/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 17-rule system { 0 1 2 1 ⟶ 1 2 1 1 0 1 2 0 1 2 , 0 1 2 1 ⟶ 1 2 1 1 0 1 2 0 1 2 0 1 2 , 0 1 2 1 ⟶ 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 ⟶ 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 ⟶ 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 ⟶ 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 ⟶ 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 ⟶ 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 ⟶ 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 ⟶ 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 ⟶ 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 ⟶ 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 ⟶ 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 ⟶ 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 ⟶ 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 ⟶ 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 ⟶ 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 } Applying sparse tiling TRFC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (1,0) ↦ 0, (0,1) ↦ 1, (1,2) ↦ 2, (2,1) ↦ 3, (1,1) ↦ 4, (2,0) ↦ 5, (2,2) ↦ 6 }, it remains to prove termination of the 102-rule system { 0 1 2 3 0 ⟶ 4 2 3 4 0 1 2 5 1 2 5 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 6 , 0 1 2 3 0 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 6 , 0 1 2 3 0 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 0 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 0 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 0 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 0 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 0 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 0 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 0 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 0 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 0 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 0 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 0 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 0 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 0 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 0 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 6: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6 }, it remains to prove termination of the 85-rule system { 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 6 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 6 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 4 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 6: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6 }, it remains to prove termination of the 68-rule system { 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 0 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 6: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6 }, it remains to prove termination of the 51-rule system { 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 6 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 6 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 4 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 3 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 6: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6 }, it remains to prove termination of the 34-rule system { 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 6 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 6 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 0 1 2 3 2 ⟶ 4 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 , 5 1 2 3 2 ⟶ 3 2 3 4 0 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 2 6 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 6: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 5 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 0 ↦ 5, 6 ↦ 6 }, it remains to prove termination of the 17-rule system { 0 1 2 3 2 ⟶ 3 2 3 4 5 1 2 0 1 2 6 , 0 1 2 3 2 ⟶ 3 2 3 4 5 1 2 0 1 2 0 1 2 6 , 0 1 2 3 2 ⟶ 3 2 3 4 5 1 2 0 1 2 0 1 2 0 1 2 6 , 0 1 2 3 2 ⟶ 3 2 3 4 5 1 2 0 1 2 0 1 2 0 1 2 0 1 2 6 , 0 1 2 3 2 ⟶ 3 2 3 4 5 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 6 , 0 1 2 3 2 ⟶ 3 2 3 4 5 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 6 , 0 1 2 3 2 ⟶ 3 2 3 4 5 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 6 , 0 1 2 3 2 ⟶ 3 2 3 4 5 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 6 , 0 1 2 3 2 ⟶ 3 2 3 4 5 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 6 , 0 1 2 3 2 ⟶ 3 2 3 4 5 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 6 , 0 1 2 3 2 ⟶ 3 2 3 4 5 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 6 , 0 1 2 3 2 ⟶ 3 2 3 4 5 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 6 , 0 1 2 3 2 ⟶ 3 2 3 4 5 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 6 , 0 1 2 3 2 ⟶ 3 2 3 4 5 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 6 , 0 1 2 3 2 ⟶ 3 2 3 4 5 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 6 , 0 1 2 3 2 ⟶ 3 2 3 4 5 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 6 , 0 1 2 3 2 ⟶ 3 2 3 4 5 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 6 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 6: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { }, it remains to prove termination of the 0-rule system { } The system is trivially terminating.