/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 48-rule system { 0 0 0 ⟶ 0 0 1 0 1 , 0 2 0 ⟶ 0 0 2 1 1 , 0 3 2 ⟶ 2 1 3 0 , 0 4 2 ⟶ 4 1 1 1 0 2 , 3 0 4 ⟶ 0 1 3 4 1 , 3 2 0 ⟶ 0 2 1 3 , 3 2 0 ⟶ 3 0 2 1 , 3 2 0 ⟶ 0 1 2 1 3 , 3 2 0 ⟶ 3 0 1 2 1 4 , 3 2 4 ⟶ 1 2 1 3 3 4 , 3 2 4 ⟶ 4 1 2 1 1 3 , 3 2 5 ⟶ 1 3 5 2 , 3 2 5 ⟶ 3 1 4 5 2 , 3 2 5 ⟶ 1 1 3 3 5 2 , 3 2 5 ⟶ 1 3 4 5 4 2 , 3 5 0 ⟶ 3 1 0 5 1 1 , 5 0 2 ⟶ 1 1 0 2 5 , 5 0 4 ⟶ 0 5 4 1 1 , 5 2 4 ⟶ 5 2 1 1 3 4 , 5 3 2 ⟶ 1 1 3 5 1 2 , 5 5 0 ⟶ 5 1 1 5 0 , 0 0 2 4 ⟶ 0 4 0 2 1 1 , 0 3 1 2 ⟶ 1 1 3 0 2 , 0 3 2 4 ⟶ 2 1 3 0 4 4 , 0 4 3 2 ⟶ 2 3 4 0 1 , 3 0 2 4 ⟶ 3 0 4 2 1 1 , 3 0 5 4 ⟶ 3 4 0 5 1 3 , 3 1 0 0 ⟶ 0 1 1 3 0 1 , 3 2 4 0 ⟶ 3 1 0 4 2 , 3 2 4 5 ⟶ 3 1 4 2 1 5 , 3 2 5 4 ⟶ 3 4 3 5 2 , 3 5 3 2 ⟶ 3 3 0 2 5 , 4 0 4 2 ⟶ 4 4 2 1 3 0 , 4 3 2 0 ⟶ 0 2 1 3 4 , 5 4 1 0 ⟶ 5 1 1 4 0 1 , 5 5 0 0 ⟶ 1 1 0 5 5 0 , 5 5 2 2 ⟶ 5 5 1 1 2 2 , 0 0 1 2 4 ⟶ 0 0 2 1 1 4 , 0 5 3 1 2 ⟶ 5 2 1 3 1 0 , 0 5 5 1 0 ⟶ 5 0 0 5 1 1 , 3 0 1 4 5 ⟶ 3 4 0 5 1 1 , 3 0 3 1 2 ⟶ 0 2 1 1 3 3 , 3 2 2 0 4 ⟶ 3 2 2 0 1 4 , 3 5 1 4 0 ⟶ 1 3 3 4 5 0 , 3 5 3 0 4 ⟶ 3 0 1 5 3 4 , 5 1 0 2 4 ⟶ 2 1 5 4 0 3 , 5 1 3 2 0 ⟶ 2 3 5 0 1 1 , 5 4 2 0 2 ⟶ 5 0 4 2 1 2 } The system was reversed. After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5 }, it remains to prove termination of the 48-rule system { 0 0 0 ⟶ 1 0 1 0 0 , 0 2 0 ⟶ 1 1 2 0 0 , 2 3 0 ⟶ 0 3 1 2 , 2 4 0 ⟶ 2 0 1 1 1 4 , 4 0 3 ⟶ 1 4 3 1 0 , 0 2 3 ⟶ 3 1 2 0 , 0 2 3 ⟶ 1 2 0 3 , 0 2 3 ⟶ 3 1 2 1 0 , 0 2 3 ⟶ 4 1 2 1 0 3 , 4 2 3 ⟶ 4 3 3 1 2 1 , 4 2 3 ⟶ 3 1 1 2 1 4 , 5 2 3 ⟶ 2 5 3 1 , 5 2 3 ⟶ 2 5 4 1 3 , 5 2 3 ⟶ 2 5 3 3 1 1 , 5 2 3 ⟶ 2 4 5 4 3 1 , 0 5 3 ⟶ 1 1 5 0 1 3 , 2 0 5 ⟶ 5 2 0 1 1 , 4 0 5 ⟶ 1 1 4 5 0 , 4 2 5 ⟶ 4 3 1 1 2 5 , 2 3 5 ⟶ 2 1 5 3 1 1 , 0 5 5 ⟶ 0 5 1 1 5 , 4 2 0 0 ⟶ 1 1 2 0 4 0 , 2 1 3 0 ⟶ 2 0 3 1 1 , 4 2 3 0 ⟶ 4 4 0 3 1 2 , 2 3 4 0 ⟶ 1 0 4 3 2 , 4 2 0 3 ⟶ 1 1 2 4 0 3 , 4 5 0 3 ⟶ 3 1 5 0 4 3 , 0 0 1 3 ⟶ 1 0 3 1 1 0 , 0 4 2 3 ⟶ 2 4 0 1 3 , 5 4 2 3 ⟶ 5 1 2 4 1 3 , 4 5 2 3 ⟶ 2 5 3 4 3 , 2 3 5 3 ⟶ 5 2 0 3 3 , 2 4 0 4 ⟶ 0 3 1 2 4 4 , 0 2 3 4 ⟶ 4 3 1 2 0 , 0 1 4 5 ⟶ 1 0 4 1 1 5 , 0 0 5 5 ⟶ 0 5 5 0 1 1 , 2 2 5 5 ⟶ 2 2 1 1 5 5 , 4 2 1 0 0 ⟶ 4 1 1 2 0 0 , 2 1 3 5 0 ⟶ 0 1 3 1 2 5 , 0 1 5 5 0 ⟶ 1 1 5 0 0 5 , 5 4 1 0 3 ⟶ 1 1 5 0 4 3 , 2 1 3 0 3 ⟶ 3 3 1 1 2 0 , 4 0 2 2 3 ⟶ 4 1 0 2 2 3 , 0 4 1 5 3 ⟶ 0 5 4 3 3 1 , 4 0 3 5 3 ⟶ 4 3 5 1 0 3 , 4 2 0 1 5 ⟶ 3 0 4 5 1 2 , 0 2 3 1 5 ⟶ 1 1 0 5 3 2 , 2 0 2 4 5 ⟶ 2 1 2 4 0 5 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (0,↑) ↦ 0, (0,↓) ↦ 1, (1,↓) ↦ 2, (2,↓) ↦ 3, (2,↑) ↦ 4, (3,↓) ↦ 5, (4,↓) ↦ 6, (4,↑) ↦ 7, (5,↑) ↦ 8, (5,↓) ↦ 9 }, it remains to prove termination of the 170-rule system { 0 1 1 ⟶ 0 2 1 1 , 0 3 1 ⟶ 4 1 1 , 0 3 1 ⟶ 0 1 , 4 5 1 ⟶ 0 5 2 3 , 4 5 1 ⟶ 4 , 4 6 1 ⟶ 4 1 2 2 2 6 , 4 6 1 ⟶ 0 2 2 2 6 , 4 6 1 ⟶ 7 , 7 1 5 ⟶ 7 5 2 1 , 7 1 5 ⟶ 0 , 0 3 5 ⟶ 4 1 , 0 3 5 ⟶ 0 , 0 3 5 ⟶ 4 1 5 , 0 3 5 ⟶ 0 5 , 0 3 5 ⟶ 4 2 1 , 0 3 5 ⟶ 7 2 3 2 1 5 , 0 3 5 ⟶ 4 2 1 5 , 7 3 5 ⟶ 7 5 5 2 3 2 , 7 3 5 ⟶ 4 2 , 7 3 5 ⟶ 4 2 6 , 7 3 5 ⟶ 7 , 8 3 5 ⟶ 4 9 5 2 , 8 3 5 ⟶ 8 5 2 , 8 3 5 ⟶ 4 9 6 2 5 , 8 3 5 ⟶ 8 6 2 5 , 8 3 5 ⟶ 7 2 5 , 8 3 5 ⟶ 4 9 5 5 2 2 , 8 3 5 ⟶ 8 5 5 2 2 , 8 3 5 ⟶ 4 6 9 6 5 2 , 8 3 5 ⟶ 7 9 6 5 2 , 8 3 5 ⟶ 8 6 5 2 , 8 3 5 ⟶ 7 5 2 , 0 9 5 ⟶ 8 1 2 5 , 0 9 5 ⟶ 0 2 5 , 4 1 9 ⟶ 8 3 1 2 2 , 4 1 9 ⟶ 4 1 2 2 , 4 1 9 ⟶ 0 2 2 , 7 1 9 ⟶ 7 9 1 , 7 1 9 ⟶ 8 1 , 7 1 9 ⟶ 0 , 7 3 9 ⟶ 7 5 2 2 3 9 , 4 5 9 ⟶ 4 2 9 5 2 2 , 4 5 9 ⟶ 8 5 2 2 , 0 9 9 ⟶ 0 9 2 2 9 , 0 9 9 ⟶ 8 2 2 9 , 7 3 1 1 ⟶ 4 1 6 1 , 7 3 1 1 ⟶ 0 6 1 , 7 3 1 1 ⟶ 7 1 , 4 2 5 1 ⟶ 4 1 5 2 2 , 4 2 5 1 ⟶ 0 5 2 2 , 7 3 5 1 ⟶ 7 6 1 5 2 3 , 7 3 5 1 ⟶ 7 1 5 2 3 , 7 3 5 1 ⟶ 0 5 2 3 , 7 3 5 1 ⟶ 4 , 4 5 6 1 ⟶ 0 6 5 3 , 4 5 6 1 ⟶ 7 5 3 , 4 5 6 1 ⟶ 4 , 7 3 1 5 ⟶ 4 6 1 5 , 7 3 1 5 ⟶ 7 1 5 , 7 9 1 5 ⟶ 8 1 6 5 , 7 9 1 5 ⟶ 0 6 5 , 7 9 1 5 ⟶ 7 5 , 0 1 2 5 ⟶ 0 5 2 2 1 , 0 1 2 5 ⟶ 0 , 0 6 3 5 ⟶ 4 6 1 2 5 , 0 6 3 5 ⟶ 7 1 2 5 , 0 6 3 5 ⟶ 0 2 5 , 8 6 3 5 ⟶ 8 2 3 6 2 5 , 8 6 3 5 ⟶ 4 6 2 5 , 8 6 3 5 ⟶ 7 2 5 , 7 9 3 5 ⟶ 4 9 5 6 5 , 7 9 3 5 ⟶ 8 5 6 5 , 7 9 3 5 ⟶ 7 5 , 4 5 9 5 ⟶ 8 3 1 5 5 , 4 5 9 5 ⟶ 4 1 5 5 , 4 5 9 5 ⟶ 0 5 5 , 4 6 1 6 ⟶ 0 5 2 3 6 6 , 4 6 1 6 ⟶ 4 6 6 , 4 6 1 6 ⟶ 7 6 , 0 3 5 6 ⟶ 7 5 2 3 1 , 0 3 5 6 ⟶ 4 1 , 0 3 5 6 ⟶ 0 , 0 2 6 9 ⟶ 0 6 2 2 9 , 0 2 6 9 ⟶ 7 2 2 9 , 0 1 9 9 ⟶ 0 9 9 1 2 2 , 0 1 9 9 ⟶ 8 9 1 2 2 , 0 1 9 9 ⟶ 8 1 2 2 , 0 1 9 9 ⟶ 0 2 2 , 4 3 9 9 ⟶ 4 3 2 2 9 9 , 4 3 9 9 ⟶ 4 2 2 9 9 , 7 3 2 1 1 ⟶ 7 2 2 3 1 1 , 7 3 2 1 1 ⟶ 4 1 1 , 4 2 5 9 1 ⟶ 0 2 5 2 3 9 , 4 2 5 9 1 ⟶ 4 9 , 4 2 5 9 1 ⟶ 8 , 0 2 9 9 1 ⟶ 8 1 1 9 , 0 2 9 9 1 ⟶ 0 1 9 , 0 2 9 9 1 ⟶ 0 9 , 0 2 9 9 1 ⟶ 8 , 8 6 2 1 5 ⟶ 8 1 6 5 , 8 6 2 1 5 ⟶ 0 6 5 , 8 6 2 1 5 ⟶ 7 5 , 4 2 5 1 5 ⟶ 4 1 , 4 2 5 1 5 ⟶ 0 , 7 1 3 3 5 ⟶ 7 2 1 3 3 5 , 0 6 2 9 5 ⟶ 0 9 6 5 5 2 , 0 6 2 9 5 ⟶ 8 6 5 5 2 , 0 6 2 9 5 ⟶ 7 5 5 2 , 7 1 5 9 5 ⟶ 7 5 9 2 1 5 , 7 1 5 9 5 ⟶ 8 2 1 5 , 7 1 5 9 5 ⟶ 0 5 , 7 3 1 2 9 ⟶ 0 6 9 2 3 , 7 3 1 2 9 ⟶ 7 9 2 3 , 7 3 1 2 9 ⟶ 8 2 3 , 7 3 1 2 9 ⟶ 4 , 0 3 5 2 9 ⟶ 0 9 5 3 , 0 3 5 2 9 ⟶ 8 5 3 , 0 3 5 2 9 ⟶ 4 , 4 1 3 6 9 ⟶ 4 2 3 6 1 9 , 4 1 3 6 9 ⟶ 4 6 1 9 , 4 1 3 6 9 ⟶ 7 1 9 , 4 1 3 6 9 ⟶ 0 9 , 1 1 1 →= 2 1 2 1 1 , 1 3 1 →= 2 2 3 1 1 , 3 5 1 →= 1 5 2 3 , 3 6 1 →= 3 1 2 2 2 6 , 6 1 5 →= 2 6 5 2 1 , 1 3 5 →= 5 2 3 1 , 1 3 5 →= 2 3 1 5 , 1 3 5 →= 5 2 3 2 1 , 1 3 5 →= 6 2 3 2 1 5 , 6 3 5 →= 6 5 5 2 3 2 , 6 3 5 →= 5 2 2 3 2 6 , 9 3 5 →= 3 9 5 2 , 9 3 5 →= 3 9 6 2 5 , 9 3 5 →= 3 9 5 5 2 2 , 9 3 5 →= 3 6 9 6 5 2 , 1 9 5 →= 2 2 9 1 2 5 , 3 1 9 →= 9 3 1 2 2 , 6 1 9 →= 2 2 6 9 1 , 6 3 9 →= 6 5 2 2 3 9 , 3 5 9 →= 3 2 9 5 2 2 , 1 9 9 →= 1 9 2 2 9 , 6 3 1 1 →= 2 2 3 1 6 1 , 3 2 5 1 →= 3 1 5 2 2 , 6 3 5 1 →= 6 6 1 5 2 3 , 3 5 6 1 →= 2 1 6 5 3 , 6 3 1 5 →= 2 2 3 6 1 5 , 6 9 1 5 →= 5 2 9 1 6 5 , 1 1 2 5 →= 2 1 5 2 2 1 , 1 6 3 5 →= 3 6 1 2 5 , 9 6 3 5 →= 9 2 3 6 2 5 , 6 9 3 5 →= 3 9 5 6 5 , 3 5 9 5 →= 9 3 1 5 5 , 3 6 1 6 →= 1 5 2 3 6 6 , 1 3 5 6 →= 6 5 2 3 1 , 1 2 6 9 →= 2 1 6 2 2 9 , 1 1 9 9 →= 1 9 9 1 2 2 , 3 3 9 9 →= 3 3 2 2 9 9 , 6 3 2 1 1 →= 6 2 2 3 1 1 , 3 2 5 9 1 →= 1 2 5 2 3 9 , 1 2 9 9 1 →= 2 2 9 1 1 9 , 9 6 2 1 5 →= 2 2 9 1 6 5 , 3 2 5 1 5 →= 5 5 2 2 3 1 , 6 1 3 3 5 →= 6 2 1 3 3 5 , 1 6 2 9 5 →= 1 9 6 5 5 2 , 6 1 5 9 5 →= 6 5 9 2 1 5 , 6 3 1 2 9 →= 5 1 6 9 2 3 , 1 3 5 2 9 →= 2 2 1 9 5 3 , 3 1 3 6 9 →= 3 2 3 6 1 9 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 119-rule system { 0 1 1 ⟶ 0 2 1 1 , 0 3 1 ⟶ 4 1 1 , 4 5 1 ⟶ 0 5 2 3 , 4 5 1 ⟶ 4 , 4 6 1 ⟶ 4 1 2 2 2 6 , 7 1 5 ⟶ 7 5 2 1 , 7 1 5 ⟶ 0 , 0 3 5 ⟶ 4 1 , 0 3 5 ⟶ 4 1 5 , 0 3 5 ⟶ 4 2 1 , 0 3 5 ⟶ 7 2 3 2 1 5 , 0 3 5 ⟶ 4 2 1 5 , 7 3 5 ⟶ 7 5 5 2 3 2 , 7 3 5 ⟶ 4 2 , 7 3 5 ⟶ 4 2 6 , 8 3 5 ⟶ 4 9 5 2 , 8 3 5 ⟶ 4 9 6 2 5 , 8 3 5 ⟶ 4 9 5 5 2 2 , 8 3 5 ⟶ 4 6 9 6 5 2 , 0 9 5 ⟶ 8 1 2 5 , 4 1 9 ⟶ 8 3 1 2 2 , 7 1 9 ⟶ 7 9 1 , 7 1 9 ⟶ 8 1 , 7 3 9 ⟶ 7 5 2 2 3 9 , 4 5 9 ⟶ 4 2 9 5 2 2 , 0 9 9 ⟶ 0 9 2 2 9 , 0 9 9 ⟶ 8 2 2 9 , 7 3 1 1 ⟶ 4 1 6 1 , 4 2 5 1 ⟶ 4 1 5 2 2 , 7 3 5 1 ⟶ 7 6 1 5 2 3 , 7 3 5 1 ⟶ 7 1 5 2 3 , 7 3 5 1 ⟶ 0 5 2 3 , 7 3 5 1 ⟶ 4 , 4 5 6 1 ⟶ 0 6 5 3 , 4 5 6 1 ⟶ 7 5 3 , 4 5 6 1 ⟶ 4 , 7 3 1 5 ⟶ 4 6 1 5 , 7 9 1 5 ⟶ 8 1 6 5 , 0 1 2 5 ⟶ 0 5 2 2 1 , 0 1 2 5 ⟶ 0 , 0 6 3 5 ⟶ 4 6 1 2 5 , 8 6 3 5 ⟶ 8 2 3 6 2 5 , 7 9 3 5 ⟶ 4 9 5 6 5 , 4 5 9 5 ⟶ 8 3 1 5 5 , 4 6 1 6 ⟶ 0 5 2 3 6 6 , 4 6 1 6 ⟶ 4 6 6 , 0 3 5 6 ⟶ 7 5 2 3 1 , 0 3 5 6 ⟶ 4 1 , 0 2 6 9 ⟶ 0 6 2 2 9 , 0 2 6 9 ⟶ 7 2 2 9 , 0 1 9 9 ⟶ 0 9 9 1 2 2 , 0 1 9 9 ⟶ 8 9 1 2 2 , 4 3 9 9 ⟶ 4 3 2 2 9 9 , 7 3 2 1 1 ⟶ 7 2 2 3 1 1 , 7 3 2 1 1 ⟶ 4 1 1 , 4 2 5 9 1 ⟶ 0 2 5 2 3 9 , 4 2 5 9 1 ⟶ 4 9 , 0 2 9 9 1 ⟶ 8 1 1 9 , 8 6 2 1 5 ⟶ 8 1 6 5 , 4 2 5 1 5 ⟶ 4 1 , 7 1 3 3 5 ⟶ 7 2 1 3 3 5 , 0 6 2 9 5 ⟶ 0 9 6 5 5 2 , 0 6 2 9 5 ⟶ 8 6 5 5 2 , 7 1 5 9 5 ⟶ 7 5 9 2 1 5 , 7 1 5 9 5 ⟶ 8 2 1 5 , 7 3 1 2 9 ⟶ 0 6 9 2 3 , 7 3 1 2 9 ⟶ 7 9 2 3 , 7 3 1 2 9 ⟶ 8 2 3 , 0 3 5 2 9 ⟶ 0 9 5 3 , 0 3 5 2 9 ⟶ 8 5 3 , 4 1 3 6 9 ⟶ 4 2 3 6 1 9 , 1 1 1 →= 2 1 2 1 1 , 1 3 1 →= 2 2 3 1 1 , 3 5 1 →= 1 5 2 3 , 3 6 1 →= 3 1 2 2 2 6 , 6 1 5 →= 2 6 5 2 1 , 1 3 5 →= 5 2 3 1 , 1 3 5 →= 2 3 1 5 , 1 3 5 →= 5 2 3 2 1 , 1 3 5 →= 6 2 3 2 1 5 , 6 3 5 →= 6 5 5 2 3 2 , 6 3 5 →= 5 2 2 3 2 6 , 9 3 5 →= 3 9 5 2 , 9 3 5 →= 3 9 6 2 5 , 9 3 5 →= 3 9 5 5 2 2 , 9 3 5 →= 3 6 9 6 5 2 , 1 9 5 →= 2 2 9 1 2 5 , 3 1 9 →= 9 3 1 2 2 , 6 1 9 →= 2 2 6 9 1 , 6 3 9 →= 6 5 2 2 3 9 , 3 5 9 →= 3 2 9 5 2 2 , 1 9 9 →= 1 9 2 2 9 , 6 3 1 1 →= 2 2 3 1 6 1 , 3 2 5 1 →= 3 1 5 2 2 , 6 3 5 1 →= 6 6 1 5 2 3 , 3 5 6 1 →= 2 1 6 5 3 , 6 3 1 5 →= 2 2 3 6 1 5 , 6 9 1 5 →= 5 2 9 1 6 5 , 1 1 2 5 →= 2 1 5 2 2 1 , 1 6 3 5 →= 3 6 1 2 5 , 9 6 3 5 →= 9 2 3 6 2 5 , 6 9 3 5 →= 3 9 5 6 5 , 3 5 9 5 →= 9 3 1 5 5 , 3 6 1 6 →= 1 5 2 3 6 6 , 1 3 5 6 →= 6 5 2 3 1 , 1 2 6 9 →= 2 1 6 2 2 9 , 1 1 9 9 →= 1 9 9 1 2 2 , 3 3 9 9 →= 3 3 2 2 9 9 , 6 3 2 1 1 →= 6 2 2 3 1 1 , 3 2 5 9 1 →= 1 2 5 2 3 9 , 1 2 9 9 1 →= 2 2 9 1 1 9 , 9 6 2 1 5 →= 2 2 9 1 6 5 , 3 2 5 1 5 →= 5 5 2 2 3 1 , 6 1 3 3 5 →= 6 2 1 3 3 5 , 1 6 2 9 5 →= 1 9 6 5 5 2 , 6 1 5 9 5 →= 6 5 9 2 1 5 , 6 3 1 2 9 →= 5 1 6 9 2 3 , 1 3 5 2 9 →= 2 2 1 9 5 3 , 3 1 3 6 9 →= 3 2 3 6 1 9 } 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 1 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎝ ⎠ 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 1 ⎟ ⎜ 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 1 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 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 1 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 116-rule system { 0 1 1 ⟶ 0 2 1 1 , 0 3 1 ⟶ 4 1 1 , 4 5 1 ⟶ 0 5 2 3 , 4 5 1 ⟶ 4 , 4 6 1 ⟶ 4 1 2 2 2 6 , 7 1 5 ⟶ 7 5 2 1 , 7 1 5 ⟶ 0 , 0 3 5 ⟶ 4 1 , 0 3 5 ⟶ 4 1 5 , 0 3 5 ⟶ 4 2 1 , 0 3 5 ⟶ 7 2 3 2 1 5 , 0 3 5 ⟶ 4 2 1 5 , 7 3 5 ⟶ 7 5 5 2 3 2 , 7 3 5 ⟶ 4 2 , 7 3 5 ⟶ 4 2 6 , 8 3 5 ⟶ 4 9 5 2 , 8 3 5 ⟶ 4 9 6 2 5 , 8 3 5 ⟶ 4 9 5 5 2 2 , 8 3 5 ⟶ 4 6 9 6 5 2 , 0 9 5 ⟶ 8 1 2 5 , 4 1 9 ⟶ 8 3 1 2 2 , 7 1 9 ⟶ 7 9 1 , 7 1 9 ⟶ 8 1 , 7 3 9 ⟶ 7 5 2 2 3 9 , 4 5 9 ⟶ 4 2 9 5 2 2 , 0 9 9 ⟶ 0 9 2 2 9 , 0 9 9 ⟶ 8 2 2 9 , 7 3 1 1 ⟶ 4 1 6 1 , 4 2 5 1 ⟶ 4 1 5 2 2 , 7 3 5 1 ⟶ 7 6 1 5 2 3 , 7 3 5 1 ⟶ 7 1 5 2 3 , 7 3 5 1 ⟶ 0 5 2 3 , 7 3 5 1 ⟶ 4 , 4 5 6 1 ⟶ 0 6 5 3 , 4 5 6 1 ⟶ 7 5 3 , 4 5 6 1 ⟶ 4 , 7 3 1 5 ⟶ 4 6 1 5 , 7 9 1 5 ⟶ 8 1 6 5 , 0 1 2 5 ⟶ 0 5 2 2 1 , 0 1 2 5 ⟶ 0 , 0 6 3 5 ⟶ 4 6 1 2 5 , 8 6 3 5 ⟶ 8 2 3 6 2 5 , 7 9 3 5 ⟶ 4 9 5 6 5 , 4 5 9 5 ⟶ 8 3 1 5 5 , 4 6 1 6 ⟶ 0 5 2 3 6 6 , 4 6 1 6 ⟶ 4 6 6 , 0 3 5 6 ⟶ 7 5 2 3 1 , 0 3 5 6 ⟶ 4 1 , 0 2 6 9 ⟶ 0 6 2 2 9 , 0 2 6 9 ⟶ 7 2 2 9 , 0 1 9 9 ⟶ 0 9 9 1 2 2 , 0 1 9 9 ⟶ 8 9 1 2 2 , 4 3 9 9 ⟶ 4 3 2 2 9 9 , 7 3 2 1 1 ⟶ 7 2 2 3 1 1 , 7 3 2 1 1 ⟶ 4 1 1 , 4 2 5 9 1 ⟶ 0 2 5 2 3 9 , 4 2 5 9 1 ⟶ 4 9 , 0 2 9 9 1 ⟶ 8 1 1 9 , 8 6 2 1 5 ⟶ 8 1 6 5 , 4 2 5 1 5 ⟶ 4 1 , 7 1 3 3 5 ⟶ 7 2 1 3 3 5 , 0 6 2 9 5 ⟶ 0 9 6 5 5 2 , 0 6 2 9 5 ⟶ 8 6 5 5 2 , 7 1 5 9 5 ⟶ 7 5 9 2 1 5 , 7 1 5 9 5 ⟶ 8 2 1 5 , 0 3 5 2 9 ⟶ 0 9 5 3 , 0 3 5 2 9 ⟶ 8 5 3 , 4 1 3 6 9 ⟶ 4 2 3 6 1 9 , 1 1 1 →= 2 1 2 1 1 , 1 3 1 →= 2 2 3 1 1 , 3 5 1 →= 1 5 2 3 , 3 6 1 →= 3 1 2 2 2 6 , 6 1 5 →= 2 6 5 2 1 , 1 3 5 →= 5 2 3 1 , 1 3 5 →= 2 3 1 5 , 1 3 5 →= 5 2 3 2 1 , 1 3 5 →= 6 2 3 2 1 5 , 6 3 5 →= 6 5 5 2 3 2 , 6 3 5 →= 5 2 2 3 2 6 , 9 3 5 →= 3 9 5 2 , 9 3 5 →= 3 9 6 2 5 , 9 3 5 →= 3 9 5 5 2 2 , 9 3 5 →= 3 6 9 6 5 2 , 1 9 5 →= 2 2 9 1 2 5 , 3 1 9 →= 9 3 1 2 2 , 6 1 9 →= 2 2 6 9 1 , 6 3 9 →= 6 5 2 2 3 9 , 3 5 9 →= 3 2 9 5 2 2 , 1 9 9 →= 1 9 2 2 9 , 6 3 1 1 →= 2 2 3 1 6 1 , 3 2 5 1 →= 3 1 5 2 2 , 6 3 5 1 →= 6 6 1 5 2 3 , 3 5 6 1 →= 2 1 6 5 3 , 6 3 1 5 →= 2 2 3 6 1 5 , 6 9 1 5 →= 5 2 9 1 6 5 , 1 1 2 5 →= 2 1 5 2 2 1 , 1 6 3 5 →= 3 6 1 2 5 , 9 6 3 5 →= 9 2 3 6 2 5 , 6 9 3 5 →= 3 9 5 6 5 , 3 5 9 5 →= 9 3 1 5 5 , 3 6 1 6 →= 1 5 2 3 6 6 , 1 3 5 6 →= 6 5 2 3 1 , 1 2 6 9 →= 2 1 6 2 2 9 , 1 1 9 9 →= 1 9 9 1 2 2 , 3 3 9 9 →= 3 3 2 2 9 9 , 6 3 2 1 1 →= 6 2 2 3 1 1 , 3 2 5 9 1 →= 1 2 5 2 3 9 , 1 2 9 9 1 →= 2 2 9 1 1 9 , 9 6 2 1 5 →= 2 2 9 1 6 5 , 3 2 5 1 5 →= 5 5 2 2 3 1 , 6 1 3 3 5 →= 6 2 1 3 3 5 , 1 6 2 9 5 →= 1 9 6 5 5 2 , 6 1 5 9 5 →= 6 5 9 2 1 5 , 6 3 1 2 9 →= 5 1 6 9 2 3 , 1 3 5 2 9 →= 2 2 1 9 5 3 , 3 1 3 6 9 →= 3 2 3 6 1 9 } 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 1 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 1 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 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 1 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 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, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 113-rule system { 0 1 1 ⟶ 0 2 1 1 , 0 3 1 ⟶ 4 1 1 , 4 5 1 ⟶ 0 5 2 3 , 4 5 1 ⟶ 4 , 4 6 1 ⟶ 4 1 2 2 2 6 , 7 1 5 ⟶ 7 5 2 1 , 7 1 5 ⟶ 0 , 0 3 5 ⟶ 4 1 , 0 3 5 ⟶ 4 1 5 , 0 3 5 ⟶ 4 2 1 , 0 3 5 ⟶ 7 2 3 2 1 5 , 0 3 5 ⟶ 4 2 1 5 , 7 3 5 ⟶ 7 5 5 2 3 2 , 7 3 5 ⟶ 4 2 , 7 3 5 ⟶ 4 2 6 , 8 3 5 ⟶ 4 9 5 2 , 8 3 5 ⟶ 4 9 6 2 5 , 8 3 5 ⟶ 4 9 5 5 2 2 , 8 3 5 ⟶ 4 6 9 6 5 2 , 0 9 5 ⟶ 8 1 2 5 , 4 1 9 ⟶ 8 3 1 2 2 , 7 3 9 ⟶ 7 5 2 2 3 9 , 4 5 9 ⟶ 4 2 9 5 2 2 , 0 9 9 ⟶ 0 9 2 2 9 , 0 9 9 ⟶ 8 2 2 9 , 7 3 1 1 ⟶ 4 1 6 1 , 4 2 5 1 ⟶ 4 1 5 2 2 , 7 3 5 1 ⟶ 7 6 1 5 2 3 , 7 3 5 1 ⟶ 7 1 5 2 3 , 7 3 5 1 ⟶ 0 5 2 3 , 7 3 5 1 ⟶ 4 , 4 5 6 1 ⟶ 0 6 5 3 , 4 5 6 1 ⟶ 7 5 3 , 4 5 6 1 ⟶ 4 , 7 3 1 5 ⟶ 4 6 1 5 , 7 9 1 5 ⟶ 8 1 6 5 , 0 1 2 5 ⟶ 0 5 2 2 1 , 0 1 2 5 ⟶ 0 , 0 6 3 5 ⟶ 4 6 1 2 5 , 8 6 3 5 ⟶ 8 2 3 6 2 5 , 7 9 3 5 ⟶ 4 9 5 6 5 , 4 5 9 5 ⟶ 8 3 1 5 5 , 4 6 1 6 ⟶ 0 5 2 3 6 6 , 4 6 1 6 ⟶ 4 6 6 , 0 3 5 6 ⟶ 7 5 2 3 1 , 0 3 5 6 ⟶ 4 1 , 0 2 6 9 ⟶ 0 6 2 2 9 , 0 2 6 9 ⟶ 7 2 2 9 , 0 1 9 9 ⟶ 0 9 9 1 2 2 , 0 1 9 9 ⟶ 8 9 1 2 2 , 4 3 9 9 ⟶ 4 3 2 2 9 9 , 7 3 2 1 1 ⟶ 7 2 2 3 1 1 , 7 3 2 1 1 ⟶ 4 1 1 , 4 2 5 9 1 ⟶ 0 2 5 2 3 9 , 4 2 5 9 1 ⟶ 4 9 , 0 2 9 9 1 ⟶ 8 1 1 9 , 8 6 2 1 5 ⟶ 8 1 6 5 , 4 2 5 1 5 ⟶ 4 1 , 0 6 2 9 5 ⟶ 0 9 6 5 5 2 , 0 6 2 9 5 ⟶ 8 6 5 5 2 , 7 1 5 9 5 ⟶ 7 5 9 2 1 5 , 7 1 5 9 5 ⟶ 8 2 1 5 , 0 3 5 2 9 ⟶ 0 9 5 3 , 0 3 5 2 9 ⟶ 8 5 3 , 4 1 3 6 9 ⟶ 4 2 3 6 1 9 , 1 1 1 →= 2 1 2 1 1 , 1 3 1 →= 2 2 3 1 1 , 3 5 1 →= 1 5 2 3 , 3 6 1 →= 3 1 2 2 2 6 , 6 1 5 →= 2 6 5 2 1 , 1 3 5 →= 5 2 3 1 , 1 3 5 →= 2 3 1 5 , 1 3 5 →= 5 2 3 2 1 , 1 3 5 →= 6 2 3 2 1 5 , 6 3 5 →= 6 5 5 2 3 2 , 6 3 5 →= 5 2 2 3 2 6 , 9 3 5 →= 3 9 5 2 , 9 3 5 →= 3 9 6 2 5 , 9 3 5 →= 3 9 5 5 2 2 , 9 3 5 →= 3 6 9 6 5 2 , 1 9 5 →= 2 2 9 1 2 5 , 3 1 9 →= 9 3 1 2 2 , 6 1 9 →= 2 2 6 9 1 , 6 3 9 →= 6 5 2 2 3 9 , 3 5 9 →= 3 2 9 5 2 2 , 1 9 9 →= 1 9 2 2 9 , 6 3 1 1 →= 2 2 3 1 6 1 , 3 2 5 1 →= 3 1 5 2 2 , 6 3 5 1 →= 6 6 1 5 2 3 , 3 5 6 1 →= 2 1 6 5 3 , 6 3 1 5 →= 2 2 3 6 1 5 , 6 9 1 5 →= 5 2 9 1 6 5 , 1 1 2 5 →= 2 1 5 2 2 1 , 1 6 3 5 →= 3 6 1 2 5 , 9 6 3 5 →= 9 2 3 6 2 5 , 6 9 3 5 →= 3 9 5 6 5 , 3 5 9 5 →= 9 3 1 5 5 , 3 6 1 6 →= 1 5 2 3 6 6 , 1 3 5 6 →= 6 5 2 3 1 , 1 2 6 9 →= 2 1 6 2 2 9 , 1 1 9 9 →= 1 9 9 1 2 2 , 3 3 9 9 →= 3 3 2 2 9 9 , 6 3 2 1 1 →= 6 2 2 3 1 1 , 3 2 5 9 1 →= 1 2 5 2 3 9 , 1 2 9 9 1 →= 2 2 9 1 1 9 , 9 6 2 1 5 →= 2 2 9 1 6 5 , 3 2 5 1 5 →= 5 5 2 2 3 1 , 6 1 3 3 5 →= 6 2 1 3 3 5 , 1 6 2 9 5 →= 1 9 6 5 5 2 , 6 1 5 9 5 →= 6 5 9 2 1 5 , 6 3 1 2 9 →= 5 1 6 9 2 3 , 1 3 5 2 9 →= 2 2 1 9 5 3 , 3 1 3 6 9 →= 3 2 3 6 1 9 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 10: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 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 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 1 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 1 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 0 ⎟ ⎜ 0 1 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 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 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 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 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 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 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 0 0 0 0 0 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 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 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 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, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 111-rule system { 0 1 1 ⟶ 0 2 1 1 , 0 3 1 ⟶ 4 1 1 , 4 5 1 ⟶ 0 5 2 3 , 4 5 1 ⟶ 4 , 4 6 1 ⟶ 4 1 2 2 2 6 , 7 1 5 ⟶ 7 5 2 1 , 7 1 5 ⟶ 0 , 0 3 5 ⟶ 4 1 , 0 3 5 ⟶ 4 1 5 , 0 3 5 ⟶ 4 2 1 , 0 3 5 ⟶ 7 2 3 2 1 5 , 0 3 5 ⟶ 4 2 1 5 , 7 3 5 ⟶ 7 5 5 2 3 2 , 7 3 5 ⟶ 4 2 , 7 3 5 ⟶ 4 2 6 , 8 3 5 ⟶ 4 9 5 2 , 8 3 5 ⟶ 4 9 6 2 5 , 8 3 5 ⟶ 4 9 5 5 2 2 , 8 3 5 ⟶ 4 6 9 6 5 2 , 0 9 5 ⟶ 8 1 2 5 , 4 1 9 ⟶ 8 3 1 2 2 , 7 3 9 ⟶ 7 5 2 2 3 9 , 4 5 9 ⟶ 4 2 9 5 2 2 , 0 9 9 ⟶ 0 9 2 2 9 , 0 9 9 ⟶ 8 2 2 9 , 7 3 1 1 ⟶ 4 1 6 1 , 4 2 5 1 ⟶ 4 1 5 2 2 , 7 3 5 1 ⟶ 7 6 1 5 2 3 , 7 3 5 1 ⟶ 7 1 5 2 3 , 7 3 5 1 ⟶ 0 5 2 3 , 7 3 5 1 ⟶ 4 , 4 5 6 1 ⟶ 0 6 5 3 , 4 5 6 1 ⟶ 7 5 3 , 4 5 6 1 ⟶ 4 , 7 3 1 5 ⟶ 4 6 1 5 , 7 9 1 5 ⟶ 8 1 6 5 , 0 1 2 5 ⟶ 0 5 2 2 1 , 0 1 2 5 ⟶ 0 , 0 6 3 5 ⟶ 4 6 1 2 5 , 8 6 3 5 ⟶ 8 2 3 6 2 5 , 7 9 3 5 ⟶ 4 9 5 6 5 , 4 5 9 5 ⟶ 8 3 1 5 5 , 4 6 1 6 ⟶ 0 5 2 3 6 6 , 4 6 1 6 ⟶ 4 6 6 , 0 3 5 6 ⟶ 7 5 2 3 1 , 0 3 5 6 ⟶ 4 1 , 0 1 9 9 ⟶ 0 9 9 1 2 2 , 0 1 9 9 ⟶ 8 9 1 2 2 , 4 3 9 9 ⟶ 4 3 2 2 9 9 , 7 3 2 1 1 ⟶ 7 2 2 3 1 1 , 7 3 2 1 1 ⟶ 4 1 1 , 4 2 5 9 1 ⟶ 0 2 5 2 3 9 , 4 2 5 9 1 ⟶ 4 9 , 0 2 9 9 1 ⟶ 8 1 1 9 , 8 6 2 1 5 ⟶ 8 1 6 5 , 4 2 5 1 5 ⟶ 4 1 , 0 6 2 9 5 ⟶ 0 9 6 5 5 2 , 0 6 2 9 5 ⟶ 8 6 5 5 2 , 7 1 5 9 5 ⟶ 7 5 9 2 1 5 , 7 1 5 9 5 ⟶ 8 2 1 5 , 0 3 5 2 9 ⟶ 0 9 5 3 , 0 3 5 2 9 ⟶ 8 5 3 , 4 1 3 6 9 ⟶ 4 2 3 6 1 9 , 1 1 1 →= 2 1 2 1 1 , 1 3 1 →= 2 2 3 1 1 , 3 5 1 →= 1 5 2 3 , 3 6 1 →= 3 1 2 2 2 6 , 6 1 5 →= 2 6 5 2 1 , 1 3 5 →= 5 2 3 1 , 1 3 5 →= 2 3 1 5 , 1 3 5 →= 5 2 3 2 1 , 1 3 5 →= 6 2 3 2 1 5 , 6 3 5 →= 6 5 5 2 3 2 , 6 3 5 →= 5 2 2 3 2 6 , 9 3 5 →= 3 9 5 2 , 9 3 5 →= 3 9 6 2 5 , 9 3 5 →= 3 9 5 5 2 2 , 9 3 5 →= 3 6 9 6 5 2 , 1 9 5 →= 2 2 9 1 2 5 , 3 1 9 →= 9 3 1 2 2 , 6 1 9 →= 2 2 6 9 1 , 6 3 9 →= 6 5 2 2 3 9 , 3 5 9 →= 3 2 9 5 2 2 , 1 9 9 →= 1 9 2 2 9 , 6 3 1 1 →= 2 2 3 1 6 1 , 3 2 5 1 →= 3 1 5 2 2 , 6 3 5 1 →= 6 6 1 5 2 3 , 3 5 6 1 →= 2 1 6 5 3 , 6 3 1 5 →= 2 2 3 6 1 5 , 6 9 1 5 →= 5 2 9 1 6 5 , 1 1 2 5 →= 2 1 5 2 2 1 , 1 6 3 5 →= 3 6 1 2 5 , 9 6 3 5 →= 9 2 3 6 2 5 , 6 9 3 5 →= 3 9 5 6 5 , 3 5 9 5 →= 9 3 1 5 5 , 3 6 1 6 →= 1 5 2 3 6 6 , 1 3 5 6 →= 6 5 2 3 1 , 1 2 6 9 →= 2 1 6 2 2 9 , 1 1 9 9 →= 1 9 9 1 2 2 , 3 3 9 9 →= 3 3 2 2 9 9 , 6 3 2 1 1 →= 6 2 2 3 1 1 , 3 2 5 9 1 →= 1 2 5 2 3 9 , 1 2 9 9 1 →= 2 2 9 1 1 9 , 9 6 2 1 5 →= 2 2 9 1 6 5 , 3 2 5 1 5 →= 5 5 2 2 3 1 , 6 1 3 3 5 →= 6 2 1 3 3 5 , 1 6 2 9 5 →= 1 9 6 5 5 2 , 6 1 5 9 5 →= 6 5 9 2 1 5 , 6 3 1 2 9 →= 5 1 6 9 2 3 , 1 3 5 2 9 →= 2 2 1 9 5 3 , 3 1 3 6 9 →= 3 2 3 6 1 9 } 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 1 ⎟ ⎜ 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 1 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 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 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, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 110-rule system { 0 1 1 ⟶ 0 2 1 1 , 0 3 1 ⟶ 4 1 1 , 4 5 1 ⟶ 0 5 2 3 , 4 5 1 ⟶ 4 , 4 6 1 ⟶ 4 1 2 2 2 6 , 7 1 5 ⟶ 7 5 2 1 , 7 1 5 ⟶ 0 , 0 3 5 ⟶ 4 1 , 0 3 5 ⟶ 4 1 5 , 0 3 5 ⟶ 4 2 1 , 0 3 5 ⟶ 7 2 3 2 1 5 , 0 3 5 ⟶ 4 2 1 5 , 7 3 5 ⟶ 7 5 5 2 3 2 , 7 3 5 ⟶ 4 2 , 7 3 5 ⟶ 4 2 6 , 8 3 5 ⟶ 4 9 5 2 , 8 3 5 ⟶ 4 9 6 2 5 , 8 3 5 ⟶ 4 9 5 5 2 2 , 8 3 5 ⟶ 4 6 9 6 5 2 , 0 9 5 ⟶ 8 1 2 5 , 4 1 9 ⟶ 8 3 1 2 2 , 7 3 9 ⟶ 7 5 2 2 3 9 , 4 5 9 ⟶ 4 2 9 5 2 2 , 0 9 9 ⟶ 0 9 2 2 9 , 0 9 9 ⟶ 8 2 2 9 , 4 2 5 1 ⟶ 4 1 5 2 2 , 7 3 5 1 ⟶ 7 6 1 5 2 3 , 7 3 5 1 ⟶ 7 1 5 2 3 , 7 3 5 1 ⟶ 0 5 2 3 , 7 3 5 1 ⟶ 4 , 4 5 6 1 ⟶ 0 6 5 3 , 4 5 6 1 ⟶ 7 5 3 , 4 5 6 1 ⟶ 4 , 7 3 1 5 ⟶ 4 6 1 5 , 7 9 1 5 ⟶ 8 1 6 5 , 0 1 2 5 ⟶ 0 5 2 2 1 , 0 1 2 5 ⟶ 0 , 0 6 3 5 ⟶ 4 6 1 2 5 , 8 6 3 5 ⟶ 8 2 3 6 2 5 , 7 9 3 5 ⟶ 4 9 5 6 5 , 4 5 9 5 ⟶ 8 3 1 5 5 , 4 6 1 6 ⟶ 0 5 2 3 6 6 , 4 6 1 6 ⟶ 4 6 6 , 0 3 5 6 ⟶ 7 5 2 3 1 , 0 3 5 6 ⟶ 4 1 , 0 1 9 9 ⟶ 0 9 9 1 2 2 , 0 1 9 9 ⟶ 8 9 1 2 2 , 4 3 9 9 ⟶ 4 3 2 2 9 9 , 7 3 2 1 1 ⟶ 7 2 2 3 1 1 , 7 3 2 1 1 ⟶ 4 1 1 , 4 2 5 9 1 ⟶ 0 2 5 2 3 9 , 4 2 5 9 1 ⟶ 4 9 , 0 2 9 9 1 ⟶ 8 1 1 9 , 8 6 2 1 5 ⟶ 8 1 6 5 , 4 2 5 1 5 ⟶ 4 1 , 0 6 2 9 5 ⟶ 0 9 6 5 5 2 , 0 6 2 9 5 ⟶ 8 6 5 5 2 , 7 1 5 9 5 ⟶ 7 5 9 2 1 5 , 7 1 5 9 5 ⟶ 8 2 1 5 , 0 3 5 2 9 ⟶ 0 9 5 3 , 0 3 5 2 9 ⟶ 8 5 3 , 4 1 3 6 9 ⟶ 4 2 3 6 1 9 , 1 1 1 →= 2 1 2 1 1 , 1 3 1 →= 2 2 3 1 1 , 3 5 1 →= 1 5 2 3 , 3 6 1 →= 3 1 2 2 2 6 , 6 1 5 →= 2 6 5 2 1 , 1 3 5 →= 5 2 3 1 , 1 3 5 →= 2 3 1 5 , 1 3 5 →= 5 2 3 2 1 , 1 3 5 →= 6 2 3 2 1 5 , 6 3 5 →= 6 5 5 2 3 2 , 6 3 5 →= 5 2 2 3 2 6 , 9 3 5 →= 3 9 5 2 , 9 3 5 →= 3 9 6 2 5 , 9 3 5 →= 3 9 5 5 2 2 , 9 3 5 →= 3 6 9 6 5 2 , 1 9 5 →= 2 2 9 1 2 5 , 3 1 9 →= 9 3 1 2 2 , 6 1 9 →= 2 2 6 9 1 , 6 3 9 →= 6 5 2 2 3 9 , 3 5 9 →= 3 2 9 5 2 2 , 1 9 9 →= 1 9 2 2 9 , 6 3 1 1 →= 2 2 3 1 6 1 , 3 2 5 1 →= 3 1 5 2 2 , 6 3 5 1 →= 6 6 1 5 2 3 , 3 5 6 1 →= 2 1 6 5 3 , 6 3 1 5 →= 2 2 3 6 1 5 , 6 9 1 5 →= 5 2 9 1 6 5 , 1 1 2 5 →= 2 1 5 2 2 1 , 1 6 3 5 →= 3 6 1 2 5 , 9 6 3 5 →= 9 2 3 6 2 5 , 6 9 3 5 →= 3 9 5 6 5 , 3 5 9 5 →= 9 3 1 5 5 , 3 6 1 6 →= 1 5 2 3 6 6 , 1 3 5 6 →= 6 5 2 3 1 , 1 2 6 9 →= 2 1 6 2 2 9 , 1 1 9 9 →= 1 9 9 1 2 2 , 3 3 9 9 →= 3 3 2 2 9 9 , 6 3 2 1 1 →= 6 2 2 3 1 1 , 3 2 5 9 1 →= 1 2 5 2 3 9 , 1 2 9 9 1 →= 2 2 9 1 1 9 , 9 6 2 1 5 →= 2 2 9 1 6 5 , 3 2 5 1 5 →= 5 5 2 2 3 1 , 6 1 3 3 5 →= 6 2 1 3 3 5 , 1 6 2 9 5 →= 1 9 6 5 5 2 , 6 1 5 9 5 →= 6 5 9 2 1 5 , 6 3 1 2 9 →= 5 1 6 9 2 3 , 1 3 5 2 9 →= 2 2 1 9 5 3 , 3 1 3 6 9 →= 3 2 3 6 1 9 } 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 1 ⎟ ⎜ 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 1 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 1 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 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 108-rule system { 0 1 1 ⟶ 0 2 1 1 , 0 3 1 ⟶ 4 1 1 , 4 5 1 ⟶ 0 5 2 3 , 4 5 1 ⟶ 4 , 4 6 1 ⟶ 4 1 2 2 2 6 , 7 1 5 ⟶ 7 5 2 1 , 7 1 5 ⟶ 0 , 0 3 5 ⟶ 4 1 , 0 3 5 ⟶ 4 1 5 , 0 3 5 ⟶ 4 2 1 , 0 3 5 ⟶ 7 2 3 2 1 5 , 0 3 5 ⟶ 4 2 1 5 , 7 3 5 ⟶ 7 5 5 2 3 2 , 7 3 5 ⟶ 4 2 , 7 3 5 ⟶ 4 2 6 , 8 3 5 ⟶ 4 9 5 2 , 8 3 5 ⟶ 4 9 6 2 5 , 8 3 5 ⟶ 4 9 5 5 2 2 , 8 3 5 ⟶ 4 6 9 6 5 2 , 0 9 5 ⟶ 8 1 2 5 , 4 1 9 ⟶ 8 3 1 2 2 , 7 3 9 ⟶ 7 5 2 2 3 9 , 4 5 9 ⟶ 4 2 9 5 2 2 , 0 9 9 ⟶ 0 9 2 2 9 , 0 9 9 ⟶ 8 2 2 9 , 4 2 5 1 ⟶ 4 1 5 2 2 , 7 3 5 1 ⟶ 7 6 1 5 2 3 , 7 3 5 1 ⟶ 7 1 5 2 3 , 7 3 5 1 ⟶ 0 5 2 3 , 7 3 5 1 ⟶ 4 , 4 5 6 1 ⟶ 0 6 5 3 , 4 5 6 1 ⟶ 7 5 3 , 4 5 6 1 ⟶ 4 , 7 3 1 5 ⟶ 4 6 1 5 , 7 9 1 5 ⟶ 8 1 6 5 , 0 1 2 5 ⟶ 0 5 2 2 1 , 0 1 2 5 ⟶ 0 , 0 6 3 5 ⟶ 4 6 1 2 5 , 8 6 3 5 ⟶ 8 2 3 6 2 5 , 7 9 3 5 ⟶ 4 9 5 6 5 , 4 5 9 5 ⟶ 8 3 1 5 5 , 4 6 1 6 ⟶ 0 5 2 3 6 6 , 4 6 1 6 ⟶ 4 6 6 , 0 3 5 6 ⟶ 7 5 2 3 1 , 0 3 5 6 ⟶ 4 1 , 0 1 9 9 ⟶ 0 9 9 1 2 2 , 0 1 9 9 ⟶ 8 9 1 2 2 , 4 3 9 9 ⟶ 4 3 2 2 9 9 , 4 2 5 9 1 ⟶ 0 2 5 2 3 9 , 4 2 5 9 1 ⟶ 4 9 , 0 2 9 9 1 ⟶ 8 1 1 9 , 8 6 2 1 5 ⟶ 8 1 6 5 , 4 2 5 1 5 ⟶ 4 1 , 0 6 2 9 5 ⟶ 0 9 6 5 5 2 , 0 6 2 9 5 ⟶ 8 6 5 5 2 , 7 1 5 9 5 ⟶ 7 5 9 2 1 5 , 7 1 5 9 5 ⟶ 8 2 1 5 , 0 3 5 2 9 ⟶ 0 9 5 3 , 0 3 5 2 9 ⟶ 8 5 3 , 4 1 3 6 9 ⟶ 4 2 3 6 1 9 , 1 1 1 →= 2 1 2 1 1 , 1 3 1 →= 2 2 3 1 1 , 3 5 1 →= 1 5 2 3 , 3 6 1 →= 3 1 2 2 2 6 , 6 1 5 →= 2 6 5 2 1 , 1 3 5 →= 5 2 3 1 , 1 3 5 →= 2 3 1 5 , 1 3 5 →= 5 2 3 2 1 , 1 3 5 →= 6 2 3 2 1 5 , 6 3 5 →= 6 5 5 2 3 2 , 6 3 5 →= 5 2 2 3 2 6 , 9 3 5 →= 3 9 5 2 , 9 3 5 →= 3 9 6 2 5 , 9 3 5 →= 3 9 5 5 2 2 , 9 3 5 →= 3 6 9 6 5 2 , 1 9 5 →= 2 2 9 1 2 5 , 3 1 9 →= 9 3 1 2 2 , 6 1 9 →= 2 2 6 9 1 , 6 3 9 →= 6 5 2 2 3 9 , 3 5 9 →= 3 2 9 5 2 2 , 1 9 9 →= 1 9 2 2 9 , 6 3 1 1 →= 2 2 3 1 6 1 , 3 2 5 1 →= 3 1 5 2 2 , 6 3 5 1 →= 6 6 1 5 2 3 , 3 5 6 1 →= 2 1 6 5 3 , 6 3 1 5 →= 2 2 3 6 1 5 , 6 9 1 5 →= 5 2 9 1 6 5 , 1 1 2 5 →= 2 1 5 2 2 1 , 1 6 3 5 →= 3 6 1 2 5 , 9 6 3 5 →= 9 2 3 6 2 5 , 6 9 3 5 →= 3 9 5 6 5 , 3 5 9 5 →= 9 3 1 5 5 , 3 6 1 6 →= 1 5 2 3 6 6 , 1 3 5 6 →= 6 5 2 3 1 , 1 2 6 9 →= 2 1 6 2 2 9 , 1 1 9 9 →= 1 9 9 1 2 2 , 3 3 9 9 →= 3 3 2 2 9 9 , 6 3 2 1 1 →= 6 2 2 3 1 1 , 3 2 5 9 1 →= 1 2 5 2 3 9 , 1 2 9 9 1 →= 2 2 9 1 1 9 , 9 6 2 1 5 →= 2 2 9 1 6 5 , 3 2 5 1 5 →= 5 5 2 2 3 1 , 6 1 3 3 5 →= 6 2 1 3 3 5 , 1 6 2 9 5 →= 1 9 6 5 5 2 , 6 1 5 9 5 →= 6 5 9 2 1 5 , 6 3 1 2 9 →= 5 1 6 9 2 3 , 1 3 5 2 9 →= 2 2 1 9 5 3 , 3 1 3 6 9 →= 3 2 3 6 1 9 } 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 1 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 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 1 1 ⎟ ⎝ ⎠ 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 1 ⎟ ⎜ 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 1 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 101-rule system { 0 1 1 ⟶ 0 2 1 1 , 0 3 1 ⟶ 4 1 1 , 4 5 1 ⟶ 0 5 2 3 , 4 5 1 ⟶ 4 , 4 6 1 ⟶ 4 1 2 2 2 6 , 0 3 5 ⟶ 4 1 , 0 3 5 ⟶ 4 1 5 , 0 3 5 ⟶ 4 2 1 , 0 3 5 ⟶ 7 2 3 2 1 5 , 0 3 5 ⟶ 4 2 1 5 , 7 3 5 ⟶ 7 5 5 2 3 2 , 7 3 5 ⟶ 4 2 , 7 3 5 ⟶ 4 2 6 , 8 3 5 ⟶ 4 9 5 2 , 8 3 5 ⟶ 4 9 6 2 5 , 8 3 5 ⟶ 4 9 5 5 2 2 , 8 3 5 ⟶ 4 6 9 6 5 2 , 0 9 5 ⟶ 8 1 2 5 , 4 1 9 ⟶ 8 3 1 2 2 , 7 3 9 ⟶ 7 5 2 2 3 9 , 4 5 9 ⟶ 4 2 9 5 2 2 , 0 9 9 ⟶ 0 9 2 2 9 , 0 9 9 ⟶ 8 2 2 9 , 4 2 5 1 ⟶ 4 1 5 2 2 , 7 3 5 1 ⟶ 7 1 5 2 3 , 4 5 6 1 ⟶ 0 6 5 3 , 4 5 6 1 ⟶ 7 5 3 , 4 5 6 1 ⟶ 4 , 7 3 1 5 ⟶ 4 6 1 5 , 7 9 1 5 ⟶ 8 1 6 5 , 0 1 2 5 ⟶ 0 5 2 2 1 , 0 1 2 5 ⟶ 0 , 0 6 3 5 ⟶ 4 6 1 2 5 , 8 6 3 5 ⟶ 8 2 3 6 2 5 , 7 9 3 5 ⟶ 4 9 5 6 5 , 4 5 9 5 ⟶ 8 3 1 5 5 , 4 6 1 6 ⟶ 0 5 2 3 6 6 , 4 6 1 6 ⟶ 4 6 6 , 0 3 5 6 ⟶ 7 5 2 3 1 , 0 3 5 6 ⟶ 4 1 , 0 1 9 9 ⟶ 0 9 9 1 2 2 , 0 1 9 9 ⟶ 8 9 1 2 2 , 4 3 9 9 ⟶ 4 3 2 2 9 9 , 4 2 5 9 1 ⟶ 0 2 5 2 3 9 , 4 2 5 9 1 ⟶ 4 9 , 0 2 9 9 1 ⟶ 8 1 1 9 , 8 6 2 1 5 ⟶ 8 1 6 5 , 4 2 5 1 5 ⟶ 4 1 , 0 6 2 9 5 ⟶ 0 9 6 5 5 2 , 0 6 2 9 5 ⟶ 8 6 5 5 2 , 0 3 5 2 9 ⟶ 0 9 5 3 , 0 3 5 2 9 ⟶ 8 5 3 , 4 1 3 6 9 ⟶ 4 2 3 6 1 9 , 1 1 1 →= 2 1 2 1 1 , 1 3 1 →= 2 2 3 1 1 , 3 5 1 →= 1 5 2 3 , 3 6 1 →= 3 1 2 2 2 6 , 6 1 5 →= 2 6 5 2 1 , 1 3 5 →= 5 2 3 1 , 1 3 5 →= 2 3 1 5 , 1 3 5 →= 5 2 3 2 1 , 1 3 5 →= 6 2 3 2 1 5 , 6 3 5 →= 6 5 5 2 3 2 , 6 3 5 →= 5 2 2 3 2 6 , 9 3 5 →= 3 9 5 2 , 9 3 5 →= 3 9 6 2 5 , 9 3 5 →= 3 9 5 5 2 2 , 9 3 5 →= 3 6 9 6 5 2 , 1 9 5 →= 2 2 9 1 2 5 , 3 1 9 →= 9 3 1 2 2 , 6 1 9 →= 2 2 6 9 1 , 6 3 9 →= 6 5 2 2 3 9 , 3 5 9 →= 3 2 9 5 2 2 , 1 9 9 →= 1 9 2 2 9 , 6 3 1 1 →= 2 2 3 1 6 1 , 3 2 5 1 →= 3 1 5 2 2 , 6 3 5 1 →= 6 6 1 5 2 3 , 3 5 6 1 →= 2 1 6 5 3 , 6 3 1 5 →= 2 2 3 6 1 5 , 6 9 1 5 →= 5 2 9 1 6 5 , 1 1 2 5 →= 2 1 5 2 2 1 , 1 6 3 5 →= 3 6 1 2 5 , 9 6 3 5 →= 9 2 3 6 2 5 , 6 9 3 5 →= 3 9 5 6 5 , 3 5 9 5 →= 9 3 1 5 5 , 3 6 1 6 →= 1 5 2 3 6 6 , 1 3 5 6 →= 6 5 2 3 1 , 1 2 6 9 →= 2 1 6 2 2 9 , 1 1 9 9 →= 1 9 9 1 2 2 , 3 3 9 9 →= 3 3 2 2 9 9 , 6 3 2 1 1 →= 6 2 2 3 1 1 , 3 2 5 9 1 →= 1 2 5 2 3 9 , 1 2 9 9 1 →= 2 2 9 1 1 9 , 9 6 2 1 5 →= 2 2 9 1 6 5 , 3 2 5 1 5 →= 5 5 2 2 3 1 , 6 1 3 3 5 →= 6 2 1 3 3 5 , 1 6 2 9 5 →= 1 9 6 5 5 2 , 6 1 5 9 5 →= 6 5 9 2 1 5 , 6 3 1 2 9 →= 5 1 6 9 2 3 , 1 3 5 2 9 →= 2 2 1 9 5 3 , 3 1 3 6 9 →= 3 2 3 6 1 9 } 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 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 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 1 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 1 ⎟ ⎜ 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 1 ⎟ ⎜ 0 0 0 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 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, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 100-rule system { 0 1 1 ⟶ 0 2 1 1 , 0 3 1 ⟶ 4 1 1 , 4 5 1 ⟶ 0 5 2 3 , 4 5 1 ⟶ 4 , 4 6 1 ⟶ 4 1 2 2 2 6 , 0 3 5 ⟶ 4 1 , 0 3 5 ⟶ 4 1 5 , 0 3 5 ⟶ 4 2 1 , 0 3 5 ⟶ 7 2 3 2 1 5 , 0 3 5 ⟶ 4 2 1 5 , 7 3 5 ⟶ 7 5 5 2 3 2 , 7 3 5 ⟶ 4 2 , 7 3 5 ⟶ 4 2 6 , 8 3 5 ⟶ 4 9 5 2 , 8 3 5 ⟶ 4 9 6 2 5 , 8 3 5 ⟶ 4 9 5 5 2 2 , 8 3 5 ⟶ 4 6 9 6 5 2 , 0 9 5 ⟶ 8 1 2 5 , 4 1 9 ⟶ 8 3 1 2 2 , 7 3 9 ⟶ 7 5 2 2 3 9 , 4 5 9 ⟶ 4 2 9 5 2 2 , 0 9 9 ⟶ 0 9 2 2 9 , 0 9 9 ⟶ 8 2 2 9 , 4 2 5 1 ⟶ 4 1 5 2 2 , 4 5 6 1 ⟶ 0 6 5 3 , 4 5 6 1 ⟶ 7 5 3 , 4 5 6 1 ⟶ 4 , 7 3 1 5 ⟶ 4 6 1 5 , 7 9 1 5 ⟶ 8 1 6 5 , 0 1 2 5 ⟶ 0 5 2 2 1 , 0 1 2 5 ⟶ 0 , 0 6 3 5 ⟶ 4 6 1 2 5 , 8 6 3 5 ⟶ 8 2 3 6 2 5 , 7 9 3 5 ⟶ 4 9 5 6 5 , 4 5 9 5 ⟶ 8 3 1 5 5 , 4 6 1 6 ⟶ 0 5 2 3 6 6 , 4 6 1 6 ⟶ 4 6 6 , 0 3 5 6 ⟶ 7 5 2 3 1 , 0 3 5 6 ⟶ 4 1 , 0 1 9 9 ⟶ 0 9 9 1 2 2 , 0 1 9 9 ⟶ 8 9 1 2 2 , 4 3 9 9 ⟶ 4 3 2 2 9 9 , 4 2 5 9 1 ⟶ 0 2 5 2 3 9 , 4 2 5 9 1 ⟶ 4 9 , 0 2 9 9 1 ⟶ 8 1 1 9 , 8 6 2 1 5 ⟶ 8 1 6 5 , 4 2 5 1 5 ⟶ 4 1 , 0 6 2 9 5 ⟶ 0 9 6 5 5 2 , 0 6 2 9 5 ⟶ 8 6 5 5 2 , 0 3 5 2 9 ⟶ 0 9 5 3 , 0 3 5 2 9 ⟶ 8 5 3 , 4 1 3 6 9 ⟶ 4 2 3 6 1 9 , 1 1 1 →= 2 1 2 1 1 , 1 3 1 →= 2 2 3 1 1 , 3 5 1 →= 1 5 2 3 , 3 6 1 →= 3 1 2 2 2 6 , 6 1 5 →= 2 6 5 2 1 , 1 3 5 →= 5 2 3 1 , 1 3 5 →= 2 3 1 5 , 1 3 5 →= 5 2 3 2 1 , 1 3 5 →= 6 2 3 2 1 5 , 6 3 5 →= 6 5 5 2 3 2 , 6 3 5 →= 5 2 2 3 2 6 , 9 3 5 →= 3 9 5 2 , 9 3 5 →= 3 9 6 2 5 , 9 3 5 →= 3 9 5 5 2 2 , 9 3 5 →= 3 6 9 6 5 2 , 1 9 5 →= 2 2 9 1 2 5 , 3 1 9 →= 9 3 1 2 2 , 6 1 9 →= 2 2 6 9 1 , 6 3 9 →= 6 5 2 2 3 9 , 3 5 9 →= 3 2 9 5 2 2 , 1 9 9 →= 1 9 2 2 9 , 6 3 1 1 →= 2 2 3 1 6 1 , 3 2 5 1 →= 3 1 5 2 2 , 6 3 5 1 →= 6 6 1 5 2 3 , 3 5 6 1 →= 2 1 6 5 3 , 6 3 1 5 →= 2 2 3 6 1 5 , 6 9 1 5 →= 5 2 9 1 6 5 , 1 1 2 5 →= 2 1 5 2 2 1 , 1 6 3 5 →= 3 6 1 2 5 , 9 6 3 5 →= 9 2 3 6 2 5 , 6 9 3 5 →= 3 9 5 6 5 , 3 5 9 5 →= 9 3 1 5 5 , 3 6 1 6 →= 1 5 2 3 6 6 , 1 3 5 6 →= 6 5 2 3 1 , 1 2 6 9 →= 2 1 6 2 2 9 , 1 1 9 9 →= 1 9 9 1 2 2 , 3 3 9 9 →= 3 3 2 2 9 9 , 6 3 2 1 1 →= 6 2 2 3 1 1 , 3 2 5 9 1 →= 1 2 5 2 3 9 , 1 2 9 9 1 →= 2 2 9 1 1 9 , 9 6 2 1 5 →= 2 2 9 1 6 5 , 3 2 5 1 5 →= 5 5 2 2 3 1 , 6 1 3 3 5 →= 6 2 1 3 3 5 , 1 6 2 9 5 →= 1 9 6 5 5 2 , 6 1 5 9 5 →= 6 5 9 2 1 5 , 6 3 1 2 9 →= 5 1 6 9 2 3 , 1 3 5 2 9 →= 2 2 1 9 5 3 , 3 1 3 6 9 →= 3 2 3 6 1 9 } 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 0 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 1 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎜ 0 0 0 1 ⎟ ⎝ ⎠ 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 1 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 1 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 1 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 1 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, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 97-rule system { 0 1 1 ⟶ 0 2 1 1 , 0 3 1 ⟶ 4 1 1 , 4 5 1 ⟶ 0 5 2 3 , 4 5 1 ⟶ 4 , 4 6 1 ⟶ 4 1 2 2 2 6 , 0 3 5 ⟶ 4 1 , 0 3 5 ⟶ 4 1 5 , 0 3 5 ⟶ 4 2 1 , 0 3 5 ⟶ 7 2 3 2 1 5 , 0 3 5 ⟶ 4 2 1 5 , 7 3 5 ⟶ 7 5 5 2 3 2 , 7 3 5 ⟶ 4 2 , 7 3 5 ⟶ 4 2 6 , 8 3 5 ⟶ 4 9 5 2 , 8 3 5 ⟶ 4 9 6 2 5 , 8 3 5 ⟶ 4 9 5 5 2 2 , 8 3 5 ⟶ 4 6 9 6 5 2 , 0 9 5 ⟶ 8 1 2 5 , 4 1 9 ⟶ 8 3 1 2 2 , 4 5 9 ⟶ 4 2 9 5 2 2 , 0 9 9 ⟶ 0 9 2 2 9 , 0 9 9 ⟶ 8 2 2 9 , 4 2 5 1 ⟶ 4 1 5 2 2 , 4 5 6 1 ⟶ 0 6 5 3 , 4 5 6 1 ⟶ 7 5 3 , 4 5 6 1 ⟶ 4 , 7 3 1 5 ⟶ 4 6 1 5 , 0 1 2 5 ⟶ 0 5 2 2 1 , 0 1 2 5 ⟶ 0 , 0 6 3 5 ⟶ 4 6 1 2 5 , 8 6 3 5 ⟶ 8 2 3 6 2 5 , 4 5 9 5 ⟶ 8 3 1 5 5 , 4 6 1 6 ⟶ 0 5 2 3 6 6 , 4 6 1 6 ⟶ 4 6 6 , 0 3 5 6 ⟶ 7 5 2 3 1 , 0 3 5 6 ⟶ 4 1 , 0 1 9 9 ⟶ 0 9 9 1 2 2 , 0 1 9 9 ⟶ 8 9 1 2 2 , 4 3 9 9 ⟶ 4 3 2 2 9 9 , 4 2 5 9 1 ⟶ 0 2 5 2 3 9 , 4 2 5 9 1 ⟶ 4 9 , 0 2 9 9 1 ⟶ 8 1 1 9 , 8 6 2 1 5 ⟶ 8 1 6 5 , 4 2 5 1 5 ⟶ 4 1 , 0 6 2 9 5 ⟶ 0 9 6 5 5 2 , 0 6 2 9 5 ⟶ 8 6 5 5 2 , 0 3 5 2 9 ⟶ 0 9 5 3 , 0 3 5 2 9 ⟶ 8 5 3 , 4 1 3 6 9 ⟶ 4 2 3 6 1 9 , 1 1 1 →= 2 1 2 1 1 , 1 3 1 →= 2 2 3 1 1 , 3 5 1 →= 1 5 2 3 , 3 6 1 →= 3 1 2 2 2 6 , 6 1 5 →= 2 6 5 2 1 , 1 3 5 →= 5 2 3 1 , 1 3 5 →= 2 3 1 5 , 1 3 5 →= 5 2 3 2 1 , 1 3 5 →= 6 2 3 2 1 5 , 6 3 5 →= 6 5 5 2 3 2 , 6 3 5 →= 5 2 2 3 2 6 , 9 3 5 →= 3 9 5 2 , 9 3 5 →= 3 9 6 2 5 , 9 3 5 →= 3 9 5 5 2 2 , 9 3 5 →= 3 6 9 6 5 2 , 1 9 5 →= 2 2 9 1 2 5 , 3 1 9 →= 9 3 1 2 2 , 6 1 9 →= 2 2 6 9 1 , 6 3 9 →= 6 5 2 2 3 9 , 3 5 9 →= 3 2 9 5 2 2 , 1 9 9 →= 1 9 2 2 9 , 6 3 1 1 →= 2 2 3 1 6 1 , 3 2 5 1 →= 3 1 5 2 2 , 6 3 5 1 →= 6 6 1 5 2 3 , 3 5 6 1 →= 2 1 6 5 3 , 6 3 1 5 →= 2 2 3 6 1 5 , 6 9 1 5 →= 5 2 9 1 6 5 , 1 1 2 5 →= 2 1 5 2 2 1 , 1 6 3 5 →= 3 6 1 2 5 , 9 6 3 5 →= 9 2 3 6 2 5 , 6 9 3 5 →= 3 9 5 6 5 , 3 5 9 5 →= 9 3 1 5 5 , 3 6 1 6 →= 1 5 2 3 6 6 , 1 3 5 6 →= 6 5 2 3 1 , 1 2 6 9 →= 2 1 6 2 2 9 , 1 1 9 9 →= 1 9 9 1 2 2 , 3 3 9 9 →= 3 3 2 2 9 9 , 6 3 2 1 1 →= 6 2 2 3 1 1 , 3 2 5 9 1 →= 1 2 5 2 3 9 , 1 2 9 9 1 →= 2 2 9 1 1 9 , 9 6 2 1 5 →= 2 2 9 1 6 5 , 3 2 5 1 5 →= 5 5 2 2 3 1 , 6 1 3 3 5 →= 6 2 1 3 3 5 , 1 6 2 9 5 →= 1 9 6 5 5 2 , 6 1 5 9 5 →= 6 5 9 2 1 5 , 6 3 1 2 9 →= 5 1 6 9 2 3 , 1 3 5 2 9 →= 2 2 1 9 5 3 , 3 1 3 6 9 →= 3 2 3 6 1 9 } 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 1 ⎟ ⎜ 0 0 1 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 1 ⎟ ⎜ 0 1 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 1 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 1 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 1 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, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 94-rule system { 0 1 1 ⟶ 0 2 1 1 , 0 3 1 ⟶ 4 1 1 , 4 5 1 ⟶ 0 5 2 3 , 4 5 1 ⟶ 4 , 4 6 1 ⟶ 4 1 2 2 2 6 , 0 3 5 ⟶ 4 1 , 0 3 5 ⟶ 4 1 5 , 0 3 5 ⟶ 4 2 1 , 0 3 5 ⟶ 7 2 3 2 1 5 , 0 3 5 ⟶ 4 2 1 5 , 8 3 5 ⟶ 4 9 5 2 , 8 3 5 ⟶ 4 9 6 2 5 , 8 3 5 ⟶ 4 9 5 5 2 2 , 8 3 5 ⟶ 4 6 9 6 5 2 , 0 9 5 ⟶ 8 1 2 5 , 4 1 9 ⟶ 8 3 1 2 2 , 4 5 9 ⟶ 4 2 9 5 2 2 , 0 9 9 ⟶ 0 9 2 2 9 , 0 9 9 ⟶ 8 2 2 9 , 4 2 5 1 ⟶ 4 1 5 2 2 , 4 5 6 1 ⟶ 0 6 5 3 , 4 5 6 1 ⟶ 7 5 3 , 4 5 6 1 ⟶ 4 , 7 3 1 5 ⟶ 4 6 1 5 , 0 1 2 5 ⟶ 0 5 2 2 1 , 0 1 2 5 ⟶ 0 , 0 6 3 5 ⟶ 4 6 1 2 5 , 8 6 3 5 ⟶ 8 2 3 6 2 5 , 4 5 9 5 ⟶ 8 3 1 5 5 , 4 6 1 6 ⟶ 0 5 2 3 6 6 , 4 6 1 6 ⟶ 4 6 6 , 0 3 5 6 ⟶ 7 5 2 3 1 , 0 3 5 6 ⟶ 4 1 , 0 1 9 9 ⟶ 0 9 9 1 2 2 , 0 1 9 9 ⟶ 8 9 1 2 2 , 4 3 9 9 ⟶ 4 3 2 2 9 9 , 4 2 5 9 1 ⟶ 0 2 5 2 3 9 , 4 2 5 9 1 ⟶ 4 9 , 0 2 9 9 1 ⟶ 8 1 1 9 , 8 6 2 1 5 ⟶ 8 1 6 5 , 4 2 5 1 5 ⟶ 4 1 , 0 6 2 9 5 ⟶ 0 9 6 5 5 2 , 0 6 2 9 5 ⟶ 8 6 5 5 2 , 0 3 5 2 9 ⟶ 0 9 5 3 , 0 3 5 2 9 ⟶ 8 5 3 , 4 1 3 6 9 ⟶ 4 2 3 6 1 9 , 1 1 1 →= 2 1 2 1 1 , 1 3 1 →= 2 2 3 1 1 , 3 5 1 →= 1 5 2 3 , 3 6 1 →= 3 1 2 2 2 6 , 6 1 5 →= 2 6 5 2 1 , 1 3 5 →= 5 2 3 1 , 1 3 5 →= 2 3 1 5 , 1 3 5 →= 5 2 3 2 1 , 1 3 5 →= 6 2 3 2 1 5 , 6 3 5 →= 6 5 5 2 3 2 , 6 3 5 →= 5 2 2 3 2 6 , 9 3 5 →= 3 9 5 2 , 9 3 5 →= 3 9 6 2 5 , 9 3 5 →= 3 9 5 5 2 2 , 9 3 5 →= 3 6 9 6 5 2 , 1 9 5 →= 2 2 9 1 2 5 , 3 1 9 →= 9 3 1 2 2 , 6 1 9 →= 2 2 6 9 1 , 6 3 9 →= 6 5 2 2 3 9 , 3 5 9 →= 3 2 9 5 2 2 , 1 9 9 →= 1 9 2 2 9 , 6 3 1 1 →= 2 2 3 1 6 1 , 3 2 5 1 →= 3 1 5 2 2 , 6 3 5 1 →= 6 6 1 5 2 3 , 3 5 6 1 →= 2 1 6 5 3 , 6 3 1 5 →= 2 2 3 6 1 5 , 6 9 1 5 →= 5 2 9 1 6 5 , 1 1 2 5 →= 2 1 5 2 2 1 , 1 6 3 5 →= 3 6 1 2 5 , 9 6 3 5 →= 9 2 3 6 2 5 , 6 9 3 5 →= 3 9 5 6 5 , 3 5 9 5 →= 9 3 1 5 5 , 3 6 1 6 →= 1 5 2 3 6 6 , 1 3 5 6 →= 6 5 2 3 1 , 1 2 6 9 →= 2 1 6 2 2 9 , 1 1 9 9 →= 1 9 9 1 2 2 , 3 3 9 9 →= 3 3 2 2 9 9 , 6 3 2 1 1 →= 6 2 2 3 1 1 , 3 2 5 9 1 →= 1 2 5 2 3 9 , 1 2 9 9 1 →= 2 2 9 1 1 9 , 9 6 2 1 5 →= 2 2 9 1 6 5 , 3 2 5 1 5 →= 5 5 2 2 3 1 , 6 1 3 3 5 →= 6 2 1 3 3 5 , 1 6 2 9 5 →= 1 9 6 5 5 2 , 6 1 5 9 5 →= 6 5 9 2 1 5 , 6 3 1 2 9 →= 5 1 6 9 2 3 , 1 3 5 2 9 →= 2 2 1 9 5 3 , 3 1 3 6 9 →= 3 2 3 6 1 9 } 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 1 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 ⎟ ⎜ 0 0 1 0 0 1 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎝ ⎠ 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 1 ⎟ ⎜ 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 1 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 1 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 1 1 ⎟ ⎝ ⎠ 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 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 93-rule system { 0 1 1 ⟶ 0 2 1 1 , 0 3 1 ⟶ 4 1 1 , 4 5 1 ⟶ 0 5 2 3 , 4 5 1 ⟶ 4 , 4 6 1 ⟶ 4 1 2 2 2 6 , 0 3 5 ⟶ 4 1 , 0 3 5 ⟶ 4 1 5 , 0 3 5 ⟶ 4 2 1 , 0 3 5 ⟶ 7 2 3 2 1 5 , 0 3 5 ⟶ 4 2 1 5 , 8 3 5 ⟶ 4 9 5 2 , 8 3 5 ⟶ 4 9 6 2 5 , 8 3 5 ⟶ 4 9 5 5 2 2 , 8 3 5 ⟶ 4 6 9 6 5 2 , 0 9 5 ⟶ 8 1 2 5 , 4 1 9 ⟶ 8 3 1 2 2 , 4 5 9 ⟶ 4 2 9 5 2 2 , 0 9 9 ⟶ 0 9 2 2 9 , 0 9 9 ⟶ 8 2 2 9 , 4 2 5 1 ⟶ 4 1 5 2 2 , 4 5 6 1 ⟶ 0 6 5 3 , 4 5 6 1 ⟶ 7 5 3 , 4 5 6 1 ⟶ 4 , 7 3 1 5 ⟶ 4 6 1 5 , 0 1 2 5 ⟶ 0 5 2 2 1 , 0 1 2 5 ⟶ 0 , 0 6 3 5 ⟶ 4 6 1 2 5 , 8 6 3 5 ⟶ 8 2 3 6 2 5 , 4 5 9 5 ⟶ 8 3 1 5 5 , 4 6 1 6 ⟶ 0 5 2 3 6 6 , 4 6 1 6 ⟶ 4 6 6 , 0 3 5 6 ⟶ 7 5 2 3 1 , 0 3 5 6 ⟶ 4 1 , 0 1 9 9 ⟶ 0 9 9 1 2 2 , 0 1 9 9 ⟶ 8 9 1 2 2 , 4 3 9 9 ⟶ 4 3 2 2 9 9 , 4 2 5 9 1 ⟶ 0 2 5 2 3 9 , 4 2 5 9 1 ⟶ 4 9 , 0 2 9 9 1 ⟶ 8 1 1 9 , 8 6 2 1 5 ⟶ 8 1 6 5 , 4 2 5 1 5 ⟶ 4 1 , 0 6 2 9 5 ⟶ 0 9 6 5 5 2 , 0 6 2 9 5 ⟶ 8 6 5 5 2 , 0 3 5 2 9 ⟶ 0 9 5 3 , 0 3 5 2 9 ⟶ 8 5 3 , 4 1 3 6 9 ⟶ 4 2 3 6 1 9 , 1 1 1 →= 2 1 2 1 1 , 1 3 1 →= 2 2 3 1 1 , 3 5 1 →= 1 5 2 3 , 3 6 1 →= 3 1 2 2 2 6 , 6 1 5 →= 2 6 5 2 1 , 1 3 5 →= 5 2 3 1 , 1 3 5 →= 2 3 1 5 , 1 3 5 →= 5 2 3 2 1 , 1 3 5 →= 6 2 3 2 1 5 , 6 3 5 →= 6 5 5 2 3 2 , 6 3 5 →= 5 2 2 3 2 6 , 9 3 5 →= 3 9 5 2 , 9 3 5 →= 3 9 6 2 5 , 9 3 5 →= 3 9 5 5 2 2 , 9 3 5 →= 3 6 9 6 5 2 , 1 9 5 →= 2 2 9 1 2 5 , 3 1 9 →= 9 3 1 2 2 , 6 1 9 →= 2 2 6 9 1 , 6 3 9 →= 6 5 2 2 3 9 , 3 5 9 →= 3 2 9 5 2 2 , 1 9 9 →= 1 9 2 2 9 , 6 3 1 1 →= 2 2 3 1 6 1 , 3 2 5 1 →= 3 1 5 2 2 , 6 3 5 1 →= 6 6 1 5 2 3 , 3 5 6 1 →= 2 1 6 5 3 , 6 3 1 5 →= 2 2 3 6 1 5 , 6 9 1 5 →= 5 2 9 1 6 5 , 1 1 2 5 →= 2 1 5 2 2 1 , 1 6 3 5 →= 3 6 1 2 5 , 9 6 3 5 →= 9 2 3 6 2 5 , 6 9 3 5 →= 3 9 5 6 5 , 3 5 9 5 →= 9 3 1 5 5 , 3 6 1 6 →= 1 5 2 3 6 6 , 1 3 5 6 →= 6 5 2 3 1 , 1 2 6 9 →= 2 1 6 2 2 9 , 1 1 9 9 →= 1 9 9 1 2 2 , 3 3 9 9 →= 3 3 2 2 9 9 , 6 3 2 1 1 →= 6 2 2 3 1 1 , 3 2 5 9 1 →= 1 2 5 2 3 9 , 1 2 9 9 1 →= 2 2 9 1 1 9 , 9 6 2 1 5 →= 2 2 9 1 6 5 , 3 2 5 1 5 →= 5 5 2 2 3 1 , 6 1 3 3 5 →= 6 2 1 3 3 5 , 1 6 2 9 5 →= 1 9 6 5 5 2 , 6 1 5 9 5 →= 6 5 9 2 1 5 , 1 3 5 2 9 →= 2 2 1 9 5 3 , 3 1 3 6 9 →= 3 2 3 6 1 9 } 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 1 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 1 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 1 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 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, 6 ↦ 5, 5 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 86-rule system { 0 1 1 ⟶ 0 2 1 1 , 0 3 1 ⟶ 4 1 1 , 4 5 1 ⟶ 4 1 2 2 2 5 , 0 3 6 ⟶ 4 1 , 0 3 6 ⟶ 4 1 6 , 0 3 6 ⟶ 4 2 1 , 0 3 6 ⟶ 7 2 3 2 1 6 , 0 3 6 ⟶ 4 2 1 6 , 8 3 6 ⟶ 4 9 6 2 , 8 3 6 ⟶ 4 9 5 2 6 , 8 3 6 ⟶ 4 9 6 6 2 2 , 8 3 6 ⟶ 4 5 9 5 6 2 , 0 9 6 ⟶ 8 1 2 6 , 4 1 9 ⟶ 8 3 1 2 2 , 0 9 9 ⟶ 0 9 2 2 9 , 0 9 9 ⟶ 8 2 2 9 , 4 2 6 1 ⟶ 4 1 6 2 2 , 7 3 1 6 ⟶ 4 5 1 6 , 0 1 2 6 ⟶ 0 6 2 2 1 , 0 1 2 6 ⟶ 0 , 0 5 3 6 ⟶ 4 5 1 2 6 , 8 5 3 6 ⟶ 8 2 3 5 2 6 , 4 5 1 5 ⟶ 0 6 2 3 5 5 , 4 5 1 5 ⟶ 4 5 5 , 0 3 6 5 ⟶ 7 6 2 3 1 , 0 3 6 5 ⟶ 4 1 , 0 1 9 9 ⟶ 0 9 9 1 2 2 , 0 1 9 9 ⟶ 8 9 1 2 2 , 4 3 9 9 ⟶ 4 3 2 2 9 9 , 4 2 6 9 1 ⟶ 0 2 6 2 3 9 , 4 2 6 9 1 ⟶ 4 9 , 0 2 9 9 1 ⟶ 8 1 1 9 , 8 5 2 1 6 ⟶ 8 1 5 6 , 4 2 6 1 6 ⟶ 4 1 , 0 5 2 9 6 ⟶ 0 9 5 6 6 2 , 0 5 2 9 6 ⟶ 8 5 6 6 2 , 0 3 6 2 9 ⟶ 0 9 6 3 , 0 3 6 2 9 ⟶ 8 6 3 , 4 1 3 5 9 ⟶ 4 2 3 5 1 9 , 1 1 1 →= 2 1 2 1 1 , 1 3 1 →= 2 2 3 1 1 , 3 6 1 →= 1 6 2 3 , 3 5 1 →= 3 1 2 2 2 5 , 5 1 6 →= 2 5 6 2 1 , 1 3 6 →= 6 2 3 1 , 1 3 6 →= 2 3 1 6 , 1 3 6 →= 6 2 3 2 1 , 1 3 6 →= 5 2 3 2 1 6 , 5 3 6 →= 5 6 6 2 3 2 , 5 3 6 →= 6 2 2 3 2 5 , 9 3 6 →= 3 9 6 2 , 9 3 6 →= 3 9 5 2 6 , 9 3 6 →= 3 9 6 6 2 2 , 9 3 6 →= 3 5 9 5 6 2 , 1 9 6 →= 2 2 9 1 2 6 , 3 1 9 →= 9 3 1 2 2 , 5 1 9 →= 2 2 5 9 1 , 5 3 9 →= 5 6 2 2 3 9 , 3 6 9 →= 3 2 9 6 2 2 , 1 9 9 →= 1 9 2 2 9 , 5 3 1 1 →= 2 2 3 1 5 1 , 3 2 6 1 →= 3 1 6 2 2 , 5 3 6 1 →= 5 5 1 6 2 3 , 3 6 5 1 →= 2 1 5 6 3 , 5 3 1 6 →= 2 2 3 5 1 6 , 5 9 1 6 →= 6 2 9 1 5 6 , 1 1 2 6 →= 2 1 6 2 2 1 , 1 5 3 6 →= 3 5 1 2 6 , 9 5 3 6 →= 9 2 3 5 2 6 , 5 9 3 6 →= 3 9 6 5 6 , 3 6 9 6 →= 9 3 1 6 6 , 3 5 1 5 →= 1 6 2 3 5 5 , 1 3 6 5 →= 5 6 2 3 1 , 1 2 5 9 →= 2 1 5 2 2 9 , 1 1 9 9 →= 1 9 9 1 2 2 , 3 3 9 9 →= 3 3 2 2 9 9 , 5 3 2 1 1 →= 5 2 2 3 1 1 , 3 2 6 9 1 →= 1 2 6 2 3 9 , 1 2 9 9 1 →= 2 2 9 1 1 9 , 9 5 2 1 6 →= 2 2 9 1 5 6 , 3 2 6 1 6 →= 6 6 2 2 3 1 , 5 1 3 3 6 →= 5 2 1 3 3 6 , 1 5 2 9 6 →= 1 9 5 6 6 2 , 5 1 6 9 6 →= 5 6 9 2 1 6 , 1 3 6 2 9 →= 2 2 1 9 6 3 , 3 1 3 5 9 →= 3 2 3 5 1 9 } 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 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 1 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 1 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 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 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 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 85-rule system { 0 1 1 ⟶ 0 2 1 1 , 0 3 1 ⟶ 4 1 1 , 4 5 1 ⟶ 4 1 2 2 2 5 , 0 3 6 ⟶ 4 1 , 0 3 6 ⟶ 4 1 6 , 0 3 6 ⟶ 4 2 1 , 0 3 6 ⟶ 7 2 3 2 1 6 , 0 3 6 ⟶ 4 2 1 6 , 8 3 6 ⟶ 4 9 6 2 , 8 3 6 ⟶ 4 9 5 2 6 , 8 3 6 ⟶ 4 9 6 6 2 2 , 8 3 6 ⟶ 4 5 9 5 6 2 , 0 9 6 ⟶ 8 1 2 6 , 4 1 9 ⟶ 8 3 1 2 2 , 0 9 9 ⟶ 0 9 2 2 9 , 0 9 9 ⟶ 8 2 2 9 , 4 2 6 1 ⟶ 4 1 6 2 2 , 7 3 1 6 ⟶ 4 5 1 6 , 0 1 2 6 ⟶ 0 6 2 2 1 , 0 1 2 6 ⟶ 0 , 0 5 3 6 ⟶ 4 5 1 2 6 , 8 5 3 6 ⟶ 8 2 3 5 2 6 , 4 5 1 5 ⟶ 0 6 2 3 5 5 , 4 5 1 5 ⟶ 4 5 5 , 0 3 6 5 ⟶ 7 6 2 3 1 , 0 3 6 5 ⟶ 4 1 , 0 1 9 9 ⟶ 0 9 9 1 2 2 , 0 1 9 9 ⟶ 8 9 1 2 2 , 4 2 6 9 1 ⟶ 0 2 6 2 3 9 , 4 2 6 9 1 ⟶ 4 9 , 0 2 9 9 1 ⟶ 8 1 1 9 , 8 5 2 1 6 ⟶ 8 1 5 6 , 4 2 6 1 6 ⟶ 4 1 , 0 5 2 9 6 ⟶ 0 9 5 6 6 2 , 0 5 2 9 6 ⟶ 8 5 6 6 2 , 0 3 6 2 9 ⟶ 0 9 6 3 , 0 3 6 2 9 ⟶ 8 6 3 , 4 1 3 5 9 ⟶ 4 2 3 5 1 9 , 1 1 1 →= 2 1 2 1 1 , 1 3 1 →= 2 2 3 1 1 , 3 6 1 →= 1 6 2 3 , 3 5 1 →= 3 1 2 2 2 5 , 5 1 6 →= 2 5 6 2 1 , 1 3 6 →= 6 2 3 1 , 1 3 6 →= 2 3 1 6 , 1 3 6 →= 6 2 3 2 1 , 1 3 6 →= 5 2 3 2 1 6 , 5 3 6 →= 5 6 6 2 3 2 , 5 3 6 →= 6 2 2 3 2 5 , 9 3 6 →= 3 9 6 2 , 9 3 6 →= 3 9 5 2 6 , 9 3 6 →= 3 9 6 6 2 2 , 9 3 6 →= 3 5 9 5 6 2 , 1 9 6 →= 2 2 9 1 2 6 , 3 1 9 →= 9 3 1 2 2 , 5 1 9 →= 2 2 5 9 1 , 5 3 9 →= 5 6 2 2 3 9 , 3 6 9 →= 3 2 9 6 2 2 , 1 9 9 →= 1 9 2 2 9 , 5 3 1 1 →= 2 2 3 1 5 1 , 3 2 6 1 →= 3 1 6 2 2 , 5 3 6 1 →= 5 5 1 6 2 3 , 3 6 5 1 →= 2 1 5 6 3 , 5 3 1 6 →= 2 2 3 5 1 6 , 5 9 1 6 →= 6 2 9 1 5 6 , 1 1 2 6 →= 2 1 6 2 2 1 , 1 5 3 6 →= 3 5 1 2 6 , 9 5 3 6 →= 9 2 3 5 2 6 , 5 9 3 6 →= 3 9 6 5 6 , 3 6 9 6 →= 9 3 1 6 6 , 3 5 1 5 →= 1 6 2 3 5 5 , 1 3 6 5 →= 5 6 2 3 1 , 1 2 5 9 →= 2 1 5 2 2 9 , 1 1 9 9 →= 1 9 9 1 2 2 , 3 3 9 9 →= 3 3 2 2 9 9 , 5 3 2 1 1 →= 5 2 2 3 1 1 , 3 2 6 9 1 →= 1 2 6 2 3 9 , 1 2 9 9 1 →= 2 2 9 1 1 9 , 9 5 2 1 6 →= 2 2 9 1 5 6 , 3 2 6 1 6 →= 6 6 2 2 3 1 , 5 1 3 3 6 →= 5 2 1 3 3 6 , 1 5 2 9 6 →= 1 9 5 6 6 2 , 5 1 6 9 6 →= 5 6 9 2 1 6 , 1 3 6 2 9 →= 2 2 1 9 6 3 , 3 1 3 5 9 →= 3 2 3 5 1 9 } 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 1 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 1 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 0 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 1 ⎟ ⎜ 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 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 81-rule system { 0 1 1 ⟶ 0 2 1 1 , 0 3 1 ⟶ 4 1 1 , 4 5 1 ⟶ 4 1 2 2 2 5 , 0 3 6 ⟶ 4 1 , 0 3 6 ⟶ 4 1 6 , 0 3 6 ⟶ 4 2 1 , 0 3 6 ⟶ 7 2 3 2 1 6 , 0 3 6 ⟶ 4 2 1 6 , 8 3 6 ⟶ 4 9 6 2 , 8 3 6 ⟶ 4 9 5 2 6 , 8 3 6 ⟶ 4 9 6 6 2 2 , 8 3 6 ⟶ 4 5 9 5 6 2 , 0 9 6 ⟶ 8 1 2 6 , 4 1 9 ⟶ 8 3 1 2 2 , 0 9 9 ⟶ 0 9 2 2 9 , 0 9 9 ⟶ 8 2 2 9 , 7 3 1 6 ⟶ 4 5 1 6 , 0 1 2 6 ⟶ 0 6 2 2 1 , 0 1 2 6 ⟶ 0 , 0 5 3 6 ⟶ 4 5 1 2 6 , 8 5 3 6 ⟶ 8 2 3 5 2 6 , 4 5 1 5 ⟶ 0 6 2 3 5 5 , 4 5 1 5 ⟶ 4 5 5 , 0 3 6 5 ⟶ 7 6 2 3 1 , 0 3 6 5 ⟶ 4 1 , 0 1 9 9 ⟶ 0 9 9 1 2 2 , 0 1 9 9 ⟶ 8 9 1 2 2 , 0 2 9 9 1 ⟶ 8 1 1 9 , 8 5 2 1 6 ⟶ 8 1 5 6 , 0 5 2 9 6 ⟶ 0 9 5 6 6 2 , 0 5 2 9 6 ⟶ 8 5 6 6 2 , 0 3 6 2 9 ⟶ 0 9 6 3 , 0 3 6 2 9 ⟶ 8 6 3 , 4 1 3 5 9 ⟶ 4 2 3 5 1 9 , 1 1 1 →= 2 1 2 1 1 , 1 3 1 →= 2 2 3 1 1 , 3 6 1 →= 1 6 2 3 , 3 5 1 →= 3 1 2 2 2 5 , 5 1 6 →= 2 5 6 2 1 , 1 3 6 →= 6 2 3 1 , 1 3 6 →= 2 3 1 6 , 1 3 6 →= 6 2 3 2 1 , 1 3 6 →= 5 2 3 2 1 6 , 5 3 6 →= 5 6 6 2 3 2 , 5 3 6 →= 6 2 2 3 2 5 , 9 3 6 →= 3 9 6 2 , 9 3 6 →= 3 9 5 2 6 , 9 3 6 →= 3 9 6 6 2 2 , 9 3 6 →= 3 5 9 5 6 2 , 1 9 6 →= 2 2 9 1 2 6 , 3 1 9 →= 9 3 1 2 2 , 5 1 9 →= 2 2 5 9 1 , 5 3 9 →= 5 6 2 2 3 9 , 3 6 9 →= 3 2 9 6 2 2 , 1 9 9 →= 1 9 2 2 9 , 5 3 1 1 →= 2 2 3 1 5 1 , 3 2 6 1 →= 3 1 6 2 2 , 5 3 6 1 →= 5 5 1 6 2 3 , 3 6 5 1 →= 2 1 5 6 3 , 5 3 1 6 →= 2 2 3 5 1 6 , 5 9 1 6 →= 6 2 9 1 5 6 , 1 1 2 6 →= 2 1 6 2 2 1 , 1 5 3 6 →= 3 5 1 2 6 , 9 5 3 6 →= 9 2 3 5 2 6 , 5 9 3 6 →= 3 9 6 5 6 , 3 6 9 6 →= 9 3 1 6 6 , 3 5 1 5 →= 1 6 2 3 5 5 , 1 3 6 5 →= 5 6 2 3 1 , 1 2 5 9 →= 2 1 5 2 2 9 , 1 1 9 9 →= 1 9 9 1 2 2 , 3 3 9 9 →= 3 3 2 2 9 9 , 5 3 2 1 1 →= 5 2 2 3 1 1 , 3 2 6 9 1 →= 1 2 6 2 3 9 , 1 2 9 9 1 →= 2 2 9 1 1 9 , 9 5 2 1 6 →= 2 2 9 1 5 6 , 3 2 6 1 6 →= 6 6 2 2 3 1 , 5 1 3 3 6 →= 5 2 1 3 3 6 , 1 5 2 9 6 →= 1 9 5 6 6 2 , 5 1 6 9 6 →= 5 6 9 2 1 6 , 1 3 6 2 9 →= 2 2 1 9 6 3 , 3 1 3 5 9 →= 3 2 3 5 1 9 } 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 1 ⎟ ⎜ 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 1 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 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 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 79-rule system { 0 1 1 ⟶ 0 2 1 1 , 0 3 1 ⟶ 4 1 1 , 4 5 1 ⟶ 4 1 2 2 2 5 , 0 3 6 ⟶ 4 1 , 0 3 6 ⟶ 4 1 6 , 0 3 6 ⟶ 4 2 1 , 0 3 6 ⟶ 7 2 3 2 1 6 , 0 3 6 ⟶ 4 2 1 6 , 8 3 6 ⟶ 4 9 6 2 , 8 3 6 ⟶ 4 9 5 2 6 , 8 3 6 ⟶ 4 9 6 6 2 2 , 8 3 6 ⟶ 4 5 9 5 6 2 , 0 9 6 ⟶ 8 1 2 6 , 4 1 9 ⟶ 8 3 1 2 2 , 0 9 9 ⟶ 0 9 2 2 9 , 0 9 9 ⟶ 8 2 2 9 , 7 3 1 6 ⟶ 4 5 1 6 , 0 1 2 6 ⟶ 0 6 2 2 1 , 0 1 2 6 ⟶ 0 , 0 5 3 6 ⟶ 4 5 1 2 6 , 8 5 3 6 ⟶ 8 2 3 5 2 6 , 0 3 6 5 ⟶ 7 6 2 3 1 , 0 3 6 5 ⟶ 4 1 , 0 1 9 9 ⟶ 0 9 9 1 2 2 , 0 1 9 9 ⟶ 8 9 1 2 2 , 0 2 9 9 1 ⟶ 8 1 1 9 , 8 5 2 1 6 ⟶ 8 1 5 6 , 0 5 2 9 6 ⟶ 0 9 5 6 6 2 , 0 5 2 9 6 ⟶ 8 5 6 6 2 , 0 3 6 2 9 ⟶ 0 9 6 3 , 0 3 6 2 9 ⟶ 8 6 3 , 4 1 3 5 9 ⟶ 4 2 3 5 1 9 , 1 1 1 →= 2 1 2 1 1 , 1 3 1 →= 2 2 3 1 1 , 3 6 1 →= 1 6 2 3 , 3 5 1 →= 3 1 2 2 2 5 , 5 1 6 →= 2 5 6 2 1 , 1 3 6 →= 6 2 3 1 , 1 3 6 →= 2 3 1 6 , 1 3 6 →= 6 2 3 2 1 , 1 3 6 →= 5 2 3 2 1 6 , 5 3 6 →= 5 6 6 2 3 2 , 5 3 6 →= 6 2 2 3 2 5 , 9 3 6 →= 3 9 6 2 , 9 3 6 →= 3 9 5 2 6 , 9 3 6 →= 3 9 6 6 2 2 , 9 3 6 →= 3 5 9 5 6 2 , 1 9 6 →= 2 2 9 1 2 6 , 3 1 9 →= 9 3 1 2 2 , 5 1 9 →= 2 2 5 9 1 , 5 3 9 →= 5 6 2 2 3 9 , 3 6 9 →= 3 2 9 6 2 2 , 1 9 9 →= 1 9 2 2 9 , 5 3 1 1 →= 2 2 3 1 5 1 , 3 2 6 1 →= 3 1 6 2 2 , 5 3 6 1 →= 5 5 1 6 2 3 , 3 6 5 1 →= 2 1 5 6 3 , 5 3 1 6 →= 2 2 3 5 1 6 , 5 9 1 6 →= 6 2 9 1 5 6 , 1 1 2 6 →= 2 1 6 2 2 1 , 1 5 3 6 →= 3 5 1 2 6 , 9 5 3 6 →= 9 2 3 5 2 6 , 5 9 3 6 →= 3 9 6 5 6 , 3 6 9 6 →= 9 3 1 6 6 , 3 5 1 5 →= 1 6 2 3 5 5 , 1 3 6 5 →= 5 6 2 3 1 , 1 2 5 9 →= 2 1 5 2 2 9 , 1 1 9 9 →= 1 9 9 1 2 2 , 3 3 9 9 →= 3 3 2 2 9 9 , 5 3 2 1 1 →= 5 2 2 3 1 1 , 3 2 6 9 1 →= 1 2 6 2 3 9 , 1 2 9 9 1 →= 2 2 9 1 1 9 , 9 5 2 1 6 →= 2 2 9 1 5 6 , 3 2 6 1 6 →= 6 6 2 2 3 1 , 5 1 3 3 6 →= 5 2 1 3 3 6 , 1 5 2 9 6 →= 1 9 5 6 6 2 , 5 1 6 9 6 →= 5 6 9 2 1 6 , 1 3 6 2 9 →= 2 2 1 9 6 3 , 3 1 3 5 9 →= 3 2 3 5 1 9 } 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 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 4 ↦ 3, 5 ↦ 4, 8 ↦ 5, 3 ↦ 6, 6 ↦ 7, 9 ↦ 8, 7 ↦ 9 }, it remains to prove termination of the 64-rule system { 0 1 1 ⟶ 0 2 1 1 , 3 4 1 ⟶ 3 1 2 2 2 4 , 5 6 7 ⟶ 3 8 7 2 , 5 6 7 ⟶ 3 8 4 2 7 , 5 6 7 ⟶ 3 8 7 7 2 2 , 5 6 7 ⟶ 3 4 8 4 7 2 , 3 1 8 ⟶ 5 6 1 2 2 , 0 8 8 ⟶ 0 8 2 2 8 , 9 6 1 7 ⟶ 3 4 1 7 , 0 1 2 7 ⟶ 0 7 2 2 1 , 0 1 2 7 ⟶ 0 , 5 4 6 7 ⟶ 5 2 6 4 2 7 , 0 1 8 8 ⟶ 0 8 8 1 2 2 , 5 4 2 1 7 ⟶ 5 1 4 7 , 0 4 2 8 7 ⟶ 0 8 4 7 7 2 , 0 6 7 2 8 ⟶ 0 8 7 6 , 3 1 6 4 8 ⟶ 3 2 6 4 1 8 , 1 1 1 →= 2 1 2 1 1 , 1 6 1 →= 2 2 6 1 1 , 6 7 1 →= 1 7 2 6 , 6 4 1 →= 6 1 2 2 2 4 , 4 1 7 →= 2 4 7 2 1 , 1 6 7 →= 7 2 6 1 , 1 6 7 →= 2 6 1 7 , 1 6 7 →= 7 2 6 2 1 , 1 6 7 →= 4 2 6 2 1 7 , 4 6 7 →= 4 7 7 2 6 2 , 4 6 7 →= 7 2 2 6 2 4 , 8 6 7 →= 6 8 7 2 , 8 6 7 →= 6 8 4 2 7 , 8 6 7 →= 6 8 7 7 2 2 , 8 6 7 →= 6 4 8 4 7 2 , 1 8 7 →= 2 2 8 1 2 7 , 6 1 8 →= 8 6 1 2 2 , 4 1 8 →= 2 2 4 8 1 , 4 6 8 →= 4 7 2 2 6 8 , 6 7 8 →= 6 2 8 7 2 2 , 1 8 8 →= 1 8 2 2 8 , 4 6 1 1 →= 2 2 6 1 4 1 , 6 2 7 1 →= 6 1 7 2 2 , 4 6 7 1 →= 4 4 1 7 2 6 , 6 7 4 1 →= 2 1 4 7 6 , 4 6 1 7 →= 2 2 6 4 1 7 , 4 8 1 7 →= 7 2 8 1 4 7 , 1 1 2 7 →= 2 1 7 2 2 1 , 1 4 6 7 →= 6 4 1 2 7 , 8 4 6 7 →= 8 2 6 4 2 7 , 4 8 6 7 →= 6 8 7 4 7 , 6 7 8 7 →= 8 6 1 7 7 , 6 4 1 4 →= 1 7 2 6 4 4 , 1 6 7 4 →= 4 7 2 6 1 , 1 2 4 8 →= 2 1 4 2 2 8 , 1 1 8 8 →= 1 8 8 1 2 2 , 6 6 8 8 →= 6 6 2 2 8 8 , 4 6 2 1 1 →= 4 2 2 6 1 1 , 6 2 7 8 1 →= 1 2 7 2 6 8 , 1 2 8 8 1 →= 2 2 8 1 1 8 , 8 4 2 1 7 →= 2 2 8 1 4 7 , 6 2 7 1 7 →= 7 7 2 2 6 1 , 4 1 6 6 7 →= 4 2 1 6 6 7 , 1 4 2 8 7 →= 1 8 4 7 7 2 , 4 1 7 8 7 →= 4 7 8 2 1 7 , 1 6 7 2 8 →= 2 2 1 8 7 6 , 6 1 6 4 8 →= 6 2 6 4 1 8 } 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 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8 }, it remains to prove termination of the 63-rule system { 0 1 1 ⟶ 0 2 1 1 , 3 4 1 ⟶ 3 1 2 2 2 4 , 5 6 7 ⟶ 3 8 7 2 , 5 6 7 ⟶ 3 8 4 2 7 , 5 6 7 ⟶ 3 8 7 7 2 2 , 5 6 7 ⟶ 3 4 8 4 7 2 , 3 1 8 ⟶ 5 6 1 2 2 , 0 8 8 ⟶ 0 8 2 2 8 , 0 1 2 7 ⟶ 0 7 2 2 1 , 0 1 2 7 ⟶ 0 , 5 4 6 7 ⟶ 5 2 6 4 2 7 , 0 1 8 8 ⟶ 0 8 8 1 2 2 , 5 4 2 1 7 ⟶ 5 1 4 7 , 0 4 2 8 7 ⟶ 0 8 4 7 7 2 , 0 6 7 2 8 ⟶ 0 8 7 6 , 3 1 6 4 8 ⟶ 3 2 6 4 1 8 , 1 1 1 →= 2 1 2 1 1 , 1 6 1 →= 2 2 6 1 1 , 6 7 1 →= 1 7 2 6 , 6 4 1 →= 6 1 2 2 2 4 , 4 1 7 →= 2 4 7 2 1 , 1 6 7 →= 7 2 6 1 , 1 6 7 →= 2 6 1 7 , 1 6 7 →= 7 2 6 2 1 , 1 6 7 →= 4 2 6 2 1 7 , 4 6 7 →= 4 7 7 2 6 2 , 4 6 7 →= 7 2 2 6 2 4 , 8 6 7 →= 6 8 7 2 , 8 6 7 →= 6 8 4 2 7 , 8 6 7 →= 6 8 7 7 2 2 , 8 6 7 →= 6 4 8 4 7 2 , 1 8 7 →= 2 2 8 1 2 7 , 6 1 8 →= 8 6 1 2 2 , 4 1 8 →= 2 2 4 8 1 , 4 6 8 →= 4 7 2 2 6 8 , 6 7 8 →= 6 2 8 7 2 2 , 1 8 8 →= 1 8 2 2 8 , 4 6 1 1 →= 2 2 6 1 4 1 , 6 2 7 1 →= 6 1 7 2 2 , 4 6 7 1 →= 4 4 1 7 2 6 , 6 7 4 1 →= 2 1 4 7 6 , 4 6 1 7 →= 2 2 6 4 1 7 , 4 8 1 7 →= 7 2 8 1 4 7 , 1 1 2 7 →= 2 1 7 2 2 1 , 1 4 6 7 →= 6 4 1 2 7 , 8 4 6 7 →= 8 2 6 4 2 7 , 4 8 6 7 →= 6 8 7 4 7 , 6 7 8 7 →= 8 6 1 7 7 , 6 4 1 4 →= 1 7 2 6 4 4 , 1 6 7 4 →= 4 7 2 6 1 , 1 2 4 8 →= 2 1 4 2 2 8 , 1 1 8 8 →= 1 8 8 1 2 2 , 6 6 8 8 →= 6 6 2 2 8 8 , 4 6 2 1 1 →= 4 2 2 6 1 1 , 6 2 7 8 1 →= 1 2 7 2 6 8 , 1 2 8 8 1 →= 2 2 8 1 1 8 , 8 4 2 1 7 →= 2 2 8 1 4 7 , 6 2 7 1 7 →= 7 7 2 2 6 1 , 4 1 6 6 7 →= 4 2 1 6 6 7 , 1 4 2 8 7 →= 1 8 4 7 7 2 , 4 1 7 8 7 →= 4 7 8 2 1 7 , 1 6 7 2 8 →= 2 2 1 8 7 6 , 6 1 6 4 8 →= 6 2 6 4 1 8 } 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 1 ⎟ ⎜ 0 0 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 1 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 1 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 0 ⎟ ⎜ 0 1 1 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 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, 7 ↦ 7, 8 ↦ 8 }, it remains to prove termination of the 61-rule system { 0 1 1 ⟶ 0 2 1 1 , 3 4 1 ⟶ 3 1 2 2 2 4 , 5 6 7 ⟶ 3 8 7 2 , 5 6 7 ⟶ 3 8 4 2 7 , 5 6 7 ⟶ 3 8 7 7 2 2 , 5 6 7 ⟶ 3 4 8 4 7 2 , 0 8 8 ⟶ 0 8 2 2 8 , 0 1 2 7 ⟶ 0 7 2 2 1 , 0 1 2 7 ⟶ 0 , 5 4 6 7 ⟶ 5 2 6 4 2 7 , 0 1 8 8 ⟶ 0 8 8 1 2 2 , 5 4 2 1 7 ⟶ 5 1 4 7 , 0 4 2 8 7 ⟶ 0 8 4 7 7 2 , 0 6 7 2 8 ⟶ 0 8 7 6 , 1 1 1 →= 2 1 2 1 1 , 1 6 1 →= 2 2 6 1 1 , 6 7 1 →= 1 7 2 6 , 6 4 1 →= 6 1 2 2 2 4 , 4 1 7 →= 2 4 7 2 1 , 1 6 7 →= 7 2 6 1 , 1 6 7 →= 2 6 1 7 , 1 6 7 →= 7 2 6 2 1 , 1 6 7 →= 4 2 6 2 1 7 , 4 6 7 →= 4 7 7 2 6 2 , 4 6 7 →= 7 2 2 6 2 4 , 8 6 7 →= 6 8 7 2 , 8 6 7 →= 6 8 4 2 7 , 8 6 7 →= 6 8 7 7 2 2 , 8 6 7 →= 6 4 8 4 7 2 , 1 8 7 →= 2 2 8 1 2 7 , 6 1 8 →= 8 6 1 2 2 , 4 1 8 →= 2 2 4 8 1 , 4 6 8 →= 4 7 2 2 6 8 , 6 7 8 →= 6 2 8 7 2 2 , 1 8 8 →= 1 8 2 2 8 , 4 6 1 1 →= 2 2 6 1 4 1 , 6 2 7 1 →= 6 1 7 2 2 , 4 6 7 1 →= 4 4 1 7 2 6 , 6 7 4 1 →= 2 1 4 7 6 , 4 6 1 7 →= 2 2 6 4 1 7 , 4 8 1 7 →= 7 2 8 1 4 7 , 1 1 2 7 →= 2 1 7 2 2 1 , 1 4 6 7 →= 6 4 1 2 7 , 8 4 6 7 →= 8 2 6 4 2 7 , 4 8 6 7 →= 6 8 7 4 7 , 6 7 8 7 →= 8 6 1 7 7 , 6 4 1 4 →= 1 7 2 6 4 4 , 1 6 7 4 →= 4 7 2 6 1 , 1 2 4 8 →= 2 1 4 2 2 8 , 1 1 8 8 →= 1 8 8 1 2 2 , 6 6 8 8 →= 6 6 2 2 8 8 , 4 6 2 1 1 →= 4 2 2 6 1 1 , 6 2 7 8 1 →= 1 2 7 2 6 8 , 1 2 8 8 1 →= 2 2 8 1 1 8 , 8 4 2 1 7 →= 2 2 8 1 4 7 , 6 2 7 1 7 →= 7 7 2 2 6 1 , 4 1 6 6 7 →= 4 2 1 6 6 7 , 1 4 2 8 7 →= 1 8 4 7 7 2 , 4 1 7 8 7 →= 4 7 8 2 1 7 , 1 6 7 2 8 →= 2 2 1 8 7 6 , 6 1 6 4 8 →= 6 2 6 4 1 8 } 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 8 ↦ 5, 7 ↦ 6, 5 ↦ 7, 6 ↦ 8 }, it remains to prove termination of the 57-rule system { 0 1 1 ⟶ 0 2 1 1 , 3 4 1 ⟶ 3 1 2 2 2 4 , 0 5 5 ⟶ 0 5 2 2 5 , 0 1 2 6 ⟶ 0 6 2 2 1 , 0 1 2 6 ⟶ 0 , 7 4 8 6 ⟶ 7 2 8 4 2 6 , 0 1 5 5 ⟶ 0 5 5 1 2 2 , 7 4 2 1 6 ⟶ 7 1 4 6 , 0 4 2 5 6 ⟶ 0 5 4 6 6 2 , 0 8 6 2 5 ⟶ 0 5 6 8 , 1 1 1 →= 2 1 2 1 1 , 1 8 1 →= 2 2 8 1 1 , 8 6 1 →= 1 6 2 8 , 8 4 1 →= 8 1 2 2 2 4 , 4 1 6 →= 2 4 6 2 1 , 1 8 6 →= 6 2 8 1 , 1 8 6 →= 2 8 1 6 , 1 8 6 →= 6 2 8 2 1 , 1 8 6 →= 4 2 8 2 1 6 , 4 8 6 →= 4 6 6 2 8 2 , 4 8 6 →= 6 2 2 8 2 4 , 5 8 6 →= 8 5 6 2 , 5 8 6 →= 8 5 4 2 6 , 5 8 6 →= 8 5 6 6 2 2 , 5 8 6 →= 8 4 5 4 6 2 , 1 5 6 →= 2 2 5 1 2 6 , 8 1 5 →= 5 8 1 2 2 , 4 1 5 →= 2 2 4 5 1 , 4 8 5 →= 4 6 2 2 8 5 , 8 6 5 →= 8 2 5 6 2 2 , 1 5 5 →= 1 5 2 2 5 , 4 8 1 1 →= 2 2 8 1 4 1 , 8 2 6 1 →= 8 1 6 2 2 , 4 8 6 1 →= 4 4 1 6 2 8 , 8 6 4 1 →= 2 1 4 6 8 , 4 8 1 6 →= 2 2 8 4 1 6 , 4 5 1 6 →= 6 2 5 1 4 6 , 1 1 2 6 →= 2 1 6 2 2 1 , 1 4 8 6 →= 8 4 1 2 6 , 5 4 8 6 →= 5 2 8 4 2 6 , 4 5 8 6 →= 8 5 6 4 6 , 8 6 5 6 →= 5 8 1 6 6 , 8 4 1 4 →= 1 6 2 8 4 4 , 1 8 6 4 →= 4 6 2 8 1 , 1 2 4 5 →= 2 1 4 2 2 5 , 1 1 5 5 →= 1 5 5 1 2 2 , 8 8 5 5 →= 8 8 2 2 5 5 , 4 8 2 1 1 →= 4 2 2 8 1 1 , 8 2 6 5 1 →= 1 2 6 2 8 5 , 1 2 5 5 1 →= 2 2 5 1 1 5 , 5 4 2 1 6 →= 2 2 5 1 4 6 , 8 2 6 1 6 →= 6 6 2 2 8 1 , 4 1 8 8 6 →= 4 2 1 8 8 6 , 1 4 2 5 6 →= 1 5 4 6 6 2 , 4 1 6 5 6 →= 4 6 5 2 1 6 , 1 8 6 2 5 →= 2 2 1 5 6 8 , 8 1 8 4 5 →= 8 2 8 4 1 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 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 1 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 1 ⎟ ⎜ 0 1 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 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 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 5 ↦ 3, 6 ↦ 4, 7 ↦ 5, 4 ↦ 6, 8 ↦ 7 }, it remains to prove termination of the 56-rule system { 0 1 1 ⟶ 0 2 1 1 , 0 3 3 ⟶ 0 3 2 2 3 , 0 1 2 4 ⟶ 0 4 2 2 1 , 0 1 2 4 ⟶ 0 , 5 6 7 4 ⟶ 5 2 7 6 2 4 , 0 1 3 3 ⟶ 0 3 3 1 2 2 , 5 6 2 1 4 ⟶ 5 1 6 4 , 0 6 2 3 4 ⟶ 0 3 6 4 4 2 , 0 7 4 2 3 ⟶ 0 3 4 7 , 1 1 1 →= 2 1 2 1 1 , 1 7 1 →= 2 2 7 1 1 , 7 4 1 →= 1 4 2 7 , 7 6 1 →= 7 1 2 2 2 6 , 6 1 4 →= 2 6 4 2 1 , 1 7 4 →= 4 2 7 1 , 1 7 4 →= 2 7 1 4 , 1 7 4 →= 4 2 7 2 1 , 1 7 4 →= 6 2 7 2 1 4 , 6 7 4 →= 6 4 4 2 7 2 , 6 7 4 →= 4 2 2 7 2 6 , 3 7 4 →= 7 3 4 2 , 3 7 4 →= 7 3 6 2 4 , 3 7 4 →= 7 3 4 4 2 2 , 3 7 4 →= 7 6 3 6 4 2 , 1 3 4 →= 2 2 3 1 2 4 , 7 1 3 →= 3 7 1 2 2 , 6 1 3 →= 2 2 6 3 1 , 6 7 3 →= 6 4 2 2 7 3 , 7 4 3 →= 7 2 3 4 2 2 , 1 3 3 →= 1 3 2 2 3 , 6 7 1 1 →= 2 2 7 1 6 1 , 7 2 4 1 →= 7 1 4 2 2 , 6 7 4 1 →= 6 6 1 4 2 7 , 7 4 6 1 →= 2 1 6 4 7 , 6 7 1 4 →= 2 2 7 6 1 4 , 6 3 1 4 →= 4 2 3 1 6 4 , 1 1 2 4 →= 2 1 4 2 2 1 , 1 6 7 4 →= 7 6 1 2 4 , 3 6 7 4 →= 3 2 7 6 2 4 , 6 3 7 4 →= 7 3 4 6 4 , 7 4 3 4 →= 3 7 1 4 4 , 7 6 1 6 →= 1 4 2 7 6 6 , 1 7 4 6 →= 6 4 2 7 1 , 1 2 6 3 →= 2 1 6 2 2 3 , 1 1 3 3 →= 1 3 3 1 2 2 , 7 7 3 3 →= 7 7 2 2 3 3 , 6 7 2 1 1 →= 6 2 2 7 1 1 , 7 2 4 3 1 →= 1 2 4 2 7 3 , 1 2 3 3 1 →= 2 2 3 1 1 3 , 3 6 2 1 4 →= 2 2 3 1 6 4 , 7 2 4 1 4 →= 4 4 2 2 7 1 , 6 1 7 7 4 →= 6 2 1 7 7 4 , 1 6 2 3 4 →= 1 3 6 4 4 2 , 6 1 4 3 4 →= 6 4 3 2 1 4 , 1 7 4 2 3 →= 2 2 1 3 4 7 , 7 1 7 6 3 →= 7 2 7 6 1 3 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 11: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 1 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 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 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 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 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 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 0 ⎟ ⎜ 0 0 0 0 0 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 0 0 0 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 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 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 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 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 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 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 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 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 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 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 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 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 51-rule system { 0 1 1 ⟶ 0 2 1 1 , 0 3 3 ⟶ 0 3 2 2 3 , 0 1 2 4 ⟶ 0 4 2 2 1 , 0 1 2 4 ⟶ 0 , 5 6 7 4 ⟶ 5 2 7 6 2 4 , 0 1 3 3 ⟶ 0 3 3 1 2 2 , 5 6 2 1 4 ⟶ 5 1 6 4 , 0 6 2 3 4 ⟶ 0 3 6 4 4 2 , 0 7 4 2 3 ⟶ 0 3 4 7 , 1 1 1 →= 2 1 2 1 1 , 1 7 1 →= 2 2 7 1 1 , 7 6 1 →= 7 1 2 2 2 6 , 6 1 4 →= 2 6 4 2 1 , 1 7 4 →= 4 2 7 1 , 1 7 4 →= 2 7 1 4 , 1 7 4 →= 4 2 7 2 1 , 1 7 4 →= 6 2 7 2 1 4 , 6 7 4 →= 6 4 4 2 7 2 , 6 7 4 →= 4 2 2 7 2 6 , 3 7 4 →= 7 3 4 2 , 3 7 4 →= 7 3 6 2 4 , 3 7 4 →= 7 3 4 4 2 2 , 3 7 4 →= 7 6 3 6 4 2 , 1 3 4 →= 2 2 3 1 2 4 , 7 1 3 →= 3 7 1 2 2 , 6 1 3 →= 2 2 6 3 1 , 6 7 3 →= 6 4 2 2 7 3 , 1 3 3 →= 1 3 2 2 3 , 6 7 1 1 →= 2 2 7 1 6 1 , 7 2 4 1 →= 7 1 4 2 2 , 6 7 1 4 →= 2 2 7 6 1 4 , 6 3 1 4 →= 4 2 3 1 6 4 , 1 1 2 4 →= 2 1 4 2 2 1 , 1 6 7 4 →= 7 6 1 2 4 , 3 6 7 4 →= 3 2 7 6 2 4 , 6 3 7 4 →= 7 3 4 6 4 , 7 6 1 6 →= 1 4 2 7 6 6 , 1 7 4 6 →= 6 4 2 7 1 , 1 2 6 3 →= 2 1 6 2 2 3 , 1 1 3 3 →= 1 3 3 1 2 2 , 7 7 3 3 →= 7 7 2 2 3 3 , 6 7 2 1 1 →= 6 2 2 7 1 1 , 7 2 4 3 1 →= 1 2 4 2 7 3 , 1 2 3 3 1 →= 2 2 3 1 1 3 , 3 6 2 1 4 →= 2 2 3 1 6 4 , 7 2 4 1 4 →= 4 4 2 2 7 1 , 6 1 7 7 4 →= 6 2 1 7 7 4 , 1 6 2 3 4 →= 1 3 6 4 4 2 , 6 1 4 3 4 →= 6 4 3 2 1 4 , 1 7 4 2 3 →= 2 2 1 3 4 7 , 7 1 7 6 3 →= 7 2 7 6 1 3 } 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 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, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 50-rule system { 0 1 1 ⟶ 0 2 1 1 , 0 3 3 ⟶ 0 3 2 2 3 , 0 1 2 4 ⟶ 0 4 2 2 1 , 5 6 7 4 ⟶ 5 2 7 6 2 4 , 0 1 3 3 ⟶ 0 3 3 1 2 2 , 5 6 2 1 4 ⟶ 5 1 6 4 , 0 6 2 3 4 ⟶ 0 3 6 4 4 2 , 0 7 4 2 3 ⟶ 0 3 4 7 , 1 1 1 →= 2 1 2 1 1 , 1 7 1 →= 2 2 7 1 1 , 7 6 1 →= 7 1 2 2 2 6 , 6 1 4 →= 2 6 4 2 1 , 1 7 4 →= 4 2 7 1 , 1 7 4 →= 2 7 1 4 , 1 7 4 →= 4 2 7 2 1 , 1 7 4 →= 6 2 7 2 1 4 , 6 7 4 →= 6 4 4 2 7 2 , 6 7 4 →= 4 2 2 7 2 6 , 3 7 4 →= 7 3 4 2 , 3 7 4 →= 7 3 6 2 4 , 3 7 4 →= 7 3 4 4 2 2 , 3 7 4 →= 7 6 3 6 4 2 , 1 3 4 →= 2 2 3 1 2 4 , 7 1 3 →= 3 7 1 2 2 , 6 1 3 →= 2 2 6 3 1 , 6 7 3 →= 6 4 2 2 7 3 , 1 3 3 →= 1 3 2 2 3 , 6 7 1 1 →= 2 2 7 1 6 1 , 7 2 4 1 →= 7 1 4 2 2 , 6 7 1 4 →= 2 2 7 6 1 4 , 6 3 1 4 →= 4 2 3 1 6 4 , 1 1 2 4 →= 2 1 4 2 2 1 , 1 6 7 4 →= 7 6 1 2 4 , 3 6 7 4 →= 3 2 7 6 2 4 , 6 3 7 4 →= 7 3 4 6 4 , 7 6 1 6 →= 1 4 2 7 6 6 , 1 7 4 6 →= 6 4 2 7 1 , 1 2 6 3 →= 2 1 6 2 2 3 , 1 1 3 3 →= 1 3 3 1 2 2 , 7 7 3 3 →= 7 7 2 2 3 3 , 6 7 2 1 1 →= 6 2 2 7 1 1 , 7 2 4 3 1 →= 1 2 4 2 7 3 , 1 2 3 3 1 →= 2 2 3 1 1 3 , 3 6 2 1 4 →= 2 2 3 1 6 4 , 7 2 4 1 4 →= 4 4 2 2 7 1 , 6 1 7 7 4 →= 6 2 1 7 7 4 , 1 6 2 3 4 →= 1 3 6 4 4 2 , 6 1 4 3 4 →= 6 4 3 2 1 4 , 1 7 4 2 3 →= 2 2 1 3 4 7 , 7 1 7 6 3 →= 7 2 7 6 1 3 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 8: 0 ↦ ⎛ ⎞ ⎜ 1 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 0 0 ⎟ ⎜ 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 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 1 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 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 ⎟ ⎝ ⎠ 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 1 ⎟ ⎜ 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 1 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 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 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 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 3 ↦ 1, 2 ↦ 2, 1 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 49-rule system { 0 1 1 ⟶ 0 1 2 2 1 , 0 3 2 4 ⟶ 0 4 2 2 3 , 5 6 7 4 ⟶ 5 2 7 6 2 4 , 0 3 1 1 ⟶ 0 1 1 3 2 2 , 5 6 2 3 4 ⟶ 5 3 6 4 , 0 6 2 1 4 ⟶ 0 1 6 4 4 2 , 0 7 4 2 1 ⟶ 0 1 4 7 , 3 3 3 →= 2 3 2 3 3 , 3 7 3 →= 2 2 7 3 3 , 7 6 3 →= 7 3 2 2 2 6 , 6 3 4 →= 2 6 4 2 3 , 3 7 4 →= 4 2 7 3 , 3 7 4 →= 2 7 3 4 , 3 7 4 →= 4 2 7 2 3 , 3 7 4 →= 6 2 7 2 3 4 , 6 7 4 →= 6 4 4 2 7 2 , 6 7 4 →= 4 2 2 7 2 6 , 1 7 4 →= 7 1 4 2 , 1 7 4 →= 7 1 6 2 4 , 1 7 4 →= 7 1 4 4 2 2 , 1 7 4 →= 7 6 1 6 4 2 , 3 1 4 →= 2 2 1 3 2 4 , 7 3 1 →= 1 7 3 2 2 , 6 3 1 →= 2 2 6 1 3 , 6 7 1 →= 6 4 2 2 7 1 , 3 1 1 →= 3 1 2 2 1 , 6 7 3 3 →= 2 2 7 3 6 3 , 7 2 4 3 →= 7 3 4 2 2 , 6 7 3 4 →= 2 2 7 6 3 4 , 6 1 3 4 →= 4 2 1 3 6 4 , 3 3 2 4 →= 2 3 4 2 2 3 , 3 6 7 4 →= 7 6 3 2 4 , 1 6 7 4 →= 1 2 7 6 2 4 , 6 1 7 4 →= 7 1 4 6 4 , 7 6 3 6 →= 3 4 2 7 6 6 , 3 7 4 6 →= 6 4 2 7 3 , 3 2 6 1 →= 2 3 6 2 2 1 , 3 3 1 1 →= 3 1 1 3 2 2 , 7 7 1 1 →= 7 7 2 2 1 1 , 6 7 2 3 3 →= 6 2 2 7 3 3 , 7 2 4 1 3 →= 3 2 4 2 7 1 , 3 2 1 1 3 →= 2 2 1 3 3 1 , 1 6 2 3 4 →= 2 2 1 3 6 4 , 7 2 4 3 4 →= 4 4 2 2 7 3 , 6 3 7 7 4 →= 6 2 3 7 7 4 , 3 6 2 1 4 →= 3 1 6 4 4 2 , 6 3 4 1 4 →= 6 4 1 2 3 4 , 3 7 4 2 1 →= 2 2 3 1 4 7 , 7 3 7 6 1 →= 7 2 7 6 3 1 } 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 1 ⎟ ⎜ 0 1 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 1 0 0 ⎟ ⎜ 0 1 1 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 1 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 3 ↦ 1, 2 ↦ 2, 4 ↦ 3, 5 ↦ 4, 6 ↦ 5, 7 ↦ 6, 1 ↦ 7 }, it remains to prove termination of the 48-rule system { 0 1 2 3 ⟶ 0 3 2 2 1 , 4 5 6 3 ⟶ 4 2 6 5 2 3 , 0 1 7 7 ⟶ 0 7 7 1 2 2 , 4 5 2 1 3 ⟶ 4 1 5 3 , 0 5 2 7 3 ⟶ 0 7 5 3 3 2 , 0 6 3 2 7 ⟶ 0 7 3 6 , 1 1 1 →= 2 1 2 1 1 , 1 6 1 →= 2 2 6 1 1 , 6 5 1 →= 6 1 2 2 2 5 , 5 1 3 →= 2 5 3 2 1 , 1 6 3 →= 3 2 6 1 , 1 6 3 →= 2 6 1 3 , 1 6 3 →= 3 2 6 2 1 , 1 6 3 →= 5 2 6 2 1 3 , 5 6 3 →= 5 3 3 2 6 2 , 5 6 3 →= 3 2 2 6 2 5 , 7 6 3 →= 6 7 3 2 , 7 6 3 →= 6 7 5 2 3 , 7 6 3 →= 6 7 3 3 2 2 , 7 6 3 →= 6 5 7 5 3 2 , 1 7 3 →= 2 2 7 1 2 3 , 6 1 7 →= 7 6 1 2 2 , 5 1 7 →= 2 2 5 7 1 , 5 6 7 →= 5 3 2 2 6 7 , 1 7 7 →= 1 7 2 2 7 , 5 6 1 1 →= 2 2 6 1 5 1 , 6 2 3 1 →= 6 1 3 2 2 , 5 6 1 3 →= 2 2 6 5 1 3 , 5 7 1 3 →= 3 2 7 1 5 3 , 1 1 2 3 →= 2 1 3 2 2 1 , 1 5 6 3 →= 6 5 1 2 3 , 7 5 6 3 →= 7 2 6 5 2 3 , 5 7 6 3 →= 6 7 3 5 3 , 6 5 1 5 →= 1 3 2 6 5 5 , 1 6 3 5 →= 5 3 2 6 1 , 1 2 5 7 →= 2 1 5 2 2 7 , 1 1 7 7 →= 1 7 7 1 2 2 , 6 6 7 7 →= 6 6 2 2 7 7 , 5 6 2 1 1 →= 5 2 2 6 1 1 , 6 2 3 7 1 →= 1 2 3 2 6 7 , 1 2 7 7 1 →= 2 2 7 1 1 7 , 7 5 2 1 3 →= 2 2 7 1 5 3 , 6 2 3 1 3 →= 3 3 2 2 6 1 , 5 1 6 6 3 →= 5 2 1 6 6 3 , 1 5 2 7 3 →= 1 7 5 3 3 2 , 5 1 3 7 3 →= 5 3 7 2 1 3 , 1 6 3 2 7 →= 2 2 1 7 3 6 , 6 1 6 5 7 →= 6 2 6 5 1 7 } 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 1 ⎟ ⎜ 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 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 1 0 1 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 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 47-rule system { 0 1 2 3 ⟶ 0 3 2 2 1 , 4 5 6 3 ⟶ 4 2 6 5 2 3 , 4 5 2 1 3 ⟶ 4 1 5 3 , 0 5 2 7 3 ⟶ 0 7 5 3 3 2 , 0 6 3 2 7 ⟶ 0 7 3 6 , 1 1 1 →= 2 1 2 1 1 , 1 6 1 →= 2 2 6 1 1 , 6 5 1 →= 6 1 2 2 2 5 , 5 1 3 →= 2 5 3 2 1 , 1 6 3 →= 3 2 6 1 , 1 6 3 →= 2 6 1 3 , 1 6 3 →= 3 2 6 2 1 , 1 6 3 →= 5 2 6 2 1 3 , 5 6 3 →= 5 3 3 2 6 2 , 5 6 3 →= 3 2 2 6 2 5 , 7 6 3 →= 6 7 3 2 , 7 6 3 →= 6 7 5 2 3 , 7 6 3 →= 6 7 3 3 2 2 , 7 6 3 →= 6 5 7 5 3 2 , 1 7 3 →= 2 2 7 1 2 3 , 6 1 7 →= 7 6 1 2 2 , 5 1 7 →= 2 2 5 7 1 , 5 6 7 →= 5 3 2 2 6 7 , 1 7 7 →= 1 7 2 2 7 , 5 6 1 1 →= 2 2 6 1 5 1 , 6 2 3 1 →= 6 1 3 2 2 , 5 6 1 3 →= 2 2 6 5 1 3 , 5 7 1 3 →= 3 2 7 1 5 3 , 1 1 2 3 →= 2 1 3 2 2 1 , 1 5 6 3 →= 6 5 1 2 3 , 7 5 6 3 →= 7 2 6 5 2 3 , 5 7 6 3 →= 6 7 3 5 3 , 6 5 1 5 →= 1 3 2 6 5 5 , 1 6 3 5 →= 5 3 2 6 1 , 1 2 5 7 →= 2 1 5 2 2 7 , 1 1 7 7 →= 1 7 7 1 2 2 , 6 6 7 7 →= 6 6 2 2 7 7 , 5 6 2 1 1 →= 5 2 2 6 1 1 , 6 2 3 7 1 →= 1 2 3 2 6 7 , 1 2 7 7 1 →= 2 2 7 1 1 7 , 7 5 2 1 3 →= 2 2 7 1 5 3 , 6 2 3 1 3 →= 3 3 2 2 6 1 , 5 1 6 6 3 →= 5 2 1 6 6 3 , 1 5 2 7 3 →= 1 7 5 3 3 2 , 5 1 3 7 3 →= 5 3 7 2 1 3 , 1 6 3 2 7 →= 2 2 1 7 3 6 , 6 1 6 5 7 →= 6 2 6 5 1 7 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 13: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 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 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 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 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 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 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 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 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 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 1 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 1 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 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 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 46-rule system { 0 1 2 3 ⟶ 0 3 2 2 1 , 4 5 6 3 ⟶ 4 2 6 5 2 3 , 4 5 2 1 3 ⟶ 4 1 5 3 , 0 5 2 7 3 ⟶ 0 7 5 3 3 2 , 0 6 3 2 7 ⟶ 0 7 3 6 , 1 1 1 →= 2 1 2 1 1 , 1 6 1 →= 2 2 6 1 1 , 6 5 1 →= 6 1 2 2 2 5 , 5 1 3 →= 2 5 3 2 1 , 1 6 3 →= 3 2 6 1 , 1 6 3 →= 2 6 1 3 , 1 6 3 →= 3 2 6 2 1 , 1 6 3 →= 5 2 6 2 1 3 , 5 6 3 →= 5 3 3 2 6 2 , 5 6 3 →= 3 2 2 6 2 5 , 7 6 3 →= 6 7 3 2 , 7 6 3 →= 6 7 5 2 3 , 7 6 3 →= 6 7 3 3 2 2 , 7 6 3 →= 6 5 7 5 3 2 , 1 7 3 →= 2 2 7 1 2 3 , 6 1 7 →= 7 6 1 2 2 , 5 1 7 →= 2 2 5 7 1 , 5 6 7 →= 5 3 2 2 6 7 , 1 7 7 →= 1 7 2 2 7 , 5 6 1 1 →= 2 2 6 1 5 1 , 6 2 3 1 →= 6 1 3 2 2 , 5 6 1 3 →= 2 2 6 5 1 3 , 5 7 1 3 →= 3 2 7 1 5 3 , 1 1 2 3 →= 2 1 3 2 2 1 , 1 5 6 3 →= 6 5 1 2 3 , 7 5 6 3 →= 7 2 6 5 2 3 , 5 7 6 3 →= 6 7 3 5 3 , 1 6 3 5 →= 5 3 2 6 1 , 1 2 5 7 →= 2 1 5 2 2 7 , 1 1 7 7 →= 1 7 7 1 2 2 , 6 6 7 7 →= 6 6 2 2 7 7 , 5 6 2 1 1 →= 5 2 2 6 1 1 , 6 2 3 7 1 →= 1 2 3 2 6 7 , 1 2 7 7 1 →= 2 2 7 1 1 7 , 7 5 2 1 3 →= 2 2 7 1 5 3 , 6 2 3 1 3 →= 3 3 2 2 6 1 , 5 1 6 6 3 →= 5 2 1 6 6 3 , 1 5 2 7 3 →= 1 7 5 3 3 2 , 5 1 3 7 3 →= 5 3 7 2 1 3 , 1 6 3 2 7 →= 2 2 1 7 3 6 , 6 1 6 5 7 →= 6 2 6 5 1 7 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 10: 0 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 0 ⎟ ⎜ 0 1 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 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 1 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 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 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 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 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 1 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 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 45-rule system { 0 1 2 3 ⟶ 0 3 2 2 1 , 4 5 6 3 ⟶ 4 2 6 5 2 3 , 4 5 2 1 3 ⟶ 4 1 5 3 , 0 5 2 7 3 ⟶ 0 7 5 3 3 2 , 0 6 3 2 7 ⟶ 0 7 3 6 , 1 1 1 →= 2 1 2 1 1 , 1 6 1 →= 2 2 6 1 1 , 6 5 1 →= 6 1 2 2 2 5 , 5 1 3 →= 2 5 3 2 1 , 1 6 3 →= 3 2 6 1 , 1 6 3 →= 2 6 1 3 , 1 6 3 →= 3 2 6 2 1 , 1 6 3 →= 5 2 6 2 1 3 , 5 6 3 →= 5 3 3 2 6 2 , 5 6 3 →= 3 2 2 6 2 5 , 7 6 3 →= 6 7 3 2 , 7 6 3 →= 6 7 5 2 3 , 7 6 3 →= 6 7 3 3 2 2 , 7 6 3 →= 6 5 7 5 3 2 , 1 7 3 →= 2 2 7 1 2 3 , 6 1 7 →= 7 6 1 2 2 , 5 1 7 →= 2 2 5 7 1 , 5 6 7 →= 5 3 2 2 6 7 , 1 7 7 →= 1 7 2 2 7 , 5 6 1 1 →= 2 2 6 1 5 1 , 6 2 3 1 →= 6 1 3 2 2 , 5 6 1 3 →= 2 2 6 5 1 3 , 5 7 1 3 →= 3 2 7 1 5 3 , 1 1 2 3 →= 2 1 3 2 2 1 , 1 5 6 3 →= 6 5 1 2 3 , 7 5 6 3 →= 7 2 6 5 2 3 , 5 7 6 3 →= 6 7 3 5 3 , 1 6 3 5 →= 5 3 2 6 1 , 1 2 5 7 →= 2 1 5 2 2 7 , 1 1 7 7 →= 1 7 7 1 2 2 , 6 6 7 7 →= 6 6 2 2 7 7 , 6 2 3 7 1 →= 1 2 3 2 6 7 , 1 2 7 7 1 →= 2 2 7 1 1 7 , 7 5 2 1 3 →= 2 2 7 1 5 3 , 6 2 3 1 3 →= 3 3 2 2 6 1 , 5 1 6 6 3 →= 5 2 1 6 6 3 , 1 5 2 7 3 →= 1 7 5 3 3 2 , 5 1 3 7 3 →= 5 3 7 2 1 3 , 1 6 3 2 7 →= 2 2 1 7 3 6 , 6 1 6 5 7 →= 6 2 6 5 1 7 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 9: 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 0 0 0 ⎟ ⎜ 0 0 0 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 ⎟ ⎜ 0 1 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 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 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 0 0 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 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 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 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 0 ⎟ ⎜ 0 0 0 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 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 0 0 0 ⎟ ⎜ 0 0 0 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 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 1 1 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 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎝ ⎠ 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 44-rule system { 0 1 2 3 ⟶ 0 3 2 2 1 , 4 5 6 3 ⟶ 4 2 6 5 2 3 , 4 5 2 1 3 ⟶ 4 1 5 3 , 0 5 2 7 3 ⟶ 0 7 5 3 3 2 , 0 6 3 2 7 ⟶ 0 7 3 6 , 1 1 1 →= 2 1 2 1 1 , 1 6 1 →= 2 2 6 1 1 , 6 5 1 →= 6 1 2 2 2 5 , 5 1 3 →= 2 5 3 2 1 , 1 6 3 →= 3 2 6 1 , 1 6 3 →= 2 6 1 3 , 1 6 3 →= 3 2 6 2 1 , 1 6 3 →= 5 2 6 2 1 3 , 5 6 3 →= 5 3 3 2 6 2 , 5 6 3 →= 3 2 2 6 2 5 , 7 6 3 →= 6 7 3 2 , 7 6 3 →= 6 7 5 2 3 , 7 6 3 →= 6 7 3 3 2 2 , 7 6 3 →= 6 5 7 5 3 2 , 1 7 3 →= 2 2 7 1 2 3 , 6 1 7 →= 7 6 1 2 2 , 5 1 7 →= 2 2 5 7 1 , 5 6 7 →= 5 3 2 2 6 7 , 1 7 7 →= 1 7 2 2 7 , 6 2 3 1 →= 6 1 3 2 2 , 5 6 1 3 →= 2 2 6 5 1 3 , 5 7 1 3 →= 3 2 7 1 5 3 , 1 1 2 3 →= 2 1 3 2 2 1 , 1 5 6 3 →= 6 5 1 2 3 , 7 5 6 3 →= 7 2 6 5 2 3 , 5 7 6 3 →= 6 7 3 5 3 , 1 6 3 5 →= 5 3 2 6 1 , 1 2 5 7 →= 2 1 5 2 2 7 , 1 1 7 7 →= 1 7 7 1 2 2 , 6 6 7 7 →= 6 6 2 2 7 7 , 6 2 3 7 1 →= 1 2 3 2 6 7 , 1 2 7 7 1 →= 2 2 7 1 1 7 , 7 5 2 1 3 →= 2 2 7 1 5 3 , 6 2 3 1 3 →= 3 3 2 2 6 1 , 5 1 6 6 3 →= 5 2 1 6 6 3 , 1 5 2 7 3 →= 1 7 5 3 3 2 , 5 1 3 7 3 →= 5 3 7 2 1 3 , 1 6 3 2 7 →= 2 2 1 7 3 6 , 6 1 6 5 7 →= 6 2 6 5 1 7 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 10: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 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 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 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 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 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 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 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 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 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 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 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 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 1 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 43-rule system { 0 1 2 3 ⟶ 0 3 2 2 1 , 4 5 6 3 ⟶ 4 2 6 5 2 3 , 4 5 2 1 3 ⟶ 4 1 5 3 , 0 5 2 7 3 ⟶ 0 7 5 3 3 2 , 0 6 3 2 7 ⟶ 0 7 3 6 , 1 1 1 →= 2 1 2 1 1 , 1 6 1 →= 2 2 6 1 1 , 6 5 1 →= 6 1 2 2 2 5 , 5 1 3 →= 2 5 3 2 1 , 1 6 3 →= 3 2 6 1 , 1 6 3 →= 2 6 1 3 , 1 6 3 →= 3 2 6 2 1 , 1 6 3 →= 5 2 6 2 1 3 , 5 6 3 →= 5 3 3 2 6 2 , 5 6 3 →= 3 2 2 6 2 5 , 7 6 3 →= 6 7 3 2 , 7 6 3 →= 6 7 5 2 3 , 7 6 3 →= 6 7 3 3 2 2 , 7 6 3 →= 6 5 7 5 3 2 , 1 7 3 →= 2 2 7 1 2 3 , 6 1 7 →= 7 6 1 2 2 , 5 1 7 →= 2 2 5 7 1 , 5 6 7 →= 5 3 2 2 6 7 , 1 7 7 →= 1 7 2 2 7 , 6 2 3 1 →= 6 1 3 2 2 , 5 6 1 3 →= 2 2 6 5 1 3 , 5 7 1 3 →= 3 2 7 1 5 3 , 1 1 2 3 →= 2 1 3 2 2 1 , 1 5 6 3 →= 6 5 1 2 3 , 7 5 6 3 →= 7 2 6 5 2 3 , 5 7 6 3 →= 6 7 3 5 3 , 1 6 3 5 →= 5 3 2 6 1 , 1 2 5 7 →= 2 1 5 2 2 7 , 1 1 7 7 →= 1 7 7 1 2 2 , 6 6 7 7 →= 6 6 2 2 7 7 , 1 2 7 7 1 →= 2 2 7 1 1 7 , 7 5 2 1 3 →= 2 2 7 1 5 3 , 6 2 3 1 3 →= 3 3 2 2 6 1 , 5 1 6 6 3 →= 5 2 1 6 6 3 , 1 5 2 7 3 →= 1 7 5 3 3 2 , 5 1 3 7 3 →= 5 3 7 2 1 3 , 1 6 3 2 7 →= 2 2 1 7 3 6 , 6 1 6 5 7 →= 6 2 6 5 1 7 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 9: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 ⎟ ⎜ 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 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 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 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 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 1 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 ⎟ ⎜ 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 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 ⎟ ⎜ 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 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 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 0 0 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 ⎟ ⎜ 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 0 0 0 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, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 41-rule system { 0 1 2 3 ⟶ 0 3 2 2 1 , 4 5 6 3 ⟶ 4 2 6 5 2 3 , 4 5 2 1 3 ⟶ 4 1 5 3 , 0 5 2 7 3 ⟶ 0 7 5 3 3 2 , 0 6 3 2 7 ⟶ 0 7 3 6 , 1 1 1 →= 2 1 2 1 1 , 1 6 1 →= 2 2 6 1 1 , 6 5 1 →= 6 1 2 2 2 5 , 5 1 3 →= 2 5 3 2 1 , 1 6 3 →= 3 2 6 1 , 1 6 3 →= 2 6 1 3 , 1 6 3 →= 3 2 6 2 1 , 1 6 3 →= 5 2 6 2 1 3 , 5 6 3 →= 5 3 3 2 6 2 , 5 6 3 →= 3 2 2 6 2 5 , 7 6 3 →= 6 7 3 2 , 7 6 3 →= 6 7 5 2 3 , 7 6 3 →= 6 7 3 3 2 2 , 7 6 3 →= 6 5 7 5 3 2 , 1 7 3 →= 2 2 7 1 2 3 , 6 1 7 →= 7 6 1 2 2 , 5 1 7 →= 2 2 5 7 1 , 5 6 7 →= 5 3 2 2 6 7 , 1 7 7 →= 1 7 2 2 7 , 5 6 1 3 →= 2 2 6 5 1 3 , 5 7 1 3 →= 3 2 7 1 5 3 , 1 1 2 3 →= 2 1 3 2 2 1 , 1 5 6 3 →= 6 5 1 2 3 , 7 5 6 3 →= 7 2 6 5 2 3 , 5 7 6 3 →= 6 7 3 5 3 , 1 6 3 5 →= 5 3 2 6 1 , 1 2 5 7 →= 2 1 5 2 2 7 , 1 1 7 7 →= 1 7 7 1 2 2 , 6 6 7 7 →= 6 6 2 2 7 7 , 1 2 7 7 1 →= 2 2 7 1 1 7 , 7 5 2 1 3 →= 2 2 7 1 5 3 , 5 1 6 6 3 →= 5 2 1 6 6 3 , 1 5 2 7 3 →= 1 7 5 3 3 2 , 5 1 3 7 3 →= 5 3 7 2 1 3 , 1 6 3 2 7 →= 2 2 1 7 3 6 , 6 1 6 5 7 →= 6 2 6 5 1 7 } 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 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 1 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 1 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 1 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 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 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 40-rule system { 0 1 2 3 ⟶ 0 3 2 2 1 , 4 5 6 3 ⟶ 4 2 6 5 2 3 , 4 5 2 1 3 ⟶ 4 1 5 3 , 0 6 3 2 7 ⟶ 0 7 3 6 , 1 1 1 →= 2 1 2 1 1 , 1 6 1 →= 2 2 6 1 1 , 6 5 1 →= 6 1 2 2 2 5 , 5 1 3 →= 2 5 3 2 1 , 1 6 3 →= 3 2 6 1 , 1 6 3 →= 2 6 1 3 , 1 6 3 →= 3 2 6 2 1 , 1 6 3 →= 5 2 6 2 1 3 , 5 6 3 →= 5 3 3 2 6 2 , 5 6 3 →= 3 2 2 6 2 5 , 7 6 3 →= 6 7 3 2 , 7 6 3 →= 6 7 5 2 3 , 7 6 3 →= 6 7 3 3 2 2 , 7 6 3 →= 6 5 7 5 3 2 , 1 7 3 →= 2 2 7 1 2 3 , 6 1 7 →= 7 6 1 2 2 , 5 1 7 →= 2 2 5 7 1 , 5 6 7 →= 5 3 2 2 6 7 , 1 7 7 →= 1 7 2 2 7 , 5 6 1 3 →= 2 2 6 5 1 3 , 5 7 1 3 →= 3 2 7 1 5 3 , 1 1 2 3 →= 2 1 3 2 2 1 , 1 5 6 3 →= 6 5 1 2 3 , 7 5 6 3 →= 7 2 6 5 2 3 , 5 7 6 3 →= 6 7 3 5 3 , 1 6 3 5 →= 5 3 2 6 1 , 1 2 5 7 →= 2 1 5 2 2 7 , 1 1 7 7 →= 1 7 7 1 2 2 , 6 6 7 7 →= 6 6 2 2 7 7 , 1 2 7 7 1 →= 2 2 7 1 1 7 , 7 5 2 1 3 →= 2 2 7 1 5 3 , 5 1 6 6 3 →= 5 2 1 6 6 3 , 1 5 2 7 3 →= 1 7 5 3 3 2 , 5 1 3 7 3 →= 5 3 7 2 1 3 , 1 6 3 2 7 →= 2 2 1 7 3 6 , 6 1 6 5 7 →= 6 2 6 5 1 7 } 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 1 0 0 0 ⎟ ⎜ 0 0 0 1 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 1 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 1 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 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 39-rule system { 0 1 2 3 ⟶ 0 3 2 2 1 , 4 5 2 1 3 ⟶ 4 1 5 3 , 0 6 3 2 7 ⟶ 0 7 3 6 , 1 1 1 →= 2 1 2 1 1 , 1 6 1 →= 2 2 6 1 1 , 6 5 1 →= 6 1 2 2 2 5 , 5 1 3 →= 2 5 3 2 1 , 1 6 3 →= 3 2 6 1 , 1 6 3 →= 2 6 1 3 , 1 6 3 →= 3 2 6 2 1 , 1 6 3 →= 5 2 6 2 1 3 , 5 6 3 →= 5 3 3 2 6 2 , 5 6 3 →= 3 2 2 6 2 5 , 7 6 3 →= 6 7 3 2 , 7 6 3 →= 6 7 5 2 3 , 7 6 3 →= 6 7 3 3 2 2 , 7 6 3 →= 6 5 7 5 3 2 , 1 7 3 →= 2 2 7 1 2 3 , 6 1 7 →= 7 6 1 2 2 , 5 1 7 →= 2 2 5 7 1 , 5 6 7 →= 5 3 2 2 6 7 , 1 7 7 →= 1 7 2 2 7 , 5 6 1 3 →= 2 2 6 5 1 3 , 5 7 1 3 →= 3 2 7 1 5 3 , 1 1 2 3 →= 2 1 3 2 2 1 , 1 5 6 3 →= 6 5 1 2 3 , 7 5 6 3 →= 7 2 6 5 2 3 , 5 7 6 3 →= 6 7 3 5 3 , 1 6 3 5 →= 5 3 2 6 1 , 1 2 5 7 →= 2 1 5 2 2 7 , 1 1 7 7 →= 1 7 7 1 2 2 , 6 6 7 7 →= 6 6 2 2 7 7 , 1 2 7 7 1 →= 2 2 7 1 1 7 , 7 5 2 1 3 →= 2 2 7 1 5 3 , 5 1 6 6 3 →= 5 2 1 6 6 3 , 1 5 2 7 3 →= 1 7 5 3 3 2 , 5 1 3 7 3 →= 5 3 7 2 1 3 , 1 6 3 2 7 →= 2 2 1 7 3 6 , 6 1 6 5 7 →= 6 2 6 5 1 7 } 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 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎜ 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 1 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 1 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 1 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 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 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 6 ↦ 4, 7 ↦ 5, 5 ↦ 6 }, it remains to prove termination of the 38-rule system { 0 1 2 3 ⟶ 0 3 2 2 1 , 0 4 3 2 5 ⟶ 0 5 3 4 , 1 1 1 →= 2 1 2 1 1 , 1 4 1 →= 2 2 4 1 1 , 4 6 1 →= 4 1 2 2 2 6 , 6 1 3 →= 2 6 3 2 1 , 1 4 3 →= 3 2 4 1 , 1 4 3 →= 2 4 1 3 , 1 4 3 →= 3 2 4 2 1 , 1 4 3 →= 6 2 4 2 1 3 , 6 4 3 →= 6 3 3 2 4 2 , 6 4 3 →= 3 2 2 4 2 6 , 5 4 3 →= 4 5 3 2 , 5 4 3 →= 4 5 6 2 3 , 5 4 3 →= 4 5 3 3 2 2 , 5 4 3 →= 4 6 5 6 3 2 , 1 5 3 →= 2 2 5 1 2 3 , 4 1 5 →= 5 4 1 2 2 , 6 1 5 →= 2 2 6 5 1 , 6 4 5 →= 6 3 2 2 4 5 , 1 5 5 →= 1 5 2 2 5 , 6 4 1 3 →= 2 2 4 6 1 3 , 6 5 1 3 →= 3 2 5 1 6 3 , 1 1 2 3 →= 2 1 3 2 2 1 , 1 6 4 3 →= 4 6 1 2 3 , 5 6 4 3 →= 5 2 4 6 2 3 , 6 5 4 3 →= 4 5 3 6 3 , 1 4 3 6 →= 6 3 2 4 1 , 1 2 6 5 →= 2 1 6 2 2 5 , 1 1 5 5 →= 1 5 5 1 2 2 , 4 4 5 5 →= 4 4 2 2 5 5 , 1 2 5 5 1 →= 2 2 5 1 1 5 , 5 6 2 1 3 →= 2 2 5 1 6 3 , 6 1 4 4 3 →= 6 2 1 4 4 3 , 1 6 2 5 3 →= 1 5 6 3 3 2 , 6 1 3 5 3 →= 6 3 5 2 1 3 , 1 4 3 2 5 →= 2 2 1 5 3 4 , 4 1 4 6 5 →= 4 2 4 6 1 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 0 1 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 1 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 1 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 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 }, it remains to prove termination of the 33-rule system { 0 1 2 3 ⟶ 0 3 2 2 1 , 0 4 3 2 5 ⟶ 0 5 3 4 , 1 1 1 →= 2 1 2 1 1 , 1 4 1 →= 2 2 4 1 1 , 4 6 1 →= 4 1 2 2 2 6 , 6 1 3 →= 2 6 3 2 1 , 1 4 3 →= 3 2 4 1 , 1 4 3 →= 2 4 1 3 , 1 4 3 →= 3 2 4 2 1 , 1 4 3 →= 6 2 4 2 1 3 , 6 4 3 →= 6 3 3 2 4 2 , 6 4 3 →= 3 2 2 4 2 6 , 1 5 3 →= 2 2 5 1 2 3 , 4 1 5 →= 5 4 1 2 2 , 6 1 5 →= 2 2 6 5 1 , 6 4 5 →= 6 3 2 2 4 5 , 1 5 5 →= 1 5 2 2 5 , 6 4 1 3 →= 2 2 4 6 1 3 , 6 5 1 3 →= 3 2 5 1 6 3 , 1 1 2 3 →= 2 1 3 2 2 1 , 1 6 4 3 →= 4 6 1 2 3 , 5 6 4 3 →= 5 2 4 6 2 3 , 1 4 3 6 →= 6 3 2 4 1 , 1 2 6 5 →= 2 1 6 2 2 5 , 1 1 5 5 →= 1 5 5 1 2 2 , 4 4 5 5 →= 4 4 2 2 5 5 , 1 2 5 5 1 →= 2 2 5 1 1 5 , 5 6 2 1 3 →= 2 2 5 1 6 3 , 6 1 4 4 3 →= 6 2 1 4 4 3 , 1 6 2 5 3 →= 1 5 6 3 3 2 , 6 1 3 5 3 →= 6 3 5 2 1 3 , 1 4 3 2 5 →= 2 2 1 5 3 4 , 4 1 4 6 5 →= 4 2 4 6 1 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 1 0 ⎟ ⎜ 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 1 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎜ 0 1 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 1 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 30-rule system { 0 1 2 3 ⟶ 0 3 2 2 1 , 0 4 3 2 5 ⟶ 0 5 3 4 , 1 1 1 →= 2 1 2 1 1 , 1 4 1 →= 2 2 4 1 1 , 4 6 1 →= 4 1 2 2 2 6 , 6 1 3 →= 2 6 3 2 1 , 1 4 3 →= 3 2 4 1 , 1 4 3 →= 2 4 1 3 , 1 4 3 →= 3 2 4 2 1 , 1 4 3 →= 6 2 4 2 1 3 , 6 4 3 →= 6 3 3 2 4 2 , 6 4 3 →= 3 2 2 4 2 6 , 1 5 3 →= 2 2 5 1 2 3 , 4 1 5 →= 5 4 1 2 2 , 6 1 5 →= 2 2 6 5 1 , 1 5 5 →= 1 5 2 2 5 , 6 5 1 3 →= 3 2 5 1 6 3 , 1 1 2 3 →= 2 1 3 2 2 1 , 1 6 4 3 →= 4 6 1 2 3 , 5 6 4 3 →= 5 2 4 6 2 3 , 1 4 3 6 →= 6 3 2 4 1 , 1 2 6 5 →= 2 1 6 2 2 5 , 1 1 5 5 →= 1 5 5 1 2 2 , 4 4 5 5 →= 4 4 2 2 5 5 , 1 2 5 5 1 →= 2 2 5 1 1 5 , 5 6 2 1 3 →= 2 2 5 1 6 3 , 1 6 2 5 3 →= 1 5 6 3 3 2 , 6 1 3 5 3 →= 6 3 5 2 1 3 , 1 4 3 2 5 →= 2 2 1 5 3 4 , 4 1 4 6 5 →= 4 2 4 6 1 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 1 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 1 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 1 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 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 29-rule system { 0 1 2 3 ⟶ 0 3 2 2 1 , 0 4 3 2 5 ⟶ 0 5 3 4 , 1 1 1 →= 2 1 2 1 1 , 1 4 1 →= 2 2 4 1 1 , 4 6 1 →= 4 1 2 2 2 6 , 6 1 3 →= 2 6 3 2 1 , 1 4 3 →= 3 2 4 1 , 1 4 3 →= 2 4 1 3 , 1 4 3 →= 3 2 4 2 1 , 1 4 3 →= 6 2 4 2 1 3 , 6 4 3 →= 6 3 3 2 4 2 , 6 4 3 →= 3 2 2 4 2 6 , 1 5 3 →= 2 2 5 1 2 3 , 4 1 5 →= 5 4 1 2 2 , 6 1 5 →= 2 2 6 5 1 , 1 5 5 →= 1 5 2 2 5 , 6 5 1 3 →= 3 2 5 1 6 3 , 1 1 2 3 →= 2 1 3 2 2 1 , 1 6 4 3 →= 4 6 1 2 3 , 1 4 3 6 →= 6 3 2 4 1 , 1 2 6 5 →= 2 1 6 2 2 5 , 1 1 5 5 →= 1 5 5 1 2 2 , 4 4 5 5 →= 4 4 2 2 5 5 , 1 2 5 5 1 →= 2 2 5 1 1 5 , 5 6 2 1 3 →= 2 2 5 1 6 3 , 1 6 2 5 3 →= 1 5 6 3 3 2 , 6 1 3 5 3 →= 6 3 5 2 1 3 , 1 4 3 2 5 →= 2 2 1 5 3 4 , 4 1 4 6 5 →= 4 2 4 6 1 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 9: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 ⎟ ⎜ 0 1 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 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 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 1 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 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 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 1 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎝ ⎠ 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 28-rule system { 0 1 2 3 ⟶ 0 3 2 2 1 , 0 4 3 2 5 ⟶ 0 5 3 4 , 1 1 1 →= 2 1 2 1 1 , 1 4 1 →= 2 2 4 1 1 , 4 6 1 →= 4 1 2 2 2 6 , 6 1 3 →= 2 6 3 2 1 , 1 4 3 →= 3 2 4 1 , 1 4 3 →= 2 4 1 3 , 1 4 3 →= 3 2 4 2 1 , 1 4 3 →= 6 2 4 2 1 3 , 6 4 3 →= 6 3 3 2 4 2 , 6 4 3 →= 3 2 2 4 2 6 , 1 5 3 →= 2 2 5 1 2 3 , 4 1 5 →= 5 4 1 2 2 , 6 1 5 →= 2 2 6 5 1 , 1 5 5 →= 1 5 2 2 5 , 6 5 1 3 →= 3 2 5 1 6 3 , 1 1 2 3 →= 2 1 3 2 2 1 , 1 6 4 3 →= 4 6 1 2 3 , 1 4 3 6 →= 6 3 2 4 1 , 1 2 6 5 →= 2 1 6 2 2 5 , 1 1 5 5 →= 1 5 5 1 2 2 , 1 2 5 5 1 →= 2 2 5 1 1 5 , 5 6 2 1 3 →= 2 2 5 1 6 3 , 1 6 2 5 3 →= 1 5 6 3 3 2 , 6 1 3 5 3 →= 6 3 5 2 1 3 , 1 4 3 2 5 →= 2 2 1 5 3 4 , 4 1 4 6 5 →= 4 2 4 6 1 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 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 0 0 ⎟ ⎜ 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 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 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 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 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 ⎟ ⎜ 0 2 0 0 0 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 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 1 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 0 0 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 21-rule system { 0 1 2 3 ⟶ 0 3 2 2 1 , 0 4 3 2 5 ⟶ 0 5 3 4 , 1 1 1 →= 2 1 2 1 1 , 6 1 3 →= 2 6 3 2 1 , 6 4 3 →= 6 3 3 2 4 2 , 6 4 3 →= 3 2 2 4 2 6 , 1 5 3 →= 2 2 5 1 2 3 , 4 1 5 →= 5 4 1 2 2 , 6 1 5 →= 2 2 6 5 1 , 1 5 5 →= 1 5 2 2 5 , 6 5 1 3 →= 3 2 5 1 6 3 , 1 1 2 3 →= 2 1 3 2 2 1 , 1 6 4 3 →= 4 6 1 2 3 , 1 2 6 5 →= 2 1 6 2 2 5 , 1 1 5 5 →= 1 5 5 1 2 2 , 1 2 5 5 1 →= 2 2 5 1 1 5 , 5 6 2 1 3 →= 2 2 5 1 6 3 , 1 6 2 5 3 →= 1 5 6 3 3 2 , 6 1 3 5 3 →= 6 3 5 2 1 3 , 1 4 3 2 5 →= 2 2 1 5 3 4 , 4 1 4 6 5 →= 4 2 4 6 1 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 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 1 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 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 4 ↦ 1, 3 ↦ 2, 2 ↦ 3, 5 ↦ 4, 1 ↦ 5, 6 ↦ 6 }, it remains to prove termination of the 20-rule system { 0 1 2 3 4 ⟶ 0 4 2 1 , 5 5 5 →= 3 5 3 5 5 , 6 5 2 →= 3 6 2 3 5 , 6 1 2 →= 6 2 2 3 1 3 , 6 1 2 →= 2 3 3 1 3 6 , 5 4 2 →= 3 3 4 5 3 2 , 1 5 4 →= 4 1 5 3 3 , 6 5 4 →= 3 3 6 4 5 , 5 4 4 →= 5 4 3 3 4 , 6 4 5 2 →= 2 3 4 5 6 2 , 5 5 3 2 →= 3 5 2 3 3 5 , 5 6 1 2 →= 1 6 5 3 2 , 5 3 6 4 →= 3 5 6 3 3 4 , 5 5 4 4 →= 5 4 4 5 3 3 , 5 3 4 4 5 →= 3 3 4 5 5 4 , 4 6 3 5 2 →= 3 3 4 5 6 2 , 5 6 3 4 2 →= 5 4 6 2 2 3 , 6 5 2 4 2 →= 6 2 4 3 5 2 , 5 1 2 3 4 →= 3 3 5 4 2 1 , 1 5 1 6 4 →= 1 3 1 6 5 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 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 1 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 1 ⎟ ⎜ 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 1 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 1 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 5 ↦ 0, 3 ↦ 1, 6 ↦ 2, 2 ↦ 3, 1 ↦ 4, 4 ↦ 5 }, it remains to prove termination of the 19-rule system { 0 0 0 →= 1 0 1 0 0 , 2 0 3 →= 1 2 3 1 0 , 2 4 3 →= 2 3 3 1 4 1 , 2 4 3 →= 3 1 1 4 1 2 , 0 5 3 →= 1 1 5 0 1 3 , 4 0 5 →= 5 4 0 1 1 , 2 0 5 →= 1 1 2 5 0 , 0 5 5 →= 0 5 1 1 5 , 2 5 0 3 →= 3 1 5 0 2 3 , 0 0 1 3 →= 1 0 3 1 1 0 , 0 2 4 3 →= 4 2 0 1 3 , 0 1 2 5 →= 1 0 2 1 1 5 , 0 0 5 5 →= 0 5 5 0 1 1 , 0 1 5 5 0 →= 1 1 5 0 0 5 , 5 2 1 0 3 →= 1 1 5 0 2 3 , 0 2 1 5 3 →= 0 5 2 3 3 1 , 2 0 3 5 3 →= 2 3 5 1 0 3 , 0 4 3 1 5 →= 1 1 0 5 3 4 , 4 0 4 2 5 →= 4 1 4 2 0 5 } The system is trivially terminating.