/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 }, it remains to prove termination of the 44-rule system { 0 0 0 1 1 0 0 0 0 1 2 0 1 ⟶ 1 1 1 1 0 0 0 1 0 0 1 1 0 1 1 0 0 , 0 0 0 1 1 2 2 1 1 0 0 1 1 ⟶ 0 1 1 0 2 0 0 0 1 0 1 1 0 0 0 1 0 , 0 1 0 0 2 1 0 0 1 0 0 1 0 ⟶ 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 , 0 1 0 1 1 0 0 0 0 2 0 1 2 ⟶ 1 1 1 0 0 0 1 0 1 1 0 1 1 1 0 0 2 , 0 1 0 1 1 0 2 1 0 1 0 0 0 ⟶ 0 1 1 0 1 1 0 0 1 0 0 1 0 1 1 1 0 , 0 1 1 0 0 1 0 2 0 0 0 2 1 ⟶ 0 0 1 1 0 1 0 1 1 0 0 1 1 2 1 1 1 , 0 1 2 1 0 0 0 0 0 2 1 1 1 ⟶ 0 1 0 1 1 2 0 1 0 0 0 1 1 0 0 0 1 , 0 2 0 0 2 1 0 0 0 0 0 0 1 ⟶ 1 0 0 1 0 1 1 1 0 0 0 1 0 0 0 0 1 , 0 2 1 0 1 2 1 2 2 0 1 1 1 ⟶ 0 2 1 1 0 0 2 1 1 0 0 1 2 1 1 1 1 , 0 2 1 0 2 1 0 1 2 0 0 0 1 ⟶ 0 0 2 1 1 1 1 0 0 1 0 2 0 0 0 1 0 , 0 2 1 1 0 0 0 0 2 1 0 1 1 ⟶ 0 0 1 0 1 0 1 0 2 1 1 1 0 0 0 0 0 , 0 2 2 0 0 1 1 0 1 0 1 0 1 ⟶ 0 0 2 1 0 0 0 1 1 0 1 1 0 1 1 0 0 , 1 0 0 0 0 1 1 1 0 0 2 1 1 ⟶ 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 0 , 1 0 0 0 1 1 1 2 0 0 1 0 2 ⟶ 0 1 1 0 1 0 0 1 0 0 0 0 0 2 0 1 2 , 1 0 0 0 1 2 0 1 0 0 0 1 0 ⟶ 1 1 1 0 0 0 1 1 0 1 1 0 1 1 1 0 1 , 1 0 0 0 2 1 0 0 2 0 1 0 1 ⟶ 2 1 0 0 1 1 0 1 0 0 0 0 0 1 0 0 0 , 1 0 0 1 0 1 2 1 0 2 1 1 0 ⟶ 0 1 1 1 0 0 0 0 0 2 1 1 1 1 1 1 0 , 1 0 0 1 0 2 1 1 0 1 1 0 1 ⟶ 1 1 1 1 0 0 1 0 1 0 1 1 1 0 0 1 1 , 1 0 0 1 2 0 0 0 2 2 1 1 2 ⟶ 1 0 0 0 1 0 2 2 0 1 1 1 0 1 2 1 2 , 1 0 0 1 2 1 1 1 0 2 0 2 1 ⟶ 1 0 0 1 0 0 1 2 0 1 0 1 2 2 1 0 0 , 1 0 0 2 0 1 0 1 0 2 1 0 2 ⟶ 0 1 0 0 0 2 0 1 1 0 1 0 1 1 0 0 2 , 1 0 1 0 1 2 1 1 2 1 0 0 1 ⟶ 1 1 0 0 1 0 0 2 2 0 0 0 0 1 1 0 0 , 1 0 1 2 0 1 2 1 0 0 1 0 0 ⟶ 0 0 0 0 0 1 1 0 1 1 1 1 0 2 1 0 1 , 1 0 2 2 1 0 1 0 1 0 1 0 2 ⟶ 1 1 1 0 0 0 0 0 0 1 1 0 0 0 2 1 2 , 1 1 0 1 0 2 2 2 1 0 0 0 0 ⟶ 1 1 0 1 1 0 1 0 1 0 1 0 0 2 0 0 1 , 1 1 0 2 2 0 0 1 2 1 2 1 2 ⟶ 1 1 2 0 0 0 1 1 0 2 2 1 1 0 1 0 2 , 1 1 1 1 0 1 0 2 1 1 0 0 2 ⟶ 1 1 1 0 0 1 0 1 1 0 1 0 0 2 1 0 2 , 1 1 1 2 1 1 0 2 0 2 0 1 0 ⟶ 1 1 1 0 0 1 0 2 2 0 0 1 2 1 0 1 0 , 1 1 2 1 1 0 1 1 0 1 1 0 2 ⟶ 1 1 2 0 1 0 1 0 0 0 0 0 1 0 0 1 2 , 1 1 2 2 1 1 1 1 0 1 1 0 0 ⟶ 1 0 1 1 1 2 1 1 0 1 0 1 1 1 0 0 0 , 1 1 2 2 2 1 0 0 1 1 0 0 0 ⟶ 0 1 0 1 1 1 1 1 0 2 1 0 1 1 0 0 0 , 1 2 0 1 0 1 1 1 0 1 1 1 1 ⟶ 1 0 0 1 1 1 0 1 0 0 1 1 0 1 1 0 1 , 1 2 1 0 0 0 0 1 0 1 1 2 1 ⟶ 0 1 0 0 1 0 0 0 1 1 0 0 2 1 0 0 0 , 1 2 1 0 1 0 0 1 0 2 0 1 1 ⟶ 1 0 0 1 0 0 0 2 2 0 0 1 0 0 1 1 0 , 1 2 2 0 1 1 1 1 1 1 0 1 0 ⟶ 1 0 0 2 2 0 0 1 0 0 1 0 0 0 1 1 0 , 2 1 0 0 1 1 0 1 2 1 0 1 1 ⟶ 1 0 0 0 1 0 0 0 0 2 1 0 0 1 1 0 1 , 2 1 0 1 2 0 0 2 0 1 0 1 2 ⟶ 2 1 0 0 1 0 0 0 0 1 0 2 2 2 0 1 2 , 2 1 0 2 2 2 2 1 0 0 0 0 1 ⟶ 1 1 1 0 0 0 1 1 2 1 0 0 1 2 0 1 0 , 2 1 1 0 1 0 1 1 1 2 1 1 0 ⟶ 1 1 2 1 1 0 0 1 1 0 0 1 1 0 1 0 0 , 2 1 1 0 2 1 0 2 1 0 0 0 1 ⟶ 0 1 0 1 1 1 1 2 0 1 0 0 1 0 0 1 0 , 2 1 1 1 1 1 1 1 0 0 2 0 1 ⟶ 1 1 1 0 1 1 1 0 1 0 0 1 0 2 0 1 0 , 2 2 1 0 0 2 1 1 1 1 1 0 1 ⟶ 2 2 2 0 0 0 1 0 1 0 1 1 0 1 1 1 0 , 2 2 1 0 0 2 1 2 0 0 1 0 2 ⟶ 2 1 0 0 0 2 1 0 0 1 2 0 0 1 1 1 2 , 2 2 2 0 0 0 1 0 0 0 0 1 0 ⟶ 0 2 0 0 1 1 1 0 1 1 0 0 0 1 0 1 0 } Applying sparse tiling TRFC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (0,0) ↦ 0, (0,1) ↦ 1, (1,1) ↦ 2, (1,0) ↦ 3, (1,2) ↦ 4, (2,0) ↦ 5, (0,2) ↦ 6, (1,4) ↦ 7, (0,4) ↦ 8, (2,1) ↦ 9, (3,0) ↦ 10, (3,1) ↦ 11, (2,2) ↦ 12, (2,4) ↦ 13, (3,2) ↦ 14 }, it remains to prove termination of the 704-rule system { 0 0 0 1 2 3 0 0 0 1 4 5 1 3 ⟶ 1 2 2 2 3 0 0 1 3 0 1 2 3 1 2 3 0 0 , 0 0 0 1 2 3 0 0 0 1 4 5 1 2 ⟶ 1 2 2 2 3 0 0 1 3 0 1 2 3 1 2 3 0 1 , 0 0 0 1 2 3 0 0 0 1 4 5 1 4 ⟶ 1 2 2 2 3 0 0 1 3 0 1 2 3 1 2 3 0 6 , 0 0 0 1 2 3 0 0 0 1 4 5 1 7 ⟶ 1 2 2 2 3 0 0 1 3 0 1 2 3 1 2 3 0 8 , 3 0 0 1 2 3 0 0 0 1 4 5 1 3 ⟶ 2 2 2 2 3 0 0 1 3 0 1 2 3 1 2 3 0 0 , 3 0 0 1 2 3 0 0 0 1 4 5 1 2 ⟶ 2 2 2 2 3 0 0 1 3 0 1 2 3 1 2 3 0 1 , 3 0 0 1 2 3 0 0 0 1 4 5 1 4 ⟶ 2 2 2 2 3 0 0 1 3 0 1 2 3 1 2 3 0 6 , 3 0 0 1 2 3 0 0 0 1 4 5 1 7 ⟶ 2 2 2 2 3 0 0 1 3 0 1 2 3 1 2 3 0 8 , 5 0 0 1 2 3 0 0 0 1 4 5 1 3 ⟶ 9 2 2 2 3 0 0 1 3 0 1 2 3 1 2 3 0 0 , 5 0 0 1 2 3 0 0 0 1 4 5 1 2 ⟶ 9 2 2 2 3 0 0 1 3 0 1 2 3 1 2 3 0 1 , 5 0 0 1 2 3 0 0 0 1 4 5 1 4 ⟶ 9 2 2 2 3 0 0 1 3 0 1 2 3 1 2 3 0 6 , 5 0 0 1 2 3 0 0 0 1 4 5 1 7 ⟶ 9 2 2 2 3 0 0 1 3 0 1 2 3 1 2 3 0 8 , 10 0 0 1 2 3 0 0 0 1 4 5 1 3 ⟶ 11 2 2 2 3 0 0 1 3 0 1 2 3 1 2 3 0 0 , 10 0 0 1 2 3 0 0 0 1 4 5 1 2 ⟶ 11 2 2 2 3 0 0 1 3 0 1 2 3 1 2 3 0 1 , 10 0 0 1 2 3 0 0 0 1 4 5 1 4 ⟶ 11 2 2 2 3 0 0 1 3 0 1 2 3 1 2 3 0 6 , 10 0 0 1 2 3 0 0 0 1 4 5 1 7 ⟶ 11 2 2 2 3 0 0 1 3 0 1 2 3 1 2 3 0 8 , 0 0 0 1 2 4 12 9 2 3 0 1 2 3 ⟶ 0 1 2 3 6 5 0 0 1 3 1 2 3 0 0 1 3 0 , 0 0 0 1 2 4 12 9 2 3 0 1 2 2 ⟶ 0 1 2 3 6 5 0 0 1 3 1 2 3 0 0 1 3 1 , 0 0 0 1 2 4 12 9 2 3 0 1 2 4 ⟶ 0 1 2 3 6 5 0 0 1 3 1 2 3 0 0 1 3 6 , 0 0 0 1 2 4 12 9 2 3 0 1 2 7 ⟶ 0 1 2 3 6 5 0 0 1 3 1 2 3 0 0 1 3 8 , 3 0 0 1 2 4 12 9 2 3 0 1 2 3 ⟶ 3 1 2 3 6 5 0 0 1 3 1 2 3 0 0 1 3 0 , 3 0 0 1 2 4 12 9 2 3 0 1 2 2 ⟶ 3 1 2 3 6 5 0 0 1 3 1 2 3 0 0 1 3 1 , 3 0 0 1 2 4 12 9 2 3 0 1 2 4 ⟶ 3 1 2 3 6 5 0 0 1 3 1 2 3 0 0 1 3 6 , 3 0 0 1 2 4 12 9 2 3 0 1 2 7 ⟶ 3 1 2 3 6 5 0 0 1 3 1 2 3 0 0 1 3 8 , 5 0 0 1 2 4 12 9 2 3 0 1 2 3 ⟶ 5 1 2 3 6 5 0 0 1 3 1 2 3 0 0 1 3 0 , 5 0 0 1 2 4 12 9 2 3 0 1 2 2 ⟶ 5 1 2 3 6 5 0 0 1 3 1 2 3 0 0 1 3 1 , 5 0 0 1 2 4 12 9 2 3 0 1 2 4 ⟶ 5 1 2 3 6 5 0 0 1 3 1 2 3 0 0 1 3 6 , 5 0 0 1 2 4 12 9 2 3 0 1 2 7 ⟶ 5 1 2 3 6 5 0 0 1 3 1 2 3 0 0 1 3 8 , 10 0 0 1 2 4 12 9 2 3 0 1 2 3 ⟶ 10 1 2 3 6 5 0 0 1 3 1 2 3 0 0 1 3 0 , 10 0 0 1 2 4 12 9 2 3 0 1 2 2 ⟶ 10 1 2 3 6 5 0 0 1 3 1 2 3 0 0 1 3 1 , 10 0 0 1 2 4 12 9 2 3 0 1 2 4 ⟶ 10 1 2 3 6 5 0 0 1 3 1 2 3 0 0 1 3 6 , 10 0 0 1 2 4 12 9 2 3 0 1 2 7 ⟶ 10 1 2 3 6 5 0 0 1 3 1 2 3 0 0 1 3 8 , 0 1 3 0 6 9 3 0 1 3 0 1 3 0 ⟶ 0 1 2 3 0 0 1 2 3 1 3 0 1 2 2 3 0 0 , 0 1 3 0 6 9 3 0 1 3 0 1 3 1 ⟶ 0 1 2 3 0 0 1 2 3 1 3 0 1 2 2 3 0 1 , 0 1 3 0 6 9 3 0 1 3 0 1 3 6 ⟶ 0 1 2 3 0 0 1 2 3 1 3 0 1 2 2 3 0 6 , 0 1 3 0 6 9 3 0 1 3 0 1 3 8 ⟶ 0 1 2 3 0 0 1 2 3 1 3 0 1 2 2 3 0 8 , 3 1 3 0 6 9 3 0 1 3 0 1 3 0 ⟶ 3 1 2 3 0 0 1 2 3 1 3 0 1 2 2 3 0 0 , 3 1 3 0 6 9 3 0 1 3 0 1 3 1 ⟶ 3 1 2 3 0 0 1 2 3 1 3 0 1 2 2 3 0 1 , 3 1 3 0 6 9 3 0 1 3 0 1 3 6 ⟶ 3 1 2 3 0 0 1 2 3 1 3 0 1 2 2 3 0 6 , 3 1 3 0 6 9 3 0 1 3 0 1 3 8 ⟶ 3 1 2 3 0 0 1 2 3 1 3 0 1 2 2 3 0 8 , 5 1 3 0 6 9 3 0 1 3 0 1 3 0 ⟶ 5 1 2 3 0 0 1 2 3 1 3 0 1 2 2 3 0 0 , 5 1 3 0 6 9 3 0 1 3 0 1 3 1 ⟶ 5 1 2 3 0 0 1 2 3 1 3 0 1 2 2 3 0 1 , 5 1 3 0 6 9 3 0 1 3 0 1 3 6 ⟶ 5 1 2 3 0 0 1 2 3 1 3 0 1 2 2 3 0 6 , 5 1 3 0 6 9 3 0 1 3 0 1 3 8 ⟶ 5 1 2 3 0 0 1 2 3 1 3 0 1 2 2 3 0 8 , 10 1 3 0 6 9 3 0 1 3 0 1 3 0 ⟶ 10 1 2 3 0 0 1 2 3 1 3 0 1 2 2 3 0 0 , 10 1 3 0 6 9 3 0 1 3 0 1 3 1 ⟶ 10 1 2 3 0 0 1 2 3 1 3 0 1 2 2 3 0 1 , 10 1 3 0 6 9 3 0 1 3 0 1 3 6 ⟶ 10 1 2 3 0 0 1 2 3 1 3 0 1 2 2 3 0 6 , 10 1 3 0 6 9 3 0 1 3 0 1 3 8 ⟶ 10 1 2 3 0 0 1 2 3 1 3 0 1 2 2 3 0 8 , 0 1 3 1 2 3 0 0 0 6 5 1 4 5 ⟶ 1 2 2 3 0 0 1 3 1 2 3 1 2 2 3 0 6 5 , 0 1 3 1 2 3 0 0 0 6 5 1 4 9 ⟶ 1 2 2 3 0 0 1 3 1 2 3 1 2 2 3 0 6 9 , 0 1 3 1 2 3 0 0 0 6 5 1 4 12 ⟶ 1 2 2 3 0 0 1 3 1 2 3 1 2 2 3 0 6 12 , 0 1 3 1 2 3 0 0 0 6 5 1 4 13 ⟶ 1 2 2 3 0 0 1 3 1 2 3 1 2 2 3 0 6 13 , 3 1 3 1 2 3 0 0 0 6 5 1 4 5 ⟶ 2 2 2 3 0 0 1 3 1 2 3 1 2 2 3 0 6 5 , 3 1 3 1 2 3 0 0 0 6 5 1 4 9 ⟶ 2 2 2 3 0 0 1 3 1 2 3 1 2 2 3 0 6 9 , 3 1 3 1 2 3 0 0 0 6 5 1 4 12 ⟶ 2 2 2 3 0 0 1 3 1 2 3 1 2 2 3 0 6 12 , 3 1 3 1 2 3 0 0 0 6 5 1 4 13 ⟶ 2 2 2 3 0 0 1 3 1 2 3 1 2 2 3 0 6 13 , 5 1 3 1 2 3 0 0 0 6 5 1 4 5 ⟶ 9 2 2 3 0 0 1 3 1 2 3 1 2 2 3 0 6 5 , 5 1 3 1 2 3 0 0 0 6 5 1 4 9 ⟶ 9 2 2 3 0 0 1 3 1 2 3 1 2 2 3 0 6 9 , 5 1 3 1 2 3 0 0 0 6 5 1 4 12 ⟶ 9 2 2 3 0 0 1 3 1 2 3 1 2 2 3 0 6 12 , 5 1 3 1 2 3 0 0 0 6 5 1 4 13 ⟶ 9 2 2 3 0 0 1 3 1 2 3 1 2 2 3 0 6 13 , 10 1 3 1 2 3 0 0 0 6 5 1 4 5 ⟶ 11 2 2 3 0 0 1 3 1 2 3 1 2 2 3 0 6 5 , 10 1 3 1 2 3 0 0 0 6 5 1 4 9 ⟶ 11 2 2 3 0 0 1 3 1 2 3 1 2 2 3 0 6 9 , 10 1 3 1 2 3 0 0 0 6 5 1 4 12 ⟶ 11 2 2 3 0 0 1 3 1 2 3 1 2 2 3 0 6 12 , 10 1 3 1 2 3 0 0 0 6 5 1 4 13 ⟶ 11 2 2 3 0 0 1 3 1 2 3 1 2 2 3 0 6 13 , 0 1 3 1 2 3 6 9 3 1 3 0 0 0 ⟶ 0 1 2 3 1 2 3 0 1 3 0 1 3 1 2 2 3 0 , 0 1 3 1 2 3 6 9 3 1 3 0 0 1 ⟶ 0 1 2 3 1 2 3 0 1 3 0 1 3 1 2 2 3 1 , 0 1 3 1 2 3 6 9 3 1 3 0 0 6 ⟶ 0 1 2 3 1 2 3 0 1 3 0 1 3 1 2 2 3 6 , 0 1 3 1 2 3 6 9 3 1 3 0 0 8 ⟶ 0 1 2 3 1 2 3 0 1 3 0 1 3 1 2 2 3 8 , 3 1 3 1 2 3 6 9 3 1 3 0 0 0 ⟶ 3 1 2 3 1 2 3 0 1 3 0 1 3 1 2 2 3 0 , 3 1 3 1 2 3 6 9 3 1 3 0 0 1 ⟶ 3 1 2 3 1 2 3 0 1 3 0 1 3 1 2 2 3 1 , 3 1 3 1 2 3 6 9 3 1 3 0 0 6 ⟶ 3 1 2 3 1 2 3 0 1 3 0 1 3 1 2 2 3 6 , 3 1 3 1 2 3 6 9 3 1 3 0 0 8 ⟶ 3 1 2 3 1 2 3 0 1 3 0 1 3 1 2 2 3 8 , 5 1 3 1 2 3 6 9 3 1 3 0 0 0 ⟶ 5 1 2 3 1 2 3 0 1 3 0 1 3 1 2 2 3 0 , 5 1 3 1 2 3 6 9 3 1 3 0 0 1 ⟶ 5 1 2 3 1 2 3 0 1 3 0 1 3 1 2 2 3 1 , 5 1 3 1 2 3 6 9 3 1 3 0 0 6 ⟶ 5 1 2 3 1 2 3 0 1 3 0 1 3 1 2 2 3 6 , 5 1 3 1 2 3 6 9 3 1 3 0 0 8 ⟶ 5 1 2 3 1 2 3 0 1 3 0 1 3 1 2 2 3 8 , 10 1 3 1 2 3 6 9 3 1 3 0 0 0 ⟶ 10 1 2 3 1 2 3 0 1 3 0 1 3 1 2 2 3 0 , 10 1 3 1 2 3 6 9 3 1 3 0 0 1 ⟶ 10 1 2 3 1 2 3 0 1 3 0 1 3 1 2 2 3 1 , 10 1 3 1 2 3 6 9 3 1 3 0 0 6 ⟶ 10 1 2 3 1 2 3 0 1 3 0 1 3 1 2 2 3 6 , 10 1 3 1 2 3 6 9 3 1 3 0 0 8 ⟶ 10 1 2 3 1 2 3 0 1 3 0 1 3 1 2 2 3 8 , 0 1 2 3 0 1 3 6 5 0 0 6 9 3 ⟶ 0 0 1 2 3 1 3 1 2 3 0 1 2 4 9 2 2 3 , 0 1 2 3 0 1 3 6 5 0 0 6 9 2 ⟶ 0 0 1 2 3 1 3 1 2 3 0 1 2 4 9 2 2 2 , 0 1 2 3 0 1 3 6 5 0 0 6 9 4 ⟶ 0 0 1 2 3 1 3 1 2 3 0 1 2 4 9 2 2 4 , 0 1 2 3 0 1 3 6 5 0 0 6 9 7 ⟶ 0 0 1 2 3 1 3 1 2 3 0 1 2 4 9 2 2 7 , 3 1 2 3 0 1 3 6 5 0 0 6 9 3 ⟶ 3 0 1 2 3 1 3 1 2 3 0 1 2 4 9 2 2 3 , 3 1 2 3 0 1 3 6 5 0 0 6 9 2 ⟶ 3 0 1 2 3 1 3 1 2 3 0 1 2 4 9 2 2 2 , 3 1 2 3 0 1 3 6 5 0 0 6 9 4 ⟶ 3 0 1 2 3 1 3 1 2 3 0 1 2 4 9 2 2 4 , 3 1 2 3 0 1 3 6 5 0 0 6 9 7 ⟶ 3 0 1 2 3 1 3 1 2 3 0 1 2 4 9 2 2 7 , 5 1 2 3 0 1 3 6 5 0 0 6 9 3 ⟶ 5 0 1 2 3 1 3 1 2 3 0 1 2 4 9 2 2 3 , 5 1 2 3 0 1 3 6 5 0 0 6 9 2 ⟶ 5 0 1 2 3 1 3 1 2 3 0 1 2 4 9 2 2 2 , 5 1 2 3 0 1 3 6 5 0 0 6 9 4 ⟶ 5 0 1 2 3 1 3 1 2 3 0 1 2 4 9 2 2 4 , 5 1 2 3 0 1 3 6 5 0 0 6 9 7 ⟶ 5 0 1 2 3 1 3 1 2 3 0 1 2 4 9 2 2 7 , 10 1 2 3 0 1 3 6 5 0 0 6 9 3 ⟶ 10 0 1 2 3 1 3 1 2 3 0 1 2 4 9 2 2 3 , 10 1 2 3 0 1 3 6 5 0 0 6 9 2 ⟶ 10 0 1 2 3 1 3 1 2 3 0 1 2 4 9 2 2 2 , 10 1 2 3 0 1 3 6 5 0 0 6 9 4 ⟶ 10 0 1 2 3 1 3 1 2 3 0 1 2 4 9 2 2 4 , 10 1 2 3 0 1 3 6 5 0 0 6 9 7 ⟶ 10 0 1 2 3 1 3 1 2 3 0 1 2 4 9 2 2 7 , 0 1 4 9 3 0 0 0 0 6 9 2 2 3 ⟶ 0 1 3 1 2 4 5 1 3 0 0 1 2 3 0 0 1 3 , 0 1 4 9 3 0 0 0 0 6 9 2 2 2 ⟶ 0 1 3 1 2 4 5 1 3 0 0 1 2 3 0 0 1 2 , 0 1 4 9 3 0 0 0 0 6 9 2 2 4 ⟶ 0 1 3 1 2 4 5 1 3 0 0 1 2 3 0 0 1 4 , 0 1 4 9 3 0 0 0 0 6 9 2 2 7 ⟶ 0 1 3 1 2 4 5 1 3 0 0 1 2 3 0 0 1 7 , 3 1 4 9 3 0 0 0 0 6 9 2 2 3 ⟶ 3 1 3 1 2 4 5 1 3 0 0 1 2 3 0 0 1 3 , 3 1 4 9 3 0 0 0 0 6 9 2 2 2 ⟶ 3 1 3 1 2 4 5 1 3 0 0 1 2 3 0 0 1 2 , 3 1 4 9 3 0 0 0 0 6 9 2 2 4 ⟶ 3 1 3 1 2 4 5 1 3 0 0 1 2 3 0 0 1 4 , 3 1 4 9 3 0 0 0 0 6 9 2 2 7 ⟶ 3 1 3 1 2 4 5 1 3 0 0 1 2 3 0 0 1 7 , 5 1 4 9 3 0 0 0 0 6 9 2 2 3 ⟶ 5 1 3 1 2 4 5 1 3 0 0 1 2 3 0 0 1 3 , 5 1 4 9 3 0 0 0 0 6 9 2 2 2 ⟶ 5 1 3 1 2 4 5 1 3 0 0 1 2 3 0 0 1 2 , 5 1 4 9 3 0 0 0 0 6 9 2 2 4 ⟶ 5 1 3 1 2 4 5 1 3 0 0 1 2 3 0 0 1 4 , 5 1 4 9 3 0 0 0 0 6 9 2 2 7 ⟶ 5 1 3 1 2 4 5 1 3 0 0 1 2 3 0 0 1 7 , 10 1 4 9 3 0 0 0 0 6 9 2 2 3 ⟶ 10 1 3 1 2 4 5 1 3 0 0 1 2 3 0 0 1 3 , 10 1 4 9 3 0 0 0 0 6 9 2 2 2 ⟶ 10 1 3 1 2 4 5 1 3 0 0 1 2 3 0 0 1 2 , 10 1 4 9 3 0 0 0 0 6 9 2 2 4 ⟶ 10 1 3 1 2 4 5 1 3 0 0 1 2 3 0 0 1 4 , 10 1 4 9 3 0 0 0 0 6 9 2 2 7 ⟶ 10 1 3 1 2 4 5 1 3 0 0 1 2 3 0 0 1 7 , 0 6 5 0 6 9 3 0 0 0 0 0 1 3 ⟶ 1 3 0 1 3 1 2 2 3 0 0 1 3 0 0 0 1 3 , 0 6 5 0 6 9 3 0 0 0 0 0 1 2 ⟶ 1 3 0 1 3 1 2 2 3 0 0 1 3 0 0 0 1 2 , 0 6 5 0 6 9 3 0 0 0 0 0 1 4 ⟶ 1 3 0 1 3 1 2 2 3 0 0 1 3 0 0 0 1 4 , 0 6 5 0 6 9 3 0 0 0 0 0 1 7 ⟶ 1 3 0 1 3 1 2 2 3 0 0 1 3 0 0 0 1 7 , 3 6 5 0 6 9 3 0 0 0 0 0 1 3 ⟶ 2 3 0 1 3 1 2 2 3 0 0 1 3 0 0 0 1 3 , 3 6 5 0 6 9 3 0 0 0 0 0 1 2 ⟶ 2 3 0 1 3 1 2 2 3 0 0 1 3 0 0 0 1 2 , 3 6 5 0 6 9 3 0 0 0 0 0 1 4 ⟶ 2 3 0 1 3 1 2 2 3 0 0 1 3 0 0 0 1 4 , 3 6 5 0 6 9 3 0 0 0 0 0 1 7 ⟶ 2 3 0 1 3 1 2 2 3 0 0 1 3 0 0 0 1 7 , 5 6 5 0 6 9 3 0 0 0 0 0 1 3 ⟶ 9 3 0 1 3 1 2 2 3 0 0 1 3 0 0 0 1 3 , 5 6 5 0 6 9 3 0 0 0 0 0 1 2 ⟶ 9 3 0 1 3 1 2 2 3 0 0 1 3 0 0 0 1 2 , 5 6 5 0 6 9 3 0 0 0 0 0 1 4 ⟶ 9 3 0 1 3 1 2 2 3 0 0 1 3 0 0 0 1 4 , 5 6 5 0 6 9 3 0 0 0 0 0 1 7 ⟶ 9 3 0 1 3 1 2 2 3 0 0 1 3 0 0 0 1 7 , 10 6 5 0 6 9 3 0 0 0 0 0 1 3 ⟶ 11 3 0 1 3 1 2 2 3 0 0 1 3 0 0 0 1 3 , 10 6 5 0 6 9 3 0 0 0 0 0 1 2 ⟶ 11 3 0 1 3 1 2 2 3 0 0 1 3 0 0 0 1 2 , 10 6 5 0 6 9 3 0 0 0 0 0 1 4 ⟶ 11 3 0 1 3 1 2 2 3 0 0 1 3 0 0 0 1 4 , 10 6 5 0 6 9 3 0 0 0 0 0 1 7 ⟶ 11 3 0 1 3 1 2 2 3 0 0 1 3 0 0 0 1 7 , 0 6 9 3 1 4 9 4 12 5 1 2 2 3 ⟶ 0 6 9 2 3 0 6 9 2 3 0 1 4 9 2 2 2 3 , 0 6 9 3 1 4 9 4 12 5 1 2 2 2 ⟶ 0 6 9 2 3 0 6 9 2 3 0 1 4 9 2 2 2 2 , 0 6 9 3 1 4 9 4 12 5 1 2 2 4 ⟶ 0 6 9 2 3 0 6 9 2 3 0 1 4 9 2 2 2 4 , 0 6 9 3 1 4 9 4 12 5 1 2 2 7 ⟶ 0 6 9 2 3 0 6 9 2 3 0 1 4 9 2 2 2 7 , 3 6 9 3 1 4 9 4 12 5 1 2 2 3 ⟶ 3 6 9 2 3 0 6 9 2 3 0 1 4 9 2 2 2 3 , 3 6 9 3 1 4 9 4 12 5 1 2 2 2 ⟶ 3 6 9 2 3 0 6 9 2 3 0 1 4 9 2 2 2 2 , 3 6 9 3 1 4 9 4 12 5 1 2 2 4 ⟶ 3 6 9 2 3 0 6 9 2 3 0 1 4 9 2 2 2 4 , 3 6 9 3 1 4 9 4 12 5 1 2 2 7 ⟶ 3 6 9 2 3 0 6 9 2 3 0 1 4 9 2 2 2 7 , 5 6 9 3 1 4 9 4 12 5 1 2 2 3 ⟶ 5 6 9 2 3 0 6 9 2 3 0 1 4 9 2 2 2 3 , 5 6 9 3 1 4 9 4 12 5 1 2 2 2 ⟶ 5 6 9 2 3 0 6 9 2 3 0 1 4 9 2 2 2 2 , 5 6 9 3 1 4 9 4 12 5 1 2 2 4 ⟶ 5 6 9 2 3 0 6 9 2 3 0 1 4 9 2 2 2 4 , 5 6 9 3 1 4 9 4 12 5 1 2 2 7 ⟶ 5 6 9 2 3 0 6 9 2 3 0 1 4 9 2 2 2 7 , 10 6 9 3 1 4 9 4 12 5 1 2 2 3 ⟶ 10 6 9 2 3 0 6 9 2 3 0 1 4 9 2 2 2 3 , 10 6 9 3 1 4 9 4 12 5 1 2 2 2 ⟶ 10 6 9 2 3 0 6 9 2 3 0 1 4 9 2 2 2 2 , 10 6 9 3 1 4 9 4 12 5 1 2 2 4 ⟶ 10 6 9 2 3 0 6 9 2 3 0 1 4 9 2 2 2 4 , 10 6 9 3 1 4 9 4 12 5 1 2 2 7 ⟶ 10 6 9 2 3 0 6 9 2 3 0 1 4 9 2 2 2 7 , 0 6 9 3 6 9 3 1 4 5 0 0 1 3 ⟶ 0 0 6 9 2 2 2 3 0 1 3 6 5 0 0 1 3 0 , 0 6 9 3 6 9 3 1 4 5 0 0 1 2 ⟶ 0 0 6 9 2 2 2 3 0 1 3 6 5 0 0 1 3 1 , 0 6 9 3 6 9 3 1 4 5 0 0 1 4 ⟶ 0 0 6 9 2 2 2 3 0 1 3 6 5 0 0 1 3 6 , 0 6 9 3 6 9 3 1 4 5 0 0 1 7 ⟶ 0 0 6 9 2 2 2 3 0 1 3 6 5 0 0 1 3 8 , 3 6 9 3 6 9 3 1 4 5 0 0 1 3 ⟶ 3 0 6 9 2 2 2 3 0 1 3 6 5 0 0 1 3 0 , 3 6 9 3 6 9 3 1 4 5 0 0 1 2 ⟶ 3 0 6 9 2 2 2 3 0 1 3 6 5 0 0 1 3 1 , 3 6 9 3 6 9 3 1 4 5 0 0 1 4 ⟶ 3 0 6 9 2 2 2 3 0 1 3 6 5 0 0 1 3 6 , 3 6 9 3 6 9 3 1 4 5 0 0 1 7 ⟶ 3 0 6 9 2 2 2 3 0 1 3 6 5 0 0 1 3 8 , 5 6 9 3 6 9 3 1 4 5 0 0 1 3 ⟶ 5 0 6 9 2 2 2 3 0 1 3 6 5 0 0 1 3 0 , 5 6 9 3 6 9 3 1 4 5 0 0 1 2 ⟶ 5 0 6 9 2 2 2 3 0 1 3 6 5 0 0 1 3 1 , 5 6 9 3 6 9 3 1 4 5 0 0 1 4 ⟶ 5 0 6 9 2 2 2 3 0 1 3 6 5 0 0 1 3 6 , 5 6 9 3 6 9 3 1 4 5 0 0 1 7 ⟶ 5 0 6 9 2 2 2 3 0 1 3 6 5 0 0 1 3 8 , 10 6 9 3 6 9 3 1 4 5 0 0 1 3 ⟶ 10 0 6 9 2 2 2 3 0 1 3 6 5 0 0 1 3 0 , 10 6 9 3 6 9 3 1 4 5 0 0 1 2 ⟶ 10 0 6 9 2 2 2 3 0 1 3 6 5 0 0 1 3 1 , 10 6 9 3 6 9 3 1 4 5 0 0 1 4 ⟶ 10 0 6 9 2 2 2 3 0 1 3 6 5 0 0 1 3 6 , 10 6 9 3 6 9 3 1 4 5 0 0 1 7 ⟶ 10 0 6 9 2 2 2 3 0 1 3 6 5 0 0 1 3 8 , 0 6 9 2 3 0 0 0 6 9 3 1 2 3 ⟶ 0 0 1 3 1 3 1 3 6 9 2 2 3 0 0 0 0 0 , 0 6 9 2 3 0 0 0 6 9 3 1 2 2 ⟶ 0 0 1 3 1 3 1 3 6 9 2 2 3 0 0 0 0 1 , 0 6 9 2 3 0 0 0 6 9 3 1 2 4 ⟶ 0 0 1 3 1 3 1 3 6 9 2 2 3 0 0 0 0 6 , 0 6 9 2 3 0 0 0 6 9 3 1 2 7 ⟶ 0 0 1 3 1 3 1 3 6 9 2 2 3 0 0 0 0 8 , 3 6 9 2 3 0 0 0 6 9 3 1 2 3 ⟶ 3 0 1 3 1 3 1 3 6 9 2 2 3 0 0 0 0 0 , 3 6 9 2 3 0 0 0 6 9 3 1 2 2 ⟶ 3 0 1 3 1 3 1 3 6 9 2 2 3 0 0 0 0 1 , 3 6 9 2 3 0 0 0 6 9 3 1 2 4 ⟶ 3 0 1 3 1 3 1 3 6 9 2 2 3 0 0 0 0 6 , 3 6 9 2 3 0 0 0 6 9 3 1 2 7 ⟶ 3 0 1 3 1 3 1 3 6 9 2 2 3 0 0 0 0 8 , 5 6 9 2 3 0 0 0 6 9 3 1 2 3 ⟶ 5 0 1 3 1 3 1 3 6 9 2 2 3 0 0 0 0 0 , 5 6 9 2 3 0 0 0 6 9 3 1 2 2 ⟶ 5 0 1 3 1 3 1 3 6 9 2 2 3 0 0 0 0 1 , 5 6 9 2 3 0 0 0 6 9 3 1 2 4 ⟶ 5 0 1 3 1 3 1 3 6 9 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0 1 3 1 2 4 9 3 ⟶ 3 1 3 0 1 3 0 0 1 2 3 0 6 9 3 0 0 0 , 2 4 9 3 0 0 0 1 3 1 2 4 9 2 ⟶ 3 1 3 0 1 3 0 0 1 2 3 0 6 9 3 0 0 1 , 2 4 9 3 0 0 0 1 3 1 2 4 9 4 ⟶ 3 1 3 0 1 3 0 0 1 2 3 0 6 9 3 0 0 6 , 2 4 9 3 0 0 0 1 3 1 2 4 9 7 ⟶ 3 1 3 0 1 3 0 0 1 2 3 0 6 9 3 0 0 8 , 9 4 9 3 0 0 0 1 3 1 2 4 9 3 ⟶ 5 1 3 0 1 3 0 0 1 2 3 0 6 9 3 0 0 0 , 9 4 9 3 0 0 0 1 3 1 2 4 9 2 ⟶ 5 1 3 0 1 3 0 0 1 2 3 0 6 9 3 0 0 1 , 9 4 9 3 0 0 0 1 3 1 2 4 9 4 ⟶ 5 1 3 0 1 3 0 0 1 2 3 0 6 9 3 0 0 6 , 9 4 9 3 0 0 0 1 3 1 2 4 9 7 ⟶ 5 1 3 0 1 3 0 0 1 2 3 0 6 9 3 0 0 8 , 11 4 9 3 0 0 0 1 3 1 2 4 9 3 ⟶ 10 1 3 0 1 3 0 0 1 2 3 0 6 9 3 0 0 0 , 11 4 9 3 0 0 0 1 3 1 2 4 9 2 ⟶ 10 1 3 0 1 3 0 0 1 2 3 0 6 9 3 0 0 1 , 11 4 9 3 0 0 0 1 3 1 2 4 9 4 ⟶ 10 1 3 0 1 3 0 0 1 2 3 0 6 9 3 0 0 6 , 11 4 9 3 0 0 0 1 3 1 2 4 9 7 ⟶ 10 1 3 0 1 3 0 0 1 2 3 0 6 9 3 0 0 8 , 1 4 9 3 1 3 0 1 3 6 5 1 2 3 ⟶ 1 3 0 1 3 0 0 6 12 5 0 1 3 0 1 2 3 0 , 1 4 9 3 1 3 0 1 3 6 5 1 2 2 ⟶ 1 3 0 1 3 0 0 6 12 5 0 1 3 0 1 2 3 1 , 1 4 9 3 1 3 0 1 3 6 5 1 2 4 ⟶ 1 3 0 1 3 0 0 6 12 5 0 1 3 0 1 2 3 6 , 1 4 9 3 1 3 0 1 3 6 5 1 2 7 ⟶ 1 3 0 1 3 0 0 6 12 5 0 1 3 0 1 2 3 8 , 2 4 9 3 1 3 0 1 3 6 5 1 2 3 ⟶ 2 3 0 1 3 0 0 6 12 5 0 1 3 0 1 2 3 0 , 2 4 9 3 1 3 0 1 3 6 5 1 2 2 ⟶ 2 3 0 1 3 0 0 6 12 5 0 1 3 0 1 2 3 1 , 2 4 9 3 1 3 0 1 3 6 5 1 2 4 ⟶ 2 3 0 1 3 0 0 6 12 5 0 1 3 0 1 2 3 6 , 2 4 9 3 1 3 0 1 3 6 5 1 2 7 ⟶ 2 3 0 1 3 0 0 6 12 5 0 1 3 0 1 2 3 8 , 9 4 9 3 1 3 0 1 3 6 5 1 2 3 ⟶ 9 3 0 1 3 0 0 6 12 5 0 1 3 0 1 2 3 0 , 9 4 9 3 1 3 0 1 3 6 5 1 2 2 ⟶ 9 3 0 1 3 0 0 6 12 5 0 1 3 0 1 2 3 1 , 9 4 9 3 1 3 0 1 3 6 5 1 2 4 ⟶ 9 3 0 1 3 0 0 6 12 5 0 1 3 0 1 2 3 6 , 9 4 9 3 1 3 0 1 3 6 5 1 2 7 ⟶ 9 3 0 1 3 0 0 6 12 5 0 1 3 0 1 2 3 8 , 11 4 9 3 1 3 0 1 3 6 5 1 2 3 ⟶ 11 3 0 1 3 0 0 6 12 5 0 1 3 0 1 2 3 0 , 11 4 9 3 1 3 0 1 3 6 5 1 2 2 ⟶ 11 3 0 1 3 0 0 6 12 5 0 1 3 0 1 2 3 1 , 11 4 9 3 1 3 0 1 3 6 5 1 2 4 ⟶ 11 3 0 1 3 0 0 6 12 5 0 1 3 0 1 2 3 6 , 11 4 9 3 1 3 0 1 3 6 5 1 2 7 ⟶ 11 3 0 1 3 0 0 6 12 5 0 1 3 0 1 2 3 8 , 1 4 12 5 1 2 2 2 2 2 3 1 3 0 ⟶ 1 3 0 6 12 5 0 1 3 0 1 3 0 0 1 2 3 0 , 1 4 12 5 1 2 2 2 2 2 3 1 3 1 ⟶ 1 3 0 6 12 5 0 1 3 0 1 3 0 0 1 2 3 1 , 1 4 12 5 1 2 2 2 2 2 3 1 3 6 ⟶ 1 3 0 6 12 5 0 1 3 0 1 3 0 0 1 2 3 6 , 1 4 12 5 1 2 2 2 2 2 3 1 3 8 ⟶ 1 3 0 6 12 5 0 1 3 0 1 3 0 0 1 2 3 8 , 2 4 12 5 1 2 2 2 2 2 3 1 3 0 ⟶ 2 3 0 6 12 5 0 1 3 0 1 3 0 0 1 2 3 0 , 2 4 12 5 1 2 2 2 2 2 3 1 3 1 ⟶ 2 3 0 6 12 5 0 1 3 0 1 3 0 0 1 2 3 1 , 2 4 12 5 1 2 2 2 2 2 3 1 3 6 ⟶ 2 3 0 6 12 5 0 1 3 0 1 3 0 0 1 2 3 6 , 2 4 12 5 1 2 2 2 2 2 3 1 3 8 ⟶ 2 3 0 6 12 5 0 1 3 0 1 3 0 0 1 2 3 8 , 9 4 12 5 1 2 2 2 2 2 3 1 3 0 ⟶ 9 3 0 6 12 5 0 1 3 0 1 3 0 0 1 2 3 0 , 9 4 12 5 1 2 2 2 2 2 3 1 3 1 ⟶ 9 3 0 6 12 5 0 1 3 0 1 3 0 0 1 2 3 1 , 9 4 12 5 1 2 2 2 2 2 3 1 3 6 ⟶ 9 3 0 6 12 5 0 1 3 0 1 3 0 0 1 2 3 6 , 9 4 12 5 1 2 2 2 2 2 3 1 3 8 ⟶ 9 3 0 6 12 5 0 1 3 0 1 3 0 0 1 2 3 8 , 11 4 12 5 1 2 2 2 2 2 3 1 3 0 ⟶ 11 3 0 6 12 5 0 1 3 0 1 3 0 0 1 2 3 0 , 11 4 12 5 1 2 2 2 2 2 3 1 3 1 ⟶ 11 3 0 6 12 5 0 1 3 0 1 3 0 0 1 2 3 1 , 11 4 12 5 1 2 2 2 2 2 3 1 3 6 ⟶ 11 3 0 6 12 5 0 1 3 0 1 3 0 0 1 2 3 6 , 11 4 12 5 1 2 2 2 2 2 3 1 3 8 ⟶ 11 3 0 6 12 5 0 1 3 0 1 3 0 0 1 2 3 8 , 6 9 3 0 1 2 3 1 4 9 3 1 2 3 ⟶ 1 3 0 0 1 3 0 0 0 6 9 3 0 1 2 3 1 3 , 6 9 3 0 1 2 3 1 4 9 3 1 2 2 ⟶ 1 3 0 0 1 3 0 0 0 6 9 3 0 1 2 3 1 2 , 6 9 3 0 1 2 3 1 4 9 3 1 2 4 ⟶ 1 3 0 0 1 3 0 0 0 6 9 3 0 1 2 3 1 4 , 6 9 3 0 1 2 3 1 4 9 3 1 2 7 ⟶ 1 3 0 0 1 3 0 0 0 6 9 3 0 1 2 3 1 7 , 4 9 3 0 1 2 3 1 4 9 3 1 2 3 ⟶ 2 3 0 0 1 3 0 0 0 6 9 3 0 1 2 3 1 3 , 4 9 3 0 1 2 3 1 4 9 3 1 2 2 ⟶ 2 3 0 0 1 3 0 0 0 6 9 3 0 1 2 3 1 2 , 4 9 3 0 1 2 3 1 4 9 3 1 2 4 ⟶ 2 3 0 0 1 3 0 0 0 6 9 3 0 1 2 3 1 4 , 4 9 3 0 1 2 3 1 4 9 3 1 2 7 ⟶ 2 3 0 0 1 3 0 0 0 6 9 3 0 1 2 3 1 7 , 12 9 3 0 1 2 3 1 4 9 3 1 2 3 ⟶ 9 3 0 0 1 3 0 0 0 6 9 3 0 1 2 3 1 3 , 12 9 3 0 1 2 3 1 4 9 3 1 2 2 ⟶ 9 3 0 0 1 3 0 0 0 6 9 3 0 1 2 3 1 2 , 12 9 3 0 1 2 3 1 4 9 3 1 2 4 ⟶ 9 3 0 0 1 3 0 0 0 6 9 3 0 1 2 3 1 4 , 12 9 3 0 1 2 3 1 4 9 3 1 2 7 ⟶ 9 3 0 0 1 3 0 0 0 6 9 3 0 1 2 3 1 7 , 14 9 3 0 1 2 3 1 4 9 3 1 2 3 ⟶ 11 3 0 0 1 3 0 0 0 6 9 3 0 1 2 3 1 3 , 14 9 3 0 1 2 3 1 4 9 3 1 2 2 ⟶ 11 3 0 0 1 3 0 0 0 6 9 3 0 1 2 3 1 2 , 14 9 3 0 1 2 3 1 4 9 3 1 2 4 ⟶ 11 3 0 0 1 3 0 0 0 6 9 3 0 1 2 3 1 4 , 14 9 3 0 1 2 3 1 4 9 3 1 2 7 ⟶ 11 3 0 0 1 3 0 0 0 6 9 3 0 1 2 3 1 7 , 6 9 3 1 4 5 0 6 5 1 3 1 4 5 ⟶ 6 9 3 0 1 3 0 0 0 1 3 6 12 12 5 1 4 5 , 6 9 3 1 4 5 0 6 5 1 3 1 4 9 ⟶ 6 9 3 0 1 3 0 0 0 1 3 6 12 12 5 1 4 9 , 6 9 3 1 4 5 0 6 5 1 3 1 4 12 ⟶ 6 9 3 0 1 3 0 0 0 1 3 6 12 12 5 1 4 12 , 6 9 3 1 4 5 0 6 5 1 3 1 4 13 ⟶ 6 9 3 0 1 3 0 0 0 1 3 6 12 12 5 1 4 13 , 4 9 3 1 4 5 0 6 5 1 3 1 4 5 ⟶ 4 9 3 0 1 3 0 0 0 1 3 6 12 12 5 1 4 5 , 4 9 3 1 4 5 0 6 5 1 3 1 4 9 ⟶ 4 9 3 0 1 3 0 0 0 1 3 6 12 12 5 1 4 9 , 4 9 3 1 4 5 0 6 5 1 3 1 4 12 ⟶ 4 9 3 0 1 3 0 0 0 1 3 6 12 12 5 1 4 12 , 4 9 3 1 4 5 0 6 5 1 3 1 4 13 ⟶ 4 9 3 0 1 3 0 0 0 1 3 6 12 12 5 1 4 13 , 12 9 3 1 4 5 0 6 5 1 3 1 4 5 ⟶ 12 9 3 0 1 3 0 0 0 1 3 6 12 12 5 1 4 5 , 12 9 3 1 4 5 0 6 5 1 3 1 4 9 ⟶ 12 9 3 0 1 3 0 0 0 1 3 6 12 12 5 1 4 9 , 12 9 3 1 4 5 0 6 5 1 3 1 4 12 ⟶ 12 9 3 0 1 3 0 0 0 1 3 6 12 12 5 1 4 12 , 12 9 3 1 4 5 0 6 5 1 3 1 4 13 ⟶ 12 9 3 0 1 3 0 0 0 1 3 6 12 12 5 1 4 13 , 14 9 3 1 4 5 0 6 5 1 3 1 4 5 ⟶ 14 9 3 0 1 3 0 0 0 1 3 6 12 12 5 1 4 5 , 14 9 3 1 4 5 0 6 5 1 3 1 4 9 ⟶ 14 9 3 0 1 3 0 0 0 1 3 6 12 12 5 1 4 9 , 14 9 3 1 4 5 0 6 5 1 3 1 4 12 ⟶ 14 9 3 0 1 3 0 0 0 1 3 6 12 12 5 1 4 12 , 14 9 3 1 4 5 0 6 5 1 3 1 4 13 ⟶ 14 9 3 0 1 3 0 0 0 1 3 6 12 12 5 1 4 13 , 6 9 3 6 12 12 12 9 3 0 0 0 1 3 ⟶ 1 2 2 3 0 0 1 2 4 9 3 0 1 4 5 1 3 0 , 6 9 3 6 12 12 12 9 3 0 0 0 1 2 ⟶ 1 2 2 3 0 0 1 2 4 9 3 0 1 4 5 1 3 1 , 6 9 3 6 12 12 12 9 3 0 0 0 1 4 ⟶ 1 2 2 3 0 0 1 2 4 9 3 0 1 4 5 1 3 6 , 6 9 3 6 12 12 12 9 3 0 0 0 1 7 ⟶ 1 2 2 3 0 0 1 2 4 9 3 0 1 4 5 1 3 8 , 4 9 3 6 12 12 12 9 3 0 0 0 1 3 ⟶ 2 2 2 3 0 0 1 2 4 9 3 0 1 4 5 1 3 0 , 4 9 3 6 12 12 12 9 3 0 0 0 1 2 ⟶ 2 2 2 3 0 0 1 2 4 9 3 0 1 4 5 1 3 1 , 4 9 3 6 12 12 12 9 3 0 0 0 1 4 ⟶ 2 2 2 3 0 0 1 2 4 9 3 0 1 4 5 1 3 6 , 4 9 3 6 12 12 12 9 3 0 0 0 1 7 ⟶ 2 2 2 3 0 0 1 2 4 9 3 0 1 4 5 1 3 8 , 12 9 3 6 12 12 12 9 3 0 0 0 1 3 ⟶ 9 2 2 3 0 0 1 2 4 9 3 0 1 4 5 1 3 0 , 12 9 3 6 12 12 12 9 3 0 0 0 1 2 ⟶ 9 2 2 3 0 0 1 2 4 9 3 0 1 4 5 1 3 1 , 12 9 3 6 12 12 12 9 3 0 0 0 1 4 ⟶ 9 2 2 3 0 0 1 2 4 9 3 0 1 4 5 1 3 6 , 12 9 3 6 12 12 12 9 3 0 0 0 1 7 ⟶ 9 2 2 3 0 0 1 2 4 9 3 0 1 4 5 1 3 8 , 14 9 3 6 12 12 12 9 3 0 0 0 1 3 ⟶ 11 2 2 3 0 0 1 2 4 9 3 0 1 4 5 1 3 0 , 14 9 3 6 12 12 12 9 3 0 0 0 1 2 ⟶ 11 2 2 3 0 0 1 2 4 9 3 0 1 4 5 1 3 1 , 14 9 3 6 12 12 12 9 3 0 0 0 1 4 ⟶ 11 2 2 3 0 0 1 2 4 9 3 0 1 4 5 1 3 6 , 14 9 3 6 12 12 12 9 3 0 0 0 1 7 ⟶ 11 2 2 3 0 0 1 2 4 9 3 0 1 4 5 1 3 8 , 6 9 2 3 1 3 1 2 2 4 9 2 3 0 ⟶ 1 2 4 9 2 3 0 1 2 3 0 1 2 3 1 3 0 0 , 6 9 2 3 1 3 1 2 2 4 9 2 3 1 ⟶ 1 2 4 9 2 3 0 1 2 3 0 1 2 3 1 3 0 1 , 6 9 2 3 1 3 1 2 2 4 9 2 3 6 ⟶ 1 2 4 9 2 3 0 1 2 3 0 1 2 3 1 3 0 6 , 6 9 2 3 1 3 1 2 2 4 9 2 3 8 ⟶ 1 2 4 9 2 3 0 1 2 3 0 1 2 3 1 3 0 8 , 4 9 2 3 1 3 1 2 2 4 9 2 3 0 ⟶ 2 2 4 9 2 3 0 1 2 3 0 1 2 3 1 3 0 0 , 4 9 2 3 1 3 1 2 2 4 9 2 3 1 ⟶ 2 2 4 9 2 3 0 1 2 3 0 1 2 3 1 3 0 1 , 4 9 2 3 1 3 1 2 2 4 9 2 3 6 ⟶ 2 2 4 9 2 3 0 1 2 3 0 1 2 3 1 3 0 6 , 4 9 2 3 1 3 1 2 2 4 9 2 3 8 ⟶ 2 2 4 9 2 3 0 1 2 3 0 1 2 3 1 3 0 8 , 12 9 2 3 1 3 1 2 2 4 9 2 3 0 ⟶ 9 2 4 9 2 3 0 1 2 3 0 1 2 3 1 3 0 0 , 12 9 2 3 1 3 1 2 2 4 9 2 3 1 ⟶ 9 2 4 9 2 3 0 1 2 3 0 1 2 3 1 3 0 1 , 12 9 2 3 1 3 1 2 2 4 9 2 3 6 ⟶ 9 2 4 9 2 3 0 1 2 3 0 1 2 3 1 3 0 6 , 12 9 2 3 1 3 1 2 2 4 9 2 3 8 ⟶ 9 2 4 9 2 3 0 1 2 3 0 1 2 3 1 3 0 8 , 14 9 2 3 1 3 1 2 2 4 9 2 3 0 ⟶ 11 2 4 9 2 3 0 1 2 3 0 1 2 3 1 3 0 0 , 14 9 2 3 1 3 1 2 2 4 9 2 3 1 ⟶ 11 2 4 9 2 3 0 1 2 3 0 1 2 3 1 3 0 1 , 14 9 2 3 1 3 1 2 2 4 9 2 3 6 ⟶ 11 2 4 9 2 3 0 1 2 3 0 1 2 3 1 3 0 6 , 14 9 2 3 1 3 1 2 2 4 9 2 3 8 ⟶ 11 2 4 9 2 3 0 1 2 3 0 1 2 3 1 3 0 8 , 6 9 2 3 6 9 3 6 9 3 0 0 1 3 ⟶ 0 1 3 1 2 2 2 4 5 1 3 0 1 3 0 1 3 0 , 6 9 2 3 6 9 3 6 9 3 0 0 1 2 ⟶ 0 1 3 1 2 2 2 4 5 1 3 0 1 3 0 1 3 1 , 6 9 2 3 6 9 3 6 9 3 0 0 1 4 ⟶ 0 1 3 1 2 2 2 4 5 1 3 0 1 3 0 1 3 6 , 6 9 2 3 6 9 3 6 9 3 0 0 1 7 ⟶ 0 1 3 1 2 2 2 4 5 1 3 0 1 3 0 1 3 8 , 4 9 2 3 6 9 3 6 9 3 0 0 1 3 ⟶ 3 1 3 1 2 2 2 4 5 1 3 0 1 3 0 1 3 0 , 4 9 2 3 6 9 3 6 9 3 0 0 1 2 ⟶ 3 1 3 1 2 2 2 4 5 1 3 0 1 3 0 1 3 1 , 4 9 2 3 6 9 3 6 9 3 0 0 1 4 ⟶ 3 1 3 1 2 2 2 4 5 1 3 0 1 3 0 1 3 6 , 4 9 2 3 6 9 3 6 9 3 0 0 1 7 ⟶ 3 1 3 1 2 2 2 4 5 1 3 0 1 3 0 1 3 8 , 12 9 2 3 6 9 3 6 9 3 0 0 1 3 ⟶ 5 1 3 1 2 2 2 4 5 1 3 0 1 3 0 1 3 0 , 12 9 2 3 6 9 3 6 9 3 0 0 1 2 ⟶ 5 1 3 1 2 2 2 4 5 1 3 0 1 3 0 1 3 1 , 12 9 2 3 6 9 3 6 9 3 0 0 1 4 ⟶ 5 1 3 1 2 2 2 4 5 1 3 0 1 3 0 1 3 6 , 12 9 2 3 6 9 3 6 9 3 0 0 1 7 ⟶ 5 1 3 1 2 2 2 4 5 1 3 0 1 3 0 1 3 8 , 14 9 2 3 6 9 3 6 9 3 0 0 1 3 ⟶ 10 1 3 1 2 2 2 4 5 1 3 0 1 3 0 1 3 0 , 14 9 2 3 6 9 3 6 9 3 0 0 1 2 ⟶ 10 1 3 1 2 2 2 4 5 1 3 0 1 3 0 1 3 1 , 14 9 2 3 6 9 3 6 9 3 0 0 1 4 ⟶ 10 1 3 1 2 2 2 4 5 1 3 0 1 3 0 1 3 6 , 14 9 2 3 6 9 3 6 9 3 0 0 1 7 ⟶ 10 1 3 1 2 2 2 4 5 1 3 0 1 3 0 1 3 8 , 6 9 2 2 2 2 2 2 3 0 6 5 1 3 ⟶ 1 2 2 3 1 2 2 3 1 3 0 1 3 6 5 1 3 0 , 6 9 2 2 2 2 2 2 3 0 6 5 1 2 ⟶ 1 2 2 3 1 2 2 3 1 3 0 1 3 6 5 1 3 1 , 6 9 2 2 2 2 2 2 3 0 6 5 1 4 ⟶ 1 2 2 3 1 2 2 3 1 3 0 1 3 6 5 1 3 6 , 6 9 2 2 2 2 2 2 3 0 6 5 1 7 ⟶ 1 2 2 3 1 2 2 3 1 3 0 1 3 6 5 1 3 8 , 4 9 2 2 2 2 2 2 3 0 6 5 1 3 ⟶ 2 2 2 3 1 2 2 3 1 3 0 1 3 6 5 1 3 0 , 4 9 2 2 2 2 2 2 3 0 6 5 1 2 ⟶ 2 2 2 3 1 2 2 3 1 3 0 1 3 6 5 1 3 1 , 4 9 2 2 2 2 2 2 3 0 6 5 1 4 ⟶ 2 2 2 3 1 2 2 3 1 3 0 1 3 6 5 1 3 6 , 4 9 2 2 2 2 2 2 3 0 6 5 1 7 ⟶ 2 2 2 3 1 2 2 3 1 3 0 1 3 6 5 1 3 8 , 12 9 2 2 2 2 2 2 3 0 6 5 1 3 ⟶ 9 2 2 3 1 2 2 3 1 3 0 1 3 6 5 1 3 0 , 12 9 2 2 2 2 2 2 3 0 6 5 1 2 ⟶ 9 2 2 3 1 2 2 3 1 3 0 1 3 6 5 1 3 1 , 12 9 2 2 2 2 2 2 3 0 6 5 1 4 ⟶ 9 2 2 3 1 2 2 3 1 3 0 1 3 6 5 1 3 6 , 12 9 2 2 2 2 2 2 3 0 6 5 1 7 ⟶ 9 2 2 3 1 2 2 3 1 3 0 1 3 6 5 1 3 8 , 14 9 2 2 2 2 2 2 3 0 6 5 1 3 ⟶ 11 2 2 3 1 2 2 3 1 3 0 1 3 6 5 1 3 0 , 14 9 2 2 2 2 2 2 3 0 6 5 1 2 ⟶ 11 2 2 3 1 2 2 3 1 3 0 1 3 6 5 1 3 1 , 14 9 2 2 2 2 2 2 3 0 6 5 1 4 ⟶ 11 2 2 3 1 2 2 3 1 3 0 1 3 6 5 1 3 6 , 14 9 2 2 2 2 2 2 3 0 6 5 1 7 ⟶ 11 2 2 3 1 2 2 3 1 3 0 1 3 6 5 1 3 8 , 6 12 9 3 0 6 9 2 2 2 2 3 1 3 ⟶ 6 12 12 5 0 0 1 3 1 3 1 2 3 1 2 2 3 0 , 6 12 9 3 0 6 9 2 2 2 2 3 1 2 ⟶ 6 12 12 5 0 0 1 3 1 3 1 2 3 1 2 2 3 1 , 6 12 9 3 0 6 9 2 2 2 2 3 1 4 ⟶ 6 12 12 5 0 0 1 3 1 3 1 2 3 1 2 2 3 6 , 6 12 9 3 0 6 9 2 2 2 2 3 1 7 ⟶ 6 12 12 5 0 0 1 3 1 3 1 2 3 1 2 2 3 8 , 4 12 9 3 0 6 9 2 2 2 2 3 1 3 ⟶ 4 12 12 5 0 0 1 3 1 3 1 2 3 1 2 2 3 0 , 4 12 9 3 0 6 9 2 2 2 2 3 1 2 ⟶ 4 12 12 5 0 0 1 3 1 3 1 2 3 1 2 2 3 1 , 4 12 9 3 0 6 9 2 2 2 2 3 1 4 ⟶ 4 12 12 5 0 0 1 3 1 3 1 2 3 1 2 2 3 6 , 4 12 9 3 0 6 9 2 2 2 2 3 1 7 ⟶ 4 12 12 5 0 0 1 3 1 3 1 2 3 1 2 2 3 8 , 12 12 9 3 0 6 9 2 2 2 2 3 1 3 ⟶ 12 12 12 5 0 0 1 3 1 3 1 2 3 1 2 2 3 0 , 12 12 9 3 0 6 9 2 2 2 2 3 1 2 ⟶ 12 12 12 5 0 0 1 3 1 3 1 2 3 1 2 2 3 1 , 12 12 9 3 0 6 9 2 2 2 2 3 1 4 ⟶ 12 12 12 5 0 0 1 3 1 3 1 2 3 1 2 2 3 6 , 12 12 9 3 0 6 9 2 2 2 2 3 1 7 ⟶ 12 12 12 5 0 0 1 3 1 3 1 2 3 1 2 2 3 8 , 14 12 9 3 0 6 9 2 2 2 2 3 1 3 ⟶ 14 12 12 5 0 0 1 3 1 3 1 2 3 1 2 2 3 0 , 14 12 9 3 0 6 9 2 2 2 2 3 1 2 ⟶ 14 12 12 5 0 0 1 3 1 3 1 2 3 1 2 2 3 1 , 14 12 9 3 0 6 9 2 2 2 2 3 1 4 ⟶ 14 12 12 5 0 0 1 3 1 3 1 2 3 1 2 2 3 6 , 14 12 9 3 0 6 9 2 2 2 2 3 1 7 ⟶ 14 12 12 5 0 0 1 3 1 3 1 2 3 1 2 2 3 8 , 6 12 9 3 0 6 9 4 5 0 1 3 6 5 ⟶ 6 9 3 0 0 6 9 3 0 1 4 5 0 1 2 2 4 5 , 6 12 9 3 0 6 9 4 5 0 1 3 6 9 ⟶ 6 9 3 0 0 6 9 3 0 1 4 5 0 1 2 2 4 9 , 6 12 9 3 0 6 9 4 5 0 1 3 6 12 ⟶ 6 9 3 0 0 6 9 3 0 1 4 5 0 1 2 2 4 12 , 6 12 9 3 0 6 9 4 5 0 1 3 6 13 ⟶ 6 9 3 0 0 6 9 3 0 1 4 5 0 1 2 2 4 13 , 4 12 9 3 0 6 9 4 5 0 1 3 6 5 ⟶ 4 9 3 0 0 6 9 3 0 1 4 5 0 1 2 2 4 5 , 4 12 9 3 0 6 9 4 5 0 1 3 6 9 ⟶ 4 9 3 0 0 6 9 3 0 1 4 5 0 1 2 2 4 9 , 4 12 9 3 0 6 9 4 5 0 1 3 6 12 ⟶ 4 9 3 0 0 6 9 3 0 1 4 5 0 1 2 2 4 12 , 4 12 9 3 0 6 9 4 5 0 1 3 6 13 ⟶ 4 9 3 0 0 6 9 3 0 1 4 5 0 1 2 2 4 13 , 12 12 9 3 0 6 9 4 5 0 1 3 6 5 ⟶ 12 9 3 0 0 6 9 3 0 1 4 5 0 1 2 2 4 5 , 12 12 9 3 0 6 9 4 5 0 1 3 6 9 ⟶ 12 9 3 0 0 6 9 3 0 1 4 5 0 1 2 2 4 9 , 12 12 9 3 0 6 9 4 5 0 1 3 6 12 ⟶ 12 9 3 0 0 6 9 3 0 1 4 5 0 1 2 2 4 12 , 12 12 9 3 0 6 9 4 5 0 1 3 6 13 ⟶ 12 9 3 0 0 6 9 3 0 1 4 5 0 1 2 2 4 13 , 14 12 9 3 0 6 9 4 5 0 1 3 6 5 ⟶ 14 9 3 0 0 6 9 3 0 1 4 5 0 1 2 2 4 5 , 14 12 9 3 0 6 9 4 5 0 1 3 6 9 ⟶ 14 9 3 0 0 6 9 3 0 1 4 5 0 1 2 2 4 9 , 14 12 9 3 0 6 9 4 5 0 1 3 6 12 ⟶ 14 9 3 0 0 6 9 3 0 1 4 5 0 1 2 2 4 12 , 14 12 9 3 0 6 9 4 5 0 1 3 6 13 ⟶ 14 9 3 0 0 6 9 3 0 1 4 5 0 1 2 2 4 13 , 6 12 12 5 0 0 1 3 0 0 0 1 3 0 ⟶ 0 6 5 0 1 2 2 3 1 2 3 0 0 1 3 1 3 0 , 6 12 12 5 0 0 1 3 0 0 0 1 3 1 ⟶ 0 6 5 0 1 2 2 3 1 2 3 0 0 1 3 1 3 1 , 6 12 12 5 0 0 1 3 0 0 0 1 3 6 ⟶ 0 6 5 0 1 2 2 3 1 2 3 0 0 1 3 1 3 6 , 6 12 12 5 0 0 1 3 0 0 0 1 3 8 ⟶ 0 6 5 0 1 2 2 3 1 2 3 0 0 1 3 1 3 8 , 4 12 12 5 0 0 1 3 0 0 0 1 3 0 ⟶ 3 6 5 0 1 2 2 3 1 2 3 0 0 1 3 1 3 0 , 4 12 12 5 0 0 1 3 0 0 0 1 3 1 ⟶ 3 6 5 0 1 2 2 3 1 2 3 0 0 1 3 1 3 1 , 4 12 12 5 0 0 1 3 0 0 0 1 3 6 ⟶ 3 6 5 0 1 2 2 3 1 2 3 0 0 1 3 1 3 6 , 4 12 12 5 0 0 1 3 0 0 0 1 3 8 ⟶ 3 6 5 0 1 2 2 3 1 2 3 0 0 1 3 1 3 8 , 12 12 12 5 0 0 1 3 0 0 0 1 3 0 ⟶ 5 6 5 0 1 2 2 3 1 2 3 0 0 1 3 1 3 0 , 12 12 12 5 0 0 1 3 0 0 0 1 3 1 ⟶ 5 6 5 0 1 2 2 3 1 2 3 0 0 1 3 1 3 1 , 12 12 12 5 0 0 1 3 0 0 0 1 3 6 ⟶ 5 6 5 0 1 2 2 3 1 2 3 0 0 1 3 1 3 6 , 12 12 12 5 0 0 1 3 0 0 0 1 3 8 ⟶ 5 6 5 0 1 2 2 3 1 2 3 0 0 1 3 1 3 8 , 14 12 12 5 0 0 1 3 0 0 0 1 3 0 ⟶ 10 6 5 0 1 2 2 3 1 2 3 0 0 1 3 1 3 0 , 14 12 12 5 0 0 1 3 0 0 0 1 3 1 ⟶ 10 6 5 0 1 2 2 3 1 2 3 0 0 1 3 1 3 1 , 14 12 12 5 0 0 1 3 0 0 0 1 3 6 ⟶ 10 6 5 0 1 2 2 3 1 2 3 0 0 1 3 1 3 6 , 14 12 12 5 0 0 1 3 0 0 0 1 3 8 ⟶ 10 6 5 0 1 2 2 3 1 2 3 0 0 1 3 1 3 8 } 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 6 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 8 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 1 ↦ 0, 3 ↦ 1, 0 ↦ 2, 2 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 9 ↦ 7, 12 ↦ 8, 13 ↦ 9, 11 ↦ 10 }, it remains to prove termination of the 40-rule system { 0 1 2 2 0 3 3 4 5 2 0 1 6 5 ⟶ 2 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 5 , 0 1 2 2 0 3 3 4 5 2 0 1 6 7 ⟶ 2 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 7 , 0 1 2 2 0 3 3 4 5 2 0 1 6 8 ⟶ 2 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 8 , 0 1 2 2 0 3 3 4 5 2 0 1 6 9 ⟶ 2 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 9 , 3 1 2 2 0 3 3 4 5 2 0 1 6 5 ⟶ 1 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 5 , 3 1 2 2 0 3 3 4 5 2 0 1 6 7 ⟶ 1 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 7 , 3 1 2 2 0 3 3 4 5 2 0 1 6 8 ⟶ 1 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 8 , 3 1 2 2 0 3 3 4 5 2 0 1 6 9 ⟶ 1 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 9 , 0 1 2 0 4 5 2 2 6 8 7 3 4 5 ⟶ 0 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 5 , 0 1 2 0 4 5 2 2 6 8 7 3 4 7 ⟶ 0 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 7 , 0 1 2 0 4 5 2 2 6 8 7 3 4 8 ⟶ 0 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 8 , 0 1 2 0 4 5 2 2 6 8 7 3 4 9 ⟶ 0 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 9 , 3 1 2 0 4 5 2 2 6 8 7 3 4 5 ⟶ 3 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 5 , 3 1 2 0 4 5 2 2 6 8 7 3 4 7 ⟶ 3 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 7 , 3 1 2 0 4 5 2 2 6 8 7 3 4 8 ⟶ 3 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 8 , 3 1 2 0 4 5 2 2 6 8 7 3 4 9 ⟶ 3 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 9 , 7 1 2 0 4 5 2 2 6 8 7 3 4 5 ⟶ 7 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 5 , 7 1 2 0 4 5 2 2 6 8 7 3 4 7 ⟶ 7 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 7 , 7 1 2 0 4 5 2 2 6 8 7 3 4 8 ⟶ 7 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 8 , 7 1 2 0 4 5 2 2 6 8 7 3 4 9 ⟶ 7 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 9 , 10 1 2 0 4 5 2 2 6 8 7 3 4 5 ⟶ 10 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 5 , 10 1 2 0 4 5 2 2 6 8 7 3 4 7 ⟶ 10 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 7 , 10 1 2 0 4 5 2 2 6 8 7 3 4 8 ⟶ 10 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 8 , 10 1 2 0 4 5 2 2 6 8 7 3 4 9 ⟶ 10 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 9 , 0 3 3 3 1 0 1 6 7 3 1 2 6 5 ⟶ 0 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 5 , 0 3 3 3 1 0 1 6 7 3 1 2 6 7 ⟶ 0 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 7 , 0 3 3 3 1 0 1 6 7 3 1 2 6 8 ⟶ 0 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 8 , 0 3 3 3 1 0 1 6 7 3 1 2 6 9 ⟶ 0 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 9 , 3 3 3 3 1 0 1 6 7 3 1 2 6 5 ⟶ 3 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 5 , 3 3 3 3 1 0 1 6 7 3 1 2 6 7 ⟶ 3 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 7 , 3 3 3 3 1 0 1 6 7 3 1 2 6 8 ⟶ 3 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 8 , 3 3 3 3 1 0 1 6 7 3 1 2 6 9 ⟶ 3 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 9 , 7 3 3 3 1 0 1 6 7 3 1 2 6 5 ⟶ 7 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 5 , 7 3 3 3 1 0 1 6 7 3 1 2 6 7 ⟶ 7 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 7 , 7 3 3 3 1 0 1 6 7 3 1 2 6 8 ⟶ 7 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 8 , 7 3 3 3 1 0 1 6 7 3 1 2 6 9 ⟶ 7 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 9 , 10 3 3 3 1 0 1 6 7 3 1 2 6 5 ⟶ 10 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 5 , 10 3 3 3 1 0 1 6 7 3 1 2 6 7 ⟶ 10 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 7 , 10 3 3 3 1 0 1 6 7 3 1 2 6 8 ⟶ 10 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 8 , 10 3 3 3 1 0 1 6 7 3 1 2 6 9 ⟶ 10 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 9 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 15: 0 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 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 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 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 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 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 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 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 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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, 10 ↦ 10 }, it remains to prove termination of the 39-rule system { 0 1 2 2 0 3 3 4 5 2 0 1 6 7 ⟶ 2 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 7 , 0 1 2 2 0 3 3 4 5 2 0 1 6 8 ⟶ 2 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 8 , 0 1 2 2 0 3 3 4 5 2 0 1 6 9 ⟶ 2 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 9 , 3 1 2 2 0 3 3 4 5 2 0 1 6 5 ⟶ 1 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 5 , 3 1 2 2 0 3 3 4 5 2 0 1 6 7 ⟶ 1 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 7 , 3 1 2 2 0 3 3 4 5 2 0 1 6 8 ⟶ 1 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 8 , 3 1 2 2 0 3 3 4 5 2 0 1 6 9 ⟶ 1 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 9 , 0 1 2 0 4 5 2 2 6 8 7 3 4 5 ⟶ 0 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 5 , 0 1 2 0 4 5 2 2 6 8 7 3 4 7 ⟶ 0 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 7 , 0 1 2 0 4 5 2 2 6 8 7 3 4 8 ⟶ 0 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 8 , 0 1 2 0 4 5 2 2 6 8 7 3 4 9 ⟶ 0 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 9 , 3 1 2 0 4 5 2 2 6 8 7 3 4 5 ⟶ 3 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 5 , 3 1 2 0 4 5 2 2 6 8 7 3 4 7 ⟶ 3 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 7 , 3 1 2 0 4 5 2 2 6 8 7 3 4 8 ⟶ 3 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 8 , 3 1 2 0 4 5 2 2 6 8 7 3 4 9 ⟶ 3 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 9 , 7 1 2 0 4 5 2 2 6 8 7 3 4 5 ⟶ 7 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 5 , 7 1 2 0 4 5 2 2 6 8 7 3 4 7 ⟶ 7 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 7 , 7 1 2 0 4 5 2 2 6 8 7 3 4 8 ⟶ 7 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 8 , 7 1 2 0 4 5 2 2 6 8 7 3 4 9 ⟶ 7 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 9 , 10 1 2 0 4 5 2 2 6 8 7 3 4 5 ⟶ 10 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 5 , 10 1 2 0 4 5 2 2 6 8 7 3 4 7 ⟶ 10 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 7 , 10 1 2 0 4 5 2 2 6 8 7 3 4 8 ⟶ 10 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 8 , 10 1 2 0 4 5 2 2 6 8 7 3 4 9 ⟶ 10 1 2 2 0 1 6 8 5 0 3 3 1 0 4 7 4 9 , 0 3 3 3 1 0 1 6 7 3 1 2 6 5 ⟶ 0 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 5 , 0 3 3 3 1 0 1 6 7 3 1 2 6 7 ⟶ 0 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 7 , 0 3 3 3 1 0 1 6 7 3 1 2 6 8 ⟶ 0 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 8 , 0 3 3 3 1 0 1 6 7 3 1 2 6 9 ⟶ 0 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 9 , 3 3 3 3 1 0 1 6 7 3 1 2 6 5 ⟶ 3 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 5 , 3 3 3 3 1 0 1 6 7 3 1 2 6 7 ⟶ 3 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 7 , 3 3 3 3 1 0 1 6 7 3 1 2 6 8 ⟶ 3 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 8 , 3 3 3 3 1 0 1 6 7 3 1 2 6 9 ⟶ 3 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 9 , 7 3 3 3 1 0 1 6 7 3 1 2 6 5 ⟶ 7 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 5 , 7 3 3 3 1 0 1 6 7 3 1 2 6 7 ⟶ 7 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 7 , 7 3 3 3 1 0 1 6 7 3 1 2 6 8 ⟶ 7 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 8 , 7 3 3 3 1 0 1 6 7 3 1 2 6 9 ⟶ 7 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 9 , 10 3 3 3 1 0 1 6 7 3 1 2 6 5 ⟶ 10 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 5 , 10 3 3 3 1 0 1 6 7 3 1 2 6 7 ⟶ 10 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 7 , 10 3 3 3 1 0 1 6 7 3 1 2 6 8 ⟶ 10 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 8 , 10 3 3 3 1 0 1 6 7 3 1 2 6 9 ⟶ 10 3 3 1 2 0 1 0 3 1 0 1 2 6 7 1 6 9 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 15: 0 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 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 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 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 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 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 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 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 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 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 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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, 8 ↦ 7, 9 ↦ 8, 7 ↦ 9, 10 ↦ 10 }, it remains to prove termination of the 38-rule system { 0 1 2 2 0 3 3 4 5 2 0 1 6 7 ⟶ 2 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 7 , 0 1 2 2 0 3 3 4 5 2 0 1 6 8 ⟶ 2 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 8 , 3 1 2 2 0 3 3 4 5 2 0 1 6 5 ⟶ 1 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 5 , 3 1 2 2 0 3 3 4 5 2 0 1 6 9 ⟶ 1 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 9 , 3 1 2 2 0 3 3 4 5 2 0 1 6 7 ⟶ 1 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 7 , 3 1 2 2 0 3 3 4 5 2 0 1 6 8 ⟶ 1 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 8 , 0 1 2 0 4 5 2 2 6 7 9 3 4 5 ⟶ 0 1 2 2 0 1 6 7 5 0 3 3 1 0 4 9 4 5 , 0 1 2 0 4 5 2 2 6 7 9 3 4 9 ⟶ 0 1 2 2 0 1 6 7 5 0 3 3 1 0 4 9 4 9 , 0 1 2 0 4 5 2 2 6 7 9 3 4 7 ⟶ 0 1 2 2 0 1 6 7 5 0 3 3 1 0 4 9 4 7 , 0 1 2 0 4 5 2 2 6 7 9 3 4 8 ⟶ 0 1 2 2 0 1 6 7 5 0 3 3 1 0 4 9 4 8 , 3 1 2 0 4 5 2 2 6 7 9 3 4 5 ⟶ 3 1 2 2 0 1 6 7 5 0 3 3 1 0 4 9 4 5 , 3 1 2 0 4 5 2 2 6 7 9 3 4 9 ⟶ 3 1 2 2 0 1 6 7 5 0 3 3 1 0 4 9 4 9 , 3 1 2 0 4 5 2 2 6 7 9 3 4 7 ⟶ 3 1 2 2 0 1 6 7 5 0 3 3 1 0 4 9 4 7 , 3 1 2 0 4 5 2 2 6 7 9 3 4 8 ⟶ 3 1 2 2 0 1 6 7 5 0 3 3 1 0 4 9 4 8 , 9 1 2 0 4 5 2 2 6 7 9 3 4 5 ⟶ 9 1 2 2 0 1 6 7 5 0 3 3 1 0 4 9 4 5 , 9 1 2 0 4 5 2 2 6 7 9 3 4 9 ⟶ 9 1 2 2 0 1 6 7 5 0 3 3 1 0 4 9 4 9 , 9 1 2 0 4 5 2 2 6 7 9 3 4 7 ⟶ 9 1 2 2 0 1 6 7 5 0 3 3 1 0 4 9 4 7 , 9 1 2 0 4 5 2 2 6 7 9 3 4 8 ⟶ 9 1 2 2 0 1 6 7 5 0 3 3 1 0 4 9 4 8 , 10 1 2 0 4 5 2 2 6 7 9 3 4 5 ⟶ 10 1 2 2 0 1 6 7 5 0 3 3 1 0 4 9 4 5 , 10 1 2 0 4 5 2 2 6 7 9 3 4 9 ⟶ 10 1 2 2 0 1 6 7 5 0 3 3 1 0 4 9 4 9 , 10 1 2 0 4 5 2 2 6 7 9 3 4 7 ⟶ 10 1 2 2 0 1 6 7 5 0 3 3 1 0 4 9 4 7 , 10 1 2 0 4 5 2 2 6 7 9 3 4 8 ⟶ 10 1 2 2 0 1 6 7 5 0 3 3 1 0 4 9 4 8 , 0 3 3 3 1 0 1 6 9 3 1 2 6 5 ⟶ 0 3 3 1 2 0 1 0 3 1 0 1 2 6 9 1 6 5 , 0 3 3 3 1 0 1 6 9 3 1 2 6 9 ⟶ 0 3 3 1 2 0 1 0 3 1 0 1 2 6 9 1 6 9 , 0 3 3 3 1 0 1 6 9 3 1 2 6 7 ⟶ 0 3 3 1 2 0 1 0 3 1 0 1 2 6 9 1 6 7 , 0 3 3 3 1 0 1 6 9 3 1 2 6 8 ⟶ 0 3 3 1 2 0 1 0 3 1 0 1 2 6 9 1 6 8 , 3 3 3 3 1 0 1 6 9 3 1 2 6 5 ⟶ 3 3 3 1 2 0 1 0 3 1 0 1 2 6 9 1 6 5 , 3 3 3 3 1 0 1 6 9 3 1 2 6 9 ⟶ 3 3 3 1 2 0 1 0 3 1 0 1 2 6 9 1 6 9 , 3 3 3 3 1 0 1 6 9 3 1 2 6 7 ⟶ 3 3 3 1 2 0 1 0 3 1 0 1 2 6 9 1 6 7 , 3 3 3 3 1 0 1 6 9 3 1 2 6 8 ⟶ 3 3 3 1 2 0 1 0 3 1 0 1 2 6 9 1 6 8 , 9 3 3 3 1 0 1 6 9 3 1 2 6 5 ⟶ 9 3 3 1 2 0 1 0 3 1 0 1 2 6 9 1 6 5 , 9 3 3 3 1 0 1 6 9 3 1 2 6 9 ⟶ 9 3 3 1 2 0 1 0 3 1 0 1 2 6 9 1 6 9 , 9 3 3 3 1 0 1 6 9 3 1 2 6 7 ⟶ 9 3 3 1 2 0 1 0 3 1 0 1 2 6 9 1 6 7 , 9 3 3 3 1 0 1 6 9 3 1 2 6 8 ⟶ 9 3 3 1 2 0 1 0 3 1 0 1 2 6 9 1 6 8 , 10 3 3 3 1 0 1 6 9 3 1 2 6 5 ⟶ 10 3 3 1 2 0 1 0 3 1 0 1 2 6 9 1 6 5 , 10 3 3 3 1 0 1 6 9 3 1 2 6 9 ⟶ 10 3 3 1 2 0 1 0 3 1 0 1 2 6 9 1 6 9 , 10 3 3 3 1 0 1 6 9 3 1 2 6 7 ⟶ 10 3 3 1 2 0 1 0 3 1 0 1 2 6 9 1 6 7 , 10 3 3 3 1 0 1 6 9 3 1 2 6 8 ⟶ 10 3 3 1 2 0 1 0 3 1 0 1 2 6 9 1 6 8 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 15: 0 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 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 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 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 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 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 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 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 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 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 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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, 8 ↦ 7, 9 ↦ 8, 7 ↦ 9, 10 ↦ 10 }, it remains to prove termination of the 37-rule system { 0 1 2 2 0 3 3 4 5 2 0 1 6 7 ⟶ 2 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 7 , 3 1 2 2 0 3 3 4 5 2 0 1 6 5 ⟶ 1 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 5 , 3 1 2 2 0 3 3 4 5 2 0 1 6 8 ⟶ 1 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 8 , 3 1 2 2 0 3 3 4 5 2 0 1 6 9 ⟶ 1 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 9 , 3 1 2 2 0 3 3 4 5 2 0 1 6 7 ⟶ 1 0 3 1 0 1 2 0 1 2 2 2 2 6 5 0 4 7 , 0 1 2 0 4 5 2 2 6 9 8 3 4 5 ⟶ 0 1 2 2 0 1 6 9 5 0 3 3 1 0 4 8 4 5 , 0 1 2 0 4 5 2 2 6 9 8 3 4 8 ⟶ 0 1 2 2 0 1 6 9 5 0 3 3 1 0 4 8 4 8 , 0 1 2 0 4 5 2 2 6 9 8 3 4 9 ⟶ 0 1 2 2 0 1 6 9 5 0 3 3 1 0 4 8 4 9 , 0 1 2 0 4 5 2 2 6 9 8 3 4 7 ⟶ 0 1 2 2 0 1 6 9 5 0 3 3 1 0 4 8 4 7 , 3 1 2 0 4 5 2 2 6 9 8 3 4 5 ⟶ 3 1 2 2 0 1 6 9 5 0 3 3 1 0 4 8 4 5 , 3 1 2 0 4 5 2 2 6 9 8 3 4 8 ⟶ 3 1 2 2 0 1 6 9 5 0 3 3 1 0 4 8 4 8 , 3 1 2 0 4 5 2 2 6 9 8 3 4 9 ⟶ 3 1 2 2 0 1 6 9 5 0 3 3 1 0 4 8 4 9 , 3 1 2 0 4 5 2 2 6 9 8 3 4 7 ⟶ 3 1 2 2 0 1 6 9 5 0 3 3 1 0 4 8 4 7 , 8 1 2 0 4 5 2 2 6 9 8 3 4 5 ⟶ 8 1 2 2 0 1 6 9 5 0 3 3 1 0 4 8 4 5 , 8 1 2 0 4 5 2 2 6 9 8 3 4 8 ⟶ 8 1 2 2 0 1 6 9 5 0 3 3 1 0 4 8 4 8 , 8 1 2 0 4 5 2 2 6 9 8 3 4 9 ⟶ 8 1 2 2 0 1 6 9 5 0 3 3 1 0 4 8 4 9 , 8 1 2 0 4 5 2 2 6 9 8 3 4 7 ⟶ 8 1 2 2 0 1 6 9 5 0 3 3 1 0 4 8 4 7 , 10 1 2 0 4 5 2 2 6 9 8 3 4 5 ⟶ 10 1 2 2 0 1 6 9 5 0 3 3 1 0 4 8 4 5 , 10 1 2 0 4 5 2 2 6 9 8 3 4 8 ⟶ 10 1 2 2 0 1 6 9 5 0 3 3 1 0 4 8 4 8 , 10 1 2 0 4 5 2 2 6 9 8 3 4 9 ⟶ 10 1 2 2 0 1 6 9 5 0 3 3 1 0 4 8 4 9 , 10 1 2 0 4 5 2 2 6 9 8 3 4 7 ⟶ 10 1 2 2 0 1 6 9 5 0 3 3 1 0 4 8 4 7 , 0 3 3 3 1 0 1 6 8 3 1 2 6 5 ⟶ 0 3 3 1 2 0 1 0 3 1 0 1 2 6 8 1 6 5 , 0 3 3 3 1 0 1 6 8 3 1 2 6 8 ⟶ 0 3 3 1 2 0 1 0 3 1 0 1 2 6 8 1 6 8 , 0 3 3 3 1 0 1 6 8 3 1 2 6 9 ⟶ 0 3 3 1 2 0 1 0 3 1 0 1 2 6 8 1 6 9 , 0 3 3 3 1 0 1 6 8 3 1 2 6 7 ⟶ 0 3 3 1 2 0 1 0 3 1 0 1 2 6 8 1 6 7 , 3 3 3 3 1 0 1 6 8 3 1 2 6 5 ⟶ 3 3 3 1 2 0 1 0 3 1 0 1 2 6 8 1 6 5 , 3 3 3 3 1 0 1 6 8 3 1 2 6 8 ⟶ 3 3 3 1 2 0 1 0 3 1 0 1 2 6 8 1 6 8 , 3 3 3 3 1 0 1 6 8 3 1 2 6 9 ⟶ 3 3 3 1 2 0 1 0 3 1 0 1 2 6 8 1 6 9 , 3 3 3 3 1 0 1 6 8 3 1 2 6 7 ⟶ 3 3 3 1 2 0 1 0 3 1 0 1 2 6 8 1 6 7 , 8 3 3 3 1 0 1 6 8 3 1 2 6 5 ⟶ 8 3 3 1 2 0 1 0 3 1 0 1 2 6 8 1 6 5 , 8 3 3 3 1 0 1 6 8 3 1 2 6 8 ⟶ 8 3 3 1 2 0 1 0 3 1 0 1 2 6 8 1 6 8 , 8 3 3 3 1 0 1 6 8 3 1 2 6 9 ⟶ 8 3 3 1 2 0 1 0 3 1 0 1 2 6 8 1 6 9 , 8 3 3 3 1 0 1 6 8 3 1 2 6 7 ⟶ 8 3 3 1 2 0 1 0 3 1 0 1 2 6 8 1 6 7 , 10 3 3 3 1 0 1 6 8 3 1 2 6 5 ⟶ 10 3 3 1 2 0 1 0 3 1 0 1 2 6 8 1 6 5 , 10 3 3 3 1 0 1 6 8 3 1 2 6 8 ⟶ 10 3 3 1 2 0 1 0 3 1 0 1 2 6 8 1 6 8 , 10 3 3 3 1 0 1 6 8 3 1 2 6 9 ⟶ 10 3 3 1 2 0 1 0 3 1 0 1 2 6 8 1 6 9 , 10 3 3 3 1 0 1 6 8 3 1 2 6 7 ⟶ 10 3 3 1 2 0 1 0 3 1 0 1 2 6 8 1 6 7 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 15: 0 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 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 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 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 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 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 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 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 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 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 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 { 3 ↦ 0, 1 ↦ 1, 2 ↦ 2, 0 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 8 ↦ 7, 9 ↦ 8, 7 ↦ 9, 10 ↦ 10 }, it remains to prove termination of the 36-rule system { 0 1 2 2 3 0 0 4 5 2 3 1 6 5 ⟶ 1 3 0 1 3 1 2 3 1 2 2 2 2 6 5 3 4 5 , 0 1 2 2 3 0 0 4 5 2 3 1 6 7 ⟶ 1 3 0 1 3 1 2 3 1 2 2 2 2 6 5 3 4 7 , 0 1 2 2 3 0 0 4 5 2 3 1 6 8 ⟶ 1 3 0 1 3 1 2 3 1 2 2 2 2 6 5 3 4 8 , 0 1 2 2 3 0 0 4 5 2 3 1 6 9 ⟶ 1 3 0 1 3 1 2 3 1 2 2 2 2 6 5 3 4 9 , 3 1 2 3 4 5 2 2 6 8 7 0 4 5 ⟶ 3 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 5 , 3 1 2 3 4 5 2 2 6 8 7 0 4 7 ⟶ 3 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 7 , 3 1 2 3 4 5 2 2 6 8 7 0 4 8 ⟶ 3 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 8 , 3 1 2 3 4 5 2 2 6 8 7 0 4 9 ⟶ 3 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 9 , 0 1 2 3 4 5 2 2 6 8 7 0 4 5 ⟶ 0 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 5 , 0 1 2 3 4 5 2 2 6 8 7 0 4 7 ⟶ 0 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 7 , 0 1 2 3 4 5 2 2 6 8 7 0 4 8 ⟶ 0 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 8 , 0 1 2 3 4 5 2 2 6 8 7 0 4 9 ⟶ 0 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 9 , 7 1 2 3 4 5 2 2 6 8 7 0 4 5 ⟶ 7 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 5 , 7 1 2 3 4 5 2 2 6 8 7 0 4 7 ⟶ 7 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 7 , 7 1 2 3 4 5 2 2 6 8 7 0 4 8 ⟶ 7 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 8 , 7 1 2 3 4 5 2 2 6 8 7 0 4 9 ⟶ 7 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 9 , 10 1 2 3 4 5 2 2 6 8 7 0 4 5 ⟶ 10 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 5 , 10 1 2 3 4 5 2 2 6 8 7 0 4 7 ⟶ 10 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 7 , 10 1 2 3 4 5 2 2 6 8 7 0 4 8 ⟶ 10 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 8 , 10 1 2 3 4 5 2 2 6 8 7 0 4 9 ⟶ 10 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 9 , 3 0 0 0 1 3 1 6 7 0 1 2 6 5 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 5 , 3 0 0 0 1 3 1 6 7 0 1 2 6 7 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 7 , 3 0 0 0 1 3 1 6 7 0 1 2 6 8 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 8 , 3 0 0 0 1 3 1 6 7 0 1 2 6 9 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 9 , 0 0 0 0 1 3 1 6 7 0 1 2 6 5 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 5 , 0 0 0 0 1 3 1 6 7 0 1 2 6 7 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 7 , 0 0 0 0 1 3 1 6 7 0 1 2 6 8 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 8 , 0 0 0 0 1 3 1 6 7 0 1 2 6 9 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 9 , 7 0 0 0 1 3 1 6 7 0 1 2 6 5 ⟶ 7 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 5 , 7 0 0 0 1 3 1 6 7 0 1 2 6 7 ⟶ 7 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 7 , 7 0 0 0 1 3 1 6 7 0 1 2 6 8 ⟶ 7 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 8 , 7 0 0 0 1 3 1 6 7 0 1 2 6 9 ⟶ 7 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 9 , 10 0 0 0 1 3 1 6 7 0 1 2 6 5 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 5 , 10 0 0 0 1 3 1 6 7 0 1 2 6 7 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 7 , 10 0 0 0 1 3 1 6 7 0 1 2 6 8 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 8 , 10 0 0 0 1 3 1 6 7 0 1 2 6 9 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 9 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 15: 0 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 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 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 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 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 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 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 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 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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, 10 ↦ 10 }, it remains to prove termination of the 35-rule system { 0 1 2 2 3 0 0 4 5 2 3 1 6 7 ⟶ 1 3 0 1 3 1 2 3 1 2 2 2 2 6 5 3 4 7 , 0 1 2 2 3 0 0 4 5 2 3 1 6 8 ⟶ 1 3 0 1 3 1 2 3 1 2 2 2 2 6 5 3 4 8 , 0 1 2 2 3 0 0 4 5 2 3 1 6 9 ⟶ 1 3 0 1 3 1 2 3 1 2 2 2 2 6 5 3 4 9 , 3 1 2 3 4 5 2 2 6 8 7 0 4 5 ⟶ 3 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 5 , 3 1 2 3 4 5 2 2 6 8 7 0 4 7 ⟶ 3 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 7 , 3 1 2 3 4 5 2 2 6 8 7 0 4 8 ⟶ 3 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 8 , 3 1 2 3 4 5 2 2 6 8 7 0 4 9 ⟶ 3 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 9 , 0 1 2 3 4 5 2 2 6 8 7 0 4 5 ⟶ 0 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 5 , 0 1 2 3 4 5 2 2 6 8 7 0 4 7 ⟶ 0 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 7 , 0 1 2 3 4 5 2 2 6 8 7 0 4 8 ⟶ 0 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 8 , 0 1 2 3 4 5 2 2 6 8 7 0 4 9 ⟶ 0 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 9 , 7 1 2 3 4 5 2 2 6 8 7 0 4 5 ⟶ 7 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 5 , 7 1 2 3 4 5 2 2 6 8 7 0 4 7 ⟶ 7 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 7 , 7 1 2 3 4 5 2 2 6 8 7 0 4 8 ⟶ 7 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 8 , 7 1 2 3 4 5 2 2 6 8 7 0 4 9 ⟶ 7 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 9 , 10 1 2 3 4 5 2 2 6 8 7 0 4 5 ⟶ 10 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 5 , 10 1 2 3 4 5 2 2 6 8 7 0 4 7 ⟶ 10 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 7 , 10 1 2 3 4 5 2 2 6 8 7 0 4 8 ⟶ 10 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 8 , 10 1 2 3 4 5 2 2 6 8 7 0 4 9 ⟶ 10 1 2 2 3 1 6 8 5 3 0 0 1 3 4 7 4 9 , 3 0 0 0 1 3 1 6 7 0 1 2 6 5 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 5 , 3 0 0 0 1 3 1 6 7 0 1 2 6 7 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 7 , 3 0 0 0 1 3 1 6 7 0 1 2 6 8 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 8 , 3 0 0 0 1 3 1 6 7 0 1 2 6 9 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 9 , 0 0 0 0 1 3 1 6 7 0 1 2 6 5 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 5 , 0 0 0 0 1 3 1 6 7 0 1 2 6 7 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 7 , 0 0 0 0 1 3 1 6 7 0 1 2 6 8 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 8 , 0 0 0 0 1 3 1 6 7 0 1 2 6 9 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 9 , 7 0 0 0 1 3 1 6 7 0 1 2 6 5 ⟶ 7 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 5 , 7 0 0 0 1 3 1 6 7 0 1 2 6 7 ⟶ 7 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 7 , 7 0 0 0 1 3 1 6 7 0 1 2 6 8 ⟶ 7 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 8 , 7 0 0 0 1 3 1 6 7 0 1 2 6 9 ⟶ 7 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 9 , 10 0 0 0 1 3 1 6 7 0 1 2 6 5 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 5 , 10 0 0 0 1 3 1 6 7 0 1 2 6 7 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 7 , 10 0 0 0 1 3 1 6 7 0 1 2 6 8 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 8 , 10 0 0 0 1 3 1 6 7 0 1 2 6 9 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 7 1 6 9 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 15: 0 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 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 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 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 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 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 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 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 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 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 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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, 8 ↦ 7, 9 ↦ 8, 7 ↦ 9, 10 ↦ 10 }, it remains to prove termination of the 34-rule system { 0 1 2 2 3 0 0 4 5 2 3 1 6 7 ⟶ 1 3 0 1 3 1 2 3 1 2 2 2 2 6 5 3 4 7 , 0 1 2 2 3 0 0 4 5 2 3 1 6 8 ⟶ 1 3 0 1 3 1 2 3 1 2 2 2 2 6 5 3 4 8 , 3 1 2 3 4 5 2 2 6 7 9 0 4 5 ⟶ 3 1 2 2 3 1 6 7 5 3 0 0 1 3 4 9 4 5 , 3 1 2 3 4 5 2 2 6 7 9 0 4 9 ⟶ 3 1 2 2 3 1 6 7 5 3 0 0 1 3 4 9 4 9 , 3 1 2 3 4 5 2 2 6 7 9 0 4 7 ⟶ 3 1 2 2 3 1 6 7 5 3 0 0 1 3 4 9 4 7 , 3 1 2 3 4 5 2 2 6 7 9 0 4 8 ⟶ 3 1 2 2 3 1 6 7 5 3 0 0 1 3 4 9 4 8 , 0 1 2 3 4 5 2 2 6 7 9 0 4 5 ⟶ 0 1 2 2 3 1 6 7 5 3 0 0 1 3 4 9 4 5 , 0 1 2 3 4 5 2 2 6 7 9 0 4 9 ⟶ 0 1 2 2 3 1 6 7 5 3 0 0 1 3 4 9 4 9 , 0 1 2 3 4 5 2 2 6 7 9 0 4 7 ⟶ 0 1 2 2 3 1 6 7 5 3 0 0 1 3 4 9 4 7 , 0 1 2 3 4 5 2 2 6 7 9 0 4 8 ⟶ 0 1 2 2 3 1 6 7 5 3 0 0 1 3 4 9 4 8 , 9 1 2 3 4 5 2 2 6 7 9 0 4 5 ⟶ 9 1 2 2 3 1 6 7 5 3 0 0 1 3 4 9 4 5 , 9 1 2 3 4 5 2 2 6 7 9 0 4 9 ⟶ 9 1 2 2 3 1 6 7 5 3 0 0 1 3 4 9 4 9 , 9 1 2 3 4 5 2 2 6 7 9 0 4 7 ⟶ 9 1 2 2 3 1 6 7 5 3 0 0 1 3 4 9 4 7 , 9 1 2 3 4 5 2 2 6 7 9 0 4 8 ⟶ 9 1 2 2 3 1 6 7 5 3 0 0 1 3 4 9 4 8 , 10 1 2 3 4 5 2 2 6 7 9 0 4 5 ⟶ 10 1 2 2 3 1 6 7 5 3 0 0 1 3 4 9 4 5 , 10 1 2 3 4 5 2 2 6 7 9 0 4 9 ⟶ 10 1 2 2 3 1 6 7 5 3 0 0 1 3 4 9 4 9 , 10 1 2 3 4 5 2 2 6 7 9 0 4 7 ⟶ 10 1 2 2 3 1 6 7 5 3 0 0 1 3 4 9 4 7 , 10 1 2 3 4 5 2 2 6 7 9 0 4 8 ⟶ 10 1 2 2 3 1 6 7 5 3 0 0 1 3 4 9 4 8 , 3 0 0 0 1 3 1 6 9 0 1 2 6 5 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 5 , 3 0 0 0 1 3 1 6 9 0 1 2 6 9 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 9 , 3 0 0 0 1 3 1 6 9 0 1 2 6 7 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 7 , 3 0 0 0 1 3 1 6 9 0 1 2 6 8 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 8 , 0 0 0 0 1 3 1 6 9 0 1 2 6 5 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 5 , 0 0 0 0 1 3 1 6 9 0 1 2 6 9 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 9 , 0 0 0 0 1 3 1 6 9 0 1 2 6 7 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 7 , 0 0 0 0 1 3 1 6 9 0 1 2 6 8 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 8 , 9 0 0 0 1 3 1 6 9 0 1 2 6 5 ⟶ 9 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 5 , 9 0 0 0 1 3 1 6 9 0 1 2 6 9 ⟶ 9 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 9 , 9 0 0 0 1 3 1 6 9 0 1 2 6 7 ⟶ 9 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 7 , 9 0 0 0 1 3 1 6 9 0 1 2 6 8 ⟶ 9 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 8 , 10 0 0 0 1 3 1 6 9 0 1 2 6 5 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 5 , 10 0 0 0 1 3 1 6 9 0 1 2 6 9 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 9 , 10 0 0 0 1 3 1 6 9 0 1 2 6 7 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 7 , 10 0 0 0 1 3 1 6 9 0 1 2 6 8 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 8 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 15: 0 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 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 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 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 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 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 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 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 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 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 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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, 8 ↦ 7, 7 ↦ 8, 9 ↦ 9, 10 ↦ 10 }, it remains to prove termination of the 33-rule system { 0 1 2 2 3 0 0 4 5 2 3 1 6 7 ⟶ 1 3 0 1 3 1 2 3 1 2 2 2 2 6 5 3 4 7 , 3 1 2 3 4 5 2 2 6 8 9 0 4 5 ⟶ 3 1 2 2 3 1 6 8 5 3 0 0 1 3 4 9 4 5 , 3 1 2 3 4 5 2 2 6 8 9 0 4 9 ⟶ 3 1 2 2 3 1 6 8 5 3 0 0 1 3 4 9 4 9 , 3 1 2 3 4 5 2 2 6 8 9 0 4 8 ⟶ 3 1 2 2 3 1 6 8 5 3 0 0 1 3 4 9 4 8 , 3 1 2 3 4 5 2 2 6 8 9 0 4 7 ⟶ 3 1 2 2 3 1 6 8 5 3 0 0 1 3 4 9 4 7 , 0 1 2 3 4 5 2 2 6 8 9 0 4 5 ⟶ 0 1 2 2 3 1 6 8 5 3 0 0 1 3 4 9 4 5 , 0 1 2 3 4 5 2 2 6 8 9 0 4 9 ⟶ 0 1 2 2 3 1 6 8 5 3 0 0 1 3 4 9 4 9 , 0 1 2 3 4 5 2 2 6 8 9 0 4 8 ⟶ 0 1 2 2 3 1 6 8 5 3 0 0 1 3 4 9 4 8 , 0 1 2 3 4 5 2 2 6 8 9 0 4 7 ⟶ 0 1 2 2 3 1 6 8 5 3 0 0 1 3 4 9 4 7 , 9 1 2 3 4 5 2 2 6 8 9 0 4 5 ⟶ 9 1 2 2 3 1 6 8 5 3 0 0 1 3 4 9 4 5 , 9 1 2 3 4 5 2 2 6 8 9 0 4 9 ⟶ 9 1 2 2 3 1 6 8 5 3 0 0 1 3 4 9 4 9 , 9 1 2 3 4 5 2 2 6 8 9 0 4 8 ⟶ 9 1 2 2 3 1 6 8 5 3 0 0 1 3 4 9 4 8 , 9 1 2 3 4 5 2 2 6 8 9 0 4 7 ⟶ 9 1 2 2 3 1 6 8 5 3 0 0 1 3 4 9 4 7 , 10 1 2 3 4 5 2 2 6 8 9 0 4 5 ⟶ 10 1 2 2 3 1 6 8 5 3 0 0 1 3 4 9 4 5 , 10 1 2 3 4 5 2 2 6 8 9 0 4 9 ⟶ 10 1 2 2 3 1 6 8 5 3 0 0 1 3 4 9 4 9 , 10 1 2 3 4 5 2 2 6 8 9 0 4 8 ⟶ 10 1 2 2 3 1 6 8 5 3 0 0 1 3 4 9 4 8 , 10 1 2 3 4 5 2 2 6 8 9 0 4 7 ⟶ 10 1 2 2 3 1 6 8 5 3 0 0 1 3 4 9 4 7 , 3 0 0 0 1 3 1 6 9 0 1 2 6 5 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 5 , 3 0 0 0 1 3 1 6 9 0 1 2 6 9 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 9 , 3 0 0 0 1 3 1 6 9 0 1 2 6 8 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 8 , 3 0 0 0 1 3 1 6 9 0 1 2 6 7 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 7 , 0 0 0 0 1 3 1 6 9 0 1 2 6 5 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 5 , 0 0 0 0 1 3 1 6 9 0 1 2 6 9 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 9 , 0 0 0 0 1 3 1 6 9 0 1 2 6 8 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 8 , 0 0 0 0 1 3 1 6 9 0 1 2 6 7 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 7 , 9 0 0 0 1 3 1 6 9 0 1 2 6 5 ⟶ 9 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 5 , 9 0 0 0 1 3 1 6 9 0 1 2 6 9 ⟶ 9 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 9 , 9 0 0 0 1 3 1 6 9 0 1 2 6 8 ⟶ 9 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 8 , 9 0 0 0 1 3 1 6 9 0 1 2 6 7 ⟶ 9 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 7 , 10 0 0 0 1 3 1 6 9 0 1 2 6 5 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 5 , 10 0 0 0 1 3 1 6 9 0 1 2 6 9 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 9 , 10 0 0 0 1 3 1 6 9 0 1 2 6 8 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 8 , 10 0 0 0 1 3 1 6 9 0 1 2 6 7 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 9 1 6 7 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 15: 0 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 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 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 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 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 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 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 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 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 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 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 { 3 ↦ 0, 1 ↦ 1, 2 ↦ 2, 4 ↦ 3, 5 ↦ 4, 6 ↦ 5, 8 ↦ 6, 9 ↦ 7, 0 ↦ 8, 7 ↦ 9, 10 ↦ 10 }, it remains to prove termination of the 32-rule system { 0 1 2 0 3 4 2 2 5 6 7 8 3 4 ⟶ 0 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 4 , 0 1 2 0 3 4 2 2 5 6 7 8 3 7 ⟶ 0 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 7 , 0 1 2 0 3 4 2 2 5 6 7 8 3 6 ⟶ 0 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 6 , 0 1 2 0 3 4 2 2 5 6 7 8 3 9 ⟶ 0 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 9 , 8 1 2 0 3 4 2 2 5 6 7 8 3 4 ⟶ 8 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 4 , 8 1 2 0 3 4 2 2 5 6 7 8 3 7 ⟶ 8 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 7 , 8 1 2 0 3 4 2 2 5 6 7 8 3 6 ⟶ 8 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 6 , 8 1 2 0 3 4 2 2 5 6 7 8 3 9 ⟶ 8 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 9 , 7 1 2 0 3 4 2 2 5 6 7 8 3 4 ⟶ 7 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 4 , 7 1 2 0 3 4 2 2 5 6 7 8 3 7 ⟶ 7 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 7 , 7 1 2 0 3 4 2 2 5 6 7 8 3 6 ⟶ 7 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 6 , 7 1 2 0 3 4 2 2 5 6 7 8 3 9 ⟶ 7 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 9 , 10 1 2 0 3 4 2 2 5 6 7 8 3 4 ⟶ 10 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 4 , 10 1 2 0 3 4 2 2 5 6 7 8 3 7 ⟶ 10 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 7 , 10 1 2 0 3 4 2 2 5 6 7 8 3 6 ⟶ 10 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 6 , 10 1 2 0 3 4 2 2 5 6 7 8 3 9 ⟶ 10 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 9 , 0 8 8 8 1 0 1 5 7 8 1 2 5 4 ⟶ 0 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 4 , 0 8 8 8 1 0 1 5 7 8 1 2 5 7 ⟶ 0 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 7 , 0 8 8 8 1 0 1 5 7 8 1 2 5 6 ⟶ 0 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 6 , 0 8 8 8 1 0 1 5 7 8 1 2 5 9 ⟶ 0 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 9 , 8 8 8 8 1 0 1 5 7 8 1 2 5 4 ⟶ 8 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 4 , 8 8 8 8 1 0 1 5 7 8 1 2 5 7 ⟶ 8 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 7 , 8 8 8 8 1 0 1 5 7 8 1 2 5 6 ⟶ 8 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 6 , 8 8 8 8 1 0 1 5 7 8 1 2 5 9 ⟶ 8 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 9 , 7 8 8 8 1 0 1 5 7 8 1 2 5 4 ⟶ 7 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 4 , 7 8 8 8 1 0 1 5 7 8 1 2 5 7 ⟶ 7 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 7 , 7 8 8 8 1 0 1 5 7 8 1 2 5 6 ⟶ 7 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 6 , 7 8 8 8 1 0 1 5 7 8 1 2 5 9 ⟶ 7 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 9 , 10 8 8 8 1 0 1 5 7 8 1 2 5 4 ⟶ 10 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 4 , 10 8 8 8 1 0 1 5 7 8 1 2 5 7 ⟶ 10 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 7 , 10 8 8 8 1 0 1 5 7 8 1 2 5 6 ⟶ 10 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 6 , 10 8 8 8 1 0 1 5 7 8 1 2 5 9 ⟶ 10 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 9 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 15: 0 ↦ ⎛ ⎞ ⎜ 1 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 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 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 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 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 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 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 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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, 10 ↦ 10 }, it remains to prove termination of the 31-rule system { 0 1 2 0 3 4 2 2 5 6 7 8 3 7 ⟶ 0 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 7 , 0 1 2 0 3 4 2 2 5 6 7 8 3 6 ⟶ 0 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 6 , 0 1 2 0 3 4 2 2 5 6 7 8 3 9 ⟶ 0 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 9 , 8 1 2 0 3 4 2 2 5 6 7 8 3 4 ⟶ 8 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 4 , 8 1 2 0 3 4 2 2 5 6 7 8 3 7 ⟶ 8 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 7 , 8 1 2 0 3 4 2 2 5 6 7 8 3 6 ⟶ 8 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 6 , 8 1 2 0 3 4 2 2 5 6 7 8 3 9 ⟶ 8 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 9 , 7 1 2 0 3 4 2 2 5 6 7 8 3 4 ⟶ 7 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 4 , 7 1 2 0 3 4 2 2 5 6 7 8 3 7 ⟶ 7 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 7 , 7 1 2 0 3 4 2 2 5 6 7 8 3 6 ⟶ 7 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 6 , 7 1 2 0 3 4 2 2 5 6 7 8 3 9 ⟶ 7 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 9 , 10 1 2 0 3 4 2 2 5 6 7 8 3 4 ⟶ 10 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 4 , 10 1 2 0 3 4 2 2 5 6 7 8 3 7 ⟶ 10 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 7 , 10 1 2 0 3 4 2 2 5 6 7 8 3 6 ⟶ 10 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 6 , 10 1 2 0 3 4 2 2 5 6 7 8 3 9 ⟶ 10 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 9 , 0 8 8 8 1 0 1 5 7 8 1 2 5 4 ⟶ 0 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 4 , 0 8 8 8 1 0 1 5 7 8 1 2 5 7 ⟶ 0 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 7 , 0 8 8 8 1 0 1 5 7 8 1 2 5 6 ⟶ 0 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 6 , 0 8 8 8 1 0 1 5 7 8 1 2 5 9 ⟶ 0 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 9 , 8 8 8 8 1 0 1 5 7 8 1 2 5 4 ⟶ 8 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 4 , 8 8 8 8 1 0 1 5 7 8 1 2 5 7 ⟶ 8 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 7 , 8 8 8 8 1 0 1 5 7 8 1 2 5 6 ⟶ 8 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 6 , 8 8 8 8 1 0 1 5 7 8 1 2 5 9 ⟶ 8 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 9 , 7 8 8 8 1 0 1 5 7 8 1 2 5 4 ⟶ 7 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 4 , 7 8 8 8 1 0 1 5 7 8 1 2 5 7 ⟶ 7 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 7 , 7 8 8 8 1 0 1 5 7 8 1 2 5 6 ⟶ 7 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 6 , 7 8 8 8 1 0 1 5 7 8 1 2 5 9 ⟶ 7 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 9 , 10 8 8 8 1 0 1 5 7 8 1 2 5 4 ⟶ 10 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 4 , 10 8 8 8 1 0 1 5 7 8 1 2 5 7 ⟶ 10 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 7 , 10 8 8 8 1 0 1 5 7 8 1 2 5 6 ⟶ 10 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 6 , 10 8 8 8 1 0 1 5 7 8 1 2 5 9 ⟶ 10 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 9 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 15: 0 ↦ ⎛ ⎞ ⎜ 1 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 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 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 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 1 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 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 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 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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, 10 ↦ 10 }, it remains to prove termination of the 30-rule system { 0 1 2 0 3 4 2 2 5 6 7 8 3 6 ⟶ 0 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 6 , 0 1 2 0 3 4 2 2 5 6 7 8 3 9 ⟶ 0 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 9 , 8 1 2 0 3 4 2 2 5 6 7 8 3 4 ⟶ 8 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 4 , 8 1 2 0 3 4 2 2 5 6 7 8 3 7 ⟶ 8 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 7 , 8 1 2 0 3 4 2 2 5 6 7 8 3 6 ⟶ 8 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 6 , 8 1 2 0 3 4 2 2 5 6 7 8 3 9 ⟶ 8 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 9 , 7 1 2 0 3 4 2 2 5 6 7 8 3 4 ⟶ 7 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 4 , 7 1 2 0 3 4 2 2 5 6 7 8 3 7 ⟶ 7 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 7 , 7 1 2 0 3 4 2 2 5 6 7 8 3 6 ⟶ 7 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 6 , 7 1 2 0 3 4 2 2 5 6 7 8 3 9 ⟶ 7 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 9 , 10 1 2 0 3 4 2 2 5 6 7 8 3 4 ⟶ 10 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 4 , 10 1 2 0 3 4 2 2 5 6 7 8 3 7 ⟶ 10 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 7 , 10 1 2 0 3 4 2 2 5 6 7 8 3 6 ⟶ 10 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 6 , 10 1 2 0 3 4 2 2 5 6 7 8 3 9 ⟶ 10 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 9 , 0 8 8 8 1 0 1 5 7 8 1 2 5 4 ⟶ 0 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 4 , 0 8 8 8 1 0 1 5 7 8 1 2 5 7 ⟶ 0 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 7 , 0 8 8 8 1 0 1 5 7 8 1 2 5 6 ⟶ 0 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 6 , 0 8 8 8 1 0 1 5 7 8 1 2 5 9 ⟶ 0 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 9 , 8 8 8 8 1 0 1 5 7 8 1 2 5 4 ⟶ 8 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 4 , 8 8 8 8 1 0 1 5 7 8 1 2 5 7 ⟶ 8 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 7 , 8 8 8 8 1 0 1 5 7 8 1 2 5 6 ⟶ 8 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 6 , 8 8 8 8 1 0 1 5 7 8 1 2 5 9 ⟶ 8 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 9 , 7 8 8 8 1 0 1 5 7 8 1 2 5 4 ⟶ 7 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 4 , 7 8 8 8 1 0 1 5 7 8 1 2 5 7 ⟶ 7 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 7 , 7 8 8 8 1 0 1 5 7 8 1 2 5 6 ⟶ 7 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 6 , 7 8 8 8 1 0 1 5 7 8 1 2 5 9 ⟶ 7 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 9 , 10 8 8 8 1 0 1 5 7 8 1 2 5 4 ⟶ 10 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 4 , 10 8 8 8 1 0 1 5 7 8 1 2 5 7 ⟶ 10 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 7 , 10 8 8 8 1 0 1 5 7 8 1 2 5 6 ⟶ 10 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 6 , 10 8 8 8 1 0 1 5 7 8 1 2 5 9 ⟶ 10 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 9 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 15: 0 ↦ ⎛ ⎞ ⎜ 1 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 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 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 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 1 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 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 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 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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, 10 ↦ 10 }, it remains to prove termination of the 29-rule system { 0 1 2 0 3 4 2 2 5 6 7 8 3 9 ⟶ 0 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 9 , 8 1 2 0 3 4 2 2 5 6 7 8 3 4 ⟶ 8 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 4 , 8 1 2 0 3 4 2 2 5 6 7 8 3 7 ⟶ 8 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 7 , 8 1 2 0 3 4 2 2 5 6 7 8 3 6 ⟶ 8 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 6 , 8 1 2 0 3 4 2 2 5 6 7 8 3 9 ⟶ 8 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 9 , 7 1 2 0 3 4 2 2 5 6 7 8 3 4 ⟶ 7 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 4 , 7 1 2 0 3 4 2 2 5 6 7 8 3 7 ⟶ 7 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 7 , 7 1 2 0 3 4 2 2 5 6 7 8 3 6 ⟶ 7 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 6 , 7 1 2 0 3 4 2 2 5 6 7 8 3 9 ⟶ 7 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 9 , 10 1 2 0 3 4 2 2 5 6 7 8 3 4 ⟶ 10 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 4 , 10 1 2 0 3 4 2 2 5 6 7 8 3 7 ⟶ 10 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 7 , 10 1 2 0 3 4 2 2 5 6 7 8 3 6 ⟶ 10 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 6 , 10 1 2 0 3 4 2 2 5 6 7 8 3 9 ⟶ 10 1 2 2 0 1 5 6 4 0 8 8 1 0 3 7 3 9 , 0 8 8 8 1 0 1 5 7 8 1 2 5 4 ⟶ 0 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 4 , 0 8 8 8 1 0 1 5 7 8 1 2 5 7 ⟶ 0 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 7 , 0 8 8 8 1 0 1 5 7 8 1 2 5 6 ⟶ 0 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 6 , 0 8 8 8 1 0 1 5 7 8 1 2 5 9 ⟶ 0 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 9 , 8 8 8 8 1 0 1 5 7 8 1 2 5 4 ⟶ 8 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 4 , 8 8 8 8 1 0 1 5 7 8 1 2 5 7 ⟶ 8 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 7 , 8 8 8 8 1 0 1 5 7 8 1 2 5 6 ⟶ 8 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 6 , 8 8 8 8 1 0 1 5 7 8 1 2 5 9 ⟶ 8 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 9 , 7 8 8 8 1 0 1 5 7 8 1 2 5 4 ⟶ 7 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 4 , 7 8 8 8 1 0 1 5 7 8 1 2 5 7 ⟶ 7 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 7 , 7 8 8 8 1 0 1 5 7 8 1 2 5 6 ⟶ 7 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 6 , 7 8 8 8 1 0 1 5 7 8 1 2 5 9 ⟶ 7 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 9 , 10 8 8 8 1 0 1 5 7 8 1 2 5 4 ⟶ 10 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 4 , 10 8 8 8 1 0 1 5 7 8 1 2 5 7 ⟶ 10 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 7 , 10 8 8 8 1 0 1 5 7 8 1 2 5 6 ⟶ 10 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 6 , 10 8 8 8 1 0 1 5 7 8 1 2 5 9 ⟶ 10 8 8 1 2 0 1 0 8 1 0 1 2 5 7 1 5 9 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 15: 0 ↦ ⎛ ⎞ ⎜ 1 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 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 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 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 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 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 { 8 ↦ 0, 1 ↦ 1, 2 ↦ 2, 0 ↦ 3, 3 ↦ 4, 4 ↦ 5, 5 ↦ 6, 6 ↦ 7, 7 ↦ 8, 9 ↦ 9, 10 ↦ 10 }, it remains to prove termination of the 28-rule system { 0 1 2 3 4 5 2 2 6 7 8 0 4 5 ⟶ 0 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 5 , 0 1 2 3 4 5 2 2 6 7 8 0 4 8 ⟶ 0 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 8 , 0 1 2 3 4 5 2 2 6 7 8 0 4 7 ⟶ 0 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 7 , 0 1 2 3 4 5 2 2 6 7 8 0 4 9 ⟶ 0 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 9 , 8 1 2 3 4 5 2 2 6 7 8 0 4 5 ⟶ 8 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 5 , 8 1 2 3 4 5 2 2 6 7 8 0 4 8 ⟶ 8 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 8 , 8 1 2 3 4 5 2 2 6 7 8 0 4 7 ⟶ 8 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 7 , 8 1 2 3 4 5 2 2 6 7 8 0 4 9 ⟶ 8 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 9 , 10 1 2 3 4 5 2 2 6 7 8 0 4 5 ⟶ 10 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 5 , 10 1 2 3 4 5 2 2 6 7 8 0 4 8 ⟶ 10 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 8 , 10 1 2 3 4 5 2 2 6 7 8 0 4 7 ⟶ 10 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 7 , 10 1 2 3 4 5 2 2 6 7 8 0 4 9 ⟶ 10 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 9 , 3 0 0 0 1 3 1 6 8 0 1 2 6 5 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 5 , 3 0 0 0 1 3 1 6 8 0 1 2 6 8 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 8 , 3 0 0 0 1 3 1 6 8 0 1 2 6 7 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 7 , 3 0 0 0 1 3 1 6 8 0 1 2 6 9 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 9 , 0 0 0 0 1 3 1 6 8 0 1 2 6 5 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 5 , 0 0 0 0 1 3 1 6 8 0 1 2 6 8 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 8 , 0 0 0 0 1 3 1 6 8 0 1 2 6 7 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 7 , 0 0 0 0 1 3 1 6 8 0 1 2 6 9 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 9 , 8 0 0 0 1 3 1 6 8 0 1 2 6 5 ⟶ 8 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 5 , 8 0 0 0 1 3 1 6 8 0 1 2 6 8 ⟶ 8 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 8 , 8 0 0 0 1 3 1 6 8 0 1 2 6 7 ⟶ 8 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 7 , 8 0 0 0 1 3 1 6 8 0 1 2 6 9 ⟶ 8 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 9 , 10 0 0 0 1 3 1 6 8 0 1 2 6 5 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 5 , 10 0 0 0 1 3 1 6 8 0 1 2 6 8 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 8 , 10 0 0 0 1 3 1 6 8 0 1 2 6 7 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 7 , 10 0 0 0 1 3 1 6 8 0 1 2 6 9 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 9 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 15: 0 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 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 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 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 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 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 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 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 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 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 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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, 10 ↦ 10 }, it remains to prove termination of the 27-rule system { 0 1 2 3 4 5 2 2 6 7 8 0 4 8 ⟶ 0 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 8 , 0 1 2 3 4 5 2 2 6 7 8 0 4 7 ⟶ 0 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 7 , 0 1 2 3 4 5 2 2 6 7 8 0 4 9 ⟶ 0 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 9 , 8 1 2 3 4 5 2 2 6 7 8 0 4 5 ⟶ 8 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 5 , 8 1 2 3 4 5 2 2 6 7 8 0 4 8 ⟶ 8 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 8 , 8 1 2 3 4 5 2 2 6 7 8 0 4 7 ⟶ 8 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 7 , 8 1 2 3 4 5 2 2 6 7 8 0 4 9 ⟶ 8 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 9 , 10 1 2 3 4 5 2 2 6 7 8 0 4 5 ⟶ 10 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 5 , 10 1 2 3 4 5 2 2 6 7 8 0 4 8 ⟶ 10 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 8 , 10 1 2 3 4 5 2 2 6 7 8 0 4 7 ⟶ 10 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 7 , 10 1 2 3 4 5 2 2 6 7 8 0 4 9 ⟶ 10 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 9 , 3 0 0 0 1 3 1 6 8 0 1 2 6 5 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 5 , 3 0 0 0 1 3 1 6 8 0 1 2 6 8 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 8 , 3 0 0 0 1 3 1 6 8 0 1 2 6 7 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 7 , 3 0 0 0 1 3 1 6 8 0 1 2 6 9 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 9 , 0 0 0 0 1 3 1 6 8 0 1 2 6 5 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 5 , 0 0 0 0 1 3 1 6 8 0 1 2 6 8 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 8 , 0 0 0 0 1 3 1 6 8 0 1 2 6 7 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 7 , 0 0 0 0 1 3 1 6 8 0 1 2 6 9 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 9 , 8 0 0 0 1 3 1 6 8 0 1 2 6 5 ⟶ 8 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 5 , 8 0 0 0 1 3 1 6 8 0 1 2 6 8 ⟶ 8 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 8 , 8 0 0 0 1 3 1 6 8 0 1 2 6 7 ⟶ 8 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 7 , 8 0 0 0 1 3 1 6 8 0 1 2 6 9 ⟶ 8 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 9 , 10 0 0 0 1 3 1 6 8 0 1 2 6 5 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 5 , 10 0 0 0 1 3 1 6 8 0 1 2 6 8 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 8 , 10 0 0 0 1 3 1 6 8 0 1 2 6 7 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 7 , 10 0 0 0 1 3 1 6 8 0 1 2 6 9 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 9 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 15: 0 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 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 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 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 1 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 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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, 10 ↦ 10 }, it remains to prove termination of the 26-rule system { 0 1 2 3 4 5 2 2 6 7 8 0 4 7 ⟶ 0 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 7 , 0 1 2 3 4 5 2 2 6 7 8 0 4 9 ⟶ 0 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 9 , 8 1 2 3 4 5 2 2 6 7 8 0 4 5 ⟶ 8 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 5 , 8 1 2 3 4 5 2 2 6 7 8 0 4 8 ⟶ 8 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 8 , 8 1 2 3 4 5 2 2 6 7 8 0 4 7 ⟶ 8 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 7 , 8 1 2 3 4 5 2 2 6 7 8 0 4 9 ⟶ 8 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 9 , 10 1 2 3 4 5 2 2 6 7 8 0 4 5 ⟶ 10 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 5 , 10 1 2 3 4 5 2 2 6 7 8 0 4 8 ⟶ 10 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 8 , 10 1 2 3 4 5 2 2 6 7 8 0 4 7 ⟶ 10 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 7 , 10 1 2 3 4 5 2 2 6 7 8 0 4 9 ⟶ 10 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 9 , 3 0 0 0 1 3 1 6 8 0 1 2 6 5 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 5 , 3 0 0 0 1 3 1 6 8 0 1 2 6 8 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 8 , 3 0 0 0 1 3 1 6 8 0 1 2 6 7 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 7 , 3 0 0 0 1 3 1 6 8 0 1 2 6 9 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 9 , 0 0 0 0 1 3 1 6 8 0 1 2 6 5 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 5 , 0 0 0 0 1 3 1 6 8 0 1 2 6 8 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 8 , 0 0 0 0 1 3 1 6 8 0 1 2 6 7 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 7 , 0 0 0 0 1 3 1 6 8 0 1 2 6 9 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 9 , 8 0 0 0 1 3 1 6 8 0 1 2 6 5 ⟶ 8 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 5 , 8 0 0 0 1 3 1 6 8 0 1 2 6 8 ⟶ 8 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 8 , 8 0 0 0 1 3 1 6 8 0 1 2 6 7 ⟶ 8 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 7 , 8 0 0 0 1 3 1 6 8 0 1 2 6 9 ⟶ 8 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 9 , 10 0 0 0 1 3 1 6 8 0 1 2 6 5 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 5 , 10 0 0 0 1 3 1 6 8 0 1 2 6 8 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 8 , 10 0 0 0 1 3 1 6 8 0 1 2 6 7 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 7 , 10 0 0 0 1 3 1 6 8 0 1 2 6 9 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 9 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 15: 0 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 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 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 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 1 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 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 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 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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, 10 ↦ 10 }, it remains to prove termination of the 25-rule system { 0 1 2 3 4 5 2 2 6 7 8 0 4 9 ⟶ 0 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 9 , 8 1 2 3 4 5 2 2 6 7 8 0 4 5 ⟶ 8 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 5 , 8 1 2 3 4 5 2 2 6 7 8 0 4 8 ⟶ 8 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 8 , 8 1 2 3 4 5 2 2 6 7 8 0 4 7 ⟶ 8 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 7 , 8 1 2 3 4 5 2 2 6 7 8 0 4 9 ⟶ 8 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 9 , 10 1 2 3 4 5 2 2 6 7 8 0 4 5 ⟶ 10 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 5 , 10 1 2 3 4 5 2 2 6 7 8 0 4 8 ⟶ 10 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 8 , 10 1 2 3 4 5 2 2 6 7 8 0 4 7 ⟶ 10 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 7 , 10 1 2 3 4 5 2 2 6 7 8 0 4 9 ⟶ 10 1 2 2 3 1 6 7 5 3 0 0 1 3 4 8 4 9 , 3 0 0 0 1 3 1 6 8 0 1 2 6 5 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 5 , 3 0 0 0 1 3 1 6 8 0 1 2 6 8 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 8 , 3 0 0 0 1 3 1 6 8 0 1 2 6 7 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 7 , 3 0 0 0 1 3 1 6 8 0 1 2 6 9 ⟶ 3 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 9 , 0 0 0 0 1 3 1 6 8 0 1 2 6 5 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 5 , 0 0 0 0 1 3 1 6 8 0 1 2 6 8 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 8 , 0 0 0 0 1 3 1 6 8 0 1 2 6 7 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 7 , 0 0 0 0 1 3 1 6 8 0 1 2 6 9 ⟶ 0 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 9 , 8 0 0 0 1 3 1 6 8 0 1 2 6 5 ⟶ 8 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 5 , 8 0 0 0 1 3 1 6 8 0 1 2 6 8 ⟶ 8 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 8 , 8 0 0 0 1 3 1 6 8 0 1 2 6 7 ⟶ 8 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 7 , 8 0 0 0 1 3 1 6 8 0 1 2 6 9 ⟶ 8 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 9 , 10 0 0 0 1 3 1 6 8 0 1 2 6 5 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 5 , 10 0 0 0 1 3 1 6 8 0 1 2 6 8 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 8 , 10 0 0 0 1 3 1 6 8 0 1 2 6 7 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 7 , 10 0 0 0 1 3 1 6 8 0 1 2 6 9 ⟶ 10 0 0 1 2 3 1 3 0 1 3 1 2 6 8 1 6 9 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 15: 0 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 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 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 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 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 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 { 8 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 0 ↦ 8, 9 ↦ 9, 10 ↦ 10 }, it remains to prove termination of the 24-rule system { 0 1 2 3 4 5 2 2 6 7 0 8 4 5 ⟶ 0 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 5 , 0 1 2 3 4 5 2 2 6 7 0 8 4 0 ⟶ 0 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 0 , 0 1 2 3 4 5 2 2 6 7 0 8 4 7 ⟶ 0 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 7 , 0 1 2 3 4 5 2 2 6 7 0 8 4 9 ⟶ 0 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 9 , 10 1 2 3 4 5 2 2 6 7 0 8 4 5 ⟶ 10 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 5 , 10 1 2 3 4 5 2 2 6 7 0 8 4 0 ⟶ 10 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 0 , 10 1 2 3 4 5 2 2 6 7 0 8 4 7 ⟶ 10 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 7 , 10 1 2 3 4 5 2 2 6 7 0 8 4 9 ⟶ 10 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 9 , 3 8 8 8 1 3 1 6 0 8 1 2 6 5 ⟶ 3 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 5 , 3 8 8 8 1 3 1 6 0 8 1 2 6 0 ⟶ 3 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 0 , 3 8 8 8 1 3 1 6 0 8 1 2 6 7 ⟶ 3 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 7 , 3 8 8 8 1 3 1 6 0 8 1 2 6 9 ⟶ 3 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 9 , 8 8 8 8 1 3 1 6 0 8 1 2 6 5 ⟶ 8 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 5 , 8 8 8 8 1 3 1 6 0 8 1 2 6 0 ⟶ 8 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 0 , 8 8 8 8 1 3 1 6 0 8 1 2 6 7 ⟶ 8 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 7 , 8 8 8 8 1 3 1 6 0 8 1 2 6 9 ⟶ 8 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 9 , 0 8 8 8 1 3 1 6 0 8 1 2 6 5 ⟶ 0 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 5 , 0 8 8 8 1 3 1 6 0 8 1 2 6 0 ⟶ 0 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 0 , 0 8 8 8 1 3 1 6 0 8 1 2 6 7 ⟶ 0 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 7 , 0 8 8 8 1 3 1 6 0 8 1 2 6 9 ⟶ 0 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 9 , 10 8 8 8 1 3 1 6 0 8 1 2 6 5 ⟶ 10 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 5 , 10 8 8 8 1 3 1 6 0 8 1 2 6 0 ⟶ 10 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 0 , 10 8 8 8 1 3 1 6 0 8 1 2 6 7 ⟶ 10 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 7 , 10 8 8 8 1 3 1 6 0 8 1 2 6 9 ⟶ 10 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 9 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 15: 0 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 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 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 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 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 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 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 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 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 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 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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, 10 ↦ 10 }, it remains to prove termination of the 23-rule system { 0 1 2 3 4 5 2 2 6 7 0 8 4 0 ⟶ 0 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 0 , 0 1 2 3 4 5 2 2 6 7 0 8 4 7 ⟶ 0 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 7 , 0 1 2 3 4 5 2 2 6 7 0 8 4 9 ⟶ 0 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 9 , 10 1 2 3 4 5 2 2 6 7 0 8 4 5 ⟶ 10 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 5 , 10 1 2 3 4 5 2 2 6 7 0 8 4 0 ⟶ 10 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 0 , 10 1 2 3 4 5 2 2 6 7 0 8 4 7 ⟶ 10 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 7 , 10 1 2 3 4 5 2 2 6 7 0 8 4 9 ⟶ 10 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 9 , 3 8 8 8 1 3 1 6 0 8 1 2 6 5 ⟶ 3 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 5 , 3 8 8 8 1 3 1 6 0 8 1 2 6 0 ⟶ 3 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 0 , 3 8 8 8 1 3 1 6 0 8 1 2 6 7 ⟶ 3 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 7 , 3 8 8 8 1 3 1 6 0 8 1 2 6 9 ⟶ 3 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 9 , 8 8 8 8 1 3 1 6 0 8 1 2 6 5 ⟶ 8 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 5 , 8 8 8 8 1 3 1 6 0 8 1 2 6 0 ⟶ 8 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 0 , 8 8 8 8 1 3 1 6 0 8 1 2 6 7 ⟶ 8 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 7 , 8 8 8 8 1 3 1 6 0 8 1 2 6 9 ⟶ 8 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 9 , 0 8 8 8 1 3 1 6 0 8 1 2 6 5 ⟶ 0 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 5 , 0 8 8 8 1 3 1 6 0 8 1 2 6 0 ⟶ 0 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 0 , 0 8 8 8 1 3 1 6 0 8 1 2 6 7 ⟶ 0 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 7 , 0 8 8 8 1 3 1 6 0 8 1 2 6 9 ⟶ 0 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 9 , 10 8 8 8 1 3 1 6 0 8 1 2 6 5 ⟶ 10 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 5 , 10 8 8 8 1 3 1 6 0 8 1 2 6 0 ⟶ 10 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 0 , 10 8 8 8 1 3 1 6 0 8 1 2 6 7 ⟶ 10 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 7 , 10 8 8 8 1 3 1 6 0 8 1 2 6 9 ⟶ 10 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 9 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 15: 0 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 1 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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, 10 ↦ 10 }, it remains to prove termination of the 22-rule system { 0 1 2 3 4 5 2 2 6 7 0 8 4 7 ⟶ 0 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 7 , 0 1 2 3 4 5 2 2 6 7 0 8 4 9 ⟶ 0 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 9 , 10 1 2 3 4 5 2 2 6 7 0 8 4 5 ⟶ 10 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 5 , 10 1 2 3 4 5 2 2 6 7 0 8 4 0 ⟶ 10 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 0 , 10 1 2 3 4 5 2 2 6 7 0 8 4 7 ⟶ 10 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 7 , 10 1 2 3 4 5 2 2 6 7 0 8 4 9 ⟶ 10 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 9 , 3 8 8 8 1 3 1 6 0 8 1 2 6 5 ⟶ 3 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 5 , 3 8 8 8 1 3 1 6 0 8 1 2 6 0 ⟶ 3 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 0 , 3 8 8 8 1 3 1 6 0 8 1 2 6 7 ⟶ 3 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 7 , 3 8 8 8 1 3 1 6 0 8 1 2 6 9 ⟶ 3 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 9 , 8 8 8 8 1 3 1 6 0 8 1 2 6 5 ⟶ 8 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 5 , 8 8 8 8 1 3 1 6 0 8 1 2 6 0 ⟶ 8 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 0 , 8 8 8 8 1 3 1 6 0 8 1 2 6 7 ⟶ 8 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 7 , 8 8 8 8 1 3 1 6 0 8 1 2 6 9 ⟶ 8 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 9 , 0 8 8 8 1 3 1 6 0 8 1 2 6 5 ⟶ 0 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 5 , 0 8 8 8 1 3 1 6 0 8 1 2 6 0 ⟶ 0 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 0 , 0 8 8 8 1 3 1 6 0 8 1 2 6 7 ⟶ 0 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 7 , 0 8 8 8 1 3 1 6 0 8 1 2 6 9 ⟶ 0 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 9 , 10 8 8 8 1 3 1 6 0 8 1 2 6 5 ⟶ 10 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 5 , 10 8 8 8 1 3 1 6 0 8 1 2 6 0 ⟶ 10 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 0 , 10 8 8 8 1 3 1 6 0 8 1 2 6 7 ⟶ 10 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 7 , 10 8 8 8 1 3 1 6 0 8 1 2 6 9 ⟶ 10 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 9 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 15: 0 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 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 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 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 1 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 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 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 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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, 10 ↦ 10 }, it remains to prove termination of the 21-rule system { 0 1 2 3 4 5 2 2 6 7 0 8 4 9 ⟶ 0 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 9 , 10 1 2 3 4 5 2 2 6 7 0 8 4 5 ⟶ 10 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 5 , 10 1 2 3 4 5 2 2 6 7 0 8 4 0 ⟶ 10 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 0 , 10 1 2 3 4 5 2 2 6 7 0 8 4 7 ⟶ 10 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 7 , 10 1 2 3 4 5 2 2 6 7 0 8 4 9 ⟶ 10 1 2 2 3 1 6 7 5 3 8 8 1 3 4 0 4 9 , 3 8 8 8 1 3 1 6 0 8 1 2 6 5 ⟶ 3 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 5 , 3 8 8 8 1 3 1 6 0 8 1 2 6 0 ⟶ 3 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 0 , 3 8 8 8 1 3 1 6 0 8 1 2 6 7 ⟶ 3 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 7 , 3 8 8 8 1 3 1 6 0 8 1 2 6 9 ⟶ 3 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 9 , 8 8 8 8 1 3 1 6 0 8 1 2 6 5 ⟶ 8 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 5 , 8 8 8 8 1 3 1 6 0 8 1 2 6 0 ⟶ 8 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 0 , 8 8 8 8 1 3 1 6 0 8 1 2 6 7 ⟶ 8 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 7 , 8 8 8 8 1 3 1 6 0 8 1 2 6 9 ⟶ 8 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 9 , 0 8 8 8 1 3 1 6 0 8 1 2 6 5 ⟶ 0 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 5 , 0 8 8 8 1 3 1 6 0 8 1 2 6 0 ⟶ 0 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 0 , 0 8 8 8 1 3 1 6 0 8 1 2 6 7 ⟶ 0 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 7 , 0 8 8 8 1 3 1 6 0 8 1 2 6 9 ⟶ 0 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 9 , 10 8 8 8 1 3 1 6 0 8 1 2 6 5 ⟶ 10 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 5 , 10 8 8 8 1 3 1 6 0 8 1 2 6 0 ⟶ 10 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 0 , 10 8 8 8 1 3 1 6 0 8 1 2 6 7 ⟶ 10 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 7 , 10 8 8 8 1 3 1 6 0 8 1 2 6 9 ⟶ 10 8 8 1 2 3 1 3 8 1 3 1 2 6 0 1 6 9 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 15: 0 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 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 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 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 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 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 { 10 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 0 ↦ 8, 8 ↦ 9, 9 ↦ 10 }, it remains to prove termination of the 20-rule system { 0 1 2 3 4 5 2 2 6 7 8 9 4 5 ⟶ 0 1 2 2 3 1 6 7 5 3 9 9 1 3 4 8 4 5 , 0 1 2 3 4 5 2 2 6 7 8 9 4 8 ⟶ 0 1 2 2 3 1 6 7 5 3 9 9 1 3 4 8 4 8 , 0 1 2 3 4 5 2 2 6 7 8 9 4 7 ⟶ 0 1 2 2 3 1 6 7 5 3 9 9 1 3 4 8 4 7 , 0 1 2 3 4 5 2 2 6 7 8 9 4 10 ⟶ 0 1 2 2 3 1 6 7 5 3 9 9 1 3 4 8 4 10 , 3 9 9 9 1 3 1 6 8 9 1 2 6 5 ⟶ 3 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 5 , 3 9 9 9 1 3 1 6 8 9 1 2 6 8 ⟶ 3 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 8 , 3 9 9 9 1 3 1 6 8 9 1 2 6 7 ⟶ 3 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 7 , 3 9 9 9 1 3 1 6 8 9 1 2 6 10 ⟶ 3 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 10 , 9 9 9 9 1 3 1 6 8 9 1 2 6 5 ⟶ 9 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 5 , 9 9 9 9 1 3 1 6 8 9 1 2 6 8 ⟶ 9 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 8 , 9 9 9 9 1 3 1 6 8 9 1 2 6 7 ⟶ 9 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 7 , 9 9 9 9 1 3 1 6 8 9 1 2 6 10 ⟶ 9 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 10 , 8 9 9 9 1 3 1 6 8 9 1 2 6 5 ⟶ 8 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 5 , 8 9 9 9 1 3 1 6 8 9 1 2 6 8 ⟶ 8 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 8 , 8 9 9 9 1 3 1 6 8 9 1 2 6 7 ⟶ 8 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 7 , 8 9 9 9 1 3 1 6 8 9 1 2 6 10 ⟶ 8 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 10 , 0 9 9 9 1 3 1 6 8 9 1 2 6 5 ⟶ 0 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 5 , 0 9 9 9 1 3 1 6 8 9 1 2 6 8 ⟶ 0 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 8 , 0 9 9 9 1 3 1 6 8 9 1 2 6 7 ⟶ 0 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 7 , 0 9 9 9 1 3 1 6 8 9 1 2 6 10 ⟶ 0 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 10 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 15: 0 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 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 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 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 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 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 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 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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, 10 ↦ 10 }, it remains to prove termination of the 19-rule system { 0 1 2 3 4 5 2 2 6 7 8 9 4 8 ⟶ 0 1 2 2 3 1 6 7 5 3 9 9 1 3 4 8 4 8 , 0 1 2 3 4 5 2 2 6 7 8 9 4 7 ⟶ 0 1 2 2 3 1 6 7 5 3 9 9 1 3 4 8 4 7 , 0 1 2 3 4 5 2 2 6 7 8 9 4 10 ⟶ 0 1 2 2 3 1 6 7 5 3 9 9 1 3 4 8 4 10 , 3 9 9 9 1 3 1 6 8 9 1 2 6 5 ⟶ 3 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 5 , 3 9 9 9 1 3 1 6 8 9 1 2 6 8 ⟶ 3 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 8 , 3 9 9 9 1 3 1 6 8 9 1 2 6 7 ⟶ 3 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 7 , 3 9 9 9 1 3 1 6 8 9 1 2 6 10 ⟶ 3 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 10 , 9 9 9 9 1 3 1 6 8 9 1 2 6 5 ⟶ 9 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 5 , 9 9 9 9 1 3 1 6 8 9 1 2 6 8 ⟶ 9 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 8 , 9 9 9 9 1 3 1 6 8 9 1 2 6 7 ⟶ 9 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 7 , 9 9 9 9 1 3 1 6 8 9 1 2 6 10 ⟶ 9 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 10 , 8 9 9 9 1 3 1 6 8 9 1 2 6 5 ⟶ 8 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 5 , 8 9 9 9 1 3 1 6 8 9 1 2 6 8 ⟶ 8 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 8 , 8 9 9 9 1 3 1 6 8 9 1 2 6 7 ⟶ 8 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 7 , 8 9 9 9 1 3 1 6 8 9 1 2 6 10 ⟶ 8 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 10 , 0 9 9 9 1 3 1 6 8 9 1 2 6 5 ⟶ 0 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 5 , 0 9 9 9 1 3 1 6 8 9 1 2 6 8 ⟶ 0 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 8 , 0 9 9 9 1 3 1 6 8 9 1 2 6 7 ⟶ 0 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 7 , 0 9 9 9 1 3 1 6 8 9 1 2 6 10 ⟶ 0 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 10 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 15: 0 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 1 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 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 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 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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, 10 ↦ 10 }, it remains to prove termination of the 18-rule system { 0 1 2 3 4 5 2 2 6 7 8 9 4 7 ⟶ 0 1 2 2 3 1 6 7 5 3 9 9 1 3 4 8 4 7 , 0 1 2 3 4 5 2 2 6 7 8 9 4 10 ⟶ 0 1 2 2 3 1 6 7 5 3 9 9 1 3 4 8 4 10 , 3 9 9 9 1 3 1 6 8 9 1 2 6 5 ⟶ 3 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 5 , 3 9 9 9 1 3 1 6 8 9 1 2 6 8 ⟶ 3 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 8 , 3 9 9 9 1 3 1 6 8 9 1 2 6 7 ⟶ 3 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 7 , 3 9 9 9 1 3 1 6 8 9 1 2 6 10 ⟶ 3 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 10 , 9 9 9 9 1 3 1 6 8 9 1 2 6 5 ⟶ 9 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 5 , 9 9 9 9 1 3 1 6 8 9 1 2 6 8 ⟶ 9 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 8 , 9 9 9 9 1 3 1 6 8 9 1 2 6 7 ⟶ 9 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 7 , 9 9 9 9 1 3 1 6 8 9 1 2 6 10 ⟶ 9 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 10 , 8 9 9 9 1 3 1 6 8 9 1 2 6 5 ⟶ 8 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 5 , 8 9 9 9 1 3 1 6 8 9 1 2 6 8 ⟶ 8 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 8 , 8 9 9 9 1 3 1 6 8 9 1 2 6 7 ⟶ 8 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 7 , 8 9 9 9 1 3 1 6 8 9 1 2 6 10 ⟶ 8 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 10 , 0 9 9 9 1 3 1 6 8 9 1 2 6 5 ⟶ 0 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 5 , 0 9 9 9 1 3 1 6 8 9 1 2 6 8 ⟶ 0 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 8 , 0 9 9 9 1 3 1 6 8 9 1 2 6 7 ⟶ 0 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 7 , 0 9 9 9 1 3 1 6 8 9 1 2 6 10 ⟶ 0 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 10 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 15: 0 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 1 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 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 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 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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, 10 ↦ 10 }, it remains to prove termination of the 17-rule system { 0 1 2 3 4 5 2 2 6 7 8 9 4 10 ⟶ 0 1 2 2 3 1 6 7 5 3 9 9 1 3 4 8 4 10 , 3 9 9 9 1 3 1 6 8 9 1 2 6 5 ⟶ 3 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 5 , 3 9 9 9 1 3 1 6 8 9 1 2 6 8 ⟶ 3 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 8 , 3 9 9 9 1 3 1 6 8 9 1 2 6 7 ⟶ 3 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 7 , 3 9 9 9 1 3 1 6 8 9 1 2 6 10 ⟶ 3 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 10 , 9 9 9 9 1 3 1 6 8 9 1 2 6 5 ⟶ 9 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 5 , 9 9 9 9 1 3 1 6 8 9 1 2 6 8 ⟶ 9 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 8 , 9 9 9 9 1 3 1 6 8 9 1 2 6 7 ⟶ 9 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 7 , 9 9 9 9 1 3 1 6 8 9 1 2 6 10 ⟶ 9 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 10 , 8 9 9 9 1 3 1 6 8 9 1 2 6 5 ⟶ 8 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 5 , 8 9 9 9 1 3 1 6 8 9 1 2 6 8 ⟶ 8 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 8 , 8 9 9 9 1 3 1 6 8 9 1 2 6 7 ⟶ 8 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 7 , 8 9 9 9 1 3 1 6 8 9 1 2 6 10 ⟶ 8 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 10 , 0 9 9 9 1 3 1 6 8 9 1 2 6 5 ⟶ 0 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 5 , 0 9 9 9 1 3 1 6 8 9 1 2 6 8 ⟶ 0 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 8 , 0 9 9 9 1 3 1 6 8 9 1 2 6 7 ⟶ 0 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 7 , 0 9 9 9 1 3 1 6 8 9 1 2 6 10 ⟶ 0 9 9 1 2 3 1 3 9 1 3 1 2 6 8 1 6 10 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 15: 0 ↦ ⎛ ⎞ ⎜ 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 0 0 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 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 { 3 ↦ 0, 9 ↦ 1, 1 ↦ 2, 6 ↦ 3, 8 ↦ 4, 2 ↦ 5, 5 ↦ 6, 7 ↦ 7, 10 ↦ 8, 0 ↦ 9 }, it remains to prove termination of the 16-rule system { 0 1 1 1 2 0 2 3 4 1 2 5 3 6 ⟶ 0 1 1 2 5 0 2 0 1 2 0 2 5 3 4 2 3 6 , 0 1 1 1 2 0 2 3 4 1 2 5 3 4 ⟶ 0 1 1 2 5 0 2 0 1 2 0 2 5 3 4 2 3 4 , 0 1 1 1 2 0 2 3 4 1 2 5 3 7 ⟶ 0 1 1 2 5 0 2 0 1 2 0 2 5 3 4 2 3 7 , 0 1 1 1 2 0 2 3 4 1 2 5 3 8 ⟶ 0 1 1 2 5 0 2 0 1 2 0 2 5 3 4 2 3 8 , 1 1 1 1 2 0 2 3 4 1 2 5 3 6 ⟶ 1 1 1 2 5 0 2 0 1 2 0 2 5 3 4 2 3 6 , 1 1 1 1 2 0 2 3 4 1 2 5 3 4 ⟶ 1 1 1 2 5 0 2 0 1 2 0 2 5 3 4 2 3 4 , 1 1 1 1 2 0 2 3 4 1 2 5 3 7 ⟶ 1 1 1 2 5 0 2 0 1 2 0 2 5 3 4 2 3 7 , 1 1 1 1 2 0 2 3 4 1 2 5 3 8 ⟶ 1 1 1 2 5 0 2 0 1 2 0 2 5 3 4 2 3 8 , 4 1 1 1 2 0 2 3 4 1 2 5 3 6 ⟶ 4 1 1 2 5 0 2 0 1 2 0 2 5 3 4 2 3 6 , 4 1 1 1 2 0 2 3 4 1 2 5 3 4 ⟶ 4 1 1 2 5 0 2 0 1 2 0 2 5 3 4 2 3 4 , 4 1 1 1 2 0 2 3 4 1 2 5 3 7 ⟶ 4 1 1 2 5 0 2 0 1 2 0 2 5 3 4 2 3 7 , 4 1 1 1 2 0 2 3 4 1 2 5 3 8 ⟶ 4 1 1 2 5 0 2 0 1 2 0 2 5 3 4 2 3 8 , 9 1 1 1 2 0 2 3 4 1 2 5 3 6 ⟶ 9 1 1 2 5 0 2 0 1 2 0 2 5 3 4 2 3 6 , 9 1 1 1 2 0 2 3 4 1 2 5 3 4 ⟶ 9 1 1 2 5 0 2 0 1 2 0 2 5 3 4 2 3 4 , 9 1 1 1 2 0 2 3 4 1 2 5 3 7 ⟶ 9 1 1 2 5 0 2 0 1 2 0 2 5 3 4 2 3 7 , 9 1 1 1 2 0 2 3 4 1 2 5 3 8 ⟶ 9 1 1 2 5 0 2 0 1 2 0 2 5 3 4 2 3 8 } 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 1 ⎟ ⎜ 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 { }, it remains to prove termination of the 0-rule system { } The system is trivially terminating.