YES After renaming modulo { 0->0, 1->1, 2->2 }, it remains to prove termination of the 100-rule system { 0 0 1 0 2 -> 0 0 1 2 2 , 0 0 1 0 2 -> 0 0 2 1 2 , 0 0 1 0 2 -> 0 1 0 2 2 , 0 0 1 0 2 -> 0 1 1 2 2 , 0 0 1 0 2 -> 0 1 2 0 2 , 0 0 1 0 2 -> 0 1 2 2 0 , 0 0 1 0 2 -> 0 1 2 2 2 , 0 0 1 0 2 -> 0 2 1 0 2 , 0 0 1 0 2 -> 0 2 1 2 2 , 0 0 1 0 2 -> 0 2 2 1 0 , 0 0 1 0 2 -> 0 2 2 1 2 , 0 0 1 0 2 -> 1 0 0 2 2 , 0 0 1 0 2 -> 1 0 2 0 2 , 0 0 1 0 2 -> 1 0 2 2 0 , 0 0 1 0 2 -> 1 0 2 2 2 , 0 0 1 0 2 -> 1 1 0 2 2 , 0 0 1 0 2 -> 1 2 0 2 2 , 0 0 1 0 2 -> 1 2 1 0 2 , 0 0 1 0 2 -> 1 2 2 0 2 , 0 0 1 0 2 -> 1 2 2 2 0 , 0 0 1 0 2 -> 2 1 0 2 2 , 0 0 1 0 2 -> 2 2 1 0 2 , 0 0 1 0 2 -> 2 2 2 1 0 , 0 1 2 0 2 -> 0 1 0 2 2 , 0 1 2 0 2 -> 0 1 1 2 2 , 0 1 2 0 2 -> 0 1 2 2 2 , 0 1 2 0 2 -> 0 2 1 0 2 , 0 1 2 0 2 -> 0 2 1 2 2 , 0 1 2 0 2 -> 0 2 2 1 0 , 0 1 2 0 2 -> 0 2 2 1 2 , 0 1 2 0 2 -> 1 0 2 2 2 , 0 1 2 0 2 -> 1 2 0 2 2 , 0 1 2 0 2 -> 1 2 2 0 2 , 0 1 2 0 2 -> 1 2 2 2 0 , 1 0 1 0 2 -> 0 1 2 2 2 , 1 0 1 0 2 -> 0 2 1 2 2 , 1 0 1 0 2 -> 1 0 0 2 2 , 1 0 1 0 2 -> 1 0 1 2 2 , 1 0 1 0 2 -> 1 0 2 0 2 , 1 0 1 0 2 -> 1 0 2 1 2 , 1 0 1 0 2 -> 1 0 2 2 0 , 1 0 1 0 2 -> 1 0 2 2 2 , 1 0 1 0 2 -> 1 1 0 2 2 , 1 0 1 0 2 -> 1 2 0 2 2 , 1 0 1 0 2 -> 1 2 1 0 2 , 1 0 1 0 2 -> 1 2 2 0 2 , 1 0 1 0 2 -> 1 2 2 2 0 , 1 0 1 0 2 -> 2 0 1 2 2 , 1 0 1 0 2 -> 2 0 2 1 2 , 1 0 1 0 2 -> 2 1 0 2 2 , 1 0 1 0 2 -> 2 1 2 0 2 , 1 0 1 0 2 -> 2 1 2 2 0 , 1 0 1 0 2 -> 2 2 0 1 2 , 1 0 1 0 2 -> 2 2 1 0 2 , 1 0 1 0 2 -> 2 2 1 2 0 , 1 0 1 0 2 -> 2 2 2 1 0 , 1 0 2 0 2 -> 1 0 2 2 2 , 1 0 2 0 2 -> 1 2 0 2 2 , 1 0 2 0 2 -> 1 2 2 0 2 , 1 0 2 0 2 -> 1 2 2 2 0 , 1 0 2 0 2 -> 2 1 0 2 2 , 1 0 2 0 2 -> 2 2 1 0 2 , 1 1 2 0 2 -> 0 1 2 2 2 , 1 1 2 0 2 -> 0 2 1 2 2 , 1 1 2 0 2 -> 0 2 2 1 2 , 1 1 2 0 2 -> 1 0 0 2 2 , 1 1 2 0 2 -> 1 0 1 2 2 , 1 1 2 0 2 -> 1 0 2 0 2 , 1 1 2 0 2 -> 1 0 2 1 2 , 1 1 2 0 2 -> 1 0 2 2 0 , 1 1 2 0 2 -> 1 0 2 2 2 , 1 1 2 0 2 -> 1 1 0 2 2 , 1 1 2 0 2 -> 1 2 0 2 2 , 1 1 2 0 2 -> 1 2 1 0 2 , 1 1 2 0 2 -> 1 2 2 0 2 , 1 1 2 0 2 -> 1 2 2 2 0 , 1 1 2 0 2 -> 2 0 1 2 2 , 1 1 2 0 2 -> 2 1 0 2 2 , 1 1 2 0 2 -> 2 1 2 0 2 , 1 1 2 0 2 -> 2 2 0 1 2 , 1 1 2 0 2 -> 2 2 1 0 2 , 1 1 2 0 2 -> 2 2 2 1 0 , 1 2 2 0 2 -> 1 0 2 2 2 , 2 0 1 0 2 -> 2 0 1 2 2 , 2 0 1 0 2 -> 2 0 2 1 2 , 2 0 1 0 2 -> 2 1 0 2 2 , 2 0 1 0 2 -> 2 1 2 0 2 , 2 0 1 0 2 -> 2 1 2 2 0 , 2 0 1 0 2 -> 2 2 0 1 2 , 2 0 1 0 2 -> 2 2 1 0 2 , 2 0 1 0 2 -> 2 2 1 2 0 , 2 0 1 0 2 -> 2 2 2 1 0 , 2 1 1 0 2 -> 2 0 1 0 2 , 2 1 1 0 2 -> 2 0 2 1 2 , 2 1 1 0 2 -> 2 1 2 0 2 , 2 1 1 0 2 -> 2 2 1 0 2 , 2 1 2 0 2 -> 2 0 1 2 2 , 2 1 2 0 2 -> 2 1 0 2 2 , 2 1 2 0 2 -> 2 2 1 0 2 , 2 1 2 0 2 -> 2 2 2 1 0 } The system was reversed. After renaming modulo { 2->0, 0->1, 1->2 }, it remains to prove termination of the 100-rule system { 0 1 2 1 1 -> 0 0 2 1 1 , 0 1 2 1 1 -> 0 2 0 1 1 , 0 1 2 1 1 -> 0 0 1 2 1 , 0 1 2 1 1 -> 0 0 2 2 1 , 0 1 2 1 1 -> 0 1 0 2 1 , 0 1 2 1 1 -> 1 0 0 2 1 , 0 1 2 1 1 -> 0 0 0 2 1 , 0 1 2 1 1 -> 0 1 2 0 1 , 0 1 2 1 1 -> 0 0 2 0 1 , 0 1 2 1 1 -> 1 2 0 0 1 , 0 1 2 1 1 -> 0 2 0 0 1 , 0 1 2 1 1 -> 0 0 1 1 2 , 0 1 2 1 1 -> 0 1 0 1 2 , 0 1 2 1 1 -> 1 0 0 1 2 , 0 1 2 1 1 -> 0 0 0 1 2 , 0 1 2 1 1 -> 0 0 1 2 2 , 0 1 2 1 1 -> 0 0 1 0 2 , 0 1 2 1 1 -> 0 1 2 0 2 , 0 1 2 1 1 -> 0 1 0 0 2 , 0 1 2 1 1 -> 1 0 0 0 2 , 0 1 2 1 1 -> 0 0 1 2 0 , 0 1 2 1 1 -> 0 1 2 0 0 , 0 1 2 1 1 -> 1 2 0 0 0 , 0 1 0 2 1 -> 0 0 1 2 1 , 0 1 0 2 1 -> 0 0 2 2 1 , 0 1 0 2 1 -> 0 0 0 2 1 , 0 1 0 2 1 -> 0 1 2 0 1 , 0 1 0 2 1 -> 0 0 2 0 1 , 0 1 0 2 1 -> 1 2 0 0 1 , 0 1 0 2 1 -> 0 2 0 0 1 , 0 1 0 2 1 -> 0 0 0 1 2 , 0 1 0 2 1 -> 0 0 1 0 2 , 0 1 0 2 1 -> 0 1 0 0 2 , 0 1 0 2 1 -> 1 0 0 0 2 , 0 1 2 1 2 -> 0 0 0 2 1 , 0 1 2 1 2 -> 0 0 2 0 1 , 0 1 2 1 2 -> 0 0 1 1 2 , 0 1 2 1 2 -> 0 0 2 1 2 , 0 1 2 1 2 -> 0 1 0 1 2 , 0 1 2 1 2 -> 0 2 0 1 2 , 0 1 2 1 2 -> 1 0 0 1 2 , 0 1 2 1 2 -> 0 0 0 1 2 , 0 1 2 1 2 -> 0 0 1 2 2 , 0 1 2 1 2 -> 0 0 1 0 2 , 0 1 2 1 2 -> 0 1 2 0 2 , 0 1 2 1 2 -> 0 1 0 0 2 , 0 1 2 1 2 -> 1 0 0 0 2 , 0 1 2 1 2 -> 0 0 2 1 0 , 0 1 2 1 2 -> 0 2 0 1 0 , 0 1 2 1 2 -> 0 0 1 2 0 , 0 1 2 1 2 -> 0 1 0 2 0 , 0 1 2 1 2 -> 1 0 0 2 0 , 0 1 2 1 2 -> 0 2 1 0 0 , 0 1 2 1 2 -> 0 1 2 0 0 , 0 1 2 1 2 -> 1 0 2 0 0 , 0 1 2 1 2 -> 1 2 0 0 0 , 0 1 0 1 2 -> 0 0 0 1 2 , 0 1 0 1 2 -> 0 0 1 0 2 , 0 1 0 1 2 -> 0 1 0 0 2 , 0 1 0 1 2 -> 1 0 0 0 2 , 0 1 0 1 2 -> 0 0 1 2 0 , 0 1 0 1 2 -> 0 1 2 0 0 , 0 1 0 2 2 -> 0 0 0 2 1 , 0 1 0 2 2 -> 0 0 2 0 1 , 0 1 0 2 2 -> 0 2 0 0 1 , 0 1 0 2 2 -> 0 0 1 1 2 , 0 1 0 2 2 -> 0 0 2 1 2 , 0 1 0 2 2 -> 0 1 0 1 2 , 0 1 0 2 2 -> 0 2 0 1 2 , 0 1 0 2 2 -> 1 0 0 1 2 , 0 1 0 2 2 -> 0 0 0 1 2 , 0 1 0 2 2 -> 0 0 1 2 2 , 0 1 0 2 2 -> 0 0 1 0 2 , 0 1 0 2 2 -> 0 1 2 0 2 , 0 1 0 2 2 -> 0 1 0 0 2 , 0 1 0 2 2 -> 1 0 0 0 2 , 0 1 0 2 2 -> 0 0 2 1 0 , 0 1 0 2 2 -> 0 0 1 2 0 , 0 1 0 2 2 -> 0 1 0 2 0 , 0 1 0 2 2 -> 0 2 1 0 0 , 0 1 0 2 2 -> 0 1 2 0 0 , 0 1 0 2 2 -> 1 2 0 0 0 , 0 1 0 0 2 -> 0 0 0 1 2 , 0 1 2 1 0 -> 0 0 2 1 0 , 0 1 2 1 0 -> 0 2 0 1 0 , 0 1 2 1 0 -> 0 0 1 2 0 , 0 1 2 1 0 -> 0 1 0 2 0 , 0 1 2 1 0 -> 1 0 0 2 0 , 0 1 2 1 0 -> 0 2 1 0 0 , 0 1 2 1 0 -> 0 1 2 0 0 , 0 1 2 1 0 -> 1 0 2 0 0 , 0 1 2 1 0 -> 1 2 0 0 0 , 0 1 2 2 0 -> 0 1 2 1 0 , 0 1 2 2 0 -> 0 2 0 1 0 , 0 1 2 2 0 -> 0 1 0 2 0 , 0 1 2 2 0 -> 0 1 2 0 0 , 0 1 0 2 0 -> 0 0 2 1 0 , 0 1 0 2 0 -> 0 0 1 2 0 , 0 1 0 2 0 -> 0 1 2 0 0 , 0 1 0 2 0 -> 1 2 0 0 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 is interpreted by / \ | 1 0 | | 0 1 | \ / 1 is interpreted by / \ | 1 1 | | 0 1 | \ / 2 is interpreted by / \ | 1 1 | | 0 1 | \ / After renaming modulo { 0->0, 1->1, 2->2 }, it remains to prove termination of the 17-rule system { 0 1 0 2 1 -> 0 0 1 2 1 , 0 1 0 2 1 -> 0 0 2 2 1 , 0 1 0 2 1 -> 0 1 2 0 1 , 0 1 0 2 1 -> 1 2 0 0 1 , 0 1 0 2 2 -> 0 0 1 1 2 , 0 1 0 2 2 -> 0 0 2 1 2 , 0 1 0 2 2 -> 0 1 0 1 2 , 0 1 0 2 2 -> 0 2 0 1 2 , 0 1 0 2 2 -> 1 0 0 1 2 , 0 1 0 2 2 -> 0 0 1 2 2 , 0 1 0 2 2 -> 0 1 2 0 2 , 0 1 0 0 2 -> 0 0 0 1 2 , 0 1 2 2 0 -> 0 1 2 1 0 , 0 1 0 2 0 -> 0 0 2 1 0 , 0 1 0 2 0 -> 0 0 1 2 0 , 0 1 0 2 0 -> 0 1 2 0 0 , 0 1 0 2 0 -> 1 2 0 0 0 } Applying sparse 2-tiling [Hofbauer/Geser/Waldmann, FSCD 2019]. After renaming modulo { (0,0)->0, (0,1)->1, (1,0)->2, (0,2)->3, (2,1)->4, (1,2)->5, (1,1)->6, (1,4)->7, (2,0)->8, (3,0)->9, (2,2)->10, (3,1)->11, (2,4)->12, (0,4)->13 }, it remains to prove termination of the 272-rule system { 0 1 2 3 4 2 -> 0 0 1 5 4 2 , 0 1 2 3 4 6 -> 0 0 1 5 4 6 , 0 1 2 3 4 5 -> 0 0 1 5 4 5 , 0 1 2 3 4 7 -> 0 0 1 5 4 7 , 2 1 2 3 4 2 -> 2 0 1 5 4 2 , 2 1 2 3 4 6 -> 2 0 1 5 4 6 , 2 1 2 3 4 5 -> 2 0 1 5 4 5 , 2 1 2 3 4 7 -> 2 0 1 5 4 7 , 8 1 2 3 4 2 -> 8 0 1 5 4 2 , 8 1 2 3 4 6 -> 8 0 1 5 4 6 , 8 1 2 3 4 5 -> 8 0 1 5 4 5 , 8 1 2 3 4 7 -> 8 0 1 5 4 7 , 9 1 2 3 4 2 -> 9 0 1 5 4 2 , 9 1 2 3 4 6 -> 9 0 1 5 4 6 , 9 1 2 3 4 5 -> 9 0 1 5 4 5 , 9 1 2 3 4 7 -> 9 0 1 5 4 7 , 0 1 2 3 4 2 -> 0 0 3 10 4 2 , 0 1 2 3 4 6 -> 0 0 3 10 4 6 , 0 1 2 3 4 5 -> 0 0 3 10 4 5 , 0 1 2 3 4 7 -> 0 0 3 10 4 7 , 2 1 2 3 4 2 -> 2 0 3 10 4 2 , 2 1 2 3 4 6 -> 2 0 3 10 4 6 , 2 1 2 3 4 5 -> 2 0 3 10 4 5 , 2 1 2 3 4 7 -> 2 0 3 10 4 7 , 8 1 2 3 4 2 -> 8 0 3 10 4 2 , 8 1 2 3 4 6 -> 8 0 3 10 4 6 , 8 1 2 3 4 5 -> 8 0 3 10 4 5 , 8 1 2 3 4 7 -> 8 0 3 10 4 7 , 9 1 2 3 4 2 -> 9 0 3 10 4 2 , 9 1 2 3 4 6 -> 9 0 3 10 4 6 , 9 1 2 3 4 5 -> 9 0 3 10 4 5 , 9 1 2 3 4 7 -> 9 0 3 10 4 7 , 0 1 2 3 4 2 -> 0 1 5 8 1 2 , 0 1 2 3 4 6 -> 0 1 5 8 1 6 , 0 1 2 3 4 5 -> 0 1 5 8 1 5 , 0 1 2 3 4 7 -> 0 1 5 8 1 7 , 2 1 2 3 4 2 -> 2 1 5 8 1 2 , 2 1 2 3 4 6 -> 2 1 5 8 1 6 , 2 1 2 3 4 5 -> 2 1 5 8 1 5 , 2 1 2 3 4 7 -> 2 1 5 8 1 7 , 8 1 2 3 4 2 -> 8 1 5 8 1 2 , 8 1 2 3 4 6 -> 8 1 5 8 1 6 , 8 1 2 3 4 5 -> 8 1 5 8 1 5 , 8 1 2 3 4 7 -> 8 1 5 8 1 7 , 9 1 2 3 4 2 -> 9 1 5 8 1 2 , 9 1 2 3 4 6 -> 9 1 5 8 1 6 , 9 1 2 3 4 5 -> 9 1 5 8 1 5 , 9 1 2 3 4 7 -> 9 1 5 8 1 7 , 0 1 2 3 4 2 -> 1 5 8 0 1 2 , 0 1 2 3 4 6 -> 1 5 8 0 1 6 , 0 1 2 3 4 5 -> 1 5 8 0 1 5 , 0 1 2 3 4 7 -> 1 5 8 0 1 7 , 2 1 2 3 4 2 -> 6 5 8 0 1 2 , 2 1 2 3 4 6 -> 6 5 8 0 1 6 , 2 1 2 3 4 5 -> 6 5 8 0 1 5 , 2 1 2 3 4 7 -> 6 5 8 0 1 7 , 8 1 2 3 4 2 -> 4 5 8 0 1 2 , 8 1 2 3 4 6 -> 4 5 8 0 1 6 , 8 1 2 3 4 5 -> 4 5 8 0 1 5 , 8 1 2 3 4 7 -> 4 5 8 0 1 7 , 9 1 2 3 4 2 -> 11 5 8 0 1 2 , 9 1 2 3 4 6 -> 11 5 8 0 1 6 , 9 1 2 3 4 5 -> 11 5 8 0 1 5 , 9 1 2 3 4 7 -> 11 5 8 0 1 7 , 0 1 2 3 10 8 -> 0 0 1 6 5 8 , 0 1 2 3 10 4 -> 0 0 1 6 5 4 , 0 1 2 3 10 10 -> 0 0 1 6 5 10 , 0 1 2 3 10 12 -> 0 0 1 6 5 12 , 2 1 2 3 10 8 -> 2 0 1 6 5 8 , 2 1 2 3 10 4 -> 2 0 1 6 5 4 , 2 1 2 3 10 10 -> 2 0 1 6 5 10 , 2 1 2 3 10 12 -> 2 0 1 6 5 12 , 8 1 2 3 10 8 -> 8 0 1 6 5 8 , 8 1 2 3 10 4 -> 8 0 1 6 5 4 , 8 1 2 3 10 10 -> 8 0 1 6 5 10 , 8 1 2 3 10 12 -> 8 0 1 6 5 12 , 9 1 2 3 10 8 -> 9 0 1 6 5 8 , 9 1 2 3 10 4 -> 9 0 1 6 5 4 , 9 1 2 3 10 10 -> 9 0 1 6 5 10 , 9 1 2 3 10 12 -> 9 0 1 6 5 12 , 0 1 2 3 10 8 -> 0 0 3 4 5 8 , 0 1 2 3 10 4 -> 0 0 3 4 5 4 , 0 1 2 3 10 10 -> 0 0 3 4 5 10 , 0 1 2 3 10 12 -> 0 0 3 4 5 12 , 2 1 2 3 10 8 -> 2 0 3 4 5 8 , 2 1 2 3 10 4 -> 2 0 3 4 5 4 , 2 1 2 3 10 10 -> 2 0 3 4 5 10 , 2 1 2 3 10 12 -> 2 0 3 4 5 12 , 8 1 2 3 10 8 -> 8 0 3 4 5 8 , 8 1 2 3 10 4 -> 8 0 3 4 5 4 , 8 1 2 3 10 10 -> 8 0 3 4 5 10 , 8 1 2 3 10 12 -> 8 0 3 4 5 12 , 9 1 2 3 10 8 -> 9 0 3 4 5 8 , 9 1 2 3 10 4 -> 9 0 3 4 5 4 , 9 1 2 3 10 10 -> 9 0 3 4 5 10 , 9 1 2 3 10 12 -> 9 0 3 4 5 12 , 0 1 2 3 10 8 -> 0 1 2 1 5 8 , 0 1 2 3 10 4 -> 0 1 2 1 5 4 , 0 1 2 3 10 10 -> 0 1 2 1 5 10 , 0 1 2 3 10 12 -> 0 1 2 1 5 12 , 2 1 2 3 10 8 -> 2 1 2 1 5 8 , 2 1 2 3 10 4 -> 2 1 2 1 5 4 , 2 1 2 3 10 10 -> 2 1 2 1 5 10 , 2 1 2 3 10 12 -> 2 1 2 1 5 12 , 8 1 2 3 10 8 -> 8 1 2 1 5 8 , 8 1 2 3 10 4 -> 8 1 2 1 5 4 , 8 1 2 3 10 10 -> 8 1 2 1 5 10 , 8 1 2 3 10 12 -> 8 1 2 1 5 12 , 9 1 2 3 10 8 -> 9 1 2 1 5 8 , 9 1 2 3 10 4 -> 9 1 2 1 5 4 , 9 1 2 3 10 10 -> 9 1 2 1 5 10 , 9 1 2 3 10 12 -> 9 1 2 1 5 12 , 0 1 2 3 10 8 -> 0 3 8 1 5 8 , 0 1 2 3 10 4 -> 0 3 8 1 5 4 , 0 1 2 3 10 10 -> 0 3 8 1 5 10 , 0 1 2 3 10 12 -> 0 3 8 1 5 12 , 2 1 2 3 10 8 -> 2 3 8 1 5 8 , 2 1 2 3 10 4 -> 2 3 8 1 5 4 , 2 1 2 3 10 10 -> 2 3 8 1 5 10 , 2 1 2 3 10 12 -> 2 3 8 1 5 12 , 8 1 2 3 10 8 -> 8 3 8 1 5 8 , 8 1 2 3 10 4 -> 8 3 8 1 5 4 , 8 1 2 3 10 10 -> 8 3 8 1 5 10 , 8 1 2 3 10 12 -> 8 3 8 1 5 12 , 9 1 2 3 10 8 -> 9 3 8 1 5 8 , 9 1 2 3 10 4 -> 9 3 8 1 5 4 , 9 1 2 3 10 10 -> 9 3 8 1 5 10 , 9 1 2 3 10 12 -> 9 3 8 1 5 12 , 0 1 2 3 10 8 -> 1 2 0 1 5 8 , 0 1 2 3 10 4 -> 1 2 0 1 5 4 , 0 1 2 3 10 10 -> 1 2 0 1 5 10 , 0 1 2 3 10 12 -> 1 2 0 1 5 12 , 2 1 2 3 10 8 -> 6 2 0 1 5 8 , 2 1 2 3 10 4 -> 6 2 0 1 5 4 , 2 1 2 3 10 10 -> 6 2 0 1 5 10 , 2 1 2 3 10 12 -> 6 2 0 1 5 12 , 8 1 2 3 10 8 -> 4 2 0 1 5 8 , 8 1 2 3 10 4 -> 4 2 0 1 5 4 , 8 1 2 3 10 10 -> 4 2 0 1 5 10 , 8 1 2 3 10 12 -> 4 2 0 1 5 12 , 9 1 2 3 10 8 -> 11 2 0 1 5 8 , 9 1 2 3 10 4 -> 11 2 0 1 5 4 , 9 1 2 3 10 10 -> 11 2 0 1 5 10 , 9 1 2 3 10 12 -> 11 2 0 1 5 12 , 0 1 2 3 10 8 -> 0 0 1 5 10 8 , 0 1 2 3 10 4 -> 0 0 1 5 10 4 , 0 1 2 3 10 10 -> 0 0 1 5 10 10 , 0 1 2 3 10 12 -> 0 0 1 5 10 12 , 2 1 2 3 10 8 -> 2 0 1 5 10 8 , 2 1 2 3 10 4 -> 2 0 1 5 10 4 , 2 1 2 3 10 10 -> 2 0 1 5 10 10 , 2 1 2 3 10 12 -> 2 0 1 5 10 12 , 8 1 2 3 10 8 -> 8 0 1 5 10 8 , 8 1 2 3 10 4 -> 8 0 1 5 10 4 , 8 1 2 3 10 10 -> 8 0 1 5 10 10 , 8 1 2 3 10 12 -> 8 0 1 5 10 12 , 9 1 2 3 10 8 -> 9 0 1 5 10 8 , 9 1 2 3 10 4 -> 9 0 1 5 10 4 , 9 1 2 3 10 10 -> 9 0 1 5 10 10 , 9 1 2 3 10 12 -> 9 0 1 5 10 12 , 0 1 2 3 10 8 -> 0 1 5 8 3 8 , 0 1 2 3 10 4 -> 0 1 5 8 3 4 , 0 1 2 3 10 10 -> 0 1 5 8 3 10 , 0 1 2 3 10 12 -> 0 1 5 8 3 12 , 2 1 2 3 10 8 -> 2 1 5 8 3 8 , 2 1 2 3 10 4 -> 2 1 5 8 3 4 , 2 1 2 3 10 10 -> 2 1 5 8 3 10 , 2 1 2 3 10 12 -> 2 1 5 8 3 12 , 8 1 2 3 10 8 -> 8 1 5 8 3 8 , 8 1 2 3 10 4 -> 8 1 5 8 3 4 , 8 1 2 3 10 10 -> 8 1 5 8 3 10 , 8 1 2 3 10 12 -> 8 1 5 8 3 12 , 9 1 2 3 10 8 -> 9 1 5 8 3 8 , 9 1 2 3 10 4 -> 9 1 5 8 3 4 , 9 1 2 3 10 10 -> 9 1 5 8 3 10 , 9 1 2 3 10 12 -> 9 1 5 8 3 12 , 0 1 2 0 3 8 -> 0 0 0 1 5 8 , 0 1 2 0 3 4 -> 0 0 0 1 5 4 , 0 1 2 0 3 10 -> 0 0 0 1 5 10 , 0 1 2 0 3 12 -> 0 0 0 1 5 12 , 2 1 2 0 3 8 -> 2 0 0 1 5 8 , 2 1 2 0 3 4 -> 2 0 0 1 5 4 , 2 1 2 0 3 10 -> 2 0 0 1 5 10 , 2 1 2 0 3 12 -> 2 0 0 1 5 12 , 8 1 2 0 3 8 -> 8 0 0 1 5 8 , 8 1 2 0 3 4 -> 8 0 0 1 5 4 , 8 1 2 0 3 10 -> 8 0 0 1 5 10 , 8 1 2 0 3 12 -> 8 0 0 1 5 12 , 9 1 2 0 3 8 -> 9 0 0 1 5 8 , 9 1 2 0 3 4 -> 9 0 0 1 5 4 , 9 1 2 0 3 10 -> 9 0 0 1 5 10 , 9 1 2 0 3 12 -> 9 0 0 1 5 12 , 0 1 5 10 8 0 -> 0 1 5 4 2 0 , 0 1 5 10 8 1 -> 0 1 5 4 2 1 , 0 1 5 10 8 3 -> 0 1 5 4 2 3 , 0 1 5 10 8 13 -> 0 1 5 4 2 13 , 2 1 5 10 8 0 -> 2 1 5 4 2 0 , 2 1 5 10 8 1 -> 2 1 5 4 2 1 , 2 1 5 10 8 3 -> 2 1 5 4 2 3 , 2 1 5 10 8 13 -> 2 1 5 4 2 13 , 8 1 5 10 8 0 -> 8 1 5 4 2 0 , 8 1 5 10 8 1 -> 8 1 5 4 2 1 , 8 1 5 10 8 3 -> 8 1 5 4 2 3 , 8 1 5 10 8 13 -> 8 1 5 4 2 13 , 9 1 5 10 8 0 -> 9 1 5 4 2 0 , 9 1 5 10 8 1 -> 9 1 5 4 2 1 , 9 1 5 10 8 3 -> 9 1 5 4 2 3 , 9 1 5 10 8 13 -> 9 1 5 4 2 13 , 0 1 2 3 8 0 -> 0 0 3 4 2 0 , 0 1 2 3 8 1 -> 0 0 3 4 2 1 , 0 1 2 3 8 3 -> 0 0 3 4 2 3 , 0 1 2 3 8 13 -> 0 0 3 4 2 13 , 2 1 2 3 8 0 -> 2 0 3 4 2 0 , 2 1 2 3 8 1 -> 2 0 3 4 2 1 , 2 1 2 3 8 3 -> 2 0 3 4 2 3 , 2 1 2 3 8 13 -> 2 0 3 4 2 13 , 8 1 2 3 8 0 -> 8 0 3 4 2 0 , 8 1 2 3 8 1 -> 8 0 3 4 2 1 , 8 1 2 3 8 3 -> 8 0 3 4 2 3 , 8 1 2 3 8 13 -> 8 0 3 4 2 13 , 9 1 2 3 8 0 -> 9 0 3 4 2 0 , 9 1 2 3 8 1 -> 9 0 3 4 2 1 , 9 1 2 3 8 3 -> 9 0 3 4 2 3 , 9 1 2 3 8 13 -> 9 0 3 4 2 13 , 0 1 2 3 8 0 -> 0 0 1 5 8 0 , 0 1 2 3 8 1 -> 0 0 1 5 8 1 , 0 1 2 3 8 3 -> 0 0 1 5 8 3 , 0 1 2 3 8 13 -> 0 0 1 5 8 13 , 2 1 2 3 8 0 -> 2 0 1 5 8 0 , 2 1 2 3 8 1 -> 2 0 1 5 8 1 , 2 1 2 3 8 3 -> 2 0 1 5 8 3 , 2 1 2 3 8 13 -> 2 0 1 5 8 13 , 8 1 2 3 8 0 -> 8 0 1 5 8 0 , 8 1 2 3 8 1 -> 8 0 1 5 8 1 , 8 1 2 3 8 3 -> 8 0 1 5 8 3 , 8 1 2 3 8 13 -> 8 0 1 5 8 13 , 9 1 2 3 8 0 -> 9 0 1 5 8 0 , 9 1 2 3 8 1 -> 9 0 1 5 8 1 , 9 1 2 3 8 3 -> 9 0 1 5 8 3 , 9 1 2 3 8 13 -> 9 0 1 5 8 13 , 0 1 2 3 8 0 -> 0 1 5 8 0 0 , 0 1 2 3 8 1 -> 0 1 5 8 0 1 , 0 1 2 3 8 3 -> 0 1 5 8 0 3 , 0 1 2 3 8 13 -> 0 1 5 8 0 13 , 2 1 2 3 8 0 -> 2 1 5 8 0 0 , 2 1 2 3 8 1 -> 2 1 5 8 0 1 , 2 1 2 3 8 3 -> 2 1 5 8 0 3 , 2 1 2 3 8 13 -> 2 1 5 8 0 13 , 8 1 2 3 8 0 -> 8 1 5 8 0 0 , 8 1 2 3 8 1 -> 8 1 5 8 0 1 , 8 1 2 3 8 3 -> 8 1 5 8 0 3 , 8 1 2 3 8 13 -> 8 1 5 8 0 13 , 9 1 2 3 8 0 -> 9 1 5 8 0 0 , 9 1 2 3 8 1 -> 9 1 5 8 0 1 , 9 1 2 3 8 3 -> 9 1 5 8 0 3 , 9 1 2 3 8 13 -> 9 1 5 8 0 13 , 0 1 2 3 8 0 -> 1 5 8 0 0 0 , 0 1 2 3 8 1 -> 1 5 8 0 0 1 , 0 1 2 3 8 3 -> 1 5 8 0 0 3 , 0 1 2 3 8 13 -> 1 5 8 0 0 13 , 2 1 2 3 8 0 -> 6 5 8 0 0 0 , 2 1 2 3 8 1 -> 6 5 8 0 0 1 , 2 1 2 3 8 3 -> 6 5 8 0 0 3 , 2 1 2 3 8 13 -> 6 5 8 0 0 13 , 8 1 2 3 8 0 -> 4 5 8 0 0 0 , 8 1 2 3 8 1 -> 4 5 8 0 0 1 , 8 1 2 3 8 3 -> 4 5 8 0 0 3 , 8 1 2 3 8 13 -> 4 5 8 0 0 13 , 9 1 2 3 8 0 -> 11 5 8 0 0 0 , 9 1 2 3 8 1 -> 11 5 8 0 0 1 , 9 1 2 3 8 3 -> 11 5 8 0 0 3 , 9 1 2 3 8 13 -> 11 5 8 0 0 13 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 is interpreted by / \ | 1 0 | | 0 1 | \ / 1 is interpreted by / \ | 1 1 | | 0 1 | \ / 2 is interpreted by / \ | 1 1 | | 0 1 | \ / 3 is interpreted by / \ | 1 1 | | 0 1 | \ / 4 is interpreted by / \ | 1 1 | | 0 1 | \ / 5 is interpreted by / \ | 1 0 | | 0 1 | \ / 6 is interpreted by / \ | 1 0 | | 0 1 | \ / 7 is interpreted by / \ | 1 0 | | 0 1 | \ / 8 is interpreted by / \ | 1 1 | | 0 1 | \ / 9 is interpreted by / \ | 1 1 | | 0 1 | \ / 10 is interpreted by / \ | 1 1 | | 0 1 | \ / 11 is interpreted by / \ | 1 0 | | 0 1 | \ / 12 is interpreted by / \ | 1 0 | | 0 1 | \ / 13 is interpreted by / \ | 1 0 | | 0 1 | \ / After renaming modulo { 0->0, 1->1, 5->2, 10->3, 8->4, 4->5, 2->6, 3->7, 13->8, 9->9 }, it remains to prove termination of the 16-rule system { 0 1 2 3 4 0 -> 0 1 2 5 6 0 , 0 1 2 3 4 1 -> 0 1 2 5 6 1 , 0 1 2 3 4 7 -> 0 1 2 5 6 7 , 0 1 2 3 4 8 -> 0 1 2 5 6 8 , 6 1 2 3 4 0 -> 6 1 2 5 6 0 , 6 1 2 3 4 1 -> 6 1 2 5 6 1 , 6 1 2 3 4 7 -> 6 1 2 5 6 7 , 6 1 2 3 4 8 -> 6 1 2 5 6 8 , 4 1 2 3 4 0 -> 4 1 2 5 6 0 , 4 1 2 3 4 1 -> 4 1 2 5 6 1 , 4 1 2 3 4 7 -> 4 1 2 5 6 7 , 4 1 2 3 4 8 -> 4 1 2 5 6 8 , 9 1 2 3 4 0 -> 9 1 2 5 6 0 , 9 1 2 3 4 1 -> 9 1 2 5 6 1 , 9 1 2 3 4 7 -> 9 1 2 5 6 7 , 9 1 2 3 4 8 -> 9 1 2 5 6 8 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 is interpreted by / \ | 1 0 | | 0 1 | \ / 1 is interpreted by / \ | 1 0 | | 0 1 | \ / 2 is interpreted by / \ | 1 0 | | 0 1 | \ / 3 is interpreted by / \ | 1 1 | | 0 1 | \ / 4 is interpreted by / \ | 1 1 | | 0 1 | \ / 5 is interpreted by / \ | 1 0 | | 0 1 | \ / 6 is interpreted by / \ | 1 0 | | 0 1 | \ / 7 is interpreted by / \ | 1 0 | | 0 1 | \ / 8 is interpreted by / \ | 1 0 | | 0 1 | \ / 9 is interpreted by / \ | 1 0 | | 0 1 | \ / After renaming modulo { }, it remains to prove termination of the 0-rule system { } The system is trivially terminating.