ 0 1 4 5 1 4 , 0 7 1 ⟶ 0 4 2 1 7 , 0 7 1 ⟶ 0 7 4 2 1 5 , 0 1 5 ⟶ 0 4 5 1 , 0 1 5 ⟶ 0 1 4 5 3 , 0 1 5 ⟶ 0 3 1 5 , 0 1 5 ⟶ 0 1 , 0 1 5 ⟶ 0 1 2 4 1 5 , 6 7 5 ⟶ 0 3 3 1 5 , 6 7 5 ⟶ 0 4 5 1 5 , 6 7 5 ⟶ 0 3 3 1 5 1 , 6 7 5 ⟶ 6 4 7 , 6 7 5 ⟶ 0 , 6 7 5 ⟶ 6 5 7 , 6 7 5 ⟶ 6 7 4 4 , 6 7 5 ⟶ 0 4 4 , 0 7 5 ⟶ 0 7 , 0 7 5 ⟶ 0 7 4 5 , 0 7 5 ⟶ 0 4 2 5 7 , 0 7 5 ⟶ 0 7 2 3 5 2 , 0 7 5 ⟶ 0 3 7 4 , 6 2 7 1 ⟶ 0 3 1 3 1 , 6 5 7 1 ⟶ 0 4 5 1 1 , 6 5 7 1 ⟶ 6 7 3 4 5 1 , 6 5 7 1 ⟶ 0 3 4 5 1 , 0 1 2 1 ⟶ 6 2 7 4 1 1 , 0 1 2 1 ⟶ 0 4 1 1 , 0 7 2 1 ⟶ 6 7 7 5 , 0 7 2 1 ⟶ 0 7 5 , 6 3 7 5 ⟶ 6 4 7 , 6 3 7 5 ⟶ 0 , 6 3 7 5 ⟶ 0 3 3 4 1 5 , 6 4 7 5 ⟶ 6 4 5 7 , 6 4 7 5 ⟶ 0 , 6 4 7 5 ⟶ 6 2 7 3 5 4 , 6 4 7 5 ⟶ 0 3 5 4 , 0 5 7 5 ⟶ 0 5 7 4 5 , 0 1 2 5 ⟶ 0 1 , 6 7 2 5 ⟶ 6 5 2 4 7 , 6 7 2 5 ⟶ 0 , 6 7 2 5 ⟶ 6 5 7 , 6 7 2 5 ⟶ 6 7 2 4 5 , 6 7 2 5 ⟶ 0 2 4 5 , 6 7 2 5 ⟶ 6 2 7 , 0 7 2 5 ⟶ 0 7 , 6 7 5 4 7 ⟶ 6 7 7 3 4 5 , 6 7 5 4 7 ⟶ 0 7 3 4 5 , 6 4 7 1 5 ⟶ 6 1 4 5 4 7 , 6 4 7 1 5 ⟶ 6 4 5 4 7 , 6 4 7 1 5 ⟶ 0 , 6 1 7 5 5 ⟶ 6 7 4 1 5 5 , 6 1 7 5 5 ⟶ 0 4 1 5 5 , 7 1 1 →= 7 1 2 3 1 , 7 1 1 →= 7 4 5 1 4 1 , 7 1 1 →= 4 2 1 1 7 3 , 7 1 1 →= 7 1 4 5 1 4 , 1 7 1 →= 3 7 3 1 1 , 1 7 1 →= 4 5 1 1 7 , 1 7 1 →= 7 4 5 3 1 1 , 7 7 1 →= 7 4 2 1 7 , 7 7 1 →= 7 7 4 2 1 5 , 7 1 5 →= 7 4 5 1 , 7 1 5 →= 7 1 4 5 3 , 7 1 5 →= 3 7 3 1 5 , 7 1 5 →= 4 4 5 4 7 1 , 7 1 5 →= 4 4 7 4 5 1 , 7 1 5 →= 7 1 2 4 1 5 , 1 7 5 →= 7 3 3 1 5 , 1 7 5 →= 7 4 5 1 5 , 1 7 5 →= 7 3 3 1 5 1 , 1 7 5 →= 2 3 5 1 4 7 , 1 7 5 →= 3 4 4 1 5 7 , 1 7 5 →= 5 4 1 7 4 4 , 7 7 5 →= 4 5 7 7 , 7 7 5 →= 7 7 4 5 , 7 7 5 →= 7 4 2 5 7 , 7 7 5 →= 4 4 5 4 7 7 , 7 7 5 →= 7 7 2 3 5 2 , 7 7 5 →= 4 5 7 3 7 4 , 1 2 7 1 →= 2 7 3 1 3 1 , 1 5 7 1 →= 5 7 4 5 1 1 , 1 5 7 1 →= 1 7 3 4 5 1 , 7 1 2 1 →= 1 2 7 4 1 1 , 1 7 2 1 →= 7 2 3 1 1 , 7 7 2 1 →= 4 2 1 7 7 5 , 1 7 1 5 →= 2 7 4 1 1 5 , 1 3 7 5 →= 3 4 5 1 4 7 , 1 3 7 5 →= 7 3 3 4 1 5 , 1 4 7 5 →= 2 3 1 4 5 7 , 1 4 7 5 →= 1 2 7 3 5 4 , 7 5 7 5 →= 7 5 7 4 5 , 7 1 2 5 →= 2 3 5 7 1 , 1 7 2 5 →= 1 5 2 4 7 , 1 7 2 5 →= 2 4 1 5 7 , 1 7 2 5 →= 1 7 2 4 5 , 1 7 2 5 →= 5 2 4 1 2 7 , 7 7 2 5 →= 4 2 2 5 7 7 , 1 5 3 7 1 →= 1 3 1 4 5 7 , 1 3 4 7 1 →= 1 2 3 4 1 7 , 1 7 5 4 7 →= 1 7 7 3 4 5 , 1 4 7 1 5 →= 1 1 4 5 4 7 , 1 1 7 5 5 →= 1 7 4 1 5 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 8: 0 ↦ ⎛ ⎞ ⎜ 1 0 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 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 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 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 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 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 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 ⎟ ⎜ 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 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 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 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 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 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 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 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 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 0 0 1 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 1 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 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 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 1 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 106-rule system { 0 1 1 ⟶ 0 1 2 3 1 , 0 1 1 ⟶ 0 4 5 1 4 1 , 0 1 1 ⟶ 6 1 7 3 , 0 1 1 ⟶ 6 7 3 , 0 1 1 ⟶ 0 3 , 0 1 1 ⟶ 0 1 4 5 1 4 , 0 7 1 ⟶ 0 4 2 1 7 , 0 7 1 ⟶ 0 7 4 2 1 5 , 0 1 5 ⟶ 0 4 5 1 , 0 1 5 ⟶ 0 1 4 5 3 , 0 1 5 ⟶ 0 3 1 5 , 0 1 5 ⟶ 0 1 , 0 1 5 ⟶ 0 1 2 4 1 5 , 6 7 5 ⟶ 0 3 3 1 5 , 6 7 5 ⟶ 0 4 5 1 5 , 6 7 5 ⟶ 0 3 3 1 5 1 , 6 7 5 ⟶ 6 4 7 , 6 7 5 ⟶ 0 , 6 7 5 ⟶ 6 5 7 , 6 7 5 ⟶ 6 7 4 4 , 6 7 5 ⟶ 0 4 4 , 0 7 5 ⟶ 0 7 , 0 7 5 ⟶ 0 7 4 5 , 0 7 5 ⟶ 0 4 2 5 7 , 0 7 5 ⟶ 0 7 2 3 5 2 , 0 7 5 ⟶ 0 3 7 4 , 6 5 7 1 ⟶ 0 4 5 1 1 , 6 5 7 1 ⟶ 6 7 3 4 5 1 , 6 5 7 1 ⟶ 0 3 4 5 1 , 0 1 2 1 ⟶ 6 2 7 4 1 1 , 0 1 2 1 ⟶ 0 4 1 1 , 0 7 2 1 ⟶ 6 7 7 5 , 0 7 2 1 ⟶ 0 7 5 , 6 3 7 5 ⟶ 6 4 7 , 6 3 7 5 ⟶ 0 , 6 3 7 5 ⟶ 0 3 3 4 1 5 , 6 4 7 5 ⟶ 6 4 5 7 , 6 4 7 5 ⟶ 0 , 6 4 7 5 ⟶ 6 2 7 3 5 4 , 6 4 7 5 ⟶ 0 3 5 4 , 0 5 7 5 ⟶ 0 5 7 4 5 , 0 1 2 5 ⟶ 0 1 , 6 7 2 5 ⟶ 6 5 2 4 7 , 6 7 2 5 ⟶ 0 , 6 7 2 5 ⟶ 6 5 7 , 6 7 2 5 ⟶ 6 7 2 4 5 , 6 7 2 5 ⟶ 0 2 4 5 , 6 7 2 5 ⟶ 6 2 7 , 0 7 2 5 ⟶ 0 7 , 6 7 5 4 7 ⟶ 6 7 7 3 4 5 , 6 7 5 4 7 ⟶ 0 7 3 4 5 , 6 4 7 1 5 ⟶ 6 1 4 5 4 7 , 6 4 7 1 5 ⟶ 6 4 5 4 7 , 6 4 7 1 5 ⟶ 0 , 6 1 7 5 5 ⟶ 6 7 4 1 5 5 , 6 1 7 5 5 ⟶ 0 4 1 5 5 , 7 1 1 →= 7 1 2 3 1 , 7 1 1 →= 7 4 5 1 4 1 , 7 1 1 →= 4 2 1 1 7 3 , 7 1 1 →= 7 1 4 5 1 4 , 1 7 1 →= 3 7 3 1 1 , 1 7 1 →= 4 5 1 1 7 , 1 7 1 →= 7 4 5 3 1 1 , 7 7 1 →= 7 4 2 1 7 , 7 7 1 →= 7 7 4 2 1 5 , 7 1 5 →= 7 4 5 1 , 7 1 5 →= 7 1 4 5 3 , 7 1 5 →= 3 7 3 1 5 , 7 1 5 →= 4 4 5 4 7 1 , 7 1 5 →= 4 4 7 4 5 1 , 7 1 5 →= 7 1 2 4 1 5 , 1 7 5 →= 7 3 3 1 5 , 1 7 5 →= 7 4 5 1 5 , 1 7 5 →= 7 3 3 1 5 1 , 1 7 5 →= 2 3 5 1 4 7 , 1 7 5 →= 3 4 4 1 5 7 , 1 7 5 →= 5 4 1 7 4 4 , 7 7 5 →= 4 5 7 7 , 7 7 5 →= 7 7 4 5 , 7 7 5 →= 7 4 2 5 7 , 7 7 5 →= 4 4 5 4 7 7 , 7 7 5 →= 7 7 2 3 5 2 , 7 7 5 →= 4 5 7 3 7 4 , 1 2 7 1 →= 2 7 3 1 3 1 , 1 5 7 1 →= 5 7 4 5 1 1 , 1 5 7 1 →= 1 7 3 4 5 1 , 7 1 2 1 →= 1 2 7 4 1 1 , 1 7 2 1 →= 7 2 3 1 1 , 7 7 2 1 →= 4 2 1 7 7 5 , 1 7 1 5 →= 2 7 4 1 1 5 , 1 3 7 5 →= 3 4 5 1 4 7 , 1 3 7 5 →= 7 3 3 4 1 5 , 1 4 7 5 →= 2 3 1 4 5 7 , 1 4 7 5 →= 1 2 7 3 5 4 , 7 5 7 5 →= 7 5 7 4 5 , 7 1 2 5 →= 2 3 5 7 1 , 1 7 2 5 →= 1 5 2 4 7 , 1 7 2 5 →= 2 4 1 5 7 , 1 7 2 5 →= 1 7 2 4 5 , 1 7 2 5 →= 5 2 4 1 2 7 , 7 7 2 5 →= 4 2 2 5 7 7 , 1 5 3 7 1 →= 1 3 1 4 5 7 , 1 3 4 7 1 →= 1 2 3 4 1 7 , 1 7 5 4 7 →= 1 7 7 3 4 5 , 1 4 7 1 5 →= 1 1 4 5 4 7 , 1 1 7 5 5 →= 1 7 4 1 5 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 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 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 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 1 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 96-rule system { 0 1 1 ⟶ 0 1 2 3 1 , 0 1 1 ⟶ 0 4 5 1 4 1 , 0 1 1 ⟶ 6 1 7 3 , 0 1 1 ⟶ 6 7 3 , 0 1 1 ⟶ 0 3 , 0 1 1 ⟶ 0 1 4 5 1 4 , 0 7 1 ⟶ 0 4 2 1 7 , 0 7 1 ⟶ 0 7 4 2 1 5 , 0 1 5 ⟶ 0 4 5 1 , 0 1 5 ⟶ 0 1 4 5 3 , 0 1 5 ⟶ 0 3 1 5 , 0 1 5 ⟶ 0 1 , 0 1 5 ⟶ 0 1 2 4 1 5 , 0 7 5 ⟶ 0 7 , 0 7 5 ⟶ 0 7 4 5 , 0 7 5 ⟶ 0 4 2 5 7 , 0 7 5 ⟶ 0 7 2 3 5 2 , 0 7 5 ⟶ 0 3 7 4 , 6 5 7 1 ⟶ 0 4 5 1 1 , 6 5 7 1 ⟶ 6 7 3 4 5 1 , 6 5 7 1 ⟶ 0 3 4 5 1 , 0 1 2 1 ⟶ 6 2 7 4 1 1 , 0 1 2 1 ⟶ 0 4 1 1 , 0 7 2 1 ⟶ 6 7 7 5 , 0 7 2 1 ⟶ 0 7 5 , 6 3 7 5 ⟶ 6 4 7 , 6 3 7 5 ⟶ 0 , 6 3 7 5 ⟶ 0 3 3 4 1 5 , 6 4 7 5 ⟶ 6 4 5 7 , 6 4 7 5 ⟶ 0 , 6 4 7 5 ⟶ 6 2 7 3 5 4 , 6 4 7 5 ⟶ 0 3 5 4 , 0 5 7 5 ⟶ 0 5 7 4 5 , 0 1 2 5 ⟶ 0 1 , 6 7 2 5 ⟶ 6 5 2 4 7 , 6 7 2 5 ⟶ 0 , 6 7 2 5 ⟶ 6 5 7 , 6 7 2 5 ⟶ 6 7 2 4 5 , 6 7 2 5 ⟶ 0 2 4 5 , 6 7 2 5 ⟶ 6 2 7 , 0 7 2 5 ⟶ 0 7 , 6 4 7 1 5 ⟶ 6 1 4 5 4 7 , 6 4 7 1 5 ⟶ 6 4 5 4 7 , 6 4 7 1 5 ⟶ 0 , 6 1 7 5 5 ⟶ 6 7 4 1 5 5 , 6 1 7 5 5 ⟶ 0 4 1 5 5 , 7 1 1 →= 7 1 2 3 1 , 7 1 1 →= 7 4 5 1 4 1 , 7 1 1 →= 4 2 1 1 7 3 , 7 1 1 →= 7 1 4 5 1 4 , 1 7 1 →= 3 7 3 1 1 , 1 7 1 →= 4 5 1 1 7 , 1 7 1 →= 7 4 5 3 1 1 , 7 7 1 →= 7 4 2 1 7 , 7 7 1 →= 7 7 4 2 1 5 , 7 1 5 →= 7 4 5 1 , 7 1 5 →= 7 1 4 5 3 , 7 1 5 →= 3 7 3 1 5 , 7 1 5 →= 4 4 5 4 7 1 , 7 1 5 →= 4 4 7 4 5 1 , 7 1 5 →= 7 1 2 4 1 5 , 1 7 5 →= 7 3 3 1 5 , 1 7 5 →= 7 4 5 1 5 , 1 7 5 →= 7 3 3 1 5 1 , 1 7 5 →= 2 3 5 1 4 7 , 1 7 5 →= 3 4 4 1 5 7 , 1 7 5 →= 5 4 1 7 4 4 , 7 7 5 →= 4 5 7 7 , 7 7 5 →= 7 7 4 5 , 7 7 5 →= 7 4 2 5 7 , 7 7 5 →= 4 4 5 4 7 7 , 7 7 5 →= 7 7 2 3 5 2 , 7 7 5 →= 4 5 7 3 7 4 , 1 2 7 1 →= 2 7 3 1 3 1 , 1 5 7 1 →= 5 7 4 5 1 1 , 1 5 7 1 →= 1 7 3 4 5 1 , 7 1 2 1 →= 1 2 7 4 1 1 , 1 7 2 1 →= 7 2 3 1 1 , 7 7 2 1 →= 4 2 1 7 7 5 , 1 7 1 5 →= 2 7 4 1 1 5 , 1 3 7 5 →= 3 4 5 1 4 7 , 1 3 7 5 →= 7 3 3 4 1 5 , 1 4 7 5 →= 2 3 1 4 5 7 , 1 4 7 5 →= 1 2 7 3 5 4 , 7 5 7 5 →= 7 5 7 4 5 , 7 1 2 5 →= 2 3 5 7 1 , 1 7 2 5 →= 1 5 2 4 7 , 1 7 2 5 →= 2 4 1 5 7 , 1 7 2 5 →= 1 7 2 4 5 , 1 7 2 5 →= 5 2 4 1 2 7 , 7 7 2 5 →= 4 2 2 5 7 7 , 1 5 3 7 1 →= 1 3 1 4 5 7 , 1 3 4 7 1 →= 1 2 3 4 1 7 , 1 7 5 4 7 →= 1 7 7 3 4 5 , 1 4 7 1 5 →= 1 1 4 5 4 7 , 1 1 7 5 5 →= 1 7 4 1 5 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 8: 0 ↦ ⎛ ⎞ ⎜ 1 0 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 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 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 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 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 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 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 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 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 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 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 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 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 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 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 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 ⎟ ⎜ 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 1 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 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 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 1 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 93-rule system { 0 1 1 ⟶ 0 1 2 3 1 , 0 1 1 ⟶ 0 4 5 1 4 1 , 0 1 1 ⟶ 6 1 7 3 , 0 1 1 ⟶ 6 7 3 , 0 1 1 ⟶ 0 3 , 0 1 1 ⟶ 0 1 4 5 1 4 , 0 7 1 ⟶ 0 4 2 1 7 , 0 7 1 ⟶ 0 7 4 2 1 5 , 0 1 5 ⟶ 0 4 5 1 , 0 1 5 ⟶ 0 1 4 5 3 , 0 1 5 ⟶ 0 3 1 5 , 0 1 5 ⟶ 0 1 , 0 1 5 ⟶ 0 1 2 4 1 5 , 0 7 5 ⟶ 0 7 , 0 7 5 ⟶ 0 7 4 5 , 0 7 5 ⟶ 0 4 2 5 7 , 0 7 5 ⟶ 0 7 2 3 5 2 , 0 7 5 ⟶ 0 3 7 4 , 0 1 2 1 ⟶ 6 2 7 4 1 1 , 0 1 2 1 ⟶ 0 4 1 1 , 0 7 2 1 ⟶ 6 7 7 5 , 0 7 2 1 ⟶ 0 7 5 , 6 3 7 5 ⟶ 6 4 7 , 6 3 7 5 ⟶ 0 , 6 3 7 5 ⟶ 0 3 3 4 1 5 , 6 4 7 5 ⟶ 6 4 5 7 , 6 4 7 5 ⟶ 0 , 6 4 7 5 ⟶ 6 2 7 3 5 4 , 6 4 7 5 ⟶ 0 3 5 4 , 0 5 7 5 ⟶ 0 5 7 4 5 , 0 1 2 5 ⟶ 0 1 , 6 7 2 5 ⟶ 6 5 2 4 7 , 6 7 2 5 ⟶ 0 , 6 7 2 5 ⟶ 6 5 7 , 6 7 2 5 ⟶ 6 7 2 4 5 , 6 7 2 5 ⟶ 0 2 4 5 , 6 7 2 5 ⟶ 6 2 7 , 0 7 2 5 ⟶ 0 7 , 6 4 7 1 5 ⟶ 6 1 4 5 4 7 , 6 4 7 1 5 ⟶ 6 4 5 4 7 , 6 4 7 1 5 ⟶ 0 , 6 1 7 5 5 ⟶ 6 7 4 1 5 5 , 6 1 7 5 5 ⟶ 0 4 1 5 5 , 7 1 1 →= 7 1 2 3 1 , 7 1 1 →= 7 4 5 1 4 1 , 7 1 1 →= 4 2 1 1 7 3 , 7 1 1 →= 7 1 4 5 1 4 , 1 7 1 →= 3 7 3 1 1 , 1 7 1 →= 4 5 1 1 7 , 1 7 1 →= 7 4 5 3 1 1 , 7 7 1 →= 7 4 2 1 7 , 7 7 1 →= 7 7 4 2 1 5 , 7 1 5 →= 7 4 5 1 , 7 1 5 →= 7 1 4 5 3 , 7 1 5 →= 3 7 3 1 5 , 7 1 5 →= 4 4 5 4 7 1 , 7 1 5 →= 4 4 7 4 5 1 , 7 1 5 →= 7 1 2 4 1 5 , 1 7 5 →= 7 3 3 1 5 , 1 7 5 →= 7 4 5 1 5 , 1 7 5 →= 7 3 3 1 5 1 , 1 7 5 →= 2 3 5 1 4 7 , 1 7 5 →= 3 4 4 1 5 7 , 1 7 5 →= 5 4 1 7 4 4 , 7 7 5 →= 4 5 7 7 , 7 7 5 →= 7 7 4 5 , 7 7 5 →= 7 4 2 5 7 , 7 7 5 →= 4 4 5 4 7 7 , 7 7 5 →= 7 7 2 3 5 2 , 7 7 5 →= 4 5 7 3 7 4 , 1 2 7 1 →= 2 7 3 1 3 1 , 1 5 7 1 →= 5 7 4 5 1 1 , 1 5 7 1 →= 1 7 3 4 5 1 , 7 1 2 1 →= 1 2 7 4 1 1 , 1 7 2 1 →= 7 2 3 1 1 , 7 7 2 1 →= 4 2 1 7 7 5 , 1 7 1 5 →= 2 7 4 1 1 5 , 1 3 7 5 →= 3 4 5 1 4 7 , 1 3 7 5 →= 7 3 3 4 1 5 , 1 4 7 5 →= 2 3 1 4 5 7 , 1 4 7 5 →= 1 2 7 3 5 4 , 7 5 7 5 →= 7 5 7 4 5 , 7 1 2 5 →= 2 3 5 7 1 , 1 7 2 5 →= 1 5 2 4 7 , 1 7 2 5 →= 2 4 1 5 7 , 1 7 2 5 →= 1 7 2 4 5 , 1 7 2 5 →= 5 2 4 1 2 7 , 7 7 2 5 →= 4 2 2 5 7 7 , 1 5 3 7 1 →= 1 3 1 4 5 7 , 1 3 4 7 1 →= 1 2 3 4 1 7 , 1 7 5 4 7 →= 1 7 7 3 4 5 , 1 4 7 1 5 →= 1 1 4 5 4 7 , 1 1 7 5 5 →= 1 7 4 1 5 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 5: 0 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 1 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 1 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 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 90-rule system { 0 1 1 ⟶ 0 1 2 3 1 , 0 1 1 ⟶ 0 4 5 1 4 1 , 0 1 1 ⟶ 6 1 7 3 , 0 1 1 ⟶ 6 7 3 , 0 1 1 ⟶ 0 3 , 0 1 1 ⟶ 0 1 4 5 1 4 , 0 7 1 ⟶ 0 4 2 1 7 , 0 7 1 ⟶ 0 7 4 2 1 5 , 0 1 5 ⟶ 0 4 5 1 , 0 1 5 ⟶ 0 1 4 5 3 , 0 1 5 ⟶ 0 3 1 5 , 0 1 5 ⟶ 0 1 , 0 1 5 ⟶ 0 1 2 4 1 5 , 0 7 5 ⟶ 0 7 , 0 7 5 ⟶ 0 7 4 5 , 0 7 5 ⟶ 0 4 2 5 7 , 0 7 5 ⟶ 0 7 2 3 5 2 , 0 7 5 ⟶ 0 3 7 4 , 0 1 2 1 ⟶ 6 2 7 4 1 1 , 0 1 2 1 ⟶ 0 4 1 1 , 0 7 2 1 ⟶ 6 7 7 5 , 0 7 2 1 ⟶ 0 7 5 , 6 4 7 5 ⟶ 6 4 5 7 , 6 4 7 5 ⟶ 0 , 6 4 7 5 ⟶ 6 2 7 3 5 4 , 6 4 7 5 ⟶ 0 3 5 4 , 0 5 7 5 ⟶ 0 5 7 4 5 , 0 1 2 5 ⟶ 0 1 , 6 7 2 5 ⟶ 6 5 2 4 7 , 6 7 2 5 ⟶ 0 , 6 7 2 5 ⟶ 6 5 7 , 6 7 2 5 ⟶ 6 7 2 4 5 , 6 7 2 5 ⟶ 0 2 4 5 , 6 7 2 5 ⟶ 6 2 7 , 0 7 2 5 ⟶ 0 7 , 6 4 7 1 5 ⟶ 6 1 4 5 4 7 , 6 4 7 1 5 ⟶ 6 4 5 4 7 , 6 4 7 1 5 ⟶ 0 , 6 1 7 5 5 ⟶ 6 7 4 1 5 5 , 6 1 7 5 5 ⟶ 0 4 1 5 5 , 7 1 1 →= 7 1 2 3 1 , 7 1 1 →= 7 4 5 1 4 1 , 7 1 1 →= 4 2 1 1 7 3 , 7 1 1 →= 7 1 4 5 1 4 , 1 7 1 →= 3 7 3 1 1 , 1 7 1 →= 4 5 1 1 7 , 1 7 1 →= 7 4 5 3 1 1 , 7 7 1 →= 7 4 2 1 7 , 7 7 1 →= 7 7 4 2 1 5 , 7 1 5 →= 7 4 5 1 , 7 1 5 →= 7 1 4 5 3 , 7 1 5 →= 3 7 3 1 5 , 7 1 5 →= 4 4 5 4 7 1 , 7 1 5 →= 4 4 7 4 5 1 , 7 1 5 →= 7 1 2 4 1 5 , 1 7 5 →= 7 3 3 1 5 , 1 7 5 →= 7 4 5 1 5 , 1 7 5 →= 7 3 3 1 5 1 , 1 7 5 →= 2 3 5 1 4 7 , 1 7 5 →= 3 4 4 1 5 7 , 1 7 5 →= 5 4 1 7 4 4 , 7 7 5 →= 4 5 7 7 , 7 7 5 →= 7 7 4 5 , 7 7 5 →= 7 4 2 5 7 , 7 7 5 →= 4 4 5 4 7 7 , 7 7 5 →= 7 7 2 3 5 2 , 7 7 5 →= 4 5 7 3 7 4 , 1 2 7 1 →= 2 7 3 1 3 1 , 1 5 7 1 →= 5 7 4 5 1 1 , 1 5 7 1 →= 1 7 3 4 5 1 , 7 1 2 1 →= 1 2 7 4 1 1 , 1 7 2 1 →= 7 2 3 1 1 , 7 7 2 1 →= 4 2 1 7 7 5 , 1 7 1 5 →= 2 7 4 1 1 5 , 1 3 7 5 →= 3 4 5 1 4 7 , 1 3 7 5 →= 7 3 3 4 1 5 , 1 4 7 5 →= 2 3 1 4 5 7 , 1 4 7 5 →= 1 2 7 3 5 4 , 7 5 7 5 →= 7 5 7 4 5 , 7 1 2 5 →= 2 3 5 7 1 , 1 7 2 5 →= 1 5 2 4 7 , 1 7 2 5 →= 2 4 1 5 7 , 1 7 2 5 →= 1 7 2 4 5 , 1 7 2 5 →= 5 2 4 1 2 7 , 7 7 2 5 →= 4 2 2 5 7 7 , 1 5 3 7 1 →= 1 3 1 4 5 7 , 1 3 4 7 1 →= 1 2 3 4 1 7 , 1 7 5 4 7 →= 1 7 7 3 4 5 , 1 4 7 1 5 →= 1 1 4 5 4 7 , 1 1 7 5 5 →= 1 7 4 1 5 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 5: 0 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 1 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 1 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 1 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 83-rule system { 0 1 1 ⟶ 0 1 2 3 1 , 0 1 1 ⟶ 0 4 5 1 4 1 , 0 1 1 ⟶ 6 1 7 3 , 0 1 1 ⟶ 6 7 3 , 0 1 1 ⟶ 0 3 , 0 1 1 ⟶ 0 1 4 5 1 4 , 0 7 1 ⟶ 0 4 2 1 7 , 0 7 1 ⟶ 0 7 4 2 1 5 , 0 1 5 ⟶ 0 4 5 1 , 0 1 5 ⟶ 0 1 4 5 3 , 0 1 5 ⟶ 0 3 1 5 , 0 1 5 ⟶ 0 1 , 0 1 5 ⟶ 0 1 2 4 1 5 , 0 7 5 ⟶ 0 7 , 0 7 5 ⟶ 0 7 4 5 , 0 7 5 ⟶ 0 4 2 5 7 , 0 7 5 ⟶ 0 7 2 3 5 2 , 0 7 5 ⟶ 0 3 7 4 , 0 1 2 1 ⟶ 6 2 7 4 1 1 , 0 1 2 1 ⟶ 0 4 1 1 , 0 7 2 1 ⟶ 6 7 7 5 , 0 7 2 1 ⟶ 0 7 5 , 0 5 7 5 ⟶ 0 5 7 4 5 , 0 1 2 5 ⟶ 0 1 , 6 7 2 5 ⟶ 6 5 2 4 7 , 6 7 2 5 ⟶ 0 , 6 7 2 5 ⟶ 6 5 7 , 6 7 2 5 ⟶ 6 7 2 4 5 , 6 7 2 5 ⟶ 0 2 4 5 , 6 7 2 5 ⟶ 6 2 7 , 0 7 2 5 ⟶ 0 7 , 6 1 7 5 5 ⟶ 6 7 4 1 5 5 , 6 1 7 5 5 ⟶ 0 4 1 5 5 , 7 1 1 →= 7 1 2 3 1 , 7 1 1 →= 7 4 5 1 4 1 , 7 1 1 →= 4 2 1 1 7 3 , 7 1 1 →= 7 1 4 5 1 4 , 1 7 1 →= 3 7 3 1 1 , 1 7 1 →= 4 5 1 1 7 , 1 7 1 →= 7 4 5 3 1 1 , 7 7 1 →= 7 4 2 1 7 , 7 7 1 →= 7 7 4 2 1 5 , 7 1 5 →= 7 4 5 1 , 7 1 5 →= 7 1 4 5 3 , 7 1 5 →= 3 7 3 1 5 , 7 1 5 →= 4 4 5 4 7 1 , 7 1 5 →= 4 4 7 4 5 1 , 7 1 5 →= 7 1 2 4 1 5 , 1 7 5 →= 7 3 3 1 5 , 1 7 5 →= 7 4 5 1 5 , 1 7 5 →= 7 3 3 1 5 1 , 1 7 5 →= 2 3 5 1 4 7 , 1 7 5 →= 3 4 4 1 5 7 , 1 7 5 →= 5 4 1 7 4 4 , 7 7 5 →= 4 5 7 7 , 7 7 5 →= 7 7 4 5 , 7 7 5 →= 7 4 2 5 7 , 7 7 5 →= 4 4 5 4 7 7 , 7 7 5 →= 7 7 2 3 5 2 , 7 7 5 →= 4 5 7 3 7 4 , 1 2 7 1 →= 2 7 3 1 3 1 , 1 5 7 1 →= 5 7 4 5 1 1 , 1 5 7 1 →= 1 7 3 4 5 1 , 7 1 2 1 →= 1 2 7 4 1 1 , 1 7 2 1 →= 7 2 3 1 1 , 7 7 2 1 →= 4 2 1 7 7 5 , 1 7 1 5 →= 2 7 4 1 1 5 , 1 3 7 5 →= 3 4 5 1 4 7 , 1 3 7 5 →= 7 3 3 4 1 5 , 1 4 7 5 →= 2 3 1 4 5 7 , 1 4 7 5 →= 1 2 7 3 5 4 , 7 5 7 5 →= 7 5 7 4 5 , 7 1 2 5 →= 2 3 5 7 1 , 1 7 2 5 →= 1 5 2 4 7 , 1 7 2 5 →= 2 4 1 5 7 , 1 7 2 5 →= 1 7 2 4 5 , 1 7 2 5 →= 5 2 4 1 2 7 , 7 7 2 5 →= 4 2 2 5 7 7 , 1 5 3 7 1 →= 1 3 1 4 5 7 , 1 3 4 7 1 →= 1 2 3 4 1 7 , 1 7 5 4 7 →= 1 7 7 3 4 5 , 1 4 7 1 5 →= 1 1 4 5 4 7 , 1 1 7 5 5 →= 1 7 4 1 5 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 5: 0 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 1 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 ⎟ ⎝ ⎠ 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 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 77-rule system { 0 1 1 ⟶ 0 1 2 3 1 , 0 1 1 ⟶ 0 4 5 1 4 1 , 0 1 1 ⟶ 6 1 7 3 , 0 1 1 ⟶ 6 7 3 , 0 1 1 ⟶ 0 3 , 0 1 1 ⟶ 0 1 4 5 1 4 , 0 7 1 ⟶ 0 4 2 1 7 , 0 7 1 ⟶ 0 7 4 2 1 5 , 0 1 5 ⟶ 0 4 5 1 , 0 1 5 ⟶ 0 1 4 5 3 , 0 1 5 ⟶ 0 3 1 5 , 0 1 5 ⟶ 0 1 , 0 1 5 ⟶ 0 1 2 4 1 5 , 0 7 5 ⟶ 0 7 , 0 7 5 ⟶ 0 7 4 5 , 0 7 5 ⟶ 0 4 2 5 7 , 0 7 5 ⟶ 0 7 2 3 5 2 , 0 7 5 ⟶ 0 3 7 4 , 0 1 2 1 ⟶ 6 2 7 4 1 1 , 0 1 2 1 ⟶ 0 4 1 1 , 0 7 2 1 ⟶ 6 7 7 5 , 0 7 2 1 ⟶ 0 7 5 , 0 5 7 5 ⟶ 0 5 7 4 5 , 0 1 2 5 ⟶ 0 1 , 0 7 2 5 ⟶ 0 7 , 6 1 7 5 5 ⟶ 6 7 4 1 5 5 , 6 1 7 5 5 ⟶ 0 4 1 5 5 , 7 1 1 →= 7 1 2 3 1 , 7 1 1 →= 7 4 5 1 4 1 , 7 1 1 →= 4 2 1 1 7 3 , 7 1 1 →= 7 1 4 5 1 4 , 1 7 1 →= 3 7 3 1 1 , 1 7 1 →= 4 5 1 1 7 , 1 7 1 →= 7 4 5 3 1 1 , 7 7 1 →= 7 4 2 1 7 , 7 7 1 →= 7 7 4 2 1 5 , 7 1 5 →= 7 4 5 1 , 7 1 5 →= 7 1 4 5 3 , 7 1 5 →= 3 7 3 1 5 , 7 1 5 →= 4 4 5 4 7 1 , 7 1 5 →= 4 4 7 4 5 1 , 7 1 5 →= 7 1 2 4 1 5 , 1 7 5 →= 7 3 3 1 5 , 1 7 5 →= 7 4 5 1 5 , 1 7 5 →= 7 3 3 1 5 1 , 1 7 5 →= 2 3 5 1 4 7 , 1 7 5 →= 3 4 4 1 5 7 , 1 7 5 →= 5 4 1 7 4 4 , 7 7 5 →= 4 5 7 7 , 7 7 5 →= 7 7 4 5 , 7 7 5 →= 7 4 2 5 7 , 7 7 5 →= 4 4 5 4 7 7 , 7 7 5 →= 7 7 2 3 5 2 , 7 7 5 →= 4 5 7 3 7 4 , 1 2 7 1 →= 2 7 3 1 3 1 , 1 5 7 1 →= 5 7 4 5 1 1 , 1 5 7 1 →= 1 7 3 4 5 1 , 7 1 2 1 →= 1 2 7 4 1 1 , 1 7 2 1 →= 7 2 3 1 1 , 7 7 2 1 →= 4 2 1 7 7 5 , 1 7 1 5 →= 2 7 4 1 1 5 , 1 3 7 5 →= 3 4 5 1 4 7 , 1 3 7 5 →= 7 3 3 4 1 5 , 1 4 7 5 →= 2 3 1 4 5 7 , 1 4 7 5 →= 1 2 7 3 5 4 , 7 5 7 5 →= 7 5 7 4 5 , 7 1 2 5 →= 2 3 5 7 1 , 1 7 2 5 →= 1 5 2 4 7 , 1 7 2 5 →= 2 4 1 5 7 , 1 7 2 5 →= 1 7 2 4 5 , 1 7 2 5 →= 5 2 4 1 2 7 , 7 7 2 5 →= 4 2 2 5 7 7 , 1 5 3 7 1 →= 1 3 1 4 5 7 , 1 3 4 7 1 →= 1 2 3 4 1 7 , 1 7 5 4 7 →= 1 7 7 3 4 5 , 1 4 7 1 5 →= 1 1 4 5 4 7 , 1 1 7 5 5 →= 1 7 4 1 5 5 } 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 0 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 1 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 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 76-rule system { 0 1 1 ⟶ 0 1 2 3 1 , 0 1 1 ⟶ 0 4 5 1 4 1 , 0 1 1 ⟶ 6 1 7 3 , 0 1 1 ⟶ 6 7 3 , 0 1 1 ⟶ 0 3 , 0 1 1 ⟶ 0 1 4 5 1 4 , 0 7 1 ⟶ 0 4 2 1 7 , 0 7 1 ⟶ 0 7 4 2 1 5 , 0 1 5 ⟶ 0 4 5 1 , 0 1 5 ⟶ 0 1 4 5 3 , 0 1 5 ⟶ 0 3 1 5 , 0 1 5 ⟶ 0 1 , 0 1 5 ⟶ 0 1 2 4 1 5 , 0 7 5 ⟶ 0 7 , 0 7 5 ⟶ 0 7 4 5 , 0 7 5 ⟶ 0 4 2 5 7 , 0 7 5 ⟶ 0 7 2 3 5 2 , 0 7 5 ⟶ 0 3 7 4 , 0 1 2 1 ⟶ 6 2 7 4 1 1 , 0 1 2 1 ⟶ 0 4 1 1 , 0 7 2 1 ⟶ 6 7 7 5 , 0 7 2 1 ⟶ 0 7 5 , 0 1 2 5 ⟶ 0 1 , 0 7 2 5 ⟶ 0 7 , 6 1 7 5 5 ⟶ 6 7 4 1 5 5 , 6 1 7 5 5 ⟶ 0 4 1 5 5 , 7 1 1 →= 7 1 2 3 1 , 7 1 1 →= 7 4 5 1 4 1 , 7 1 1 →= 4 2 1 1 7 3 , 7 1 1 →= 7 1 4 5 1 4 , 1 7 1 →= 3 7 3 1 1 , 1 7 1 →= 4 5 1 1 7 , 1 7 1 →= 7 4 5 3 1 1 , 7 7 1 →= 7 4 2 1 7 , 7 7 1 →= 7 7 4 2 1 5 , 7 1 5 →= 7 4 5 1 , 7 1 5 →= 7 1 4 5 3 , 7 1 5 →= 3 7 3 1 5 , 7 1 5 →= 4 4 5 4 7 1 , 7 1 5 →= 4 4 7 4 5 1 , 7 1 5 →= 7 1 2 4 1 5 , 1 7 5 →= 7 3 3 1 5 , 1 7 5 →= 7 4 5 1 5 , 1 7 5 →= 7 3 3 1 5 1 , 1 7 5 →= 2 3 5 1 4 7 , 1 7 5 →= 3 4 4 1 5 7 , 1 7 5 →= 5 4 1 7 4 4 , 7 7 5 →= 4 5 7 7 , 7 7 5 →= 7 7 4 5 , 7 7 5 →= 7 4 2 5 7 , 7 7 5 →= 4 4 5 4 7 7 , 7 7 5 →= 7 7 2 3 5 2 , 7 7 5 →= 4 5 7 3 7 4 , 1 2 7 1 →= 2 7 3 1 3 1 , 1 5 7 1 →= 5 7 4 5 1 1 , 1 5 7 1 →= 1 7 3 4 5 1 , 7 1 2 1 →= 1 2 7 4 1 1 , 1 7 2 1 →= 7 2 3 1 1 , 7 7 2 1 →= 4 2 1 7 7 5 , 1 7 1 5 →= 2 7 4 1 1 5 , 1 3 7 5 →= 3 4 5 1 4 7 , 1 3 7 5 →= 7 3 3 4 1 5 , 1 4 7 5 →= 2 3 1 4 5 7 , 1 4 7 5 →= 1 2 7 3 5 4 , 7 5 7 5 →= 7 5 7 4 5 , 7 1 2 5 →= 2 3 5 7 1 , 1 7 2 5 →= 1 5 2 4 7 , 1 7 2 5 →= 2 4 1 5 7 , 1 7 2 5 →= 1 7 2 4 5 , 1 7 2 5 →= 5 2 4 1 2 7 , 7 7 2 5 →= 4 2 2 5 7 7 , 1 5 3 7 1 →= 1 3 1 4 5 7 , 1 3 4 7 1 →= 1 2 3 4 1 7 , 1 7 5 4 7 →= 1 7 7 3 4 5 , 1 4 7 1 5 →= 1 1 4 5 4 7 , 1 1 7 5 5 →= 1 7 4 1 5 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 6: 0 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 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 1 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 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 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 74-rule system { 0 1 1 ⟶ 0 1 2 3 1 , 0 1 1 ⟶ 0 4 5 1 4 1 , 0 1 1 ⟶ 6 1 7 3 , 0 1 1 ⟶ 6 7 3 , 0 1 1 ⟶ 0 3 , 0 1 1 ⟶ 0 1 4 5 1 4 , 0 7 1 ⟶ 0 4 2 1 7 , 0 7 1 ⟶ 0 7 4 2 1 5 , 0 1 5 ⟶ 0 4 5 1 , 0 1 5 ⟶ 0 1 4 5 3 , 0 1 5 ⟶ 0 3 1 5 , 0 1 5 ⟶ 0 1 , 0 1 5 ⟶ 0 1 2 4 1 5 , 0 7 5 ⟶ 0 7 , 0 7 5 ⟶ 0 7 4 5 , 0 7 5 ⟶ 0 4 2 5 7 , 0 7 5 ⟶ 0 7 2 3 5 2 , 0 7 5 ⟶ 0 3 7 4 , 0 1 2 1 ⟶ 6 2 7 4 1 1 , 0 1 2 1 ⟶ 0 4 1 1 , 0 7 2 1 ⟶ 6 7 7 5 , 0 7 2 1 ⟶ 0 7 5 , 0 1 2 5 ⟶ 0 1 , 0 7 2 5 ⟶ 0 7 , 7 1 1 →= 7 1 2 3 1 , 7 1 1 →= 7 4 5 1 4 1 , 7 1 1 →= 4 2 1 1 7 3 , 7 1 1 →= 7 1 4 5 1 4 , 1 7 1 →= 3 7 3 1 1 , 1 7 1 →= 4 5 1 1 7 , 1 7 1 →= 7 4 5 3 1 1 , 7 7 1 →= 7 4 2 1 7 , 7 7 1 →= 7 7 4 2 1 5 , 7 1 5 →= 7 4 5 1 , 7 1 5 →= 7 1 4 5 3 , 7 1 5 →= 3 7 3 1 5 , 7 1 5 →= 4 4 5 4 7 1 , 7 1 5 →= 4 4 7 4 5 1 , 7 1 5 →= 7 1 2 4 1 5 , 1 7 5 →= 7 3 3 1 5 , 1 7 5 →= 7 4 5 1 5 , 1 7 5 →= 7 3 3 1 5 1 , 1 7 5 →= 2 3 5 1 4 7 , 1 7 5 →= 3 4 4 1 5 7 , 1 7 5 →= 5 4 1 7 4 4 , 7 7 5 →= 4 5 7 7 , 7 7 5 →= 7 7 4 5 , 7 7 5 →= 7 4 2 5 7 , 7 7 5 →= 4 4 5 4 7 7 , 7 7 5 →= 7 7 2 3 5 2 , 7 7 5 →= 4 5 7 3 7 4 , 1 2 7 1 →= 2 7 3 1 3 1 , 1 5 7 1 →= 5 7 4 5 1 1 , 1 5 7 1 →= 1 7 3 4 5 1 , 7 1 2 1 →= 1 2 7 4 1 1 , 1 7 2 1 →= 7 2 3 1 1 , 7 7 2 1 →= 4 2 1 7 7 5 , 1 7 1 5 →= 2 7 4 1 1 5 , 1 3 7 5 →= 3 4 5 1 4 7 , 1 3 7 5 →= 7 3 3 4 1 5 , 1 4 7 5 →= 2 3 1 4 5 7 , 1 4 7 5 →= 1 2 7 3 5 4 , 7 5 7 5 →= 7 5 7 4 5 , 7 1 2 5 →= 2 3 5 7 1 , 1 7 2 5 →= 1 5 2 4 7 , 1 7 2 5 →= 2 4 1 5 7 , 1 7 2 5 →= 1 7 2 4 5 , 1 7 2 5 →= 5 2 4 1 2 7 , 7 7 2 5 →= 4 2 2 5 7 7 , 1 5 3 7 1 →= 1 3 1 4 5 7 , 1 3 4 7 1 →= 1 2 3 4 1 7 , 1 7 5 4 7 →= 1 7 7 3 4 5 , 1 4 7 1 5 →= 1 1 4 5 4 7 , 1 1 7 5 5 →= 1 7 4 1 5 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 7 ↦ 6 }, it remains to prove termination of the 70-rule system { 0 1 1 ⟶ 0 1 2 3 1 , 0 1 1 ⟶ 0 4 5 1 4 1 , 0 1 1 ⟶ 0 3 , 0 1 1 ⟶ 0 1 4 5 1 4 , 0 6 1 ⟶ 0 4 2 1 6 , 0 6 1 ⟶ 0 6 4 2 1 5 , 0 1 5 ⟶ 0 4 5 1 , 0 1 5 ⟶ 0 1 4 5 3 , 0 1 5 ⟶ 0 3 1 5 , 0 1 5 ⟶ 0 1 , 0 1 5 ⟶ 0 1 2 4 1 5 , 0 6 5 ⟶ 0 6 , 0 6 5 ⟶ 0 6 4 5 , 0 6 5 ⟶ 0 4 2 5 6 , 0 6 5 ⟶ 0 6 2 3 5 2 , 0 6 5 ⟶ 0 3 6 4 , 0 1 2 1 ⟶ 0 4 1 1 , 0 6 2 1 ⟶ 0 6 5 , 0 1 2 5 ⟶ 0 1 , 0 6 2 5 ⟶ 0 6 , 6 1 1 →= 6 1 2 3 1 , 6 1 1 →= 6 4 5 1 4 1 , 6 1 1 →= 4 2 1 1 6 3 , 6 1 1 →= 6 1 4 5 1 4 , 1 6 1 →= 3 6 3 1 1 , 1 6 1 →= 4 5 1 1 6 , 1 6 1 →= 6 4 5 3 1 1 , 6 6 1 →= 6 4 2 1 6 , 6 6 1 →= 6 6 4 2 1 5 , 6 1 5 →= 6 4 5 1 , 6 1 5 →= 6 1 4 5 3 , 6 1 5 →= 3 6 3 1 5 , 6 1 5 →= 4 4 5 4 6 1 , 6 1 5 →= 4 4 6 4 5 1 , 6 1 5 →= 6 1 2 4 1 5 , 1 6 5 →= 6 3 3 1 5 , 1 6 5 →= 6 4 5 1 5 , 1 6 5 →= 6 3 3 1 5 1 , 1 6 5 →= 2 3 5 1 4 6 , 1 6 5 →= 3 4 4 1 5 6 , 1 6 5 →= 5 4 1 6 4 4 , 6 6 5 →= 4 5 6 6 , 6 6 5 →= 6 6 4 5 , 6 6 5 →= 6 4 2 5 6 , 6 6 5 →= 4 4 5 4 6 6 , 6 6 5 →= 6 6 2 3 5 2 , 6 6 5 →= 4 5 6 3 6 4 , 1 2 6 1 →= 2 6 3 1 3 1 , 1 5 6 1 →= 5 6 4 5 1 1 , 1 5 6 1 →= 1 6 3 4 5 1 , 6 1 2 1 →= 1 2 6 4 1 1 , 1 6 2 1 →= 6 2 3 1 1 , 6 6 2 1 →= 4 2 1 6 6 5 , 1 6 1 5 →= 2 6 4 1 1 5 , 1 3 6 5 →= 3 4 5 1 4 6 , 1 3 6 5 →= 6 3 3 4 1 5 , 1 4 6 5 →= 2 3 1 4 5 6 , 1 4 6 5 →= 1 2 6 3 5 4 , 6 5 6 5 →= 6 5 6 4 5 , 6 1 2 5 →= 2 3 5 6 1 , 1 6 2 5 →= 1 5 2 4 6 , 1 6 2 5 →= 2 4 1 5 6 , 1 6 2 5 →= 1 6 2 4 5 , 1 6 2 5 →= 5 2 4 1 2 6 , 6 6 2 5 →= 4 2 2 5 6 6 , 1 5 3 6 1 →= 1 3 1 4 5 6 , 1 3 4 6 1 →= 1 2 3 4 1 6 , 1 6 5 4 6 →= 1 6 6 3 4 5 , 1 4 6 1 5 →= 1 1 4 5 4 6 , 1 1 6 5 5 →= 1 6 4 1 5 5 } 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 0 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 1 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 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 ⎟ ⎝ ⎠ 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 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 1 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6 }, it remains to prove termination of the 69-rule system { 0 1 1 ⟶ 0 1 2 3 1 , 0 1 1 ⟶ 0 4 5 1 4 1 , 0 1 1 ⟶ 0 3 , 0 1 1 ⟶ 0 1 4 5 1 4 , 0 6 1 ⟶ 0 4 2 1 6 , 0 6 1 ⟶ 0 6 4 2 1 5 , 0 1 5 ⟶ 0 4 5 1 , 0 1 5 ⟶ 0 1 4 5 3 , 0 1 5 ⟶ 0 3 1 5 , 0 1 5 ⟶ 0 1 , 0 1 5 ⟶ 0 1 2 4 1 5 , 0 6 5 ⟶ 0 6 , 0 6 5 ⟶ 0 6 4 5 , 0 6 5 ⟶ 0 4 2 5 6 , 0 6 5 ⟶ 0 6 2 3 5 2 , 0 6 5 ⟶ 0 3 6 4 , 0 6 2 1 ⟶ 0 6 5 , 0 1 2 5 ⟶ 0 1 , 0 6 2 5 ⟶ 0 6 , 6 1 1 →= 6 1 2 3 1 , 6 1 1 →= 6 4 5 1 4 1 , 6 1 1 →= 4 2 1 1 6 3 , 6 1 1 →= 6 1 4 5 1 4 , 1 6 1 →= 3 6 3 1 1 , 1 6 1 →= 4 5 1 1 6 , 1 6 1 →= 6 4 5 3 1 1 , 6 6 1 →= 6 4 2 1 6 , 6 6 1 →= 6 6 4 2 1 5 , 6 1 5 →= 6 4 5 1 , 6 1 5 →= 6 1 4 5 3 , 6 1 5 →= 3 6 3 1 5 , 6 1 5 →= 4 4 5 4 6 1 , 6 1 5 →= 4 4 6 4 5 1 , 6 1 5 →= 6 1 2 4 1 5 , 1 6 5 →= 6 3 3 1 5 , 1 6 5 →= 6 4 5 1 5 , 1 6 5 →= 6 3 3 1 5 1 , 1 6 5 →= 2 3 5 1 4 6 , 1 6 5 →= 3 4 4 1 5 6 , 1 6 5 →= 5 4 1 6 4 4 , 6 6 5 →= 4 5 6 6 , 6 6 5 →= 6 6 4 5 , 6 6 5 →= 6 4 2 5 6 , 6 6 5 →= 4 4 5 4 6 6 , 6 6 5 →= 6 6 2 3 5 2 , 6 6 5 →= 4 5 6 3 6 4 , 1 2 6 1 →= 2 6 3 1 3 1 , 1 5 6 1 →= 5 6 4 5 1 1 , 1 5 6 1 →= 1 6 3 4 5 1 , 6 1 2 1 →= 1 2 6 4 1 1 , 1 6 2 1 →= 6 2 3 1 1 , 6 6 2 1 →= 4 2 1 6 6 5 , 1 6 1 5 →= 2 6 4 1 1 5 , 1 3 6 5 →= 3 4 5 1 4 6 , 1 3 6 5 →= 6 3 3 4 1 5 , 1 4 6 5 →= 2 3 1 4 5 6 , 1 4 6 5 →= 1 2 6 3 5 4 , 6 5 6 5 →= 6 5 6 4 5 , 6 1 2 5 →= 2 3 5 6 1 , 1 6 2 5 →= 1 5 2 4 6 , 1 6 2 5 →= 2 4 1 5 6 , 1 6 2 5 →= 1 6 2 4 5 , 1 6 2 5 →= 5 2 4 1 2 6 , 6 6 2 5 →= 4 2 2 5 6 6 , 1 5 3 6 1 →= 1 3 1 4 5 6 , 1 3 4 6 1 →= 1 2 3 4 1 6 , 1 6 5 4 6 →= 1 6 6 3 4 5 , 1 4 6 1 5 →= 1 1 4 5 4 6 , 1 1 6 5 5 →= 1 6 4 1 5 5 } 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 0 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 1 0 1 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 ⎟ ⎝ ⎠ 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 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 ⎟ ⎜ 0 0 0 1 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6 }, it remains to prove termination of the 68-rule system { 0 1 1 ⟶ 0 1 2 3 1 , 0 1 1 ⟶ 0 4 5 1 4 1 , 0 1 1 ⟶ 0 3 , 0 1 1 ⟶ 0 1 4 5 1 4 , 0 6 1 ⟶ 0 4 2 1 6 , 0 6 1 ⟶ 0 6 4 2 1 5 , 0 1 5 ⟶ 0 4 5 1 , 0 1 5 ⟶ 0 1 4 5 3 , 0 1 5 ⟶ 0 3 1 5 , 0 1 5 ⟶ 0 1 , 0 1 5 ⟶ 0 1 2 4 1 5 , 0 6 5 ⟶ 0 6 , 0 6 5 ⟶ 0 6 4 5 , 0 6 5 ⟶ 0 4 2 5 6 , 0 6 5 ⟶ 0 6 2 3 5 2 , 0 6 5 ⟶ 0 3 6 4 , 0 1 2 5 ⟶ 0 1 , 0 6 2 5 ⟶ 0 6 , 6 1 1 →= 6 1 2 3 1 , 6 1 1 →= 6 4 5 1 4 1 , 6 1 1 →= 4 2 1 1 6 3 , 6 1 1 →= 6 1 4 5 1 4 , 1 6 1 →= 3 6 3 1 1 , 1 6 1 →= 4 5 1 1 6 , 1 6 1 →= 6 4 5 3 1 1 , 6 6 1 →= 6 4 2 1 6 , 6 6 1 →= 6 6 4 2 1 5 , 6 1 5 →= 6 4 5 1 , 6 1 5 →= 6 1 4 5 3 , 6 1 5 →= 3 6 3 1 5 , 6 1 5 →= 4 4 5 4 6 1 , 6 1 5 →= 4 4 6 4 5 1 , 6 1 5 →= 6 1 2 4 1 5 , 1 6 5 →= 6 3 3 1 5 , 1 6 5 →= 6 4 5 1 5 , 1 6 5 →= 6 3 3 1 5 1 , 1 6 5 →= 2 3 5 1 4 6 , 1 6 5 →= 3 4 4 1 5 6 , 1 6 5 →= 5 4 1 6 4 4 , 6 6 5 →= 4 5 6 6 , 6 6 5 →= 6 6 4 5 , 6 6 5 →= 6 4 2 5 6 , 6 6 5 →= 4 4 5 4 6 6 , 6 6 5 →= 6 6 2 3 5 2 , 6 6 5 →= 4 5 6 3 6 4 , 1 2 6 1 →= 2 6 3 1 3 1 , 1 5 6 1 →= 5 6 4 5 1 1 , 1 5 6 1 →= 1 6 3 4 5 1 , 6 1 2 1 →= 1 2 6 4 1 1 , 1 6 2 1 →= 6 2 3 1 1 , 6 6 2 1 →= 4 2 1 6 6 5 , 1 6 1 5 →= 2 6 4 1 1 5 , 1 3 6 5 →= 3 4 5 1 4 6 , 1 3 6 5 →= 6 3 3 4 1 5 , 1 4 6 5 →= 2 3 1 4 5 6 , 1 4 6 5 →= 1 2 6 3 5 4 , 6 5 6 5 →= 6 5 6 4 5 , 6 1 2 5 →= 2 3 5 6 1 , 1 6 2 5 →= 1 5 2 4 6 , 1 6 2 5 →= 2 4 1 5 6 , 1 6 2 5 →= 1 6 2 4 5 , 1 6 2 5 →= 5 2 4 1 2 6 , 6 6 2 5 →= 4 2 2 5 6 6 , 1 5 3 6 1 →= 1 3 1 4 5 6 , 1 3 4 6 1 →= 1 2 3 4 1 6 , 1 6 5 4 6 →= 1 6 6 3 4 5 , 1 4 6 1 5 →= 1 1 4 5 4 6 , 1 1 6 5 5 →= 1 6 4 1 5 5 } 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 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 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 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎜ 0 1 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6 }, it remains to prove termination of the 66-rule system { 0 1 1 ⟶ 0 1 2 3 1 , 0 1 1 ⟶ 0 4 5 1 4 1 , 0 1 1 ⟶ 0 3 , 0 1 1 ⟶ 0 1 4 5 1 4 , 0 1 5 ⟶ 0 4 5 1 , 0 1 5 ⟶ 0 1 4 5 3 , 0 1 5 ⟶ 0 3 1 5 , 0 1 5 ⟶ 0 1 , 0 1 5 ⟶ 0 1 2 4 1 5 , 0 6 5 ⟶ 0 6 , 0 6 5 ⟶ 0 6 4 5 , 0 6 5 ⟶ 0 4 2 5 6 , 0 6 5 ⟶ 0 6 2 3 5 2 , 0 6 5 ⟶ 0 3 6 4 , 0 1 2 5 ⟶ 0 1 , 0 6 2 5 ⟶ 0 6 , 6 1 1 →= 6 1 2 3 1 , 6 1 1 →= 6 4 5 1 4 1 , 6 1 1 →= 4 2 1 1 6 3 , 6 1 1 →= 6 1 4 5 1 4 , 1 6 1 →= 3 6 3 1 1 , 1 6 1 →= 4 5 1 1 6 , 1 6 1 →= 6 4 5 3 1 1 , 6 6 1 →= 6 4 2 1 6 , 6 6 1 →= 6 6 4 2 1 5 , 6 1 5 →= 6 4 5 1 , 6 1 5 →= 6 1 4 5 3 , 6 1 5 →= 3 6 3 1 5 , 6 1 5 →= 4 4 5 4 6 1 , 6 1 5 →= 4 4 6 4 5 1 , 6 1 5 →= 6 1 2 4 1 5 , 1 6 5 →= 6 3 3 1 5 , 1 6 5 →= 6 4 5 1 5 , 1 6 5 →= 6 3 3 1 5 1 , 1 6 5 →= 2 3 5 1 4 6 , 1 6 5 →= 3 4 4 1 5 6 , 1 6 5 →= 5 4 1 6 4 4 , 6 6 5 →= 4 5 6 6 , 6 6 5 →= 6 6 4 5 , 6 6 5 →= 6 4 2 5 6 , 6 6 5 →= 4 4 5 4 6 6 , 6 6 5 →= 6 6 2 3 5 2 , 6 6 5 →= 4 5 6 3 6 4 , 1 2 6 1 →= 2 6 3 1 3 1 , 1 5 6 1 →= 5 6 4 5 1 1 , 1 5 6 1 →= 1 6 3 4 5 1 , 6 1 2 1 →= 1 2 6 4 1 1 , 1 6 2 1 →= 6 2 3 1 1 , 6 6 2 1 →= 4 2 1 6 6 5 , 1 6 1 5 →= 2 6 4 1 1 5 , 1 3 6 5 →= 3 4 5 1 4 6 , 1 3 6 5 →= 6 3 3 4 1 5 , 1 4 6 5 →= 2 3 1 4 5 6 , 1 4 6 5 →= 1 2 6 3 5 4 , 6 5 6 5 →= 6 5 6 4 5 , 6 1 2 5 →= 2 3 5 6 1 , 1 6 2 5 →= 1 5 2 4 6 , 1 6 2 5 →= 2 4 1 5 6 , 1 6 2 5 →= 1 6 2 4 5 , 1 6 2 5 →= 5 2 4 1 2 6 , 6 6 2 5 →= 4 2 2 5 6 6 , 1 5 3 6 1 →= 1 3 1 4 5 6 , 1 3 4 6 1 →= 1 2 3 4 1 6 , 1 6 5 4 6 →= 1 6 6 3 4 5 , 1 4 6 1 5 →= 1 1 4 5 4 6 , 1 1 6 5 5 →= 1 6 4 1 5 5 } 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 0 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 1 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 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 ⎟ ⎝ ⎠ 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 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 1 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6 }, it remains to prove termination of the 65-rule system { 0 1 1 ⟶ 0 1 2 3 1 , 0 1 1 ⟶ 0 4 5 1 4 1 , 0 1 1 ⟶ 0 3 , 0 1 1 ⟶ 0 1 4 5 1 4 , 0 1 5 ⟶ 0 4 5 1 , 0 1 5 ⟶ 0 1 4 5 3 , 0 1 5 ⟶ 0 3 1 5 , 0 1 5 ⟶ 0 1 , 0 1 5 ⟶ 0 1 2 4 1 5 , 0 6 5 ⟶ 0 6 , 0 6 5 ⟶ 0 6 4 5 , 0 6 5 ⟶ 0 4 2 5 6 , 0 6 5 ⟶ 0 6 2 3 5 2 , 0 6 5 ⟶ 0 3 6 4 , 0 6 2 5 ⟶ 0 6 , 6 1 1 →= 6 1 2 3 1 , 6 1 1 →= 6 4 5 1 4 1 , 6 1 1 →= 4 2 1 1 6 3 , 6 1 1 →= 6 1 4 5 1 4 , 1 6 1 →= 3 6 3 1 1 , 1 6 1 →= 4 5 1 1 6 , 1 6 1 →= 6 4 5 3 1 1 , 6 6 1 →= 6 4 2 1 6 , 6 6 1 →= 6 6 4 2 1 5 , 6 1 5 →= 6 4 5 1 , 6 1 5 →= 6 1 4 5 3 , 6 1 5 →= 3 6 3 1 5 , 6 1 5 →= 4 4 5 4 6 1 , 6 1 5 →= 4 4 6 4 5 1 , 6 1 5 →= 6 1 2 4 1 5 , 1 6 5 →= 6 3 3 1 5 , 1 6 5 →= 6 4 5 1 5 , 1 6 5 →= 6 3 3 1 5 1 , 1 6 5 →= 2 3 5 1 4 6 , 1 6 5 →= 3 4 4 1 5 6 , 1 6 5 →= 5 4 1 6 4 4 , 6 6 5 →= 4 5 6 6 , 6 6 5 →= 6 6 4 5 , 6 6 5 →= 6 4 2 5 6 , 6 6 5 →= 4 4 5 4 6 6 , 6 6 5 →= 6 6 2 3 5 2 , 6 6 5 →= 4 5 6 3 6 4 , 1 2 6 1 →= 2 6 3 1 3 1 , 1 5 6 1 →= 5 6 4 5 1 1 , 1 5 6 1 →= 1 6 3 4 5 1 , 6 1 2 1 →= 1 2 6 4 1 1 , 1 6 2 1 →= 6 2 3 1 1 , 6 6 2 1 →= 4 2 1 6 6 5 , 1 6 1 5 →= 2 6 4 1 1 5 , 1 3 6 5 →= 3 4 5 1 4 6 , 1 3 6 5 →= 6 3 3 4 1 5 , 1 4 6 5 →= 2 3 1 4 5 6 , 1 4 6 5 →= 1 2 6 3 5 4 , 6 5 6 5 →= 6 5 6 4 5 , 6 1 2 5 →= 2 3 5 6 1 , 1 6 2 5 →= 1 5 2 4 6 , 1 6 2 5 →= 2 4 1 5 6 , 1 6 2 5 →= 1 6 2 4 5 , 1 6 2 5 →= 5 2 4 1 2 6 , 6 6 2 5 →= 4 2 2 5 6 6 , 1 5 3 6 1 →= 1 3 1 4 5 6 , 1 3 4 6 1 →= 1 2 3 4 1 6 , 1 6 5 4 6 →= 1 6 6 3 4 5 , 1 4 6 1 5 →= 1 1 4 5 4 6 , 1 1 6 5 5 →= 1 6 4 1 5 5 } 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 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎜ 0 1 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 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 4 ↦ 2, 5 ↦ 3, 3 ↦ 4, 2 ↦ 5, 6 ↦ 6 }, it remains to prove termination of the 62-rule system { 0 1 1 ⟶ 0 1 2 3 1 2 , 0 1 3 ⟶ 0 2 3 1 , 0 1 3 ⟶ 0 1 2 3 4 , 0 1 3 ⟶ 0 4 1 3 , 0 1 3 ⟶ 0 1 , 0 1 3 ⟶ 0 1 5 2 1 3 , 0 6 3 ⟶ 0 6 , 0 6 3 ⟶ 0 6 2 3 , 0 6 3 ⟶ 0 2 5 3 6 , 0 6 3 ⟶ 0 6 5 4 3 5 , 0 6 3 ⟶ 0 4 6 2 , 0 6 5 3 ⟶ 0 6 , 6 1 1 →= 6 1 5 4 1 , 6 1 1 →= 6 2 3 1 2 1 , 6 1 1 →= 2 5 1 1 6 4 , 6 1 1 →= 6 1 2 3 1 2 , 1 6 1 →= 4 6 4 1 1 , 1 6 1 →= 2 3 1 1 6 , 1 6 1 →= 6 2 3 4 1 1 , 6 6 1 →= 6 2 5 1 6 , 6 6 1 →= 6 6 2 5 1 3 , 6 1 3 →= 6 2 3 1 , 6 1 3 →= 6 1 2 3 4 , 6 1 3 →= 4 6 4 1 3 , 6 1 3 →= 2 2 3 2 6 1 , 6 1 3 →= 2 2 6 2 3 1 , 6 1 3 →= 6 1 5 2 1 3 , 1 6 3 →= 6 4 4 1 3 , 1 6 3 →= 6 2 3 1 3 , 1 6 3 →= 6 4 4 1 3 1 , 1 6 3 →= 5 4 3 1 2 6 , 1 6 3 →= 4 2 2 1 3 6 , 1 6 3 →= 3 2 1 6 2 2 , 6 6 3 →= 2 3 6 6 , 6 6 3 →= 6 6 2 3 , 6 6 3 →= 6 2 5 3 6 , 6 6 3 →= 2 2 3 2 6 6 , 6 6 3 →= 6 6 5 4 3 5 , 6 6 3 →= 2 3 6 4 6 2 , 1 5 6 1 →= 5 6 4 1 4 1 , 1 3 6 1 →= 3 6 2 3 1 1 , 1 3 6 1 →= 1 6 4 2 3 1 , 6 1 5 1 →= 1 5 6 2 1 1 , 1 6 5 1 →= 6 5 4 1 1 , 6 6 5 1 →= 2 5 1 6 6 3 , 1 6 1 3 →= 5 6 2 1 1 3 , 1 4 6 3 →= 4 2 3 1 2 6 , 1 4 6 3 →= 6 4 4 2 1 3 , 1 2 6 3 →= 5 4 1 2 3 6 , 1 2 6 3 →= 1 5 6 4 3 2 , 6 3 6 3 →= 6 3 6 2 3 , 6 1 5 3 →= 5 4 3 6 1 , 1 6 5 3 →= 1 3 5 2 6 , 1 6 5 3 →= 5 2 1 3 6 , 1 6 5 3 →= 1 6 5 2 3 , 1 6 5 3 →= 3 5 2 1 5 6 , 6 6 5 3 →= 2 5 5 3 6 6 , 1 3 4 6 1 →= 1 4 1 2 3 6 , 1 4 2 6 1 →= 1 5 4 2 1 6 , 1 6 3 2 6 →= 1 6 6 4 2 3 , 1 2 6 1 3 →= 1 1 2 3 2 6 , 1 1 6 3 3 →= 1 6 2 1 3 3 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 0 ↦ ⎛ ⎞ ⎜ 1 0 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 1 ⎟ ⎜ 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 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 3 ↦ 2, 2 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6 }, it remains to prove termination of the 61-rule system { 0 1 2 ⟶ 0 3 2 1 , 0 1 2 ⟶ 0 1 3 2 4 , 0 1 2 ⟶ 0 4 1 2 , 0 1 2 ⟶ 0 1 , 0 1 2 ⟶ 0 1 5 3 1 2 , 0 6 2 ⟶ 0 6 , 0 6 2 ⟶ 0 6 3 2 , 0 6 2 ⟶ 0 3 5 2 6 , 0 6 2 ⟶ 0 6 5 4 2 5 , 0 6 2 ⟶ 0 4 6 3 , 0 6 5 2 ⟶ 0 6 , 6 1 1 →= 6 1 5 4 1 , 6 1 1 →= 6 3 2 1 3 1 , 6 1 1 →= 3 5 1 1 6 4 , 6 1 1 →= 6 1 3 2 1 3 , 1 6 1 →= 4 6 4 1 1 , 1 6 1 →= 3 2 1 1 6 , 1 6 1 →= 6 3 2 4 1 1 , 6 6 1 →= 6 3 5 1 6 , 6 6 1 →= 6 6 3 5 1 2 , 6 1 2 →= 6 3 2 1 , 6 1 2 →= 6 1 3 2 4 , 6 1 2 →= 4 6 4 1 2 , 6 1 2 →= 3 3 2 3 6 1 , 6 1 2 →= 3 3 6 3 2 1 , 6 1 2 →= 6 1 5 3 1 2 , 1 6 2 →= 6 4 4 1 2 , 1 6 2 →= 6 3 2 1 2 , 1 6 2 →= 6 4 4 1 2 1 , 1 6 2 →= 5 4 2 1 3 6 , 1 6 2 →= 4 3 3 1 2 6 , 1 6 2 →= 2 3 1 6 3 3 , 6 6 2 →= 3 2 6 6 , 6 6 2 →= 6 6 3 2 , 6 6 2 →= 6 3 5 2 6 , 6 6 2 →= 3 3 2 3 6 6 , 6 6 2 →= 6 6 5 4 2 5 , 6 6 2 →= 3 2 6 4 6 3 , 1 5 6 1 →= 5 6 4 1 4 1 , 1 2 6 1 →= 2 6 3 2 1 1 , 1 2 6 1 →= 1 6 4 3 2 1 , 6 1 5 1 →= 1 5 6 3 1 1 , 1 6 5 1 →= 6 5 4 1 1 , 6 6 5 1 →= 3 5 1 6 6 2 , 1 6 1 2 →= 5 6 3 1 1 2 , 1 4 6 2 →= 4 3 2 1 3 6 , 1 4 6 2 →= 6 4 4 3 1 2 , 1 3 6 2 →= 5 4 1 3 2 6 , 1 3 6 2 →= 1 5 6 4 2 3 , 6 2 6 2 →= 6 2 6 3 2 , 6 1 5 2 →= 5 4 2 6 1 , 1 6 5 2 →= 1 2 5 3 6 , 1 6 5 2 →= 5 3 1 2 6 , 1 6 5 2 →= 1 6 5 3 2 , 1 6 5 2 →= 2 5 3 1 5 6 , 6 6 5 2 →= 3 5 5 2 6 6 , 1 2 4 6 1 →= 1 4 1 3 2 6 , 1 4 3 6 1 →= 1 5 4 3 1 6 , 1 6 2 3 6 →= 1 6 6 4 3 2 , 1 3 6 1 2 →= 1 1 3 2 3 6 , 1 1 6 2 2 →= 1 6 3 1 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 0 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 1 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 ⎟ ⎜ 0 1 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 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 1 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6 }, it remains to prove termination of the 60-rule system { 0 1 2 ⟶ 0 3 2 1 , 0 1 2 ⟶ 0 1 3 2 4 , 0 1 2 ⟶ 0 4 1 2 , 0 1 2 ⟶ 0 1 , 0 1 2 ⟶ 0 1 5 3 1 2 , 0 6 2 ⟶ 0 6 , 0 6 2 ⟶ 0 6 3 2 , 0 6 2 ⟶ 0 3 5 2 6 , 0 6 2 ⟶ 0 6 5 4 2 5 , 0 6 2 ⟶ 0 4 6 3 , 6 1 1 →= 6 1 5 4 1 , 6 1 1 →= 6 3 2 1 3 1 , 6 1 1 →= 3 5 1 1 6 4 , 6 1 1 →= 6 1 3 2 1 3 , 1 6 1 →= 4 6 4 1 1 , 1 6 1 →= 3 2 1 1 6 , 1 6 1 →= 6 3 2 4 1 1 , 6 6 1 →= 6 3 5 1 6 , 6 6 1 →= 6 6 3 5 1 2 , 6 1 2 →= 6 3 2 1 , 6 1 2 →= 6 1 3 2 4 , 6 1 2 →= 4 6 4 1 2 , 6 1 2 →= 3 3 2 3 6 1 , 6 1 2 →= 3 3 6 3 2 1 , 6 1 2 →= 6 1 5 3 1 2 , 1 6 2 →= 6 4 4 1 2 , 1 6 2 →= 6 3 2 1 2 , 1 6 2 →= 6 4 4 1 2 1 , 1 6 2 →= 5 4 2 1 3 6 , 1 6 2 →= 4 3 3 1 2 6 , 1 6 2 →= 2 3 1 6 3 3 , 6 6 2 →= 3 2 6 6 , 6 6 2 →= 6 6 3 2 , 6 6 2 →= 6 3 5 2 6 , 6 6 2 →= 3 3 2 3 6 6 , 6 6 2 →= 6 6 5 4 2 5 , 6 6 2 →= 3 2 6 4 6 3 , 1 5 6 1 →= 5 6 4 1 4 1 , 1 2 6 1 →= 2 6 3 2 1 1 , 1 2 6 1 →= 1 6 4 3 2 1 , 6 1 5 1 →= 1 5 6 3 1 1 , 1 6 5 1 →= 6 5 4 1 1 , 6 6 5 1 →= 3 5 1 6 6 2 , 1 6 1 2 →= 5 6 3 1 1 2 , 1 4 6 2 →= 4 3 2 1 3 6 , 1 4 6 2 →= 6 4 4 3 1 2 , 1 3 6 2 →= 5 4 1 3 2 6 , 1 3 6 2 →= 1 5 6 4 2 3 , 6 2 6 2 →= 6 2 6 3 2 , 6 1 5 2 →= 5 4 2 6 1 , 1 6 5 2 →= 1 2 5 3 6 , 1 6 5 2 →= 5 3 1 2 6 , 1 6 5 2 →= 1 6 5 3 2 , 1 6 5 2 →= 2 5 3 1 5 6 , 6 6 5 2 →= 3 5 5 2 6 6 , 1 2 4 6 1 →= 1 4 1 3 2 6 , 1 4 3 6 1 →= 1 5 4 3 1 6 , 1 6 2 3 6 →= 1 6 6 4 3 2 , 1 3 6 1 2 →= 1 1 3 2 3 6 , 1 1 6 2 2 →= 1 6 3 1 2 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 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 1 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 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 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎜ 0 0 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6 }, it remains to prove termination of the 55-rule system { 0 1 2 ⟶ 0 3 2 1 , 0 1 2 ⟶ 0 1 3 2 4 , 0 1 2 ⟶ 0 4 1 2 , 0 1 2 ⟶ 0 1 , 0 1 2 ⟶ 0 1 5 3 1 2 , 6 1 1 →= 6 1 5 4 1 , 6 1 1 →= 6 3 2 1 3 1 , 6 1 1 →= 3 5 1 1 6 4 , 6 1 1 →= 6 1 3 2 1 3 , 1 6 1 →= 4 6 4 1 1 , 1 6 1 →= 3 2 1 1 6 , 1 6 1 →= 6 3 2 4 1 1 , 6 6 1 →= 6 3 5 1 6 , 6 6 1 →= 6 6 3 5 1 2 , 6 1 2 →= 6 3 2 1 , 6 1 2 →= 6 1 3 2 4 , 6 1 2 →= 4 6 4 1 2 , 6 1 2 →= 3 3 2 3 6 1 , 6 1 2 →= 3 3 6 3 2 1 , 6 1 2 →= 6 1 5 3 1 2 , 1 6 2 →= 6 4 4 1 2 , 1 6 2 →= 6 3 2 1 2 , 1 6 2 →= 6 4 4 1 2 1 , 1 6 2 →= 5 4 2 1 3 6 , 1 6 2 →= 4 3 3 1 2 6 , 1 6 2 →= 2 3 1 6 3 3 , 6 6 2 →= 3 2 6 6 , 6 6 2 →= 6 6 3 2 , 6 6 2 →= 6 3 5 2 6 , 6 6 2 →= 3 3 2 3 6 6 , 6 6 2 →= 6 6 5 4 2 5 , 6 6 2 →= 3 2 6 4 6 3 , 1 5 6 1 →= 5 6 4 1 4 1 , 1 2 6 1 →= 2 6 3 2 1 1 , 1 2 6 1 →= 1 6 4 3 2 1 , 6 1 5 1 →= 1 5 6 3 1 1 , 1 6 5 1 →= 6 5 4 1 1 , 6 6 5 1 →= 3 5 1 6 6 2 , 1 6 1 2 →= 5 6 3 1 1 2 , 1 4 6 2 →= 4 3 2 1 3 6 , 1 4 6 2 →= 6 4 4 3 1 2 , 1 3 6 2 →= 5 4 1 3 2 6 , 1 3 6 2 →= 1 5 6 4 2 3 , 6 2 6 2 →= 6 2 6 3 2 , 6 1 5 2 →= 5 4 2 6 1 , 1 6 5 2 →= 1 2 5 3 6 , 1 6 5 2 →= 5 3 1 2 6 , 1 6 5 2 →= 1 6 5 3 2 , 1 6 5 2 →= 2 5 3 1 5 6 , 6 6 5 2 →= 3 5 5 2 6 6 , 1 2 4 6 1 →= 1 4 1 3 2 6 , 1 4 3 6 1 →= 1 5 4 3 1 6 , 1 6 2 3 6 →= 1 6 6 4 3 2 , 1 3 6 1 2 →= 1 1 3 2 3 6 , 1 1 6 2 2 →= 1 6 3 1 2 2 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 0 ↦ ⎛ ⎞ ⎜ 1 0 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 ⎟ ⎜ 0 1 0 0 1 0 0 ⎟ ⎜ 0 0 0 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 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 1 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 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 1 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 6 ↦ 0, 1 ↦ 1, 5 ↦ 2, 4 ↦ 3, 3 ↦ 4, 2 ↦ 5 }, it remains to prove termination of the 50-rule system { 0 1 1 →= 0 1 2 3 1 , 0 1 1 →= 0 4 5 1 4 1 , 0 1 1 →= 4 2 1 1 0 3 , 0 1 1 →= 0 1 4 5 1 4 , 1 0 1 →= 3 0 3 1 1 , 1 0 1 →= 4 5 1 1 0 , 1 0 1 →= 0 4 5 3 1 1 , 0 0 1 →= 0 4 2 1 0 , 0 0 1 →= 0 0 4 2 1 5 , 0 1 5 →= 0 4 5 1 , 0 1 5 →= 0 1 4 5 3 , 0 1 5 →= 3 0 3 1 5 , 0 1 5 →= 4 4 5 4 0 1 , 0 1 5 →= 4 4 0 4 5 1 , 0 1 5 →= 0 1 2 4 1 5 , 1 0 5 →= 0 3 3 1 5 , 1 0 5 →= 0 4 5 1 5 , 1 0 5 →= 0 3 3 1 5 1 , 1 0 5 →= 2 3 5 1 4 0 , 1 0 5 →= 3 4 4 1 5 0 , 1 0 5 →= 5 4 1 0 4 4 , 0 0 5 →= 4 5 0 0 , 0 0 5 →= 0 0 4 5 , 0 0 5 →= 0 4 2 5 0 , 0 0 5 →= 4 4 5 4 0 0 , 0 0 5 →= 0 0 2 3 5 2 , 0 0 5 →= 4 5 0 3 0 4 , 1 2 0 1 →= 2 0 3 1 3 1 , 1 5 0 1 →= 5 0 4 5 1 1 , 1 5 0 1 →= 1 0 3 4 5 1 , 0 1 2 1 →= 1 2 0 4 1 1 , 1 0 2 1 →= 0 2 3 1 1 , 0 0 2 1 →= 4 2 1 0 0 5 , 1 0 1 5 →= 2 0 4 1 1 5 , 1 3 0 5 →= 3 4 5 1 4 0 , 1 3 0 5 →= 0 3 3 4 1 5 , 1 4 0 5 →= 2 3 1 4 5 0 , 1 4 0 5 →= 1 2 0 3 5 4 , 0 5 0 5 →= 0 5 0 4 5 , 0 1 2 5 →= 2 3 5 0 1 , 1 0 2 5 →= 1 5 2 4 0 , 1 0 2 5 →= 2 4 1 5 0 , 1 0 2 5 →= 1 0 2 4 5 , 1 0 2 5 →= 5 2 4 1 2 0 , 0 0 2 5 →= 4 2 2 5 0 0 , 1 5 3 0 1 →= 1 3 1 4 5 0 , 1 3 4 0 1 →= 1 2 3 4 1 0 , 1 0 5 4 0 →= 1 0 0 3 4 5 , 1 4 0 1 5 →= 1 1 4 5 4 0 , 1 1 0 5 5 →= 1 0 4 1 5 5 } The system is trivially terminating.