/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2 }, it remains to prove termination of the 72-rule system { 0 0 0 0 1 2 1 2 2 1 1 2 0 ⟶ 0 0 0 0 1 2 2 2 2 2 2 2 2 2 1 2 0 , 0 0 0 2 1 0 2 1 1 2 0 2 1 ⟶ 0 0 0 1 0 1 2 2 2 2 2 2 2 2 1 2 2 , 0 0 0 2 2 1 0 2 2 0 1 1 0 ⟶ 0 1 2 0 0 0 0 0 0 0 0 0 0 0 2 1 1 , 0 0 1 0 1 0 0 2 0 2 1 0 1 ⟶ 0 0 0 0 0 1 2 1 0 0 0 0 1 2 2 2 0 , 0 0 1 1 1 0 2 1 2 0 2 1 2 ⟶ 0 0 0 1 0 0 0 0 0 1 2 2 0 2 1 2 1 , 0 0 2 0 1 1 0 2 1 1 0 2 2 ⟶ 0 0 0 2 0 2 2 2 2 2 2 1 0 2 2 0 2 , 0 0 2 1 1 2 1 0 2 2 1 0 0 ⟶ 1 0 0 0 0 0 1 2 2 2 0 2 2 2 2 2 0 , 0 0 2 2 0 0 2 2 1 0 2 0 0 ⟶ 2 0 1 0 0 0 0 0 0 0 0 0 1 2 2 0 0 , 0 1 0 0 1 1 1 1 2 1 0 1 1 ⟶ 0 1 0 1 1 0 0 0 2 2 0 2 2 2 2 0 0 , 0 1 0 1 0 1 1 1 1 0 0 1 2 ⟶ 0 0 2 2 2 2 2 0 2 1 0 0 0 1 2 2 2 , 0 1 0 1 1 0 2 0 2 2 1 0 2 ⟶ 0 2 2 0 2 2 2 2 0 2 2 2 2 0 2 2 2 , 0 1 0 1 1 1 2 2 2 1 1 0 2 ⟶ 0 0 1 0 0 0 0 2 1 0 0 0 0 0 0 1 0 , 0 1 1 0 1 2 2 0 1 1 0 1 0 ⟶ 0 0 0 0 0 0 1 2 2 1 0 0 0 1 2 0 0 , 0 1 1 0 2 1 0 0 1 2 1 1 2 ⟶ 1 0 0 1 0 1 2 0 2 2 2 2 2 2 2 2 2 , 0 1 1 2 2 2 1 1 2 2 2 2 1 ⟶ 1 2 2 2 2 0 2 2 2 0 2 2 2 2 2 0 2 , 0 1 2 0 0 1 0 0 1 2 1 2 1 ⟶ 0 0 1 2 0 2 2 2 2 2 2 2 0 1 0 0 2 , 0 1 2 0 1 2 2 2 1 1 2 0 2 ⟶ 0 0 0 0 0 0 0 0 0 1 2 0 2 0 0 2 2 , 0 1 2 2 0 2 0 2 1 0 1 1 1 ⟶ 0 0 1 2 2 0 2 0 0 0 0 0 0 0 0 0 1 , 0 2 0 1 0 2 2 1 0 0 2 2 0 ⟶ 0 0 0 0 1 2 0 0 0 0 0 0 0 0 1 2 0 , 0 2 0 1 1 0 2 2 1 0 2 2 0 ⟶ 2 0 0 0 2 2 2 2 2 2 2 2 1 0 0 0 1 , 0 2 0 1 1 1 2 1 0 2 0 1 1 ⟶ 0 2 0 2 2 2 0 1 0 0 1 0 0 0 0 1 0 , 0 2 1 0 1 2 0 1 0 0 0 1 1 ⟶ 2 1 2 2 2 2 2 2 2 2 0 2 2 0 2 1 0 , 0 2 1 0 2 1 2 2 2 1 0 0 2 ⟶ 0 0 0 0 0 0 0 0 0 0 0 1 0 2 0 1 2 , 0 2 1 2 1 2 2 0 2 0 1 1 0 ⟶ 2 0 1 0 0 0 1 2 0 0 0 0 0 0 0 2 0 , 1 0 0 0 1 1 1 1 0 1 0 0 1 ⟶ 2 2 0 1 0 0 0 0 0 0 0 0 0 0 1 2 0 , 1 0 0 1 0 2 1 2 1 0 1 2 0 ⟶ 0 0 2 0 0 0 0 0 1 0 0 0 0 0 2 2 0 , 1 0 0 2 0 0 2 2 1 2 1 1 1 ⟶ 0 1 0 2 1 0 0 0 0 0 0 1 2 0 2 0 0 , 1 0 0 2 0 1 2 2 1 0 0 1 2 ⟶ 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 2 , 1 0 0 2 1 1 0 1 2 0 0 2 1 ⟶ 1 0 0 2 0 0 0 1 2 2 2 0 2 2 2 0 1 , 1 0 0 2 2 0 0 1 1 0 2 1 1 ⟶ 1 2 0 2 2 2 2 2 2 0 2 2 0 2 0 0 1 , 1 0 1 2 1 2 0 2 0 1 2 1 0 ⟶ 2 0 0 1 0 0 0 0 0 1 2 2 2 2 1 2 0 , 1 0 2 0 1 1 1 0 0 0 1 0 1 ⟶ 1 0 0 0 0 1 0 0 0 0 0 1 2 2 1 2 0 , 1 0 2 0 2 0 1 0 1 1 1 1 1 ⟶ 1 1 0 2 2 2 2 2 2 0 2 0 2 1 0 0 1 , 1 0 2 1 1 0 0 2 1 0 2 0 1 ⟶ 0 1 0 0 0 0 0 0 0 0 1 1 2 2 1 1 1 , 1 0 2 1 2 1 0 1 0 0 2 1 0 ⟶ 2 1 0 0 0 0 0 1 2 2 2 1 0 0 2 0 0 , 1 0 2 1 2 1 1 1 0 2 0 1 2 ⟶ 1 1 0 0 0 0 1 0 0 0 0 2 0 0 2 0 2 , 1 0 2 1 2 2 0 0 0 1 2 2 0 ⟶ 0 0 2 2 2 2 2 2 2 2 2 2 2 0 1 0 2 , 1 1 0 0 1 0 1 1 2 1 0 1 2 ⟶ 1 0 0 0 0 0 0 1 2 2 0 0 1 0 2 0 2 , 1 1 0 1 1 0 2 2 1 0 2 0 0 ⟶ 0 1 0 0 0 0 1 2 2 2 2 1 2 0 0 0 1 , 1 1 0 2 0 1 0 2 1 0 2 1 2 ⟶ 1 2 2 2 2 2 2 1 1 2 2 2 2 2 2 1 2 , 1 1 0 2 0 1 1 0 0 2 2 1 2 ⟶ 0 2 1 0 0 0 0 0 1 2 2 0 0 0 0 0 2 , 1 1 1 0 0 0 1 1 0 1 1 0 2 ⟶ 0 0 0 2 2 2 2 2 0 2 2 0 2 2 2 1 2 , 1 1 1 0 0 2 0 2 1 0 2 0 2 ⟶ 1 0 2 0 1 2 2 2 2 0 2 2 0 2 2 2 2 , 1 1 1 2 2 2 2 1 0 0 2 1 2 ⟶ 0 0 0 0 0 0 1 0 0 0 0 1 2 2 1 1 2 , 1 1 2 0 2 0 2 0 1 0 0 2 0 ⟶ 2 0 1 0 0 0 0 1 2 2 2 0 2 2 2 2 1 , 1 1 2 1 0 0 0 1 0 2 0 2 2 ⟶ 1 0 2 1 0 0 0 0 0 0 0 1 2 2 2 2 2 , 1 1 2 2 0 0 2 0 1 1 1 1 0 ⟶ 2 0 0 0 0 1 2 1 0 1 2 2 2 2 1 0 0 , 1 2 0 0 1 1 2 2 1 2 2 0 0 ⟶ 1 0 2 0 2 0 0 0 0 0 0 0 0 0 1 2 2 , 1 2 0 0 2 1 0 2 2 2 0 0 1 ⟶ 0 0 0 0 0 1 2 2 2 0 2 0 2 2 1 0 1 , 1 2 0 1 0 0 1 1 1 1 0 1 1 ⟶ 0 0 0 0 2 0 1 0 0 0 0 0 1 2 1 1 0 , 1 2 0 1 2 1 2 2 1 0 1 0 0 ⟶ 1 0 0 0 1 2 0 2 2 2 2 2 1 0 1 0 0 , 1 2 1 1 2 0 0 2 2 1 0 0 1 ⟶ 1 1 0 0 0 0 0 0 1 0 0 1 2 2 2 2 0 , 1 2 2 0 0 1 2 0 2 1 0 0 1 ⟶ 0 1 0 2 2 2 2 2 0 2 2 2 2 2 2 2 2 , 1 2 2 0 1 2 2 2 0 1 2 1 1 ⟶ 0 0 0 2 2 2 2 2 0 2 2 1 2 0 0 0 1 , 2 0 0 0 0 1 1 2 0 0 1 0 2 ⟶ 0 1 0 0 0 1 2 2 2 2 0 2 0 2 2 2 2 , 2 0 0 2 1 2 1 0 1 0 1 1 1 ⟶ 1 0 0 0 0 0 0 0 1 1 2 2 2 0 2 0 0 , 2 0 1 0 1 0 0 0 2 0 1 2 1 ⟶ 2 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 , 2 0 2 0 1 1 0 0 2 1 1 0 2 ⟶ 2 2 0 0 0 1 0 0 0 0 1 0 0 0 1 0 2 , 2 0 2 2 0 2 2 0 2 1 2 0 1 ⟶ 2 2 2 2 0 0 0 0 0 0 0 0 0 1 2 2 0 , 2 1 0 2 1 2 1 2 1 2 2 2 1 ⟶ 2 1 0 0 0 1 2 0 2 0 0 0 0 0 1 2 1 , 2 1 1 2 1 1 2 0 1 0 1 0 2 ⟶ 2 2 0 2 0 0 1 2 0 2 2 2 2 2 2 1 2 , 2 1 1 2 2 0 2 0 0 2 0 1 0 ⟶ 2 0 1 0 1 2 2 2 2 2 2 2 0 2 2 0 0 , 2 1 2 0 0 2 2 0 2 0 1 2 2 ⟶ 2 0 0 0 0 2 0 0 1 2 2 2 2 2 2 0 2 , 2 1 2 0 2 2 2 1 1 0 0 1 2 ⟶ 1 2 2 0 0 2 2 2 2 2 0 2 2 2 2 2 2 , 2 1 2 1 0 1 0 0 2 1 2 2 0 ⟶ 2 2 1 1 0 0 0 0 0 0 1 2 1 0 0 0 1 , 2 1 2 2 1 2 2 2 2 1 2 1 0 ⟶ 2 0 0 2 0 0 0 0 1 0 0 0 0 0 0 0 0 , 2 2 0 0 1 2 1 0 2 2 1 0 0 ⟶ 2 2 2 2 2 2 2 0 2 2 0 2 2 0 2 2 0 , 2 2 0 0 2 1 2 1 2 0 1 2 0 ⟶ 2 2 2 2 2 2 0 1 0 0 0 0 1 0 0 2 1 , 2 2 0 2 2 0 1 1 2 1 0 1 0 ⟶ 2 2 2 2 2 2 2 2 2 2 1 0 0 2 0 0 1 , 2 2 0 2 2 1 2 1 2 2 0 2 0 ⟶ 2 2 2 2 0 2 2 2 0 2 2 0 1 0 0 0 1 , 2 2 1 0 2 1 1 2 2 0 1 0 1 ⟶ 2 2 2 2 0 1 0 0 0 1 0 0 0 0 0 1 2 , 2 2 1 1 2 1 0 1 1 2 0 0 2 ⟶ 2 2 2 0 0 0 0 1 2 0 0 0 2 2 2 2 2 } Applying sparse tiling TRFC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (0,0) ↦ 0, (0,1) ↦ 1, (1,2) ↦ 2, (2,1) ↦ 3, (2,2) ↦ 4, (1,1) ↦ 5, (2,0) ↦ 6, (0,2) ↦ 7, (0,4) ↦ 8, (1,0) ↦ 9, (3,0) ↦ 10, (1,4) ↦ 11, (2,4) ↦ 12, (3,1) ↦ 13, (3,2) ↦ 14 }, it remains to prove termination of the 1152-rule system { 0 0 0 0 1 2 3 2 4 3 5 2 6 0 ⟶ 0 0 0 0 1 2 4 4 4 4 4 4 4 4 3 2 6 0 , 0 0 0 0 1 2 3 2 4 3 5 2 6 1 ⟶ 0 0 0 0 1 2 4 4 4 4 4 4 4 4 3 2 6 1 , 0 0 0 0 1 2 3 2 4 3 5 2 6 7 ⟶ 0 0 0 0 1 2 4 4 4 4 4 4 4 4 3 2 6 7 , 0 0 0 0 1 2 3 2 4 3 5 2 6 8 ⟶ 0 0 0 0 1 2 4 4 4 4 4 4 4 4 3 2 6 8 , 9 0 0 0 1 2 3 2 4 3 5 2 6 0 ⟶ 9 0 0 0 1 2 4 4 4 4 4 4 4 4 3 2 6 0 , 9 0 0 0 1 2 3 2 4 3 5 2 6 1 ⟶ 9 0 0 0 1 2 4 4 4 4 4 4 4 4 3 2 6 1 , 9 0 0 0 1 2 3 2 4 3 5 2 6 7 ⟶ 9 0 0 0 1 2 4 4 4 4 4 4 4 4 3 2 6 7 , 9 0 0 0 1 2 3 2 4 3 5 2 6 8 ⟶ 9 0 0 0 1 2 4 4 4 4 4 4 4 4 3 2 6 8 , 6 0 0 0 1 2 3 2 4 3 5 2 6 0 ⟶ 6 0 0 0 1 2 4 4 4 4 4 4 4 4 3 2 6 0 , 6 0 0 0 1 2 3 2 4 3 5 2 6 1 ⟶ 6 0 0 0 1 2 4 4 4 4 4 4 4 4 3 2 6 1 , 6 0 0 0 1 2 3 2 4 3 5 2 6 7 ⟶ 6 0 0 0 1 2 4 4 4 4 4 4 4 4 3 2 6 7 , 6 0 0 0 1 2 3 2 4 3 5 2 6 8 ⟶ 6 0 0 0 1 2 4 4 4 4 4 4 4 4 3 2 6 8 , 10 0 0 0 1 2 3 2 4 3 5 2 6 0 ⟶ 10 0 0 0 1 2 4 4 4 4 4 4 4 4 3 2 6 0 , 10 0 0 0 1 2 3 2 4 3 5 2 6 1 ⟶ 10 0 0 0 1 2 4 4 4 4 4 4 4 4 3 2 6 1 , 10 0 0 0 1 2 3 2 4 3 5 2 6 7 ⟶ 10 0 0 0 1 2 4 4 4 4 4 4 4 4 3 2 6 7 , 10 0 0 0 1 2 3 2 4 3 5 2 6 8 ⟶ 10 0 0 0 1 2 4 4 4 4 4 4 4 4 3 2 6 8 , 0 0 0 7 3 9 7 3 5 2 6 7 3 9 ⟶ 0 0 0 1 9 1 2 4 4 4 4 4 4 4 3 2 4 6 , 0 0 0 7 3 9 7 3 5 2 6 7 3 5 ⟶ 0 0 0 1 9 1 2 4 4 4 4 4 4 4 3 2 4 3 , 0 0 0 7 3 9 7 3 5 2 6 7 3 2 ⟶ 0 0 0 1 9 1 2 4 4 4 4 4 4 4 3 2 4 4 , 0 0 0 7 3 9 7 3 5 2 6 7 3 11 ⟶ 0 0 0 1 9 1 2 4 4 4 4 4 4 4 3 2 4 12 , 9 0 0 7 3 9 7 3 5 2 6 7 3 9 ⟶ 9 0 0 1 9 1 2 4 4 4 4 4 4 4 3 2 4 6 , 9 0 0 7 3 9 7 3 5 2 6 7 3 5 ⟶ 9 0 0 1 9 1 2 4 4 4 4 4 4 4 3 2 4 3 , 9 0 0 7 3 9 7 3 5 2 6 7 3 2 ⟶ 9 0 0 1 9 1 2 4 4 4 4 4 4 4 3 2 4 4 , 9 0 0 7 3 9 7 3 5 2 6 7 3 11 ⟶ 9 0 0 1 9 1 2 4 4 4 4 4 4 4 3 2 4 12 , 6 0 0 7 3 9 7 3 5 2 6 7 3 9 ⟶ 6 0 0 1 9 1 2 4 4 4 4 4 4 4 3 2 4 6 , 6 0 0 7 3 9 7 3 5 2 6 7 3 5 ⟶ 6 0 0 1 9 1 2 4 4 4 4 4 4 4 3 2 4 3 , 6 0 0 7 3 9 7 3 5 2 6 7 3 2 ⟶ 6 0 0 1 9 1 2 4 4 4 4 4 4 4 3 2 4 4 , 6 0 0 7 3 9 7 3 5 2 6 7 3 11 ⟶ 6 0 0 1 9 1 2 4 4 4 4 4 4 4 3 2 4 12 , 10 0 0 7 3 9 7 3 5 2 6 7 3 9 ⟶ 10 0 0 1 9 1 2 4 4 4 4 4 4 4 3 2 4 6 , 10 0 0 7 3 9 7 3 5 2 6 7 3 5 ⟶ 10 0 0 1 9 1 2 4 4 4 4 4 4 4 3 2 4 3 , 10 0 0 7 3 9 7 3 5 2 6 7 3 2 ⟶ 10 0 0 1 9 1 2 4 4 4 4 4 4 4 3 2 4 4 , 10 0 0 7 3 9 7 3 5 2 6 7 3 11 ⟶ 10 0 0 1 9 1 2 4 4 4 4 4 4 4 3 2 4 12 , 0 0 0 7 4 3 9 7 4 6 1 5 9 0 ⟶ 0 1 2 6 0 0 0 0 0 0 0 0 0 0 7 3 5 9 , 0 0 0 7 4 3 9 7 4 6 1 5 9 1 ⟶ 0 1 2 6 0 0 0 0 0 0 0 0 0 0 7 3 5 5 , 0 0 0 7 4 3 9 7 4 6 1 5 9 7 ⟶ 0 1 2 6 0 0 0 0 0 0 0 0 0 0 7 3 5 2 , 0 0 0 7 4 3 9 7 4 6 1 5 9 8 ⟶ 0 1 2 6 0 0 0 0 0 0 0 0 0 0 7 3 5 11 , 9 0 0 7 4 3 9 7 4 6 1 5 9 0 ⟶ 9 1 2 6 0 0 0 0 0 0 0 0 0 0 7 3 5 9 , 9 0 0 7 4 3 9 7 4 6 1 5 9 1 ⟶ 9 1 2 6 0 0 0 0 0 0 0 0 0 0 7 3 5 5 , 9 0 0 7 4 3 9 7 4 6 1 5 9 7 ⟶ 9 1 2 6 0 0 0 0 0 0 0 0 0 0 7 3 5 2 , 9 0 0 7 4 3 9 7 4 6 1 5 9 8 ⟶ 9 1 2 6 0 0 0 0 0 0 0 0 0 0 7 3 5 11 , 6 0 0 7 4 3 9 7 4 6 1 5 9 0 ⟶ 6 1 2 6 0 0 0 0 0 0 0 0 0 0 7 3 5 9 , 6 0 0 7 4 3 9 7 4 6 1 5 9 1 ⟶ 6 1 2 6 0 0 0 0 0 0 0 0 0 0 7 3 5 5 , 6 0 0 7 4 3 9 7 4 6 1 5 9 7 ⟶ 6 1 2 6 0 0 0 0 0 0 0 0 0 0 7 3 5 2 , 6 0 0 7 4 3 9 7 4 6 1 5 9 8 ⟶ 6 1 2 6 0 0 0 0 0 0 0 0 0 0 7 3 5 11 , 10 0 0 7 4 3 9 7 4 6 1 5 9 0 ⟶ 10 1 2 6 0 0 0 0 0 0 0 0 0 0 7 3 5 9 , 10 0 0 7 4 3 9 7 4 6 1 5 9 1 ⟶ 10 1 2 6 0 0 0 0 0 0 0 0 0 0 7 3 5 5 , 10 0 0 7 4 3 9 7 4 6 1 5 9 7 ⟶ 10 1 2 6 0 0 0 0 0 0 0 0 0 0 7 3 5 2 , 10 0 0 7 4 3 9 7 4 6 1 5 9 8 ⟶ 10 1 2 6 0 0 0 0 0 0 0 0 0 0 7 3 5 11 , 0 0 1 9 1 9 0 7 6 7 3 9 1 9 ⟶ 0 0 0 0 0 1 2 3 9 0 0 0 1 2 4 4 6 0 , 0 0 1 9 1 9 0 7 6 7 3 9 1 5 ⟶ 0 0 0 0 0 1 2 3 9 0 0 0 1 2 4 4 6 1 , 0 0 1 9 1 9 0 7 6 7 3 9 1 2 ⟶ 0 0 0 0 0 1 2 3 9 0 0 0 1 2 4 4 6 7 , 0 0 1 9 1 9 0 7 6 7 3 9 1 11 ⟶ 0 0 0 0 0 1 2 3 9 0 0 0 1 2 4 4 6 8 , 9 0 1 9 1 9 0 7 6 7 3 9 1 9 ⟶ 9 0 0 0 0 1 2 3 9 0 0 0 1 2 4 4 6 0 , 9 0 1 9 1 9 0 7 6 7 3 9 1 5 ⟶ 9 0 0 0 0 1 2 3 9 0 0 0 1 2 4 4 6 1 , 9 0 1 9 1 9 0 7 6 7 3 9 1 2 ⟶ 9 0 0 0 0 1 2 3 9 0 0 0 1 2 4 4 6 7 , 9 0 1 9 1 9 0 7 6 7 3 9 1 11 ⟶ 9 0 0 0 0 1 2 3 9 0 0 0 1 2 4 4 6 8 , 6 0 1 9 1 9 0 7 6 7 3 9 1 9 ⟶ 6 0 0 0 0 1 2 3 9 0 0 0 1 2 4 4 6 0 , 6 0 1 9 1 9 0 7 6 7 3 9 1 5 ⟶ 6 0 0 0 0 1 2 3 9 0 0 0 1 2 4 4 6 1 , 6 0 1 9 1 9 0 7 6 7 3 9 1 2 ⟶ 6 0 0 0 0 1 2 3 9 0 0 0 1 2 4 4 6 7 , 6 0 1 9 1 9 0 7 6 7 3 9 1 11 ⟶ 6 0 0 0 0 1 2 3 9 0 0 0 1 2 4 4 6 8 , 10 0 1 9 1 9 0 7 6 7 3 9 1 9 ⟶ 10 0 0 0 0 1 2 3 9 0 0 0 1 2 4 4 6 0 , 10 0 1 9 1 9 0 7 6 7 3 9 1 5 ⟶ 10 0 0 0 0 1 2 3 9 0 0 0 1 2 4 4 6 1 , 10 0 1 9 1 9 0 7 6 7 3 9 1 2 ⟶ 10 0 0 0 0 1 2 3 9 0 0 0 1 2 4 4 6 7 , 10 0 1 9 1 9 0 7 6 7 3 9 1 11 ⟶ 10 0 0 0 0 1 2 3 9 0 0 0 1 2 4 4 6 8 , 0 0 1 5 5 9 7 3 2 6 7 3 2 6 ⟶ 0 0 0 1 9 0 0 0 0 1 2 4 6 7 3 2 3 9 , 0 0 1 5 5 9 7 3 2 6 7 3 2 3 ⟶ 0 0 0 1 9 0 0 0 0 1 2 4 6 7 3 2 3 5 , 0 0 1 5 5 9 7 3 2 6 7 3 2 4 ⟶ 0 0 0 1 9 0 0 0 0 1 2 4 6 7 3 2 3 2 , 0 0 1 5 5 9 7 3 2 6 7 3 2 12 ⟶ 0 0 0 1 9 0 0 0 0 1 2 4 6 7 3 2 3 11 , 9 0 1 5 5 9 7 3 2 6 7 3 2 6 ⟶ 9 0 0 1 9 0 0 0 0 1 2 4 6 7 3 2 3 9 , 9 0 1 5 5 9 7 3 2 6 7 3 2 3 ⟶ 9 0 0 1 9 0 0 0 0 1 2 4 6 7 3 2 3 5 , 9 0 1 5 5 9 7 3 2 6 7 3 2 4 ⟶ 9 0 0 1 9 0 0 0 0 1 2 4 6 7 3 2 3 2 , 9 0 1 5 5 9 7 3 2 6 7 3 2 12 ⟶ 9 0 0 1 9 0 0 0 0 1 2 4 6 7 3 2 3 11 , 6 0 1 5 5 9 7 3 2 6 7 3 2 6 ⟶ 6 0 0 1 9 0 0 0 0 1 2 4 6 7 3 2 3 9 , 6 0 1 5 5 9 7 3 2 6 7 3 2 3 ⟶ 6 0 0 1 9 0 0 0 0 1 2 4 6 7 3 2 3 5 , 6 0 1 5 5 9 7 3 2 6 7 3 2 4 ⟶ 6 0 0 1 9 0 0 0 0 1 2 4 6 7 3 2 3 2 , 6 0 1 5 5 9 7 3 2 6 7 3 2 12 ⟶ 6 0 0 1 9 0 0 0 0 1 2 4 6 7 3 2 3 11 , 10 0 1 5 5 9 7 3 2 6 7 3 2 6 ⟶ 10 0 0 1 9 0 0 0 0 1 2 4 6 7 3 2 3 9 , 10 0 1 5 5 9 7 3 2 6 7 3 2 3 ⟶ 10 0 0 1 9 0 0 0 0 1 2 4 6 7 3 2 3 5 , 10 0 1 5 5 9 7 3 2 6 7 3 2 4 ⟶ 10 0 0 1 9 0 0 0 0 1 2 4 6 7 3 2 3 2 , 10 0 1 5 5 9 7 3 2 6 7 3 2 12 ⟶ 10 0 0 1 9 0 0 0 0 1 2 4 6 7 3 2 3 11 , 0 0 7 6 1 5 9 7 3 5 9 7 4 6 ⟶ 0 0 0 7 6 7 4 4 4 4 4 3 9 7 4 6 7 6 , 0 0 7 6 1 5 9 7 3 5 9 7 4 3 ⟶ 0 0 0 7 6 7 4 4 4 4 4 3 9 7 4 6 7 3 , 0 0 7 6 1 5 9 7 3 5 9 7 4 4 ⟶ 0 0 0 7 6 7 4 4 4 4 4 3 9 7 4 6 7 4 , 0 0 7 6 1 5 9 7 3 5 9 7 4 12 ⟶ 0 0 0 7 6 7 4 4 4 4 4 3 9 7 4 6 7 12 , 9 0 7 6 1 5 9 7 3 5 9 7 4 6 ⟶ 9 0 0 7 6 7 4 4 4 4 4 3 9 7 4 6 7 6 , 9 0 7 6 1 5 9 7 3 5 9 7 4 3 ⟶ 9 0 0 7 6 7 4 4 4 4 4 3 9 7 4 6 7 3 , 9 0 7 6 1 5 9 7 3 5 9 7 4 4 ⟶ 9 0 0 7 6 7 4 4 4 4 4 3 9 7 4 6 7 4 , 9 0 7 6 1 5 9 7 3 5 9 7 4 12 ⟶ 9 0 0 7 6 7 4 4 4 4 4 3 9 7 4 6 7 12 , 6 0 7 6 1 5 9 7 3 5 9 7 4 6 ⟶ 6 0 0 7 6 7 4 4 4 4 4 3 9 7 4 6 7 6 , 6 0 7 6 1 5 9 7 3 5 9 7 4 3 ⟶ 6 0 0 7 6 7 4 4 4 4 4 3 9 7 4 6 7 3 , 6 0 7 6 1 5 9 7 3 5 9 7 4 4 ⟶ 6 0 0 7 6 7 4 4 4 4 4 3 9 7 4 6 7 4 , 6 0 7 6 1 5 9 7 3 5 9 7 4 12 ⟶ 6 0 0 7 6 7 4 4 4 4 4 3 9 7 4 6 7 12 , 10 0 7 6 1 5 9 7 3 5 9 7 4 6 ⟶ 10 0 0 7 6 7 4 4 4 4 4 3 9 7 4 6 7 6 , 10 0 7 6 1 5 9 7 3 5 9 7 4 3 ⟶ 10 0 0 7 6 7 4 4 4 4 4 3 9 7 4 6 7 3 , 10 0 7 6 1 5 9 7 3 5 9 7 4 4 ⟶ 10 0 0 7 6 7 4 4 4 4 4 3 9 7 4 6 7 4 , 10 0 7 6 1 5 9 7 3 5 9 7 4 12 ⟶ 10 0 0 7 6 7 4 4 4 4 4 3 9 7 4 6 7 12 , 0 0 7 3 5 2 3 9 7 4 3 9 0 0 ⟶ 1 9 0 0 0 0 1 2 4 4 6 7 4 4 4 4 6 0 , 0 0 7 3 5 2 3 9 7 4 3 9 0 1 ⟶ 1 9 0 0 0 0 1 2 4 4 6 7 4 4 4 4 6 1 , 0 0 7 3 5 2 3 9 7 4 3 9 0 7 ⟶ 1 9 0 0 0 0 1 2 4 4 6 7 4 4 4 4 6 7 , 0 0 7 3 5 2 3 9 7 4 3 9 0 8 ⟶ 1 9 0 0 0 0 1 2 4 4 6 7 4 4 4 4 6 8 , 9 0 7 3 5 2 3 9 7 4 3 9 0 0 ⟶ 5 9 0 0 0 0 1 2 4 4 6 7 4 4 4 4 6 0 , 9 0 7 3 5 2 3 9 7 4 3 9 0 1 ⟶ 5 9 0 0 0 0 1 2 4 4 6 7 4 4 4 4 6 1 , 9 0 7 3 5 2 3 9 7 4 3 9 0 7 ⟶ 5 9 0 0 0 0 1 2 4 4 6 7 4 4 4 4 6 7 , 9 0 7 3 5 2 3 9 7 4 3 9 0 8 ⟶ 5 9 0 0 0 0 1 2 4 4 6 7 4 4 4 4 6 8 , 6 0 7 3 5 2 3 9 7 4 3 9 0 0 ⟶ 3 9 0 0 0 0 1 2 4 4 6 7 4 4 4 4 6 0 , 6 0 7 3 5 2 3 9 7 4 3 9 0 1 ⟶ 3 9 0 0 0 0 1 2 4 4 6 7 4 4 4 4 6 1 , 6 0 7 3 5 2 3 9 7 4 3 9 0 7 ⟶ 3 9 0 0 0 0 1 2 4 4 6 7 4 4 4 4 6 7 , 6 0 7 3 5 2 3 9 7 4 3 9 0 8 ⟶ 3 9 0 0 0 0 1 2 4 4 6 7 4 4 4 4 6 8 , 10 0 7 3 5 2 3 9 7 4 3 9 0 0 ⟶ 13 9 0 0 0 0 1 2 4 4 6 7 4 4 4 4 6 0 , 10 0 7 3 5 2 3 9 7 4 3 9 0 1 ⟶ 13 9 0 0 0 0 1 2 4 4 6 7 4 4 4 4 6 1 , 10 0 7 3 5 2 3 9 7 4 3 9 0 7 ⟶ 13 9 0 0 0 0 1 2 4 4 6 7 4 4 4 4 6 7 , 10 0 7 3 5 2 3 9 7 4 3 9 0 8 ⟶ 13 9 0 0 0 0 1 2 4 4 6 7 4 4 4 4 6 8 , 0 0 7 4 6 0 7 4 3 9 7 6 0 0 ⟶ 7 6 1 9 0 0 0 0 0 0 0 0 1 2 4 6 0 0 , 0 0 7 4 6 0 7 4 3 9 7 6 0 1 ⟶ 7 6 1 9 0 0 0 0 0 0 0 0 1 2 4 6 0 1 , 0 0 7 4 6 0 7 4 3 9 7 6 0 7 ⟶ 7 6 1 9 0 0 0 0 0 0 0 0 1 2 4 6 0 7 , 0 0 7 4 6 0 7 4 3 9 7 6 0 8 ⟶ 7 6 1 9 0 0 0 0 0 0 0 0 1 2 4 6 0 8 , 9 0 7 4 6 0 7 4 3 9 7 6 0 0 ⟶ 2 6 1 9 0 0 0 0 0 0 0 0 1 2 4 6 0 0 , 9 0 7 4 6 0 7 4 3 9 7 6 0 1 ⟶ 2 6 1 9 0 0 0 0 0 0 0 0 1 2 4 6 0 1 , 9 0 7 4 6 0 7 4 3 9 7 6 0 7 ⟶ 2 6 1 9 0 0 0 0 0 0 0 0 1 2 4 6 0 7 , 9 0 7 4 6 0 7 4 3 9 7 6 0 8 ⟶ 2 6 1 9 0 0 0 0 0 0 0 0 1 2 4 6 0 8 , 6 0 7 4 6 0 7 4 3 9 7 6 0 0 ⟶ 4 6 1 9 0 0 0 0 0 0 0 0 1 2 4 6 0 0 , 6 0 7 4 6 0 7 4 3 9 7 6 0 1 ⟶ 4 6 1 9 0 0 0 0 0 0 0 0 1 2 4 6 0 1 , 6 0 7 4 6 0 7 4 3 9 7 6 0 7 ⟶ 4 6 1 9 0 0 0 0 0 0 0 0 1 2 4 6 0 7 , 6 0 7 4 6 0 7 4 3 9 7 6 0 8 ⟶ 4 6 1 9 0 0 0 0 0 0 0 0 1 2 4 6 0 8 , 10 0 7 4 6 0 7 4 3 9 7 6 0 0 ⟶ 14 6 1 9 0 0 0 0 0 0 0 0 1 2 4 6 0 0 , 10 0 7 4 6 0 7 4 3 9 7 6 0 1 ⟶ 14 6 1 9 0 0 0 0 0 0 0 0 1 2 4 6 0 1 , 10 0 7 4 6 0 7 4 3 9 7 6 0 7 ⟶ 14 6 1 9 0 0 0 0 0 0 0 0 1 2 4 6 0 7 , 10 0 7 4 6 0 7 4 3 9 7 6 0 8 ⟶ 14 6 1 9 0 0 0 0 0 0 0 0 1 2 4 6 0 8 , 0 1 9 0 1 5 5 5 2 3 9 1 5 9 ⟶ 0 1 9 1 5 9 0 0 7 4 6 7 4 4 4 6 0 0 , 0 1 9 0 1 5 5 5 2 3 9 1 5 5 ⟶ 0 1 9 1 5 9 0 0 7 4 6 7 4 4 4 6 0 1 , 0 1 9 0 1 5 5 5 2 3 9 1 5 2 ⟶ 0 1 9 1 5 9 0 0 7 4 6 7 4 4 4 6 0 7 , 0 1 9 0 1 5 5 5 2 3 9 1 5 11 ⟶ 0 1 9 1 5 9 0 0 7 4 6 7 4 4 4 6 0 8 , 9 1 9 0 1 5 5 5 2 3 9 1 5 9 ⟶ 9 1 9 1 5 9 0 0 7 4 6 7 4 4 4 6 0 0 , 9 1 9 0 1 5 5 5 2 3 9 1 5 5 ⟶ 9 1 9 1 5 9 0 0 7 4 6 7 4 4 4 6 0 1 , 9 1 9 0 1 5 5 5 2 3 9 1 5 2 ⟶ 9 1 9 1 5 9 0 0 7 4 6 7 4 4 4 6 0 7 , 9 1 9 0 1 5 5 5 2 3 9 1 5 11 ⟶ 9 1 9 1 5 9 0 0 7 4 6 7 4 4 4 6 0 8 , 6 1 9 0 1 5 5 5 2 3 9 1 5 9 ⟶ 6 1 9 1 5 9 0 0 7 4 6 7 4 4 4 6 0 0 , 6 1 9 0 1 5 5 5 2 3 9 1 5 5 ⟶ 6 1 9 1 5 9 0 0 7 4 6 7 4 4 4 6 0 1 , 6 1 9 0 1 5 5 5 2 3 9 1 5 2 ⟶ 6 1 9 1 5 9 0 0 7 4 6 7 4 4 4 6 0 7 , 6 1 9 0 1 5 5 5 2 3 9 1 5 11 ⟶ 6 1 9 1 5 9 0 0 7 4 6 7 4 4 4 6 0 8 , 10 1 9 0 1 5 5 5 2 3 9 1 5 9 ⟶ 10 1 9 1 5 9 0 0 7 4 6 7 4 4 4 6 0 0 , 10 1 9 0 1 5 5 5 2 3 9 1 5 5 ⟶ 10 1 9 1 5 9 0 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1 2 3 9 7 4 3 9 0 1 ⟶ 7 4 4 4 4 4 4 6 7 4 6 7 4 6 7 4 6 1 , 7 4 6 0 1 2 3 9 7 4 3 9 0 7 ⟶ 7 4 4 4 4 4 4 6 7 4 6 7 4 6 7 4 6 7 , 7 4 6 0 1 2 3 9 7 4 3 9 0 8 ⟶ 7 4 4 4 4 4 4 6 7 4 6 7 4 6 7 4 6 8 , 2 4 6 0 1 2 3 9 7 4 3 9 0 0 ⟶ 2 4 4 4 4 4 4 6 7 4 6 7 4 6 7 4 6 0 , 2 4 6 0 1 2 3 9 7 4 3 9 0 1 ⟶ 2 4 4 4 4 4 4 6 7 4 6 7 4 6 7 4 6 1 , 2 4 6 0 1 2 3 9 7 4 3 9 0 7 ⟶ 2 4 4 4 4 4 4 6 7 4 6 7 4 6 7 4 6 7 , 2 4 6 0 1 2 3 9 7 4 3 9 0 8 ⟶ 2 4 4 4 4 4 4 6 7 4 6 7 4 6 7 4 6 8 , 4 4 6 0 1 2 3 9 7 4 3 9 0 0 ⟶ 4 4 4 4 4 4 4 6 7 4 6 7 4 6 7 4 6 0 , 4 4 6 0 1 2 3 9 7 4 3 9 0 1 ⟶ 4 4 4 4 4 4 4 6 7 4 6 7 4 6 7 4 6 1 , 4 4 6 0 1 2 3 9 7 4 3 9 0 7 ⟶ 4 4 4 4 4 4 4 6 7 4 6 7 4 6 7 4 6 7 , 4 4 6 0 1 2 3 9 7 4 3 9 0 8 ⟶ 4 4 4 4 4 4 4 6 7 4 6 7 4 6 7 4 6 8 , 14 4 6 0 1 2 3 9 7 4 3 9 0 0 ⟶ 14 4 4 4 4 4 4 6 7 4 6 7 4 6 7 4 6 0 , 14 4 6 0 1 2 3 9 7 4 3 9 0 1 ⟶ 14 4 4 4 4 4 4 6 7 4 6 7 4 6 7 4 6 1 , 14 4 6 0 1 2 3 9 7 4 3 9 0 7 ⟶ 14 4 4 4 4 4 4 6 7 4 6 7 4 6 7 4 6 7 , 14 4 6 0 1 2 3 9 7 4 3 9 0 8 ⟶ 14 4 4 4 4 4 4 6 7 4 6 7 4 6 7 4 6 8 , 7 4 6 0 7 3 2 3 2 6 1 2 6 0 ⟶ 7 4 4 4 4 4 6 1 9 0 0 0 1 9 0 7 3 9 , 7 4 6 0 7 3 2 3 2 6 1 2 6 1 ⟶ 7 4 4 4 4 4 6 1 9 0 0 0 1 9 0 7 3 5 , 7 4 6 0 7 3 2 3 2 6 1 2 6 7 ⟶ 7 4 4 4 4 4 6 1 9 0 0 0 1 9 0 7 3 2 , 7 4 6 0 7 3 2 3 2 6 1 2 6 8 ⟶ 7 4 4 4 4 4 6 1 9 0 0 0 1 9 0 7 3 11 , 2 4 6 0 7 3 2 3 2 6 1 2 6 0 ⟶ 2 4 4 4 4 4 6 1 9 0 0 0 1 9 0 7 3 9 , 2 4 6 0 7 3 2 3 2 6 1 2 6 1 ⟶ 2 4 4 4 4 4 6 1 9 0 0 0 1 9 0 7 3 5 , 2 4 6 0 7 3 2 3 2 6 1 2 6 7 ⟶ 2 4 4 4 4 4 6 1 9 0 0 0 1 9 0 7 3 2 , 2 4 6 0 7 3 2 3 2 6 1 2 6 8 ⟶ 2 4 4 4 4 4 6 1 9 0 0 0 1 9 0 7 3 11 , 4 4 6 0 7 3 2 3 2 6 1 2 6 0 ⟶ 4 4 4 4 4 4 6 1 9 0 0 0 1 9 0 7 3 9 , 4 4 6 0 7 3 2 3 2 6 1 2 6 1 ⟶ 4 4 4 4 4 4 6 1 9 0 0 0 1 9 0 7 3 5 , 4 4 6 0 7 3 2 3 2 6 1 2 6 7 ⟶ 4 4 4 4 4 4 6 1 9 0 0 0 1 9 0 7 3 2 , 4 4 6 0 7 3 2 3 2 6 1 2 6 8 ⟶ 4 4 4 4 4 4 6 1 9 0 0 0 1 9 0 7 3 11 , 14 4 6 0 7 3 2 3 2 6 1 2 6 0 ⟶ 14 4 4 4 4 4 6 1 9 0 0 0 1 9 0 7 3 9 , 14 4 6 0 7 3 2 3 2 6 1 2 6 1 ⟶ 14 4 4 4 4 4 6 1 9 0 0 0 1 9 0 7 3 5 , 14 4 6 0 7 3 2 3 2 6 1 2 6 7 ⟶ 14 4 4 4 4 4 6 1 9 0 0 0 1 9 0 7 3 2 , 14 4 6 0 7 3 2 3 2 6 1 2 6 8 ⟶ 14 4 4 4 4 4 6 1 9 0 0 0 1 9 0 7 3 11 , 7 4 6 7 4 6 1 5 2 3 9 1 9 0 ⟶ 7 4 4 4 4 4 4 4 4 4 3 9 0 7 6 0 1 9 , 7 4 6 7 4 6 1 5 2 3 9 1 9 1 ⟶ 7 4 4 4 4 4 4 4 4 4 3 9 0 7 6 0 1 5 , 7 4 6 7 4 6 1 5 2 3 9 1 9 7 ⟶ 7 4 4 4 4 4 4 4 4 4 3 9 0 7 6 0 1 2 , 7 4 6 7 4 6 1 5 2 3 9 1 9 8 ⟶ 7 4 4 4 4 4 4 4 4 4 3 9 0 7 6 0 1 11 , 2 4 6 7 4 6 1 5 2 3 9 1 9 0 ⟶ 2 4 4 4 4 4 4 4 4 4 3 9 0 7 6 0 1 9 , 2 4 6 7 4 6 1 5 2 3 9 1 9 1 ⟶ 2 4 4 4 4 4 4 4 4 4 3 9 0 7 6 0 1 5 , 2 4 6 7 4 6 1 5 2 3 9 1 9 7 ⟶ 2 4 4 4 4 4 4 4 4 4 3 9 0 7 6 0 1 2 , 2 4 6 7 4 6 1 5 2 3 9 1 9 8 ⟶ 2 4 4 4 4 4 4 4 4 4 3 9 0 7 6 0 1 11 , 4 4 6 7 4 6 1 5 2 3 9 1 9 0 ⟶ 4 4 4 4 4 4 4 4 4 4 3 9 0 7 6 0 1 9 , 4 4 6 7 4 6 1 5 2 3 9 1 9 1 ⟶ 4 4 4 4 4 4 4 4 4 4 3 9 0 7 6 0 1 5 , 4 4 6 7 4 6 1 5 2 3 9 1 9 7 ⟶ 4 4 4 4 4 4 4 4 4 4 3 9 0 7 6 0 1 2 , 4 4 6 7 4 6 1 5 2 3 9 1 9 8 ⟶ 4 4 4 4 4 4 4 4 4 4 3 9 0 7 6 0 1 11 , 14 4 6 7 4 6 1 5 2 3 9 1 9 0 ⟶ 14 4 4 4 4 4 4 4 4 4 3 9 0 7 6 0 1 9 , 14 4 6 7 4 6 1 5 2 3 9 1 9 1 ⟶ 14 4 4 4 4 4 4 4 4 4 3 9 0 7 6 0 1 5 , 14 4 6 7 4 6 1 5 2 3 9 1 9 7 ⟶ 14 4 4 4 4 4 4 4 4 4 3 9 0 7 6 0 1 2 , 14 4 6 7 4 6 1 5 2 3 9 1 9 8 ⟶ 14 4 4 4 4 4 4 4 4 4 3 9 0 7 6 0 1 11 , 7 4 6 7 4 3 2 3 2 4 6 7 6 0 ⟶ 7 4 4 4 6 7 4 4 6 7 4 6 1 9 0 0 1 9 , 7 4 6 7 4 3 2 3 2 4 6 7 6 1 ⟶ 7 4 4 4 6 7 4 4 6 7 4 6 1 9 0 0 1 5 , 7 4 6 7 4 3 2 3 2 4 6 7 6 7 ⟶ 7 4 4 4 6 7 4 4 6 7 4 6 1 9 0 0 1 2 , 7 4 6 7 4 3 2 3 2 4 6 7 6 8 ⟶ 7 4 4 4 6 7 4 4 6 7 4 6 1 9 0 0 1 11 , 2 4 6 7 4 3 2 3 2 4 6 7 6 0 ⟶ 2 4 4 4 6 7 4 4 6 7 4 6 1 9 0 0 1 9 , 2 4 6 7 4 3 2 3 2 4 6 7 6 1 ⟶ 2 4 4 4 6 7 4 4 6 7 4 6 1 9 0 0 1 5 , 2 4 6 7 4 3 2 3 2 4 6 7 6 7 ⟶ 2 4 4 4 6 7 4 4 6 7 4 6 1 9 0 0 1 2 , 2 4 6 7 4 3 2 3 2 4 6 7 6 8 ⟶ 2 4 4 4 6 7 4 4 6 7 4 6 1 9 0 0 1 11 , 4 4 6 7 4 3 2 3 2 4 6 7 6 0 ⟶ 4 4 4 4 6 7 4 4 6 7 4 6 1 9 0 0 1 9 , 4 4 6 7 4 3 2 3 2 4 6 7 6 1 ⟶ 4 4 4 4 6 7 4 4 6 7 4 6 1 9 0 0 1 5 , 4 4 6 7 4 3 2 3 2 4 6 7 6 7 ⟶ 4 4 4 4 6 7 4 4 6 7 4 6 1 9 0 0 1 2 , 4 4 6 7 4 3 2 3 2 4 6 7 6 8 ⟶ 4 4 4 4 6 7 4 4 6 7 4 6 1 9 0 0 1 11 , 14 4 6 7 4 3 2 3 2 4 6 7 6 0 ⟶ 14 4 4 4 6 7 4 4 6 7 4 6 1 9 0 0 1 9 , 14 4 6 7 4 3 2 3 2 4 6 7 6 1 ⟶ 14 4 4 4 6 7 4 4 6 7 4 6 1 9 0 0 1 5 , 14 4 6 7 4 3 2 3 2 4 6 7 6 7 ⟶ 14 4 4 4 6 7 4 4 6 7 4 6 1 9 0 0 1 2 , 14 4 6 7 4 3 2 3 2 4 6 7 6 8 ⟶ 14 4 4 4 6 7 4 4 6 7 4 6 1 9 0 0 1 11 , 7 4 3 9 7 3 5 2 4 6 1 9 1 9 ⟶ 7 4 4 4 6 1 9 0 0 1 9 0 0 0 0 1 2 6 , 7 4 3 9 7 3 5 2 4 6 1 9 1 5 ⟶ 7 4 4 4 6 1 9 0 0 1 9 0 0 0 0 1 2 3 , 7 4 3 9 7 3 5 2 4 6 1 9 1 2 ⟶ 7 4 4 4 6 1 9 0 0 1 9 0 0 0 0 1 2 4 , 7 4 3 9 7 3 5 2 4 6 1 9 1 11 ⟶ 7 4 4 4 6 1 9 0 0 1 9 0 0 0 0 1 2 12 , 2 4 3 9 7 3 5 2 4 6 1 9 1 9 ⟶ 2 4 4 4 6 1 9 0 0 1 9 0 0 0 0 1 2 6 , 2 4 3 9 7 3 5 2 4 6 1 9 1 5 ⟶ 2 4 4 4 6 1 9 0 0 1 9 0 0 0 0 1 2 3 , 2 4 3 9 7 3 5 2 4 6 1 9 1 2 ⟶ 2 4 4 4 6 1 9 0 0 1 9 0 0 0 0 1 2 4 , 2 4 3 9 7 3 5 2 4 6 1 9 1 11 ⟶ 2 4 4 4 6 1 9 0 0 1 9 0 0 0 0 1 2 12 , 4 4 3 9 7 3 5 2 4 6 1 9 1 9 ⟶ 4 4 4 4 6 1 9 0 0 1 9 0 0 0 0 1 2 6 , 4 4 3 9 7 3 5 2 4 6 1 9 1 5 ⟶ 4 4 4 4 6 1 9 0 0 1 9 0 0 0 0 1 2 3 , 4 4 3 9 7 3 5 2 4 6 1 9 1 2 ⟶ 4 4 4 4 6 1 9 0 0 1 9 0 0 0 0 1 2 4 , 4 4 3 9 7 3 5 2 4 6 1 9 1 11 ⟶ 4 4 4 4 6 1 9 0 0 1 9 0 0 0 0 1 2 12 , 14 4 3 9 7 3 5 2 4 6 1 9 1 9 ⟶ 14 4 4 4 6 1 9 0 0 1 9 0 0 0 0 1 2 6 , 14 4 3 9 7 3 5 2 4 6 1 9 1 5 ⟶ 14 4 4 4 6 1 9 0 0 1 9 0 0 0 0 1 2 3 , 14 4 3 9 7 3 5 2 4 6 1 9 1 2 ⟶ 14 4 4 4 6 1 9 0 0 1 9 0 0 0 0 1 2 4 , 14 4 3 9 7 3 5 2 4 6 1 9 1 11 ⟶ 14 4 4 4 6 1 9 0 0 1 9 0 0 0 0 1 2 12 , 7 4 3 5 2 3 9 1 5 2 6 0 7 6 ⟶ 7 4 4 6 0 0 0 1 2 6 0 0 7 4 4 4 4 6 , 7 4 3 5 2 3 9 1 5 2 6 0 7 3 ⟶ 7 4 4 6 0 0 0 1 2 6 0 0 7 4 4 4 4 3 , 7 4 3 5 2 3 9 1 5 2 6 0 7 4 ⟶ 7 4 4 6 0 0 0 1 2 6 0 0 7 4 4 4 4 4 , 7 4 3 5 2 3 9 1 5 2 6 0 7 12 ⟶ 7 4 4 6 0 0 0 1 2 6 0 0 7 4 4 4 4 12 , 2 4 3 5 2 3 9 1 5 2 6 0 7 6 ⟶ 2 4 4 6 0 0 0 1 2 6 0 0 7 4 4 4 4 6 , 2 4 3 5 2 3 9 1 5 2 6 0 7 3 ⟶ 2 4 4 6 0 0 0 1 2 6 0 0 7 4 4 4 4 3 , 2 4 3 5 2 3 9 1 5 2 6 0 7 4 ⟶ 2 4 4 6 0 0 0 1 2 6 0 0 7 4 4 4 4 4 , 2 4 3 5 2 3 9 1 5 2 6 0 7 12 ⟶ 2 4 4 6 0 0 0 1 2 6 0 0 7 4 4 4 4 12 , 4 4 3 5 2 3 9 1 5 2 6 0 7 6 ⟶ 4 4 4 6 0 0 0 1 2 6 0 0 7 4 4 4 4 6 , 4 4 3 5 2 3 9 1 5 2 6 0 7 3 ⟶ 4 4 4 6 0 0 0 1 2 6 0 0 7 4 4 4 4 3 , 4 4 3 5 2 3 9 1 5 2 6 0 7 4 ⟶ 4 4 4 6 0 0 0 1 2 6 0 0 7 4 4 4 4 4 , 4 4 3 5 2 3 9 1 5 2 6 0 7 12 ⟶ 4 4 4 6 0 0 0 1 2 6 0 0 7 4 4 4 4 12 , 14 4 3 5 2 3 9 1 5 2 6 0 7 6 ⟶ 14 4 4 6 0 0 0 1 2 6 0 0 7 4 4 4 4 6 , 14 4 3 5 2 3 9 1 5 2 6 0 7 3 ⟶ 14 4 4 6 0 0 0 1 2 6 0 0 7 4 4 4 4 3 , 14 4 3 5 2 3 9 1 5 2 6 0 7 4 ⟶ 14 4 4 6 0 0 0 1 2 6 0 0 7 4 4 4 4 4 , 14 4 3 5 2 3 9 1 5 2 6 0 7 12 ⟶ 14 4 4 6 0 0 0 1 2 6 0 0 7 4 4 4 4 12 } 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 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 4 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 5 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 1 ↦ 0, 2 ↦ 1, 6 ↦ 2, 0 ↦ 3, 7 ↦ 4, 3 ↦ 5, 9 ↦ 6, 4 ↦ 7, 5 ↦ 8, 11 ↦ 9 }, it remains to prove termination of the 4-rule system { 0 1 2 3 4 5 6 4 7 7 2 3 0 6 ⟶ 3 3 3 3 3 0 1 7 7 2 4 2 4 7 5 6 0 6 , 0 1 2 3 4 5 6 4 7 7 2 3 0 8 ⟶ 3 3 3 3 3 0 1 7 7 2 4 2 4 7 5 6 0 8 , 0 1 2 3 4 5 6 4 7 7 2 3 0 1 ⟶ 3 3 3 3 3 0 1 7 7 2 4 2 4 7 5 6 0 1 , 0 1 2 3 4 5 6 4 7 7 2 3 0 9 ⟶ 3 3 3 3 3 0 1 7 7 2 4 2 4 7 5 6 0 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 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 1 0 ⎟ ⎜ 0 0 0 0 0 0 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 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 3-rule system { 0 1 2 3 4 5 6 4 7 7 2 3 0 8 ⟶ 3 3 3 3 3 0 1 7 7 2 4 2 4 7 5 6 0 8 , 0 1 2 3 4 5 6 4 7 7 2 3 0 1 ⟶ 3 3 3 3 3 0 1 7 7 2 4 2 4 7 5 6 0 1 , 0 1 2 3 4 5 6 4 7 7 2 3 0 9 ⟶ 3 3 3 3 3 0 1 7 7 2 4 2 4 7 5 6 0 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 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 1 0 ⎟ ⎜ 0 0 0 0 0 0 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 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 9 ↦ 8 }, it remains to prove termination of the 2-rule system { 0 1 2 3 4 5 6 4 7 7 2 3 0 1 ⟶ 3 3 3 3 3 0 1 7 7 2 4 2 4 7 5 6 0 1 , 0 1 2 3 4 5 6 4 7 7 2 3 0 8 ⟶ 3 3 3 3 3 0 1 7 7 2 4 2 4 7 5 6 0 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 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 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 1 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 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 1 0 ⎟ ⎜ 0 0 0 0 0 0 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 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8 }, it remains to prove termination of the 1-rule system { 0 1 2 3 4 5 6 4 7 7 2 3 0 8 ⟶ 3 3 3 3 3 0 1 7 7 2 4 2 4 7 5 6 0 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 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 1 0 ⎟ ⎜ 0 0 0 0 0 0 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 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { }, it remains to prove termination of the 0-rule system { } The system is trivially terminating.